------------------------------------------------------------------------ From: "Neven" bilic@physci.uct.ac.za To: gcnews@aoc.nrao.edu Date: Thu, 29 Mar 2001 11:44:53 GMT+0200 MIME-Version: 1.0 Content-transfer-encoding: 7BIT 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 * *********************************** ------------- End Forwarded Message -------------