Combined Solar System and rotation curve constraints on MOND
MMNRAS , 1–10 (2015) Preprint 11 September 2018 Compiled using MNRAS L A TEX style file v3.0
Combined Solar System and rotation curve constraints onMOND
Aur´elien Hees (cid:63) , Benoit Famaey , Garry W. Angus and Gianfranco Gentile , Department of Mathematics, Rhodes University, 6140 Grahamstown, South Africa Observatoire astronomique de Strasbourg, Universit´e de Strasbourg, CNRS, UMR 7550, 11 rue de l’Universit´e, F-67000 Strasbourg, France Department of Astronomy and Astrophysics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium Sterrenkundig Observatorium, Universiteit Gent, Krijgslaan 281, 9000, Gent, Belgium
11 September 2018
ABSTRACT
The Modified Newtonian Dynamics (MOND) paradigm generically predicts that theexternal gravitational field in which a system is embedded can produce effects on itsinternal dynamics. In this communication, we first show that this External Field Effectcan significantly improve some galactic rotation curves fits by decreasing the predictedvelocities of the external part of the rotation curves. In modified gravity versions ofMOND, this External Field Effect also appears in the Solar System and leads to avery good way to constrain the transition function of the theory. A combined analysisof the galactic rotation curves and Solar System constraints (provided by the Cassinispacecraft) rules out several classes of popular MOND transition functions, but leavesothers viable. Moreover, we show that LISA Pathfinder will not be able to improvethe current constraints on these still viable transition functions.
Key words:
Galaxy: kinematics and dynamics – Solar System
With only six free parameters, the standard ΛCDM cos-mological model fits no less than 2500 multipoles in theCosmic Microwave Background (CMB) angular power spec-trum (Planck Collaboration XVI 2014), the Hubble dia-gram of Type Ia supernovae, the large-scale structure mat-ter power spectrum, and even the detailed scale of baryonicacoustic oscillations. It thus provides the current basis forsimulations of structure formation, and is extremely success-ful down to the scale of galaxy clusters and groups. Never-theless, it still faces numerous challenges on galaxy scales.Among these, the most important ones are the too-big-to-failproblem (Boylan-Kolchin et al. 2011) and the satellite-planeproblem (e.g. Pawlowski et al. 2012; Ibata et al. 2014) fordwarf galaxies, the tightness of the baryonic Tully-Fisher re-lation (McGaugh 2012; Vogelsberger et al. 2014), or the un-expected diversity of rotation curve shapes at a given mass-scale (Oman et al. 2015). The latter problem is actually asubset of a more general problem, i.e. that the shapes of ro-tation curves indeed do not depend on the DM halo mass,contrary to what would be expected in ΛCDM, but ratheron the baryonic surface density, as has long been noted (e.g.,Zwaan et al. 1995). This makes the problem even worse, since (cid:63)
[email protected] the rotation curve shapes are not only diverse at a givenmass-scale, but uniform at a given baryonic surface den-sity scale, implying a completely ununderstood fine-tuningof putative feedback mechanisms. On the other hand, thisbehaviour of rotation curves is an a priori prediction of theformula proposed by Milgrom more than 30 years ago (Mil-grom 1983b,a), relating the total gravitational field to theNewtonian field generated by baryons alone, and which canbe interpreted as a modification of Newtonian dynamics ongalaxy scales below a characteristic acceleration (MOND, fora review see Famaey & McGaugh 2012; Milgrom 2014). Withthis simple formula, High Surface Brightness (HSB) galax-ies are predicted to have rotation curves that rise steeplybefore becoming essentially flat, or even falling somewhat tothe not-yet-reached asymptotic circular velocity, while LowSurface Brightness (LSB) galaxies are predicted to have ro-tation curves that rise slowly to the asymptotic velocity. Thisis precisely what is observed, and was predicted by Milgromlong before LSB galaxies were even known to exist. The for-mula also predicts the tightness of the baryonic Tully-Fisherrelation.Since the original formulation of the MOND paradigm,a lot of relativistic theories of gravitation reproducing theMOND regime in very weak fields have been developed.Usually, these GR extensions imply the presence of addi-tional scalar or vector fields in addition to the standard met- c (cid:13) a r X i v : . [ a s t r o - ph . GA ] O c t A. Hees et al. ric to mediate the gravitational interaction. These relativis-tic MOND theories include the original Bekensetein tensor-vector-scalar (TeVeS) theory (Bekenstein 2004; Sanders1997, 2005), Einstein-Aether theories (Jacobson & Mat-tingly 2001; Zlosnik et al. 2006, 2007), bimetric theo-ries (Milgrom 2009a), or non-local theories (Deffayet et al.2014). Reviews of the relativistic extensions of the MONDparadigm can be found in Bruneton & Esposito-Far`ese(2007) and in Famaey & McGaugh (2012). More recently,new interpretations of MOND in terms of a non-standardDM fluid have been developed (Blanchet 2007; Blanchet &Le Tiec 2008, 2009; Bernard & Blanchet 2015; Blanchet &Heisenberg 2015; Khoury 2015; Berezhiani & Khoury 2015),in which case Milgrom’s formula is akin to an effective mod-ification of gravity on galaxy scales. These latter theorieshave the advantage of naturally reproducing the CMB powerspectrum, and to basically differ from ΛCDM only on galaxyscales and below.In the non-relativistic regime on galaxy scales and be-low, almost all these theories boil down to two types ofmodified Poisson equations, which we explicitly discuss inSec. 2. One feature of MOND is that it generically (at leastfor all modified gravity theories) predicts a violation of thestrong equivalence principle. This implies that the internalgravitational dynamics of a system depends on the externalgravitational field in which the system is embedded (Mil-grom 1983a). This External Field Effect (EFE) occurs evenfor a constant external gravitational field , and it can haveobservational effects, in particular for computing the escapespeed from galaxies (Famaey et al. 2007; Wu et al. 2008), inthe rotation curve of the outskirts of galaxies, and even in theSolar System (Milgrom 2009b; Blanchet & Novak 2011b).The latter can put stringent constraints on the transitionbehaviour between the high-acceleration Newtonian regimeand the low-acceleration MOND regime, which we investi-gate in details in the present contribution. Another questionis whether deviations from General Relativity could be de-tected close to the saddle point of the gravitational poten-tial in the Solar System (e.g. Bekenstein & Magueijo 2006),thereby putting additional constraints on MOND. Here wecheck in particular whether measurements from the LISApathfinder mission could add new constraints to existentones from other Solar System tests.In Sec. 2, we review the basics of MOND, in Sec. 3we produce rotation curve fits to a sample of galaxies withvarious transition functions, including for the first time theEFE in the fits, in Sec. 4 we combine the best-fit values ofthe rotation curve MOND fits with existing Solar Systemconstraints to exclude a large range of transition functions,and check whether improved constraints could be obtainedwith LISA pathfinder. We conclude in Sec. 5. For instance, in the case of non-standard DM theories repro-ducing MOND, this can nevertheless depend on the presence ornot of the DM fluid in the systems under consideration. Of course, if the external field is not constant, it will produceadditional standard tidal effects.
The original idea of the MOND paradigm is to modify thestandard Newtonian gravitation law a = g N (where a is theacceleration of a body and g N is the Newtonian gravitationalfield) by the relation a = g with g determined by the relation µ (cid:18) ga (cid:19) g = g N . (1)or ν (cid:18) g N a (cid:19) g N = g . (2)In these expressions, µ or ν is the MOND interpolating func-tion or transition function. The MOND regime appears inweak gravitational fields ( g << a ) where the transitionfunction needs to satisfy µ ( x ) → x or ν ( y ) → y − / in orderto explain the galactic rotation curves (Milgrom 1983a,b).On the other hand, in order to recover the very well con-strained Newtonian regime in the Solar System, the MONDtransition function has to satisfy µ ( x ) → ν ( y ) → g >> a .An equation such as Eq. (1) or Eq. (2) cannot be validoutside of spherical symmetry for any type of orbit (Felten1984). A first approach for a more fundamental underly-ing theory is known as Modified Inertia. implying that theparticle equations of motion are modified while the gravi-tational potential is still given by the standard Newtonianpotential (Milgrom 1994, 2011). These theories are typicallynonlocal and Eq. (1) or Eq. (2) is then valid only for circularorbits.All relativistic theories of MOND are rather ModifiedGravity theories (or effective modified gravity in the case ofnon-standard DM), and in the non-relativistic regime theybasically reduce to two types of modified Poisson equation: • The first one takes the non-linear form (Bekenstein &Milgrom 1984) ∇ . (cid:20) µ (cid:18) | ∇ Φ | a (cid:19) ∇ Φ (cid:21) = 4 πGρ = ∇ Φ N , (3)with G the Newtonian constant, ρ the matter density, Φ N the Newtonian gravitational potential solution of the stan-dard Poisson equation. The gravitational potential Φ is theMONDian gravitational potential that enters the particle’sequations of motion a = − ∇ Φ. This is typically the weak-field limit of MOND-inspired Einstein-Aether theories (Zlos-nik et al. 2007). • The second is called quasi-linear MOND (orQUMOND) (Milgrom 2010). In QUMOND, the gravi-tational field is the solution of the equation ∇ Φ = ∇ . (cid:20) ν (cid:18) | ∇ Φ N | a (cid:19) ∇ Φ N (cid:21) . (4)This approach requires solving two linear Poisson equationsto find the gravitational potential Φ (for the previous ap-proach, we had to solve a non-linear Poisson equation). Thiscan be the weak-field limit of bimetric MOND theories (Mil-grom 2009a).It is known that these two equations are fully equivalentin spherically symmetric situations (Milgrom 2010; Zhao &Famaey 2010). In that case, the transition functions µ and MNRAS , 1–10 (2015) ombined constraints on MOND Ν Α (cid:72) y (cid:76) Ν Ν Ν (cid:144) y Ν (cid:72) y (cid:76) Ν Ν Ν(cid:142) Ν(cid:96) Figure 1.
Representation of different MOND transition functions ν (see Eqs. (5) for their expression). ν are related by ν ( y ) = 1 /µ ( x ) with x and y related through xµ ( x ) = y (Milgrom 2010).Different types of MOND transition function have beenused in the literature, the most common families of functionsbeing (Famaey & McGaugh 2012) ν α ( y ) = (cid:34) (cid:0) y − α (cid:1) / (cid:35) /α , (5a)˜ ν α ( y ) = (cid:0) − e − y (cid:1) − / + α e − y , (5b)¯ ν α ( y ) = (cid:16) − e − y α (cid:17) − / α + (1 − / α ) e − y α , (5c)ˆ ν α ( y ) = (cid:16) − e − y α/ (cid:17) − /α . (5d)For instance, ν is the so-called “simple” interpolating func-tion (Famaey & Binney 2005; Zhao & Famaey 2006), ν is the “standard” one, and ¯ ν . has been extensively usedin Famaey & McGaugh (2012). Fig. 1 represents all thesedifferent transition functions. The family of functions ¯ ν α ispresented for different values of α as we will see in Sec. 4 thatthis family is the most promising one to fit rotation curvesand to satisfy Solar System constraints simultaneously.The EFE mentioned in the previous section is due to thefact that the MOND equations (3) and (4) are non-linear andinvolve the total gravitational acceleration with respect to apre-defined frame (e.g. the CMB frame). Decomposing thetotal gravitational field ∇ Φ into an internal part g and anexternal field g e and using a similar decomposition for theNewtonian gravitational acceleration ( ∇ Φ N = g N + g Ne )allows us to solve the equations by taking into account theexternal field. This must typically be done with a numericalPoisson solver (Wu et al. 2008; Angus et al. 2012; L¨ughausenet al. 2015). Nevertheless, fits to rotation curves in MONDusually neglect the small corrections due to the non-sphericalsymmetry of the problem, in order to allow for a direct fit of the rotation curve. In the same spirit, and in order toget a first glimpse of the influence of the EFE on rotationcurves, we generalize the one-dimensional solution, by usingthe following formula to fit rotation curves, namely Eq.(60)from Famaey & McGaugh (2012): g = ν (cid:18) | g N + g Ne | a (cid:19) ( g N + g Ne ) − ν (cid:18) g Ne a (cid:19) g Ne . (6)The 1D version of this formula has been shown to be a goodapproximation of the true 3D solution from a numerical Pois-son solver for a random orientation of the external field, atleast for computing the Galactic escape speed (Famaey et al.2007; Wu et al. 2008). Further work should investigate therange of variation of the actual rotation curve compared tothe one obtained in this way, for full numerical solutions ofthe modified Poisson equation and various orientations ofthe EFE. As mentioned in Famaey & McGaugh (2012), theEFE is negligible if g e << g but can play a significant rolewhen the gravitational field g ∼ g e < a . This conditionis always reached at some point in the external part of thegalaxies. In this case, the relation (6) shows that the EFEwill induce a decrease in the internal gravitational field. Inother words, the EFE can lead to a decrease of the externalpart of the rotation curves. We will study this effect morecarefully in Sec. 3.On the other hand, the EFE also subtly affects the in-ternal dynamics of the Solar System. It has been shown thatwithin the MOND paradigm, the external field of our galaxyproduces a quadrupolar modification of the Newtonian po-tential (Milgrom 2009b; Blanchet & Novak 2011b) which ispresent even in the case of a rapidly vanishing transitionfunction. As mentioned in (Milgrom 2009b; Blanchet & No-vak 2011b,a; Hees et al. 2012), planetary ephemerides anal-ysis (in particular from Saturn) is sensitive to this effect. Anestimation of this quadrupolar modification of the Newto-nian potential has been performed using Cassini radiosciencedata (Hees et al. 2014). We will use this estimation here toconstrain the transition functions in Sec. 4. In this section, we produce traditional MOND fits to rota-tion curves (Begeman et al. 1991; Sanders & Verheijen 1998;de Blok & McGaugh 1998; Sanders & Noordermeer 2007;Gentile et al. 2011) using different transition functions. Inparticular, we determine how the best-fit value of a changeswith the adopted transition. Furthermore, the influence ofthe EFE on galactic rotation curves will be assessed for thefirst time.We use rotation curve data from 27 dwarf and low sur-face brightness galaxies, for which the MOND effect is im-portant, and that have low-enough accelerations in the outerparts for the EFE to perhaps play a role. The dataset usedis thoroughly described in Swaters et al. (2010, hereafterSSM10). In the following, we will study the influence ofthe chosen MOND transition function ν and of the corre-sponding MOND acceleration scale a . Moreover, we willalso allow some freedom on local galactic parameters: theindividual R -band stellar mass to light (M/L) ratio Υ g , arescaling of the distance to the different galaxies ( d g ) anda hypothetical external Newtonian gravitational field g Neg
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A. Hees et al. (the indices g refer to a particular galaxy and indicate thatthe parameters are local parameters).As stated above, the gravitational field is given by the1D version of Eq. (6). The predicted rotation velocity isgiven by V M ( R i d g ; a , Υ g , d g , g Neg ) = (cid:112) R i d g g ( R i ; a , Υ g , g Neg ) , (7)where V M is the predicted MONDian velocity at radius R i ,Υ g is the stellar M/L ratio, d g is a distance scale factor d g = D g, MOND /D g, where D g, are the distances given inTab. 1 of SSM10 and g Neg is the Newtonian external field.The norm of the gravitational field g is determined by Eq. (6)where the Newtonian gravitational field is given by g N ( R i , Υ g ) = V i R i + Υ g V (cid:63)i R i , (8)where V gas i and V (cid:63)i are the contribution of the gas and of thestellar disk (at radius R i ) to the rotation curves calculatedin the Newtonian regime. In what precedes, we have used thefact that the Newtonian observed velocities due to the gasand to the stellar disk are rescaled as ∝ √ d with a distancerescaling. Similarly, the measured radial distances R i arerescaled proportionally to d . The procedure then consists ofthe two following steps:(i) Step 1: Using a subset of 19 galaxies from SSM10, weperform a least-squares fit of the global MOND accelerationscale a and of the local Υ g and d g parameters neglecting theexternal field g Neg = 0. The galaxies not considered in thispart of the analysis are the galaxies that seem to experiencea potentially non-negligible EFE, i.e. where the MOND fit isslightly too large for the external part of the rotation curves(this first MOND fit was actually made in a previous step,Step 0, where all galaxies are taken into account). The goalof this first step is to find a robust estimation of the MONDacceleration scale a that is not influenced by the EFE.(ii) Step 2: Using the optimal value of a obtained fromthe first step, we perform a local fit of the parameters Υ g , d g and g Neg for each of the 27 galaxies from the dataset ofSSM10. This fit is done using a standard Bayesian inver-sion with a Metropolis-Hasting Monte Carlo Markov Chain(MCMC) algorithm (Gregory 2010). The marginalized pos-terior distribution of the parameter g Neg allows us to iden-tify the galaxies with a significantly non-vanishing externalfield.During the analysis, we always impose a constraint thatthe stellar M/L ratios must have values included between 0.3and 5 (in units of (
M/L ) (cid:12) ). Similarly, we require the scal-ing of the distance d to be between 0.7 and 1.3 which corre-sponds to the standard uncertainties on the distances (Swa-ters & Balcells 2002). Furthermore, we also include a Gaus-sian prior (characterized by a mean of 1 and a standarddeviation of 0.1) on the parameters d g . In this analysis, weconsider a large range of MOND transition functions fromall the families ν α , ¯ ν α , ˆ ν α and ˜ ν α . We perform a global fit of the global MOND accelerationscale a and the local parameters Υ g and d g using all the 27 galaxies (Step 0). The EFE is neglected at this stage. Forthe function ¯ ν , which we take as a representative examplethroughout this analysis, this first global fit leads to an op-timal value of a = 7 . × − m/s . The purpose of thisfirst fit is only to identify which galaxies seem to experiencea non negligible EFE. We identify these galaxies as beingthe ones where the MOND fit statistically produces a toohigh velocity on the last points of the rotation curves. Theso identified galaxies are: UGC 4173, UGC 4325, UGC 7559,UGC 7577, UGC 11707, UGC 11861, UGC 12060, F574-1.A new global fit using the 19 other galaxies leads to a newoptimal value a = 8 . × − m/s . This new value is morerobust and less influenced by the EFE. The local optimal pa-rameters obtained for each galaxy for this optimal MONDacceleration scale are given in Tab. 1 (let us note here againthat the EFE is neglected in this first part) and the obtainedrotation curves for ¯ ν are shown in Fig. 2. The same pro-cedure is repeated for a large class of transition functionsand the resulting best-fit a for each function is presentedin Tab. 2. In the second step, we use the fixed value of a obtainedpreviously (i.e. a = 8 . × − m/s for ¯ ν ) and we performlocal fits of Υ g , d g and g Neg for each of the 27 galaxies fromthe dataset. This part of the analysis is performed using aMCMC algorithm. Let us remind the reader that we use aflat prior on
M/L between 0.3 and 5. Moreover, a Gaussianprior is used on d g (with mean 1 and standard deviation 0.1).In addition, we force d g to have a value included between 0.7and 1.3. This approach allows us to find realistic confidenceintervals for the three parameters and to assess correlations.The marginalized 68 % Bayesian confidence intervals onthe parameters are presented in Tab. 1 and represented inFig. 3. For the EFE parameters, we only present the esti-mations that produce a non-vanishing g Neg . The obtainedrotation curves are also presented in Fig. 2. In addition, thisanalysis shows a correlation between the d g and the Υ g pa-rameters. A higher estimation of the M/L ratio will lead toa lower estimation of the distance ratio. Moreover, as can beseen from Fig. 3, taking into account the EFE produces es-timations of d that are slightly higher, while the estimationsof the stellar M/L ratios do not change significantly.The EFE improves spectacularly a few of the rota-tion curves fits. In particular, rotation curves for galaxiesUGC4173 and UGC7577 are now very well fitted whereasthe quality of the fit was quite poor when neglecting theEFE. The case of the galaxy UGC12060 is also very inter-esting since the fit is also improved at low radii because ofan increase of the inner part of the rotation curves, whichis mainly due to an increase of the optimal M/L. A similarsituation is encountered for UGC 11707. The fits of UGC731, UGC5005, UGC 7559, UGC 9211, F568-V1, F574-1 areslightly improved by the addition of the EFE.In Appendix A, in addition to ¯ ν , we present fits usingtwo other transition functions: ν and ˆ ν . We will show inthe next section that these functions (¯ ν , ν and ˆ ν ), whileproduce good fits to rotation curves, are not rejected bySolar System constraints. MNRAS , 1–10 (2015) ombined constraints on MOND R o t a t i on v e l o c i t y (cid:64) k m (cid:144) s (cid:68) (cid:45) V10 2 4 6 8 10 12 140501000 2 4 6 8 10 12F574 (cid:45) (cid:45) (cid:45) MOND int. function : Ν Optimal a (cid:61) (cid:180) (cid:45) m (cid:144) s Radius (cid:64) kpc (cid:68)
Figure 2.
Results of the fit using the MOND transition function ¯ ν , which is compatible with Solar System constraints (see Sec. 4), forthe optimal value a = 8 . × − m/s . The dashed (red) thick lines represent the optimal fit without any EFE ; the thick (green) solidline represents the optimal fit with EFE ; the thin solid line represents the Newtonian contributions of the stars and the thin dashed linerepresents the gas contribution. Since the optimal fits with and without EFE do not necessarily produce the same distance scale factor,the radial scales may not be the same. On the top of the plots we mention the radial scale obtained without EFE (corresponding to thedashed red thick lines), at the bottom of the plots we mention the radial scale obtained with EFE (corresponding to the thick green solidlines).MNRAS000
Results of the fit using the MOND transition function ¯ ν , which is compatible with Solar System constraints (see Sec. 4), forthe optimal value a = 8 . × − m/s . The dashed (red) thick lines represent the optimal fit without any EFE ; the thick (green) solidline represents the optimal fit with EFE ; the thin solid line represents the Newtonian contributions of the stars and the thin dashed linerepresents the gas contribution. Since the optimal fits with and without EFE do not necessarily produce the same distance scale factor,the radial scales may not be the same. On the top of the plots we mention the radial scale obtained without EFE (corresponding to thedashed red thick lines), at the bottom of the plots we mention the radial scale obtained with EFE (corresponding to the thick green solidlines).MNRAS000 , 1–10 (2015) A. Hees et al.
Table 1.
Best-fit parameters obtained for the MOND transition function ¯ ν and for a = 8 . × − m/s . Cols. 2 and 3 are the optimalvalues obtained in the case where the EFE is neglected. Cols. 4, 5 and 6 are optimal values and 68 % Bayesian confidence intervals forthe parameters when the EFE is taken into account. The values of the external gravitational field are mentioned only when differentfrom 0. Name No EFE With EFEΥ g d g Υ g d g log g ep ( M/L ) (cid:12) ( M/L ) (cid:12) [m/s ]UGC 731 5.0 0.82 5 . +0 . − . . +0 . − . − . +0 . − . UGC 3371 3.2 0.86 3 . +0 . − . . +0 . − . –UGC 4173 0.3 0.70 0 . +0 . − . . +0 . − . − . +0 . − . UGC 4325 3.1 0.94 3 . +0 . − . . +0 . − . − . +0 . − . UGC 4499 0.3 0.97 0 . +0 . − . . +0 . − . –UGC 5005 0.6 0.93 0 . +0 . − . . +0 . − . − . +0 . − . UGC 5414 0.6 0.84 1 . +0 . − . . +0 . − . − . +1 . − . UGC 5721 2.4 1.23 2 . +0 . − . . +0 . − . –UGC 5750 0.3 1.02 0 . +0 . − . . +0 . − . –UGC 6446 1.8 0.72 1 . +0 . − . . +0 . − . − . +0 . − . UGC 7232 0.8 1.03 0 . +0 . − . . +0 . − . –UGC 7323 0.6 1.01 0 . +0 . − . . +0 . − . –UGC 7399 5.0 1.30 5 . +0 . − . . +0 . − . –UGC 7524 1.8 0.70 1 . +0 . − . . +0 . − . − . +0 . − . UGC 7559 0.0 0.79 0 . +0 . − . . +0 . − . − . +0 . − . UGC 7577 0.0 0.76 0 . +0 . − . . +0 . − . − . +0 . − . UGC 7603 0.4 1.17 0 . +0 . − . . +0 . − . –UGC 8490 1.4 1.30 1 . +0 . . . +0 . − . –UGC 9211 2.3 0.94 3 . +0 . − . . +0 . − . − . +0 . − . UGC 11707 2.6 0.70 3 . +0 . − . . +0 . − . − . +0 . − . UGC 11861 2.4 0.77 2 . +0 . − . . +0 . − . − . +0 . − . UGC 12060 2.8 0.73 4 . +0 . − . . +0 . − . − . +0 . − . UGC 12632 4.7 0.75 5 . +0 . − . . +0 . − . − . +0 . − . F568-V1 4.9 0.91 5 . +0 . − . . +0 . − . − . +0 . − . F574-1 3.7 0.78 5 . +0 . − . . +0 . − . − . +0 . − . F583-1 2.3 0.95 2 . +0 . − . . +0 . − . –F583-4 1.7 0.99 3 . +0 . − . . +0 . − . − . +0 . − . As shown in the previous section, the MOND EFE can havea non negligible effect on the outer parts of some galaxyrotation curves. This effect turns out to be crucial withinthe Solar System. Indeed, within the MOND paradigm, theexternal gravitational field of our galaxy produces interest-ing modifications in the internal dynamics of the Solar Sys-tem (Milgrom 2009b; Blanchet & Novak 2011b). The maineffect consists in a quadrupole correction to the Newto-nian potential. This correction can phenomenologically beparametrized by Q and the gravitational potential can bewritten asΦ = − GMr − Q x i x j (cid:18) e i e j − δ ij (cid:19) , (9)where e is a unitary vector pointing towards the Galacticcenter and − GM/r is the standard Newtonian potential dueto the Sun. This correction produces an anomalous forcewhich, along the Galactic external field direction, rises lin-early with distance from the Sun, whilst it decreases linearlyalong the two other cartesian axes (for a positive value of Q ). From a theoretical point of view, the value of Q de- pends on the MOND transition function, on the value of theexternal gravitational field g e and on the value of the MONDacceleration scale a . Instead of working with Q , one canintroduce a dimensionless parameter q defined by (Milgrom2009b) q = − Q ( GM ) / a / . (10)This dimensionless parameter depends only on the MONDtransition function and on the ratio η = g e a (11)between the external field and the MOND acceleration scale.In the context of the QUMOND formulation (4), Mil-grom (2009b) has derived an exact expression for the q pa-rameter given by q ( η ) = 32 (cid:90) ∞ dv (cid:90) − dξ ( ν − (cid:2) η N (3 ξ − ξ ) + v (1 − ξ ) (cid:3) , (12)with ν = ν (cid:104)(cid:112) η N + v + 2 η N v ξ (cid:105) and η N = ηµ ( η ) (orequivalently η N is solution of η N ν ( η N ) = η ). As mentionedby Milgrom (2009b), the term -1 in ν − MNRAS , 1–10 (2015) ombined constraints on MOND (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) M (cid:144) L (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) d (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45)
12 Num. of galaxy
Log g N e (cid:64) m (cid:144) s (cid:68) Figure 3.
Blue: 68 % Bayesian confidence intervals obtained withlocal fits on each of the 27 galaxies with the transition function ¯ ν and a = 8 . × − m/s . The blue dots represent the best fit.The red squares represent the optimal values obtained withoutany external field effect. by − ν (cid:104)(cid:112) ν N + v (cid:105) or by − ν (cid:2) | η N ± v | (cid:3) to improve the nu-merical convergence of the integral.In the case of the Bekenstein approach (3), the aboveintegral leads only to an approximate value for the q parame-ter. In this approach, the q parameter can only be computedby numerically solving the non-linear Poisson equation asdone in (Milgrom 2009b; Blanchet & Novak 2011b).From an observational point of view, the modificationof the Newtonian potential (9) will modify the trajectoriesof planets, asteroids and comets (Milgrom 2009b; Blanchet& Novak 2011b,a; Hees et al. 2012; Maquet & Pierret 2015).Using 9 years of Cassini range and Doppler tracking mea-surements, the value of the parameter Q , for an externalfield assumed to point towards the Galactic enter, has beenestimated by (Hees et al. 2014) Q = (3 ± × − s − . (13)In the following, we will use the expression from Eq. (12)to estimate the value of the q parameter for different MONDtransition functions and different values of the ratio η . Then,using the optimal value of a obtained from the fit to galaxyrotation curves from Sec. 3, we estimate the value of the Q parameter using the relation from Eq. (10). This value of Q characterizes the Solar System deviation from Newtoniangravity predicted by MOND for values of a that optimallyexplain galactic rotation curves. Finally, the obtained valueof Q can be compared to the Cassini estimations from (Heeset al. 2014) to assess what transition functions are compat-ible with galactic rotation curves and with Solar Systemobservations simultaneously. First of all, we have reproduced Tab. I from (Milgrom2009b) to validate our calculation of q using Eq. (12). Then,we have computed q for a wide range of MOND transitionfunctions ν and values of η . The corresponding results areshown in Tab. B1 in the Appendix.Our main result consists of a combined analysis usingboth galactic and Solar System observations and is presentedin Tab. 2. For different MOND transition functions ν , theoptimal MOND acceleration scale a has been estimatedwith galactic rotation curves using the procedure describedin Sec. 3 (see the second column from Tab. 2). The reducedchi-square obtained for the global fit of all galactic rota-tion curves is also presented. Then, using the optimal valueof a , we have computed the value of Q using Eqs.(12)and (10). This estimation of Q has been done using twodifferent values of the external gravitational field. The twovalues of g e used correspond to current estimations of galac-tic parameters (McMillan & Binney 2010; McMillan 2011): g e = 1 . × − m/s and g e = 2 . × − m/s . The esti-mated values of Q are exact in the framework of QUMONDbut are only approximate estimations in the framework ofthe Bekenstein approach. Note that, as can be seen fromTab. I from (Milgrom 2009b), the values obtained with theformulas from QUMOND slightly underestimate the corre-sponding values of Q in the Bekenstein approach. The re-sults from Tab. 2 are therefore otpimistic in the Bekensteinapproach. The estimated values of Q presented in Tab. 2can be compared to the estimation (13) obtained with theCassini radioscience tracking data (Hees et al. 2014). Thevalues of Q within the 1 σ estimation are mentioned in bold-face in Tab. 2.Several conclusions may be drawn from this combinedanalysis. First of all, the class of transition functions ˜ ν α seems to be completely excluded by this combined analy-sis. The functions ν α and ˆ ν α are excluded for low values of α but begins to be marginally acceptable for large values of α . The only class of functions that seem to be able to pro-duce a satisfactory fit to the galactic rotation curves withoutproducing a too large deviation in the Solar System is ¯ ν α for α ≥ Bekenstein & Magueijo (2006); Bevis et al. (2010); Trenkelet al. (2012); Magueijo & Mozaffari (2012); Trenkel &Wealthy (2014) have proposed to redirect the Laser Inter-ferometer Space Antenna (LISA) pathfinder towards theEarth-Sun saddle point to constrain MOND in a low gravita-tional field. The LISA pathfinder project (McNamara et al.2008) is a space mission designed to test the technology tobe used in the eLISA project. This mission allows the veryaccurate measurement of tidal stresses by measuring the rel-ative motion of two test masses separated by 35 cm. The ideaproposed by, e.g., Bevis et al. (2010); Trenkel et al. (2012);Magueijo & Mozaffari (2012) is to measure the tidal stressesvery close to the saddle point where the Newtonian gravita-tional field is very low and where MOND effects are expectedto show up. In this section, we will assess the order of magni-tude of the tidal stresses produced by the MOND transitionfunctions used in the previous section and show that theyare far too small to be detected by LISA pathfinder even inthe most optimistic scenario.
MNRAS000
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A. Hees et al.
Table 2.
Col. 2: optimal value of the MOND acceleration scale a found by rotation curve fits. Col. 3: reduced χ computed onall 27 galactic rotation curves presented in Sec. 3. Col. 4: value ofthe ratio η = g e /a with g e = 1 . × − m/s . Col. 5: value of − q computed with Eq. (12) for the value of η from Col. 4. Col.6: value of Q obtained using Eq. (10) and the value of − q fromCol. 5. Col. 7: value of η for g e = 2 . × − m/s . Col. 8: valueof − q computed with Eq. (12) for the value of η from Col. 7. Col.9: value of Q obtained using Eq. (10) and the value of − q fromCol. 8. The values of Q that are in bold are included in the 1 σ Cassini estimation 0 ≤ Q ≤ × − s − . g e = g e min g e = g e max a χ η − q Q η − q Q − − − − − [m/s ] [s − ] [s − ] ν .
60 2 .
02 1 .
20 10 . . .
50 11 . .ν .
55 1 .
97 1 .
20 8 .
29 21 . .
50 7 .
82 20 .ν .
51 1 .
94 1 .
30 6 .
76 16 . .
60 5 .
34 13 .ν .
49 1 .
93 1 .
30 5 .
51 13 . .
60 3 .
71 8 . ν .
46 1 .
92 1 .
30 4 .
55 11 . .
60 2 .
67 6 . ν .
45 1 .
92 1 .
30 3 .
82 8 . .
70 2 . ν .
44 1 .
92 1 .
30 3 .
27 7 . .
70 1 . ˜ ν . .
48 2 .
16 1 .
30 14 . . .
60 18 . . ˜ ν .
38 2 .
12 1 .
40 18 . . .
70 25 . . ˜ ν . .
18 2 .
16 1 .
60 24 . . .
00 34 . . ˜ ν .
815 2 .
24 2 .
30 44 . . .
90 47 . . ˜ ν . .
977 2 .
23 1 .
90 33 . . .
50 51 . . ˜ ν .
743 1 .
07 2 .
60 56 . . .
20 65 . . ˜ ν .
723 2 .
01 2 .
60 54 . . .
30 85 . . ˜ ν .
715 1 .
97 2 .
70 48 . . .
40 94 . . ¯ ν . .
48 2 .
15 1 .
30 13 . . .
60 17 . . ¯ ν .
38 2 .
12 1 .
40 16 . . .
70 19 . . ¯ ν . .
18 2 .
16 1 .
60 19 . . .
00 15 . . ¯ ν .
815 2 .
24 2 .
30 6 . .
90 2 . ν .
743 2 .
07 2 .
60 1 . .
20 0 . ν .
723 2 .
01 2 .
60 1 . .
30 0 . ν .
715 1 .
97 2 .
70 1 . .
40 0 . ν .
713 1 .
95 2 .
70 1 . .
40 0 . ν .
729 1 .
95 2 .
60 1 . .
30 0 . ˆ ν .
48 2 .
15 1 .
30 13 . . .
60 17 . . ˆ ν .
59 2 .
01 1 .
20 10 . . .
50 11 . . ˆ ν .
55 1 .
96 1 .
20 8 .
32 21 . .
60 7 .
49 19 . ˆ ν .
51 1 .
94 1 .
30 6 .
66 16 . .
60 4 .
79 12 . ˆ ν .
48 1 .
93 1 .
30 5 .
34 13 . .
60 3 . . ˆ ν .
46 1 .
92 1 .
30 4 .
31 9 . .
60 2 . ν .
45 1 .
92 1 .
30 3 .
55 8 . .
70 1 . The simulations that have been performed use similarassumptions as in Magueijo & Mozaffari (2012). The space-craft is supposed to move along the X axis, which is definedby the Sun-Earth direction. Moreover, it is assumed that thedirection of the observed tidal stress would be perpendicularto this axis. In order words, the anomalous observed tidalstress produced by MOND is given by S yy with S ij = ∂ Φ ∂x i ∂x j − ∂ Φ N ∂x i ∂x j , (14)where Φ represents the MOND gravitational potential andΦ N the Newtonian potential. Moreover, we will assume acase where the spacecraft misses the saddle point by 1 kilo-meter. This situation is very optimistic since as mentionedby Magueijo & Mozaffari (2012), the saddle point can bepinpointed to about a kilometer and the spacecraft locationcan be determined to about 10 kilometers.Fig. 4 represents the evolution of the MOND tidal stress (cid:45) (cid:45)
20 0 20 40 (cid:45) (cid:64) km (cid:68) S yy (cid:64) (cid:45) s (cid:45) (cid:68) Figure 4.
Signature of the transverse MOND tidal stress pro-duced by the MOND transition function ν (see Eq. (5a)). In thissimulation, the spacecraft misses the saddle point by 1 kilometer(the origin of the X axis corresponding to the closest approachwith the saddle point). S yy produced by the MOND transition function ν (seeEq. (5a)). In this simulation, the impact factor with respectto the saddle point is 1 kilometer. The maximal amplitudeof the absolute value of S yy is of the order of 10 − s − . Theaccuracy of LISA Pathfinder is expected to be of the orderof 10 − s − (Magueijo & Mozaffari 2012). Therefore, LISAPathfinder will not be able to detect this MOND transitionfunction. One might argue that the internal self-gravity ofthe spacecraft might however increase the effect (Trenkel &Wealthy 2014). However, let us remember that ν is actu-ally excluded by our present analysis. The situation is ac-tually much worse for the other functions: the signal for ν is two orders of magnitude smaller than then one pro-duce by ν while the one for ν is 4 orders of magnitudesmaller. Also recall that, for that ν α family, only α > ν . for instanceproduces a deviation of the order of 10 − s − , while theother transition functions ˜ ν α , ¯ ν α and ˆ ν α lead to even smallertidal stresses. These very small numbers reflect the exponen-tial convergence towards the Newtonian regime provided bythese transition functions. Even taking into account the in-ternal self-gravity of the spacecraft as in Trenkel & Wealthy(2014) will thus not provide the necessary correction of lit-erally multiple thousands of orders of magnitude for makingthe effect detectable with an acceptable transition functionsuch as ¯ ν .In conclusion, LISA pathfinder does not offer any pos-sibility to constrain the transition functions considered inthis analysis. The Cassini constraint from Hees et al. (2014)using the External Field Effect is much more efficient. The non-linearity inherent to the MOND paradigm leads tothe fact that the internal dynamics of a system is influencedby the external gravitational field in which it is embedded.In this communication, we use this EFE to derive constraintson the various MOND transition functions with a combinedanalysis of galactic rotation curves and of the Solar System.First of all, we have derived the best-fit value of a for a large class of transition functions, and we have shownthat, at the galactic level, the EFE can lead to a velocitydecrease in the external part of the rotation curves. This MNRAS , 1–10 (2015) ombined constraints on MOND helps to improve several galactic rotation curves in our an-alyzed dataset, the most impressive being UGC 4173, UGC7577 and UGC 12060. The typical range of optimal valuesfor the external gravitational field (ranging between 10 − and 10 − m/s , see Tab. 1) is a priori realistic. It will beextremely interesting to investigate whether a source of non-negligible external field can be found in the environment ofthese galaxies. Nevertheless, it is not a trivial task becausea massive source at large distance can contribute more thana low mass one at close distance. This also depends on theMOND cosmology (e.g. Blanchet & Le Tiec 2008, 2009; An-gus et al. 2013). For instance an external field of 10 − m/s can be produced by a 7 × M (cid:12) galaxy at a distance of100 kpc, by a 3 × M (cid:12) group/cluster at 2 Mpc or by alarge attractor of 2 × M (cid:12) at 50 Mpc (the typical distancefrom the Great Attractor to the Milky way). For instance, inthe case of UGC 7577, we note that there are ∼
50 galaxiesat a projected distance of less than 40 kpc, which is roughlyenough to produce an external field effect of 10 − m/s (seethe estimated value from Tab. 1, also shown in Fig. 3).In the Solar System, the EFE produces non negligibleeffects even for transition functions that present an expo-nential transition towards the Newtonian regime. This al-lows us to test MOND in the Solar System as mentioned byMilgrom (2009b); Blanchet & Novak (2011b). Cassini ob-servations have provided the estimation (13). We have per-formed a combined analysis of a sample of galactic rotationcurves and of the Cassini estimation to constrain the MONDtransition function. The galactic rotation curves provide anestimation of the MOND acceleration scale a that is usedto estimate the Q parameters. This estimation is comparedwith the observational estimation of Q provided by Cassinidata (Hees et al. 2014). The results are presented in Tab. 2.The functions ˜ ν α are completely rejected by this analysis.The transition functions ν α and ˆ ν α can, on the other hand,still be viable for large values of α . The only class of func-tions that is compatible with both types of observations foralmost all α is ¯ ν α , for α ≥
2. We note however that theseconstraints do not apply to, e.g., modified inertia theories.Finally, we have shown that for these classes of accept-able transition functions, the space mission LISA pathfinderwill not be able to detect or to constrain them.
ACKNOWLEDGEMENTS
A.H. acknowledges support from “Fonds Sp´ecial deRecherche” through a FSR-UCL grant. We are grateful tothe authors of SSM10 for sharing their data, and we ac-knowledge insightful discussions about the present workwith Stacy McGaugh. We also acknowledge interesting dis-cussions with Michele Armano about LISA pathfinder.
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APPENDIX A: FIT TO ROTATION CURVESWITH ν AND ˆ ν Here we present fits using the transition functions ν andˆ ν which, like ¯ ν , are not rejected by Solar System observa-tions. It is interesting to notice that, at the level of galacticrotation curves, fits using ν α and ˆ ν α are very similar. Thisis explained by the similarity in the profile of the transitionfunction as can be seen in Fig. 1 and can be noticed fromthe first columns of Tab. 2 where the optimal values for theMOND acceleration scales and the χ of rotation curves ispresented for different transition functions. It can be seenthat values for ν α and ˆ ν α are very similar.Fig. A1 represents the rotation curve fits for ν andFig. A2 for ˆ ν . The optimal values and confidence intervalsfor Υ g , d g and the external field effects are presented inFig. A3. The fits for both of these transition functions arequalitatively similar. The EFE improves more fits with ν and ˆ ν than with ¯ ν . As with ¯ ν , the quality of the fits ofUGC 4173 and UGC 11707 and UGC 7577 are significantlyimproved and the fit for UGC 12060 is improved for all theradii. In addition, the EFE improves the quality of the fitsfor UGC 731, UGC 4325, UGC 5005, UGC 6446, UGC 7524,UGC 7559, UGC 9211, F568-V1, F574-1, F583-1 and F583-4. More fits are improved using ν and ˆ ν compared to ¯ ν and for some galaxies, the improvement is more significantas well (see for example UGC 4325). APPENDIX B: COMPUTATION OF THEQUADRUPOLAR EFE IN THE SOLAR SYSTEM
Here, we have computed q using Eq. (12) for a wide rangeof MOND transition functions ν and values of the externalfield η . The results are shown in Tab. B1. MNRAS , 1–10 (2015) ombined constraints on MOND R o t a t i on v e l o c i t y (cid:64) k m (cid:144) s (cid:68) (cid:45) V10 2 4 6 8 10 12 140501000 2 4 6 8 10 12F574 (cid:45) (cid:45) (cid:45) Ν Optimal a (cid:61) (cid:180) (cid:45) m (cid:144) s Radius (cid:64) kpc (cid:68)
Figure A1.
Results of the fits using the MOND transition function ν for the optimal value a = 1 . × − m/s . The dashed (red)thick lines represent the optimal fit without any EFE ; the thick (green) solid line represents the optimal fit with EFE ; the thin solidline represents Newtonian the contributions of the stars and the thin dashed line represents the gas contribution. Since the optimal fitswith and without EFE do not necessarily produce the same distance scale factor, the radial scales may not be the same. On the top ofthe plots we mention the radial scale obtained without EFE, at the bottom of the plots we mention the radial scale obtained with EFE.MNRAS000
Results of the fits using the MOND transition function ν for the optimal value a = 1 . × − m/s . The dashed (red)thick lines represent the optimal fit without any EFE ; the thick (green) solid line represents the optimal fit with EFE ; the thin solidline represents Newtonian the contributions of the stars and the thin dashed line represents the gas contribution. Since the optimal fitswith and without EFE do not necessarily produce the same distance scale factor, the radial scales may not be the same. On the top ofthe plots we mention the radial scale obtained without EFE, at the bottom of the plots we mention the radial scale obtained with EFE.MNRAS000 , 1–10 (2015) A. Hees et al. R o t a t i on v e l o c i t y (cid:64) k m (cid:144) s (cid:68) (cid:45) V10 2 4 6 8 10 12 140501000 2 4 6 8 10 12F574 (cid:45) (cid:45) (cid:45)
Ν(cid:96) Optimal a (cid:61) (cid:180) (cid:45) m (cid:144) s Radius (cid:64) kpc (cid:68)
Figure A2.
Results of the fits using the MOND transition function ˆ ν for the optimal value a = 1 . × − m/s . The dashed (red)thick lines represent the optimal fit without any EFE ; the thick (green) solid line represents the optimal fit with EFE ; the thin solidline represents Newtonian the contributions of the stars and the thin dashed line represents the gas contribution. Since the optimal fitswith and without EFE do not necessarily produce the same distance scale factor, the radial scales may not be the same. On the top ofthe plots we mention the radial scale obtained without EFE, atalso the bottom of the plots we mention the radial scale obtained withEFE. MNRAS , 1–10 (2015) ombined constraints on MOND (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) M (cid:144) L (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) d (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45)
12 Num. of galaxy
Log g N e (cid:64) m (cid:144) s (cid:68) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) M (cid:144) L (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) d (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45)
12 Num. of galaxy
Log g N e (cid:64) m (cid:144) s (cid:68) Figure A3.
Left: fits with ν and a = 1 . × − m/s . Right: fits with ˆ ν and a = 1 . × − m/s . Blue: 68 % Bayesian confidenceintervals obtained with local fits to each of the 27 galaxies with the transition function. The blue dots represent the best fit. The redsquares represent the optimal values obtained without any external field effect. Table B1.
Value of the parameter q computed using Eq. (12) for different MOND interpolating function ν and value of η . η . .
25 1 . .
75 2 . .
25 2 . .
75 3 ν . × − . × − . × − . × − . × − . × − . × − . × − . × − ν . × − . × − . × − . × − . × − . × − . × − . × − . × − ν . × − . × − . × − . × − . × − . × − . × − . × − . × − ν . × − . × − . × − . × − . × − . × − . × − . × − . × − ν . × − . × − . × − . × − . × − . × − . × − . × − . × − ν . × − . × − . × − . × − . × − . × − . × − . × − . × − ν . × − . × − . × − . × − . × − . × − . × − . × − . × − ˜ ν . . × − . × − . × − . × − . × − . × − . × − . × − . × − ˜ ν . × − . × − . × − . × − . × − . × − . × − . × − . × − ˜ ν . . × − . × − . × − . × − . × − . × − . × − . × − . × − ˜ ν . × − . × − . × − . × − . × − . × − . × − . × − . × − ˜ ν . . × − . × − . × − . × − . × − . × − . × − . × − . × − ˜ ν . × − . × − . × − . × − . × − . × − . × − . × − . × − ˜ ν . × − . × − . × − . × − . × − . × − . × − . × − . × − ˜ ν . × − . × − . × − . × − . × − . × − . × − . × − . × − ¯ ν . × − . × − . × − . × − . × − . × − . × − . × − . × − ¯ ν . . × − . × − . × − . × − . × − . × − . × − . × − . × − ¯ ν . × − . × − . × − . × − . × − . × − . × − . × − . × − ¯ ν . × − . × − . × − . × − . × − . × − . × − . × − . × − ¯ ν . × − . × − . × − . × − . × − . × − . × − . × − . × − ¯ ν . × − . × − . × − . × − . × − . × − . × − . × − . × − ¯ ν . × − . × − . × − . × − . × − . × − . × − . × − . ¯ ν . × − . × − . × − . × − . × − . × − . × − . . ˆ ν . × − . × − . × − . × − . × − . × − . × − . × − . × − ˆ ν . × − . × − . × − . × − . × − . × − . × − . × − . × − ˆ ν . × − . × − . × − . × − . × − . × − . × − . × − . × − ˆ ν . × − . × − . × − . × − . × − . × − . × − . × − . × − ˆ ν . × − . × − . × − . × − . × − . × − . × − . × − . × − ˆ ν . × − . × − . × − . × − . × − . × − . × − . × − . × − ˆ ν . × − . × − . × − . × − . × − . × − . × − . × − . × − MNRAS000