1 Introduction

The nuclei of normal galaxies are thought to harbor massive dark objects (MDOs), which could be supermassive black holes. These central regions often possess dense agglomerations of stars, whose structural and kinematical properties appear to be correlated with global galaxy properties (see Gebhardt et al. 1996; Ferrarese & Merritt 2000; Gebhardt et al. 2000). The imprint of galaxy formation is surely recorded in the nature of stellar orbits. No more unusual examples are, perhaps, known than the nuclei of the galaxies, NGC 4486B (in the Virgo cluster) and M 31 (our nearest large neighbor). The proximity of M 31 has enabled detailed photometric and kinematic observations of its nucleus, beginning with the detection of its asymmetrical shape by Stratoscope II (Light et al. 1974), and its resolution into a double-peaked structure by the HST images of Lauer et al. (1993). The central peak (P2) lies close to the presumed location of the MDO, located in a small region of UV-bright stars (King et al. 1995; Lauer et al. 1998; Kormendy & Bender 1999). Tremaine (1995) proposed that the off-centered peak (P1) marks the region in a disk of stars, where lie the apoapses of many eccentric orbits. This lopsided structure is expected to rotate steadily with some pattern speed, and remain locked in place by the self-gravity of all the stars. We construct numerical stellar dynamical models, wherein the disk potential is derived directly - after bulge subtraction - from the HST photometry of Lauer et al. (1998). Model construction and comparisons with data make many demands on computational resources. Hence it was not practical to explore the effects of varying values of many of the parameters concerning the bulge and disk; we take many of these values from Kormendy & Bender (1999). However, we do explore the effect of varying the pattern speed. We state our assumptions and give an outline of our method below.

We assumed that the bulge-subtracted light emanated from a steadily rotating, inclined, razor-thin, flat, disk of stars, in orbit about the MDO. The stars compose a collisionless, self-gravitating system. Hence the orbits of individual stars are governed by the combined gravitational attractions of the MDO, and the smooth, self-gravitational potential of all the stars. A bulge-disk decomposition of the V-band image of Lauer et al. (1998) yielded the disk surface density, from which the smooth disk potential was computed. For some chosen value of the pattern speed ($\Omega$), orbits of test stars were integrated numerically in the rotating frame. A selection of prograde and retrograde (quasi-periodic) loop orbits of various sizes composed an orbit library. The orbits were populated with "stars'' ($\sim$237 000 in all), spaced uniformly in time, and the disk light (in a central region) partitioned into many cells, with more cells than orbits. Determination of orbit masses, from the known luminosities of the cells, required solving an overdetermined problem, involving positive quantities. This was achieved through $\sim$5000 iterations of a RL algorithm (Richardson 1972; Lucy 1974). The entire procedure was repeated for several values of $\Omega$. Comparisons with the kinematic maps of Bacon et al. (2001) followed. Central line-of-sight velocity distributions were calculated to emphasize the regions in velocity space, where retrograde orbits contribute. Following Kormendy & Bender (1999), we assumed a distance to M 31 of 770 kpc (on the sky, $1\arcsec$ corresponds to $\simeq$$3.73\pc$), mass of the MDO, $M=3.3 \times 10^7~M_{\odot}$, and mass-to-light ratio of the (bulge-subtracted) nuclear disk equal to $\Upsilon_V = 5.73$.