tokens: capture times for the SG model are always shorter, due to higher gas densities; thus, an SG disc captures all stars with a 10 4 R g at i 0 = 45 ○ , while a TQM disc captures only red giants at most radii, and captures other stars only at very small disc radii. Fig. 3 shows T cap as function of i 0 for different stellar types assuming a 0 = 10 4 R g . The curves in Fig. 3 correspond to the analytic expression in equation ( 4 ), which is very weakly dependent on i , hence the flatness of the curves. However, Fig. 3 also shows that numerical integration is required for highly inclined orbiters, and for disc models with highly variable radial height profiles (e.g. TQM ). Counter-intuitively, highly inclined stellar orbiters are captured faster by the SG disc since relative velocity is largest for high i 0 and therefore F GEO is largest. The TQM disc model is less efficient than the SG disc model at star capture at a 0 = 10 4 R g for all stellar types, with only red giants ending up within the disc for plausible AGN lifteimes. For simplicity, we assume no change in stellar mass due to disc crossings. For red giants in particular, this is likely incorrect since Kieffer Bogdanović ( 2016 ) find that RG can lose up to 10 per cent of their mass during a disc crossing (though for column densities at least two orders of magnitude greater than any considered here). Further complicating the expected capture time of giants, Armitage, Zurek Davies ( 1996 ) find that mass-loss due to repeated impacts with the disc leads to an expansion of the star and a subsequent increase in cross-section. Stars with a larger surface area are captured by the disc faster since they experience a stronger drag from equation ( 1 ), and therefore including the mass-loss in our calculations could reduce capture time for giants in both SG and TQM . In addition, Armitage finds that a main-sequence star can survive repeated disc passages while RG are destroyed much more quickly; this could explain the observed absence of RG in the NSC of the Milky Way. Some variations in T cap as a function of i for the TQM disc appear to be due to changes in a across the sharp (unphysical) density and scale height changes at a ∼ 5 × 10 3 R g . Figure 3. Capture time T cap due to geometric drag as a function of initial orbital inclination ( i 0 ) with notation as for Fig. 2 . We assumed Keplerian prograde orbits with a 0 = 10 4 R g . Curves correspond to the analytic estimate for T cap from equation ( 4 ) and star symbols indicate T cap from numerical integration of equation ( 10 ). Conservation of energy causes higher orbital inclinations to produce large changes in a , as the number of orbits needed to reach i crit increases. This results in a lower T cap for higher inclinations at the same initial semimajor axis. Changing a also causes the anomalies seen in the TQM panel, due to orbiters crossing the (unphysical) density discontinuity at approximately 5 × 10 3 R g . From the top panels of Figs 2
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