Abstract
Using $N$-body simulations we study the buckling instability in a galactic
bar forming in a Milky Way-like galaxy. The galaxy is initially composed of an
axisymmetric, exponential stellar disk embedded in a spherical NFW dark matter
halo. The parameters of the model are chosen so that the galaxy is mildly
unstable to bar formation and the evolution is followed for 10 Gyr. A strong
bar forms slowly over the first few Gyr and buckles after 4.5 Gyr from the
start of the simulation becoming much weaker and developing a pronounced
boxy/peanut shape. We measure the properties of the bar at the time of buckling
in terms of the distortion and streaming velocity in the vertical direction.
The maps of these quantities in face-on projections reveal characteristic
quadrupole patterns which wind up over a short time-scale. We also detect a
secondary buckling event lasting much longer and occurring only in the outer
part of the bar. We then study the orbital structure of the bar in periods
before and after the first buckling. We find that most of the buckling orbits
originate from x1 orbits supporting the bar. During buckling the ratio of the
vertical to horizontal frequency of the stellar orbits decreases dramatically
and after buckling the orbits obey a very tight relation between the vertical
and circular frequency: $3 = 4 Ømega$. We propose that buckling is
initiated by the vertical resonance of the x1 orbits creating the initial
distortion of the bar that later evolves as kinematic bending waves.
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