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My understanding is that there is a gravitationally stable region between the Sun and Mercury where asteroids could - in theory - orbit for long periods without being perturbed. However these hypothetical vulcanoids are seemingly non-existent; it being likely that their orbits decay through other means besides the influence of a planet, such as the Yarkovsky effect. We also have trojan asteroids, which share the orbits of planets such as Jupiter but spend their time in the leading and trailing Lagrange Points, keeping their orbits stable.

If a Hot Jupiter such as 51 Pegasi b were to host trojan asteroids, would their proximity to the planet allow their orbits to remain stable for longer, or would their orbits decay regardless? If a Hot Jupiter migrated inwards towards its star, is it more likely to have swept away any possible trojans?

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    $\begingroup$ Small point, and more of a pet peeve of mine than a correction, but "stable for longer" are you talking years or orbital periods. The orbital periods obviously get considerably shorter as the orbiting objects get closer to the star. It seems to me that orbital stability should be counted in number of orbits, not in amount of time. (Maybe I should make that a separate question). $\endgroup$ – userLTK Mar 9 '18 at 19:49
  • $\begingroup$ I'd say 'long period' meaning as long as the planet has been around, or at least from the time the hot jupiter settles into its current orbit to the present. $\endgroup$ – user10106 Mar 12 '18 at 8:38
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Yes, hot Jupiters can have stable Trojans.

A great analysis of this very problem is given by Kislyakova et al. (2015). They suggest that super-hot outgassing Trojans may be an explanation for the ultraviolet anomalies seen in the WASP-12 and HD 189733 systems. As part of their analysis, they explore the photogravitational stability of Trojans of various size, and while they show that meter-sized bodies can be ejected on timescales of around a million years, Io-sized objects can be stable over the lifetime of the system. It may also be worth noting that although they found many stable tadpole orbits, they did not find any stable horse-shoe orbits (although they do not show that such an orbit cannot exist).

As for the actual formation and capture of such a Trojan, Beauge' et al. (2007) modeled the inward migration of a massive planet, with its semi-major axis decaying over a time scale of $10^5 - 10^7~\mathrm{yr}$. Their results suggest that a Trojan satellite with a tadpole orbit close to $L_4$ will be maintained during the migration.

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    $\begingroup$ A nice find! Will definitely have to go over the linked papers in more depth. I like possibility that a migrating planet can drag its trojans along with it! $\endgroup$ – user10106 Mar 14 '18 at 8:47

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