chornedsnorkack

2010-Mar-27, 06:28 PM

Does anyone have analyses and rules of thumb for 4 body Trojan stability?

With 3 bodies, tertiary of negligible mass compared to secondary is stable if the mass of secondary is under about 1/26 of primary mass. If the mass of tertiary is not negligible, the combined mass of secondary and tertiary must be under the 1/26 primary limit (slightly less).

But what about 4 body system?

The secondary will stabilize 2 Lagrange points at 60 degrees, Trojan and Achaean. If both are negligible in mass compared to secondary then each of them can ignore the other.

But Achaean are 120 degrees from Trojan. So if Trojan body has significant mass compared to the secondary, the stabilizing force 60 degrees from secondary competes with the perturbation of Trojan 120 degrees away.

At which mass ratio between Trojan and secondary does the stability of Achaean point vanish?

With 3 bodies, tertiary of negligible mass compared to secondary is stable if the mass of secondary is under about 1/26 of primary mass. If the mass of tertiary is not negligible, the combined mass of secondary and tertiary must be under the 1/26 primary limit (slightly less).

But what about 4 body system?

The secondary will stabilize 2 Lagrange points at 60 degrees, Trojan and Achaean. If both are negligible in mass compared to secondary then each of them can ignore the other.

But Achaean are 120 degrees from Trojan. So if Trojan body has significant mass compared to the secondary, the stabilizing force 60 degrees from secondary competes with the perturbation of Trojan 120 degrees away.

At which mass ratio between Trojan and secondary does the stability of Achaean point vanish?