Tug of war (astronomy)

According to Isaac Newton's law of universal gravitation

${\displaystyle F=G\cdot {\frac {m_{1}\cdot m_{2}}{d^{2}}}}$ 

In this equation

F is the force of attraction

G is the gravitational constant

m₁ and m₂ are the masses of two bodies

d is the distance between the two bodies

The two main attraction forces on a satellite are the attraction of the Sun and the satellite's primary (the planet the satellite orbits). Therefore, the two forces are

${\displaystyle F_{p}={\frac {G\cdot m\cdot M_{p}}{d_{p}^{2}}}}$ 

${\displaystyle F_{s}={\frac {G\cdot m\cdot M_{s}}{d_{s}^{2}}}}$ 

where the subscripts p and s represent the primary and the sun respectively, and m is the mass of the satellite.

The ratio of the two is

${\displaystyle {\frac {F_{p}}{F_{s}}}={\frac {M_{p}\cdot d_{s}^{2}}{M_{s}\cdot d_{p}^{2}}}}$ 

Callisto is a satellite of Jupiter. The parameters in the equation are ^[2]

• Callisto–Jupiter distance (d_p) is 1.883 · 10⁶ km.
• Mass of Jupiter (M_p) is 1.9 · 10²⁷ kg
• Jupiter–Sun distance (i.e. mean distance of Callisto from the Sun, d_s) is 778.3 · 10⁶ km.
• The solar mass (M_s) is 1.989 · 10³⁰ kg

${\displaystyle {\frac {F_{p}}{F_{s}}}={\frac {1.9\cdot 10^{27}\cdot (778.3)^{2}}{1.989\cdot 10^{30}\cdot (1.883)^{2}}}\approx 163}$ 

The ratio 163 shows that the solar attraction is much weaker than the planetary attraction.

Asimov lists tug-of-war ratio for 32 satellites (then known in 1963) of the Solar System. The list below shows one example from each planet.

Unlike other satellites of the solar system, the solar attraction on the Moon is more than that of its primary. According to Asimov, the Moon is a planet moving around the Sun in careful step with the Earth.^[1]