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The Milky Way (MW) and M31 both harbor massive satellite galaxies, the LargeMagellanic Cloud (LMC) and M33, which may comprise up to 10 per cent of theirhost's total mass. Massive satellites can change the orbital barycentre of thehost-satellite system by tens of kiloparsecs and are cosmologically expected toharbor dwarf satellite galaxies of their own. Assessing the impact of theseeffects depends crucially on the orbital histories of the LMC and M33. Here, werevisit the dynamics of the MW-LMC system and present the first detailedanalysis of the M31-M33 system utilizing high precision proper motions andstatistics from the dark matter-only Illustris cosmological simulation. Withthe latest Hubble Space Telescope proper motion measurements of M31, wereliably constrain M33's interaction history with its host. In particular, likethe LMC, M33 is either on its first passage (t_{inf} < 2 Gyr ago) or if M31 ismassive (>=2x10^12 Msun), it is on a long period orbit of about 6 Gyr.Cosmological analogs of the LMC and M33 identified in Illustris support thispicture and provide further insight about their host masses. We conclude that,cosmologically, massive satellites like the LMC and M33 are likely completingtheir first orbits about their hosts. We also find that the orbital energies ofsuch analogs prefer a MW halo mass ~1.5x10^12 Msun and an M31 halo mass>=1.5x10^12 Msun. Despite conventional wisdom, we conclude it is highlyimprobable that M33 made a close (< 100 kpc) approach to M31 recently (t_{peri} < 3 Gyr ago). Such orbits are rare (< 1 per cent) within the 4$\sigma$error space allowed by observations. This conclusion cannot be explained byperturbative effects through four body encounters between the MW, M31, M33, andthe LMC. This surprising result implies that we must search for a newexplanation for M33's strongly warped gas and stellar discs.

Orbits of Massive Satellite Galaxies: I. A Close Look at the Large Magellanic Cloud and a New Orbital History for M33