Monday, August 19, 2024

Von Zeipel meets Hilda

As we're looking into orbits subject to von Zeipel, let's check the Sol-Venus innermost sector of our System. I'd start with Mercury but relativistic effects scotch such research; so we'll scale up to SV-Hilda 3:2 resonance, where - if we don't have asteroids or stations - we should.

Here we've already calculated that Jacobi, but we'll normalise that to 1. SV-Hilda's constant at flat inclination is Math.Sqrt(1-0.112*0.112) = 0.99371. Luckily we've already forced eccentricity to maximum at zero inclination so any switches to higher inclination should just make it all more circular. Semimajor 0.55175 AU, perihelion at Mercury-L2 0.49 AU. So where inclination is zero: 0.49 = 0.55175(1-ε), meaning maximum eccentricity be 0.1119 for aphelion 0.6135 AU.

As to inclination at a circular orbit, cos i = 0.99371 so maximum inclination 0.1122355 radians for 6.43°, not all that large. We also need timescales. Relative to Venus: 1.5 (1-e2)1.5 here with Venereal e=0.00677323. That eccentricity won't undercut the 1.5 Venereal year baseline noticeably. So, it's just 337 days. That's short enough we'll disregard when Venus starts shifting its own eccentricity.

It is all somewhat "ideal" given that Venus runs its own resonance with Earth (13:8), which may allow a 3:2 with that former, but not on von Zeipel's terms.

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