The dynamics of three nearby E0 galaxies in refracted gravity
AAstronomy & Astrophysics manuscript no. Elliptical_Galaxies_SLUGGS_All © ESO 2021February 26, 2021
The dynamics of three nearby E0 galaxies in refracted gravity
V. Cesare , , A. Diaferio , , and T. Matsakos Dipartimento di Fisica, Università di Torino, via P. Giuria 1, 10125, Torino, Italy, e-mail: [email protected] Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Torino, Torino, ItalyReceived < date > / Accepted < date > ABSTRACT
We test whether refracted gravity (RG), a modified theory of gravity that describes the dynamics of galaxies without the aid of darkmatter, can model the dynamics of the three massive elliptical galaxies, NGC 1407, NGC 4486, and NGC 5846, out to „ R e ,where the stellar mass component fades out and dark matter is required in Newtonian gravity. We probe these outer regions withthe kinematics of the globular clusters provided by the SLUGGS survey. RG mimics dark matter with the gravitational permittivity,a monotonic function of the local mass density depending on three paramaters, (cid:15) , ρ c , and Q , that are expected to be universal. RGsatisfactorily reproduces the velocity dispersion profiles of the stars and red and blue globular clusters, with stellar mass-to-light ratiosin agreement with stellar population synthesis models, and orbital anisotropy parameters consistent with previous results obtained inNewtonian gravity with dark matter. The sets of three parameters of the gravitational permittivity found for each galaxy are consistentwith each other within „ σ . We compare the mean (cid:15) , ρ c , and Q found here with the means of the parameters required to model therotation curves and vertical velocity dispersion profiles of 30 disk galaxies from the DiskMass survey (DMS): ρ c , and Q are within1 σ from the DMS values, whereas (cid:15) is within 2.5 σ from the DMS value. This result suggests the universality of the permittivityfunction, despite our simplified galaxy model: we treat each galaxy as isolated, when, in fact, NGC 1407 and NGC 5846 are membersof galaxy groups and NGC 4486 is the central galaxy of the Virgo cluster. Key words.
Gravitation - Galaxies: kinematics and dynamics - Galaxies: individual: NGC 1407, NGC 4486, NGC 5846 - Darkmatter - Surveys - Methods: statistical
1. Introduction
The observed kinematical properties of the luminous matter ingalaxies suggest that „ ´
90% of the mass of the galaxiesis dark (Del Popolo 2014). This result consistently fits withinthe standard cosmological model, where the properties of theUniverse on large scales, including the temperature anisotropiesof the cosmic microwave background (CMB) (Planck Collabo-ration et al. 2020), the large-scale distribution of galaxies (Do-delson et al. 2002) and the abundance of light elements (Cyburtet al. 2016) imply that the matter content of the Universe is dom-inated by cold dark matter.The standard cosmological model, however, su ff ers fromsome tensions both on large scales and on the scale of galaxies.On large scales, the discrepancy between the value of the Hub-ble constant, H , derived from observations in the local Universeand the value derived from the CMB (Verde et al. 2019), the un-likely features in the CMB temperature anisotropies (Schwarzet al. 2016), and the deficiency of Li (Mathews et al. 2020) haverecently challenged the standard model. On the scale of galax-ies, the tensions have been long-lasting: in disk galaxies, fine-tuned interactions between baryonic and dark matter are nec-essary to describe the radial acceleration relation or the bary-onic Tully-Fisher relation (McGaugh 2020); the missing satel-lite problem, the plane of satellite galaxies around the MilkyWay, and the cusp-core problem in dwarf galaxies pose addi-tional challenges (see e.g. Kroupa 2012; Del Popolo & Le Del-liou 2017; de Martino et al. 2020, for reviews). Moreover, thesearch for the elementary particles constituting the dark matterremains inconclusive (Tanabashi et al. 2018). On the scale of galaxies, a number of models beyondthe standard model have been suggested. Dark matter par-ticles might not be cold and collisionless, but rather self-interacting (Tulin & Yu 2018), fuzzy (Hui et al. 2017), axion-like (Marsh 2016), or a superfluid (Ferreira et al. 2019). Al-ternatively, the theory of gravity might break down on galaxyand larger cosmic scales and dark matter might thus be unnec-essary. Modified gravity theories proposed in the literature arenumerous: for example, f p R q gravity, where the Ricci scalar R in the Einstein-Hilbert action is replaced by an arbitrary func-tion of R (Nojiri & Odintsov 2011) or conformal gravity, where R is replaced by the contraction of the fourth-rank Weyl ten-sor (Mannheim 2019). However, both modified dark matter mod-els and modified gravity theories have encountered a numberof challenges. For example, neither f p R q nor conformal grav-ity appear to be able to yield the correct abundance of light ele-ments (Azevedo & Avelino 2018; Elizondo & Yepes 1994). Con-formal gravity also appears to su ff er from a severe fine tuningin the description of the kinematics of disk galaxies (Campig-otto et al. 2019) and requires a too hot intergalactic medium ingalaxy clusters (Horne 2006; Diaferio & Ostorero 2009). Mod-ified Newtonian dynamics (MOND) has been proved to be themost successful phenomenological model on the scale of galax-ies (Sanders & McGaugh 2002; McGaugh 2020), but attemptsto build a covariant formulation of MOND remain challeng-ing (Famaey & McGaugh 2012; Milgrom 2015; Hernandez et al.2019).Matsakos & Diaferio (2016) suggested Refracted Gravity(RG), a theory of gravity where the Poisson equation is modi-fied by the introduction of the gravitational permittivity, a mono-tonic increasing function of the local mass density. RG appears Article number, page 1 of 22 a r X i v : . [ a s t r o - ph . GA ] F e b & A proofs: manuscript no. Elliptical_Galaxies_SLUGGS_All to properly describe the observed phenomenology on the scaleof galaxies and clusters of galaxies, with the role of dark mattermimicked by the gravitational permittivity (Matsakos & Diafe-rio 2016; Cesare et al. 2020). In addition, RG provides a straight-foward covariant extension within the family of the scalar-tensortheories (Sanna et al., in preparation). In the covariant extension,the role of the gravitational permittivity is played by the scalarfield. This same scalar field is also responsible for the acceleratedexpansion of the Universe. Thus, RG has the attractive feature ofunifying the two dark sectors of the Universe into a single scalarfield (Carneiro 2018; Bertacca et al. 2010).In the weak-field limit, RG has two distinctive properties:in flat systems, like disk galaxies, the gravitational permittivitybends the lines of the gravitational field towards the system, thusdetermining a boost of the gravitational field in the plane perpen-dicular to the minor axis of the system. In spherically symmet-ric systems, the field lines remain radial, but the gravitationalfield is enhanced by the inverse of the gravitational permittiv-ity. In Cesare et al. (2020), we investigate the dynamics of 30disk galaxies from the DiskMass Survey (DMS, Bershady et al.2010). These galaxies are almost face-on and they thus enablethe measurements of both the rotation curve and the profile of thestellar velocity dispersion perpendicular to the plane of the disk.Cesare et al. (2020) prove that the bending of the gravitationalfield lines occurring in RG is su ffi cient to describe the kinemat-ics of these 30 DMS galaxies without resorting to dark matter.This description depends on three parameters appearing in thegravitational permittivity. A single set of parameters describesthe entire sample of 30 galaxies, supporting the expectation thatthese parameters should be universal. These encouraging resultson flat systems motivate us to test whether RG is also able todescribe the dynamics of spherical systems with the same set ofparameters. In other words, we wish to probe that the boost ofthe gravitational field can be set by the gravitational permittivityalone, independently of the redirection of the field lines.Here, we illustrate the result of this test on three E0 galax-ies, that are approximately spherical. We consider NGC 1407,NGC 4486, alias M87, and NGC 5846 from the SLUGGS sur-vey (Pota et al. 2013; Brodie et al. 2014; Forbes et al. 2017).In elliptical galaxies, the baryonic matter within 1 R e , the ef-fective radius at half projected luminosity, su ffi ces to describethe galaxy dynamics with Newtonian gravity (e.g. Dekel et al.2005; Mamon & Łokas 2005; Proctor et al. 2009; Cappellariet al. 2013). To probe RG, we thus need kinematic informationin the outer regions, where Newtonian gravity breaks down un-less dark matter is assumed to exist. X-ray emitting gas, plane-tary nebulae, and globular clusters (GCs) have been adopted askinematic tracers in these regions, out to „ R e , where the stel-lar luminosity fades out (Danziger 1997; Mathews & Brighenti2003; Werner et al. 2019; Pota et al. 2013; Pulsoni et al. 2018).In addition, GCs usually appear separated into two distinct pop-ulations, according to their colour. Red GCs are generally morespatially concentrated and have smaller velocity dispersions thanblue GCs (Geisler et al. 1996; Ashman & Zepf 1998; Bassinoet al. 2006; Brodie & Strader 2006; Faifer et al. 2011; Straderet al. 2011; Forbes et al. 2012; Lee et al. 2008; Pota et al. 2013).By adopting GCs as tracers of the velocity field in the outer re-gions of ellipticals, we thus actually have two tracers rather thanone and they can thus more strongly constrain the model. TheSLUGGS survey provides the ideal dataset for our test of RG.Studying the outer regions of ellipticals might be, however,particularly insidious. Here, we will model each galaxy as an https://sluggs.swin.edu.au/Start.html isolated system. However, ellipticals tend to live in dense en-vironments: NGC 1407 and NGC 5846 are within groups andNGC 4486 is the central galaxy of the Virgo cluster. Therefore,their outer regions are subject to the gravitational field of nearbygalaxies and of the hosting system as a whole. Nevertheless, wewill see that our RG model can satisfactorily describe the overalldynamics of these elliptical galaxies.Section 2 summarises the main features of RG. Sections 3and 4 describe the photometric and spectroscopic data thatwe use in our analysis. In Sect. 5, we illustrate our RG dy-namical model, and in Sect. 6 we report our results. Weconclude in Sect. 7. We adopt a Hubble constant H “
73 km s ´ Mpc ´ (Riess et al. 2016) throughout the paper.
2. Refracted gravity
RG is a classical theory of modified gravity where dark matteris mimicked by a gravitational permittivity, (cid:15) p ρ q , a monotonicincreasing function of the local mass density ρ . The gravitationalpermittivity appears in the modified Poisson equation ∇ ¨ r (cid:15) p ρ q ∇ φ s “ π G ρ, (1)where φ is the gravitational potential and G the gravitational con-stant; the permittivity (cid:15) p ρ q has the asymptotic values (cid:15) p ρ q “ " , ρ " ρ c (cid:15) , ρ ! ρ c , (2)where 0 ă (cid:15) ď ρ c is a critical density.According to Eqs. (1) and (2), when the density is muchlarger than ρ c , we recover the Newtonian Poisson equation ∇ φ “ π G ρ . On the contrary, in low-density regions, the grav-itational permittivity enhances the gravitational field. RG pre-dicts a di ff erent behaviour for the gravitational field in sphericaland flattened systems. For spherical systems, the integration ofEq. (1) yieldsd φ d r “ GM p r q (cid:15) p ρ q r , (3)where M p r q is the total mass of the system within radius r . TheRG gravitational field has the same direction and r -dependenceas in the Newtonian case, but it is enhanced by the inverse of thepermittivity. For flat systems, the lines of the gravitational field are fo-cussed towards the mid-plane of the source: the expansion of thedivergence of Eq. (1), B (cid:15) B ρ ∇ ρ ¨ ∇ φ ` (cid:15) p ρ q ∇ φ “ π G ρ, (4) Equation (3) shows that the gravitational field is r ´ in the vac-uum, where both M p r q and (cid:15) p ρ q are constant. This behaviour of the RGgravitational field does not necessarily contradict the observed velocitydispersion profiles of elliptical galaxies (Jiménez et al. 2013; Durazoet al. 2017, 2018) and of GCs (Hernandez & Jiménez 2012; Hernandezet al. 2013a; Durazo et al. 2017; Hernandez & Lara-D I 2020), or theexternal ρ r ´ mass density profiles of spherical stellar systems (Her-nandez et al. 2013b) that suggest a gravitational field falling o ff as r ´ .No astrophysical system is completely isolated and exactly satisfies theconditions M p r q “ const and (cid:15) p ρ q “ const in its outer regions. There-fore, in principle, the permittivity (cid:15) p ρ q can determine an r ´ behaviourof the gravitational field in each of these systems depending on the ac-tual density field of its environment.Article number, page 2 of 22. Cesare, A. Diaferio, and T. Matsakos: The dynamics of three nearby E0 galaxies in refracted gravity shows that the gravitational potential φ depends on both the den-sity ρ and on its variation ∇ ρ . RG predicts that, when interpretedin Newtonian gravity, the flatter the system, the larger its massdiscrepancy, because the redirection of the field lines increaseswith the flatness of the source. This prediction might be consis-tent with the claimed correlation between the dark matter con-tent of elliptical galaxies and their ellipticity (Deur 2014, 2020).Similarly, although it still requires a quantitative confirmation,this prediction might also be connected with the di ff erent darkmatter content of dwarf galaxies and GCs: albeit of similar stel-lar mass, the dwarfs are spheroidal and dark matter-dominatedin Newtonian gravity, whereas the GCs are spherical and devoidof dark matter (e.g. Strigari et al. 2008; Sollima & Nipoti 2010).In this work, we solve Eq. (3) for three approximately spher-ical galaxies to determine whether the gravitational permittivitycan describe the dynamics of spherical systems independentlyof the redirection of the field lines. In Eq. (3), the mass den-sity ρ and, consequently, the cumulative mass profile M p r q ofeach galaxy are associated with the baryonic matter alone. Forthe gravitational permittivity, according to Matsakos & Diaferio(2016) and Cesare et al. (2020), we adopt the smooth step func-tion (cid:15) p ρ q “ (cid:15) ` p ´ (cid:15) q tanh « ln ˆ ρρ c ˙ Q ff ` + . (5) Q regulates the steepness of the transition between the Newto-nian and the RG regimes: the larger Q , the steeper the transition.With this formulation, RG has three free parameters that we ex-pect to be universal: (cid:15) , ρ c , and Q .
3. Photometric data
In this section, we describe the observables that enter our massmodel of each galaxy: the surface brightness of the stars, the sur-face number densities of the GCs populations, the mass densityof the hot X-ray emitting gas, and the mass of the central super-massive black hole (SMBH). Table 1 lists the quantities charac-terising the three galaxies and the colour cut that we adopt inSect. 4.2 to separate the GC samples into the red and blue popu-lations.NGC 1407, NGC 4486, and NGC 5846 are at redshift z ă . k -correction in the photometricmeasures and in the distance modulus m ´ M “ r D p pc qs´
5, with m and M the apparent and absolute magnitudes of thesource. Similarly, to convert the radial coordinate R projected onthe sky from angular units, in arcsec, to physical units, in kpc, weadopt the relation, valid in a non-expanding Euclidean geometry, R p kpc q “ . ˆ ´ D p kpc q R p arcsec q , (6)with D the distance to the galaxy. For elliptical galaxies, the ra-dial coordinate R is R “ d qx ` y q , (7)where q is the minor-to-major axis ratio of the galaxy on the sky,and the x and y coordinates are oriented along the galaxy majorand minor axes, respectively (e.g. Cappellari 2008; Cappellariet al. 2013; Napolitano et al. 2009; Pota et al. 2013, 2015). For the surface brightness of the stars, we adopt the models de-rived by Pota et al. (2015) for NGC 1407 and by Scott et al.(2013) for NGC 4486 and NGC 5846.
The stellar surface brightness profile of NGC 1407 is mea-sured in the B -band with the Hubble Space telescope / ACS (Spo-laor et al. 2008; Rusli et al. 2013) and in the g -band with theSubaru / Suprime-Cam (Pota et al. 2013). The B and g -bandslargely overlap; therefore, Pota et al. (2015) derive a unique sur-face brightness profile in the B -band by a proper transformationof the g -band data. NGC 1407 has ellipticity ε “ ´ q “ . I ˚ p R q “ I e exp ´ b n s «ˆ RR e ˙ n s ´ ff+ , (8)where the e ff ective radius R e encloses half of the total luminosityof the stellar distribution, I e is the surface brightness at radius R e , n s is the Sérsic index, and b n s “ n s ´ ` n s ` n (9)(Ciotti & Bertin 1999). After deconvolving for the seeing, Potaet al. (2015) derive R e “ p ˘ q arcsec, I e “ . ˆ L d , B arcsec ´ , and n s “ . ˘ . Scott et al. (2013) measure the surface brightness of NGC 4486and NGC 5846 in the ugriz bands from the Sloan Digital SkySurvey DR7 (Abazajian et al. 2009) and the Wide Field Cam-era on the 2.5-m Isaac Newton Telescope at the Roque delos Muchachos observatory. Scott et al. (2013) model the two-dimensional map of the surface brightness in the r -band; thisband reduces the dust contamination and provides images withthe optimal signal-to-noise value (Cappellari et al. 2013). Theyadopt the axisymmetric Multi-Gaussian Expansion (MGE) ap-proach (Emsellem et al. 1994), which yields the surface bright-ness map (Cappellari 2008; Cappellari et al. 2013) I ˚ p x , y q “ N ÿ k “ L k πσ k q k exp « ´ σ k ˜ x ` y q k ¸ff , (10)where N is the number of Gaussian components in the MGE fit,and L k , σ k , and 0 ď q k ď x and y are the coordinateson the plane of the sky, where the x -axis is oriented along themajor axis of the galaxy. Scott et al. (2013) find q k “ ε “ ´ q “ .
14, and NGC 5846 has ε “ .
08. We thus model these galaxies as spherical systems, set q k “ L k and σ k determined by Scott et al. (2013) for Eq. (10) in the surfacebrightness profile (Cappellari 2008) I ˚ p R q “ N ÿ k “ L k πσ k exp ˜ ´ R σ k ¸ . (11) Article number, page 3 of 22 & A proofs: manuscript no. Elliptical_Galaxies_SLUGGS_All
Table 1: General properties of NGC 1407, NGC 4486, and NGC 5846.NGC
D z q N
GCs N GCs , Blue N GCs , Red V hel p g ´ i q TH [Mpc] “ km s ´ ‰ (1) (2) (3) (4) (5) (6) (7) (8) (9)1407 28 .
05 0 . .
95 379 153 148 1779 ˘ . . . .
86 737 480 199 1284 ˘ . . . .
92 195 91 102 1712 ˘ . Notes.
Column 1: galaxy name; Col. 2: distance; Col. 3: redshift; Col. 4: minor-to-major axis ratio; Col. 5: number of confirmed GCs; Cols.6–7: number of blue and red GCs in the final sample (see text). Col. 8: heliocentric velocity; Col. 9: p g ´ i q colour threshold separating the GCpopulation. The distance and the number of confirmed GCs of NGC 1407 are from Pota et al. (2015), whereas the distance and the number ofconfirmed GCs of NGC 4486 and NGC 5846 are from Pota et al. (2013). The number of blue and red GCs of NGC 1407 are from Pota et al.(2015), whereas the number of blue and red GCs of NGC 4486 and NGC 5846 are determined from our analysis. The minor-to-major axis ratioand the heliocentric velocities of the three galaxies are from Pota et al. (2013), as well as the p g ´ i q colour thresholds for NGC 1407 and NGC5846. The p g ´ i q colour threshold for NGC 4486 is from Strader et al. (2011). Scott et al. (2013) find N “ N “ L k and σ k after deconvolving for the seeing. Finally, the totalstellar luminosity of the galaxy is (Cappellari et al. 2013) L ˚ , tot “ N ÿ k “ L k . (12) For NGC 1407, the three-dimensional (3D) luminosity densityof the stars is the Abel integral, valid in spherical symmetry, ν ˚ p r q “ ´ π ż `8 r d I ˚ d R d R ? R ´ r , (13)where I ˚ p R q is given by Eq. (8) and r is the radial coordinatein three dimensions. For NGC 4486 and NGC 5846, the depro-jected MGE 3D luminosity density is (Cappellari 2008) ν ˚ p r q “ N ÿ k “ L k p ? πσ k q exp ˜ ´ r σ k ¸ . (14)The 3D luminosity density of the stars, ν ˚ p r q , yields the stel-lar luminosity profile L ˚ p r q “ π ż r ν ˚ p r q r d r . (15)The total stellar luminosity, L ˚ , tot , is formally obtained by set-ting r “ 8 . However, L ˚ p r q rapidly converges to L ˚ , tot . Bysetting R “ L ˚ p R q “ . ˆ L d for NGC 1407, L ˚ p R q “ . ˆ L d for NGC 4486, and L ˚ p R q “ . ˆ L d for NGC 5846. These total luminosi-ties are consistent with the total galaxy luminosities L ˚ , tot “ . ˆ L d , found by Pota et al. (2015) for NGC 1407,and L ˚ , tot “ . ˆ L d and L ˚ , tot “ . ˆ L d ,found by Scott et al. (2013) with Eq. (12), for NGC 4486 andNGC 5846, respectively. This result supports our spherical ap-proximation and our choice of R “ The number density profiles of GCs are from Pota et al. (2015)for NGC 1407, from Strader et al. (2011) for NGC 4486, andfrom Pota et al. (2013) for NGC 5846. Each GC sample is sepa-rated into a blue and a red sample according to the colour thresh-olds listed in Table 1. To ensure the magnitude completeness ofthe GCs samples in all the three galaxies, Pota et al. (2013) re-move the sources fainter than M i “ ´ . i -band. Thismagnitude limit corresponds to the peak of the GC luminosityfunction that has a roughly Gaussian shape (e.g. Harris et al.1991; Whitmore et al. 1995; Kundu & Whitmore 1998). More-over, the samples might be contaminated by ultracompact dwarfgalaxies (UCDs), that are a kinematically and spatially distinctpopulation from the GCs. Therefore, Pota et al. (2013) removethe sources brighter than M i “ ´ .
6. This luminosity is 1 magbrighter than ω Cen, the brightest GC in the Milky Way, andbrighter objects are likely to be UCDs. NGC 1407, NGC 4486,and NGC 5846 have 379, 737, and 195 confirmed GCs, respec-tively (Table 1). Imposing the magnitude range ´ . ă M i ă´ . ff ected.In addition to this photometric selection, Pota et al. (2013,2015) apply a kinematic selection on the two separated popula-tions of blue and red GCs. Despite Galactic stars and GCs aretwo populations whose velocity distributions are usually welldistinct, sometimes the population of stars might have a low-velocity tail so that a Galactic star could erroneously be identi-fied as a GC. Pota et al. (2013, 2015) thus remove the sourceswhose velocity is more than 3 σ discrepant from the mean veloc-ity of the 20 closest neighbours in the GC sample, where σ is thevelocity dispersion of these 20 neighbours (Merrett et al. 2003).This kinematic criterion removes 6, 5, and 2 objects from theNGC 1407, NGC 4486, and NGC 5846 samples, respectively.The sizes of the final samples of blue and red GCs for eachgalaxy are listed in Table 1.Each GC sample is binned in circular annuli around thegalaxy centre. The background-subtracted surface number den-sity profiles, along with their Poissonian uncertainties, are shownin Fig. 1. Article number, page 4 of 22. Cesare, A. Diaferio, and T. Matsakos: The dynamics of three nearby E0 galaxies in refracted gravity
Table 2: Parameters of the stellar surface brightness profile of NGC 4486. L k σ k r L d s [arcsec](1) (2)3 . ˆ . ˆ . ˆ . ˆ . ˆ . ˆ . ˆ . ˆ . ˆ L k σ k r L d s [arcsec](1) (2)3 . ˆ . . ˆ . . ˆ . . ˆ . . ˆ . . ˆ . . ˆ . We model the profiles of the blue and red GCs in NGC 5846with the Sérsic profile N GC p R q “ N e exp ´ b n s «ˆ RR e ˙ n s ´ ff+ , (16)where the parameters R e , N e , and n s have similar meaning ofthose in Eq. (8).We estimate the free parameters of the surface number den-sity profiles of the blue and red GCs with a Monte Carlo MarkovChain (MCMC) method with a Metropolis–Hastings acceptancecriterion. Details of this algorithm are in Sect. 5.2. We run theMCMC for 2 ˆ steps with 10 burn-in elements to achieve theconvergence of the chains, according to the Geweke (1992) diag-nostics. We adopt flat priors on the free parameters of the modelin the ranges listed in Table 4. The upper limit of R e is the radiusof the most external data point of the surface number density; theupper limit of N e is slightly larger than the surface number den-sity at the smallest R . Therefore, the upper limits of the priors for R e and N e are di ff erent for the two GC populations. Our posteriordistributions are single-peaked. We thus adopt their medians asour parameter estimates and the range between the 15.9 and 84.1percentiles, which includes 68% of the posterior cumulative dis-tribution, as our parameter uncertainties. These parameters arelisted in Table 5. They provide the curves shown in Fig. 1. We model the surface number densities of the two GC popula-tions in NGC 1407 and NGC 4486 with a Sérsic profile, adoptingthe parameters obtained by Pota et al. (2015) and Strader et al.(2011) for the two galaxies, respectively. Pota et al. (2015) model their data with Eq. (16). Strader et al. (2011) adopt the slightlydi ff erent parametrisation N GC p R q “ N exp « ´ ˆ RR s ˙ m ff . (17)Table 5 lists the parameters of the models shown in Fig. 1.For the three galaxies, the 3D number density of the GCs is ν GC p r q “ ´ π ż `8 r d N GC d R d R ? R ´ r . (18) For the 3D mass density of the hot X-ray emitting gas, we adoptthe model parameters derived by Zhang et al. (2007). They as-sume spherical symmetry and model the 3D number density pro-file of the gas, n g p r q , with the two- β function n p r q “ n , « ` ˆ rR c , ˙ ff ´ β ` n , « ` ˆ rR c , ˙ ff ´ β . (19)If we assume hydrodynamic equilibrium and an ideal andthermalised gas, n p r q enters the observed X-ray surface bright-ness profile S g p R q “ S ż `8 R Λ p T , Z q n p r q r d r ? r ´ R ` S bkg , (20)where T p r q and Z p r q are the temperature and the metallicity pro-files, Λ p T , Z q is the cooling function, and S bkg is the background Article number, page 5 of 22 & A proofs: manuscript no. Elliptical_Galaxies_SLUGGS_All
Table 4: Priors of the parameters of the GC surface number density models of NGC 5846.GC sample N e R e n s ” GCsarcmin ı [arcsec](1) (2) (3) (4)Blue U p . , . s U p . , s U p . , . s Red U p . , . s U p . , s U p . , . s Notes. U stands for uniform distribution. Table 5: Parameters of the models of the surface number density of the GCs.NGC GC sample N e , N R e , R s n s , m χ ,ν ” GCsarcmin ı [arcsec](1) (2) (3) (4) (5) (6)1407 Blue 7 ˘ ˘
29 1 . ˘ . . ˘ ˘ . ˘ . . p . ˘ . q ˆ ˘ . ˘ .
47 0 . p . ˘ . q ˆ p . ˘ . q ˆ ´ . ˘ .
24 0 . . ` . ´ . ` ´ . ` . ´ . . . ` . ´ . ` ´ . ` . ´ . . Notes.
Column 1: galaxy name; Col. 2: colour of the GC population; Cols. 3–5: parameters of the model of the surface number density; Col. 6:reduced chi-square, χ ,ν , for ν degrees of freedom. The parameters t N e , R e , n s u , adopted for NGC 1407 and NGC 5846, refer to Eq. (16) and arefrom Pota et al. (2015) and our MCMC analysis, respectively; the parameters t N , R s , m u , adopted for NGC 4486, refer to Eq. (17) and are fromStrader et al. (2011). (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
50 100 500 10000.1110100 7 14 68 136R [ arcsec ] N G C ( R ) [ G C s a r c m i n - ] R [ kpc ] (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
50 100 500 10000.1110100 R [ arcsec ] N G C ( R ) [ G C s a r c m i n - ] (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
100 200 500 10000.010.050.100.501510 8 17 42 83R [ arcsec ] N G C ( R ) [ G C s a r c m i n - ] R [ kpc ] (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
100 200 500 10000.010.050.100.501510 R [ arcsec ] N G C ( R ) [ G C s a r c m i n - ] (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)
100 200 5000.010.050.100.501 12 23 59R [ arcsec ] N G C ( R ) [ G C s a r c m i n - ] R [ kpc ] (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)
100 200 5000.010.050.100.501 R [ arcsec ] N G C ( R ) [ G C s a r c m i n - ] NGC 1407 NGC 4486 NGC 5846
Fig. 1: Surface number density profiles of blue (blue symbols and blue lines) and red (red symbols and red lines) GCs. Blue and reddots with error bars show the measured profiles; the solid lines are the models.contamination level. The observed S g p R q was measured with Chandra
ACIS in the 0.7–7 keV band and with
ROSAT
PSPCin the 0.2–2 keV band. Zhang et al. (2007) fit Eq. (20) to thedata and obtain n g , “ . ´ , n g , “ . ˆ ´ cm ´ , R c , “ p . ˘ . q arcsec, R c , “ p . ˘ . q arcsec, β “ . ˘ .
01, and β “ . ˘ .
01. They report n g , and n g , without uncertainty.The 3D density profile of the gas entering our model is ρ g p r q “ µ m H n g p r q , (21)where m H “ . ˆ ´ kg is the atomic unit mass and µ “ . The normalisations n g , and n g , yield ρ g , “ . ˆ ´ g cm ´ and ρ g , “ . ˆ ´ g cm ´ . We derive our analytic model of the gas mass density profilefrom the measurements of Paggi et al. (2017). They use thedata from the X-ray observations of the
Chandra
ACIS and
XMM-Newton
MOS instruments. Paggi et al. (2017) model thegalaxy as a spherical system and fit the galaxy spectrum of Zhang et al. (2007) assume a non-null metallicity, in the range r . , . s Z d , where Z d “ . µ is basically insensitive to Z . Indeed, adopting Z “ . Z d yields µ “ . each concentric circular annulus around the galaxy centre withthe Astrophysical Plasma Emission Code (APEC) (Smith et al.2001). Each circular annulus has a minimum width of 1 arcsec,for Chandra data, and of 30 arcsec, for
XMM -MOS data; thischoice yields a finer grid in the innermost galaxy region wheremost of the
Chandra data are. The inner and the outer radii ofeach annulus are chosen to achieve a minimum signal-to-noise S { N “ Chandra
ACIS data, and S { N “
50, for
XMM -MOS data. The spectrum in each circular annulus is the integralof the emission from all the 3D spherical shells along the lineof sight. For a source at redshift z and angular diameter distance D A , the APEC parameters include the temperature of the plasmain each spherical shell, its element abundances, and its normali-sationEM “ ´ π rp ` z q D A s ż n e p r q n H p r q d V , (22)where the integral is taken over the volume of the 3D sphericalshell. The equation above returns EM in cm ´ , when D A is incm, n e and n H in cm ´ , and the volume V in cm . In Eq. (22), n e and n H are the electron and hydrogen number densities. For afully ionised gas, η “ n e { n H “ .
2. In addition, for NGC 5846,Paggi et al. (2017) assume that n e and n H are constant withinthe shell. Therefore, 4 π rp ` z q D A s EM “ ´ η n V , where V is the volume of the spherical shell. In each spherical shell, themass density is thus ρ g “ µ m H n H “ µ m H p ` z q D A „ π EM η V { r g cm ´ s , (23)where µ “ .
62 accounts for the metallicity Z ą We model the set of data points of the gas mass density ineach spherical shell provided by Paggi et al. (2017) with the two- β function in Eq. (19), where the two normalisation constants arenow ρ g , and ρ g , . We estimate the parameters of the gas massdensity with the MCMC method described in Sect. 5.2 adoptingthe Metropolis-Hasting acceptance criterion. Running the chainsfor 2 ˆ steps with 10 burn-in elements is su ffi cient to reachthe convergence of the chains, according to the Geweke (1992)diagnostics. We adopt uniform priors in the intervals listed in Ta-ble 6. The upper limits of R c , and R c , are the radii of the mostexternal data point of the gas mass density profile in Eq. (23).The upper limits of ρ g , and ρ g , are slightly larger than the datapoint with the smallest R . Our posterior distributions are single-peaked, and we thus adopt the medians of the posterior distribu-tions as our parameter estimates and the range between the 15.9and 84.1 percentiles as our parameter uncertainties. The param-eters of the model, along with their uncertainties, are listed inTable 6. We adopt the model derived by Fabricant et al. (1980). Theymodel the X-ray surface brightness profile of the gas, S g p R q ,measured in the 0.7–3.0 keV energy band with the Einstein Ob-servatory , with S g p R q “ S p ` bR ` cR ` dR q n , (24) In fact, assuming a null metallicity, and thus µ “ .
6, leaves ourresults una ff ected. where b “ .
009 arcmin ´ , c “ . ˆ ´ arcmin ´ , d “ . ˆ ´ arcmin ´ , and n “ .
68. To obtain the 3Dmass density profile, they numerically invert Eq. (24), assum-ing an isothermal gas. They adopt zero metallicity, with a meanmolecular weight µ “ .
6, and model the numerical deprojecteddensity distribution with the function ρ p r q “ ρ p ` b r ` c r ` d r q n , (25)where b “ . ˆ ´ arcmin ´ , c “ . ˆ ´ arcmin ´ , d “ . ˆ ´ arcmin ´ , n “ .
59, and ρ “ . ˆ ´ g cm ´ . These parameters are provided without uncer-tainty. We include the contribution of the central SMBH in our massmodel. We adopt the SMBH masses provided by Rusli et al.(2013): M ‚ “ . ` . ´ . ˆ M d , 6 . ` . ´ . ˆ M d , and1 . ` . ´ . ˆ M d for NGC 1407, NGC 4486, and NGC 5846,respectively. We model the mass density of the SMBH as ρ ‚ p r q “ M ‚ π r δ p r q (26)where δ is the Dirac δ function. The cumulative mass profile isthus M p r q “ M ‚ .
4. Spectroscopic data
For NGC 1407, we use the root-mean-square velocity disper-sion profile of the stars measured by Pota et al. (2015). Theyderive the profile from two di ff erent data sets, depending on theradial range. For radii in the range r , s arcsec, they use thelong-slit data, along the major axis of the galaxy, from the Euro-pean Southern Observatory Faint Object Spectrograph and Cam-era (v.2) (EFOSC2) (Proctor et al. 2009). For radii in the range r , s arcsec, they use the multislit Keck / DEIMOS data (Fos-ter et al. 2016), again along the galaxy major axis. In the over-lapping range r , s arcsec, they keep both data sets.The root-mean-square velocity dispersion profile of the starsat the circularised projected radius R is V rms p R q “ “ V p R q ` σ p R q ‰ { , (27)where σ p R q is the stellar velocity dispersion and V rot p R q “ V p R q´ V sys is the major axis stellar rotation velocity in the galaxyframe, properly corrected for the angle between the major rota-tion axis and the galaxy region considered (see Proctor et al.2009 for details), with V p R q the observed rotation velocity, and V sys the galaxy systemic velocity due to the Hubble flow. V rot p R q and σ p R q are estimated from the mean and the stan-dard deviation of the Gaussian distribution that best fits the line-of-sight velocity distribution (LOSVD) of the stars derived fromthe spectral line profile (Cappellari 2008; Cappellari et al. 2013). Pota et al. (2015) and Pota et al. (2013) approximate V sys with thegalaxy heliocentric velocity V hel listed in Table 1. V hel actually dependson the date of the observation, but the di ff erence between V sys and V hel is small enough to leave the results for the dynamical model una ff ected.We adopt the same approximation throughout the paper.Article number, page 7 of 22 & A proofs: manuscript no. Elliptical_Galaxies_SLUGGS_All
Table 6: Priors and parameters of the 3D mass density model of the hot X-ray emitting gas of NGC 5846. ρ g , ρ g , R c , R c , β β “ ´
26 gcm ‰ “ ´
26 gcm ‰ [arcsec] [arcsec](1) (2) (3) (4) (5) (6) U p , . s U p , . s U p , s U p , s U p , s U p , s . ` . ´ . . ` . ´ . ` ´ ` ´ . ` . ´ . . ` . ´ . Notes. U stands for uniform distribution. Cappellari et al. (2006) and Cappellari et al. (2007) use semi-analytic dynamical models to show that V rms “ p V ` σ q { obtained from this Gaussian fit is a better estimator of the secondmoment of the velocity distribution than the second moment es-timated with the integral of the LOSVD modelled with a Gauss-Hermite parametrisation (van der Marel & Franx 1993; Gerhard1993). This result is due to the sensitivity of V rot and σ to thetails of the LOSVD, which are a ff ected by large observationaluncertainties (Cappellari 2008; Cappellari et al. 2013).The analysis of Proctor et al. (2009) confirms that NGC 1407is a slow rotator, namely V rot ! σ ; therefore, Eq. (27) yields V rms p R q ≈ σ p R q . To obtain the final V rms p R q profile, Pota et al.(2015) fold and average the data with respect to the galaxy cen-tre. In our dynamical model described in Sect. 5, we assume anorbital anisotropy parameter β independent of 3D radius r , asPota et al. (2015) do. However, by fitting Schwarzschild orbit-superposition models to their kinematic data, Thomas et al.(2014) infer a variable β p r q within R “ V rms profile be-yond R “ = .
272 kpc. The top-left panel of Fig. 2shows the V rms p R q profile of the stars obtained by Pota et al.(2015). For NGC 4486 and NGC 5846, we derive V rms p R q from theATLAS survey (Cappellari et al. 2011). All the data are pub-licly available on the ATLAS website. The available data provide the two-dimensional map of V and σ of the galaxy on the plane of the sky, where V and σ are the observed rotation velocity and velocity dispersion of thestars. In NGC 4486 and NGC 5846, 99.6% and 97.9% of the datapoints, respectively, have V { σ ď .
2. Therefore these galaxies,like NGC 1407, are slow rotators, and we have V rms ≈ σ .We rotate the galaxy map counter-clockwise in a new Carte-sian coordinates p x rot , y rot q , such that the photometric major axisof the galaxy is aligned with the east-west direction (Krajnovi´cet al. 2011). We iteratively remove the values of V rms p x rot , y rot q that deviate more than 3 standard deviations from the mean V rms of the entire map (Cappellari et al. 2013). For NGC 4486 andNGC 5846, the procedure converges after 7 and 6 iterations andremoves 155 and 201 data points, respectively. This removal isnecessary because some data points could be spurious, becauseof the presence of Milky Way stars or problematic bins at theedge of the field of view (Cappellari et al. 2013). Finally, wetransform the two-dimensional map V rms p x rot , y rot q into a one-dimensional profile, V rms p R q , by folding and averaging the datawith respect to the galaxy centre.This procedure yields V rms p R q profiles with 2444 and 1752data points for NGC 4486 and NGC 5846, respectively. These profiles are shown in the upper middle and right panels of Fig. 2.These numbers are „ V rms p R q of the stars and are basically insensitiveto the GC kinematic profiles. To overcome this problem, we binthe stellar kinematic data. Each bin contains a constant numberof data points: N “
111 for NGC 4486 and N “
103 for NGC5846. NGC 4486 and NGC 5846 have now 22 and 17 binneddata, respectively. These binned data are the medians, both inthe V rms and R directions, of the values of the N data points inthe bin. The uncertainties are the semi-interval between the 15.9and the 84.1 percentiles of the distributions, which are nearlysymmetric, of the data points in the bin.In general, massive early-type galaxies, like these two galax-ies, show negative orbital anisotropy parameters in their centralregions, indicating tangential orbits; on the contrary, in the outerregions, the orbits appear radial (Thomas et al. 2014; Rantalaet al. 2019). Numerical simulations show that this di ff erencein the stellar orbits might originate if massive elliptical galax-ies form from the merging of two progenitors with mass ratioslarger than 1 { β independent ofradius r , we exclude the innermost 2 arcsec in the kinematic pro-files of NGC 4486 and NGC 5846. This projected radius corre-sponds to 0 .
167 kpc and 0 .
235 kpc for NGC 4486 and NGC5846, respectively.
We derive the V rms profiles of the GCs in NGC 4486 and NGC5846 following the procedure of Pota et al. (2013) and Pota et al.(2015). As already specified in Sect. 3.2, our samples of GCscontain 480 blue GCs and 199 red GCs in NGC 4486 and 91blue GCs and 102 red GCs in NGC 5846.We bin each GC population into circular annuli centred onthe galaxy centre. Each annulus contains the same number ofGCs. Specifically, each bin contains N “
30 and N “
25 GCsfor the blue and red populations in NGC 4486 and N “
26 GCsfor both populations in NGC 5846. These N ’s are a trade-o ff be-tween too poor bins and a too coarse profile. For NGC 4486, weobtain 16 and 8 bins for the blue and the red GCs, respectively.In NGC 5846, we obtain 6 and 7 bins for the blue and the redGCs, respectively. In NGC 5846, some GCs are common to con-tiguous bins, so that we have a su ffi cient number of bins despitethe small total number of GCs. The radius R of each bin is themedian of the radial coordinates of the GCs in the bin. The V rms Article number, page 8 of 22. Cesare, A. Diaferio, and T. Matsakos: The dynamics of three nearby E0 galaxies in refracted gravity ●●●●●●●● ● ● ● ●● ● ● ● ● [ arcsec ] v r m s [ k m / s ] R [ kpc ] ●●●●●●●● ● ● ● ●● ● ● ● ● [ arcsec ] v r m s [ k m / s ] ● ● ● ● ● ●● ● ● ● ● [ arcsec ] v r m s [ k m / s ] R [ kpc ] ● ● ● ● ● ●● ● ● ● ● [ arcsec ] v r m s [ k m / s ] ●●●●●●●● ● ● ● ● ● ● ● ●●●● ●● ● ● ● [ arcsec ] v r m s [ k m / s ] R [ kpc ] ●●●●●●●● ● ● ● ● ● ● ● ●●●● ●● ● ● ● [ arcsec ] v r m s [ k m / s ] ● ● ● ● ● ●● ● ● ● ● ● ● [ arcsec ] v r m s [ k m / s ] R [ kpc ] ● ● ● ● ● ●● ● ● ● ● ● ● [ arcsec ] v r m s [ k m / s ] NGC 1407 ● ● ● ● ● ● ●●●●●●●●●●●●●● ● ● [ arcsec ] v r m s [ k m / s ] R [ kpc ] ● ● ● ● ● ●●●●●●●● ● ● ● ● [ arcsec ] v r m s [ k m / s ] R [ kpc ] NGC 4486 NGC 5846 ● ● ● ● ● ● ●●●●●●●●●●●●●● ● ● [ arcsec ] v r m s [ k m / s ] ● ● ● ● ● ●●●●●●●● ● ● ● ● [ arcsec ] v r m s [ k m / s ] Fig. 2: Root-mean-square velocity dispersion profiles of the stars (upper panels) and of the GCs (lower panels). Each column refersto the galaxy indicated in the upper panels. In the upper middle and right panels, the grey dots and the black dots with error barsshow the unbinned and binned profiles, respectively. In the lower panels, the red and blue dots with error bars refer to the blue andred GC populations, respectively. Note the di ff erent radial and velocity ranges of the panels.of the GCs in each bin is (Pota et al. 2013, 2015) V “ N N ÿ i “ rp V rad , i ´ V sys q ´ p ∆ V rad , i q s , (28)where V rad , i is the radial velocity of the i -th GC in the radial binand ∆ V rad , i is its uncertainty (Jackson 1973; Danese et al. 1980).We take V rad and ∆ V rad from Strader et al. (2011), for NGC 4486,and Pota et al. (2013), for NGC 5846. To estimate the errors oneach value of V rms we use the procedure reported in Danese et al.(1980), as performed by Pota et al. (2015). For convenience, wereport this procedure in Appendix A.The kinematic profiles of the GCs are shown in Fig. 2. Theseprofiles are „
48 and „
13 times more extended than the kine-matic profiles of the stars for NGC 4486 and NGC 5846, respec-tively. In NGC 4486, the root-mean-square velocity dispersionprofiles of the GC populations show a bell-like shape that peaksat large radii. This feature suggests that the GC kinematics isinfluenced by the gravitational potential well of the Virgo clus-ter whose central giant elliptical galaxy is exactly NGC 4486(Strader et al. 2011). This shape is common to the stellar ve-locity dispersion profiles of other elliptical galaxies located atthe centre of galaxy clusters, like the brightest cluster galaxy ofAbell 383 (Geller et al. 2014).
We extract the V rms p R q profiles of the two populations of GCsin NGC 1407 from Fig. 5 of Pota et al. (2015). They derive theprofiles with the same procedure that we adopt here for NGC4486 and NGC 5846. Pota et al. (2015) obtain their kinematicdata from nine DEIMOS masks in Pota et al. (2013) and an ad-ditional DEIMOS mask in Pota et al. (2015). They have a totalsample of 379 GCs. After applying the photometric and kine-matic selection criteria detailed in Sect. 3.2, Pota et al. (2015)obtain a sample with 153 blue GCs and 148 red GCs. The kine-matic profiles of the blue and red GC populations in NGC 1407are shown in Fig. 2 and are „
5. Dynamical model
We model the kinematics of the three E0 galaxies by assumingspherical symmetry and no net rotation, as suggested by the ob-served slow rotation of the galaxies. We model the velocity dis-persion profile of each dynamical tracer t “ t˚ , R , B u , namely Article number, page 9 of 22 & A proofs: manuscript no. Elliptical_Galaxies_SLUGGS_All stars, red GCs, and blue GCs, as V , t p R q “ I t p R q ż `8 R K ´ β t , rR ¯ ν t p r q d φ d r r d r , (29)which is the solution to the spherical Jeans equations (Jeans1915; Cappellari 2008; Mamon & Łokas 2005; Pota et al. 2015).In Eq. (29), R is the circularised radius projected onto the skyaccording to Eq. (7), whereas r is the 3D radius; I t p R q is thesurface brightness of the stars or the surface number density ofGCs; ν t p r q is the 3D luminosity density of the stars or the 3Dnumber density of GCs; φ p r q is the gravitational potential; and β t “ ´ σ θ { σ r is the orbital anisotropy parameter, with σ θ and σ r , the velocity dispersions in the tangential and the radial direc-tions, respectively. We assume β t to be independent of r . K is thekernel K ´ β t , rR ¯ “ ´ rR ¯ β t ´ „ˆ ´ β t ˙ ? π Γ p β t ´ q Γ p β t q` β t B R r ˆ β t ` , ˙ ´ B R r ˆ β t ´ , ˙ , (30)where Γ p z q “ ş `8 t z ´ e ´ t d t is the Euler Γ function and B x p a , b q “ ş x t a ´ p ´ t q b ´ d t is the incomplete beta function(see Eq. A16 in Mamon & Łokas 2005). By inserting the RGgravitational field, Eq. (3), into Eq. (29), we obtain V , t p R q “ GI t p R q ż `8 R K ´ β t , rR ¯ ν t p r q M p r q (cid:15) p ρ q d rr . (31)In our model, M p r q is the baryonic mass alone, namely M p r q “ M ˚ p r q ` M g p r q ` M ‚ p r q ` M GC p r q , (32)where M ˚ p r q , M g p r q , M ‚ p r q , and M GC p r q are the cumulativemass profiles of the stars, X-ray emitting gas, SMBH, and GCs,respectively. We estimate the mass profile of the stars as M ˚ p r q “ Υ L ˚ p r q , (33)where L ˚ p r q is the luminosity profile, Eq. (15), and Υ is thestellar mass-to-light ratio that we assume to be independentof r . In our dynamical model, Υ is a free parameter in theranges predicted by the stellar population synthesis (SPS) mod-els of Humphrey et al. (2006) and Zhang et al. (2007), in the B -band, and Bell et al. (2003) and Zibetti et al. (2009), in the r -band. These ranges depend on the initial mass function. Forthe B -band, the range is r . , . s M d { L d and the lower andupper limits are set by the Kroupa (2001) and Salpeter (1955)initial mass functions, respectively. For the r -band, the rangeis r . , . s M d { L d and the lower and upper limits are set bythe Bottema (1993) and Salpeter (1955) initial mass functions,respectively.The mass profile of the gas, M g p r q , derives from the inte-gration of its density profile. This density profile is indicated inEq. (19), for NGC 1407 and NGC 5846, and in Eq. (25) for NGC4486. For each galaxy, the mass of the SMBH, M ‚ , is the valuereported in Sect. 3.4. For the cumulative mass profile of GCs, weadopt M GC p r q “ π M GC ż r ν GC p r q r d r , (34)where ν GC p r q is the 3D number density of GCs, Eq. (18), and M GC is a constant in the range of r ´ s M d , according to the typical mass function of GCs (Kimmig et al. 2015; Baumgardt2017).Figure 3 shows the mass profile of each component. The starsprovide the largest contribution to the total mass of each galaxyin most radial range, except in the very centre, where the SMBHdominates, and in the outskirts of NGC 4486 and NGC 5846,where the gas contribution overcomes the star contribution. Thecontribution of GCs to the total galaxy mass is always smallerthan 1% and we thus ignore it hereafter. Therefore, in the veloc-ity dispersion profile of Eq. (31), we also ignore the GC contri-bution to the density profile ρ p r q “ ρ ˚ p r q ` ρ g p r q ` ρ ‚ p r q , (35)which appears as the argument of the permittivity of Eq. (5).We solve Eq. (31) by adopting two independent linear gridsin the R and the r coordinates. Both grids cover the range r ´ , s arcsec, with a step of 4 arcsec. For each galaxy, the dynamical model has seven free parameters:four parameters, Υ , (cid:15) , Q and ρ c , contribute to the gravitationalpotential well of the galaxy and are common to the three trac-ers, and three parameters, β ˚ , β B and β R , are specific for eachtracer. Note that the contributions of the X-ray emitting gas andthe SMBH to the gravitational potential have no free parame-ters. For convenience, hereafter, we use the parameters B ˚ “´ log p ´ β ˚ q , B B “ ´ log p ´ β B q , B R “ ´ log p ´ β R q ,and P c “ log r ρ c p g { cm qs . Tangential and radial orbits corre-spond to B t ă B t ą
0, respectively, where t “ p˚ , B , R q .We explore this seven-dimensional parameter spacewith a MCMC algorithm, where we adopt a Metropolis-Hastings acceptance criterion: the random variate x i ` “ p Υ , (cid:15) , Q , P c , B ˚ , B B , B R q , at step i ` G p x i ` | x i q ,which depends on the random variate x i at the previous step. For G p x | x i q , we adopt a multi-variate Gaussian density distributionwith mean value x i . We choose the standard deviation of thisGaussian distribution according to the priors: when we adoptuniform priors, the standard deviation is 10% of the range of theprior; when we adopt Gaussian priors, the standard deviation is1 { L p x q “ exp « ´ χ , tot p x q ff , (36)where χ , tot p x q “ χ ˚ p x q ` χ p x q ` χ p x q n dof , tot , (37)and χ p x q “ N t ÿ i “ r V rms , mod , t p R i , x q ´ V rms , data , t p R i qs ∆ V , data , t p R i q , (38)where t “ p˚ , B , R q , N t is the number of data points of the tracer, V rms , data , t and ∆ V rms , data , t are the measured root-mean-square ve-locity dispersions at the projected distances R i , and their uncer-tainties, and V rms , mod , t is the root-mean-square velocity disper-sion model derived from Eq. (31). The total number of degreesof freedom is n dof , tot “ N ˚ ` N B ` N R ´ N par , (39) Article number, page 10 of 22. Cesare, A. Diaferio, and T. Matsakos: The dynamics of three nearby E0 galaxies in refracted gravity [ arcsec ] M ( r ) [ M ⊙ ] r [ kpc ] NGC 1407 NGC 4486 NGC 5846 r [ arcsec ] M ( r ) [ M ⊙ ] r [ arcsec ] M ( r ) [ M ⊙ ] r [ arcsec ] M ( r ) [ M ⊙ ] [ arcsec ] M ( r ) [ M ⊙ ] r [ kpc ] [ arcsec ] M ( r ) [ M ⊙ ] r [ kpc ] Fig. 3: Cumulative mass profiles of the baryonic components of each galaxy: stars (black solid line), blue GCs (blue solid line), redGCs (red solid line), hot X-ray emitting gas (magenta solid line), and SMBH (green solid line). The purple solid line shows the sumof all the mass contributions. The stellar mass profiles assume Υ “ . M d { L d for NGC 1407, and Υ “ . M d { L d for NGC 4486and NGC 5846. The grey shaded area around the star mass profiles shows the mass variation by adopting mass-to-light ratios in therange r . , . s M d { L d , for NGC 1407, and r . , . s M d { L d , for NGC 4486 and NGC 5846, in the B - and r -band, respectively.The mass profiles of blue and red GCs assume a GC mass M GC “ M d ; the blue and red shaded areas show the mass variationby adopting M GC in the range r , s M d . The gas mass profile of NGC 4486 has no shaded area because the uncertainties onthe adopted mass density profile are unavailable. The green solid lines and shaded areas are the masses of the SMBH and theiruncertainties. The purple shaded areas show the possible range of the total baryonic mass profile.where N par “ A “ p p x q ˆ L p x q p p x i q ˆ L p x i q G p x | x i q G p x i | x q , (40)where p p x q is the product of the priors of the components of x . If A ě
1, we accept the proposed combination of free parametersand we thus set x i ` “ x ; otherwise, we either set x i ` “ x , withprobability A , or x i ` “ x i , with probability 1 ´ A . We adopt2 ˆ steps for our MCMC, a number su ffi cient to achieve thechain convergence, according to the Geweke (1992) diagnostics.The first 10 elements are discarded as the burn-in chain.
6. Results
To model each galaxy with our MCMC analysis, we need toadopt a prior for each of our seven free parameters. For NGC1407, the photometric information available in the B -band sug-gests the uniform prior r . , . s M d { L d for Υ . For NGC 4486and NGC 5846, with photometric information in the r -band, weadopt the uniform prior r . , . s M d { L d . We choose both priorranges according to the ranges expected from the SPS modelsmentioned in Sect. 5 (Humphrey et al. 2006; Zhang et al. 2007;Bell et al. 2003; Zibetti et al. 2009). For all the three orbitalanisotropy parameters, B ˚ , B B , and B R , we adopt the uniformprior r´ . , . s , thus including almost all possible orbits, fromvery tangential to very radial. Except for the prior of Υ in the r -band for NGC 4486 and NGC 5846, our priors coincide withthose adopted by Pota et al. (2015), who model the kinematicsof NGC 1407 in Newtonian gravity with a generalised Navarro-Frenk-White dark matter halo (Hernquist 1990; Zhao 1996).For the three permittivity parameters, we adopt uniform pri-ors in the ranges p , s , r . , s , and r´ , ´ s , for (cid:15) , Q , and P c , respectively. These priors are those adopted in Cesare et al.(2020), except for (cid:15) : our range here is wider, compared to therange r . , s of Cesare et al. (2020). Figures 4–6 show the posterior distributions of the parametersof the model for each galaxy and Fig. 7 shows the models ofthe V rms profiles whose parameters are the medians of the pos-terior distributions. The description of the kinematics of NGC4486 and NGC 5846 shown in Fig. 7 appears satisfactory. How-ever, this result is not shared by NGC 1407, despite the conver-gence of its MCMC: the model of the blue GCs severely under-estimates the corresponding kinematic data. The corner plots inFig. 4 show that the nearly flat posterior distribution of Q , thebroad posterior distributions of (cid:15) , B B , and B R without distinctpeaks, and the substantial uncorrelation between several pairs ofparameters, complicate the identification of the best values of (cid:15) , Q , B B , and B R ; therefore, adopting the medians of the posteriordistribution for these values is unconvincing.This conclusion also applies to NGC 4486 and NGC 5846,despite the fact that the estimated parameters appear to providea proper modelling of their kinematic profiles. Indeed, Figs. 5and 6 show that the one-dimensional posterior distributions of Q and B R of both galaxies are broad and their medians do notcoincide with their peaks. In addition, the posterior distributionsshow that most parameters are independent of each other.We thus perform a second MCMC analysis with refined priordistributions as follows. For NGC 1407, the first MCMC analysisshows that the values (cid:15) “ .
13 and Q “ .
38 yield a reasonabledescription of the kinematic data, almost independently of thevalues of P c and of the other parameters. In our second MCMCanalysis, for (cid:15) and Q , we thus adopt Gaussian priors with meansand standard deviations (cid:15) “ . ˘ .
05 and Q “ . ˘ . σ smaller than the me-dians of the posterior distributions of the first MCMC analysis Article number, page 11 of 22 & A proofs: manuscript no. Elliptical_Galaxies_SLUGGS_All
NGC 1407
Fig. 4: Posterior distributions of the seven parameters of our first MCMC analysis adopting uniform priors for NGC 1407. Theparameters are the stellar mass-to-light ratio, Υ , the anisotropy parameters of the three tracers, B ˚ , B B , B R , and the three permittivityparameters, (cid:15) , Q , and P c . The red squares with error bars show the medians and their uncertainites, set by the 15 . . σ , 2 σ , and 3 σ regions, respectively. Themedians and their uncertainites are also reported above each column and in the one-dimensional posterior distributions in the toppanels of each column as red solid and black dashed lines, respectively. The magenta solid lines in the one-dimensional posteriordistributions of (cid:15) and Q show the means of the Gaussian priors that we adopt in our second MCMC analysis.(Fig. 4). For the other parameters, we keep the flat priors of thefirst analysis.For NGC 4486 and NGC 5846, the medians of the poste-rior distributions estimated from the first MCMC analysis repro-duce the kinematic data well. We thus simply adopt Gaussianpriors peaked on these values for these two galaxies. We set thestandard deviations of the Gaussians for all the parameters, ex-cept Υ and P c , to the mean uncertainties estimated from the firstanalysis; for Υ and P c , we set the standard deviations to threetimes the mean uncertainties. We adopt larger dispersions for these two parameters because the relative widths of their pos-terior distributions derived in the first analysis are in the nar-row range r . , s %: setting the second moment to this rangewould thus limit the exploration of the parameter space in thesecond MCMC analysis. On the contrary, the relative widths ofthe other parameters are in the range r , s % and setting thesecond moment of the Gaussians to this width provides su ffi -ciently broad priors. The Gaussian priors are set to zero outsidethe ranges listed in Table 7. This choice prevents the explorationof unphysical or unreasonable values. Article number, page 12 of 22. Cesare, A. Diaferio, and T. Matsakos: The dynamics of three nearby E0 galaxies in refracted gravity
NGC 4486
Fig. 5: Same as Fig. 4 for NGC 4486.
The posterior distributions determined by our second MCMCanalysis are shown in Figs. 8–10 and have well-identified peaks.We can thus adopt the medians of these distributions as the pa-rameters of our model. As uncertainties on these parameters, weadopt the range between the 15.9 and the 84.1 percentiles of theposterior distributions. Table 8 lists the parameters that we findand that we use to plot the curves of the models shown in Fig. 11.Table 8 also lists the reduced χ ’s, Eq. (37), that quantify theagreement between these curves and the data.The refinement of the priors improves the data descriptionfor all the three galaxies. For NGC 1407, the kinematics of thestars, red and blue GCs appear to be satisfactorily described byRG. The underestimation of the profile of the blue GCs disap-pears. For NGC 4486 and NGC 5846, the kinematic model of the stars and red GCs also appears appropriate. This result is notshared, however, by the blue GCs. In NGC 5846, RG underesti-mates the outer points of the measured profile, whereas, in NGC4486, RG cannot interpolate the peak of the velocity dispersionprofile. These data are the main cause of the large values of the χ : if we remove these data points, the reduced χ ’s decreasefrom 1 .
71 to 0 .
99 and from 3 .
30 to 2 .
37, for NGC 5846 andNGC 4486, respectively. These poor fits might originate fromthe assumption that these two galaxies are isolated rather thanembedded in a larger system, as we discuss below. The fits mightalso partly improve if we remove our assumption that the veloc-ity anisotropy parameters β are independent of radius r or thatthe galaxies have null net rotation: albeit weak, the net galaxyrotation is not completely absent in the real galaxies.Nevertheless, RG overall provides a global good descriptionof the kinematic data of the three tracers for all the three galaxies Article number, page 13 of 22 & A proofs: manuscript no. Elliptical_Galaxies_SLUGGS_All
NGC 5846
Fig. 6: Same as Fig. 4 for NGC 5846.with stellar mass-to-light ratios consistent with the SPS models.For NGC 1407, the values of Υ , B ˚ , B B , and B R that we ob-tain are at ` . σ , ` . σ , ` . σ , and ´ . σ , respectively,from the parameters found by Pota et al. (2015) in their analysiswith Newtonian gravity and a dark matter halo.The RG models require radial orbits for the stars and tangen-tial orbits for the blue GCs in all the three galaxies. The red GCshave tangential orbits in NGC 4486 and NGC 5846, and radialorbits in NGC 1407, in agreement with the result found by Potaet al. (2015) in Newtonian gravity for NGC 1407. The orange squares with error bars in Fig. 12 show the per-mittivity parameters estimated for the three E0 galaxies: theparameters are consistent with each other within „ σ , sug- gesting their universality. Their mean values are t (cid:15) , Q , P c u “t . ` . ´ . , . ` . ´ . , ´ . ` . ´ . u . Figure 12 also shows thepermittivity parameters derived from 30 disk galaxies of theDMS sample (Cesare et al. 2020): the purple squares with er-ror bars show the means of the parameters estimated for eachindividual DMS galaxy, whereas the green dots at the centre ofthe shaded areas show the global values derived with an approx-imate procedure from the entire 30 galaxy sample.Our estimates of P c and Q are within 1 σ from the DMS meanvalues, whereas the DMS permittivity of the vacuum (cid:15) is within2.5 σ from our values. This marginal discrepancy might origi-nate from our simplistic modelling of the E0 galaxies. Indeed,here we assume that these galaxies have not net rotation and arerelaxed and isolated, whereas, in fact, these galaxies are in clus-ters or groups and show sign of interactions with nearby galax-ies: NGC 1407 is at the centre of the Eridanus A group (Brough Article number, page 14 of 22. Cesare, A. Diaferio, and T. Matsakos: The dynamics of three nearby E0 galaxies in refracted gravity
NGC 1407 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ● ● ● ●●●● ●● ● ● ● [ arcsec ] v r m s [ k m / s ] NGC 4486 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ● ● ● ●●●● ●● ● ● ● [ arcsec ] v r m s [ k m / s ] R [ kpc ] ●●●●●●●●●●●●●●●●● ● ● ● ● ● ●● ● ● ● ● ● ● [ arcsec ] v r m s [ k m / s ] R [ kpc ] NGC 5846 ●●●●●●●●●●●●●●●●● ● ● ● ● ● ●● ● ● ● ● ● ● [ arcsec ] v r m s [ k m / s ] ● ● ● ● ● ● ●●●●●●●●●●●●●● ● ● [ arcsec ] v r m s [ k m / s ] R [ kpc ] ● ● ● ● ● ● ●●●●●●●●●●●●●● ● ● [ arcsec ] v r m s [ k m / s ] ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● [ arcsec ] v r m s [ k m / s ] R [ kpc ] ● ● ● ● ● ● ● ●●●● ● ● ● ● ● ● [ arcsec ] v r m s [ k m / s ] ●●●●●●●●●●●●●● ●●● ● ● ● ● ● ●● ● ● ● ● [ arcsec ] v r m s [ k m / s ] R [ kpc ] ●●●●●●●●●●●●●● ●●● ● ● ● ● ● ●● ● ● ● ● ●●●●●●●● ● ● ● ●● ● ● ● ● [ arcsec ] v r m s [ k m / s ] R [ kpc ] ●●●●●●●● ● ● ● ●● ● ● ● ● [ arcsec ] v r m s [ k m / s ] Fig. 7: Models of the root-mean-square velocity dispersion profiles emerging from our first MCMC analysis. In the upper panels,the green, blue, and red solid lines show the model profiles for the stars, blue GCs, and red GCs, respectively. The lower panelsshow a zoom-in of the stellar profiles. The dots with error bars are the measured profiles from Fig. 2.et al. 2006) and certainly feels the gravitational influence of itsneighbour galaxy NGC 1400. NGC 4486, namely M87, is thecentral giant elliptical galaxy of the Virgo cluster (Strader et al.2011) and NGC 5846 is the central and brightest galaxy of agalaxy group (Mahdavi et al. 2005). Moreover, the GC sampleof NGC 5846 might be contaminated by the GCs of its neighbourNGC 5846A (Pota et al. 2013). The environment has a relevante ff ect on the intensity of the gravitational field in RG (Matsakos& Diaferio 2016); this e ff ect clearly propagates into the actualvalues of the permittivity parameters. The environmental e ff ectscan also be responsible for the poor fitting of the velocity dis-persion profiles of the blue GCs of NGC 4486 and NGC 5846shown in Fig. 11. It would thus be necessary to refine our modelby taking into account the e ff ects of both rotation and the massdistribution surrounding the galaxies.The tension between our permittivity parameters and thosederived from the DMS sample might become more severe ifwe consider the DMS global values, namely the green dots inFig. 12. Our P c and Q are still within 3 σ from the DMS pa-rameters, whereas (cid:15) shows a „ σ tension, clearly driven bythe small width of the posterior distribution of the DMS globalvalue. This discrepancy might be due to the approximate proce-dure adopted by Cesare et al. (2020) to estimate this DMS globalvalue: with their MCMC analysis, Cesare et al. (2020) only ex-plore the space of the three permittivity parameters while keep-ing fixed the mass-to-light ratio Υ and the disk-scale height h z of each of the 30 individual galaxies. This approach is a short-cut to the appropriate, but computationally overwhelming, pro-cedure of exploring the 63-dimensional parameter space of thefull sample, namely the properties, Υ and h z , of all the galaxiesand the three permittivity parameters. The set of the permittiv-ity global parameters returned by this shortcut still describes thekinematic profiles of each galaxy, but the agreement is poorerthan the agreement obtained by modelling each galaxy individu-ally. Therefore these values of the permittivity parameters shouldbe considered with caution.
7. Discussion and conclusion
We use the kinematic information of the GCs in the outer regionsof three E0 galaxies from the SLUGGS survey (Pota et al. 2013;Brodie et al. 2014; Forbes et al. 2017) to show that RG can de-scribe the kinematics of spherical systems without requiring theexistence of dark matter. Our results complement the ability ofRG to describe the rotation curves and the vertical velocity dis-persion profiles of 30 disk galaxies from the DMS survey (Ber-shady et al. 2010) shown in Cesare et al. (2020). In RG, the darkmatter is mimicked by the gravitational permittivity, a mono-tonic function of the local mass density that is expected to beuniversal. For the permittivity, according to Matsakos & Diafe-rio (2016) and Cesare et al. (2020), we adopt a smooth function
Article number, page 15 of 22 & A proofs: manuscript no. Elliptical_Galaxies_SLUGGS_All
Table 7: Priors of the parameters of the kinematic model, Eq. (31), adopted in our final analysis.NGC Υ (cid:15) Q P c B ˚ B B B R ” M d L d ı (1) (2) (3) (4) (5) (6) (7) (8)1407 U r . , . s G p . , . q G p . , . q U r´ , ´ s U r´ . , . s U r´ . , . s U r´ . , . sP p , s P r . , s G p . , . q G p . , . q G p . , . q G p´ . , . q G p . , . q G p´ . , . q G p´ . , . qP p , `8q P p , s P r . , s P r´ , ´ s P r´ . , . s P r´ . , . s P r´ . , . s G p . , . q G p . , . q G p . , . q G p´ . , . q G p . , . q G p´ . , . q G p´ . , . qP p , `8q P p , s P r . , s P r´ , ´ s P r´ . , . s P r´ . , . s P r´ . , . s Notes.
Column 1: galaxy name; Cols. 2–8: free parameters of the kinematic model, Eq. (31). U and G stand for uniform and Gaussian distributions,respectively. For the uniform distributions, we list the entire range. For the Gaussian distributions, we list the mean, the standard deviation and, onthe second line of each galaxy, the range explored by the MCMC. Table 8: Parameters estimated from the three dynamical tracers.NGC Υ (cid:15) Q P c B ˚ B B B R χ , tot ” M d L d ı (1) (2) (3) (4) (5) (6) (7) (8) (9)1407 8 . ` . ´ . . ` . ´ . . ` . ´ . ´ . ` . ´ . . ` . ´ . ´ . ` . ´ . . ` . ´ . . . ` . ´ . . ` . ´ . . ` . ´ . ´ . ` . ´ . . ` . ´ . ´ . ` . ´ . ´ . ` . ´ . . . ` . ´ . . ` . ´ . . ` . ´ . ´ . ` . ´ . . ` . ´ . ´ . ` . ´ . ´ . ` . ´ . . Notes.
Column 1: galaxy name; Cols. 2–8: medians of the posterior distributions with their uncertanties; Col. 9: reduced chi-square, χ , tot , fromEq. (37). that depends on three parameters: the permittivity of the vacuum (cid:15) , the critical density ρ c , and the transition slope Q .We find that the sets of parameters of the three E0 galax-ies are consistent with each other within „ σ . Their values av-eraged over the three galaxies are t (cid:15) , Q , log r ρ c p g { cm qsu “t . ` . ´ . , . ` . ´ . , ´ . ` . ´ . u . With this permittivity, RGrequires stellar mass-to-light ratios in agreement with SPS mod-els and tangential or radial orbits for the GCs, depending onthe galaxy and the colour of the GCs. In particular, for NGC1407, the mass-to-light ratio and the orbital anisotropy parame-ters are within 1 σ from the values found by Pota et al. (2015),who model this galaxy with Newtonian gravity and a dark matterhalo.By modelling the kinematic properties of each individualDMS disk galaxy, Cesare et al. (2020) derive the mean val-ues of the permittivity parameters t (cid:15) , Q , log r ρ c p g { cm qsu “t . ˘ . , . ˘ . , ´ . ˘ . u . The mean values of Q and log ρ c are within 1 σ from the parameters we find here,whereas (cid:15) is within 2.5 σ . These combined results thus supportthe universality of the parameters of the RG permittivity.Cesare et al. (2020) also derive a set ofglobal parameters t (cid:15) , Q , log r ρ c p g { cm qsu “t . ` . ´ . , . ` . ´ . , ´ . ` . ´ . u with a MCMC analy-sis where the mass-to-light ratio and the disk thickness ofeach individual galaxy are kept fixed. This simplified approachreturns an approximate estimate of the unique set of the per-mittivity parameters capable of describing the entire sample ofthe 30 DMS disk galaxies. This analysis of Cesare et al. (2020)is suggestive of the possible universality of the permittivityparameters, but their actual values should be taken with caution.The global DMS values of log ρ c and Q found by Cesareet al. (2020) are within 3 σ from the values we find here andare thus still consistent with the expected universality, whereas the global DMS (cid:15) is „ σ discrepant. This discrepancy ismostly driven by the width of the posterior distribution foundby Cesare et al. (2020) which is small when compared to theuncertainties of the corresponding mean values (Fig. 12). If thiswidth is not severely underestimated, this discrepancy mighthighlight at least three di ff erent possible problems: a simplisticmodel of the ellipticals, an incorrect form of the permittivity, ora fundamental fault of RG.The model of the ellipticals that we adopt here is oversimpli-fied, because we treat them as isolated systems and completelyneglect the galaxy net rotation. Indeed, NGC 1407 and NGC5846 are within galaxy groups and NGC 4486 is the centralgalaxy of the Virgo cluster. Neglecting the galaxy environmentcan clearly overlook relevant interactions with neighbour galax-ies that could actually a ff ect the parameters of the permittivity.Including these environmental e ff ects requires modelling all thegalaxies within the system at the same time, with each galaxymodelled according to its morphological type and shape. Prop-erly modelling such a system is arduous and its non-linear e ff ectsare di ffi cult to assess at the present stage. We cannot thus excludethat they might be part, if not all, of the cause of the possible ten-sion on (cid:15) that we find here. Alternatively, the explicit form ofthe permittivity (cid:15) p ρ q (Eq. (5)), that is chosen arbitrarily, mightactually be inappropriate, especially in the low-density regime:hints from the weak-field limit of the covariant version of RG(Sanna et al., in preparation) might suggest a better motivatedpermittivity that might alleviate the tension. Finally, all of theabove might turn out to play a negligible role, and this possibletension on the permittivity of the vacuum might simply betray afundamental fault of RG. Acknowledgements.
We sincerely thank Xavier Hernández and Federico Lellifor their precious advice and detailed comments that improved the presentationof this work, and Alessandro Paggi and Valentina Missaglia, for their help inthe modelling of the gas of NGC 5846 and of the number density profiles of
Article number, page 16 of 22. Cesare, A. Diaferio, and T. Matsakos: The dynamics of three nearby E0 galaxies in refracted gravity
NGC 1407
Fig. 8: Same as Fig. 4 for our second MCMC analysis. the GCs. We also thank Michele Cappellari, for his help in the derivation ofthe stellar root-mean-square velocity dispersion profiles of NGC 4486 and NGC5846. We thank Compagnia di San Paolo (CSP) for funding the graduate-studentfellowship of VC. We also acknowledge partial support from the INFN grantInDark and the Italian Ministry of Education, University and Research (MIUR)under the
Departments of Excellence grant L.232 / References
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Fig. 9: Same as Fig. 5 for our second MCMC analysis.
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Article number, page 20 of 22. Cesare, A. Diaferio, and T. Matsakos: The dynamics of three nearby E0 galaxies in refracted gravity - . - . - . - . Q . . . . . Q . . . . . - . - . - . - . ε P c Fig. 12: Permittivity parameters estimated from the three E0 galaxies in our sample (orange squares with error bars) comparedwith the permittivity paramaters estimated by Cesare et al. (2020) from the DMS disk galaxies: the purple squares with error barsshow the means of the permittivity parameters found for the individual DMS galaxies, whereas the light blue shaded areas show theposterior distributions of the three permittivity parameters found with an approximate procedure from the entire DMS sample at thesame time, with the green dots indicating their median values and the yellow, red, and black contours indicating the 1 σ , 2 σ , and 3 σ levels, respectively. Article number, page 21 of 22 & A proofs: manuscript no. Elliptical_Galaxies_SLUGGS_All
Appendix A: Derivation of the uncertainties on theroot-mean-square velocity dispersions of GCs
We adopt the procedure of Danese et al. (1980) to estimate theuncertainties ∆ V rms p R q on the values of the root-mean-square ve-locity dispersion profile V rms p R q that we compute with Eq. (28).In the reference frame of the centre of mass of the galaxy, theline-of-sight velocity of the i -th GC is v || , i “ V rad , i ´ V sys ` V sys { c , (A.1)where V rad , i is the observed radial velocity of the GC, V sys isthe systemic velocity of the galaxy, and c “ ˆ km s ´ is the speed of light. The three E0 galaxies of our analysis haveredshift z ă .
007 and we set to 1 the denominator of Eq. (A.1).The variance of the radial velocities of GCs in each radial bin ofradius R is σ || p R q “ N ´ N ÿ i “ v || , i , (A.2)where N is the number of GCs in the bin. This equation neglectsthe errors on V rad and V sys .The quantity S p R q “ p N ´ q σ || p R q V p R q , (A.3)is a χ ν random variate with ν “ N ´ S ´ and S ` with the probability α of having S in the range p S ´ , S ` q . The value α “ .
68 defines the upper, indicated withthe ` sign, and lower, indicated with the ´ sign, 1 σ uncertaintyon the dispersion of the radial velocities derived with Eq. (A.2): ∆ σ || , ˘ p R q “ »–˜ ν S ¯ p R q ¸ { ´ fifl σ || p R q` Ď δ ˚ N ˜ ` Ď δ ˚ σ || p R q ¸ , (A.4)where Ď δ ˚ is Ď δ ˚ “ ∆ V sys ` V sys { c (A.5)and ∆ V sys is the uncertainty on V sys ; Ď δ ˚ “ ∆ V sys in our analysis.Finally, the upper and lower 1 σ uncertainties on V rms p R q are ∆ V rms , ˘ p R q “ V rms p R q « N ` ˆ ∆ σ || , ˘ p R q σ || p R q ˙ ff { . (A.6)In our analysis, we use the symmetrised uncertainty ∆ V rms p R q “ ∆ V rms , ´ p R q ` ∆ V rms , ` p R q . (A.7)(A.7)