The phase space coordinates of individual halo stars obtained by Galactic surveys enable the computation of their full 3-dimensional orbits. Spectral analysis of halo orbits can be used to construct frequency maps which provide a compact representati
on of the 6-dimensional phase space distribution function. Frequency maps identify important major orbit families, and the orbital abundances reflect the shape and orientation of the dark matter halo relative to the disk. We apply spectral analysis to halo orbits in a series of controlled simulations of disk galaxies. Although the shape of the simulated halo varies with radius, frequency maps of local samples of halo orbits confined to the inner halo contain most of the information about the global shape of the halo and its major orbit families. Quiescent or adiabatic disk formation results in significant trapping of halo orbits in resonant orbit families (i.e. orbits with commensurable frequencies). If a good estimate of the Galactic potential in the inner halo (within ~50kpc) is available, the appearance of strong, stable resonances in frequency maps of halo orbits will allow us to determine the degree of resonant trapping induced by the disk potential. The locations and strengths of these resonant families are determined both by the global shape of the halo and its distribution function. Identification of such resonances in the Milky Ways stellar halo would therefore provide evidence of an extended period of adiabatic disk growth. If the Galactic potential is not known exactly, a measure of the diffusion rate of large sample of 10^4 halo orbits can help distinguish between the true potential and an incorrect potential. The orbital spectral analysis methods described in this paper provide a strong complementarity to existing methods for constraining the potential of the Milky Way halo and its stellar distribution function (ABRIDGED).
We evaluate the orbital evolution and several plausible origins scenarios for the mutually inclined orbits of Upsilon Andromedae c and d. These two planets have orbital elements that oscillate with large amplitudes and lie close to the stability boun
dary. This configuration, and in particular the observed mutual inclination, demands an explanation. The planetary system may be influenced by a nearby low-mass star, Upsilon And B, which could perturb the planetary orbits, but we find it cannot modify two coplanar orbits into the observed mutual inclination of ~30 deg. However, it could incite ejections or collisions between planetary companions that subsequently raise the mutual inclination to >30 deg. Our simulated systems with large mutual inclinations tend to be further from the stability boundary than Upsilon And, but we are able to produce similar systems. We conclude that scattering is a plausible mechanism to explain the observed orbits of Upsilon And c and d, but we cannot determine whether the scattering was caused by instabilities among the planets themselves or by perturbations from Upsilon And B. We also develop a procedure to quantitatively compare numerous properties of the observed system to our numerical models. Although we only implement this procedure to Upsilon And, it may be applied to any exoplanetary system.
