Theoretical considerations lead us to expect a relaxed stellar cusp around the MBH in the Galactic Center. Does such a cusp indeed exist there? The answer depends critically on the problem of identifying which of the observed stars are dynamically relaxed, since only those faithfully trace the underlying old stellar population. The analysis presented here shows that it is possible to interpret the available observations self consistently in the framework of a high density cusp. However, the reader should keep in mind that the issue is an empirical one and, as such, may be subject to revisions when more and better data are obtained about the stars near the MBH.

Direct evidence for the existence of a cusp comes from the analysis of star maps, which show a concentration of stars toward the center. Assuming a 3D density distribution of the form n_* ∝ r^−α, the corresponding projected 2D surface density can be compared to the observed distribution to find the most likely value of α. Figure 1.1 shows the likelihood curves for α based on three independent star maps, after all the stars that were spectroscopically identified as young were taken out of the sample (the faint blue stars nearest to Sgr A^* are included only in the Keck data set, but not in the other two). The most likely value for the density power-law index α lies in the range ∼1.5-1.75. A flat core (α ∼ 0), such as exists in globular clusters, is decisively rejected. Similarly, a likelihood test for the maximal size of a flat inner core indicates that such a core, if it exists, is smaller than ∼0.1 pc (2.5''). It can be shown that extinction by interstellar dust is unlikely to bias these results by a significant amount.

Additional evidence for the existence of a very high density cusp comes from the observed gradual depletion of the luminous giants toward the MBH in the inner 0.1 pc (figure 1.2). Luminous red giants have very large extended envelopes, and therefore a large cross section for collisions with other stars. When the impact parameter is a small fraction of the giant's radius, the envelope may be stripped, leaving behind an almost bare burning core. This will make the star effectively invisible in the infrared (IR) because the IR spectral range lies in the Rayleigh- Jeans part of the stellar blackbody spectrum, and so the IR luminosity scales as L_IR ∝ R²_* T_eff while the total luminosity scales as L_* ∝ R²_*T ⁴_eff. Suppose that

Figure 1.1. A maximum likelihood analysis of the surface density distribution of stars near Sgr A^* for a 3D stellar density distribution n_* ∝ r^−α (Alexander 1999). Three different data sets (Blum et al 1996, Genzel et al 1996, Eckart and Genzel 1997, Ghez et al 1998) indicate that the most likely value for α lies in the range ∼3/2 to ∼7/4, which is the theoretically predicted range for a relaxed stellar system around a MBH (Bahcall and Wolf 1977). Order of magnitude estimates suggest that the stellar system around the MBH in the Galactic Center has undergone two-body relaxation. (Reprinted with permission from The Astrophysical Journal.)

the collision disperses the envelope of a ∼100 R_ּ red supergiant and leaves a ∼1 R_ּ burning core. In order to maintain the total stellar luminosity, the effective temperature will have to rise by a factor of 10, which will result in a decrease of the IR luminosity by a factor of 1000 (7.5 magnitudes).

Figure 1.2 compares a theoretical prediction for the collisional depletion of luminous giants with the data. The match with the observed trend is remarkably good, given the fact that no attempt was made to fit the data. The calculation is based on detailed modeling of expected numbers, sizes, luminosities and lifetimes of giants in the population, on cross sections for envelope disruption that were calibrated by hydrodynamical simulations, and on a stellar density cusp that is normalized by dynamical estimates of the enclosed mass.

It should be noted that the total mass loss rate from these collisions is smaller than that supplied by the strong stellar winds of the blue super giants in the inner few arcseconds, and so stellar collisions are not a dominant source of mass supply to the MBH at this time.

Figure 1.2. Evidence for collisional destruction of bright giant envelopes in a high-density stellar cusp around the MBH in the Galactic Center (Alexander 1999). The apparent stellar K-band magnitude is plotted against the projected angular distance from the black hole, p (Keck data from Ghez et al 1998). The ages of the stars marked by circles are unknown, but it is likely that most of them are old, and therefore dynamically relaxed. Stars marked by ‘L’ are spectroscopically identified as old stars. Stars marked by ‘H’ are spectroscopically identified as young stars and are not dynamically relaxed. Such stars are not expected to be affected by collisions because of their short lifetimes. The stars marked by ‘E’ have featureless blue spectra and are either young stars or old stars that were affected by the extreme conditions very near the black hole. The three contour lines represent detailed model predictions for the decrease in surface density of bright stars due to collisional destruction in a high density n_* ∝ r^−3/2 stellar cusp. The stellar density reaches a value of ∼4 × 10⁸ M_ּ pc⁻³ at r = 0.25'' (0.01 pc), which is nine orders of magnitude higher than in the Solar Neighborhood, and almost three orders of magnitude higher than in the densest globular cluster core. The model predicts, on average, 1.5 (top contour), 1.0 (central contour), and 0.5 (bottom contour) dynamically relaxed stars per 0.25 arcsecond bin that are brighter than the contour level. This is consistent with the observed trend in the surface density distribution. (Reprinted with permission from The Astrophysical Journal.)

The self-consistent picture that emerges from this analysis is that the stars near the MBH in the Galactic Center, which are expected to be dynamically relaxed, are indeed concentrated in a stellar cusp of the form predicted by theory for a relaxed system. The very high stellar density in the inner few 0.01 pc leads to frequent collisions that destroy the envelopes of giant stars, thereby explaining the gradual depletion in the number of luminous giants toward the center. The central cluster of faint blue stars in the inner 0.5'' coincides with the collisionally dominated region. It is therefore relevant to consider dynamical explanations for their nature and appearance as an alternative to assuming that they are newly formed, unrelaxed stars. The concentration of such a distinct population in a small volume is consistent with the tightly bound orbits that are typical of a steep cusp.