Fig. A.1.

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Stellar orbit distribution, p(r, λz), calculated in three different ways for an example galaxy from TNG50 at z = 0. In the left, r and λz are calculated from the instantaneous positions and velocities of particles at z = 0. In the middle, r and λz are phase-space-averaged. In the right, r and λz are time-averaged by integrating the orbits in the frozen potential. Most of the orbits in the bulge and in the hot inner stellar halo are on box orbits, and particles on the box orbits span a wide range of λz but with time-averaged values of zero. By doing a phase-space averaging, we can narrow down the λz distribution of particles on box orbits but cannot shrink them to zero.
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