Longitude Systems and Conversion

P₁ = reference rotational period of system #1
P₂ = reference rotational period of system #2
L₁(t-t₀) is a known system #1 longitude relative to time t₀
L₂(t-t₀) is the desired longitude value in system #2 relative to time t₀

L₂(t-t₀) = L₁(t-t₀) + L₂₁ + (t-t₀) · [ (24 · 360/P₂) - (24 · 360/P₁) ]
L₂₁ is a reference longitude. It is defined as:

L₂₁= L₂(t=t₀) - L₁(t=t₀)

Jupiter

S_I = 9h 50m 30.003s = 877.90°/24h. Used for equatorial motions.
S_II = 9h 55m 40.632s = 870.27°/24h. Used for mid-latitudes and GRS.
S_III = 9h 55m 29.711s ± 0.04s = 9.9249197h ± 0.000011h = 870.536°/24h ± 0.000966°/24h

_I = 67.10 + 877.900 · (t-t₀)

_II = 43.30 + 870.270 · (t-t₀)

_III = 284.95 + 870.536 · (t-t₀)

(S_III) = (S_I) + 217.85 - 7.364 · (t-t₀)

(S_III) = (S_II) + 241.65 + 0.266 · (t-t₀)
where t₀ = 2451545.0 TDB or Jan. 1.5, 2000.

U_III(S_I) = 106.35 m/s

U_III(S_II) = -3.4 m/s

Saturn

S_III = 10h 39m 22.4s ± 7s = 10.656222h ± 0.00194h = 810.7939024°/24h ± 0.148°/24h

_III = 38.90 + 810.793902 · (t-t₀)

_I = 227.2037 + 844.3 · (t-t₀)

(S_III) = (S_I) + 171.6963 - 33.5061 · (t-t₀)

U_III(S_I) = 410.36 m/s

For the Voyager encounters the reference time should be set to the closest approach time. Other possibilities are:

	P1	P2	FDS(t=0)	L21
Neptune	17.866	16.11	11200.0	-334.5365