A Vertical Resonance Heating Model for X- or Peanut-Shaped Galactic
  Bulges
  We explore a second order Hamiltonian vertical resonance model for X-shaped
or peanut-shaped galactic bulges. The X-shape is caused by the 2:1 vertical
Lindblad resonance with the bar, with two vertical oscillation periods per
orbital period in the bar frame. We examine N-body simulations and find that
due to the bar slowing down and disk thickening during bar buckling, the
resonance and associated peanut-shape moves outward. The peanut-shape is
consistent with the location of the vertical resonance, independent of whether
the bar buckled or not. We estimate the resonance width from the potential m=4
Fourier component and find that the resonance is narrow, affecting orbits over
a narrow range in the angular momentum distribution, dL/L ~ 0.05. As the
resonance moves outward, stars originally in the mid plane are forced out of
the mid plane into orbits just within the resonance separatrix. The height of
the separatrix orbits, estimated from the Hamiltonian model, is approximately
consistent with the peanut-shape height. The X-shape is comprised of stars in
the vicinity of the resonance separatrix. The velocity distributions from the
simulations illustrate that low inclination orbits are depleted within
resonance. Within resonance, the vertical velocity distribution is broad,
consistent with resonant heating caused by the passage of the resonance through
the disk. In the Milky Way bulge we relate the azimuthally averaged mid-plane
mass density near the vertical resonance to the rotation curve and bar pattern
speed. At an estimated vertical resonance galactocentric radius of ~1.3 kpc, we
confirm a mid-plane density of ~5x10^8 Msol/kpc^3, consistent with recently
estimated mass distributions. We find that the rotation curve, bar pattern
speed, 2:1 vertical resonance location, X-shape tips, and mid-plane mass
density, are all self-consistent in the Milky Way galaxy bulge.
