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Constraining String Gauge Field by
Galaxy Rotation Curve and Planet
Perihelion Precession
Yeuk-Kwan Edna Cheung,aFeng Xua
aDepartment of Physics, Nanjing University,
22 Hankou Road, Nanjing, China
E-mail: cheung@nju.edu.cn,schyfeng@gmail.com
Abstract. We discuss a cosmological model in which the string gauge field coupled univer-
sally to matter gives rise to an extra centripetal force and will have effects on cosmological
and astronomical observations. Several tests are performed using data including galaxy ro-
tation curves of twenty-two spiral galaxies of varied luminosities and sizes, and perihelion
precessions of planets in the solar system.
Remarkable fit of galaxy rotation curves is achieved using the one-parameter string
model as contrasted to the three-parameter model of dark matter model with the Navarro-
Frenk-White profile. The rotation curves of the same group of galaxies are independently
fit using dark matter model with the generalized Navarro-Frenk-White profile and using the
string model. The average χ2of the NFW fit is 9% better than that of the string model at a
price of two more free parameters. From the string model we give a dynamical explanation
of Tully-Fisher relation. We are able to derive a relation between field strength, galaxy size
and luminosity, which can be verified with data of the 22 galaxies.
To test the hypothesis of the universal existence of string gauge field, the string model is
also used to explain the anomalous perihelion precession of planets in the solar system. The
required field strength is deduced from the data. The field distribution resembles a dipole
field originated from the Sun. A refined model assumes that the string gauge field is partly
contributed by the Sun and is partly contributed by the stellar matter in the Milky Way.
The field strength needed to explain the excess precession is of similar magnitudes as the
field strength needed to sustain the rotational speed of the sun inside the Milky Way.
Keywords: Testing GR at low frequencies, Dark Matter Model, String Cosmology
arXiv:1108.5459v1 [hep-th] 27 Aug 2011
Contents
Introduction 1
1 Galaxy rotation curve 2
1.1 The string model 2
1.2 The Dark Matter Model 3
1.3 Stellar matter 4
1.4 Fitting Procedures: 5
1.5 Analysis 5
1.5.1 Visible Mass to Light vs B-magnitude 6
1.5.2 Tully-Fisher relation 7
1.5.3 A relation obtained from the string model fitting result 9
1.6 Discussion 10
2 Perihelion precession 12
2.1 Field strength from precession in solar system 12
2.2 Profile fitting of precession in solar system 14
2.3 Field strength from Milky Way rotation curve 15
2.4 Discussion 16
3 Summary 17
A Solar system magnetic field and its effect on planet perihelion precession 18
B Different fitting models give different best fittings 18
C Detailed Fitting Results of 22 galaxies using dark matter model and using
string model 21
Introduction
Recent cosmological and astronomical observations are becoming increasingly interesting
laboratories for precision tests of new physical theories aiming at extending the standard
paradigms. On the one hand high energy completion of gravity theory should leave signatures
sufficiently different from the low energy effective theories the like of general relativity and
its modifications. These will be detectable with the advances in detector technologies. On
the other hand, string theory and other quantum gravity candidate theories when applied to
cosmology or astronomy, should shed new light on old problems.
In this work we will confine our interest to two problems. The first one is the “missing
mass problem” in galaxies–the mass discrepancy between the one measured by rotational
speeds of stars inside a spiral galaxy and the one predicted from its stellar matter distribution.
Another one is the anomalous precession of the planets inside the solar system recently
reported, the explanation of which cannot be found within the standard framework. The
model we propose to solve these two problems at the same time is a very special string
model. The model, first discovered by Nappi and Witten [1], falls into a general class of
– 1 –
exactly solvable string models but it has the added merit that all effects due to the finite
size of the strings are taken into account. In other words it is a bona fide string model.
And it lives in four dimension spacetime which resembles as closely as possible to Minkowski
spacetime but with the presence of the string gauge field. Due to the existence of the string
gauge field, the geodesics in this four dimensional space-time are concentric circles instead
of the usual straight lines in Minkowski spacetime. This is analogous to the Laudau orbits
of an electron in the presence of a magnetic field. Thus Cheung et al [2] proposed the model
to explain the galaxy rotation curves in spiral galaxies in lieu of Cold Dark Matter (CDM).
In this paper we are going to extend their work with a direct comparison of goodness of fit
of the string model with the CDM model in galaxy rotation curves fitting. Furthermore we
are going to apply the idea to explain the anomalous precession of planets’ perihelions in the
solar system.
1 Galaxy rotation curve
While the Cold Dark Matter cosmology has been accepted by many as the correct theory
for structure formation at large scale and the solution to the missing mass problem at the
galactic scale there still lacks hard proof for the existence of Dark Matter particles. Until
the coming Linear Hadron Collider and future experiments–see exciting development in this
direction [3,4]–tells us definitely what constitute Dark Matter a more natural and universal
explanation in lieu of dark matter cannot be excluded. Here we entertain the possibility that
a higher-rank gauge field universally coupled to strings can give rise to a Lorentz force in
four dimensions providing an extra centripetal acceleration for matter towards the center of
a galaxy in addition to the gravitational attraction due to stellar matter. If not properly
accounted for, it would appear as if there were extra invisible matter in a galaxy. A salient
feature of this Lorentz-like force is that it fits galaxies with an extended region of linearly
rising rotational velocity significantly better than the dark matter model. This feature also
endows the model with testability: in the region where the gravitational attraction of the
visible matter completely gives way to the linear rising Lorentz force, typically in the region
r∼20Rd, should we still observe linear rising rotation velocity, say for the satellites of the
host galaxy, it would be a strong support for the model. Otherwise if all rotation curves are
found to fall off beyond 20Rdfor all galaxies then the model is proven wrong. Furthermore
this is the first of such attempts to verify directly the validity of string theory as a description
of low energy physics. Given this last reason alone we regard it a worthwhile endeavour.
1.1 The string model
Consider a four-dimensional string model proposed by Nappi and Witten [1] in which the
string theory is exactly integrable. Furthermore tree-level correlation functions–describing an
arbitrary number of interacting particles–have been computed, which capture all finite-sized
effect of the strings to all orders in string scale [5]. This is valid for all energy scales as long
as the string coupling constant is weak. This is the region of interest when we extrapolate
to our low energy world. The three-form background gauge field, H(3), coupled uniquely to
the worldsheet of the strings, is constant.
Because we are no longer approximating strings as point particles, this coupling between
the two-form gauge potential B(2) and the two dimensional worldsheet of the string produces
a net force on the string when it is viewed as a point particle [5]. The center of mass of the
– 2 –
closed string executes Landau orbits:
a=a0+reiH t (1.1)
where ais the complex coordinate of the plane in which the time-like part of the three-form
field has non-zero components. This phenomenon is analogous to the behaviour of an electron
in a constant magnetic field.
For the purpose of data fitting we only need to retain a component of the tensor gauge
field which is perpendicular to the galactic plane of the spiral galaxies. We denote the
strength of this gauge field by H, which together with the “charge-mass” ratio, forms the
only free parameter of this model, denoted by Ω ≡Q
mH. This is to be contrasted with the
three free parameters one needs in the dark matter model using the celebrated Navarro-
Frenk-White profile [6] for an iso-thermal and isotropic dark matter distribution, rs(the
characteristic radius of the dark matter distribution), σ0(the central density), and α(the
steepness parameter). A general profile with a free parameter αis used because of the need
to fit rotation curves from dwarf galaxies as well as galaxies with a varied surface brightness,
from high surface brightness to low surface brightness.
The simplistic property of the string model is, in fact, a favoured approach from the
string theory point of view, because the coupling of the field to matter has universal strength,
i.e. all matter is charged, rather than neutral. Therefore if all matter is indeed made up
of fundamental strings and hence couples universally to the tensor gauge field, H, each star
in a galaxy will then execute the circular motion in concentric landau orbits on the galactic
plane. Effectively there is an additional Lorentz force term in the equation of motion for a
test star:
mv2
r=q H v +m F∗(1.2)
where the field, H, is generated by the rotating stellar matter and the halo of gases alike.
1.2 The Dark Matter Model
According to the Cold Dark Matter (CDM) paradigm there is approximately ten times more
dark matter than visible matter. The fluctuations of the primordial density perturbations
of the universe get amplified by gravitational instabilities. Hierarchical clustering models
furthermore predict that dark matter density traces the density of the universe at the time
of collapse and thus all dark matter halos have similar density. Baryons then fall into the
gravitational potential created by the clusters of dark matter particles, forming the visible
part of the galaxies. In a galaxy the dark matter exists in a spherical halo engulfing all of
the visible matter and extending much further beyond the stellar disc. To describe the dark
matter component we use the generalized Navarro-Frenk-White profile:
σ=σ0
(r
rs)α(1 + r
rs)3−α,(1.3)
where α= 1 corresponds to the NFW profile, and rsis the characteristic radius of the dark
matter halo. In the Dark Matter fitting routine we allow the rsto vary from 3Rdto 30Rd.
We further require that the dark matter density be strictly smaller than the visible mass
density, σ0< ρ0. (The data can in fact be fit equally well with the roles of dark matter and
visible matter inverted.) Here we treat σ0,rsand αas free parameters. Together with ρ0
– 3 –
and Rdfrom the visible component, the Dark Matter model utilizes five free parameters. All
in all the rotation velocity of a test star is given by
v2
r=F∗+FDM (1.4)
in the Dark Matter model.
1.3 Stellar matter
To describe the visible matter we use the parametric distribution with exponential fall off in
density due to van der Kruit and Searle in both models:
ρ(r, z) = ρ0e−r
Rdsech2(z
Zd
) (1.5)
with ρ0being the central matter density, Rdthe characteristic radius of the stellar disc and
Zdthe characteristic thickness. Following a common practice we choose Zdto be 1
6Rd; the
dependence of the final results on this choice is very weak [7].
Gravitational attraction due to the visible matter is henceforth given by
F∗(r) = GNρ0Rd˜
F(˜r) (1.6)
where after rescaling ˜r≡r
Rd
˜
F(˜r) = ∂
∂˜rZall space
e−r0sech2(6z0)
|~
˜r−~
r0|r0dr0dz0dφ0
becomes a universal function for all galaxies.
Summary of equations in both models: We are now ready to fit the galaxy rotation
curves data. In the string model the three free parameters, Ω, Rd, and ρ0are defined by the
following equation:
v2
r=GNρ0Rd˜
F+ 2 Ω v . (1.7)
The fundamental charge-to-mass ratio and the strength of the gauge field is encoded alto-
gether in one free parameter Ω ≡qH
2m.
The five free parameters in CDM model are defined by
v2
r=GNρ0Rd˜
F+FDM .(1.8)
The two free parameter ρ0and Rdare common in both models, describing stellar contribution
to the rotational velocities of stars about the center of galaxy. FD M (r) is given by the
following expression:
FDM (r)=4πZr
dr0σ0
(r0
rs)α(1 + r0
rs)3−α.
All in all the CDM profile carries another three free parameters, namely, the central density of
dark matter halo, σ0, the characteristic length scale of the halo, Rs, as well as the “steepness”
parameter of the halo, α.
– 4 –
1.4 Fitting Procedures:
A few remarks concerning our fitting procedures are in order. The Dark Matter model with
its five free parameters and the string model with its three free parameters are independently
fit to the data to obtain the best fit values for each set of the parameters for each of the
twenty-two spiral galaxies. Under no circumstances, the best fit values from one model are
fed into the other model as prior values. Except restricting the rsto vary from 3Rdto 30Rd
in CDM model and letting Zdto be 1
6Rdin the stellar distribution as conventionally done
(see for example [7]) to save computing time there are no other simplifications. We then use
the best fit values for these two parameters (and three others in the dark matter model) to
compute the total mass, as well as the mass-to-light ratios, for these galaxies. These will
serve as sanity check for the best fit values of the parameters in both models.
Independent of any galaxy rotation modelling, ρand Rdcan be fit with photometric
data and hence are not really free parameters. Since we are only interested in comparing the
dark matter model and our string model in fitting the galaxy rotation curves we are treating
them as free parameters in each model. We instead choose to use our best fit values for these
parameters, from Dark Matter model and String model in turn, to compute the total mass in
each galaxy and cross check with independent astronomical observations. We also compute
the percentages of baryonic matter in the galaxies for the CDM model as usually done by
astronomers. This serves as another check of our methodology.
We are ignoring the gas contributions from our fitting; because putting in more free
parameters will no doubt improve the fit for both models. For the same reason we do not
allow for any correction for star extinctions and supernova feedback as they would not affect
any conclusion we draw concerning the relative quality of the fit between the CDM model and
string model. Keeping this simplistic spirit we do not allow for any dark matter component
at all in the string model and we also assume that the strength of the string field be constant
throughout the span of each galaxy. Local back reaction of spacetime to the presence of the
string field is also ignored.
1.5 Analysis
The rotation velocity of a given test star is solved from equations (1.7) and (1.8) for the string
and the dark matter model, respectively. The data are then fit to the dark matter model
and string model independently. The best fit values of the parameters from each model are
obtained by minimizing the χ2functionals. 1We obtained our rotation curve data of the
twenty-two galaxies in the SINGS sample from the FaNTOmM website. Using the best fit
values of these parameter of the dark matter model, we can compute the mass of the dark
matter halo, the mass of stellar mass, and then the ratio of luminosity to the total mass.
These values are tabulated in Table Cof Appendix C. From the string model we can likewise
determine the set of values for the strength of the string gauge field, the mass of stellar mass
and then finally the ratio of luminosity to the stellar mass. These values are tabulated in
Table 9in Appendix C.
1Notice that in the observations the distance measurement from the center of the galaxy is assumed to be
exactly. The uncertainty is attributed, instead, to the velocity measurements. During fitting, however, we
discovered that uncertainty in determine the “center of the galaxy” affects the quality of the fit significantly.
According to both models the rotation velocity at the “center” of the galaxy should be exactly zero. If we
could shift some data by a linear translation to make the zero velocity point coincide with the r= 0 point
by hand, we would have obtained much lower χ2values for both models. Therefore this linear shift is better
attributed to the error in distance determination.
– 5 –
0 1 2 3 4 5 6
0
25
50
75
100
125
150
175
0 1 2 3 4 5 6
0
25
50
75
100
125
150
175
Figure 1. Rotation curve of NGC2403 fit with the dark matter model (left) and with the string
model (right). The χ-squared value per degree of freedom using the dark matter model with a NFW
profile is 4.515 while that using the string model is 4.304. The X-axis is radius in kpc and Y-axis is
velocity in kms−1.
In Figure 1rotation curve of galaxy NGC2403 fitted using NFW profile (left) and
string model (right) are plotted side by side for comparison. Squares with error bars are
observational data. Theoretical predictions are indicated by the solid lines with stars in the
NFW fit (left) and with triangles in the string fit (right). The string model clearly gives a
better fit.
The χ-squared value, per degree of freedom, from the string fit is 4.304 whereas that
from the NFW fit is 4.515. Overall string model gives a χ-squared value of 1.656 averaged
over the 22 galaxies while the dark matter model gives 1.594. The fitting results of 22 galaxies
using dark matter model and string model are detailed in Appendix C. The best fit values
of the free parameters are tabulated in Table Cfor Dark Matter model and in Table 9for
String model, respectively, in Appendix C. We can see that the NFW profile fits marginally
better at a price of two more free parameters.
After we obtain the best fit values for the free parameters we can compute the (total)
masses for the galaxies.
String Model: For this model there is only visible matter whose mass can be straightfor-
wardly computed by integrating (1.5) with the best fit values of Rdand ρ0for each galaxy.
NFW profile: Matter in this model consists of the visible matter, same as that in the
string model, and the dark matter which assumes the generalized NFW density profile (1.3).
The NFW profile gives divergent mass if the radius is integrated to infinity. We therefore
adopt the usual cutoff and compute the mass only up to the virial radius within which the
average density is 200 times the critical density for closure.
1.5.1 Visible Mass to Light vs B-magnitude
Using the measured B-band absolute magnitudes we compute the visible mass to light ratios
for the galaxies. In string model these ratios fall between 0.11 ∼5.6 and centred around 1
as shown in Fig. 3. The same ratios from the NFW model span five orders of magnitude
– 6 –
Figure 2. The total-mass-to-light ratios derived from the best fit values and the measured B-band
luminosity of the 22 galaxies in dark matter model.
Figure 3. The visible-mass-to-light ratios derived from the best fit values and the measured B-band
luminosity of the 22 galaxies in the string model.
(see Fig. 2) with a lot of them falling much below 1. For the NFW model we also compute
the percentage of baryonic matter in the total mass. According to the CDM paradigm this
number should be around 10%. However the actual results come with a wild scatter. The
scatter in the mass-to-light ratios and the baryon fractions clearly indicate that NFW profile
is not capturing the underlying physics correctly.
1.5.2 Tully-Fisher relation
A Tully-Fisher relation can be derived from the string model which relates the rotation
velocity in the “flat” region of the rotation curves to the product of the total luminous mass,
– 7 –
M∗, and the parameter, Ω ,
v3=GM∗Ω.(1.9)
From the equation of motion (1.2) we solve for v,
v= Ω r+pΩ2r2+F∗r . (1.10)
We then look for a balance of falling Newtonian attraction and rising Lorentz force,
resulting in ∂ v
∂r ∼0. Because we know that the turning point is at r∼2.2rd, setting ∂ v
∂r ∼0
yields a relation between rdand Ω:
8 Ω2∼GM∗
r3.(1.11)
Inside the orbit r∼2.2Rdlies most of the visible mass. We can therefore use the point-
mass approximation when computing F∗and ∂F∗
∂r . Upon substituting (1.11) into (1.10) our
Tully-Fisher relation follows. The string model therefore provides a dynamic origin of this
well-tested rule of thumb.
Figure 4. The luminous mass and velocity relation of the 22 galaxies fit by the Tully-Fisher relation
derived from the string model.
We plot our best fit values of G M∗Ω against vin Fig. 1.5.2. The representative velocity,
v, is selected to be the maximal observed velocity in the entire curve for each galaxy, to
eliminate man-made bias. This no doubt introduces more scatter than necessary. Despite
that the data obey the relation remarkably well.
Note that this is a nontrivial relation for it relates two parameters from two additive
force terms to an observed quantity, the rotational speed. Furthermore if one can determine
the luminous mass of the galaxy, M∗and the field strength, H, we can determine a funda-
mental ratio, the charge-to-mass ratio, universal for all matter according to string theory, by
measuring the rotating speed of a test star. Furthermore our model provides a dynamical
explanation to the Tully-Fisher relation.
– 8 –
1.5.3 A relation obtained from the string model fitting result
By dimensional analysis, together with some common results from astronomy, we can find
a simple relationship between the field strength Ω, galaxy luminosity and size. This serves
as a consistency check for the string model. According to the string model, considering its
analogy with electromagnetism, it is reasonable to expect the average field strength to be
proportional to MαR−βwhere Mis the total luminous matter in the galaxy and Ris the size
scale of the galaxy. 2To be more specific we will heavily use the electromagnetism analogy in
the following discussion. Consider a group of electrons azimuthal symmetrically distributed
and in rotation around the zaxis. Let us look at the magnetic field at the center of this
distribution, the determining physical quantities are: the magnetic constant µ0, mass density
scale ρ0, distribution size scale R0and rotational angular velocity scale ω0.3
(Other determining factors include the shape and the spatial dependence of the mass
distribution, the spacial distribution of the angular velocity. These factors do not change
the result of dimensional analysis but change the proportional constant.) By dimensional
analysis, we have
B∝µ0·ρ0·ω0·R2
0(1.12)
Using the total charge Q∝ρ0·R3
0, and defining v0=ω0·R0, we have
B∝µ0·Q·R−2
0·v0(1.13)
Note that v0is the rotational velocity scale for the galaxy. Translating to the language of
the string model, it is
Ω∝Q·R−2
0·v0(1.14)
The proportional constant here depends only on the mass and angular velocity distributions,
or abstractly on the galaxy type. 4Thus galaxies with similar mass distribution profile and
rotation curve shapes should have similar constants of proportionality. Now recall M∝Q,
where the proportional constant is universal and thus the same for all galaxies. Furthermore
we also use the assumption L∝M.5Thus
Ω∝L·R−2
0·v0(1.15)
To relate v0to Lwe use the Tully-Fisher relation which says L∝∆Vαwhere αis around
2.5±0.3 and ∆Vis the velocity width of the galaxy[8]. The proportional constant in the
Tully-Fisher relation is galaxy type independent. Since v0is the overall scale for ∆V, we
have also L∝(v0)α. Using this in (1.15), we get
Ω·R2
0∝L1+ 1
α(1.16)
2By the Tully-Fisher relation [8] velocity is related to the total mass of the galaxy, therefore we do not
need a separate term for the velocity dependence.
3What we really want to check is the averaged field over the galaxy, but it is proportional to the field
strength at the center.
4More precisely, it is not just the galaxy morphology type. The velocity distribution also matters. It
is possible galaxies of the same morphology type but very different velocity distributions will have different
proportional constants.
5This relation is independent of the galaxy type. For more discussion about the mass luminosity-relation
among galaxies of different types, see [9].
– 9 –
108
109
1010
1011
L
0.1
1
10
h R2
Figure 5. luminosity to ΩR2log-log plot
which after taking logarithm
ln ΩR2
0=1 + 1
αln (L) + ln κ(1.17)
where κis the proportional constant in the relation (1.16). The log-log diagram is shown in
Fig.5.
There seems to be a trend of a linear relation in the “main” part of the diagram. The
slope of the line is about 3
2, quite close to the 1 + 1
2.5±0.3derived theoretically. Two points
seems to lie outside of the “main” part, i.e one at the lower left corner for the dwarf galaxy
m81dwb, another one for NGC4236 at the left of the upper right group. In terms of galaxy
type these two galaxies are “exotic” among the 22, and thus perhaps their ln κdeviated more
from those in the “main” part. Actually in terms of morphology type, NGC4236 is SBdm,
which is the most irregular one among the regular types, while m81dwb is the only dwarf
galaxy among the 22 galaxies [10], all others are more or less regular galaxies. Presumably
these two lie on another line for irregular galaxies which is parallel to the line passing the rest.
It is possible that there are series of parallel lines for different types of galaxies. However,
statistical error from the small size of this data set makes above arguments weak. Analysis
of this kind for a larger number and more types of galaxies could make the situation clearer.
But this exercise is beyond the scope of the current project.
1.6 Discussion
The original appeal of the NFW profile based on the ideas of hierarchical clustering was its
universality. One simple NFW profile was expected to explain structure formation, rotation
curves of galaxies, giant or dwarf, from high to low surface brightness. This promise has
been undermined by the cusp and core debate in dwarf galaxies as well as in the low surface
brightness galaxies (see for example [7]). The fact that light does not follow dark matter–well
established by detailed observation and analysis in the Milky Way (see [11] for a summary),
in addition to a clear deficit of satellite galaxies in MW have only served to thicken the
– 10 –
plot. Recently this debate has been taken to broader context by observational progress: The
simplicity of the galaxies [12] and the early assemble of the most massive galaxies [13] are
at odd with the hierarchical clustering paradigm. While we are not claiming that our string
toy model can answer all these questions in one stroke we merely show that it pans out just
as well as the Dark Matter model in fitting the galaxy rotation curves while using two fewer
parameters. Moreover by tuning the ratio of the strength of the string field to stellar mass
density galaxies with a wide range of surface brightness and sizes can be accommodated.
We have one dwarf and several LSB galaxies in our sample. At the same time the model,
based as it is on a tractable physical principle consistent with laws of mechanics and special
relativity, does not suffer from the arbitrariness and puzzling inconsistencies of MOND.
In order to describe a universal galaxy rotation curve [14,15] one at most needs three
parameters–to specify the initial slope, where it bends, and the final slope. Any more pa-
rameter is redundant. In this regard the string model utilizes just the right number of them.
The fact that it fits well on par with the dark matter model which employs two extra param-
eters should be taken seriously. However one should guard against reading too much into the
game of fitting. For example, one cannot obtain a unique decomposition of the mass compo-
nents of a galaxy using the rotation curve data alone, a difficulty having been encountered
in the context of comparing different dark matter halo profiles. Acceptable fits (defined as
χ<χmin + 1 [16]) can be gotten with dark matter alone without any stellar matter in CDM
model. And the roles of dark matter and stellar matter can be completely reverted in the
fitting routine. The physical difference is, on the other hand, dramatic. This degeneracy
is less severe in the string model in the sense that string field cannot be completely traded
off in favour of stellar matter, or vice versa. However there still exists a range of values for
Rd, ρ0and H, where “acceptable” fits can be obtained. Therefore given the quality of the
presently available data rotation curve fitting alone cannot distinguish between dark matter
and string field in galaxies.
However precision measurements extended to radii r∼20Rdcan distinguish string
model from the other models: a gently rising rotation curve in this region is a signature
prediction of this string toy model. At this moment we are, nevertheless, encouraged by this
inchoate results to pursue further. In a separate article we shall subject our string model
to other reality checks, and we shall report on how this simple string model accounts for
gravitational lensing which is often quoted as another strong evidence for the existence of
dark matter at intergalactic scales.
At this point it is worth mentioning that a critical reanalysis of available data on velocity
dispersion of F-dwarfs and K-giants in the solar neighborhood, with more plausible models,
performed by Kuijken and Gilmore concluded that the data provided no robust evidence for
the existence of any missing mass associated with the galactic disc in the neighborhood of the
Sun [17]. Instead a local volume density of ρ0= 0.10Msunpc−3is favored, which agrees with
the value obtained by star counting. Dark matter would have to exist outside the galactic
disk in the form of a gigantic halo. Their pioneer work was later corroborated by [18–21]
using other sets of A-star, F-star and G-giant data. Note that this observation can be nicely
explained by our model as the field only affects the centripetal motion on the galactic plane
but does not affect the motion perpendicular to the galactic plane.
We presented a simple string toy model with only one free parameter and we showed
that it can fit the galaxy rotation curves equally well as the dark matter model with the
generalized NFW profile. The latter employs two more free parameters compared with the
string model. The string model respects all known principles of physics and can be derived
– 11 –
from first principle using string theory, which in turn unifies gravity with other interactions.
Our model has an unambiguous prediction concerning the rotation dynamics of satellites and
stars far away from the center of the (host) galaxy. The ability to test the validity of string
theory as a description of low energy physics makes the exercise worthwhile.
2 Perihelion precession
In this section we test the string model with planetary precession data in the solar system.
In [22], anomalous precessions for Mars and Saturn was reported, for which no explanation
within the the standard paradigm seems to exist. It is thus an interesting laboratory to
test our string model and ask if the string gauge field can explain the reported anomalous
precession. Furthermore, this also serves as a check of the string model independent of
the galaxy rotation curves. It is because we can use the data of anomalous precession to
determine the profile as well as the strength of the string gauge field inside the solar system.
As it turns out, the extra field needed to generate the extra centripetal force to account for
the anomalous precession has a profile of a dipole field generated by the Sun. It is then
important to compare the magnitudes of field strength as obtained by different methods and
observations. A consistent model should give similar values in the field strength for the same
object.
2.1 Field strength from precession in solar system
Here we will use the anomalous precession data to determine the string field profile and field
strength in the solar system. We attribute all the anomalous precession to the magnetic like
force due to the string field. We are interested in the quantity [2]
Ω = QH
m(2.1)
where Q
mand Hare the string charge-to-mass ratio and the field strength, respectively. 6
Ω has dimension, s−1, that of frequency. In other words we are testing the validity of
Newtonian gravity in the extremely low frequency regime. The corresponding quantity in
electromagnetism is eB
m. In these units it is easy for us to compare it with strengths of other
magnetic-like forces, e.g eB
m, in electromagnetism. Precession from a magnetic-like force
perturbation has been worked out in detail from first principle in [23]
δ˙ω=−qB
m1
√GM
π a3
2
T!.(2.2)
To calculate the strength of Ω using precession data we replace qB
mwith Ω = QH
min the above
formula (2.2) and invert it to get
Ω=(−δ˙ω)pGM
T
πa 3
2
(2.3)
Relative errors from other contributions are extremely small compared to the one of the
precession rate, thus the error of Ω can be computed by
Err(Ω) = Err(δ˙ω)pGM
T
πa 3
2
,(2.4)
6Note that here we dropped the 1
2factor in Ω as defined when discussing GRC fitting as constants of order
one are not important.
– 12 –
2. ´108
4. ´108
6. ´108
8. ´108
1. ´109
1.2 ´109
1.4 ´109
r
-0.5
0.5
1.0
1.5
2.0
2.5
W
Figure 6. Ω(1 ×10−17s−1) versus distance r(km) to the Sun
or simply
Err(Ω)
Ω
=
Err(δ˙ω)
δ˙ω
.(2.5)
The value of Ω determined from precession data are shown in the following table, Table 1. And
the fitting results are shown in Fig. 6. Orbital data used in fitting are from HORIZON [24].
Precession data are obtained from [22].
The central values of Ω for the inner planets exhibit a decreasing pattern with respect
to r, although big error bars allow also for the case of vanishing string field. 7At Saturn, Ω
is nonzero within one σ. However, as mentioned in [22], the error bar at Saturn may actually
be bigger than quoted, in which case the value of precession, or Ω, may vanish. More precise
measurements on precession are needed to determine definitely the existence of Ω (in other
words, anomalous precession rate) in the solar system. Inspired by this decreasing pattern in
the section 2.2 we will fit data of inner planets with a (nearly) power law profile. No matter
how critically we take the profile and magnitude of Ω here, nevertheless it is certain that the
upper limit of Ω at the solar system is on the order of 10−17s−1.
As a comparison, let us note that for the real magnetic field near earth, B∼10−9T esla,
and with e
m= 1.76 ×1011C/kg, we have Ωem ∼100s−1. (See Appendix Afor details.) In
that sense, the string field strength is, naively, 1019 times smaller than the strength of the
magnetic field near the earth, assuming the charge to mass ratio under the string field is
the same as that of an electron. We may wonder why this “strong” magnetic field has not
affected precession of planets, specifically we can ask if it is related to the observed anomalous
precession of planets. One reason why we do not have to worry about the real magnetic field is
that planets are electrically neutral, but charged under the string gauge field as postulated.
The real magnetic field can act on neutral matter only through dipole-dipole interaction,
which, as explained in the Appendix A, for several reasons, does not contribute to planet
precession.
7The same argument applies to precession rate as well, which is proportional to Ω.
– 13 –
Name Mercury Venus Earth Mars Saturn
δ˙ω(10−400 /cy)−36 ±50 −4±5−2±4 1 ±5−60 ±20
T(y) 0.240846 0.6151970 1.0000175 1.8808 29.4571
a(km) 57909100 108942109 152097071 249209300 1513325783
e 0.205630 0.0068 0.016710219 0.093315 0.055723219
Ω(10−17s−1) 1.11 1.22 ×10−15.99 ×10−2−2.69 ×10−21.68794
Table 1. : Planets data and values of Ω
There is however another question we may ask about the string model: now we are
assuming matter is charged under string field and being acted on by the corresponding
magnetic part for this charge, then is there an electrical part of interaction between stringly
charged matter? After all, so far we seem to be assuming all matter take the “same” kind
of charge. This question lies outside of our current model and calls for more theoretical
investigations into the nature of this string charge. As for the model used here, we are
considering a magnetic interaction in the form of a Lorentz force. 8
A few words on the Saturn are warranted. One aspect special about Saturn is that it
belongs to the gas giant group while all other planets considered here are small and solid
planets of the inner planet family of the solar system. As in the electromagnetic theory, the
content and structures of planets may affect their interactions with the string field, which
might explains the somewhat anomalous behavior of the Saturn. This speculation could be
supported if anomalous precession behaviors of other gas giants similar to the Saturn can be
measured in the future.
2.2 Profile fitting of precession in solar system
The decreasing pattern of Ω for inner planets with respect to rindicates it might be useful
to fit these values with a power law term. Presumably we could attribute this rdependent
term to the Sun from which ris measured. On the other hand, note that Ω at the Mars
is negative, although it seems also sitting on the same curve passing the first three inner
planets.
One possible configuration for Ω compatible with these 2 facts is then that, in addition
to a power law term, there is also a weak constant background Ω with opposite direction to
the Ω from the Sun. The constant background might be provided by all other matter in the
universe. The Milky Way should be the most important source of this influence. Thus here
we try to fit field strength at different inner planets with the following profile,
Ω(r) = A+Br−a(2.6)
For the actual fitting on computer, the profile used is
Ω(r)=Ω0+ Ω1r
107km −a(2.7)
8It is amusing to find out the extra acceleration due to the string field for objects on Earth. Since the field
strength is gotten for the rest frame relative to the Sun, the velocity of objects on the earth should be nearly
the same with the velocity of Earth relative to Sun, i.e 30km/s. The corresponding acceleration produced is
therefore ∼10−13m·s−2.
– 14 –
Planet Mercury Venus Earth Mars
ratio 50.3 7.1 2.5 0.54
Table 2. relative strength: power law term/constant term
The best fitting parameters for this profile is
Ω0=−0.0223787 ×10−17s−1(2.8)
Ω1= 260.504 ×10−17s−1
a= 3.09953
It is helpful to know the relative strength of the two components from the Sun and the
constant background, which is shown in Table 2. The power law term decreases with r
quickly relative to the background. We can safely say that in most areas in the solar system,
the string field would just be around that background, which is on the order of 10−19Hz.
And that ais found to be near 3, which is exactly the power for a dipole field. It means that
the string field interaction between the sun and planets is similar to that between a magnetic
dipole and charged particles.
Also note that the fitting result tells us that the constant background in the solar system
is negative. This is good news for the string model. As in electromagnetism, we expect the
string gauge field in a galaxy to be generated by the rotation of (stringly charged) matter in
the galaxy, just like rotating electric charge would generate a magnetic field. Since the Sun
and planets in the solar system rotate in the opposite direction of that of stars’ rotation in
the Milky Way, it is therefore reasonable to expect the background field to be negative if we
consider the one from the Sun as positive. This is exactly what the profile fitting has told
us! Here only data of the solar system was used, but the conclusion is for the whole Milky
Way, specifically for its rotation direction.
However, this conclusion should be taken with a grain of salt, as we will explain below.
Firstly, the long error bars of precession weaken the conclusion from this profile fitting.
Secondly, there exists possibilities for the field from the Sun to be in the same direction with
that of the Milky way. As in ordinary electromagnetic theory, for a right handed current disk
the magnetic field is downward outside of the major current distribution, but upward if we go
into the current disk somewhere, specifically at the center of the disk. There is a place within
the concentration where the magnetic field changes its sign. 9Thus even if the Milky Way is
rotating in opposite direction from the solar system, if the Sun is too close inside the major
mass concentration of the Milky Way, the background therefore should still be positive. To
our advantage, it is known that the Sun contains 99% of the total mass in the solar system,
so it is reasonable to assume all planets are well outside of most mass in the solar system.
And the solar system lies somewhat outside the majority of mass concentration of the Milky
Way.
2.3 Field strength from Milky Way rotation curve
In section 2.1 we obtained Ω in the solar system from anomalous precession. The constant
background value, found by the profile fitting in section 2.2, should mainly comes from other
matter in the Milky Way altogether. On the other hand, by the same idea used in [25], we
9Considering this, the Ω in [2,25] are position-averaged one over the galaxy.
– 15 –
can also estimate Ω at the Solar system due into the Milky way’s rotation. Thus a natural
check of the string model is to compare these two values of Ω: one of the constant background
from precession in the solar system, another one from the rotation curve of the Milky Way.
Directly using the idea in [25], we can make a rough estimate of Ω in the Milky way
as follows. In the string model, the total force on the galaxy mass is composed of only the
gravitational attraction from visible mass in the Milky Way and the magnetic-like force from
the string field. With increasing r, the gravitational force decreases quickly, and the magnetic
like force always increases. At the position of the Sun the rotation curve is well into the flat
region [26–28]. Therefore on the Sun the gravitational force should be negligible with respect
to the magnetic-like force from the string field. Then [2]
mv2
r≈QHv (2.9)
and thus
Ω≈v
r(2.10)
For the sun [26–28], v≈200km ·s−1and r≈7.6kpc. So
Ω≈26s−1·km
kpc = 8.5×10−16s−1(2.11)
This is the upper limit on Ω due to the Milky way at the position of the Sun. Since
there is still a portion of distance further out to nearly 20kpc where the curve is quite flat
(with velocity ∼200km ·s−1)[26–28], we could have used these distances instead of 7.6kpc
and the value of Ω will be reduced by a factor of about three. In any case, it is safe to say
the upper limit of Ω is on the order 10−16s−1. This is the strength of the field component
perpendicular to the galactic plane. To convert it to the solar system, notice that the north
galactic pole and the north ecliptic pole form an angle of 60.2◦(which means the field in
the solar system would be reduced almost by half), and the milky way rotates clockwise
when viewed from north galactic pole (which means the field is negative in the solar system
since planets in the solar system rotate counterclockwise if viewed from the same direction).
Therefore from the rotation curve of the milky way the upper limit of the effective field
strength in the solar system due to matter in the milky way is −10−17 ∼ − 10−16s−1.
This magnitude is close to the one found by direct precession calculation without the profile
assumption, but it is not the case for field direction. The precession indicates the field in
the solar system is positive, while rotation curve of the milky way says it is negative. One
possible explanation is that at places near the Sun the field is dominated by the positive field
generated by the Sun, and the Milky way provides only the constant background in the solar
system. The profile fitting of precession, apart from a dipole like part due to the Sun, indeed
gives a negative background (−0.02 ×10−17s−1). However the magnitude there is smaller by
almost two orders of magnitude.
2.4 Discussion
With anomalous precession data, we calculated the upper limit on the field strength at several
places in the solar system. It is found to be on the order of 10−17s−1. The distribution of
the field in the solar system looks like a superposition of a constant background and a r−3
decreasing “dipole” component in (2.6). We discussed a possible configuration that the Milky
– 16 –
way produced the background and the Sun sources the “dipole” component. A profile fitting
is done for this configuration. The background is found to be negative, which, when combined
with analysis using electromagnetism analogy, correctly matches with the fact that the solar
system and the Milky Way rotate in opposite directions (assuming the solar system is far
enough away enough from the Milky way center).
As a comparison with the field strength gotten from precession in the solar system, we
estimated the upper limit on the field strength of the string gauge field in the Milky way
directly from rotation curve of the Milky Way at the position of the Sun. This estimate is
of a similar order of magnitude with the one gotten from precession. But the direction is
opposite. If we consider the background plus dipole field configuration used for profile fitting
to be correct, then we should consider only the background value when comparing with the
one estimated by Milky way rotation curve. In that case, although the background direction
agrees with the one predicted by Milky Way rotation curve, but the magnitude is smaller by
2 orders of magnitude.
3 Summary
We discussed the possibility that the rotational speed of the stars about the center of a spiral
galaxy is supported by the presence of string gauge field which couples universally to all forms
of matter. We compared the good of fit of the string model with the commonly accepted
Cold Dark Matter model with the generalized Navarro-Frenk-White profile for Dark Matter
distribution. 10 We fit rotational speed data of 22 spiral galaxies of varied range in size
and luminosity. Dark Matter model fit marginally, 9%, better at the price of two more free
parameter than the string model. A Tully-Fisher relation relating the visible mass and field
strength of the string gauge field to velocity GMstarΩ = v3can be derived dynamically from
the string model, which is obeyed fairly well by all 22 galaxies of varied sizes and luminosities.
The existence of string gauge field is taken a step further and is applied to explain the
anomalous precession of planets in the solar system. The extra field needed to generate the
extra centripetal force to account for the anomalous precession has a profile of a dipole field
generated by the Sun.
The values of string fields in the following cases
•galaxy rotation curves of a set of 22 galaxies (without the Milky way),
•perihelion precession of some planets in the solar system and
•Milky way rotation curve
are summarized in Table 3. Interestingly, and also luckily for the string model, results from
these different methods lead to similar order of magnitude for Ω.
Acknowledgments
We would like to thank Yuran Chen and Youhua Xu for collaboration at the early stage of the
precession project. We also thank Konstantine Savvidy for helpful discussions related to the
rotation curve project. We have also benefited from discussions with Gang Chen, Zheyong
10The NFW profile with α= 1 is known to have difficulty fitting dwarf galaxies as well as galaxies with low
surface brightness. Hence the generalized profile which leaves αa free parameter is called for.
– 17 –
Solar System ⊂Milky Way 22 other galaxies
Rotation Curve |Ω|.10−16 6×10−18 .|Ω|.10−15
Precession |Ω|.10−17
Table 3. Dimension for Ω is s−1. The first column indicates the kind of observation used, and the
first row indicates the object on the which the observation is done.
Fan, Anke Knauf, Changhong Li, Congxing Qiu, Lingfei Wang, Tianheng Wang, Zhen Yuan
and Yun Zhang. This work is supported in parts by the National Science Foundation of China
under the Grant No. 10775067 as well as Research Links Programme of Swedish Research
Council under contract No. 348-2008-6049.
A Solar system magnetic field and its effect on planet perihelion precession
References for this appendix are [29–33]. The Sun and most planets in the solar system have
magnetic field due to dynamo effect. If we treat both the Sun and the planet as magnetic
dipoles interacting in vacuum (which leads to a central force with n=−4), using data of
magnetic fields of the Sun (around 1 ∼2 gauss at the polar region) and the Earth (around
0.6 gauss at the polar region), we can find the corresponding precession produced is nearly
10−6arcsec/cy, which is 2 orders of magnitude smaller than the observed one. In fact the
magnetic field in the solar system is much more complicated than those produced by several
dipoles in vacuum. First, the solar system is not empty but filled with particles emitted from
the Sun, i.e the solar wind. Charged particles lock with it the magnetic field of the Sun and
spread it all around in the solar system. From the Sun to about the position of the Earth,
the magnetic force line is parallel to the radial stream of solar wind particle and falls off
by r−2. From the position of the Earth to about position of the Mars is a field free region
with B < 10−6gauss. Further out to the position of the Jupiter is a region with disordered
magnetic field with B∼10−5gauss. For precession, the most important feature of the solar
system field is that it is oscillating. First, for the Sun the magnetic dipole axis is inclined
relative to the rotational axis. This leads to an oscillating neutral current sheet. Therefore
planets on the ecliptic plane is above the neutral current sheet for half of solar self rotation
period, below for another half. Since field directions above and below the neutral current
sheet is opposite, the magnetic force experienced by the planets also change directions within
one self rotation of the Sun. Second, the magnetic field of the Sun also changes direction every
22 years due to its differential rotation, which leads to another oscillation of magnetic field
on the planets. Altogether these two oscillations make the magnetic field effect on precession
neglectable with respect to other accumulating effects.
B Different fitting models give different best fittings
Fitting a galaxy consists of following steps:
•assume a density profile (a parametric description of the density),
•adjust the values of free parameters to minimize an “error function.”
The corresponding resulting parameters are called best fitting parameters. For a single
galaxy we can define different error functions. It can be defined by rotation curve, by surface
– 18 –
brightness or some other observation data. The point of this appendix is that we usually
get different best fitting parameters when using different error functions. In particular, best
fitting parameters for rotation curve are different from those for surface brightness. Below
we provide a simple and idealized example to illustrate this point.
Real Suppose the real density distribution is a linearly decreasing function of radius and
becomes zero outside of a cutoff radius,
ρ(r) = nρ0(1 −r
R),(0 ≤r≤R)
0.(r≥R)(B.1)
where ρ0and Rare two fixed constants for this particular galaxy. 11 The corresponding
velocity, using Newtonian gravitation theory, is
v(r) = (q2πρ0GN(1
2−1
3
r
R)r, (0 ≤r≤R),
q2πρ0GN1
6
R2
r,(r≥R)
(B.2)
and brightness is (assuming light is proportional to mass) where γis light to mass ratio.
Which gravity theory we use does not affect the conclusion of this appendix, so long as we
use the same theory for any profile to derive the velocity.
B(r) = nγρ0(1 −r
R),(0 ≤r≤R),
0,(r≥R)(B.3)
Guessed Without knowing the real distribution, suppose we assumed for this galaxy a
constant distribution profile
ρ(r) = nσ, (0 ≤r≤D)
0.(r≥D)(B.4)
Here σand Dare two parameters, rather than constants, being fitted to get best fitting
values, while ρ0and Rhave particular fixed values for this galaxy. This density profile
produces following velocity profile
v(r) = (√πσGNr, (0 ≤r≤D),
qπσGND21
r,(r≥D),(B.5)
and brightness profile
B(r) = nγσ, (0 ≤r≤D)
0,(r≥D).(B.6)
Fitting As said before, we can define different error functions to do the fitting. We can use
either rotation velocity or brightness for fitting. Ideally the error functions can be defined
respectively for rotation velocity and surface brightness by
χ2
v(D, σ) = Z∞
0
dr0[vmodel(r0, D, σ)−vreal(r0)]2(B.7)
χ2
B(D, σ) = Z∞
0
dr0[Bmodel(r0, D, σ)−Breal(r0)]2(B.8)
11Here we assume the galaxy is a disk and the distribution has only rdependence. So it is actually more
appropriate to call it surface density.
– 19 –
After minimizing these two error functions we get two sets of best fitting parameters for D
and σ. Let’s denote the best fitting value for velocity error function χ2
vby (DA, σA), for
brightness error function χ2
Bby (DB, σB). Below we show these two sets of values do not
coincide.
If we use the error function for brightness, obviously the best fitting parameters are
DB=R, σB=1
2ρ0(B.9)
To show (DB, σB)6= (DA, σA) it is sufficient to show that (DB, σB) does not minimize χ2
v.
For simplicity we take units such that
ρ0= 1, R = 1, γ = 1, πGN= 1,(B.10)
then
DB= 1, σB=1
2.(B.11)
In this unit system the real velocity is
v(r) = (pr
2,(0 ≤r≤1),
q1
2r,(r≥1).(B.12)
Taking 100000 as the upper limit of the integration, we find
χ2
v(DB= 1, σB= 0.5) = 0.2,(B.13)
χ2
v(D= 0.9, σ = 0.5) = 0.05 (B.14)
Thus (DB, σB)6= (DA, σA). The best fitting parameters for brightness do not best fit the
velocity curve.
Remarks Above we considered a simple and idealized galaxy and showed that best fitting
parameters for different error functions are in general different, although we were doing the
fitting for the same galaxy. Another point worth noting here is that if we had guessed at the
correct density profile (which linearly decreases and vanishes beyond some cutoff radius), the
two best fitting parameters will be the same. We get different fitting results because we used
a “wrong” profile for this galaxy. In real life the mass distribution for a galaxy is extremely
complicated and can not be exactly described by any simple “profile function.” Hence after
we assume the profile, define the error function and then do the fitting, we will always get
different fitting results for “independent” error functions (e.g velocity and brightness). If the
guessed profile is closer to the real distribution we get closer results for fittings by different
error functions. In other words, a big difference between fitting results by different error
functions means the profile we guessed at is very different from the real one.
Given different profile assumptions for the galaxy distribution, we thus have a way to
judge in some sense which one is more “correct”. With each profile we can derive corre-
sponding distributions of velocity, brightness and so on, and with each of these distributions
we can compute( if we have the data) an error function χ2, which is dependent on a set of
parameters owned by this profile,
Density Profile −→ (Velocity χ2
v
Brightness χ2
B
.
.
..
.
.
– 20 –
!"#"$%&'(")*+('(,-
."!(&/
0 name B magnitude
1 m81dwb -12.5 3.5 N/A
2 ngc0628 -20.60 9.77 29.95
3 ngc0925 -20.05 9.33 29.85
4 ngc2403 -19.56 3.44 27.68
5 ngc2976 -18.12 4.17 28.1
6 ngc3031 -21.54 5.18 28.57
7 ngc3184 -19.88 10.96 30.2
8 ngc3198 -20.44 12.13 30.42
9 ngc3521 -21.08 11.43 30.29
10 ngc3621 -20.51 8.24 29.58
11 ngc3938 -20.01 14.52 30.81
12 ngc4236 -18.10 2.61 27.08
13 ngc4321 -22.06 23.99 31.9
14 ngc4536 -21.79 26.42 32.11
15 ngc4569 -21.10 12.71 30.52
16 ngc4579 -21.68 22.91 31.8
17 ngc4625 -17.63 11.75 30.35
18 ngc4725 -21.76 19.50 31.45
19 ngc5055 -21.20 10.42 30.09
20 ngc5194 -20.51 10.05 30.01
21 ngc6946 -20.89 6.67 29.12
22 ngc7331 -21.58 14.13 30.75
Distance(Mpc) mucin
Figure 7. Relevant galaxy observation data for determining luminosity and distance.
If the profile perfectly match the real one, there exists a single set of parameters simultane-
ously making all χ2zero. On the other hand, if the profile differs too much from the real one,
even if we can make one of the χ2small, the corresponding parameters will usually make
other χ2’s very large. Therefore it makes sense to use, e.g. the average of several χ2’s as
the error function to be minimized. This profile, with the corresponding minimizing param-
eters of the average error function, best describes the distribution averagely, i.e. considering
distributions whose χ2is averaged.
However we cannot use this to find the real distribution. We can only compare profiles
given their profile assumptions and say which is better in describing velocity, brightness and
so on, or if we use some averaged error function, also say which is better considering their
general performances in describing various properties simultaneously.
C Detailed Fitting Results of 22 galaxies using dark matter model and
using string model
In this appendix we present all the results on the data fitting. Figure 7summarizes the
relevant data for luminosity as well as for distance determination. In Figure Cand Figure 9
the best fit values for the five free parameters in the dark matter fitting and three free
parameter in the string model fitting are presented. Mass of the stellar mass and dark matter
halos (in the case of dark matter model) as well as the mass to light ratios are computed from
the best fit values for each galaxy. These values are tabulated in Figure C(dark matter) and
in Figure 11 (string). The rest of the appendix presents 22 graphs of dark matter fit and
string fit side by side for each of the 22 galaxies. In each of the graph small cubes with error
bars represent observational data the other symbols, and the curves, represent theoretical
values. The X-axis is radius in kpc and Y-axis is velocity in kms−1.
– 21 –
best fit DM
Page 1
0 galaxy rs/Rd
1 m81dwb 0.158 0.24 8.2 1.54 613 19.2
2 ngc0628 4.398 1.88 9.9 0.75 8409 98.4
3 ngc0925 2.392 0.79 27.6 0.22 859 56.6
4 ngc2403 4.467 0.75 21.1 0.76 744 84.6
5 ngc2976 0.504 2.52 18.3 0.38 998 16.8
6 ngc3031 5.614 3.99 5.9 0.96 705 57.7
7 ngc3184 0.573 3.97 6.1 0.58 784 42.1
8 ngc3198 0.441 0.51 18.2 1.09 291 99.95
9 ngc3521 0.167 0.57 18.8 1.56 1738 99.5
10 ngc3621 0.125 6.85 1.8 1.80 903 5.6
11 ngc3938 0.905 2.45 5.5 1.33 747 60.6
12 ngc4236 1.180 4.40 6.5 0.83 603 1.4
13 ngc4321 0.956 0.87 19.6 1.02 1581 80.5
14 ngc4536 0.598 0.99 10.7 0.86 105 97.2
15 ngc4569 0.651 0.74 18.1 0.95 1350 145.4
16 ngc4579 0.719 7.90 4.3 1.80 1200 3.4
17 ngc4625 0.874 1.91 14.1 0.20 191 42.6
18 ngc4725 0.481 4.00 7.5 0.77 225 53.9
19 ngc5055 1.133 2.25 4.5 1.66 2044 67.0
20 ngc5194 0.442 1.45 4.4 1.50 1518 49.9
21 ngc6946 5.670 2.62 29.4 0.27 2491 42.6
22 ngc7331 0.839 0.50 19.6 1.32 1325 197.7
liklihood Rd(kpc) α ρ σ
Figure 8. Best fit values for the five free parameters in the Dark Matter model with generalized
NFW profile.
– 22 –
best fit string
Page 1
0 galaxy likelihood
1 m81dwb 0.125356 1.11931 15919 0.146964
2 ngc0628 4.34675 8.87156 11738.1 1.80288
3 ngc0925 2.45182 5.07681 500 2.47328
4 ngc2403 4.24683 13.4183 16496.6 0.518048
5 ngc2976 0.400153 0.1 2135.04 1.86658
6 ngc3031 5.69772 2 3187 4.3893
7 ngc3184 0.574027 0.655271 1552.44 4.68515
8 ngc3198 0.274847 1.96846 1970.96 3.52157
9 ngc3521 0.587508 6.04622 26677.8 1.479
10 ngc3621 0.365828 10.3377 7412.54 1.08406
11 ngc3938 1.03036 9.72158 16198.8 1.23998
12 ngc4236 0.325479 10.3577 109.939 2.02158
13 ngc4321 2.07034 5.20774 3752.43 3.14755
14 ngc4536 0.739433 1.46501 733.762 5.86703
15 ngc4569 0.662054 16.3703 12261.7 1.04165
16 ngc4579 0.688 7.385 5164 3
17 ngc4625 0.547835 0.1 1140 1.44574
18 ngc4725 0.268336 1.37848 1523.28 6.72621
19 ngc5055 3.52084 4.06423 24581.1 1.54384
20 ngc5194 0.80673 3.18615 10718 1.10159
21 ngc6946 6.03455 7.44028 4979.54 1.80496
22 ngc7331 0.52206 8.46459 18613.1 1.68007
Ω(Hz*km/kpc) ρ Rd (kpc)
Figure 9. Best fit values for the three free parameters in the string model.
DM mass summary
Page 1
0 name visible mass (M_sun) dark mass (M_sun) total mass (M_sun) B magnitude log mass/light mass/light
1 m81dwb 4.13E+006 7.85E+008 7.89E+008 -12.5 1.71 50.72
2 ngc0628 2.72E+010 3.31E+012 3.34E+012 -20.60 2.09 123.36
3 ngc0925 2.06E+008 2.26E+012 2.26E+012 -20.05 2.14 138.71
4 ngc2403 1.53E+008 1.70E+012 1.70E+012 -19.56 2.22 164.12
5 ngc2976 7.78E+009 4.78E+012 4.79E+012 -18.12 3.24 1738.70
6 ngc3031 2.18E+010 3.81E+012 3.83E+012 -21.54 1.78 59.63
7 ngc3184 2.39E+010 2.44E+012 2.46E+012 -19.88 2.25 176.74
8 ngc3198 1.88E+007 4.71E+011 4.71E+011 -20.44 1.30 20.18
9 ngc3521 1.57E+008 8.70E+011 8.70E+011 -21.08 1.32 20.68
10 ngc3621 1.41E+011 5.29E+010 1.94E+011 -20.51 0.89 7.81
11 ngc3938 5.35E+009 8.81E+011 8.86E+011 -20.01 1.75 56.45
12 ngc4236 2.50E+010 4.80E+010 7.30E+010 -18.10 1.43 27.01
13 ngc4321 5.07E+008 2.20E+012 2.20E+012 -22.06 1.33 21.25
14 ngc4536 4.96E+007 6.24E+011 6.24E+011 -21.79 0.89 7.71
15 ngc4569 2.66E+008 2.09E+012 2.09E+012 -21.10 1.69 48.82
16 ngc4579 2.88E+011 6.08E+011 8.96E+011 -21.68 1.09 12.26
17 ngc4625 6.48E+008 2.95E+012 2.95E+012 -17.63 3.23 1683.06
18 ngc4725 7.01E+009 6.75E+012 6.76E+012 -21.76 1.93 85.89
19 ngc5055 1.13E+010 4.88E+011 5.00E+011 -21.20 1.03 10.64
20 ngc5194 2.25E+009 8.05E+010 8.27E+010 -20.51 0.52 3.32
21 ngc6946 2.18E+010 7.09E+013 7.09E+013 -20.89 3.30 2006.87
22 ngc7331 8.07E+007 1.33E+012 1.33E+012 -21.58 1.30 19.92
Figure 10. Summary of stellar mass and mass of dark matter halos computed from the best fit
values, as well as mass to light ratios of the 22 galaxies in the dark matter model. The mass is in
units of Msun. Mass to light ratio is relative to the mass of light ratio of the Sun.
– 23 –
string mass summary
Page 1
0 name B magnitude M/L log(M/L)
1 m81dwb -12.5 1.11931 2.46E+07 1.5814 0.199039
2 ngc0628 -20.60 8.87156 3.35E+10 1.2388 0.092988
3 ngc0925 -20.05 5.07681 3.68E+09 0.2261 -0.645720
4 ngc2403 -19.56 13.4183 1.12E+09 0.1076 -0.968005
5 ngc2976 -18.12 0.1 6.76E+09 2.4549 0.390036
6 ngc3031 -21.54 2 1.31E+11 2.0420 0.310058
7 ngc3184 -19.88 0.655271 7.77E+10 5.5805 0.746676
8 ngc3198 -20.44 1.96846 4.19E+10 1.7963 0.254377
9 ngc3521 -21.08 6.04622 4.20E+10 0.9990 -0.000455
10 ngc3621 -20.51 10.3377 4.60E+09 0.1848 -0.733383
11 ngc3938 -20.01 9.72158 1.50E+10 0.9577 -0.018783
12 ngc4236 -18.10 10.3577 4.42E+08 0.1636 -0.786285
13 ngc4321 -22.06 5.20774 5.70E+10 0.5492 -0.260279
14 ngc4536 -21.79 1.46501 7.22E+10 0.8919 -0.049700
15 ngc4569 -21.10 16.3703 6.75E+09 0.1575 -0.802794
16 ngc4579 -21.68 7.385 6.79E+10 0.9286 -0.032160
17 ngc4625 -17.63 0.1 1.68E+09 0.9565 -0.019334
18 ngc4725 -21.76 1.37848 2.26E+11 2.8680 0.457581
19 ngc5055 -21.20 4.06423 4.40E+10 0.9373 -0.028102
20 ngc5194 -20.51 3.18615 6.98E+09 0.2803 -0.552336
21 ngc6946 -20.89 7.44028 1.43E+10 0.4037 -0.393918
22 ngc7331 -21.58 8.46459 4.30E+10 0.6446 -0.190709
Ω(Hz*km/kpc) total mass(Msun)
Figure 11. Summary of stellar mass as well as mass to light ratios as computed from the best fit
values in string model. The mass of the galaxies is in units of Msun. Mass to light ratio is relative to
the mass of light ratio of the Sun.
– 24 –
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Figure 12. Rotation curve of dwarf galaxy m81dwb fit with the dark matter model (left) and with
the string model (right). The χ-squared value per degree of freedom using the dark matter model
with a NFW profile is 0.158 while that using the string model is 0.1254.
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Figure 13. Rotation curve of ngc0628 fit with the dark matter model (left) and with the string model
(right). The χ-squared value per degree of freedom using the dark matter model with a NFW profile
is 4.398 while that using the string model is 4.3468.
– 25 –
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Figure 14. Rotation curve of ngc0925 fit with the dark matter model (left) and with the string model
(right). The χ-squared value per degree of freedom using the dark matter model with a NFW profile
is 2.392 while that using the string model is 2.4518.
0 1 2 3 4 5 6
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Figure 15. Rotation curve of NGC2403 fit with the dark matter model (left) and with the string
model (right). The χ-squared value per degree of freedom using the dark matter model with a NFW
profile is 4.467 while that using the string model is 4.2468.
– 26 –
0.5 1 1.5 2 2.5
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Figure 16. Rotation curve of ngc2976 fit with the dark matter model (left) and with the string
model (right). The χ-squared value per degree of freedom using the dark matter model with a NFW
profile is 0.504 while that using the string model is 0.4001.
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Figure 17. Rotation curve of ngc3031 fit with the dark matter model (left) and with the string
model (right). The χ-squared value per degree of freedom using the dark matter model with a NFW
profile is 5.614 while that using the string model is 5.6977.
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Figure 18. Rotation curve of NGC3184 fit with the dark matter model (left) and with the string
model (right). The χ-squared value per degree of freedom using the dark matter model with a NFW
profile is 0.573 while that using the string model is 0.5740.
– 27 –
0 2 4 6 8 10 12 14
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Figure 19. Rotation curve of ngc3198 fit with the dark matter model (left) and with the string model
(right). The χ-squared value per degree of freedom using the dark matter model with a NFW profile
is 0.441 while that using the string model is 0.2748.
0 2.5 5 7.5 10 12.5 15 17.5
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Figure 20. Rotation curve of ngc3521 fit with the dark matter model (left) and with the string model
(right). The χ-squared value per degree of freedom using the dark matter model with a NFW profile
is 0.167 while that using the string model is 0.5875.
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Figure 21. Rotation curve of NGC3621 fit with the dark matter model (left) and with the string
model (right). The χ-squared value per degree of freedom using the dark matter model with a NFW
profile is 0.125 while that using the string model is 0.3658.
– 28 –
0 2 4 6 8
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Figure 22. Rotation curve of ngc3938 fit with the dark matter model (left) and with the string model
(right). The χ-squared value per degree of freedom using the dark matter model with a NFW profile
is 0.905 while that using the string model is 1.0304.
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Figure 23. Rotation curve of ngc4236 fit with the dark matter model (left) and with the string model
(right). The χ-squared value per degree of freedom using the dark matter model with a NFW profile
is 1.180 while that using the string model is 0.3255.
0 2.5 5 7.5 10 12.5 15 17.5
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Figure 24. Rotation curve of NGC4321 fit with the dark matter model (left) and with the string
model (right). The χ-squared value per degree of freedom using the dark matter model with a NFW
profile is 0.956 while that using the string model is 2.0703.
– 29 –
5 10 15 20 25 30 35
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Figure 25. Rotation curve of NGC4536 fit with the dark matter model (left) and with the string
model (right). The χ-squared value per degree of freedom using the dark matter model with a NFW
profile is 0.598 while that using the string model is 0.7394.
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Figure 26. Rotation curve of ngc4569 fit with the dark matter model (left) and with the string model
(right). The χ-squared value per degree of freedom using the dark matter model with a NFW profile
is 0.651 while that using the string model is 0.6621.
2 4 6 8 10
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Figure 27. Rotation curve of NGC4579 fit with the dark matter model (left) and with the string
model (right). The χ-squared value per degree of freedom using the dark matter model with a NFW
profile is 0.719 while that using the string model is 0.688.
– 30 –
0.5 1 1.5 2
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Figure 28. Rotation curve of ngc4625 fit with the dark matter model (left) and with the string
model (right). The χ-squared value per degree of freedom using the dark matter model with a NFW
profile is 0.874 while that using the string model is 0.5478.
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300
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200
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Figure 29. Rotation curve of ngc4725 fit with the dark matter model (left) and with the string model
(right). The χ-squared value per degree of freedom using the dark matter model with a NFW profile
is 0.481 while that using the string model is 0.2683.
– 31 –
0 2.5 5 7.5 10 12.5 15
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150
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Figure 30. Rotation curve of NGC5055 fit with the dark matter model (left) and with the string
model (right). The χ-squared value per degree of freedom using the dark matter model with a NFW
profile is 1.133 while that using the string model is 3.5208.
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125
150
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Figure 31. Rotation curve of ngc5194 fit with the dark matter model (left) and with the string model
(right). The χ-squared value per degree of freedom using the dark matter model with a NFW profile
is 0.442 while that using the string model is 0.8067.
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100
150
200
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50
100
150
200
Figure 32. Rotation curve of ngc6946 fit with the dark matter model (left) and with the string model
(right). The χ-squared value per degree of freedom using the dark matter model with a NFW profile
is 5.670 while that using the string model is 6.0346.
– 32 –
2 4 6 8 10
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300
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Figure 33. Rotation curve of NGC7331 fit with the dark matter model (left) and with the string
model (right). The χ-squared value per degree of freedom using the dark matter model with a NFW
profile is 0.839 while that using the string model is 0.5221.
– 33 –
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