arXiv:2005.11316v1 [astro-ph.GA] 22 May 2020 to ytevsbebroi asadKpeindynamics. Keplerian and under- mass baryonic be ro- visible cannot outer the which flat by nearly stood disk the of be (RCs) will curves study tation this of topic main The INTRODUCTION 1 MNRAS ( † ⋆ observations. the provide might for DM explanation than modificatio an rather and a potential composition Even gravitational its discovered. the that as be of states well to time, as have DM still same of distribution the amount at the but, on details exist, halos dark interpretations, their and high. ies very is light-density to mass-density outer of the 3115, ratio new NGC the In that the required. or of be cause parts the might He be small. considerations could too dynamical was absorption outer coefficient that indicat velocity the mass-luminosity suggested angular of the constant rotation that with a ing disk 31) the (M of Nebula portions Andromeda the for (DM; matter dark of concept a enapidt xli h a C fds aais(cf. galaxies disk of RCs flat the concept explain e.g. DM to the cluster, applied galaxy been Coma has the distributions in velocity and the Galaxy our in discovered problems resolve nterda ceeaino ikgalaxies disk of acceleration radial the On c 2 1 2008 lu Wilhelm, Klaus eateto hsc,Ida nttt fTcnlg (Ban Technology of Institute Indian Physics, of Department a-lnkIsiu f Max-Planck-Institut -al [email protected] E-mail: -al [email protected] E-mail: 00TeAuthors The 2020 ui 1983 Rubin Since Burkert nadtie umr fteosrain fgalax- of observations the of summary detailed a In ); 000 olnKlhn ulc Kaplinghat & Bullock Boylan-Kolchin, Oort , 1 ( 1995 – 13 , ( 1986 1932 22)Pern 6My22 oplduigMRSL MNRAS using Compiled 2020 May 26 Preprint (2020) ); ru&Koopmans & Treu .Ee before, Even ). rSnessefrcug(P) utsvnLei-e 3 Justus-von-Liebig-Weg (MPS), Sonnensystemforschung ur ¨ and ) 1 ⋆ hl .Dwivedi N. Bhola Zwicky Rubin ukeMaterie dunkle Oort a enpsil ofidgo t ihu Mfrteosre ikv disk physics observed astroparticle 3198. the NGC for of words: NG DM matter Key and extraplanar without 3205 the fits UGC for good 1090, also NGC find example, 2403, to NGC as possible 3198, NGC been galaxie galaxies description has more a the five for considered of have gravitation data and of curves theory velocity question and impact bu open celeration W the process. an studies, use physical remains to the a matter attempts without of concept baryonic mathematical e most with a are in interaction formulates (MOND) its curves dynamics and rotation Newtonian DM non-Keplerian of the modification t a model gravitational or consistent t over a st scenarios within literature are (DM) understanding the galaxies an in at disk appeared arriving have of out articles dynamics of the Hundreds defining stood. processes physical The ABSTRACT ( ( Babcock ( 1940 1933 2000 ( 2004 lofudthat found also ) the introduced ) ocue that concludes ) nodrto order in ) ( ); 1939 aais prl–glxe:knmtc n yais–gaiain– gravitation – dynamics and kinematics galaxies: – spiral galaxies: McGaugh found ) ( 2011 2 rsHnuUiest) aaai210,India Varanasi-221005, University), Hindu aras of ); n † - D rudglxe . stedrc aietto fone of manifestation direct the as ... galaxies around “DM on otene o eiino h urn Mparadigm” DM current the of revision a ( for need the ( to energy point dark and DM etl ta.2009 al. et Gentile are candidates DM more many Many that by open. mentioned conclude component but remain main mass, distribution. questions increased gaslike a with Grijs de a flatten have & disks with Kruit der profiles one van and Kregel, DM halo”) “coupled the (the dark that the shows between between ratios are mass components the baryonic that and and cores central Pesta˜na & Bottema 2016 ( ( al. et Weinberg fisms xrodnr ytre”( mysteries” extraordinary most its of in ihtehl fetne C f1 ikgalaxies disk 12 of RCs extended of help the with tions akmte SD)i osdrda osblt ( possibility self-interacti as considered bosonic is by on (SIDM) challenges matter formation are dark there that scales. but galactic Universe, the it in successful although cosmic unknown, large-scale still of dynamics is structures. the formation dominates galaxy apparently in role ld mseiu aeil ( material” “mysterious clude ml n ag aais respectively. galaxies, large and small el ta.2017 al. et Lelli 2019 2018 ti hsntsrrsn htsaeet bu Min- DM about statements that surprising not thus is It .I nivsiaino uiosadD distribu- DM and luminous of investigation an In ). tt httecl akmte CM ocp is concept (CDM) matter dark cold the that state ) exact its and DM of nature the that conclude also ) 77 G 37077 , Freeman cnie tal. et Schneider ( 2015 .I s“. rubydeetmseyof mystery deepest arguably “... is It ). ; aue auc 2017 Salucci & Karukes ( tign Germany ¨ ottingen, ); 2015 ( eesen2010 Bekenstein atgir tal. et Battaglieri 2008 ocueta Mhlshave halos DM that conclude ) and ) iny a srkr1987 Ostriker & May Binney, ( 2017 ( 2002 ). alnht e Yu & Ren Kaplinghat, els eae with- decades last he aecniudour continued have e er.Dr matter Dark heory. n rn that trend a find ) .“u eut may results “Our ). ftepcla ac- peculiar the of oaoe l 2009 al. et Donato .Uigpublished Using s. ( l oryunder- poorly ill A ≈ 2017 h aueof nature the t T E Giraud ,mseisof mysteries ), tl l v3.0 file style X to 9 lcte and, elocities and ) n MOND and poe to mployed 75 it 1705, C b tal. et Eby ≈ Salucci ( 2000 for 5 ng ), ) ; 2 Klaus Wilhelm and Bhola N. Dwivedi modern physics” (Ghari et al. 2019); de Blok (2018) asks based on gravitons which lose energy and momentum when whether there is a universal alternative and at the begin- interacting with baryonic matter. We have invoked the basic ning of their abstract Rodrigues et al. (2018) write: “Dark idea of impacting gravitons – originally called quadrupoles – matter is currently one of the main mysteries of the Uni- with no mass and a speed of light c0. They are absorbed by − verse.” Salucci (2019) concludes his paper with reference to massive particles and re-emitted with reduced energy TG − the new observational opportunities: “... if one believes or according to T = TG (1 Y ), where TG is the energy G − not that this will lead to a solution of the old mystery of (very small) of a graviton in the background flux and Y dark matter.” with 0

MNRAS 000, 1–13 (2020) On the radial acceleration of disk galaxies 3

19 4 −4 3 SPIRAL GALAXIES where Af = 9.94 10 kg s m is the best fit for the co- × efficient A (cf. McGaugh et al. 2019). It is valid over many MOND provides reasonable predictions both for galaxies f decades in mass (McGaugh 2008) and describes a tight rela- with low masses and rotation velocities (Milgrom & Sanders tion between rotation speed and the luminosity or the bary- 2007) as well as for massive baryonic systems, whereas galac- onic mass of disk galaxies (cf. e.g. Navarro 1998; Bekenstein tic bulges pose a complication (Sanders & Noordermeer 2010; Lelli et al. 2017), although its relation to MOND is 2007). Modified gravity thus is an alternative for DM unclear (McGaugh 2012). (Bekenstein 2010; Sanders 2019). Li, Tang & Lin (2017) Low surface brightness (LSB) galaxies have larger compare DM models, but cannot prefer one model and ex- mass discrepancies between visible and Newtonian clude others. Since DM particle have not been found, MOND dynamical mass than high surface brightness (HSB) is the best fit for RCs. It breaks down, however, for very low 36 6 1 ones (Sanders & Noordermeer 2007; McGaugh 2014; masses of Mbar < 2 10 kg (i.e. 10 M⊙) (Dutton et al. × ≈ McGaugh et al. 2019). We feel that these observations 2019). provide important support for our interaction model as will There is general agreement that a close relationship ex- be discussed in Section 4. ists between baryonic mass and the observed radial accel- The RC shapes also depend on the surface brightness: eration g (McGaugh 2005a). It can be described by the obs LSB galaxies have slowly rising RCs, whereas in HSB sys- radial acceleration relation (RAR) tems the speed rises sharply and stays flat or even declines gbar beyond the optical radius (Navarro 1998). gobs = (3) 1 e−√gbar /gt In the abstract Noordermeer et al. (2007) write: “At in- − termediate radii, many RCs decline, with the asymptotic according to McGaugh, Lelli & Schombert (2016) and −10 −2 rotation velocity typically 10 to 20 percent cent lower than McGaugh et al. (2019), where gt = 1.2 10 m s . × the maximum. The strength of the decline is correlated with Lelli et al. (2017) point out that there is no room for a the total luminosity of the galaxies, more luminous galax- substantial variation of g . Assuming the DM concept, t ies having on average more strongly declining RCs. At large the DM halo contribution is then fully specified by that radii, however, all declining RCs flatten out, indicating that of the baryons (see e.g. Tenneti et al. 2018; Ghari et al. substantial amounts of dark matter must be present in these 2019). The authors add that black holes are not important galaxies.” These findings might be directly relevant for our in this context and Richards et al. (2018) find it remark- study, cf. Sections 4.1, 4.3 and 4.4. able that the tight relationship between DM and baryonic mass is not affected by the different distributions of the baryons in galaxies. Many more studies demonstrate that DM halos and stellar disks are indeed strongly correlated 4 MULTIPLE INTERACTIONS OF (e.g. Sancisi 2004; Fraternali, Sancisi & Kamphuis 2011; GRAVITONS IN GALACTIC DISKS Salucci & Turini 2017; Wechsler & Tinker 2018; Li et al. For isolated masses Newton’s law is valid for the impact 2019a,b). This may imply new physics for DM or a mod- model, however, in large mass conglomerations multiple in- ification of gravity (McGaugh 2005a). Recent observations, teractions of the proposed gravitons happen and lead to their related to improved observation and evaluation techniques multiple energy decreases. In a spherical configuration, this (cf. e.g. Behroozi et al. 2019), have revealed very low halo does not significantly change the 1/r2 dependence of the ac- masses relative to the disk mass adding more constraints on celeration field, but in a flat geometry, such as that of disk the galaxy–DM halo connection (cf. Sofue 2016; Posti et al. galaxies, the multiply affected gravitons spread out predom- 2019). inantly in the plane of the disk and thus display a 1/r de- The mass discrepancy-acceleration relation (MDAR) pendence consistent with the flat RCs. It must be noted, V 2 (R) however, that a disk galaxy has a certain thickness, pre- obs = (4) V 2 (R) D sumably related to its surface brightness. LSB galaxies will bar thus, in general, comply with the 1/r rule better than HBS describes a force law in disk galaxies at radius R (McGaugh galaxies that are expected to have a 1/rβ shape with β > 1. 2014; Lelli et al. 2017; Navarro et al. 2017; Desmond 2017). We assumed in table 2 of Wilhelm & Dwivedi (2018) McGaugh (2008) discusses this relation and MOND and con- that the mean free path length of gravitons λ0 is equal to cludes that the physics behind these empirical relations is Rgal and found that an amplification factor F of 1.75 to 2 unclear. In the inner regions of dwarf galaxies the baryonic could be achieved by 5 to 6 interactions. With λ0 = Rgal/4 mass distribution cannot be responsible for the RCs and thus the calculation has been extended for this study and gives a MDAR is not valid there according to Santos-Santos et al. factor of about 5 for 10 iterations. (2019). Details on the disk galaxies NGC 3198, NGC 2403, The Tully-Fisher relation (TFR) between luminosity NGC 1090, UGC 3205 and NGC 1705 will be presented in and flat rotation velocity (Tully & Fisher 1977) as well as the following subsections. the corresponding baryonic Tully-Fisher relation (BTFR) (McGaugh 2005b) depend on the fourth power of the flat 4.1 NGC 3198 rotation velocity Vf at large R:

4 Mbar = Af Vf , (5) The spiral disk galaxy NGC 3198 is shown in Fig. 1. Begeman (1987) observed a large discrepancy between the actual RC and that predicted from the light distribution in 1 a Solar mass: M⊙, cf. Table 1 ( ). this galaxy. Karukes, Salucci & Gentile (2015) state that it

MNRAS 000, 1–13 (2020) 4 Klaus Wilhelm and Bhola N. Dwivedi

Figure 1. Spiral galaxy NGC 3198. Type: SB(rs)c. The ap- parent size is 8.5′ × 3.3′ (Bratton 2011), which corresponds to about (22.0 × 8.5) × 1020 m (i.e. 71.3 kpc × 27.5 kpc) at a distance of 14.4 Mpc (O’Meara 2011). Image: NGC3198-SDSS- DR14(panorama).jpg. Credit: (SDSS), DR14 with SciServer. provides spectacular evidence of a dark force in action as baryons are unable to account for the kinematics. NGC 3198 Figure 2. Spiral Galaxy NGC 3198 (cf. Fig. 1). The Vobs and V data in the upper section are taken from table 1 of has been studied by many groups (e.g. van Albada et al. star Karukes, Salucci & Gentile (2015) (combined from various refer- 1985; Taga & Iye 1994; Blais-Ouellette, Amram & Carignan ences) and Vgas from their figure 3. Rotation velocities Vmult2D 2001; Kassin, de Jong & Weiner 2006; Kostov 2006; are calculated from the acceleration gmult2D in the lower sec- Lovas & Kielkopf 2014; Karukes & Salucci 2017; tion (see Section 4). The plots gobs, gstar and ggas are ob- de Almeida, Amendola & Niro 2017; Daod & Zeki 2019) tained with equation (1) from the corresponding velocities. The and we compiled some of the results in Figs. 1 to 3 as amplification factors f1 to F and gfit are explained in the well as in Table 1. We did, however, not refer to the many text. Three-dimensional fits in the outer reaches of the galaxy star gas statements related to DM as we feel that our multiple provide estimates on M3D and M3D (cf. Table 1). Charac- 20 teristic scales are: RD = 1.14 × 10 m (i.e. 3.69 kpc) (p. interaction model can explain the flat RCs without a DM 20 5, Karukes, Salucci & Gentile 2015), λ0 = 2.82 × 10 m component. 20 (i.e. 7.39 kpc), Rfit = 3.39 × 10 m (i.e. 11.0 kpc) and Rgal = The RCs published by Karukes, Salucci & Gentile 20 3.65 × 10 m (i.e. 11.8 kpc). The acceleration gmult2D(R) is de- (2015) are plotted in the upper panel of Fig. 2 and converted 1.05 clining with (Rfit/R) for R ≥ Rfit. The superimposed trian- to accelerations in the lower panel with equation (1). Using gles will be explained in Section 6 equation (3), i.e. Eq. (4) of McGaugh, Lelli & Schombert (2016), the curve gobs was approximated by the shaded area enclosing the data points. The required gfit as a modification free path λ0 of the gravitons is comparable to Rgal. A near of gt is given in Table 2 and the uncertainties are indicated perfect fit to the observations at larger radii is then obtained on the right-hand scale of the diagram. For large radii three- by star gas dimensional fits gave mass estimates of M3D and M3D . β Under the assumption that most of the baryonic mass Rfit gmult2D(R)= gmult2D(Rfit) (7) is inside a certain radius R = λ0, we can write  R  with an exponent β = 1.05. The flat and slightly declin- GN Mbar λ0 , (6) ing RC in the upper panel, i.e. Vmult2D(R), then follows ≈ r gfit with equation (1). The physical interpretation of an expo- −11 3 −1 −2 where GN = 6.674 30(15) 10 m kg s is the New- nent β > 1 obviously is that the disk of the galaxy has a × tonian constant of gravitation (Source: 2018 CODATA) and certain thickness and the graviton interactions do not ex- gfit is the acceleration near λ0. actly occur in a two-dimensional plane. This conclusion is On the abscissa of Fig. 2, we have plotted RD, λ0 and also supported by the slower rotation velocity of the extra- Rgal and notice that at Rfit (near Rgal) the factor between planar gas observed by Gentile et al. (2013) as summarized gstar and gobs is close to 1.75, the value expected, if the mean in their conclusions: “We revealed for the first time in this

MNRAS 000, 1–13 (2020) On the radial acceleration of disk galaxies 5

Figure 3. Spiral Galaxy NGC 3198 shown with explicit bulge Figure 4. The diagram is the same as in Fig. 3, but the bulge 2 2 0.5 contribution. The Vobs, Vstar, Vgas and Vbulge data in the up- baryons are not included in Vdisk = (Vstar + Vgas) . This elim- per section are taken from figure 37 (ISO, fixed) of de Blok et al. ination obviously has a dramatic impact at small R, however, (2008). The baryonic rotation velocities in the upper panel are did not influence the RC Vmult2D at large R. The characteristic 2 2 2 0.5 20 20 calculated by Vbar = (Vstar + Vgas + Vbulge) . Vmult2D and the scales λ0 = 3.52×10 m (i.e. 11.4 kpc) and Rfit = 3.70×10 m other quantities are obtained as in Fig. 2 and in later diagrams. (i.e. 12.0 kpc) changed slightly, but the amplification factor F and 20 The characteristic scales are: RD = 1.14× 10 m (i.e. 3.69 kpc) the exponent β are not affected. 20 (p. 5, Karukes, Salucci & Gentile 2015), λ0 = 3.83 × 10 m 20 (i.e. 12.4 kpc), Rfit = 4.32 × 10 m (i.e. 14.0 kpc) and 20 Rgal = 3.65 × 10 m (i.e. 11.8 kpc). The accelera- NGC 3198 indicates the presence of a bulge. In Fig. 3 this 1.05 tion gmult2D(R) is declining with (Rfit/R) for R ≥ Rfit. component is explicitly included with data of de Blok et al. (2008). The treatment of the data is very similar to that of Fig. 2. As a result only λ and R are shifted to some- galaxy the presence of extraplanar gas over a thickness of a 0 fit what larger values and g decreased, but still is close to the few ( 3) kpc. Its amount is approximately 15 % of the total fit ∼ acceleration, where the amplification factor F is 1.75 at Rfit. mass, and one of its main properties is that it appears to be Since the increased accelerations appear to be directly rotating more slowly than the gas close to midplane, with a related to the disk-like distribution of the baryonic mat- (rather uncertain) rotation velocity gradient in the vertical − − ter, the more spheroidal bulge component should not signif- direction (lag) of (7 to 15) km s 1 kpc 1”. Other observa- icantly contribute to the effect. In order to test this expec- tions of extraplanar gas showing a decrease in velocity above tation, we eliminated in Fig. 4 the bulge before calculating the galactic plane are mentioned in the next subsection. V (R). This obviously had a major effect on the rotation In Section 6 we will discuss this configuration in more disk speeds and accelerations at small radii, however, had nearly detail taking into account energy and angular momentum no influence on Vmult2D(R) for R Rfit. conservation. ≥ The acceleration gmult2D(Rfit) also provides a starting point for approximations at smaller radii. The multiple inter- 4.2 NGC 2403 action scenario implies that inside the disk gravitons travel back and forth after multiple energy reductions. The ampli- The spiral galaxy NGC 2403 is shown in Fig. 5. This fication factor should, therefore, be close to zero at R = 0. LSB galaxy is also the subject of many publications (e.g. A linear interpolation between Rfit and R = 0 gives a very Jarrett et al. 2003; Lovas & Kielkopf 2014; McGaugh 2014). good fit. The values of four of the factors are plotted in the It is discussed by Jalocha et al. (2011) in the con- diagram. text of “a global thin-disc model of spiral galaxies” as a The irregular behaviour of gobs near the centre of system with vertical gradients of the azimuthal velocity.

MNRAS 000, 1–13 (2020) 6 Klaus Wilhelm and Bhola N. Dwivedi

Figure 5. Spiral galaxy NGC 2403. Type: SAB(s)cd, apparent size 21.9′ × 12.3′, which corresponds to about (10.6 × 5.9) × 1020 m (i.e. 34.3 kpc × 19.1 kpc) at a distance of 2.69 Mpc (NASA/IPAC Extragalactic Database). Credit and Copyright Adam Block/Mount Lemmon SkyCenter/University of Arizona - www.caelumobservatory.com/gallery/n2403.shtml.

Figure 7. Spiral galaxy NGC 1090. Type: SB(rs)bc, apparent size 3.9′ × 1.8′ , which corresponds to about (2.6 × 1.2) × 1021 m (i.e. 84.3 kpc×38.9 kpc) at a distance of 37.5 Mpc (Bratton 2011). Image: NGC1090-SDSS-DR14.jpg. Credit: Sloan Digital Sky Sur- vey (SDSS), DR14.

Based on data from Fraternali et al. (2002), they find a near linear decrease at heights above the plane between (0.6 and 3.0) kpc of ( 10 4) km s−1 kpc−1. This decrease − ± is consistent with other observations of peculiar kinemat- ics (Schaap, Sancisi & Swaters 2000; Fraternali et al. 2001; Fraternali, Oosterloo & Sancisi 2004; Fraternali & Binney 2008; Fraternali et al. 2010), but Fraternali & Binney (2006) point out that their model does not predict the ob- served inflow. In Fig. 6 the RCs taken from figure 1 of Frigerio Martins & Salucci (2007) display a perfectly flat Vobs that could best be fitted with β 1. For a thin, ≈ LSB galaxy such a result could be expected as most of the graviton interactions have to occur in the plane of the disk. Outside this plane, the extraplanar gas will experience normal gravitational attraction by Mbar with a Keplerian rotation velocity, i.e. slower than the Vdisk.

4.3 NGC 1090

In Fig. 8 we have modelled the observed RC of spiral galaxy NGC 1090 shown in Fig. 7 quite successfully as Vmult2D(R) under the assumption of an amplification factor F = 1.5 Figure 6. Spiral Galaxy NGC 2403 (cf. Fig. 5). The and an exponent β = 1.15. In this case, the high exponent Vobs, Vbar and Vgas velocity data are taken from figure 1 might be caused by the small amplification factor F and a of Frigerio Martins & Salucci (2007). Characteristic scales are: relatively great λ0 indicating fewer multiple interactions of 19 RD = 6.42× 10 m (i.e. 2.08 kpc) (McGaugh 2005b), rb,half = gravitons. 20 1.21 × 10 m (i.e. 3.92 kpc) (Santos-Santos et al. 2019), λ0 = Lovas & Kielkopf (2014) write that there is a clear 20 20 1.64 × 10 m (i.e. 5.31 kpc), Rgal = 2.05 × 10 m (i.e. 6.64 kpc) need for CDM for explaining the RCs of NGC 1090 and R = 2.08 × 1020 m (i.e. 6.74 kpc). The flat RC is not fit (and other galaxies), however, its distribution is un- declining, i.e. β ≈ 1. clear. Kassin, de Jong & Weiner (2006) define a radius RX, where the relative contribution of DM to the ob-

MNRAS 000, 1–13 (2020) On the radial acceleration of disk galaxies 7

Figure 8. Spiral Galaxy NGC 1090 (cf. Fig. 7). The Figure 9. Spiral Galaxy UGC 3205 of Type Sab. The veloci- Vobs, Vbar and Vgas data are taken from figure 1 of ties Vobs, Vstar, Vgas and Vbulge data in the upper section are Frigerio Martins & Salucci (2007). The characteristic scales are: taken from figure 1 of Sanders & Noordermeer (2007) as well as 20 2 2 2 0.5 RD = 1.05 × 10 m (i.e. 3.40 kpc) (Frigerio Martins & Salucci “the Newtonian sum” Vbar = (Vstar + Vgas + Vbulge) . Char- 20 20 20 2007), Rgal = 3.36 × 10 m (i.e. 10.9 kpc), λ0 = 3.72 × 10 m acteristic scales are: rb,half = 1.92 × 10 m (i.e. 6.22 kpc) 20 20 (i.e. 12.1 kpc), Rfit = 3.86 × 10 m (i.e. 12.5 kpc) and (Santos-Santos et al. 2019), λ0 = 3.34 × 10 m (i.e. 10.8 kpc) 20 RX = 23.3 kpc) with 0.1 dex uncertainties (see table 2 in and Rfit = 4.37 × 10 m (i.e. 14.2 kpc). The accelera- 1.05 arXiv:astro-ph/0602027v1, Kassin, de Jong & Weiner 2006). The tion gmult2D(R) is declining with (Rfit/R) for R ≥ Rfit, re- amplification factor at Rfit is F = 1.50, whereas a value of sulting in ∆Vmult2D ≈ 6 km/s at R = 40 kpc relative to a flat RC. 1.75 was found for NGC 3198 and NGC 2403. The accelera- 1.15 tion gmult2D(R) is declining with (Rfit/R) for R ≥ Rfit. The dashed line in the upper panel indicates a constant flat RC. R2 will be discussed in the text. included and in Fig. 11 the bulge component is distributed over the disk. The observed RC is declining at large radii. In this con- served RC Vobs equals that of the baryonic mass Mbar. text, it may be of interest that many galaxies with declin- For NGC 1090 they find RX = 23.2 kpc and empha- ing RCs are found at z > 1, interpreted as possibly indi- size the findings of e.g. Persic & Salucci (1990) and cating smaller DM contributions (e.g. Genzel et al. 2017). Persic, Salucci & Stel (1996) that for LSB galaxies, such Drew et al. (2018), however, published a counterexample. as NGC 1090 as extreme case, cf. (Gentile et al. 2004) and Sanders & Noordermeer (2007) present a fit of UGC 3205 also http://www.kopernik.org/images/archive/n1090.htm, with MOND and find deviations from the observed veloci- the RCs deviate from the baryonic contribution at smaller ties below 10 kpc, tentatively attributed to the weak bar of radii than for HSB ones. The radius R2, where the value of this galaxy, but quite good agreement at large radii. gmult2D is twice that of gbar, can be compared to RX indi- We apply our multiple interaction model in Fig. 9 and cating that the multiple interaction process can explain the obtain a good fit even at small radii. This can be seen, in apparent mass discrepancy. particular, in the acceleration diagram in the lower panel. The similarity between the distribution of the bulge com- 4.4 UGC 3205 ponent of this galaxy with that of NGC 3198 in Fig. 3 mo- tivated us to apply the same procedure of eliminating the In Figs. 9 to 11, we plot RCs of the early-type bulge contribution in Fig. 10. The result again is that the (HSB) disk galaxy UGC 3205 with data taken from presumably spheroidal bulge contribution does not affect the Sanders & Noordermeer (2007), which are slightly modified RC at large radii. The claim of McGaugh (2014) that the in the next two figures. In Fig. 10 the bulge component is not BTFR is insensitive to the distribution of the baryonic mass

MNRAS 000, 1–13 (2020) 8 Klaus Wilhelm and Bhola N. Dwivedi

Figure 10. These diagrams are based on the same data as in Figure 11. These diagrams are again based on the same data as Fig. 9 (Sanders & Noordermeer 2007, figure 1), but the bulge in Fig. 9 taken from figure 1 of Sanders & Noordermeer (2007), component is not included. An important result of the Vmult2D but the bulge component is now distributed over the disk by as- mod mod and gmult2D calculations is that the elimination of the bulge suming a modified gdisk = gdisk ×Mdisk /Mdisk (cf. Table 1). This mod baryons does not effect the RC at large radii. The characteristic has a major impact on the extended RC with Vmult2D higher than 20 20 −1 scales λ0 = 3.20×10 m (i.e. 10.4 kpc) and Rfit = 3.63×10 m Vmult2D by more than 10 km s . The characteristic scales are 20 20 (i.e. 11.8 kpc) changed only very little, but the amplification fac- λ0 = 3.34 × 10 m (i.e. 10.8 kpc) and Rfit = 3.62 × 10 m tor F and β remain the same. (i.e. 11.7 kpc). in a galaxy, has been tested in Fig. 11, where the bulge mass RCs is – stated explicitly for NGC 1705 – that they are diffi- has been distributed over the disk. This has a major impact cult to reproduce with DM scenarios, if the central baryon on the extended RC. The relative increase of the resulting component is small. mod If we use the same procedure for this galaxy as for the Vmult2D is greater than 5 % and thus would give a fractional gain in baryonic mass of more than 20 % in equation (5). other ones, we find a very large gfit and a small λ0 relative Santos-Santos et al. (2019) also found that the RCs are af- to 2 rb,half . As mentioned in Sect. 4 this configuration allows fected by the distribution of the baryonic mass. many multiple interactions and produces an amplification factor of 5 for 10 iterations. This factor gives a good fit for R > Rfit 2 rb,half . Inside Rfit the amplification factor has ≈ 4.5 NGC 1705 to remain high down to λ0. The data do, however, not allow any conclusion for R<λ0. The four galaxies discussed in the preceding sections are all characterized by regular disk shapes. The irregular dwarf galaxy NGC 1705 we want to include in our sample in or- 5 SUMMARY der to test the applicability of our procedure on a broader scale. This compact galaxy is much smaller and contains in As a summary of the evaluations of the five galaxies, we com- its nucleus a luminous super with a mass of pile in Table 1 all mass values found in the literature and 35 5 ≈ 2 10 kg (i.e. 10 M⊙) and a significant cold dust compo- those obtained through the various fits. Uncertainty ranges × nent (Tosi et al. 2001; Annibali et al 2003; O’Halloran et al. are not included, but the large variations both of the pub- 2010). lished and the deduced values indicate significant uncertain- The RCs in Fig. 13 demonstrate that there is a very ties. In Table 2 the observed and derived velocities of the large discrepancy between Vbar and Vobs. A conclusion of RCs are listed together with the fit parameter gfit and the Santos-Santos et al. (2019) on the shapes of dwarf galaxy exponent β.

MNRAS 000, 1–13 (2020) On the radial acceleration of disk galaxies 9

Table 1. Baryonic mass values of the five disk galaxies from the literature (without uncertainty margins) and from fits in the figures.

a Mass( ) Mbulge/ Mstar/ Mgas/ Mdisk/ Mbar/ M3D/ Galaxies Figures 1040 kg 1040 kg 1040 kg 1040 kg 1040 kg 1040 kg References

NGC 3198 6.76 0.994 Begeman (1989) 12.9 Kostov (2006) 2.62 3.88 Stark, McGaugh & Swaters (2009) Fig. 2 (6.48)(b) (3.59) 8.75 (10.1) Karukes, Salucci & Gentile (2015) Fig. 3 0.573 5.60 (11.0) de Blok et al. (2008), figure 37 Fig. 4 { - }(c) (6.18)

NGC 2403 Fig. 6 0.887 2.41 [3.29](d) (3.97) Frigerio Martins & Salucci (2007) 2.19 0.934 [3.12] [3.12] McGaugh (2005b) 1.84 Santos-Santos et al. (2019) 2.41 0.887 0.801 3.97 de Blok et al. (2014) 1.04 Fraternali et al. (2002) 0.810 1.17 Stark, McGaugh & Swaters (2009)

NGC 1090 Fig. 8 [1.68](e) 9.34 [11.3] (13.1) Frigerio Martins & Salucci (2007) 1.69 Gentile et al. (2004) 10.7 Kassin, de Jong & Weiner (2006)

UGC 3205 13.9 Santos-Santos et al. (2019) Fig. 9 (1.66) (20.1) Fig. 10 { - }(c) (15.1) (3.32) (18.4) Fig. 11 { - }(f ) [20.1]

NGC 1705 ≥ 0.0557 ≈ 0.0239 Annibali et al (2003) Fig. 13 0.0897 (0.0897) Santos-Santos et al. (2019)

a 30 ( ): Mass of Sun (1.98847 ± 0.00007) × 10 kg=1 M⊙ (International Astronomical Union); b c ( ): Results of three-dimensional fits in figures are given in parentheses; ( ): Mbulge neglected; d e ( ): Values in brackets with modifications explained in text; ( ): 18 % of Mbar (Frigerio Martins & Salucci 2007); f ( ): Mbulge distributed over the disk.

Table 2. Observed rotation velocities of the five disk galaxies together with derived mass and acceleration values.

max max a min min b Galaxy Vobs / Vf / Afit/ ( )[Mfit,Mf ]/ Vobs / gmult2D/ gfit/ β( ) Figures km s−1 km s−1 kg s4 m−4 1040 kg km s−1 m s−2 m s−2

NGC 3198 Fig. 2 161.0 154.2 1.76 × 1020 [9.98, 5.63] 148.7 1.49 × 10−11 8.50 × 10−11 1.05 Fig. 3 158.7 151.0 3.00 × 1020 [15.6, 5.17] 147.3 1.85 × 10−11 5.00 × 10−11 1.05 Fig. 4 158.7 152.8 2.72 × 1020 [14.9, 5.42] 148.5 1.88 × 10−11 5.50 × 10−11 1.05

NGC 2403 Fig. 6 136.2 133.3 2.50 × 1020 [7.88, 3.13] 133.3 3.01 × 10−11 6.00 × 10−11 ≈ 1.00 NGC 1090 Fig. 8 177.2 172.7 3.33 × 1020 [29.6, 8.84] 161.7 2.83 × 10−11 4.50 × 10−11 1.15

UGC 3205 Fig. 9 242.9 221.2 1.25 × 1020 [29.9, 23.8] 215.7 3.86 × 10−11 1.20 × 10−10 1.05 Fig. 10 242.9 223.3 1.25 × 1020 [31.0, 24.7] 216.7 3.89 × 10−11 1.20 × 10−10 1.05 Fig. 11 242.9 233.2 (c) (c) 226.2 4.24 × 10−11 1.20 × 10−10 1.05

NGC 1705 Fig. 13 73.44 73.31 3.33 × 1019 [0.0962, 0.287] 73.31 2.89 × 10−11 4.50 × 10−10 ≈ 1.00

a 19 4 −4 b ( ): Coefficient Af = 9.94 × 10 kg s m in equation (5) (McGaugh 2008, 2012); ( ): Exponent of declining RCs; (c): Unrealistic test.

6 DISCUSSION AND CONCLUSIONS that the masses are in reasonable agreement for NGC 3198, NGC 2404, UGC 3205, and even for NGC 1705, but not for With the exception of the irregular galaxy NGC 1705, the NGC 1090. other acceleration diagrams in the lower panels indicate that The observed non-Keplerian behaviour of RCs could be for a good fit an amplification factor F between 1.5 and modelled for five disk galaxies without difficulties under the 1.8 was required to raise the g values to g . According bar obs assumption of multiple interactions of gravitons in the planes to equation (6) the corresponding galactic radius has to be of the disks. This can be considered as a substantial support Rfit λ0 provided most of the baryonic mass is within this ≈ for the graviton impact theory – a modification of the old limit. Equations (1), (6) and (7) then give for the flat RCs impact theory proposed by Nicolas Fatio de Duillier in 1690. with Vf an estimate of Afit 1/(gfit GN) for the coefficients ≈ in equation (5) and the corresponding baryonic masses Mfit As mentioned in Subsection 4.1, the physics of the ex- in Table 2. A comparison with mass values in Table 1 shows traplanar gas observations needs to be studied. This can

MNRAS 000, 1–13 (2020) 10 Klaus Wilhelm and Bhola N. Dwivedi

only be done in a very crude approximation at this stage. It will be helpful to visualize the gravitational potential at R>λ0 Rfit both for the case of Newton’s law outside the ≈ galactic plane (a) and near that plane (b). The situation will be considered explicitly at four radii R = Rfit [1, 2, 3, 4]. × Case (a) gives the usual gravitational potential of G M U(R)= N bar (8) − R

with U0 = 0 at . In case (b), we cannot assume a per- ∞ fectly flat case, corresponding to g(R) 1/R, because this ∝ would lead to Uplane(R) = ln(R), which is not zero at infin- β ity. However, gmod(R) 1/R with 1 < β 2 is a good ∝ ≪ approximation of the acceleration in the plane and can be integrated to give the potential

(β−1) GN Mbar Rfit Umod(R)= F (9) − (β 1) Rfit  R  Figure 12. Irregular dwarf galaxy NGC 1705, Type: SAO - pec. − ′ ′ Apparent size: 1.9 ×1.4 (Wikipedia), which corresponds to about when we require an amplification factor F at Rfit. The shape (1.8 × 1.3) × 1020 m (i.e. 5.83 kpc × 4.21 kpc) at a distance of of the potential near the galactic plane can be compared to (5.1±0.6) Mpc (O’Halloran et al. 2010). Credit: NASA, ESA and a canyon, if β is increased to 2 just above the plane. the Hubble Heritage Team (STScI/AURA), Acknowledgment: M. We will apply these results to NGC 3198 in Fig. 2 Tosi (INAF, Osservatorio Astronomico di Bologna) 40 and get with a baryonic mass of Mbar = 6.91 10 kg 10 × (i.e. 3.46 10 M⊙) the accelerations indicated by the tri- × angle marks and with equation (1) the velocities marked in the upper panel, where the slow speeds correspond to the Keplerian case outside the plane. The ratio of the potentials is Umod(Rfit)/U(Rfit) = 35, i.e. the “depth” of the canyon at Rfit. The extraplanar material cannot directly “fall” into the canyon, as this would violate the angular momentum con- servation (assuming a closed system). Energy and angular momentum can, however, be conserved, if the falling ma- terial moves inwards at the same time, because the radius decreases and the speed is nearly constant. Our model thus provides consistent pictures both for the in plane and the extraplanar material. Nevertheless, many questions remain, in particular, on the physical properties of the gravitons. Since it was possible to explain the anoma- lous RCs without assuming any DM that is also a complete mystery, there is hope that further studies will clarify the situation. Considering that MOND models are good fits for many galaxies, the multiple interaction might even point to a physical process behind the mathematical modification of the gravitational acceleration at small values.

ACKNOWLEDGEMENTS

This research has made extensive use of the Astrophysics Data System (ADS). Administrative support has been pro- vided by the Max-Planck-Institute for Solar System Re- search and the Indian Institute of Technology (Banaras Hindu University). We thank the anonymous referee for his/her constructive and valuable comments which improved the presentation of the manuscript.

Figure 13. Spiral Galaxy NGC 1705 (cf. Fig. 12). The Vobs and Vbar data in the upper section are taken from figure 5 of Santos-Santos et al. (2019). Characteristic scales are: λ0 = 1.15× 19 19 10 m (i.e. 0.373 kpc), rb,half = 2.90 × 10 m (i.e. 0.940 kpc) 19 and Rfit = 4.87 × 10 m (i.e. 1.58 kpc). Note the large amplifi- cation factor F = 5 and β ≈ 1.

MNRAS 000, 1–13 (2020) On the radial acceleration of disk galaxies 11

REFERENCES Dutton, A. A., Macci`o, A.V.,˙ Obreja, A., Buck, T., 2019, NIHAO - XVIII. Origin of the MOND phenomenology of galactic ro- van Albada, T. S., Bahcall, J. N., Begeman, K., Sancisi, R., 1985, tation curves in a ΛCDM Universe, MNRAS, 485, Issue 2, Distribution of dark matter in the spiral galaxy NGC 3198, 1886 ApJ, 295, 305 Dyson, F. W., Eddington, A. S., Davidson, C., 1920. A determina- ´ de Almeida, A., Amendola, L., Niro, V., 2017, Galaxy rotation tion of the deflection of light by the Sun’s gravitational field, curves in modified gravity models, J. Cosmology Astroparticle from observations made at the total eclipse of May 29, 1919, Phys., Issue 08, id. 012 Phil. Trans. R. astr. Soc. Lond. A, 220, 291 Angus, G. W., van der Heyden, K. J., Diaferio, A., 2012, The dy- Eby, J., Kouvaris, C., Nielsen, N. G., Grønlund, N., Wijeward- namics of the bulge dominated galaxy NGC 7814 in MOND, hana, L. C. R., 2016, Boson from self-interacting dark A&A, 543, id. A76 matter, J. High Ener. Phys., 2016, id.28 Annibali, F., Greggio, L., Tosi, M., Aloisi, A., Leitherer, C., 2003, Einstein, A., 1916, Die Grundlage der allgemeinen Relativit¨ats- The star formation history of NGC 1705: A poststarburst theorie, Ann. Phys. (Leipzig), 354, Issue 7, 769 galaxy on the verge of activity, AJ, 126, Issue 6, 2752 Fatio de Duilleir, N., 1690, De la cause de la pesanteur. Not. Rec. Babcock, H. W., 1939, The rotation of the Andromeda Nebula, Roy. Soc. London, 6, 2, 125 Lick Observatory Bull., 498, 41 Fraternali, F., 2010, Gas circulation and galaxy evolution , AIP Battaglieri, M., Belloni, A., Chou, A., and 248 authors, 2017, US Conf. Proc., 1240, 135 Cosmic Visions: New ideas in Dark Matter 2017: Community Fraternali, F., Binney, J. J., 2006, A dynamical model for the Report, arXiv170704591 extraplanar gas in spiral galaxies, MNRAS, 366, Issue 2, 449 Begeman, K. G., 1987, HI rotation curves of spiral galaxies, Ph.D. Fraternali, F., Binney, J. J., 2008, Accretion of gas on to nearby thesis, Kapteyn Institute spiral galaxies, MNRAS, 386, Issue 2, 935 Begeman, K. G., 1989, HI rotation curves of spiral galaxies. I Fraternali, F., Oosterloo, T., Sancisi, R., van Moorsel, G., 2001, NGC 3198, A&A, 223, 47 A new, kinematically anomalous H I component in the spiral Behroozi, P., Becker, M., van den Bosch, F. C. and 20 authors, galaxy NGC 2403, ApJ, 562, Issue 1, L47 2019, Empirically constraining galaxy evolution, BAAS, 51, Fraternali, F., van Moorsel G., Sancisi, R., Oosterloo, T., 2002, Issue 3, id. 125 The HI halo of NGC 2403, AJ, 123, 3124 Bekenstein, J. D., 2010, Alternatives to dark matter: Modified Fraternali, F., Oosterloo, T., Sancisi, R., 2004, Kinematics of the gravity as an alternative to dark matter, arXiv:1001.3876 ionised gas in the spiral galaxy NGC 2403, A&A, 424, 485 Binney, J., May, A., Ostriker, J. P., 1987, On the flattening of Fraternali, F., Sancisi, R., Kamphuis, P., 2011, A tale of two dark haloes, MNRAS, 226, 149 galaxies: Light and mass in NGC 891 and NGC 7814, A&A, Blais-Ouellette, S., Amram, P., Carignan, C., 2001, Accurate de- 531, A64 termination of the mass distribution in spiral galaxies. II. Freeman, K. C., 1970, On the disks of spiral and S0 galaxies, ApJ, Testing the shape of dark halos , AJ, 121, Issue 4, 1952 160, 811 de Blok, W. J. G., 2018, Is there a universal alternative to dark Freeman, K. C., 2008, Formation and evolution of galaxy disks: matter? Nature Astron., 2, 615 Overview, ASP Conf. Ser. 396, 3 de Blok, W. J. G., Walter, F., Brinks, E., Trachternach, C., Frigerio Martins, C., Salucci, P., 2007, Analysis of rotation curves Oh, S.-H., Kennicutt, R. C., Jr., 2008, High-resolution rota- in the framework of Rn gravity, MNRAS, 381, 1103 tion curves and galaxy mass models from THINGS , AJ, 136, Gentile, G., Salucci, P., Klein, U., Vergani, D., Kalberla, P., 2004, Issue 6, 2648 The cored distribution of dark matter in spiral galaxies, MN- de Blok, W. J. G., Keating, K. M., Pisano, D. J., Fraternali, F., RAS, 351, Issue 3, 903 Walter, F., Oosterloo, T., Brinks, E., Bigiel, F., Leroy, A., Gentile, G., Famaey, B., Zhao, H., Salucci, P., 2009, Universality 2014, A low H I column density filament in NGC 2403: signa- of galactic surface densities within one dark halo scale-length, ture of interaction or accretion , A&A, 569, id. A68 Nature, 461, 7264, 627 Bopp, K., (Ed.), 1929, Nicolas Fatio de Duillier: De la cause de Gentile, G., J´ozsa, G. I. G., Serra, P., Heald, G. H., la pesanteur, Schriften der Straßburger Wiss. Ges. de Blok, W. J. G., Fraternali, F., Patterson, M. T., Heidelberg, 10, 19 Walterbos, R. A. M., Oosterloo, T., 2013, HALOGAS: Extra- Bottema, R., Pesta˜na, J. L. G., 2015, The distribution of dark planar gas in NGC 3198, A&A, 554, id.A125 and luminous matter inferred from extended rotation curves, Genzel, R., F¨orster Schreiber, N. M., Ubler,¨ H. and 18 authors, MNRAS, 448, 2566 2017, Strongly baryon-dominated disk galaxies at the peak Boylan-Kolchin, M., Bullock, J. S., Kaplinghat, M., 2011, of galaxy formation ten billion years ago, Nature, 543, Issue Too big to fail? The puzzling darkness of massive milky way 7645, 397 subhaloes, MNRAS, 415, L40 Ghari, A., Haghi, H., Zonoozi, A. H., 2019, The radial acceleration Bratton, M., 2011, The complete guide to the Herschel objects, relation and dark baryons in MOND, MNRAS, 487, 2, 2148 Cambridge University Press, Cambridge, UK Ghari, A., Famaey, B., Laporte, C., Haghi, H., 2019, Dark matter- Burkert, A., 1995, The structure of dark matter halos in dwarf baryon scaling relations from Einasto halo fits to SPARC galaxies, ApJ, 447, L25 galaxy rotation curves , A&A, 623 id. A123 Daod, N. A., Zeki, M. K., 2019, Density and mass distribution of Giraud, E., 2000, A universal coupling relation between luminous spiral galaxy NGC 3198, ApJ, 870, id. 107,4 and dark matter surface densities in disk rotating galaxies, Desmond, H., 2017, A statistical investigation of the mass ApJ, 531, 701 discrepancy-acceleration relation, MNRAS, 464, Issue 4, 4160 Hague, P. R., Wilkinson, M. I., 2014, Dark matter in disc galax- Donato, F., Gentile, G., Salucci, P., Frigerio Martins, C., Wilkin- ies – II. Density profiles as constraints on feedback scenarios, son, M.I., Gilmore, G., Grebel, E.K.; Koch, A., Wyse, R., MNRAS, 443, 3712 2009, A constant dark matter halo surface density in galax- Jalocha, J., Bratek,L., Kutschera, M., Skindzier, P., 2011, ies, MNRAS, 397, Issue 3, 1169 Vertical gradients of azimuthal velocity in a global thin-disc Drew, P.M., Casey, C.M., Burnham, A.D., Hung, C.-L., model of spiral galaxies NGC 2403, NGC 4559, NGC 4302 and Kassin, S A., Simons, R. C., Zavala, J. A., 2018, Evidence of a NGC 5775, MNRAS, 412, 331 flat outer rotation curve in a star-bursting disk galaxy at Jarrett, T. H., Chester, T., Cutri, R., Schneider, S. E., z = 1.6, ApJ, 869, Issue 1, id. 58 Huchra, J. P., 2003, The 2MASS large galaxy atlas, AJ, 125,

MNRAS 000, 1–13 (2020) 12 Klaus Wilhelm and Bhola N. Dwivedi

525 Milgrom, M., Sanders, R. H., 2007, Modified Newtonian Dynam- Kaplinghat, M., Ren, T., Yu, H.-B., 2019, Dark matter ics Rotation curves of very low mass spiral galaxies, ApJ, 658, cores and cusps in spiral galaxies and their explanations, Issue 1, L17 arXiv:1911.00544 Navarro, J. F., 1998, The cosmological significance of disk galaxy Karukes, E. V., Salucci, P., Gentile, G., 2015, The dark matter rotation curves, arXiv:astro-ph/9807084 distribution in the spiral NGC 3198 out to 0.22 Rvir, A&A, Navarro, J. F., Ben´ıtez-Llambay, A., Fattahi, A., Frenk, C.S., 578, id.A13 Ludlow, A. D., Oman, K. A., Schaller, M., Theuns, T., 2017, Karukes, E. V., Salucci, P., 2017, The universal rotation curve of The origin of the mass discrepancy-acceleration relation in dwarf disc galaxies, MNRAS, 465, 4703 ΛCDM , MNRAS, 471, Issue 2, 1841 Kassin, S. A., de Jong, R. S., Weiner, B. J., 2006, Dark and bary- Nijhoff, M., (Ed.), 1901, Correspondance, 1685–1690, Nos. 2570 onic matter in bright spiral galaxies. II. Radial distributions et 2582. Œuvres Compl`etes de Christiaan Huygens, IX, 381 for 34 galaxies , ApJ, 643, 804 (attn.: table 2 of arXiv:astro- and 407, La Haya ph/0602027v1) Noordermeer, E., van der Hulst, J. M., Sancisi, R., Swaters, R.S., Kostov, V., 2006, Mass distribution of spiral galaxies in a thin van Albada, T. S., 2007, The mass distribution in early-type disk model with velocity curve extrapolation, arXiv:astro- disc galaxies: declining rotation curves and correlations with ph/0604395 optical properties, MNRAS, 376, Issue 4, 1513 Kregel, M., van der Kruit, P. C., de Grijs, R., 2002, O’Halloran, B., Galametz, M., Madden, S. C. and 56 authors, Flattening and truncation of stellar discs in edge-on spiral 2010, Herschel photometric observations of the low metallicity galaxies, MNRAS, 334, 646 dwarf galaxy NGC 1705, A&A, 518, id. L58 Kroupa, P., Pawlowski, M., Milgrom, M., 2012, O’Meara, S. J., 2011, Deep-sky companions: The secret deep, The failures of the Standard Model of Cosmology require a Cambridge University Press, Cambridge, UK new paradigm, Int. J. Mod. Phys. D, 21, id. 1230003 Oort, J. H., 1932, The force exerted by the stellar system in the von Laue, M., 1959, Geschichte der Physik, 4. erw. Aufl. Ullstein direction perpendicular to the galactic plane and some related Taschenbucher-Verlag,¨ Frankfurt/Main problems, Bull. Astron. Inst. Netherlands, 6, 249 Lelli, F., McGaugh, S. S., Schombert, J. M., Pawlowski, M. S., Oort, J. H., 1940, Some problems concerning the structure and 2017, One law to rule them all: The radial acceleration re- dynamics of the galactic system and the elliptical nebulae lation of galaxies, ApJ, 836, Issue 2, id. 152 NGC 3115 and 4494, ApJ, 91, 273 Li, X., Tang, L., Lin, H.-N., 2017, Comparing dark matter mod- Persic, M., Salucci, P., 1990, The disc contribution to rotation els, modified Newtonian dynamics and modified gravity in curves of spiral galaxies, MNRAS, 247, 349 accounting for galaxy rotation curves, Chin. Phys. C, 41, Is- Persic, M., Salucci, P., Stel, F., 1996, The universal rotation curve sue 5, id. 055101 of spiral galaxies – I. The dark matter connection, MNRAS, Li, P., Lelli, F., McGaugh, S., Pawlowski, M. S., Zwaan, M. A., 281, Issue 1, 27 (Note Erratum: MNRAS, 283, 1102) Schombert, J., 2019a, The halo mass function of late-type Posti, L., Marasco, A., Fraternali, F., Famaey, B., 2019, Galaxy galaxies from H I kinematics, ApJ, 886,1, id L11 disc scaling relations: A tight linear galaxy-halo connection Li, P., Lelli, F., McGaugh, S., Starkman, N., Schombert, J., challenges abundance matching, A&A, 629, id. A59 2019b, A constant characteristic volume density of dark mat- Richards, E. E., van Zee, L., Barnes, K. L., Staudaher, S., ter haloes from SPARC rotation curve fits, MNRAS, 482, Is- Dale, D. A., Braun, T. T., Wavle, D. C., Dalcanton, J. J., Bul- sue 4, 5106 lock, J. S., Chandar, R., 2018, Baryonic distributions in galaxy Lovas, S., Kielkopf, J. F., 2014, Distribution of dark and luminous dark matter haloes - II. Final results . MNRAS, 476, Issue 4, mass in galaxies, AJ, 147, id. 135,5 5127 McGaugh, S. S., 2005a, Balance of dark and luminous mass in Rodrigues, D. C., Marra, V., del Popolo, A., Davari, Z., 2018, Ab- rotating galaxies, Phys. Rev. Lett., 95, id. 171302 sence of a fundamental acceleration scale in galaxies, Nature McGaugh, S. S., 2005b, The baryonic Tully-Fisher relation of Astron., 2, 668 galaxies with extended rotation curves and the stellar mass Rubin, V. C., 1983, Dark matter in spiral galaxies, Scientific of rotating galaxies, ApJ, 632, 859 American, 248, 96 McGaugh, S. S., 2008, Balance of dark and luminous mass in ro- Rubin, V. C., 1986, Dark matter in the Universe, Highlights of tating galaxies, Phys. Rev. Lett., 95, Issuee 17, id. 171302 Astronomy, 7, 27 McGaugh, S. S., 2012, The baryonic Tully-Fisher relation of gas- Rubin, V. C., 2000, One hundred years of rotating galaxies, PASP, rich galaxies as a test of ΛCDM and MOND, 112, 747 AJ, 143, id. 40 Salucci, P., 2018, Dark Matter in galaxies: Evidences and chal- McGaugh, S. S., 2014, The third law of galactic rotation , Galax- lenges, Found. Phys. 48, Issue 10, 1517 ies, 2, 601 Salucci, P., 2019, The distribution of dark matter in galaxies, McGaugh, S. S., Lelli, F., Schombert, J. M., 2016, Astron. Astrophys. Rev., 27, Issue 1, id. 2 Radial acceleration relation in rotationally supported galax- Salucci, P., Turini, N., 2017, Evidences for collisional dark matter ies, Phys. Rev. Lett., 117, id. 201101 in galaxies? arXiv:1707.01059 McGaugh, S. S., Lelli, F., Li, P., Schombert, J. M., 2019, Dynam- Samurovi´c, S., 2016, The Newtonian and MOND dynamical mod- ical regularities in galaxies, arXiv:1909.02011 els of NGC 5128: Investigation of the dark matter contribu- Milgrom, M., 1983, A modification of the Newtonian dynamics tion, Serbian Astron. J., 192, 9 as a possible alternative to the hidden mass hypothesis ApJ, Sancisi, R., 2004, The visible matter – dark matter coupling, 270, 365 IAUS, 220, 233 Milgrom, M., 1994, Dynamics with a nonstandard inertia-accel- Sanders, R. H., 2019, The prediction of rotation curves in gas- eration relation: An alternative to dark matter in galactic dominated dwarf galaxies with modified dynamics, MNRAS, systems, Ann. Phys., 229, 384 485, Issue 1, 513 Milgrom, M., 2015, MOND theory, Can. J. Phys., 93, 107 Sanders, R. H., Noordermeer, E., 2007, Confrontation of MOdified Milgrom, M., 2016, MOND impact on and of the recently updated Newtonian Dynamics with the rotation curves of early-type mass-discrepancy-acceleration relation, arXiv:1609.06642 disc galaxies, MNRAS, 379, 702 Milgrom, M., 2020, The a0 – cosmology connection in MOND, Santos-Santos, I. M. E., Navarro, J. F., Robertson, A., Ben´ıtez- arXiv:2001.09729 Llambay, A., Oman, K. A., Lovell, M. R., Frenk, C. S.,

MNRAS 000, 1–13 (2020) On the radial acceleration of disk galaxies 13

Ludlow, A. D., Fattahi, A., Ritz, A., 2019, Baryonic clues to the puzzling diversity of dwarf galaxy rotation curves, arXiv:1911.09116 Schaap, W. E., Sancisi, R., Swaters, R. A., 2000, The vertical ex- tent and kinematics of the HI in NGC 2403, A&A, 356, L49 Schneider, A., Trujillo-Gomez, S., Papastergis, E., Reed, D.S., Lake, G., 2017, Hints against the cold and collisionless nature of dark matter from the galaxy velocity function, MNRAS, 470, 1542 Sofue, Y., 2016, Rotation curve decomposition for size-mass re- lations of bulge, disk, and dark halo components in spiral galaxies, Publ. Astron. Soc. Japan, 68, Issue 1, id.2 Sofue, Y., Rubin, V., 2001, Rotation curves of spiral galaxies, Ann. Rev. Astron. Astrophys., 39, 137 Soldner, J., 1804, Ueber die Ablenkung eines Lichtstrals von seiner geradlinigen Bewegung durch die Attraktion eines Weltk¨orpers, an welchem er nahe vorbei geht, Berliner As- tron. Jahrb. 161 Stark, D. V., McGaugh, S. S., Swaters, R. A., 2009, A first at- tempt to calibrate the baryonic Tully-Fisher relation with gas-dominated galaxies, AJ, 138, Issue 2, 392 Taga, M., Iye, M., 1994, Halo model of spiral galaxy NGC 3198, MNRAS, 271, 427 Tenneti, A., Mao, Y.-Y., Croft, R.A.C., Di Matteo, T., Kosowsky, A., Zago, F., Zentner, A. R., 2018, The radial ac- celeration relation in disc galaxies in the MassiveBlack-II sim- ulation, MNRAS, 474, Issue 3, 3125 Tosi, M., Sabbi, E., Bellazzini, M., Aloisi, A., Greggio, L., Leitherer, C., Montegriffo, P., 2001, The resolved stellar pop- ulations in NGC 1705, AJ, 122, 1271 Treu, T., Koopmans, L. V. E., 2004, Massive dark matter halos and evolution of early-type galaxies to z ≈ 1, ApJ, 611, 739 Tully, R. B., Fisher, J. R., 1977, A new method of determining distances to galaxies, A&A, 54 (3), 661 Tully, R. B., Courtois, H. M., Sorce, J. G., 2016, Cosmicflows-3 , AJ, 152, Issue 2, id. 50 Le Verrier, U., 1859, Lettre de M. Le Verrier ´aM. Faye sur la th´eorie de Mercure et sur le mouvement du p erih´elie de cette plan`ete, Compt. rend. hebdomad. s´eanc. Acad. sci., 49, 379 Wechsler, R. H., Tinker, J. L., 2018, The connection between galaxies and their Dark Matter halos, Ann. Rev. Astron. As- trophys., 56, 435 Weinberg, D. H., Bullock, J. S., Governato, F., Kuzio de Naray, R., Peter, A. H. G., 2015, Cold dark matter: Controversies on small scales, Proc. Nat. Acad. Sci., 112, 12249 Wilhelm, K., Wilhelm, H., Dwivedi, B.N., 2013, An impact model of Newton’s law of gravitation, Ap&SS, 343, 135 Wilhelm, K., Dwivedi, B. N., 2014, Secular perihelion advances of the inner planets and Asteroid Icarus, New Astron., 31, 5 Wilhelm, K., Dwivedi, B. N., 2015a, On the potential energy in a gravitationally bound two-body system, New Astron., 34, 250 Wilhelm, K., Dwivedi, B. N., 2015b, Anomalous Earth flybys of spacecraft, Ap&SS, 358, id. 18 Wilhelm, K., Dwivedi, B. N., 2018, A physical process of the radial acceleration of disc galaxies, MNRAS, 474, Issue 4, 4723 Wilhelm, K., Dwivedi, B. N., 2020, Impact models of gravitational and electrostatic forces, Chapter 5 in Planetology (ed. B. Palaszewski), DOI: 10.5772/intechopen.75213 Zehe, H., 1983, Die Gravitationstheorie des Nicolas Fatio de Duil- lier. Arch. Hist. Exact Sci., 28, 1 Zwicky, F., 1933, Die Rotverschiebung von extragalaktischen Nebeln, Helvetica Physica Acta, 6, 110

This paper has been typeset from a TEX/LATEX file prepared by the author.

MNRAS 000, 1–13 (2020)