Tug of war (astronomy)

The tug of war in astronomy is the ratio of planetary and solar attractions on a natural satellite. The term was coined by Isaac Asimov in The Magazine of Fantasy and Science Fiction in 1963.

Law of universal gravitation

According to Isaac Newton's law of universal gravitation

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

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

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

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

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

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

Example

Callisto is a satellite of Jupiter. The parameters in the equation are

• 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

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

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

The table of planets

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.

Primary	Satellite	Tug-of-war ratio
Neptune	Triton	8400
Uranus	Titania	1750
Saturn	Titan	380
Jupiter	Ganymede	490
Mars	Phobos	195
Earth	Moon	0.46

The special case of the Moon

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.