We explore a second order Hamiltonian vertical resonance model for X-shaped or peanut-shaped galactic bulges .
The X- or peanut-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 2:1 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 \sim 0.05 .
As the resonance moves outward , stars originally in the mid plane are forced out of the mid plane and 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 peanut- or 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 \sim 1.3 kpc , we confirm a mid-plane density of \sim 5 \times 10 ^ { 8 } M _ { \odot } { 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 .