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Mass Distribution, Rotation Curves and Gravity Theories
Rahul Datta
Space Applications Centre, Indian Space Research Organization,
Ahmedabad-380015, India
Dilip G. Banhatti
School of Physics, Madurai-Kamaraj University, Madurai 625021, India
[Prepared for “Dark Matter Conference”, 9 - 11 Feb 2009, Arcetri, Florence, Italy.]
ABSTRACT
{Comparison of mass density profiles of galaxies of varying sizes based on some
gravity theories from observed galaxy rotation curves and the need for dark matter.}
We present an analysis of the rotation curves of five galaxies of varying galactic radii:
NGC6822 (4.8 kpc), Large Magellanic Cloud (9 kpc), The Milky Way (17 kpc),
NGC3198 (30 kpc) and UGC9133 (102.5 kpc). The mass and mass density profiles of
these galaxies have been computed using the scientific computing s/w package
MATLAB taking the already available velocity profiles of the galaxies as the input,
and without considering any Dark Matter contribution. We have plotted these profiles
after computing them according to three different theories of gravity (and dynamics):
Newtonian (black line), Modified Newtonian Dynamics (MoND) (green line) and
Vacuum Modified Gravity (red line). We also consider how the profile due to the
Newtonian theory would modify if we take into account a small negative value of the
Cosmological Constant (5 x 10-56 cm-2 from theory) (blue line). Comparing these mass
and mass density profiles, we try to form an idea regarding what could be a realistic
theory of gravity and whether we need Dark Matter to explain the results.
Keywords : disk galaxy rotation curves, galaxy mass, mass density profile, dark
matter, Newtonian theory, MoND, Vacuum Modified Gravity, negative cosmological
constant
MOTIVATION
“Missing mass” was first inferred via dynamics for clusters of mostly elliptical
galaxies using virial theorem. It was soon found to be dynamically present, although it
was the light that was missing, same as in clusters, on even single galaxy scale,
especially for spirals, most naturally taken to be balanced between rotation and
gravity. Assigning mass-to-light ratios on various scales via putative astrophysical
light emission processes on those scales, Dark Matter, which gravitated (and hence
participated in dynamics), but emitted little or no light, became firmly established in
astrophysics and cosmology.
In this paper, we consider the single galaxy scale, and use five galaxies, either disk
spirals or dynamically similar galaxies, to examine the connection between mass
distribution and rotation curve in a few representative models, without assuming any
Dark Matter, on scales from 5 kpc (size of the smallest of the five galaxies) to 100 kpc
(size of the largest of the 5).
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CALCULATION / COMPUTATION SCHEME
We first define the geometry and the notation for our calculation (computed using
MATLAB s/w package). The galaxy disk is assumed axisymmetric.
FIG.1: Top View of the Galaxy. The semicircular arcs represent
the rings constituting the disc of the galaxy. The other half
forming the full circular arc is implied.
FIG.2: Edge-on view of a fundamental segment of a ring in an axisymmetric disk
galaxy. Cf FIG.1.
Definitions & notation (referring to FIGs. 1 & 2):
pc = parsec, 3.08568E13 kms, 3.26151 light years
pi = 3.14159265
dr = radial thickness of a ring, pc
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dth = (pi /180) radians
th = angle at the galaxy center, between galaxy radial line to test
mass and radial line to ring segment, radians
alpha = angle subtended at the test mass by the center of the
fundamental segment mass and the center of the galaxy,
radians
beta = angle subtended at the test mass by the center of the
fundamental segment mass and the elemental thickness
dz of the fundamental segment mass, radians
pytks = multiplier to change v(pcs/yr) to v(kms/sec)
= 9.7778E5 km/pc*yr/sec
msun = mass of our sun, 1.989E33 gms
htype indicates type of the galaxy
r(i) = radius to centerline circle of ith ring
= rm - dr/2 pc
rr(i) = radius to rod used to represent fundamental segment mass
for the ith ring
= rm – (dr / 2) * (rm - 0.575 dr) / (rm - dr / 2) pc
Nr = number of rings
G = gravitational constant, 4.498E(-15) pc3/(msuns*yr2)
rm = radius to outer edge of a ring, pc
rmax = galaxy maximum/rim radius, pc
rt = radius to test mass from galaxy center, pcs
the test mass is located at the outer edge of each ring
h(i) = galaxy thickness at radius r corresponding to ith ring, pc
mr(i) = mass contained only inside the ith ring, msuns
Mr(i) = galaxy mass contained within the outer edge of the ith
ring, i.e., sum of masses inside each ring starting from
the galaxy center upto the ith ring, msuns
M = galaxy total mass, msuns
dv(i) = volume of a fundamental segment of the ith ring, pc3
= h(i)*r(i)*dth*dr
rv(i) = volume of the ith ring, pc3
= 360*dv(i)
tvol = galaxy total volume, pc3
rho(i) = density of the ith ring, msuns/pc3
rhoav = average density of galaxy
= M / tvol, msuns/pc3
SMD(i) = surface mass density of ith ring, msuns / pc2
SMDav = average surface mass density of galaxy
= M / (π ∗ rmax2 ) msuns / pc2
rSMD(i) = rd(i) * SMD(i)
v(i) = computed orbital speed at test mass radius rt at the outer
edge of ith ring, kms/sec
vkrim = sqrt(GM / rmax) , the Kepler speed at the rim, kms/sec
vm = measured speed, km/sec
vmax = maximum measured speed, km/sec
mrnor(i) = mr(i) / M
Mrnor(i) = Mr(i) / M
hd(i), rd(i) = (h(i), r(i)) / rmax
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rhonor(i) = rho(i) / rhoav
SMDnor(i) = SMD(i) / SMDav
rSMDnor(i) = rd(i) * SMDnor(i)
vnor(i) = v(i) / vkrim
FORWARD PART: Computation of the rotation profile from a given mass
distribution
Referring to FIG.2,
Acceleration due to one fundamental segment mass of a ring:
From Newton’s law,
Centripetal acceleration = GM/r2
Here, for a fundamental segment, rho is a constant.
Hence, M/r2 is equivalent to rho ∫(S dz/b2) cos(beta), taking the
component of the acceleration due to the mass of the elemental thickness, dz of the
fundamental segment of a ring along the direction of the plane of the galactic disc,
where S = cross-sectional area of the fundamental segment cut along the plane of the
galaxy disc.
Therefore,
Acc = G (rho) ∫(S dz/b2) cos(beta)
= G (rho) ∫(S dz/b2) (c/b)
But, z = c tan(beta) and b = c sec(beta)
or, dz = c sec2(beta) d(beta)
Therefore,
Acc = G (rho) S ∫(c sec2(beta) d(beta)/b2) (c/b)
= G (rho) S ∫(c2 sec2(beta)/b3) d(beta)
= G (rho) S ∫(c2 sec2(beta)/c3 sec3(beta)) d(beta)
= G (rho) (S/c) ∫d(beta)/sec(beta)
= G (rho) (S/c) ∫cos(beta) d(beta)
z=+h/2
= G (rho) (S/c) [sin(beta)]
z=-h/2
z=+h/2
= G (rho) (S/c) [z/(c2+z2)0.5]
z=-h/2
= G (rho) (S/c) [(h/2)/(c2+(h/2)2)0.5 + (h/2)/(c2+(h/2)2)0.5]
= G (rho) (S/c) [h/(c2+(h/2)2)0.5]
Referring to FIG.1,
Each ring consists of 360 such fundamental segments of mass and angle th varies
from 0 to π radians or 0 to 180 degrees, both in the clockwise and the anticlockwise
direction from the line joining the galaxy center to the test mass. The centers of the
fundamental segments correspond to 0.5 degrees, 1.5 degrees, 2.5 degrees and so on
until 179.5 degrees both in the clockwise and anticlockwise direction.
Therefore the centripetal acceleration of the test mass due to each ring:
th= π
v2/rt = 2 (pytks)2 ∫ G (rho) (S/c) [h/(c2+(h/2)2)0.5] cos(alpha) dth
th=0
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the cos(alpha) term being multiplied for taking the component of the acceleration of
the test mass due to each fundamental segment of the ring along the radial direction
towards the galactic center and the term 2 being multiplied to account for both the
clockwise and anticlockwise directions.
cos(alpha) = (rt - rr cos(th))/c
Hence, the squared orbital velocity due to each ring:
th= π
v2 = 2 rt (pytks)2 ∫ G (rho) (S/c2) [h/(c2+(h/2)2)0.5] (rt – rr cos(th)) dth
th=0
Discretizing the above formula by converting the integral to summation for the
purpose of computation, we have
i=180
v2 = 2 rt (pytks)2 ∑ G (rho) (S/c2) [h/(c2+(h/2)2)0.5] (rt – rr cos((i-0.5)dth))
i=0
where dth = (pi/180) radians and
rr = (j dr) - (dr/2) {(j dr)-(0.575 dr)/(j dr) - (0.5 dr)}
dr = radial thickness of each ring
S = r(j) dth dr
Summing up over all the rings, we have
j=Nr i=180
v2 = rt (pytks)2 ∑ 2 ∑ G (rho) r(j) dth dr/c2)[h(j)/(c2+(h(j)/2)2)0.5][rt - rr cos((i-0.5)dth)]
j=0 i=0
REVERSE PART: Finding the mass distribution for a given rotation profile
The reverse part is done by repeated applications of the forward part using the
equivalent thickness distribution, an arbitrary density distribution is input, and the
speed profile computed using the above formula. The speed errors (measured minus
computed) at each of the ring outer radii are then used to correct the densities of the
rings and the process repeated until all speed errors are small enough to be neglected.
errv = (vm - v)/vmax ; f = 0.75*errv
if all errv < 1E-6 then quit
else
rho(i) = (1+f) rho(i-1) for each cycle i
where vm = measured speed, and vmax = maximum computed speed
To compute the mass and mass density profiles after taking into account the
contribution of a Cosmological Constant, λ taken from theory to be equal to 5 x 10-56
cm-2 in magnitude can be found out by using a new velocity profile as input, which is
obtained after subtracting a quantity equal to c*r(i)*sqrt(λ/3) from the observed
velocity profile (astro-ph/9911485 November 1999 Steven B Whitehouse and George
V. Kraniotis). Note that we have considered a negative Cosmological Constant
scenario.
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To compute the mass and mass density profiles as per the Modified Newtonian
Dynamics (MoND), we use the following equation (astro-ph/0506370v4 J. R.
Brownstein and J. W. Moffat) for the FORWARD Algorithm and the REVERSE
Algorithm, which incorporates the results of the Forward Algorithm, remains
otherwise the same as earlier.
vMoND
2
= v2 (1 + sqrt[(M0/Mr1(j)) (1 - exp(-rt/r0) (1+rt/r0))])
j=Nr i=180
vMoND
2
= rt (pytks)2 ∑ 2 ∑ G (rho) r(j) dth dr/c2)[h(j)/(c2+(h(j)/2)2)0.5][rt - rr cos((i-
j=0 i=0
0.5)dth)] (1 + sqrt[(M0/Mr1(j)) (1 - exp(-rt/r0) (1+rt/r0))])
where
M0 = 9.6e+11 msuns for rmax>12000pc
= 2.4e+11 msuns for rmax=<12000pc
r0 = 13920 pc for rmax>12000pc
= 6960 pc for rmax=<12000pc
Mr1(j) = the quantity of mass contained within the radius at
which the velocity is being computed.
To compute the mass and mass density profiles as per Vacuum Modified Gravity
(VMG), we use the following equation (astro-ph/0712.1110v1 R. Van Nieuwenhove)
for the FORWARD Algorithm and the REVERSE Algorithm, which incorporates the
results of the Forward Algorithm, remains otherwise the same as earlier.
j=Nr i=180
v2 = rt (pytks)2{ ∑ 2 ∑ (G rho r(j) dth dr/c2) [h(j)/(c2+(h(j)/2)2)0.5] [rt - rr cos((i-
j=0 i=0
j=Nr i=180
0.5)dth)](cos(rt/rc1(x))+sin(rt/rc1(x))) – ∑ 2 ∑ (G rho r(j) dth dr) [atan(4 c h(j)/(4 c2-
j=0 i=0
h(j)2))/c][rt – rr cos((i-0.5)dth)] (cos(rt/rc1(j))-sin(rt/rc1(j)))/rc1(j) }
where
rhov1(j)=(1/3)*(vel)4/4/3.14159265/Mr1(j)/G/beta
rc1(j)=(vel)2*sqrt(1/(3*4*3.14159265*rhov1(j)*G))
Mr1(j) = the quantity of mass contained within the radius at
which the velocity is being computed.
beta=1.22*1.22*(1.989E+30)/(3.08568E+16)2 pc/sqrt(msuns)
vel=3/9.7778 pc/year
RESULTS
Newtonian (black line)
Modified Newtonian Dynamics (green line)
Vacuum Modified Gravity (red line)
Negative Cosmological Constant (blue line)
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NGC 6822 (r_max = 4.8 kpc)
20
Rings
4800 pc
Total
Volume
(pc3 )
Total Mass
(solar mass,
msuns)
Average
Density
(msuns/pc3 )
Average
Surface
Mass
Density
(msuns/pc2)
Keplerian
Velocity at
Galaxy
Rim
(Kms/sec)
Newtonian
Dyynamics
7.6436
e+009
1.0032
E+009
0.1312 13.8593 29.9796
Newtonian
Dynamics with
Negative
Cosmological
constant
7.6436
e+009
9.8781
E+008
0.1292
13.6472
29.7493
MoND 1.1309
e+010
3.2500
E+008
0.0287 4.4901 17.0641
Vacuum
Modified
Gravity
1.1309
e+010
9.0237
E+008
0.0798
12.4667
28.4335
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Large Magellanic Cloud (r_max = 9 kpc)
20
Rings
9000 pc
Total
Volume
(pc3 )
Total Mass
(solar mass,
msuns)
Average
Density
(msuns/pc3 )
Average
Surface
Mass
Density
(msuns/pc2)
Keplerian
Velocity at
Galaxy
Rim
(Kms/sec)
Newtonian
Dyynamics
8.0439
e+010
5.3772
E+009
0.0668 21.1309 50.6891
Newtonian
Dynamics with
Negative
Cosmological
constant
8.0439
e+010
5.2885
E+009
0.0657
20.7824
50.2694
MoND 8.0439
e+010
1.9989
E+009
0.0248 7.8551 30.9052
Vacuum
Modified
Gravity
8.0439
e+010
3.5654
E+009
0.0443
14.0113
41.2758
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The Milky Way (r_max = 17 kpc)
20
Rings
17000 pc
Total
Volume
(pc3 )
Total Mass
(solar mass,
msuns)
Average
Density
(msuns/pc3 )
Average
Surface
Mass
Density
(msuns/pc2)
Keplerian
Velocity at
Galaxy
Rim
(Kms/sec)
Newtonian
Dyynamics
5.4211
e+011
1.4955
E+011
0.2759 164.7222 194.5070
Newtonian
Dynamics with
Negative
Cosmological
constant
5.4211
e+011
1.4842
E+011
0.2738
163.4686
193.7654
MoND 5.4211
e+011
1.0694
E+011
0.1973 117.7889 164.4793
Vacuum
Modified
Gravity
5.4211
e+011
1.1782
E+011
0.2173
129.7735
172.6443
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NGC 3198 (r_max = 30 kpc)
20
Rings
30000 pc
Total
Volume
(pc3 )
Total Mass
(solar mass,
msuns)
Average
Density
(msuns/pc3 )
Average
Surface
Mass
Density
(msuns/pc2)
Keplerian
Velocity at
Galaxy
Rim
(Kms/sec)
Newtonian
Dyynamics
2.9792
e+012
8.7450
E+010
0.0294 30.9290 111.9639
Newtonian
Dynamics with
Negative
Cosmological
constant
2.9792
e+012
8.5459
E+010
0.0287
30.2250
110.6824
MoND 2.9792
e+012
4.1890
E+010
0.0141 14.8154 77.4911
Vacuum
Modified
Gravity
2.9792
e+012
5.4961
e+010
0.0184
19.4384
88.7617
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UGC 9133 (r_max = 102.5 kpc)
20
Rings
102500 pc
Total
Volume
(pc3 )
Total Mass
(solar mass,
msuns)
Average
Density
(msuns/pc3 )
Average
Surface
Mass
Density
(msuns/pc2)
Keplerian
Velocity at
Galaxy
Rim
(Kms/sec)
Newtonian
Dyynamics
7.4429
e+013
8.2719
e+011
0.0111 25.0616 186.2949
Newtonian
Dynamics with
Negative
Cosmological
constant
7.4429
e+013
7.9198
e+011
0.0106
23.9949
182.2871
MoND 7.4429
e+013
4.1810
e+011
0.0056 12.6673 132.4460
Vacuum
Modified
Gravity
7.4429
e+013
5.5632
e+011
0.0075
16.8548
152.7774
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CONCLUSIONS
We can observe from the figures & tables, infer and conclude as follows:
1. As we move from smaller to larger galaxies, there is an increasing similarity
between the mass and mass density profiles predicted as per different theories.
For small galaxies, NGC6822 and LMC, there is a significant variation
between what the different theories predict. As we move to the Milky Way
and NGC3198, we see much more similar looking curves and for UGC9133,
the curves appear strikingly similar.
2. The well-studied luminosity profiles and mass-to-light ratios of galaxies
suggest that there is an exponential decrease in the mass of visible matter as
we move away from the centre of the galaxy. But, considering that ordinary
matter in the outer reaches of a galaxy should be much cooler and may escape
detection and the matter near the galactic centre should be very hot, we can
probably explain why we see a very sharp fall in the mass and mass densities
near the galactic centre and an almost constant profile as we move further
outwards.
3. The mass and mass density profiles show an unexpected rising trend towards
the outer edges of the galaxies and this trend is more prominent for smaller
galaxies and the extent of the rise decreases as we move towards larger
galaxies and becomes almost unnoticeable for very large galaxies.
4. For the larger galaxies, all the existing and proposed theories of gravity show
nearly identical predictions, which appear intuitively satisfying considering
what we already know regarding structures of galaxies. But for smaller
galaxies, the predictions from different theories appear to vary significantly
and also the results appear to contradict our expectations.
5. For example, MoND seems to give predictions that are very close to what is
expected for galaxies of every size that we have examined whereas the well
established Newtonian Dynamics gives bizarre predictions for small galaxies,
though as we move to larger galaxies the predictions are closer to our
expectations. Similar is the situation with Vacuum Modified Gravity.
6. Hence, what we may conclude is that there is a need to study smaller galaxies
extensively in order to arrive at a conclusion regarding whether at all we need
dark matter and whether MoND can be considered as a valid theory of gravity.
7. From the trend that is observed it appears that either MoND is correct or else
there is a need to invoke Dark Matter along with Newtonian Dynamics in
order to explain the observations. And this Dark Matter, if there, seems to be
concentrated near the outer edges of the galaxies with its gravitational effect
on visible matter increasing as we move towards the galactic rim. And also the
effect of Dark Matter compared to ordinary matter is more pronounced for
smaller galaxies than the larger ones and is almost negligible for the very large
ones.
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REFERENCES
1. astro-ph/0006140
Disk-Galaxy Density Distribution from Orbital Speeds using Newton's Law
Kenneth F. Nicholson
2. astro-ph/0006330 23 Jun 2000
Disk-Galaxy Density Distribution from Orbital Speeds using Newton's Law
Version 1.1
Kenneth F. Nicholson
3. astro-ph/0101401 24 Jan 2001
Galaxy Mass Distributions for Some Extreme Orbital-Speed Profiles
Kenneth F. Nicholson
4. astro-ph/0309762 v2, revised 1 Mar 2007
Galactic mass distribution without dark matter or modified Newtonian
mechanics
Kenneth F Nicholson
5. astro-ph/0506370v4
Galaxy Rotation Curves Without Non-Baryonic Dark Matter
J. R. Brownstein and J. W. Moffat
6. asto-ph/0806.1131
Newtonian mechanics & gravity fully model disk galaxy rotation curves
without dark matter
Dilip G. Banhatti
[For IAU Symposium 254 on Galaxy Disk in Cosmological Context, 9-14 June 2008,
Copenhagen, Denmark]
7. astro-ph/0703430v7 [25 Apr 2008 Current Science 94 (8) 960+986-95]
Disk galaxy rotation curves and dark matter distribution
Dilip G. Banhatti
8. astro-ph/0712.1110v1
Vacuum Modified Gravity as an explanation for flat galaxy rotation curves
R. Van Nieuwenhove
9. RHCPP99-20T astro-ph/9911485 November 1999
A possible explanation of Galactic Velocity Rotation Curves in terms of a
Cosmological Constant
Steven B Whitehouse and George V. Kraniotis
