Mon. Not. R. Astron. Soc. 000, 000–000 (2007) Printed 8 November 2018 (MN LATEX style file v2.2)
Vertical dynamics of disk galaxies in MOND
Carlo Nipoti1, Pasquale Londrillo2, Hong Sheng Zhao3, and Luca Ciotti1
1Astronomy Department, University of Bologna, via Ranzani 1, 40127 Bologna, Italy
2INAF-Bologna Astronomical Observatory, I-40127 Bologna, Italy
3SUPA, University of St. Andrews, KY16 9SS, Fife, UK
Accepted 2007 April 3. Received 2007 March 28; in original form 2007 February 12.
ABSTRACT
We investigate the possibility of discriminating between Modified Newtonian Dy-
namics (MOND) and Newtonian gravity with dark matter, by studying the vertical
dynamics of disk galaxies. We consider models with the same circular velocity in the
equatorial plane (purely baryonic disks in MOND and the same disks in Newtonian
gravity embedded in spherical dark matter haloes), and we construct their intrinsic
and projected kinematical fields by solving the Jeans equations under the assumption
of a two-integral distribution function. We found that the vertical velocity dispersion
of deep-MOND disks can be much larger than in the equivalent spherical Newtonian
models. However, in the more realistic case of high-surface density disks this effect
is significantly reduced, casting doubts on the possibility of discriminating between
MOND and Newtonian gravity with dark matter by using current observations.
Key words: gravitation — stellar dynamics — galaxies: kinematics and dynamics
1 INTRODUCTION
The flatness of the rotation curves of disk galaxies in their
external regions has been the primary focus in the last-
ing on dark matter (DM) and Modified Newtonian Dy-
namics (MOND). As a consequence of the formalisation
of MOND in fundamental physics (Bekenstein 2004; Zlos-
nik, Ferreira & Starkman 2007), combined with some dif-
ficulties of the cold dark matter scenario on galaxy scales
(e.g. Binney 2004, and references therein), the debate ex-
tended to other astrophysical contexts. For example, MOND
models have been recently tested against the Cosmic Back-
ground Radiation (Skordis et al. 2006, Skordis 2006), grav-
itational lensing (Chen & Zhao 2006; Zhao et al. 2006;
Angus et al. 2007; Takahashi & Chiba 2007), star cluster
and galaxy dynamics (Zhao & Tian 2006; Sanchez-Salcedo,
Reyes-Iturbide & Hernandez 2006; Haghi, Rahvar & Hasani-
Zonooz 2006; Nipoti, Londrillo & Ciotti 2007; Scarpa et
al. 2007; Tiret & Combes 2007), and the solar system
(Bekenstein & Magueijo 2006, Sanders 2006, Sereno & Jet-
zer 2006). Due to the surprising ability of MOND to repro-
duce the kinematics of different systems (e.g., Milgrom 2002;
Sanders & McGaugh 2002; Bekenstein 2006), it is of obvious
interest to look for other tests to discriminate between DM
and MOND.
In this paper we explore the possibility of differentiat-
ing DM and MOND by studying the vertical dynamics of
disk galaxies. While it is often taken for granted that—as
far as disk kinematics is concerned—the DM and MOND in-
terpretations are nearly degenerate, in fact this is not true.
An important issue is the choice of the Newtonian system
used for comparison with the MOND results. In the present
context, the proper comparison is between a baryonic disk in
MOND, and the same disk, in Newtonian gravity, immersed
in a DM halo (which for simplicity we assume spherically
symmetric), such that the circular velocity in the equatorial
plane is the same as in the MOND model1. In particular, we
compare the kinematical fields of equivalent models under
the assumption of two-integral distribution function. A work
in some respect complementary to ours, though with impor-
tant differences, was done by Milgrom (2001), who studied,
for a given baryonic disk, the shape of the DM halo of the
Newtonian model with exactly the same dynamics (not only
the same rotation curves) as the disk in MOND gravity. Re-
cently, a test of MOND based on the comparison of disk
circular velocity and vertical velocity dispersion has been
proposed also by Stubbs & Garg (2005)2.
The paper is organised as follows. In Section 2 we de-
scribe the general method adopted, and we perform a pre-
liminary investigation of the problem. In Section 3 we study
in detail the projected kinematical fields of Miyamoto-Nagai
and thick exponential disks in deep MOND regime, and we
compare them with their equivalent Newtonian models with
DM, for different values of flattening/thickness. We also ad-
1 Other cases are known in which MOND and Newtonian systems
are equivalent with respect to some dynamical properties, but
different with respect to others (e.g. Ciotti & Binney 2004).
2 The presented test is affected by the unjustified assumption
that the vertical velocity dispersion depends only on the local
gravitational field.
c© 2007 RAS
http://arxiv.org/abs/0704.0740v2
2 Nipoti et al.
dress the problem of MOND dynamics in Milky-Way like
galaxy models. Our results are summarised in Section 4.
2 THE METHOD
In the present work we consider MOND in Bekenstein &
Milgrom’s (1984) formulation, in which the Poisson equation
∇2φN = 4πGρ, (1)
where φN is the Newtonian gravitational potential gener-
ated by the density distribution ρ, is substituted by the
non-relativistic field equation
= 4πGρ. (2)
In equation above ‖...‖ is the standard Euclidean norm, φ is
the MOND gravitational potential produced by ρ, and for
finite mass systems ∇φ → 0 for ‖x‖ → ∞. The interpolating
function µ(t) is not constrained by theory except that it
must run smoothly from µ(t) ∼ t at t ≪ 1 to µ(t) ∼ 1 at t ≫
1, with a dividing acceleration scale a0 ≃ 1.2× 10−10ms−2.
In the so-called ‘deep MOND regime’ (hereafter dMOND),
describing low-acceleration systems (‖∇φ‖ ≪ a0), µ(t) = t
and equation (2) reduces to
∇ · (‖∇φ‖∇φ) = 4πGa0ρ. (3)
As well known, equation (2) can be combined with equa-
tion (1) so that the MOND g = −∇φ and Newtonian
N = −∇φN gravitational fields are linked as
µ(g/a0)g = g
+ S, (4)
where S is a solenoidal field dependent on the specific ρ con-
sidered. It can be proved that the equation above reduces to
Milgrom’s (1983) empirical relation (i.e., S = 0) in case of
one-dimensional symmetries and in the special case of the
razor-thin Kuzmin (1956) disk density distribution (Brada
& Milgrom 1995), but in general one cannot impose S = 0.
The presented results are derived by solving the field equa-
tion (2) with the numerical potential solver presented in
Ciotti, Londrillo & Nipoti (2006).
2.1 Two-integral dynamics
In the following we compare the dynamics of two disk galaxy
models with the same circular velocity curve in the equato-
rial plane
c (R) = R
∂φT(R, 0)
, (5)
where φT(R, z) is the total potential of the model, and
(R, z, ϕ) are the standard cylindrical coordinates. One model
is built in MOND for a baryonic disk galaxy described by
the density distribution ρ(R, z). The associated Newtonian
model, which we call equivalent, consists of the same bary-
onic disk plus a spherical DM halo, whose radial density
profile is determined by the condition of matching the ro-
tation curve obtained in the MOND model. In practise, the
density distribution of the DM halo of the equivalent model
is given by
ρDM(r) =
4πGr2
c (r)− rv2cN(r)
, (6)
where vc and vcN(R) are the disk circular velocity in MOND
and Newtonian gravity, respectively. Note that the positiv-
ity of ρDM is not guaranteed for a generic disk density, so
we always check the positivity of the equivalent DM halo
density distribution.
For a two-integral distribution function, i.e., f =
f(E, Jz), the Jeans equations for the disk are
= −ρ∂φT
, (7)
ρ(σ2 − v2ϕ)
= −ρ∂φT
, (8)
where σ2 ≡ σ2R = σ2z , v2ϕ = u2ϕ + σ2ϕ, uϕ = vϕ, and a
bar over a symbol indicates its phase-space mean over the
velocity space (e.g., see Binney & Tremaine 1987, hereafter
BT). Note that in a MOND model φT is the solution of
equation (2) for the disk density distribution, while in the
equivalent Newtonian model φT = φ
N(R, z) + φDM(r), with
φDM determined from the density profile in equation (6).
The velocity dispersion σ2(R, z) is obtained from inte-
gration of equation (7) with boundary condition ρσ2 = 0 for
z → ∞, i.e.
. (9)
In the equivalent model
, (10)
where r =
R2 + z′2 and dφDM/dr = (v
c − v2cN)/r.
To split v2ϕ into streaming motion uϕ (that for simplicity
we assume nowhere negative), and azimuthal dispersion σ2ϕ,
we adopt the Satoh (1980) k–decomposition
ϕ = k
(v2ϕ − σ2), (11)
ϕ = σ
+ (1− k2)(v2ϕ − σ2), (12)
with 0 ≤ k ≤ 1; this procedure can be applied only when
v2ϕ − σ2 ≥ 0 everywhere. For k = 0 no ordered motions are
present, and the velocity dispersion tensor is maximally tan-
gentially anisotropic, while in the isotropic rotator (k = 1)
the galaxy flattening is due to azimuthal streaming velocity.
In principle, by allowing for k = k(R, z), even more rotation-
ally supported models can be constructed, up to the maxi-
mum rotation case considered in Ciotti & Pellegrini (1996),
where σ2ϕ = 0 everywhere.
The explicit projection formulae for the kinematical
fields of axisymmetric two-integral systems viewed along a
generic direction n of the line of sight (los) can be found
elsewhere (e.g., see Lanzoni & Ciotti 2003, Ciotti & Bertin
2005, Riciputi et al. 2005): here we just recall that at any
given place in the galaxy the los component of the stream-
ing velocity field is un ≡ 〈v,n〉 = uini, where 〈, 〉 is the
standard inner product, the repeated index convention is
applied, and Cartesian coordinates are assumed. The analo-
gous quantity associated with the velocity dispersion tensor
is σ2n ≡ 〈v − u〉2 = σijninj , and the corresponding (mass–
weighted) kinematical projected fields are obtained by inte-
gration along the los as
Σvp ≡
ρundl, (13)
c© 2007 RAS, MNRAS 000, 000–000
Vertical dynamics of disk galaxies in MOND 3
Figure 1. Density profile of spherical DM haloes of Newto-
nian models equivalent to dMOND MN disks (left) and thick
exponential disks (right). The dotted lines in the left panel are
given by the distribution (24), obtained in the S = 0 approxi-
mation. Density is in units of (M/4πb3)
a0b2/GM (left) and
(M/4πR2
/GM (right).
ndl, (14)
ndl, (15)
where Σ =
ρ dl is the disk density projected along n.
In general σp is not the velocity dispersion measured in ob-
servations: in the presence of a non–zero projected velocity
field vp, this quantity is
los ≡
ρ (〈v,n〉 − vp)2dl = Σ×
p + V
p − v2p
. (16)
For simplicity, in this paper we focus on the limit cases
of disk observed face-on and edge-on. The face-on projected
velocity dispersion is given by
p = 2
dz = 2
zdz, (17)
where Σ(R) = 2
ρ(R, z)dz is the face-on disk density. In
the edge-on case, assuming the los directed along the x axis,
the los direction in the natural coordinate system is given
by3 n = (1, 0, 0) and the projection plane is (y, z), so that
un = −uϕ sinϕ, (18)
n = σ
+ (1− k2)(v2ϕ − σ2) sin2 ϕ, (19)
3 The los vector points towards the observer, and so positive ve-
locities correspond to a blue–shift.
Figure 2. Ratio of the vertical accelerations of dMOND (gz,disk)
and equivalent Newtonian (gN
) models as a function of the
height z above the disk plane, at fixed cylindrical radius.
where sinϕ = y/R = y/
y2 + x2. Note that, after projec-
tion, the y coordinate can been identified with R.
2.2 A preliminary analysis
As an introductory exercise we compare, as a function of
disk flattening, the vertical force field near the equatorial
plane of a Miyamoto & Nagai (1975, hereafter MN) disk
in dMOND regime and its equivalent Newtonian model. In
order to carry out the calculations analytically we assume
S = 0, while in Section 3 the exact MOND gravitational
field with the appropriate S field is used. The MN density
distribution with scale length b and flattening parameter
s ≡ a/b is
ρ(R, z) =
sR̃2 + (s+ 3ζ)(s+ ζ)2
[R̃2 + (s+ ζ)2]5/2ζ3
, (20)
where M is the total disk mass, ζ ≡
1 + z̃2, z̃ ≡ z/b, and
R̃ ≡ R/b. The corresponding Newtonian potential is (BT)
φ(R, z) = −GM
R̃2 + (s+ ζ)2
. (21)
For a = 0 the MN density distribution is a Plummer (1911)
sphere with scale radius b, while for b = 0 one obtains the
Kuzmin (1956) disk. The Newtonian disk circular velocity
is given by equations (5) and (21) as
cN(R) =
[R̃2 + (s+ 1)2]3/2
. (22)
In the S = 0 approximation the dMOND circular velocity is
obtained from equation (4) as
c (R) =
GMa0R̃
[R̃2 + (s+ 1)2]3/4
, (23)
and according to equation (6), ρDM ∝ A(r̃) −
GM/a0b2B(r̃), where A and B are two dimensionless
c© 2007 RAS, MNRAS 000, 000–000
4 Nipoti et al.
Figure 3. Radial trend of velocity ratios between dMOND and equivalent Newtonian models (quantities with label N) in the fully
isotropic case. From top to bottom: z = 0 vertical velocity dispersion (9), face-on projected velocity dispersion (17), z = 0 edge-on
projected streaming velocity (13), and z = 0 edge-on projected velocity dispersion (16).
functions, and r̃ ≡ r/b. In the dMOND limit
GM/a0b2 =
0 and so
ρDM(r) ≃
2r̃2 + 5(s+ 1)2
r̃[r̃2 + (s+ 1)2]7/4
, (24)
which is positive everywhere and has the expected r−2 be-
haviour for r → ∞. Note that equation (24) implies that in
the equivalent Newtonian model the MN disk is just a tracer
of the DM gravitational field.
The obtained formulae are useful because they allow to
evaluate the vertical gravitational field near the disk plane
in the two models. In the S = 0 approximation the dMOND
vertical field is
gz,disk =
||gN||
z , (25)
where gNz is the vertical Newtonian gravitational field of the
MN disk, and the vertical gravitational field of the equivalent
Newtonian DM halo is
z,DM = −
∂φDM(r)
= −zv
c (r)
. (26)
Simple algebra then shows that for R → ∞ and z → 0,
gNz,DM
gz,disk
1 + s
2(1 + s)2
s(5 + s)
4(1 + s)R̃2
, (27)
i.e., the vertical field is stronger in dMOND than in DM
models, and the discrepancy between the two cases is larger
for flatter models (i.e. for larger values of s). Thus, we ex-
pect σ2 in the disk to be different in the two models. In
turn, equations (8) and (11) imply that also the streaming
velocity field in the disk equatorial plane is different in the
two models, even though the circular velocity is the same by
construction.
3 RESULTS
We now obtain the kinematical fields (for simplicity in the
fully isotropic case) of a few disk galaxy models by comput-
ing the exact MOND and Newtonian potentials with the nu-
merical code presented in Ciotti et al. (2006). The potential
solver is based on a spherical grid of coordinates (r, ϑ, ϕ),
with Nr ×Nϑ ×Nϕ points, uniformly spaced in (arctan r, ϑ,
ϕ). To compute the intrinsic and projected velocity disper-
sions, we interpolate the potential from the spherical grid to
a cylindrical grid with NR×Nz points, uniform in (arctanR,
arctan z), on which we evaluate the integrals (9) and (13-15).
The numerical integration routines are verified by deriving
the intrinsic and projected velocity dispersions of MN disks
in Newtonian gravity, and comparing the results with the an-
alytical expressions reported in Ciotti & Pellegrini (1996).
For instance, with Nr = Nϑ = NR = Nz = 128 the analyti-
cal results are reproduced with relative errors ∼ 0.1 %.
3.1 Miyamoto-Nagai disks
We consider two MN galaxy models with flattening param-
eter s = 1 (i.e., an almost spherical system), and s = 5.
Their exact dMOND fields are computed by solving numer-
ically equation (3). The density distributions of the spheri-
cal DM haloes of the equivalent Newtonian models are then
c© 2007 RAS, MNRAS 000, 000–000
Vertical dynamics of disk galaxies in MOND 5
Figure 4. Density distribution in the equatorial plane for the
baryonic and dark components of the considered Milky-Way like
galaxy models (top), and the corresponding rotation curves (bot-
tom).
obtained from equation (6) with vcN = 0, as appropriate
for the dMOND case, in which the disk is just a tracer of
the halo potential. As expected, the resulting halo profiles
(Fig. 1, left panel; solid and dashed lines) decrease as r−2
in the outer regions, matching the analytical formula (24),
obtained in the S = 0 approximation (dotted lines). How-
ever, for r<∼b the exact halo density profiles are consider-
ably flatter than those predicted by equation (24), having
inner logarithmic slope d log ρDM/d log r>∼− 1/2. The differ-
ence (which is larger in the flatter disk) indicates that the
contribution of the field S is non-negligible in the central
regions, as also found by Ciotti et al. (2006) for triaxial and
axisymmetric Hernquist (1990) models in dMOND regime.
However, in the outer regions the agreement is perfect.
In Fig. 2 (left panel), we plot the ratio gNz,DM/gz,disk as
a function of z at the representative radius R = b. Again,
the results nicely confirm the asymptotic estimate of equa-
tion (27) when z = 0: the effect is bigger in the flatter disk,
and just above the disk plane gz,disk is significantly stronger
than gNz,DM. We also note that for z ≫ b the vertical New-
tonian field exceeds the dMOND one, though slightly. As a
consequence, also the intrinsic and projected velocity disper-
sions are different in the equivalent Newtonian and dMOND
MN disks, while the circular velocities in the equatorial plane
are the same by construction. The midplane vertical veloc-
ity dispersion σ(R, 0) and the projected face-on velocity dis-
persion σlos are larger in the dMOND systems than in the
equivalent Newtonian systems at all radii R < 5b, with the
larger discrepancies (up to a factor of 2) in the flatter disk
(Fig. 3, left column). The situation is more complicate in
the edge-on projection: the projected streaming velocity vp
in the Newtonian models are larger than in the dMOND
cases, while the edge-on projected velocity dispersion σlos in
the galactic plane is higher in dMOND than in Newtonian
models.
Figure 5. Vertical force as a function of z, at R = 8kpc, for the
considered Milky-Way like galaxy models.
3.2 Thick exponential disks
We now investigate the more realistic thick exponential disk
ρ(R, z) =
4πR20z0
, (28)
where M is the total disk mass, and R0 and z0 are character-
istic scale-lengths. As for the MN case, we consider here two
dMOND models (and so in the equivalent Newtonian model
the disk does not contribute to the potential): a thicker disk
with R0 = 3z0, and a thinner disk with R0 = 15z0.
As in MN disks, the spherical DM haloes density pro-
files of the equivalent Newtonian models are rather flat in the
central regions and ∝ r−2 at large radii (Fig. 1, right panel).
Interestingly, these DM haloes “predicted” by MOND for
low-surface brightness disks are not characterised by the
steep central cusps expected in the context of cold dark mat-
ter. The dMOND vertical force near the disk is significantly
stronger than in the equivalent Newtonian models (Fig. 2,
right panel): at R = R0 in the thinner model the dMOND
vertical force ratio is ∼ 10 just above the plane. This differ-
ence reflects in the velocity fields (Fig. 3, right panels), which
behave qualitatively as the MN models, but the discrepan-
cies are larger: for example, the ratio of intrinsic velocity
dispersion is as high as ∼ 4 in the thinner model.
3.3 Milky-Way like galaxies
The previous analysis restricted to models in the dMOND
regime, in order to estimate the largest possible differences
in the kinematical fields of equivalent disk galaxies. We now
focus on a Milky-Way like galaxy model, with the purpose of
quantifying the MOND kinematical effects in realistic high
surface-brightness galaxies. We will not enter the challeng-
ing problem of finding the best-fitting MOND model of the
Milky Way (which is beyond the aim of the present paper;
see Famaey & Binney 2005); instead, we apply our method
to one of the currently available mass models of the Milky
c© 2007 RAS, MNRAS 000, 000–000
6 Nipoti et al.
Figure 6. Left (from top to bottom): z = 0 vertical velocity dispersion, face-on projected velocity dispersion, z = 0 edge-on projected
streaming velocity and z = 0 edge-on projected velocity dispersion for model DBMOND (solid lines) and its equivalent Newtonian model
DBEQ (dashed lines), in units of km s−1. Right: percentage absolute differences between the corresponding MOND and Newtonian
velocity fields in the left panels. Thin and thick lines refer to thin-disk and thick-disk stars, respectively.
Way that have been derived in the context of DM. In par-
ticular, we consider Model 1 of Dehnen & Binney (1998;
hereafter model DB), which consists of a truncated oblate
spheroidal power-law bulge, a stellar disk made of a thin and
a thick component, a gaseous exponential disk with a cen-
tral hole, and finally an oblate spheroidal DM distribution.
Model DB represents a good Newtonian dynamical model of
the solar neighbourhood and the Galaxy, and can be classi-
fied as a disk dominated model, because in Newtonian grav-
ity the disk provides the main contribution to the rotation
curve in the central regions.
As we solve now equation (2), we need to specify the in-
terpolating function µ. Following Famaey & Binney (2005),
we fix a0 ≃ 1.2 × 10−10ms−2 and
µ(t) =
1 + t
. (29)
This function gives a better fit to the terminal velocity
curve of the Milky Way than the commonly adopted func-
tion µ(t) = t/
1 + t2 and fits extremely well the rotation
curve of NGC3198, also being consistent with Bekenstein’s
(2004) TeVeS theory (Zhao & Famaey 2006; see also Famaey
et al. 2007). We consider two models with the same bary-
onic distribution as model DB: model DBMOND (a purely
baryonic MOND model), and model DBEQ (the equiva-
lent Newtonian model with spherical DM halo). Clearly,
the DM halo density profile in model DBEQ is not iden-
tical to that of model DB: Fig. 4 (top panel) shows that in
the equatorial plane the two distributions are very similar
in the radial range R ∼ 5 − 15 kpc, while the halo profile
of DBEQ is steeper than that of DB at small radii. As a
consequence, the rotation curve of models DBMOND and
DBEQ (which is the same by construction) differs from the
rotation curve of DB, being systematically higher at small
radii and lower at large radii (Fig. 4, bottom). However,
the differences in the radial range 0 − 15 kpc are within
∼ 5%, and vc(8 kpc) = 226 km s−1 in model DBMOND (and
DBEQ), consistent with the observational constraint at the
solar neighbourhood 220± 15 kms−1 (solid symbol and ver-
tical bar in the diagram; e.g., BT).
In contrast with the dMOND cases discussed in Sec-
tions 3.1 and 3.2, now the contribution of the baryonic com-
ponent to the gravitational field is important also in the
Newtonian model, and this results in smaller differences be-
tween the MOND and Newtonian equivalent models. For
instance, considering the vertical force as a function of z
at R = 8kpc (Fig. 5), we find that the MOND force is
stronger than the Newtonian one, with a difference of ∼ 30%
at z ∼1 kpc. The solid dot marks an observational estimate
(and associated uncertainty) of the vertical force 1.1 kpc
above the plane at the solar radius (Kuijken & Gilmore 1989,
1991). Remarkably, the difference between the vertical forces
at z = 1.1 kpc predicted by the MOND and Newtonian mod-
els is larger than the observational error. So, taken at face
value, this result would favour DM rather than MOND mod-
els, though we stress again that we did not attempt to build
the best-fit MOND model of the Milky Way, and we can-
not exclude the possibility of finding a MOND model of
the Milky Way satisfying this and other observational con-
straints (see also Famaey & Binney 2005). In any case, this
c© 2007 RAS, MNRAS 000, 000–000
Vertical dynamics of disk galaxies in MOND 7
Figure 7. Isodensity contours in the meridional plane of the fully equivalent DM haloes associated with dMOND MN (top panels) and
thick exponential disks (bottom panels). Density increases towards the equatorial plane.
result shows that in principle it could be possible to try to
falsify MOND by using the observational constraints on the
vertical force above the plane.
Following the treatment of Sections 3.1 and 3.2 we
computed the kinematical fields of models DBMOND and
DBEQ, again under the same assumptions of two-integral
distribution function4 and full isotropy. For both models we
computed the intrinsic and projected velocity fields of the
thin-disk (thin curves) and thick-disk (thick curves) stars
(Fig. 6, left panels), finding that the discrepancy between
the two models is up to 20-25% for intrinsic and face-on
velocity dispersion, while just few per cent in case of edge-
on projection (Fig. 6, right panels). These results show that
the differences in the vertical kinematical fields in a realis-
tic high-surface density galaxy are quite small, and it is not
obvious that one can discriminate between MOND and New-
tonian gravity (plus DM) by using the information currently
derived by the observations.
4 DISCUSSION AND CONCLUSIONS
In this paper we quantified the differences in the vertical
force and kinematical fields between MOND disk galaxy
models and equivalent (i.e., having identical circular veloc-
ity in the equatorial plane) Newtonian models with spherical
4 Because of this assumption our results on the velocity fields
cannot be intended, strictly speaking, to represent the specific
case of the Milky Way, whose distribution function is not two-
integral, being σ2
> σ2z in the solar neighbourhood.
DM haloes. We showed that, in principle, MOND and New-
tonian gravity with DM can be differentiated with accurate
measurements of the vertical force near the disk and of the
projected kinematical fields. In particular, MOND models
have stronger vertical force near the plane and higher ver-
tical velocity dispersion than equivalent Newtonian models
with DM. Our results are not inconsistent with the finding of
Read & Moore (2005) that satellite orbits in DM and MOND
behave very similarly, because far from the disk plane the
force fields predicted by the two theories are very similar.
From the observational point of view, the robust dis-
crimination between MOND and Newtonian gravity with
DM is still challenging due to several factors: (i) the
strongest effects are expected in low-acceleration (low-
surface-brightness) systems, in which measurements of the
velocity fields are difficult; (ii) there are often several stel-
lar populations with different scale-lengths and heights, and
the mass-to-light of the stellar component is tunable; (iii)
most disks have gaseous components whose density has non-
smooth features which could make differences in any local
volume of kpc scale; (iv) the models should be compared to
data with noise so our comparisons of MOND and DM mod-
els of exactly same circular velocity curves remains a theo-
retical exercise; (v) DM haloes are not necessarily spherically
symmetric.
With regard to point (v), we briefly discuss how our re-
sults would change by relaxing the assumption of an equiva-
lent spherical DM halo. Following Milgrom (2001), we start
by constructing, for a given disk distribution ρ, the “fully
equivalent” DM density distribution as ρDM = ∇2φ/4πG−ρ,
where φ is the MOND potential; as well known, ρDM can be
c© 2007 RAS, MNRAS 000, 000–000
8 Nipoti et al.
negative in some region of space (Milgrom 1986; Ciotti et
al. 2006). By construction, the fully equivalent Newtonian
model has gravitational field identical to the MOND disk
field in all the space, not just the same midplane circular
velocity (as the equivalent model with spherical halo). In-
terestingly, we found that the density distributions ρDM of
the fully equivalent DM haloes of the dMOND models pre-
sented in Sections 3.1 and 3.2 are nowhere negative (in these
cases ρDM = ∇2φ/4πG, because the baryonic disk is just a
tracer in dMOND-equivalent models). Figure 7 shows that
ρDM presents a significant disk structure (more pronounced
in the case of more flattened baryonic disks). This result
clearly indicates that it is not possible to obtain a Newto-
nian model with spheroidal DM distribution strictly equiva-
lent to a MOND disk model (see also Milgrom 2001). How-
ever, we found that the discrepancy in the vertical dynamics
in the galactic plane between MOND and DM models can
be significantly reduced considering spheroidal DM haloes:
for instance, a dMOND MN disk with s = 5 and an oblate
spheroidal halo with axis ratio ∼ 0.25 − 0.3 (and the same
asymptotic circular velocity as the baryonic dMOND disk)
would produce very similar vertical velocity dispersion and
rotational velocity in the galactic plane (excluding the very
central regions).
For all these reasons, it appears difficult to differenti-
ate between MOND and DM using the currently available
observational measures of stellar kinematics of external disk
galaxies (e.g., Bottema 1993; Kregel & van der Kruit 2005,
and references therein). The most promising application of
our method seems to be the case of the Milky Way: in partic-
ular, for the specific baryonic mass distribution considered in
this paper the difference between the MOND and DM ver-
tical forces above the plane is larger than the uncertainty
on the observational estimates. The estimates of both the
baryonic mass distribution and the vertical force above the
plane are expected to be greatly improved in the near future
thanks to data from the GAIA mission.
ACKNOWLEDGMENTS
We thank Renzo Sancisi for helpful discussions, and the Ref-
eree, Robert Sanders, for his useful suggestions. L.C. and
P.L. were partially supported by a MIUR grant Cofin 2004.
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	Introduction
	The method
	Two-integral dynamics
	A preliminary analysis
	Results
	Miyamoto-Nagai disks
	Thick exponential disks
	Milky-Way like galaxies
	Discussion and conclusions
	REFERENCES

  We investigate the possibility of discriminating between Modified Newtonian
Dynamics (MOND) and Newtonian gravity with dark matter, by studying the
vertical dynamics of disk galaxies. We consider models with the same circular
velocity in the equatorial plane (purely baryonic disks in MOND and the same
disks in Newtonian gravity embedded in spherical dark matter haloes), and we
construct their intrinsic and projected kinematical fields by solving the Jeans
equations under the assumption of a two-integral distribution function. We
found that the vertical velocity dispersion of deep-MOND disks can be much
larger than in the equivalent spherical Newtonian models. However, in the more
realistic case of high-surface density disks this effect is significantly
reduced, casting doubts on the possibility of discriminating between MOND and
Newtonian gravity with dark matter by using current observations.

Introduction
	The method
	Two-integral dynamics
	A preliminary analysis
	Results
	Miyamoto-Nagai disks
	Thick exponential disks
	Milky-Way like galaxies
	Discussion and conclusions
	REFERENCES
