In Milgroms modified Newtonian dynamics MOND framework the dynamical mass of a galaxy is fully determined by its baryonic matter distribution. We fit the distribution of cold and hot gas halos focusing on hot gas around SDSS central galaxies using weak lensing signals from the DECaLS survey in MOND. The central galaxies are classified into two samples the total galaxies and starforming galaxies. We find that hot gas halo densities nearly follow Plummers profile for both samples across all mass bins. The rotation curves of the galaxy samples are also demonstrated. The efficiency of converting gas into stars MM  Mrm g is between 0.3 and 0.8 in all mass bins of the starforming sample which is higher than in the total sample. We also calculate gas mass using the satellite kinematics method in MOND. A constant mildly radial anisotropy or isotropy in satellite motion leads to good agreement between the satellite kinematics and weak lensing methods. Combining both methods we observe a baryonic mass to lineofsight velocity dispersion of satellites Mrm bsigmarm s relation. In addition we examine more sophisticated models in MOND including external field effects and OsipkovMerritt anisotropy profiles and find them unnecessary. A simple mildly radial constant anisotropic model or an isotropic model even when isolated already aligns with observations. The strong concordance between the two methods suggests that weak lensing signals reliably measure the dynamical mass of central galaxies and can constrain the distribution of missing baryons in galaxy clusters.
['Astrophysics - Astrophysics of Galaxies']
Measuring Mass of Gas in Central Galaxies using weak lensing and satellite kinematics in MOND
{'Measuring Mass of Gas in Central Galaxies using weak lensing and satellite kinematics in MOND': "<!-- image -->  \n<!-- image -->  \nLi Ma a , b Ziwen-Zhang a , b Huiyuan-Wang a , b and Xufen Wu a , b  \n<!-- image -->  \n<!-- image -->  \na Department of Astronomy, University of Science and Technology of China, Hefei 230026, China b School of Astronomy and Space Science, University of Science and Technology of China, Hefei 230026, China  \nAbstract. In Milgrom's modified Newtonian dynamics (MOND) framework, the dynamical mass of a galaxy is fully determined by its baryonic matter distribution. We fit the distribution of cold and hot gas halos, focusing on hot gas, around SDSS central galaxies using weak lensing signals from the DECaLS survey in MOND. The central galaxies are classified into two samples, the total galaxies and star-forming galaxies. We find that hot gas halo densities nearly follow Plummer's profile for both samples across all mass bins. The rotation curves of the galaxy samples are also demonstrated. The e ffi ciency of converting gas into stars, M ∗ / ( M ∗ + M g), is between 0.3 and 0.8 in all mass bins of the star-forming sample, which is higher than in the total sample. We also calculate gas mass using the satellite kinematics method in MOND. A constant, mildly radial anisotropy or isotropy in satellite motion leads to good agreement between the satellite kinematics and weak lensing methods. Combining both methods, we observe a baryonic mass to line-of-sight velocity dispersion of satellites ( M bσ s) relation. In addition, we examine more sophisticated models in MOND, including external field e ff ects and Osipkov-Merritt anisotropy profiles, and find them unnecessary. A simple, mildly radial constant anisotropic model or an isotropic model, even when isolated, already aligns with observations. The strong concordance between the two methods suggests that weak lensing signals reliably measure the dynamical mass of central galaxies and can constrain the distribution of missing baryons in galaxy clusters.", '1 Introduction': "1  \nObservations revealed that only less than 10% baryons of our Universe lie in galaxies, which is the so-called 'missing baryons' problem [see the review of 1]. The stellar component including main sequence stars, neutron stars, white dwarfs, substellar objects, and black holes only constitutes about 6% of baryons [2]. The baryons in gaseous form are thus important to understand the formation and evolution of galaxies and large-scale structures. On the other hand, the multiphase of gaseous components in galaxies are ideal tracers of the gravitational potential of galaxies. The measurements of the emission lines from the cold gas, including H α , HI, CO and maser emissions, are the most important methods to study the rotation curves of the spiral galaxies [see the reviews by 3, 4]. The cold molecular and atomic gas contributes a large portion to the circular velocities of galaxies. In the  \ncentral few kilo-parsecs of spiral galaxies, over 90% of the cold gas is molecular gas [5]. In contrast, at larger radii, the cold gas is predominantly HI gas [6]. Observations of HI gas have shown that rotation curves remain flat at these larger radii, extending to a few times beyond the optical disks [e.g., 6, 7]. Thus the distribution of HI gas in galaxies significantly influences the rotation curves at large radii. There are many e ff orts on mapping the radial distribution of HI gas in galaxies [e.g., 8, 9]. In these works, a homogeneous radial profile has been found in the outer regions of dwarf and spiral galaxies [10, 11].  \nIn addition to cold baryons such as stars, cold gas, and black holes within galaxies, observations indicate that the primary baryonic component in the low red-shift universe is the hot intergalactic medium [12]. Numerous groups have studied the radial distribution of X-ray emitting hot gas around the Milky Way and other external galaxies [13-15]. The X-ray profiles of the star-forming and quiescent galaxies have been measured out to a few hundred kilo-parsecs, revealing distinct di ff erences in their radial projected luminosity profiles [16]. These X-ray profiles are important for studying galaxy mass distribution and may explain a significant portion of the missing baryons [17-19].  \nDespite the missing baryons problem, the Λ cold dark matter (CDM) paradigms, supported by the Planck mission observations[20], indicate that the baryons only contribute 4 . 9% to the critical density of the universe. In contrast, the CDM contributes 26 . 4%, with the rest provided by dark energy. In the framework of Λ CDM, galaxies are hosted by dark matter halos. Galaxy-galaxy lensing (hereafter GGL) provides direct measurement of the total mass distribution of galaxies including the dark matter halos [21-23]. Recently, GGL signals have been utilized to derive the radial acceleration relation (RAR) of galaxies [24, 25]. Alternatively, the kinematics of satellite galaxies can also be used to probe the total mass distribution of central galaxies [e.g., 26-33]. Ref. [23] has introduced a method combining GGL and satellite kinematics to investigate the gas-to-stars conversion e ffi ciency in central galaxies. Scaling relationships between halo masses and galaxy properties including stellar masses have been further studied using the same method [33].  \nOn the other hand, as an alternative to the Λ CDM models, Milgrom's MOdified Newtonian Dynamics (MOND; 34) is proposed to solve the mass discrepancy problem without dark matter. The MOND theory has been tested on various scales ranging from a few parsecs to a few Gpc, and is found to be favored by many observations [e.g., 35-37]. In the framework of MOND, galaxies only consist of baryons and the dynamical mass of a galaxy is fully predictive once the baryonic distribution is known. Consequently, the galaxy-galaxy weak lensing shear signals directly relate to the baryonic mass distribution in MOND. Weak lensing can be served as a test for MOND [38-40] and its relativistic versions, such as TeVeS (proposed by ref. [41] and tested using GGL signals by ref. [42]). Using the GGL signals, ref. [43] found that the rotation curves for a sample of isolated galaxies are flat out to hundreds of kilo-parsecs, which is a strong support to MOND. Moreover, MONDhas also been tested using satellite kinematics [44-48].  \nGiven the direct relation between the dynamical and the baryonic masses in MOND, it is possible to predict the distribution of the undetected baryons by the observed dynamical mass. These baryons can be the hot gas around galaxies. Since both the GGL signals and the motion of satellite galaxies reflect the dynamical masses of the central galaxies, these two methods can be used to explore the hot gas distribution within the framework of MOND. In this work, we will use these two methods to model the distribution of gas components in galaxies in MOND, using the observational data from ref. [23]. The baryonic mass model for the galaxies, including stellar, cold, and hot gaseous components, is adapted to fit the GGL data for two samples of galaxies, i.e., the star-forming sample and the total sample containing quiescent and star-forming galaxies. The fraction between the gas and overall baryon mass is calculated for both samples in §2. Moreover, the hot gas mass in both samples  \nis also calculated by the line-of-sight (LoS) velocity dispersion profiles of the satellite galaxies in §3. A relation between the baryonic mass and the global LoS velocity dispersion of the satellites is studied in §4. More sophisticated models are incorporated in §5. We summarize our results in §6.", '2.1 Sample selection': 'The galaxy samples in ref. [23] are selected from the New York University Value Added Galaxy Catalog (NYU-VAGC; 49) of the Sloan Digital Sky Survey Data Release 7 (SDSS DR7; 50). The halo-based group catalog [51] identifies the central galaxies. Two samples of central galaxies are selected. One is the total population of 304,162 galaxies including the quiescent and star-forming galaxies. The other contains 129,278 star-forming galaxies. The Excess Surface Density (ESD) profiles are measured by the Dark Energy Camera Legacy Survey (DECaLS; 52) DR8 imaging data. The stellar masses of the galaxies are obtained from the SDSS photometry data. Each galaxy sample is divided into six sub-samples binned in stellar masses, M ∗ . The binning method segments the logarithm of M ∗ into equal bins of 0.5 dex. The ESD profiles of each sub-sample are obtained by stacking the ESD profiles of individual galaxies within that sub-sample. The mean stellar mass of each sub-sample is then easily calculated.', '2.2 Three-component baryonic mass model': "Abaryonic mass model for the central galaxies is required to reproduce the ESD profiles of the galaxy samples in MOND. We present a three-component spherically symmetric model here, including stars, a cold gas halo and a hot gas halo.  \nAs for the stellar component, we use Hernquist's mass density model as follow [53],  \nρ ∗ ( r ) = M ∗ 2 π ( r / a )( r + a ) 3 , (2.1)  \nwhere M ∗ is the stellar mass of a galaxy, a is the scale length which can be estimated from the e ff ective radius of the stellar component by a ≈ R e / 1 . 8153. The stellar mass and the e ff ective radius are measured from observations.  \nFor the HI gas component, the surface density profile was suggested by Eq. 9 in ref. [10]. Here we derive the three-dimensional mass density, ρ cg( r ), from the surface density distribution. It follows  \nρ cg( r ) = Σ 0 K 0( r / R s) π R s . (2.2)  \nIn the above equations, Σ 0 and R s are the central surface density and the scale length of the projected radius for the HI gas, respectively. K 0 is the zeroth-order modified Bessel function of the second kind.  \nThe mass density distribution for the hot gas halo, ρ hg( r ), follows a parametric power law proposed by ref. [54],  \nρ hg( r ) = ρ 0 h 1 + ( r / r c) 2 i -3 2 γ , (2.3)  \nwhere ρ 0 and r c represent the central density and core radius of the hot gas halo, respectively. The parameter γ is the power law index for the hot gas halo profile.  \nThe total baryonic mass distribution is the sum of the three components,  \nρ b = ρ ∗ + ρ cg + ρ hg . (2.4)  \nThe parameters of both HI gas and the hot gas halo models, including Σ 0 , R s , ρ 0 , r c , γ , will be fitted from the ESD profiles of the galaxy samples in the following §2.4.", '2.3 MONDfomulations': "As an alternative to the standard Λ CDM framework, MOND [34] predicts precisely the rotation curves of galaxies in the absence of dark matter halos by a modification to gravity, which follows  \ng N = µ ( X ) g , (2.5)  \nwhere X = g a 0 . When the strength of actual acceleration, g = | g | , is much greater than Milgrom's constant, a 0 = 3700 km 2 s -2 kpc -1 , the interpolating function µ → 1 . 0. In this limit, the actual acceleration approaches the Newtonian acceleration induced by the baryonic matter. In the weak field limit where g << a 0, µ → X . The formulation becomes  \ng N = µ ( X )( g -g ext) , X = | g -g ext | a 0 (2.6)  \nwhen there is an external field, g ext [34].  \nDespite the rotation curves, MOND has been tested from star cluster scale to large scales. The LoS velocity dispersion profiles for globular clusters in the Milky Way are flat [55-57], which has been predicted in MOND [58]. MOND o ff ers influential alternative interpretations of the kinematics and dynamics of galaxies [e.g., 59-61]. Moreover, MOND points to a radial acceleration relation (RAR) which links the dynamical acceleration and the Newtonian acceleration introduced by the pure baryonic matter [34]. Consequently, RAR indicates that the shape of rotation curves of galaxies has a tight correlation with the Newtonian circular velocity induced by baryons ( v b ) in MOND, since  \nv v b ! 2 = g r g N r ≃ 1 µ ( X ) . (2.7)  \nSuch a relation has been validated in various observations in both rotation-supported [62, 63] and early-type systems [64]. The RAR is a natural prediction in MOND but at odds in the Λ CDMframeworks [65, 66]. Ref. [24] applied GGL data from the Kilo-Degree Survey (KiDS-1000, 4 th release) to study the RAR relation out to hundred kpc scale, finding agreement with MOND predictions. Ref. [25] combined GGL with spatially resolved kinematic data of galaxies, validating the RAR out to a scale of several Mpc.  \nOn the scale of the cluster of galaxies, MOND faces several challenges. The lensing data requires supplementary dark matter even in the context of modified gravity, for clusters including the Bullet Cluster 1E0647-56 and the dark matter ringed cluster Cl0024 + 17 [67-69]. In the original relativistic MOND framework, the third peak of the angular power spectrum of the cosmic microwave background (CMB) radiation was a significant hurdle [70]. However, the new relativistic MOND theory [71, 72] has successfully matched the CMB temperature and polarization angular power spectra.  \nSeveral MOND formulations satisfy the conservation laws, such as AUQAL [73] and QUMOND [74]. The latter formulation is quasi-linear. The modified Poisson's equation reads  \n∇ 2 Φ = ∇ · [ ν (Y) ∇ Φ N ] , (2.8)  \nwhere Milgromian and Newtonian gravitational accelerations are denoted as Φ and Φ N, and Y = |∇ ϕ N | / a 0. Here ν (Y) is an interpolating function. When |∇ ϕ N | >> a 0, ν → 1 and when |∇ ϕ N | << a 0, ν → 1 / √ Y . The right-hand side of Eq. 2.8 is related only to the Newtonian gravitation induced by the observed baryonic matter. In the following sections, we will adopt the QUMOND formulation to investigate the distribution of gaseous components from the left-hand side of Eq. 2.8. We shall use the 'simple' form for the ν -function:  \nν ( Y ) = 1 / 2 + p 1 / 4 + 1 / Y . (2.9)  \nThe Poisson's equation can be reformulated as follows,  \n∇ 2 Φ = 4 π G ( ρ b + ρ PDM ) , ρ PDM = 1 4 π G ∇ · [( ν -1) ∇ Φ N] , (2.10)  \nwhere ρ b represents the baryonic density, and ρ PDM denotes the density of the phantom dark matter, which is determined by ρ b.", '2.4.1 GGL Method': "In the framework of QUMOND theory, the ρ b( r ) and ρ PDM( r ) profiles can be calculated from the modified Poisson's equation, Eq. 2.10. In spherically symmetric coordinates, the radial direction is segmented with scale intervals of 0 . 1 kpc within the radius of R ( i ) ∈ (0 , 2000] kpc and the polar direction is divided with angular intervals of 0.01 rad within the angle of θ ( j ) ∈ [ -1 . 57 1 . 57] rad. Once the baryonic density ρ ( r ) with undetermined parameters is given, the distribution of ρ PDM can be derived from Eq. 2.10.  \nIn a galaxy, Σ ( R ) can be obtained by integrating the 3D density profile ρ b along the LoS direction using Simpson's method, with the gas parameters Σ 0 , R s , ρ 0 , r c and γ to be determined  \nΣ ( R ) = Z ∞ -∞ GLYPH<2> ρ b( r ) + ρ PDM( r ) GLYPH<3> dz = R Z π/ 2 -π/ 2 ρ b( R , θ ) + ρ PDM( R , θ ) cos 2 θ d θ. (2.11)  \nThe ESD profile of a lens central galaxy is defined as  \n∆Σ ( R ) ≡ Σ ( R ) -¯ Σ ( R ) = Σ ( R ) -2 R 2 Z R 0 Σ ( R ' ) R ' d R ' (2.12)  \nwhere Σ ( R ) is the projected surface density of the galaxy at a projected radius R . ¯ Σ ( R ) is the average surface density within R .  \nWith the above method, the ESD profiles can be described as a function of the model parameters,  \n∆Σ ( R ) = f ( R , M ∗ , R e , Σ 0 , R s , ρ 0 , r c , γ ) . (2.13)  \nFigure 1 . The ESD profiles of the galaxy samples and the best fitting results using the three-component baryonic mass model. The blue and green symbols show the ESD profiles of the star-forming galaxies (first and third columns) and the total galaxy samples (second and fourth columns). The ESD profiles obtained from the best-fitting models, including our model (solid), the isothermal hot gas halo model (dashed), and the model with the existence of an external field (dotted), are shown by the blue and green lines for di ff erent galaxy samples.  \n<!-- image -->  \nAs aforementioned, the central galaxies are classified into two samples: the star-forming sample and the overall sample including both star-forming and quiescent galaxies. Each sample is further divided into six sub-samples binned in stellar masses. For each sub-sample of galaxies, we numerically fit their parameterized ESD profiles in MOND using a Bayesian method combined with MCMC sampling. We sample the posterior distribution of the parameters with the MCMC sampler emcee [75]. By incorporating the value of the logarithmic posterior probability density function into the code emcee , setting the sampling parameters ndim = 5, nwalkers = 50, and nsteps = 3000, we determine that the fitted value of γ is approximately 1 . 6 for all models. We use only the results from the last 1000 steps as the posterior samples of the parameters, and the Bayesian estimation of the uncertainty of derived quantities such as M g is obtained from the calculation for each set of sampled parameters.  \nT able 1 . The properties of star -forming and total g alaxy samples. T welv e properties, including (a) the stellar mass range, (b) the mean stellar mass, (c) the g as mass log M g / M ⊙ obtained from the GGL method, (d) the e ffi cienc y M ∗ / ( M ∗ + M g ) obtained from the GGL method, (e) the central surf ace density Σ 0 of HI g as in the unit of 10 10 M ⊙ kpc -2 , (f) the scale length R s of HI g as in the unit of kpc, (g) the central density ρ 0 of the hot g as halo in the unit of 10 5 M ⊙ kpc -3 , (h) the core radius r c of the hot g as halo profile in the unit of kpc, (i) the po wer la w inde x γ of the hot g as halo profile obtained from the GGL method, (j) the g as mass log M g / M ⊙ obtained from the satellite kinematics method, (k) the e ffi cienc y M ∗ / M b obtained from the satellite kinematics method, (l) the constant anisotrop y parameter β obtained from the satellite kinematics method, of each sample are listed in each column, respecti v ely . The errors for each property of the 12 sub-samples indicate the 16th and  \n.  \nproperty  \nthe  \nof  \nution  \ndistrib  \nposterior  \nthe  \nof  \npercentiles  \n84th  \nSublog M ∗ -range (a) log M ∗ (b) log M g , GGL (c) E GGL (d) Σ 0 (e) R s (f) ρ 0 (g) r c (h) γ (i) log M g , SK (j) E SK (k) β (l) sample (M ⊙ ) (M ⊙ ) (M ⊙ ) -(10 10 M ⊙ / kpc 2 ) ( kpc) (10 5 M ⊙ / kpc 3 ) ( kpc) -(M ⊙ ) --  \nSF 1 [8.84, 9.34] 9.14 9 . 52 + 0 . 41 -0 . 48 0 . 29 + 0 . 26 -0 . 15 0 . 03 + 0 . 06 -0 . 03 0 . 39 + 0 . 78 -0 . 27 0 . 13 + 0 . 43 -0 . 11 38 . 60 + 8 . 38 -12 . 66 1 . 62 + 0 . 03 -0 . 08 10 . 51 + 0 . 25 -0 . 45 0 . 04 + 0 . 07 -0 . 02 0 . 26 + 0 . 17 -0 . 18 SF 2 [9.34, 9.84] 9.64 10 . 03 + 0 . 26 -0 . 35 0 . 29 + 0 . 19 -0 . 11 0 . 10 + 0 . 21 -0 . 09 0 . 43 + 1 . 09 -0 . 26 0 . 18 + 0 . 65 -0 . 13 48 . 78 + 12 . 17 -17 . 39 1 . 62 + 0 . 04 -0 . 10 9 . 91 + 0 . 25 -0 . 44 0 . 35 + 0 . 25 -0 . 12 0 . 23 + 0 . 19 -0 . 16 SF 3 [9.84, 10.34] 10.13 9 . 76 + 0 . 29 -0 . 43 0 . 70 + 0 . 16 -0 . 16 0 . 37 + 0 . 63 -0 . 34 0 . 35 + 0 . 95 -0 . 23 0 . 02 + 0 . 08 -0 . 02 61 . 54 + 14 . 13 -21 . 55 1 . 61 + 0 . 04 -0 . 07 10 . 44 + 0 . 22 -0 . 40 0 . 33 + 0 . 22 -0 . 10 0 . 09 + 0 . 11 -0 . 06 SF 4 [10.34, 10.84] 10.61 10 . 45 + 0 . 14 -0 . 25 0 . 59 + 0 . 13 -0 . 08 0 . 82 + 1 . 74 -0 . 74 0 . 62 + 1 . 26 -0 . 29 0 . 03 + 0 . 20 -0 . 03 75 . 31 + 18 . 84 -34 . 90 1 . 62 + 0 . 03 -0 . 08 10 . 58 + 0 . 24 -0 . 48 0 . 52 + 0 . 25 -0 . 13 0 . 08 + 0 . 13 -0 . 06 SF 5 [10.84, 11.34] 11.03 10 . 55 + 0 . 26 -0 . 37 0 . 75 + 0 . 12 -0 . 13 2 . 18 + 5 . 53 -2 . 07 0 . 32 + 1 . 29 -0 . 22 0 . 04 + 0 . 08 -0 . 03 103 . 79 + 19 . 10 -32 . 56 1 . 62 + 0 . 03 -0 . 07 10 . 50 + 0 . 48 -nan 0 . 77 + 0 . 44 -0 . 24 0 . 12 + 0 . 15 -0 . 08 SF 6 [11.34, 11.82] 11.43 11 . 53 + 0 . 30 -0 . 40 0 . 44 + 0 . 22 -0 . 16 4 . 32 + 13 . 64 -4 . 09 0 . 49 + 1 . 84 -0 . 32 0 . 29 + 0 . 97 -0 . 22 128 . 35 + 31 . 25 -50 . 26 1 . 62 + 0 . 03 -0 . 06 12 . 06 + 0 . 81 -1 . 68 0 . 19 + 0 . 73 -0 . 15 0 . 27 + 0 . 16 -0 . 18 T otal 1 [8.84, 9.34] 9.14 9 . 84 + 0 . 40 -0 . 49 0 . 17 + 0 . 21 -0 . 09 0 . 04 + 0 . 06 -0 . 03 0 . 38 + 0 . 97 -0 . 27 0 . 29 + 1 . 01 -0 . 21 38 . 75 + 8 . 03 -13 . 50 1 . 61 + 0 . 04 -0 . 09 10 . 56 + 0 . 19 -0 . 36 0 . 04 + 0 . 04 -0 . 01 0 . 26 + 0 . 15 -0 . 18 T otal 2 [9.34, 9.84] 9.64 10 . 29 + 0 . 22 -0 . 31 0 . 18 + 0 . 13 -0 . 06 0 . 08 + 0 . 19 -0 . 07 0 . 50 + 1 . 02 -0 . 33 0 . 43 + 1 . 89 -0 . 28 46 . 58 + 11 . 61 -19 . 59 1 . 62 + 0 . 03 -0 . 07 10 . 29 + 0 . 19 -0 . 21 0 . 18 + 0 . 08 -0 . 06 0 . 16 + 0 . 17 -0 . 12 T otal 3 [9.84, 10.34] 10.15 10 . 27 + 0 . 19 -0 . 20 0 . 43 + 0 . 11 -0 . 10 0 . 44 + 0 . 64 -0 . 41 0 . 52 + 1 . 44 -0 . 23 0 . 19 + 4 . 23 -0 . 15 47 . 65 + 18 . 25 -28 . 05 1 . 62 + 0 . 04 -0 . 07 10 . 62 + 0 . 19 -0 . 32 0 . 25 + 0 . 16 -0 . 08 0 . 09 + 0 . 13 -0 . 07 T otal 4 [10.34, 10.84] 10.64 10 . 92 + 0 . 06 -0 . 07 0 . 35 + 0 . 03 -0 . 03 0 . 41 + 1 . 66 -0 . 36 0 . 96 + 1 . 84 -0 . 64 54 . 01 + 459 . 38 -51 . 09 13 . 03 + 18 . 73 -6 . 88 1 . 60 + 0 . 05 -0 . 10 11 . 02 + 0 . 15 -0 . 20 0 . 29 + 0 . 10 -0 . 07 0 . 05 + 0 . 08 -0 . 04 T otal 5 [10.84, 11.34] 11.09 11 . 45 + 0 . 04 -0 . 04 0 . 30 + 0 . 02 -0 . 02 0 . 64 + 3 . 15 -0 . 59 1 . 24 + 3 . 51 -0 . 77 37 . 38 + 94 . 65 -30 . 44 22 . 54 + 15 . 39 -7 . 77 1 . 62 + 0 . 04 -0 . 09 11 . 58 + 0 . 11 -0 . 14 0 . 25 + 0 . 06 -0 . 04 0 . 03 + 0 . 04 -0 . 02 T otal 6 [11.34, 11.84] 11.47 12 . 63 + 0 . 03 -0 . 03 0 . 06 + 0 . 00 -0 . 00 0 . 10 + 0 . 58 -0 . 09 4 . 53 + 13 . 88 -2 . 94 15 . 92 + 6 . 24 -4 . 59 77 . 68 + 13 . 40 -14 . 73 1 . 54 + 0 . 07 -0 . 18 12 . 53 + 0 . 08 -0 . 10 0 . 08 + 0 . 02 -0 . 01 0 . 06 + 0 . 08 -0 . 04", '2.4.2 Profile parameters': 'The best-fitting profiles are shown in solid curves in Fig. 1. The observed ESD profiles for 12 subsamples of galaxies are compared with the theoretically predicted ESD profiles in Fig. 1. We find that the ESD profiles predicted in MOND agree well with the observations, especially in the outer regions of the galaxies. Generally, the concordance between models and observations is more pronounced in the sample of total galaxies. A possible reason could be the much smaller errors in observations in this sample. Moreover, the parameters of the profiles of the cold and hot gas halos are determined. We present the gas profile parameters in the 6 th -10 th columns in Table 1.  \nFor the cold gas halos, in the sample of star-forming galaxies, the central surface density ( Σ 0) of the cold gas component exhibits an increase with the growth of stellar mass. The scale of the cold gas halo ( R s) varies little with changes in the stellar mass. In the total sample, the fitted parameters for cold gas density profiles exhibit similar trends to the stellar masses, except for the sub-sample Total 6. The central surface density of cold gas Σ 0 declines to 1 . 0 + 0 . 58 -0 . 09 × 10 9 M ⊙ kpc -2 . This indicates that the cold gas of this sub-sample is rather di ff use.  \nOur main focus is the hot gas halos. In the sample of star-forming galaxies, the core radius ( r c) increases as the stellar mass grows. However, we find that the power law index of the hot gas density profile, γ , is almost a constant of approximately 1 . 62 in all mass ranges of the star-forming galaxies. Unlike the star-forming sample, r c of the hot gas in the total sample does not monotonically increase with the growth of M ∗ . However, the value of γ remains ≈ 1 . 62 for the total sample of galaxies. This leads to ρ hg ∝ r -4 . 8 , which is close to a Plummer model [76] where ρ ∝ (1 + ( r 2 / r 2 p )) -5 / 2 ∝ r -5 . In the sub-sample of Total 6, γ ≃ 1 . 54. The marginally lower value of γ in this sub-sample implies a somewhat more extended hot gas distribution in the outer regions. However, the di ff erence in the density power law index is not distinct.  \nTo conclude, the density power law indices for both samples of galaxies (star-forming and total) remain nearly the same constant, irrespective of the stellar mass. This consistency suggests that the density distribution of the hot gas halo has a universal form in the outer regions, regardless of whether the galaxies are star-forming or quiescent, or their stellar masses.  \nIn the X-ray observations, the hot gas halo around the Milky Way is di ff use and extended, with γ = 0 . 64 -0 . 68 and peaked at γ = 2 / 3 [19], which indicates a much more extended hot gas halo within 115 kpc, and ρ X -ray ∝ r -2 when r >> r c . A MOND predicted gas halo has a more sharply declining profile outside the core radius, ρ hg ∝ r -5 . However, it is still too early to conclude that MOND has been ruled out. For comparison, we perform the ESD fitting again using an isothermal hot gas halo model (i.e., a fixed γ = 2 / 3) and show the results in dashed curves in Fig. 1. We find that the isothermal hot gas halos also predict ESD profiles that agree well with most subsamples of observations within the 1 σ error range, including the sub-samples with the similar stellar mass of the Milky Way. In some sub-samples, we cannot distinguish the fitted ESD profiles from a Plummer-like to an isothermal gas halo, such as sub-sample SF 3-6, and Total 3. A possible reason for the MOND predicted values of γ significantly di ff ering from the observations is that the large uncertainties in the observed ESD data allow for a wide range of γ values. In several sub-samples with small observational errors, particularly in total 6, our sharply declining γ = 1 . 6 models exhibit better agreement with the data than the γ = 2 / 3 models. In addition, another possible reason could be that the virial radius in MOND is much further than 115 kpc of the X-ray observations. We fit the ESD profiles within 115 kpc of the 12 sub-samples, and the density power law indices become significantly smaller. For instance, in sub-samples such as SF 2, Total 2, and Total 3, ρ hg ∝ r -2 within this truncation radius. This suggests that the distribution of hot gas halos is relatively extended within  \n100 kpc, while the density rapidly decreases at larger radii. Our prediction o ff ers a new probe to test gravity. Future observations on the distribution of X-ray gas halos around central galaxies will provide a valuable comparison.', '2.4.3 Gas masses and conversion e ffi ciency': 'Once the parameters for the ρ b( r ) profiles are determined, the probability distribution of the gas mass M g can be derived from the posterior distribution of the model parameters. The mass of the gaseous component is defined as the total mass of HI gas and the hot gas halo within the virial radius, r vir, of the central galaxy. The relation between the virial mass of the phantom dark matter halo plus the baryonic mass of a galaxy in the MOND framework can be expressed as [77]  \nM vir ≡ pr 3 vir = v 2 c r vir / G , (2.14)  \nwith v c being the circular velocity at the infinity radius of a galaxy. In MOND, v c = ( GM b a 0) 1 / 4 where M b = M ∗ + M g is the total mass of baryons in a galaxy. Thus the virial mass of a central galaxy reads  \nM vir = ( Ga 0 M b) 3 / 4 p -1 / 2 G -3 / 2 , (2.15)  \nwhere p = 4 3 π × 200 ρ crit. ρ crit = 3 H 2 8 π G is the critical density of the Universe with a Hubble constant of H = 67 . 8 km s -1 Mpc -1 . Thus M g and r vir are calculated iteratively by solving Eq. 2.8. Once the residual of ∆ M g = | M g , i + 1 -M g , i | / M g , i is less than 0 . 1%, the calculation is converged. Here M g , i and M g , i + 1 are the masses of gas computed from the i th and ( i + 1) th iteration.  \nWe listed the median values of gas masses and the corresponding errors of the 12 sub-samples of galaxies in the 4 th columns of Table 1. In general, M g increases as the growth of the stellar mass in both samples of galaxies. The e ffi ciency of gas converted into stars, denoted as E , can be calculated by E = M ∗ / ( M ∗ + M g). The values of E for the 12 sub-samples computed from the GGL are presented in the 5 th column of Table 1. For the star-forming galaxies, E ∈ [0 . 29 , 0 . 75], and for the total sample, E ∈ [0 . 06 , 0 . 43]. The values of E here di ff er from those in ref. [23], where the baryonic mass of a central galaxy is defined as a product of a constant cosmic fraction f b and the dark matter halo mass M h. In MOND, the fraction between baryonic mass and dynamical mass, say, the mass discrepancy, is not a constant for all galaxies. This is the reason why the values of E are di ff erent from the results in ref. [23]. However, we find that the E values for the star-forming sample of galaxies are higher than those for the total sample at the same stellar mass. Such a conclusion agrees well with that in ref. [23].', '2.4.4 Rotation curves': 'The rotation curves of the 12 sub-samples of galaxies can be calculated from the GGL-fitted baryonic density, ρ b. We demonstrate the rotation curves in Fig. 2. The error range of 1 σ confidence interval is induced by the uncertainties of the five parameters in the ESD fitting, computed using a Monte Carlo method. The rotation curves are almost flat from a few tens of kilo-parsec for all the star-forming sub-samples and most of the total sub-samples. The flat rotation curves in large radii are in good agreement with the observations [e.g., 25]. There is one exception, namely sub-sample Total 6. The rotation curve for this sub-sample is mildly rising out to 100 kpc. However, within the radial range of 100-200 kpc, the rotation curve becomes asymptotically flat. The reason for the rising rotation curve within 100 kpc in sub-sample Total 6 might be that this sub-sample is the most massive, thus the matter distribution is more extended compared to other sub-samples.  \nFigure 2 . Rotation curves calculated within the framework of MOND for the 12 sub-samples of galaxies. The colored shadow areas represent the error range of 1 σ confidence interval.  \n<!-- image -->  \nAs aforementioned, the value of γ is smaller in this sub-sample, implying that hot gas distribution declines more gradually in the outer region compared to other sub-samples. Moreover, E = 0 . 06 + 0 . 00 -0 . 00 for this sub-sample, indicating a gas-dominated system. Additionally, the central projected density of cold gas in this sub-sample is also significantly lower compared to sub-samples Total 4 and Total 5. In summary, the above results suggest that sub-sample Total 6 is likely dominated by a more extended hot gas halo. This more extended distribution of hot gas is consistent with the gradually rising rotation curve observed in the outer region of galaxies in this sub-sample.', '3 Satellite kinematics for determining galactic baryonic mass': 'Aside from gravitational lensing, satellite kinematics can also probe into the total mass distribution of central galaxies. In the deep MOND regime, a simple formula  \nσ 2 LoS = 2 9 ( GM b a 0) 1 / 2 (3.1)  \nexists for the LoS dispersion of an isolated low surface density system. A modified formula is proposed [78] in the presence of an external field. In a more realistic context, the LoS velocity dispersion profiles of satellites, σ LoS, are anisotropic. The anisotropic velocity profiles have been introduced in both MOND and CDM frameworks to compare with the observations [44-48]. The observations revealed that anisotropic velocity dispersion is required to agree with the observed kinematics of the satellites in both frameworks. Here we will begin with constant anisotropies for the satellites around their central galaxies in the 12 sub-samples.  \nFigure 3 . The PDFs of the LoS velocity di ff erence of satellite galaxies and the best fitting results. The blue and green symbols with error bars represent the PDFs of satellites around the star-forming galaxies (upper row), and around the total galaxies (lower row), respectively. The grey symbols represent the data points excluded by the | ∆ v | < 1 . 7 v 200 cut. The dotted lines represent the best-fitting results of the PDFs using our modified Gaussian model in Eq. 3.5. The results of star-forming and total samples in di ff erent stellar mass bins are shown in di ff erent columns.  \n<!-- image -->', '3.1 Method to calculate the LoS velocity dispersion': 'There are some pioneering works on testing MOND by the σ LoS profiles of satellite galaxies [45, 48]. Wewill follow their method to calculate the σ LoS profiles of the satellites in the framework of MOND. Weneed to solve the Jeans equation. Under the assumption of spherical symmetry, the Jeans equation can be written in the form of  \nd σ 2 d r + σ 2 2 β + α r = -g ( r ) , (3.2)  \nwhere σ ( r ) is the radial velocity dispersion, β = 1 -σ 2 ⊥ / 2 σ 2 is the anisotropy parameter, α = | d ln n ( r ) / d ln r | is the slope of the satellite number density, n ( r ). Assuming the satellites are test particles, the general solution of the Jeans equation can be expressed as [45, 48]  \nσ 2 ( r ) = 1 χ ( r ) n ( r ) Z ∞ r χ ( r ) n ( r ) g ( r )d r , (3.3)  \nwhere χ ( r ) = exp h 2 R r 0 β ( r ) i r -1 d r . The Milgromian gravitational acceleration g ( r ) here is calculated by g ( r ) = -∇ Φ from Eq. 2.8, and approximately the Newtonian acceleration g N = GM b / r 2 with the central galaxy being a point mass particle. Such an approximation is reasonable since the distances of satellites are much larger than the typical size of central galaxies. Taking the anisotropy β as a constant, we obtain χ ( r ) = r 2 β . The σ LoS profile can be obtained by integrating σ ( r ) along the LoS direction. The projected pressure of a cluster of galaxies can be written as [Eq. A7 in 79]  \nΣ n ( R ) σ 2 LoS ( R ) = 2 Z ∞ R 1 -β R 2 r 2 ! n ( r ) σ 2 ( r ) r d r √ r 2 -R 2 , (3.4)  \nwhere Σ n ( R ) is the projected number density of the satellites along the LoS. Now we can derive the σ LoS profile from Eqs. 3.3-3.4 with the undetermined M b.  \nFigure 4 . The baryonic mass estimation from GGL and satellite kinematics. In each panel, The symbols represent the baryonic mass estimation of the SF (blue, filled) and total (green, open) galaxy samples from the GGLmethod. The error bars of the symbols represent the 16th and 84th percentiles of the posterior distribution. The solid lines represent the baryonic mass estimation of the SF (blue) and total (green) galaxy samples from the satellite kinematics method. The shaded regions represent the 16th and 84th percentiles of the posterior distribution. The first three panels represent the results for di ff erent cut values: | ∆ v | < 2 v 200, | ∆ v | < 1 . 5 v 200, | ∆ v | < 1 . 7 v 200, respectively. The rightmost panel displays the mass relation when the σ LoS of satellites follows the Osipkov-Merritt anisotropy.  \n<!-- image -->  \n∗  \n/circledot  \n∗  \n/circledot  \n∗  \n/circledot  \n∗  \n/circledot', '3.2.1 Method to fit the PDFs of relative velocities': 'The baryonic mass of each galaxy sub-sample can be estimated from the PDFs of the LoS velocity di ff erence of the satellites. We used the PDFs of ∆ v of the satellites for the 12 sub-samples in ref. [23]. The fitting for the PDFs of ∆ v follows  \nA √ 2 πσ LoS Σ n ( R ) exp        -∆ v 2 2 σ 2 LoS ( R )        + d . (3.5)  \nHere the Gaussian term with variable dispersion stands for the real satellites, and the constant term d represents the foreground interlopers which are not gravitationally bound to the cluster. In the previous work in the framework of Λ CDM [23], they selected satellites around the central galaxies following the criteria of | ∆ v | ≤ 3 v 200, R p ≤ r 200 and M s < M ∗ . Here, ∆ v and R p are the LoS relative velocity and the relative projected distance between the satellite and the central galaxy, respectively. The circular velocity at the virial radius, r 200, of the Navarro-Frenk-White (NFW; 80) dark matter halo, derived from lensing data, is denoted as v 200. They then used a Gaussian profile plus a constant to fit the probability distribution functions (PDFs) of ∆ v of the satellites as observed results.  \nHowever, note that the criterion | ∆ v | ≤ 3 v 200 in ref. [23] is not strict enough to exclude all the interlopers, as further pointed out by ref. [33]. A new criterion | ∆ v | < | ∆ v | cut was suggested to reduce the contamination of the interlopers. Stable fitting results were achieved with | ∆ v | cut in the range of 1 . 3 v 200 to 1 . 8 v 200 [33].  \nIn this work, we test three di ff erent values of | ∆ v | cut: | ∆ v | cut = 2 v 200 , 1 . 5 v 200 and 1 . 7 v 200. We take the truncation of | ∆ v | cut at 1 . 7 v 200 for the PDFs of the LoS velocity di ff erence of the satellites as an example. The PDFs of ∆ v can be expressed as a function of 3 parameters p [ ∆ v | ⃗ θ ( M b , α, β )]. We use the MCMC sampler emcee again to sample the posterior distribution of the baryonic mass, M b, and the anisotropy parameter, β , with a fixed satellite number density slope α = 2 obtained from  \nobservations [81]. The best fitting results to the PDFs of ∆ v are shown in Fig. 3. The fitted PDFs go through precisely the observed data points except for three sub-samples, SF 1, Total 1, and Total 6, due to the relatively poor quality of data in these three sub-samples. With the best fit to the PDFs of ∆ v , we obtain the mass of the gaseous component using the method of satellite kinematics, denoted as M g , SK = M b -M ∗ and shown in the 11 th column of Table 1. The conversion e ffi ciency using this method can also be calculated and is listed in the 12 th column of Table 1.  \nThe e ffi ciency E calculated using satellite kinematics is generally higher for the star-forming galaxies than the total galaxies in all mass bins. Similar to the results obtained from the GGL method, the global trend is that E peaks around M ∗ = 10 10 . 5 M ⊙ and decreases towards both the low and high mass ends. The maximal e ffi ciency is 0 . 751 + 0 . 125 -0 . 128 for the SF 5 sample, and the E values for the massive star-forming galaxies with M ∗ > 10 10 M ⊙ all exceed 0.4, which indicates that the star formation in these galaxies is very e ffi cient. The stellar component is the dominant mass component in these massive star-forming galaxies, which agrees with ref. [23].', '3.2.2 Gas mass obtained from the GGL and SK methods': 'The masses of gas and E obtained from two independent methods, i.e., the GGL and the satellite kinematics, can be cross-checked. The values from the two methods show good agreement for subsamples with small errors in their observed ESDs. However, the deviation between M g , GGL and M g , SK becomes significant when the errors of the observed ESDs are large, particularly in sub-samples SF 1, SF 3, SF 6, Total 1 and Total 3. In addition, the errors of the observed PDFs are large in subsamples of SF 1, SF 6, and Total 1. These combined errors result in discrepancies in the baryonic masses of these sub-samples when using the two methods, with deviations in the logarithmic gas mass exceeding 0 . 3dex. In the rest of the sub-samples, the deviations are less than 0 . 13dex. This agreement suggests that when the quality of the observed ESDs and PDFs is su ffi cient, the gas mass estimation using the GGL method is reliable. However, when the errors in the observed ESDs are significant, alternative methods such as satellite kinematics are necessary to verify the reasonableness of the gas mass estimation.  \nFurthermore, the values of anisotropy for the 12 sub-samples of galaxies have been fitted individually in §3.2.1 while computing M g , SK and σ LoS. The values of β are displayed in the 13 th column of Table 1. For the star-forming galaxies, β ∈ [0 . 08 , 0 . 27]; while for the total sample of galaxies, β ∈ [0 . 03 , 0 . 26]. The 12 sub-samples show mildly radial anisotropy. The galaxies are nearly isotropic or mildly radially anisotropic for the total sample of galaxies with M ∗ > 10 10 . 34 M ⊙ , where β ≤ 0 . 06. The star-forming galaxies are more radially anisotropic within the same stellar mass range. In the earlier works by [45, 48], unnaturally strongly radial anisotropy ( β ≥ 0 . 6) at r > 200 kpc in MOND is required to agree with the SDSS observations, which is di ff erent to our findings in this work.  \nAs aforementioned, the validity of the satellite kinematics method largely depends on the selection of true satellites. In the first three panels of Fig. 4, we show the comparison of overall baryonic mass M b calculated from the GGL and satellite kinematics methods. The baryonic mass estimated from the latter method is computed with relative velocities between the satellites and central galaxies cut in three ranges: | ∆ v | < 2 v 200, | ∆ v | < 1 . 5 v 200 and | ∆ v | < 1 . 7 v 200, following the procedure introduced in §3.2.1. The values of the reduced chi-squared statistics between the two methods, χ 2 n , within the three velocity cuts, are 6 . 95 , 34 . 16 and 9 . 21, respectively. The fitting to the PDFs of relative velocities agrees better when the observations are truncated at | ∆ v | < 1 . 7 v 200 than at | ∆ v | < 1 . 5 v 200, and the results remain stable when adopting the | ∆ v | < 2 v 200 truncation although more observed data  \npoints of the PDFs of | ∆ v | are included. Thus in the following section, the values of M g , SK will use the ones calculated within the velocity truncation of | ∆ v | < 1 . 7 v 200.  \nFurthermore, the M ∗ -M b relation in Fig. 4 is approximately monolithic in both methods. While the gas mass predicted through GGL does not exhibit a monolithic increase with stellar mass, the overall baryonic mass does. In Fig. 4, we find that the baryonic mass in the total sample tends to be higher than that in the star-forming sample at a given M ∗ by using both methods, implying a lower fraction of gas in the star-forming galaxies. This is consistent with the conclusion in the Λ CDM framework [23, 33].  \nIn summary, the central galaxies have three main components, stars, cold and hot gas halos. The baryonic mass distribution fully determines the gravitational field of the galaxies. In the GGL method, light from the source galaxies is deflected by the dynamical mass, including baryons and the accompanied phantom dark matter halos of the central galaxies. In the method of satellite kinematics, the satellites move in the Milgromian gravitation field dominated by the central galaxies. Our results indicate that the above two methods are consistent when predicting the mass of overall baryonic matter, thereby obtaining the yet-to-be-observed gas mass.  \nThe agreement between the above two methods indicates that, within the MOND framework, GGLsignals are reliable predictors of the dynamical mass of central galaxies. Quite recently, ref. [25] developed a new method to study the rotation curves from the ESDs of GGL. They also found that the RAR obtained from GGL is consistent with the RAR derived from the HI kinematic. Their GGL method extended the study of RAR to much lower acceleration regimes of galaxies. Thus, GGL can provide a new way to probe the gravitational field around galaxies and is also a good test of the MONDtheory.', '4 The baryonic mass-to-LoS velocity dispersion relation': "A halo-mass-to-satellite-velocity-dispersion ( M h -σ s) relation was shown in ref. [23]. In MOND, there is no dark matter halo in a galaxy; however, a similar scaling relation is expected to exist. This scaling relation is between the baryonic mass of the central galaxy and the σ LoS of the satellites, denoted as the M b -σ s relation. In this section, we will compare our model predicted σ s values with the observed values obtained from ref. [23]. The baryonic masses of galaxies M b are obtained by using the method of GGL in §2. The reason for using these masses is that the three-component baryonic model for galaxies is more precise in the GGL method, especially in small radii. In contrast, a central galaxy is simplified to a point mass particle when using the satellite kinematics method.  \nAccording to Eq. 3.4, once the form of anisotropy is given, σ LoS( R ) can be obtained. The mean value of σ LoS for satellites, σ s, around a central galaxy within r vir follows  \nσ 2 s = R r vir δ Σ n ( R ' ) σ 2 Los ( R ' )dR ' R r vir δ Σ n ( R ' )dR ' , (4.1)  \nwhere δ = 10 kpc.  \nWeaim to demonstrate the relationship of M bσ s in Fig. 5. In the left panel, initially, we present the observed, model-independent values of σ s and their errors for 12 sub-samples, as reported by [23], fitted using the Gaussian distribution of the PDFs of | ∆ v | . The GGL-predicted M b and these observed σ s values suggest M b ∝ σ 3 . 88 s , approximately M b ∝ σ 4 s . Next, we use the satellite kinematics method, using Eqs 3.3-3.4, to calculate the σ s and the corresponding errors to the given M b 1 .  \nFigure 5 . Baryonic mass-satellite velocity dispersion relation. The open and filled symbols in the vertical axes represent observed satellite velocity dispersion for the star-forming and total galaxy samples, respectively, from ref. [23]. The baryonic masses in the horizontal axes are calculated from the GGL method. Di ff erent colors indicate samples of di ff erent stellar mass bins the same as those adopted in Fig. 2. The error for M b represents the 16th and 84th percentiles of the posterior distribution from MCMC sampling. The blue curve with shade represents the σ s versus M b predicted in MOND constant anisotropic models. The Osipkov-Merritt anisotropy profiles are introduced when computing σ s in the middle and right panels, incorporating an isolated environment and an external field of 0 . 01 a 0, respectively. Isotropic models in isolation (the left and middle panels) and an external field (the right panel) are displayed as black curves.  \n<!-- image -->  \n/circledot  \n/circledot  \n/circledot  \nThese kinematics-fitted σ s and errors to the given M b are presented with a blue curve and shadow area, respectively. The model-predicted M bσ s relationship aligns well with the model-independent observed σ s versus the GGL-predicted M b relation.  \nIn addition, an isotropic model has been introduced into the satellite kinematics method (Eqs. 3.3-3.4) to calculate σ s. This M bσ s relationship is shown in a black solid curve, which also agrees with the observed σ s at the given M b.  \nNotably, in determining the M bσ s relationship, we assume constant anisotropies for satellites in each sub-sample and consider galaxy clusters in isolation. Despite these simplifications, the fitted relation aligns well with observations, making the simplifications reasonable. We will further introduce more sophisticated models, including an Osipkov-Merritt anisotropy profile [82, 83] and an external field for σ LoS of satellites, in the following section.", '5.1 The Osipkov-Merritt anisotropy model': 'A widely used model for the anisotropy profile to describe the motion of satellite galaxies is the Osipkov-Merritt anisotropy model [82, 83],  \nβ ( r ) = r 2 r 2 + r 2 a , (5.1)  \nwhere r a is the anisotropy radius. In this model, β ≈ 0 when r ≪ r a and β ≈ 1 when r ≫ r a , representing an isotropic central region and a purely radially anisotropic outer region of a system.  \nUsing the method described in §3.1-3.2, with β following the Osipkov-Merritt anisotropy profile, we can derive the radial velocity dispersion profiles of the satellites. The PDFs of ∆ v can then be expressed as a function of 3 parameters p [ ∆ v | ⃗ θ ( M b , α, r a )]. After MCMC sampling, the baryonic mass estimates are shown in the fourth panel of Fig. 4, with a PDF fitting result in the upper left sub-panel. Introducing Osipkov-Merritt anisotropy results in a sharp PDF peak near zero velocity due to radial satellite orbits at large radii. Generally, the masses calculated from GGL and satellite kinematics methods are consistent.  \nThe M bσ s relation for the Osipkov-Merritt models is studied in the second panel of Fig. 5. Compared to constant β models, Osipkov-Merritt models predict higher σ s for a given M b due to radial orbits at large radii. Thus the predicted σ s do not agree as well with the observed data as the constant anisotropy models. In our best fit Osipkov-Merritt models, r a ∈ [500 , 1000] kpc, beyond which orbits become radially anisotropic for the 12 sub-samples. The observed σ s suggests simple, constant, mildly radial anisotropy for the motion of satellites, rather than the Osipkov-Merritt models with strongly radial anisotropy at large radii. Our findings di ff er from refs. [45, 48], which require strongly radial anisotropy beyond 200 kpc in MOND.', '5.2 External field e ff ect': "In the framework of MOND, the internal acceleration is determined by both the internal baryonic distribution and the presence of a uniform external field [e.g., 34]. That is known as the violation of the strong equivalence principle (SEP), as shown in Eq. 2.6. According to this equation, internal acceleration is deduced by the existence of a constant external field, leading to a sharply declining rotation curve in the external field-dominated region [84]. Consequently, the phantom dark matter halo is truncated at a certain radius in systems where the external field is dominant [65]. Such an external field e ff ect (EFE) has been tested in various works, such as explaining the absence of dark matter in open clusters [34] and dwarf galaxies [e.g., 85-87]. Moreover, simulations have demonstrated that the mass of phantom dark matter halo in a satellite galaxy varies as it moves along a radial orbit, due to the spatial dependence of the external field induced by the central galaxy [65, 88]. As a result, in a collision, the gravitation perturbation introduced by a dwarf galaxy is weakened by the gravitational field from the target galaxy [89]. The external field causes σ LoS to decrease in large radii, thus it is important to study the EFE in this work.  \nTo include the external field, the equation of QUMOND needs to be rewritten as follows,  \ng ( r ) + g ext = ν (Y ' ) GLYPH<2> g N(r) + g N , ext GLYPH<3> , with Y ' = | g N(r) + g N , ext | a0 , (5.2)  \nwhere g N , ext and g ext are the Newtonian and Milgromian accelerations induced by the source of the external field, respectively.  \nThe samples selected in this work are central galaxies, and the external fields are introduced by their nearby massive galaxies or large-scale structures. Considering the typical scale of a cluster is a few Mpc, the acceleration from an external field, g ext, caused by a massive nearby central galaxy with M b = 6 × 10 11 M ⊙ (which is the upper limit of baryonic mass in our samples of galaxies), is approximately √ GM b a 0 / 2 Mpc ≃ 0 . 01 a 0. The g ext caused by the nearby massive objects is generally weaker than 0 . 01 a 0. For instance, the Milky Way galaxy su ff ers an external field from the M31 of this strength.  \nWe fit the ESD profiles by introducing a g ext of 0 . 01 a 0 for the central galaxies and display them with dotted curves in Fig. 1. The presence of an external field causes the ESD profiles to drop faster  \nin the outermost regions of galaxies in the low-mass sub-samples, including SF 1-3 and Total 1-3, opposite to the e ff ect of the two-halo term in the Λ CDMframework. In these low-mass sub-samples, the external fields are typically much weaker than 0 . 01 a 0. If the external fields were stronger, it would imply the presence of a massive galaxy with a stellar mass of ≈ 10 12 M ⊙ within 1 -2 Mpc of these low-mass 'central galaxies'. This scenario contradicts the definition of central galaxies, as they would instead be identified as satellites. Therefore, such an external field is not necessary for the lowmass sub-samples. In sub-sample SF 4, the ESD profile behaves the same as those in the low-mass sub-samples when g ext is introduced. The stellar mass of this sub-sample is similar to the Milky Way. The rapidly dropping ESD profile at large radii agrees better with the observations, compared to the case of the isolated model. On the other hand, the external field does not change the model-predicted ESD profiles in the rest of the sub-samples.  \nDue to the violation of the SEP, the strength of the internal acceleration, g , depends on the angles θ between the external and internal acceleration vectors. In this study, we follow the methodology outlined by ref. [48], where the angles θ for a galaxy embedded in an external field are averaged to maintain spherical symmetry. Consequently, the equations used to compute σ LoS in §3 can be used while incorporating the external field:  \ng N( r ) = Z π 0 q ( g N + g N , ext cos θ ) 2 + ( g N , ext sin θ ) 2 d θ. (5.3)  \nThe strength of internal acceleration, g , is solved by Eqs. 5.2-5.3. Then both σ Los and M b are obtained following the same method in §3.1-3.2.  \nWe present the M bσ s relationship in a model incorporating an external field of g ext = 0 . 01 a 0 and an Osipkov-Merritt anisotropy (refer to the right panel of Fig. 5). When compared to the isolated Osipkov-Merritt anisotropy model (the middle panel), the σ s values for the low-mass sub-samples are significantly suppressed. Consequently, the fitted M bσ s relation aligns well with the observations in the low-mass sub-samples. However, at the high-mass end, the external field does not su ffi ciently decrease σ s for a given M b. Therefore, an external field of g ext = 0 . 01 a 0 combined with an OsipkovMerritt anisotropy can reproduce most of the observed M bσ s relation 2 , except at the high mass end. But this is only theoretically consistent, as introducing an external field at the low-mass end is meaningless, as previously mentioned. In addition, the σ s values in the same mass range are calculated for an isotropic model combined with g ext = 0 . 01 a 0 (the solid black curve in the right panel of Fig. 5). The isotropic model within an external field reproduces the observed M bσ s relation in all mass ranges. Similarly, this is only theoretically meaningful.  \nIn summary, isolated models with constant mildly radial anisotropies or isotropy predict an M bσ s relation that aligns well with observations. When incorporating an external field, an isotropic model also agrees with observations but is only meaningful solely from a theoretical perspective. However, introducing Osipkov-Merritt anisotropy to isolated models does not match the observed M bσ s relation. While the addition of an external field induced by nearby massive objects improves the fit, discrepancies remain at the high-mass end. Thus sophisticated models such as the OsipkovMerritt anisotropy or the external fields are unnecessary. The simple, isolated models with mildly radial anisotropy or isotropy in §3 are su ffi cient to explain the observed data in MOND.", '6 Conclusions': "In the framework of MOND, the baryonic mass distribution of galaxies fully determines the gravitational fields. In this study, we use two methods - galaxy-galaxy weak lensing and satellite kinematics - to estimate the baryonic mass of the galaxy sub-samples in MOND, based on observational data across a wide radial range of galaxies. The samples of galaxies are divided into two: the star-forming and the total samples. Each sample of galaxies is further segmented into six sub-samples according to their stellar mass ranges.  \nUsing the GGL method, we have successfully obtained the density profile parameters for cold and hot gaseous halos for each sub-sample. We have found that in all of the sub-samples of galaxies, the power law index of the hot gas density distribution is almost a constant, γ = 1 . 6. Thus the density of the hot gas halos is close to the Plummer density distribution. This seems distinct from the X-ray observations of the hot gas halo around the Milky Way Galaxy, in which γ ≈ 2 / 3 [19]. However, after fitting the ESD profiles with a fixed value of γ ≈ 2 / 3, we find that it is indistinguishable from the fitted ESD profiles with γ = 1 . 6 for the sub-samples of galaxies in similar stellar mass ranges of the Milky Way, due to the large uncertainties of the ESD data. Thus in these stellar mass ranges of subsamples, our fitted density profiles agree with the X-ray observations. A di ff erent prediction of the hot gas halo density profile is at the high mass end of the total sample of galaxies, sub-sample Total 6. In this sub-sample, the power law indices of γ = 1 . 6 and γ = 2 / 3 predict significantly di ff erent ESD profiles (solid and dashed curves in Fig. 1, respectively), and only the former one agree with the observations with small uncertainties. Thus MOND predicts a more sharply declining density profile for the hot gas halos in massive central galaxies compared to those in galaxies with stellar masses similar to the Milky Way.  \nAccording to the density distribution profiles, the M g and the r vir of the central galaxies can be computed iteratively. The conversion e ffi ciency from gas into stars, E , in the 12 sub-samples has been derived. We have found that the values of E are generally higher for the star-forming galaxies than those for the total sample of galaxies at the same stellar mass. This agrees with what has been found in the Λ CDM framework [23]. Moreover, we have calculated the rotation curves of the 12 sub-samples of galaxies. The rotation curves tend to be flat beyond a few tens of kpc for most subsamples, except for sub-sample Total 6. In Total 6, the rotation curve appears to be flat from a larger radius of hundred kpc, since E is low for this sample and the average core radius for gas halos in this sub-sample is out to approximately 80 kpc. Since our rotation curves are theoretical predictions in MOND, they naturally follow the RAR relation described in Eq. 2.7.  \nFurthermore, we have calculated the M b for the 12 sub-samples of galaxies using the satellite kinematics. Meanwhile, we have obtained the σ LoS( r ) profiles with constant anisotropy for each subsample. The baryonic masses obtained from the satellite kinematics agree with those from the GGL method. Combining the two methods, we have demonstrated that a M bσ s relation exists in MOND.  \nFinally, more sophisticated models including an EFE and the Osipkov-Merritt anisotropy profile have been studied. We have found that if there are nearby massive objects that provide a g ext of 0 . 01 a 0, M b predicted from the satellite kinematics methods still agree with those from the GGL method. But this holds significance only in theoretical terms. When introducing the Osipkov-Merritt anisotropy, the M bσ s relations do not match the observations. A simple isotropic model in isolation or an external field of 0 . 01 a 0 aligns well with observations. However, the external field of 0 . 01 a 0 introduced to the low-mass central galaxies contradicts the definition of 'central galaxies' and is therefore meaningless. Thus complex models such as incorporating the Osipkov-Merritt anisotropy  \nor an external field are not necessary. A simple, isolated MOND model with mildly radial anisotropy or isotropy is su ffi cient to explain the observed data.  \nThe strong concordance between these two methods suggests that, within the MOND framework, GGL signals serve as reliable indicators of the dynamical mass of central galaxies. The GGL can be used to constrain the distribution of missing baryons around galaxies, and thus provide a probe to test the gravity.", 'Acknowledgments': "We thank Dr. HongSheng Zhao for the suggestions for the Osipkov-Merritt anisotropy models in the early stage of this work. X.W. thanks the financial support from the Natural Science Foundation of China (Number NSFC-12073026 and NSFC-12433002) and 'the Fundamental Research Funds for the Central Universities' (WK3440000004), which made this research possible. HYW is supported by CAS Project for Young Scientists in Basic Research (No. YSBR-062). 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