aa r X i v : . [ a s t r o - ph ] M a y Draft version November 11, 2018
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MOND AND THE UNIVERSAL ROTATION CURVE: SIMILAR PHENOMENOLOGIES
Gianfranco Gentile
Draft version November 11, 2018
ABSTRACTThe Modified Newtonian Dynamics (MOND) and the Universal Rotation Curve (URC) are twoways to describe the general properties of rotation curves, with very different approaches concerningdark matter and gravity. Phenomenological similarities between the two approaches are studied bylooking for properties predicted in one framework that are also reproducible in the other one.First, we looked for the analogous of the URC within the MOND framework. Modifying in anobservationally-based way the baryonic contribution V bar to the rotation curve predicted by the URC,and applying the MOND formulas to this V bar , leads to a “MOND URC” whose properties are re-markably similar to the URC.Second, it is shown that the URC predicts a tight mass discrepancy - acceleration relation, whichis a natural outcome of MOND. With the choice of V bar that minimises the differences between theURC and the “MOND URC” the relation is almost identical to the observational one.This similarity between the observational properties of MOND and the URC has no implicationsabout the validity of MOND as a theory of gravity, but it shows that it can reproduce in detail thephenomenology of disk galaxies’ rotation curves, as described by the URC. MOND and the URC, eventhough they are based on totally different assumptions, are found to have very similar behaviours andto be able to reproduce each other’s properties fairly well, even with the simple assumptions made onthe luminosity dependence of the baryonic contribution to the rotation curve. Subject headings: galaxies: spiral – galaxies: kinematics and dynamics – galaxies: fundamentalparameters – dark matter INTRODUCTION
Rotation curves of spiral galaxies have been studied forseveral decades now, and they are a useful tool to showthe mass discrepancy in galaxies: the observed kinemat-ics and that predicted from the observed baryonic dis-tribution do not match. Either an additional mass com-ponent or a modification of gravity are needed. In therecent past, rotation curves have been exploited to testthe predictions of cosmological models of structure for-mation in the Universe, such as the currently favouredΛ Cold Dark Matter (ΛCDM). Most observations showthat the baryons are the main kinematic component inthe inner parts and that the so-called “cuspy” halos pre-dicted by ΛCDM (Navarro, Frenk & White 1996, Mooreet al. 1999, Navarro et al. 2004) fail to reproduce ob-served rotation curves (see e.g. de Blok et al. 2001; deBlok & Bosma 2002; Gentile et al. 2004, 2005, 2007a,b;McGaugh et al. 2007).Rotation curves have been shown to have some gen-eral properties that can be described by a stellar disk +dark matter halo model (the Universal Rotation Curve,URC; Persic, Salucci & Stel 1996, hereafter PSS). In thisframework, the circular velocity of a spiral galaxy at acertain radius depends only on one parameter, e.g. thetotal luminosity of the stellar disk.MOND, the Modified Newtonian Dynamics (Milgrom1983) is another successful prescription to predict the ro-tation curve of a spiral galaxy. According to MOND, themass discrepancy is not due to an unseen mass compo- Sterrenkundig Observatorium, Ghent University, Krijgslaan281, S9, B-9000 Ghent, Belgium University of New Mexico, Department of Physics and Astron-omy, 800 Yale Blvd NE, Albuquerque, NM 87131, USA nent (the dark matter halo), but it is instead the sig-nature of the failure of Newtonian gravity to describethe observed kinematics at the low accelerations foundin the outer parts of galaxies. In MOND, below a cer-tain critical acceleration a the Newtonian gravity is nolonger valid. Earlier concerns about the inconsistency ofMOND with General Relativity are now overcome withthe TeVeS theory (Bekenstein 2004). MOND has a re-markable predictive power for the kinematics of galaxies:it fits the kinematics of small dwarf galaxies (Gentileet al. 2007a,c), of the Milky Way (Famaey & Binney2005), of early-type spiral galaxies (Sanders & Noorder-meer 2007), of massive ellipticals (Milgrom & Sanders2003), and it naturally explains observed tight scalingrelations in spiral galaxies (McGaugh 2004, 2005). MOND
The Modified Newtonian Dynamics (Milgrom 1983, seeSanders & McGaugh 2002 for a review) can be invokedas an alternative to dark matter to explain the observedkinematics of disk galaxies. Within the MOND frame-work, the true gravitational acceleration g is linked tothe Newtonian one g N through the following relation: g = g N µ ( g/a ) (1)where µ ( x ) is an interpolation function whose asymp-totic values are µ ( x ) = 1 when g ≫ a and µ ( x ) = g/a when g ≪ a . a is the critical acceleration belowwhich the Newtonian gravity is no longer valid; previ-ous studies (Begeman, Broeils & Sanders 1991) foundthat a ∼ . × − cm s − . Even though in a generalcase a modified version of the Poisson equation should besolved, eq. 2 can be shown to be a good approximationfor axisymmetric disks (Brada & Milgrom 1995). The in-terpolation function has been given usually the followingfunctional form: µ orig ( x ) = x √ x (2)However, it is obvious that a whole family of functionsare compatible with the required asymptotic behaviours.For instance, Famaey & Binney (2005) proposed that theform: µ FB ( x ) = x x (3)could be a better choice, since, contrary to eq. 2,it is compatible with the relativistic theory of MOND(TeVeS) put forward by Bekenstein (2004). Famaey etal. (2007) showed that using eq. 3 leads to a slightlydifferent value of a : a = 1 . × − cm s − .If eq. 2 is used as the interpolation function, thenwithin the MOND framework the circular velocity veloc-ity V obs ( r ) can be expressed as a function of a and theNewtonian baryonic contribution to the rotation curve V bar ( r ) at radius r : V ( r ) = V ( r )+ V ( r ) vuuut r (cid:16) ra V ( r ) (cid:17) − (4)where V bar ( r ) = q V ( r ) + V ( r ) (ignoring thecontribution of the bulge), V stars ( r ) and V gas ( r ) are theNewtonian contributions to the rotation curve of the stel-lar and gaseous disks, respectively (see Milgrom 1983).The amplitude of V stars ( r ) (whose shape is fixed byphotometric observations), can be scaled according tothe chosen, or fitted, stellar mass-to-light ( M/L ) ratio. V gas ( r ) is derived from HI observations, when they areavailable. Note that the second term of the right-handside of eq. 4 acts as a “pseudo-dark matter halo” termand that it is completely determined by the baryonicterms.If eq. 3 is chosen as the interpolation function insteadof eq. 2, the equivalent of eq. 4 becomes: V ( r ) = V ( r ) + V ( r ) q a rV ( r ) − (5)(see e.g. Richtler et al. 2008). As expected, in botheqs. 4 and 5 the “pseudo-dark matter halo” term van-ishes in the limit a →
0. We note MOND has a re-markable predictive power for the general properties ofrotation curves, and in many cases it is able to fit ob-served individual rotation curves (Kent 1987; Milgrom1988; Begeman, Broeils & Sanders 1991; Sanders 1996;de Blok & McGaugh 1998) . MOND correctly predictsgeneral scaling relations linked to rotation curves, suchas the baryonic Tully-Fisher relation (see McGaugh 2005,even though we note that other authors find different
Fig. 1.—
Gas scaling factor versus (log of) B-band luminosity.The data are taken from Swaters (1999) and Hoekstra et al. (2001),and the solid line is the result of a linear fit to the points. slopes, mainly due to different choices of the stellar
M/L ratio) or the mass discrepancy-acceleration relation (Mc-Gaugh 2004). THE UNIVERSAL ROTATION CURVE (URC)
PSS, from the analysis of a sample of more than 1000rotation curves, showed that the circular velocity of adisk galaxies at a given radius (note that they consideredrotation curves extended up to 2 optical radii r opt ) canbe described as a function of only one parameter, e.g. thetotal luminosity. The function which describes the rota-tion velocity depending only on radius and luminosityis given the name Universal Rotation Curve (URC). Wenote that the kinematics of real galaxies depend on otherfactors too (such as the surface brightness), but the gen-eral phenomenological properties of rotation curves canbe described well with the luminosity as only parameter.In detail, the circular velocity V URC ( r ) at radius r canbe expressed as a function of V opt (the circular veloc-ity at r opt ), and of the stellar and dark contributions V disk , N and V halo , N normalised at V opt (i.e., V disk , N ( r opt )+ V halo , N ( r opt ) = 1): V URC ( r ) = V opt q V , N ( r ) + V , N ( r ) (6) V opt , V disk , N , and V halo , N can be in turn expressed asfunctions of r , λ = L/L ∗ ( L is the total blue luminosityof the galaxy in question and log L ∗ =10.4 in solar units),and r opt ≃ λ . kpc: V opt = 200 λ . h .
80 + 0 .
49 log λ + .
75 exp( − . λ )0 . . λ . i / km s − (7) V , N = (0 .
72 + 0 .
44 log λ ) 1 . r/r opt ) . [( r/r opt ) + 0 . . (8) Fig. 2.—
Comparison between the Universal Rotation Curve (thick black curves), at various luminosities, and the equivalent for MOND(thin red curves, using µ orig ( x )), with the stars and gas contributions defined in the text. Dotted curves indicate the uncertainties. V , N = 1 . − . λ ) ( r/r opt ) ( r/r opt ) + 2 . λ . (9)According to PSS, eqs. 6 to 9 predict the circular ve-locity of a spiral galaxy at radius r with an uncertaintyof 4%. Choosing r opt ≃ λ . kpc hides any variation ofsurface brightness for a given luminosity, which results indifferent rotation curve shapes for galaxies with differentsurface brightnesses (see e.g. McGaugh & de Blok 1998).However, we note that the original formulation of theURC (with the radius r in units of the stellar exponen-tial scale length r D ) takes this effect into account, sincethe rotation curves of low- and high-surface brightnessgalaxies are virtually indistinguishable once the radius isexpressed in units of r D (e.g., Verheijen & de Blok 1999).Also, in the URC the contribution of the gaseous disk is ignored. This is acceptable in the range of luminositiesconsidered by PSS because a) the gasous disk never dom-inates the kinematics and b) because of its scaling withthe dark matter contribution, its contribution can be im-plicitely accounted by V halo .An important feature of the URC is the strong depen-dence of the logarithmic slope of the rotation curve ∇ on the luminosity. They define ∇ in the range 0 . r opt The question that this paper is addressing is: sinceboth MOND and the URC are successful ways to pre-dict the general kinematical properties of disk galaxies, Fig. 3.— Comparison between the Universal Rotation Curve (thick black curves), at various luminosities, and the equivalent for MOND(thin red curves, using µ FB ( x )), with the stars and gas contributions defined in the text. Dotted curves indicate the uncertainties. to what extent do they have similar properties? One wayto compare their properties is to try to derive somethinglike the URC within the framework of MOND. In orderto achieve this, both V stars ( r ) and V gas ( r ) are needed.For the former term, the URC provides us with a recipeto estimate it, while the latter is ignored in the URC. Arough estimate of V gas ( r ) can be made through the scal-ing property of V halo ( r ) compared to V gas ( r ) noticed bye.g. Bosma (1981) and Hoekstra, van Albada & Sancisi(2001): V ( r ) ≈ n × V ( r ). The factor n was calcu-lated by performing a linear fit to the observational datain the scaling factor versus luminosity plot (see Fig. 1).The data were taken from the samples of Swaters (1999)and Hoekstra et al. (2001). The former work gives R-band luminosities, whereas the URC uses B-band. TheB-band luminosities of the Swaters (1999) sample werecalculated from the LEDA database, when available, and from NED. The linear fit gives n = 1 . λ − . 4. There-fore, we estimated V gas ( r ) through the following rela-tion: V gas ( r ) ∼ V halo , URC ( r ) / √ . λ − . 4. This turnsout to be in line with the findings of Swaters (1999),who finds n to be correlated with surface brightness, andhence with luminosity. Such a scaling of the actual sur-face density with the gas surface density would happenin the case of a conspicuous amount of disk dark mat-ter, e.g. H in the form of clumps (Pfenniger, Combes &Martinet 1994) or in a cold neutral medium phase (Pa-padopoulos, Thi & Viti 2002).The MOND formulas (eqs. 4 and 5) were then appliedto the V stars ( r ) and V gas ( r ) derived above. However, us-ing the face value V stars ( r ) coming from the URC for-mulas leads to a MOND Universal Rotation Curve thatis quite different from the original URC, especially forgalaxies where the stellar disk dominates the kinematics.In the lower panels of Figs. 2 and 3 the MOND curveswould lie significantly above the URC curves. Thus, aparameter η such that V , MOND ( r ) = ηV , URC ( r )was fitted in order to have the best possible agreementbetween the URC and the ”MOND URC”, assuming thatthe scaling of the stellar mass-to-light ratio with luminos-ity predicted by the URC is correct.This parameter η corresponds to the ratio of the stellarM/L ratios required in MOND and the URC frameworks,respectively. A best-fit value of η = 0 . 77 was found forthe standard interpolation function (eq. 2) and a value of η = 0 . 57 for the simple one (eq. 3). This is in agreementwith Famaey et al. (2007), who compare rotation curvefits made with the two µ functions considered here areconclude that the simple µ function gives stellar M/L ratios that are on average ∼ 30% lower than with thestandard µ .The stellar mass-to-light ratios are generally not wellknown in spiral galaxies. Here we considered a conserva-tive estimate of the uncertainties to be 50% (see de Jong& Bell 2006). The uncertainty on the gas scaling factor n was taken to be 3.4, from the scatter in the distribu-tion of n shown in Fig. 1. This scatter is large, but it isnot possible to find more a accurate way to estimate thescaling factor within the context of the URC, becauseone would need to consider other parameters (e.g. theHubble type) than the luminosity, which are much moredifficult to take into account: for instance, in early-typedisk galaxies usually there is a central depression in theHI distribution, but its size and magnitude are variable.Moreover, the URC, by construction, has the luminosityas the only parameter varying from galaxy to galaxy. Theuncertainties on the baryonic contribution to the rotationcurve result in uncertainties on the calculated rotationcurves. They are only rough estimates, since other (lesseasily quantifiable) sources of errors were ignored, suchas the presence of the bulge. These uncertainties on thestellar M/L ratios mean that the disagreement betweenthe MOND and URC stellar M/L ratios (i.e., η being = 1) does not necessarily imply that the approach of thepresent paper is incorrect. Moreover, this is in qualita-tive agreement with the fact that 1) in McGaugh (2005)the ratio of the “maximum” M/L ratio and the MOND M/L ratio has a median value of 1.8, where “maximum”in McGaugh (2005) is admittedly loosely defined but itis generally taken as a fit where the peak of the rotationcurve can be entirely explained by the stellar disk; and2) the URC has disks close to “maximum” (following theabove definition) for most galaxies apart from the leastmassive ones.The URC and the “MOND URC” (using µ orig ( x )) arecompared in Fig. 2: for the whole range of circular ve-locities and radii usually sampled by rotation curves, thetwo datasets display almost identical properties. Notsurprisingly, the best agreement is found for the range100 km s − . V opt . 250 km s − , where most observedrotation curves lie. The worst disagreement is found forthe largest distances, in particular in the most massivegalaxies. This has to be expected, though, as the num-ber of data points in this region of the parameter spaceis quite small: the sample of PSS has very few data forlarge radii of massive galaxies. Indeed, only a minorityof the curves come from HI observations (which trace the Fig. 4.— Logarithmic slope of the rotation curve versuslog( V opt ). The thick black lines represents the URC, the (lessthick) red and (thin) blue lines represent the MOND curves, withthe standard and simple µ functions, respectively. kinematics typically out to 2-3 r opt ), and few galaxies areearly-type large spirals (only two galaxies in their samplehave type Sab or earlier). However, using the uncertain-ties on the baryonic contribution discussed above, theURC and the MOND URC are always in agreement witheach other within the errorbars. Also, note that in thispaper we ignored the contribution of the bulge, since itwas ignored in the URC. This is likely to have a strongeffect on the rotation curves of the largest galaxies.The same considerations can be made if one uses µ BF ( x ) instead of µ orig ( x ). Fig. 3 shows that the URCand the “MOND URC” using µ BF ( x ) agree with eachother as well as if one uses µ orig ( x ).Using either interpolation function, the trend in loga-rithmic slope (see Section 3) is also reproduced: in theouter parts of the galaxy, the faintest galaxies have a ris-ing rotation curve while the brightest galaxies have a de-clining rotation curve. This is illustrated in Fig. 4, wherethe logarithmic slope as a function of V opt is displayed.Again, within the uncertainties the two formalisms agreewith each other. MOND does not reproduce particularlywell the slopes of the most massive galaxies, mainly be-cause of the fact that the bulge is not taken into account.For consistency with PSS we first performed a linear fit ofthe rotation curves between 0.5 and 1 r opt , then we tookthe slope at 1 r opt . The uncertainties on the slopes werecomputed as the maximum between the propagation ofthe uncertainties on the rotation curves and a minimumvalue of 0.1 (see PSS). THE MASS DISCREPANCY-ACCELERATION RELATIONFOR THE URC Now that it has been shown that MOND can repro-duce something which is quite close to the URC, in or-der to further compare the two phenomena let us have alook if the URC can reproduce one of the main proper-ties linked to MOND: the mass discrepancy-accelerationrelation. McGaugh (2004) showed that rotation curvedata seem to organise themselves along a relation linkingthe mass discrepancy (defined as D ( r ) = V ( r ) /V ( r ),where V ( r ) is the rotation velocity at radius r ) to thegravitational acceleration (here the Newtonian acceler-ation is used, g N = V /r ). In the same paper, it isalso shown that the stellar M/L ratios arising from theMOND fits minimise the scatter in the mass discrepancy-acceleration relation.Fig. 5 shows that using the URC formulas one findssuch a relation, whatever choice of baryonic mass ismade. Since we are looking for such a relation in theURC context, the mass discrepancy is always computedwith respect to the URC rotation curve (i.e., in the abovedefinition of D ( r ), V ( r ) = V ( r )). Obviously, using V ( r ) = V ( r ) (whichever µ is used) leads to a zero-scatter relation, by construction. A priori the tightnessof the relations shown in Fig. 5 comes a bit as a sur-prise: while in MOND a critical acceleration is inbuilt inthe theory, in the URC framework there is no such thing,at least in an explicit way. Semi-analytical dark mattermodels (e.g., van den Bosch & Dalcanton 2000) can alsoreproduce a mass discrepancy-acceleration relation usingthe right choice of parameters. However, the main thingto note is that in the present paper the URC gives a massdiscrepancy-acceleration relation without any parameterto adjust.From the lower four panels Fig. 5 one can see that themass discrepancy-acceleration relation from the URC for-mulas is consistent with that found by McGaugh (2004)with observational data if the baryonic contribution tothe rotation curve is chosen to be the same as in Sec-tion 4, with η = 0 . 57 or 0.77. The former value of η slightly overestimates the mass discrepancy (comparedto McGaugh’s points) for a given Newtonian accelera-tion g N : this is to be expected because η = 0 . 57 cor-responds to the “simple” interpolation function, wherethe full MOND regime is reached faster than with the“standard” function. Using η = 0 . 77 gives an excellentagreement: it corresponds to the “standard” interpola-tion function, which is what McGaugh used. Taking theface-value V bar ( r ) predicted by the URC yields a remark-ably tight relation; the scaling is obviously different, sincethe stellar M/L ratio is different and the gaseous contri-bution is ignored. We also considered different rangesof luminosities: since the URC is defined for V opt & − (see PSS), the predicted scaling relations are notexpected to hold with high accuracy. Indeed, the leftpanels Fig. 5 show that for V opt . 80 km s − the inner-most few points of the rotation curves start to becomediscrepant. DISCUSSION One of the main differences between the URC and theMOND fits is the stellar M/L ratio: while in the formerdisks are mostly close to maximum (following the defini-tion in Section 4), the latter yields slightly submaximaldisks (see McGaugh 2005), which results in a very differ-ent scaling in the mass discrepancy - acceleration plane.Again, we note that a limitation of the present approachis that these statements ignore any surface brightnessvariation at equal luminosity. A robust, precise and in-controvertible method to determine stellar M/L ratios isyet to be found. A number of methods have been sug-gested (see e.g. de Jong & Bell 2006), which suggest M/L ratios ranging roughly between half-maximum to maxi- mum disks. Therefore, a distinction between the URCand MOND based on the predicted stellar M/L ratios isnot viable at the moment. The fact that the simple µ function gives lower stellar M/L than the standard onehas already been shown by Famaey et al. (2007).In the present paper the discussion is focussed only onthe general properties of spiral galaxies’ rotation curves.Both MOND and the URC are successful ways of de-scribing the overall behaviour of rotation curves, but in-evitably there are cases where they fail to reproduce thedetails in the rotation curves. Also, only a rough es-timate of V bar ( r ) was made in the context of MOND,in particular for V gas ( r ). The observed V halo ( r )/ V gas ( r )scaling shows considerable variations around the valueof √ . λ − . η = 0 . 57 or 0.77), thefunctional form of the halo contribution in the URC con-text (eq. 9) is completely different from the functionalform of the “pseudo-halo” contribution in MOND (sec-ond term of the right-hand side of eq. 4).As pointed out by Salucci & Gentile (2006), any alter-native theory of gravity (such as MOND or MOG, seeMoffat 2006) should be able to account for the observedphenomenology of rotation curves. In the present paperwe have demonstrated that MOND does fairly well, de-spite the naive expectation of having only flat rotationcurves because of the asymptotically flat behaviour of theMOND rotation curves. Hence, the MOND frameworknot only usually gives good fits to rotation curves, butit also predicts the correct scaling of the rotation curvesproperties with luminosity, as shown by the URC. * Log L/L : 9.3 10.9 * Log L/L : 9.0 10.9 Fig. 5.— Full circles: mass discrepancy-acceleration (MDA) relation using the URC formulas. Open circles: the MDA data fromMcGaugh (2004), using the stellar M/L ratios arising from the MOND rotation curve fits. The horizontal axis is the log of the Newtonianacceleration (in units of km s − kpc − ) and the mass discrepancy is defined with respect to the URC rotation curve. The ranges ofB-band luminosities are indicated at the top (in solar units) and the radii range from 0.05 r opt to 2 r opt , in steps of 0.05 r opt . In the upperpanel the face-value URC prescription for the baryons was considered, while for the middle and bottom panel the values for V disk and V gas were taken as in Section 4. CONCLUSIONS Starting from the consideration that both MOND andthe Universal Rotation Curve (URC) are valid ways todescribe the general properties of rotation curves, the twoapproaches were compared by trying to reproduce one’spredictions using the prescriptions of the other one.The first comparison was made by attempting to re-produce something like the URC in the MOND frame-work. Some (observationally-based) assumptions on thestellar and gaseous contributions to the rotation curvewere made. It turns out that it is possible to build a“MOND URC” with very similar properties to the URC.The MOND URC also exhibits the trend in logarithmicslope versus luminosity seen in the URC. Both the “stan-dard” MOND interpolation function and the simple oneproposed by Famaey & Binney (2005) were tested, withalmost identical results. The second comparison was made by looking for a massdiscrepancy - acceleration relation using the URC for-mulas. With all three choices of baryonic contributions( V bar ) considered in this paper a tight (unexpected a pri-ori) relation arises. Using the V bar values that matchbest the “MOND URC” to the URC one finds a massdiscrepancy - acceleration relation like the observed one.While these results have no implications as to whetherMOND is a valid theory of gravity, MOND and the URC,even though they are based on totally different assump-tions, are found to display similar properties and to re-produce each other’s predictions well.I wish to thank the referee for constructive commentsthat improved the quality of the paper, and Stacy Mc-Gaugh for providing his data on the mass discrepancy -acceleration relation. GG is a postdoctoral fellow withthe National Science Fund (FWO-Vlaanderen).values that matchbest the “MOND URC” to the URC one finds a massdiscrepancy - acceleration relation like the observed one.While these results have no implications as to whetherMOND is a valid theory of gravity, MOND and the URC,even though they are based on totally different assump-tions, are found to display similar properties and to re-produce each other’s predictions well.I wish to thank the referee for constructive commentsthat improved the quality of the paper, and Stacy Mc-Gaugh for providing his data on the mass discrepancy -acceleration relation. GG is a postdoctoral fellow withthe National Science Fund (FWO-Vlaanderen).