The hot phase of the circumgalactic medium CGM allows us to probe the inflow and outflow of gas within a galaxy which is responsible for dictating the evolution of the galaxy. Studying the hot CGM sheds light on a better understanding of gas physics which is crucial to inform and constrain simulation models. With the recent advances in observational measurements probing the hot CGM in Xrays and tSZ we have a new avenue for widening our knowledge of gas physics and feedback by exploiting the information from currentfuture observations. In this paper we use the TNG300 hydrodynamical simulations to build a fully selfconsistent forward model for the hot CGM. We construct a lightcone and generate mock Xray observations. We quantify the projection effects namely the locally correlated largescale structure in Xrays and the effect due to satellite galaxies misclassified as centrals which affects the measured hot CGM galactocentric profiles in stacking experiments. We present an analytical model that describes the intrinsic Xray surface brightness profile across the stellar and halo mass bins. The increasing stellar mass bins result in decreasing values of beta the exponent quantifying the slope of the intrinsic galactocentric profiles. We carry forward the current stateoftheart by also showing the impact of the locally correlated environment on the measured Xray surface brightness profiles. We also present for the first time the effect of misclassified centrals in stacking experiments for three stellar mass bins 1010.511 Modot 101111.2 Modot and 1011.211.5 Modot. We find that the contaminating effect of the misclassified centrals on the stacked profiles increases when the stellar mass decreases.
['Astrophysics - Astrophysics of Galaxies', 'Astrophysics - High Energy Astrophysical Phenomena']
Quantifying Observational Projection Effects with a Simulationbased hot CGM model
{'Quantifying Observational Projection Effects with a Simulation-based hot CGM model': 'Soumya Shreeram 1 ⋆ , Johan Comparat 1 , Andrea Merloni 1 , Yi Zhang 1 , Gabriele Ponti 1 , 2 , Kirpal Nandra 1 , John ZuHone 3 , Ilaria Marini 4 , Stephan Vladutescu-Zopp 5 , Paola Popesso 3 , Ruediger Pakmor 6 , Riccardo Seppi 7 , Celine Peroux 3 , and Daniele Sorini 8  \n- 1 Max Planck Institute for Extraterrestrial Physics (MPE), Gießenbachstraße 1, 85748 Garching, Munich, Germany\n- 2 INAF-Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807 Merate (LC), Italy\n- 3 Smithsonian Astrophysical Observatory, Observatory Building E, 60 Garden St, Cambridge, MA 02138, United States\n- 4 European Southern Observatory, Karl-Schwarzschild-Straße 2, 85748 Garching, Munich, Germany\n- 5 Universitäts-Sternwarte München, Scheinerstraße 1, 81679 München, Germany\n- 6 Max Planck Institute for Astrophysics, Karl-Schwarzschild-Straße 1, 85748 Garching, Munich, Germany\n- 7 University of Geneva, 1205 Geneva, Switzerland\n- 8 Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham, DH1 3LE, United Kingdom  \nReceived ZZZZ, ZZZZ; accepted XXXX, XXXX; published YYYY, YYYY', 'ABSTRACT': 'Aims. The hot phase of the circumgalactic medium (CGM) allows us to probe the inflow and outflow of gas within a galaxy, which is responsible for dictating the evolution of the galaxy. Studying the hot CGM sheds light on a better understanding of gas physics, which is crucial to inform and constrain simulation models. With the recent advances in observational measurements probing the hot CGM in X-rays and tSZ, we have a new avenue for widening our knowledge of gas physics and feedback by exploiting the information from current / future observations.  \nMethods. In this paper, we use the TNG300 hydrodynamical simulations to build a fully self-consistent forward model for the hot CGM. We construct a lightcone and generate mock X-ray observations. We quantify the projection e ff ects, namely the locally correlated large-scale structure in X-rays and the e ff ect due to satellite galaxies misclassified as centrals, which a ff ect the measured hot CGM galactocentric profiles in stacking experiments.  \nResults. We present an analytical model that describes the intrinsic X-ray surface brightness profile across the stellar and halo mass bins. The increasing stellar mass bins result in decreasing values of β , the exponent quantifying the slope of the intrinsic galactocentric profiles. We carry forward the current state-of-the-art by also showing the impact of the locally correlated environment on the measured X-ray surface brightness profiles. We also present, for the first time, the e ff ect of misclassified centrals in stacking experiments for three stellar mass bins: 10 10 . 5 -11 M ⊙ , 10 11 -11 . 2 M ⊙ , and 10 11 . 2 -11 . 5 M ⊙ . We find that the contaminating e ff ect of the misclassified centrals on the stacked profiles increases when the stellar mass decreases. When stacking galaxies of Milky-Way-like stellar mass, this e ff ect is non-negligible already at a low level of contamination: in particular, misclassified centrals contributing 30%, 10%, or 1% of a sample dominate the measured surface brightness profile at radii ≥ 0 . 06 × R 200 m , ≥ 0 . 2 × R 200 m , and ≥ 0 . 3 × R 200 m , respectively.  \nKey words. Hot circumgalactic medium - galaxies: evolution - methods: numerical', '1. Introduction': "The Circumgalactic medium (CGM) plays a crucial role at interfacing the gas between the interstellar medium (ISM) within galaxies and the external intergalactic medium (IGM) outside them. The CGM is commonly defined as the gas within the virial radii of the galaxy and outside their galactic disks. It is the gravitationally bound gas that encompasses the fossil imprints of the physical mechanisms, such as outflows, inflows, and feedback processes, dictating the evolution of the galaxy (see Tumlinson et al. 2017 for a review). These mechanisms are sensitive to the galaxy's environment and its halo properties, as shown by the complex Stellar-to-Halo-Mass-Relation (SHMR); see the review from Wechsler & Tinker (2018). According to most models, the SHMR indicates that the low-halo-mass end is sensitive  \nto stellar and supernovae (SN) driven feedback (Dekel & Woo 2003; Benson et al. 2003), causing heating of the gas, hot bubbles (McKee & Ostriker 1977), galactic wind outflows (Dekel & Silk 1986), and turbulence (Ostriker & Shetty 2011; Strickland & Heckman 2009). In contrast, Active Galactic Nuclei (AGN) should be the main drivers of feedback in the high-halo-mass end (Silk & Rees 1998; Fabian 2012; Eckert et al. 2021). Of particular interest is the peak of the SHMR relation at the pivotal halo mass M h ∼ 10 12 M ⊙ , similar to our Milky-Way (MW) mass, corresponding to the mass scale where star formation e ffi ciency reaches its maximum. Thus, studying the volume-filling hot gas component of the CGM, especially in the MW halo mass regime, encapsulating a range of physical processes, is crucial for testing galaxy formation models (Faucher-Giguère & Oh 2023).  \nThe most general scenario for the presence of such hot CGM assumes that infalling gas within halos ≳ 10 12 M ⊙ is shock heated  \nup to the virial temperature, T vir ≳ 10 6 K, resulting in X-ray emission (White & Rees 1978). This hot gas is probed with the thermal Sunyaev-Zeldovich (Sunyaev & Zeldovich 1972, tSZ) e ff ect at mm-wavelengths (Lim et al. 2021; Das et al. 2023; Oren et al. 2024) and in X-rays via absorption and emission studies. X-ray absorption studies use sightlines with bright background sources to probe the hot CGM via absorption lines like OVI or OVII K α (Galeazzi et al. 2007; Bhattacharyya et al. 2023; Mathur et al. 2023). The major challenge in this technique is constructing statistical samples and mapping the largescale extent of the CGM, which will improve with future X-ray telescopes with microcalorimeters (Wijers et al. 2020; Bogdán et al. 2023). On the emission side, studies use narrow-band and broad-band observations. The narrow-band emission comprising of OVII and OVIII metal lines dominates the lower halo mass regime 10 12 -10 13 M ⊙ (Bertone et al. 2010; van de Voort 2013; Wijers 2022) and is well-studied within our MW (Zheng et al. 2024; Locatelli et al. 2024; Ponti et al. 2023; Koutroumpa et al. 2007). However, for extragalactic studies, future high-spectral resolution instruments are required to distinguish the extragalactic emission from that of the MW foreground (Nelson et al. 2023; Truong et al. 2023; Schellenberger et al. 2024; ZuHone et al. 2024).  \nMultiple extragalactic single-object broad-band X-ray emission studies have been published (Bogdán et al. 2013a,b; Anderson et al. 2016; Bogdán et al. 2017; Li et al. 2017; Das et al. 2019). However, due to the X-ray emissivity scaling with the square of the density, detections are limited to the densest parts of the CGM, within 40 -150 kpc from the galaxy centre. To map the extended CGM out to the halo virial radius, R vir, Anderson et al. (2015) stacked SDSS galaxies with the full-sky X-ray data from the ROSAT survey. With the advent of SRG / eROSITA, we have unprecedented statistics for studying the hot CGM in Xrays with stacking analysis, as first shown by Oppenheimer et al. (2020), who generated mock observations with IllustrisTNG and EAGLE hydrodynamical simulations. Comparat et al. (2022) and Chadayammuri et al. (2022), and more recently, Zhang et al. (2024, hereafter Z + 24) conduct such experiments with the eROSITA data and detect the hot CGM out to the virial radius of Milky-Way and M31-like galaxies. X-ray stacking analysis provides the most sensitive state-of-the-art observations of the hot CGM in the current observational landscape. Still, they are subject to various observational e ff ects that a ff ect the interpretation of the hot CGM. Consequently, we need a complete theoretical framework for describing the intrinsic hot CGM emission to disentangle the intrinsic signal from other observational e ff ects.  \nA plethora of theoretical models (e.g., Faerman et al. 2017; Voit et al. 2019; Pal Choudhury et al. 2019; Stern et al. 2024; Singh et al. 2021; Pandya et al. 2023; Faerman & Werk 2023) have been used to describe and predict the intrinsic properties of the hot CGM; Singh et al. (2024) discuss the comparison of some idealised cases. Similarly, Oppenheimer et al. (2020); Vladutescu-Zopp et al. (2024) made explicit predictions of the intrinsic hot CGM profiles obtained by assigning mock X-ray emission to the hot halos found in cosmological hydrodynamical simulations. However, before testing theoretical and numerical predictions of the hot CGM against state-of-the-art observations, we must quantify all the e ff ects that influence the hot CGMobservations in stacking experiments. Among the primary sources of contamination in X-ray stacking experiments, inhibiting us from retrieving the physical properties of the detected CGM emission, are (1) the unresolved AGN and X-ray binaries (XRB) population of galaxies (Bi ffi et al. 2018; VladutescuZopp et al. 2023), and (2) the projection e ff ects within the hot gas  \nemission of the cosmic web. These projection e ff ects include the contribution from the Large-Scale Structure (LSS) halo environment (locally correlated X-ray emission), the e ff ect of satellite galaxies being misclassified as central due to limitations in the (photometric) redshift accuracy for the galaxies in large surveys (see e.g., Weng et al. 2024), and the Line-of sight (LoS) projection of uncorrelated X-ray emission of fore- and background structures. Z + 24 tried to model these e ff ects empirically for the first time; however, we need forward models based on cosmological hydrodynamical simulations for a complete description and full understanding of these contaminating e ff ects.  \nIn this paper, we analyse numerical simulations to empirically quantify the e ff ects of the locally correlated X-ray emission and the misclassified centrals, also called the satellite-boost e ff ect, relevant for measuring the hot CGM in stacking experiments. To do this, we use the TNG300 hydrodynamical simulations (Pillepich et al. 2018a; Marinacci et al. 2018; Naiman et al. 2018; Nelson et al. 2015; Springel et al. 2018) to construct a lightcone and generate mock X-ray observations. We focus here on the projection e ff ects of hot gas a ff ecting the X-ray surface brightness profiles. The structure of the paper is as follows. Sec. 2 details the construction of the TNG300 lightcone (LCTNG300) used for modelling the hot CGM in this study. We explain all the projection e ff ects in detail in Sec. 2.1. In Sec. 3, we describe the process for generating mock X-ray observations that mimic observational data and provide the machinery for computing the surface brightness profiles. Sec. 4 presents the main results of this work done with the TNG300 lightcone. Lastly, Sec. 5 discusses the main findings and Sec. 6 provides an overview with prospects.", '2. Simulated lightcone with IllustrisTNG: LC-TNG300': "Since we are interested in studying the impact of the locally correlated X-ray emission within the LSS environment of the halo, we require a simulation that contains all the complexities introduced by feedback and cooling, which are imprinted in X-ray measurements. We also require a large box size to encompass the e ff ects of the cosmological LSS itself. Therefore, we use cosmological hydrodynamical simulations that self-consistently predict the LSS and its impact on gas dynamics within a halo. We use the IllustrisTNG cosmological hydrodynamical simulation with the box of side length 302 . 6 Mpc (TNG300; Nelson et al. 2019) 1 . IllustrisTNG accounts for many physical processes, among which the most notable ones are star formation regulated by a subgrid ISM model (Springel & Hernquist 2003), metal enrichment (Naiman et al. 2018), radiative gas cooling, galactic wind outflows (Pillepich et al. 2018b), magnetic fields and diffuse radio emissions (Marinacci et al. 2018), supermassive black hole growth with Bondi accretion and mergers, thermal and kinetic modes for black hole feedback (Weinberger et al. 2018). IllustrisTNG also reproduces correlation functions and power spectra of particles and halos (Springel et al. 2018). TNG300 contains 2500 3 dark matter particles, with a baryonic mass resolution of 1 . 1 × 10 7 M ⊙ , a comoving value of the adaptive gas gravitational softening length for gas cells of 370 comoving parsec, gravitational softening of the collisionless component of 1 . 48 kpc, and dark matter mass resolution of 5 . 9 × 10 7 M ⊙ . The TNG simulations adopt the Planck Collaboration XIII 2016 cosmological parameters, with the matter density parameter Ω m = Ω dm + Ω b = 0 . 3089, baryonic density parameter Ω b = 0 . 0486,  \nTable 1. For every stellar mass bin, we present the mean stellar mass within twice the stellar half mass radius, the total number of central galaxies, the total number of satellite galaxies, mean halo mass, the minimum and maximum values of halo masses for the given stellar mass bin, the mean R 200 m , and the mean R 500 c .Table 2. For every halo mass bin used to generate the X-ray surface brightness profiles, we present the mean halo mass, the total number of distinct halos, the mean stellar mass, the mean R 200m and the mean R 500c, for every halo mass bin.  \nFig. 1. Illustration of the lightcone built using TNG300 in the x-y plane. The figure shows the subhalos within LC-TNG300 at 0 . 03 ≤ z ≤ 0 . 3 remapped using boxremap (Carlson & White 2010). The observer is set at the (0 , 0 , 0) location. The lightcone goes out to 1231 cMpc along the x-axis, subtending an area of 47 . 28 deg 2 on the sky in the y-z plane. The subhalos are colour-coded with their stellar masses.  \n<!-- image -->  \nMpc  \nHubble constant H0 = 100 h km / s / Mpc with h = 0 . 6774, and ΩΛ = 0 . 6911.  \nThe Friends-of-Friends (FoF) algorithm is applied to the dark matter particles with linking length b = 0 . 2 to obtain the halos. The subhalos are retrieved with subfind (Springel et al. 2001; Dolag et al. 2009) that detects gravitationally bound substructures, which are equivalent to galaxies in observations. Additionally, subfind also classified the subhalos into centrals and satellites, where centrals are the most massive substructure within a distinct halo.  \nTo model the hot gas emission up to z ∼ 0 . 3, as done in observations (e.g. Comparat et al. 2022; Chadayammuri et al. 2022; Zhang et al. 2024), we need four boxes lined up behind each other to arrive at the comoving distance, dC(z = 0 . 3), of 1231 Mpc. For a full-sky lightcone going up to z = 0 . 3, we must replicate and stack 512 boxes; however, in this technique, the large-scale projection e ff ects cannot be estimated properly; see Merson et al. (2013). Therefore, we box remap 2 all snapshots into a configuration where the longest length of one of  \nthe sides is ∼ 4 × the original length (Carlson & White 2010). This technique ensures that the new elongated box has a oneto-one remapping, remains volume-preserving, and keeps local structures intact. For a box whose original dimensions are normalized to (1 , 1 , 1), the unique solution for the transformed box sides is (4 . 1231 , 0 . 7276 , 0 . 3333), respectively. We remap the coordinates of particles (gas, dark matter, and stars) and the halo and their subhalo catalogues. Therefore, we obtain 22 remapped particle cuboids and 22 remapped galaxy catalog cuboids at redshift z < 0 . 3. We define the observer's position as being at a corner of the smallest face. The opening angles for the observer are ( θ LC obs , ϕ LC obs) = (10 . 16 , 4 . 64) degrees. The area subtended on the sky for a given observer is 47 . 28 deg 2 . To study the impact of the projection e ff ects for MW-like galaxies (M ⋆ = 10 10 . 5 -11 M ⊙ ), we have 5 , 109 centrals and 7 , 826 satellites in the light cone, which is ample to statistically model the projection e ff ects within Milky-Way-sized halos. We illustrate the lightcone constructed with the box remap technique in Fig. 1. We also publically release LightGen , the code used for generating the lightcone in this work.  \n-  \n-  \n2  \n·  \n00  \n1  \n·  \n00  \n0  \n·  \n00  \n01  \n·  \n00  \n02  \n·  \n00  \n-  \n2  \n0  \n02  \n-  \n<!-- image -->  \nh  \n23 h 40 23 50 0 h 00 0 10 0 h 20 Fig. 2. The projected rest-frame X-ray events from the TNG300 ligthcone in the 0 . 5 -2 . 0 keV band for a telescope with energy-independent collecting area 1000 cm 2 and exposure time of 1000 ks. The events are generated using the hot gas cells within the TNG300 lightcone at redshifts 0 . 03 -0 . 3 using pyXsim (ZuHone & Hallman 2016). The centre for the projection onto the sky is chosen as R.A., Dec. ≡ (0 ., 0 . ) degrees. The contours represent the two X-ray surface brightness levels of 5 × 10 -14 kev s -1 cm -2 arcmin -2 and 2 × 10 -16 kev s -1 cm -2 arcmin -2 .  \n-  \nh  \n<!-- image -->  \nh  \n23 40 23 50 0 00 0 10 0 h 20 Fig. 3. The projected halos ( centrals ) are overplotted with their corresponding scaled R 500 c at z = 0 . 3; the R 500 c of the halos are represented by the size of the circles. We also show the projected rest-frame X-ray events from the TNG300 ligthcone in the 0 . 5 -2 . 0 keV band for a telescope with energy-independent collecting area 1000 cm 2 and exposure time of 1000 ks at the redshift slice of 0 . 284 ≤ z ≤ 0 . 3. The halos with M 200 m ∈ 10 14 -14 . 5 M ⊙ are shown in red, M 200 m ∈ 10 13 . 5 -14 M ⊙ in grey, M 200 m ∈ 10 13 -13 . 5 M ⊙ in cyan, and M 200 m ∈ 10 12 . 5 -13 M ⊙ in blue.  \nh  \nThe distinct halos, CEN sim halo , and subhalos within the LCTNG300 are cataloged with their physical properties and binned in stellar mass and halo mass bins, as shown in Tab. 1 and Tab. 2. This paper defines the stellar mass used from TNG300 as the mass within twice the stellar half-mass radius. We present quantities relative to both critical and mean densities because R 500 c 3 is the radius most commonly used by X-ray astronomers (see e.g., Lyskova et al. (2023) and references therein), and R 200 m 4 repre-  \nh  \nh  \nsents the halo's viral radius and is theoretically more relevant as it presents quantities within the virialized halo. The galaxy catalog is divided into central, CEN sim , and satellites, SAT sim for the concerned stellar mass bins. To construct the SAT sim catalogue, we refer to the halo / subhalo classification, where we match all the central galaxies with the distinct halos, thereby leaving behind all the secondary subhalos or satellites.", '2.1. Contributions to observed X-ray surface brightness profiles': 'The main objective of this study is to model the various sources of contamination that come into play when measuring the hot gas component of the CGM. Here, we list all the components contributing to the observed X-ray surface brightness profile.  \n- 1. Intrinsic emission from the halo. This corresponds to the emission within the radius, R 200m, of the central halo in 3D. We associate this X-ray emission with being intrinsic to the galaxy CGM. It must be noted that the definition of a halo boundary at which the gas is bound is non-trivial; we refer to Diemer et al. (2017) and references therein for more details.\n- 2. Locally correlated environment. This component encapsulates the surrounding emission of the galaxy with the LSS in which it resides. Also known as the 2-halo term (Cooray & Sheth 2002; Kravtsov et al. 2004), this corresponds to the contribution arising due to the local background changing with the size of the halo. This e ff ect is explored in tSZE studies, where Vikram et al. (2017), Lim et al. (2021) show the impact of the two-halo component, where the one-halo contribution at mass scales M 200 ≤ 10 13 -13 . 5 h -1 M ⊙ is swamped by the two-halo term due to nearby massive systems dominating the measured signal. Given our focus on MW-size halos, modelling the contribution from the locally-correlated background is crucial for disentangling intrinsic hot CGM emission from the local background.\n- 3. Intrinsic emission from satellites. The X-ray emission from the satellite galaxy contributes to the total CGM emission. As the host galaxy is more massive, with a deeper potential well, the contribution of this e ff ect is negligible in stacking experiments (see Rohr et al. (2024) for further insights on the detectability of the satellite emission).\n- 4. Contamination from misclassified centrals. In observations that use photometric surveys, classifications between centrals and satellites are inhibited due to limitations in the redshift accuracy. This e ff ect is large and unavoidable for photometric surveys ( ∼ 30% for MW-like galaxies; Sec 3.5 in Z + 24) and significantly mitigated for spectroscopic surveys ( ∼ 1% for MW-like galaxies; Sec 3.7 in Z + 24), but not completely removed due to survey incompleteness or residual uncertainty in the central / satellite classification for systems with a low number of galaxies (see also Weng et al. 2024 who quantify the e ff ect for absorbers in cold gas with TNG50). This implies that the measured X-ray surface brightness profiles in stacked samples of galaxies classified as central contain the intrinsic emission around truly central galaxies but are contaminated by the emission measured around misclassified centrals. In conclusion, including the emission around satellites in the stacking analysis alters the recovered profiles and, therefore, must be modelled.\n- 5. Contamination from other X-ray sources. Other X-ray emitting sources like XRB and AGN contaminate the measured hot CGM. XRB emission is distributed on the scale of the stellar body of a galaxy; however, for an instrument with a Point Spread Function (PSF) like eROSITA, it is unresolved and appears as a point source. Z + 24 take special precautions to account for this by masking the eROSITA detected point sources within the X-ray data. They also model additional contributions from the unresolved XRB and AGN. For this study, we leave the modelling of the AGN and XRB resolved / unresolved emissions with the LC-TNG300 for future work.\n- 6. Line of Sight (LoS) projection. Objects along the line of sight, which do not reside near the galaxy, also contaminate the detected signal. Nevertheless, this uncorrelated contamination is well-modelled as the large-scale background and foreground in observations and, therefore, is not discussed further in this work.  \nThis work focuses on modelling the locally correlated environment and the e ff ect due to misclassified-centrals (the satelliteboost e ff ect).', '3. Method': 'We detail the process to create mock X-ray observations in Sec. 3.1 and introduce the formalism used to fit the X-ray surface brightness profiles in Sec. 3.2. We detail the data products generated to study the projection e ff ects in Sec. 3.3.', '3.1. Mock X-ray observation': 'The photons are simulated in the 0 . 5 -2 . 0 keV intrinsic band with pyXsim (ZuHone & Hallman 2016), which is based on phox (Bi ffi et al. 2013; Bi ffi et al. 2018), by assuming an input emission model where the hot X-ray emitting gas is in collisional ionization equilibrium. The spectral model computations of hot plasma use the Astrophysical Plasma Emission Code, apec 5 code (Smith et al. 2001) with atomic data from atomdb v3.0.9 (Foster et al. 2012). This model requires the plasma temperature of the gas cells (in keV), the metal abundances, the redshift z and the normalization,  \nN = 10 -14 4 π [ D A(1 + z )] 2 Z n e n HdV , (1)  \nwhere D A is the angular diameter distance to the source (cm), dV is the volume element (cm 3 ), n e and n H are the electron and hydrogen densities (cm -3 ), respectively. The temperature is calculated from the internal energy u and the electron abundance x e( = n e / n H) of the gas cells 6 . The temperature T for every gas cell is defined as  \nT = ( γ -1) u k B µ (2)  \nwhere k B is Boltzmann\'s constant in CGS units and γ = 5 / 3 is the adiabatic index. The mean molecular weight µ is given as  \nµ = 4 1 + 3 X H + 4 X H x e m p (3)  \nwhere X H = 0 . 76 is the hydrogen mass fraction and m p is the proton mass in grams. The metal abundances within TNG are provided for the snapshots at redshift intervals of every 0 . 1. As a result, 19 of the 22 snapshots within the lightcone constructed in this work lack metallicity information. Given the lack of metallicity information and the inaccuracies introduced by extrapolation of metallicity values between the 0.1 redshift intervals due to the evolution of gas particles, we assume a constant metallicity of 0 . 3 Z ⊙ ; this is consistent with measurements for our MW(Miller & Bregman 2015; Bregman et al. 2018; Kaaret et al. 2020; Ponti et al. 2023). This work uses the solar abundance values from Anders & Grevesse (1989).  \nFig. 4. Mean X-ray surface brightness profiles in the stellar mass bins: M ⋆ = 10 10 . 5 -11 M ⊙ , corresponding to MW-like galaxies ( left ), M ⋆ = 10 11 -11 . 2 M ⊙ ( centre ), and M ⋆ = 10 11 . 2 -11 . 5 M ⊙ ( right ). The vertical dashed line is the mean R 500 c at 242 . 75 kpc ( left ), 369 . 01 kpc ( centre ), and 484 . 52 kpc ( right ) of the respective galaxy stellar mass bin with the shaded area corresponding to the minimum and maximum values. The black line is the analytical model, shown in Eq. 7, fit to the LC-TNG300 mean X-ray surface brightness profiles. The best-fitting parameters for the model are given in Tab. 3. We find decreasing values for the slope, β , with increasing stellar mass bins.  \n<!-- image -->  \nFig. 5. Mean X-ray surface brightness profiles in the halo mass bins: M 200m = 10 12 . 5 -13 M ⊙ , corresponding to MW-like galaxies ( left ), M 200m = 10 13 -13 . 5 M ⊙ ( centre ), and M 200m = 10 13 . 5 -14 M ⊙ ( right ). The vertical dashed line is the mean R 500 c at 246 . 35 kpc ( left ), 353 . 35 kpc ( centre ), and 519 . 34 kpc ( right ) of the respective halo mass bin with the shaded area corresponding to the minimum and maximum values. The black line is the analytical model, shown in Eq. 7, that fits the LC-TNG300 mean X-ray surface brightness profiles. The best-fitting parameters for the model are given in Tab. 4. We find decreasing values for the slope, β , with increasing halo mass bins.  \n<!-- image -->  \nThe TNG star formation model is based on the subgrid twophase model proposed by Springel & Hernquist (2003), with some modifications (see Pillepich et al. 2018a, and references therein). The gas cells that emit e ffi ciently in the soft X-ray band (i.e. [0.3-5] keV) are due to the SN-driven kinetic decoupled winds, which ultimately deposit energy into non-star-forming gas cells. Additionally, the hot component of this two-phase model exhibits typically high temperature ( > 10 5 K). As further explained in Truong et al. (2020) (see Appendix B-C), because this multiphase structure within the TNG model is not resolved, and it is instead modelled by a simplistic two-phase structure with unrealistic assumptions, we cannot make a sensible estimate of the X-ray emission from the unresolved phases of the ISM. Therefore, by excluding the parameter space of the warmneutral ISM (Le Brun et al. 2014; Rahmati et al. 2016; Wijers et al. 2019), namely: (1) excluding star-forming gas cells, (2) ignoring gas cells below 10 5 K, and (3) ignoring gas cells with densities above 10 -25 g / cm 3 , we ensure the gas particles used in this work are physically emitting X-rays.  \nWithin pyXSim , the number of photons generated depends on the specified collecting area of the assumed X-ray telescope, its exposure time, and redshift. We generate su ffi cient photons by  \nassuming a telescope with an energy-independent collecting area of 1000 cm 2 and an exposure time of 1000 ks. The photon-list is generated in the observed frame of the X-ray emitting gas cells and is corrected to rest frame energies. The LoS direction determines the event\'s position in the sky. We define the LoS along the x-axis within the lightcone. By applying a θ LC obs / 2 = 5 . 08 deg rotation along the z-axis and -ϕ LC obs / 2 = -2 . 32 deg rotation along the y-axis, we centre the y-z plane at (0 , 0) degrees. The photons generated by the gas particles are projected onto the sky; the resulting image is shown in Fig. 2.  \nWe also project the halo and subhalo positions on the sky, as shown in Fig. 3 for a redshift slice of z = 0 . 3. The X-ray events are then stacked around the projected halo / subhalo positions. The surface brightness profiles 7 are calculated S ( r ) as a function of the projected radius on the sky within the 0 . 5 -2 keV energy band. The number of photons with rest frame energy E [erg] in each radially outward bin r , N ( E , r ), is weighted by the area, A [kpc 2 ], of the 2D shell between r and r + dr, the fraction of photons collected from the source, f A [cm 2 ], and the exposure  \nTable 3. For every stellar mass bins, we present the best-fitting parameters of the model (see Eq. 7): S 0,:the central surface brightness; r c, the core radius; β , the exponent quantifying the slope of the profile; rs , the scale radius at which the slope changes to ϵ .  \n.  \n.  \n.  \n.  \nTable 4. For every halo mass bins, we present the best-fitting parameters of the model (see Eq. 7): S 0, the central surface brightness; r c, the core radius; β , the exponent quantifying the slope of the profile; rs , the scale radius at which the slope changes to ϵ .  \ntime, t exp [seconds],  \nS X( r ) = P 2 keV Ei = 0 . 5 keV N ( Ei , r ) Ei f A t exp A ( r ) " erg sec kpc 2 # . (4)  \nHere, f A is the fraction of the source photons collected by the synthesized telescope that has a collecting area A :  \nf A = A 4 π d L( z ) 2 (5)  \nwhere d L( z ) is the luminosity distance to the source.', '3.2. X-ray surface brightness profiles and their analytic modelling': 'The hot CGM surface brightness profile can be analytically described by the β model (Cavaliere & Fusco-Femiano 1976);  \n¯ S X , 0 . 5 -2 . 0 keV[ r ] = S 0        1 + r r c ! 2        -3 β + 1 2 , (6)  \nwhere S 0 is the central surface brightness, r c is the core radius at which the profile slope becomes steeper, and β is the exponent quantifying the slope of the profile. In cases where the outskirt steepens more than the slope defined for the inner radii by the β -model, following Vikhlinin et al. (2006), we introduce a new slope-parameter as follows:  \n¯ S X , 0 . 5 -2 . 0 keV[ r ] = S 0        1 + r r c ! 2        -3 β + 1 2 × " 1 + r r s ! γ # -ϵ/γ , (7)  \nwhere rs is the scale radius at which the slope changes to ϵ and γ defines the width of the transition region. We fix γ = 3 and restrict the priors on rs > rc and ϵ < 5, as suggested by Vikhlinin et al. (2006), in the fitting procedure.', '3.3. Prerequisite data products for quantifying projection effects': 'To obtain contributions from the large-scale structure, the locally correlated X-ray emission, we generate cubes and profiles with the CEN sim catalog for the two cases as follows:  \n- 1. R 200m cubes and profiles: every central galaxy within the halo catalog is assigned X-ray events within R 200m of the parent halo. We define these profiles as the intrinsic hot gas emission profiles.\n- 2. R ± 3Mpc, R ± 9Mpc and R ± 27Mpc cubes and profiles: all the X-ray events, intrinsic and locally extrinsic, irrespective of whether they belong to the galaxy but within 3 Mpc, 9 Mpc, and 27 Mpc of the source centre, are selected in 3D. We construct the profiles and cubes to quantify the impact of contamination from the local vicinity on the intrinsic source emission.  \nTo quantify contributions from the emission associated with misclassified centrals, we stack the emission around the galaxies in the SAT sim catalogue in the same stellar mass bin as the central galaxies. For galaxies in the three stellar mass bins: 10 10 . 5 -11 M ⊙ , 10 11 -11 . 25 M ⊙ , and 10 11 . 25 -11 . 5 M ⊙ , we construct a total galaxy sample, N T = N CEN + N SAT with varying fractions of satellites; namely f sat = 0 . 01 , 0 . 1 , and 0 . 3.', '4. Results': 'Here, we present the main results of this work. In Sec. 4.1, we discuss the outcome of fitting the analytical model to the stacked LC-TNG300 intrinsic profiles. In Sec. 4.2, we carry forward the current state-of-the-art to understand the impact of the locally correlated environment on the intrinsic X-ray surface brightness profile. Lastly, in Sec. 4.3, we show the prominence of the e ff ect due to misclassified centrals.', '4.1. Fitting analytic models to the intrinsic X-ray surface brightness profiles': 'Fig. 4 and Fig. 5 show the mean X-ray surface brightness profiles in three stellar mass and halo mass bins, respectively. We fit the surface brightness profiles with the analytic model introduced in Eq. 7 and provide the best-fit parameters in Tab. 3 and Tab. 4 for the stellar and halo mass bins, respectively. A simple β model, described by Eq. 6, describes the lowest stellar and halo mass bins. However, the β model does not describe the more massive stellar and halo mass bins. Therefore, we implement Eq. 7, which very well describes the LC-TNG300 profiles in the stellar and halo mass bins across all masses.  \nThe model has two break radii, the core radius rc and the scale radius rs , where rc ≪ rs . The scale radius rs a ff ects the profile at radii beyond R 500c, the mean R 500c of the stacked mass bin. The model also introduces two slope parameters, β , which influences the X-ray surface brightness profile at radii r < rs and ϵ , the slope that a ff ects the profile at r > rs . As R 500c is the radius used most commonly in observations, and β quantifies the shape  \nFig. 6. Mean X-ray surface brightness profiles in the stellar mass bin M ⋆ = GLYPH<2> 10 10 . 5 , 10 11 GLYPH<3> M ⊙ , corresponding to MW-like galaxies. The locally correlated large-scale structure contributions are shown by comparing the profiles obtained with photons selected within ± R 200 m ( purple ) and those obtained within ± 3 Mpc ( solid black ), ± 9 Mpc ( solid grey ), ± 27 Mpc ( dashed grey ) away from the halo centre. The crosses are from previous work by Oppenheimer et al. (2020), where they generate mock X-ray observations using TNG-100. The vertical dashed line at 242 kpc and the dotted line at 525 kpc is the mean R 500 c and R 200 m , respectively, of the 5 , 109 galaxies used in the mass bin with the shaded area corresponding to the minimum and maximum R 500 c values. The shaded uncertainties on the profiles represent the variance obtained by bootstrapping.  \n<!-- image -->  \nof the profile at radii ≤ R 500c, we discuss here the variations in β across the stellar and halo mass bins.  \nThe increasing stellar mass bins 10 10 . 5 -11 M ⊙ , 10 11 . 2 -11 . 5 M ⊙ , and 10 11 . 2 -11 . 5 M ⊙ result in decreasing values of β from 0 . 4, 0 . 28, and 0 . 25, respectively. The same trend holds from the halo mass bins, where increasing halo mass bins: 10 12 . 5 -13 M ⊙ , 10 13 -13 . 5 M ⊙ , and 10 13 . 5 -14 M ⊙ have decreasing values of β from 0 . 57, 0 . 30, to 0 . 24, respectively. One of the reasons for the decrease is attributed to the mass-dependence of the feedback prescriptions in the TNG300 model, i.e., stellar mode dominating at lower masses, and the kinetic and thermal modes of energy injection by AGN dominating galaxies with M ⋆ ≥ 10 10 . 5 M ⊙ (Weinberger et al. 2016). As discussed in Weinberger et al. (2018), the two AGN-related feedback channels depend on the black-hole (BH) accretion rates. For the high accretion rates, the thermal mode causes the gas cells close to the galactic centre to heat, eventually releasing the energy radiatively in X-rays. However, for lower accretion rates, the kinetic mode kicks in, depositing energy through winds and jets in random directions away from the BH. This causes gas heating and, hence, X-ray emission via cooling at a larger distance away from the galaxy centre, overall causing a flattening of the radial X-ray surface brightness profile as shown by the slope β in Fig. 4 and 5. This reasoning is further consolidated with studies detailing the e ff ects of the kinetic mode of BH feedback in the TNG model (Terrazas et al. 2020), its impact on the gas properties, i.e., temperature, entropy, density, and CGM frac-  \n020; Davies et al. 2020), and its correlations with X-ray emission (Truong et al. 2020; Oppenheimer et al. 2020; Truong et al. 2021; Ayromlou et al. 2023). Sorini et al. (2024) show a similar trend of decreasing slope for the gas density profiles with the SIMBA (Davé et al. 2019) suite of simulations. With simulation comparison projects like camels simulations (Villaescusa-Navarro et al. 2021), it is possible to study the correlation between X-ray emission around galaxies with di ff erent feedback implementations, the exploration of which we leave for a future study.  \nIn the following two sections, we focus on how the projection e ff ects result in deviations from the intrinsic profile. We discuss the e ff ects on the stellar mass bins as it is an observationally available mass proxy.', '4.2. Locally correlated environment': "For the MW-mass bin, M ⋆ = 10 10 . 5 -11 M ⊙ , with mean R 200m = 525 kpc, we generate cubes within ± 3 Mpc, ± 9 Mpc and ± 27 Mpc from the galaxy center. This corresponds to photons selected within ∼ 5 . 7 × R 200m, ∼ 17 × R 200m, and ∼ 51 . 4 × R 200m, respectively. Fig. 6 presents the impact of the locally correlated LSS for MW-like halos. We show the intrinsic profile that is obtained by selecting events within ± R 200 m of the galaxy centre in purple. We compare our result with previous work from Oppenheimer et al. (2020), shown with the crosses in Fig. 6. They predict the profile between 10 -242 kpc for a stellar mass sample  \nFig. 7. E ff ect of centrals ( purple ) and misclassified-centrals ( green ), in the stellar mass bin M ⋆ = GLYPH<2> 10 10 . 5 , 10 11 GLYPH<3> M ⊙ , on the total X-ray surface brightness profiles ( black ). The total sample is constructed such that there are 1% satellites and 99% centrals ( left panel ), 10% satellites and 90% centrals ( middle panel ), and 30% satellites and 70% centrals ( right panel ). The vertical dashed line at 242 kpc is the mean R 500 c of the 5109 galaxies used in the mass bin with the shaded area corresponding to the minimum and maximum values. The shaded green region is the uncertainty on the misclassified central profile obtained by bootstrapping.  \n<!-- image -->  \nFig. 8. E ff ect of centrals ( purple ) and misclassified-centrals ( green ), in the stellar mass bin M ⋆ = GLYPH<2> 10 11 , 10 11 . 25 GLYPH<3> M ⊙ , on the total X-ray surface brightness profiles ( black ). The total sample is constructed with 1% ( left panel ), 10% ( middle panel ), and 30% ( right panel ) satellites. The vertical dashed line at 369 kpc is the mean R 500 c of the 680 galaxies used in the mass bin with the shaded area corresponding to the minimum and maximum values.  \n<!-- image -->  \nFig. 9. E ff ect of centrals ( purple ) and misclassified-centrals ( green ), in the stellar mass bin M ⋆ = GLYPH<2> 10 11 . 25 , 10 11 . 5 GLYPH<3> M ⊙ , on the total X-ray surface brightness profiles ( black ). The total sample is constructed with 1% ( left panel ), 10% ( middle panel ), and 30% ( right panel ) satellites. The vertical dashed line at 484 . 5 kpc is the mean R 500 c of the 305 galaxies used in the mass bin with the shaded area corresponding to the minimum and maximum values.  \n<!-- image -->  \nin the mass range 10 8 . 2 -11 . 39 M ⊙ containing ∼ 400 galaxies. Their stellar mass sample is divided into low sSFR and high sSFR. We take the mean of the low sSFR and high sSFR profiles generated in their work and compare them with ours. The halo mass range corresponding to their stellar mass bin is 10 ∼ 12 . 3 -13 M ⊙ , which entails the mean halo mass M 200m = 10 12 . 7 M ⊙ , same as that of the MW mass bin used in our work. Oppenheimer et al. (2020) use the TNG-100 simulation and synthesize mock X-ray observations for individual halos as opposed to this work that uses TNG300 and synthesizes mock X-ray observations within the lightcone. Despite these di ff erences, this work's predicted X-ray intrinsic emission profiles are in good agreement with Oppenheimer et al. (2020).  \nWe find that increasing the volume by 4 π 3 (5 . 7 3 -1 3 ) R 3 200 m = 771 . 5 × R 3 200 m , i.e., by including events in ± 3 Mpc, boosts the Xray surface brightness profile at R 200 m by a factor of 5.2. We show this with the thick black line in Fig. 6. We find deviations from the true intrinsic profile due to events selected within ± 3 Mpc at ∼ 150 kpc, which corresponds to ≈ 0 . 6 × R 500 c and ≈ 0 . 3 × R 200 m for the MW-stellar mass bin.  \nWhen considering the events in ± 9 Mpc, i.e., increasing the volume used to compute the X-ray surface brightness profiles by 4 π 3 (17 3 -1 3 ) R 3 200 m = (2 . 0 × 10 4 ) R 3 200 m , we find that the X-ray surface brightness profile at R 200 m is boosted by a factor of 16.8 This is shown by the grey curve in Fig. 6. We find deviations from the true intrinsic profile due to events selected within ± 9 Mpc, at ∼ 100 kpc, which corresponds to ≈ 0 . 4 × R 500 c and ≈ 0 . 2 × R 200 m for the MW-stellar mass bin.  \nFor events selected in ± 27 Mpc or by including events in a volume of (8 . 2 × 10 4 ) R 3 200 m , the X-ray surface brightness profile at R 200 m is boosted by 47 . 3. This is shown by the light grey dashed curve in Fig. 6. The profile, in this case, remains unchanged only at radii ≤ 40 kpc, which corresponds to ≈ 0 . 2 × R 500 c and ≈ 0 . 08 × R 200 m for the MW-stellar mass bin.  \nIn conclusion, we show for the first time the e ff ect of the local environment on the true intrinsic profile of a mean MWlike stacked X-ray surface brightness profile. Namely, we find deviations from the true profile at ≈ 0 . 3 × R 200 m , ≈ 0 . 2 × R 200 m , and ≈ 0 . 08 × R 200 m by including events out to ∼ 5 . 7 × R 200m, ∼ 17 × R 200m, and ∼ 51 . 4 × R 200m, respectively. The black, grey, and dashed-grey curves in Fig. 6 show that increasing the integration volume swamps the features of the intrinsic profile at radii closer to the galaxy centre.  \nIn observations, one is sensitive to the complete line of sight towards the observer. Therefore, this e ff ect can be corrected by subtracting a background emission level determined empirically from the observed surface brightness at a large distance from the halos of interest (assuming spherical symmetry in the largescale emission). Nevertheless, given the setup used in this work, we can, for the first time, probe this e ff ect locally around the halo and show its stark impact in stacking experiments.", '4.3. The Effect of Misclassified Centrals': 'In simulations, we completely and accurately classify central and satellite galaxies within a given stellar mass bin. Using this to our advantage, here, we make a precise prediction of the average emission arising from stacking around satellite galaxies by considering the following fraction of satellite (or misclassified centrals) contaminating the total galaxy sample: 0 . 01, 0 . 1 and 0 . 3. We select these contamination fractions of satellites by bootstrapping over the entire satellite galaxy sample. Tab. 1 details the number of centrals and satellites in the three stellar mass  \nbins presented here. We show our findings in Fig. 7, 8, and 9, corresponding to stellar mass bins 10 10 . 5 -11 M ⊙ , 10 11 -11 . 2 M ⊙ , and 10 11 . 2 -11 . 5 M ⊙ , respectively.  \nFig. 7 quantifies the e ff ect of misclassified centrals on the total surface brightness profile, shown with the solid black line, in the MW-stellar mass bin of 10 10 . 5 -11 M ⊙ . We show the integrated profile obtained by stacking only the satellite galaxies misclassified as centrals in green and the intrinsic profile due to central galaxies in purple. The point at which the profile due to misclassified centrals ( green ) intersects the intrinsic central galaxy ( purple ) profile represents the point at which the e ff ect due to misclassified centrals contributes ∼ 50% to the total (centrals and satellites) emission. For a sample with 1% ( left panel ) and 10% ( central panel ) contaminating satellites, the e ff ect due to misclassified centrals dominates over the intrinsic central galaxy emission at radii ≥ 1 . 2 × R 500 c ( ∼ 300 kpc) and ≥ 0 . 2 × R 500 c ( ∼ 84 kpc), respectively. In a sample with 30% satellites ( right panel ), the e ff ect due to misclassified centrals dominates over the intrinsic central galaxy emission at radii ≥ 0 . 06 × R 500 c ( ∼ 34 kpc).  \nAnalogously, Fig. 8 quantifies the e ff ect of misclassified centrals on the total surface brightness profile, shown with the solid black line, for galaxies in the stellar mass bin 10 11 -11 . 2 M ⊙ . For a sample with 1% ( left panel ) contaminating satellites, we find that the e ff ect of misclassified centrals has negligible impact within radii ≤ R 500 c . For 10% satellite contamination ( central panel ), the e ff ect due to misclassified centrals dominates over the intrinsic central galaxy emission at radii ≥ 1 . 6 × R 500 c ( ∼ 600 kpc) and for 30% satellites contamination ( right panel ), the e ff ect due to misclassified centrals dominates over the intrinsic central galaxy emission at radii ≥ 0 . 8 × R 500 c ( ∼ 300 kpc).  \nFinally, Fig. 9 quantifies this e ff ect for the largest mass bin we are considering, 10 11 . 2 -11 . 5 M ⊙ . For a sample with 1% ( left panel ), 10% ( centre panel ) and 30% satellite contamination ( right panel ), the e ff ect due to misclassified centrals remains negligible at all radii ≤ R 500 c . More precisely, the 1% satellite contamination has a negligible impact on the total surface brightness profile. The 10% and 30% satellite contamination dominate over the intrinsic central galaxy profile at ≥ 2 × R 500 c ( ∼ 1000 kpc) and ≥ 1 . 4 × R 500 c ( ∼ 700 kpc), respectively.  \nAs expected, we find from Fig. 7, 8, and 9 that the e ff ect due to misclassified centrals becomes increasingly important as we probe lower stellar mass bins in stacking experiments. As we go to higher mass bins, the decreasing impact of the e ff ect of misclassified centrals is attributed to the satellites - of the same stellar mass bin - residing in less massive parent halos (see further explanation in Sec. 5).  \nFig. 10 presents the radial fraction of the X-ray emission from the intrinsic CGM - for four di ff erent levels of misclassified centrals contamination - in three stellar mass bins: 10 10 . 5 -11 M ⊙ , 10 11 -11 . 25 M ⊙ , and 10 11 . 25 -11 . 5 M ⊙ , respectively. We complement the conclusions from Fig. 7, 8, and 9 by showing that for MWlike galaxies, the e ff ect of misclassified centrals at the lowest contamination fraction of 0 . 01 results in the intrinsic central galaxy emission contributing only ∼ 62% of the total emission at R 500 c . This further deteriorates with the increasing fraction of misclassified centrals in the galaxy sample, where fractions of 0 . 1, 0 . 3, or 0 . 5 result in the intrinsic emission contributing ∼ 14%, ∼ 5% and ∼ 3% of the total emission at R 500 c , respectively. This e ff ect is less pronounced for the intermediate and most massive stellar mass bins of 10 11 -11 . 25 M ⊙ and 10 11 . 25 -11 . 5 M ⊙ compared to the MW-like stellar mass bin. In the case of the 10 11 -11 . 25 M ⊙ mass bin, for a fraction of misclassified centrals of 0 . 01, 0 . 1, 0 . 3, or 0 . 5, the central intrinsic emission  \nFig. 10. The fraction of X-ray emission from the intrinsic hot CGM for di ff erent levels of satellite contamination (misclassified centrals) in the total galaxy sample. The fraction of X-ray emission from the intrinsic hot CGM for misclassified central contamination fractions of 0 . 01 (dotted line), 0 . 1 (dash-dotted line), 0 . 3 (dashed line), and 0 . 5 (solid line) are shown for three di ff erent mass bins: MW-like galaxies with stellar-masses in 10 10 . 5 -11 M ⊙ (left panel), 10 11 -11 . 2 M ⊙ (middle panel), and 10 11 . 2 -11 . 5 M ⊙ (right panel). The shaded regions around each curve correspond to the uncertainty of the mean profile obtained by bootstrapping. The vertical black solid line at 242 kpc in the left panel, 369 kpc in the middle panel, and 484 . 5 kpc in the right panel corresponds to the mean R 500 c of the respective mass bins with the shaded area signifying the minimum and maximum values. We find that the contaminating e ff ect of the misclassified centrals on the stacked profiles increases when the stellar mass decreases.  \n<!-- image -->  \ncontributes ∼ 96%, ∼ 71%, ∼ 45% and ∼ 33% of the total emission at R 500 c , respectively. Analogously, for the stellar mass bin of 10 11 . 25 -11 . 5 M ⊙ , for a fraction of misclassified centrals of 0 . 01, 0 . 1, 0 . 3, or 0 . 5, the central intrinsic emission contributes ∼ 99%, ∼ 87%, ∼ 68% and ∼ 57% of the total emission at R 500 c , respectively. Therefore, we present a clear trend of the increasing importance of the e ff ect of misclassified centrals, not only due to the increasing fraction of satellite contamination in the galaxy sample but also due to the decreasing stellar mass bins.', '5. Discussion': 'This work uses a TNG300-based forward model for the hot CGM emission and presents, for the first time, the e ff ect of locally correlated large-scale structure around a halo and the e ff ect of misclassified centrals in stacked hot CGM galactocentric profiles. Our findings are vital in light of our explorations for studying X-ray emission around lower mass galaxies with stacking experiments. We divide the discussion of our results by focusing on the projection e ff ects due to the locally correlated environment, first, in Sec. 5.1 and second, the e ff ect of misclassified centrals in Sec. 5.2.', '5.1. Locally correlated environment': 'The locally correlated environment for MW-like stellar mass galaxies boosts the mean galactocentric X-ray emission at radii ≤ R 200 m . More precisely, at R 200 m the X-ray emission is boosted 5 . 2 × , 16 . 8 × and 47 . 3 × by including emission ± 3 Mpc, ± 9 Mpc, ± 27 Mpc away from the halo centre, respectively.  \nIncreasing the volume over which the intrinsic galactocentric profiles are measured leads to increased X-ray emission at smaller radii. This increased X-ray emission, in projection, is attributed to the local environment in which the galaxy resides. The upturn in the black line in Fig. 6, corresponding to including events within ± 3 Mpc ( ∼ 5 . 7 R 200 m ) of the galaxy centre, signifies the presence of other X-ray emitting halos ≳ 1 Mpc away from MW-like galaxies, i.e., ≈ 2 × R 200 m or ≈ 4 × R 500 c . This "upturn"  \nfeature is washed out as we start including events in even larger volumes, as shown by grey and dashed-grey lines (corresponding to events selected within ± 9 Mpc and ± 27 Mpc, respectively) in Fig. 6. This supports our finding that an observer starts probing the averaged hot gas emission from all galaxies in projection along the line of sight, drowning out the features due to the local environment when integrating events over larger volumes around a galaxy centre.  \nFurther improvements to study this e ff ect would involve including events to even larger radii around the galaxy. Given the limited area of the lightcone, 47 . 28 deg 2 , we are constrained in studying this e ff ect out to ± 27 Mpc. To quantify the effects of the locally correlated large-scale environment to even larger distances than those explored here, we need larger-volume lightcones, which is possible with larger cosmological hydrodynamical simulations like Magneticum (Dolag 2015), MillenniumTNG (Pakmor et al. 2023; Hernández-Aguayo et al. 2023), and FLAMINGO (Schaye et al. 2023). Additionally, future studies could study how di ff erent simulation feedback prescriptions can impact the trend observed in the X-ray surface brightness profiles by including events in larger volumes. It is particularly interesting to understand how the physics implemented in hydrodynamical simulations impacts the baryon spread around galaxies, i.e., the radius at which all the emissions converge to the mean background level in X-rays. We leave the study of these aspects to future work, which is made possible with the advent of projects like the camels simulations (Villaescusa-Navarro et al. 2021); see, e.g., Gebhardt et al. (2024) and Sorini et al. (2022).', '5.2. The Effect of Misclassified Centrals': 'We quantify the impact of the misclassified centrals (satelliteboost) residing in the same stellar mass bin as the central galaxies in Fig. 7, 8, and 9 for the stellar mass bins 10 10 . 5 -11 , 10 11 -11 . 2 , and 10 11 . 2 -11 . 5 , respectively. For satellite galaxies with stellar masses of MW-like galaxies, the emission around satellites dominates over central emission because the satellites of the stellar mass range 10 10 . 5 -11 M ⊙ reside in distinct halos with masses  \nM 200m = 10 11 . 9 -15 . 1 , corresponding to a mean halo mass M 200 m = 10 13 . 7 M ⊙ . We find that the satellite subhalos probe the emission of their parent halos, contributing to the stack with o ff -centred surface brightness profiles of their more massive parent halo. This contribution causes shallower slopes for the total emission (central and satellites) in the MW-stellar mass bin. In cases where 30%, 10%, or 1% of the satellites contribute to the total emission, we have shown that they dominate the measured total profile at radii ≥ 0 . 06 × R 200 m , ≥ 0 . 2 × R 200 m , and ≥ 0 . 3 × R 200 m . We further investigate the e ff ect of misclassified centrals in the more massive stellar mass bins of 10 11 -11 . 2 M ⊙ , and 10 11 . 2 -11 . 5 M ⊙ in Fig. 8 and Fig. 9, respectively. We find the e ff ect of misclassified centrals, with contamination fractions of 0 . 01, 0 . 1, and 0 . 3, less a ff ects the 10 11 -11 . 2 M ⊙ , and 10 11 . 2 -11 . 5 M ⊙ mass bins as opposed to the MW-stellar mass bin. We rea ffi rm this with the trend shown in Fig. 10, where the fraction of intrinsic emission at the mean R 500 c is lowest for the MW stellar mass bin for all contamination levels, as opposed to the more massive stellar mass bins.  \nTo understand why the MW-stellar mass bin is most a ff ected by the e ff ect of misclassified centrals, we first look at the mean halo masses of the satellites in the higher stellar mass bins of 10 11 -11 . 2 M ⊙ and 10 11 . 2 -11 . 5 M ⊙ . The satellites reside in mean distinct halos of ∼ 10 13 . 8 M ⊙ and ∼ 10 13 . 9 M ⊙ , respectively. This is a direct implication arising from the di ff erence in the stellarto-halo-mass relation for central and satellite galaxies (Shuntov et al. 2022). Given that the mean halo mass of the central galaxies in the LC-TNG300 stellar mass bins of 10 11 -11 . 2 M ⊙ , and 10 11 . 2 -11 . 5 M ⊙ is 10 13 . 3 M ⊙ and 10 13 . 6 M ⊙ , respectively, we require large fraction of contamination to see the e ff ect of misclassified centrals compared to that for the MW-stellar mass bin. More precisely, we find that the e ff ect of misclassified centrals dominates over the central galaxy emission at R 500 c with satellite contamination fractions, f cont , sat, of 0 . 02 , 0 . 28, and 0 . 72 for the stellar mass bins 10 10 . 5 -11 M ⊙ , 10 11 -11 . 25 M ⊙ , and 10 11 . 25 -11 . 5 M ⊙ , respectively. Hence, the contamination fractions probed here ( f cont , sat = 0 . 01 , 0 . 1 , and 0 . 3), which are closer to the realistic f cont , sat in observations, do not significantly a ff ect the stellar mass bins 10 11 -11 . 25 , and 10 11 . 25 -11 . 5 , respectively.  \nGiven these findings, we conclude that the contribution of satellites in MW-like galaxy samples containing satellites must not be neglected in stacking analysis. Observationally, to disentangle the contamination of the misclassified centrals, we must fit the profiles contributing to the intrinsic central galaxy jointly with the component capturing the e ff ect of misclassified centrals for a given fraction of satellite contamination. The analytical model encapsulating the central profile and the e ff ect of misclassified centrals will be addressed in future work.', '6. Summary': 'The main conclusions from this work are summarised as follows.  \n- 1. We present an analytical model (Eq. 7) that well-describes the intrinsic S X -profile in LC-TNG300 across the stellar mass bins 10 10 . 5 -11 M ⊙ , 10 11 -11 . 2 M ⊙ , and 10 11 . 2 -11 . 5 M ⊙ and halo mass bins of 10 12 . 5 -13 M ⊙ , 10 13 -13 . 5 M ⊙ , and 10 13 . 5 -14 M ⊙ . We provide the best-fitting parameters for the analytical model in Tab. 3 and Tab. 4 for the stellar and halo mass bins explored in this work.\n- 2. We carry forward the current state-of-the-art modelling analysis presented in Oppenheimer et al. (2020) by also showing the impact of the locally correlated environment on the measured X-ray surface brightness profiles.  \n- 3. We present, for the first time, the e ff ect of misclassified centrals in stacking experiments for three stellar mass bins: 10 10 . 5 -11 M ⊙ , 10 11 -11 . 2 M ⊙ , and 10 11 . 2 -11 . 5 M ⊙ . We find that the contaminating e ff ect of the misclassified centrals on the stacked profiles increases when the stellar mass decreases.\n- 4. For the MW-like galaxies, we conclude that the contribution of satellites (or misclassified centrals) can not be neglected in stacking analysis (see Fig. 10). In cases where 30%, 10%, or 1% of the satellites contribute to the total emission of MWlike galaxies, we have shown that they dominate the measured total S X profile at radii ≥ 0 . 06 × R 200 m , ≥ 0 . 2 × R 200 m , and ≥ 0 . 3 × R 200 m , respectively.  \nModelling observed CGM profiles and comparing them with simulations is crucial to constrain the di ff erent galaxy formation models and to understand how the feedback and physics prescriptions a ff ect the hot CGM profile. Current state-of-the-art cosmological hydrodynamical simulations are calibrated on the stellar mass function and successfully reproduce realistic galaxy populations. Despite this, Davies et al. (2020) show that EAGLE and IllustrisTNG predict di ff erent total gas mass fractions, a ff ecting the observed CGM properties at MW-masses. Similarly, Khrykin et al. (2024) shows how the hot gas is sensitive to the di ff erent feedback variants within SIMBA. The model and methodology presented here provide the machinery to compare the measured hot CGM profiles among current simulations in future works.  \nFuture X-ray missions, on the observation side, like Athena (Nandra et al. 2013), HUBS (Cui et al. 2020), LEM (Kraft et al. 2022) will push our current detection limits and provide us with (i) the spatial resolution to reach higher redshifts, (ii) better quantify point-like source contamination within the hot CGM, (iii) the spectral resolution to allow disentangling components via spectral fitting and (iv) the grasp to reach even fainter surface brightness levels. This work, focusing on modelling the projection e ff ects, is a step towards exploiting the information provided by the next-generation telescopes to better understand the intrinsic hot CGM emission. The prospects of this framework would be to extend it to interpret hot X-ray CGM measurements in stacking experiments by accounting for all projection e ff ects, i.e., not only the impact due to the local environment of the halo and the e ff ect due to misclassified centrals but also the emission from other X-ray sources such as AGN and XRB.  \nAcknowledgements. SS would like to thank Fulvio Ferlito, Matteo Guardiani, Christian Garrel, Emre Bahar, and Jeremy Sanders for all the useful scientific discussions.  \nforzamento delle Eccellenze (FARE) per la ricerca in Italia (R20L5S39T9).  \nComputations were performed on the HPC system Raven at the Max Planck Computing and Data Facility. We acknowledge the project support by the Max Planck Computing and Data Facility.', 'Data Availability': 'The lightcone generation code, LightGen , will be publicly available on GitLab 8 upon acceptance of this paper. 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