Hamiltonians of spherical galaxies in action-angle coordinates

Abstract

We present a simple formula for the Hamiltonian in terms of the actions for spherically symmetric, scale-free potentials. The Hamiltonian is a power law or logarithmic function of a linear combination of the actions. Our expression reduces to the well-known results for the familiar cases of the harmonic oscillator and the Kepler potential. For other power laws, as well as for the singular isothermal sphere, it is exact for the radial and circular orbits, and very accurate for general orbits. Numerical tests show that the errors are always very small, with mean errors across a grid of actions always <1 per cent and maximum errors <2.5 per cent. Simple first-order corrections can reduce mean errors to <0.6 per cent and maximum errors to <1 per cent. We use our new result to show that: (1) the misalignment angle between debris in a stream and a progenitor is always very nearly zero in spherical scale-free potentials, demonstrating that streams can sometimes be well approximated by orbits, (2) the effects of an adiabatic change in the stellar density profile in the inner region of a galaxy weaken any existing 1/r dark matter density cusp, which is reduced to 1/r1/3. More generally, we derive the full range of adiabatic cusp transformations and show that a 1/rγ0 density cusp may be changed to 1/rγ1 only if γ0/(4-γ0) ≤ γ1 ≤ (9-2γ0)/(4-γ0). It follows that adiabatic transformations can never completely erase a dark matter cusp. © 2014 The Authors. Published by Oxford University Press on behalf of the Royal Astronomical Society.