Disk galaxy rotation curves and dark matter distribution

CURRENT SCIENCE, VOL. 94, NO. 8, 25 APRIL 2008 960
In this issue

Antidiabetic activity of
Aloe vera

Ayesha Noor et al. have carried out
(page 1070) a study of the antidiabetic
activity of standardized Aloe vera
extract fed to streptozotocin-induced
diabetic rats, with histological evi-
dences to understand the mechanism.
In diabetic induced rats fed with Aloe
vera (300 mg/kg body weight) the
fasting plasma glucose (FPG) levels
were reduced to normal and body
weight was found to be increased. In
the histological sections of pancreas
of diabetic rats the islets were
shrunken and reduced. After feeding








with Aloe vera, the islets were com-
parable to normal rats. In liver, the
changes caused after induction of
diabetes are granular cytoplasm, di-
lated sinusoids, shrunken nuclei and
inflammation, which was reduced
after feeding with Aloe vera. Excess
proliferation of epithelium in small
intestine was observed in diabetic
rats, which was reduced after feeding
with Aloe vera to diabetic rats. No
significant changes were observed
with regard to kidney and stomach
sections. The administration of Aloe
vera extract lowers the FPG levels to
normal levels in diabetic induced rats
and has regenerating effect on the
pancreas, liver and small intestine of
diabetic induced rats.
Delineation of concealed
lineaments

In hard rock areas, the nature and ex-
tent of weathering vary considerably,
depending on the degree of fractures
at depth and the geomorphological
features. Often these fractured zones
are found to be good groundwater
potential zones. Hence, in view of
groundwater exploration, identifica-
tion and analysis of fractures and
concealed lineaments are crucial in
hard rock terrains. Conventionally,
apparent resistivity values are used to
detect such underground structures.
The true resistivity values are also
used and intern resistivity contour
map for different depths are prepared
from Electrical Resistivity Imaging
(ERI) data indicating different resis-
tivity zones, which could be used to
delineate low resistivity zones repre-
senting concealed lineament. One
such map is obtained where the con-
cealed lineament lies in NE–SW
direction at the depth of 12.8 m and
extended up to 30 m at Ghattupal vil-
lage, Wailapalli watershed of Andhra
Pradesh, India. There is no surface
signature of this buried lineament.
This underground lineament with
favourable hydrogeological condi-
tions allows large-scale migration of
groundwater in granitic terrain. The
existence of water-bearing concealed
lineament and the tiny fractures
located below this lineament are con-
firmed by the EC-logs. It shows that
the lineament is extended up to 30 m
depth and there are also tiny frac-
tures at 42 and 49 m depths. There-
fore, this lineament can be used as
suitable sites for the drilling new
boreholes for sustainable water sup-
ply. See page 1023.

Spiral galaxies and dark matter

Initially called ‘missing mass’ in the
1930s when Fritz Zwicky thought
there was insufficient mass to provide
the gravity needed in galaxy clusters
to balance the measured speeds, and
renamed ‘missing light’ four decades
later, when it was realized that the
clusters are relaxed structures most
likely in or near virial equilibrium,
so that it was light that was not ob-
served while the gravity was felt, the
presence of ‘dark matter’ was soon
inferred on all scales from individual
galaxies to groups to clusters. In the
solar system, Neptune was found at
the spot predicted from anomalous
motion of Uranus, while the anoma-
lous part of the precession of Mer-
cury did not lead to Vulcan, rather it
was explained by general relativity,
Einstein’s improvement over Newton
of the description of gravity. Rotation
about the centre of a spiral galaxy, in
analogy with planets moving around
the Sun, has been used to get the dis-
tribution of mass inside the orbits of
test particles. Assuming that the sums
are done right, the variation of circu-
lar speed with galactocentric dis-
tance, called the ‘rotation curve’,
gives the distribution of mass to pro-
vide the gravity needed to hold the
test particles in orbits. On page 986,
Banhatti presents the current status
of these observations and calculations
for disk galaxy rotation curves, and
the inferred dark matter, for the Milky
Way as well as other spiral galaxies.

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CURRENT SCIENCE, VOL. 94, NO. 8, 25 APRIL 2008 986
*Based on a pedagogic/didactic seminar given at the Graduate College
‘High Energy and Particle Astrophysics’ at Karlsruhe, Germany on 20
January 2006.
Dilip G. Banhatti is in the School of Physics, Madurai Kamaraj Univer-
sity, Madurai 625 021, India.
Present address: Solid State Theory group at University of Münster,
Germany. e-mail: dilip.g.banhatti@gmail.com
Disk galaxy rotation curves and dark matter
distribution*

Dilip G. Banhatti

After explaining the motivation for this article, I briefly recapitulate the methods used to determine,
somewhat coarsely, the rotation curves of our Milky Way Galaxy and other spiral galaxies, especially
in their outer parts, and the results of applying these methods. Recent observations and models of
the very inner central parts of galaxian rotation curves are only briefly described. I then present
the essential Newtonian theory of (disk) galaxy rotation curves. The next two sections present two
numerical simulation schemes and brief results. Application of modified Newtonian dynamics to the
outer parts of disk galaxies is then described. Finally, attempts to apply Einsteinian general relativity
to the dynamics are summarized. The article ends with a summary and prospects for further work in
this area.

Keywords: Dark matter, Milky Way, rotation curves, spiral galaxies.

Motivation
Extensive radio observations determined the detailed rota-
tion curve of our Milky Way Galaxy as well as other (spi-
ral) disk galaxies to be flat, much beyond their extent as
seen in the optical band. Assuming a balance between the
gravitational and centrifugal forces within Newtonian
mechanics, the orbital speed V is expected to fall with the
galactocentric distance r as V2 = GM/r, beyond the physical
extent of the galaxy of mass M, G being the gravitational
constant. The run of V against r, for distances less than
the physical extent, then leads to the distribution M(r) of
mass within radius r. The observation V ≈ constant for large
enough r, up to the largest r, up to 100 kpc, thus shows
that there is substantial amount of matter beyond even
this largest distance. A spherically symmetric matter den-
sity ρ(r) ∝ 1/r2, characteristic of an isothermal ideal gas
sphere, naturally leads to V = constant. Since this matter
does not emit radiation, it is called dark matter. In gen-
eral, the existence of dark matter is, by astrophysical
definition, inferred solely from its gravitational effects.
(Astro)particle physicists1 hope to change this by directly
detecting dark-matter particles. Considering the evidence
for different types of dark matter on scales from our solar
system out to the observable universe2, the detailed struc-
ture on the sky of cosmic microwave background radia-
tion3, simulations for large-scale structure formation4, and
big-bang nucleosynthesis calculations5, such gravitation-
ally and otherwise normally interacting dark matter may
be made of very light normal particles, like (massive) neu-
trinos, which are relativistic (and hence called hot dark
matter) or much heavier (about a GeV/c2 or more) exotic
nonrelativistic particles (hence called cold dark matter)6.
By ‘exotic’ here I mean not any of the particle zoo of the
standard model of particle physics6. For an exposition re-
garding our current picture of the composition of the uni-
verse and other details, see e.g. Grupen1 and references
therein. Here, we confine our attention to the scale of indi-
vidual galaxies.
One often assumes an isothermal dark matter halo,
although ρ (r) ∝ 1/r2 is only one of the many density pro-
files leading to V = constant, the others being disk-like. For
disk-like mass distributions, in contrast to spherically
symmetric ones, the circular speed at a given r is deter-




Figure 1. A schematic picture of Sun’s location in the Milky Way
Galaxy, illustrating galactic coordinates b (latitude) and l (longitude).
Rotation is indicated by an arrow.

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CURRENT SCIENCE, VOL. 94, NO. 8, 25 APRIL 2008 987


Figure 2. a, Different types of clouds in a given direction. b, Neutral hydrogen line profile showing typical regions as
numbered in (a).




Figure 3. Resolving distance ambiguity by comparing widths in Galactic latitude.




Figure 4. Milky Way rotation curve in the outer parts.


mined by matter distributed from 0 to r and also beyond r,
as can be easily seen by applying Gauss’ integral theorem
(or law) to appropriately shaped closed volumes. Some
textbooks make the error of integrating only up to r, leading
to wrong results, as pointed out by Méra et al.7.
Recently, de Boer et al.8 reanalysed the public domain
data from the 0.1 to 10 GeV all-sky γ-ray survey, which
was done by the satellite-borne telescope EGRET. They
found that excess diffuse γ rays of the same spectrum are
observed in all the sky directions, and that the spectral
shape can be interpreted as annihilation of (dark matter)
particles (and antiparticles) into intermediaries like π0
mesons and then to γ-ray photons, implying a mass of 60–
70 MeV/c2 for the annihilating (dark matter weakly inter-
acting massive) particle (or antiparticle). From the inten-
sity variation of the diffuse γ-ray excess with respect to
Galactic longitude and latitude (Figure 1), and assuming
a spherically symmetric component in the mass distribu-
tion, they derived an almost spherical isothermal profile
plus substructure in the Galactic Plane in the form of two
toroidal rings at 4.2 and 14 kpc from the Galactic Centre.
(The absolute normalization of the dark-matter profile is
tied to the local rotation speed 220 km/s at 8.3 kpc, the
Sun’s distance from the Galactic Centre. This may need
renormalization for consistency with better and more recent
determinations, e.g. like those of Xu et al.9 and Hachi-
suka et al.10.) The two rings produce, within Newtonian
dynamics, the observed bumps in the detailed shape of
the Galactic rotation curve. These rings are actually bro-
ken segments disposed around the Galactic Centre in ring-
like structures, as also seen for OB stellar associations via
their distribution and kinematics11. In general, nonaxi-
symmetric structures like spiral arms and bars should also

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CURRENT SCIENCE, VOL. 94, NO. 8, 25 APRIL 2008 988
be taken into account12. Dekel13 cautions to choose care-
fully test particles to measure rotation curves in general,
giving an example where low stellar speeds turn out to be
a red herring, detailed disks’ merger model for the elliptical
galaxies in question showing orbits consistent with dark
matter halo and low stellar speeds. Aharonian et al.14 de-
scribe the discovery of TeV γ-rays from the Galactic Cen-
tre Ridge. Ando15 treats cosmic γ-rays as being from dark
matter annihilation. In this article, I restrict myself to
outer parts of galaxies, i.e. to sufficiently large r, where the
rotation curve has stabilized to a flat shape on average.
Measurements in the Milky Way Galaxy16,17
In the kinematic method of determining distances in the
Galactic Plane in a given direction (Figure 2), the radial
speeds of neutral hydrogen (i.e. H I ≡ H0) clouds are meas-
ured by the Doppler shifts of the λ21 cm line correspond-
ing to the electronic spin-flip transition of the H atom.
The highest radial speed in the line profile in Galactic
longitude direction A gives the circular rotation speed V at
distance r = rSun sin A from the Galactic Centre. The dis-
tance ambiguity for clouds of types 1 and 3 is removed
(or resolved) by measuring their extent in Galactic lati-
tude (Figure 3). Another way is to measure hydrogen ab-
sorption at radio frequencies, since distant sources show
wider velocity range18. Such measurements give the Milky
Way rotation curve in its outer parts, up to about the Solar
Circle, as shown in Figure 4. Relative to the Sun (i.e. a frame
of reference rotating with angular speed Ω(rSun)), the cir-
cular speed of a gas cloud at radius r is r[Ω(r) – Ω(rSun)]
(Figure 5). The line of sight speed V|| at a given Galactic
latitude A is V|| = r[Ω(r) – Ω(rSun)]sin(θ + A) = rSun[Ω(r) –
Ω(rSun)]sin A.
Thus V(r) = rΩ(r) = Vmax + rSunΩ(rSun)sin A, where Vmax
is the maximum value of V|| from the line profile. Beyond




Figure 5. Geometry to calculate kinematic distances in the Galactic
Plane.
Solar Circle r = rSun, giant H II ≡ H+ complexes are used
in place of H I (≡ H0) regions. The tracers for these com-
plexes are the CO line (at λ ≈ 1–3 mm) and H109α (at
λ 6 cm), among others. Figure 6 shows the Milky Way ro-
tation curve beyond the Solar Circle, with the ‘expected’
curves for uniform (i.e. rigid body) rotation in the inner
part and Keplerian behaviour in the outer part. Coarse ob-
servations, when interpolated across the nucleus, corre-
spond roughly to uniform or rigid-body rotation in the
inner part, although the recently discovered finer-scale
structure there has other implications19. Briefly, molecu-
lar and maser line spectroscopy of massive inner nuclear
disks has shown the presence of order of magnitude larger
Keplerian speeds than the outer ‘flat’ value, indicating a
possible supermassive nuclear black hole. Bar kinematics
has been inferred from detailed observations. Even counter-
rotating nuclear disks, possibly resulting from mergers,
are present. Perhaps the nuclear black hole has ‘sucked in’
these peculiarities of structure and kinematics to leave
more orderly (flat) rotational kinematics in the outer parts
of (spiral) galaxies. The reader can refer to the literature
for recent observational details20. In the outer part, the ro-
tation speed is clearly super-Keplerian, on average flat,
with some modulations, in this plot (Figure 6) from
Shu16. Binney and Tremaine17 summarize the detailed
disk surface mass density model of Caldwell and Os-
triker21, which is essentially a fit with 13 observationally
determined parameters:

σ (R) = σ0[exp(–R/R1) – exp(–R/R2)].

The central model density is zero, which cannot accom-
modate a central mass concentration, for which there is
ample independent evidence19. However, the model has
toroidal rings at R1 and R2 for which there is separate ob-
servational evidence8,11 as mentioned earlier. So a super-



Figure 6. Milky Way rotation curve beyond the Solar Circle. Uni-
form rotation means rigid-body rotation v = ω ⋅ r, with uniform angular
speed ω, independent of r.

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CURRENT SCIENCE, VOL. 94, NO. 8, 25 APRIL 2008 989
position of such a model with another component having
centrally concentrated mass should fit the observed rota-
tion curve for inner as well as outer parts of the Milky
Way Galaxy. For other detailed models of (especially the
outer parts of) the Milky Way rotation curve, see Cowsik
et al.22 and Dehnen and Binney23.
Rotation curves of other galaxies
Assuming Newtonian gravitation to predominantly deter-
mine the dynamics, the mass M may be estimated from
orbital speed V in ellipticals as well as spirals. The mass
M interior to r is roughly M(r) = rV2/G, with r the depro-
jected distance and V the (spread in) random speeds for
ellipticals, while in spirals it refers to the circular speed
about the galaxian centre. For a disk galaxy it is more
meaningful to take M(R) as the mass within a cylinder of
radius R, while for a spheroidal galaxy M(r) is more con-
veniently the mass within a sphere of radius r. Observa-
tionally, measurements are possible only along the total
line-of-sight through a galaxy (Figure 7). Thus the cylin-
drical radius is more pertinent for observations. However,
the practical difference between the two is not large for
actual galaxies, as is seen from the following example.
Take ρ(r) = C/r2. Then the surface mass density is

σ(R) = ∫all z dz ρ(r), where r2 = R2 + z2 (Figure 7)
= π C/R.

Hence the masses within a sphere of radius r and a cylinder
of radius R are

Msph(r) = 2
0
d 4 ( ) 4
r
r r r Crπ ρ π=∫ and

Mcyl(R) = 2
0
d 2 ( ) = 2 .
R
R R R CRπ σ π∫




Figure 7. Line-of-sight through a galaxy showing the relation bet-
ween spherical radius r and cylindrical radius R.
For r = R, Msph/Mcyl = 2/π ~ 1. This is only relevant for
observations. Dynamically, even for disk geometry, the
relevant quantity is Msph(r), as one can see by applying
Gauss’ law to appropriately shaped closed volumes, as
emphasized earlier.
In general, (radio) line observations give data on the
intensity (and possibly polarization) of the targetted emit-
ting matter in very many velocity (as implied by shifted
frequency) channels in each pixel ≡ restoring beam. These
data can be viewed and plotted in many different ways.
The two most popular presentations give (1) a (polarized
or total) intensity contour map with colour-coded iso-
velocity contours superposed (Figure 8; called spider dia-
gram), and (2) a cut through such a map showing a plot of
velocity vs distance along the cut (Figure 9; called posi-
tion–velocity or PV plot).
The velocity is always corrected for projection using
an estimate of the inclination angle of the disk to the line-
of-sight. There are other ways of getting the rotation
curve, defined as such a position–velocity plot19. A neutral
hydrogen (i.e. H0) study24 of NGC 6744 out to 40 kpc is
shown in Figure 10. UGC2885, NGC 5533 and NGC6674
rotation curves are measured25 to ~70 kpc.
In the radio band, rotation curves can be measured out
to about 2–3 Holmberg radii27, while in the optical band
it is possible to measure only to about 0.5 times the
Holmberg radius. The Holmberg radius corresponds to
the isophote at a specific well-defined low surface bright-
ness, about 1–2% above the background sky brightness17.
Other scales used to gauge the extent of disk galaxies are
exponential disk scale length rd from I-band photometry
and the radius Ropt encompassing 83% of the total inte-
grated light, fruitfully used by Catinella et al.28, who con-
structed template rotation curves by combining data on
about 2200 disk galaxies, fitting for the amplitude V0, expo-
nential scale rpe of the inner region and slope α of the
outer part:

Vpe(r) = V0[1 – exp(–r/rpe)](1 + αr/rpe),

pe, representing ‘polyex’, the name of the model, and r
and rpe expressed in units of rd or Ropt. Detailed mapping



Figure 8. Schematic representation of iso-velocity contours super-
posed on intensity contour map (spider diagram).

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CURRENT SCIENCE, VOL. 94, NO. 8, 25 APRIL 2008 990
of radial distribution of visible and dark matter in disk
galaxies requires use of luminosity profiles and extended
H0 rotation curves in addition to optical rotation curves.
Martín27 has collated H0 maps from the literature published
between 1953 and 1995, into a uniform catalogue of about
1400 disk galaxies, and has analysed some of the data in
an attempt to derive features and relations common to
most galaxies. It is sufficient to use angular distances
(arcminutes or arcseconds) for the radial distances r, rpe,
rd and Ropt for such compilations. A translation to physi-
cal units (kpc) is needed only for going to mass models.
From a study of a homogeneous sample of about 1100
optical and radio rotation curves and relative surface pho-
tometry, Persic et al.29 found that a single property like
total luminosity dictates the rotation velocity at any ra-
dius for any galaxy, revealing the existence of a universal
rotation curve, which they derived, confirming their re-
sult from an earlier study based on 58 rotation curves.
However, Bosma30 cautions that derivation of any univer-
sal rotation curve may not be warranted. See also recent
evidence for two halo components in a galaxy, one of
them retrograde20. I also mention Narayan and Jog31 in
this connection, who show how the convenience of pre-
senting intensity vs radius as a log–linear plot led to a
spurious observational cut-off to the luminous disk,
promptly taken up by theorists to construct elaborate






Figure 9. a, Schematic galaxy rotation curve (PV plot), such as may
be derived from a spider diagram like in Figure 8 by integrating or tak-
ing a one-dimensional section. b, An observed galaxy rotation curve26
for UGC4779 = NGC2742.
models! Different ways of presenting astrophysical data
are essential for many purposes, as, for example, done by
Kundt32, who presented rotation curves in a novel way.
(Newtonian) theory of (disk) galaxy rotation
curves
Binney and Tremaine17 have given a detailed treatment of
observations, models and theory of galaxian dynamics
within Newtonian gravitational and dynamical frame-
work. Saslaw33 has treated gravitational systems in gen-
eral from a much broader perspective, and, in the process,
given a concise summary of the essential Newtonian
methodology. The reader can refer to these (and other
possibly more recent) studies for details. Here only a few
glimpses are given, hopefully enough to whet the appetite
for more.
Relation between Φ (r) and ρ (r) [or V2(r) and M(r)]
The gravitational potential energy per unit mass is called
Newtonian gravitational potentials Φ. This is only a con-
venient mathematical quantity and has no physical exis-
tence in Newtonian theory, which is a simultaneous far-
action theory rather than (special) relativistically propa-
gating field theory like (classical) electromagnetism. (See
Graneau and Graneau34 for an elaboration of this point.
See also Banhatti and Banhatti35.)
For a test particle in a circular orbit at radius r in a
spherically symmetric mass distribution ρ (r), the circular
speed V(r) is found from
V2(r) = rdΦ/dr = rF = GM(r)/r = (4πG/r) 2
0
d ( ),
r
r r rρ′ ′ ′∫
where F is the (radial) force/unit mass, and M(r) the mass
within a sphere of radius r. The escape speed Ve =
(2|Φ(r)|)1/2, since the kinetic and potential energies are
just balanced at this speed.




Figure 10. Rotation curve of NGC6744 to large galactocentric dis-
tance. (Velocity in km/s vs deprojected distance in kpc; from Ryder
et al.24).

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CURRENT SCIENCE, VOL. 94, NO. 8, 25 APRIL 2008 991
Potential ↔ density pairs (and other quantities)
Spherically symmetric: For illustration, some simple po-
tentials are listed.
Point mass M: Φ(r) = –GM/r, V(r) = (GM/r)1/2 and
Ve(r) = (2GM/r)1/2.
Homogeneous sphere: M(r) = (4/3)πr3ρ, with ρ uni-
form (i.e. independent of r) and V(r) = (4π Gρ/3)1/2r, ris-
ing linearly with radius. The orbital period is T =
2πr/V = (3π/Gρ)1/2, independent of radius r. A test mass
released at r oscillates harmonically around r = 0, where
it reaches after tdyn = T/4 = (3π /16Gρ)1/2, the dynamical
time of a system of mean density ρ. For radial size a,


2 2
3
2 ( / 3),
( ) .
4 / 3 ,
G a r r a
r
G a r r a
π ρ
π ρ
⎡ ⎤− − ≤Φ = ⎢ ⎥− ≥⎢ ⎥⎣ ⎦


Other systems of interest are17 isochrone potential, modi-
fied Hubble profile and power-law density.
Flattened systems: Plummer–Kuzmin, Toomre’s n
and logarithmic. For details, see Binney and Tremaine17.
Poisson’s equation for thin disks: For an axisymmet-
ric system with density ρ (R, z),

∂2Φ/∂z2 = 4πGρ (R, z) + (1/R)(∂/∂R)(RFR); FR ≡ –∂Φ/∂R

being the radial force.
Near z = 0, the first term on the RHS p the second
term, so that ∂2Φ/∂z2 = 4πGρ (R, z).
So Poisson’s equation for a thin disk can be solved in
two steps: (1) Using surface density (zero thickness), de-
termine Φ(R, 0). (2) At each radius R, solve this simplified
Poisson’s equation for the structure normal to the disk.
Disk potentials: By separating variables in cylindrical
polar coordinates, surface density σ (R) and potential
Φ(R, z) are related by

Φ(R, z) = –2πG 0
0
d exp( | |) ( )k k z J kR
∞
−∫
× 0
0
d ( ) ( ),R R R J kRσ
∞
′ ′ ′ ′∫

where J0 is a Bessel function. Writing
S(k) = −2πG 0
0
d ( ) ( ),R R R J kRσ
∞
∫
the circular speed is given by
V2(R) = R(∂Φ/∂R)z=0 = –R 1
0
d ( ) ( ),k kS k J kR
∞
∫ using

dJ0(x)/dx = –J1(x).
Examples applying these formulae: Rotation curve of
Mestel’s36 disk: σ (R) = σ0R0/R.
Calculation gives uniform, i.e. R-independent, circular
speed: V2 = 2π Gσ0R0.
Since
M(R) = 2π
0
d ( ),
R
R R Rσ′ ′ ′∫
this can also be written V2 = GM(R)/R, as for a spherical
system, which is true only for Mestel’s disk.
Exponential disk: σ (R) = σ0exp(–R/Rd).
Calculation gives V2(R) = 4πGσ0Rdy2[I0(y)K0(y) –
I1(y)K1(y)], where y ≡ R/2Rd, and I0, K0, I1, K1 are Bessel
functions.
Deducing σ (R) given V(R) is formally possible, but in-
volves differentiating the noisy observed function V2(R),
which numerically worsens the error and is unstable.
Fourier (numerical) method in cylindrical polar
coordinates37
With u ∝ AnR, a rectangular grid in the (u, ϕ) plane,
where −∞ < u < ∞ and 0 ≤ ϕ (the azimuthal angle) ≤ 2π,
generates, in the (R, ϕ) plane, cells that become smaller
as R → 0. This is well suited to (numerical simulations
of) centrally concentrated disks. A particularly efficient
implementation is Miller’s code38 using ‘leapfrog’ nu-
merical scheme, called Verlet method in molecular dy-
namics, extensively used at Turku (Finland) and later39
for various aspects of disk galaxies, including spiral arms,
tidal interactions, Seyfert activity, etc. The N-body code
uses 60,000 particles in a smoothed potential. The parti-
cles are distributed on a 24 × 36 standard grid (part of
which is shown in Figure 11), determined by r = L exp(λu);
λ = 2π /36 and u ranging from 0 to 23.5, so that r ranges
from L to 60.4L.
The initial disk has

σ (R) =
2 1
0[ / ]cos ( / ) for ;
0 for
V GR R A R A
R A
π −⎡ ⎤≤⎢ ⎥≥⎣ ⎦





Figure 11. Two-dimensional polar coordinate grid used in Miller’s
numerical code38.

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CURRENT SCIENCE, VOL. 94, NO. 8, 25 APRIL 2008 992
where A ≡ disk radius. (See Grupen1, eq. (13.7), p. 268,
where one finds a concise summary of disk galaxy dyna-
mics vis-à-vis motivation for dark matter.)
Thus the circular speed V0 is, by integrating σ (R) to
get the total galaxy mass M, given by

V20 = π GM/2A. The potential/unit mass is

Φ(R) =
2
0
2 3 (2 1)
0
0
( /2 ) for
(2 / ) [1·3·5···(2 1) /{2 !(2 1) }]( / ) .
for
V n R A R A
V A R
R A
π ∞ +
=
⎡ ⎤≤⎢ ⎥⎢ ⎥− + +⎢ ⎥⎢ ⎥⎢ ⎥>⎣ ⎦
∑ A A
A
A
A A A

The action of a spherical halo of density ∝ 1/r2 is the
same as if it was projected on the disk plane, as can be
verified by calculation that the surface density of the pro-
jected halo has the same form as assumed. In fact, this
is the motivation for using this form. The halo acts on
the disk, but is not acted on, either by itself or by the
disk. This is somewhat puzzling and may give spurious
effects.
Disk halo break-up and a new calculation
scheme7
Poisson’s equation is numerically solved for an axially
symmetric disk in cylindrical polar coordinates, using
250,000 points = particles distributed along 500 rings of
radii ∝ i2, where i is the ring number from the centre
outward, up to Rg, the radius of the finite disk. The force
F ∝ ∫[σ(R)/R3]R dR acting on a given particle is discre-
tized to

Fi = 3( / ) ,i j ij ij
j i
G m m d
≠
∑ d where dij = xj − xi.
also
= mi(v2i /di)(xi/di) to give the rotation curve, i = 0, 1, . . . , n.

d2i j = d2i + d2j − 2didjcos(θij),
so that with
Fij = G(di – dj cos θij)/d3i j,

the set of equations reduces to ∑j≠i miFij = v2i /di. This is a
system of n linear equations, with n + 1 unknowns mi
(since we seek to invert V2 → σ). The additional equation
needed is provided by the total mass Mg of the galaxy:
Mg = ∑i mi. Writing μi = mi/Mg ≡ ω mi, the n + 1 equations
for n + 1 unknowns μi, for each value of ω, are (for
i = 1, . . . , n) (i = 0 gives 0 = 0):
2
0 ( ) 0
/ , 1,
n n
i ij i i j
j j i j
m F v dω μ
= ≠ =
= =∑ ∑
with the constraint μi ≥ 0.
The constraint restricts ω between ωmin and ωmax,
which are close to each other (within 10–2 or less). The
physical significance of the existence of the free parameter
ω is that the rotation curve is known only up to Rg, the
radius of the finite disk. The range allowed for ω corre-
sponds to all possible extensions of the rotation curve be-
yond Rg. Since the range of ω is narrow, the mass of the
galaxy from the known rotation curve is naturally found.
The method has been tested successfully for exponential
disk, point mass and Mestel’s disk. Figure 12 shows the
surface density σ(R) for the Milky Way derived in this
way from the rotation curve of Vallée40. Rg = 14 kpc is
used from Robin et al.41. Comparison with other results
and taking into account MACHO gravitational lensing
candidates toward LMC, a halo of the same radius as the
disk is needed.
However, as Méra et al.42 (and references therein)
show in detail using star counts, microlensing observa-
tions and kinematics, the model eventually to be found
consistent with all constraints is not yet determined,
although both maximal halo-type models with non-
baryonic (i.e. exotic) dark matter and maximal disk-type
models with all matter baryonic are possible.
Gentile et al.43 use H0 and Hα data on five spirals to
decompose the rotation curve into stellar, gaseous and dark
matter contributions, pointing toward halos with constant
density cores.
MoND and disk galaxy rotation curves44
MoND was proposed in 1983 as a phenomenological
model to fit gross features of rotation velocity data for
spirals, especially Tully–Fisher relation M(mass) ∝ V4, V
denoting the flat value of the outermost part of the rota-
tion curve. The idea behind MoND was to seek a low field
value a0 (≈10–8cm/s2, as it turns out), which modifies
Newtonian gravitational acceleration (or field) from gN to
(gNa0)1/2 for low gravitational fields. As it happens,
a0 ≈ cH0/6, with c the speed of light and H0 the Hubble
constant. Detailed variation of spiral rotation curves fits
well with only one additional parameter giving the mass-
to-light ratio across the whole spiral disk. Thus evidence
for dark matter from spiral rotation curves can equally
well be interpreted as evidence for MoND, which may at
worst be taken to be a parametrization of rotation curve
data, which a more fundamental model or theory should
account for. The search is still on for such a physical basis
(in the form of a field theory) for this acceleration (i.e.
gravitational field)-based modification of Newtonian dyna-
mics (and inertia). Recently45, a unification of dark matter
and dark energy into a dark fluid described by a tensor–

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CURRENT SCIENCE, VOL. 94, NO. 8, 25 APRIL 2008 993


Figure 12. Surface mass density for Milky Way Galaxy compared with the
model of Méra et al.7 (dotted). (The same figure is also available in Méra et
al.42 as figure 2.)


vector–scalar (TeVeS) field theory has consonance with
MoND. For details, see Sanders and McGaugh42 and Zhao45.
General relativity vs Newtonian gravity and
dynamics
Newtonian gravity and dynamics together form a simul-
taneous far-action theory. Newtonian gravitational poten-
tial (energy) is essentially a force field, giving the force/
unit mass in space. This is true despite application of
sophisticated mathematical techniques, as there is no
(special) relativistic field propagation in Newtonian the-
ory, which is not Lorentz-invariant, but Galilean, with
absolute simultaneity, and space and time independent of
each other. General relativity, on the other hand, is locally
Lorentz-invariant, and is a genuine field theory with rela-
tivistic field propagation built into its structure.
Galaxy rotation curves
A galaxy is modelled as a stationary axially symmetric
pressure-free fluid in general relativity. The rotation
curve is derived by tracing paths of test particles, i.e. de-
termining geodesics. The recent claim by Cooperstock
and Tieu46 that such a procedure leads to a good fit to ob-
served galaxy rotation curves, without having to assume
the existence of exotic dark matter, has been shown to be
misguided by Vogt and Letelier47, who infer that matter
of negative energy density is implied in the disk. Korzyń-
ski48 has shown that a singular disk is implied. Cross49
found that correcting an internal inconsistency no longer
leads to a flat rotation curve. Fuchs and Phelps50 have
pointed out the inadequacy of the model to reproduce
local mass density and vertical density profile of the
Milky Way. However, the idea of treating the nonlinear
galactodynamical problem using general relativity rather
than Newtonian gravity, due to the inherent nonlinearity
of self-gravity and (general relativistic) dragging of iner-
tial frames in the rotating model spacetimes, is well
worth pursuing further. An example is Vogt and Letel-
ier51. Another is Balasin and Grumiller52, who point out
that the Newtonian approximation breaks down globally,
even though it is valid locally everywhere, confirming
Fuchs and Phelps’ failed attempt50 to apply the model to
the Milky Way Galaxy. Letelier53 has summarized the
position, also referring to an earlier relevant study by
González and Letelier54. Finally, it is worth mentioning
Gödel’s study of rotating spacetimes, examined from a
broader perspective by Yourgrau55. See also Banhatti56.
Summary/conclusion
Galaxy rotation curves are flat to the greatest extent they
can be measured. Precise observations have delineated
the undulations in this overall flat structure, especially for
the Milky Way Galaxy. The low-field theory MoND phe-
nomenologically fits details of disk rotation curves sur-
prisingly well, with the same mass-to-light ratio across
the whole disk as the only galaxian parameter, along with
a constant low-field value, the constant acceleration of
about 10–8 cm/s2, intrinsic to MoND.
There is a direct relation between rotation curves and
mass models for galaxies. Dark matter is probably needed
to fit the observations. In particular, a break-up of the
mass distribution into a disk + a halo around the disk is
probably needed for both luminous and dark matter com-
ponents.

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CURRENT SCIENCE, VOL. 94, NO. 8, 25 APRIL 2008 994
In addition to analytic approaches, numerical calcula-
tions/simulations in 2D polar coordinates are fruitful in
relating mass models to observations.
It is not clear if general relativistic treatment, normally
needed for high gravitational fields, gives anything more
than Newtonian dynamics, but is worth exploring further.

Notes added in proof: (1) For an exposition on how a
rotation curve is derived from the observations of a disk
galaxy velocity field on the sky, deprojecting for disk in-
clination, see Teuben57. (2) Over the years there have
been attempts to derive radial disk mass distributions
from observed rotation curves. Two numerical schemes
are described in the article. Another semi-analytic attempt
is by Feng and Gallo58, who also refer to a series of nu-
merical calculations by K. F. Nicholson. All these inde-
pendent studies do not need dark matter. (3) For an
update on MoND after Sanders and McGaugh44, see
Bekenstein59. (4) Finally, a fundamental theory purport-
ing to show that dark matter, dark energy and the acceler-
ating universe are artefacts of our 3 + 1 dimensional
perspective on a pre-geometric structure based on ‘proc-
ess physics’ has been developed by Cahill60.


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ACKNOWLEDGEMENTS. I thank Prof. Dr Wim de Boer for invit-
ing me to Karlsruhe, Germany for a visit and a seminar. University of
Münster, Germany provided general facilities and use of library. Part of
the work was done while visiting Institute of Mathematical Sciences
(MatScience), Chennai, where I also gave a seminar on the topic. De-
partment of Theoretical Physics, University of Madras also arranged
my seminar on this topic. Nora Loiseau’s comments at various stages
improved clarity. Prof. Wolfgang Kundt, Bonn, Germany provided
valuable comments. Prof. Peter Boschan, Münster encouraged me by
discussing relevant matters and lending books when needed. I also
thank the referees for their valuable suggestions. Finally, I thank Dr
Radha D. Banhatti for encouragement and critical comments on various
versions of the article.

Received 9 March 2007; revised accepted 24 February 2008