diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzflia" "b/data_all_eng_slimpj/shuffled/split2/finalzzflia" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzflia" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nOver past decades, much attentions focus on multitemporal Synthetic Aperture Radar (SAR) image change detection, since a SAR is capable of working in all-time and all-weather without the influence of extremely bad weather and the cloud. \nIn a past decade, most traditional SAR image change detection methods are developed how to extract changed areas from a difference image (DI), which suppose to include the information of changed regions. The DI calculated by the log-ratio (LR) \\cite{bazi2006automatic} is usually subject to the speckle and it is challenging to extract the accurate and clear information on the changed region. To tackle this issue, sparse learning\\cite{wang2016sar} was recently proposed learning robust features from the noisy DI. Wang et al.\\cite{wang2019can} analyzed the affects of the SAR image speckle on change detection and proposed a sparse learning-based method for SAR image change detection.\n\nRecently, deep neural networks have been successfully employed to computer vision and remote sensing image analysis due to its ability to exploiting essential and robust structural features on categories of objects. It also has been introduced into the field of change detection. Gong et al\\cite{Gong2017Change} proposed a deep neural network for SAR image change detection. Gao et al. \\cite{gao2016automatic} proposed a simple convolutional neural network, known as PCA-Net, exploring robust features on the changed regions from the noisy DI. However, the performance of these two unsupervised methods are limited without the correct guidance. To tackle this issue, Wang et al. \\cite{wang2018imbalanced} proposed a supervised PCA-Net to improve the performance by carefully collecting typical training samples, which obtains state-of-arts performance of bitemporal SAR image change detection. However, this two-layer convolutional neural network is low efficient, since the convolutional kernels are trained or generated by a Principle Component Analysis (PCA) decomposition. Recently, Li et al. \\cite{li2019deep} proposed a convolutional neural network (CNN) for SAR image change detection based on both unsupervised and supervised learning. Zhao et al. \\cite{zhao2017novel} proposed a bitemporal PolSAR image change detection by a joint classification and a similarity measurement. Currently, it is still an open problem to extract the changed regions from the noisy DI. \n{Nowadays, a large volume of SAR images are acquired by satellites and it is imperative to develop an efficient model that can produce promising results of SAR image interpretation. Most above networks are too heavy and cost much computational burden. It is strongly required to develop a lightweight convolutional neural network.}\n\n {Recently, several lite networks are proposed to improve the inference efficiency. Howard et al. and Sandler et al. proposed two lite networks MobileNetV1 \\cite{howard2017mobilenets} and MobileNetV2 \\cite{sandler2018mobilenetv2} for visual category. Recently, Howard et al. proposed to search for MobileNetV3 \\cite{howard2019searching}. Tan et al. \\cite{tan2019efficientnet} proposed an efficient network for visual category. These lite networks have been extensively employed to visual category and the experimental results show that they can achieve comparable performance with heavy networks, but with low latency and network capacity. It can be potentially performed on edge devices with low power. Most recently, Chen et al.\\cite{chen2020a} proposed a lightweight multiscale spatial pooling network for bitemporal SAR image change detection. \n\nFollowing the idea of the lightweight neural network, in this letter, we focus on the application of lite networks in SAR image change detection. To achieve this, we propose a lite CNN for SAR image change detection. In the proposed network, bottleneck layers are introduced to reduce the number of output channel. Furthermore, the dilated convolutional layers\\cite{li2018csrnet} are introduced to enlarge receptive field with a few of non-zero entries in the kernel, which reduces the number of network parameters. We verify the proposed network by comparing other conventional CNN. Compared with the lightweight network in \\cite{chen2020a}, the proposed network is more robust with the residual and bottleneck structure. Experimental results on four sets of bitemporal SAR image show that our proposed method obtain comparable performance with CNN, while being much more efficient than CNNs. }\n\nThe rest of paper will be organized as follows. We will introduce our proposed method in Section 2. Then the proposed method will be verified on four datasets in Section 3. Finally, we will draw a conclusion in Section 4. \n\n\n\\section{Proposed Method}\n{Given bitemporal SAR images ${\\bf I}_1$ and ${\\bf I}_2$, the DI can be generated as follows}\n \\begin{equation}\\label{di}\n {\\bf I}_{DI} = {\\bf I}_1 \\ominus {\\bf I}_2\n \\end{equation}\n {where $\\ominus$ denote the difference operator. However, most existing difference operator is subject to the speckle. \nThen we will propose a lightweight convolutional neural network to exploit the changed regions from the noisy DI.}\n\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=\\textwidth]{fig1.png}\n\\caption{The framework of the proposed network.(a)Network Architecture. (b) Bottleneck for encoder. (c)Bottleneck for decoder.}\n\\label{framework}\n\\end{figure}\n\n\\begin{table}[!htbp]\n\\centering\n\\caption{The tensors of all the layers.}\n\\label{tab:network}\n\\begin{tabular}{cccccccc}\n\\toprule\n Layer Name & Tensor Size & & Layer Name & Tensor Size & & Layer Name & Tensor Size \\\\\n \\hline\n \\multicolumn{8}{c}{Initial Block} \\\\\nInput& (32,32,1) \\\\ \n Conv&(16,16,13) & & \n Max-pooling & (16,16,1) && \n Concatenation & (16,16,14) \\\\\n \\hline\n \\multicolumn{8}{c}{Group 1} \\\\\nBottleNeck 1.0&(8,8,64) & &\nBottleNeck 1.1&(8,8,64) & &\nBottleNeck 1.2&(8,8,64)\\\\\nBottleNeck 1.3&(8,8,64) & &\nBottleNeck 1.4&(8,8,64) \\\\\n\\hline\n \\multicolumn{8}{c}{Group 2} \\\\\nBottleNeck 2.0&(4,4,128) & &\nBottleNeck 2.1&(4,4,128) & &\nBottleNeck 2.2&(4,4,128)\\\\\nBottleNeck 2.3&(4,4,128) & &\nBottleNeck 2.4&(4,4,128) & &\nBottleNeck 2.5&(4,4,128) \\\\\nBottleNeck 2.6&(4,4,128) & &\nBottleNeck 2.7&(4,4,128)& &\nBottleNeck 2.8&(4,4,128) \\\\\n\\hline\n \\multicolumn{8}{c}{Group 3} \\\\\nBottleNeck 3.0&(4,4,128) & &\nBottleNeck 3.1&(4,4,128) & &\nBottleNeck 3.2&(4,4,128)\\\\\nBottleNeck 3.3&(4,4,128) & &\nBottleNeck 3.4&(4,4,128) & &\nBottleNeck 3.5&(4,4,128)\\\\\nBottleNeck 3.6&(4,4,128) & &\nBottleNeck 3.7&(4,4,128)\\\\\n\\hline\n\\multicolumn{8}{c}{Group 4} \\\\\nBottleNeck 4.0&(8,8,64) & &\nBottleNeck 4.1&(8,8,64) & &\nBottleNeck 4.2&(8,8,64)\\\\\n\\hline\n\\multicolumn{8}{c}{Group 5} \\\\\nBottleNeck 5.0&(16,16,16) & &\nBottleNeck 5.1&(16,16,16) \\\\\n\\hline\n\\multicolumn{8}{c}{Output} \\\\\nConv&(32,32,2) \\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\nThe whole framework of the proposed network can illustrated in Fig.\\ref{framework}(a). It is shown that the network consists of five groups of bottleneck layers \\cite{lin2013network} with an 1$\\times$1 kernels, illustrated by variety of colorful bars, among which the former three ones work as the encoder, and the latter two ones as the decoder. {The tensors of all layers are listed in Table \\ref{tab:network}.}\n\n\nIn the forward process, the network takes a patch of DI as the input. Firstly, the input data go through a normal convolutional layer and a max-pooling (MP) layer, respectively and then the outputs are concatenated. Next, the contact activations go through the decoder with three groups bottleneck layers. The essential structure of a bottleneck of encoder can be illustrated in Fig.\\ref{framework}(b). It is shown that a bottleneck layer is constructed by a small residual block, including a maxpooling path and a convolutional path. More specifically, the convolutional path consists of two 1$\\times$1 convolutions and one main convolution. The main convolution will vary with the various function of the bottleneck. It can be a normal convolution, a dilated convolution \\cite{li2018csrnet} or an asymmetrical convolution\\cite{szegedy2016rethinking}. {The tensors inside the encoding bottleneck are listed in Table \\ref{tab:bo_en}.} \n\\begin{table}[!htbp]\n\\centering\n\\caption{The tensors inside an encoding bottleneck layer.}\n\\label{tab:bo_en}\n\\begin{tabular}{ccccc}\n\\toprule\n Layer Name & Tensor Size & & Layer Name & Tensor Size \\\\\n \\hline\n Input& (16,16,14) \\\\\n \\hline\n \\multicolumn{5}{c}{Branch 1} \\\\\n Conv1& (8,8,16) &&\nConv2&(8,8,16) \\\\\nConv3&(8,8,64) &&\nDropout&(8,8,64) \\\\\n\\hline\n\\multicolumn{5}{c}{Branch 2} \\\\\nMax-pooling&(8,8,14) & &\nPadding&(8,8,64)\\\\\n\\multicolumn{5}{c}{Output} \\\\\n\\hline\nAddition & (8,8,64)\\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\\begin{table}[!htbp]\n\\centering\n\\caption{The tensors inside a decoding bottleneck layer.}\n\\label{tab:bo_de}\n\\begin{tabular}{ccccc}\n\\toprule\n Layer Name & Tensor Size & & Layer Name & Tensor Size \\\\\n \\hline\n Input& (4,4,128) \\\\\n \\hline\n \\multicolumn{5}{c}{Branch 1} \\\\\nConv1& (4,4,16) &&\nConv2&(8,8,16) \\\\\nConv3&(8,8,64) \\\\\n\\hline\n\\multicolumn{5}{c}{Branch 2} \\\\\nConv&(4,4,64) & &\nUp-Sampling&(8,8,64)\\\\\n\\multicolumn{5}{c}{Output} \\\\\n\\hline\nAddition & (8,8,64)\\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\n The first group consist a down-sampling bottleneck and four normal convolutional bottleneck layers. In the normal bottleneck layers, the main convolution component is default set as the main normal convolutional layer. Especially, in the down-sampling bottleneck, the main convolutional component is set as a normal convolution kernel with 3$\\times$3 and the 1$\\times$1 convolution component is replaced by a 2$\\times$2 one. In the next two groups, to exploit the spatial context, we insert the bottleneck layers with asymmetrical and dilated convolution layers with various kernel sizes among the normal bottleneck, where the kernel sizes are set 2, 4, 8 and 16, respectively, as shown the digits below the green bars. In these bottleneck layers, the main convolutional layers are replaced by the dilated convolutional layers, where the kernels are sparse and most entries are zeros. Furthermore, the bottleneck layers with asymmetric convolution layers are also insert between the dilated and normal convolutional layer, illustrated by the blue bars. After encoding, the sizes of feature maps decrease as the one fourth as the original image. \n\nIn the decoding part, the context information collected by the encoder will be propagated to the pixel level. To achieve this, inspired by the idea of U-Net \\cite{ronneberger2015u}, we schedule a upsampling layer and two bottleneck layers. The essential structure of bottleneck for decoder can be illustrated in Fig.\\ref{framework}(c). Similar to the bottleneck for the encoder, the bottleneck for the decoder contains a pooling path and a convolution path. The former includes a maxpooling layer and a 1$\\times$1 convolution, while the latter includes two 1$\\times$1 convolutional layers and a 3$\\times$3 convolutional layer. Especially, when the bottleneck layer is used for upsampling, the 3$\\times$3 convolution component, illustrated by the yellow bar, will be replaced by a 3$\\times$3 transpose convolution \\cite{dumoulin2016guide}. Then the bottleneck will do the 2x upsampling. Trough two groups of decoding bottleneck layers, the feature map will be recover the same size as the input image. {The tensors inside the decoding bottleneck are listed in Table \\ref{tab:bo_de}.} Finally, we put a 2$\\times$2 convolutional layer to get the probability map of two categories. \n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=\\textwidth]{fig2.png}\n\\caption{Four sets of bitemporal SAR images. The first two rows are bitemporal images and the last row is the DIs. (a) YR-A. (b) YR-B. (c) Sendai-A. (d) Sendai-B. }\n\\label{fig2}\n\\end{figure}\n\n\\section{Experimental Results}\n\\label{sec:experiment}\n\\subsection{Experiment Datasets}\nIn this paper, the proposed method is verified on four sets of bitemporal SAR images. Two scenes (YR-A and YR-B) are from bitemporal Yellow River SAR images \\cite{Gong2017Change} acquired by the Radarsat-2 satellite in 2008 and 2009, respectively. Their image sizes are 306 $\\times$ 291 and 400 $\\times$ 350, respectively. Other two are parts of TerraSAR-X images acquired prior to (on Oct. 20, 2010) and after (on May 6, 2011) the Sendai earthquake in Japan \\cite{Cui2016A}. Their sizes (Sendai-A and Sendai-B) are 590 $\\times$687 and 689 $\\times$ 734, respectively. These four datasets are shown in Fig.\\ref{fig2}). These four datasets are quite challenging, such as the linear-shape changed regions in YR-B dataset and complex scene in both Sendai-A and Sendai-B datasets. \n\n\\begin{table}[!htbp]\n\\centering\n\\caption{The number of training samples.}\n\\label{tab:nos}\n\\begin{tabular}{ccccc}\n\\toprule\n Dataset & YR-A & YR-B & Sendai-A & Sendai-B \\\\\n \\hline\n No. Samples& 3596 & 6205 & 15375 & 20294 \\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{fig3.png}\n\\caption{The variations of loss and accuracies.}\n\\label{fig:loss}\n\\end{figure}\n\n\\subsection{Implementations}\n{\nWe have introduced a lite CNN for change detection for bitemporal SAR images. We first generate the DI by the Eq.(\\ref{di}) and the difference operator is implemented by the neighborhood-based LR operator \\cite{gong2011neighborhood}. \nTo train the network, we collect a training dataset according to the method in \\cite{wang2018imbalanced}. The numbers of training samples for each dataset are listed in Table \\ref{tab:nos}.\n\nMore specifically, the patchsize of each sample is set as 32 $\\times$ 32 and 8 samples are fed in each training step. Additionally, the network is trained by an end-to-end back-propagation manner and the loss is set as {the binary entropy function defined in \\cite{sadowski2016notes}}. \nThe training is optimized by the Adam algorithm \\cite{kingma2014adam} in the training stage, where the initial learning rate is set as 0.005. \nThe training is performed on the PyTorch platform built on the Ubuntu 16.04 installed in a PC with a 16 GB DDR memory and an NVIDIA TITAN Xp Graphics Processing Unit of 11 GB memory. The training process will converge at around 15 epochs. We show the variations of loss values and accuracies of training process in Fig.\\ref{fig:loss}.\n}\n\\subsection{Comparison Experiments}\nTo verify the benefits of the proposed method, it is compared with the unsupervised PCA-Net (U-PCA-Net)\\cite{gao2016automatic}, the supervised PCA-Net (S-PCA-Net) \\cite{wang2018imbalanced} which achieves the state-of-arts performance on SAR image change detection. We also compare the proposed method with the deep neural network (DNN) method \\cite{Gong2017Change} and CNN \\cite{li2019deep}. Among these methods, DNN and U-PCA-Net are unsupervised methods,while S-PCA-Net, CNN and the proposed method are supervised ones. \n\nThe performance of the compared methods is evaluated by probabilistic Missed Alarm (pMA), probabilistic False Alarm (pFA) and kappa coefficient, where pFA (pMA) are calculated by the ratios between FA (MA) and the number of Non-Changed pixels (NC)\\cite{wang2019can}. \n\n\\begin{figure}[!thb]\n\\centering\n\\includegraphics[width=\\textwidth]{fig4.png}\n\\caption{The visual comparison results. (a)U-PCA-Net. (b)S-PCA-Net. (c)DNN. (d) CNN. (e)Lite CNN. (f)Reference. }\n\\label{fig3}\n\\end{figure}\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=\\textwidth]{fig5.png}\n\\caption{The quantitative evaluations of compared methods.(a) MA. (b)FA. (c) Kappa. }\n\\label{eva}\n\\end{figure}\n\\subsection{Experiment Results on Individual Dataset Change Detection}\nIn this experiment, for the supervised learning methods, we collect the training samples from an individual dataset, which covers 30\\% areas of the whole image frame. The rest part is employed for testing. \n\nThe visual comparison results are shown in Fig.\\ref{fig3}. It is shown that for the YR-A dataset, S-PCA-Net, DNN and Lite CNN get less noisy but more completed changed regions. Lite CNN can get more clear boundary of changed regions than DNN. For the YR-B dataset, Lite CNN can get more completed changed regions, especially the line at the bottom of the image. Other methods can not get the completed changed regions. For the Sendai-A dataset with complex scene, S-PCA-Net and the Lite CNN are less subject to the speckle and the background and get more clear changed regions, while other compared methods are subjected to the speckle and background and they are almost failed. Moreover, compared with S-PCA-Net, Lite CNN get better inner regional consistence. For the Sendai-B dataset, Lite CNN gets more accurate changed regions than other methods. \n\n\nMoreover, we show the quantitative evaluations in Fig.\\ref{eva}. {It is shown in Fig.\\ref{eva} (a) that the proposed Lite CNN gets a lower pMA on YR-A, Sendai-A and Sendai-B dataset, while the CNN gets a lower pMA on YR-B dataset. It is shown in Fig.\\ref{eva} (b) that the proposed Lite CNN gets a lower pFA on Sendai-B dataset, while S-PCA-Net gets a lower pFA on YR-A, YR-B and Sendai-A dataset. }\nIt is shown in Fig.\\ref{eva} (c) that on the YR-A dataset, S-PCA-Net gets the best kappa among all the methods, while the Lite CNN gets the comparable kappas with other methods, except S-PCA-Net. On the YR-B dataset, Lite CNN gets the comparable kappas with other methods. However, on both Sendai-A and Sendai-B datasets, the Lite CNN performs better than other methods in terms of pMA and kappas. \n\n\\begin{figure*}[!hbt]\n\\centering\n\\includegraphics[width=\\textwidth]{fig6.png}\n\\caption{The visual comparison results. (a) S-PCA-Net. (b) CNN. (c) Lite CNN. (d) Reference. }\n\\label{fig4}\n\\end{figure*}\n\\begin{figure*}[!htb]\n\\centering\n\\includegraphics[width=\\textwidth]{fig7.png}\n\\caption{The quantitative evaluations of compared methods.(a) MA. (b)FA. (c) Kappa.}\n\\label{eva2}\n\\end{figure*}\n\\subsection{Experiment Results on Cross-dataset Change Detection}\nTo further compare the proposed method with other supervised learning methods, S-PCA-Net and CNN, we perform the comparisons on the cross-dataset change detection, where the network trained on several datasets is applied to an unknown testing dataset. More specifically, to achieve this, this experiment is conducted through the leave-one-out manner, i.e. each dataset alternative is selected as the testing dataset and others as training datasets. \n\nThe visual comparisons are shown in Fig.\\ref{fig4}. It is shown that on the YR-A dataset Lite CNN gets more clear visual result with less noisy spots. On the YR-B dataset, Lite CNN performs better than CNN, but not better than S-PCA-Net. There is many miss alarms in the results of Lite CNN. On both Sendai-A and Sendai-B datasets, Lite CNN gets better results than other two methods. \n\n{The quantitative evaluations in terms of pFA, pMA and Kappa are shown in Fig.\\ref{eva2}. It is shown in Fig.\\ref{eva2}(a) that the S-PCA-Net gets a lower pMA on all datasets. It is shown in Fig.\\ref{eva2}(b) that the Lite CNN gets a lower pFA on the Sendai-A dataset, while S-PCA-Net gets a lower pFA on the YR-A dataset and CNN gets a lower pFA on the YR-B and Sendai-B dataset.} It is shown in Fig.\\ref{eva2}(c) that Lite CNN performs better than CNN but comparable with S-PCA-Net on YR-A and YR-B datasets. However, Lite CNN shows great advantages over other two methods on Sendai-A and Sendai-B datasets. It indicates that the Lite CNN performs better than other two methods in model generalization, especially on challenging datasets with complex scenes.\n\\begin{table}[!htbp]\n\\centering\n\\caption{The training times of compared methods.}\n\\label{time}\n\\begin{tabular}{lccc}\n\\toprule\n Methods & S-PCA-Net & CNN& Lite CNN \\\\\n \\hline\nTimes& ~3 h & 30 mins. &15 mins. \\\\ \n \\bottomrule\n\\end{tabular}\n\\end{table}\n\\subsection{Discussion}\nFrom the above comparisons, it is shown that Lite CNN can obtain comparable performance with other method on the YR-A and YR-B datasets. On the challenging datasets, e.g. Sendai-A and Sendai-B, Lite CNN outperforms other methods. Moreover, it has better ability to model generalization. Moreover, Lite CNN is more computationally efficient than S-PCA-Net and CNN. The training times of three supervised learning methods are compared in Table \\ref{time}. It is shown that Lite CNN is easy to train and take less time than S-PCA-Net and CNN, while S-PCA-Net takes longer time, since the convolutional kernel is generated by the principle component analysis decomposition. \n\nOverall, Lite CNN can obtain comparable or even better performance than S-PCA-Net and CNN. It is also more computationally efficient than other two methods. It is expected that Lite CNN is more practical in change detection, especially for the requirement of real-time detection.\n\n\\section{Conclusion}\n\\label{sec:conlusion}\nIn this paper, we develop a lightweight convolutional neural network for bitemporal SAR image change detection. The proposed network consists of groups of bottlenecks layers which exploit the image feature. To verify the benefits of our proposed method, we compare it with several traditional neural networks and the comparisons are performed on fours sets of bitemporal SAR images. The experimental results show that our proposed method Lite CNN performs better than other two methods on cross-dataset change detection, especially when the scene is complex. Furthermore, Lite CNN is a lightweight network, which is more computationally efficient than CNN and S-PCA-Net. In the future, we will further optimize Lite CNN and make it more efficient on the edge device. \n\n \\section*{Acknowledgment}\n This work was supported by the State Key Program of National Natural Science of China (No. 61836009), the National Natural Science Foundation of China (No. 61701361, 61806154), the Major Research Plan of National Natural Science Foundation of China (No. 91838303), the Open Fund of Key Laboratory of Intelligent Perception and Image Understanding of Ministry of Education, Xidian University (Grant No. IPIU2019006), the Natural Science Basic Research Plan in Shaanxi Province of China (No.2018JM6083) and the Open Fund of State Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan University (No. 17E02).\n\n\n\\bibliographystyle{spiejour} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\nA function $g:\\mathcal{I}\\subseteq\\mathbb{R}\\to\\mathbb{R}$ is said to be convex on the\ninterval $\\mathcal{I}$, if the inequality\n\\begin{align}\\label{eq:11}\ng(\\eta\\,x+(1-\\eta)y)\\leq \\eta\\,g(x)+(1-\\eta)g(y)\n\\end{align}\nholds for all $x,y\\in\\mathcal{I}$ and $\\eta\\in[0,1]$. We say that $g$ is concave, provided $-g$ is convex.\n\nFor convex functions \\eqref{eq:11}, many equalities and inequalities have been established, {\\em e.g.},\nOstrowski type inequality \\cite{7}, Opial inequality \\cite{Farid}, Hardy type inequality \\cite{8}, Olsen type \ninequality \\cite{9}, Gagliardo-Nirenberg type inequality \\cite{10}, midpoint and trapezoidal type inequalities \n\\cite{6,Mohammed9} and the Hermite--Hadamard type (HH-type) inequality \\cite{5} that will be used in our study, \nwhich is defined by:\n\\begin{align}\\label{eq:12}\ng\\left(\\frac{u+v}{2}\\right)&\\leq \\frac{1}{v-u}\\int_u^v g(x)dx\\leq \\frac{g(u)+g(v)}{2},\n\\end{align}\nwhere $g:\\mathcal{I}\\subseteq\\mathbb{R}\\to\\mathbb{R}$ is assumed to be a convex function on \n$\\mathcal{I}$ where $a, b\\in \\mathcal{I}$ with $u0$:\n\\begin{align}\\label{eq:13}\ng\\left(\\frac{u+v}{2}\\right)&\\leq \\frac{\\Gamma(\\mu+2)}{2(v-u)^\\mu}\\left[I^{\\mu}_{u^+}g(v)\n+I^{\\mu}_{v^-}g(u)\\right]\\leq \\frac{g(u)+g(v)}{2},\n\\end{align}\nwhere $I^{\\mu}_{u^+}$ and $I^{\\mu}_{v^-}$ denote left-sided and right-sided \nRiemann-Liouville fractional integrals of order $\\mu>0$, respectively, defined as~\\cite{11}:\n\\begin{align}\\label{eq:14}\n\\begin{aligned}\nI^{\\mu}_{u^+}g(x)=\\frac{1}{\\Gamma(\\mu)}\\int_u^x (x-t)^{\\mu-1}g(t)dt, \\quad x>u,\n\\\\\nI^{\\mu}_{v^-}g(x)=\\frac{1}{\\Gamma(\\mu)}\\int_x^v (t-x)^{\\mu-1}g(t)dt, \\quad x0$. Let $\\psi(x)$ be an increasing and positive monotone function on the interval $(u,v]$ with a \ncontinuous derivative $\\psi'(x)$ on the interval $(u,v)$. Then the left and right-sided \n$\\psi$-Riemann--Liouville fractional integrals of a function $g$ with respect to another function $\\psi(x)$ \non $[u,v]$ are defined by \\cite{11,19,20}:\n\\begin{align}\\label{eq:15}\n\\begin{aligned}\nI^{\\mu:\\psi}_{u^+}g(x)=\\frac{1}{\\Gamma(\\mu)}\\int_{u}^{x} \\psi'(t)(\\psi(x)-\\psi(t))^{\\mu-1}g(t)dt,\n\\\\\nI^{\\mu:\\psi}_{v^-}g(x)=\\frac{1}{\\Gamma(\\mu)}\\int_{x}^{v} \\psi'(t)(\\psi(t)-\\psi(x))^{\\mu-1}g(t)dt.\n\\end{aligned}\n\\end{align}\nIt is important to note that if we set $\\psi(x)=x$ in \\eqref{eq:15}, then $\\psi$-Riemann--Liouville \nfractional integral reduces to Riemann--Liouville fractional integral \\eqref{eq:14}.\n\\end{definition}\n\nAs we said, in this study we investigate several inequalities of midpoint type for Riemann--Liouville \nfractional integrals of twice differentiable convex functions with respect to increasing functions.\n\\section{Main Results}\nOur main results follow the following lemma:\n\\begin{lemma}\\label{lem:31}\nLet $g:[u,v]\\subseteq\\mathbb{R}\\to\\mathbb{R}$ be a differentiable function and $g''\\in L_1[u,v]$ with \n$0\\leq u1$. Using inequality of \\eqref{eq:36}, convexity of $|g''|^q$ and the power--mean's \ninequality for $q>1$, we have\n\\begin{align}\\label{eq:314}\n\\int_{0}^{1}t^{\\mu+1}\\left|g''\\left(\\frac{t}{2}u+\\frac{2-t}{2}v\\right)\\right|dt\n&=\\int_{0}^{1}t^{\\mu+1-\\frac{\\mu+1}{q}}\\left[t^{\\frac{\\mu+1}{q}}\n\\left|g''\\left(\\frac{t}{2}u+\\frac{2-t}{2}v\\right)\\right|\\right]dt\n\\nonumber \\\\\n&\\leq \\left(\\int_{0}^{1}t^{\\mu+1}\\right)^{1-\\frac{1}{q}}\n\\left(\\int_{0}^{1}t^{\\mu+1}\\left|g''\\left(\\frac{t}{2}u+\\frac{2-t}{2}v\\right)\\right|^{q}dt\n\\right)^{\\frac{1}{q}}\n\\nonumber \\\\\n&\\leq \\left(\\frac{1}{\\mu+2}\\right)^{1-\\frac{1}{q}}\n\\left(\\int_{0}^{1}\\left(\\frac{t^{\\mu+2}}{2}|g''(u)|^{q}+\\frac{2t^{\\mu+1}-t^{\\mu+2}}{2}|g''(v)|^{q}\n\\right)dt\\right)^{\\frac{1}{q}}\n\\nonumber \\\\\n&=\\left(\\frac{1}{\\mu+2}\\right)^{1-\\frac{1}{q}}\n\\left[\\frac{1}{2(\\mu+3)}|g''(u)|^q+\\left(\\frac{1}{\\mu+2}-\\frac{1}{2(\\mu+3)}\\right)\n|g''(v)|^q\\right]^{\\frac{1}{q}}.\n\\end{align}\nIn the same manner, we get\n\\begin{align}\\label{eq:315}\n&\\int_{0}^{1}t^{\\mu+1}\\left|g''\\left(\\frac{2-t}{2}u+\\frac{t}{2}v\\right)\\right|dt\n\\leq \\left(\\frac{1}{\\mu+2}\\right)^{1-\\frac{1}{q}}\n\\left[\\left(\\frac{1}{\\mu+2}-\\frac{1}{2(\\mu+3)}\\right)|g''(u)|^q\n+\\frac{1}{2(\\mu+3)}|g''(v)|^q\\right]^{\\frac{1}{q}}.\n\\end{align}\nUsing \\eqref{eq:314} and \\eqref{eq:315} in \\eqref{eq:36} we obtain \\eqref{eq:34} for $q>1$.\nThus the proof of theorem \\ref{th:31} is completed.\n\\end{proof}\n\\begin{corollary} \\label{cor:32}\nWith the similar assumptions of Theorem \\ref{th:31} if\n\\begin{enumerate}\n\\item $\\psi(x)=x$, we have\n\\begin{align*}\n&\\left|\\frac{2^{\\mu-1}\\Gamma(\\mu+2)}{(v-u)^\\mu}\\left[I^{\\mu}_{\\left(\\frac{u+v}{2}\\right)^+}g(v)\n+I^{\\mu}_{\\left(\\frac{u+v}{2}\\right)^-}g(u)\\right]-(\\mu+1)g\\left(\\frac{u+v}{2}\\right)\\right|\n\\\\\n&\\leq\\frac{(v-u)^2}{8}\\left(\\frac{1}{\\mu+2}\\right)^{1-\\frac{1}{q}}\n\\Biggl\\{\\left[\\frac{1}{2(\\mu+3)}|g''(u)|^q+\\left(\\frac{1}{\\mu+2}-\\frac{1}{2(\\mu+3)}\\right)\n|g''(v)|^q\\right]^{\\frac{1}{q}}\n\\\\\n&+\\left[\\left(\\frac{1}{\\mu+2}-\\frac{1}{2(\\mu+3)}\\right)|g''(u)|^q\n+\\frac{1}{2(\\mu+3)}|g''(v)|^q\\right]^{\\frac{1}{q}}\\Biggr\\},\n\\end{align*}\nwhich is obtained by Tomar et al. \\cite{21}.\n\\item $\\psi(x)=x$ and $\\mu=1$, we have\n\\begin{align*}\n&\\left|\\frac{1}{v-u}\\int_{u}^{v}g(x)dx-g\\left(\\frac{u+v}{2}\\right)\\right|\n\\leq\\frac{(v-u)^2}{48}\\Biggl[\\left(\\frac{3|g''(u)|^q+5|g''(v)|^q}{8}\\right)^{\\frac{1}{q}}\n+\\left(\\frac{5|g''(u)|^q+3|g''(v)|^q}{8}\\right)^{\\frac{1}{q}}\\Biggr],\n\\end{align*}\nwhich is obtained by Sarikaya et al. \\cite{23}.\n\\item $\\psi(x)=x$ and $q=1$, we have\n\\begin{align*}\n&\\left|\\frac{2^{\\mu-1}\\Gamma(\\mu+2)}{(v-u)^\\mu}\\left[I^{\\mu}_{\\left(\\frac{u+v}{2}\\right)^+}g(v)\n+I^{\\mu}_{\\left(\\frac{u+v}{2}\\right)^-}g(u)\\right]-(\\mu+1)g\\left(\\frac{u+v}{2}\\right)\\right|\n\\leq\\frac{(v-u)^2}{8(\\mu+2)}\\biggl(|g''(u)|+|g''(v)|\\biggr),\n\\end{align*}\nwhich is obtained by Tomar et al. \\cite{21}.\n\\item $\\psi(x)=x, \\mu=1$ and $q=1$, we have\n\\begin{align*}\n&\\left|\\frac{1}{v-u}\\int_{u}^{v}g(x)dx-g\\left(\\frac{u+v}{2}\\right)\\right|\n\\leq\\frac{(v-u)^2}{24}\\left(\\frac{|g''(u)|+|g''(v)}{2}\\right),\n\\end{align*}\n\\end{enumerate}\nwhich is obtained by Sarikaya et al. \\cite{23}.\n\\end{corollary}\n\\begin{theorem}\\label{th:32}\nLet $g:[u,v]\\subseteq\\mathbb{R}\\to\\mathbb{R}$ be a differentiable function and $g''\\in L_1[u,v]$ with \n$0\\leq u1$ and $\\frac{1}{p}+\\frac{1}{q}=1$.\n\\end{theorem}\n\\begin{proof}\nBy using the Holder's inequality, we have\n\\begin{align}\\label{eq:38}\n\\int_{0}^{1}t^{\\mu+1}\\left|g''\\left(\\frac{t}{2}u+\\frac{2-t}{2}v\\right)\\right|dt\n&\\leq \\left(\\int_{0}^{1}t^{(\\mu+1)p}\\right)^{\\frac{1}{p}}\n\\left(\\int_{0}^{1}\\left|g''\\left(\\frac{t}{2}u+\\frac{2-t}{2}v\\right)\\right|^{q}dt\\right)^{\\frac{1}{q}}\n\\nonumber \\\\\n&\\leq \\left(\\frac{1}{(\\mu+1)p+1}\\right)^{\\frac{1}{p}}\n\\left(\\int_{0}^{1}\\left(\\frac{t}{2}|g''(u)|^{q}+\\frac{2-t}{2}|g''(v)|^{q}\\right)\ndt\\right)^{\\frac{1}{q}}\n\\nonumber \\\\\n&=\\left(\\frac{1}{(\\mu+1)p+1}\\right)^{\\frac{1}{p}}\n\\left(\\frac{|g''(u)|^{q}+3|g''(v)|^{q}}{4}\\right)^{\\frac{1}{q}}.\n\\end{align}\nSimilarly, we have\n\\begin{align}\\label{eq:39}\n\\int_{0}^{1}t^{\\mu+1}\\left|g''\\left(\\frac{2-t}{2}u+\\frac{t}{2}v\\right)\\right|dt\n&\\leq \\left(\\frac{1}{(\\mu+1)p+1}\\right)^{\\frac{1}{p}}\n\\left(\\frac{3|g''(u)|^{q}+|g''(v)|^{q}}{4}\\right)^{\\frac{1}{q}}.\n\\end{align}\nThus, the inequalities \\eqref{eq:36}, \\eqref{eq:38} and \\eqref{eq:39} complete the proof of\nthe first inequality of \\eqref{eq:37}.\n\nTo prove the second inequality of \\eqref{eq:37}, we apply the formula \n\\begin{align*}\n\\sum_{i=1}^{n}\\left(c_i+d_i\\right)^m\\leq \\sum_{i=1}^{n}c_i^m+\\sum_{i=1}^{n}+d_i^m, \\quad 0\\leq m<1\n\\end{align*}\nfor $c_{1}=3|g''(u)|^{q}, c_{2}=|g''(u)|^{q}, d_{1}=|g''(v)|^{q}, d_{2}=3|g''(v)|^{q}$ and $m=\\frac{1}{q}$.\nThen \\eqref{eq:36} gives\n\\begin{align*}\n\\left|\\sigma_{\\mu,\\psi}(g;u,v)\\right|\n&\\leq\\frac{(v-u)^2}{8}\\left(\\frac{1}{(\\mu+1)p+1}\\right)^{\\frac{1}{p}}\n\\Biggl[\\left(\\frac{|g''(u)|^q+3|g''(v)|^q}{4}\\right)^{\\frac{1}{q}}\n+\\left(\\frac{3|g''(u)|^q+|g''(v)|^q}{4}\\right)^{\\frac{1}{q}}\\Biggr]\n\\\\\n&\\leq\\frac{(v-u)^2\\left(3^{\\frac{1}{q}}+1\\right)}{16}\\left(\\frac{1}{(\\mu+1)p+1}\\right)^{\\frac{1}{p}}\n\\bigl[|g''(u)|+|g''(v)|\\bigr]\n\\\\\n&\\leq\\frac{(v-u)^2}{8}\\left(\\frac{1}{(\\mu+1)p+1}\\right)^{\\frac{1}{p}}\\bigl(|g''(u)|+|g''(v)|\\bigr).\n\\end{align*}\nHence the proof of Theorem \\ref{th:32} is completed.\n\\end{proof}\n\\begin{corollary}\\label{cor:33}\nWith the similar assumptions of Theorem \\ref{th:32}, if\n\\begin{enumerate}\n\\item $\\psi(x)=x$, we have\n\\begin{align*}\n&\\left|\\frac{2^{\\mu-1}\\Gamma(\\mu+2)}{(v-u)^\\mu}\\left[I^{\\mu}_{\\left(\\frac{u+v}{2}\\right)^+}g(v)\n+I^{\\mu}_{\\left(\\frac{u+v}{2}\\right)^-}g(u)\\right]-(\\mu+1)g\\left(\\frac{u+v}{2}\\right)\\right|\n\\\\\n&\\leq\\frac{(v-u)^2}{8}\\left(\\frac{2}{(\\mu+1)p+1}\\right)^{\\frac{1}{p}}\n\\Biggl[\\left(\\frac{|g''(u)|^q+3|g''(v)|^q}{4}\\right)^{\\frac{1}{q}}\n+\\left(\\frac{3|g''(u)|^q+|g''(v)|^q}{4}\\right)^{\\frac{1}{q}}\\Biggr]\n\\\\\n&\\leq\\frac{(v-u)^2}{8}\\left(\\frac{2}{(\\mu+1)p+1}\\right)^{\\frac{1}{p}}\\bigl(|g''(u)|+|g''(v)|\\bigr),\n\\end{align*}\nwhich is obtained by Tomar et al. \\cite{21}.\n\\item $\\psi(x)=x$ and $\\mu=1$, we have\n\\begin{align*}\n\\left|\\frac{1}{v-u}\\int_{u}^{v}g(x)dx-g\\left(\\frac{u+v}{2}\\right)\\right|\n&\\leq\\frac{(v-u)^2}{16(2p+1)^{\\frac{1}{p}}}\n\\Biggl[\\left(\\frac{|g''(u)|^q+3|g''(v)|^q}{4}\\right)^{\\frac{1}{q}}\n+\\left(\\frac{3|g''(u)|^q+|g''(v)|^q}{4}\\right)^{\\frac{1}{q}}\\Biggr]\n\\\\\n&\\leq\\frac{(v-u)^2}{2^{2+\\frac{2}{q}}(2p+1)^{\\frac{1}{p}}}\\bigl(|g''(u)|+|g''(v)|\\bigr),\n\\end{align*}\n\\end{enumerate}\nwhich is obtained by Sarikaya et al. \\cite{23}.\n\\end{corollary}\n\\begin{corollary}\\label{cor:34}\nFrom Theorems \\ref{th:31}--\\ref{th:32}, we obtain the following inequality for $\\psi(x)=x, \\mu=1$ and $q>1$:\n\\begin{align*}\n\\left|\\frac{1}{v-u}\\int_{u}^{v}g(x)dx-g\\left(\\frac{u+v}{2}\\right)\\right|\n&\\leq(v-u)^2\\min\\{\\delta_{1},\\delta_{2}\\}\\bigl(|g''(u)|+|g''(v)|\\bigr),\n\\end{align*}\nwhere $\\delta_{1}=\\frac{1}{24}$ and $\\delta_{2}=\\frac{1}{2^{2+\\frac{2}{q}}(2p+1)^{\\frac{1}{p}}}$\nsuch that $p=\\frac{q}{q-1}$.\n\\end{corollary}\n\\section{Applications}\nIn this section some applications are presented to demonstrate usefulness of our obtained results in the\nprevious sections. \n\\subsection{Applications to special means}\nLet $u$ and $v$ be two arbitrary positive real numbers, then consider the following special means:\n\\begin{enumerate}\n\\item[(i)] The arithmetic mean:\n\\[A=A(u,v)=\\frac{u+v}{2}.\\]\n\\item[(ii)] The inverse arithmetic mean:\n\\[H=H(u,v)=\\frac{2}{\\frac{1}{u}+\\frac{1}{v}}, \\quad u,v\\neq 0.\\]\n\\item[(iii)] The geometric mean:\n\\[G=G(u,v)=\\sqrt{u\\,v}.\\]\n\\item[(iv)] The logarithmic mean:\n\\[L(u,v)=\\frac{v-u}{\\log(v)-\\log(u)}, \\quad u\\neq v.\\]\n\\item[(v)] The generalized logarithmic mean:\n\\[L_{n}(u,v)=\\left[\\frac{v^{n+1}-u^{n+1}}{(v-u)(n+1)}\\right]^{\\frac{1}{n}}, \n\\quad n\\in\\mathbb{Z}\\setminus\\{-1,0\\}.\\]\n\\end{enumerate}\n\\begin{proposition}\\label{prop:1}\nLet $|n|\\geq 3$ and $u, v\\in\\mathbb{R}$ with $0-1$ we have\n\\begin{align}\\label{eq:prop64}\n&\\left|\\frac{\\mathcal{I}_{p}(v)-\\mathcal{I}_{p}(u)}{v-u}\n-\\frac{a+b}{4(p+1)}\\mathcal{I}_{p+1}\\left(\\frac{u+v}{2}\\right)\\right|\n\\leq (v-u)^{2}\\min\\{\\delta_{1},\\delta_{2}\\}\\, 2^{3-2p}\\sqrt{\\pi}\\Gamma(p+1)\n\\nonumber \\\\\n&\\times\\Biggl(\\left|a\\right|^{p-3}\n\\left|\\,_{2}F_{3}\\left(\\frac{p+1}{2},\\frac{p+2}{2};\\frac{p+1-n}{2},\\frac{p+2-n}{2},p+1;\\frac{a^2}{4}\\right)\n\\right|\n\\nonumber \\\\\n&+\\left|b\\right|^{p-3}\n\\left|\\,_{2}F_{3}\\left(\\frac{p+1}{2},\\frac{p+2}{2};\\frac{p+1-n}{2},\\frac{p+2-n}{2},p+1;\\frac{b^2}{4}\\right)\n\\right|\\Biggr).\n\\end{align}\n\\end{proposition}\n\\begin{proof}\nLet $g(x)=\\mathcal{I}'_{p}(x)$. Note that the function $x\\mapsto\\mathcal{I}'''_{p}(x)$ is convex on the \ninterval $[0,\\infty)$ for each $p>-1$. Using Corollary \\ref{cor:34} and \\eqref{eq:prop61}--\\eqref{eq:prop62}, \nwe obtain the desired inequality \\eqref{eq:prop64} immediately.\n\\end{proof}\n\\section{Conclusion}\nIn this paper, we established some new integral inequalities of midpoint type for convex functions with \nrespect to increasing functions involving Riemann--Liouville fractional integrals. It can be noted from \nCorollary \\ref{cor:31}--\\ref{cor:33} that our results are a generalization of all obtained results in\n\\cite{21,22,23}.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nIn the current paradigm of galaxy formation, it is believed that virtually all\ngalaxies initially form as `disks' owing to the cooling of gas with non-zero\nangular momentum in virialized dark matter haloes. This smooth gas accretion\ndominates the galactic gas supply and hence the fuel for star formation.\nGalaxies that reside in the centers of lower mass halos, those with masses\nless than $M_{halo} \\lower.7ex\\hbox{\\ltsima} 3\\times 10^{11} \\>h^{-1}\\rm M_\\odot$, accrete gas through very\nefficient cold mode accretion, i.e. gas that is never heated (Keres et\nal. 2005, Keres et al. 2008). The central galaxies that reside in larger\nhalos accrete their gas through the classic, but less efficient, hot mode of\naccretion where the gas is shock heated to near the virial temperature near\nthe virial radius and then must cool to be accreted by the central galaxy.\nHence the naive expectation would be that dwarf galaxies should be actively\nstar forming and blue.\n\nHowever, when a small halo is accreted by a larger halo, i.e. when it becomes\na subhalo, the central galaxy that formed in the small halo becomes a\nsatellite galaxy and may experience a number of environmental effects that may\nchange its properties. For instance, the diffuse gas originally associated\nwith the subhalo may be stripped, thus removing the fuel for future\nstar-formation (e.g. Larson, Tinsley \\& Caldwell 1980). This process,\nreferred to as strangulation (Balogh \\& Morris 2000), can result in a gradual\ndecline of the star formation rate in the satellite galaxy, making it redder\nwith the passage of time. If the external pressure is sufficiently high,\nram-pressure stripping may also be able to remove the entire cold gas\nreservoir of the satellite (e.g. Gunn \\& Gott 1972), causing a fast quenching\nof its star formation. A satellite galaxy is also subject to tidal heating\nand stripping and galaxy harassment (Moore et al 1996), which may also cause\nthe satellite to lose its fuel for star formation. These processes are\nbelieved to have played an important role in the evolution of satellite\ngalaxies, and to be responsible, to a large extent, for the relation between\ngalaxy properties and their environment. Indeed, satellite galaxies are\ngenerally found to be redder and somewhat more concentrated than central\ngalaxies with similar stellar masses (e.g. van den Bosch et al. 2008a;\nWeinmann et al. 2008; Yang et al. 2008b; c; Guo et al. 2009).\n\nFurthermore, recent investigations based on cosmological $N$-body simulations\nhave shown that a significant fraction of dark matter haloes that are close\nto, but beyond the virial radius of, a more massive neighboring halo, are\nphysically connected to their neighbor. As shown by Lin et al. (2003) and\nmore recently by Ludlow et al. (2008), some low-mass halos are {\\it\nphysically} associated with more massive halos, in the sense that they were\nonce subhalos within the virial radii of these more massive progenitors and\nhave subsequently been ejected. This population of halos was found to extend\nbeyond three times the virial radii of their host halos, and represents about\n$10\\%$ of the entire population of low-mass halos (Wang, Mo \\& Jing 2008). If\ngalaxies have managed to form in the progenitors of these ejected halos, it is\nlikely that the same environmental processes operating on satellite galaxies\nmay also have affected the properties of these galaxies. In particular, we\nwould expect the presence of a red population of faint galaxies that are\nclosely associated with massive halos that once hosted them.\n\nGalaxies are observed to be bimodal in the color-magnitude plane: red galaxies\nwith very little star formation (the red sequence) and blue star forming\ngalaxies that are typically disky (the blue cloud) (e.g. Kauffman et al. 2003,\nBaldry et al. 2004) Extrapolating the observed division line (Yang et\nal. 2008a) to dwarf galaxies we surprisingly find that for central dwarf\ngalaxies in the SDSS, with $r$-band magnitudes between -14.46 and -17.05, just\nover 1\/4 are red. In this paper we will investigate the nature of these red\ndwarf galaxies. Quantifying the spatial distribution of this population of\ngalaxies is clearly important, because it allows us to determine whether or\nnot they can be explained as a population of satellite galaxies that were\nejected from larger halos.\n\nIn this paper, we use the galaxy group catalogue constructed by Yang et al.\n(2007) from the Sloan Digital Sky Survey Data Release 4 (SDSS DR4;\nAdelman-McCarthy {et al.~} 2006) to study the distribution of central dwarf\ngalaxies around massive halos. The structure of this paper is as follows. In\n\\S\\ref{sec_data} we briefly describe the criteria used to select galaxies and\ngalaxy groups. In \\S\\ref{sec_analyze} we study the radial distribution of\ndwarf galaxies around their nearest neighbor halos and its dependence on\ngalaxy color and concentration. Some systematic effects that may change our\nresults are discussed in \\S\\ref{sec_systematics}. In \\S\\ref{sec_mock}, we use\nmock catalogues to test the reliability of our results and to quantify their\nimplications. Finally, in \\S\\ref{sec_discussion}, we present some further\ndiscussion regarding our results.\n\n\\section{Observational Data}\n\\label{sec_data}\n\n\\subsection{Samples of Galaxy Groups}\n\nOur analysis uses the galaxy group catalogues of Yang {et al.~} (2007), which were\nconstructed from the New York University Value-Added Galaxy Catalog (NYU-VAGC,\nsee Blanton {et al.~} 2005b) based on the Sloan Digital Sky Survey Data Release 4\n(SDSS DR4; Adelman-McCarthy {et al.~} 2006). Only galaxies in the Main Galaxy\nSample with redshifts in the range $0.01 \\leq z \\leq 0.20$ and with a redshift\ncompleteness ${\\cal C} > 0.7$ were used. Three sets of group catalogues were\nconstructed using a modified version of an adaptive halo-based group finder,\nwhich was optimized to assign galaxies into groups according to their common\ndark matter halos (Yang {et al.~} 2005). For our study here, we use group sample\nII, in which only galaxies with spectroscopic redshifts (either provided by\nthe SDSS or taken from alternative surveys) are used. We have tested, though,\nthat using group sample III, which also includes galaxies that have been\nmissed owing to fiber collisions, does not have a significant impact on any of\nour results.\n\nFor each group in the catalogue, Yang {et al.~} (2007) estimated the corresponding\nhalo mass using either the ranking of its characteristic luminosity (this mass\nis denoted by $M_L$) or using the ranking of its stellar mass (this mass is\ndenoted by $M_S$). Throughout this paper, we use $M_S$ as our halo masses.\nWe have also tested that using $M_L$ instead does not change any of our\nresults. As described in Yang {et al.~} (2007), the characteristic luminosity and\nstellar mass of a group are defined to be the total luminosity and total\nstellar mass of all group members, respectively, with $\\>^{0.1}{\\rm M}_r-5\\log h \\leq -19.5$.\nThus, groups whose member galaxies are all fainter than $\\>^{0.1}{\\rm M}_r-5\\log h = -19.5$ cannot\nbe assigned halo masses according to the ranking. For these groups, the halo\nmasses are estimated in the following way. In Yang {et al.~} (2008b) it is shown\nthat the stellar masses of central galaxies are tightly correlated with the\nmasses of their host haloes. The mean of this relation is well described by\n\\begin{equation}\\label{eq:Ms_fit}\nM_{\\ast} = M_0\n\\frac { (M_h\/M_1)^{\\alpha +\\beta} }{(1+M_h\/M_1)^\\beta } \\,,\n\\end{equation}\nwhere $M_{\\ast}$ and $M_h$ are the central galaxy stellar mass and the host\nhalo mass of the group, respectively, and ($\\log M_0$, $\\log M_1$, $\\alpha$,\n$\\beta$) = (10.306, 11.040, 0.315, 4.543). For groups that cannot be assigned\na halo mass according to the stellar-mass (luminosity) ranking, we use the\nabove relation to obtain $M_h$ through the stellar masses of their central\ngalaxies.\n\n\n\\subsection{Galaxy Samples}\n\\label{sec:samples}\n\\begin{deluxetable*}{lccccccccc}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{Galaxy Samples \\label{tab1}} \\tablewidth{0pt} \\tablehead{ID &\n $\\>^{0.1}{\\rm M}_r-5\\log h$ & $N_{\\rm total}$ & $N_{\\rm cent}$ & $N_{\\rm sat}$ &\n $f_{\\rm red,cent}$ & $f_{\\rm red,sat}$ & $f^b_{\\rm red,cent}$ & $f^b_{\\rm\n red,sat}$ & $f^b(r_{\\rm p}\/R_{180})\\le 3$ \\\\\n (1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9) & (10)} \\startdata\nS1 & (-14.46,-16.36] & 1500 & 1103 & 397 & 13.69\\% & 37.53\\% & 33.79\\% &\n 53.42\\% & 34.60\\%\\\\\nS2 & (-16.36,-16.78] & 1500 & 1081 & 419 & 13.69\\% & 36.28\\% & 22.50\\% &\n 47.02\\% & 47.46\\%\\\\\nS3 & (-16.78,-17.05] & 1500 & 1008 & 492 & 10.20\\% & 40.24\\% & 19.51\\% &\n 52.64\\% & 47.72\\%\\\\\nS1+S2+S3 & (-14.46,-17.05] & 4500 & 3192 & 1308 & 12.59\\% & 38.15\\% & 25.49\\% &\n 51.08\\% & 41.63\\% \\\\\n\\cline{1-10}\\\\\nS4 & & 1500 & 1080 & 420 & 13.51\\% & 37.02\\% \\enddata\n\n\n\\tablecomments{Column 1 indicates the sample ID. Column 2 lists the absolute\n magnitude range of each sample. Columns 3 to 5, indicate the number of\n total, central and satellite galaxies in each sample, respectively. Columns\n 6 and 7 list the red fractions among central and satellite galaxies,\n respectively, where the red galaxies are defined to be the reddest 20\\% of\n all the galaxies. Column 8 lists the red fraction of central dwarf galaxies\n where the red galaxies are defined by extrapolating the division between red\n sequence and blue cloud galaxies from Yang et al. (2008a). Column 9 lists\n also this red fraction, but for the satellite dwarf galaxies. Column 10\n lists the fraction of those (in Column 8) central red dwarf galaxies that\n have $r_{\\rm p}\/R_{180}\\le 3$. }\n\\end{deluxetable*}\n\n\nGroup catalogue II consists of 369447 galaxies, which are assigned into 301237\ngroups. The majority, 271420, of the groups contain only one member, i.e.,\nall of them are the central galaxies of the groups. The remaining 98027\ngalaxies are in groups with more than one member, and 29817 of them are {\\it\ncentral} galaxies (the brightest one in each group). We refer to the other\n68210 galaxies as {\\it satellites} (Yang et al. 2007).\n\nFrom our galaxy sample, we select three subsamples of dwarf galaxies as\nfollows. We rank order all galaxies according to their absolute magnitudes\n(in the $r$-band, $K$- and evolution- corrected to redshift $z=0.1$), starting\nwith the faintest galaxy. The 1500 galaxies with the highest rank (i.e. the\n1500 faintest galaxies) make up our first sample, called S1, The galaxies with\nranks 1501-3000 make up sample S2, and those with ranks 3001-4500 sample\nS3. Table~\\ref{tab1} lists the (sequential) absolute magnitude ranges of all\nthree samples, as well as the numbers of central and satellite galaxies. Note\nthat all the galaxies in S1, S2 and S3 are fainter than $\\>^{0.1}{\\rm M}_r-5\\log h = -17.05$. In\nwhat follows we refer to all galaxies in these three samples as dwarf\ngalaxies.\n\n\\begin{figure} \\plotone{f1.eps}\n \\caption{The color-magnitude distribution of dwarf galaxies (including\n central and satellite galaxies). In four panels the dwarf galaxies are\n separated into 20\\%, 30\\%, 40\\%, 50\\% red galaxies as indicated,\n respectively. The vertical dashed lines in the upper-left panel show the\n separation criteria of samples S1, S2 and S3.} \\label{fig:col}\n\\end{figure}\n\nTo study how the spatial distribution of dwarf galaxies depends on galaxy\ncolor, we separate each of the samples, S1, S2, and S3, into red and blue\nsubsamples. In particular, we define a color cut\n\\begin{equation}\\label{colcut}\n^{0.1}(g-r) = a + b~(\\>^{0.1}{\\rm M}_r-5\\log h)\\,,\n\\end{equation}\nand we adjust the parameters $a$ and $b$ such that samples S1, S2, and S3\nroughly have the same fractions of galaxies, $f_{\\rm red}$, redder than this\nparticular cut. We consider four values for $f_{\\rm red}$: 20\\%, 30\\%, 40\\%,\nand 50\\%, for which we obtain [a, b]=[-0.421,-0.060], [-0.423,-0.055],\n[-0.375,-0.049] and [-0.313,-0.043], respectively. Thus, if $f_{\\rm red} =\n20$\\% it means that the red subsamples of S1, S2 and S3 each consist of the\n20\\% reddest galaxies in their particular samples, etc. Fig~\\ref{fig:col}\nshows the color-magnitude relations of galaxies in S1, S2 and S3 (delineated\nby vertical dashed lines). The four panels correspond to the four different\nvalues of $f_{\\rm red}$, as indicated, and the solid line in each panel\ncorresponds to the color cut of Eq.~(\\ref{colcut}) used in each case.\n\n\nIn Table~\\ref{tab1} we list, for each sample, the red fractions of central\n($f_{\\rm red,cent}$) and satellite galaxies ($f_{\\rm red,sat}$). Here red\ngalaxies are defined to be the reddest 20\\% of all galaxies (both centrals and\nsatellites) in our sample of dwarf galaxies. Clearly, dwarf galaxies that are\nsatellites have a much higher red fraction than central dwarf galaxies. For\ncomparison, extrapolating the observed division line between the red sequence\nand the blue cloud from Yang et al. (2008a) down to dwarf galaxies one would\nfind that 33.8\\%, 22.5\\%, and 19.5\\% of the central galaxies in the S1, S2,\nand S3 samples, respectively, were red. While in total, there are more red\ncentral dwarfs than red satellite dwarfs, which is so far not predicted,\ne.g. by halo occupation models (Brown et al. 2008). Note that different\ndefinition of red galaxies may change these fractions (e.g. with respect to\nthe galaxies of similar stellar masses), however not the general results we\nfind in this paper.\n\n\n\n\\section {The distribution of central dwarfs}\n\\label{sec_analyze}\n\n\\begin{figure} \\plotone{f2.eps}\n \\caption{Fractions of the red central galaxies near more massive halos as a\n function of projected distance $r_{\\rm p}\/R_{180}$. The four panels show the\n results of the color subsamples with 20\\%, 30\\%, 40\\% and 50\\% red\n galaxies, respectively. The related color subsample separation criteria\n are shown in Fig. \\ref{fig:col}. The different lines correspond to the\n different samples as indicated. The fit long-dashed, green line is\n described in the text. For comparison, we show the fraction of red {\\it\n satellite} galaxies that are within the same luminosity ranges as the\n central galaxies at $r_{\\rm p}\/R_{180}=0$. The small window within each panel\n shows the number counts of dwarf central (or satellite) galaxies in S1 as\n a function of $r_{\\rm p}\/R_{180}$: black histogram for all galaxies and shaded,\n red histogram for red galaxies. } \\label{fig:f_r}\n\\end{figure}\n\n\nIn this section, we investigate how central dwarf galaxies are distributed\nwith respect to their nearest more massive halo (i.e. more massive than their\nown halo). Since the distances of galaxies based on redshifts suffer from\nredshift distortions, we separate the distance between a central dwarf galaxy\nand its nearest more massive halo into two components: $\\pi$, which is the\nseparation along the line-of-sight, and $r_{\\rm p}$, which is the separation in the\nperpendicular direction.\n\nFor each group in the catalogue, we use the assigned halo mass, $M_S$, to\nestimate its halo radius, $R_{180}=[3M_{S}\/(4\\pi*180\\bar{\\rho})]^{1\/3}$, which\nfollows from defining the mean mass density within a halo as $180$ times the\naverage density of the universe, $\\bar{\\rho}$. We search around each central\ndwarf galaxy, within a line-of-sight separation $|\\pi| = 15\\>h^{-1}{\\rm {Mpc}}$,\n\\footnote{Tests have shown that changing the line-of-sight separation for the\nsearch from $|\\pi| \\le 15\\>h^{-1}{\\rm {Mpc}}$ to $|\\pi| \\le 10\\>h^{-1}{\\rm {Mpc}}$ or to $|\\pi| \\le\n20\\>h^{-1}{\\rm {Mpc}}$ does not have a significant impact on any of our results.} for the\ngroup that has (i) a halo mass larger than that of the dwarf galaxy in\nquestion and (ii) the lowest value of $r_{\\rm p}\/R_{180}$ (with $R_{180}$ the halo\nradius of the group). The central galaxy is then said to be at a scaled\n`distance' $r_{\\rm p}\/R_{180}$ from a massive halo, and the halo is referred to as\nthe nearest halo of the galaxy. We use this scaled distance because $R_{180}$\nis the only important length scale related to the dynamics of a virialized\nhalo.\n\n\\subsection{Color Dependence}\n\nFig.~\\ref{fig:f_r} shows the fraction $N_{\\rm red}\/N_{\\rm total}$ of central\ndwarf galaxies that are red as a function of the scaled distance,\n$r_{\\rm p}\/R_{180}$, to their nearest halos. Here $N_{\\rm total}$ is the total\nnumber of dwarf galaxies in each sample (S1, S2 or S3) in that bin of\n$r_{\\rm p}\/R_{180}$, and $N_{\\rm red}$ is the number of those galaxies that are red\naccording to the criterion used. The numbers are shown in the small window in\neach panel (the black histogram for $N_{\\rm total}$, and the red, hatched\nhistogram for $N_{\\rm red}$). The four panels show the results for $f_{\\rm\nred} = 20$\\%, 30\\%, 40\\% and 50\\%. In each panel, the different lines show\nthe results obtained for the three samples, S1, S2, and S3, as indicated in\nthe upper-left panel. The error-bars are obtained using 100 bootstrap\nresamplings (Barrow, Bhavsar, \\& Sonoda 1984; Mo, Jing \\& B\\\"orner 1992).\nGalaxies are counted in bins specified by $N-0.5 \\leq r_{\\rm p}\/R_{180} \\leq N+0.5$\n(for $N=2, 3,... ,7$), and $0\\leq r_{\\rm p}\/R_{180} \\leq N+0.5$ for $N=1$. For\ncomparison, the data point at $r_{\\rm p}\/R_{180} =0$, indicates the fraction of red\n{\\it satellite} galaxies within the same luminosity range as the central\ngalaxies. Note that the results obtained for S1, S2, and S3 are almost\nidentical, indicating that the spatial distribution of dwarf galaxies around\nmassive haloes does not depend on their luminosities. However, if we consider\nmuch brighter galaxies, e.g. at $\\>^{0.1}{\\rm M}_r-5\\log h \\sim -18.5$, the radial dependence\nstarts to level off.\n\nThere is a clear trend that the fraction of red central dwarf galaxies\nincreases with decreasing scaled distance to the nearest halo. The fraction\nof the 20\\% reddest population at $r_{\\rm p}\/R_{180}\\ga 4$ is around 5\\% to 10\\%,\nincreases systematically to $\\sim 25$\\% at $r_{\\rm p}\/R_{180}\\sim 1$, and to $\\sim\n40$\\% at $r_{\\rm p}\/R_{180} = 0$ for the satellite galaxies. For the other three\ncases (with $f_{\\rm red} =30$\\%, 40\\% and 50\\%), the fraction also decreases\nwith $r_{\\rm p}\/R_{180}$, but reaches a higher level at large $r_{\\rm p}\/R_{180}$. This\nindicates that the less red galaxies in these subsamples are not strongly\nassociated with massive halos. We quantify the association of red dwarf\ngalaxies with massive halos by fitting the data obtained from S3 shown in the\nfour panels simultaneously with a function $f=a+b\\times\\exp(-x\/2)$, where\n$x=r_{\\rm p}\/R_{180}$. In the fit we subtract a constant of, $0.1$, $0.2$ and\n$0.3$, from the data for the 30\\%, 40\\% and 50\\% reddest subsamples,\nrespectively, to account for the component that is not closely associated with\nmassive halos. The best fit results, with $a=0.045$ and $b=0.356$, are shown\nin each panel of Fig. \\ref{fig:f_r} as the green, long-dashed lines. We have\nalso checked the radial distribution of the central dwarf galaxies when using\n$f_{\\rm red} = 15$\\% to define the subsample of red dwarfs. In this case, we\nfind less than a 5\\% decrease in the red fraction at large $r_{\\rm p}\/R_{180}$,\nindicating that the 5\\% least red galaxies in the 20\\% reddest subsample are\nnot randomly but more closely distributed relative to the massive halos. Thus\nthe overall results suggest that the $15\\%$ - $20\\%$ reddest dwarf galaxies\nare quite distinct from the other dwarfs, in that they reveal a clear\npreference to reside close to their nearest more massive dark matter halo. In\naddition, the non-zero red fraction at large $r_{\\rm p}\/R_{180}$ indicates that\nthere is a $\\sim 5\\%$ tail of red dwarfs randomly distributed throughout the\nbackground population, especially in the voids, due to some processes that\nshut down the star formation. A similar trend has also been found by Cooper et\nal. (2007) from the DEEP2 survey, however for more massive galaxies in low\ndensity regions at higher redshifts. In \\S\\ref{sec_mock}, we use mock galaxy\nredshift surveys constructed from cosmological $N$-body simulations to\nquantify this connection.\n\n\n\\subsection{Concentration Dependence}\n\nApart from a color dependence, we also check whether the radial distribution\nof dwarf galaxies with respect to their nearest more massive dark matter halos\ndepends on their surface brightness profiles. To this end, we split our dwarf\ngalaxies into two subsamples according to the value of their concentration\nparameter $C=r_{90}\/r_{50}$. Here $r_{90}$ and $r_{50}$ are the radii that\ncontain 90 and 50 percent of the Petrosian $r$-band flux, respectively. As\nshown in Strateva {et al.~} (2001), $C$ is a reasonable proxy for the Hubble type,\nwith $C>2.6$ corresponding to early-type galaxies. We, therefore, separate\ngalaxies into high ($C>2.6$) and low ($C\\le 2.6$) concentrations, as\nillustrated in the lower-right panel of Fig.~\\ref{fig:concen}. Roughly 20\\%\nof the dwarf galaxies thus end up in the high-concentration subsample.\n\nThe lower-right panel of Fig.~\\ref{fig:f_con} shows the fraction of galaxies\nin this high-concentration subsample as a function of the scaled distance to\nthe nearest more massive halo. Unlike the reddest galaxies, the most\nconcentrated galaxies have a radial distribution that is similar to that of\nthe total population of central dwarf galaxies. Note however, for brighter\ngalaxies, especially in and around clusters, there is a so called\nmorphology-radius relation (e.g., Dressler et al. 1997), which according to\nPark \\& Hwang (2008) may be largely induced by the interaction of the target\ngalaxy with its nearest (early-type) neighbor galaxy.\n\n\n\\section{Systematics}\n\\label{sec_systematics}\n\nBefore we proceed to study the origin of the central red dwarf galaxies, there\nare a number of issues that need to be addressed. An obvious worry is that the\ngroup finder is not perfect, and has misclassified a number of satellite\ngalaxies as central galaxies. We will discuss this issue in more detail with\nthe help of mock galaxy and group catalogs in \\S\\ref{sec_mock}. In this\nsection we discuss systematics that can be addressed without the need for mock\ncatalogs.\n\n\n\\subsection{Stellar Mass Dependence}\n\\begin{figure} \\plotone{f3.eps}\n \\caption{Upper-left panel: the color-stellar mass distribution of the 20\\%\n red (`x'-crosses) and 80\\% blue (`+'-crosses) dwarf galaxies in terms of\n {\\it similar stellar masses}. Upper-right panel: the same set of dwarf\n galaxies as in the upper-left panel, but separated into 20\\% blue\n (`x'-crosses) and 80\\% red (`+'-crosses). Lower-left panel: the same set\n of dwarf galaxies as in the upper-left panel, but randomly selected and\n separated into 20\\% (`x'-crosses) and 80\\% (`+'-crosses) populations.\n Lower-right panel: the concentration-magnitude distribution of the dwarf\n galaxies, which are separated into $\\sim 20\\%$ high and $\\sim 80\\%$ low\n concentration populations by $C=2.6$. } \\label{fig:concen}\n\\end{figure}\n\n\\begin{figure} \\plotone{f4.eps}\n \\caption{Similar to Fig. \\ref{fig:f_r}, but with related subsample\n separation criteria shown in Fig. \\ref{fig:concen}. Upper-left panel:\n fraction of the red central galaxies near more massive halos as a function\n of $r_{\\rm p}\/R_{180}$, where the 20\\% red population is defined with respect to\n the similar stellar mass galaxies. Upper-right panel: similar to the\n upper-left panel but for the 20\\% blue population. Lower-left panel:\n similar to the upper-left panel but for the 20\\% random population.\n Lower-right panel: fraction of the high concentration $C>2.6$ central\n galaxies near more massive halos as a function of\n $r_{\\rm p}\/R_{180}$. } \\label{fig:f_con}\n\\end{figure}\n\nFor a given luminosity, redder galaxies are expected to have a larger stellar\nmass. The color separation used above may thus introduce a bias in the sense\nthat galaxies in the redder subsample are systematically more massive. To\ncheck whether or not the color distribution we obtained in the previous\nsection is robust when the dwarf galaxies are selected in a similar stellar\nmass bin, we construct a controlled subsample S4, where stellar masses for\ngalaxies are estimated using the relation between the stellar mass-to-light\nratio and color obtained by Bell et al.(2003). Note that the survey {\\it\nmagnitude limit} of the SDSS observation corresponds to a higher (lower) {\\it\nstellar mass limit} for the redder (bluer) galaxies. In general, one can\nconstruct a stellar mass limit sample (and hence the subsample S4) for all\n(including both red and blue) galaxies by adopting the stellar mass limit (as\na function of redshift) for the reddest galaxies (see Appendix of van den\nBosch et al. 2008b), which, however, may significantly reduce the number of\ndwarf galaxies in our sample. Instead, as a rough approximation, we construct\nsubsample S4 as follows (with the hidden assumption that if the galaxies are\ncomplete in both luminosity and stellar mass they have similar color\ndistributions and thus similar red and blue fractions). First, we separate all\nthe dwarf galaxies into red and blue populations: the reddest 20\\% being red\nand the rest being blue, using the separation line shown in the upper-left\npanel of Fig ~\\ref{fig:col}. Next, for each of the (1500$\\times$20\\%$=300$)\nred galaxies in S1, we randomly select four blue galaxies from the blue\npopulation with stellar masses within $\\Delta \\log M_{\\ast}=0.025$ of the red\ngalaxy. This yields a blue control sample of (1500$\\times$80\\%$=1200$) dwarf\ngalaxies, which has the same stellar mass distribution as the 20\\% reddest\ngalaxies. The control subsample S4 so constructed has exactly the same red\nfraction of dwarf galaxies as S1, but now with respect to blue galaxies with\nsimilar stellar masses. The upper-left panel of Fig.~\\ref{fig:concen} shows\nthe color-stellar mass relation for S4, split into red and blue galaxies.\n\n\nFor comparison, we also form the following two subsamples from S4. In one, we\nrandomly select 20\\% of the galaxies from S4; in the other, we select the 20\\%\nbluest galaxies that have the same stellar mass distribution as all the\ngalaxies in sample S4. The color-stellar mass relations of these two\nsubsamples are shown in the lower-left and upper-right panels of Fig.\n\\ref{fig:concen}, respectively.\n\nThe upper-left panel of Fig. \\ref{fig:f_con} shows $N_{\\rm red}\/N_{\\rm total}$\nas a function of $r_{\\rm p}\/R_{180}$ obtained using sample S4. Fitting the data\nagain with the function $f=a+ b \\times\\exp(-x\/2)$, we obtain $a=0.053$ and\n$b=0.309$, and the corresponding model is shown as the long-dashed curve.\nThis dependence on the scaled distance, $x\\equiv r_{\\rm p}\/R_{180}$, is only\nslightly weaker than that for the corresponding luminosity sample S1,\nindicating that the bias caused by the stellar-mass difference between the red\nand blue subsamples is not important.\n\nThe upper-right and lower-left panels of Fig. \\ref{fig:f_con} show the\nresults obtained for the 20\\% bluest galaxies and for the 20\\% random galaxies\n(as defined above). For these two cases there is no significant radial\ndependence. Although one expects such a lack of radial dependence for the\nrandom subsample, it does indicate that there are no significant systematic\nerrors in our analysis. The lack of a radial dependence for the 20\\% bluest\nsubsample is due to the fact that only the $\\sim 15-20\\%$ reddest galaxies\nreveal a radial distribution that is peaked towards smaller $r_{\\rm p}\/R_{180}$.\n\n\n\\subsection{Dependence on the Mass of the Nearest Neighbor}\n\n\\begin{figure} \\plotone{f5.eps}\n \\caption{The nearest neighbor to host halo mass ratio $M_n\/M_h$ - projected\n distance $r_{\\rm p}\/R_{180}$ distribution of the dwarf central galaxies in\n samples S1+S2+S3. The triangles and crosses show the red and blue central\n galaxies, respectively, where the red population is defined to be the 20\\%\n reddest all galaxies. } \\label{fig:mass_rp}\n\\end{figure}\n\\begin{figure} \\plotone{f6.eps}\n \\caption{Similar to Fig. \\ref{fig:f_r}, but for all galaxies in samples\n S1+S2+S3 within different central-nearest halo systems. In each panel the\n selection criteria, $M_n\/M_h$, is indicated. Here the results are shown\n for the fraction of the 20\\% red population. For comparison, in each\n panel, we also show as the symbols with solid line the results for all\n central-nearest halo systems. } \\label{fig:M_n}\n\\end{figure}\n\nIn our analysis above, the ``nearest more massive halo'' of a central dwarf\ngalaxy is defined as the halo with a line-of-sight separation $|\\pi| \\le\n15\\>h^{-1}{\\rm {Mpc}}$ which has (i) a mass that is more massive than that of the dwarf\ngalaxy, and (ii) the smallest value of $r_{\\rm p}\/R_{180}$ (see\nsection~\\ref{sec_analyze}). This implies that some of these nearest more\nmassive halos may have masses that are only slightly larger than that of the\ndwarf galaxy under consideration.\n\nIn what follows we use $M_h$ to refer to the halo mass of the central dwarf\ngalaxy, and $M_n$ to refer to the mass of its nearest more massive halo. For\nour combined sample (S1 + S2 + S3) the average value of $M_h$ is about\n$10^{10.9}\\>h^{-1}\\rm M_\\odot$. Fig.~\\ref{fig:mass_rp} shows the ratio $M_n\/M_h$ as a\nfunction of the scaled distance $r_{\\rm p}\/R_{180}$ for all central dwarf galaxies\nin S1+S2+S3. Here the results for the 20\\% reddest galaxies are shown as red\ntriangles, while the other 80\\% are indicated by blue crosses. Note that there\nis a very large amount of scatter in $M_n\/M_h$, ranging from unity to well in\nexcess of 1000.\n\nIt is interesting to investigate whether the color dependence of the radial\ndistribution of central dwarfs with respect to their nearest more massive halo\ndepends on $M_n$. This can provide valuable insight into the actual origin of\nthis color dependence. We therefore proceed as follows. We first combine\nsamples S1+S2+S3, and then calculate the fraction of central red dwarf\ngalaxies that belong to the 20\\% reddest subsample as we did in\n\\S\\ref{sec_analyze}. However, now we only select systems for which $M_n\/M_h$\nis restricted to [1,2], [2,4], [4,8] or $\\log [M_n\/ \\>h^{-1}\\rm M_\\odot]\\ge 12.0$,\nrespectively. The results are shown in the four panels of Fig.~\\ref{fig:M_n}\nas indicated. For comparison we also show, in each panel, the results obtained\nfor all systems (i.e. $M_n\/M_h > 1$). A comparison of all four panels shows\nthat there is a clear, albeit somewhat weak, dependence of the trend on\n$M_n\/M_h$. Overall, the colors of central dwarf galaxies are most strongly\naffected by nearest neighbor halos that are more massive. In the case of $1 <\nM_n\/M_h \\leq 2$ (upper-left panel), the central dwarf galaxies have a $N_{\\rm\nred}\/N_{\\rm total}$ that is almost independent of $r_{\\rm p}\/R_{180}$, and much\nlower than that of the dwarf satellites. On the other hand, the central\ndwarfs that are distributed around halos more massive than $10^{12.0} \\>h^{-1}\\rm M_\\odot$\n(lower-right panel), have a radial dependence that is somewhat stronger than\nthat for all systems.\n\n\\subsection{Survey Edge Effect}\n\\begin{figure} \\plotone{f7.eps}\n \\caption{Similar to Fig. \\ref{fig:f_r}, but here we compare the results for\n all galaxies in samples S1+S2+S3, with and without edge effects by\n removing the central dwarf galaxies that are near the survey\n edge. } \\label{fig:edge}\n\\end{figure}\n\nSince the SDSS is not a full-sky survey, and since our group catalogue is\nconstructed using only galaxies with redshifts $0.01 \\leq z \\leq 0.2$, our\nresults may be influenced by edge effects of the survey: for central dwarf\ngalaxies near an edge of the survey, there is an enhanced probability that it\nis actually a satellite (or central) galaxy in a more massive group (halo),\nbut for which all other members just happen to lie beyond the edges of the\nsurvey. Although we tried to take these effects into account when assigning\nhalo masses to our groups (see Yang et al. 2007 for details), it could still\nbe that a significant fraction of our central dwarf galaxies are in reality\nmisclassified centrals or satellites owes to the survey geometry.\n\nTo check the impact of these edge effects, we follow Yang et al. (2007) by\nmeasuring the edge parameter $f_{\\rm edge}$. For each central dwarf galaxy in\nS1+S2+S3, we randomly distribute $500$ points within a radius $1\\>h^{-1}{\\rm {Mpc}}$. Next\nwe apply the SDSS DR4 survey mask and remove those random points that fall\noutside of the region where the completeness ${\\cal C} > 0.7$. For each central\ndwarf galaxy we then compute the number of remaining points, $N_{\\rm remain}$,\nand we define $f_{\\rm edge}=N_{\\rm remain}\/500$ as a measure for the volume\naround the central dwarf galaxy that lies within the survey edges. To test\nthe impact of edge effects on our measurements in \\S\\ref{sec_analyze}, we\nremove those central dwarf galaxies with $f_{\\rm edge}\\le 0.8$ (about 13\\%)\nand recalculate the radial distribution of the remaining central galaxies. The\nresult is shown in Fig. \\ref{fig:edge} (dashed line), compared to the results\nfor all the central dwarfs, independent of their value of $f_{\\rm edge}$\n(solid line). Clearly, the two curves are almost indistinguishable,\nindicating that our results are not an artifact of survey edge effects.\n\n\\section{Test with mock samples}\n\\label{sec_mock}\n\n\\begin{figure} \\plotone{f8.eps}\n \\caption{Similar to Fig \\ref{fig:f_r}, but here we compare the observational\n and mock results. The upper-left, upper-right and lower-left panels show\n the results for all host-nearest halo systems using different color\n models: Case I, II and III as indicated (see text). In the lower-right\n panel, we show results for those central-nearest halo systems with $M_n\\ge\n 10^{12.0}\\>h^{-1}\\rm M_\\odot$ using color model Case III but with different parameters.\n In each panel, the symbols connected with dashed lines are results\n obtained from the mock galaxy and group catalogues where the halos are\n assumed to be spherical. The symbols connected with dot-dashed lines are\n results obtained from the mock galaxy and group catalogues where the halos\n are assumed to follow a triaxial Jing \\& Suto (2002) profile. For\n reference, in each panel we also show, as the dots connected with solid\n lines, the results we obtained for the SDSS samples S1+S2+S3. See text\n for details. } \\label{fig:mock}\n\\end{figure}\n\nOne potential problem with the results presented above is that the group\nfinder used to identify galaxy groups is not perfect. Hence, some of the\ndwarf galaxies classified as central galaxies may in fact be satellite\ngalaxies. To test the severity of such effects and to quantify the true\nassociation between central dwarf galaxies and their nearby massive halos, we\napply the same analysis to mock samples and compare the results with the\nobservational data that we have obtained. Here we use the mock SDSS DR4\ngalaxy and group catalogues that are constructed by Yang et al. (2007) to\ntest the performance of the group finder. Following Yang et al. (2004), the\nmock galaxy catalogue is constructed by populating dark matter haloes in\nnumerical simulations of the standard $\\Lambda$CDM model with galaxies of\ndifferent luminosities, using the conditional luminosity function (CLF) model\nof Cacciato et al. (2008). The cosmological parameters adopted here are\nconsistent with the three-year data release of the WMAP mission: $\\Omega_m =\n0.238$, $\\Omega_{\\Lambda}=0.762$, $n_s=0.951$, $h=0.73$ and $\\sigma_8=0.75$\n(Spergel et al. 2007). This CLF describes the halo occupation statistics of\nSDSS galaxies, and accurately matches the SDSS luminosity function, as well as\nthe clustering and galaxy-galaxy lensing data of SDSS galaxies as a function\nof their luminosity. Next a mock redshift survey is constructed mimicking the\nsky coverage of the SDSS DR4 and taking detailed account of the angular\nvariations in the magnitude limits and completeness of the data (see Li {et al.~}\n2007 for details). Finally we construct a group catalogue from this mock\nredshift survey, using the same halo-based group finder as for the real SDSS\nDR4.\n\nTo test the impact of contamination on our observational results obtained\nabove, we consider three models for the distribution of red dwarf galaxies:\n\\begin{itemize}\n\\item Case I: Here we assume that a fraction, $f_{\\rm red, sat}$, of true {\\it\n satellite} dwarf galaxies are red, but that all true central dwarf\n galaxies are blue.\n\\item Case II: Same as Case I, but here we assume that a fraction, $f_{\\rm\n red, cent}$, of true central dwarf galaxies are also red and have the same\n spatial distribution as blue central dwarfs.\n\\item Case III: Similar to Case II, but here we assume that $f_{\\rm red,\n cent}$ depends on the distance of the central galaxy to its nearest more\n massive halo, according to $f_{\\rm red, cent}(r)= a+b\\times\\exp(-(y-1)\/2)$.\n Here $a$ and $b$ are constants and $y=r\/R_{180}$.\n\\end{itemize}\nThese models are used to assign a color to each of the mock dwarf galaxies\naccording to their positions in real space and their true membership of host\nhalos.\n\nAs for the observational data, we select the 4500 faintest galaxies from the\nmock group catalogue, using the same criteria as described in\nSubsection~\\ref{sec:samples}. We choose the observational result for the\ncentral galaxies in S1+S2+S3 (shown as the dots with error bars in\nFig.~\\ref{fig:M_n} to compare with our models. As discussed in the previous\nsection, this result is representative of the distribution of red dwarf\ngalaxies with respect to their nearest more massive halos. For all Cases (I,\nII and III), we adopt $f_{\\rm red, sat}=0.38$ so that the red fraction of the\nsatellite galaxies is consistent with the SDSS data (repeated in Fig.\n\\ref{fig:mock} as a solid line). In Case II we set $f_{\\rm red, cent}=0.07$,\nso that the red fraction of the central galaxies at large projected distance,\n$r_{\\rm p}\/R_{180}\\ge 4$, is roughly the same as the observational data. The\ncorresponding results obtained from Case I and Case II are shown in the\nupper-left and upper-right panels of Fig. \\ref{fig:mock}, respectively. It\nis clear that in Case I, in which the only red dwarfs are satellites, the\nfraction of false central galaxies in the mock catalogue is too small too\nmatch the observational data at $r_{\\rm p}\/R_{180}\\ga 1$. For Case II, although by\nconstruction the fraction of red central galaxies matches the observational\nresults at large $r_{\\rm p}\/R_{180}\\ge 4$, the model underestimates the fraction of\nred central galaxies at intermediate $r_{\\rm p}\/R_{180}$. These results indicate\nthat (i) not all red dwarf galaxies are satellites, and (ii) the central red\ndwarf galaxies have a different distribution than the total central dwarf\npopulation.\n\nNow, let us look at Case III. We have experimented with different values for\n$a$ and $b$, and found that the following set of parameters matches the\nobservational data reasonably well: $(a,b)=(0.05, 0.36)$. The results\nobtained from the mock catalogue using this set of model parameters are shown\nin the lower-left panel of Fig. \\ref{fig:mock}. This shows that the central\nred dwarf galaxies are correlated with massive halos on scales given by $(y-1)\n\\la 2$ (i.e. $r\\la 3 R_{180}$).\n\nFinally, as we did for the observational sample, we can also obtain $f_{\\rm\nred, cent}(r)$ for dwarf galaxies near halos of different masses using the\nmock sample Case III. As an illustration, we consider the case with $M_n\\ge\n10^{12.0}\\>h^{-1}\\rm M_\\odot$. The model for $f_{\\rm red, cent}(r)$ that best matches the\nobservational result has $(a,b)=(0.05, 0.45)$ and is slightly steeper than\nthat obtained for the case without any restrictions on $M_n$. The model\nprediction obtained from the mock sample is shown in the lower-right panel of\nFig. \\ref{fig:mock} along with the corresponding observational data.\n\nIn the mock catalogue considered above, the distribution of satellite galaxies\nin individual halos is assumed to be spherically symmetric and to follow the\nNFW (Navarro, Frenk \\& White 1997) profile (see Yang et al. 2004 for\ndetails). In reality, the distribution of satellite galaxies in individual\nhalos may not be spherical, which may cause further contaminations of the\ngroup memberships selected by the group finder. To test this, we have\nconstructed a mock catalogue assuming that the distribution of satellite\ngalaxies in individual halos is triaxial, with axis ratios given by the model\nof Jing \\& Suto (2002) for CDM halos. We found a slightly higher level of\ncontamination in this new mock catalogue, but it does not change any of our\nresults significantly. As an illustration, in each panel of\nFig. \\ref{fig:mock} we also show the results for non-spherical halos using\nsymbols connected with dot-dashed lines. We obtain for Case III slightly\nweaker radial dependence with $(a,b)=(0.05,0.34)$ and $(a,b)=(0.05,0.43)$, for\nthe cases shown in the lower-left and lower-right panels, respectively.\n\n\\section{Discussion}\n\\label{sec_discussion}\n\n\nWe set out to understand why about 1\/4 of {\\it central} dwarf galaxies, those\nwith $r$-band magnitudes between -14.46 and -17.05, are red when one defines a\ndwarf galaxy to be red by extrapolating the division between red sequence\ngalaxies and the blue cloud, as determined by Yang et al. (2008a) for brighter\ngalaxies, down to dwarf magnitudes. Current models of galaxy formation would\nnaively expect such galaxies to be blue since they would be efficiently\naccreting gas through cold mode accretion and rapidly converting it into\nstars.\n\nIn a recent study, Ludlow et al. (2008; see also Lin et al. 2003) analysed\nthe properties of subhalos in galaxy-sized cold dark matter halos using a\nsuite of cosmological N-body simulations. The subhalos in their definition\nrefer to the whole population of subhalos physically associated with the main\nsystem, including both subhalos that are found within the virial radius of the\nhost halo at the present time, and halos that were once within the virial\nradius of the main progenitor of the host and have survived as self-bound\nentities until $z=0$. They found that such populations can extend beyond {\\it\nthree times} the virial radius, and contain objects on extreme orbits, with\nsome approaching the nominal escape speed from the system. On average the\nsubhalos identified within the virial radius represent only about {\\it one\nhalf} of all associated subhalos, and many relatively central halos may have\nactually been ejected in the past from a more massive system. Since galaxies\nare assumed to form in dark matter halos, it is interesting to see if the\nresults we obtain here can be understood in terms of galaxy formation in this\npopulation of subhalos.\n\nAccording to the current theory of galaxy formation, satellite galaxies can\nexperience various environmental effects that can quench their star formation\nand make them red (e.g., van den Bosch et al. 2008b and references therein).\nBecause the galaxies in ejected subhalos have also been satellite galaxies, at\nleast for some period of time, they are likely to have been subjected to\nsimilar environmental effects, and thus to have experienced some quenching of\ntheir star formation rates. It is thus likely that the association of red\ndwarf galaxies with massive halos presented here is produced by the\nassociation of ejected subhalos with their (former) hosts.\n\nAs shown in Table~\\ref{tab1}, about 30\\% of the dwarf galaxies are satellite\ngalaxies. According to the results obtained by Ludlow et al. (2008), there\nshould thus also be a significant fraction of dwarf galaxies that are\nphysically associated with nearby more massive halos out to about three times\nthe virial radius. If these associated galaxies (now outside their hosts)\nhave properties similar to the satellite galaxies, we would expect an enhanced\nfraction of red dwarf galaxies that are distributed outside massive halos.\nThis is qualitatively consistent with our findings presented above and also as\nshown in Table~\\ref{tab1}. However, since the observational data are obtained\nin redshift space and based on galaxy groups that may contain interlopers and\nmay be incomplete, a detailed comparison between the data and the models\nrequires the construction of mock catalogues that make use of the subhalo\npopulation and contain all the observational selection effects.\n\nThe fact that central dwarf galaxies have concentrations that are independent\nof their distances to the nearest massive halos indicates that the processes\nthat causes them to become red does not have a significant impact on their\nstructure. This is similar to the results obtained by van den Bosch et al.\n(2008a) who found that the transformation mechanisms operating on satellites\naffect color more than structure (see also Kauffmann et al. 2004; Blanton et\nal. 2005a; Ball, Loveday \\& Brunner 2008; Weinmann et al. 2008). Once again,\nthis similarity between red dwarfs that are satellites and those that are\ncentrals suggests that both populations may have experienced similar kinds of\nenvironmental effects.\n\nHowever, as shown in Table~\\ref{tab1}, in the S1+S2+S3 sample less than 42\\%\nof the red dwarf central galaxies have $r_{\\rm p}\/R_{180}\\le 3$. In other words,\nmore than 58\\% of the red dwarf central galaxies are not close enough to a\nlarger halo so that they could have been preprocessed there, becoming red and\nthen subsequently being ejected. The origin of this population of central red\ndwarf galaxies, which is almost 10\\% of the combined S1+S2+S3 dwarf sample,\nstill remains a mystery within the standard paradigm of galaxy formation.\nFurthermore, if we had used stellar mass instead of $r$-band magnitude to\ndefine our dwarf sample, the percentage of galaxies in this population would\nlikely increase. This population of isolated red dwarfs are not merely a dust\nreddened star forming population seen near edge-on because their axis ratios\nare consistent with a randomly oriented population. Although as Croton \\&\nFarrar (2008) probed the origin of the red dwarfs in voids using the\nsemi-analytical models, they only found $\\sim 0.4\\%$ of the dwarfs in the\ntotal population are red centrals in voids. While here we find that $\\sim\n10\\%$ dwarfs are red centrals without close neighbours with $r_{\\rm p}\/R_{180}\\le\n3$.\n\nAn outstanding problem for all galaxy formation models concerns the low mass\nslope of the galaxy mass function. CDM models in general predict too many\nlow-mass dark matter haloes compared to the number of low mass galaxies. The\nmass function of dark matter haloes, $n(M)$, scales with halo mass roughly as\n$n(M)\\propto M^{-2}$ at the low-mass end. This is in strong contrast with the\nobserved luminosity function of galaxies, $\\Phi (L)$, which has a rather\nshallow shape at the faint end, with $\\Phi(L) \\propto L^{-1}$. To reconcile\nthis difference one usually invokes some form of feedback within these low\nmass halos. If the feedback mechanism were to prevent gas from entering these\nhalos at late times, such galaxies would appear red. For example, the\npreheating mechanism of Mo et al. (2005), where gas is preheated by gas shocks\nwithin the forming large scale structures in which the low mass dark matter\nhalos themselves are forming, has this feature. Therefore, this population of\nisolated, red, central dwarf galaxies could represent the tail of the process\nthat prevents the vast majority of low mass dark matter halos from forming\ngalaxies and their further study could shed new light on the mechanism\nresponsible.\n\n\n\\acknowledgements We thank the referee Darren Croton for helpful comments that\nimproved the presentation of this paper. YW acknowledges the support of China\nPostdoctoral Science Foundation. This work is supported by the {\\it One\nHundred Talents} project, Shanghai Pujiang Program (No. 07pj14102), 973\nProgram (No. 2007CB815402), the CAS Knowledge Innovation Program (Grant No.\nKJCX2-YW-T05) and grants from NSFC (Nos. 10533030, 10673023, 10821302). HJM\nwould like to acknowledge the support of NSF AST-0607535, NASA AISR-126270 and\nNSF IIS-0611948. NSK and D.H.M. would like to acknowledge the support of NASA\nLTSA NAG5-13102.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Figures}\n\n\\begin{figure}[H]\n \\includegraphics[width=1.\\linewidth]{overlay}\n \\caption{\n Panels \\textbf{(a) and (b)} are all-sky views\n in Mollweide projection in\n Galactic coordinates of longitude $\\ell$ and latitude $b$ with east to the left, with the ROI marked by the dotted box.\n Panels \\textbf{(c)--(e)} are in a cylindrical projection and zoom in on the ROI.\n %\n \\textbf{{\\bf The Fermi Bubbles, including the Cocoon sub-structure, and the Sgr dSph galaxy}.\n Panels (a) and (c)} display \n the $\\gamma$-ray spatial template for the Fermi Bubbles\\cite{Ackermann2014} in arbitrary units with linear colour scale, highlighting the cocoon.\n \\textbf{Panels (b) and (d)}\n show the angular density of RR Lyrae stars with line-of-sight distances $>20$ kpc from the {\\it Gaia} Data Release 2 (DR2), in arbitrary units with logarithmic scaling; the Sgr dSph, Sgr stream, and the Large and Small Magellanic Clouds are clearly visible. The proper motion of the {Sgr~dSph~} is upwards in this figure. The dashed ellipses in panels (a)-(d) mark the same coordinates in each panel, and highlight both the cocoon and the Sgr dSph.\n \\textbf{Panel (e)} shows contours of RR Lyrae surface density overlaid on the Fermi Bubbles template shown as the coloured background.\n }\n \\label{fig:SgrdSphOverlay}\n\\end{figure}\n\n\n\n\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=\\linewidth]{plotSgrSpectrumDataPlusFit}\n\\caption{ \n{\\bf Measured $\\gamma$-ray spectral brightness distributions of the {Sgr~dSph~} and the surrounding Fermi Bubbles}. \nThe black, dashed line \nshows a differential number flux obeying\n$dN_\\gamma\/dE_\\gamma \\propto E_\\gamma^{-2.1}$.\nThese data \nare as obtained by us in our {{\\em Fermi}-LAT} \\ data analysis as described in Methods. \nWe have converted luminosities to surface brightnesses \nadopting source solid angles of $\\Omega_{\\rm Sgr \\ dSph} = 9.6 \\times 10^{-3}$ sr, \nand $\\Omega_{\\rm FB} = 0.49$ sr, with the latter set by the $40^\\circ \\times 40^\\circ$ region of interest (ROI), not the intrinsic sizes of the Bubbles (which are larger than the ROI).\nError bars show $1\\sigma$ errors; for the Sgr dSph, the error bars incorporate both statistical and systematic errors added in quadrature.\nThe smooth blue curves show (solid) the best fit combined (magnetospheric + IC) and (dashed) the best fit magnetospheric spectra.\n}\n\\label{fig:luminosities}\n\\end{figure}\n\n\\begin{figure}[H]\n \\centering\n\\includegraphics[width=\\linewidth]{plotLgammaOvrMstar.pdf}\n\\caption{{\\bf {$\\gamma$-ray} \\ luminosity\nnormalised to stellar mass for various structures whose emission is plausibly dominated by MSPs.}\nThe `Sgr magneto.' datum shows our\nbest-fit magnetospheric luminosity per stellar mass\n(the spectrum shown as the dashed blue curve in \\autoref{fig:luminosities})\nwhile the `Sgr tot' datum is the total, directly-measured luminosity. Globular cluster (`GC') measurements are from ref.~\\cite{Song2021}, while the remaining data (collated by ref.~\\cite{Song2021}) are from ref.~\\cite{Macias2019} (nuclear bulge of the Milky Way, `NB'), ref.~\\cite{Ackermann2017} (M31), and ref.~\\cite{Bartels2018} (Milky Way disc). \nError bars show 1$\\sigma$ errors.\nThe horizontal, dashed, grey curves show\nthe predicted total \n{$\\gamma$-ray} \\ luminosity per unit stellar mass\nat the nominated efficiencies, $f_{\\rm \\gamma,tot} = \\{0.1, 0.9\\}$, \ngiven an MSP spin-down power per unit stellar mass of\n$2 \\times 10^{28}$ erg\/s\/$M_{\\odot}$ as\nwe infer from ref.~\\cite{Sudoh2020}.}\n\\label{fig:LgammaOvrMstar}\n\\end{figure}\n\n\n\\begin{table}\n \\centering\n \\small\n \\begin{tabular}{llll@{\\qquad\\qquad}rrrr}\n \\hline\\hline\n \\multicolumn{4}{c}{Template choices} & \\multicolumn{4}{c}{Results} \\\\\n Hadr. \/ Bremss. & IC & FB & Sgr dSph &\n $-\\log(\\mathcal{L}_{\\rm Base})$ & $-\\log(\\mathcal{L}_{{\\rm Base}+{\\rm Sgr}})$ & $\\mbox{TS}_{\\rm Source}$& Significance \\\\[0.5ex] \\hline \n \\multicolumn{8}{c}{Default model} \\\\[0.5ex]\n HD & 3D & S & Model I & 866680.6 &866633.0 & 95.2 & $8.1\\;\\sigma$ \\\\[0.5ex] \\hline\n \\multicolumn{8}{c}{Alternative background templates} \\\\[0.5ex]\n HD & 2D A & S & Model I & 866847.1 & 866810.9 & 72.3 & $6.9\\;\\sigma$ \\\\\n HD & 2D B & S & Model I & 867234.9 &867192.1 & 85.8 & $7.8\\;\\sigma$ \\\\\n HD & 2D C & S & Model I & 866909.4 & 866868.5 & 81.7 & $7.4\\;\\sigma$ \\\\\n Interpolated & 3D & S & Model I & 867595.4 & 867567.4 & 56.0 & $5.8\\;\\sigma$ \\\\\n GALPROP & 3D & S & Model I & 866690.5 & 866640.8 & 99.5 & $8.3\\;\\sigma$ \\\\[0.5ex] \\hline\n \\multicolumn{8}{c}{Flat FB template} \\\\[0.5ex]\n HD & 3D & U & Model I & 867271.7 & 867060.1 & 423.2 & $19.1\\;\\sigma$ \\\\\n HD & 2D A & U & Model I & 867284.2 &867122.9 & 322.5 & $16.5\\;\\sigma$ \\\\\n HD & 2D B & U & Model I & 867624.3 & 867464.0& 320.7 & $16.4\\;\\sigma$ \\\\\n HD & 2D C & U & Model I & 867322.7 &867158.2 &329.0 & $16.6\\;\\sigma$ \\\\\n Interpolated & 3D & U & Model I & 867287.4 & 867081.2& 412.4 & $18.9\\;\\sigma$ \\\\\n GALPROP & 3D & U & Model I & 868214.6 & 868040.9& 347.6 & $17.2\\;\\sigma$ \\\\[0.5ex]\\hline\n \\multicolumn{8}{c}{Alternative Sgr dSph templates} \\\\[0.5ex]\n HD & 3D & S & Model II & 866680.6 & 866626.3 & 108.5 & $8.7\\;\\sigma$ \\\\\n HD & 3D & S & Model III & 866680.6 & 866647.5 & 66.1 & $6.4\\;\\sigma$ \\\\\n HD & 3D & S & Model IV & 866680.6 & 866678.2 & 4.8 & $0.4\\;\\sigma$ \\\\\n HD & 3D & S & Model V & 866680.6 &866644.9 & 71.5 & $6.7\\;\\sigma$ \\\\\n HD & 3D & U & Model II & 867271.7 & 866970.7 & 602.1 & $23.2\\;\\sigma$ \\\\\n HD & 3D & U & Model III & 867271.7 & 866994.1 & 555.3 & $22.2\\;\\sigma$ \\\\\n HD & 3D & U & Model IV & 867271.7 & 867152.2 & 239.1 & $14.0\\;\\sigma$ \\\\\n HD & 3D & U & Model V & 867271.7 & 866993.3 & 556.9 & $22.2\\;\\sigma$ \\\\\n \\hline\\hline\n \\end{tabular}\n \\caption{Template analysis results comparing {\\it baseline} to {\\it baseline + Sgr dSph} models. \n Columns (1) - (3) specify the {baseline} templates used for Galactic hadronic \/ bremsstrahlung emission, inverse Compton emission, and the Fermi Bubbles, respectively.\n %\n Column (4) specifies {source} templates\n describing the Sgr dSph (see Methods for details). Columns (5) and (6) give the log likelihood for the baseline model (without the Sgr dSph) and the baseline + Sgr dSph model, and columns (7) and (8) give the test statistic with which the baseline + Sgr dSph model is preferred, and the corresponding statistical significance of that preference. \n %\n The improvement in TS going from $\\{\\rm HD, 3D, U , Model \\ I\\}$ to $\\{\\rm HD, 3D, S , Model \\ I\\}$ is $\\Delta$TS = 854.2, equivalent to 28.0 $\\sigma$.\n Note that Sgr dSph model IV -- which generates a statistically insignificant improvement to the baseline for one particular combination in the last cluster -- is the sparsest stellar template, containing only 675 stars.\n }\n \\label{tab:loglikelihood}\n\\end{table}\n\n\n\\clearpage\n\n\\newrefsegment\n\n\\section*{Methods}\n\\label{sec:Methods}\n\nOur analysis pipeline consists of three steps: (1) data and template selection, (2) fitting, and (3) spectral modeling. \n\n\\subsection*{\nData and template selection\n}\n\\label{sec:fermidata}\n\nWe use eight years of LAT data, selecting \\texttt{Pass 8 UltraCleanVeto} class events in the energy range from 500 MeV to 177.4 GeV. We choose the limit at low energy to mitigate both the impact of {$\\gamma$-ray} \\ leakage from the Earth's limb and the increasing width of the point-spread function at lower energies.\nWe spatially bin the data to a resolution of $0.2^\\circ$, and divide it into 15 energy bins; the 13 lowest-energy of these are equally spaced in log energy, while the 2 highest-energy are twice that width in order to improve the signal to noise.\nWe select data obtained over the same observation period as that used in the construction of the Fourth Fermi Catalogue (4FGL)\\cite{Fermi-LAT:4FGL} (August 4, 2008 to August 2, 2016).\nThe region of interest (ROI) of our analysis is a square region defined by $-45^\\circ\\leq b \\leq -5^\\circ$, and $30^\\circ \\geq \\ell \\geq -10^\\circ$ (\\autoref{fig:SgrdSphOverlay}). \nThis sky region fully contains the Fermi cocoon substructure but avoids the Galactic plane ($|b|\\leq 5^\\circ$) where uncertainties are largest. Because the ROI is of modest size, we allow the Galactic diffuse emission (GDE)\ntemplates greater freedom to reproduce potential features in the data. \nWe carry out all data reduction and analysis using the standard \n\\textsc{Fermitools v1.0.1}\\footnote{\\url{https:\/\/github.com\/fermi-lat\/Fermitools-conda\/wiki}} \nsoftware package.\nWe model the performance of the LAT with the \\texttt{P8R3\\_ULTRACLEANVETO\\_V2} Instrument Response Functions (IRFs).\n\nWe fit the spatial distribution of the ROI data as the sum of a series of templates for different components of the emission. For all the templates we consider, we define a ``baseline'' model that includes only known point and diffuse emission sources, to which we compare a ``baseline + Sgr dSph'' model that includes those templates plus the Sgr dSph. Our baseline models, following the approach of Ref.~\\cite{Abazajian:2020}, contain the following templates: (1) diffuse isotropic emission, (2) point sources, (3) emission from the Sun and Moon, (4) Loop I, (5) the Galactic Centre Excess, (6) Galactic cosmic ray-driven hadronic and bremsstrahlung emission, (7) inverse Compton emission, and (8) the \\textit{Fermi} Bubbles; baseline + Sgr dSph models also include a Sgr dSph template.\n\n\nOur templates for the first five emission sources are straightforward, and we adopt a single template for each of them throughout our analysis. Since our data selection is identical to that used to construct the 4FGL, we adopt the standard isotropic background and point source models provided as part of the catalogue \\cite{Fermi-LAT:4FGL}, \\texttt{iso$_{-}$P8R3$_{-}$ULTRACLEANVETO$_{-}$V2$_{-}$v1.txt}, and \\texttt{gll\\_psc\\_v20.fit}, respectively; the latter includes 177 $\\gamma$ ray point sources within our ROI. We similarly adopt the standard Sun and Moon templates provided. For the foreground structure Loop I, we adopt the model of Ref.~\\cite{Wolleben:2007}. Finally, given that the low-latitude boundary of our ROI overlaps with the spatial tail of the Galactic Centre Excess (GCE), we include the `Boxy Bulge' template of Ref.~\\cite{Freudenreich1998}, which has been shown \\cite{Macias2018,Bartels2018,Macias2019} to provide a good description of the observed GCE away from the nuclear Bulge region (which is outside our ROI). The inclusion of this template in our ROI model has only a small impact on our results.\n\nThe remaining templates require more care. The dominant source of $\\gamma$-rays within the ROI is hadronic and bremsstrahlung emission resulting from the interaction of Milky Way cosmic ray (CR) protons and electrons with interstellar gas; the emission rate is proportional to the product of the gas density and the CR flux. We model this distribution using three alternative approaches. Our preferred approach follows that described in Ref.~\\cite{Macias2018}. We assume that the spatial distribution of $\\gamma$-ray emission traces the gas distribution from the hydrodynamical model of Ref.~\\cite{Pohl2008}, which gives a more realistic description of the inner Galaxy than alternatives. To normalise the emission, we divide the Galaxy into four rings spanning the radial ranges $0-3.5$ kpc, $3.5-8.0$ kpc, $8.0-10.0$ kpc, and $10.0-50.0$ kpc, within which we treat the emission per unit gas mass in each of our 15 energy bins as a constant to be fit. We refer to the template produced in this way as the ``HD'' model. Our first alternative is to use the same procedure of dividing the Galaxy into rings, but describe the gas distribution within those rings using a template constructed from interpolated maps of Galactic H~\\textsc{i} and H$_2$, following the approach described in Appendix B of Ref.~\\cite{Ackermann2012}; we refer to this as the ``Interpolated'' approach. Our third alternative, the ``GALPROP'' model, is the SA50 model described by Ref.~\\cite{Johannesson:2018bit}, which prescribes the full-sky hadronic CR emission distribution.\n\nWe similarly need a model for diffuse, Galactic IC emission -- the second largest source of background -- which is a product of the CR electron flux and the interstellar radiation field (ISRF). As with hadronic emission, we consider four alternative distributions. Our default choice is the SA50 model described by Ref.~\\cite{Johannesson:2018bit}, which includes 3D models for the ISRF~\\cite{Porter:2017vaa}. We therefore refer to this as the ``3D'' model. However, unlike in Ref.~\\cite{Johannesson:2018bit}, we use this model only to obtain the spatial distribution of the emission, not its normalisation or energy dependence. Instead, we obtain these in the same way as for our baseline hadronic emission model, i.e., we divide the Galaxy into four rings and leave the total amount of emission in each ring at each energy as a free parameter to be fit to the data; this approach reduces the sensitivity of our results to uncertainties in the electron injection spectrum and ISRF normalisation. Our three alternatives to this are models ``2D A'', ``2D B'', and ``2D C'', corresponding to models A, B, and C as described by Ref.~\\cite{Ackermann:2014usa}, which model IC emission over the full sky under a variety of assumptions about CR injection and propagation, but rely on a 2D model for the ISRF.\n\nThe final component of our baseline template is a model for the Fermi Bubbles themselves, which are one of the strongest sources of foreground emission in high latitude regions of the ROI. The FBs are themselves defined as highly statistically-significant and spatially-coherent residuals in the inner Galaxy that remain once other sources are modelled out in all-sky {$\\gamma$-ray} \\ analyses. The FBs are not reliably traced by emission at any other wavelength, so we do not have an {\\it a priori} model with which to guide the construction of a spatial template of these structures. However, one characteristic that renders the FBs distinct from other large angular scale diffuse $\\gamma$-ray structures is their hard {$\\gamma$-ray} \\ spectrum. Indeed, the state-of-the-art, {\\it structured} spatial template for them generated by the {\\it Fermi} Collaboration\\cite{Ackermann2014} -- the templates one would normally employ in large ROI, inner Galaxy {{\\em Fermi}-LAT} \\ analyses -- were constructed using a spectral component analysis. That study recovered a number of regions of apparent substructure within the solid angle of the FBs, most notably substructure overlapping the previously-discovered\\cite{Su2012,Selig2015} ``cocoon'' which, as we have discussed here, is largely coincident with the Sgr dSph. Of course, a potential issue with constructing a phenomenological, spectrally-defined model for the FBs is that, if there happens to be an extended, spectrally-similar source coincident with the FBs, it will tend to be incorporated into the template. For this reason Ref.~\\cite{Ackermann2014} suggest using a flat FB template when searching for new structures. Despite this proposal, our default analysis uses the more conservative choice of a structured FB template. However, we also run tests using an unstructured template for comparison, and to understand the systematic uncertainties associated with the choice of template. We refer to these two cases as the ``U'' (Unstructured) and ``S'' (Structured) FB templates, respectively.\n\nFinally, our baseline + Sgr dSph models require a template for the Sgr dSph. Our templates trace the distribution of bright stars in the dwarf, which we construct from five alternative stellar catalogues, all based on different selections from \\textit{Gaia} Data Release 2; we refer to the resulting templates as models I - V, and show them in E.D.~\\autoref{fig:Stellartemplates}. Full details on how we construct each of these templates are provided in S.I.~sec.~2. Model I, our default choice, comes from the catalogue of $2.26\\times 10^5$ Sgr dSph candidate member stars from Ref.~\\cite{Vasiliev2020}; the majority of the catalogue consists of red clump stars. Model II uses the catalogue of RR Lyrae stars in the Sagittarius Stream from Ref.~\\cite{Ibataetal:2020}, which we have down-selected to a sample of 2369 stars whose kinematics are consistent with being members of the Sgr dSph itself. Model III uses the catalogue of $1.31\\times 10^4$ RR Lyrae stars belonging to the Sgr dSph provided by Ref.~\\cite{Iorio2019}. Finally, models IV and V come from the nGC3 and Strip catalogues of RR Lyrae stars from Ref.~\\cite{Ramosetal:2020}; the former contains 675 stars with higher purity but lower completeness, while the latter contains 4812 stars of higher completeness but lower purity.\n\n\\subsection*{Fitting procedure}\n\nOur fitting method follows that introduced in Refs.~\\cite{Macias2018, Macias2019}, and treats each of the 15 energy bins as independent, thereby removing the need to assume any particular spectral shape for each component and allowing the spectra to be determined solely by the data. Our data to be fit consist of the observed $\\gamma$-ray photon counts in each spatial pixel $i$ and energy bin $n$, which we denote $\\Phi_{n,i,\\rm obs}$, where\n$n$ goes from 1 to 15, and the index $i$ runs over the positions $(\\ell_i,b_i)$ of all spatial pixels within the ROI. For a given choice of template, we write the corresponding model-predicted $\\gamma$-ray counts as $\\Phi_{n,i,\\rm mod} = \\sum_c \\mathcal{N}_{n,c} R_{n,i} \\Phi_{c,i}$, where $R_{n,i}$ is the instrument response for each pixel and energy bin (computed assuming an $E^{-2}$ spectrum within the bin), and $\\Phi_{c,i}$ is the value of template component $c$ evaluated at pixel $i$; for baseline models, we have a total of 8 components, while for baseline + Sgr dSph models we have 9. Note that $\\Phi_{c,i}$ is a function of $i$ but not of $n$, i.e., we assume that the spatial distribution of each template component is the same at all energies, except for the IC templates, for which an energy-dependent morphology is predicted by our GALPROP simulations. Without loss of generality we further normalise each template component as $\\sum_i \\Phi_{c,i} = 1$, in which case $\\mathcal{N}_{n,c}$ is simply the total number of photons contributed by component $c$ in energy bin $n$, integrated over the full ROI; the values of $\\mathcal{N}_{n,c}$ are the parameters to be fit. We find the best fit by maximising the usual Poisson likelihood function\n\\begin{equation}\n \\ln\\mathcal{L}_n = \\sum_{i} \\frac{\\Phi_{n,i,\\rm mod}^{\\Phi_{n,i,\\rm obs}} e^{-\\Phi_{n,i,\\rm mod}}}{\\Phi_{n,i,\\rm obs}!},\n \\label{eq:log-likelihood}\n\\end{equation}\nusing the \\texttt{pylikelihood} routine, the standard maximum-likelihood method in \\texttt{FermiTools}. Note that, since each energy bin $n$ is independent, we carry out the likelihood maximisation bin-by-bin.\n\n\nWe perform all fits in pairs, one for a baseline model containing only known emission sources, and one for a baseline + Sgr dSph model containing the same known sources plus a component tracing the Sgr dSph. The set of paired fits we perform in this manner is shown in \\autoref{tab:loglikelihood}. We compare the quality of these baseline and baseline + Sgr dSph fits by defining the test statistic $\\mathrm{TS}_n = -2\\ln(\\mathcal{L}_{n,\\rm base}\/\\mathcal{L}_{n,\\rm base+Sgr})$; the total test statistic for all energy bins is simply $\\mathrm{TS} = \\sum_n \\mathrm{TS}_n$. We can assign a $p$-value to a particular value of the TS by noting that baseline + Sgr dSph models have 15 additional degrees of freedom compared to baseline models: the value of $\\mathcal{N}_{n,c}$ for the component $c$ corresponding to the Sgr dSph, evaluated at each of the \n15 energy bins. In this case, the mixture distribution formula gives\\cite{Macias2018}\n\\begin{equation}\n p(\\mathrm{TS}) = 2^{-N} \\left[\\delta(\\mathrm{TS}) + \\sum_{n=1}^N \\binom{N}{n} \\chi^2_n(\\mathrm{TS})\\right],\n\\end{equation}\nwhere $N = 15$ is the difference in number of degrees of freedom, \n$\\binom{N}{n}$ is the binomial coefficient, $\\delta$ is the Dirac delta function, and $\\chi^2_n$ is the usual $\\chi^2$ distribution with $n$ degrees of freedom. The corresponding statistical significance (in $\\sigma$ units) is\\cite{Macias2018}:\n\\begin{equation}\n\\label{eq:numberofsigmas}\n\\mbox{Number of $\\sigma$}\\equiv \\sqrt{\\rm InverseCDF\\left(\\chi_1^2,{\\rm CDF}\\left[p(\\mbox{TS}),\\hat{{\\rm TS}}\\right]\\right)},\n\\end{equation}\nwhere (InverseCDF) CDF is the (inverse) cumulative distribution function and the first argument of each of these functions is the distribution function, the second is the value at which the CDF is evaluated, and the total TS is denoted by $\\hat{\\rm TS}$. For 15 extra degrees of freedom, a 5$\\sigma$ detection corresponds to $\\mbox{TS}=46.1$. (Additional details of these formulae are given in S.I. Sec.~2 of Ref.~\\cite{Macias2018}.) We report values of $\\mathcal{L}_{\\rm base}$, $\\mathcal{L}_{\\rm base+Sgr}$, $\\mathrm{TS}$, and the significance level for all the templates we try in \\autoref{tab:loglikelihood}.\n\nA final step in our fitting chain is to assess the uncertainties. For our default choice of baseline + Sgr dSph model (first row in \\autoref{tab:loglikelihood}), our maximum likelihood analysis returns the central value $\\mathcal{N}^{\\rm def}_n$ on the total $\\gamma$-ray flux in the $n$th energy bin attributed to the Sgr dSph, and also yields an uncertainty $\\sigma^{\\rm def}_{\\mathcal{N},n}$ on this quantity. This represents the statistical error arising from measurement uncertainties. However, there are also systematic uncertainties stemming from our imperfect knowledge of the templates characterising the other emission sources. To estimate these, we examine the five alternative models listed in \\autoref{tab:loglikelihood} as ``Alternative background templates'', where we use different templates for the hadronic plus bremsstrahlung and inverse Compton backgrounds. Each of these models $m$ also returns a central value $\\mathcal{N}_n^m$ and an uncertainty $\\sigma_{\\mathcal{N},n}^m$ on the Sgr dSph flux. We use the uncertainty-weighted dispersion of these models as an estimate of the systematic uncertainty (e.g.,ref.~\\cite{Ackermann2018}):\n\\begin{equation}\n \\delta\\mathcal{N}_n = \\sqrt{\\frac{1}{\\sum_m \\left(\\sigma_{\\mathcal{N},n}^m\\right)^{-2}} \\sum_m \\left(\\sigma_{\\mathcal{N},n}^m\\right)^{-2} \\left(\\mathcal{N}_n^{\\rm def} - \\mathcal{N}^m_n\\right)^2},\n\\end{equation}\nwhere the sums run over the $m=6-1$ alternative models. We take the total uncertainty on the Sgr dSph flux in each energy bin to be a quadrature sum of the systematic and statistical uncertainties, i.e., $(\\sigma^{\\rm def,tot}_{\\mathcal{N},n})^2 = (\\sigma^{\\rm def}_{\\mathcal{N},n})^2 + \\delta\\mathcal{N}_n^2$. We plot the central values and uncertainties of the fluxes for the default model derived in this manner in \\autoref{fig:luminosities}.\n\nWe have carried out several validation tests of this pipeline, which we describe in the Supplementary Information (SI).\n\n\n\n\n\\subsection*{Spectral modelling}\n\n\nWe model the observed Sgr dSph $\\gamma$-ray spectrum as a combination of prompt magnetospheric MSP emission and IC emission from {$e^\\pm$}~escaping MSP magnetospheres. We construct this model as follows. The prompt component is due to curvature radiation from {$e^\\pm$}~in within MSP magnetospheres. \nThe {$e^\\pm$}~energy distribution can be approximated as an exponentially-truncated power law \\cite{Abdo2013,Song2021}\n\\begin{equation}\n \\frac{dN_{\\mathrm{MSP},e^\\pm}}{dE_{e^\\pm}} \\propto E_{e^\\pm}^{\\gamma_\\mathrm{MSP}} \\exp\\left(-\\frac{E_{e^\\pm}}{E_{\\mathrm{cut},e^\\pm}}\\right),\n\\label{eq:promptSpec}\n\\end{equation}\nand curvature radiation from these particles has a rate of photon emission per unit energy per unit time\n\\begin{equation}\n \\frac{d\\dot{N}_{\\rm \\gamma,prompt}}{dE_\\gamma} = \\mathcal{N}\\left(L_{\\gamma,\\mathrm{prompt}}\\right) E_\\gamma^{\\alpha} \\exp\\left(-\\frac{E_\\gamma}{E_{\\rm cut, prompt}} \\right),\n\\label{eq:cutoffpwrlaw}\n\\end{equation}\nwhere $E_\\gamma$ is the photon energy, $\\mathcal{N}(L_{\\gamma,\\mathrm{prompt}})$ is a normalisation factor chosen so that the prompt component has total luminosity $L_{\\gamma,\\mathrm{prompt}}$, the index $\\alpha$ is related to that of the {$e^\\pm$}~distribution by $\\alpha = (\\gamma_{\\rm MSP} - 1)\/3$, and the photon cutoff energy is related to the {$e^\\pm$}~cutoff energy by \\cite{Baring2011}\n\\begin{equation}\nE_{\\rm cut,prompt} = \\frac{3 \\hbar c}{2 \\rho_c} \\left(\\frac{E_{\\rm cut,e^\\pm}}{m_e}\\right)^3 \\simeq 2.0 \\ {\\rm GeV} \\ \\left(\\frac{\\rho_c}{\\rm 30 \\ km}\\right)^{-1} \\left(\\frac{E_{\\rm cut,e^\\pm}}{\\rm 3 \\ TeV}\\right)^3\n\\label{eq:EcutPrompt}\n\\end{equation}\nwhere $m_e$ is the electron mass, $\\rho_c$ is the radius of curvature of the magnetic field lines, and the other symbols have the usual meanings. Given the rather small magnetospheres, we expect $\\rho_c$ to be a small multiple of the $\\sim$ 10 km neutron star characteristic radius; henceforth we set $\\rho_c =$ 30 km. Empirically, $L_{\\gamma,\\rm prompt}$ is $\\sim 10\\%$ of the total MSP spin-down power \\cite{Abdo2013}.\n\n\n\nA larger proportion of the spin-down power goes into a wind of {$e^\\pm$}~escaping the magnetosphere. In the ultra-low density environment of the Sgr dSph, ionization and bremmstrahlung losses for this population, which occur at a rate proportional to the gas density, are negligible. Synchrotron losses, which scale as the magnetic energy density, will also be negligible; as noted in the main text, observed magnetic fields in dwarf galaxies are very weak \\cite{Regis2015}, and we can also set a firm upper limit on the Sgr dSph magnetic field strength simply by noting that the magnetic pressure cannot exceed the gravitational pressure provided by the stars since, if it did, that magnetic field, and the gas to which it is attached, would blow out of the galaxy in a dynamical time. The gravitational pressure is $P \\approx (\\pi\/2) G \\Sigma^2$, where $\\Sigma = M\/\\pi R^2$ is the surface density, and using our fiducial numbers $M = 10^8$ M$_{\\odot}$ and $R = 2.6$ kpc gives an upper limit on the magnetic energy density 0.06 eV \/ cm$^3$; non-zero gas or cosmic ray pressure would lower this estimate even further. This is a factor of four smaller than the energy density of the CMB, implying that synchrotron losses are at most a 20\\% effect, and can therefore be neglected.\n\nThis analysis implies that the only significant loss mechanism for these {$e^\\pm$}~is IC emission, resulting in a steady-state {$e^\\pm$}~energy distribution\n\\begin{equation}\n\\frac{dN_{e^\\pm}}{dE_{e^\\pm}} \\propto E_{e^\\pm}^\\gamma \\exp\\left(-\\frac{E_{e^\\pm}}{E_{\\mathrm{cut},e^\\pm}}\\right),\n\\end{equation}\nwhere $\\gamma = \\gamma_{\\rm MSP}-1$. We compute the IC photon distribution produced by these particles following ref.~\\cite{Khangulyan2014}, assuming that ISRF of the {Sgr~dSph~} is the sum of the CMB \nand two subdominant contributions, one consisting of light escaping from the Milky Way and the other a dilute stellar blackbody radiation field due to the stars of the dwarf.\nWe estimate the Milky Way contribution to the photon field at position of the dwarf using GalProp \\cite{Porter:2017vaa},\nwhich predicts a total energy density of $0.095$ eV\/cm$^3$ (compared to 0.26 eV\/cm$^{-3}$ for the CMB), comprised of 5 dilute black bodies\nwith colour temperatures and dilution factors\n$\\{ T_{\\rm rad},\\kappa \\}$ as follows: \n$\\{40 \\ {\\rm K }, 1.4 \\times 10^{-6} \\},\n\\{430 \\ {\\rm K }, 3.0 \\times 10^{-11} \\},\n\\{3400 \\ {\\rm K }, 4.3 \\times 10^{-14} \\},\n\\{6400 \\ {\\rm K }, 4.0 \\times 10^{-15} \\},$\nand\n$ \\{26000 \\ {\\rm K }, 8.0 \\times 10^{-18} \\}\n$.\nWe characterise the intrinsic light field of the dwarf as having a\ncolour temperature 3500 K and dilution factor of $7.0\\times 10^{-15}$ (giving energy density $0.005$ eV cm$^{-3}$; these choices are those expected for a spherical region of radius 2.6 kpc and stellar luminosity $2\\times 10^8$ $L_\\odot$, the approximate parameters of the Sgr dSph).\nThis yields an IC spectrum\n\\begin{equation}\n \\frac{d\\dot{N}_{\\gamma,\\mathrm{IC}}}{dE_\\gamma} = \\mathcal{N}\\left(L_{\\gamma,\\rm IC}\\right) F\\left(\\gamma,E_{\\mathrm{cut},e^\\pm}\\right),\n\\end{equation}\nwhere $\\mathcal{N}\\left(L_{\\gamma,\\rm IC}\\right)$ is again a normalisation chosen to ensure that the total IC luminosity is $L_{\\gamma,\\rm IC}$, and $F\\left(\\gamma,E_{\\mathrm{cut},e^\\pm}\\right)$ is the functional form given by equation 14 of ref.~\\cite{Khangulyan2014}, which depends on the {$e^\\pm$}~spectral index $\\gamma$ and cutoff energy $E_{\\mathrm{cut},e^\\pm}$.\n\n\n\nCombining the prompt and IC components, we may therefore write the complete emission spectrum as\n\\begin{equation}\n \\frac{d\\dot{N}_\\gamma}{dE_\\gamma} = \\mathcal{N}\\left(L_{\\gamma,\\mathrm{prompt}}\\right) E_\\gamma^{\\alpha} \\exp\\left(-\\frac{E_\\gamma}{E_{\\rm cut, prompt}} \\right) + \\mathcal{N}\\left(L_{\\gamma,\\rm IC}\\right) F\\left(\\gamma,E_{\\mathrm{cut},e^\\pm}\\right).\n\\end{equation}\nThis model is characterised by four free parameters: the total prompt plus IC luminosity $L_{\\gamma,\\rm tot} = L_{\\gamma,\\rm prompt} + L_{\\gamma,\\rm IC}$, the ratio of the prompt and IC luminosities $f = L_{\\gamma,\\rm prompt}\/L_{\\gamma,\\rm IC}$, the spectral index $\\alpha$ of the prompt component (which in turn fixes the other two spectral indices $\\gamma_{\\rm MSP}$ and $\\gamma$), and the cutoff energy for the prompt component $E_{\\rm cut, prompt}$ (which then fixes the {$e^\\pm$}~cutoff energy $E_{\\mathrm{cut},e^\\pm}$).\nNote that we make the simplest assumption that $\\alpha$ and $E_{\\rm cut, prompt}$ are uniform across the MSP population.\nIn reality, there may be a distribution of these properties but the parameteric form of \\autoref{eq:promptSpec}\nprovides a good description, in general, \nof both individual MSP spectra and the\naggregate spectra of GC MSP populations \\cite{Song2021}.\n\n\nWe fit the observed {Sgr~dSph~}~spectrum to this model using a standard $\\chi^2$ minimisation, using the combined statistical plus systematic uncertainty.\nWe obtain an excellent fit: the minimum $\\chi^2$ is 7.7 for 15 (data points) - 4 (fit parameters) = 11 (degrees of freedom, dof) or a reduced $\\chi^2$ of 0.70.\nWe report the best-fitting parameters in E.D.~\\autoref{tab:table1}, and plot the result best-fit spectra over the data in \\autoref{fig:luminosities}; we show the best-fit estimate (with $\\pm 1\\sigma$ confidence region) for the magnetospheric luminosity per stellar mass of the {Sgr~dSph~} MSPs in \\autoref{fig:LgammaOvrMstar}. \n\nWe also carry out an additional consistency check, by comparing our best-fit parameters describing the prompt emission -- $\\alpha$ and $E_{\\rm cut,prompt}$ -- to direct measurements of the prompt component from nearby, resolved MSPs \\cite{Abdo2013,Song2021}, and to measurements of GCs, whose emission is likely dominated by unresolved MSPs \\cite{Song2021}. We carry out this comparison in E.D.~\\autoref{fig:FitContours}. In this figure, we show joint confidence intervals on $\\alpha$ and $E_{\\rm cut,prompt}$ from our fit. For comparison, we construct confidence intervals for $\\alpha$ and $E_{\\rm cut,prompt}$ from observations using the sample of ref.~\\cite{Song2021}, who fit the prompt emission from 40 GCs and 110 individually-resolved MSPs. We draw 100,000 Monte Carlo samples from these fits, treating the stated uncertainties as Gaussian, and construct contours in the $(E_{\\rm cut,prompt}, \\alpha)$ plane containing 68\\%, 95\\%, and 99\\% of the sample points. As the plot shows, the confidence region from our fit is fully consistent with the confidence regions from the observations, indicating that our best-fit parameters are fully consistent with those typically observed for MSPs and GCs.\n\n\n\\section*{Data availability}\n\nAll data analysed for this study are publicly available.\nIn particular, {{\\em Fermi}-LAT} \\ data are available from \\url{https:\/\/fermi.gsfc.nasa.gov\/ssc\/data\/} and Gaia data are available from \\url{https:\/\/gea.esac.esa.int\/archive\/}.\nThe statistical pipeline, astrophysical templates, and gamma-ray observations necessary to reproduce our main results are publicly available in the following zenodo repository: \\url{10.5281\/zenodo.6210967}.\n\n\n\\section*{Code availability}\n\n{{\\em Fermi}-LAT} \\ data used in our study were reduced and analysed using the standard \n\\textsc{Fermitools v1.0.1}\nsoftware package available from \\url{https:\/\/github.com\/fermi-lat\/Fermitools-conda\/wiki}.\nThe performance of the {{\\em Fermi}-LAT} \\ was modelled with the \\texttt{P8R3\\_ULTRACLEANVETO\\_V2} Instrument Response Functions (IRFs).\nSpectral analysis and fitting was performed using custom \\textsc{MATHEMATICA}\ncode created by the authors which is available upon reasonable request.\n\n\\clearpage\n\n\n\\section*{Acknowledgements}\n\nRMC acknowledges \nsupport from the Australian Government through the Australian Research Council, award\nDP190101258 (shared with MRK)\nand hospitality from the Virginia Institute of Technology, the Max-Planck Institut f\\\"ur Kernphysik, and the GRAPPA Institute at the University of Amsterdam supported by the Kavli IPMU at the University of Tokyo. \nO.M. is supported by the GRAPPA Prize Fellowship and JSPS KAKENHI Grant Numbers JP17H04836, JP18H04340, JP18H04578, and JP20K14463.\nThis work was supported by World Premier International Research Centre Initiative (WPI Initiative), MEXT, Japan. \nADM acknowledges support from the Australian Government through a Future Fellowship from the Australian Research Council, award FT160100206.\nM.R.K. acknowledges support from the Australian Government through the Australian Research Council, award\nDP190101258 (shared with RMC) and\nFT180100375.\nThe work of S.H.\\ is supported by the U.S.\\ Department of Energy Office of Science under award number DE-SC0020262 and NSF Grant No.\\ AST-1908960 and No.\\ PHY-1914409. The work of DS is supported by the U.S.\\ Department of Energy Office of Science under award number DE-SC0020262. \nT.V. and A.R.D. acknowledge the support of the Australian Research Council's Centre of Excellence for Dark Matter Particle Physics (CDM) CE200100008.\nAJR acknowledges support from the Australian Government through the Australian Research Council, award FT170100243.\nRMC thanks Elly Berkhuijsen, Rainer Beck, Ron Ekers, Matt Roth, and Thomas Siegert for useful communications.\n\n\n\\section*{Author contributions statement}\n\nR.M.C. initiated the project and led the spectral analysis and theoretical interpretation. \nO.M constructed the astrophysical templates, designed the analysis pipeline, and performed the data analysis of $\\gamma$ ray observations.\nD.M., M.R.K., C.G., R.J.T., F.A., J.A.H., S.A., S.H., A.G., M.R., L.F., and A.R. provided theoretical insights and interpretation, and advice about statistical analysis.\nT.V. and A.R.D. provided insights on the expected distribution of dark matter.\nR-Z.Y. performed an initial {$\\gamma$-ray} \\ data analysis.\nM.D.F helped with radio data.\nThe main text was written by RMC, MRK, and O.M. and the Methods section was written by O.M., R.M.C., and MRK.\nAll authors were involved in the interpretation of the results and all reviewed the manuscript.\n\n\n\n\\section*{Additional information}\n\nTo include, in this order: \\textbf{Accession codes} (where applicable); . \n\n\\subsection*{Competing interests}\n\nThe authors declare no competing interests.\n\n\n\n\\clearpage\n\n\n\\section*{Extended Data}\n\\setcounter{figure}{0}\n\\setcounter{table}{0}\n\n\\renewcommand{\\figurename}{Extended Data Figure}\n\\renewcommand{\\tablename}{Extended Data Table}\n\n\n\\begin{figure*}[ht!]\n\\centering\n\\begin{tabular}{lll}\n\\includegraphics[width=0.33\\textwidth]{Sgr_Stream_VasilevBelokurov.pdf} & \\includegraphics[width=0.33\\textwidth]{Sgr_Stream_Strasbourg.pdf} & \\includegraphics[width=0.33\\textwidth]{Sgr_Stream_Iorio.pdf}\\\\\n\\includegraphics[width=0.33\\textwidth]{Sgr_Stream_BarcelonaI.pdf} & \\includegraphics[width=0.33\\textwidth]{Sgr_Stream_BarcelonaII.pdf} & \\\\\n\\end{tabular}\n\\caption{The stellar density templates for the Sgr dSph used in this study. Each map has been normalized, so the units are arbitrary; the color scale is logarithmic. Morphological differences among the templates are due to different stellar candidates (red clump or RR Lyrae), search algorithms, and search target (the dwarf remnant or the stream). \nData sources are as follows: \nModel I, ref.~\\cite{Vasiliev2020};\nModel II, ref.~\\cite{Ibataetal:2020};\nModel III, ref.~\\cite{Iorio2019};\nModel IV and Model V, ref.~\\cite{Ramosetal:2020}.\nDetailed descriptions of these templates are given in the S.I.~sec.~2.}\n\\label{fig:Stellartemplates}\n\\end{figure*}\n\n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[scale=0.3]{grid_MC_loglikes.pdf} \\\\\n\\caption{Goodness of fit computation for the best-fitting baseline + Sgr dSph model using our preferred set of templates (first entry in \\autoref{tab:loglikelihood}). In each of the 15 panels, one for each of the energy bins in our analysis pipeline, the blue histograms show the distribution of $-\\ln\\mathcal{L}$ values produced in 100 Monte Carlo trials where we use our pipeline to fit a mock data set produced by drawing photons from the same set of templates used in the fit; orange dashed vertical lines show the 68\\% confidence range of this distribution, and black dashed vertical lines show the mean. Under the null hypothesis that our best-fitting model for the real \\textit{Fermi} observations is a true representation of the data, and that disagreements between the model and the data are solely the result of photon counting statistics, the log-likelihood values for our best-fitting model should be drawn from the distributions shown by the blue histograms. For comparison, the red vertical line shows the actual measured log likelihoods for our best fit. The fact that these measured values are well within the range spanned by the Monte Carlo trials\nindicates that we cannot rule out the null hypothesis, indicating that our model is as good a fit to the data as could be expected given the finite number of photons that \\textit{Fermi} has observed.\n}\\label{fig:fitvalidation}\n\\end{figure}\n\n\n\\begin{figure*}[ht!]\n\\centering\n\\begin{tabular}{lll}\n\\includegraphics[width=0.32\\textwidth]{row_1a.pdf} & \\includegraphics[width=0.294\\textwidth]{row_1b.pdf} & \\includegraphics[width=0.294\\textwidth]{row_1c.pdf}\\\\\n\\includegraphics[width=0.32\\textwidth]{row_2a.pdf} & \\includegraphics[width=0.294\\textwidth]{row_2b.pdf} & \\includegraphics[width=0.294\\textwidth]{row_2c.pdf}\\\\\n\\includegraphics[width=0.32\\textwidth]{row_3a.pdf} & \\includegraphics[width=0.294\\textwidth]{row_3b.pdf} & \\includegraphics[width=0.294\\textwidth]{row_3c.pdf}\n\\end{tabular}\n\\caption{Measured photon counts (left), best-fit baseline + {Sgr~dSph~} model (middle), and the fractional residuals $(Data-Model)\/Model$ (right). The images were constructed by summing the corresponding energy bins over the energy ranges displayed on top of each panel: [0.5, 1.0] GeV, [1.0, 4.0] GeV, [4.0, 15.8] GeV, from top to bottom. The maps have been smoothed with Gaussian filters of radii $1.0^\\circ$, $0.8^\\circ$, and $0.5^\\circ$ for each energy range displayed, respectively\n(where these angular scales are determined by the {{\\em Fermi}-LAT} \\ point spread function at the low-edge of the energy interval for the former two, while the latter is determined by the angular resolution of the gas maps).\nThe spectrum of baseline + {Sgr~dSph~} model components shown here can be seen in ~\\autoref{fig:totalspectra}. The 4FGL~\\cite{Fermi-LAT:4FGL} {$\\gamma$-ray} \\ point sources included in the baseline model are represented by the red circles.\n}\n\\label{fig:Residuals}\n\\end{figure*}\n\n\\begin{figure}\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[scale=0.7]{Injection_mismodeling_GDE.pdf} &\\includegraphics[scale=0.7]{Injection_MC_StrucFB_Fit_FlatFB.pdf}\\\\\n\\includegraphics[scale=0.7]{RecoveredSpectra_GDE.pdf} & \\includegraphics[scale=0.7]{RecoveredSpectra_FBs.pdf} \n\\end{tabular}\\caption{\nResults from our template mismatch tests. Each of the coloured lines shows the results of a test where we generate synthetic data with one set of templates, and attempt to recover the Sgr dSph in those data using a different set. In the upper two panels, the horizontal axis shows the true, energy-integrated Sgr dSph photon flux in the synthetic data, while the vertical axis shows the value (with $1 \\sigma$ statistical error bars) retrieved by our pipeline; the black dashed lines indicate perfect recovery of the input, and the vertical bands show the photon flux we measure for the Sgr dSph in the real \\textit{Fermi} data. In the bottom two panels we plot the recovered energy flux in each energy bin (with $1 \\sigma$ statistical error bars), for the case where the injected photon flux most closely matches the real Sgr dSph flux; the black dashed line again shows perfect recovery of the injected signal. The left panels show experiments where we mismatch the Galactic hadronic and IC templates, while the right panels show experiments where we mismatch the FB templates; see Methods for details.\n}\\label{fig:injectionrecovery}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\begin{tabular}{lll}\n\\includegraphics[scale=0.65]{Results_rotation_around_center.pdf} & \\includegraphics[scale=0.65]{Results_rotation_around_GC.pdf} & \\includegraphics[scale=0.123]{translation_StructFBs.pdf}\n\\end{tabular}\\caption{\nResults of our rotation and translation tests. \\textit{Left:} change in TS when repeating the analysis using the default baseline + Sgr dSph model, but with the Sgr dSph rotated about its centre by the indicated angle (blue points); TS values $>0$ indicate an improved fit (dashed grey line), with $\\mbox{TS} = 46.1$ corresponding to a $5\\sigma$-significant improvement (red dashed line). \n\\textit{Centre:} same as the left panel, but for tests with the Sgr dSph template rotated about the Milky Way centre, rather than its own centre. \\textit{Right:} tests for translation of the Sgr dSph template. The true position of the Sgr dSph centre is the center of the plot, and the colour in each pixel indicates the change in TS if we displace the Sgr dSph centre to the indicated position; the maximum shown, at a displacement $\\Delta b \\approx -4^\\circ$, has $\\mbox{TS} = 40.8$, corresponding to $4.5\\sigma$ significance. For comparison, white contours show the original, unshifted Sgr dSph template, and the green arrow shows the direction anti-parallel to the Sgr dSph's proper motion, back along its past trajectory; red arrows show the projection of the green arrow in the $\\ell$ and $b$ directions.\n}\\label{fig:rotationAndTranslationTests} \n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=1]{Sgr_Sph_StructuredFBs_spectra_systematics.pdf}\\\\\n\\caption{{Sgr~dSph~} spectra derived from template analysis using different Galactic diffuse emission models; in all cases the spectrum shown is the flux averaged over the entire ROI, not the flux within the footprint of the Sgr dSph template. The fiducial model is our default choice (first entry in Table~\\ref{tab:loglikelihood}), while other lines correspond to alternate foregrounds -- models 2D A (red), 2D B (black), and 2D C (blue) for the Galactic IC foreground, and models Interpolated (dark green) and GALPROP 3D-gas (light green) for the Galactic hadronic + bremsstrahlung foreground. The error bars display $1\\sigma$ statistical errors. See Table~\\ref{tab:loglikelihood} and text for details.}\n\\label{fig:SgrSpecVar}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=1]{Total_measured_spectra_RoI.pdf}\\\\ \n\\caption{\nContribution of each template component to the $\\gamma$-ray spectrum averaged over the entire ROI, for our default baseline + {Sgr~dSph~} model. Components shown are as follows: $\\pi^0+\\mbox{brems}$ is the Galactic hadronic plus bremsstrahlung foreground, ICS is the Galactic inverse Compton foreground, 4FGL indicates point sources from the 4th \\textit{Fermi} catalogue, Fermi Bubbles indicates the structured Fermi Bubble template, isotropic is the isotropic $\\gamma$-ray background, ``other'' includes the Sun and Moon, Loop I, and the Galactic Centre Excess, and Sgr stream indicates the Sgr dSph.\nThe error bars display $1\\sigma$ statistical errors.\n}\n\\label{fig:totalspectra}\n\\end{figure}\n\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.5\\linewidth]{plotSpectralFitContours.pdf}\n\\caption{ \nFilled contours indicate the best-fit region for the spectral parameters $E_{\\rm cut,prompt}$ and $\\alpha$\nthat determine the shape of the magnetospheric emission from the Sgr dSph; the outer, coloured region shows the 2$\\sigma$ region, the inner shows the 1$\\sigma$ region, and the red point marks the best fit.\nThe dotted and dashed contours describe the 1, 2, and 3 $\\sigma$ confidence regions measured in ref~\\cite{Song2021} for globular clusters (GCs) and individual resolved MSPs, respectively, constructed from the observations as described in Methods.\n}\n\\label{fig:FitContours}\n\\end{figure}\n\n\\begin{figure}\n \\includegraphics[width=1.\\linewidth]{plotLgammaOvrMstarVstSimple}\n \\caption{\n{\\bf Data:} $\\gamma$-ray luminosity per stellar mass for a number of stellar systems (cf.~fig.~3 main text) versus mean stellar age of those systems. The mean stellar ages have been determined from empirically-determined star formation histories for all these objects (data sources as follows: {Sgr~dSph~} \\cite{Weisz2014}, M31 \\cite{Williams2017}, Galactic Bulge \\cite{Bernard2018}, NB \\cite{Nogueras-Lara2020} and ref.~\\cite{Weisz2013} for the LMC). The globular cluster datum (`GCs') is plotted at the mean measured {$\\gamma$-ray} \\ luminosity for the 27 systems analysed in ref.~\\cite{Song2021} divided by their stellar masses, and the age is the luminosity-weighted mean age for the 31 systems analysed in ref.~\\cite{Wu2022} (while the error bars for this datum show the standard deviations of these measurements for each population). The purple datum shows the secondary electron plus positron luminosity of the Milky Way (`disk $e^\\pm$') as inferred in ref.~\\cite{Strong2010} and adopting a disk stellar mass of $5.2 \\times 10^{10} \\ M_{\\odot}$\\cite{Bland_Hawthorn_2016}. {\\bf Model curve:} The solid blue curve shows the evolution with time (since the initial, burst-like star formation event) of the total spin-down power generated by a population of MSPs (normalised to the stellar mass expected to host that same population) according to the recent binary population synthesis modelling presented in ref.~\\cite{Gautam2021} (with the blue band indicating the estimated the $\\pm 1 \\sigma$ error on this quantity dominated by the uncertainties in the overall stellar binarity fraction). The {\\bf dashed red line} is an approximate fit to the solid blue line described by $5.0 \\times 10^{28}$ $ \\exp(-t\/t_{\\rm decay})$ \\ erg$\/s\/M_{\\odot}$ with $t_{\\rm decay} = 3$ Gyr. The dashed blue curve shows 10\\% of the mass-normalised spin-down power (with the error band suppressed for clarity). The {\\bf brown, dashed, horizontal line} shows the total power (per unit stellar mass) from MSP spin-down we infer from the study by Sudoh et al.~\\cite{Sudoh2020} of radio continuum emission from massive, quiescent galaxies (with expected mean stellar ages $>8-10$ Gyr; see the main text for more details).\n}\n\\label{fig:plotLgammaOvrMstarVstSimple}\n\\end{figure}\n\n\\begin{table}[ht!]\n \\centering\n \\begin{tabular}{ccccc}\n \\hline\n quantity & best-fit & 68\\% c.l. & units & literature \\\\\n &&&& value(s) \\\\\n \\hline\n $l_0 \\equiv L_{\\rm \\gamma,tot}\/M_\\star$ & $5.2 $ & $[4.4 , 6.0]$ & $10^{28}$ erg\/s\/$M_{\\odot}$ \n & $\\sim (1-10)$ \\cite{Sudoh2020} \\\\\n $f = L_{\\rm \\gamma,prompt}\/L_{\\rm \\gamma,IC}$ & $0.83 $ & $[0.59, 1.3]$ & --- & $\\sim 0.1$\\cite{Sudoh2020} \\\\\n $\\alpha$ & $0.039 $ & $[-0.38,0.62]$ & --- & $-0.88 \\pm 0.44$\\cite{Song2021} \\\\\n $E_{\\rm cut, prompt}$ & $1.0 $ & $[0.74, 1.3]$ & GeV & \n $1.91^{+0.85}_{-0.59} \\pm 0.44$ \\cite{Song2021} \\\\\n \\hline\n \\end{tabular}\n \\caption{Best fit spectral parameters with $\\pm 1\\sigma$ confidence regions as determined from $\\chi^2$ fitting to the measured {$\\gamma$-ray} \\ spectrum of the Sgr dSph. \n %\n The parameter $l_0$ is calculated using a stellar mass\n $M_\\star = 10^8 M_{\\odot}$ \\cite{Vasiliev2021} for the Sgr dSph.\n %\n See also E.D. \\autoref{fig:FitContours}\n %\n }\n \\label{tab:table1}\n\\end{table}\n\n\n\n\n\\clearpage\n\n\n\\printbibliography[segment=\\therefsegment,title={Methods and Extended Data References}, check=onlynew]\n\n\\clearpage\n\n\\newrefsegment\n\n\\renewcommand{\\figurename}{Supplementary Information Figure}\n\\renewcommand{\\tablename}{Supplementary Information Table}\n\n\\setcounter{figure}{0}\n\\setcounter{table}{0}\n\n\\section*{Supplementary Information}\n\n\\section{Chance overlap calculation}\n\\label{sec:overlap}\n\nIn the main text we estimate the probability of a chance overlap between cocoon {$\\gamma$-ray} \\ structure and {Sgr~dSph~} to be $\\approx 1\\%$.\nThis follows simply from noting that the solid angle of the Bubbles is around 0.7 sr\\cite{Ackermann2014} and the cocoon covers $\\lesssim$20\\% of this solid angle, so the chance probability for an overlap if these objects were placed randomly on the sky is $\\lesssim 0.2 \\times 0.7\/(4 \\pi) \\sim 0.012$. However, this is a generous upper limit; it does not take into account that, as revealed by the template analysis, there is a much more detailed correspondence between the {$\\gamma$-ray} \\ substructure and the stellar distribution not accounted for here. Moreover, the naive 1\\% estimate does include a `look-elsewhere' correction: the Milky Way is surrounded by satellite galaxies and there are apparently other regions of sub-structure within the Fermi Bubbles.\nHowever, not only is the cocoon the brightest and first-discovered region of sub-structure \\cite{Su2012}, it is also the only region that has been reliably detected by independent analyses \\cite{Selig2015,Ackermann2014}, and is visibly-evident in \nindependently-produced\n{$\\gamma$-ray}\\ maps \\cite{Yang2014,deBoer2015}.\nThe {Sgr~dSph~} is also a special object: it is the\nbrightest MW satellite not yet (prior to this work) detected in $\\gamma$-rays. \n(In fact, not only is the {Sgr~dSph~} the brightest satellite undiscovered in $\\gamma$-rays, it is substantially brighter than the next brightest galaxy\\footnote{The list of all the MW satellites with apparent magnitude $m<10$ includes 8 objects,\nthe brightest two, the LMC and SMC, with $m \\sim 0.3$ and $\\sim 2.1$, respectively, are already detected in $\\gamma$-rays.\nThe next brightest is the {Sgr~dSph~} with $m \\sim 3$; after that come Fornax, Sculptor, and Leo I with $m \\sim 7.3, 8.7$ and 10.0\nand angular diameters of $0.24^\\circ, 0.51^\\circ$ and $0.11^\\circ$, respectively, which, even assuming they could be detected, would at best only appear marginally extended to {{\\em Fermi}-LAT}.}.)\nOverall, we have a spatial overlap\n(and detailed morphological correspondence as argued elsewhere) between \nthe brightest region of substructure within the {\\it Fermi} Bubbles\nand the Sgr dSph, the\nsecond closest, third-most massive, third brightest, and third most angularly extended satellite galaxy of the MW.\n\n\\begin{comment}\n\\section{Full description of templates}\n\\label{sec:templates}\n\n\n\\begin{table*}[!htbp]\\caption{{\\bf List of (baseline) spatial templates considered in our maximum likelihood runs. \\label{Tab:templates}}}\n\\begin{adjustbox}{width=1.0\\textwidth, center}\n\\centering\\begin{threeparttable}\n \\scriptsize\n\\begin{tabular}{llr}\n\\hline\\hline\nTemplates & Summary description & Reference\\\\\\hline \nHadronic and Bremsstrahlung & Three alternative models: (i) 3D templates predicted by \\texttt{GALPROP v56} (``{\\bf GALPROP}'')\n & \\\\\n$\\gamma$ rays & (ii) hydrodynamical gas (``{\\bf HD}''), and (iii) interpolated gas templates (``{\\bf Interpolated}''). & \\\\\n & These consist of H{\\tiny I}, H\\boldmath$_{2}$, and dust correction column density maps. & \\\\\n& The H{\\tiny I} and H\\boldmath$_{2}$ maps are divided in four rings each. In the case of the dust maps,\\\\\n& we use two total residual maps with different $E(B-V)$ magnitude cuts. & ~\\cite{Macias2018, Macias2019}\\\\\n &&\\\\\nInverse Compton & Used various alternatives models: (i) three kinds of 2D ICS maps; a standard one (``{\\bf 2D A}''), & \\\\\n$\\gamma$ rays & another that assumes spatially variable diffusion (``{\\bf 2D B}''), and one including a central source &\\\\\n& of electrons (``{\\bf 2D C}''), (ii) a 3D ICS map$^\\dagger$ divided in four rings (``{\\bf 3D}'') & \\cite{Porter:2017vaa,Ackermann:2014usa}\\\\\n&&\\\\\n\\textit{Fermi} bubbles& (i) Flat\/unstructured FBs template (``{\\bf U}''), and (ii) structured FBs template (``{\\bf S}'') &~\\cite{Macias2019}\\\\\n&&\\\\\nLoop I& Analytical model & \\cite{Wolleben:2007}\\\\\nGalactic centre excess& Stellar distribution model based on Freudenreich 1998 (F98) & \\cite{Freudenreich1998}\\\\\nPoint sources & 4FGL catalogue of $\\gamma$-ray point sources \n({\\it gll\\textunderscore psc\\textunderscore v20.fit})& \\cite{Fermi-LAT:4FGL}\\\\\nSun and Moon & Models constructed in the 4FGL catalog&\\cite{Fermi-LAT:4FGL} \\\\\nIsotropic emission& \\texttt{iso$_{-}$P8R3$_{-}$ULTRAC.L.EANVETO$_{-}$V2$_{-}$v1.txt}&\\\\\n\\hline\\hline\n\\end{tabular}\n\\begin{tablenotes}\n\\item The interstellar gas maps are divided in four rings of sizes ($0-3.5$, $3.5-8.0$, $8.0-10.0$, and $10.0-50.0$ kpc). The 3D ICS map are divided in rings of the same size as the gas maps. The 2D ICS maps correspond to ICS (Model A), ICS (Model B), and ICS (Model B) introduced in Ref.~\\cite{Ackermann:2014usa}. The baseline model considered in this work includes: (a) the hydrodynamical gas maps divided in rings, (b) the 3D ICS maps divided divided in rings, (c) the structured FBs template, (d) Loop I, (e) the F98 stellar template, (f) tailor-made maps for the Sun and the Moon, (g) an isotropic emission template, and (h) the 4FGL point sources. Note that in our bin-by-bin analysis procedure, only the normalisation of each template is varied in the fit. Since the energy bins are small, the fit results are independent of the assumed template spectra. \\end{tablenotes}\n\\end{threeparttable}\n\\end{adjustbox}\n\\end{table*}\n\n\n\n\n\nHere we provide full details for how we construct all the templates that we use in our analysis.\n\n\\subsection{Hadronic plus bremsstrahlung models}\n\\label{sec:GDEdescrip}\n\nAs discussed in Methods, the dominant source of {$\\gamma$-ray} s from the ROI is hadronic and bremsstrahlung emission resulting from the interaction of CR protons and electrons with interstellar gas.\nWe model this component with three alternative templates: HD (hydrodynamic), Interpolated, and 3D (GALPROP); \nof these, the hydrodynamical template provides the best fit to the inner Galaxy {$\\gamma$-ray} \\ sky and is used in our baseline analysis.\n\nIn detail, the models we investigated for the gas-correlated {$\\gamma$-ray} \\ emission are as follows:\n\\begin{itemize}\n \\item {\\bf Interpolated gas maps}: these assume that the hadronic and bremsstrahlung components can be phenomenologically modelled with a linear combination of atomic hydrogen, molecular hydrogen, and dust residual maps. These are divided in four Galactocentric rings (four rings of H$_{\\rm I}$, four rings of H$_{2}$, and two residual dust maps---see also Table~\\ref{Tab:templates}) to account for potential uncertainties in the CR densities. The method used to create the interpolated gas maps is given in Appendix B of Ref.~\\cite{Ackermann2012}, which we have faithfully reproduced in~\\cite{Macias2018}. \n %\n The main objective of this method is to estimate the gas column density in the direction of the inner Galaxy\n where the non-axisymmetric gravitational potential of the Galactic bar induces non-circular gas orbits.\n %\n Such interpolated gas maps are the standard interstellar gas distribution models employed in most studies by the Fermi team.\n \n \\item {\\bf Hydrodynamical gas maps}: these were constructed using a suite of hydrodynamical simulations of interstellar gas flow~\\cite{Pohl2008} that generate physically-motivated solutions for the gas kinematics in the direction of the inner Galaxy. \n The hydrodynamical gas maps provide a much better fit to the Galactic centre data than the interpolated gas maps\\cite{Macias2018}. Nevertheless, the main purpose of using alternative gas models in our study is evaluating the impact that these have in the inferred properties of the Sgr dSph. We note that the hydrodynamical maps are divided in the same ring scheme (four rings of H$_{\\rm I}$, four rings of H$_{2}$, and two residual dust maps) as the interpolated gas maps.\n \n \n \\item {\\bf 3D gas (GALPROP)}: to generate alternative templates for the hadronic and bremsstrahlung gamma-ray emission, we reproduced one of the models proposed in Ref.~\\cite{Johannesson:2018bit} (Model SA50 in Table 5 of that reference) using \\texttt{GALPROP V56}~\\cite{Porter:2017vaa,Johannesson:2018bit}. The latest release of this software contains new 3D spatial density models of atomic and molecular hydrogen. These include the effects of several Galactic structures such as the spiral arms, and the Galactic disk. Note that in this case, we do not divide the resulting hadronic and bremsstrahlung maps in rings so that we are able to explore the effects that a less flexible Galactic diffuse emission model has in our results. \n\\end{itemize}\n\nFor the IC background component, we again generated and tested alternative maps, in particular, 2D and 3D variants.\nUncertainties in the IC component become more important in the high latitude regions \\cite{Ackermann:2014usa}. \n\nIn detail, models we investigated for diffuse IC emission are:\n\\begin{itemize}\n \\item {\\bf 3D IC} maps divided in four rings: for these we utilized \\texttt{GALPROP~V56}, the propagation parameter setup SA50 (see Table 5 in Ref.~\\cite{Johannesson:2018bit}), and the same ring subdivisions as those of the interstellar gas maps (see Table~\\ref{Tab:templates}). The advantage of the new 3D IC maps (over the ones constructed with previous versions of the code), is that it now incorporates fully 3D models for the interstellar radiation fields (ISRF)~\\cite{Porter:2017vaa}; hence avoiding potential biases introduced by the previously implicitly-assumed Galactocentric symmetry. Also, since the 3D IC maps can be divided in rings, we are able to reduce the impact of modelling assumptions such as the characteristics of the electron injection spectrum, and the normalisation of the ISRF.\n \n \\item {\\bf 2D IC} maps: we used the three different IC maps constructed in Ref.~\\cite{Ackermann:2014usa} (Model A, B, and C). These were computed with an older version of the CR propagation code (\\texttt{GALPROP V54}), assume Galactocentric symmetry of the CR halo, and are monolithic (i.e., not divided in rings). These models encapsulate a wide range of uncertainties in the CR source distribution, CR injection spectra, the diffusion coefficient, Galactic magnetic fields, and a central source of electrons.\n\\end{itemize}\n\\end{comment}\n\n\n\\section{Construction of the Sgr dSph templates}\n\\label{sec:stellarmapsDetails}\n\nHere we provide detailed descriptions of \nhow we construct the Sgr dSph templates shown in E.D.~\\autoref{fig:Stellartemplates}.\n\n\\subsubsection*{Model I:} \n\nWe extract this template from the stellar catalogue constructed in Ref.~\\cite{Vasiliev2020}, which was derived using photometric and astrometric data from {\\it Gaia} Data Release 2 (DR2), and kinematic measurements from various other surveys. The catalogue consists of a list of $2.26\\times 10^5$ candidate member stars of the {Sgr~dSph~} remnant, which are reliably separated from the field stars. Every object in the catalogue has an extinction-corrected G-band magnitude larger than 18, and more than half of the objects in this catalogue are classified as red clump stars. Note that Ref.~\\cite{Vasiliev2020} adapted their procedure to reproduce the observed properties of the {Sgr~dSph~} remnant, not the stream, which is why the first panel of E.D.~\\autoref{fig:Stellartemplates} only shows the remnant. We show profiles of stellar number count along the long and short axes of the dwarf for this template in S.I.~\\autoref{fig:profile_longaxis}. \n\n\\begin{figure*}[t!]\n \\centering\n \\includegraphics[scale=0.15]{Long_axis_star_profile.pdf}\n \\includegraphics[scale=0.15]{Short_axis_star_profile.pdf}\n \\caption{Star count profiles for Model I, showing number of stars measured in bins of angular distance from the gravitational centre of the dwarf along its long (top) and short (bottom) axes. In the inset images (which are identical to the first panel of \\autoref{fig:Stellartemplates}), we mark the gravitational centre of the dwarf with a cyan circle, and show the long and short axes along which we measure the profiles as white bands.}\n \\label{fig:profile_longaxis}\n\\end{figure*}\n\n\n\\subsubsection*{Model II:}\n\nOur second template comes from ref.~\\cite{Ibataetal:2020}. Instead of red clump stars, this study selected a sample of RR Lyrae stars from {\\it Gaia} DR2 data, for which distances are accurately measured. Also, rather than focusing on member stars of the {Sgr~dSph~} remnant, Ref.~\\cite{Ibataetal:2020} used the \\textsc{streamfinder} algorithm to single out stars with high probability of belonging to the Sagittarius Stream. By using the kinematic properties of the stars in that study, we constructed a template containing 2369 RR Lyrae stars (cf. \\autoref{fig:Stellartemplates}) in our ROI. Note that the stellar number count in this map is approximately two orders of magnitude smaller than that in Model I. \n\n\\subsubsection*{Model III:}\n\nRef.~\\cite{Iorio2019} performed an all-sky analysis of RR Lyrae stars (in {\\it Gaia} DR2 data) belonging to globular clusters, dwarf spheroidal galaxies, streams, and the Magellanic Clouds. Our Model III template is a subset of their data identified as belonging to the Sgr dSph, selected to reproduce their Fig.~1 (bottom-right). It includes $1.31\\times 10^4$ RR Lyrae stars in our ROI. \n\n\\subsubsection*{Model IV and Model V:}\n\nRef.~\\cite{Ramosetal:2020} developed two empirical catalogues of RR Lyrae stars in {\\it Gaia} DR2 data, which form the basis for our final two templates. The first (Model IV), corresponds to the nGC3 sample, which is characterized for its lower-completeness and higher-purity. This template contains 675 stars in our ROI. The second (Model V), is the Strip sample, containing higher-completeness, but lower purity. The total number of stars in our ROI for this model is 4812.\n\n\n\n\n\n\n\\section{Validation tests}\n\\label{ssec:validation}\n\n\nWhile our template analysis indicates a strong statistical preference for emission tracing the Sgr dSph, we also carry out five further validation tests to check the robustness of the result.\n\nFirst, we check whether the residuals between the baseline + {Sgr~dSph~} source model and the {\\it Fermi} data from our ROI are consistent with the level expected simply as a result of photon counting statistics, using a method similar to that of Ref.~\\cite{Buschmann2020}. Under the null hypothesis that the {\\it Fermi} data are a Poisson draw from our best-fit baseline + {Sgr~dSph~} model (i.e., that our model is correct, and any differences between it and the actual data are simply due to shot noise), we can determine the expected distribution of $\\ln\\mathcal{L}_n$ values via Monte Carlo. For each Monte Carlo trial, we draw a set of mock photon counts $\\Phi_{n,i,\\rm mock}$ in each pixel and energy bin from our best-fitting model (multiplied by the instrument response function), and then compute the energy-dependent log-likelihood for this mock data set using the same pipeline we use on the real data. We repeat this procedure 100 times, and plot the distribution of log-likelihood values it produces \nas the blue histograms\nin E.D. \\autoref{fig:fitvalidation}. These\nhistograms represent the expected log likelihood in each energy bin under the null hypothesis. We then compare this to the actual value of $\\ln\\mathcal{L}_n$ we measure for our model as compared to the real \\textit{Fermi} data. The plot shows that our measured log-likelihood falls squarely within the range expected under the null hypothesis, and we therefore conclude that the residuals between our model and the real data are consistent with being solely the result of photon counting statistics. \n\nIn addition to testing whether the residuals between model and data are consistent with simply being shot noise when we sum over all pixels (which is what the likelihood measures), we can also examine the residuals as a function of position. We do so in E.D.~\\autoref{fig:Residuals}, which shows the measured \\textit{Fermi} counts in our ROI (summed in three energy bins) in the first column, our best-fitting baseline + {Sgr~dSph~} model in the second column, and fractional residuals [$(\\rm{Data}-\\rm{Model})\/\\rm{Model}$] in the third column. The images are smoothed with a $0.5^\\circ$ Gaussian kernel, since this is roughly the resolution of our interstellar gas maps~\\cite{Macias2018,Macias2019}. The plot shows that, on a point-by-point basis, our models reproduce the data within $\\sim 10\\%$ over most of the ROI. \nThere are, however, a few small patches of correlated residuals, which are only at the $\\sim 30\\%$ level, and are far from the Sgr dSph region. \nThis points to the existence of real structure in the Fermi Bubbles that is not yet perfectly modelled, but given the small level of the residuals and the distance between them and the signal in which we are interested, this modelling imperfection has little impact on our results.\n\n\n\n\n\nAs our second validation test, we evaluate the sensitivity of our pipeline to uncertainties in our templates for Galactic diffuse emission, and we verify that our pipeline can recover synthetic signals similar to the Sgr dSph even when our templates are imperfect. Recall that we have three components of Galactic diffuse emission for which the templates are at least somewhat uncertain: hadronic + bremsstrahlung emission (for which our template can be HD, Interpolated, or GALPROP), Galactic IC emission (for which the template can be 3D, 2D A, 2D B, or 2D C), and the Fermi Bubbles (for which the template can be S, structured, or U, unstructured). \nWe test the sensitivity of our fits to these template choices as follows. First, we generate a set of mock background data by drawing a random realisation of photon counts from one combination of these templates, and on top of this we add a synthetic Sgr dSph signal; the Sgr dSph photons follow the spatial morphology of our Sgr dSph model I template, have a spectral shape $dN_\\gamma\/dE_\\gamma \\propto E_\\gamma^{-2}$, and have a normalisation that we vary systematically from $\\approx 10^{-11}$ ph cm$^{-2}$ s$^{-1}$ (integrated over all energies) to $\\approx 10^{-5}$ ph cm$^{-2}$ s$^{-1}$; our best-fit Sgr dSph photon flux falls in the middle of this range, $\\approx 2\\times 10^{-8}$ ph cm$^{-2}$ s$^{-1}$. Then we use our pipeline to recover the flux of the Sgr dSph from the synthetic map, but using a \\textit{different} set of templates for Galactic diffuse emission to the ones used to generate the synthetic data. Comparing the recovered Sgr dSph spectrum to the injected one reveals how well our pipeline performs when the input diffuse emission templates are not exactly correct. We carry out this experiment with four diffuse emission template combinations: (1) synthetic data generated from GALPROP + 3D + S, analysed using HD + 3D + S; (2) synthetic data generated from HD + 2D A + S, analysed using HD + 3D + S; (3) synthetic data generated from HD + 3D + S, analysed using HD + 3D + U; (4) synthetic data generated using HD + 3D + S, analysed using HD + 3D but no template for the FBs at all.\n\nWe show the results for the first two of these experiments in the two left panels of E.~D.~\\autoref{fig:injectionrecovery}; the top left panel shows the recovered energy-integrated photon flux compared to the injected flux, while the bottom left shows the recovered spectra when the input flux is $\\approx 2 \\times 10^{-8}$ ph cm$^{-2}$ s$^{-1}$. The plot shows that our pipeline yields excellent agreement between the injected and recovered signals for both the integrated flux and the spectrum unless the Sgr dSph signal is $\\sim 1$ order of magnitude weaker than our estimate. In no circumstance does our pipeline produce a false signal comparable in magnitude to our observed one. The two right panels of E.~D.~\\autoref{fig:injectionrecovery} show the third and fourth tests, where we mismatch the FB template. Here the effects are somewhat larger, but still relatively minor: if we create synthetic data with the S Fermi Bubble template (so that there is structure corresponding to the cocoon), and then analyse it using either the U template or no FB template at all, then we make a factor of $\\sim 2-3$ level error in the absolute flux, but no substantial error in the spectral shape. This test suggests that our detection of the Sgr dSph is very robust, but that we have a factor of $\\sim 2-3$ uncertainty in its absolute flux, stemming from our imperfect knowledge of the foreground FBs.\n\n\n\n\nThe third validation test we perform is to check whether a fit using the observed stellar distribution of the Sgr dSph as a template performs better than one using a purely geometric template placed at the same position; if the emission really is tracing the stars of the dwarf, and is not merely a chance overlap, a template matching the shape of the dwarf should perform better than a purely geometric distribution. For this purpose we consider disc-shaped templates of varying radii, centred at Galactic coordinates $(\\ell,b)=(5.61^\\circ, -14.09^\\circ)$ --- the dynamical centre of the Sgr dSph -- and repeat our standard procedure of comparing baseline models to baseline + Sgr dSph models, using these geometric templates in place of the Sgr dSph stellar templates. We use our fiducial choices for all other templates (hadronic and bremmstrahlung emission, galactic IC emission, and the Fermi Bubbles).\n\nWe show the results of this experiment in S.I.~\\autoref{tab:geometric_templates}. We find that geometric templates do perform better than baseline models with no Sgr dSph component, but, as expected, even the best geometric template (for a disc of radius $r=2.0^\\circ$) yields significantly less fit improvement ($TS=63.8$) than our fiducial stellar template ($TS=95.2$); this difference in test statistic, $\\Delta\\, TS = 31.4$, corresponds to the Sgr dSph template being preferred at $3.7\\sigma$ significance. Moreover, this result becomes even stronger if we notice two additional points. First, because we tried a wide range of radii for the geometric models, the geometric templates effectively provide an extra degree of freedom that the Sgr dSph template, which is fixed by observations, lacks. Because we fix the template radius while performing each fit, we do not treat the varying radius as an extra degree of freedom when computing the test statistic, but if we did so, then the difference in performance between the geometric and stellar templates would be even larger. Second, the geometric model that gives the best fit to the data is in fact the one whose radius most closely approximates the actual size of the core of the Sgr dSph. Indeed, Fig.~\\ref{fig:profile_longaxis} (bottom) shows that, in the direction of the short axis, the Sgr dSph stellar profile falls off steeply $\\sim 2^\\circ-3^\\circ$ away from the Sgr dSph centre. Thus the geometric template that gives the best match to the observations happens to be the one that most closely approximates the actual distribution of stars in the Sgr dSph.\n\n\\begin{table}[h!]\n \\centering\n \\small\n \\begin{tabular}{llll@{\\qquad\\qquad}rrrr}\n \\hline\\hline\n \\multicolumn{4}{c}{Template choices} & \\multicolumn{4}{c}{Results} \\\\\n Hadr. \/ Bremss. & IC & FB & Sgr dSph &\n $-\\log(\\mathcal{L}_{\\rm Base})$ & $-\\log(\\mathcal{L}_{{\\rm Base}+{\\rm Sgr}})$ & $\\mbox{TS}_{\\rm Source}$& Significance \\\\[0.5ex] \\hline \n \\multicolumn{8}{c}{Default model} \\\\[0.5ex]\n HD & 3D & S & Model I & 866680.6 &866633.0 & 95.2 & $8.1\\;\\sigma$\n \\\\[0.5ex]\n HD & 3D & S & Disc ($r=0.5^\\circ$) & 866680.6 & 866666.1 & 28.9 & $3.5\\;\\sigma$\n \\\\[0.5ex]\n HD & 3D & S & Disc ($r=1.0^\\circ$) & 866680.6 & 866661.3 & 38.6 & $4.4\\;\\sigma$\n \\\\[0.5ex]\n HD & 3D & S & Disc ($r=2.0^\\circ$) & 866680.6 & 866648.7 & 63.8 & $6.3\\;\\sigma$\n \\\\[0.5ex]\n HD & 3D & S & Disc ($r=3.0^\\circ$) & 866680.6 & 866654.9 & 51.4 & $5.4\\;\\sigma$\n \\\\[0.5ex]\n HD & 3D & S & Disc ($r=4.0^\\circ$) & 866680.6 & 866658.1 & 45.0 & $4.9\\;\\sigma$\n \\\\[0.5ex]\n HD & 3D & S & Disc ($r=5.0^\\circ$) & 866680.6 & 866661.3 & 38.6 & $4.4\\;\\sigma$\n \\\\[0.5ex]\n HD & 3D & S & Disc ($r=6.0^\\circ$) & 866680.6 & 866669.3 & 22.7 & $2.8\\;\\sigma$\n \\\\[0.5ex]\n HD & 3D & S & Disc ($r=7.0^\\circ$) & 866680.6 & 866670.4 & 20.4 & $2.6\\;\\sigma$\n \\\\[0.5ex]\n \n HD & 3D & S & Disc ($r=9.0^\\circ$) & 866680.6 & 866664.9 & 31.4 & $3.7\\;\\sigma$ \\\\[0.5ex]\n HD & 3D & S & Disc ($r=11.0^\\circ$) & 866680.6 & 866665.8 & 29.6 & $3.6\\;\\sigma$ \\\\[0.5ex]\n HD & 3D & S & Disc ($r=13.0^\\circ$) & 866680.6 & 866673.0 & 15.2 & $1.9\\;\\sigma$ \\\\[0.5ex]\n HD & 3D & S & Disc ($r=15.0^\\circ$) & 866680.6 & 866676.5 & 8.3 & $0.9\\;\\sigma$ \\\\[0.5ex]\n \\hline\\hline\n \\end{tabular}\n \\caption{Same as \\autoref{tab:loglikelihood} in the main letter, except that here we compare the results for our fiducial stellar template for the Sgr dSph (Model I, top row) to results using disc templates of various angular radii centred at the dynamical centre of the Sgr dSph. }\n \\label{tab:geometric_templates}\n\\end{table}\n\n\nOur \nfourth\nvalidation test is to check whether our fit degrades if we artificially rotate or translate the Sgr dSph template; if the signal we are detecting really does come from the Sgr dSph, the best fit should be for a template that traces its actual orientation and position, while rotated or shifted templates should produce progressively worse fits. This check is significant in part because Ref.~\\cite{Ackermann2014} performed similar rotation analysis for the hypothesis that the cocoon is tracing a jet from Sgr A$^*$, and found that there was no preference for a jet oriented toward Sgr A$^*$ over one oriented in some other way; they took this as evidence against the jet hypothesis. To check if the Sgr dSph template performs better on this test, we first rerun our analysis pipeline for our default set of templates (first line in \\autoref{tab:loglikelihood}), but with the Sgr dSph template rotated about its core. For each rotation angle we compute the TS, and compare to the TS of the original, unrotated model. We plot the result of this experiment in the left panel of E.D.~\\autoref{fig:rotationAndTranslationTests}. It is clear that, as expected, the fit is best when we use the actual orientation of the Sgr dSph, and degrades as we increase the rotation. Next, we carry out a similar procedure, but this time rather than rotating the Sgr dSph template about its core, we rotate around the centre of the Galaxy, thereby both translating and rotating the template. (This latter test was motivated by the particular alignment of the Sgr Stream with the previously claimed collimated jets from the Galaxy's supermassive black hole~\\cite{Su2012}.) We show the results in the middle panel of E.D.~\\autoref{fig:rotationAndTranslationTests}, and, again as expected, the TS strongly favours the true location and orientation of the Sgr dSph. Finally, we translate the Sgr dSph while leaving its orientation unchanged. We show the TS for displaced Sgr dSph in the right panel of E.D.~\\autoref{fig:rotationAndTranslationTests}. In this case the fit improves if we do displace the Sgr dSph from its true position by $\\approx 4^\\circ$ south. The amount by which the shift is favoured is fairly significant -- the TS improved by 40.8, which corresponds to $4.5\\sigma$ significance. Interestingly, the direction of the displacement is within a few degrees of the direction anti-parallel to the Sgr dSph's proper motion, suggesting that the dwarf's $\\gamma$-ray signal trails it slightly on its orbit.\nIf IC-emitting CR {$e^\\pm$} \\ are largely responsible for the observed {Sgr~dSph~} {$\\gamma$-ray} \\ signal as suggested by our spectral modelling, a systematic displacement of this signal southward by $\\sim 4^\\circ$ from the stars of {Sgr~dSph~} is quite reasonable as we have explained elsewhere (and see \\autoref{sec:CRtrans}).\n\n\n\n\n\n\n\\section{Transport of IC-emitting CR $e^\\pm$}\n\\label{sec:CRtrans}\n\nWe have seen that, while our pipeline detects a signal from the Sgr dSph at very high statistical significance, the fit improves even more (by $\\approx 4.5\\sigma$) if we displace the Sgr dSph template $\\approx 4^\\circ$ from its actual position (corresponding to 1.9 kpc at the distance of the Sgr dSph), in a direction very close to anti-parallel to the dwarf's proper motion. Here we demonstrate that a displacement of this type is expected in a model where the $\\gamma$-ray signal from the Sgr dSph is powered by MSPs. Part of the MSP signal emerges directly from the MSP magnetospheres, and thus traces the stellar component of the Sgr dSph. However, the majority of the observed signal is, in our model, IC emission powered by {$e^\\pm$}~escaping MSP magnetospheres and interacting with the CMB. The time between when {$e^\\pm$}~leave MSPs and when they IC scatter to produce $\\gamma$-ray photons is non-negligible: the CMB is dominated by photons with energies $\\sim k_{\\rm B}T_{\\rm CMB}$ (with $T_{\\rm CMB}=2.7$ K), so IC photons with energies of $\\sim 1-100$ GeV must be produced by {$e^\\pm$}~with energies $E_{e^\\pm} \\sim 0.6-6$ TeV. The characteristic IC loss time for such particles is\n\\begin{equation}\n t_{\\rm IC} = \\frac{3 m_e^2 c^3}{4 \\sigma_{\\rm T} E_{e^\\pm} U_{\\rm CMB}} = 1.2\\left(\\frac{E_{e^\\pm}}{\\mbox{TeV}}\\right)^{-1}\\mbox{ Myr},\n\\end{equation}\nwhere $m_e$ is the electron mass, $c$ is the speed of light, $\\sigma_{\\rm T}$ is the Thomson cross section, and $U_{\\rm CMB} = a_R T_{\\rm CMB}^4 = 0.25$ eV cm$^{-3}$ is the energy density of the CMB.\n\nDuring this time, the {$e^\\pm$}~will have the opportunity to move a significant distance prior to producing $\\gamma$-rays, due to both bulk gas motion and CR flow relative to the gas. \nWith regard to bulk advection, we note that the proper speed of the {Sgr~dSph~} is $\\approx 260$ km s$^{-1}$, and we therefore expect an effective wind of Galactic halo gas to be blowing through (or, at least, around) the dwarf at approximately this speed.\nThis wind would advect the IC-radiating {$e^\\pm$} \\ southward.\nQuantitatively, the extent of the angular displacement of an IC {$\\gamma$-ray} \\ signal at $E_\\gamma$\n\\begin{equation}\n \\Delta \\theta_{\\rm adv}(E_\\gamma) \\simeq 1.0^\\circ \\left(\\frac{E_\\gamma}{\\rm GeV}\\right)^{-1} \n \\left(\\frac{v_{\\rm prop}}{\\rm 260 \\ km\/s}\\right)\n\\end{equation}\nwhere $v_{\\rm prop}$ is the proper motion on the sky.\nThus advection is expected to generate a \nsouthward displacement of $\\sim 1^\\circ$.\n\n\nThis is less than the displacement we observe, but advection is also likely less important than CR transport through the gas. While the diffusion coefficient for CRs in the galactic halo is very poorly known, we can make an order of magnitude estimate by adopting\nthe functional form for the diffusion coefficient given in ref~\\cite{Gabici2007} which is normalised to $3 \\times 10^{27}$ cm$^2$ s$^{-1}$ for a 1 GeV CR in a 3 $\\mu$G field. Then the expected diffusive displacement of the IC-radiating {$e^\\pm$}~is\n\\begin{equation}\n \\Delta \\theta_{\\rm diff}(E_\\gamma) \\simeq 3.5^\\circ \\left(\\frac{E_\\gamma}{\\rm GeV}\\right)^{-0.12} \n \\left(\\frac{B}{\\rm 0.1 \\ \\mu G}\\right)^{-0.27} .\n\\end{equation}\nWhile this is roughly the correct amount of displacement to reproduce what we observe, if the diffusion were isotropic then we would still not have explained the systematic offset between the dwarf and the displaced location picked out by our template analysis. However, we do not expect isotropic diffusion in the environment of the Sgr dSph. Simulations of objects plunging through diffuse halo gas indicate that a generic outcome of such interactions is the development of a coherent magneto-tail back along the objects' direction of motion \\cite{Dursi2008}. Such a structure formed by the Sgr dSph plunging through the Milky Way's halo would naturally explain why, rather than being isotropic, the diffusive transport is primarily backwards along the dwarf's trajectory.\n\n\n\\section{Energetics of the Sgr dSph MSP population}\n\\label{ssec:energetics}\n\n\nAs discussed in the main text, \nthe {$\\gamma$-ray} \\ luminosity per stellar mass we measure for the {Sgr~dSph~} \\ is substantially brighter than we measure for the Galactic Bulge, Galactic disk, or M31, but is substantially dimmer than is observed for globular clusters.\nIndeed, in \\autoref{fig:LgammaOvrMstar}, {Sgr~dSph~} \\ appears as a transition object between gas-poor, low metallicity, low star formation rate, and relatively low stellar mass systems on the left side and relatively gas rich and massive systems (some with appreciable star formation) on the right side.\nIn order to investigate more deeply how the $\\gamma$-ray luminosity of the Sgr dSph compares to that of other observed systems, and to theoretical expectations, in \\autoref{fig:plotLgammaOvrMstarVstSimple} we collect measurements of $\\gamma$-ray luminosity per unit stellar mass versus approximate age for a range of observed systems, and compare these measurements to model predictions. For the observed systems we include M31, the Milky Way bulge and nuclear bulge, the mean of Milky Way globular clusters, and the Milky Way disc; for the latter we have included both the $\\gamma$-ray emission directly measured from MSPs, and the observed {$e^\\pm$}~luminosity of the disc, which may include a significant MSP contribution. As in \\autoref{fig:LgammaOvrMstar}, we see that the Sgr dSph is intermediate between the metal-rich galactic systems -- M31, the Milky Way disc and bulge -- and the globular clusters (GCs). However, the figure also reveals a clear trend that galactic systems dim as a function of age, with Sgr dSph as both the youngest and the most luminous of the galactic systems.\n\n\nThe trend with age is consistent with theoretical expectations, indicated by the blue band in \\autoref{fig:plotLgammaOvrMstarVstSimple} which shows the prediction of a binary population synthesis (BPS) model \\cite{Gautam2021} for the total spin-down power per unit stellar mass liberated by magnetic braking of MSPs. Some of this power should emerge as prompt emission, and some as {$e^\\pm$} \\ injected into the ISM; the lower dashed blue line shows 10\\% of the total spin-down power, a rough estimate for the prompt component. In this particular calculation, the MSPs derive from Accretion Induced Collapse, the population is assumed to be of Solar metallicity, and each binary evolves independently (i.e., the `field star' limit is assumed). Based on the predictions of this model, and the estimated age of the Sgr dSph, we estimate that the $\\gamma$-ray signal we have detected can be explained by the presence of $\\approx 650$ MSPs in the galaxy. Given that the overall {$\\gamma$-ray} \\ luminosity of an MSP population is bounded by the spin-down power, it is evident from the figure that the expected energetics appear to be elegantly sufficient to power the signal from {Sgr~dSph~} \\ given the (relatively young) mean age of its stars; this age difference naturally explains why the Sgr dSph should be more luminous per unit mass than M31 or components of the Milky Way.\n\n\nIt is also noteworthy that the GCs are considerably more luminous per unit mass than both the BPS model and the Sgr dSph. The extremely high brightness of GCs is plausibly explained by some combination of dynamical effects, which lead to dynamical hardening of binaries and thence a higher production rate of MSPs, and metallicity effects, which lead to higher MSP production because metal-poor stars have weaker winds and thus experience less mass loss during their main sequence lifetimes than Solar-metallicity stars \\cite{Ruiter2019}. The former effect would not occur in the Sgr dSph, but the latter would, since the Sgr dSph has a metallicity $\\log_{10}(Z_{\\rm Sgr}\/Z_\\odot) \\simeq -0.9$ \\cite{Vasiliev2020}, where $Z_\\odot$ is the solar metallicity, which is comparable to typical GC metallicities.\n\n\n\n\nThe final comparison we show in \\autoref{fig:plotLgammaOvrMstarVstSimple} is with the MSP power inferred by Sudoh et al.\\cite{Sudoh2020} in massive, quiescent galaxies ($M_*>10^{9.5}$ M$_\\odot$, star formation rate $< 0.1$ M$_\\odot$ yr$^{-1}$). Such galaxies typically have stellar population ages $\\gtrsim 8-10$ Gyr \\cite{McDermid2015,Pacifici2016,Sudoh2020}, and Sudoh et al.~show that they produce anomalously-large synchrotron emission, which they attribute to radiation from {$e^\\pm$} \\ injected by MSPs; they infer an injection power $1.8\\times 10^{28}$ erg$\/$s$\/M_{\\odot}$, which we show as the brown dashed line in \\autoref{fig:plotLgammaOvrMstarVstSimple}. We see that this estimate is consistent both with the BPS model and comparable to the luminosity we infer for the Sgr dSph.\n\n\nOur overall conclusion is that the MSP luminosity we have derived for the Sgr dSph is fully consistent with both theoretical expectations and with a wide variety of observed systems. The Sgr dSph is more luminous per unit mass than the Milky Way or M31, but this is easily explained by its youth and low metallicity, and it is comparably- or less-luminous than other observed systems that are of comparable age or metallicity.\n\n\n\n\n\\section{Astrophysical {$\\gamma$-ray} \\ emission from other dSphs}\n\nOn the basis of the normalisation ($L_{\\gamma}\/M_\\star$) supplied by the Sgr dSph $\\gamma$-ray detection, we can make \nrough predictions for the astrophysical $\\gamma$-ray fluxes from a number of other dSph systems, simply by assuming this normalisation applies to them as well; future work should be based on full theoretical models including metallicity and age effects, but the simple calculation we present here can serve as a guide to the system for which such investigations are likely to be fruitful.\nOur predictions can, in turn, be compared to i) actual observational upper limits to the {$\\gamma$-ray} \\ fluxes from these dSphs and ii) (model-dependent) predictions for the (WIMP) dark-matter-driven {$\\gamma$-ray} \\ fluxes from the same satellite galaxies. For this purpose we use the data assembled in Winter et al.~\\cite{Winter2016} for the distances, stellar masses, \nand MSP- and DM-driven fluxes for a population of 30 dSphs satellites of the Milky Way. These authors derive their MSP fluxes by extrapolating the $\\gamma$-ray luminosity function of resolved Milky Way MSPs; their result implies that at energies above 500 GeV, galaxies should produce an MSP photon flux per unit stellar mass of $\\approx 6.3\\times 10^{29}$ s$^{-1}$ M$_\\odot^{-1}$, roughly a factor of 40 smaller than the $\\approx 2.5\\times 10^{31}$ s$^{-1}$ M$_\\odot^{-1}$ we detect for the Sgr dSph. We report our revised estimates dwarf spheroidals' MSP luminosity in S.I.~\\autoref{tab:dsph_fluxes}. This finding has two implications, which we explore below: first, for some dwarfs this brings the predicted {$\\gamma$-ray}~flux close to current observational upper limits, suggesting that a more detailed analysis of \\textit{Fermi}-LAT data might yield a detection. Second, in some dSph galaxies, the predicted MSP flux is comparable to or exceeds the {$\\gamma$-ray}~fluxes that might be expected from dark matter annihilation.\n\n\n\n\\begin{table}\n\\begin{tabular}{lcc}\n\\hline\nGalaxy name & Predicted MSP flux & Predicted DM flux \\\\\n& (cm$^{-2}$ s$^{-1}$) & (cm$^{-2}$ s$^{-1}$) \\\\\n\\hline\nFornax & $2.38\\times 10^{-10}$ & $2.06\\times 10^{-11}$ \\\\\nSculptor & $1.10\\times 10^{-10}$ & $5.18\\times 10^{-11}$ \\\\\nSextans & $1.96\\times 10^{-11}$ & $3.27\\times 10^{-11}$ \\\\\nUrsa Minor & $1.95\\times 10^{-11}$ & $8.20\\times 10^{-11}$ \\\\\nLeo I & $1.59\\times 10^{-11}$ & $6.52\\times 10^{-12}$ \\\\\nDraco & $1.18\\times 10^{-11}$ & $8.20\\times 10^{-11}$ \\\\\nCarina & $8.12\\times 10^{-12}$ & $1.64\\times 10^{-11}$ \\\\\nLeo II & $4.54\\times 10^{-12}$ & $5.18\\times 10^{-12}$ \\\\\nBootes I & $1.36\\times 10^{-12}$ & $2.06\\times 10^{-11}$ \\\\\nCanes Ven. I & $1.33\\times 10^{-12}$ & $6.52\\times 10^{-12}$ \\\\\nUrsa Major II & $1.10\\times 10^{-12}$ & $2.59\\times 10^{-10}$ \\\\\nReticulum II & $5.27\\times 10^{-13}$ & $2.59\\times 10^{-10}$ \\\\\nComa Ber. & $5.19\\times 10^{-13}$ & $1.30\\times 10^{-10}$ \\\\\nHercules & $4.48\\times 10^{-13}$ & $1.64\\times 10^{-11}$ \\\\\nUrsa Major I & $4.25\\times 10^{-13}$ & $2.59\\times 10^{-11}$ \\\\\nTucana III & $2.67\\times 10^{-13}$ & $2.59\\times 10^{-10}$ \\\\\nGrus II & $2.53\\times 10^{-13}$ & $6.52\\times 10^{-11}$ \\\\\nTucana IV & $1.99\\times 10^{-13}$ & $6.52\\times 10^{-11}$ \\\\\nTucana II & $1.88\\times 10^{-13}$ & $8.20\\times 10^{-11}$ \\\\\nEridanus II & $1.60\\times 10^{-13}$ & $2.59\\times 10^{-12}$ \\\\\nWillman I & $1.45\\times 10^{-13}$ & $1.64\\times 10^{-10}$ \\\\\nSegue 1 & $1.34\\times 10^{-13}$ & $4.11\\times 10^{-10}$ \\\\\nLeo IV & $7.53\\times 10^{-14}$ & $1.03\\times 10^{-11}$ \\\\\nHorologium I & $6.65\\times 10^{-14}$ & $3.27\\times 10^{-11}$ \\\\\nPhoenix II & $6.55\\times 10^{-14}$ & $3.27\\times 10^{-11}$ \\\\\nCanes Ven. II & $6.51\\times 10^{-14}$ & $1.03\\times 10^{-11}$ \\\\\nReticulum III & $4.95\\times 10^{-14}$ & $2.06\\times 10^{-11}$ \\\\\nColumba I & $3.91\\times 10^{-14}$ & $5.18\\times 10^{-12}$ \\\\\nIndus I & $3.50\\times 10^{-14}$ & $2.59\\times 10^{-11}$ \\\\\nIndus II & $2.24\\times 10^{-14}$ & $3.27\\times 10^{-12}$ \\\\\n\\hline\n\\end{tabular}\n\\caption{\n\\label{tab:dsph_fluxes}\nPredicted MSP photon and dark matter annihilation photon fluxes at energies $E_\\gamma > 500$ MeV from nearby dSph galaxies, taken from the sample of ref.~\\cite{Winter2016}. Column 1: galaxy name; column 2: predicted MSP photon flux based on the Sgr dSph (see SI for details); column 3: DM annihalation flux predicted by ref.~\\cite{Winter2016}.\n}\n\\end{table}\n\n\n\n\n\n\n\n\\subsection{omparison with existing upper bounds}\n\n\n\nTo estimate whether other dwarf spheroidals might be detectable, we compare our differential flux predictions (incorporating both prompt and IC emission where, for simplicity, we make the approximation that the CMB-dominated ISRF of the {Sgr~dSph~} \\ also pertains in each other dSph under consideration)\nagainst the results from ref.~\\cite{Mazziotta2012}\\footnote{This is the most recent publication we can find that explicitly tabulates bin-by-bin, numerical 95\\% confidence upper limits on the differential flux received by a number of dSphs that also appear in the compilation of ref.~\\cite{Winter2016}}. On the basis of this comparison, we do not predict {$\\gamma$-ray} \\ emission from any dSph that surpasses the upper limits from ref.~\\cite{Mazziotta2012}. However two dSphs reach a significant fraction of the relevant upper limit in at least one energy bin (of width $log_{10}(\\Delta E$\/GeV) = 0.5): Fornax (which reaches 0.24 of the upper limit for the energy bin centred at 1.36 GeV) and Sculptor (which reaches 0.09 of the upper limit for the energy bin centred at 2.46 GeV). Furthermore, the results of ref.~\\cite{Mazziotta2012} were obtained using Pass7 {{\\em Fermi}-LAT} \\ data accumulated over only the first 3 years of {{\\em Fermi}-LAT} \\ operation. On the basis of, e.g., the results of ref.~\\cite{Ackermann2015} we expect that updated upper limits (Pass8, 15 years data) should be at least a factor of 4 more stringent. This makes Fornax and Sculptor both very interesting targets for a future study, though we remind the reader that our predictions are predicated on a normalisation obtained from the {Sgr~dSph~} \\ detection that may be somewhat over-optimistic because it ignores the stellar age effects evidenced in \\autoref{fig:plotLgammaOvrMstarVstSimple}\\footnote{The stellar population of Sculptor, in particular, is significantly older \\cite{Bettinelli2019} than that of Sagittarius, giving the MSP population more time to have spun down, though Fornax, on the other, has experienced some significant and relatively recent star formation \\cite{Rusakov2021}, like Sagittarius, qualifying it as a particularly compelling target for {$\\gamma$-ray} \\ observation.}. After these two, the brightest expected dSphs are Sextans, Ursa Minor, Leo I, and Draco. These may also be interesting targets, though we note that we expect that they are almost one order of magnitude dimmer than Fornax and Sculptor.\n\n\n\\subsection{Comparison with predicted DM annihilation fluxes}\n\n\nWinter et al.\\cite{Winter2016} estimate DM annihilation fluxes for nearby dSphs using a DM annihilation cross section derived by assuming that the the Galactic Centre Excess (GCE) is a DM signal. We caution that this is likely only an upper limit, since of course our finding for the Sgr dSph suggests that some or all of the GCE is in fact due to MSPs (see also ref.~\\cite{Gautam2021}). Nonetheless, we proceed with our calculation using the Winter et al.~estimate precisely because it represents an upper limit on the DM signal. Comparing the MSP and DM signals estimated in S.I.~\\autoref{tab:dsph_fluxes} leads us to the important finding that, in contrast to the results obtained by Winter et al., there are three dSphs for which the MSP-driven $> 500$ MeV photon number flux exceeds the predicted DM flux (viz., Fornax by $\\sim$12; Leo I by $\\sim$2.4; and Sculptor by $\\sim$2.1) and three more where it exceeds $\\sim 1\/2$ the DM flux (viz., Leo \\\nII with 0.89; Sextans with 0.60; and Carina with 0.50 of the DM flux).\nA clear implication of these, albeit preliminary, results is that these targets should be avoided in the quest to better constrain putative WIMP DM self-annihilation cross-sections. By contrast, there remain other dSphs where the expected DM signal remains comfortably much larger than the MSP signal; these are more promising targets.\n\n\n\n\n\n\n\\clearpage\n\n\\printbibliography[segment=\\therefsegment,title={Supplementary Information References}, check=onlynew]\n\n\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nObservations of high-redshift quasars probe the growth of supermassive\nblack holes (SMBH) and their connection to galaxy formation at the earliest\ncosmic epochs. \nThe discovery of strong submillimeter\/millimeter [(sub)mm] dust \ncontinuum in about 30\\% of the quasars known at z$\\sim$6 provides the \nfirst evidence of active star formation in young quasar host \ngalaxies at the end of the reionization era \\citep{bertoldi03a,bertoldi03b,petric03,priddey03,\nrobson04,wang07,wang08}. The star formation rates \nestimated from the FIR luminosities (a few $\\rm 10^{12}$ \nto $\\rm 10^{13}\\, L_{\\odot}$) are on the order of $\\rm 10^{2}$ \nto $\\rm 10^{3}\\,M_{\\odot}\\,yr^{-1}$, which are comparable to the typical \nvalues found in so-called submillimeter galaxies at $\\rm z=2\\sim3$ \n\\citep{scott02,greve05,kovacs06}. The spatially resolved [C {\\small II}] \nline emission from one of the most FIR luminous z$\\sim$6 quasars, SDSS \nJ114816.64+525150.3 (hereafter J1148+5251), further suggests a high star formation surface density \nof $\\rm \\sim1000\\,M_{\\odot}\\,yr^{-1}\\,kpc^{-2}$ over the central \n1.5 kpc region of the quasar host galaxy \\citep{maiolino05,walter09}. \n\nMolecular CO (6-5) line emission has been detected in ten of the \nFIR luminous z$\\sim$6 quasars (\\citealp{bertoldi03b,walter03,carilli07,\nwang10}, 2011, in prep.), \nindicating the existence of highly-excited molecular \ngas in the quasar hosts. The CO (3-2), (6-5), and (7-6) transitions \ndetected in the z=6.42 quasar J1148+5251 reveal a molecular gas \ncomponent on scales of $\\sim$5 kpc in the host galaxy with CO \nexcitation conditions similar to those found in local starburst \ngalaxies and CO-detected quasars at lower redshifts \\citep{bertoldi03b,walter04,riechers09}. \n\nEmission in the low-order CO transitions ($\\rm J\\leq2$) from the z$\\sim$6 quasar host \ngalaxies is poorly constrained due to the limited sensitivity and \nfrequency coverage of the previous instruments \\citep{wagg08,wang10}. \nThe new Ka band receivers on the Expanded Very Large Array (EVLA, \\citealp{perley11}) open an \nimportant frequency window for studies of the cold molecular gas in high-redshift \ngalaxies (e.g., \\citealp{ivison10,ivison11,riechers10}). In this paper, we report \nour EVLA observations of the CO (2-1) line emission in five z$\\sim$6 \nquasars \\citep{fan04,fan06,willott10a,willott10b} \nThree of them are from the Sloan Digital Sky Survey (SDSS, \\citealp{fan04,fan06}), \nwith two objects, SDSS J084035.09+562419.9 and SDSS J092721.82+200123.7, \npreviously detected in strong ($\\rm >3\\,mJy$) 250 \nGHz dust continuum and molecular CO (6-5) and (5-4) line emission. \nAnother object, SDSS J162331.81+311200.5, was detected in the [C {\\small II}] \nline, but undetected in millimeter dust \ncontinuum and high-J CO transitions (Bertoldi et al. 2011, in prep.).\nThe other two objects are from the Canada-France High-z Quasar\nSurvey (CFHQS, \\citealp{willott10a,willott10b}) and do not have published CO observations yet. \nOne of them, CFHQS 142952.17+544717.6, was detected in 250 GHz dust continuum \n(Omont et al. 2011, in prep.). We describe the observations in Section 2, present \nthe results in Section 3, and discuss the CO excitation and host galaxy \nevolution properties of the detections in Section 4. A $\\rm \\Lambda$-CDM\ncosmology with $\\rm H_{0}=71km\\ s^{-1}\\ Mpc^{-1}$, $\\rm\n\\Omega_{M}=0.27$ and $\\rm \\Omega_{\\Lambda}=0.73$ is adopted throughout this\npaper \\citep{spergel07}.\n\n\\section{Observations}\n\nThe observations were carried out using the Ka-band receiver on the EVLA \nin 2010 in the D, DnC, and C configurations. \nThe WIDAR correlator in Open Shared Risk Observing mode provided a maximum \nbandwidth of 128 MHz and a resolution of 2 MHz in each of the two \nbasebands (A\/C and B\/D intermediate frequency [IF] bands). The A\/C IFs \ncould not be tuned below 32 GHz. The redshifts and observing frequencies \nof the CO (2-1) line of the five targets are estimated with previous detections of the CO (6-5), \n[C {\\small II}], or quasar UV lines (\\citealt{carilli07,wang10}; \nBertoldi et al 2011, in prep.). For the \nthree sources with redshifts of $\\rm z\\leq6.2$ [corresponding to redshifted CO (2-1) line \nfrequencies of $\\rm \\nu_{obs}\\geq32$ GHz], we use the two 128 MHz IF pairs overlapped \nby 30 MHz and cover a total bandwidth of 226 MHz (i.e., $\\sim$2000 $\\rm km\\,s^{-1}$ \nin velocity and $\\rm \\sim0.05$ in redshift at $\\rm z=6$)\\footnote{10 MHz overlap and a total \nbandwidth of 246 MHz for J1429+5447.}. \nFor the other two objects with $\\rm z>6.2$, we \ncentered the 128 MHz window of the B\/D IF pairs on the line frequency \nand observed the continuum at $\\geq$32 GHz with the other window. \nThe observing time is 15 to 20 hours for each of the five targets (see Table 1).\nFlux calibrations were performed using the standard VLA \ncalibrators, 3C286 and 3C48, and we use 5-minute \nscan loops between targets and phase calibrators to calibrate the phase. \nThe data were reduced with AIPS, and the \nspatial resolutions (FWHM) of the final images are typically 2$''$ for data taken in the \nD configuration and 0.7$''$ for the C configuration. \n\n\\section{Result}\n\nCO (2-1) line emission has been detected in two of the \nfive z$\\sim$6 quasars, J0927+2001 and J1429+5447, and marginally \ndetected in J0840+5624. We present all the observing parameters and \nmeasurements in Table 1. The detailed results are listed below.\n\n{\\bf J0927+2001} Toward this source strong dust continuum \nat 850 GHz, 250 GHz, and 85 GHz, and CO (6-5) and (5-4) line emission \nwere detected \\citep{carilli07,wang10}. We have \ndetected the CO (2-1) line and the emission distribution (averaged \nover a velocity range of 880 $\\rm km\\,s^{-1}$) along with a spectrum is \nshown in Figure 2. The line peak emission centeroid is consistent with the optical \nquasar position and the peaks of the high-J CO lines. \nThe line width (FWHM) and redshift fitted with a single Gaussian \nprofile are $\\rm 590\\pm130\\,km\\,s^{-1}$ and $\\rm 5.7716\\pm0.0012$\nwhich are in good agreement with the measurements \nfrom the high-order CO transitions ($\\rm z=5.7722\\pm0.0006$ and \n$\\rm FWHM=600\\pm70\\,km\\,s^{-1}$, \\citealt{carilli07}). \nThe line emission appears marginally \nresolved by the $\\rm 2.19''\\times1.96''$ synthesized beam with a peak \nsurface brightness of $\\rm 147\\pm21\\, \\mu Jy\\,beam^{-1}$ and a total \nintensity of $\\rm 230\\pm45\\,\\mu Jy$, with a source size of \n$\\rm (2.7''\\pm0.4'')\\times(2.4''\\pm0.3'')$ determined from a fit with a two-dimensional \nGaussian distribution (the deconvolved source size \nis about $\\rm 1.7''\\times 1.4''$, or $\\rm 10\\,kpc\\times8\\,kpc$). \nThe corresponding line fluxes and luminosities (Table 1) are higher than the upper limits estimated from\nprevious GBT observations, but are still consistent given the large\nuncertainties and baseline feature contamination in the GBT data \\citep{wagg08,wang10}.\n\n{\\bf J1429+5447} Toward this object strong radio continuum \nemission was detected in the FIRST survey \\citep{becker95} \nand recent VLBI observations \\citep{frey11}, making it the\nstrongest radio source among the known z$\\sim$6 quasars and the \nmost distant radio-loud quasar. It has also been detected in \ndust continuum at 250 GHz with a flux density of $\\rm \\sim3$ mJy \n(Omont et al. 2011, in prep.). \nWe have detected both CO (2-1) line emission and \ncontinuum emission at the line frequency. The \ncontinuum source is unresolved by the $\\rm 0.71''\\times0.67''$ synthesized \nbeam and the flux density averaged over the line-free channels at 32 GHz \nis $\\rm 257\\pm15$ $\\mu$Jy. We subtract the continuum by performing \nlinear fitting to the visibility data, \nusing the UVLIN task in AIPS. The CO line emission is resolved into two \npeaks with a spatial separation of $\\sim$1.2$''$ (6.9 kpc at the\nquasar redshift), and the optical and radio quasar positions are \nconsistent with the west peak (Figure 3). \nA Gaussian fit to the spectra yields a redshift \nof $\\rm z=6.1831\\pm0.0007$ and a line width \nof $\\rm FWHM=280\\pm70\\,km\\,s^{-1}$ for the west source, \nand $\\rm z=6.1837\\pm0.0015$ and $\\rm FWHM=400\\pm140\\,km\\,s^{-1}$ for the east source. \nThe line fluxes estimated with the peak surface brightness on \nthe velocity-averaged map averaging over a velocity range of \n$\\rm \\sim450\\,km\\,s^{-1}$ are $\\rm 0.065\\pm0.011\\,Jy\\,km\\,s^{-1}$ \nand $\\rm 0.050\\pm0.013\\,Jy\\,km\\,s^{-1}$\nfor the west and east components, respectively. However, \na two-dimensional Gaussian distribution fitted to the east component suggest \npossible extension with a source size of $\\rm (1.1''\\pm0.2'')\\times(0.7''\\pm0.2'')$, \nwhich should be checked with deeper observations at higher spatial resolution.\n\n{\\bf J0840+5624} This source was detected\nin (sub)mm dust continuum emission and CO\n(6-5) and (5-4) line emission; it has the broadest line width,\n$\\rm FWHM=860\\,km\\,s^{-1}$, among the CO-detected\nz$\\sim$6 quasars \\citep{wang07,wang10}. We observed the line\nat the redshift of $\\rm z=5.8441\\pm0.0013$ derived\nfrom the high-order CO detections and find no clear detection\nin a velocity-averaged map averaging over 1070 $\\rm km\\,s^{-1}$ made at the\nfull resolution of $\\rm 1.09''\\times0.76''$. At a lower resolution\nof $\\rm 2.19''\\times1.96''$, marginal signal ($\\rm 2.8\\sigma$)\nappears on the map (Figure 1), with a double-peaked\nmophology along the east-west direction. The optical quasar position\nis 0.8$''$ away from the east peak. We plot the spectrum at the position\nof east peak in the right panel of Figure 1, and there is only very\nmarginal signal (1 to 2$\\sigma$) over $\\rm \\sim-500$\nto $\\rm 500\\,km\\,s^{-1}$, i.e., the typical velocity range of the CO (6-5)\nand (5-4) line emission \\citep{wang10}.\nThe CO (2-1) line flux estimated with the surface\nbrightness of the east peak is $\\rm 0.062\\pm0.022\\,Jy\\,km\\,s^{-1}$ (Table 1).\nHowever, the signal is indeed marginal and deeper observations with a wider\nbandwidth are required to improve the measurement.\n\n{\\bf J0210$-$0456} This object is the highest\nredshift quasar known to date with $\\rm z=6.438\\pm0.004$ determined\nfrom the object's $\\rm Mg\\,{\\small II}\\,\\lambda2798\\AA$ line emission \\citep{willott10b}.\nWe searched for CO (2-1) line emission in the 128 MHz window centered at\nthe $\\rm Mg\\,{\\small II}$ redshift but did not detect it.\nHere we assume a line width of $\\rm 800\\,km\\,s^{-1}$, which is the\ntypical full width at zero intensity ($\\rm v_{FWZI}$) value found with\nsamples of high-z CO-detected quasars \\citep{coppin08,wang10}\nto estimate the upper limit of the line intensity.\nThe $\\rm 1\\sigma$ rms noise level on the map averaged over this velocity range\nis $\\rm \\sigma_{rms} =16\\,\\mu Jy\\,beam^{-1}$, and the $\\rm 3\\sigma$ upper limit of the line\nflux is estimated as $\\rm 3\\sigma_{rms} v_{FWZI}=0.038\\,Jy\\,km\\,s^{-1}$.\nThe corresponding 3$\\sigma$ upper limit of\nthe line luminosity is $\\rm {L'}_{CO(2-1)}<1.28\\times10^{10}\\,K\\,km\\,s^{-1}\\,pc^2$ \n(see equation (3) in \\citealt{solomon05}). However, we cannot rule out that \nthe $\\rm Mg\\,{\\small II}$ line emission is significantly offset \nfrom the quasar host galaxy redshift and CO (2-1) line falls outside the 128 MHz window. \nThe continuum emission is \nalso undetected with the other window centered at 32.1 GHz, and the \nchannel-averaged map yields a 3$\\sigma$ upper limit of $\\rm <54\\,\\mu Jy$.\n\n{\\bf J1623+3112} This object is detected in $\\rm [C\\,{\\small II}]$ 158$\\mu$m \nfine structure line emission by Bertoldi et al. (2011, in prep.), \nbut undetected in 250 GHz dust continuum \\citep{wang07}. We searched for the CO (2-1) line in \nthe 128 MHz-bandwidth window centered at the $\\rm [C\\,{\\small II}]$ \nredshift of $\\rm z=6.2605\\pm0.0005$ and did not detect the line. \nThe rms on the map averaged over a velocity range of \n$\\rm 800\\,km\\,s^{-1}$ is $\\rm 26\\,\\mu Jy\\,beam^{-1}$. This yields \na 3$\\sigma$ upper limit of $\\rm <0.062\\,Jy\\,km\\,s^{-1}$ for the line flux \nand $\\rm <2.0\\times10^{10}\\,K\\,km\\,s^{-1}\\,pc^2$ for the line luminosity. \nThe 3$\\sigma$ upper limit of the continuum emission at 35 GHz \nmeasured with A\/C IFs is $\\rm <75\\,\\mu Jy$. \n\n\\section{Discussion}\n\nWe have observed molecular CO (2-1) line emission toward five quasars \nat z$\\sim$6 using the EVLA, and detections\/marginal detection have been \nobtained from the three objects that have strong FIR dust \ncontinuum emission. This is consistent with the picture of massive star \nformation fueling by huge amount of molecular gas in these young \nquasar hosts. The detection of $\\rm [C\\,{\\small II}]$ in J1623+3112 is \nalso likely to be a sign of star formation, but the current sensitivity \nof our EVLA observations cannot detect molecular CO from the host galaxy. \nJ0927+2001 and J0840+5624, were \npreviously detected strongly in the CO (6-5) and (5-4) transitions. \nCO (2-1) line emission has been detected and marginally \nresolved in the host galaxy of J0927+2001 over a scale of $\\sim$10 kpc. \nThe molecular gas masses ($\\rm M_{gas}$) estimated from the CO (2-1) line\npeak surface brightness and the total intensity on the velocity-averaged\nmap are listed in Table 1, assuming a CO luminosity-to-gas mass conversion factor\nof $\\rm \\alpha=0.8 M_{\\odot}\\,(K\\,km\\,s^{-1}\\,pc^{2})^{-1}$ appropriate\nfor local ultraluminous infrared galaxies \\citep{solomon97,downes98}.\nThese estimates are 1.7 and 2.5 times higher than the \nvalue of $\\rm (1.8\\pm0.3)\\times10^{10}\\,M_{\\odot}$ estimated\nfrom the high-order CO transitions \\citep{carilli07,wang10}. \nWe plot the CO excitation ladder of this source in Figure 4,\ntogether with the results of Large Velocity Gradient (LVG) modeling\nof the highly-excited molecular gas components \n(gas densities of \norder $\\rm 10^{4}\\,cm^{-3}$, kinetic temperatures of 50 to 60 K, \nand peak at $\\rm J\\geq6$) \nfound in other high-z FIR and CO luminous\nquasars and nearby starburst galaxies \\citep{riechers06,riechers09,gusten06}.\nWe normalize the models to the high-order CO transitions. \nThe CO (2-1) line flux measured with the peak surface \nbrightness on the velocity-averaged map is consistent\/marginally consistent \nwith the values expected by these single-component models, while the total line flux \nintegrated over the line-emitting area falls above all the models. This may suggest \nthe exsitence of additional low excitation gas in the central $\\rm \\sim10\\,kpc$ region \nas was found in the submillimeter galaxy AzTEC-3\nat z=5.3 \\citep{riechers10} and the nearby starburst \ngalaxy M82 \\citep{weiss05}. However, there are still large \nuncertainties in the measurements of all the three transitions, \nand observations of other CO transitions are necessary to \naddress if there are multiple CO excitation components in the quasar host galaxy. \nOur observations show no evidence of excess CO (2-1) line emission and additional \nlow excitation component in the host galaxy of J0840+5624. \n\nThe C array imaging of the CO (2-1) line emission from J1429+5447 has \nresolved the molecular gas into two distinct peaks, with a spatial separation \nof $\\sim$6.9 kpc; the quasar position is consistent with the West peak. \nThere is no clear velocity offset ($\\rm 26\\pm60\\,km\\,s^{-1}$) between \nthe two components. These results suggest a gas-rich, major merging \nsystem with two distinct components that are comparable in CO luminosity and \nmolecular gas mass. The west component of this system \nis in a radio-loud quasar phase. Similar quasar-starburst systems with \nmultiple CO emission peaks were previously found in \nthe CO luminous quasars BRI 1202$-$0725 at z=4.7 \\citep{omont96,carilli02}, \nBRI 1335$-$0417 at z=4.4 \\citep{riechers08} \nand J1148+5251 at z=6.42 \\citep{walter04}. These systems demonstrate \nthe early phase of quasar-galaxy formation in which both AGN and starburst \nactivities are triggered by major mergers and the molecular gas in the \nnuclear region is not fully coalesced \\citep{narayanan08}. We will \nexpect further high-resolution observations with the EVLA in C or B \narray to constrain the gas surface density and dynamics, \nand with ALMA or the PdBI to resolve the dust continuum and distributed \nstar formation in these young quasar host galaxies.\n\n\\acknowledgments \nThis work is based on observations carried out with the Expanded Very Large \nArray (NRAO). The National\nRadio Astronomy Observatory (NRAO) is a facility of the National\nScience Foundation operated under cooperative agreement by Associated\nUniversities, Inc. We acknowledge support from the Max-Planck Society\nand the Alexander von Humboldt Foundation through the Max-Planck-Forschungspreis\n2005. Dominik A. Riechers acknowledges support from NASA through Hubble\nFellowship grant HST-HF-51235.01 awarded by the Space Telescope Science\nInstitute, which is operated by the Association of Universities for\nResearch in Astronomy, Inc., for NASA, under contract NAS 5-26555.\nM. A. Strauss acknowledges the support of NSF grant Ast-0707266.\n{\\it Facilities:} \\facility{EVLA}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}