\\
Institute of Theoretical Physics and Astrophysics, \\
Department of Physics, University of Cape Town, \\
Private Bag, Rondebosch 7701, South Africa \\
}
\maketitle
\begin{abstract}
We consider a self-gravitating ideal
fermion gas at nonzero temperature as a model for the
Galactic halo.
The Galactic halo of mass
$\sim 2 \times 10^{12} M_{\odot}$ enclosed within
a radius of $\sim 200$
kpc implies the existence of a
supermassive compact dark object at
the Galactic center that is in hydrostatic and
thermal equilibrium with the halo.
The central object has a maximal mass of $\sim 2.3 \times 10^{6}
M_{\odot}$ within a minimal radius of $\sim 18$ mpc for
fermion masses $\sim 15$ keV.
\end{abstract}
%\pacs{PACS Nos.: 98.35.Gi, 98.35.Jk, 95.35.+d}
In the past,
self-gravitating neutrino matter has been suggested as a model
for quasars,
with neutrino
masses in the $0.2 {\rm keV} \simlt m \simlt 0.5 {\rm MeV}$ range
\cite{1}.
More recently,
supermassive compact
objects consisting of weakly interacting degenerate
fermionic matter, with fermion masses
in the $10 \simlt m/{\rm keV}\simlt 20$
range, have been proposed
\cite{2,3,4,5,6}
as an alternative to the
supermassive black holes that are
believed to reside at the centers of many
galaxies.
So far the masses of $\sim 20$ supermassive
compact dark objects at the galactic
centers have been measured
\cite{kor}.
The most massive compact dark object ever observed is
located at the center of M87 in the Virgo cluster,
and it has a mass of
$\sim 3 \times 10^9 M_{\odot}$ \cite{7}.
If we identify this object
of maximal mass with a degenerate fermion star
at the Oppenheimer-Volkoff (OV) limit \cite{opp},
i.e., $M_{\rm{OV}} = 0.54 M_{\rm Pl}^3\,
m^{-2} g^{-1/2} \simeq 3 \times 10^9
M_{\odot}$
\cite{4},
where $M_{\rm{Pl}}=\sqrt{\hbar c/G}$,
this allows us to fix the fermion mass to $m \simeq 15$ keV
for a spin and particle-antiparticle degeneracy factor of $g=2$.
Such a relativistic object
would have a radius of $R_{\rm{OV}}= 4.45
R_{\rm{S}} \simeq$ 1.5 light-days,
where $R_{\rm{S}}$ is the Schwarzschild
radius of the mass $M_{\rm{OV}}$.
It would thus be virtually indistinguishable from
a black hole of the same mass,
as the closest stable orbit around a black hole
has a radius of 3 $R_{\rm{S}}$ anyway.
Near the lower end of the observed mass range
is the compact dark object
located at the Galactic center
\cite{eck}
with a mass of $M_{\rm c} \simeq 2.6 \times
10^6 M_{\odot}$.
Interpreting this object as a degenerate fermion star consisting of
$m \simeq 15$ keV and $g= 2$ fermions,
the radius is $R_{\rm c} \simeq 21$ light-days
$\simeq 7 \times 10^4 R_{\rm S}$
\cite{2},
$R_{\rm S}$ being the Schwarzschild
radius of the mass $M_{\rm c}$.
Such a nonrelativistic object is far from being a black hole.
The observed motion of stars within a projected distance of
$\sim$ 6 to $\sim$ 50 light-days from Sgr A$^{*}$
\cite{eck}
yields, apart from the
mass, an upper limit for
the radius of the fermion star $R_{\rm c} \simlt 22$
light-days.
The required weakly
interacting fermion of $\sim$ 15 keV mass cannot be an
active neutrino,
as it would overclose the Universe by orders of magnitude
\cite{11}.
However, the $\sim\! 15$ keV fermion could very well be a sterile
neutrino, contributing
$\Omega_{\rm d} \simeq$ 0.3 to the dark matter
fraction of the critical density today.
Indeed, as has been shown for an
initial lepton asymmetry of
$\sim 10^{-3}$, a sterile neutrino of mass
$\sim$ 10 keV may be
resonantly produced in the early Universe with near
closure density, i.e., $\Omega_{\rm d} \sim$ 1
\cite{shi}.
As an alternative possibility,
the required $\sim$ 15 keV
fermion could be the axino
\cite{cov}
or the gravitino
\cite{lyt}
in soft supersymmetry breaking scenarios.
In the recent past,
galactic halos have been successfully modeled as
a self-gravitating
isothermal gas of particles of arbitrary mass,
the density of which scales asymptotically as
$r^{-2}$, yielding flat rotation curves
\cite{col}.
As the supermassive compact
dark objects at the galactic centers
are well described by a gas of fermions
of mass $m \sim 15$ keV at $T = 0$,
it is tempting to explore the
possibility that one could describe
both the supermassive compact dark objects
and their galactic halos in
a unified way in terms of a fermion gas
at finite temperature.
We will show in this letter that this
is indeed the case, and that
the observed dark matter distribution in the
Galactic halo is consistent with
the existence of a supermassive compact dark
object at the center of
the Galaxy which has about the right mass and size.
Degenerate fermion stars are well understood
in terms of the Thomas-Fermi theory applied
to self-gravitating fermionic matter at $T = 0$
\cite{2}.
Extending this theory to
nonzero temperature
\cite{16,bil1,bil2,17},
it has been shown
that at some critical
temperature $T = T_{\rm c}$,
a self-gravitating ideal fermion gas, having a mass
below the OV limit enclosed in a
sphere of radius $R$, may
undergo a first-order gravitational
phase transition from a diffuse state to a
condensed state.
However, this first-order
phase transition can only take place
if the Fermi gas is able to get rid
of the large latent heat.
As the short-range interactions
of the fermions are negligible, the gas cannot
release its latent heat;
it will thus be trapped for temperatures $T <
T_{\rm c}$ in a thermodynamic
quasi-stable supercooled state
close to the point of gravothermal
collapse.
The formation of a supercooled state
close to the point of
gravothermal collapse,
may be understood as a process similar
to that of violent relaxation
\cite{lynd,bin}.
Through the gravitational collapse of an
overdense fluctuation, $\sim$ 1 Gyr after the Big Bang,
part of gravitational energy transforms into the kinetic energy
of random motion of small-scale density fluctuations.
The resulting virialized
cloud will thus be well
approximated by a gravitationally stable thermalized
halo.
In order to
estimate the particle mass-temperature ratio,
we assume that a cold overdense cloud of
the mass of the Galaxy $M$,
stops expanding at the time $t_{\rm m}$,
reaching its maximal radius
$R_{\rm m}$ and minimal
average density $\rho_{\rm{m}}= 3 M/(4 \pi R_{\rm{m}}^3)$.
The total energy per particle is just the gravitational energy
\begin{equation}
E=-\frac{3}{5}\frac{GM}{R_{\rm{m}}} \, .
\label{eq001}
\end{equation}
Assuming spherical collapse
\cite{padma}
one arrives at
\begin{equation}
\rho_{\rm{m}}=\frac{9\pi^2}{16} \bar{\rho}(t_{\rm{m}})
=\frac{9\pi^2}{16} \Omega_{\rm{d}} \rho_0 (1+z_{\rm{m}})^3,
\label{eq002}
\end{equation}
where $\bar{\rho}(t_{\rm{m}})$ is the background density
at the time $t_{\rm{m}}$ or cosmological
redshift $z_{\rm{m}}$, and
$\rho_0\equiv 3 H_0^2/(8\pi G)$ is the present critical
density.
We now
approximate the virialized cloud by
a singular isothermal sphere
\cite{bin}
of mass $M$ and
radius $R$,
characterized by
a constant circular velocity
$ \Theta=(2 T/m)^{1/2}$
and the density profile
$ \rho(r)=\Theta^2/4\pi G r^2 .$
% \begin{equation}
% \rho(r)=\frac{\Theta^2}{4\pi G r^2}\, .
% \label{eq003}
% \end{equation}
Its total energy per particle is the sum of gravitational
and thermal energies, i.e.
\begin{equation}
E=-\frac{1}{4}\frac{GM}{R}
=-\frac{1}{4}\Theta^2 .
\label{eq004}
\end{equation}
Combining Eqs. (\ref{eq001}), (\ref{eq002}),
and (\ref{eq004}),
we find
\begin{equation}
\Theta^2=\frac{6\pi}{5} G
(6 \Omega_{\rm{d}} \rho_0 M^2)^{1/3}(1+z_{\rm{m}}) .
\label{eq005}
\end{equation}
Taking $\Omega_{\rm{d}}=0.3$, $M=2\times 10^{12} M_{\odot}$,
$z_{\rm{m}}=4$, and $H_0=65\,{\rm km\, s^{-1} Mpc^{-1}}$, we find
$\Theta \simeq 220\, {\rm km\, s^{-1}}$, which corresponds to the
mass-temperature ratio $m/T\simeq 4\times 10^6$.
We now briefly discuss the general-relativistic
Tho\-mas-Fer\-mi theory
\cite{bil1,bil2}
for a self-gravitating gas
of $N$ fermions with mass $m$
at the temperature $T$ enclosed in a sphere of radius $R$.
We denote
by $p$, $\rho$, and $n$
the pressure,
energy density, and particle number density
of the gas,
respectively.
The metric is assumed to be
static, spherically symmetric, and asymptotically
flat, i.e.,
\begin{equation}
ds^2=\xi^2 dt^2 -\frac{dr^2}{1-2{\cal{M}}/r} -
r^2(d\theta^2+\sin \theta d\phi^2).
\label{eq13}
\end{equation}
For numerical convenience,
we introduce the parameter
$\alpha=\mu/T$
and the substitution
$\xi=(\varphi+1)^{-1/2} \mu/m$,
where $\mu$ is the chemical potential associated with the
conserved fermion number $N$.
The equation
of state for a self-gravitating gas may be represented
in parametric form \cite{bil2}
with appropriate mass and length scales
$b$ and $a$, respectively,
$b=
(2/g)^{1/2}
M^3_{\rm{Pl}}/ m^2$,
$a=
b\hbar /
(c M_{\rm{Pl}}^2)$.
Einstein's equations for the metric (\ref{eq13})
are given by
\begin{equation}
\frac{d\varphi}{dr} =
-2(\varphi+1)\frac{{\cal M}+4\pi r^3 p}{r(r-2{\cal{M}})} \, ,
\label{eq88}
\end{equation}
\begin{equation}
\label{eq89}
\frac{d{\cal{M}}}{dr}=4\pi r^2 \rho.
\end{equation}
To these two equations we add
\begin{equation}
\frac{d{\cal N}}{dr}=4\pi r^2 (1-2{\cal{M}}/r)^{-1/2} n
\label{eq93}
\end{equation}
imposing
particle-number conservation as
a condition at the boundary
${\cal N}(R)=N$.
Equations (\ref{eq88})-(\ref{eq93})
should be integrated using
the bo\-un\-da\-ry conditions at the origin, i.e.,
\begin{equation}
\varphi(0)=\varphi_0 > -1
\, ; \;\;\;\;\;
{\cal{M}}(0)=0
\, ; \;\;\;\;\;
{\cal{N}}(0)=0.
\label{eq90}
\end{equation}
It is useful to introduce the degeneracy
parameter
$\eta=\alpha \varphi/2$.
As $\varphi$ is monotonously decreasing with increasing
$r$, the strongest degeneracy is obtained at the center
with $\eta_0=\alpha\varphi_0/2$.
The parameter $\eta_0$,
uniquely related to the central
density and pressure,
will eventually be fixed
by
${\cal N}(R)=N$.
For $r\geq R$, the function $\varphi$ yields
the usual empty-space Schwarzschild
solution
\begin{equation}
\varphi(r)=\frac{\mu^2}{m^2}
\left(1-\frac{2 M}{r}\right)^{-1}-1\, ,
\label{eq91}
\end{equation}
with
\begin{equation}
M={\cal M}(R)=\int_0^R dr 4\pi r^2 \rho(r) .
\label{eq92}
\end{equation}
The set of self-consistency
equations (\ref{eq88})-(\ref{eq93}), with the boundary
conditions (\ref{eq90})-(\ref{eq92})
defines the general-relativistic Thomas-Fermi
equation.
The numerical procedure is now straightforward.
For a fixed, arbitrarily chosen
$\alpha$,
we first integrate
Eqs.
(\ref{eq88}) and (\ref{eq89})
numerically
on the interval $[0,R]$ to find
the solutions
for various central values
$\eta_0$.
Integrating (\ref{eq93}) simultaneously
yields
${\cal N}(R)$ as a function of $\eta_0$.
We then select the
value of $\eta_0$
for which
${\cal N}(R)=N$.
The chemical potential $\mu$
corresponding to
this particular solution
is given by
Eq. (\ref{eq91})
which in turn yields
the
parametric dependence on the temperature
through $\alpha=\mu/T$.
The quantities
$N$, $T$,and $R$ are free parameters in our model
and their
values are dictated
by physics.
At $T=0$
the number of fermions $N$ is
restricted by the OV limit
$N_{\rm OV}=2.89
\times 10^{9}\,
\sqrt{2/g}
(15\,{\rm keV}/m)^2
M_{\odot}/m $.
However, at nonzero temperature, stable solutions exist
$N>N_{\rm OV}$, depending on temperature
and radius.
In the following $N$ is required to be
of the order $2\times 10^{12} M_{\odot}/m$,
so that for any $m$, the total mass
is close to the estimated mass of the halo
\cite{wilk}.
As we have demonstrated,
the expected temperature of the halo
is given by $\alpha\simeq m/T=4\times 10^4$.
Our choice of
$R=200$ kpc
is based on the estimated size of the Galactic halo.
The only remaining free parameter is the fermion
mass, which we shall fix at
$m=15$ keV, and justify its choice {\em a posteriori}.
For fixed $N$, there is a range of $\alpha$
where the Thomas-Fermi equation has multiple solutions.
For example, for $N=2\times 10^{12}$ and
$\alpha=4\times 10^6$ we find six solutions,
which we denote by
(1), (2), (3), (3'), (2'), and (1')
corresponding to the values $\eta_0 =$
-30.53,
-25.35,
-22.39,
29.28,
33.38, and
40.48, respectively.
In Fig.\ \ref{fig1}
we plot the density profiles.
For negative central value $\eta_0$,
for which the degeneracy parameter is negative everywhere,
the system behaves basically as a
Maxwell-Boltzmann isothermal sphere.
Positive values of the central degeneracy parameter $\eta_0$
are characterized by a pronounced central core
of mass of about $2.5 \times 10^6 M_{\odot}$
within a radius of about 20 mpc.
The presence of the core is obviously due to
the degeneracy pressure.
A similar structure was obtained in
collisionless stellar systems modeled as
a nonrelativistic Fermi gas
\cite{chav}.
\begin{figure}[p]
\centering
\epsfig{file=fig1.ps,width=10cm}
\caption{
The density profile of the halo
for $\eta_0=0$ (dotted line) and for
the six $\eta_0$-values discussed in the text.
Configurations with negative $\eta_0$
((1)-(3)) are depicted by the dashed
and those with positive $\eta_0$
((1')-(3')) by the solid line.
}
\label{fig1}
\end{figure}
Fig.\ \ref{fig1}
shows two important features.
First,
a galactic halo at a given temperature
may or may not have a central core
depending whether the central degeneracy papameter $\eta_0$
is positive or negative.
Second,
the closer to zero $\eta_0$ is,
the smaller the radius at which the
$r^{-2}$ asymptotic behavior of the density begins.
The flattening of the Galactic rotation curve
begins in the range $1 \simlt r/{\rm kpc} \simlt 10$,
hence the solution $3'$ most likely describes the
Galactic halo.
This may be verified by calculating the rotation
curves in our model.
We know already from our estimate (\ref{eq005})
that our model
yields the correct asymptotic circular velocity of
220 km/s.
In order to make a more realistic comparison
with the observed Galactic rotation curve,
we must include
two additional matter components: the bulge and
the disk.
The bulge is modeled as a spherically symmetric matter distribution
of the form
\cite{you}
\begin{equation}
\rho_{\rm b}(s)=\frac{e^{-hs}}{2s^3}
\int_0^{\infty} du
\frac{e^{-hsu}}{[(u+1)^8-1]^{1/2}} \, ,
\label{eq006}
\end{equation}
where $s=(r/r_0)^1/4$, $r_0$ is the effective radius of the bulge
and $h$ is a parameter.
We adopt $r_0=2.67$ kpc
and $h$ yielding a bulge mass
$M_{\rm b}= 1.5 \times 10^{10} M_{\odot}$
\cite{suc}.
\begin{figure}[p]
\centering
\epsfig{file=fig2.ps,width=10cm}
\caption{
Enclosed mass of the halo plus bulge versus
radius for $\eta_0$ =
24 (dashed),
28 (solid),
and
32 (dot-dashed line).
}
\label{fig2}
\end{figure}
In Fig.\ \ref{fig2} the mass
of halo and bulge enclosed within
a given radius is plotted for various $\eta_0$.
The data points, indicated by squares, are
the mass
$M_{\rm c}=2.6 \times 10^6 M_{\odot}$ within
18 mpc, estimated from the motion of the stars
near Sgr A$^*$ \cite{eck},
and the mass
$M_{50}=5.4^{+0.2}_{-3.6}\times 10^{11}$
within 50 kpc,
estimated from
the motion
of satellite galaxies and globular clusters
\cite{wilk}.
Variation of the central degeneracy parameter
$\eta_0$ between 24 and 32 does not change
the essential halo features.
In Fig.\ \ref{fig3} we plot
the circular velocity components:
the halo, the bulge, and the disk.
The contribution of the disk
is modeled as \cite{per}
\begin{equation}
\Theta_{\rm d}(r)^2=
\Theta_{\rm d}(r_{\rm o})^2
\frac{1.97 (r/r_{\rm o})^{1.22}}{
[(r/r_{\rm o})^2+0.78^2]^{1.43}} \, ,
\label{eq007}
\end{equation}
where we take
$r_{\rm o}=13.5$ kpc and
$\Theta_{\rm d}=100$ km/s.
Here it is assumed for simplicity that
the disc does not influence the mass distribution
of bulge and halo.
Choosing the central degeneracy
$\eta_0=28$ for the halo, the data
by Merrifield and Olling \cite{oll} are reasonably well
fitted.
We now turn to the discussion of
our choice of the fermion mass $m=15$ keV
for the degeneracy factor $g=2$.
To that end
we investigate how the mass of the
central object,
i.e., the mass $M_{\rm c}$ within 18 mpc,
depends on $m$ in the interval
5 to 25 keV,
for various
$\eta_0$.
We find that $m\simeq15$ gives always the maximal value of
$M_{\rm c}$
ranging between 1.7 and 2.3 $\times 10^6 M_{\odot}$
for $\eta_0$ between 20 and 28.
Hence, with $m\simeq 15$ keV we get the
value closest to the mass of the central object
$M_{\rm c}$
estimated from the motion of the stars
near Sgr A$^*$ \cite{eck}.
\begin{figure}[p]
\centering
%\epsfig{file=fig3.ps,width=10cm,rheight=8.0cm}
\epsfig{file=fig3.ps,width=10cm}
\caption{
Fit to the Galactic rotation curve.
The data points are
by Olling and Merrifield \protect\cite{oll},
for $R_0=8.5$ kpc and $\Theta_0=220$ km/s.
}
\label{fig3}
\end{figure}
In summary,
using the Thomas-Fermi
theory, we have shown that
a weakly interacting
self-gravitating
fermionic gas at finite temperature
yields a mass distribution that
successfully describes both the center and the halo
of the Galaxy.
For a fermion mass
$m \simeq 15$ keV,
a reasonable fit to the rotation
curve is achieved with the
temperature $T = 3.75$ meV and
the degeneracy parameter
at the center $\eta_0=28$.
With the same parameters,
the masses enclosed
within 50 and 200 kpc are
$M_{50} = 5.04\times 10^{11} M_{\odot}$
and
$M_{200} = 2.04\times 10^{12} M_{\odot}$,
respectively.
These values agree quite well with the mass estimates
based on the motion
of satellite galaxies and globular clusters
\cite{wilk}.
Moreover, the mass
of $M_{\rm c} \simeq 2.27\times 10^6 M_{\odot}$,
enclosed within 18 mpc,
agrees reasonably well
with the observations of the compact dark object at
the center of the Galaxy.
We thus conclude that both the galactic halo and center
could be made of the same fermions.
%________________________
%{\bf Acknowledgement}
This
research is in part supported by the Foundation of Fundamental
Research (FFR) grant number PHY99-01241 and the Research Committee of
the University of Cape Town.
The work of N.B.\ is supported in part by
the Ministry of Science and Technology of the Republic of Croatia
under Contract No.\ 00980102.
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%\newpage
\end{document}
***********************************
* 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 *
***********************************