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Toomre Stability of Disk Galaxies in Quasi-Linear MOND

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Toomre Stability of Disk Galaxies in Quasi-Linear MOND MNRAS 000,1{13 (2018) Preprint 23 January 2019 Compiled using MNRAS LATEX style file v3.0 Toomre stability of disk galaxies in quasi-linear MOND Indranil Banik1;2∗, Mordehai Milgrom3 and Hongsheng Zhao1 1Scottish Universities Physics Alliance, University of Saint Andrews, North Haugh, Saint Andrews, Fife, KY16 9SS, UK 2Helmholtz-Institut fur¨ Strahlen und Kernphysik (HISKP), University of Bonn, Nussallee 14−16, D-53115 Bonn, Germany 3Department of Particle Physics and Astrophysics, Weizmann Institute, Rehovot 7610001, Israel 23 January 2019 ABSTRACT We consider disk stability in the quasi-linear formulation of MOND (QUMOND), the basis for some N-body integrators. We generalize the Toomre criterion for the stability of disks to tightly wound, axisymmetric perturbations. We apply this to a family of thin exponential disks with different central surface densities. By numerically calculating their QUMOND rotation curves, we obtain the minimum radial velocity dispersion required for stability against local self-gravitating collapse. MOND correctly predicts much higher rotation speeds in low surface brightness galaxies (LSBs) than does Newtonian dynamics without dark matter. Newtonian mod- els thus require putative very massive halos, whose inert nature implies they would strongly stabilize the disk. MOND also increases the stability of galactic disks, but in contradistinction to Newtonian gravity, this extra stability is limited to a factor of 2. MOND is thus rather more conducive to the formation of bars and spiral arms. Therefore, observation of such features in LSBs could be problematic for Newtonian galaxy models. This could constitute a crucial discriminating test. We quantitatively account for these facts in QUMOND. We also compare numerical QUMOND rotation curves of thin exponential disks to those predicted by two algebraic expressions commonly used to calculate MOND rotation curves. For the choice that best approximates QUMOND, we find the circular velocities agree to within 1.5% beyond ≈ 0:5 disk scale lengths, regardless of the central surface density. The other expression can underestimate the rotational speed by up to 12.5% at one scale length, though rather less so at larger radii. Key words: gravitation { instabilities { dark matter { galaxies: kinematics and dynamics { methods: analytical { methods: numerical 1 INTRODUCTION crepancies1 are not due to the presence of DM but arise from a breakdown of Newtonian dynamics that is reckoned MOND (Milgrom 1983) is an alternative to the dark matter without in analyzing the dynamics of these systems. MOND (DM) hypothesis in accounting for the observed dynam- is extensively reviewed in Famaey & McGaugh(2012) and arXiv:1808.10545v5 [astro-ph.GA] 20 Jan 2019 ical discrepancies in galactic systems, especially between Milgrom(2014). The M31 timing argument in MOND was their measured rotation curves (RCs) and those predicted discussed by Zhao et al.(2013) and detailed calculations by Newtonian gravity (e.g. Rubin & Ford 1970; Rogstad & were presented in section 2 of Banik et al.(2018). Shostak 1972). A discrepancy is also apparent in the `timing MOND introduces a as a fundamental acceleration argument': in Newtonian gravity, the visible masses of the 0 scale of nature. When the gravitational field strength g Galaxy and M31 are insufficient to turn their initial post- a , standard dynamics is restored. However, when g a , Big Bang recession around to the observed extent (Kahn 0 0 the dynamical equations become scale-invariant (Milgrom & Woltjer 1959). MOND posits that these acceleration dis- 2009b). In this deep-MOND regime, an inevitable conse- quence of scale invariance is that, asymptotically far outside ∗Email: [email protected] (Indranil Banik) [email protected] (Mordehai Milgrom) 1 sometimes called mass discrepancies, though the reason for the [email protected] (Hongsheng Zhao) discrepancy may not be missing mass c 2018 The Authors 2 Indranil Banik, Mordehai Milgrom and Hongsheng Zhao a distribution of total mass M, the rotational speed of a with the collisionless DM halos of the ΛCDM paradigm (e.g. test particle becomes independent of its distance R from the Salucci & Turini 2017; Desmond 2017a,b). p mass. This occurs when R rM ≡ GM=a0 , where rM is These halos were originally introduced to boost the RCs the MOND radius of the mass M. Therefore, in a modified of disk galaxies. If real, they would also endow their embed- gravity formulation of MOND, g / R−1 beyond the MOND ded disks with added stability because the halo contribution radius. Applying dimensional arguments to the fact that a0 to the gravitational field responds very little to density per- is the only additional constant in MOND, we must have that turbations in the disk, making the disk more like a set of s test particles. This fact forms the basis of another argument pGMa 0 GM originally adduced for the presence of DM halos around disk g / for R rM ≡ : (1) R a0 galaxies (Ostriker & Peebles 1973) − without such halos, ob- served galactic disks would be deleteriously unstable (Hohl The normalization of a0 is taken so that the proportionality −10 2 1971). here becomes an equality. Empirically, a0 ≈ 1:2×10 m/s to match galaxy RCs (e.g. McGaugh 2011; Li et al. 2018). A crucial prediction of MOND was that low surface Note that MOND gravity must be non-linear in the matter brightness galaxies (LSBs) would show large acceleration p discrepancies at all radii because g a everywhere in distribution because g / M. 0 the galaxy (Milgrom 1983). This prediction was thoroughly It was noticed early on (e.g. Milgrom 1983, 1999) that, vindicated by later observations (e.g. de Blok & McGaugh remarkably, this value is similar to accelerations of cosmo- 1997; McGaugh & de Blok 1998). logical significance. For example, In ΛCDM, such LSBs must be assigned a halo much c2 more massive than the disk. Such a halo would cause the 2πa0 ≈ aH (0) ≡ cH0 ≈ aΛ ≡ ; (2) `Λ disk to become very stable, stymieing the formation of bars and spiral arms which are believed to result from disk insta- where H0 is the present-day value of the Hubble constant −1=2 bilities (Lin & Shu 1964). and `Λ = (Λ=3) is the de Sitter radius corresponding to the observed value of the cosmological constant Λ (e.g. DES Since MOND posits that dark halos are absent, the Collaboration 2018). question arises regarding the degree of stability of disks, especially LSB disks in which MOND is predicted to have Stated another way, a is similar to the acceleration at 0 significant effects. This issue was discussed in some detail which the classical energy density in a gravitational field by Milgrom(1989), who showed that MOND generally does (Peters 1981, equation 9) becomes comparable to the dark add to the stability of low-acceleration disks with a given energy density u ≡ ρ c2 implied by Λ. Λ Λ mass distribution and velocity dispersion. The reason is that 2 g instead of the Newtonian relation gd / Σ between surface < uΛ , g . 2πa0 : (3) 8πG mass density Σ and the acceleration gpd it produces in the This association of local MOND with cosmology suggests disk, the deep-MOND relation is gd / Σ. This means that that MOND may arise from quantum gravity effects (e.g. in Newtonian disks without a DM halo, density perturba- Milgrom 1999; Pazy 2013; Verlinde 2016; Smolin 2017). tions δΣ produce acceleration perturbations δg δΣ Regardless of its underlying microphysical explanation, d ∼ : (4) MOND correctly predicted the RCs of a wide variety of both gd Σ spiral and elliptical galaxies across a vast range in mass, Adding a non-responsive DM halo with contribution gh, we surface brightness and gas fraction (e.g. Lelli et al. 2017; can write gd = gb + gh, where gb is the Newtonian contribu- Li et al. 2018). Although internal accelerations are harder tion of the baryonic disk that satisfies Equation4. Because to measure within ellipticals, this can sometimes be done density perturbations in the disk do not affect gh, we have accurately when they have a surrounding X-ray emitting that gas envelope in hydrostatic equilibrium (Milgrom 2012) or a δg δΣ g −1 thin rotation-supported gas disk (e.g. den Heijer et al. 2015; d ∼ 1 + h : (5) Serra et al. 2016). The success of MOND extends down to gd Σ gb pressure-supported galaxies as faint as the satellites of M31 The reduced response of gd implies increased disk stability. (McGaugh & Milgrom 2013) and the Milky Way, though The analogous result in the deep-MOND regime is in the latter case one must be careful to exclude galaxies δg 1 δΣ where MOND predicts significant tidal distortion (McGaugh d ∼ (6) & Wolf 2010). For a recent overview of how well MOND gd 2 Σ p works in several different types of galaxy across the Hubble because gd / Σ and gh = 0. The added degree of sta- sequence, we refer the reader to Lelli et al.(2017). bility in deep MOND is thus similar to that endowed by a It is worth emphasizing that MOND does all this based halo with gh ∼ gb. As MOND effects are strongest in this solely on the distribution of luminous matter. It is clear regime, it is clear that this is the limit to how much MOND that these achievements are successful a priori predictions enhances the stability of disk galaxies, even those with very because most of these RCs − and sometimes even the baryon low surface densities. However, the massive halos required distributions − were measured in the decades after the first by ΛCDM would increase the stability indefinitely as the MOND field equation was put forth (Bekenstein & Milgrom surface density is reduced. This makes deep-MOND LSBs 1984) and its new fundamental constant a0 was determined develop spiral arms and fast-rotating bars more readily than (Begeman et al.
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