------------------------------------------------------------------------
From: "Neven" bilic@physci.uct.ac.za
To: gcnews@aoc.nrao.edu
Date: Thu, 29 Mar 2001 11:44:53 GMT+0200
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Subject: astro-ph/0103466 - the motion of stars near the Galactic cen
%astro-ph/0103466
\documentstyle[aasms4,12pt]{article}
\begin{document}
\title{ The motion of stars near the Galactic center:
A comparison of the black hole and fermion ball scenarios}
\author{Faustin Munyaneza and Raoul D. Viollier}
\affil{Institute of Theoretical Physics and Astrophysics\\
Department of Physics, University of Cape Town\\
Private Bag, Rondebosch 7701, South Africa\\
fmunyaneza@hotmail.com, viollier@physci.uct.ac.za
}
\begin{abstract}
After a discussion of the properties of degenerate fermion balls, we
analyze the orbits of the stars S0-1 and S0-2, which have the
smallest
projected distances to Sgr A$^{*}$, in the supermassive black hole as
well
as in the fermion ball scenarios of the Galactic center. It is shown
that
both scenarios are consistent with the data, as measured during the
last six
years by Genzel et al. and Ghez et al. The free parameters of the
projected
orbit of a star are the unknown components of its velocity $v_{z}$
and distance $z$ to Sgr A$^{*}$ in 1995.4, with the $z$-axis being in
the line
of sight. We show, in the case of S0-1 and S0-2, that the $z-v_{z}$
phase-
space, which fits the data, is much larger for the fermion ball than
for the
black hole scenario. Future measurements of the positions or radial
velocities
of S0-1 and S0-2 could reduce this allowed phase-space and eventually
rule out
one of the currently acceptable scenarios. This may shed some light
into the
nature of the supermassive compact dark object, or dark matter in
general at
the center of our Galaxy.
\end{abstract}
\keywords{black hole physics-celestial mechanics, stellar dynamics
dark matter - elementary particles - Galaxy: center}
\section{Introduction}
There is strong evidence for the existence of a supermassive compact
dark
object near the enigmatic radio source Sagittarius A$^{*}$ (Sgr
A$^{*}$) which
is located
at or close to the dynamical center of the Galaxy (Rogers et al.
1994; Genzel et
al. 1997; Lo et al. 1998; Ghez et al. 1998). Stars observed in the
2.2 $\mu$m
infrared K-band at projected distances $\gtrsim$ 5 mpc from Sgr
A$^{*}$, and
moving with projected velocities
$\lesssim$ 1400 km~s$^{-1}$, indicate that a mass
of (2.6 $\pm$ 0.2) $\times$ 10$^{6} M_{\odot}$ must be concentrated
within a
radius $\sim$ 15 mpc from Sgr A$^{*}$ (Haller et al. 1996; Eckart and
Genzel
1996, 1997; Genzel \& Townes 1987; Genzel et al. 1994, 1996, 1999,
2000;
Ghez et al. 1998, 2000). VLBA radio
interferometry measurements at 7 mm wavelength constrain the size of
the radio wave emitting region of Sgr A$^{*}$ to $\lesssim$ 1 AU in
E-W
direction and $\sim$ 3.6 AU in N-S direction
(Rogers et al. 1994, Bower and Backer 1998, Krichbaum et al. 1994, Lo
et al.
1998), and the proper motion of Sgr A$^{*}$ relative to the quasar
background
to
$\lesssim$ 20 km~s$^{-1}$
(Baker 1996; Reid et al. 1999; Baker and Sramek 1999). As the fast
moving
stars of the central cluster interact gravitationally with Sgr
A$^{*}$, the
proper motion of the radio source cannot remain as small as it is now
for $\sim$ 200 kyr unless
Sgr A$^{*}$ is attached to some mass $\gtrsim$ 10$^{3} M_{\odot}$.
In spite of these well-known stringent facts, the enigmatic radio
source Sgr
A$^{*}$, as well as the supermassive compact dark object that is
perhaps
associated with it, are still two of the most challenging
mysteries of modern astrophysics.
It is currently believed that the enigmatic radio source Sgr A$^{*}$
coincides
in position with a supermassive black hole (BH) of (2.6 $\pm$ 0.2)
$\times$
10$^{6} M_{\odot}$ at the dynamical center of the Galaxy. Although
standard
thin
accretion disk theory fails to explain the peculiar low
luminosity $\lesssim$ 10$^{37}$ erg s$^{-1}$ of the Galactic center
(Goldwurm et al. 1994), many
models have been developed that describe the spectrum of Sgr A$^{*}$
fairly
well, based on the assumption that it is a BH. The models proposed
for the
radio emission, range from quasi-spherical inflows (Melia 1994;
Narayan and
Mahadevan 1995; Narayan et al. 1998; Mahadevan 1998)
to a jet-like outflow (Falcke, Mannheim and
Biermann 1993; Falcke and Biermann 1996; Falcke and Biermann 1999).
Yet, as some of these models appear to contradict each other, not all
of them can represent the whole truth.
We also
note that the Galactic center is a weak source of diffuse emission in
the 2-10
keV energy range and in the lines of several ions (Sunyaev et al.
1993; Koyama
et al. 1996; Sidoli and Mereghetti 1999). Thus, apart from earthbound
VLBA radio interferometers, space missions such as the European
Multi-Mirror satellite (XMM) and Chandra X-ray satellite,
may eventually
provide conclusive evidence for the nature of Sgr A$^{*}$ and the
supermassive
compact dark
object at the Galactic center. In fact, the Chandra X-ray satellite
has
recently detected a point source at the location of Sgr A$^{*}$
(Baganoff et
al. 1999) with a luminosity two times smaller than the upper limit
set by the
ROSAT satellite some years ago (Predehl and Tr\"umper 1994). For more
detailed recent reviews on the Galactic center we refer to Morris and
Serabyn
1996, Genzel and Eckart 1999, Kormendy and Ho 2000, and Yusef-Zadeh
et al.
2000.
Supermassive compact dark objects have also been inferred at the
centers of many
other
galaxies, such as M87 (Ford et al. 1994; Harms et al. 1994; Macchetto
et al.
1997) and NGC 4258 (Greenhill et al. 1995; Myoshi et al. 1995).
For recent reviews we refer to Richstone et al. 1998,
Ho and Kormendy 2000, and Kormendy 2000. In fact,
perhaps with the exception of dwarf galaxies, all galaxies may harbor
such
supermassive compact dark objects at their centers. However, only a
small
fraction of these
show strong radio emission similar to that of the enigmatic radio
source Sgr
A$^{*}$ at the center of our Galaxy. For instance M31 does not have
such a
strong compact radio source, although the supermassive compact dark
object at
the center of M31 has
a much larger mass ($\sim$ 3 $\times$ 10$^{7} M_{\odot}$) than that
of our
Galaxy
(Dressler and Richstone 1988; Kormendy
1988). It seems, therefore, prudent not to take for granted that the
enigmatic radio source Sgr A$^{*}$ and the supermassive compact dark
object at
the center of our Galaxy are necessarily one and the same
object.
An unambiguous proof for the existence of a BH requires the
observation of
stars moving at relativistic velocities near the event horizon.
However, in
the case of our Galaxy, the stars S0-1 and S0-2, that are presumably
closest
to the suspected BH, reach
projected velocities $\lesssim$ 1400 km~s$^{-1}$.
Assuming a radial velocity of $v_{z}$ = 0, this corresponds to the
escape
velocity at a distance $\gtrsim$ 5 $\times$ 10$^{4}$ Schwarzschild
radii
from the
BH. Thus any dark object,
having a mass $\sim$ 2.6 $\times$ 10$^{6} M_{\odot}$ and a
radius $\lesssim$ 5 $\times$ 10$^{4}$ Schwarzschild radii,
would fit the current
data on the proper motion of the stars of the central cluster as well
as the
BH scenario. One of the reasons why the BH scenario of Sgr A$^{*}$ is
so
popular, is that the only baryonic alternative to a BH that we can
imagine, is
a cluster of dark stars (e.g. brown dwarfs, old white dwarfs, neutron
stars,
etc.), having a total mass of $\sim$ 2.6 $\times$ 10$^{6} M_{\odot}$
concentrated within a radius of $\sim$ 15 mpc. However, such a star
cluster
would disintegrate through gravitational ejection of stars on a
time scale $\lesssim$
100 Myr, which is much too short to explain why this object still
seems to be
around today $\sim$ 10 Gyr after its likely formation together with
the
Galaxy (Sanders 1992; Haller et al. 1996; Maoz 1995, 1998).
Nevertheless, in order to test the validity of the BH hypothesis
meaningfully, we definitely need an alternative and consistent finite
size
model of the supermassive compact dark objects at the galactic
centers.
\section{ The case for degenerate fermion balls}
It is well known that our Galactic halo is dominated by dark matter,
the bulk part of which must be nonbaryonic (Alcock 2000). Numerical
simulations show that dark matter in the form of a gas of weakly
interacting
massive particles, will eventually produce a high-density spike at
the
center of the Galaxy (Navarro et al. 1997; Gondolo \& Silk 1999). It
is
therefore conceivable that
the supermassive compact dark object at the center of our Galaxy is
made of
the same dark matter that dominates the Galactic halo at large. In
fact, some
years
ago, we suggested that the supermassive compact dark object at the
Galactic
center may be a gravitationally stable ball of weakly interacting
fermions in
which the degeneracy pressure balances the gravitational attraction
of the
massive fermions (Viollier et al. 1992, 1993; Viollier 1994;
Tsiklauri \&
Viollier 1996; Bili\'{c}, Munyaneza \& Viollier 1999).
Such degenerate fermion balls (FBs) could
have been formed in the early universe during a first-order
gravitational
phase transition (Bili\'{c} \& Viollier 1997, 1998, 1999a,b). A
further
formation mechanism of FBs that is based on gravitational ejection of
degenerate matter has recently
been discussed in Bili\'{c} et al. 2000.
There are three main reasons why it is worthwhile to study such
degenerate FBs
as an alternative to BHs at the center of the galaxies, in particular
our own:
\begin{itemize}
\item[(i)] Introducing a weakly interacting fermion
in the $\sim$ 13 keV/$c^{2}$ to $\sim$ 17 keV/$c^{2}$ mass range, one
can
explain the full range of the masses and radii of the supermassive
compact
dark objects, that have been observed so far at the galactic centers,
in terms
of degenerate FBs
with
masses ranging from 10$^{6}$ to 10$^{9.5} M_{\odot}$ (Kormendy and
Richstone
1995; Richstone et al. 1998). The maximal mass allowed for a FB
composed of
degenerate fermions of a given mass $m_{f}$ and degeneracy factor
$g_{f}$ is
the Oppenheimer-Volkoff (OV) limit $M_{OV}$ = 0.54195 $M_{P
\ell}^{3}$ $m_{f}^{-
2}$ $g_{f}^{- \frac{1}{2}}$ = 2.7821 $\times$ 10$^{9} M_{\odot}$
(15 keV/$m_{f}c^{2}$)$^{2}$(2/$g_{f}$)$^{\frac{1}{2}}$,
where $M_{P \ell} = (\hbar c/G)^{\frac{1}{2}}$ is the Planck mass
(Bili\'{c},
Munyaneza \& Viollier 1999). It is tempting to identify the mass of
the most
massive compact dark object ever observed at a center of a galaxy
(Kormendy \& Ho 2000), e.g. that
of the center of M87, with the OV-limit, i.e. $M_{OV}$ = (3.2 $\pm$
0.9)
$\times$ 10$^{9} M_{\odot}$ (Macchetto et al. 1997). This requires a
fermion
mass of 12.4 keV/$c^{2}$ $\lesssim$ $m_{f}$ $\lesssim$ 16.5
keV/$c^{2}$ for
$g_{f}$ = 2,
or 10.4 keV/$c^{2}$ $\lesssim$ $m_{f}$ $\lesssim$ 13.9 keV/$c^{2}$
for $g_{f}$ =
4.
For $M_{OV}$ = 3.2 $\times$ 10$^{9} M_{\odot}$ such a
relativistic FB would have a radius of $R_{OV}$ = 4.45 $R_{OV}^{s}$ =
1.36
mpc, where $R_{OV}^{s}$ is the Schwarzschild radius of the mass
$M_{OV}$.
It would thus be virtually indistinguishable from
a BH, as the radius of the last stable orbit around a BH is 3
$R_{OV}^{s}$ =
0.92 mpc
anyway. The situation is quite different for a nonrelativistic FB of
mass
$M$ = (2.6 $\pm$ 0.2) $\times$ 10$^{6} M_{\odot}$, which for the
upper limit
of the allowed fermion mass ranges, $m_{f}$ = 16.5 keV/$c^{2}$ for
$g_{f}$ = 2,
or $m_{f}$ = 13.9 keV/$c^{2}$ for $g_{f}$ = 4, would have a radius
bound by
16.7 mpc $\lesssim$ $R$ $\lesssim$ 17.6 mpc, corresponding to $\sim$
7 $\times$
10$^{4}$ Schwarzschild radii, as the FB radius scales
nonrelativistically like
$R \propto m_{f}^{-8/3}$ $g_{f}^{- 2/3}$ $M^{-1/3}$. Such an object
is far from
being a black hole: its escape velocity from the center is
$\sim$ 1,700 km~s$^{-1}$. As the fermions interact only weakly with
the
baryons,
baryonic stars could also move inside a FB without experiencing
noticeable
friction with the fermions
(Tsiklauri and
Viollier 1998a,b; Munyaneza, Tsiklauri and Viollier, 1998, 1999).
Since the potential within $\sim$ 10 mpc from the center is rather
shallow,
star formation in this region will be less inhibited
by tidal forces than in the BH case.
\item[(ii)] A FB with mass $M$ = (2.6 $\pm$ 0.2) $\times$ 10$^{6}
M_{\odot}$ and radius $R$ $\lesssim$ 18.4 mpc is consistent with the
current
data on the proper motion of the stars in the central cluster around
Sgr
A$^{*}$. This implies lower limits for the fermion masses of $m_{f}$
$\gtrsim$
15.9 keV/$c^{2}$ for
$g_{f}$ = 2 and $m_{f}$ $\gtrsim$ 13.4 keV/$c^{2}$ for $g_{f}$ = 4,
which partly
overlap with the fermion mass ranges derived for M87. By increasing
the
fermion mass, one can interpolate between the FB and the BH
scenarios.
However, for fermion masses $m_{f}$ $\gtrsim$ 16.5 keV/$c^{2}$, for
$g_{f}$ = 2 and $m_{f} \gtrsim$ 13.9 keV/$c^{2}$ for $g_{f}$ = 4, the
interpretation of some of the most massive compact dark objects in
terms of
degenerate FBs is no longer possible. It is quite remarkable that we
can
describe the two extreme cases, the supermassive compact dark object
at the
center of M87 and that of our Galaxy, in terms of self-gravitating
degenerate
FBs using a single fermion mass. This surprising fact is a
consequence of the
equation of state of degenerate fermionic matter; this would
not be the case for degenerate bosonic matter. Indeed, for a
supermassive object consisting of nonrelativistic self-gravitating
degenerate
bosons, mass and radius would scale, for a constant boson mass, as $R
\propto
M^{-1}$, rather than $R \propto M^{-1/3}$, as for a supermassive
object
consisting of nonrelativistic self-gravitating degenerate fermions,
for a
constant fermion mass. The ratio of the radii of the supermassive
objects with
10$^{6.5} M_{\odot}$ and 10$^{9.5} M_{\odot}$ would be 10$^{3}$ in
the boson
case, instead of 10 as in the fermion case. Thus it would not be
possible
to fit mass and radius of both the supermassive compact dark object
at the
center of M87 and that of our Galaxy, in the boson case. We therefore
conclude
that,
if we want to describe all the supermassive compact dark objects in
terms of
self-gravitating degenerate particles of the same kind and mass,
these objects
cannot be composed of bosons, they must consist of fermions.
\item[(iii)] The FB scenario provides a natural cut-off of the
emitted
radiation at infrared frequencies $\gtrsim$ 10$^{13}$ GHz, as is
actually observed in the spectrum of the Galactic center (Bili\'{c},
Tsiklauri
and Viollier 1998; Tsiklauri and Viollier 1999; Munyaneza and
Viollier 1999).
This is because matter, e.g. in the form of stars, gas, dust or
dark matter, etc.
falling from infinity at rest towards the FB, cannot
acquire velocities larger than the escape velocity from the center of
the FB,
i.e. $\sim$ 1,700 km~s$^{-1}$. Consequently, there is also a natural
cut-off of
the high-frequency tail of the radiation emitted by the accreted
baryonic
matter. This is quite a robust prediction of the FB scenario, because
it is
virtually independent of the details of the accretion model. In a
thin disk
accretion model, the radiation at the observed cut-off is emitted at
distances $\sim$ 10 mpc from the center of the FB. This is also the
region,
where the gravitational potential becomes nearly harmonic due to the
finite
size of the FB. The fermion masses required for a cut-off at the
observed
frequency $\sim$ 10$^{13}$ GHz
depend somewhat on the accretion rate and the inclination angle of
the disk assumed,
but
$m_{f}$ $\lesssim$ 20 keV/$c^{2}$ for
$g_{f}$ = 2 or $m_{f}$ $\lesssim$ 17 keV/$c^{2}$ for $g_{f}$ = 4
seem to be reasonable conservative upper limits (Tsiklauri and
Viollier 1999,
Munyaneza and Viollier 1999).
\end{itemize}
Summarizing the preceding arguments (i) to (iii), we can constrain
the
allowed fermion masses for the supermassive compact dark objects in
our Galaxy
to 15.9 keV/$c^{2}$ $\lesssim$ $m_{f}$ $\lesssim$ 16.5 keV/$c^{2}$
for
$g_{f}$ = 2 or 13.4 keV/$c^{2}$ $\lesssim$ $m_{f}$ $\lesssim$ 13.9
keV/$c^{2}$ for $g_{f}$ = 4, where the lower limits are determined
from
the proper motion of stars in
the central cluster of our Galaxy, while the upper limits arise from
the
supermassive compact dark object at the center of M87.
This fermion mass
range is
also consistent with the infrared cut-off of the radiation emitted by
the
accreted baryonic matter at the Galactic center. Of course, one of
the major
challenges will be to accommodate, within the FB scenario, the
properties
of Sgr A$^{*}$ which is perhaps peculiar to our galaxy.
We now would like to identify a suitable candidate for the postulated
weakly
interacting
fermion. This particle should have been either already observed, or
its
existence
should have been at least predicted
in recent elementary particle theories.
The required fermion cannot be the gaugino-like neutralino,
i.e. a linear combination of the bino, wino and the two higgsinos,
as its mass is expected to be in the $\sim$ 30 GeV/$c^{2}$ to
$\sim$ 150 GeV/$c^{2}$ range (Roszkowski 2001).
It cannot be a standard
neutrino either (however, see Giudice et al. 2000), as this would
violate the cosmological bound on neutrino mass
and, more seriously, it would contradict the Superkamiokande data
(Fukuda et al. 2000). However, the
required fermion could be the sterile neutrino that has been recently
suggested as a cold dark matter candidate in the mass range between
$\sim$ 1
keV/$c^{2}$ to
$\sim$ 10 keV/$c^{2}$ (Shi and Fuller 1999; Chun \& Kim 1999; Tupper
et al.
2000), although one would have to stretch the mass range a little bit
and
worry about the (possibly too rapid) radiative decay into a standard
neutrino.
This sterile neutrino is resonantly produced with a cold spectrum and
near
closure density,
if the initial lepton asymmetry is $\sim$ 10$^{-3}$.
Alternatively, it could be either the gravitino, postulated in
supergravity
theories with a mass in the $\sim$ 1 keV/$c^{2}$
to $\sim$ 100 GeV/$c^{2}$ range (Lyth 1999), or the axino, with a
mass
in the range between $\sim$ 10 keV/$c^{2}$ and $\sim$ 100
keV/$c^{2}$, as
predicted by the
supersymmetric extensions of the Peccei-Quinn solution to the strong
CP-problem (Goto \& Yamaguchi 1992). In this scenario, the axino mass
arises
quite naturally as a radiative correction
in
a model with a no-scale superpotential. In summary, there are at
least three
promising candidates which have been recently predicted for
completely
different reasons
in elementary particle theories. One of these particles could play
the role of
the weakly interacting
fermion required for the supermassive compact dark objects at the
centers of
the galaxies and for cold or warm dark matter at large, if its mass
is in the
range between
$\sim$ 13 keV/$c^{2}$ and $\sim$ 17 keV/$c^{2}$ and its contribution
to the
critical density is $\Omega_{f} \sim$ 0.3.
\section{Outline of the paper}
The purpose of this paper is to compare the predictions of the BH and
FB
scenarios of the Galactic center, for the stars with the smallest
projected
distances to Sgr A$^{*}$, based on the measurements of their
positions during
the last six years (Ghez et al. 2000). The projected orbits of three
stars,
S0-1 (S1), S0-2 (S2) and S0-4 (S4), show deviations from uniform
motion on a
straight line during the last six years, and they thus may contain
nontrivial
information about the potential.
We do not rely on the accelerations
determined directly from the data by Ghez et al. 2000, as this was
done in the constant acceleration approximation which we think is not
reliable. Indeed, the Newtonian predictions for the acceleration vary
substantially, both in
magnitude and direction, during the six years of observation.
In view of this fact, we prefer to work with the raw data directly,
trying
to fit
the projected positions in right ascension (RA) and declination
of the stars in the BH and FB scenarios. For our analysis we have
selected only
two stars,
S0-1 and S0-2, because their projected distances from SgrA$^{*}$ in
1995.53,
4.42 mpc and 5.83 mpc, respectively, make it most likely that
these could be orbiting within a FB of radius $\sim$ 18 mpc. We thus
may
in principle distinguish between the BH and FB scenarios for these
two stars.
The third star, S0-4, that had in 1995.53 a projected distance
of 13.15 mpc from Sgr A$^{*}$, and was moving away from Sgr A$^{*}$
at a
projected
velocity of $\sim$ 990 km~s$^{-1}$, is now definitely
outside a FB with a radius $\sim$ 18 mpc. One would thus not be
able to distinguish the two scenarios for a large part of the
orbit of S0-4.
In the following, we perform a detailed analysis of the orbits of the
stars S0-1
and S0-2, based
on the Ghez et al. 1998 and 2000 data, including the error bars of
the
measurements, and varying the unknown components of the position and
velocity
vectors of the stars in 1995.4, $z$ and $v_{z}$.
For simplicity, we assume throughout this paper that the supermassive
compact
dark object has a
mass of 2.6 $\times$ 10$^{6} M_{\odot}$, and is centered at the
position of
Sgr A$^{*}$ which is taken to be at a distance of 8 kpc from the sun.
In fact,
because of the small proper motion $\lesssim$ 20 km~s$^{-1}$ of Sgr
A$^{*}$,
there are strong dynamical reasons to assume in the BH scenario, that
Sgr
A$^{*}$ and the supermassive BH are at the same position, while in
the FB
scenario, Sgr A$^{*}$ and the FB could be off-center by a few mpc
without
affecting
the results. We do not vary the mass of the supermassive compact dark
object,
as the calculations are not very sensitive to this parameter, as long
as the
mass
is within the range of the error bar inferred from the
statistical data on the proper motion of the stars in the central
cluster (Ghez et al. 1998).
This paper is organized as follows: In section 4, we present the main
equations
for the description of the supermassive compact dark object as a FB,
as well as the formalism for the description of the dynamics of the
stars
in the gravitational field of a FB
or a BH.
We then investigate, in section 5,
the dynamics of
S0-1 and S0-2, based on the Ghez et al. 2000 data,
and conclude with a summary and outlook in section 6.
\section{The dynamics of the stars near the Galactic center}
As the stars near the Galactic center have projected velocities
$\lesssim$
1,400 km~s$^{-1}$, one may very well describe their dynamics
in terms of Newtonian mechanics for both the BH and the FB scenarios.
Similarly, fermions of mass $m_{f} \sim$ 13 keV/$c^{2}$ to $\sim$ 17
keV/$c^{2}$,
which are condensed
in a degenerate FB of (2.6 $\pm$ 0.2) $\times$ 10$^{6} M_{\odot}$,
are
nonrelativistic,
since their local Fermi velocity is certainly smaller than the escape
velocity of $\sim$ 1,700 km~s$^{-1}$ from the center of the FB. The
fermions will, therefore, obey the
equation of hydrostatic equilibrium, the Poisson equation and the
nonrelativistic equation of state of degenerate fermionic matter
\begin{equation}
P_{f} = K n_{f}^{5/3}
\end{equation}
with
\begin{equation}
K = \frac{\hbar^{2}}{5 m_{f}} \; \left( \frac{6 \pi^{2}}{g_{f}}
\right)^{2/3}
\; \; .
\end{equation}
Here, $P_{f}$ and $n_{f}$, denote the local pressure and particle
number density
of the fermions, respectively. FBs have been discussed extensively in
a number
of
papers (e.g. Viollier 1994; Bili\'{c}, Munyaneza and Viollier 1999;
Tsiklauri and Viollier 1999). Here we merely quote the equations
that we need further below, in order to make this paper
self-contained.
The gravitational potential of a degenerate FB
is given by
\begin{eqnarray}
\Phi (r) =
\left\{
\begin{array}{l}
\displaystyle{\frac{GM_{\odot}}{a} \; \left( v'(x_{0}) -
\frac{v(x)}{x}
\right) \; \; , \; \; x \leq x_{0} } \\ [.5cm]
\displaystyle{- \frac{GM}{ax} \hspace{3.35cm} , \; \; x > x_{0} }
\; \; ,\\
\end{array} \right.
\end{eqnarray}
where $a$ is an appropriate unit of length
\begin{equation}
a \; = \; \left( \frac{3 \pi \hbar^{3}}{4 \sqrt{2}\; m_{f}^{4}\;
g_{f}\; G^{3/2}
\; M_{\odot}^{1/2}} \right)^{2/3} \; = \; 0.94393\;\mbox{pc} \;
\left( \frac{15 \; \mbox{keV}}{m_{f} c^{2}} \right)^{8/3} \; g_{f}^{-
2/3} \;
\; ,
\end{equation}
$r = ax$ is the distance from the center of the FB and $R = ax_{0}$
the
radius of the FB.
The dimensionless quantity $v(x)$, that is related to the
gravitational
potential $\Phi (r)$ through eq.(3),
obeys the Lan\'{e}-Emden differential
equation
\begin{equation}
\frac{d^{2} v}{d x^{2}} \; = \; - \; \frac{v^{3/2}}{x^{1/2}} \; \; ,
\end{equation}
with polytropic index $n = 3/2$. For a pure FB without a
gravitational point source at the center, the boundary conditions at
the
center and the surface of the FB are $v(0) = v(x_{0}) = 0$.
All the relevant quantities
of the FB can be expressed in terms of $v$ and $x$, e.g. the matter
density as
\begin{equation}
\rho \; = \; \frac{\sqrt{2}}{3} \; \frac{m_{f}^{4}\; g_{f}}{\pi^{2}
\hbar^{3}} \; \left( \frac{GM_{\odot}}{a} \right)^{3/2} \; \left(
\frac{v}{x}
\right)^{3/2} \; \; ,
\end{equation}
where $m_{f}$ and $g_{f}$ are the mass and the spin degeneracy factor
of
the fermions and antifermions, respectively,
i.e. $g_{f}$ = 2 for Majorana and $g_{f}$ = 4 for Dirac fermions and
antifermions. Based on eqs.(5) and (6), the mass enclosed within a
radius $r$
in a FB is given by
\begin{equation}
M(r) \; = \; \int_{0}^{r} \; 4 \pi \; \rho \; r^{2} \; dr \; = \;
- \; M_{\odot} \; \left(v'(x) x - v(x) \right) \; \; ,
\end{equation}
and the total mass of the FB by
\begin{equation}
M \; = \; M(R) \; = \; - \; M_{\odot} \; v'(x_{0}) \; x_{0} \; \; .
\end{equation}
From eq.(5), one can derive a scaling relation for the mass and
radius of a
nonrelativistic FB, i.e.
\begin{eqnarray}
M R^{3} &=& x_{0}|v'(x_{0})|x_{0}^{3} \; a^{3} \; M_{\odot} \; = \;
\frac{91.869 \; \hbar^{6}}{G^{3} m_{f}^{8}} \; \left( \frac{2}{g_{f}}
\right)^{2} \nonumber \\
&=& 27.836 \; M_{\odot} \; \left( \frac{15 \; \mbox{keV}}{m_{f}
c^{2}}
\right)^{8}
\;
\left( \frac{2}{g_{f}} \right)^{2} \; (\mbox{pc})^{3} \; \; .
\end{eqnarray}
Here $v(x)$ is the solution of eq.(5) with $v(0)$ = 0 and $v'(0)$ =
1,
yielding $v(x_{0})$ = 0 again at $x_{0}$ = 3.65375, and $v'(x_{0})$ =
--
0.742813. The precise index of the power law of the scaling
relationship (9)
depends on the
polytropic index of the equation of state (1). As the mass of the FB
approaches
the OV
limit, this scaling law is no longer valid, because the degenerate
fermion gas
has
to be described by the correct relativistic equation of state and
Einstein's equations for the gravitational field and hydrostatic
equilibrium
(Bili\'{c}, Munyaneza \& Viollier 1999).
We now turn to the description of the dynamics of the stars near the
Galactic
center. The mass of the BH and FB is taken to be $M$ = 2.6 $\times$
10$^{6}
M_{\odot}$. In order to emphasize the differences between the FB and
the BH
scenarios, we choose the fermion masses $m_{f}$ = 15.92 keV/$c^{2}$
for
$g_{f}$ = 2 or $m_{f}$ = 13.39 keV/$c^{2}$ for $g_{f}$ = 4. These are
the
minimal fermion masses consistent with the mass distribution inferred
from
the statistics of proper motions of the stars in the central cluster
(Munyaneza,
Tsiklauri and Viollier, 1999; Ghez et al. 1998). The dynamics of the
stars in
the gravitational field of the supermassive compact dark object can
be
calculated solving Newton's equations of motion
\begin{eqnarray}
\ddot{\vec{r}} \; = \; - \; \frac{GM(r)}{r^{3}} \; \vec{r} \; \; ,
\end{eqnarray}
taking into account the position and velocity vectors at e.g. $t_{0}$
=
1995.4 yr, i.e. $\vec{r}(t_{0}) \equiv (x,y,z)$ and
$\dot{\vec{r}}(t_{0})
\equiv (v_{x}, v_{y}, v_{z})$. For the FB scenario, $M(r)$ is given
by eq.(7),
while in the
BH case it is replaced by $M$ of eq.(8). The $x$-axis is chosen in
the direction
opposite to the right ascension (RA), the $y$-axis in the direction
of the
declination, and the $z$-axis points towards the sun. The BH and the
center of
the FB are assumed to be at the position of Sgr A$^{*}$ which is also
the
origin of the coordinate system at an assumed distance of 8 kpc from
the sun.
\section{Analysis of the orbits of S0-1 and S0-2}
In 1995.4, the projected positions and velocities of S0-1 reported by
Ghez et al. 1998,
were $x = - 0.107''$, $y = 0.039''$, $v_{x}$ = (470 $\pm$ 130)
km~s$^{-1}$ and
$v_{y}$ = (-1330 $\pm$ 140) km~s$^{-1}$. We now investigate how the
projected orbits,
calculated using eq.(10), are affected by (i) the error bars of
$v_{x}$ and
$v_{y}$ of S0-1 measured in 1995.4, (ii) the lack of knowledge of $z$
of S0-1
in
1995.4, (iii) the lack of information on $v_{z}$ of S0-1 in 1995.4.
We then
compare the results with the S0-1 data recently reported by Ghez et
al. 2000.
Fig.1 shows the RA of S0-1 as a function of time, taking into account
the error bars of $v_{x}$ and
$v_{y}$ and choosing $z = v_{z}$ = 0 in 1995.4.
The top panel represents the RA of S0-1 in the BH scenario, while the
bottom
panel illustrates the same quantities in the FB case. From Fig.1 we
conclude
that, for $z = v_{z}$ = 0 in 1995.4, the error bars of
$v_{x}$ and $v_{y}$ of 1995.4 do not allow for
***********************************
* Dr Neven Bilic *
* Department of Physics *
* University of Cape Town *
* Private Bag Rondebosch 7700 *
* South Africa *
* phone: +27-21-650 3344 *
* fax: +27-21-650 3352 *
***********************************
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