Callisto's Rotation Equations Peyman Parsa November 15, Updated July 2012

Transcription

1 Callisto's Rotation Equations Peyman Parsa November 5, Updated July 0 Constants f R i M i St i Ma i Sp Free Rotation Constant Maximum Influenced Rotation Constant (for planets and moons only) Start Influenced Rotation Distance Constant Maximum Influenced Rotation Distance Constant Stop Rotation Distance Constant Facts Callisto Jupiter Sun Mass (kg) m m m M Density (g/cm 3 ) m.834 s.408 Axis Tilt (deg) t m t 3.3 t s 7.5 Semi-major Axis (km) a m a Orbit Eccentricity (deg) e m e Orbit Inclination (deg), i m 0.87 i 6.09 with respect to equator 6 F f R mm m F Callisto's Free Rotational Speed (per day)

2 Part Callisto's Influenced Rotation by the influence of the Jupiter q a m e m q Q a m e m Q Callisto's Perihelion Distance (km) Callisto's Aphelion Distance (km) i r cos i m 80 i r Callisto's Orbit Inclination Reduction Factor Mi 6 mm i r 6 m m i M 3 M m Mi Callisto's Maximum Influenced Rotational Speed by the Jupiter (p.d.) S t 6 mm i r m i St 3 M m S t Callisto's Start Influenced Rotation Distance to the Jupiter (km) M a 6 mm i r m i Ma 3 M m M a Callisto's Maximum Influenced Rotation Distance to the Jupiter (km) S p 6 mm i r m i Sp 3 M m S p Callisto's Stop Influenced Rotation Distance to the Jupiter (km)

3 Calculating Callisto's average distance to the Jupiter, (q S p Q ) S p a m x q S p S p Q 0 0 e m x 0 b a m e m X value at Callisto's orbit intersection with S p Boundary (km) b Callisto's Semi-minor Axis (km) y b a m x a m y x atan y Y value at the Callisto's orbit intersection with S p Boundary (km) Half-angle of the Callisto's orbit out of S p Boundary (rad) P o a m 0 P o e m sin d Callisto's Orbital Perimeter (km) s a m 0 e m sin d s Half of the Callisto's orbit out of S p Boundary (km) e m cos( E) de e m cos( E) sa m a a q S p S p Q a m 0 0 e m cos( E) de sa m a a 0 Callisto's average distance outside S p Boundary (km)

4 n s P o n 0 3 a a a 3 Temporal ratio of the Callisto's orbit out of S p Boundary to the whole orbit a a S p a m S p d q S p S p Q M a S p M a S p d a m S p M a S p Callisto's corresponding distance on x axis of the graph Non-proportional Rotation Graph Rotation 0.9 D0 D 0.9 D 0.9 D S t M a M a S p 0.9 D 0D Callisto's position, d S p M a D Distance S t

5 Rotation 0.9 D0 D 0.9 D 0.9 D S t M a M a S p 0.9 D 0D Left end of the Rotation Graph Callisto's position, d S p D Distance M a ( d) 0.9 d 0 d 0.9 d d d S t M a d M a S p ( d) Callisto's corresponding Influenced Rotation by the Jupiter on the Y axis of the graph t m F t m Mi t r a m M a Mi F ( d) Mi t r c a m M a c 0 ( d) Mi F t m F 90 t m ( d) Mi 90 Callisto's Maximum and Free Rotational Speed Reduction by Axis Tilt Degree F t r ( d) Mi F t r q S p Q S p n 0 Callisto's end result Influenced Rotation by the Jupiter (p.d.)

6 Part Callisto's Influenced Rotation by the influence of the Sun q a( e) q Callisto/Jupiter's Perihelion Distance to the Sun (km) Q a( e) Q Callisto/Jupiter's Aphelion Distance to the Sun (km) i r cos i 80 i r Callisto/Jupiter's Orbit Inclination Reduction Factor Mi 6 mm i r M i M 6 m Mi Callisto's Maximum Influenced Rotational Speed by the Sun (p.d.) S t 6 mm i r i St M S t Callisto's Start Influenced Rotation Distance to the Sun (km) M a 6 mm i r i Ma M M a Callisto's Maximum Influenced Rotation Distance to the Sun (km) S p 6 mm i r i Sp M S p Callisto's Stop Influenced Rotation Distance to the Sun (km)

7 d a S p M a S p d Callisto's corresponding distance to the Sun relative to S p on the X axis of the graph Non-proportional Rotation Graph Rotation 0.9 D0 D 0.9 D 0.9 D S t M a M a S p 0.9 D 0D Callisto's position, d S p M a D Distance S t Left end of the Rotation Graph Rotation 0.9 D0 D 0.9 D 0.9 D S t M a M a S p 0.9 D 0D 0.5 Callisto's position, d 4.3 S p D Distance M a

8 ( d) 0.9 d 0 d 0.9 d d d S t M a d M a S p ( d) Callisto's corresponding Influenced Rotational Speed by the Sun on Y axis of the graph t m t m t t m Callisto's Axis Tilt with respect to the Sun (deg) t m F t m Mi t r a M a Mi F ( d) Mi t r F t m F 90 t m ( d) Mi 90 Callisto's Maximum and Free Rotational Speed Reduction by Axis Tilt c a M a ( d) Mi t r q S t ( d) Mi t r 0 c Callisto's end result Influenced Rotation by the Sun (p.d.) (Negative number means the reduction amount from Callisto's Free Rotation)

9 Part 3 Callisto's Total Rotation s c i i s Callisto's Total Rotation (p.d.) T c 0 0 t 90 s s T Callisto's Sidereal Rotation Period (day) If (T = 0, Callisto's Synchronous Tropical Rotation) Observation T o Callisto's Sidereal Rotation Period (day) If (T = 0, Callisto's Synchronous Tropical Rotation).%Df T T o 00 T T o %Df Percentage deference between the calculation and the observation