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+{"text":"\\section{Introduction}\\label{sec:intro}\n\nClose binaries containing compact objects span a wide range of interesting and exotic stars, such as millisecond pulsars, galactic black hole candidates, detached white dwarf (WD) binaries, \nneutron star binaries, and interacting binaries, such as cataclysmic variables and low-mass X-ray binaries. The small binary separations of all these compact binaries imply that the radius of \nthe progenitor of the compact object must have exceeded the current orbital separation quite far. How such close-compact binary systems could form was outlined more than 30 years ago by \n\\citet{paczynski76-1}. The progenitors of close-compact binaries were initially relatively close binary systems ($a_{\\mathrm{i}}\\sim\\,100-1000\\Rsun$) consisting of two main-sequence (MS) stars. \nOnce the primary, i.e. the more massive star, evolved off the MS and filled its Roche lobe during the first giant branch (FGB) or asymptotic giant branch (AGB), dynamically unstable mass transfer \nwas generated, and the less massive star (from now on the secondary) could not accrete the transferred material, which thus started to accumulate around it and quickly formed a common envelope (CE); \ni.e., the envelope of the primary surrounded the core of the primary and the secondary star. Owing to drag forces between the envelope and the two stars, orbital energy was transferred from the binary \n(consisting of the core of the primary and the secondary) to the envelope, causing the binary separation to be significantly reduced and the CE to be ejected. After the envelope ejection, \nthe system appears as a close but detached post-common-envelope binary (PCEB) consisting of a compact object, i.e. the core of the primary, and a MS star. \nAmong the most numerous compact binaries are those containing a WD primary, and the stellar parameters are most easily measured if both stars are in a detached orbit. Such white dwarf + main sequence (WD+MS) \nPCEBs are therefore ideal systems for providing observational constraints on models of CE evolution.  \n\nBinary population studies of PCEBs have been performed since the early nineties \\citep[][]{dekool+ritter93-1}. \nThe most important and, at the same time, least understood phase of compact binary evolution is CE evolution. \nThe outcome of the CE phase is generally approximated by equating the binding energy of the envelope and the change in orbital energy scaled with an efficiency $ \\alpha_{\\mathrm{CE}}$, i.e., \n\\begin{equation}\\label{eq:alpha}\nE_\\mathrm{bind} = \\alpha_{\\mathrm{CE}}\\Delta E_\\mathrm{orb}.\n\\end{equation}\nThe most basic assumption is to approximate the binding energy only by the gravitational energy of the envelope:\n\\begin{equation}\\label{eq:Egr}\nE_\\mathrm{bind} = E_\\mathrm{grav} = \u2212\\frac{G M_\\mathrm{1} M_\\mathrm{1,e}}{\\lambda R_\\mathrm{1}},\n\\end{equation}\nwhere $M_\\mathrm{1}$, $M_\\mathrm{1,e}$, and $R_\\mathrm{1}$ are the total mass, envelope mass, and radius of the primary star, and $\\lambda$ is a binding energy parameter that depends on the structure \nof the primary star. Previous simulations of PCEBs \\citep{dekool+ritter93-1,willems+kolb04-1,politano+weiler06-1,politano+weiler07-1} have been performed using different values of the CE efficiency \n$\\alpha_{\\mathrm{CE}}$ but assuming $\\lambda = 0.5$ or $\\lambda = 1.0$, or assuming different fixed values for $\\alpha_{\\mathrm{CE}}\\lambda$ \\citep{toonen+nelemans13-1}. However, keeping $\\lambda$ \nconstant is not a very realistic assumption for all types of possible primaries, as pointed out by \\citet{dewi+tauris00-1} and \\citet{podsiadlowskietal03-1}. Very loosely bound envelopes in more evolved \nstars, e.g. if the primary is close to the tip of the AGB, can reach much higher values of $\\lambda$. This is especially true if other sources of energy of the envelope, \nthe most important being recombination energy,\nsupport the ejection process. If a fraction $\\alpha_{\\mathrm{rec}}$ of the recombination energy of the envelope contributes to the ejection process, the binding energy equation becomes \n\\begin{equation}\\label{eq:Eball}\nE_\\mathrm{bind}=\\int_{M_\\mathrm{1,c}}^{M_\\mathrm{1}}-\\frac{G m}{r(m)}dm + \\alpha_{\\mathrm{rec}}\\int_{M_\\mathrm{1,c}}^{M_\\mathrm{1}}U_\\mathrm{rec}(m)\n\\end{equation}\nwhere $M_\\mathrm{1,c}$ is the core mass of the primary and $r(m)$  the radius that encloses the mass $m$. The effects of the additional energy source $U_{\\mathrm{rec}}$ can \nbe included in the $\\lambda$ parameter by equating Eqs.\\,(\\ref{eq:Egr}) and\\,(\\ref{eq:Eball}). While it is clear that $\\lambda$ is not constant, the contributions from other sources of energy, such as recombination, \nare very uncertain. On one hand, the existence of the long orbital-period PCEB IK\\,Peg \\citep{wonnacottetal93-1} might imply that there are missing \nterms in the energy equation, and the most promising candidate is indeed recombination energy available in the envelope \\citep[see][ for a more detailed discussion]{webbink08-1}. \nOn the other hand, it has been claimed that the opacity in the envelope is too low for an efficient use of recombination energy \\citep{soker+harpaz03-1}. \n\nA first fairly rough attempt was made to investigate the impact of possible contributions of the recombination energy on the predictions of binary population models \\citep{davisetal10-1}. \nHowever, the parameter space evaluated by these authors  was rather limited. First, they assumed $\\alpha_{\\mathrm{CE}} = 1.0$. Second, \nthe values of $\\lambda$ were obtained by interpolating the very sparse grid of \\citet{dewi+tauris00-1}, which covered only eight primary masses and only the extreme cases \nof recombination energy contribution, i.e., $\\alpha_{\\mathrm{rec}}=0$ or $\\alpha_{\\mathrm{rec}}=1$. \n\nIn this paper we simulate the population of detached WD+MS PCEBs with different values of the CE efficiency and with the inclusion of different fractions of recombination energy \n($\\alpha_{\\mathrm{rec}}$) in order to explore how these crucial parameters affect the properties of the predicted PCEB population. \n \n\n\\section{The simulations}\\label{sec:sim}\n\nWe generate an initial MS+MS binary population of $10^7$ systems. The primary masses are distributed according to the initial mass function (IMF) of \\citet{kroupaetal93-1}:\n\\begin{equation}\nf(M_\\mathrm{1}) = \\left\\{\\begin{array}{l l}\n  0 & \\quad \\mbox{$M_\\mathrm{1}\/\\Msun<0.1,$}\\\\\n  0.29056M_\\mathrm{1}^{-1.3} & \\quad \\mbox{$0.1\\leq{M_\\mathrm{1}\/\\Msun}<0.5,$} \\\\\n  0.15571M_\\mathrm{1}^{-2.2} & \\quad \\mbox{$0.5\\leq{M_\\mathrm{1}\/\\Msun}<1.0,$} \\\\\n  0.15571M_\\mathrm{1}^{-2.7} & \\quad \\mbox{$1.0\\leq{M_\\mathrm{1}\/\\Msun}.$} \\\\\n  \\end{array}\n  \\right.\n\\label{M1dist}\n\\label{eq:IMF}\n\\end{equation}\nFor the mass of the secondary star we assume a flat initial-mass-ratio distribution (IMRD), i.e., $n(q)$ = constant, where $q = M_\\mathrm{2}\/M_\\mathrm{1}$.\nThe initial orbital separation $a_\\mathrm{i}$ follows the distribution\n\\begin{equation}\nh(a_\\mathrm{i}) = \\left\\{\\begin{array}{l l}\n  0 & \\quad \\mbox{$a_{\\mathrm{i}}\/\\Rsun<3$ or $a_{\\mathrm{i}}\/\\Rsun>10^{4},$}\\\\\n  0.078636a_{\\mathrm{i}}^{-1} & \\quad \\mbox{$3\\leq a_{\\mathrm{i}}\/\\Rsun \\leq{10^4}$}\\\\ \n  \\end{array}\n  \\right.\n\\label{adist}\n\\end{equation}\n\\citep{davisetal08-1}\\footnote{\\citet{davisetal08-1} give an upper limit of $10^{6}\\Rsun$ for the distribution of initial separations. \nWe cut the distribution at $10^{4}\\Rsun$ because in systems with larger initial separations, the primary will never fill the Roche lobe.}.\nWe assumed solar metallicity for all the systems.\nFinally we assign a ``born time'' ($t_{\\mathrm{born}}$) to all the systems, corresponding to the age of the Galaxy when the system was born, assuming a \nconstant star formation rate between $0$ and the age of the Galaxy ($t_{\\mathrm{Gal}} \\sim\\,13.5$\\,Gyr).\n\nWe use the latest version of the binary-star evolution (BSE) code from \\citet{hurleyetal02-1}, updated as in \\citet{zorotovic+schreiber13-1},  to evolve all \nthe systems during $t_{\\mathrm{evol}} = t_{\\mathrm{Gal}} - t_{\\mathrm{born}}$, in order to obtain the {\\em{current}} orbital and stellar parameters. \nDisrupted magnetic braking is assumed. As discussed in detail in \\citet{zorotovicetal10-1}, the latest version of the BSE code allows one to compute the binding \nenergy of the envelope, including not only the gravitational energy but also a fraction $\\alpha_{\\mathrm{rec}}$ of the recombination energy of the envelope. The two \nfree parameters in our simulations are then the CE efficiency ($\\alpha_{\\mathrm{CE}}$) and the fraction of recombination energy that is used to expel the envelope \n($\\alpha_{\\mathrm{rec}}$). \n\n\\begin{table}\n\\caption{\\label{tab:mod} Different models analyzed in this work.}\n\\begin{center}\n\\begin{tabular}{lccc}\n\\hline\\hline\nModel  & $\\alpha_{\\mathrm{CE}}$ & $\\alpha_{\\mathrm{rec}}$\\\\\n\\hline\na & 0.25 & 0.00\\\\\nb & 0.25 & 0.02\\\\\nc & 0.25 & 0.25\\\\\nd & 0.50 & 0.00\\\\\ne & 0.50 & 0.02\\\\\nf & 0.50 & 0.25\\\\\ng & 1.00 & 0.00\\\\\nh & 1.00 & 0.02\\\\\ni & 1.00 & 0.25\\\\\n\\hline\n\\noalign{\\smallskip}\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nWe assume that the fraction of recombination energy that contributes to the envelope ejection process cannot exceed the efficiency of using the orbital \nenergy of the binary. This is reasonable because the recombination energy is probably radiated away much more easily. Table\\,\\ref{tab:mod} shows the combination of \nthe two efficiency parameters for the nine different models we studied in this work.\n\nOur simulated PCEB sample contains all the WD+MS binaries that went through a CE phase but did not yet reach the second phase of mass transfer, which \nwould probably make them cataclysmic variables.\n\n\\section{Results} \n\nIn what follows we describe and explain the characteristics of the predicted parameter distributions for the nine models listed in Table\\,\\ref{tab:mod}. \n\n\\subsection{Number of PCEBs}\n\nTable\\,\\ref{tab:res} lists the total number of detached WD+MS PCEBs predicted by each model\\footnote{The total number of systems obtained for each model\nis not a prediction of what should be expected observationally, and should not be used to estimate space densities. It is only listed to show how increasing \nboth efficiencies allows more systems to survive the CE phase.}, \nas well as the fractions of systems containing He-core and C\/O-core WDs. The total number of systems increases noticeably with the value of $\\alpha_{\\mathrm{CE}}$ and also slightly with \nthe value of $\\alpha_{\\mathrm{rec}}$. This is easy to understand: a higher value of the CE efficiency implies a more efficient use of orbital energy and thus a \nsmaller reduction of the binary separation, which allows more systems to survive. In addition, systems that survive with a low  CE efficiency emerge from the CE \nphase at longer periods when we increase the efficiency, and therefore stay longer as detached PCEBs. The same occurs if an increasing fraction of recombination energy \nis assumed to contribute. However, $\\alpha_{\\mathrm{rec}}$ does not affect all the systems in the same way, because the relative contribution of recombination energy \ndepends on the mass and evolutionary state of the primary. For example, for less evolved primaries on the FGB, the contribution of recombination energy to the binding \nenergy remains small compared to the contribution of gravitational energy even for high values of $\\alpha_{\\mathrm{rec}}$, because the envelope is not as extended \nas in the AGB and is still tightly bound to the core.\n\n\n\\begin{table}\n\\caption{\\label{tab:res} Results for $n(q)$ = constant.}\n\\begin{center}\n\\begin{tabular}{lccccc}\n\\hline\\hline\nModel & $\\alpha_{\\mathrm{CE}}$ & $\\alpha_{\\mathrm{rec}}$ & $N_{\\mathrm{sys}}$ & He (\\%) & C\/O (\\%)\\\\\n\\hline\na & 0.25 & 0.00 & 33\\,917 & 44.6 & 55.4 \\\\\nb & 0.25 & 0.02 & 36\\,098 & 42.8 & 57.2 \\\\\nc & 0.25 & 0.25 & 45\\,279 & 41.3 & 58.7 \\\\\nd & 0.50 & 0.00 & 60\\,444 & 51.3 & 48.7 \\\\\ne & 0.50 & 0.02 & 61\\,745 & 50.6 & 49.4 \\\\\nf & 0.50 & 0.25 & 68\\,215 & 49.7 & 50.3 \\\\\ng & 1.00 & 0.00 & 88\\,039 & 56.1 & 43.9 \\\\\nh & 1.00 & 0.02 & 88\\,886 & 55.7 & 44.3 \\\\\ni & 1.00 & 0.25 & 92\\,726 & 55.2 & 44.8 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\tiny Total number of detached PCEBs obtained with each model and percentage of systems with He WDs and C\/O WDs. From the $10^7$ initial MS+MS binaries simulated with this distribution, $\\sim40.7$\\% entered a CE phase.\n\\end{table}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=270,width=0.59\\textwidth]{Mwd.ps}\n\\caption{WD mass distribution for the different models. Gray shaded histograms represent the entire distribution, while the color histograms are for systems with He WDs \n(red) and with C\/O WDs (blue). \n}\n\\label{fig:Mwd}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=270,width=0.59\\textwidth]{Pf.ps}\n\\caption{Orbital period distribution for the different models described in Table\\,\\ref{tab:mod}. Colors are the same as in Fig.\\,\\ref{fig:Mwd}. \n}\n\\label{fig:Pf}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=270,width=0.59\\textwidth]{M2.ps}\n\\caption{Secondary mass distribution for the different models. Colors are the same as in Fig.\\,\\ref{fig:Mwd}. \n}\n\\label{fig:M2}\n\\end{figure*}\n\n\\subsection{The WD mass distribution}\n\nFigure\\,\\ref{fig:Mwd} and Table\\,\\ref{tab:res} show the WD mass distribution for all the models. The gap separating systems with He WDs from those containing C\/O WDs is caused by the stellar \nradius at the tip of the FGB being larger than at the beginning of the AGB, while the core mass still increases from $\\sim0.48$ to $0.51\\Msun$. In this range of \ncore masses, the primary star cannot fill its Roche lobe because it would have done so before on the FGB. \n\nFigure\\,\\ref{fig:Mwd} clearly shows that the relative number of systems with He WDs increases and that the distribution extends towards lower mass systems for higher \nvalues of $\\alpha_{\\mathrm{CE}}$. Less evolved systems, like those in which the primary star is a (low-mass) He WD, are initially closer and therefore a lower value \nof $\\alpha_{\\mathrm{CE}}$ implies an increased merger rate for these systems, while the progenitors of systems with C\/O WDs are initially more separated and can survive \nthe CE evolution even if more orbital energy is required to expel the envelope (small $\\alpha_{\\mathrm{CE}}$). Therefore, the shape of the WD mass distribution for \nsystems containing high-mass C\/O WDs is almost unaffected by the value of the CE efficiency. For a fixed value of $\\alpha_{\\mathrm{CE}}$, on the other hand, \nthe percentage of systems containing a He WD remains nearly constant (with a very slight decrease) for different values of $\\alpha_{\\mathrm{rec}}$. This is because \nthe recombination energy becomes more important than the gravitational energy\nonly for very advanced evolutionary stages, especially later on the AGB. For those systems, the initial separation is generally \nlarge enough to avoid a merger even without including this additional energy.  \n\n\\subsection{The orbital period distribution}\n\nThe orbital period distributions predicted by our nine models are shown in Fig.\\,\\ref{fig:Pf}. The orbital periods for systems containing C\/O WDs are on average \nlonger than those of systems containing He WDs in all the models. The peak of the period distributions for the entire sample shifts toward longer periods \nif $\\alpha_{\\mathrm{CE}}$ is increased. Also, by increasing the value of $\\alpha_{\\mathrm{CE}}$, the distribution becomes slightly wider. This \nis because greater CE efficiency implies a smaller reduction of the orbital period, moving the distributions toward longer orbital periods but also adding new \nsystems with short periods that mainly contain He WDs. These systems merge for low values of $\\alpha_{\\mathrm{CE}}$ but can survive the CE phase if the orbital \nenergy is used efficiently. The effect of increasing the fraction of recombination energy mostly affects systems with longer periods and C\/O WDs that descend from \nevolved primaries where the contribution of the recombination energy of the envelope becomes important. A tail toward longer orbital periods appears in the distribution \nfor systems with C\/O WDs with increasing $\\alpha_{\\mathrm{rec}}$, while the shape of the distribution for systems with He WDs remains nearly constant for a fixed \nvalue of $\\alpha_{\\mathrm{CE}}$. Almost all the systems with periods longer than about ten days can only be obtained when a fraction of the recombination energy is taken into account. \n\n\\subsection{The secondary mass distribution}\n\nIn Fig.\\,\\ref{fig:M2} we show the distributions obtained for the secondary masses. The relative number of systems increases with increasing secondary mass, with a steep \ndecline at $M_2\\sim\\,0.35$\\Msun. This corresponds to the boundary for fully convective secondaries where, according to the disrupted magnetic braking theory, \nangular momentum loss due to magnetic braking becomes inefficient. A PCEB evolves toward shorter orbital periods because of orbital angular momentum loss through \ngravitational radiation and the much stronger magnetic wind braking. Below $M_2\\sim\\,0.35$\\Msun\\, PCEBs get closer only thanks to gravitational radiation, which is \nmuch less efficient than magnetic braking, causing these systems to spend more time as detached PCEBs before the secondary fills its Roche lobe and becomes a \ncataclysmic variable, and therefore increasing the relative number of systems with low-mass secondaries. This behavior has already been predicted by \\citet{politano+weiler07-1} \nand observationally confirmed by \\citet{schreiberetal10-1}. The effect of increasing $\\alpha_{\\mathrm{CE}}$ is that this decline becomes less apparent. This is because the \ndistributions are normalized for each model, and as already mentioned, increasing $\\alpha_{\\mathrm{CE}}$ rapidly increases the number of systems obtained \n(see Table\\,\\ref{tab:res}) and moves the orbital period distribution toward longer periods. More systems therefore stay as detached PCEBs for very long periods of time, \nup to several Hubble times, even when magnetic braking is efficient ($M_2\\gappr\\,0.35$\\Msun\\,). The effect of increasing $\\alpha_{\\mathrm{rec}}$ is similar but much less \npronounced because recombination energy mainly affects systems with more evolved primaries. The drop of systems toward masses higher than $\\sim\\,1$\\Msun\\, is the imprint \nof the IMF for the primary, because $M_2$ is related to $M_1$ through the IMRD. \n\n\\subsection{Relating the final parameters}\\label{sec:relate}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=270,width=0.59\\textwidth]{PMwd.ps}\n\\caption{Relation between WD mass and orbital period. The intensity of the gray scale represents the density of objects in each bin, on a linear scale, and normalized \nto one for the bin that contains most systems. \n} \n\\label{fig:PMwd}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=270,width=0.59\\textwidth]{PM2.ps}\n\\caption{Relation between secondary mass and orbital period. The intensity of the gray scale means the same as in Fig.\\,\\ref{fig:PMwd}. \n}\n\\label{fig:PM2}\n\\end{figure*}\n\n\\begin{figure*} \n\\centering\n\\includegraphics[angle=270,width=0.59\\textwidth]{MwdM2.ps}\n\\caption{Relation between WD and secondary mass. The intensity of the gray scale means the same as in Fig.\\,\\ref{fig:PMwd}. \n}\n\\label{fig:MwdM2}\n\\end{figure*}\n\nIn addition to inspecting distributions of a single parameter, it is instructive to investigate possible relations between the orbital and stellar parameters. \nFigure\\,\\ref{fig:PMwd} shows the relation between the WD mass and the orbital period. The gap separating systems with He WDs from systems with C\/O WDs is evident. \n\nAmong the systems with He WDs, there is a correlation between the orbital period and the WD mass, a trend that becomes more apparent by increasing\n$\\alpha_{\\mathrm{CE}}$ as systems with lower mass He WDs survive. In contrast, no clear\ntrend can be identified\nfor systems with C\/O WDs. This difference agrees with the observations \\citep{zorotovicetal11-2} and can be understood as follows. \nThe C\/O WDs in PCEBs descend from a wider range of progenitor masses and initial separations \\citep[see][ their figure 2]{zorotovic+schreiber13-1},\nwhich also results in a wider range of masses for the companion.\nThis translates into a wider range of initial energies (orbital and binding) and values of the binding energy parameter $\\lambda$ (especially if the effects of recombination energy are included). \nThis wider range of initial conditions naturally transfers into a wider range of final orbital periods for systems containing C\/O WDs with similar masses.\nIn particular, the strong impact of potential contributions of recombination energy on the final periods of PCEBs containing C\/O WDs is clearly visible in Fig.\\,\\ref{fig:PMwd}.\nIncreasing the fraction of recombination energy that is used to expel the envelope mainly affects those systems with more massive C\/O WDs, \nwhere the value of $\\lambda$ can become extremely high, moving them toward longer periods. Therefore, as pointed out previously by \\citet{rebassa-mansergasetal12-1}, \nclear observational constraints on the role of the recombination energy could be derived eventually if the orbital periods of a large and \nrepresentative sample of PCEBs containing high-mass WDs could be measured. \n\nIn Fig.\\,\\ref{fig:PM2} we show the relation between the mass of the secondary star and the orbital period. There is a tendency to predict longer periods for \nsystems with more massive secondaries in agreement with the observational analysis of \\citet{zorotovicetal11-2}. The reason for this is \ntwofold. First, for a given primary mass and orbital period, more initial orbital energy is available for systems with more massive secondaries, and therefore the fraction of this \nenergy that is needed to unbind the envelope is smaller, leading to longer orbital periods. Second, for a given WD mass, the minimum period at which a\nPCEB remains detached decreases with secondary mass. Since lower mass secondaries have smaller radii, they can remain within their Roche lobes at smaller\nseparations (shorter orbital periods).  \n\nThe previously mentioned paucity of systems with $M_2\\sim0.35-0.5$\\Msun\\,is also evident in Fig.\\,\\ref{fig:PM2}.\nOwing to the assumption of disrupted magnetic braking in our simulations, PCEBs with masses exceeding $\\sim0.35$\\Msun\\, become closer not only because of gravitational \nradiation but also due to magnetic braking, which is supposed to be much more efficient. This causes much shorter evolutionary time scales from the \nCE to the CV phase. This explains the reduction of systems with secondary masses exceeding the fully convective boundary located at $0.35$\\Msun. \nIn the range $M_2\\sim0.35-0.5$\\Msun\\, almost all systems with long ($\\log\\Porb[d]>0.5$) and short orbital periods ($\\log\\Porb[d]<-0.5$) disappeared. \nAt $\\log\\Porb[d]\\sim0$, a significant number of systems with $M_2\\sim0.35-0.5$\\Msun\\  remain despite the efficient angular momentum loss due\nto magnetic braking because of the very large number of PCEBs formed with these parameters (for a flat IMRD as assumed here). \n\nFigure\\,\\ref{fig:PM2} also nicely shows that increasing the values of\n$\\alpha_{\\mathrm{CE}}$ or $\\alpha_{\\mathrm{rec}}$ reduces the paucity of\nsystems with \n$\\sim0.35-0.5$\\Msun\\, \nsecondary stars (caused by assuming disrupted magnetic braking). \nThis is because the PCEBs emerge from CE evolution with longer orbital periods and remain longer as detached systems,\nwhich increases the total number of PCEBs, even if the mass of the secondary star implies magnetic braking to be efficient. \nIt can also be seen that the increase in long-period systems due to higher values of $\\alpha_{\\mathrm{rec}}$ is independent of secondary mass. \n\nFinally, Fig.\\,\\ref{fig:MwdM2} shows the relation between the masses of the WD and the secondary star. The three previously mentioned features can be identified as well, i.e. the increase \nin the total number of systems with increasing $\\alpha_{\\mathrm{CE}}$, the increase of systems with low-mass He WDs for increasing $\\alpha_{\\mathrm{CE}}$, and the less apparent \ndecline in the number of systems with masses  $\\sim0.35-0.5$\\Msun\\, as $\\alpha_{\\mathrm{CE}}$ or $\\alpha_{\\mathrm{rec}}$ are increased. In agreement with the observational \nfindings from \\citet{zorotovicetal11-2}, there seems to be no relation between the two stellar masses.\n\n\\subsection{The initial-mass-ratio distribution}\n\nTo test whether the IMRD has any effect on the period and mass\ndistributions, we decided to repeat our full set of simulations assuming \ndifferent IMRDs. \nAnd assuming two extreme cases, i.e. $n(q)\\propto q$, in addition to $n(q)\\propto q^{-1}$, we obtained the following results. \n\n\nThe total number of detached WD+MS PCEBs predicted by each model and the fractions of systems containing He and C\/O WDs are shown in\nTables\\,\\ref{tab:res2} and\\,\\ref{tab:res3} for the additional IMRDs. The fraction of systems entering a CE phase is virtually independent\nof the IMRD, because the assumed initial mass function for the primary and the distribution of initial separations are identical in all simulations \nand dominate the weak dependence of the Roche-lobe radius of the primary on the secondary mass.\nFor the two new IMRDs, the total number of systems increases markedly with \n$\\alpha_{\\mathrm{CE}}$ and also somewhat with $\\alpha_{\\mathrm{rec}}$, as in the case of a flat distribution (see Table\\,\\ref{tab:res}).\nThe simulations that assume an IMRD inversely proportional to $q$ generate more WD+MS PCEBs than in the case of a flat distribution, while\nsimulations assuming $n(q)\\propto q$ generate less systems. This can be explained as a combination of two effects. Assuming $n(q)\\propto q^{-1}$ favors \nthe formation of systems with low-mass secondary stars, which take longer to evolve and therefore remain longer as MS stars. On the other hand, \nmore massive secondaries may have enough time to evolve, and then the system will no longer be a WD+MS PCEB. Also, if\nthe mass of the secondary is smaller than $\\sim\\,0.35\\,\\Msun$ the system remains detached after the CE phase for longer,\nbecause magnetic braking is not acting (or at least not efficiently acting) and angular momentum loss is driven mainly due to gravitational radiation.\n\nFor both distributions, the fraction of systems with He or C\/O WDs behaves in the same way as for a flat IMRD; i.e., the fraction of systems with He WDs \nincreases notably by increasing $\\alpha_{\\mathrm{CE}}$ and slightly decreases by increasing $\\alpha_{\\mathrm{rec}}$.\nThe fraction of systems with He WDs is greater for the distribution favoring more massive secondary stars ($n(q)\\propto q$).\nThis is for several reasons. First and most important, systems with more massive secondaries have a \nhigher initial orbital energy, before the CE phase, and therefore have more energy available to unbind the envelope. Systems where the \nenvelope relatively tightly bound, such as the progenitors of He WDs, can survive the CE phase\nmore easily if they have a massive companion. \nSecond, systems with more massive secondaries emerge from the CE phase with a longer orbital periods and therefore \nremain detached PCEBs for longer. This increases the fraction of systems with He WDs because these are the ones that end up closer after \nthe CE phase and start a second phase of mass transfer faster earlier. Finally, there is also a tendency to produce slightly less massive WDs \nin systems with more massive secondaries because, for a given primary mass, the Roche lobe of the primary is smaller when \nthe secondary star is more massive.\n\n\\begin{table}\n\\caption{\\label{tab:res2} Results for $n(q)\\propto q$.}\n\\begin{center}\n\\begin{tabular}{lccccc}\n\\hline\\hline\nModel & $\\alpha_{\\mathrm{CE}}$ & $\\alpha_{\\mathrm{rec}}$ & $N_{\\mathrm{sys}}$ & He (\\%) & C\/O (\\%)\\\\\n\\hline\na & 0.25 & 0.00 & 30\\,195 & 51.6 & 48.4 \\\\\nb & 0.25 & 0.02 & 31\\,188 & 50.9 & 49.1 \\\\\nc & 0.25 & 0.25 & 36\\,712 & 50.4 & 49.6 \\\\\nd & 0.50 & 0.00 & 50\\,090 & 59.2 & 40.8 \\\\\ne & 0.50 & 0.02 & 50\\,087 & 58.7 & 41.3 \\\\\nf & 0.50 & 0.25 & 54\\,589 & 58.6 & 41.4 \\\\\ng & 1.00 & 0.00 & 70\\,035 & 64.2 & 35.8 \\\\\nh & 1.00 & 0.02 & 70\\,490 & 63.9 & 36.1 \\\\\ni & 1.00 & 0.25 & 72\\,474 & 63.6 & 36.4 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\tiny Same as in Table\\,\\ref{tab:res} but for the IMRD proportional to the mass ratio. From the $10^7$  initial MS+MS binaries simulated with this distribution, $\\sim40.4$\\% entered a CE phase.\n\\end{table}\n\n\\begin{table}\n\\caption{\\label{tab:res3} Results for $n(q)\\propto q^{-1}$.}\n\\begin{center}\n\\begin{tabular}{lccccc}\n\\hline\\hline\nModel & $\\alpha_{\\mathrm{CE}}$ & $\\alpha_{\\mathrm{rec}}$ & $N_{\\mathrm{sys}}$ & He (\\%) & C\/O (\\%)\\\\\n\\hline\na & 0.25 & 0.00 & 38\\,680 & 39.0 & 61.0 \\\\\nb & 0.25 & 0.02 & 42\\,515 & 36.0 & 64.0 \\\\\nc & 0.25 & 0.25 & 55\\,572 & 34.4 & 65.6 \\\\\nd & 0.50 & 0.00 & 72\\,625 & 45.5 & 54.5 \\\\\ne & 0.50 & 0.02 & 74\\,989 & 44.3 & 55.7 \\\\\nf & 0.50 & 0.25 & 83\\,632 & 43.1 & 56.9 \\\\\ng & 1.00 & 0.00 & 109\\,711 & 49.6 & 50.4 \\\\\nh & 1.00 & 0.02 & 110\\,629 & 49.3 & 50.7 \\\\\ni & 1.00 & 0.25 & 115\\,953 & 49.0 & 51.0 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\tiny Same as in Table\\,\\ref{tab:res} but for the IMRD inversely proportional to the mass ratio. From the $10^7$ initial MS+MS binaries simulated with this distribution, $\\sim41.0$\\% entered a CE phase.\n\\end{table}\n\nThe WD mass distribution is almost unaffected by the assumption of a different IMRD. The shape of the two distributions for systems containing\nHe and C\/O WDs remains almost identical with the only variation being their relative contributions to the whole population.\nThis was expected because, as mentioned in Sect.\\,\\ref{sec:relate}, both masses do not appear to be related (see also Fig.\\,\\ref{fig:MwdM2}). \n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=270,width=0.59\\textwidth]{Pf_propq.ps}\n\\caption{Same as in Fig.\\,\\ref{fig:Pf} but for $n(q)\\propto q$. \n}\n\\label{fig:Pf_prop}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=270,width=0.59\\textwidth]{Pf_invq.ps}\n\\caption{Same as in Fig.\\,\\ref{fig:Pf} but for $n(q)\\propto q^{-1}$. \n}\n\\label{fig:Pf_inv}\n\\end{figure*}\n\nThe period distributions are shown in Figs.\\,\\ref{fig:Pf_prop} and\\,\\ref{fig:Pf_inv} for the IMRD proportional to the mass ratio\nand for the one in which the secondary mass depends inversely on the mass ratio, respectively.\nThe shape of the distributions does not change dramatically by using a different IMRD. However, the entire distributions move slightly\ntoward longer (shorter) orbital periods when we favor the formation of systems with more (less) massive secondaries, respectively. This is because, as we show in Fig.\\,\\ref{fig:PM2},\nthere is a relation between the mass of the secondary and the orbital period; i.e., systems with more massive secondaries tend to have longer\nperiods.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=270,width=0.59\\textwidth]{M2_propq.ps}\n\\caption{Same as in Fig.\\,\\ref{fig:M2} but for $n(q)\\propto q$. \n}\n\\label{fig:M2_prop}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=270,width=0.59\\textwidth]{M2_invq.ps}\n\\caption{Same as in Fig.\\,\\ref{fig:M2} but for $n(q)\\propto q^{-1}$. \n}\n\\label{fig:M2_inv}\n\\end{figure*}\n\nFigures\\,\\ref{fig:M2_prop} and\\,\\ref{fig:M2_inv} show the distributions of secondary masses for the cases in which\n$n(q)\\propto q$ and $n(q)\\propto q^{-1}$, respectively.  \nWhile for the case of a flat IMRD the two peaks in this distributions have approximately the same height (see Fig.\\,\\ref{fig:M2}), \nit is evident from these two figures that we are favoring the formation of systems with high- and low-mass secondaries, respectively.\nAs in the case of the flat IMRD, the steep decline at the boundary for fully convective secondaries is more pronounced for low \nvalues of $\\alpha_{\\mathrm{CE}}$, and it becomes almost indistinguishable when we increase the value of $\\alpha_{\\mathrm{CE}}$\nfor the models in which we assume $n(q)\\propto q$.\nIf one could have a homogeneous and unbiased sample of WD+MS PCEBs, covering the whole range of masses for the companion star, the distribution\nof secondary masses would be very useful for deriving constraints on the IMRD. \n\n\\section{Discussion}\n\nWe have performed detailed binary population simulations of detached WD+MS binaries that evolved through CE evolution for different CE efficiencies $\\alpha_{\\mathrm{CE}}$. \nFor the first time we have done a systematic and comprehensive study of the effects of the recombination energy parametrized with $\\alpha_{\\mathrm{rec}}$. In what follows we discuss the \npredictions of our model in the context of previous model calculations. \n\nThe first detailed simulations of WD+MS PCEBs were performed by \\citet{dekool+ritter93-1}, and several of their predictions are still valid; for example, the decrease in the relative \nnumber of PCEBs with He WDs for lower values of the CE efficiencies goes back to this early work. However, \\citet{dekool+ritter93-1} just used a relatively small set of \ndifferent parameters and assumed a constant binding energy parameter $\\lambda=0.5,$ which is not always a realistic assumption \\citep{dewi+tauris00-1}. More than a decade later, \n\\citet{willems+kolb04-1} updated and extended the early work of \\citet{dekool+ritter93-1} by covering a larger parameter space and using more recent fits to stellar evolutionary \nsequences \\citep{hurleyetal00-1}. The predictions presented in these early papers are, however, difficult to compare with the observations because only current zero-age PCEB \ndistributions were calculated; i.e., the evolution of PCEBs toward shorter orbital periods was not taken into account. \n\n\\citet{politano+weiler07-1} were the first to present a predicted present-day population of PCEBs (their figures 2-5) to investigate the impact of assuming very low values \nof the CE efficiency (i.e., $\\alpha_{\\mathrm{CE}}<0.2$) and a dependence of $\\alpha_{\\mathrm{CE}}$ on the mass of the secondary star. Our simulations agree with their predictions\nwith respect to the reduced number of He WD primaries for low CE efficiencies and to the existence of less massive He WDs for higher values \u200b\u200bof $\\alpha_{\\mathrm{CE}}$ \n(bottom panels in their Fig.\\,3 and our Fig.\\,1) and with the more pronounced decrease at the fully convective boundary in the distribution of the secondary masses \n(top panels in their Fig.\\,3 and our Fig.\\,3). Later, \\citet{davisetal10-1} performed comprehensive binary population simulations of PCEBs for \nthe first time taking into account that the binding energy parameter is probably not a constant. They find that the predicted distributions agree reasonably well with the observed populations \nfor a constant value of $\\alpha_{\\mathrm{CE}}$ but predict a tail of long orbital period systems that was not present in the observed sample available to them. \n\nFinally, in \na very recent work, \\citet{toonen+nelemans13-1} simulated the current population of PCEBs in the Galaxy taking observational biases specific to the Sloan\nDigital Sky Survey (SDSS) into account. They find a better fit to the observations by using a low value of $\\alpha_{\\mathrm{CE}}$ (0.25), which is consistent with the results from \n\\citet{zorotovicetal10-1}. However, the fraction of systems containing He WD primaries is too high in their simulations. They suggest that this can be solved by using a \nhigher value of $\\alpha_{\\mathrm{CE}}$ when the CE phase begins during the AGB. However, this study also did not include the effects of recombination energy and adopted a constant \nvalue for $\\alpha_{\\mathrm{CE}}\\lambda$, which as outlined above, is not always a good approximation because $\\lambda$ depends on the properties of the star, \nin particular on its mass and radius \\citep[see,\n  e.g.,][]{dewi+tauris00-1}. Although a constant value might be a good\napproximation for most systems, this becomes completely unrealistic\nfor systems where the primary filled the Roche \nlobe at a more advanced evolutionary stage, with a less tightly bound \nenvelope.\nWe emphasize at this point that one therefore needs to be careful when \ndrawing conclusions based on the assumption of $\\alpha_{\\mathrm{CE}}\\lambda$ \nconstant.\n\nHere we extended the study of \\citet{politano+weiler07-1}, \\citet{davisetal10-1}, and \\citet{toonen+nelemans13-1} by presenting the first systematic investigation that \nincludes the contribution from recombination energy to the energy budget of CE evolution. \n\n\\section{Conclusions}\n\nWe have performed binary population synthesis models of PCEBs that include the possible contribution of recombination energy during CE evolution. The main features that characterize \nthe distributions of the orbital parameters for the different models can be summarized as follows:\n\\begin{itemize}\n\\item The orbital period distributions become slightly wider by increasing the value of $\\alpha_{\\mathrm{CE}}$. \n\\item Including a fraction of the recombination energy mainly affects systems with the more massive C\/O WDs by producing a tail in the period distribution toward longer orbital periods.\n\\item The fraction of systems with He WDs increases by increasing $\\alpha_{\\mathrm{CE}}$, and the distribution extends toward lower mass systems ($\\lappr\\,0.3\\,\\Msun$).\n\\item The distribution of secondary masses has a steep decline at $M_2\\sim\\,0.35\\,\\Msun$, as a consequence of assuming disrupted magnetic braking, which is more pronounced \nfor low values of $\\alpha_{\\mathrm{rec}}$ and especially of $\\alpha_{\\mathrm{CE}}$. \n\\item Systems with more massive secondaries tend to have longer periods after\n  the CE phase in all models. \n\\item The predicted distribution of secondary masses is very similar for different WD masses. The distribution changes with the IMRD; i.e., if initially high mass ratios are favored, all WDs have \nlarger numbers of relatively massive companions. If instead low initial mass ratios dominate, all WDs (independent of their mass) are more frequently found to have low-mass companions.\n\\item The relation between the period and the mass of the secondary means that\n  the period distribution moves slightly toward longer orbital periods when\n  we assumed an IMRD that favors the formation of systems with massive companions.\n\\item The mass distribution of the secondaries is strongly affected by the choice of the IMRD.\n\n\\end{itemize}\n\nSome of these features may be used in combination with a large observational sample to put constraints on the values of $\\alpha_{\\mathrm{CE}}$ and\/or $\\alpha_{\\mathrm{rec}}$,\nas well as on the IMRD. \nA detailed analysis of the selection effects that affect the sample of WD+MS PCEBs obtained from the SDSS, as well as a thorough comparison with the observed sample of these systems, \nwas recently presented by \\citet{camachoetal2014-1}. While the best agreement between observations and theory has been found for low values of $\\alpha_{\\mathrm{CE}}\\sim0.25$, the observed sample\nis still too small to derive robust constraints. This is mostly for three reasons. First, the spectroscopic SDSS survey allows one to identify only low-mass companions (spectral type M) \nto WDs. Second, the performed radial velocity survey somewhat favors the detection of short orbital period systems. Third, after taking the observational biases and selection\neffects into account, a relatively small sample of observed systems remained. Once a large and homogeneous sample of PCEBs is known, we recommend the following diagnostics to constrain currently unknown parameters. \n\n\\begin{itemize}\n\n\\item The value of $\\alpha_{CE}$ is most sensitive to the measured fraction of He-core WDs among systems with short orbital period (below one day). \n\n\\item If recombination energy plays a significant role, the orbital period distribution of PCEBs containing massive WDs should extend to very long periods (up to several hundred days). \n\n\\item The secondary mass distribution for a given WD mass should reflect the IMRD. \n\n\\end{itemize}\n\nWe are admittedly relatively far from reaching these goals. For example, we have just one observed PCEB with a massive companion (IK\\,Peg). Because it might well be that the CE\nefficiencies depend on the mass of the secondary star \\citep{politano+weiler07-1,davisetal10-1,demarcoetal11-1}, it is not only required that we measure more orbital periods of PCEBs from\nSDSS, but it is also urgent that observational surveys be extended to higher secondary masses.\n\n\n\\begin{acknowledgements}\nMZ acknowledges support from CONICYT\/FONDECYT\/POSTDOCTORADO\/3130559. MRS \nthanks FONDECYT (project 1100782 and 114126) and the Millennium Science Initiative, Chilean Ministry of Economy, Nucleus P10-022-F. The work of EG--B, ST, and JC was partially supported by the AGAUR, by MCINN grant AYA2011--23102, by the European Union FEDER funds, by the ESF \nEUROGENESIS project, and by the AECI grant A\/023687\/09. ARM acknowledges financial support from a LAMOST fellowship, from the Postdoctoral Science Foundation of China \n(grant 2013M530470), and from the Research Fund for International Young Scientists by the National Natural Science Foundation of China (grant 11350110496).\nThe research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme\n(FP\/2007-2013) \/ ERC Grant Agreement n. 320964 (WDTracer). BTG was supported in part by the UK's Science and Technology Facilities Council (ST\/I001719\/1).\n\\end{acknowledgements}\n\n\n\\bibliographystyle{aa}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe Weierstrass-Enneper representation~\\cite{W1866} is a fundamental tool in differential geometry for creating interesting examples of minimal surfaces in Euclidean 3-space. A similar representation was developed by Bryant~\\cite{B1987}, and later~\\cite{UY1992}, for surfaces of constant mean curvature (CMC) $H\\equiv c$ in hyperbolic 3-space of constant sectional curvature $-c^{2}$. Many other Weierstrass-type representations exist for various surface classes, for example: \n\\begin{itemize}\n\\item maximal surfaces in Lorentz 3-space~\\cite{K1983},\n\\item CMC surfaces $H\\equiv c$ in de Sitter 3-space of constant sectional curvature $c^{2}$~\\cite{AA1998},\n\\item flat surfaces in hyperbolic 3-space~\\cite{GMM2000}, \n\\item linear Weingarten surfaces of Bryant type in hyperbolic 3-space~\\cite{GMM2004}, \n\\item linear Weingarten surfaces of Bianchi type in de Sitter 3-space~\\cite{AE2007}.\n\\end{itemize}\nThe ingredients for such representations are always the same: a meromorphic function and a holomorphic 1-form. Thus one might expect that these representations are related in some way. A unification of some of these representations was achieved in~\\cite{AGM2005} under the umbrella of marginally trapped surfaces in Minkowski space. \n\nOn the other hand, geometric interpretations of these representations have been sought in various works. The classical Weierstrass-Enneper representation can be understood using the Christoffel transformation of isothermic surfaces~\\cite{H2003}. The Umehara-Yamada perturbation~\\cite{UY1992} deforms minimal surfaces in Euclidean 3-space into CMC-1 surfaces in hyperbolic 3-space. This perturbation was given a M\\\"{o}bius geometric interpretation in~\\cite{HMN2001}. A Laguerre geometric interpretation of this perturbation was studied in~\\cite{MN2016}, extending this notion to a wider class of surfaces.  \n \nMotivated by an observation in~\\cite{BHR2012}, we seek to understand Weierstrass-type representations using transformations of $\\Omega$-surfaces. In~\\cite{D1911iii}, using the Christoffel transformation for isothermic surfaces in Minkowski space, Demoulin develops a notion of dual surfaces for $\\Omega$-surfaces, yielding a transformation for this surface class. We show that L-isothermic surfaces can be characterised as \n$\\Omega$-surfaces admitting a special dual surface, akin to how minimal surfaces in Euclidean space can be characterised as Christoffel transformations of their Gauss map. We then give an invariant explanation of all of the aforementioned Weierstrass-type representations as an application of Demoulin's dual transformation to a prescribed Gauss map.  \n\n\\textit{Acknowledgements.} The author would like to thank F. Burstall, J. Cho, U. Hertrich-Jeromin, W. Rossmann and M. Yasumoto for fruitful and pleasant conversations about this topic. This work was supported by the Austrian Science Fund (FWF) through the research project P28427-N35 ``Non-rigidity and symmetry breaking\".\n\n\\section{$\\Omega$-surfaces}\n\nLet $x:\\Sigma\\to \\mathbb{R}^{3,1}$ be a spacelike immersion. As defined in~\\cite{C1867,P1988} a smooth map $x^{*}:\\Sigma\\to \\mathbb{R}^{3,1}$ is called a \\textit{Christoffel dual} of $x$ if $x$ and $x^{*}$\n\\begin{itemize}\n\\item have parallel tangent planes, \n\\item induce the same conformal structure on $T\\Sigma$, and\n\\item induce opposite orientations on $T\\Sigma$.\n\\end{itemize}\nIt was shown\\footnote{In fact, it was shown using Clifford algebra that this relation can be expressed by the vanishing of a single wedge.} in~\\cite{B2006} that these conditions can be encapsulated by\n\\begin{equation}\n\\label{eqn:cdual}\n(dx\\wedge dx^{*})=0 \\quad \\text{and}\\quad dx\\curlywedge dx^{*}=0, \n\\end{equation}\nwhere $(dx\\wedge dx^{*})$ is a symmetric 2-form defined by \n\\[ (dx\\wedge dx^{*})(X,Y) = (dx(X),dx^{*}(Y))-(dx(Y),dx^{*}(X)),\\]\nand $dx\\curlywedge dx^{*}$ is a symmetric 2-form with values in $\\wedge^{2}\\underline{\\mathbb{R}}^{3,1}$ (with $\\underline{\\mathbb{R}}^{3,1}$ denoting the trivial bundle $\\Sigma\\times\\mathbb{R}^{3,1}$) defined by \n\\[ dx\\curlywedge dx^{*}(X,Y) = dx(X)\\wedge dx^{*}(Y)-dx(Y)\\wedge dx^{*}(X).\\]\nA surface which possesses a Christoffel dual is then said to be \\textit{isothermic}. By the symmetric nature of~\\eqref{eqn:cdual}, $x^{*}$ is itself isothermic with Christoffel dual $x$. \n\nIn Laguerre geometry, one uses isotropy projection to identify points in $\\mathbb{R}^{3,1}$ with spheres (see for example~\\cite{C2008}). Therefore a spacelike immersion $x:\\Sigma\\to\\mathbb{R}^{3,1}$ represents a congruence of spheres. Given a spacelike immersion $x:\\Sigma\\to\\mathbb{R}^{3,1}$, we may write the normal bundle of $x$ as $dx(T\\Sigma)^{\\perp}=G\\oplus \\tilde{G}$, where $G$ and $\\tilde{G}$ are null line subbundles of $dx(T\\Sigma)^{\\perp}$. We call $G$ and $\\tilde{G}$ the \\textit{lightlike Gauss maps} of $x$. We may then construct null affine line bundles $L:=x+G$ and $\\tilde{L}:=x+\\tilde{G}$ of $\\underline{\\mathbb{R}}^{3,1}$. In Laguerre geometry, these represent the envelopes of the sphere congruence $x$ (see for example~\\cite{MN2000}). \n\nAs classically defined by Demoulin~\\cite{D1911iii,D1911i,D1911ii}, the envelopes of isothermic (spacelike) sphere congruences are called $\\textit{$\\Omega$-surfaces}$. Therefore, $\\Omega$-surfaces are the null affine line bundles $L$ of $\\underline{\\mathbb{R}}^{3,1}$, for which we may write $L=x+G$ for some isothermic surface $x:\\Sigma\\to\\mathbb{R}^{3,1}$. Given Christoffel dual maps $x$ and $x^{*}$, we say that envelopes $L$ and $L^{*}$ of $x$ and $x^{*}$, respectively, are \\textit{$\\Omega$-dual} if $L$ is parallel\\footnote{It is always possible to arrange $L$ and $L^{*}$ to be parallel, since the normal bundles of $x$ and $x^{*}$ are the same.} to $L^{*}$. \n\n\\section{Marginally trapped surfaces}\n\\label{sec:margtrap}\n\nA spacelike immersion $x:\\Sigma\\to \\mathbb{R}^{3,1}$ is called \\textit{marginally trapped} if its mean curvature vector\n\\[ \\textbf{H} = \\frac{1}{2}(d_{X}d_{X}x + d_{Y}d_{Y}x)^{\\perp}\\]\nis lightlike, where $X,Y\\in \\Gamma T\\Sigma$ is an orthonormal basis with respect to the induced metric of $x$ and $(.)^{\\perp}$ denotes projection onto the normal bundle of $x$. Given $p\\in \\Sigma$, $\\textbf{H}(p)$ is then lightlike if and only if $\\textbf{H}(p)\\in G(p)$ or $\\textbf{H}(p)\\in \\tilde{G}(p)$, where $G$ and $\\tilde{G}$ are the lightlike Gauss maps of $x$. We say that a marginally trapped surface $x$ is \\textit{regular} if either $\\textbf{H}(p)\\in G(p)$ for all $p\\in\\Sigma$ or $\\textbf{H}(p)\\in \\tilde{G}(p)$ for all $p\\in\\Sigma$, i.e., $\\textbf{H}\\in \\Gamma G$ or $\\textbf{H}\\in \\Gamma \\tilde{G}$. \n\nAssume now that the normal bundle of $x$ is flat. Then there locally exists parallel sections $g\\in \\Gamma G$ and $\\tilde{g}\\in \\Gamma \\tilde{G}$ with $(g,\\tilde{g})=-1$. Since $g$ is parallel, we may write $dg=dx\\circ S$ for some $S\\in \\Gamma \\End(T\\Sigma)$. Since $(dx,dg)$ is symmetric, we have that $S$ is symmetric with respect to the induced metric of $x$. Therefore, $S$ admits a basis $X,Y\\in \\Gamma T\\Sigma$ of orthonormal (with respect to the induced metric of $x$) eigenvectors with respective eigenvalues $\\alpha$ and $\\beta$. Now the mean curvature vector field is given by \n\\begin{align*}  \n\\textbf{H}=- \\frac{1}{2}((d_{X}d_{X}x + d_{Y}d_{Y}x,g)\\tilde{g} + (d_{X}d_{X}x + d_{Y}d_{Y}x,\\tilde{g})g)= \\frac{1}{2}(\\alpha + \\beta)\\tilde{g} \\, mod\\, G.\n\\end{align*}\nThus, $\\textbf{H}\\in \\Gamma G$ if and only if $\\alpha+\\beta=0$, i.e., $S$ is trace-free. On the other hand \n\\[ dx\\curlywedge dg(X,Y) = (\\beta+ \\alpha) d_{X}x\\wedge d_{Y}x.\\]\nThus $S$ is trace-free if and only if $dx\\curlywedge dg=0$. \n\nIf $dx\\curlywedge dg=0$ for some $g\\in \\Gamma G$, then $(ddx,g) =- (dx\\wedge dg)=0$. Thus, $dx\\curlywedge dg=0$ implies that $x$ and $g$ are Christoffel dual. We thus arrive at the following theorem:\n\n\\begin{theorem}\nA spacelike immersion in Minkowski space is a regular marginally trapped surface with flat normal bundle if and only if it is Christoffel dual to a section of one of its lightlike Gauss maps. \n\\end{theorem}\n\nSuppose that $x:\\Sigma\\to\\mathbb{R}^{3,1}$ is Christoffel dual to $g\\in \\Gamma G$. By defining $L:=x+G$, we see that $L$ is $\\Omega$-dual to $L^{*}:=G$. Moreover, since $x$ has the same induced conformal structure on $T\\Sigma$ as $G$, one identifies $x$ as the \\textit{middle sphere congruence} of $L$, see~\\cite{B1929,MN2018}. One deduces from~\\cite{BHPR2018} that such $L$ are the envelopes of \\textit{L-isothermic surfaces}, that is, surfaces in Euclidean 3-space that admit curvature line coordinates which are conformal with respect to the third fundamental form. This characterisation gives an analogue of minimal surfaces being Christoffel dual to their Gauss map in Euclidean space: \n\n\\begin{corollary}\nEnvelopes of L-isothermic surfaces are the $\\Omega$-surfaces that are $\\Omega$-dual to their lightlike Gauss map. \n\\end{corollary}\n\nNow, the condition $dx\\curlywedge dg =0$, implies that\\footnote{Throughout this paper we shall use the well-known identification $\\wedge^{2}\\mathbb{R}^{3,1}\\cong \\mathfrak{so}(3,1)$, via $(a\\wedge b)v = (a,v)b-(b,v)a$.} $\\zeta:= g\\wedge dx\\in \\Omega^{1}(G\\wedge G^{\\perp})$ is a closed 1-form. Moreover, since $G\\wedge G^{\\perp}$ is an abelian subbundle of $\\wedge^{2}\\underline{\\mathbb{R}}^{3,1}$, one has that $[\\zeta\\wedge \\zeta]=0$. Therefore, $\\{d+t\\zeta\\}_{t\\in\\mathbb {R}}$ is a 1-parameter family of flat metric connections. Let $T_{t}:\\Sigma\\to O(3,1)$ be the local orthogonal trivialising gauge transformations of $d+t\\zeta$, i.e., \n\\[ T_{t}(d+t\\zeta)T_{t}^{-1} = d.\\]\nSuch transformations are unique up to premultiplication by a constant endomorphism $A\\in O(3,1)$. Now $T_{t}dx\\in \\Omega^{1}(\\underline{\\mathbb{R}}^{3,1})$ is a closed 1-form: \n\\[ d(T_{t}dx) = T_{t}(d+t\\zeta)dx = T_{t}(ddx + t\\zeta\\wedge dx) =0\\]\nsince $\\zeta\\wedge dx =-(dx\\wedge dx) g = 0$. Thus we may integrate to obtain a new surface $x_{t}:\\Sigma\\to\\mathbb{R}^{3,1}$ satisfying $dx^{t} = T_{t}dx$. Define $G_{t}:= T_{t}G$ and set $g_{t}:=T_{t}g$. Then $dg_{t} = T_{t}(d+t\\zeta)g=T_{t}dg$, since $\\zeta g=0$, and one deduces that \n\\[ dx_{t}\\curlywedge dg_{t} = T_{t} \\, dx\\curlywedge dg \\, T_{t}^{-1} = 0.\\]\nThus $x^{t}$ is a marginally trapped surface and $L_{t}:= x_{t}+G_{t}$ is the envelope of an L-isothermic surface. This is the \\textit{T-transform} of L-isothermic surfaces (see~\\cite{MN2016,MN2018}). Notice that we obtain a new closed 1-form \n\\[ \\zeta_{t}:= g_{t}\\wedge dx_{t} = Ad_{T_{t}}\\zeta\\in \\Omega^{1}(G_{t}\\wedge G_{t}^{\\perp}) .\\]\nWe therefore obtain local orthogonalising gauge transformations $T^{t}_{s}:\\Sigma\\to O(3,1)$ of the 1-parameter family of flat connections $\\{d+s\\zeta_{t}\\}_{s\\in\\mathbb{R}}$. In~\\cite[Section 5.5.9]{H2003} the following property was shown for iterating these transformations: \n\\begin{equation}\n\\label{eqn:iterate}\nT^{t}_{s}T_{t} = T_{t+s},\\quad \\text{and thus}\\quad  T^{t}_{-t} = (T_{t})^{-1}. \n\\end{equation}\nThis property will be useful for us in the following section. \n\n\\section{Weierstrass-type representations}\n\nThe ingredients for Weierstrass-type representations are:\n\\begin{itemize}\n\\item \\textbf{A simply connected Riemann surface $\\Sigma$}. Equivalently, $\\Sigma$ is a simply connected 2-dimensional manifold \nequipped with a conformal structure and an orientation. \n\n\\item \\textbf{A meromorphic function $\\phi:\\Sigma\\to \\mathbb{C}\\cup \\{\\infty\\}$}. Equivalently, since $\\mathbb{C}\\cup \\{\\infty\\} \\cong S^{2}\\cong \\mathbb{P}(\\mathcal{L})$, where $\\mathcal{L}\\subset\\mathbb{R}^{3,1}$ denotes the lightcone, we can identify $\\phi$ with a smooth map $G:\\Sigma\\to \\mathbb{P}(\\mathcal{L})$. $\\phi$ being meromorphic is equivalent to $G$ being an orientation preserving map whose induced conformal structure is weakly equivalent to the conformal structure on $\\Sigma$. \n\n\\item \\textbf{A holomorphic 1-form $\\omega$}. Alternatively, one may prescribe a holomorphic quadratic differential\\footnote{That is, we may write $q^{2,0} = h dz^{2}$ for some local holomorphic coordinate $z$ on $\\Sigma$ and some holomorphic function $h$.} $q$. We then have the relation $q=d\\phi\\, \\omega + d\\bar{\\phi}\\,\\bar{\\omega}$. \n\\end{itemize}\nWe make the assumption\\footnote{Since $\\phi$ is meromorphic, this excludes the case that $G$ is constant and isolated points of $\\Sigma$ where $G$ does not immerse.} that $\\phi$ has no critical points, and thus $G$ is an immersion. Now for any non-zero lift $g\\in \\Gamma G$, we may write $q = (dg,dg\\circ Q)$ for some endomorphism $Q\\in \\Gamma \\End(T\\Sigma)$. $q$ is then a holomorphic quadratic differential if and only if\n\\[ \\zeta := g\\wedge dg\\circ Q\\in \\Omega^{1}(G\\wedge G^{\\perp})\\]\nis a closed 1-form (see~\\cite{BS2012,S2008}). \n\nExplicitly, one may identify a meromorphic function $\\phi$ with the map $G:\\Sigma\\to\\mathbb{P}(\\mathcal{L})$ spanned by the lift\\begin{equation}\n\\label{eqn:secG}\n g=(1+\\phi\\bar{\\phi})e_{0} + (\\phi +\\bar{\\phi})e_{1} - i(\\phi - \\bar{\\phi}) e_{2} + (-1+\\phi\\bar{\\phi})e_{3}\\in \\Gamma G,\n\\end{equation}\nwhere $\\{e_{0},...,e_{3}\\}$ is a pseudo-orthonormal basis for $\\mathbb{R}^{3,1}$ with $e_{0}$ timelike and $e_{1},e_{2},e_{3}$ spacelike. The induced metric of $g$ is $(dg,dg) = 4 d\\phi d\\bar{\\phi}$. By defining \n\\[ Q = \\frac{1}{2}\\left( \\bar{\\omega}\\otimes \\frac{\\partial}{\\partial \\phi}  +  \\omega\\otimes\\frac{\\partial}{\\partial \\bar{\\phi}}\\right),\\]\nwe have that $(dg,dg\\circ Q) = d\\phi \\omega + d\\bar{\\phi}\\bar{\\omega} =q$. One then computes\n\\begin{align}\n\\label{eqn:zeta}\n\\zeta = g\\wedge dg\\circ Q &= \\frac{1}{2}\\{(e_{0}+\\phi e_{1}-i\\phi e_{2}-e_{3})\\wedge (\\phi e_{0} + e_{1} + i e_{2} + \\phi e_{3})\\omega\\\\\n&+ (e_{0}+\\bar{\\phi} e_{1}+i\\bar{\\phi} e_{2}-e_{3})\\wedge (\\bar{\\phi} e_{0} + e_{1} - i e_{2} + \\bar{\\phi} e_{3})\\bar{\\omega}\\}\\nonumber.\n\\end{align}\nNotice that given Weierstrass data $(\\phi, \\omega)$, the right hand side of~\\eqref{eqn:zeta} yields a closed 1-form with values in $G\\wedge G^{\\perp}$, regardless of whether $\\phi$ has critical points or not. \n\n\\subsection{Affine hyperplanes in $\\mathbb{R}^{3,1}$}\n\\label{subsec:affine}\nLet $\\mathfrak{p}\\in\\mathbb{R}^{3,1}$ be a non-zero vector. Then hyperplanes with normal $\\mathfrak{p}$ are flat 3-dimensional affine spaces. Given a closed $\\zeta\\in \\Omega^{1}(G\\wedge G^{\\perp})$ we have that $\\zeta\\mathfrak{p}\\in \\Omega^{1}(\\underline{\\mathbb{R}}^{3,1})$ is a closed 1-form and we may locally integrate it to obtain \n\\[ x:\\Sigma\\to \\mathbb{R}^{3,1}\\quad \\text{satisfying} \\quad dx = - \\zeta\\mathfrak{p}.\\] \nNow $d(x,\\mathfrak{p})= (\\zeta\\mathfrak{p},\\mathfrak{p})=0$, since $\\zeta$ is skew-symmetric. Thus $x$ takes values in an affine hyperplane with normal $\\mathfrak{p}$. If $G(p)\\perp \\mathfrak{p}$ for some $p\\in \\Sigma$ then $d_{p}x\\in G(p)$ and thus $x$ does not immerse at $p$. Away from such points, we may write $\\zeta = g\\wedge \\omega$, where $g\\in \\Gamma G$ satisfies $(g,\\mathfrak{p})=-1$ and $\\omega\\in \\Omega^{1}(G^{\\perp})$. Then $dx=-\\zeta\\mathfrak{p} = \\omega \\, mod\\, G$ implies that $\\zeta = g\\wedge dx$ and the closedness of $\\zeta$ implies that $dg\\curlywedge dx=0$. Hence, $L:=x+G$ is $\\Omega$-dual to $L^{*}:= G$ and $x$ is marginally trapped. In fact, since $\\mathfrak{p}$ lies in $dx(T\\Sigma)^{\\perp}$, one deduces that $\\textbf{H}=0$. Hence, $x$ has zero mean curvature. We thus have the following 3 cases:\n\\begin{enumerate}\n\\item if $\\mathfrak{p}$ is timelike then $x$ is a minimal surface in a Euclidean 3-space,\n\\item if $\\mathfrak{p}$ is spacelike then $x$ is a maximal surface in a Lorentzian 3-space, \n\\item if $\\mathfrak{p}$ is lightlike then $x$ has zero mean curvature in an isotropic 3-space. \n\\end{enumerate}\nIn cases (1) and (2) we obtain a unit (spacelike or timelike, respectively) normal of $x$ by setting $n:= g-\\mathfrak{p} \\in \\Gamma \\mathfrak{p}^{\\perp}$. \n\nBy choosing $\\mathfrak{p}=e_{0}$, one deduces from~\\eqref{eqn:zeta} that \n\\[ dx = -\\zeta e_{0} = \\text{Re}\\{((1- \\phi^{2})e_{1} + i(1 + \\phi^{2})e_{2} +2\\phi e_{3})\\omega\\}\\]\nand we thus recover the Weierstrass-Enneper representation~\\cite{W1866}. By choosing $\\mathfrak{p}=e_{3}$, we have that \n\\[ dx = - \\zeta e_{3} = \\text{Re}\\{(2\\phi e_{0} + (1+ \\phi^{2})e_{1} + i(1 -\\phi^{2})e_{2})\\omega\\},\\]\nrecovering the representation of~\\cite{K1983} for maximal surfaces in Minkowski 3-space. Choosing $\\mathfrak{p}= \\frac{e_{0}+e_{3}}{2}$ we obtain a representation of zero mean curvature surfaces in isotropic 3-space:\n\\[ dx = -\\zeta\\tfrac{e_{0}+e_{3}}{2}= \\text{Re}\\left\\{\\left( e_{1}+ie_{2} + 2\\phi\\tfrac{e_{0}+e_{3}}{2}\\right)\\omega\\right\\}.\\]\n\\begin{remark}\nIn~\\cite{MN2016} it was shown that surfaces in Euclidean space that are simultaneously L-minimal and L-isothermic are those surfaces whose middle sphere congruence is one of the three cases above. On the other hand, a Weierstrass-type representation was developed for such surfaces in~\\cite{S2009}. One can recover this representation by intersecting the envelopes $L$ of the three cases above with appropriate affine Euclidean 3-spaces. \n\\end{remark}\n\n\n\n\\subsection{Quadrics in $\\mathbb{R}^{3,1}$}\n\\label{subsec:quadrics}\nGiven a closed 1-form $\\zeta\\in \\Omega^{1}(G\\wedge G^{\\perp})$ we have that $\\{d+t\\zeta\\}_{t\\in\\mathbb{R}}$ is a 1-parameter family of flat connections. Therefore, there locally exists parallel sections of these connections. Suppose that $x:\\Sigma\\to \\mathbb{R}^{3,1}$ satisfies \n\\[ (d+m\\zeta)x = 0\\]\nfor some\\footnote{The choice of $m$ here amounts to a constant scaling of the Hopf differential. In many works this scaling is fixed by choosing $m$ appropriately.} non-zero $m\\in\\mathbb{R}$. Then $d(x,x) = -2m(\\zeta x,x)=0$, since $\\zeta$ is skew-symmetric. Hence, $(x,x)$ is constant. If $x(p)\\perp G(p)$ for some $p\\in \\Sigma$, then $d_{p}x\\in G(p)$ and thus $x$ does not immerse at $p$. Away from these points, we may write $\\zeta = g\\wedge \\omega$ where $g\\in \\Gamma G$ such that $(g,x)=-1$ and $\\omega\\in \\Omega^{1}(G^{\\perp})$. Then the condition $dx= -m\\zeta x$ implies that $\\zeta = \\frac{1}{m}g\\wedge dx$. The closedness of $\\zeta$ then implies that $dx\\curlywedge dg=0$. Hence, $L:=x+G$ is $\\Omega$-dual to $L^{*}=G$ and $x$ is marginally trapped. From~\\cite{HI2015} we then obtain the following 3 cases:\n\\begin{enumerate}\n\\item if $(x,x)=-c^{2}$ then $x$ is a CMC-c surface in $\\mathbb{H}^{3}(-c^{2})$,\n\\item if $(x,x)=c^{2}$ then $x$ is a CMC-c surface in $\\mathbb{S}^{2,1}(c^{2})$,\n\\item if $(x,x)=0$ then $x$ is an intrinsically flat surface in $\\mathcal{L}$. \n\\end{enumerate}\nIn cases (1) and (2) $G$ then has a geometric interpretation as the hyperbolic Gauss map of $x$. An important observation is that parallel sections $x$ of $d+m\\zeta$ are given by $x = T^{-1}_{m}\\mathfrak{c}$, where $\\mathfrak{c}\\in \\mathbb{R}^{3,1}$ and $T_{m}$ is a local trivialising orthogonal gauge transformations of $d+m\\zeta$, i.e., $T_{m}(d+m\\zeta)T_{m}^{-1}=d$ (see Section~\\ref{sec:margtrap}). \n\nThe Hermitian model of $\\mathbb{R}^{3,1}$ identifies points in $\\mathbb{R}^{3,1}$ with $2\\times 2$ Hermitian matrices via the isometry\n\\[ x_{0}e_{0} + x_{1}e_{1} + x_{2} e_{2} + x_{3}e_{3} \\mapsto \\begin{pmatrix} x_{0}+ x_{3} & x_{1} + ix_{2}\\\\ x_{1}-ix_{2} & x_{0}-x_{3}\\end{pmatrix},\\]\nwhere the metric on the space of Hermitian matrices is given by $(A,A)=-\\det A$. Our basis vectors become\n\\[ e_{0} = \\begin{pmatrix} 1&0\\\\0&1\\end{pmatrix}, \\, e_{1} = \\begin{pmatrix} 0&1\\\\1&0\\end{pmatrix},\\, e_{2} = \\begin{pmatrix} 0&i\\\\-i&0\\end{pmatrix},\\, e_{3}=\\begin{pmatrix} 1&0\\\\0&-1\\end{pmatrix}.\\]\nWe identify $SL(2,\\mathbb{C})$ with the orthogonal group $O(3,1)$ by its action on Hermitian matrices\n\\[ A\\cdot v = AvA^{*}.\\]\nSkew-symmetric endomorphisms, i.e., elements of $\\mathfrak{o}(3,1)$, are then identified with elements of $\\mathfrak{sl}(2,\\mathbb{C})$ via \n\\[ B\\cdot v = Bv + vB^{*}.\\]\nIn particular, the skew-symmetric endomorphisms $e_{i}\\wedge e_{j}$ are identified with $e_{ij}\\in \\mathfrak{sl}(2,\\mathbb{C})$, where \n\\[ e_{01} = -\\frac{1}{2}e_{1}, \\, e_{02} = -\\frac{1}{2}e_{2}, \\, e_{03} = -\\frac{1}{2}e_{3}, \\, e_{12} = \\frac{i}{2} e_{3}, \\, e_{13} = -\\frac{i}{2}e_{2}, \\, e_{23} = \\frac{i}{2}e_{1}.\\]\nOne then computes that $\\zeta$ from~\\eqref{eqn:zeta} is identified with \n\\[ \\xi = \\frac{1}{2}\\{ -(1-\\phi^{2}) e_{1} - i(1+\\phi^{2})e_{2} - 2\\phi e_{3} \\} \\omega = \\begin{pmatrix} -\\phi & \\phi^{2} \\\\ -1 &\\phi\\end{pmatrix}\\omega \\in \\Omega^{1}(\\mathfrak{sl}(2,\\mathbb{C})).\\]\nSince $\\xi$ is a closed 1-form, there exists $F_{t}:\\Sigma\\to SL(2,\\mathbb{C})$ so that \n\\[ F_{t}^{-1}dF_{t} = t\\xi.\\]\nOne quickly deduces that the orthogonal transformations identified with $F_{t}$ are in fact local trivialising orthogonal gauge transformations $T_{t}$ of $d+t\\zeta$. Thus parallel sections of $d+m\\zeta$ are given by $F^{-1}_{m}\\mathfrak{c}(F^{-1}_{m})^{*}$ for $\\mathfrak{c}\\in\\mathbb{R}^{3,1}$. Defining $\\Psi_{m} := F^{-1}_{m}$ we have that \n\\begin{equation}\n\\label{eqn:Psi}\nd\\Psi_{m} = -m\\xi \\Psi_{m} =m \\begin{pmatrix} \\phi & -\\phi^{2} \\\\ 1 &-\\phi\\end{pmatrix}\\omega \\Psi_{m}.\n\\end{equation}\nNow, by setting $\\mathfrak{c} :=\\frac{1}{2}((1-\\mu)e_{0}+(1+\\mu)e_{3})= \\begin{pmatrix} 1 & 0\\\\ 0&-\\mu\\end{pmatrix}$ for $\\mu\\in \\mathbb{R}$ and \n\\begin{equation} \n\\label{eqn:hyprep}\nx = \\Psi_{m}\\mathfrak{c}(\\Psi_{m})^{*}\n\\end{equation}\nwe obtain \n\\begin{itemize}\n\\item CMC $H\\equiv \\frac{1}{\\sqrt{-\\mu}}$ surfaces in $\\mathbb{H}^{3}(\\frac{1}{\\mu})$ when $\\mu<0$; \n\\item CMC $H\\equiv \\frac{1}{\\sqrt{\\mu}}$ surfaces in  $\\mathbb{S}^{2,1}(\\frac{1}{\\mu})$ when $\\mu>0$;\n\\item intrinsically flat surfaces in $\\mathcal{L}$ when $\\mu=0$. \n\\end{itemize}\nWhen $\\mu = -1$ we see that this coincides with the representation of CMC $H\\equiv 1$ in hyperbolic 3-space surfaces given in~\\cite[Corollary 2.4]{RUY1997} and when $\\mu=1$ this coincides with the representation of CMC $H\\equiv 1$ in de-Sitter 3-space given in~\\cite[Theorem 1]{F2007}. \n\n\\subsection{The Umehara-Yamada perturbation}\nSuppose that $x$ is a parallel section of $d+m\\zeta$ for some non-zero $m\\in \\mathbb{R}$, and thus $x= T^{-1}_{m}\\mathfrak{c}$ for some non-zero $\\mathfrak{c}\\in \\mathbb{R}^{3,1}$. Then the T-transform $x_{m}$ of $x$ satisfies \n\\[ dx_{m} = T_{m}dx= -T_{m}m\\zeta x = - m\\zeta_{m} T_{m}x = - m\\zeta_{m}\\mathfrak{c} .\\] \nHence, from Subsection~\\ref{subsec:affine} we know that $x_{m}$ is a zero mean curvature surface in an affine hyperplane of $\\mathbb{R}^{3,1}$. Therefore, we recover the result of~\\cite{MN2016} that the T-transform of L-isothermic surfaces perturbs the cases (1), (2), (3) of Subsection~\\ref{subsec:quadrics} into the cases (1), (2), (3) of Subsection~\\ref{subsec:affine}, respectively. This generalises the Umehara-Yamada perturbation~\\cite{UY1992} and gives a Laguerre geometric analogue of its interpretation in~\\cite{HMN2001}. \n\nWe call $G_{m}=T_{m}G$ the \\textit{secondary Gauss map} of $x$. Since $G_{m}$ and $G$ induce the same conformal structures on $T\\Sigma$, there exists a holomorphic function $\\psi$ such that\n\\begin{equation} \n\\label{eqn:secGauss}\ng_{m} = (1+\\psi\\bar{\\psi})e_{0} + (\\psi +\\bar{\\psi})e_{1} - i(\\psi - \\bar{\\psi}) e_{2} + (-1+\\psi\\bar{\\psi})e_{3}\n\\end{equation}\nis a lift of $G_{m}$. We then define a holomorphic 1-form $\\eta$ so that $q = d\\psi \\eta + d\\bar{\\psi}\\bar{\\eta}$. The closed 1-form $\\zeta_{m}= Ad_{T_{m}}\\zeta$ is then identified with \n\\[ \\xi_{m}:= \\begin{pmatrix} -\\psi & \\psi^{2} \\\\ -1 &\\psi\\end{pmatrix}\\eta.\\]\nNow parallel sections $x$ of $d+m\\zeta$ are of the form $x=T^{-1}_{m}\\mathfrak{c} = T^{m}_{-m}\\mathfrak{c}$, using~\\eqref{eqn:iterate}. $T^{m}_{-m}$ satisfies $dT^{m}_{-m}= -mT^{m}_{-m}\\zeta_{m}$ and correspondingly $\\Psi_{m}= F^{-1}_{m}$ satisfies \n\\begin{equation}\n\\label{eqn:secondrep}\nd\\Psi_{m} = -m\\Psi_{m}\\xi_{m} = m\\Psi_{m}\\begin{pmatrix} \\psi & -\\psi^{2} \\\\ 1 &-\\psi\\end{pmatrix}\\eta.\n\\end{equation}\nThis formulation shows that the representation~\\eqref{eqn:hyprep} coincides with the representation of~\\cite{UY1992} when $\\mu=-1$ and with the representation of~\\cite{AA1998} when $\\mu=1$. The duality between~\\eqref{eqn:Psi} and~\\eqref{eqn:secondrep} has been remarked upon in~\\cite{BPS2003,RUY1997}. \n\n\\subsection{Linear Weingarten surfaces of Bryant type}\nA surface $x:\\Sigma\\to \\mathbb{H}^{3}$ is a \\textit{linear Weingarten surface of Bryant type} if the mean curvature $H$ and Gauss curvature $K$ of $x$ satisfy a relation\n\\begin{equation} \n\\label{eqn:brytype}\n(\\mu+1)K-2\\mu H + \\mu-1=0\n\\end{equation}\nfor some $\\mu\\in\\mathbb{R}$. A Weierstrass-type representation was developed for these surfaces in~\\cite{GMM2004}. The middle sphere congruence of such surfaces is given by $x^{M} = x-\\frac{\\mu+1}{2}\\tilde{g}$, where $\\tilde{g}\\in \\Gamma G$ such that $(\\tilde{g},x)=-1$. Three cases emerge\\footnote{This analysis is analogous to that performed in~\\cite[Section 4.6]{BHR2014} for parallel families of linear Weingarten surfaces in hyperbolic space.}:\n\\begin{itemize}\n\\item if $\\mu<0$ then $x^{M}$ is a CMC-$\\frac{1}{\\sqrt{-\\mu}}$ surface in $\\mathbb{H}^{3}(\\frac{1}{\\mu})$, \n\\item if $\\mu>0$ then $x^{M}$ is CMC-$\\frac{1}{\\sqrt{\\mu}}$ surface in $\\mathbb{S}^{2,1}(\\frac{1}{\\mu})$, \n\\item if $\\mu=0$ then $x^{M}$ is an intrinsically flat surface in $\\mathcal{L}$.\n\\end{itemize}\nWe thus have that \n\\[x^{M} = \\Psi_{m}\\begin{pmatrix} 1 & 0\\\\ 0&-\\mu\\end{pmatrix}\\Psi^{*}_{m}\\]\nwhere $\\Psi_{m}:\\Sigma\\to SL(2,\\mathbb{C})$ satisfies~\\eqref{eqn:secondrep}. \nUsing the Hermitian model, the lift $g_{m}$ from~\\eqref{eqn:secGauss} of the secondary Gauss map is given by\n\\[ g_{m} = 2\\begin{pmatrix}\n|\\psi|^{2} & \\psi\\\\ \\bar{\\psi} & 1\n\\end{pmatrix}.\\]\nNow, since $\\tilde{g}\\in \\Gamma G$ satisfies $(\\tilde{g},x)=-1$ and $T^{-1}_{m}g_{m}\\in \\Gamma G$, \n\\[\n\\tilde{g} =- \\frac{T_{m}^{-1}g_{m}}{(T_{m}^{-1}g_{m},x)} =- \\frac{ \\Psi_{m}g_{m}\\Psi_{m}^{*}}{\\left(\\Psi_{m}g_{m}\\Psi_{m}^{*}, \\Psi_{m}\\begin{pmatrix} 1 & 0\\\\ 0&-\\mu\\end{pmatrix}\\Psi^{*}_{m}\\right)}= \\frac{\\Psi_{m}g_{m}\\Psi_{m}^{*}}{1-\\mu|\\psi|^{2}}.\n\\]\nThus\\footnote{Note that in order for this expression to be well defined, one must assume that $1-\\mu|\\psi|^{2}$ is nowhere zero. This coincides with the assumption of~\\cite{GMM2004}.}, \n\\begin{align*}\n x = x^{M} + \\frac{\\mu+1}{2}\\tilde{g} &= \\Psi_{m}\\left( \\begin{pmatrix} 1 & 0\\\\ 0&-\\mu\\end{pmatrix} + \\frac{\\mu+1}{1-\\mu|\\psi|^{2}} \\begin{pmatrix}\n|\\psi|^{2} & \\psi \\\\ \\bar{\\psi} & 1\n\\end{pmatrix} \\right) \\Psi^{*}_{m}\\\\\n&= \\frac{1}{1-\\mu|\\psi|^{2}} \\Psi_{m}\\begin{pmatrix} 1+|\\psi|^{2} & (\\mu+1)\\psi \\\\ (\\mu+1)\\bar{\\psi} & 1+\\mu^{2}|\\psi|^{2}\\end{pmatrix}\\Psi_{m}^{*}. \n\\end{align*}\nBy setting $H:= \\Psi_{m}\\begin{pmatrix} i\\psi & i\\\\i & 0\\end{pmatrix}$ we have that\n\\[ x = H\\begin{pmatrix} \\frac{1+\\mu^{2}|\\psi|^{2}}{1-\\mu|\\psi|^{2}} & \\mu\\bar{\\psi}\\\\ \\mu \\psi & 1-\\mu |\\psi|^{2}\\end{pmatrix}H^{*}.\\]\nWe deduce from~\\eqref{eqn:secondrep} that \n\\[ H^{-1}dH= \\begin{pmatrix} 0 & m \\eta\\\\ d\\psi & 0\\end{pmatrix}.\\]\nHence, we have recovered the representation of~\\cite{GMM2004}. Moreover, in the case that $\\mu=0$, we obtain the representation of~\\cite{GMM2000} for flat surfaces in $\\mathbb{H}^{3}$ . \n\n\\begin{remark}\n\\textit{Linear Weingarten surfaces of Bianchi type}, that is surfaces satisfying~\\eqref{eqn:brytype} in $\\mathbb{S}^{2,1}$, were shown to admit a Weierstrass-type representation in~\\cite{AE2007}. An analogous analysis can be performed for these surfaces as above. \n\\end{remark}\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Sec:Introduction}\n\n\nUltraluminous X-ray sources (ULXs) are empirically defined as bright, non-nuclear, point-like, {mainly} extragalactic sources with an X-ray luminosity $ \\rm L_X > 10^{39} \\, \\rm erg \\, s^{-1}$ (\\citealt{Kaaret_2017}). They are brighter than the Eddington luminosity limit for a $10 M\\textsubscript{\\(\\odot\\)}$ black hole (BH) and {can even reach luminosities in excess of}  $10^{41} \\, \\rm erg \\, s^{-1}$ in the canonical X-ray band (0.3-10 keV, \\citealt{Walton_2021}).\n\nMany conjectures have been made to account for the high luminosity of these sources. Initially, ULXs were thought to be powered by black holes with masses greater than 10\\,$M\\textsubscript{\\(\\odot\\)}$ and, potentially, in the intermediate-mass regime ($10^{\\,2-5}\\,M\\textsubscript{\\(\\odot\\)}$, \\citealt{Miller_2004}) {with the best IMBH candidate being HLX-1} (\\citealt{Webb2012}). An alternative scenario suggested stellar-mass black holes whose light was beamed along the line of sight (LOS) of the observer by a thick disc-wind cone (e.g. \\citealt{King_2001, Poutanen_2007}). However, the discovery of coherent X-ray pulsations from M\\,82 ULX-2 with NuSTAR provided unambiguous evidence in support of a ULX hosting a neutron star (NS, \\citealt{Bachetti_2014}). Other notable examples are NGC 7793 P13 and NGC 5907 ULX-1 with similar properties (\\citealt{Fuerst_2016,Israel_2017a}). \nNGC 7793 P13, the second discovered pulsating ULX (or pULX), is also the only ULX for which an upper limit to the mass of the compact object was dynamically obtained ($15 M\\textsubscript{\\(\\odot\\)}$, \\citealt{Motch2014}). The pulsating source NGC 5907 ULX-1 reaches $L_{X} \\sim 10^{41} \\, \\rm erg \\, s^{-1}$ (\\citealt{Fuerst_2017}), which corresponds to 500 times the Eddington limit of a NS, \\textit{de facto} making it the brightest NS known. As of today, only 10 pulsating ULXs (including transient ULXs, see e.g. \\citealt{King2020} and references therein) are confirmed among the {$\\sim$ 1800} ULXs known (\\citealt{Walton_2021}). However, the fraction of pulsating neutron stars might be higher in ULXs as only about 30 ULXs have data with sufficiently good quality to search for pulsations. This suggests that neutron stars might power $\\gtrsim 30$\\,\\% of the nearby, bright, ULXs (see e.g. \\citealt{Rodriguez_2020}).\n\nThe so-called \\textit{ultraluminous state} \\citep{Gladstone_2009} is characterized by a strong curvature between 2--10\\, keV (e.g. \\citealt{Walton_2020} and references therein), and often a soft excess below 2 keV (see Fig. \\ref{fig: Comparison X-ray spectra of the brightest ULX known}). \\citet{Sutton_2013} classified ULXs into three main regimes according to their spectral slope in the 0.3--10 keV band (soft ultraluminous\/SUL for $\\Gamma>2$ or hard ultraluminous\/HUL for $\\Gamma<2$). In the latter case, if the X-ray spectrum has a single peak and is dominated by a blackbody-like component in the 2--5 keV band, it is classified as in the broadened disc (BD) regime. ULXs often switch between different spectral regimes (\\citealt{Walton_2020}). The presence of the {low-energy} spectral turnover rules out models of sub-Eddington accretion onto intermediate mass black holes {because the temperature of the soft component would require black hole masses of up to 1000s $M\\textsubscript{\\(\\odot\\)}$ with ${\\dot M}$ (accretion rate) $\\sim 0.01 {\\dot M_{Edd}}$, which disagree with their soft spectra.}\n\nShort time-scales variability from seconds to hours has been observed in several ULXs \\citep{Heil_2009} {rarely associated to} quasi-periodic patterns \\citep{Alston_2021,Strohmayer_2003, Strohmayer_2007,Gurpide_2021a} but is generally at a lower level than in sub-Eddington AGN and black hole X-ray binaries. Long-term variability is also observed on time-scales of a few months and could be associated with super-orbital motions due to precession  and is most common in the HUL regime and, particularly, in pulsating ULXs \\citep{Walton_2016,Fuerst_2018,Brightman_2019,Gurpide_2021a}.\n\\citet{Heil_2010} and \\citet{DeMarco_2013} showed that the soft energy band lagged the hard band in the ULX NGC 5408 ULX-1 at frequencies of $\\sim$10 mHz.\nMore recently, very long soft lags of the order of a few ks were found in NGC 55 ULX-1, NGC 1313 ULX-1 and NGC 7456 ULX-1 which show a broad range of spectral hardness (\\citealt{Pinto_2017, Kara_2020, Pintore_2020}). The time lags range between a few seconds to ks and it is not clear whether they are produced by the same mechanism.{ They could be produced by down scattering of hard X-ray photons through a thick disc wind cone in the LOS. \nIn fact, super-Eddington accretion predicts the launch of powerful, relativistic ($\\sim 0.1c$), winds (\\citealt{Takeuchi2013}).}\n\nEvidence in support of winds in ULXs was found through the presence of strong, although unresolved, features at soft X-ray energies ($<$ 2 keV) in CCD low-resolution spectra \\citep{Strohmayer_2007}. Their time variability and correlation with the source spectral hardness suggested that they might be produced by the ULX itself in the form of a wind (\\citealt{Middleton2015b}). This scenario was unambiguously confirmed through the first detection and identification of emission and absorption lines in the high-resolution XMM-\\textit{Newton} (here and after XMM) \/RGS spectra of NGC 1313 ULX-1 and NGC 5408 ULX-1 (\\citealt{Pinto_2016}). In particular, the line-emitting gas is generally close to rest (see, e.g., \\citealt{Kosec_2021}) with some exceptions such as NGC 5204 ULX-1 ($v_{\\rm LOS}\\sim0.3  c$, \\citealt{Kosec_2018a}) and in NGC 55 ULX-1 ($0.01-0.08 c$, \\citealt{Pinto_2017}). The absorption lines are highly blueshifted  ($0.1-0.3 c$). The ionisation state increases with the outflow velocity and the source spectral hardness, which indicates a detection of hotter and faster phases coming from the inner region at lower inclinations (\\citealt{Pinto_2020a}). \n\nAn interesting sub-class of ULXs are ultraluminous supersoft sources (ULSs or SSUL regime). These objects are identified with X-ray spectra dominated by a cool blackbody-like component with $kT \\sim 0.1 \\ \\rm keV$, a bolometric luminosity $\\gtrsim 10^{39} \\ \\rm erg \\ s^{-1} $, and very little emission above 1 keV. The discovery of similar winds in archetypal, persistent, ULXs (\\citealt{Pinto_2016}) and very recently in NGC 247 ULX-ULS 1 (\\citealt{Pinto_2021}), and the presence of a variable hard X-ray tail in some ULSs suggested that they are similar super-Eddington accretors but viewed at a different angle with respect to the disc-wind cone. The fainter hard tail and strong variability such as the presence of dips in the lightcurves of ULSs may be the consequence of a higher inclination angle or a higher accretion rate and a thicker (and variable) wind in the LOS (see, e.g., \\citealt{Urquhart2016,Pinto_2017,Alston_2021,D'Ai_2021}).\n\\\\\n\\\\\nDespite two decades of dedicated studies several open questions remain unanswered. \n\\textit{Are ULX spectral transitions driven by stochastic changes in the wind or variations in the accretion rate \/ geometry? What is the fraction of matter lost into the wind and, therefore, the net accretion rate onto the compact object? What is the fraction of NS-powered ULXs?}\nULXs which exhibit strong spectral variability are the ideal targets to tackle them. \n\n\\begin{figure\n\t\t\\centering\n\t\t\\includegraphics[width=0.48\\textwidth]{Images\/ULX_sequence_all_edi_2020.pdf}\n        \\vspace{-0.5cm}\n\t\t\\caption{{\\small X-ray spectra of some brightest ULXs with the hardness increasing from the bottom to top. Note how the high- and low-flux spectra of NGC 55 ULX-1 link the ULX spectra with soft and intermediate hardness. Adapted from \\citet{Pinto_2017}. }} \n\t\n\t\t\\label{fig: Comparison X-ray spectra of the brightest ULX known}\n\t\t\\end{figure}\n\t\t\n\n\\subsection{NGC 55 ULX-1}\nULX-1 is the brightest X-ray source in the NGC 55 galaxy (see Fig.\\,\\ref{fig:NGC 55 composite}). At a distance of 1.94 Mpc\\footnote{https:\/\/ned.ipac.caltech.edu}, this source has an X-ray luminosity peak of about $4 \\times 10^{39} \\ \\rm erg \\ s^{-1}$ (see, e.g., \\citealt{Gurpide_2021a}). The X-ray light curve exhibits sharp drops and 100s-long dips, during which the source flux is quenched in the 2.0 - 4.5 keV band (\\citealt{Stobbart_2004}). The spectrum is very soft (if modelled with a powerlaw it yields a slope $\\Gamma = 4$, \\citealt{Pinto_2017}) and similar to the brightest ULSs, but with a stronger hard tail above 1 keV. It is possible to see from Fig. \\ref{fig: Comparison X-ray spectra of the brightest ULX known} that the X-ray spectrum of NGC 55 ULX-1 fits just in between the spectra of bright ULSs and the soft-intermediate spectra of ULXs and, therefore, the source can be considered as a link between these subclasses of ULXs.\nNGC 55 ULX-1 is the ideal target for our study as it is very bright, has a spectral curvature above 1 keV similar to (although less severe than) ULSs and both broadband and wind properties are halfway between ULSs and soft-intermediate ULXs (see \\citealt{Pinto_2020a}).\nMoreover, its X-ray luminosity often crosses the  $ 10^{39} \\ \\rm erg \/ s $ threshold, giving an opportunity to test disc structural changes around the Eddington limit, {assuming the accretor is a stellar mass BH or a NS}.\nFinally, its flux and spectral variability provide the workbench necessary to {fit L-T trends and break model degeneracies}.\n\n\n\n\nIn order to study the spectral variability of NGC 55 ULX-1 and the wind response to changes in the broadband flux, our team has requested and has been awarded with three full XMM orbits in different AOs (about 390 ks, PI: Pinto).\nThe first observation occurred in 2018 and caught the source in a low state, while the two latter ones were triggered in 2020 and 2021 in order to obtain well-exposed RGS spectra during intermediate-high states. More detail on the triggering and the RGS analysis will be provided in a forthcoming paper.\n\nThis paper is the first in a series and will focus on the broadband spectral variability. This paper is organized as follows. In Sect. \\ref{Observations and Spectral modelling} we report the details on the observations, the spectral modeling  in Sect. \\ref{Spectral modelling}. In Sect. \\ref{Discussion} we discuss our results and provide our conclusions in Sect. \\ref{Conclusions}. All uncertanties are at 1$\\sigma$ (68 \\% level).\n\n\n\n\n\t\t\n\n\\section{Observations and data reduction}\n\\label{Observations and Spectral modelling}\n\nXMM-\\textit{Newton} observed NGC 55 ULX-1 ten times over a period of 20 years. \nAs we can see from Table \\ref{table:observations log}, XMM-\\textit{Newton} observed NGC 55 ULX-1 six times with reasonably long ($> 30 $ ks) observations, which enabled us to study the evolution of the spectra over two decades. Four additional short ($< 10$ ks net) observations were taken in the recent years. This permits us to probe both short-term (hours-days) and long-term (months-years) variability time scales.\n\n\\begin{center}\n\t\\begin{table\n\\caption{Table of observations of the source NGC 55 ULX-1.}  \n \\renewcommand{\\arraystretch}{1.}\n \\small\\addtolength{\\tabcolsep}{0pt}\n \\vspace{0.1cm}\n\t\\centering\n\t\\scalebox{0.9}{%\n\t\\begin{tabular}{ccccccc}\n    \\toprule\n  \n  \n   \n    {{Obs. ID}}  &\n    {{Date}}  &\n    {{t$\\,_{\\rm tot}$ [s]}} &\n    {{t$\\,_{\\rm net, \\, EPIC-PN}$ [s]}} &\n    {{{{CR}$\\,_{\\rm EPIC-PN}$ [c\/s]}}} \\\\\n    \n    \\midrule\n0028740201 & 2001-11-14  & 33619  & 27223  & 1.197 $\\pm$ 0.007  \\\\\\midrule\n0028740101 & 2001-11-15  & 31518  & 24718  & 0.578 $\\pm$ 0.005 \\\\\\midrule\n0655050101 & 2010-05-24  & 127437 & 95295  & 0.772 $\\pm$ 0.003  \\\\\\midrule\n0824570101 & 2018-11-17  & 139800 & 90733  & 0.513 $\\pm$ 0.002 \\\\\\midrule\n0852610101 & 2019-11-27  & 11000  &  4079  & 1.10 $\\pm$ 0.02  \\\\\\midrule\n0852610201 & 2019-12-27  &  8000  &  4055  & 1.09 $\\pm$ 0.02  \\\\\\midrule\n0852610301 & 2020-05-11  &  9000  &  4909  & 0.422 $\\pm$ 0.009  \\\\\\midrule\n0852610401 & 2020-05-19  &  8000  &  3969  & 0.152 $\\pm$ 0.006  \\\\\\midrule\n0864810101 & 2020-05-24  & 132800 & 102158 & 0.707 $\\pm$ 0.003  \\\\\\midrule\n0883960101 & 2021-12-12  & 130200 & 92294  & 0.872 $\\pm$ 0.003  \\\\\\midrule\n    \\bottomrule\n    \\end{tabular}}\\label{table:observations log}\n       \\vspace{0.3cm}\n      \\begin{quotation}\\footnotesize\n      {t$\\,_{\\rm net}$ is the exposure time after the removal of periods with high solar flares and {CR}$\\,_{\\rm EPIC-PN}$ is the net source count rate.}\n      \\end{quotation}\n\n\\end{table}\n\\end{center}\n\nThe data were reduced with the \\textit{Science Analysis System} ({\\scriptsize{SAS}}) version 18.0.0\\footnote{https:\/\/www.cosmos.esa.int\/web\/XMM-\\textit{Newton}}.\nThe raw data were obtained from the XMM-\\textit{Newton} Science Archive (XSA)\\footnote{https:\/\/www.cosmos.esa.int\/web\/XMM-\\textit{Newton}\/xsa}. \nWe used recent calibration files (February 2021).\nWe ran the \\textit{epproc} and \\textit{emproc} tasks to build the EPIC-PN and EPIC-MOS 1,2 event files, respectively. \nThese are subsequently filtered for the flaring particle background. We chose the recommended cutting threshold in the lightcurves above 10 keV (count rate $<$ 0.5 c\/s for EPIC-PN and $<$ 0.35 c\/s for EPIC-MOS 1 and 2).\n\nWe extracted EPIC MOS 1-2 and PN images in three energy bands to create a false-color RGB image (red 0.3-1 keV, green 1-2 keV, blue 2-10 keV)\\footnote{https:\/\/sites.google.com\/cfa.harvard.edu\/saoimageds9}. The images from the same energy band and different observations were stacked to increase the statistics. The final RGB mosaic is shown in Fig.\\,\\ref{fig:NGC 55 composite} (bottom panel). ULX-1 is the yellow-white, brightest object, in the central-left region. Some fainter and harder (blue) X-ray binaries are present near the galaxy centre. The soft (red) source outside the galaxy contours is a field star in our Galaxy. Overlaid are the contours of surface brightness of the optical image obtained from the Digitized Sky Survey (DSS)\\footnote{DSS, https:\/\/irsa.ipac.caltech.edu\/data\/DSS\/} (top panel).\n\n\\begin{figure\n\t\t\\centering\n\t\t\\includegraphics[width=0.45\\textwidth]{Images\/ngc55dss.pdf}\n\t\t\\includegraphics[width=0.45\\textwidth]{Images\/ngc55x.pdf}\n       \n\t\t\\caption{\\small Upper panel: Optical image of NGC 55 obtained from the Digitized Sky Survey. \n\t\t\\ Lower panel: false-color RGB X-ray image obtained by extracting images in three energy bands (red 0.3-1 keV, green 1-2 keV, blue 2-10 keV) and stacking over all observations. The contours of the optical image are overlaid to show the ULX position in the galaxy.}\n\t\t\\label{fig:NGC 55 composite}\n\t\\end{figure}\n\n\nWe also extracted background-corrected lightcurves for ULX-1 and each individual observation after carefully selecting the source and the background regions using the \\textit{epiclccorr} task. For the source we selected a circular region of 0.5 arcmin radius centered on {the \\textit{Chandra} X-ray estimated position} (RA: 00$^{h}$ 15$^{m}$ 28.89$^{s}$ DEC: -39$^{\\circ}$ 13' 18.8''), while for the background we chose a larger circular region a few arc minutes away from the source and avoiding contamination from the copper ring (wherever it was possible) in the same chip of the source region and away from of chip gaps.\nThere might be a very small loss of counts in the observations, especially for Obs. ID: 0028740101, with the ULX-1 off-axis but this should be marginal given the softness of its spectra. Indeed, despite the larger off-axis angle, its spectrum appears as the brightest and hardest (see Sect. \\ref{Sec:spectra}).\n\t\n\t\n       \n\t\nThe 0.3-10\\,keV EPIC-PN lightcurves of the individual observations were joined with {{\\scriptsize PYTHON}} and shown in Fig. \\ref{fig: lightcurves} (top-left panel). The XMM lightcurves confirm that the source flux changed dramatically as anticipated in Sect. \\ref{Sec:Introduction}. In the early XMM observations - when the source was brighter - the lightcurve exhibited flux dips lasting hundreds of seconds where its flux dropped by a factor of 2. In the top-right panel we also show the histogram of the count rate for the all-time lightcurve.\nIn particular, defining the hardness ratio (HR) as the ratio between the counts in the 1-10\\,keV and the 0.3-10\\,keV energy band, respectively, we show that the HR is generally higher when the source is brighter and that\ndecreases in proximity of the dips (Fig. \\ref{fig: lightcurves} bottom-left panel).\n\nWe extracted EPIC PN and MOS spectra for each observations using the same source and background regions chosen for the lightcurves extraction. The \\textit{rmfgen} and \\textit{arfgen} tasks were used to generate response matrices and effective area files. The EPIC-PN spectra of all the observations are shown in Fig. \\ref{fig: EPIC SPECTRA}.\n\nThe individual observations have exposure times which differ by up to an order of magnitude and do not show dramatic changes in the spectral hardness despite their substantial flux variability in agreement with the lightcurve. The spectra extracted for some observations are also nearly superimposable (see Fig. \\ref{fig: EPIC SPECTRA}). In order to compare spectra with similar statistics, we also extracted spectra in ranges of count rate selected ad-hoc according to the count rate histogram (see also \\citealt{Pinto_2017}). We split the XMM all-time lightcurve in eight regimes of count rate as shown in Fig. \\ref{fig: lightcurves}. These regimes were chosen in order to balance the total counts for each level (see Table \\ref{frs table} for more detail) and some spikes appearing in the count-rate histogram (Fig. \\ref{fig: lightcurves} top-right panel). The count-rate selected spectra were then stacked among the different observations in order to obtain eight time-averaged flux-resolved spectra (one per EPIC camera). This selection criterium allow us to probe variability mechanisms at different time scales. The flux-selected EPIC-PN spectra are plotted in Fig. \\ref{fig: EPIC SPECTRA FRS}. \n\n\n\n\\begin{figure*\n\t\t\\centering\n\t\t\\includegraphics[height=0.47\\textwidth]{Images\/NGC55_lc_HR.pdf}\n\t\t\\includegraphics[height=0.476\\textwidth]{Images\/NGC55_histo.pdf}\n        \\vspace{-0.3cm}\n\t\t\\caption{{\\small Left panel: XMM EPIC-PN lightcurves (top left) of NGC 55 ULX-1 and hardness ratio (bottom left, defined as the 1-10 keV \/ 0.3-10 keV counts ratio) from 2001 to 2021 {with time bins of 1 ks}. Vertical dotted lines separate the individual XMM observations, which have be attached for displaying purposes. Right panel: count-rate histogram.\n\t\tThe horizontal red lines indicate the levels chosen to extract eight count-rate resolved spectra.}}\n\t\t\\label{fig: lightcurves}\n\\end{figure*}\n\n\n\\begin{figure\n\t\t\\centering\n\t\t\\includegraphics[width=0.45\\textwidth]{Images\/pn_spectra_all_10obs.pdf}\n        \\vspace{-0.3cm}\n\t\t\\caption{{\\small EPIC-PN spectra for all observations.}}\n\t\t\\label{fig: EPIC SPECTRA}\n\t\t\\includegraphics[width=0.45\\textwidth]{Images\/pn_spectra_FRS8_final.pdf}\n        \\vspace{-0.3cm}\n\t\t\\caption{{\\small EPIC-PN spectra extracted for the eight flux-resolved interval.}}\n\t\t\\label{fig: EPIC SPECTRA FRS}\n\t\\end{figure}\n\n\t\\begin{center}\n\t\\begin{table\n\\caption{Total counts and net exposure times of EPIC flux-selected spectra. }  \n \\renewcommand{\\arraystretch}{1.}\n \\small\\addtolength{\\tabcolsep}{-2pt}\n \\vspace{0.1cm}\n\t\\centering\n\t\\scalebox{1}{%\n\t\\begin{tabular}{ccccccc}\n    \\toprule\n  \n  \n   \n    {{ {Range}}}  &\n    {{EPIC-PN}}  &\n    {{MOS1}}  & {{MOS2}} &\n    {{EPIC-PN}} & {{MOS1}} & {{MOS2}} \\\\\n    {{[c\/s]}} & {{Cnts}} & {{Cnts}} & {{Cnts}} & {{Exp. [s]}} & {{Exp. [s]}} & {{Exp. [s]}} \\\\\n    \n\\midrule\n       0.50-0.71 & 38392 & 13274 & 13121 & 76795 & 95051 & 95322    \\\\\\midrule\n       0.71-0.84  & 43729 & 13977 & 14160 & 72282 & 84108 & 84206    \\\\\\midrule\n       0.84-0.93 & 43845 & 14728 & 14699 & 64371 & 79211 & 78945  \\\\\\midrule\n       0.93-1.03 & 41879 & 14215 & 13847 & 55327 & 66468 & 66286  \\\\\\midrule\n       1.03-1.11 & 42008 & 13816 & 13925 & 50550 & 61191 & 60917  \\\\\\midrule\n       1.11-1.24 & 40366 & 12964 & 13021 & 44413 & 53436 & 53513 \\\\\\midrule\n       1.24-1.73 & 31926 & 10640 & 10737 & 36211 & 42301 & 42391\n       \\\\\\midrule\n       1.73-2.25 & 29716 & 10735 & 10657 & 28954 & 32309 & 32332\n       \\\\\\midrule\n    \\bottomrule\n    \\label{frs table}\n    \\end{tabular}}\n\\end{table}\n\\end{center}\n\n\n\n\n\n\\section{Spectral modelling}\n\\label{Spectral modelling}\n\nThe spectra were modelled with the {\\scriptsize{SPEX}} fitting package v3.06 \\citep{Kaastra_1996}. In order to use $\\chi^{2}$ statistics, we rebinned the EPIC PN and MOS 1,2 spectra with bins of at least 1\/3 of the spectral resolution and with at least 25 counts using the {\\scriptsize{SAS}} task \\textit{specgroup}. All emission components were corrected for absorption from the circumstellar and interstellar medium with the \\textit{hot} model (freezing the temperature of the gas to $10^{-4}$ keV, which provides a neutral gas in {\\scriptsize{SPEX}}). For all emitting and absorbing plasma components we adopted the recommended Solar abundances of \\citet{Lodders2009} which are the default abundances in {\\scriptsize{SPEX}}. The spectral models accounted for the source redshift as well ($z=0.00043$\\footnote{https:\/\/ned.ipac.caltech.edu}). Each spectral model is fitted simultaneously to the EPIC MOS 1,2 and PN spectra as they overlap in the 0.3-10 keV energy band. \n\n\\subsection{Testing different spectral models for Obs. ID 0655050101}\nIt is common to use multiple thermal models (e.g. blackbody emission components) to reproduce ULX spectra (see, e.g., \\citealt{Stobbart_2006,Pintore_2015,Walton_2018,Gurpide_2021a}).\nAt first, we tested several models with the spectra from the observation 0655050101. On the one hand we wanted to compare the new \\textsc{SPEX} code used for this spectral modelling with the previous versions and other codes, {e.g. {\\scriptsize{XSPEC}} or other {\\scriptsize{SPEX}} versions}, used in the recent years for the same source (see, e.g., \\citealt{Pinto_2017}, \\citealt{Pintore_2015}). On the other hand, the comparison of the $\\chi^{2}$ values (and degrees of freedom or d.o.f.) from different spectral models constrains the best fitting continuum model. Finally, we aimed at understanding the systematic effects {when we have not modelled the features.}\n\nThe spectral models that we tested were based on different combinations of the following continuum components: blackbody emission (\\textit{bb} model in {\\scriptsize{SPEX}}), blackbody modified by coherent \\footnote{ {\\scriptsize{SPEX}} manual for more detail at  https:\/\/personal.sron.nl\/~jellep\/spex\/manual.pdf} Compton scattering (\\textit{mbb}), multi-temperature disc blackbody (\\textit{dbb}), and Comptonization of soft photons in a hot plasma (\\textit{comt}). The components used to fit emission and absorption lines are: gaussian line (\\textit{gaus}), collisional-ionisation equilibrium emission (\\textit{cie}) and photoionisation-equilibrium absorption (\\textit{xabs}, see {\\scriptsize{SPEX}} manual for more details).\n\nHere there is a summary of the models tested. \\\\\n- {RHB}: simple blackbody emission (B) corrected for redshift (R) and neutral interstellar absorption (H).\\\\\n- {RHBB}: two blackbody components are used to account for a different structure in the inner and outer regions of the accretion disc.\\\\\n- {RHBD}: a cool blackbody describes the outer disc and a disc blackbody (D) reproduces the inner disc.\\\\\n- {RHBM}: a modified blackbody model (M) describes the inner disc.\\\\\n- {RHBCom}: a Comptonization emission is added to the RHB model.\\\\\n- {RHBMC}: a \\textit{cie} emission model (C) is added to the RHBM model.\\\\\n- {RHBMCX}: The \\textit{xabs} absorption model is added to the RHBMC.\\\\\n- {RHBMCG(G)}: One or more gaussian lines (G) are used to fit emission and absorption lines.\n\nThe first 5 models were primarily used to reproduce the broadband shape and continuum spectrum of the source.  All the continuum models leave strong residuals around 1 keV (positive) and 0.7-0.8 keV and 1.2-1.3 keV (negative), see e.g. Fig. \\ref{fig: Spectral fit for 06555050101 observation using the RHBM model}. These were resolved in groups of narrow lines with the high-resolution RGS spectrometers \\citep{Pinto_2017,Pinto_2021}.\n\nThe other 4 models mentioned above were therefore used to account for any wind features. The \\textit{cie} emission model was used mainly to describe the broad emission feature at around 1 keV, which is the most intense. CCD detectors cannot resolve the small velocity shift of the emission lines and we therefore considered the \\textit{cie} to be at rest. The dominant absorption component of the wind was found to be outflowing at $0.2c$ (\\citealt{Pinto_2017}) which is a shift large enough to be detected even with the EPIC detectors. { However, we chose to fix the outflow velocity for the \\textit{xabs} absorption component to this value, in order to avoid model degeneracies with the ionization parameter $\\xi$}. Both absorption and emission lines were found to be narrow ($\\sigma_{V} \\lesssim1000$ km\/s) in RGS, which cannot be resolved by EPIC and, therefore, we adopted the default velocity dispersion of 100 km\/s for the \\textit{cie} and \\textit{xabs} components (EPIC resolution $\\gg$ 1000 km\/s around 1 keV). The gaussian model provided an alternative phenomenological approach to measure the strength of the dominant spectral emission feature around 1 keV and the absorption features around 0.75 keV and 1.25 keV.\n\nIn Fig. \\ref{fig: Spectral fit for 06555050101 observation using the RHBM model} we show three representative examples of our fits for Obs. ID 06555050101 showing a single \\textit{bb} component continuum model (top panel), a \\textit{bb+mbb} model (middle panel) and a model which also accounts for wind emission lines (\\textit{bb+mbb+cie}, bottom panel). As expected, a single thermal component provided a very poor description of the spectral continuum. Two thermal models were able to fit the broadband shape, while the inclusion of the \\textit{cie} accounts for the dominant wind emission feature at 1 keV.\n\nIn Table \\ref{table: XMM 0655050101 spectral fits} we report the results from our fits of Obs. ID 06555050101. Among all pure-continuum models the best-fit one turned out to be the RHBM model, composed of a cool ($\\sim0.16$ keV) blackbody and a hot ($\\sim0.7$ keV) blackbody modified by coherent Compton scattering (see {\\scriptsize{SPEX}} manual for more details), although comparable results were achieved by a hot disc-blackbody or Comptonization component. The results obtained with the RHBM model on the observation 0655050101 were fully consistent with \\citet{Pinto_2017}. We therefore decided to keep such model as the baseline continuum model for the rest of the analysis.\n\nThe total column density {of cold gas in the LOS towards the source}, $N_{H}$, was about $2.5 \\times 10^{21} \\, \\rm cm^{-2}$, a few times larger than the Galactic value ($7 \\times 10^{20} \\, \\rm cm^{-2}$)\\footnote{https:\/\/heasarc.gsfc.nasa.gov\/cgi-bin\/Tools\/w3nh\/w3nh.pl}. This suggested that a substantial amount of gas found along the LOS is located in the circumstellar medium around the ULX or in the host galaxy.\n\nThe inclusion of the wind (both emission and absorption models) significantly improved the overall quality of the fits. The addition of the wind emission corresponded to a decrease in the overall $\\chi^2$ of 73 for 2 additional d.o.f. (normalisation and temperature of the \\textit{cie}) which flattened most residuals around 1 keV (see Fig. \\ref{fig: Spectral fit for 06555050101 observation using the RHBM model}). The addition of the wind absorption yields a further $\\Delta \\chi^2 = 18$ for 2 more d.o.f. (column density and ionisation parameter of the \\textit{xabs}). \nHowever, it is important to \nnotice that these additional line components did not strongly affect the continuum parameters nor the total bolometric luminosity as previously found by \\citet{Pinto_2020b} and \\citet{Walton_2020}.\n\n\t\\begin{center}\n\t\\begin{table*\n\\caption{Results from the modeling of the XMM-\\textit{Newton} spectrum of NGC 55 ULX-1 with the data of the observation 0655050101. }  \n \\renewcommand{\\arraystretch}{1.}\n \\small\\addtolength{\\tabcolsep}{-4pt}\n \\vspace{0.1cm}\n\t\\centering\n\t\\scalebox{1}{%\n\t\\begin{tabular}{ccccccccccc}\n    \\toprule\n  \n  \n   \n    {{Parameter \/}}  &\n    {{RHB }}  &\n    {{RHBB}}  & {{RHBD}} &\n    {{RHBM}} & {{RHBCom}} & {{RHBMC}} & {{RHBMCX}} & {{RHBMG}} & {{RHBMGG}} \\\\\n    {{component}} &{{Model}}  &  {{Model}}   & {{Model}}  & {{Model}} & {{Model}} & {{Model}} & {{Model}} & {{Model}} & {{Model}}\\\\ \n    \n    \\midrule\n       $L_{X\\,bb1}$   &  0.72 $\\pm$ 0.01 & 0.98 $\\pm$ 0.09 &  0.9 $\\pm$ 0.1 & 0.93 $\\pm$ 0.02  & 0.8 $\\pm$ 0.1 & 0.8 $\\pm$ 0.1 & 1.0 $\\pm$ 0.3 & 0.8 $\\pm$ 0.1  & 1.3 $\\pm$ 0.3   \\\\\\midrule\n       $L_{X\\,bb2}$  &  --- \\par & 0.36 $\\pm$ 0.02 & --- \\par  & --- \\par & --- \\par & --- \\par & --- \\par & --- \\par & --- \\par  \\\\\\midrule\n       $L_{X\\,mbb}$  &  --- \\par & --- \\par & --- \\par & 0.57 $\\pm$ 0.02  & --- \\par & 0.56 $\\pm$ 0.03 & 0.6 $\\pm$ 0.1 & 0.61 $\\pm$ 0.03 &  0.63 $\\pm$ 0.03  \\\\\\midrule\n       $L_{X\\,dbb}$  &  --- \\par & --- \\par & 0.58 $\\pm$ 0.04  & --- \\par  & --- \\par & --- \\par & --- \\par & --- \\par & --- \\par  \\\\\\midrule\n       $L_{X\\,comt}$  & --- \\par & --- \\par & --- \\par & --- \\par & 0.7 $\\pm$ 0.1 & --- \\par & --- \\par & --- \\par & --- \\par  \\\\\\midrule \n       $L_{X\\,CIE}$  & --- \\par & --- \\par & --- \\par & --- \\par & --- \\par  & 0.07 $\\pm$ 0.01 & 0.08 $\\pm$ 0.02 & --- \\par & --- \\par  \\\\\\midrule \n       $L_{X\\,gauss}$  & --- \\par & --- \\par & --- \\par & --- \\par &  --- \\par & --- \\par & --- \\par & 0.025 $\\pm$ 0.004 & 0.008 $\\pm$ 0.003   \\\\\\midrule\n       kT$_{bb1}$  & 0.287 $\\pm$ 0.001 &  0.179 $\\pm$ 0.002 & 0.164 $\\pm$ 0.002 & 0.164 $\\pm$ 0.001 & 0.158 $\\pm$ 0.005 & 0.159 $\\pm$ 0.003 & 0.164 $\\pm$ 0.004 & 0.159 $\\pm$ 0.003 & 0.147 $\\pm$ 0.003  \\\\\\midrule \n       kT$_{bb2}$  & --- \\par &  0.482 $\\pm$ 0.005 & --- \\par & --- \\par & --- \\par & --- \\par & --- \\par & --- \\par & --- \\par \\\\\\midrule\n       kT$_{mbb}$  & --- \\par & --- \\par & --- \\par & 0.680 $\\pm$ 0.007 & --- \\par & 0.668 $\\pm$ 0.008 & 0.673 $\\pm$ 0.008 & 0.663 $\\pm$ 0.008 & 0.654 $\\pm$ 0.008 \\\\\\midrule\n       kT$_{dbb}$  & --- \\par & --- \\par & 1.2 $\\pm$ 0.1 & --- \\par & --- \\par & --- \\par & --- \\par & --- \\par & --- \\par \\\\\\midrule \n       kT$_{seed}$  & --- \\par & --- \\par & --- \\par & --- \\par & 0.158 (coupled) & --- \\par & --- \\par & --- \\par & --- \\par \\\\\\midrule \n       kT$_{e}$  & --- \\par & --- \\par & --- \\par & --- \\par & 0.58 $\\pm$ 0.02 & --- \\par & --- \\par & --- \\par & --- \\par  \\\\\\midrule\n       $\\tau$  & --- \\par & --- \\par & --- \\par & --- \\par & 12 $\\pm$ 2 & --- \\par & --- \\par & --- \\par & --- \\par  \\\\\\midrule \n       kT$_{CIE}$  & --- \\par & --- \\par & --- \\par & --- \\par & --- \\par & 1.11 $\\pm$ 0.05 & 1.10 $\\pm$ 0.05 &  --- \\par & --- \\par  \\\\\\midrule \n       $N_{H}$  & 0.733 $\\pm$ 0.001 & 2.23 $\\pm$ 0.07 & 2.52 $\\pm$ 0.07 & 2.53 $\\pm$ 0.07 & 2.5 $\\pm$ 0.1 & 2.44 $\\pm$ 0.09 &  2.3 $\\pm$ 0.1 & 2.40 $\\pm$ 0.09 & 3.0 $\\pm$ 0.2 \\\\\\midrule\n       ${N_H}_{Xabs}$  &  --- \\par &  --- \\par &  --- \\par &  --- \\par &  --- \\par &  --- \\par & 0.14 $\\pm$ $^{0.26}_{0.10}$  &  --- \\par &  --- \\par   \\\\\\midrule\n       $ \\rm Log \\ \\xi $ &  --- \\par &  --- \\par &  --- \\par &  --- \\par &  --- \\par &  --- \\par &  3.7 $\\pm$ 0.1  &  --- \\par &  --- \\par  \\\\\\midrule\n       $E_{0}^{1}$  & --- \\par & --- \\par & --- \\par & --- \\par & --- \\par & --- \\par & --- \\par & 0.98 $\\pm$ 0.01 & 1.01 $\\pm$ 0.02  \\\\\\midrule\n       $E_{0}^{2}$  & --- \\par & --- \\par & --- \\par & --- \\par & --- \\par  & --- \\par & --- \\par & --- \\par & 0.76 $\\pm$ 0.01 \\\\\\midrule\n        FWHM  & --- \\par & --- \\par & --- \\par & --- \\par & --- \\par & --- \\par & --- \\par &  0.22 $\\pm$ 0.03 & 0.12 $\\pm$ 0.03 \\\\\\midrule\n       $\\rm Norm_{1}$  & --- \\par & --- \\par & --- \\par & --- \\par & --- \\par & --- \\par & --- \\par & 1.6 $\\pm$ 0.3 & 0.5 $\\pm$ 0.2 \\\\\\midrule\n       $\\rm Norm_{2}$  & --- \\par & --- \\par & --- \\par & --- \\par & --- \\par & --- \\par & --- \\par & --- \\par & -2.7 $\\pm$ $^{0.7}_{1.0}$   \\\\\\midrule\n       ${\\chi}^2$\/d.o.f & 4537\/287 & 463\/285 & 437\/285 &  428\/285 & 438\/284 & 360\/283 & 342\/281 & 346\/282 & 325\/279 \\\\\\midrule \n       ${\\chi}^2_{PN}$ & 2368 & 214 & 204 & 202 & 206 & 145 & 133 & 132 & 120  \\\\\\midrule\n       ${\\chi}^2_{MOS1}$ & 1064 & 141 & 125 & 122 & 122 & 118 & 117 & 123  & 112  \\\\\\midrule\n       ${\\chi}^2_{MOS2}$ & 1104 & 108 & 108 & 109 & 110 & 96 & 91 & 91 & 92 \\\\\\midrule\n    \\bottomrule\n    \\label{table: XMM 0655050101 spectral fits}\n    \\end{tabular}}\n    \n    \\begin{quotation}\\footnotesize\n    Parameter units: $E_{0}^{1}$ and $E_{0}^{2}$ (in keV unit) refer to the centroids of the first and second gaussian, respectively. \n    $\\rm Norm_{1}$ and $\\rm Norm_{2}$ (in $10^{46} \\rm ph\/s $ units) refer to the normalisations of the first and second gaussian, respectively. The temperatures kT (for each model) and FWHM are expressed in keV unit. The X-ray and bolometric luminosities $L_{X}$ and $L_{bol}$  (always intrinsic or unabsorbed) are calculated, respectively, between the 0.3 - 10 keV and 0.001 - 1000 keV bands, and are expressed in $10^{39}$ erg\/s unit. The ionisation parameter $ \\rm Log \\ \\xi $ is in  erg\/s cm. The column density of the cold gas $N_{H}$ is in 10$^{21}$\/cm$^{2}$ unit, while ${N_H}_{Xabs}$ is in 10$^{24}$\/cm$^{2}$ unit. $\\tau$ is the optical depth of the Comptonization component.\n\\end{quotation}\n \\vspace{-0.3cm}\n\\end{table*}\n\\end{center}\n\n\\begin{figure\n\t\t\\centering\n\t\t\\includegraphics[width=0.45\\textwidth]{Images\/PLT_0655050101_all_final_bb.pdf}\n\t\t\\includegraphics[width=0.45\\textwidth]{Images\/PLT_0655050101_all_final_bbmbb.pdf}\n\t\t\\includegraphics[width=0.45\\textwidth]{Images\/PLT_0655050101_all_final_bbmbbcie.pdf}\n        \\vspace{-0.3cm}\n\t\t\\caption{{\\small NGC 55 ULX-1 EPIC MOS and PN spectral fits performed using the RHB model (top panel), RHBM model (middle panel) and the RHBMC model (bottom panel) for the Obs. ID: 06555050101. {The dashed-dotted lines represent the background spectra.}}}\n\t\t\\label{fig: Spectral fit for 06555050101 observation using the RHBM model}\n\t\t\\end{figure}\n\n\n\\subsection{Spectral modelling of individual observations}\n\\label{Sec:spectra}\n\nIn Fig. \\ref{fig: EPIC SPECTRA} we show the EPIC-PN spectra of the 10 XMM observations (the other two EPIC MOS 1,2 cameras are not shown here for clarity). We fitted the EPIC PN and MOS 1,2 from all observations using only continuum models because in some observations the residuals were too weak to constrain the parameters of the wind model. Besides the inclusion of the wind does not alter the continuum parameters even for the high quality spectra.\n\nAs for observation 06555050101, the spectral modelling was performed again simultaneously to the EPIC-PN, MOS1,2 spectra for each observation and testing all models mentioned before (blackbody, modified blackbody, powerlaw, and Comptonization). However, for clarity purposes and means of comparison with previous work, we focussed on the results obtained with a double thermal model. A warm ($kT_{bb}$ $\\sim$ 0.2 keV) blackbody component reproduces the soft X-ray emission, presumably from the outer disc and the wind photosphere.\nA hot ($kT_{mbb}$ $\\sim$ 0.7 keV) blackbody component modified by coherent Compton scattering, accounts for the inner disc emission. The results are shown in detail in Table \\ref{Results RHBM model} and the plots for the four observations with longest ($\\gtrsim 90$ ks net) exposure and, therefore, the spectral fits are shown in Fig. \\ref{fig: Spectral fit for 06555050101 observation using the RHBM model} (middle panel) and Fig. \\ref{fig: Fit model for all observations using the RHBM model}.\n\nThis model provided a good description of the data except for the usual strong, narrow, residuals around 1 keV due to the well-known winds. Indeed, by adding a wind photoionisation absorption or collisionally ionised components - as previously found with the high-resolution RGS detectors - we could get rid of most spectral residuals (see observation 06555050101 as an example in Fig. \\ref{fig: Spectral fit for 06555050101 observation using the RHBM model} bottom panel and table \\ref{table: XMM 0655050101 spectral fits}). Our results are fully consistent with previous work and, in particular, seem to indicate that both thermal components become increasingly hotter (i.e. with a higher kT) at higher luminosities (\\citealt{Pintore_2015,Pinto_2017}). For a comparison with results from the literature see Sect. \\ref{Discussion}. \n\nThe neutral column density was consistent among all high-quality, deep ($\\gtrsim 100$ ks) exposures. The shortest ($\\lesssim 30$ ks) exposures did not have sufficient signal-to-noise ratio to constrain both the spectral shape and the neutral ISM absorption. In particular, testing several models from single to multiple components model (RHB, RHBB, RHBM, etc.), we obtained values ranging from $(2-3) \\times 10^{21} \\rm cm^{-2}$. This was likely due to degeneracy produced by the lower statistics. Therefore, whilst fitting the spectra of these six short exposures we preferred to fix the $N_{H}$ to the average value obtained in the latter observations as previously done in e.g. \\citet{Robba2021} for NGC 1313 ULX-2.\n\n\t\t\n\t\t\n\t\t\\begin{center}\n\t\\begin{table*\n\\caption{Results from the spectral modeling for the individual observations (RHBM model). }  \n \\renewcommand{\\arraystretch}{1.}\n \\small\\addtolength{\\tabcolsep}{0pt}\n \\vspace{0.1cm}\n\t\\centering\n\t\\scalebox{0.8}{%\n\t\\begin{tabular}{ccccccccccc}\n    \\toprule\n  \n  \n   \n    {{Parameter \/}}  &\n    {{0028740201}}  &\n    {{0028740101}}  & {{0655050101}} &\n    {{0824570101}} & {{0852610101}} & {{0852610201}} & {{0852610301}} & {{0852610401}} & {{0864810101}} & {{0883960101}}\\\\\n    {{component}} & {{Obs}}  &  {{Obs}}   & {{Obs}}   & {{Obs}}   & {{Obs}}   & {{Obs}}  & {{Obs}} & {{Obs}} & {{Obs}} & {{Obs}}\\\\ \n    \n    \\midrule\n       $L_{X\\,bb}$      & 1.50 $\\pm$ 0.06   & 1.12 $\\pm$ 0.08   &  0.93 $\\pm$ 0.02   &  0.71 $\\pm$ 0.02   & 1.5 $\\pm$ 0.2     & 1.6 $\\pm$ 0.2     & 0.7 $\\pm$ 0.1     & 1.1 $\\pm$ 0.2     &  0.91 $\\pm$ 0.02  & 1.15 $\\pm$ 0.03 \\\\\\midrule\n       $L_{X\\,mbb}$     & 1.37 $\\pm$ 0.06   & 1.48 $\\pm$ 0.08   &   0.57 $\\pm$ 0.02  & 0.38 $\\pm$ 0.01    & 1.2 $\\pm$ 0.2     & 1.2 $\\pm$ 0.2     & 0.46 $\\pm$ 0.03   & 0.6 $\\pm$ 0.2     & 0.58 $\\pm$ 0.02   & 0.67 $\\pm$ 0.02 \\\\\\midrule\n       kT$_{bb}$        & 0.172 $\\pm$ 0.002 & 0.174 $\\pm$ 0.003 & 0.164  $\\pm$ 0.001 & 0.162 $\\pm$ 0.001  & 0.168 $\\pm$ 0.004 & 0.162 $\\pm$ 0.004 & 0.153 $\\pm$ 0.005 & 0.163 $\\pm$ 0.006 & 0.167 $\\pm$ 0.001 & 0.166 $\\pm$ 0.001\\\\\\midrule\n       kT$_{mbb}$       & 0.82 $\\pm$ 0.01   & 0.90 $\\pm$ 0.02   & 0.680 $\\pm$ 0.007  & 0.755 $\\pm$ 0.008  & 0.71 $\\pm$ 0.03   & 0.69 $\\pm$ 0.03   & 0.67 $\\pm$ 0.04   & 0.68 $\\pm$ 0.05   & 0.713 $\\pm$ 0.007 & 0.680 $\\pm$ 0.006 \\\\\\midrule \n       $N_{H}$          & 2.5               & 2.5               & 2.53 $\\pm$ 0.07    & 2.57 $\\pm$ 0.09    & 2.5               & 2.5               & 2.5               & 2.5               & 2.52 $\\pm$ 0.07   &  2.5 \\\\\\midrule\n       ${\\chi}^2$\/d.o.f & 334\/275           & 294\/240           & 428\/285            & 413\/279            & 192\/153           & 139\/137           & 135\/97            & 87\/87             & 463\/285           & 378\/294 \\\\\\midrule\n       $L_{bol}$        & 3.19              & 2.85              & 1.71               & 1.26               & 3.15              & 3.17              & 1.32              & 2.01              & 1.69              & 2.08 \\\\\\midrule\n    \\bottomrule\n    \\label{Results RHBM model}\n    \\end{tabular}}\n    \\begin{quotation}\\footnotesize\n       Units are the same as in the Table \\ref{table: XMM 0655050101 spectral fits}.\n\\end{quotation}\n        \\vspace{-0.3cm}\n\\end{table*}\n\\end{center}\n\n\n\\begin{figure\n\t\t\\centering\n\t\t\\renewcommand{\\arraystretch}{1.}\n\n\n        \\includegraphics[width=0.45\\textwidth]{Images\/PLT_0824570101_all_final.pdf}\n        \\includegraphics[width=0.45\\textwidth]{Images\/PLT_0864810101_all_final.pdf}\n        \\includegraphics[width=0.45\\textwidth]{Images\/PLT_0883960101_all_final.pdf}\n        \n        \\vspace{-0.3cm}\n\t\t\\caption{{\\small Spectral fits for the other three long ($\\gtrsim 90$ ks net) observations of NGC 55 ULX-1 using the RHBM model (blackbody + modified blackbody). }}\n\t\t\\label{fig: Fit model for all observations using the RHBM model}\n        \\vspace{-0.3cm}\n\t\t\\end{figure}\n\n\\subsection{Spectral modelling of different flux levels}\n\nIn order to confirm and corroborate any trends between spectral hardness and source flux we fit the eight flux-resolved spectra shown in Sect. \\ref{Observations and Spectral modelling} with the RHBM model as done before for the individual observations. The results of the spectral fits, fixing the column density $N_{H}$ to the average value obtained in the previous results ($ 2.5 \\times 10^{21} \\rm cm^{-2}$), are shown in table \\ref{Results RHBM model-FRS}. At higher luminosities, the temperature of the hot modified blackbody component increased from 0.70 to 0.86 keV while the one for the cooler component increased by just 0.01 keV. An exception was the lowest flux level 1, which was largely dominated by Obs. ID: 0824570101, where there seemed to be an increase in the \\textit{mbb} temperature. An interesting result is the consistency between the parameters of the cool \\textit{bb} component between the highest level 8 and the bright, dipping, level 7. At odds, there was instead a clear drop in the luminosity and temperature of the hot \\textit{mbb} component during level 7. The results from these spectral fits are discussed later on in Sect. \\ref{Discussion}.\n\nIn Sect. \\ref{sec:wind_variability} we show how gaussian lines can describe the spectral residuals in these high-quality flux-resolved spectra, discuss their variability and how they may provide insights on the overall disc evolution and structure.\n\n\t\t\\begin{center}\n\t\\begin{table*\n\\caption{Results from the modeling with the RHBM model of the eight flux-resolved spectra.}  \n \\renewcommand{\\arraystretch}{1.}\n \\small\\addtolength{\\tabcolsep}{0pt}\n \\vspace{0.1cm}\n\t\\centering\n\t\\scalebox{0.9}{%\n\t\\begin{tabular}{ccccccccc}\n    \\toprule\n  \n  \n   \n    {{Parameter \/}}  &\n    {{Level}}  &\n    {{Level}}  & {{Level}} &\n    {{Level}} & {{Level}} & {{Level}} & {{Level}} & {{Level}} \\\\\n    {{component}} & {{1}}  &  {{2}}   & {{3}}   & {{4}}   & {{5}}   & {{6}}  & {{7}} & {{8}} \\\\ \n    \n    \\midrule\n     $L_{X\\,bb}$  & 0.70 $\\pm$ 0.02 & 0.81 $\\pm$ 0.03   &  0.90 $\\pm$ 0.03 &  1.00 $\\pm$ 0.03 &  1.11 $\\pm$ 0.03   & 1.20 $\\pm$ 0.04 & 1.39 $\\pm$ 0.05 & 1.40 $\\pm$ 0.07 \\\\\\midrule\n    $L_{X\\,mbb}$  & 0.38 $\\pm$ 0.02 & 0.50 $\\pm$ 0.02   &   0.55 $\\pm$ 0.03 & 0.60 $\\pm$ 0.03 & 0.65 $\\pm$ 0.03 & 0.70 $\\pm$ 0.04 & 0.89 $\\pm$ 0.05  & 1.65 $\\pm$ 0.07 \\\\\\midrule\n       kT$_{bb}$ & 0.162 $\\pm$ 0.001 & 0.162 $\\pm$ 0.001  & 0.165  $\\pm$ 0.001 & 0.166 $\\pm$ 0.001 & 0.166 $\\pm$ 0.001 & 0.167 $\\pm$ 0.001 & 0.170 $\\pm$ 0.001 & 0.175 $\\pm$ 0.002 \\\\\\midrule\n       kT$_{mbb}$ & 0.75 $\\pm$ 0.01 & 0.704 $\\pm$ 0.008  & 0.705 $\\pm$ 0.009  & 0.694 $\\pm$ 0.009 & 0.692 $\\pm$ 0.009 & 0.669 $\\pm$ 0.009 & 0.74 $\\pm$ 0.01 & 0.89 $\\pm$ 0.01 \\\\\\midrule \n       $N_{H}$ & 2.5 & 2.5 & 2.5 & 2.5 & 2.5 & 2.5 & 2.5 & 2.5 \\\\\\midrule\n       ${\\chi}^2$\/d.o.f & 452\/282 & 431\/276  & 390\/279 & 350\/274 & 366\/271 & 295\/262 & 324\/270 & 390\/295 \\\\\\midrule\n       $L_{bol}$ & 1.24 & 1.49 & 1.65 & 1.82 & 2.00 & 2.17 & 2.58 & 3.36 \\\\\\midrule\n    \\bottomrule\n    \\label{Results RHBM model-FRS}\n    \\end{tabular}\n    }\n    \\begin{quotation}\\footnotesize\n     Units are the same as in the Table \\ref{table: XMM 0655050101 spectral fits}.\n\\end{quotation}\n        \\vspace{-0.3cm}\n\\end{table*}\n\\end{center}\n\n\\subsection{Variability of the wind features}\n\\label{sec:wind_variability}\n\n\nIn order to study the strength of the spectral features and their variability in NGC 55 ULX-1, we performed a quick fit of the flux resolved spectra modifying our baseline \\textit{bb} + \\textit{mbb} model by adding three gaussian lines (RHBMGGG model). Following \\citet{Pinto_2021}, we fixed the line width to 1 eV for all three gaussian lines since EPIC spectra lack the necessary spectral resolution around 1 keV. The energy centroids were free to vary however they agreed within the error bars with 0.8 and 1.2 keV for the absorption lines and 1.0 keV for the emission line.\n\nThe normalisations of the gaussian lines for the eight spectra are shown in Fig. \\ref{fig: Normalization_vs_Bolometric Luminosity plot}. They do not show strong trends with the flux, possibly due to fairly large uncertainties, with the exception of the lowest-energy line at 0.8 keV, which seems to get stronger at high fluxes in agreement with NGC 1313 ULX-1 (\\citealt{Pinto_2020b}) and NGC 247 ULX-1 (\\citealt{Pinto_2021}). These trends were also attributed to an increasing $\\dot M$.\n\n\n\t\t\\begin{figure\n\t\t\\centering\n\t\n\t\t\\includegraphics[width=0.45\\textwidth]{Images\/Norm_Lum_Bol_3gaussian_8spectra_final.pdf}\n        \\vspace{-0.1cm}\n\t\t\\caption{{Normalisation of the gaussian lines vs Bolometric luminosity for the 3-gaussian model fits of the flux resolved spectra. The negative normalisations refer to absorption lines.}}\n\t\t\\label{fig: Normalization_vs_Bolometric Luminosity plot}\n\t\t\\end{figure}\n\t\t\n\n\\section{Discussion}\n\\label{Discussion}\n\n\nThe aim of this work is to {investigate} the processes that trigger the spectral transitions in ULXs, particularly between the soft and intermediate hardness regimes. The main question is whether they are driven by stochastic variability in the wind (e.g. \\citealt{Kobayashi2018}) or variations in the accretion rate, which in turn produce variability in the wind and in the obscuration of the innermost, hottest, regions (e.g. \\citealt{Urquhart2016}). Geometrical effects may also play a relevant role in systems where the accretion disc precesses (e.g. \\citealt{Middleton_2015a}).\n\nAccording to {radiation-magnetohydrodynamic (r-MHD)} simulations of super-Eddington accretion discs (e.g. \\citealt{Takeuchi2013}), the winds are expected to become optically thick enough to block and reprocess a fraction of the disc X-ray photons, making the source appear as a soft thermal emitter or ultraluminous (super)soft X-ray source (e.g. \\citealt{Guo2019}) {at moderately high inclination angles}. A comprehensive study would require to measure the properties of the winds \nand compare them to the characteristics of the spectral continuum \n(similarly to the \\citealt{Pinto_2020b} for the intermediate-to-hard source NGC 1313 ULX-1). In order to do this, we used the XMM-\\textit{Newton} observations of one of the most variable and brightest (in flux) nearby source NGC 55 ULX-1. The X-ray spectra of this ULX fit in just between the softest and intermediate ULX spectra (see Fig. \\ref{fig: Comparison X-ray spectra of the brightest ULX known}). Indeed, the temperatures of the \\textit{bb} fits are among the lowest in ULXs (e.g. \\citealt{Sutton_2013}). Moreover, we benefited from three new deep observations that enabled us to achieve more statistics in the low-flux regime and 4 new short observations which fill the gaps between the high- and low-flux regimes. In this work we focused on the evolution of the spectral continuum throughout the different epochs. A follow-up work will focus on the study of the high-resolution X-ray spectra which are however available only for the low-intermediate flux on-axis observations and the evolution of the wind in detail.\n\n\n\\subsection{X-ray broadband properties}\n\nThe XMM-\\textit{Newton}\/EPIC lightcurves showed evidence for strong variability by up to a factor 4 in flux over timescales of a few days (see Fig. \\ref{fig: lightcurves}, top panel) along with dips lasting a few 100s to a few ks, which have been interpreted as due to wind clumps that temporarily obscured the central object (see, e.g., \\citealt{Stobbart_2004,Pintore_2015}). In fact, during the dips the spectral hardness decreased and reached levels comparable to the low-flux observations although the dip flux was 2-3 times higher than that of the low-flux observations (Fig. \\ref{fig: lightcurves}, bottom panel). The dips appear mainly when the source is bright indicating that the accretion rate increases thereby launching optically-thick wind clouds into the LOS. This suggests the presence of at least two different ongoing variability processes (see also below). \\textit{Swift}\/XRT long-term lightcurves taken between 2013 and 2021 confirm the general source behavior with flux variations by a factor of up to 6 with the spectrum of the source appearing harder when brighter \\citep{Jithesh_2021}.\n\nWe extracted spectra for individual observations to probe variations on timescales of a few days to years as well as flux-resolved spectra to probe the nature of on {different timescales} variability (see Fig. \\ref{fig: EPIC SPECTRA} and \\ref{fig: EPIC SPECTRA FRS}). \nFlux level 1 mainly sampled the lowest flux observation (Obs. ID:082457), while flux levels 7 and 8 described the dip and no-dip time intervals in the bright epochs (Obs. ID: 002874-0201, from 2001, see Fig. \\ref{fig: lightcurves}). The other levels (2-6) traced the intermediate-flux epochs (Obs. ID: 065505,086481). All spectra peaked around 1 keV with the peak slightly shifting towards higher energies at higher fluxes. Interestingly, the lowest-flux spectrum appeared harder than the intermediate-flux ones {in contradiction with the trend sees at higher fluxes.}\n\nOur analysis confirmed that the spectrum requires at least two components with a blackbody-like shape. A double blackbody model could indeed reproduce the overall spectral shape with the hotter component being broader and likely modified by Compton scattering in the wind (see Fig. \\ref{fig: Spectral fit for 06555050101 observation using the RHBM model} and Table \\ref{table: XMM 0655050101 spectral fits}). This agrees with the general picture and the typical properties of soft ULXs (see, e.g., \\citealt{Sutton_2013}). The inclusion of wind features in the model in the form of gaussian lines or physical components\nproduced significant improvements to the spectral fits in terms of $\\Delta \\chi^2$ but did not affect the continuum parameters\nalbeit slightly increasing the uncertainties (see Table \\ref{table: XMM 0655050101 spectral fits}). \n\nThe spectral fits of both the individual observations and the eight flux levels showed that both the soft and hard components became hotter (i.e. with a higher kT) at higher luminosities (see Table \\ref{Results RHBM model} and \\ref{Results RHBM model-FRS}), which indicates that either the accretion rate is increasing or that we are progressively having a clearer view of the inner accretion flow through a less dense wind photosphere. However, the results on the wind features (e.g. the more intense 0.8 keV line at high fluxes) would favor an increase in the ${\\dot M}$ (see Sect. \\ref{sec:wind_variability}).\nThis agrees with a scenario in which variations in the local accretion rate are driving the transition from soft\/fainter to hard\/brighter spectra. The brightest ones are associated to epochs in which a surplus of matter may launch optically-thick clouds that obscure the innermost region thereby producing the flux dips.\n\n\\subsection{Comparison with disc models}\n\nThe nature of the compact object powering NGC 55 ULX-1 is not yet known and therefore the accretion in Eddington units is uncertain. Although the best-fit values of temperatures might not exactly correspond to gas temperatures due to the unknown structure of the emitting regions {(because we have modelled a thick disk with a thin disk template)}, it is still very useful to compare them with the bolometric luminosities of the blackbody components. This is indeed a commonly adopted procedure to understand the behaviour of the thermal components and, in particular, to place some constraints on the disc structure and accretion regime, possibly providing some information on the nature of the compact object, especially for not too high accretion rates or luminosities ($L_{\\rm BOL}\\sim10^{39}$ erg\/s, e.g., \\citealt{Urquhart2016,Earnshaw2017,Walton_2020,Gurpide_2021a,Robba2021,D'Ai_2021}). \n\nIt is useful to compare the Luminosity--Temperature ($L-T$) trends measured for NGC 55 ULX-1 with those expected from theoretical models such as the thin disc in a sub-Eddington regime (L $\\propto$ $T^{4}$) with a constant emitting area (SS73, \\citealt{SS1973}) and the advection-dominated disc model with L $\\propto$ $T^{2}$ (\\citealt{10.1093\/pasj\/53.5.915}).\nIn Fig. \\ref{fig: L-T plot} (top panel) we show the trends between the temperature and the bolometric luminosity for both the cool (blue points) and warm (orange points) blackbody components from the spectral fits of the ten individual XMM observations.\nOverlaid are the best-fitting least-squares regression line (solid blue, $L \\propto T^{\\alpha}$), the L $\\propto$ $T^{4}$ (dashed red) and the L $\\propto$ $T^{2}$ (dotted black) trends. In Fig. \\ref{fig: L-T plot} (bottom panel) we show the same results obtained with the eight spectra extracted in the count rate ranges of the XMM lightcurve (see Fig. \\ref{fig: lightcurves}).\nThe bolometric (or total) luminosities of the cool blackbody and warm modified blackbody components are computed between 0.001-1000 keV by extrapolation, although {most of the flux is emitted in the 0.1-10 keV range}. Both plots show that the measured L--T relationship is in broad agreement with the thin SS73 disc.\n{From the regression lines we obtain a power index, with the time resolved spectroscopy $\\alpha_{cool}= 4.2 \\pm 1.9 $ and $\\alpha_{hot}= 6.1 \\pm 2.3 $, for the cool and hot components, respectively. Instead, with the flux resolved spectroscopy, the power indices are $\\alpha_{cool}= 5.4 \\pm 1.7 $  and $\\alpha_{hot}= 5.4 \\pm 1.8 $. The possible correlation of the points in the L--T plot, for the cool and hot components, can be established with the Pearsons and Sperman correlation coefficients. These are reported in the Table \\ref{pearsons and spearman correlation coefficients} by using \\textit{scipy.stats.pearsonsr} and \\textit{scipy.stats.spearmanr} routine in {{\\scriptsize PYTHON}. They do not show always strong correlations.}}\n\\begin{center}\n\t\\begin{table}\n\t\\caption{Pearsons and Spearman coefficients for the L-T trends, for both cool and hot components, by using the best fit values of the RHBM model.}  \n\t \\renewcommand{\\arraystretch}{1.}\n \\small\\addtolength{\\tabcolsep}{0pt}\n \\vspace{0.1cm}\n\t\\centering\n\t\\scalebox{0.85}{%\n  \\begin{tabular}{c c c c c}\n    \\hline\n    \\multirow{2}{*}{Correlation coefficient} &\n      \\multicolumn{2}{c}{TRS}  & \n      \\multicolumn{2}{c}{FRS} \\\\ \\cline{2-5}\n    &  ${(L-T)}_{cool}$ & ${(L-T)}_{hot}$ & ${(L-T)}_{cool}$ & ${(L-T)}_{hot}$  \\\\\n    \\hline\n    Pearsons & 0.48 & 0.65 & 0.90 & 0.83 \\\\\n    \\hline\n    Spearman & 0.39 & 0.55 & 0.93 & 0.05  \\\\\n    \\hline\n  \\end{tabular}}   \\label{pearsons and spearman correlation coefficients}\n  \\begin{quotation}\\footnotesize\nTRS and FRS acronyms stand for time-resolved spectroscopy and flux-resolved spectroscopy, respectively.\n\\end{quotation}\n        \\vspace{-0.3cm}\n\\end{table}\n\\end{center}\nLocally there are small deviations in Fig. \\ref{fig: L-T plot} from the tight correlations. \nIn particular, at high luminosities the temperature of the cool component is lower than the predictions from the L $\\propto$ $T^{4}$ model or the regression line, which suggests that the disc is expanding and, perhaps, that the wind starts to contribute to the emission {or the radius of the thermal component is not constant}. The lowest-flux observation also shows a notable deviation, especially for the warm disc component, with a temperature higher than as predicted from the $L-T$ trends. This could be a hint of low\/hard - high\/soft behavior seen in Galactic X-ray binaries and might indicate a spectral transition below $10^{39}$ erg\/s, {although the spectra of NGC 55 ULX-1 never get as hard as the Galactic X-Ray Binaries (XRBs) likely due to its persistently high accretion rate} (\\citealt{Koljonen_2010}).\n\n\n Fig.\\,\\ref{fig: L-T plot} shows that there are deviations for the luminosity and temperatures from the best-fit regression line, approximately for {total} luminosity greater than $2 \\times 10^{39} \\rm erg\/s$. This fact can be used to give a rough estimate of the mass of the compact object. For instance, if we assume that the deviation is due to the total luminosity reaching the Eddington limit ($L_{\\rm tot} \\sim L_{\\rm Edd}$) and that the apparent luminosity is comparable to the intrinsic luminosity, using the definition $L_{\\rm Edd} = 1.4 \\times 10^{38} \\frac{M}{M\\textsubscript{\\(\\odot\\)} }$ erg\/s, we estimate that the mass of the compact object is about 14 $M\\textsubscript{\\(\\odot\\)}$. Instead, if we assume that the deviation is due to the disc becoming supercritical ($L_{\\rm tot}\\sim L_{\\rm critical} = 9\/4 \\  L_{\\rm Edd}$, \\citealt{Poutanen_2007}), then we obtain a value for the mass of the compact object of about 6 $M\\textsubscript{\\(\\odot\\)}$. Of course, should the intrinsic luminosity be greater than the apparent luminosity, which is possible given the weakness of the hard component, then the mass of the compact object would be larger. Bright ULXs with hard spectra typically have $L_{\\rm Bol} = (0.5-1) \\times 10^{40}$ erg\/s, which would result in 10--30 $M\\textsubscript{\\(\\odot\\)}$ for NGC 55 ULX-1. Therefore, the  mass range obtained would suggest that the compact object is a black hole. This agrees with the results obtained by \\citet{Fiacconi_2017} using wind arguments. {Interestingly, some deviations are also seen in Galactic XRBs above $0.3 \\ L_{Edd}$ (e.g. \\citealt{Steiner_2009}). If the deviations that we see refer to such threshold, then a larger mass of the compact object, i.e. a heavier BH, is forecast.}\n \n The radius of the cool component has a value of the order of $3000 \\rm \\ km$; instead the radius of the hot component is of the order of $100 \\rm \\ km$, ({estimated with the relations between luminosity and temperature and blackbody definition}). These correspond to, respectively, 200 $R_{G}$ and 10 $R_{G}$, assuming a black hole of 10 $M\\textsubscript{\\(\\odot\\)}$. The radii of both components from the first to the eighth flux level do not vary significantly possibly due to the much larger uncertainties with respect to the temperature. Such results show a slight tension with \\citet{Jithesh_2021}\nin which a tentative anti-correlation between the radius and the temperature of the cool component was found albeit at large uncertainties. This is probably due to the fact that we fixed the column density $\\rm N_{H} $ since we do not expect a strong variation of the neutral gas within a few hours, {which are the timescale of the dips and the separation between consecutive observations}.\n\nBy considering that the spherisation radius is $R_{\\rm sph}= 27\/4 \\ \\dot{m} \\ R_{G}$ and assuming that $\\dot{m}$ $\\sim$ $\\dot{m}_{Edd}$, the radius of the hot blackbody component is comparable to the spherisation radius ($R_{\\rm sph}\\sim 7 R_{G}$) (\\citealt{Poutanen_2007}), instead the radius of the cool component is significantly larger than $R_{\\rm sph}$ identifying the outer disc. However, these results are correct if the intrinsic luminosity is comparable with the observed one or if there are not large losses due to the occultation of the inner portion of the disc by the wind. \\citet{Jithesh_2021} also used \\textit{NuSTAR} data to better constrain the hard band; {we do no think the use of these data might alter our conclusions, since there is only marginal flux in the NuSTAR band which is not covered by the EPIC-PN}. Besides, there are only a few simultaneous  XMM\/\\textit{NuSTAR} observations.\n\n\\subsection{Comparison with other ULXs}\n\\label{sec:comparison_with_other_ulxs}\n\nIn the brighter ($L_X$ up to $10^{40}$ erg\/s) and pulsating source NGC 1313 ULX-2 there is a clear anti-correlation between the bolometric luminosity and the temperature of the cool blackbody-like component (\\citealt{Robba2021}). In that source the trend is in agreement with the L $\\propto$ $T^{-4}$ relationship predicted by X-ray emission from a wind photosphere rather than a disc (e.g. \\citealt{Qiu_2021}, \\citealt{Kajava2009}, \\citealt{King_2009}), and is likely due to the higher observed luminosity and possibly higher accretion rate for the pulsating neutron star. In the even brighter source in same galaxy, NGC 1313 ULX-1, \\citet{Walton_2020} found a deviation in the L--T relationship for the warm disc-like component which has been attributed to an intervening wind or a higher disc scale height. The cool component seems fairly constant {in luminosity} in NGC 1313 ULX-1. The debate {on the nature of the different L-T trends} is still open.\n\nThe lightcurve of NGC 55 ULX-1 shows a dipping behavior during the high-flux epochs which is very similar to that one shown by supersoft ULXs such as NGC 247 ULX-1 (\\citealt{Feng_2016}). There are much more data available for NGC 247 ULX-1 thanks to a recent deep XMM-Newton campaign (PI: Pinto). The results from the broadband analysis were shown by \\citet{D'Ai_2021} who reported on a complex hardness-intensity diagram characterised by two main branches; here at high luminosities the source enters the dipping branch in which the spectrum becomes progressively softer. It is possible that the deviation of the temperature of the cool component shown by NGC 55 ULX-1 at high luminosities (see above and Fig. \\ref{fig: L-T plot}) is the first stage of the dipping behavior seen in NGC 247 ULX-1 but with the wind not optically-thick enough to completely obscure the emission above 1 keV. This could be due to either a lower accretion rate or inclination angle in NGC 55 ULX-1. Given the lower luminosity and harder spectrum of NGC 55 ULX-1 as compared with the NGC 247 ULX-1 (see, e.g., \\citealt{Pinto_2021}), it is reasonable to speculate that the former is at a lower accretion rate with a thinner wind in the line of sight.\nIndeed, the dominant emission line {blend} at 1 keV from the wind is much stronger in NGC 247 ULX-1. More precisely, when modelled it with a \\textit{cie} component we obtained $L_{\\rm X \\, [0.3-10 \\, keV]} \\sim 0.7 \\times 10^{38}$ erg\/s in NGC 55 ULX-1 (see Table \\ref{table: XMM 0655050101 spectral fits}), while for NGC 247 ULX-1 it was $\\sim 1.4 \\times 10^{38}$ erg\/s \\citep{Pinto_2021}, i.e. twice as strong. \nA more powerful wind in NGC 247 ULX-1 is likely a consequence of a higher accretion rate.\n\n\\begin{figure\n\t\t\\centering\n\t\t\\includegraphics[width=0.475\\textwidth]{Images\/plot_LT_BB12_10obs_final.pdf}\n\t\t\\includegraphics[width=0.475\\textwidth]{Images\/plot_LT_BB12_FRS8_final.pdf}\n        \\vspace{-0.1cm}\n\t\t\\caption{{\\small Bolometric luminosity estimated in the 0.001-1000 keV energy band versus temperature for the cool blackbody (in blue) and hot modified blackbody (in orange) components. Top panel: Luminosity-temperature plot for the time resolved spectra (TRS). Bottom panel: Luminosity-temperature plot for the flux resolved spectra (FRS). The blue solid line, the black-dotted and the red dashed lines represent the regression line and the two theoretical models of the slim disc and Shakura-Sunyaev models, respectively. }}\n\t\t\\label{fig: L-T plot}\n\t\t\\end{figure}\n\t\t\n\\section{Conclusions}\n\\label{Conclusions}\n\nIn this work we have performed a spectral analysis to understand the structure and evolution of the accretion disc in the ultraluminous X-ray source NGC 55 ULX-1. The archival data was enriched with three deep XMM-\\textit{Newton} observations that we obtained in 2018, 2020 and 2021. This enabled to follow up the {continuum changes and their relation} with the source luminosity. As for most ULXs, it is necessary to use at least two spectral components to reproduce the spectral shape. The spectrum can be well fit with a two-blackbody model composed of a cool blackbody component, that describes the softer X-ray emission coming from the outer and cool part of the accretion disc and the wind photosphere, and a hotter blackbody modified by Compton scattering, that instead accounts for the emission from the hot inner disc.  Both components become hotter at higher luminosities, indicating either a better and clearer view of the inner disc, due to e.g. a reduction of the wind photosphere density, or to an increase in the accretion rate. The variability of the wind features would favour the latter case. The trends between the bolometric luminosity and temperature of each component broadly agree with the $L \\propto T^{4}$ relationship expected from constant area such as a thin disc. This suggests that the intrinsic luminosity of the source is not extremely high and likely close to the Eddington limit of a 10 $M\\textsubscript{\\(\\odot\\)}$ black hole. At high luminosities the cool component is cooler than the predictions from the thin-disc model and the best-fit regression line. This would imply an expansion of the disc and a contribution to the emission from the wind. If the deviation occurs between the Eddington limit and the supercritical accretion rate, a black hole within 6-14 $M\\textsubscript{\\(\\odot\\)}$ is foreseen.\n\n\\section{Data availability}\n\nAll of the data, with the exception of the Obs. ID 0883960101, and software used in this work are publicly available from ESA's XMM-\\textit{Newton} Science Archive (XSA\\footnote{https:\/\/www.cosmos.esa.int\/web\/XMM-\\textit{Newton}\/xsa}) and NASA's HEASARC archive\\footnote{https:\/\/heasarc.gsfc.nasa.gov\/}. \n\n\\section*{Acknowledgements}\n\nThis work is based on observations obtained with XMM-\\textit{Newton}, an ESA science mission funded by ESA Member States and USA (NASA). This work has been partially supported by the ASI-INAF program I\/004\/11\/4 from the agreement ASI-INAF n.2017-14-H.0 and from the INAF mainstream grant. We acknowledge the XMM-\\textit{Newton} SOC for the great support in scheduling our observations.\nAM acknowledges a financial support from the agreement ASI-INAF n.2017-14-H.0 (PI: T. Belloni, A. De Rosa), the HERMES project by the Italian Space Agency (ASI) n. 2016\/13 U.O, the H2020 ERC Consolidator Grant \"MAGNESIA\" No. 817661 (PI: Rea) and National Spanish grant PGC2018-095512-BI00.\n\n\n\n \n\n\n\n\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIndustrial robots typically work well for tasks where accurate position control is sufficient, and where work spaces and robot programs have been carefully prepared, so that hardware configurations can be foreseen a priori by robot programmers in each step of the tasks. Such preparation is very time consuming, and introduces high costs in terms of engineering work. Further, the arrangements are sensitive to variations, \\textit{e.g.}, uncertainties in work object positions, small differences between individual work objects, \\textit{etc.} This has prohibited the automation of a range of tasks, including seemingly repetitive ones such as assembly tasks and short-series production.\n\nIt would therefore be beneficial if the capabilities of robots to adapt to their surroundings could be improved. The framework of dynamical movement primitives (DMPs), used to model robot movement, has an emphasis on such adaptability \\cite{ijspeert2013dynamical}. For instance, the time scale and goal position of a movement can be adjusted through one parameter each. The fundamentals of DMPs have been described in \\cite{ijspeert2013dynamical}, and earlier versions have been introduced in \\cite{schaal2000nonlinear,ijspeert2002humanoid,ijspeert2003learning}. DMPs have been used to modify robot movement based on moving targets in the context of object handover \\cite{prada2014handover}, and based on demonstrations by humans \\cite{karlsson2017autonomous,karlsson2017motion,chiara2018passivity,karlsson2019human}. In most of the previous research, it has been assumed that the robot configuration space is a real coordinate space, such as joint space or Cartesian position space; see, \\textit{e.g.}, \\cite{prada2014handover,karlsson2017autonomous,chiara2018passivity,papageorgiou2018sinc,yang2018learning}. However, in \\cite{ude2014orientation} DMPs were formulated for orientation in Cartesian space.\n\nTemporal coupling for DMPs enables robots to recover from unforeseen events, such as disturbances or detours based on sensor data. This concept was introduced in \\cite{ijspeert2013dynamical}, was made practically realizable in \\cite{karlsson2017dmp}, and proven exponentially stable in \\cite{karlsson2018convergence}. However, these previous results are applicable only if the robot state space is Euclidean, which is not true for orientation in Cartesian space. Higher levels of robot control typically operate in Cartesian space, for instance to control the pose of a robot end-effector or an unmanned aerial vehicle.\n\n\nIn this paper, we therefore address the question of whether the control algorithm in \\cite{karlsson2017dmp} could be extended also to incorporate orientations. Because a contractible state space is necessary for design and analysis of a continuous globally asymptotically stable control law (see \\cref{sec:contractible}), we first investigate the contractibility properties of the quaternion set used to represent orientations. A space is contractible if and only if it is homotopy equivalent to a one-point space \\cite{hatcher2002algebraic}, which intuitively means that the space can be deformed continuously to a single point; see, \\textit{e.g.}, \\cite{hatcher2002algebraic} for a definition of homotopy equivalence.\n\n\\subsection{Contribution}\nThis paper provides a control algorithm for DMPs with temporal coupling in Cartesian space. It extends our previous research in \\cite{karlsson2017dmp,karlsson2018convergence} by including orientation in Cartesian space. Equivalently, it extends \\cite{ude2014orientation} by including temporal coupling. Furthermore, it is shown that the quaternion set minus one single point is contractible, which is a necessary property for design of a continuous and globally asymptotically stable control algorithm. Finally, the theoretical results are verified experimentally on an ABB YuMi robot; see \\cref{fig:yumi_gore_tex} and \\cite{yumi}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.98\\columnwidth]{figs\/robot_intro\/yumi_gore_tex.JPG}\n\\caption{The ABB YuMi robot \\cite{yumi} used in the experiments.}\n\\label{fig:yumi_gore_tex}\n\\end{figure}\n\n\\pagebreak\n\n\\section{A Contractible Subset \\\\ of the Unit Quaternion Set}\n\\label{sec:contractible}\nThe fundamentals of mathematical topology and set theory are described in, \\textit{e.g.}, \\cite{hatcher2002algebraic,crossley2006essential,schwarz2013topology}. As noted in \\cite{mayhew2011quaternion}, the rotation group SO(3) is not contractible, and therefore it is not possible for any continuous state-feedback control law to yield a globally asymptotically stable equilibrium point in SO(3) \\cite{bhat2000topological,koditschek1988application}. Contractibility is also necessary to apply the contraction theory from \\cite{lohmiller1998contraction}, as done in \\cite{karlsson2018convergence}. In this paper, unit quaternions are used to parameterize SO(3). Similarly to SO(3), the unit quaternion set, $\\mathbb{H}$, is not contractible. In this section however, is is shown that it is sufficient to remove one point from $\\mathbb{H}$ to yield a contractible space. \\cref{table:notation} lists some of the notation used in this paper.\n\n\n\n\n\n\\begin{table}\n\\begin{center}\n\\caption{Notation used in this paper. All quaternions represent orientations and are therefore of unit length. For such quaternions, the inverse is the same as the conjugate.} \\label{table:notation}\n\\begin{tabular}{l l l}\nNotation  &  & Description \\\\\n\\hline\n$\\mathbb{H}$ &   & Unit quaternion set \\\\\n$\\mathbb{S}^n$ &  $\\in \\mathbb{R}^{n+1}$ & Unit sphere of dimension $n$ \\\\\n$y_a$ & $\\in \\mathbb{R}^3$ & Actual robot position \\\\\n$g$ & $\\in \\mathbb{R}^3$ & Goal position \\\\\n$y_c$ & $\\in \\mathbb{R}^3$ & Coupled robot position \\\\\n$q_a$ & $\\in \\mathbb{H}$ & Actual robot orientation \\\\\n$q_g$ & $\\in \\mathbb{H}$ & Goal orientation \\\\\n$q_c$ & $\\in \\mathbb{H}$ & Coupled robot orientation \\\\\n$\\omega_c$ & $\\in \\mathbb{R}^3$ & Coupled angular velocity\\\\\n$q_0$ & $\\in \\mathbb{H}$ & Initial robot orientation \\\\\n$\\mathfrak{h}$  & & Quaternion difference space \\\\\n$d_{cg}$ & $\\in \\mathfrak{h}$  & Difference between $q_c$ and $q_g$   \\\\\n$z, \\omega_z$ & $\\in \\mathbb{R}^3$ & DMP states \\\\\n$\\alpha_z, \\beta_z, k_v, k_p$ & $\\in \\mathbb{R}^+$ & Constant control coefficients \\\\\n$\\tau$ & $\\in \\mathbb{R}^+$ & Nominal DMP time constant \\\\\n$\\tau_a$ & $\\in \\mathbb{R}^+$ & Adaptive time parameter \\\\\n$x$ & $\\in \\mathbb{R}^+$ & Phase variable \\\\\n$\\alpha_x, \\alpha_e, k_c$ & $\\in \\mathbb{R}^+$ & Positive constants \\\\\n$f(x)$ & $\\in \\mathbb{R}^6$  & Learnable virtual forcing term \\\\\n$f_p(x), f_o(x)$ & $\\in \\mathbb{R}^3$ & Position and orientation components \\\\\n$N_b$ & $\\in \\mathbb{Z}^+$ & Number of basis functions \\\\\n$\\Psi_j(x)$ & $\\in \\mathbb{R}^6$ & The $j$:th basis function vector \\\\\n$w_j$ & $\\in \\mathbb{R}^6$ & The $j$:th weight vector \\\\\n$e$ & $\\in \\mathbb{R}^3 \\times \\mathfrak{h} $ & Low-pass filtered pose error \\\\\n$e_p$ & $\\in \\mathbb{R}^3$ & Position component of $e$ \\\\\n$e_o$ & $\\in \\mathfrak{h}$ & Orientation component of $e$ \\\\\n$\\ddot{y}_r, \\dot{\\omega}_r$ & $\\in \\mathbb{R}^3$ & Reference robot acceleration \\\\\n$\\xi$ &  $ \\in \\mathbb{R}^{22} \\times \\mathfrak{h}^3$ & DMP state vector \\\\\n$\\bar{q}$ & $\\in \\mathbb{H}$ & Inverse of quaternion $q$ \\\\\n$\\simeq $ & & Homotopy equivalence \\\\\n$\\cong$ & & Homeomorphic relation \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\n\\subsection{Preliminary topology}\nWe will use that homeomorphism (defined in, \\textit{e.g.}, \\cite{crossley2006essential}) is a stronger relation than homotopy equivalence.\n\\begin{lemma}\n\\label{lm:homeomorphism}\nIf two spaces $X$ and $Y$ are homeomorphic, then they are homotopy equivalent.\n\\end{lemma}\n\\begin{proof}\nSee Lemma 6.11 in \\cite{crossley2006essential}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lm:remove_point}\nAssume that $X \\cong Y$, with a homeomorphism $f: X \\rightarrow Y$. Then $X$ minus a point $p \\in X$, denoted $X\\setminus p$, is homeomorphic to $Y\\setminus f(p)$.\n\\end{lemma}\n\\begin{proof}\nConsider the function $f_2: X\\setminus p \\rightarrow Y\\setminus f(p)$, and let $f_2(x) = f(x) \\hspace{2mm} \\forall x \\in X\\setminus p$. It can be seen that $f_2$ is a restriction of $f$. Since a restriction of a homeomorphism is also a homeomorphism \\cite{lehner1964discontinuous}, $f_2$ is a homeomorphism, and hence $X\\setminus p \\cong Y\\setminus f(p)$.\n\\end{proof}\n\nWe will also use that homeomorphism preserves contractibility.\n\\begin{lemma}\n\\label{lm:contract_homeomorphism}\nIf $X \\cong Y$, and $X$ is contractible, then $Y$ is also contractible.\n\\end{lemma}\n\\begin{proof}\nSince $X \\cong Y$, they are homotopy equivalent according to \\cref{lm:homeomorphism}. In turn, $X$ is contractible and therefore homotopy equivalent to a one-point space. Hence $Y$ is also homotopy equivalent to a one-point space, and therefore contractible.\n\\end{proof}\n\n\n\\subsection{The quaternion set minus one point is contractible}\nFirst, it will be shown that the unit sphere $\\mathbb{S}^n$ (see \\cref{def:unit_sphere}) minus a point is contractible. This will then be applied to $\\mathbb{H}$, which is homeomorphic to $\\mathbb{S}^3$ \\cite{lavalle2006planning}.\n\\begin{definition}\n\\label{def:unit_sphere}\nLet $n$ be a non-negative integer. The unit sphere with dimension $n$ is defined as\n\\begin{equation}\n\\mathbb{S}^n = \\left\\lbrace p \\in \\mathbb{R}^{n+1} \\hspace{2mm} \\mid \\hspace{2mm}    \\left\\lVert p \\right\\rVert _2 = 1 \\right\\rbrace\n\\end{equation}\n\\end{definition}\n\n\n\\begin{theorem}\n\\label{thm:sphere_n}\nThe unit sphere $\\mathbb{S}^n$ minus a point $p \\in \\mathbb{S}^n$, denoted $\\mathbb{S}^n\\setminus{p}$, is contractible. \n\\end{theorem}\n\n\\begin{proof}\nConsider first the case $n \\geq 1$. There exists a mapping from $\\mathbb{S}^n\\setminus{p}$ to $\\mathbb{R}^n$ called stereographic projection from $p$, which is a homeomorphism. Thus, $\\mathbb{S}^n\\setminus{p} \\cong \\mathbb{R}^n$ \\cite{huggett2009topological,schwarz2013topology}. See \\cref{fig:sphere_vs_plane} for a visualization of these spaces. Since $\\mathbb{R}^n$ is a Euclidean space it is contractible, and it follows from \\cref{lm:contract_homeomorphism} that $\\mathbb{S}^n\\setminus{p}$ is also contractible.\n\nConsider now the case $n=0$. The sphere $\\mathbb{S}^0$ consists of the pair of points $\\{-1,1\\}$ according to \\cref{def:unit_sphere}. Thus $\\mathbb{S}^0\\setminus{p}$ consists of one point only, and homotopy equivalence with a one-point space is trivial. Hence $\\mathbb{S}^0\\setminus{p}$ is contractible.\n\\end{proof}\n\n\\begin{remark}\nAlbeit we consider unit spheres in this paper, it is not necessary to assume radius 1 in \\cref{thm:sphere_n}. Further, it is arbitrary which point $p \\in \\mathbb{S}^n$ to remove.\n\\end{remark}\n\n\\begin{figure}\n\n\t\\input{figs\/sphere_vs_plane.tex}\n\t\\caption{Visualization of $\\mathbb{S}^n\\setminus p$ (left) and $\\mathbb{R}^n$ (right) for \\\\ $n=0,1,2$. The red cross marks a point $p$ removed from the unit sphere. Each space to the left is homeomorphic to the corressponding space to the right, \\emph{i.e.}, $\\mathbb{S}^n\\setminus p \\cong \\mathbb{R}^n$. In turn, $\\mathbb{R}^n$ is homotopy equivalent to a point (for instance $\\hat{p}$ marked by a purple dot in each plot to the right) and therefore $\\mathbb{S}^n\\setminus p$ is contractible according to \\cref{lm:contract_homeomorphism}. Higher dimensions are difficult to visualize, and therefore $\\mathbb{S}^2$ is commonly used to visualize parts of the quaternion set, as done in \\cref{fig:result_quat_spheres}.} \\label{fig:sphere_vs_plane}\n\\end{figure}\n\n\n\n\\begin{theorem}\n\\label{thm:quat_contractible}\nThe set of unit quaternions $\\mathbb{H}$ minus a point $\\tilde{q} \\in \\mathbb{H}$, denoted $\\mathbb{H}\\setminus{\\tilde{q}}$, is contractible. \n\\end{theorem}\n\\begin{proof}\nThe set $\\mathbb{H}$ is homeomorphic to $\\mathbb{S}^3$ \\cite{lavalle2006planning}. Therefore $\\mathbb{H}\\setminus{\\tilde{q}} \\cong \\mathbb{S}^3\\setminus{p}$ for some point $p \\in \\mathbb{S}^3$, according to \\cref{lm:remove_point}. \\cref{thm:sphere_n} with $n=3$ yields that $\\mathbb{S}^3\\setminus{p}$ is contractible, and because of the homeomorphic relation, \\cref{lm:contract_homeomorphism} yields that $\\mathbb{H}\\setminus{\\tilde{q}}$ is also contractible.\n\\end{proof}\n\nIt is noteworthy that the contractible subset $\\mathbb{H}\\setminus{\\tilde{q}}$ is the largest possible subset of $\\mathbb{H}$, because one point is the smallest possible subset to remove. Hence, it is guaranteed that no unnecessary restriction is made in \\cref{thm:quat_contractible}, though there are other, more limited, subsets of $\\mathbb{H}$ that are also contractible. Sometimes only half of $\\mathbb{H}$, for instance the upper half of the quaternion hypersphere, is used to represent orientations. However, instead of continuous transitions between the half spheres this results in discontinuities within the upper half sphere \\cite{lavalle2006planning}. In the context of DMPs and automatic control such discontinuities would cause severe obstructions, which motivates the search for the largest possible contractible subset of $\\mathbb{H}$. One of the experiments (Setup~3 in \\cref{sec:experiments}) provides an example of when both half spheres are necessary for a continuous representation of the robot orientation.\n\n\n\n\\section{Control Algorithm}\n\\label{sec:control_algorithm}\nIn this section, we augment the controller in \\cite{karlsson2017dmp,karlsson2018convergence} to incorporate orientation in Cartesian space. The resulting algorithm can also be seen as a temporally coupled version of the Cartesian DMPs proposed in \\cite{ude2014orientation}. The pose in Cartesian space consists of position and orientation. The position control in this paper is the same as described in \\cite{karlsson2017dmp,karlsson2018convergence}, except that it is also affected by the orientation through the shared time parameter $\\tau_a$ in this paper.\n\nSimilar to the approaches in \\cite{ude2014orientation,ude1999filtering}, we define a difference between two quaternions, $q_1$ and $q_2$, as \n\\begin{equation}\n\\label{eq:quat_diff}\nd(q_1\\bar{q}_2) = 2 \\cdot \\text{Im}[ \\log(q_1\\bar{q}_2)] \\in \\mathfrak{h}\n\\end{equation}\nwhere $\\mathfrak{h}$ is the orientation difference space, defined as the image of $d$, and Im denotes the imaginary quaternion part, assuming for now that $q_1\\bar{q}_2 \\neq (-1,0,0,0)$. \nThis is elaborated on in \\cref{sec:discussion}. Further, we will use a shorter notation, so that for instance \n\\begin{equation}\nd_{cg} = d(q_c\\bar{q}_g) = 2 \\cdot \\text{Im}[ \\log(q_c\\bar{q}_g)]\n\\end{equation}\nrepresents the difference between coupled and goal orientations. This mapping preserves the contractibility concluded in \\cref{sec:contractible}, as established by \\cref{thm:h_contractible}.\n\\begin{theorem}\n\\label{thm:h_contractible}\nThe orientation difference space $\\mathfrak{h}$ is contractible.\n\\end{theorem}\n\n\\begin{proof}\nThe mapping \n\\begin{equation}\nd \\hspace{2mm} : \\hspace{2mm} \\mathbb{H}\\setminus (-1,0,0,0) \\rightarrow \\mathfrak{h}\n\\end{equation}\nhas the properties necessary to qualify as a homeomorphism. It is one-to-one \\cite{ude1999filtering} and onto, continuous (since the point $(-1,0,0,0)$ has been removed), and its inverse (division by 2 followed by the exponential map) is also continuous. Further, its domain $\\mathbb{H}\\setminus (-1,0,0,0)$ is contractible (see \\cref{thm:quat_contractible}), and therefore its image $\\mathfrak{h}$ is contractible (see \\cref{lm:contract_homeomorphism}).\n\\end{proof}\n\n\nUsing the function $d$, a coupled DMP pose trajectory is modeled by the dynamical system\n\\begin{align}\n\\label{eq:dotz}\n\\tau_a \\dot{z} &= \\alpha_z(\\beta_z(g-y_c)-z) + f_p(x) \\\\\n\\tau_a \\dot{y}_c &= z  \\\\\n\\tau_a \\dot{\\omega}_z &= \\alpha_z(\\beta_z (-d_{cg})-\\omega_z) + f_o(x) \\\\\n\\tau_a \\omega_c &= \\omega_z\n\\end{align}\nHere, $x$ is a phase variable that evolves as \n\\begin{align}\n\\tau_a \\dot{x} =& -\\alpha_x x\n\\label{eq:x}\n\\end{align}\nFurther, $f_o(x)$ is a virtual forcing term in the orientation domain, and each element $i$ of $f_o(x)$ is given by\n\\begin{align}\nf^i_o(x) =& \\frac{\\sum_{j=1}^{N_b} \\Psi_{i,j}(x)w_{i,j}}{\\sum_{j=1}^{N_b} \\Psi_{i,j}(x)} x \\cdot d_i(q_g \\bar{q}_0)\n\\label{eq:f}\n\\end{align}\nwhere each basis function, $\\Psi_{i,j}(x)$, is determined as\n\\begin{align}\n\\Psi_{i,j}(x) =& \\exp \\left(-\\frac{1}{2\\sigma_{i,j}^2}(x-c_{i,j})^2 \\right)\n\\label{eq:psi}\n\\end{align}\nHere, $\\sigma$ and $c$ denote the width and center of each basis function, respectively. The forcing term $f_p(x)$ is determined accordingly, see \\cite{karlsson2017dmp,karlsson2018convergence}. Further, the parameters of $f(x)$ can be determined based on a demonstrated trajectory by means of locally weighted regression \\cite{atkeson1997locally}, as described in \\cite{ijspeert2013dynamical}. \n\nAll dimensions of the robot pose are temporally coupled through the shared adaptive time parameter $\\tau_a$. Denote by $y_a$ the actual position of the robot, and by $q_a$ the actual orientation. The adaptive time parameter $\\tau_a$ is determined based on the low-pass filtered difference between the actual and coupled poses as follows.\n\\begin{align}\n\\dot{e}_p &= \\alpha_e(y_a - y_c - e_p) \\\\\n\\dot{e}_o &= \\alpha_e(d_{ac} - e_o)  \\\\\n\\label{eq:edot} \ne &= [e_p^T \\hspace{2mm}  e_o^T]^T \\\\\n\\tau_a &= \\tau(1 + k_c e^Te)\n\\label{eq:taua}\n\\end{align}\nThis causes the evolution of the coupled system to slow down in case of configuration deviation; see \\cite{ijspeert2013dynamical,karlsson2017dmp}. Moreover, the controller below is used to drive $y_a$ to $y_c$, and $q_a$ to $q_c$.\n\\begin{align}\n\\label{eq:our_ddoty}\n\\ddot{y}_r &= k_p(y_c-y_a) + k_v(\\dot{y}_c - \\dot{y}_a) + \\ddot{y}_c \\\\\n\\dot{\\omega}_r &= -k_p d_{ac} - k_v(\\omega_a-\\omega_c)+\\dot{\\omega}_c\n\\label{eq:our_dotomega}\n\\end{align}\nThis can be seen as a pose PD controller together with the feedforward terms $\\ddot{y}_c$ and $\\dot{\\omega}_c$. Here, $\\ddot{y}_r$ and $\\dot{\\omega}_r$ denote reference accelerations sent to the internal controller of the robot, after conversion to joint values using the robot Jacobian \\cite{spong2006robot}. We let $k_p = k_v^2\/4$, so that \\cref{eq:our_ddoty}~--~\\cref{eq:our_dotomega} represent a critically damped control loop. Similarly, $\\beta_z=\\alpha_z \/ 4$ \\cite{ijspeert2002humanoid}. The control system is schematically visualized in \\cref{fig:coupling_scheme}. We model the 'Robot' block as a double integrator, so that $\\ddot{y}_a=\\ddot{y}_r$ and $\\dot{\\omega}_a= \\dot{\\omega}_r$, as justified in \\cite{karlsson2018convergence} for accelerations with moderate magnitudes and changing rates. In summary, the proposed control system is given by\n\\begin{align}\n\\label{eq:cartesian_entire_control_a}\n\\ddot{y} &= -k_\\text{p}(y-y_\\text{c}) - k_\\text{v} (\\dot{y}-\\dot{y}_\\text{c}) + \\ddot{y}_\\text{c} \\\\\n\\label{eq:cartesian_entire_control_b}\n\\dot{\\omega}_\\text{a} &= -k_\\text{p} d_\\text{ac} - k_\\text{v}(\\omega_\\text{a}-\\omega_\\text{c})+\\dot{\\omega}_\\text{c} \\\\\n\\label{eq:cartesian_entire_control_c}\n\\dot{e} &= \\alpha_e\\left(\\left[ \\left[ y-y_\\text{c} \\right]^T \\phantom{d} d_\\text{ac}^T\\right]^T - e\\right) \\\\\n\\tau_\\text{a} &= \\tau (1 + k_\\text{c} e^Te) \\\\\n\\tau_\\text{a} \\dot{x} &= -\\alpha_x x \\\\\n\\tau_\\text{a} \\dot{y}_\\text{c} &= z  \\\\\n\\tau_\\text{a} \\dot{z} &= \\alpha(\\beta(g-y_\\text{c})-z) + f_\\text{p}(x) \\\\\n\\tau_\\text{a} \\omega_\\text{c} &= \\omega_z \\\\\n\\label{eq:cartesian_entire_control_last}\n\\tau_\\text{a} \\dot{\\omega}_z &= \\alpha(\\beta(-d_{\\text{c}g})-\\omega_z) + f_\\text{o}(x)\n\\end{align}\nWe introduce a state vector $\\xi$ as\n\\begin{equation}\n\\xi = \n\\begin{pmatrix}\ny - y_\\text{c} \\\\[3pt] \\dot{y}-\\dot{y}_\\text{c} \\\\[3pt] d_\\text{ac} \\\\[3pt] \\omega_\\text{a}-\\omega_\\text{c} \\\\[3pt] e \\\\[3pt] x \\\\[3pt] y_\\text{c}-g \\\\[3pt] z \\\\[3pt]d_{\\text{c}g} \\\\[3pt] w_z\n\\end{pmatrix}\n\\in \\mathbb{R}^{22} \\times \\mathfrak{h}^3\n\\label{eq:cartesian_states_first_time}\n\\end{equation}\n\n\\begin{figure}\n\t\\centering\n    \\input{figs\/coupling_scheme.tex}\n    \\caption{The control structure for temporally coupled Cartesian DMPs. The block denoted 'Robot' includes the internal controller of the robot, together with transformations between Cartesian and joint space for low-level control. The 'DMP' block corresponds to the computations in \\cref{eq:dotz}~--~\\cref{eq:taua}. The PD controller and the feedforward terms are specified in \\cref{eq:our_ddoty}~--~\\cref{eq:our_dotomega}. This forms a cascade controller, with the DMP as outer controller and the PD as the inner.}\n\\label{fig:coupling_scheme}\n\\end{figure}\n\n\n\n\n\n\n\n\n\\section{Experiments}\n\\label{sec:experiments}\nThe control law in \\cref{sec:control_algorithm} was implemented in the Julia programming language \\cite{BEKS14}, to control an ABB YuMi \\cite{yumi} robot. The Julia program communicated with the internal robot controller through a research interface version of Externally Guided Motion (EGM) \\cite{egm,bagge2017yumi} at a sampling rate of \\SI{250}{Hz}. \n\nThree different setups were used to investigate the behavior of the controller. As preparation for each setup, a temporally coupled Cartesian DMP had been determined from a demonstration by means of lead-through programming, which was available in the YuMi product by default. In each trial, the temporally coupled DMP was executed while the magnitudes of the states in \\cref{eq:cartesian_states_first_time} were logged.\n\nPerturbations were introduced by physical contact with a human. This was enabled by estimating joint torques induced by the contact, and mapping these to Cartesian contact forces and torques using the robot Jacobian. A corresponding acceleration was then added to the reference acceleration $\\ddot{y}_r$ as a load disturbance. However, we emphasize that this paper is not focused on how to generate the perturbations themselves. Instead, that functionality was used only as an example of unforeseen deviations, and to investigate the stability properties of the proposed control algorithm.\n\nA video of the experimental arrangement is available as an attachment to this paper, and a version with higher resolution is available in \\cite{cartesian_dmp_youtube}. The setups were as follows.\n\n\n\\begin{figure}\n\\centering\n\\begin{minipage}{.48\\columnwidth}\n\\centering\n\\includegraphics[width=\\columnwidth,height=28mm]{figs\/setup1\/setup1a_white.JPG}\n\\subcaption{}\n\\end{minipage}\\hfill\n\\begin{minipage}{.48\\columnwidth}\n\\includegraphics[width=\\columnwidth]{figs\/setup1\/setup1b_white.JPG}\n\\subcaption{}\n\\end{minipage}\n\\caption{Photographs of a trial of Setup 1. The robot was initially released from the pose in (a), with an offset to the goal pose. In (b), the goal pose was reached.}\n\\label{fig:setup1}\n\\end{figure}\n\n\n\\textbf{Setup 1.} This setup is visualized in \\cref{fig:setup1}. Prior to the experiment, a test DMP that did not perform any particular task was executed, and the robot then converged to the goal pose, \\emph{i.e.}, to $y_a=y_c=g$ and $d_{ac}=d_{cg}=0$. Thereafter, the operator pushed the end-effector, so that the actual pose deviated from the coupled and goal poses. The experiment was initialized when the operator released the robot arm. The purpose of this procedure was to examine the stability of the subsystem in (\\ref{eq:cartesian_entire_control_a})~--~(\\ref{eq:cartesian_entire_control_c}). A total of 100 perturbations were conducted.\n\n\n\n\n\\textbf{Setup 2.} See \\cref{fig:setup2}. The task of the robot was to reach a work object (in this case a gore-tex graft used in cardiac and vascular surgery) from its home position. A DMP defined for this purpose was executed, and the operator introduced two perturbations during the robot movement. The purpose of this setup was to investigate the stability of the entire control system in (\\ref{eq:cartesian_entire_control_a})~--~(\\ref{eq:cartesian_entire_control_last}). A total of 10 trials were conducted.\n\n\\textbf{Setup 3.} See \\cref{fig:setup3}. The task of the robot was to hand over the work object from its right arm to its left.  The movement was specifically designed to require an end-effector rotation angle of more than $\\pi$, thus requiring both the upper and the lower halves of the quaternion hypersphere (see \\cref{fig:result_quat_spheres}), and not only one of the halves which is sometimes used \\cite{lavalle2006planning}. Such movements motivate the search for the largest possible contractible subset of $\\mathbb{H}$ in \\cref{sec:contractible}. Similar to Setup~2, the purpose was to investigate the stability of (\\ref{eq:cartesian_entire_control_a})~--~(\\ref{eq:cartesian_entire_control_last}), and 10 trials were conducted.\n\n\n\n\n\n\n\n\n\n\\begin{figure}\n\\centering\n\\begin{minipage}{.48\\columnwidth}\n\\centering\n\\includegraphics[width=\\columnwidth,height=28mm]{figs\/setup2\/setup2a_white.JPG}\n\\subcaption{}\n\\end{minipage}\\hfill\n\\begin{minipage}{.48\\columnwidth}\n\\includegraphics[width=\\columnwidth]{figs\/setup2\/setup2b_white.JPG}\n\\subcaption{}\n\\end{minipage}\n\\par\\vspace{3mm}\n\\begin{minipage}{.48\\columnwidth}\n\\includegraphics[width=\\columnwidth]{figs\/setup2\/setup2c_white.JPG}\n\\subcaption{}\n\\end{minipage}\\hfill\n\\begin{minipage}{.48\\columnwidth}\n\\centering\n\\includegraphics[width=\\columnwidth]{figs\/setup2\/setup2d_white.JPG}\n\\subcaption{}\n\\end{minipage}\n\\caption{Photographs of a trial of Setup 2. The DMP was executed from the home position (a), and was perturbed twice on its way toward the goal (b). It recovered from these perturbations (c), and reached the goal at the work object (d).}\n\\label{fig:setup2}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\centering\n\\begin{minipage}{.48\\columnwidth}\n\\centering\n\\begin{tikzpicture}\n\\node[anchor=south west,inner sep=0] at (0,0) {\\includegraphics[width=\\columnwidth,height=28mm]{figs\/setup3\/setup3a_white.JPG}};\n\\draw [->,line width=1.5pt,red] (1.4,.5) arc[x radius=.5cm, y radius =.5cm, start angle=270, end angle=90];\n\\end{tikzpicture}\n\\subcaption{}\n\\end{minipage}\\hfill\n\\begin{minipage}{.48\\columnwidth}\n\\includegraphics[width=\\columnwidth]{figs\/setup3\/setup3b_white.JPG}\n\\subcaption{}\n\\end{minipage}\n\\par\\vspace{3mm}\n\\begin{minipage}{.48\\columnwidth}\n\\begin{tikzpicture}\n\\node[anchor=south west,inner sep=0] at (0,0) {\\includegraphics[width=\\columnwidth,height=28mm]{figs\/setup3\/setup3c_white.JPG}};\n\\draw [->,line width=1.5pt,red] (2,2.3) arc[x radius=0.6cm, y radius =.6cm, start angle=90, end angle=0];\n\\end{tikzpicture}\\subcaption{}\n\\end{minipage}\\hfill\n\\begin{minipage}{.48\\columnwidth}\n\\centering\n\\includegraphics[width=\\columnwidth]{figs\/setup3\/setup3d_zoom.JPG}\n\\subcaption{}\n\\end{minipage}\n\\caption{Photographs of a trial of Setup 3. The robot started its movement from the configuration in (a). The end-effector was rotated as indicated by the red arrows, which resulted in a rotation larger than $\\pi$ from start to goal. The robot was perturbed twice by the operator (b), recovered and continued its movement (c), and accomplished the handover (d).}\n\\label{fig:setup3}\n\\end{figure}\n\n\n\n\\begin{figure}[h]\n\t\\centering\t\n\t\\input{figs\/setup1_plot.tex}\n\t\\caption{Data from a trial of Setup 1. The notation $\\Vert\\cdot\\Vert$ represents the 2-norm, and the unit symbol [1] indicates dimensionless quantity. The experiment was initialized with some position error $y_a-y_c$ and orientation error $d_{ac}$. The operator released the robot at $t=0$. It can be seen that each state converged to 0.}\n\t\\label{fig:setup1_plot}\n\\end{figure}\n\n\n\\section{Results}\n\\label{sec:results}\nFigures~\\ref{fig:setup1_plot}--\\ref{fig:result_quat_spheres} display data from the experiments. \\Cref{fig:setup1_plot} shows the magnitude of the states during a trial of Setup~1, and it can be seen that each state converged to 0 after the robot had been released. Similarly, \\cref{fig:grasp_states,fig:handover_states} show data from Setup~2 and 3 respectively, and it can be seen that the robot recovered from each of the perturbations. Further, each state subsequently converged to 0. All trials in a given setup gave similar results. Further, these results suggest that the control system (\\ref{eq:cartesian_entire_control_a})~--~(\\ref{eq:cartesian_entire_control_last}) is exponentially stable.\n\n\n\\begin{figure}\n\t\\centering\t\n\t\\input{figs\/grasp_states.tex}\n\t\\caption{Data from a trial of Setup 2. Consider first the upper plot. The two perturbations are clearly visible, and these were recovered from as the states converged to 0. In the lower plot, it can be seen that the time evolution of the states was slowed down in the presence of perturbations. It can further be seen that each of the states converged to 0.}\n\t\\label{fig:grasp_states}\n\\end{figure}\n\n\n\\Cref{fig:result_quat_spheres} shows orientation data from Setup~2 (left) and Setup~3 (right). The upper plots show quaternions for the demonstrated paths, $q_d$, determined using lead-through programming prior to the experimental trials, relative to the goal quaternions $q_g$. The middle plots show coupled orientations $q_c$ relative to $q_g$. It can be seen that the paths of $q_d$ and $q_c$ were similar for each of the setups, which was expected given a sufficient number of DMP basis functions. The perturbations can be seen in the bottom plots, which show $q_a$ relative to $q_c$. Though $q_a \\bar{q}_c$ was very close to the identity quaternion for most of the time, it deviated significantly twice per trial as a result of the perturbations. Setup~3 is an example of a movement where it would not be possible to restrict the quaternions to the upper half sphere, without introducing discontinuities. This is shown in \\Cref{fig:result_quat_spheres}, as quaternions were present not only on the upper half sphere, but also on the lower, for Setup~3.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\nIn each of the experiments, the robot recovered from the perturbations and subsequently reached the goal pose, which was the desired behavior. Further, the behavior corresponded to that in \\cite{karlsson2017dmp,karlsson2018convergence}, except that orientations in Cartesian space are now supported. Most of the discussion in \\cite{karlsson2017dmp,karlsson2018convergence} is therefore valid also for these results, and is not repeated here. \n\n\nA mathematical proof that the proposed control system is exponentially stable would enhance the contribution of this paper, but remains as future research. Nevertheless, is has now been shown that the topology of $\\mathfrak{h}$ does not prohibit a globally exponentially stable control system. One may object that this topological result is not directly necessary for the control design in \\cref{sec:control_algorithm}. However, it is still useful because it rules out the otherwise possible obstruction of a non-contractible state space. This result is relevant not only for DMP applications, but for any control application where quaternions are used to represent orientation. Furthermore, the experimental results indicate exponential stability, since in practice the DMP states converged to 0.\n\n\n\n\n\nThe magnitude of the difference between two quaternions, $||d(q_1 \\bar{q}_2)||$, corresponds to the length of a geodesic curve connecting $q_1$ and $q_2$ \\cite{ude1999filtering}. This results in proper scaling between orientation difference and angular velocity in the DMP control algorithm, as explained in \\cite{ude2014orientation}. This is the reason why the quaternion difference in \\cref{eq:quat_diff} was used in \\cite{ude2014orientation} and in this paper. \n\nIn \\cref{sec:contractible}, the largest possible contractible subset of $\\mathbb{H}$ was found as $\\mathbb{H} \\setminus \\tilde{q}$. Hence, it is not necessary to remove a large proportion of the quaternion set, which is sometimes done. For instance, sometimes the lower half of the quaternion hypersphere is removed \\cite{lavalle2006planning}, which is unnecessarily limiting. The results from Setup~3 show that this proposed method works also when it is necessary to use both half spheres, see \\cref{fig:result_quat_spheres}. In \\cref{sec:control_algorithm}, the removed point $\\tilde{q}$ was chosen as $(-1,0,0,0)$, which corresponds to a full $2 \\pi$ rotation from the identity quaternion. A natural question is therefore how to handle the case where $(-1,0,0,0)$ is visited by $q_a \\bar{q}_c$ or $q_c \\bar{q}_g$. In theory, almost any control signal could be used to move the orientations away from this point, and in practice a single point would never be visited because it is infinitely small. However, in practice some care should be taken in a small region around $(-1,0,0,0)$, because of possible numerical difficulties and rapidly changing control signals.\n\n\nIn this paper, the same control gains were used in the position domain as in the orientation domain. This was done in order to limit the notation, but is not actually required.\n\nAn interesting direction of future work is to use the proposed controller to warm start reinforcement learning approaches for robotic manipulation. Reinforcement learning with earlier DMP versions has been investigated in, \\textit{e.g.}, \\cite{stulp2011learning,stulp2012reinforcement,stulp2012model,li2018reinforcement}.\n\n\n\n\\section{Conclusion}\nIn this paper, it was first shown that the unit quaternion set minus one point is contractible, thus allowing for continuous and asymptotically stable control systems. This was used to design a control algorithm for DMPs with temporal coupling in Cartesian space. The proposed DMP functionality was verified experimentally on an industrial robot.\n\nA video that shows the experiments is provided as an attachment to this paper, and a version with higher resolution is available in \\cite{cartesian_dmp_youtube}.\n\n\\begin{figure}[h]\n\t\\centering\t\n\t\\input{figs\/handover_states.tex}\n\t\\caption{Data from a trial of Setup 3. The organization is the same as in \\cref{fig:grasp_states}, and similar conclusions can be drawn. In addition, the required rotation angle from start to goal was larger than $\\pi$ in this setup, which corresponds to $||d_{cg}||$ being larger than $\\pi$ initially.}\n\t\\label{fig:handover_states}\n\\end{figure}\n\n\n\\begin{figure}\n\t\\centering\t\n\t\\input{figs\/result_quat_spheres.tex}\n\t\\caption{Orientation data from Setup~2 (left) and Setup~3 (right). Quaternions have been projected on $\\mathbb{S}^2$ for the purpose of visualization. Vertical axes represent quaternion real parts, and horizontal axes represent the first two imaginary elements with magnitudes adjusted to yield unit length of the resulting projection. The bottom plots show the quaternion set seen from above, and hence their real axes are directed out from the figure.}\n\t\\label{fig:result_quat_spheres}\n\\end{figure}\n\n\n\n\\bibliographystyle{IEEEtran}\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nNaive estimates seem to show that the triple pomeron diagram contributes\nsignificantly to the scattering cross-section at high energies. \nTake the\ndiagram of Fig. 1$a$ as an example. Let the total c.m. energy squared be\n$s=\\exp Y$,  those for the upper and lower pomerons $s_{1}=\\exp y_{1}$ and\n$s_{2}=\\exp y_{2}$, respectively, $y_{1}+y_{2}=Y$,  and $\\Delta$ the pomeron\nintercept. Then the contribution of the diagram seems to be\n\\begin{equation}\n\\gamma_{1}\\gamma_{2}\\gamma_{3P}\\int_{0}^{Y}dy_{1}\\exp (\\Delta y_{1})\n\\exp (2\\Delta (Y-y_{1}))=\\gamma_{1}\\gamma_{3}\\gamma_{3P}\\Delta^{-1}\n\\exp(2\\Delta Y)(1-\\exp(-\\Delta Y)\n\\end{equation} \nwhere $\\gamma_{1}$ and $\\gamma_{2}$ are the couplings of the colliding\nparticle (the same for the projectile and target, for simplicity) to one\nand two pomerons and $\\gamma_{3P}$ is the triple pomeron coupling.\nThis contribution is of the same order as the one from the \"pure\" \ntwo pomeron exchange, corresponding to the diagram of Fig. 1$b$\n\\begin{equation}\n\\gamma_{2}^{2}\\exp (2\\Delta Y)\n\\end{equation}\neven in the small coupling limit, when one should take into account that\n$\\gamma_{3P}\\sim\\gamma_{2}$.\n\nHowever inspecting the result (1) one notices that the whole contribution\nto the righthand side at large $Y$ comes from the region of integration on\nthe left $y_{1}<<Y$, where the existence of the upper pomeron becomes\nquestionable. So the conclusion that at large energies the triple pomeron\ngives a sizable contribution is not at all obvious. In fact, in our earlier\npaper [1] we assumed that it does not contribute, which lead us to the\ncross-section essentially of the eikonal form.\n\nIn this note we try to study the triple pomeron contribution in\nthe framework of the BFKL hard pomeron [2]  in some detail. We shall study\nonly the simplest diagram of Fig. 1$a$ and even for this diagram we shall\nnot be able to obtain results  in a form, amenable for numerical\ncalculations. However, what we shall find is sufficient to draw definite\nconclusions as to the relative importance of the triple pomeron\ncontribution. Since the derivation is very technical, we present here the\nmain result together with its simple explanation.\n\nAs it turns out, the relative importance of the triple pomeron contribution\ndepends on the magnitude of the colliding particle virtuality (or mass)\n$Q^{2}$. If it is high enough (e.g. for the structure function) then the\ncontribution of the triple pomeron is neglegible. However, with $Q^{2}$\ndiminishing, the triple pomeron contribution becomes more important and at\nlow $Q^{2}$ it results at least of the same magnitude as the pure two\npomeron exchange (Fig. 1$b$).\n\nTo understand this result note that the hard pomeron theory with a small\n(fixed) coupling constant $g$ possesses an intrinsic large rapidity (or\nenergy) scale\n\\begin{equation} Y_{0}=1\/g^{2} \\end{equation}\nThis should be contrasted to the \"external\" rapidity $Y$ and colliding\nparticle virtuality $Q^{2}=\\ln\\zeta$. Evidently the pomeron theory applies\nonly when\n\\begin{equation}\n1<<Y_{0}<<Y,\\ \\ \\zeta <<Y\n\\end{equation}\nHowever the relation between $\\zeta$ and $Y_{0}$ may be arbitrary.\n\n Now take\nthe generic diagram of Fig. 2 corresponding to  both of the diagrams in Fig.\n1, with the upper blob $B$ taken at the energy $s_{1}=\\exp y_{1}$. Its\ncontribution is given by the integral similar to (1)\n\\begin{equation}\n\\gamma_{2}\\int_{0}^{Y}dy_{1}B(y_{1},\\zeta)\n\\exp (2\\Delta (Y-y_{1}))\n\\end{equation} \nThe integration in (5) is evidently limited to  values of $y_{1}$\nsmaller or of the order of $\\Delta^{-1}\\sim Y_{0}$. Then we may distinguish\nbetween two possibilities.\n\nIf $Q^{2}$ is large, so that $\\zeta$ is of the order (or greater) than\n$Y_{0}$\n\\begin{equation} \\zeta\\sim Y_{0} \\end{equation}\nthen  blob $B$ is, in fact,  not in the low-$x$ but in a pure\nperturbative regime (recall that the coupling constant is supposed to be\nfixed and small). Then, to the accuracy adopted in the BFKL theory, we have\nto retain only the lowest order contribution to it. This leads to the\npure double pomeron exchange. For the photon projectile it means that the two\npomerons should be directly coupled to the  the $q\\bar q$ loop. Thus in the\nregion (6) the triple pomeron does not contribute: it simply does not exist.\n\nOn the other hand for small $Q^{2}$, i.e. when\n\\begin{equation}\n\\zeta<<Y_{0}\n\\end{equation}\nblob $B$ turns out to be in the low-$x$ regime and we can take a pomeron \nfor its leading contribution $B(y,\\zeta)=\\gamma_{1}\\gamma_{3P}P(g^{2}y)$.\nThe integral (5) then gives\n\\begin{equation}\n\\gamma_{1}\\gamma_{2}\\gamma_{3P}g^{-2}\\exp(2\\Delta Y)\n\\int_{0}^{Y\/g^{2}}dzP(z)\\exp(-2z)\n\\end{equation} \nThis contribution now does not reduce to the pure two-pomeron exchange and\ncorresponds to the triple p[omeron of Fig. 1$a$. Note that in (8) finite\nvalues of $z$ contribute, so that the upper pomeron of Fig. 1$a$ enters\nhere not in its asymptotic regime (as in (1)) but at finite values of\n$g^{2}\\ln s$ and so has to be taken in its full complexity.\n\nAs a result, we find that the hard pomeron theory allows for a clear\ndistinction between colliding \"particles\", with $g^{2}\\ln Q^{2}<<1$, and\nvirtual probes, with $g^{2}\\ln Q^{2}\\sim 1$. For the former the triple\npomeron dominates the cross-section. For the latter it does not contribute.\n\nTurning to the literature, we recall that our results for summing\nmultipomeron exchanges in [1] referred to the photon structure function at\nhigh $Q^{2}$. So the omission of the triple (and multiple) pomeron\ninteractions, which was made there, results justified. On the other hand,\nin a series of papers of A.Mueller {\\it et al.} [3-5] the calculation of\nthe double (and multiple) pomeron exchange seems to be made under the\nassumption of a fixed mass $M$ of colliding \"onia\" and the coupling $g$\ngoing to zero. Then one inevitably arrives at the region (7) where the triple\npomeron seems to give the bulk of the contribution. This may be a part of the\nexplanation why the resulting amplitudes do not eikonalize in A.Mueller's\napproach.\n\nWe have to note, however, that to move closer to the realistic QCD, one has\nto take the onia mass $M$ large enough to make the effective coupling\nconstant small. This latter condition, translated into the fixed coupling\nconstant language, means that $g^{2}\\ln M^{2}\\sim 1$, so that the region (6)\nshould rather be considered, where no triple pomeron contribution is to be\nexpected.\n\nIn the rest of this note we present the calculation of the triple pomeron\ncontribution (Fig. 1$a$). As mentioned we shall not be able to find an\nexpression for it which admits, say, numerical estimates. However we shall\nclearly see how and why this contribution goes to zero when $Q^{2}$ rises.\nAlso a part (dominant at high $Q^{2}$) will be found in a form which can be\nstudied numerically. Our calculations will be heavily based on the results\nobtained in our previous paper [6], where the triple pomeron vertex was\nstudied in the asymptotic regime.\n\n\\section{The triple pomeron contribution. The target part}\nThe cross-section corresponding to Fig. 1$a$. in the hard pomeron approach\nof [2] is explicitly given by\n\\[\n\\sigma=-\\frac{g^{10}N^{3}}{32(2\\pi)^{9}}\\int d^{2}l\\int\n(ds_{1}\/s_{1})\n\\int \\prod_{i=1}^{3}(d^{2}r_{i}d^{2}r'_{i}\n(2\\pi )^{-2})K_{l}(r_{1},r_{2},r_{3})\\]\n\\begin{equation}\\exp (-il(r_{2}+r_{3})\/2)\nG_{0}(s_{1},r_{1},r'_{1})G_{l}(s_{2},r_{2},r'_{2})\nG_{-l}(s_{2},-r_{3},r'_{3})\\rho_{1}(r'_{1})\\rho_{2}(r'_{2},r'_{3})\n\\end{equation}\nHere $N$ is the number of flavours; $s_{2}=s\/s_{1}$ where $s$ is the overall\nc.m. energy squared; $r'_{i}$ and $r_{i}$, $i=1,2,3$, are the initial and\nfinal transverse dimensions of the upper ($i=1$) and lower ($i=2,3$)\npomerons as they propagate from the sources (colliding particles) to the\ntriple pomeron junction. The sources colour densities are $\\rho_{1}$ for the\nprojectile and $\\rho_{2}$ for the target. The BFKL Green function\n$G_{l}(s,r,r')$ describes propagation of the pomeron with the total momentum\n$l$ and energy $s$. $K_{l}(r_{1},r_{2},r_{3})$ is the triple pomeron\ninteraction vertex. It originates from the Bartels' vertex for the\ntransition of 2 reggeized gluons into 4 [7]. Its explicit form  in the\ncoordinate space was found in [6]. Taking into account that the Green\nfunctions $G_{l}$ vanish if $r=0$ or $r'=0$, it can be reduced to\n\\begin{equation}\nK_{l}(r_{1},r_{2},r_{3})=\n-\\frac{2}{(2\\pi)^{2}}\\frac{(r_{2}r_{3})}{r_{2}^{2}r_{3}^{2}}\\nabla_{1}^{4}\n\\delta^{2}(r_{1}+r_{2}+r_{3})\n\\end{equation}\nThe numerical coefficient in (9) combines factors $g^{2}N\/2$ and\n$g^{4}N\/2$ separated from the colour densities $\\rho_{1}$ and\n$\\rho_{2}$, respectively, a factor $g^{4}N$ from the triple pomeron\nvertex, a factor $1\/2$ for the identity of two lower pomerons and some\nnumerical factors which relate the cross-section to the amplitude\ncorresponding to the diagram of Fig. 1$a$.\n\n\nFor our purpose the explicit form of the target density\n$\\rho_{2}(r'_{2},r'_{3})$ will be inessencial, except that it should be\nwell behaved  and restrict $r'_{2}$ and $r'_{3}$ to some average value\n$r_{20}$ of the order of the target transverse dimension. To simplify, we\nthen take\n$\\rho_{2}(r'_{2},r'_{3})=\n\\delta^{2}(r'_{2}-r'_{3})\\delta(r'_{2}-r_{20})\/(2\\pi\nr_{20})$, which means that we simply substitute in the two Green functions\nfor the lower pomerons $r'_{2}=r'_{2}=r_{20}$ where the latter vector has\nlength $r_{20}$ and an arbitrary direction.\nAfter this substitution, and with the explicit form of $K_{l}$ given by\n(10), the cross-section (9) transforms into\n\\[\n\\sigma=-\\frac{g^{10}N^{3}}{16(2\\pi)^{13}}\\int d^{2}l\\int\n(ds_{1}\/s_{1})\\int d^{2}r_{1}d^{2}q\\rho_{1}(r_{1})\\]\\begin{equation}\n\\chi_{1}(s_{1},q+l\/2,r_{1})\n(\\nabla_{q}\\chi_{2}(s_{2},l,q))^{2}\n\\end{equation}\nwhere\n\\begin{equation}\n\\chi_{1}(s,q,r_{1})=\\int d^{2}r\\nabla^{4}G_{0}(s,r,r_{1})\\exp iqr\n\\end{equation}\nand\n\\begin{equation}\n\\chi_{2}(s,l,q)=\\int d^{2}r r^{-2}G_{l}(s,r,r_{20})\\exp iqr\n\\end{equation}\n\nAs mentioned in the Introduction, the triple pomeron interaction which\nenters (11) has been studied in [6] in the asymptotic region\n$s_{1}\\sim s_{2}\\rightarrow\\infty$. In our case, we rather have to\nintegrate over all $s_{1}$, so that the upper pomeron will stay far from\nits asymptotic regime and we shall have to take the exact form for its\nGreen function. However the target part $\\chi_{2}$ enters (11) in its\nasymptotic region, so that we may borrow its form from [6], to which paper we\nrefer the reader for more detail in this respect.\n\n\nThe leading contribution\nto the BFKL Green function at $l\\neq 0$, which appears in (13), has the form\n[8] \\begin{equation}\nG_{l}(s,r,r')=(1\/4\\pi^{2})\\int\\frac{d\\nu \\nu^{2}}{(\\nu^{2}+1\/4)^{2}}\ns^{\\omega(\\nu)}E_{l}^{\\nu}(r)E_{l}^{-\\nu}(r')\n\\end{equation}\nwhere\n\\begin{equation}\n\\omega(\\nu)=(g^{2}N\/2\\pi^{2})(\\psi(1)-{\\rm Re}\\psi(1\/2+i\\nu))\n\\end{equation}\nis the pomeron intercept and\n\\begin{equation}\nE_{l}^{\\nu}(r)=\\int d^{2}R\\exp(ilR)(\\frac{r}{|R+r\/2||R-r\/2|})^{1+2i\\nu}\n\\end{equation}\n The Green function (14), transformed into momentum\nspace, contains terms proportional to $\\delta^{2}(l\/2\\pm q)$, which\nshould be absent in the physical solution (this circumstance was first\nnoted by A.H.Mueller and W.-K.Tang [9]). For that, (14) goes to zero\nat $r=0$. Terms proportional to $\\delta^{2}(l\/2+q)$ are not dangerous\nto us: they are killed by the vertex $K_{l}$. \n To remove the dangerous singularity at $q=l\/2$\nand simultaneously preserve a good behaviour at $r=0$, as in [6], we  make\na subtraction in $E$, changing it to\n\\begin{equation}\n\\tilde{E}^{\\nu}_{l}(r)=\n\\int d^{2}R\\exp(ilR)((\\frac{r}{|R+r\/2||R-r\/2|})^{1+2i\\nu}-\n|R+r\/2|^{-1-2i\\nu}+|R-r\/2|^{-1-2i\\nu})\n\\end{equation}\n\nIntegration over $r$ leads to the integral\n\\begin{equation}\nJ_{2}(l,q)=\\int (d^{2}r\/(2\\pi)^{2})(1\/r^{2})\\tilde{E}^{\\nu}_{l}(r)\n\\end{equation}\nThis integral is convergent at any values of $\\nu$, the point $\\nu=0$\nincluded, when the convergence at large values of $r$ and $R$ is\nprovided by the exponential factors. So in the limit\n$s\\rightarrow\\infty$, when small values of $\\nu$ dominate, we can put\n$\\nu=0$ in $J_{2}$:\n\\begin{equation}\nJ_{2}(l,q)=\\int(d^{2}Rd^{2}r\/(2\\pi)^{2})\\frac{\\exp(ilR+iqr)}\n{r|R+r\/2||R-r\/2|}+(1\/l)\\ln\\frac{|l\/2-q|}{|l\/2+q|}\n\\end{equation}\nThe second term comes from the subtraction terms in (17). Passing to the\nFourier transform of the function $1\/r$ one can represent (19) as an\nintegral in the momentum space\n\\begin{equation}\nJ_{2}(l,q)=(1\/l)\\int \\frac{d^{2}p (l+|l\/2+p|-|l\/2-p|)\n}{(2\\pi)|l\/2+p||l\/2-p||q+p|}\n\\end{equation}\n\nIn the same manner we can put $\\nu=0$ in the function $E^{-\\nu}_{l}(r_{20})$\nobtaining \n\\begin{equation}\n\\int \\frac{d^{2}R\\exp(ilR)r_{20}}{(2\\pi)^{2}|R+r_{20}\/2||R-r_{20}\/2|}\\equiv\nF(l)\n\\end{equation}\nThe rest of the Green function is easily calculated by\nthe stationary point method to finally give\n\\begin{equation}\n\\chi_{2}(s,l,q)=8\\sqrt{\\pi}s^{\\Delta}(a\\ln s)^{-3\/2}F_{2}(l)J_{2}(l,q)\n\\end{equation}\nwhere\n\\begin{equation}\n\\Delta= \\omega_{0}=(g^{2}N\/\\pi^{2})\\ln 2,\\ \\\na=(7g^{2}N\/2\\pi^{2})\\zeta (3)\n\\end{equation}\n\n\\section{The projectile part}\nThe BFKL Green function at $l=0$, which enters (12)\nis given by the expression [8]\n\\begin{equation}\nG_{0}(s,r,r')=(1\/8)rr'\\int_{-\\infty}^{\\infty}\\frac{d\\nu s^{\\omega(\\nu)}}\n{(\\nu^{2}+1\/4)^{2}}(r\/r')^{-2i\\nu}\n\\end{equation}\n Applying to (24) the operator $\\nabla^{4}$ we obtain\n\\begin{equation}\n\\nabla^{4}G_{0}(s,r,r')=(2r'\/r^{3})\\int_{-\\infty}^{\\infty}\nd\\nu s^{\\omega(\\nu)}(r\/r')^{-2i\\nu}\n\\end{equation}\n\nBefore integrating over $r$ in (12) we shall integrate over\nthe energy $s_{1}$, as indicated in (11). Part of the $s_{1}$-dependence\ncomes from  factors $\\chi_{2}$. Recall that $s_{2}=s\/s_{1}$ and that also\n$s>>s_{1}$. This latter condition allows to neglect the dependence on\n$s_{1}$ in the logarithmic factor in (22), so that the two factors\n$\\chi_{2}$ will only contribute a factor $s_{1}^{-2\\Delta}$. We then obtain\nan integral\n\\begin{equation}\n\\int ds_{1}s_{1}^{-1-2\\Delta}\\nabla^{4}G_{0}(s,r,r')=\n(2r'\/r^{3})\\int_{-\\infty}^{\\infty}\nd\\nu (r\/r')^{-2i\\nu}(2\\Delta-\\omega(\\nu))^{-1}\n\\end{equation}\n\nNow we have to finally integrate over $r$ to obtain the function $\\chi_{1}$,\nEq. (12), integrated over $s_{1}$. This integration requires some care due\nto a high singularity of the righthand side of (26) at $r=0$. To do it, we\nconsider $\\Delta$ in (26) as a variable and first take $\\Delta<0$. Then the\ndenominator in (26) will not vanish in the strip of the upper half-plane of\n$\\nu$ with ${\\mbox Im}\\,\\nu<\\frac{1}{2}+\\epsilon$, $\\epsilon>0$. This\nallows to shift the integration contour in (26) to a line\n${\\mbox Im}\\,\\nu=\\frac{1}{2}+\\epsilon$. Then the singularity in $r$ will be\ndiminished to make the integration over $r$ possible. We get\n\\begin{equation}\nJ_{1}(q,r_{1})\\equiv\\int ds_{1}s_{1}^{-1-2\\Delta}\\chi_{1}(s_{1},q,r_{1})=\n-\\pi q r_{1}\\int_{{\\mbox Im}\\,\\nu=\\frac{1}{2}+\\epsilon}\nd\\nu (qr_{1}\/2)^{2i\\nu}\\frac{1}{2\\Delta-\\omega(\\nu))}\\frac\n{\\Gamma(1\/2-i\\nu)}{\\Gamma(1\/2+i\\nu)}\n\\end{equation}\nWith $\\Delta<0$ the integrand on the righthand side has no singularities in\nthe strip ${\\mbox Im}\\,\\nu<\\frac{1}{2}+\\epsilon$, since the pole of the\n$\\Gamma$ function at $\\nu=i\/2$ is compensated by the pole of the function\n$\\psi$ in $\\omega(\\nu)$ at the same point. So we can return to the\nintegration over the real $\\nu$ and subsequently pass to the physical\nvalue $\\Delta>0$. Then finally \n\\begin{equation}\nJ_{1}(q,r_{1})=\n-\\pi q r_{1}\\int_{-\\infty}^{+\\infty}\nd\\nu (qr_{1}\/2)^{2i\\nu}\\frac{1}{2\\Delta-\\omega(\\nu))}\\frac\n{\\Gamma(1\/2-i\\nu)}{\\Gamma(1\/2+i\\nu)}\n\\end{equation}\n\nThis integral can be calculated as a sum of residues of the integrand at\npoints $\\nu=\\pm ix_{k}$, $0<x_{1}<x_{2}<...$, at which\n\\begin{equation}\n2\\Delta-\\omega(\\nu)=0\n\\end{equation}\nResidues in the upper semiplane are to be taken if $qr_{1}\/2>1$ and those\nin the lower semiplane if $qr_{1}\/2<1$. Thus we obtain\n\\begin{equation}\nJ_{1}(q,r_{1})=-\\frac{(2\\pi)^{4}}{g^{2}N}\\sum_{k}a_{k}\n(qr_{1}\/2)^{1\\pm 2x_{k}}\\frac{\\Gamma(1\/2\\mp x_{k})}{\\Gamma(1\/2\\pm x_{k})}\n\\end{equation}\nwhere \n\\begin{equation}\na_{k}=\n\\frac{1}{\\psi'(1\/2-x_{k})-\\psi'(1\/2+x_{k})}\n\\end{equation}\nand the signs should be chosen to always have $(qr_{1}\/2)^{\\pm 2x_{k}}<1$.\n\nThe first three roots of Eq. (29) are\n\\begin{equation} \nx_{1}=0.2648,\\ \\ x_{2}=1.3505,\\ \\ x_{3}=2.3704\n\\end{equation}\nwith the corresponding coefficients $a_{k}$\n\\begin{equation}\na_{1}=0.05944,\\ \\  \na_{2}=0.02139,\\ \\  \na_{3}=0.01610,\\ \\ \n\\end{equation}\n\n\\section{The triple pomeron cross-section}\nAs we have found, the projectile part gives a contribution to the\ncross-section (12) in the form of a sum of powers $(qr_{1})^{1\\pm\n2x_{k}}$, with rising values of $x_{k}$. For large $Q^{2}$ we expect that\n$r_{1}\\sim 1\/Q$, so that the product $qr_{1}$ is small. Then the plus sign\nshould be taken in (30) and the bulk of the contribution is expected to\ncome from the lowest power $x_{1}$. For this reason in the following we\nshall study the contribution from only the nearest pole $\\nu=-ix_{1}$ to\n(28), taking\n\\begin{equation}\nJ_{1}(q,r_{1})\\simeq -\\frac{(2\\pi)^{4}}{g^{2}N}a_{1}\n(qr_{1}\/2)^{1+ 2x_{1}}\\frac{\\Gamma(1\/2- x_{1})}{\\Gamma(1\/2+ x_{1})}\n\\end{equation}\n\nCombining all the factors, in this approximation we find for the cross-section $\\sigma$, Eq.\n(11)\n\\begin{equation}\n\\sigma=\\frac{g^{8}N^{2}}{(2\\pi)^{6}}2^{-2x_{1}}a_{1}\nB_{1}\\frac{\\Gamma(1\/2- x_{1})}{\\Gamma(1\/2+ x_{1})}\n\\frac{s^{2\\Delta}}{(a\\ln s)^{3}}\\langle r_{1}^{1+2x_{1}}\\rangle\n\\int d^{2}l \\,l^{-1+2x_{1}}F^{2}(l)\n\\end{equation}\nHere we have introduced the average value of $r_{1}^{1+2x_{1}}$ in the\nprojectile \n\\begin{equation}\n\\langle r_{1}^{1+2x_{1}}\\rangle=\\int d^{2}r_{1}\nr_{1}^{1+2x_{1}}\\rho_{1}(r_{1})\n\\end{equation}\n$B_{1}$ is a number defined as\nas a result of the $q$ integration\n\\begin{equation}\nB_{1}=l^{1-2x_{1}}\n\\int (d^{2}q\/(2\\pi)^{2})|l\/2+q|^{1+2x_{1}}(\\nabla_{q}J(l,q))^{2}\n\\end{equation}\nIt does not depend on $l$ and can be represented as an integral over\nthree momenta\n\\begin{equation}\nB_{1}=\\frac{1}{(2\\pi )^{4}l^{1+2x_{1}}}\\int d^{2}qd^{2}pd^{2}p'\n\\frac{|l\/2+q|^{1+2x_{1}}(q+p)(q+p')(l+p_{+}-p_{-})(l+p'_{+}-p'_{-})}\n{p_{+}p_{-}p'_{+}p'_{-}|q+p|^{3}|q+p'|^{3}}\n\\end{equation}\nwhere\n \\begin{equation} p_{\\pm}=|p\\pm l\/2|;\\ \\ p'_{\\pm}=|p'\\pm l\/2|\\end{equation}\nIt is a well-defined integral.  $B_{1}$\ngeneralizes a similar constant $B$ which appears in the asymptotic triple\npomeron vertex [6], in which $x_{1}$ is absent. Calculations give\n\\[ B_{1}=6.84 \\]\n\nThe explicit form of $F(l)$ is given by (21). So its square introduces\ntwo more integrations, over $R$ and $R'$. Integration over $l$  then gives\n\\begin{equation}\n\\int d^{2}l\\,l^{-1+2x_{1}}\\exp il(R-R')=\n 2^{1+2x_{1}}\\pi\\frac{\\Gamma(1\/2+ x_{1})}{\\Gamma(1\/2- x_{1})}\n|R-R'|^{-1-2x_{1}}\n\\end{equation} \nWe are left with the final integral over $R$ and $R'$\n\\begin{equation}\n\\int \\frac{d^{2}Rd^{2}R'}\n{(2\\pi)^{4}|R-R'|^{1+2x_{1}}}\\frac{1}\n{|R+r_{20}\/2||R-r_{20}\/2||R'+r_{20}\/2||R'-r_{20}\/2|}=\n\\frac{D}{r_{20}^{1+2x_{1}}}\n\\end{equation}\nwhich defines another numerical constant $D$. Numerical integration gives\n\\[ D=0.566 \\]\n\nPutting this into (35) we obtain the final cross-section\n\\begin{equation}\n\\sigma=\\frac{g^{8}N^{2}}{(2\\pi)^{5}}a_{1}B_{1}D\n\\frac{s^{2\\Delta}}{(a\\ln s)^{3}}r_{20}^{1-2x_{1}}\\langle\nr_{1}^{1+2x_{1}}\\rangle\n\\end{equation}\nwith the known numerical constants $a_{1}$, $B_{1}$ and\n$D$.\n\n The $Q^{2}$-dependence of the\ncross-section (42) is concentrated in the average value \n$\\langle r_{1}^{1+2x_{1}}\\rangle$. At large $Q^{2}$ this average has the\norder $Q^{-1-2x_{1}}$, which determines the order of the cross-section\n$\\sigma$ to be\n\\begin{equation} \\sigma\\sim (r_{20}\/Q)(Qr_{20})^{-2x_{1}}\\end{equation}\nThis should be compared with the cross-section which results from the \npure two-pomeron exchange, Fig. 1$b$. As found in [1], it has the same\ndependence on $s$ but falls only as $1\/Q$ at large $Q^{2}$. Therefore for\nlarge $Q^{2}$ the triple pomeron contribution is neglegible relative to the\npure two-pomeron exchange, due to the reduction of the anomalous dimension\nby $2x_{1}$. This is the main result of our study.\n\nTwo comments are to be added in conclusion. First, one might think that the\nobtained result is only a consequence of different scales of the projectile\nand target. Taking the virtality of the target of the same order $Q^{2}$,\none might argue that $\\sigma\\sim 1\/Q^{2}$ on dimensional grounds, which is\nof the same order as for the pure two pomeron exchange. However this\nargument would be wrong. With a highly virtual target, one cannot simply\nput $r'_{2}=r'_{3}=r_{20}$, but has to consider the perturbative density\n$\\rho_{2}$, which is singular at the origin. Then one has to regularize the\nintegrations in $r'_{2}$ and $r'_{3}$ by introducing a finite  mass $m$\nfor quarks inside the target. As a result, the average values of\n$r'_{2,3}$ will not have the order $1\/Q$ but rather $1\/m$. Then the final\nconclusion will remain the same: the contribution will be damped by the\nfactor $(m\/Q)^{2x_{1}}$.\n\nSecond, the procedure followed here for the nearest pole at $\\nu=-ix_{1}$\ncannot be trivially generalized to other poles. The point is that with\n$x_{k}>1\/2$ neither the integral (38) nor the integral (41) converge. For\nsuch $x_{k}$ one has to use the general form (30), taking different signs in\ndifferent parts of the ($q, r_{1}$) phase space. Then the integrals over\n$r_{1},\\,q,\\,R$ and $R'$ do not decouple and the cross-section turns out to\nbe represented by a very complicated 12-dimensional integral over\n$r_{1},\\,q,\\,R,\\,R',\\,p$ and $p'$. However one can estimate the resulting\n$Q^{2}$ dependence by noting that for $x_{k}>1\/2$ the extra dimension in\n(38) will be supplied by the corresponding power of $Q$. One then finds that\nfor all $x_{k}>1\/2$ the cross-section has the same order $1\/Q^{2}$. Thus,\nalthough the contributions of all $x_{k}>1\/2$ are definitely smaller than\nthe one from $x_{1}$, they all have to be calculated simultaneously.\nTherefore our derivation is only practical for the dominant contribution\ncorresponding to the pole at $\\nu=-ix_{1}$.\n\n\\section{Conclusions.}\nWe have shown that for highly virtual probes the triple (and hopefully\nmultiple) pomeron interaction does not contribute to the cross-section, so\nthat it can be calculated as a sum of independent multipomeron exchanges,\nas has been done in [1]. The amplitude then aquires an essentially eikonal\nform.\n\nFor colliding low-mass particles the triple pomeron does contribute\nsignificantly. We have not been able to find this contribution in a form\nsuitable for practical calculations. However we would like to stress that\neven if we had succeded, that would not have solved the problem. On the one\nhand, when\na higher number of pomerons is exchanged\nhigher order multipomeron interactions evidently come into play, whose\ncalculation is still more hopeless. On the other hand, for low-mass\nparticles the coupling to a pomeron (or to several pomerons) is\nessentially non-perturbative. So even without any multipomeron interactions\ncalculation of multipomeron exchanges becomes hardly possible.\nMultipomeron interactions then  do not make the situation significantly\nworse, only adding a contribution which can be calculated perurbatively, in\nprinciple, but not in practice. \n\n\\section{Acknowledgements}\nThe author expresses his deep gratitude to Prof. Carlos Pajares for his\nconstant interest in the present work and helpful discussions. He is\nespecially thankful to Dr. Gavin Salam, whose comments have initiated this\nstudy. He also thanks IBERDROLA, Spain,  for financial support.\n\n\n\\newpage\n\\section{References}\n\\noindent 1. M.A.Braun, Univ. of St. Petersburg preprint SPbU-IP-1995\/3\n(hep-ph\/9502403, to be published in Z. Phys. {\\bf C}).\\\\\n2.  V.S.Fadin, E.A.Kuraev and L.N.Lipatov, Phys. Lett. {\\bf B60} (1975)\n50; I.I.Balitsky and L.N.Lipatov, Sov. J. Nucl. Phys. {\\bf 15} (1978) 438.\\\\\n3. A.H. Mueller, Nucl. Phys. {\\bf B415} (1994) 373.\\\\\n4. A.H.Mueller and B.Patel, Nucl. Phys., {\\bf B425} (1994) 471.\\\\\n5. A.H. Mueller, Nucl. Phys. {\\bf B437} (1995) 107.\\\\\n6. M.A.Braun, Univ. of St. Petersburg preprint SPbU-IP-1995\/10\n(hep-ph\/9506245, to be published in Z.Physik {\\bf C}).\\\\\n7. J.Bartels, Nucl. Phys. {\\bf B175} (1980) 365.\\\\\n8. L.N.Lipatov, in {\\it Perturbative Quantum Chromodynamics},\nEd. A.H.Mueller, Advanced Series\non Directions in High Energy Physics, World Scientific, Singapore 1989.\\\\\n9. A.H.Mueller and W.K.-Tang, Phys. Lett. {\\bf B 284} (1992) 123.\\\\\n\n\\newpage\n\\section{Figure captions}\n\\noindent 1. Triple pomeron ($a$) and \"pure\" double pomeron exchange ($b$)\ncontributions to the scattering amplitude.\\\\\n2. The generic double pomeron exchange diagram for the scattering amplitude.\n\n\n\n\n\n\n\\newpage\n\\begin{picture}(300,300)(0,-50)\n\\thicklines\n\\put (25,225){\\line (1,0){100}}\n\\put (25,25){\\line (1,0){100}}\n\\put (73,225){\\line (0,-1){100}}\n\\put (77,225){\\line (0,-1){100}}\n\\put (75,75){\\oval(50,100)}\n\\put (75,75){\\oval(42,92)}\n\\put(75,225){\\circle*{8}}\n\\put(75,125){\\circle*{8}}\n\\put(75,25){\\circle*{8}}\n\\put(75,0){\\makebox(0,0){\\Large a}}\n\\put(150,-20){\\makebox(0,0){\\Large Fig. 1}}\n\n\\put (175,225){\\line (1,0){100}}\n\\put (175,25){\\line (1,0){100}}\n\\put (225,125){\\oval(50,200)}\n\\put (225,125){\\oval(42,192)}\n\\put(225,225){\\circle*{8}}\n\\put(225,25){\\circle*{8}}\n\\put(225,0){\\makebox(0,0){\\Large b}}\n\\end{picture}\n\n\n\\newpage\n\\begin{picture}(150,300)(0,-50)\n\\thicklines\n\\put (15,225){\\line (1,0){120}}\n\\put (25,25){\\line (1,0){100}}\n\\put (75,125){\\oval(50,200)[b]}\n\\put (75,125){\\oval(42,192)[b]}\n\\put(35,125){\\framebox(80,100){\\Large\\bf B}}\n\\put(75,25){\\circle*{8}}\n\\put(75,-20){\\makebox(0,0){\\Large Fig. 2}}\t\n\\end{picture}\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}