On the relationship between MOND and DM

0.1 MB·English

by  Jörn Dunkel

On the relationship between MOND and DM
J¨orn Dunkel
Institut fu¨r Physik, Humboldt-Universita¨t zu Berlin, Newtonstraße 15, 12489 Berlin, Germany
[email protected]
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ABSTRACT

n
Numerous astrophysicalobservations have shown that classical Newtonian dynamics fails on
a
J galactic scales and beyond, if only visible matter is taken into account. The two most popular
8 theoretical concepts dealing with this problem are Dark Matter (DM) and Modified Newtonian
2 Dynamics (MOND). In the first part of this paper it is demonstrated that a generalized MOND
equation can be derived in the framework of Newtonian Dark Matter theory. For systems satis-

1
fying a fixed relationship between the gravitationalfields caused by DM and visible matter, this
v
generalizedMOND equation reduces to the traditionalMOND law,first postulated by Milgrom.
1
9 Therefore, we come to the conclusion that traditional MOND can also be interpreted as special
5 limit case of DM theory. In the second part, a formal derivation of the Tully-Fisher relation is
1 discussed.
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4 Subject headings: dark matter — galaxies: kinematics and dynamics
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/ 1. Introduction piricallyverysuccessful,DMhas become awidely
h
p accepted cornerstone of the contemporary cosmo-
Seventyyearsago,Zwicky(1933,1937)wasthe
- logical standard model (Sadoulet 1999; van den
o first to notice that the speed of galaxies in large
Bergh 2001; Ostriker and Steinhardt 2003). Nev-
r
clusters is much too great to keep them gravi-
t ertheless, it must also be emphasized that un-
s tationally bound together, unless they are much
a til now DM has been detected only indirectly by
: heavier than one would estimate on the basis of means of its gravitational effects on the visible
v
visible matter. Since those daysnumerousfurther
i matter or the light.
X astrophysicalobservations,e.g.,Dopplermeasure-
Aiming to avoid the introduction of invisible
r ments of rotation velocities in disk galaxies, have
a matter, an alternative phenomenological concept
confirmed the failure of the classical Newtonian
was proposed by Milgrom (1983a,b,c). Instead of
theory,ifonly visible matteris takeninto account
adapting the mass distribution, his approach re-
(Combes et al. 1995; Bertin and Lin 1996; Field
quires a modified Newtonian dynamics (MOND)
1999; Sanders and McGaugh 2002). Historically,
in the limit of small accelerations. As extensively
theoretical concepts addressing this problem can
reviewed by Sanders and McGaugh (2002), this
be subdivided in two categories. The first cat-
theory can explain galaxy data, such as the flat
egory comprises the Dark Matter (DM) theories
rotation curves, in a very compelling way. On
(Binney and Tremaine 1994; Sadoulet 1999; van
the other hand, there also have been some indi-
den Bergh 2001; Ostriker and Steinhardt 2003),
cations in the past that MOND might be an ef-
whereas the second group assumes that Newton’s
fectiveorapproximatetheory,applicabletoalim-
gravitational law requires modification (Milgrom
itedrangeofastrophysicalproblemsonly(Aguirre
1983a,b,c).
2003). Thishypothesisissupportedbyfundamen-
DM theories are based on the hypothesis that
tal difficulties associated with relativistic gener-
there exist significant amounts of invisible (non-
alizations of Milgrom’s theory (Sanders and Mc-
baryonic)matter in the universe,interacting with
Gaugh 2002; Soussa and Woodard 2003; Aguirre
ordinaryvisiblematteronlyviagravity. Sinceem-
1
2003). Also, according to Aguirre et al. (2001), where ρ (x) is the corresponding mass density
v/d
MOND seems to become less effective on larger andGdenotesthegravitationalconstant. Forcon-
scales; e.g., it cannot account for cluster densities venience, we define the accelerations
and temperature profiles in detail.
g (x):=−∇Φ (x). (3)
The fact that, to some extend, both DM and v/d v/d
MOND can successfully explain galactic dynam-
Thus, Eq. (1) simplifies to
icsfavorsthe possibilitythatthereexistsadeeper
connection between these two theories [for a gen- x¨ =g +g =:g. (4)
v d
eralcomparison,see (Aguirre 2003)]. Among oth-
Now let us additionally assume that the accelera-
ers, this idea was formulatedby McGaugh and de
tionvectorsg andg pointinthesamedirection,
Blok(1998),andlaterpursuedbyKaplinghatand v v
Turner (2002). Using arguments based on galaxy denoted by
formation processes in the early universe, the lat-
g ↑↑g . (5)
ter authors claim that MOND follows from cold v d
DMtheory. Inhisresponse,Milgrom(2002)ques- Note that in this case also g ↑↑ g. Roughly
v/d
tions these results. Among others, he argues that speaking,the assumptions(5) meansthatthe vis-
the predictions made by Kaplinghat and Turner ible massdistributionρ andthe DMdistribution
v
(2002) would not only conflict with astronomical ρ behave very similar. Next, we rewrite Eq. (4)
d
observations of pairs of galaxies (McGaugh and as
de Blok1998),butalsowith numericalresultsob-
g
tainedforDMmodels(Navarroetal.1997). Thus, x¨ = 1+ d g , (6)
unclaritystillseemstoexistaboutwhetherornot (cid:18) gv(cid:19) v
MOND can in fact be understood in the frame-
where g :=|g | with
work of DM (Aguirre 2003). v/d v/d
It is therefore the main purpose of the present g =g−g ≥0, (7)
v d
paper to explicitely demonstrate that the MOND
equations (if considered as modified Newtonian ifcondition(5)holds. Insertingthisinto(6)yields
gravity) can be derived from classical Newtonian
1
dynamics,providedonealsotakesintoaccountthe x¨ = 1+ g . (8)
g/g −1 v
gravitational influence of a DM component. In (cid:18) d (cid:19)
particular,it willbe shownthat the characteristic Thus, by virtue of (4), we find that
threshold acceleration, a0 ≈ 1.2·10−10 m/s2, be-
low which MOND effects begin to dominate, can ǫ
g = g =:µ˜(ǫ)g, (9)
also be interpreted as the asymptotic value of a v ǫ+1
(cid:18) (cid:19)
more general accelerationfield, characterizing the
where we have introduced
difference between the gravitational forces caused
by visible matter and dark matter, respectively. g(x)
ǫ(x):= −1≥0. (10)
g (x)
d
2. MONDfromNewtoniandynamicswith
DM Equation(9)canbecomparedtothefundamental
MOND formula (Milgrom 1983a,b,c; Sanders and
Asstartingpoint,considertheNewtonianEOM McGaugh 2002)
of a point-like test particle
g
mx¨ =−m∇[Φv(x)+Φd(x)], (1) gv =µ a0 g, (11)
(cid:18) (cid:19)
where Φ (x) and Φ (x) denote the gravitational
v d where,due toempiricalreasons,the functionµ(ξ)
potentials due to visible and dark matter, respec-
is postulated to have the asymptotic behavior
tively. Both potentials are solutions of Poisson
equations,
1, ξ ≫1;
µ(ξ)= (12)
2
∇ Φv/d =4πGρv/d, (2) (ξ, ξ ≪1.
2
One readily observes, that this is exactly the nat- Note that the second equality holds, only if one
ural asymptotic behavior of µ˜(ǫ) for ǫ → 0 and additionally assumes that ǫ(r) ≪ 1 for r → ∞.
ǫ →∞, respectively. Hence, if we identify µ with The reason is that, according to (9), only in this
µ˜ and introduce an acceleration field a(x) by very case the approximation g2 ≈ ag is valid.
v
Physically,the conditionǫ(r)≪1reflects adomi-
g(x) g(x)
=ǫ(x)= −1, (13) nating DM influence, as implied by (10)and(13),
a(x) gd(x) respectively.
then it becomes obvious that (9) is the natural The Tully-Fisher law (16) follows directly from
generalization of the MOND postulate (11). The the rhs. of (17). Assuming that a(r) → a∞ for
onlydifferenceis thatwehavealocalacceleration r →∞ and, in agreement with the standard pro-
field a(x) in (9), whereas a0 = const was postu- cedure, a Keplerian behavior gv(r) ≃ GM/r2 for
lated in the MOND formula (11). Note, that Eq. r →∞, we find the desired result
(13) can also be written in the equivalent form
4
v∞ =a∞GM. (18)
1 1 1
a(x) = g (x) − g(x) For the special case a∞ = a0, this is the well-
d
known MOND formula. Note that according to
1 1
= − . (14) our approach Eq. (18) represents, at least for-
g (x) g (x)+g (x)
d v d mally, a derived result, whereas it plays the role
Thus, the special MOND case of a postulate in the originalMOND papers (Mil-
grom 1983a; Sanders and McGaugh 2002). It
a(x)≡a0 (15) mightbeworthwhiletoemphasizehereonceagain
the crucial aspect, which is that the function µ˜
implies a fixed relation between the acceleration
from (9) naturally satisfies the MOND postulates
fieldsduetovisibleanddarkmatter. Inparticular,
(12).
since the characteristic MOND acceleration a0 is
Nevertheless,onemustbeawareofthefactthat
relatively small, one can further infer from (14)
the above derivation of Eq. (18) was essentially
that galaxies satisfying the MOND limit are DM
guided by the knowledge of the empirical Tully-
dominated.
Fisherlaw(16). Moreprecisely,theDMparadigm
in its current form does not provide any explana-
3. Axisymmetric disk galaxies and Tully-
tionfor the factthat inmany disk galaxiesvisible
Fisher law
anddarkmatterhavearrangedinsuchawaythat
In the following, let us concentrate on the a(r)rapidlyconvergestoaconstantnon-vanishing
quasi-two-dimensional problem of axisymmetric value.
disk galaxies. It is an experimental observation Since g and g reflect the distributions of vis-
v d
that for many such systems the Tully-Fisher rela- ible and dark matter, and because of
tionholds(SandersandMcGaugh2002;McGaugh
1 1 1
and de Blok 1998)
= lim − , (19)
4 4 a∞ r→∞(cid:26)gd(r) gv(r)+gd(r)(cid:27)
v := lim v (r)∝L∝M, (16)
∞
r→∞ the quantity a∞ gives us information about the
where L denotes the luminosity and M is the vis- asymptoticmassdistributions. Accordingto(Mil-
ible (baryonic) mass of the galaxy. The quantity grom 1983a,b,c; Sanders and McGaugh 2002), for
v(r) is the absolute velocity of stars or gaseous several disk galaxies the experimental value is
components,rotatinginthediskplanearoundthe given by the MOND value, a∞ = a0. From the
galacticcenter (r is the distance fromthe galactic point ofview adopted inthis paper, this indicates
center, defining the origin of the coordinate sys- that the compositionofthese galaxiesis generally
tem). Equating centripetal acceleration v2/r and similar.
g(r), we find In contrast, at least for some clusters of galax-
ies the actual value of a(x) seems to essentially
2
v∞ = lim rg(r)= lim r a(r)gv(r). (17) deviate from the MOND value a0. As mentioned
r→∞ r→∞
p
3
earlier, Aguirre et al. (2001) have shown that the only formaldifference consists inthe factthat the
experimentally observed, radial temperature pro- constantthresholdvaluea0isreplacedbythemore
files of Coma, Abell 2199 and Virgo can not be general acceleration field a(x) from (14). In the
fitted if one assumes a globally constant value DM picture, a(x) reflects the local difference be-
a(x)≡a0. Furthermore,theseauthorsreportsat- tween the gravitationalforces caused by dark and
isfactoryagreementwhentheyapplystandardDM visible matter, respectively. In order to exactly
models instead. With regardto our above consid- regain the traditional MOND law (11), one addi-
erations, the latter procedure simply corresponds tionally has to demand that a(x) ≡ a0. Thus,
to using a locally varying field a(x) 6≡ a0. On MOND can in principle also be interpreted as a
the one hand, this supports the hypothesis that DM theory, satisfying the two additional condi-
MOND should be viewed as a special limit case tions (5) and (15).
of DM theory; on the other hand, one is led to Therefore, it seems reasonable to assume that
ask, why a(x) is approximately constant in disk the traditional MOND theory represents a spe-
galaxies,but seems to vary in clusters. According ciallimitcaseofNewtonianDMtheory. Adopting
to the author’s opinion, the answer to this ques- this point of view, one can further conclude that
tioncanonlybegivenbyanimprovedDMtheory, MOND successfully explains the rotation curves
yet to be developed. In particular, such a the- ofdiskgalaxiesbecauseforsuchobjectstheabove
orymustpredict the dynamics ofdarkandvisible conditions(5)and(15)arefulfilled. Ifthisistrue,
mass components in detail. then,asalsodiscussedabove,theMONDconstant
Finally, we still note that if gv(x) ≪ gd(x) a0 can be interpreted as the asymptotic value of
holds, then one can expand (14) yielding the field a(r) as r →∞.
More generally speaking, whenever there is a
g (x)2
a(x)≈ d . (20) fixed relationship between gd and gv (or ρd and
gv(x) ρv, respectively) such that a(x) ≈ a0, then the
traditionalMONDtheoryshouldcontinuetowork
Forsphericalmatterdistributionsthismeansthat
successfully. In turn, if a disk galaxy is in the
[GM (r)/r2]2 MOND regime,then Eq. (14)canbe usedto esti-
d
a(r)≈ , (21)
GM (r)/r2 mate the DM distributionρd,providedthe visible
v
matterdistributionρ isknownfromobservations.
v
whereM (r)denotesthevisible/darkmasscon- Furthermore, it was shown that µ(ξ) = ξ/(ξ+1)
v/d
tainedwithinradiusr. Forthespecialcasea(r)≈ is the natural candidate for the MOND func-
a0, this is equivalent to tion. Another result of this paper was the formal
derivation of the Tully-Fisher law (18) in Sec. 3.
2
1 Md(r) a0 kg 3 M⊙ This relation should hold whenever the two con-
≈ ≈2 ≈10 , (22)
Mv(r)(cid:20) r (cid:21) G m2 pc2 ditions gv ≪ gd and a∞ > 0 are satisfied, where
a∞ := limr→∞a(r). In this context it must be
whichimpliesastrongcorrelationbetweenthedis-
stressed that the current DM model cannot ex-
tributionsofvisibleanddarkmatterintheMOND
plain,inwhichsituationsthesetwoconditionsare
limit. It should be mentioned here that the possi-
fulfilled,and,ifso,whythisisthecase. Therefore,
bility of such a connection was already suggested
modificationsoftheconventionalDMtheoryseem
by McGaugh and de Blok (1998) and, later, also
inevitably necessary.
more extensively discussed by McGaugh (2000).
We conclude this short paper with a more gen-
eral remark. In principle, there seems to be an
4. Summary and conclusions
agreementthatNewton’stheoryappliedtovisible
ItwasshownthatthegeneralizedMONDequa- matter does not give a generally correct descrip-
tion(9)canbederivedfromNewtoniandynamics, tionofthedynamicsofgalaxiesand,therefore,has
if one adds a DM contribution Φ to the (bary- to be modified. A first way to do this is to simply
d
onic)NewtonianpotentialΦv,suchthatΦv/d lead consider an additional potential Φd and, follow-
to equally directed accelerations g = −∇Φ . ing the standardstrategy,to attach a ”generating
v/d v/d
Comparedto the traditionalMONDlaw (11), the object” called DM to this potential. As shown
4
above, Milgrom’s concept (if considered as mod- Milgrom, M. 1983,ApJ, 270, 365
ification of gravity) is in fact very similar, even
Milgrom, M. 1983,ApJ, 270, 371
though it seems quite different at first glance. In
particular,theMONDequationscanalsobetrans-
Milgrom, M. 1983,ApJ, 270, 384
formedinto amodificationofthe formerpotential
type,bystartingwitha(x)≡a0 andreversingthe Milgrom, M. 2002,ApJ, 571, L81
above manipulations. The ”generating object” of
Navarro, J. F., Frenk, C. S., and White, S. D. M.
the related potential can then be named DM as
1997,ApJ, 490, 493
well.
Ostriker, J. P., and Steinhardt, P. 2003, Science,
The author is very grateful to Christian Theis 300, 1909
for his encouraging support and careful reading
Sadoulet, B. 1999, Rev. Mod. Phys., 71, S197
of the manuscript. He also wants to thank Ste-
fan Hilbert for numerous, very helpful discussions
Sanders, R. H., and McGaugh, S. S. 2002,
and Stacy McGaugh for valuable comments. This
ARA&A, 40, 263, arXiv:astro-ph/0204521
work was, in parts, financially supported by the
Studienstiftung des deutschen Volkes. Soussa, M. E., and Woodard, R. P. 2001,
arXiv:astro-ph/030/0307358v1
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