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We only evaluate “ceutral” SZ decrement from the pressure profiles of our models.
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We only evaluate “central” SZ decrement from the pressure profiles of our models.
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In this case. the inteeral in equation (19)) reduces to Tn the Ravleigh-Jeaus part of the ΝΤΟ spectrum. the deviation from the black-body spectrum results iu a decrement of the CMD temperature. We use the pressure profiles resulting from our model to calculate the ceutral SZ decremoeut in the temperature of the CAB.
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In this case, the integral in equation \ref{eq:y_sph_sym}) ) reduces to In the Rayleigh-Jeans part of the CMB spectrum, the deviation from the black-body spectrum results in a decrement of the CMB temperature, We use the pressure profiles resulting from our model to calculate the central SZ decrement in the temperature of the CMB.
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The angular two-point correlation functiou of the SZ temperature distribution iu the sky ds convoeutionallv expanded into the Legeudre polvuouials: Since we consider diserete sources. we cam write €;=on|CO where on is the contribution from the Poisson. noise. aud C,AUCI) is. the correlation. among clusters (Peebles 1980. 11).
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The angular two-point correlation function of the SZ temperature distribution in the sky is conventionally expanded into the Legendre polynomials: Since we consider discrete sources, we can write $C_{l}=
C_{l}^{(P)} + C_{l}^{(C)}$ , where $C_{l}^{(P)}$ is the contribution from the Poisson noise and $C_{l}^{(C)}$ is the correlation among clusters (Peebles 1980, 41).
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We— define the frequency independent pat in the power spectrmu as C(BP)=2 ο).
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We define the frequency independent part in the power spectrum as $C_{l}^{*(P)}\,\equiv\,C_{l}/g^{2}(x)$ .
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The: integral. expression. of SCIENC, can be derived. following+. Cole Waiser (1988) as where V(:) is the co-moving volume and gj ds the aueular Fourier trausforiu of g(0) oeven bv where Jy is the Bessel function of the first kind of the integral order 0.
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The integral expression of $C_{l}^{*(P)}$ can be derived following Cole Kaiser (1988) as where $V(z)$ is the co-moving volume and $y_{l}$ is the angular Fourier transform of $y(\theta)$ given by where $J_{0}$ is the Bessel function of the first kind of the integral order $0$.
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Iu equation (21)). tace is the redshift of photon decoupling and diafdAl is the mass function of clusters which is computed in the Press-Schechter formis (Press Schechter 1971).
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In equation \ref{eq:Cl}) ), $z_{\rm dec}$ is the redshift of photon decoupling and $dn/dM$ is the mass function of clusters which is computed in the Press-Schechter formalism (Press Schechter 1974).
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The mass function has been computed using the power spectrum for a ACDM model with normalization of σς=0.9.
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The mass function has been computed using the power spectrum for a $\Lambda$ CDM model with normalization of $\sigma_{8}=0.9$.
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We choose Mg,=5SLOPAL. aud Aus=2«107 aud inteexate till redshift of +=5 instead of το.
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We choose $M_{\rm min}=5\times 10^{13}M_{\odot}$ and $M_{\rm max}
=2\times 10^{15}M_{\odot}$ and integrate till redshift of $z=5$ instead of $z_{\rm dec}$.
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This is done because tle integral in equation (21)) is found to be insensitive to the upper limit in redshift bevoud +—I the reason being that the mass function is expoucutially suppressed bevoud that redshift in this mass ranee.
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This is done because the integral in equation \ref{eq:Cl}) ) is found to be insensitive to the upper limit in redshift beyond $z=4$, the reason being that the mass function is exponentially suppressed beyond that redshift in this mass range.
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Tn this section. we discuss our results for cluster evolution due to heating. cooling aud conduction.
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In this section, we discuss our results for cluster evolution due to heating, cooling and conduction.
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We also disctiss our results for the central SZ decrement for clusters with masses ranging from Ma=5«Lote2101?AJ.
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We also discuss our results for the central SZ decrement for clusters with masses ranging from $M_{\rm cl}=5\,\times
10^{13}\hbox{--}2\times 10^{15}M_{\rm \odot}$.
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The eas is heated for a time f,,, and cooled simultaneously.
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The gas is heated for a time $t_{\rm\scriptscriptstyle heat}$ and cooled simultaneously.
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After this time. the heating source is switched off.
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After this time, the heating source is switched off.
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The gas is then allowed to cool raciatively until a total sinmilation time of £,,21.35 «1019. years has elapsed.
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The gas is then allowed to cool radiatively until a total simulation time of $t_{\rm \scriptscriptstyle
H}$ $\times$ $^{10}$ years has elapsed.
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The final eutropy. values at Ο.Ε and roy are compared with the observed oues.
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The final entropy values at $0.1r_{\rm \scriptscriptstyle 200}$ and $r_{\rm \scriptscriptstyle 500}$ are compared with the observed ones.
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In Figure (1). the evolution of the density aud temperature profiles of the ICM. are shown for a cluster of mass 6s1014 AF, and for a luninosity of =5.25«1077 +.
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In Figure \ref{fig:den_temp_cond}) ), the evolution of the density and temperature profiles of the ICM are shown for a cluster of mass $6\times10^{14}$ $M_{\rm \scriptscriptstyle\odot}$ and for a luminosity of $ = 5.25\times
10^{45}$ $^{-1}$.
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The eas deusity decreases with tine during the heating epoch. aud increases due to radiative cooling aud conduction after the heating source is switched off.
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The gas density decreases with time during the heating epoch, and increases due to radiative cooling and conduction after the heating source is switched off.
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It is interesting to note that the chanecs iu density are minimal bevoud 0.5r54,,. as compared to (.2regy in Figure (3) iu RRBNOL and that couduction plavs a very important role in regulating the density profiles after the heating source is switched off.
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It is interesting to note that the changes in density are minimal beyond $0.5r_{\rm\scriptscriptstyle 200}$, as compared to $0.2r_{\rss 200}$ in Figure (3) in RRBN04, and that conduction plays a very important role in regulating the density profiles after the heating source is switched off.
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It is secu that couduction actually decreases he deusitv of the gas at larger radii (bevoud 0.57599) by conducting heat out from the ceutral regions.
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It is seen that conduction actually decreases the density of the gas at larger radii (beyond $0.5r_{\rss 200}$ ) by conducting heat out from the central regions.
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This is secu nore clearly if one studies the evolution of the temperature xofiles.
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This is seen more clearly if one studies the evolution of the temperature profiles.
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After the heating source is switched off. it is seen hat the temperature of the central regious fall very rapidly since conduction pumps out heat from the ceutral regions and redistiibutes it iu the outer regious of the cluster.
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After the heating source is switched off, it is seen that the temperature of the central regions fall very rapidly since conduction pumps out heat from the central regions and redistributes it in the outer regions of the cluster.
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Thus he temperature profiles do not rise towards the centre as compared to what is seen in Figure (3) in RRBNOL.
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Thus the temperature profiles do not rise towards the centre as compared to what is seen in Figure (3) in RRBN04.
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On the other haud. their evolution shows a rise iu the outer regions (hevond 0.565599) due to thermal conduction even after the reat source has been switched off.
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On the other hand, their evolution shows a rise in the outer regions (beyond $0.5r_{\rss 200}$ ) due to thermal conduction even after the heat source has been switched off.
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Thus couductiou acts ike a heating source for larger radii.
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Thus conduction acts like a heating source for larger radii.
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Figure (2)) shows the time evolution of scaled cutropy xofiles of a cluster of mass A,ον1011, for = 5.25.10 eres 1;
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Figure \ref{fig:ent_cond}) ) shows the time evolution of scaled entropy profiles of a cluster of mass $M_{\rm \scriptscriptstyle cl} = 6\times10^{14}
M_{\rm \scriptscriptstyle \odot}$ for = $\times 10^{45}$ erg $^{-1}$.
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We use the same method of cuissivity weighting as in Rovchowdluwy Nath (2003) to calculate he average quantities.
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We use the same method of emissivity weighting as in Roychowdhury Nath (2003) to calculate the average quantities.
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The entropy profiles are plotted iu inic-steps of 5«105 veurs.
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The entropy profiles are plotted in time-steps of $5\times10^{8}$ years.
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They are ποσα to rise withtime as the ICAL is heated.
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They are seen to rise withtime as the ICM is heated.
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Then. after the heating is switched off (after fí,,=5<109 vears). tle eas loses eutropy due o cooling auc the profiles are seeu to fall progressively.
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Then, after the heating is switched off (after $t_{\rm \scriptscriptstyle heat} = 5\times 10^9$ years), the gas loses entropy due to cooling and the profiles are seen to fall progressively.
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The inclusion of conduction removes the negative eradieut
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The inclusion of conduction removes the negative gradient
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edees of even FR I radio sources (c.g.. MeNaimara O'Connell 1993).
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edges of even FR I radio sources (e.g., McNamara O'Connell 1993).
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Maintaining our couservative approach. we shall ignore this additional contribution.
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Maintaining our conservative approach, we shall ignore this additional contribution.
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Now. dividing Poo(L|D) bythe (fia) gives the actual proper deusity at 2=2.5 of powerful radio sources born in an interval 7. (Willott et 22001).
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Now, dividing $\rho_{\rm obs}(1+z)^3$ bythe ${\langle f_d \rangle}$ gives the actual proper density at $z = 2.5$ of powerful radio sources born in an interval $T$, (Willott et 2001).
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To obtain the inteerated density of radio sources we consider the width of the relevant logP54 biu. which is about [1.25. 1.5| dex.
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To obtain the integrated density of radio sources we consider the width of the relevant $ {\log} P_{151}$ bin, which is about [1.25, 1.5] dex.
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Thus the total proper density of galaxies with bean powers sufficient to produce FR II sources (whether or not they are detected iu the survey) is o(P)=[5.1.3.1]«10PTS|iMpe5
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Thus the total proper density of galaxies with beam powers sufficient to produce FR II sources (whether or not they are detected in the survey) is $\phi(T) = [5.1,3.1] \times 10^{-5}
T_5 (1+z)^3 {\rm Mpc}^{-3}$.
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We μας finally account for the fact that the epoch during which the nuuber deusitv of sources is roughly constant at the above value exteuds from :zc1.5 to Dom3. with characteristic 2oz2.5 (Jarvis Rawlines 2000: Rawlings 2001).
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We must finally account for the fact that the epoch during which the number density of sources is roughly constant at the above value extends from $z \simeq 1.5$ to $z \simeq 3$, with characteristic $z \simeq 2.5$ (Jarvis Rawlings 2000; Rawlings 2001).
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This corresponds to a quasar cra of lougth for~2 Cyr which cucompasses several generations of radio sources.
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This corresponds to a quasar era of length $t_{\rm QE} \sim 2$ Gyr which encompasses several generations of radio sources.
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The values of fog vary with Qa; so as to compensate for the difference due to cosmology iu the definition of ρω. so we finally find that the total proper density. 9. of intrinsically powerful radio sources is essentially independent of T.(as long as it exceeds ~107 vr) aud ij: P=o(T)ftog/T)-—17.4«107!Mpe
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The values of $t_{\rm QE}$ vary with $\Omega_M$ so as to compensate for the difference due to cosmology in the definition of $\rho_{\rm obs}$, so we finally find that the total proper density, $\Phi$, of intrinsically powerful radio sources is essentially independent of $T$ (as long as it exceeds $\sim 10^8$ yr) and $\Omega_M$: $\Phi = \phi(T) (t_{\rm QE}/ T) = 7.7 \times 10^{-3} ~{\rm Mpc}^{-3}$.
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Recent high-resolution hywdrodyuanuüc simulations of ACDM models sugeest that at the present epoch roughly of the barvous exist in a web of fibuuents as wari eas and cimibedded galaxies and clusters. altogether occupying about of the volume of the universe (Con Ostriker 1999: Davé et 22001).
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Recent high-resolution hydrodynamic simulations of $\Lambda$ CDM models suggest that at the present epoch roughly of the baryons exist in a web of filaments as warm-hot gas and embedded galaxies and clusters, altogether occupying about of the volume of the universe (Cen Ostriker 1999; Davé et 2001).
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However. at the network of flamecents occupied only around colmoving volune. and their mass content has steadily grown since that epoch from about20%.. at the expeuse of the surrounding wari immediun (the eas cooler than ~ QUK. responsible for the Lyauau-a. absorption).
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However, at $z \simeq
2.5$, the network of filaments occupied only around of the comoving volume, and their mass content has steadily grown since that epoch from about, at the expense of the surrounding warm medium (the gas cooler than $\sim~10^5$ K, responsible for the ${\alpha}$ absorption).
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Since massive galaxies. the progenitors of powerful radio sources, lie near theexpectsjunctions of the filaments. their radio jets aud lobes are to directly interact with the cool circtmealactic material as well as the warin-hot aud iof eas contained iu the filaments.
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Since massive galaxies, the progenitors of powerful radio sources, lie near the junctions of the filaments, their radio jets and lobes are expected to directly interact with the cool circumgalactic material as well as the warm-hot and hot gas contained in the filaments.
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Significant amounts of star formation are trigecred by the shocks aud high xessure associated with the radio ciuitting features (83).
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Significant amounts of star formation are triggered by the shocks and high pressure associated with the radio emitting features 3).
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Thus. if a good fraction of this volue of the universe was permeated by radio lobes iu the quasar era. he lobes could play a substautial role iu trigecrine the intense star formation activity seen iu the universe at D—]l]2
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Thus, if a good fraction of this volume of the universe was permeated by radio lobes in the quasar era, the lobes could play a substantial role in triggering the intense star formation activity seen in the universe at $z \sim 1-2$.
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We can now examinethe viability of this proposal.
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We can now examinethe viability of this proposal.
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The effective. volume of relevance here ds just that of the filaments contaimiug the galaxies aud overdcusc protogalactic gas at 22.5. whichis only the fraction ij of the total volume.
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The effective volume of relevance here is just that of the filaments containing the galaxies and overdense protogalactic gas at $z \sim 2.5$, which is only the fraction $\eta$ of the total volume.
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The volume occupied by the svuchrotrou cluitting lobes of a powerful radio source (actually a lower Iuuit to the volume encompassed by the outer bow shock) at au age f ds where Rr. the ratio of the source’s leusth. D. to its width. 28. is typically ~5 LLeaby Williams 1981).
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The volume occupied by the synchrotron emitting lobes of a powerful radio source (actually a lower limit to the volume encompassed by the outer bow shock) at an age $t$ is where $R_T$, the ratio of the source's length, $D$, to its width, $2R$, is typically $\sim 5$ Leahy Williams 1984).
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By iutegratiug equation (3). weighted by the distribution function p(Qu) (sce 82.2). aud usine cquation (1) for D(f.Qu). we compute the average volume filled by these sources at their maxinnun ages to be (ΓΗ=5244Tz"'Mpe"Is?73
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By integrating equation (3), weighted by the distribution function $p(Q_0)$ (see 2.2), and using equation (1) for $D(t, Q_0)$, we compute the average volume filled by these sources at their maximum ages to be $\langle V(T) \rangle = 2.1~ T_5^{18/7} {\rm Mpc}^3$ .
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Putting together the results from 82. the fractional relevant volume which radio lobes born during the quasar era eumulativelv cover is. for our canonical choice of T (DEW99).
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Putting together the results from 2, the fractional relevant volume which radio lobes born during the quasar era cumulatively cover is, for our canonical choice of $T$ (BRW99).
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We emphasize that this Πιο factor is the stu of the lobe volumes created during the eutire quasar era: this is relevant for estimatiue the domain of star formation triggered bv the lobes.
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We emphasize that this filling factor is the sum of the lobe volumes created during the entire quasar era; this is relevant for estimating the domain of star formation triggered by the lobes.
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In contrast. only one generation of sources is considered in estimating the contribution to the euergv density. «4. of svuchrotron plana injected iuto the cosmic web bv their lobes. suce the left-over contribution frou. previous generations of lobes should be mareinal. due to severe expansion losses.
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In contrast, only one generation of sources is considered in estimating the contribution to the energy density, $u$, of synchrotron plasma injected into the cosmic web by their lobes, since the left-over contribution from previous generations of lobes should be marginal, due to severe expansion losses.
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This leads to «z2.7TQ,00D)x24.10Jin7 within the filamcuts.
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This leads to $u \simeq 2.7~ T Q_m \phi(T) \approx 2 \times 10^{-16} {\rm J~m}^{-3}$ within the filaments.
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See Table 1 for specific values.
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See Table 1 for specific values.
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Thus. the main result is that. quite plausible a very sjeuificaut fraction of the relevant volume of the universe was hapiuged upon by the erowius radio lobes durius the redshift interval when radio source production was at its peak (2z2.5).
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Thus, the main result is that, quite plausibly, a very significant fraction of the relevant volume of the universe was impinged upon by the growing radio lobes during the redshift interval when radio source production was at its peak $z \simeq 2.5$ ).
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Radio lobes propagating through this protogalactic medium mainly euncouuter the hot (f> LOTS). vohuue filling. lower deusity eus. but when they M the embedded cooler chuups (P.~10! FK: Fall mRees 1985). the initial bow shock compression will rlarge-scale star formation. which is sustained by the persistent overpressure from the engul&ug radio cocoon.
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Radio lobes propagating through this protogalactic medium mainly encounter the hot $T > 10^6$ K), volume filling, lower density gas, but when they envelop the embedded cooler clumps $T \sim 10^4\,$ K; Fall Rees 1985), the initial bow shock compression will trigger large-scale star formation, which is sustained by the persistent overpressure from the engulfing radio cocoon.
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Note that the cocoon pressure is likely to be well above the equipartition estimate (Blundell Rawhnes 2000).
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Note that the cocoon pressure is likely to be well above the equipartition estimate (Blundell Rawlings 2000).
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This scenario is supported by many models. both analytical BBeechuan Ciofi 1989: Rees 1989: Daly 1990). aud lydrodvuamiuical DDe Young 1989: Cioffi Dlondin 1992). aud provides an explanation for the remarkable radiooptical aligumeut effect. exhibited by hieh-: radio galaxies υπ(c.e..MeC'arthy et 11987: Chambers. Milev van LOSS).
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This scenario is supported by many models, both analytical Begelman Cioffi 1989; Rees 1989; Daly 1990), and hydrodynamical De Young 1989; Cioffi Blondin 1992), and provides an explanation for the remarkable radio–optical alignment effect exhibited by $z$ radio galaxies (e.g., McCarthy et 1987; Chambers, Miley van Breugel 1988).
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Additional support for jet or lobe-induced star formation comes from the TST images of 2~1 radio galaxies (Best. Lousair Rotttecring 1996). and of some radio sources at üeher 2 iaportantMMiley. et 11992: Bickuell et 22000).
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Additional support for jet or lobe-induced star formation comes from the HST images of $z\sim 1$ radio galaxies (Best, Longair Rötttgering 1996), and of some radio sources at higher $z$ Miley et 1992; Bicknell et 2000).
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It is to check if the overpressure of the expanding lobes over the ambient mecdinu persists hroughout the active lifetime. of the radio source.
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It is important to check if the overpressure of the expanding lobes over the ambient medium persists throughout the active lifetime of the radio source.
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From BRW99 FFale 19901): piiX££L17000. but Dx£O 7, so oyexDEte)
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From BRW99 Falle 1991): $p_{\rm lobe} \propto
t^{(-4-\beta)/(5-\beta)}$, but $D \propto t^{3/(5-\beta)}$ , so $p_{\rm lobe} \propto D^{(-4-\beta)/3}$.
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The external xessure declines less rapidly. pasXD. 80 posPeeDi 1127/3,
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The external pressure declines less rapidly, $p_{\rm ext} \propto D^{-\beta}$, so $p_{\rm lobe}/p_{\rm ext} \propto D^{(-4+2\beta)/3}$ .
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For 3=3/2. monetpoeXDYO. while for >=Ll. which might be more reasonable at laree radial distances. Plobe/porXD7%.
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For $\beta = 3/2$, $p_{\rm lobe}/p_{\rm ext} \propto D^{-1/3}$, while for $\beta = 1$, which might be more reasonable at large radial distances, $p_{\rm lobe}/p_{\rm ext} \propto
D^{-2/3}$.
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For the ranges of Qu. po and eg cousidered here. appropriate for FR II sources. overpressures at D=50 kpe will amount to factors of LO? 101, corresponding to Mach unmmbers of 10.100 (BRW99) for the |)owshock.
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For the ranges of $Q_0$ , $\rho_0$ and $a_0$ considered here, appropriate for FR II sources, overpressures at $D = 50$ kpc will amount to factors of $10^2$ $10^4$ , corresponding to Mach numbers of 10–100 (BRW99) for the bowshock.
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Thus. overpressure should persist even for activityD<>1 AIpc. sustaining lobe expansion even after the jet COnses,
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Thus, overpressure should persist even for $D \gg 1$ Mpc, sustaining lobe expansion even after the jet activity ceases.
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Supersonic CXpausioniuto a two-phase circunigalactie medina will compress many of the cooler
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Supersonic expansioninto a two-phase circumgalactic medium will compress many of the cooler
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The Israel Space Agency (ISA) issued a call for xe-xoposals im 1988 for "scientific experiments to be flow1 on an Israch satellite”.
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The Israel Space Agency (ISA) issued a call for pre-proposals in 1988 for “scientific experiments to be flown on an Israeli satellite”.
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The call was answered »* ipproxiniatelv 50 pre-proposals ranging from space astrononiv experinents, to characterizing f1ο behavior ofeectronic devices in the space environment, to the vchavior of fishes in zero-g. All the pre-proposals were evaluated by an internal ISA panel ann two were selected and funded to subit Phase A proposals.
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The call was answered by approximately 50 pre-proposals ranging from space astronomy experiments, to characterizing the behavior of electronic devices in the space environment, to the behavior of fishes in zero-g. All the pre-proposals were evaluated by an internal ISA panel and two were selected and funded to submit Phase A proposals.
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Among these, the oue from Tel Aviv University proposed to orbit wo small telescopes to provide relativelv wide-fiek inaegine in the space-ultraviole (UW) domain.
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Among these, the one from Tel Aviv University proposed to orbit two small telescopes to provide relatively wide-field imaging in the space-ultraviolet (UV) domain.
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The susequent stibinission stage at the comction ic Pase A. porfmined together with a coinercial ractor (ELOp Electro-Optical 1idustiies low part 16 ELBIT Systems company), prodiced a detailed study of the iiussioji.
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The subsequent submission stage at the completion of the Phase A, performed together with a commercial contractor (El-Op Electro-Optical Industries now part of the ELBIT Systems company), produced a detailed study of the mission.
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ELOp was selec‘ted as Prime Contracor since it had a strong heritage «of sopjisticatedli clectro-opical pavloads for eround, naval. and airOTE inaeime while develooue substantial infrasticture for space iuaegiug pavloads.
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El-Op was selected as Prime Contractor since it had a strong heritage of sophisticated electro-optical payloads for ground, naval, and airborne imaging while developing substantial infrastructure for space imaging payloads.
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Iu. particular, it operated a thermnal-vacumn clauber equipped wit1 à collimator that allowed a paxoad to be eacd-to-eud ested in vacuum and at extrene temperatures.
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In particular, it operated a thermal-vacuum chamber equipped with a collimator that allowed a payload to be end-to-end tested in vacuum and at extreme temperatures.
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The TAUVEN Ph:ie A result was a doesign ΓΙΝΕ in Table 1. with tree 20«πα co-aliered. telescopes mounted within a cvΠιο lia ColId fit he inucr space of an OFEQ-class satellite
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The TAUVEX Phase A result was a design summarized in Table 1, with three 20-cm co-aligned telescopes mounted within a cylinder that could fit the inner space of an OFEQ-class satellite.
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T1C field «X view (FOV) of cach telescope was choscu o be approximately one degree.
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The field of view (FOV) of each telescope was chosen to be approximately one degree.
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Erving on the cojscrvative sice. we decided to use onlv space-proven echuiqes and components, and to require the Prime Contractor to include fully-reduudant svstenis in this first natioial astronomy experiment.
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Erring on the conservative side, we decided to use only space-proven techniques and components, and to require the Prime Contractor to include fully-redundant systems in this first national astronomy experiment.
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For UV «letectors we selectec sealed photoclectric§f detectors with Cacsin Telluride ποni-transparent
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For UV detectors we selected sealed photoelectric detectors with Caesium Telluride semi-transparent
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We beein by assunune a spherical source of radius 7 containing a tangled magnetic feld of streneth B.
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We begin by assuming a spherical source of radius $R$ containing a tangled magnetic field of strength $B$.
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We also assune that monocnerectic 5— ravs of energv €. (iu units of ο) ave uniformly produced by some uuspecified mechanigi throughout the volume of the source.
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We also assume that monoenergetic $\gamma-$ rays of energy $\eg$ (in units of $\melec c^2$ ) are uniformly produced by some unspecified mechanism throughout the volume of the source.
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If these are injected with a bhnuninositv LP! one can define the injected > ray conipactness as) where op is ἳthe Thomsou cross section.
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If these are injected with a luminosity $L_\gamma^{\rm inj}$ , one can define the injected $\gamma-$ ray compactness as, where $\sigma_T$ is the Thomson cross section.
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Without any substantial soft photon population inside the source. the ravs will escape without auv attenuation iu one crossing time.
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Without any substantial soft photon population inside the source, the $\gamma-$ rays will escape without any attenuation in one crossing time.
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Towever. as SIS showed. the injected 5 rav conrpactuess cannot become arbitrarily hieh because ifa critical value is reached. the following loop starts operating l.
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However, as SK showed, the injected $\gamma-$ ray compactness cannot become arbitrarily high because if a critical value is reached, the following loop starts operating 1.
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Gamunaravs pair-produce on soft photons. which can be arbitrarily low iuside the source.
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Gamma-rays pair-produce on soft photons, which can be arbitrarily low inside the source.
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2.
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2.
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The produced clectrou-positron pairs cool bx cluitting svuchrotron photons. thus acting as a source of soft photons.
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The produced electron-positron pairs cool by emitting synchrotron photons, thus acting as a source of soft photons.
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3.
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3.
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The soft photous serve as targets for more 35 luteractions.
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The soft photons serve as targets for more $\ggabs$ interactions.
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There are two conditions that should be satisfied sinultauneouslv for his uetwork to occi: The first. which is afeedback condition. requires that the svuchrotrou yhotous emitted from the pairs have sufficicut energv O pair-produce on the 3 ravs.
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There are two conditions that should be satisfied simultaneously for this network to occur: The first, which is a condition, requires that the synchrotron photons emitted from the pairs have sufficient energy to pair-produce on the $\gamma-$ rays.
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By auaking suitable siupliiug assuniptions. one can derive an analytic relation forif — see also SE.
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By making suitable simplifying assumptions, one can derive an analytic relation forit – see also SK.
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Thus. combining (1) the ireshold condition for 35-absorption e.ey—2. (2) re fact that there is equipartition of energv a1nong 1ο create electrou-positroun pairs +,=οσο3€-/2 and (3) the assuniptiouthat the required soft yhotous of energv ερ are the svuchrotron photons that je clectrons/positrous radiate. he. ey=bs? where BiBoi aud Dagnz(ο)zLEslobe 6 is the critical value of the maguetie field. one derives ie nmuininmun value of the maguetie feld required for quenching to become relevantBag.
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Thus, combining (1) the threshold condition for $\ggabs$ -absorption $\eg\epsilon_0=2$, (2) the fact that there is equipartition of energy among the created electron-positron pairs $\gamma_p=\gamma_e=\gamma=\eg/2$ and (3) the assumptionthat the required soft photons of energy $\epsilon_0$ are the synchrotron photons that the electrons/positrons radiate, i.e., $\epsilon_0=b\gamma^2$ where $b=B/B_{\rm crit}$ and $B_{\rm crit}=(\melec^2 c^3)/(e\hbar)
\simeq 4.4\times 10^{13}$ G is the critical value of the magnetic field, one derives the minimum value of the magnetic field required for quenching to become relevant.
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Thus for B>D, the eedback criterion is satisfied.
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Thus for $B\ge B_q$ the feedback criterion is satisfied.
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Note that this is a coudition that only coutaius the eiitted ~ raw eucrev and the magnetic field strength: moreover. it can casily be satisfied. at least if one is to use the values interred from typical modelling of the sources (2?)..
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Note that this is a condition that only contains the emitted $\gamma-$ ray energy and the magnetic field strength; moreover, it can easily be satisfied, at least if one is to use the values inferred from typical modelling of the sources \citep{boettcher07, boettcher09}.
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Along with the feedback criterion. a second criterion must be imposed for the quenching to be fully operative.
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Along with the feedback criterion, a second criterion must be imposed for the quenching to be fully operative.
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A simple wav to see this is with the following consideration.
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A simple way to see this is with the following consideration.
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Asstuue that the raves pair-produce on sole soft photon and that the created electrou-positron pairs cool bx cluitting svuchrotron photous.
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Assume that the $\gamma-$ rays pair-produce on some soft photon and that the created electron-positron pairs cool by emitting synchrotron photons.
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Because an electrou cuits several such pliotous before cooling the critical condition occurs if the πιάνο deusitv of the + ravs is such that at least one of the svuchrotrou plotous pair-produces ou a 5 ταν instead of escaping from the source.
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Because an electron emits several such photons before cooling the critical condition occurs if the number density of the $\gamma-$ rays is such that at least one of the synchrotron photons pair-produces on a $\gamma-$ ray instead of escaping from the source.
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Thus the condition for criticality can be written as /3)ntes 8. 1. where ne.) ds the nuuuber deusitv of + ravs. σε ds the cross section for 55 interactions aud A45) is the uunuber of svuchrotrou photons enmütted bv an electron with Lorentz factor 5 before it cools.
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Thus the condition for criticality can be written as /2) ) 1, where $n(\eg)$ is the number density of $\gamma-$ rays, $\sigma_\ggabs$ is the cross section for $\ggabs$ interactions and $\cal{N}_{\rm s}(\gamma)$ is the number of synchrotron photons emitted by an electron with Lorentz factor $\gamma$ before it cools.
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Asstuing that the pair-produciug collisions occur close to threshold. approxinatius the cross section there by σ.-2o7/3 and usineSe MG)°mqn: and (62)=v—áÀvUCerLU4.og the critical condition can be written as l|. where the feedback condition (2)) aud eq. (1))
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Assuming that the pair-producing collisions occur close to threshold, approximating the cross section there by $\sigma_\ggabs\simeq\sigma_T/3$ and using ${\cal{N}}_{\rm s}(\gamma)\simeq\frac{\gamma}{b\gamma^2}$ and $n(\eg)=\frac{L_{\gamma}^{\rm inj}}{V\eg \melec c^2}\frac{R}{c}$, the critical condition can be written as 4, where the feedback condition \ref{bcrit}) ) and eq. \ref{lgg}) )
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were also used.
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were also used.
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Taken as equalities. relatious (2)) and (1)) define the criterion. which csscutially isa condition for the ταν Duuinosity.
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Taken as equalities, relations \ref{bcrit}) ) and \ref{lcr0}) ) define the criterion, which essentially is a condition for the $\gamma-$ ray luminosity.
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We beein by writing the kinetic equations that describe the distributions of οταν photons. soft photons. and electrons in the source.
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We begin by writing the kinetic equations that describe the distributions of $\gamma$ -ray photons, soft photons, and electrons in the source.
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These are respectively laOrsay Hay = and =τον. where s. ng and. nm, are the differential οταν, soft xioton. and electrou number densities. respectively. aud €.. c, aud 5 are the corresponding energies normalized Hi ο undts.
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These are respectively +n= +n_0 = and =, where $n$ , $n_0$ and, $n_e$ are the differential $\gamma$ -ray, soft photon, and electron number densities, respectively, and $\egamma$, $x$, and $\gamma$ are the corresponding energies normalized in $\melec c^2$ units.
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The densities refer to the uunmber of articles contained in a volume element oH.
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The densities refer to the number of particles contained in a volume element $\sth R$.
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