# 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.[1]

## Law of universal gravitation

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

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

## Example

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

• Callisto–Jupiter distance (dp) is 1.883 · 106 km.
• Mass of Jupiter (Mp) is 1.9 · 1027 kg
• Jupiter–Sun distance (i.e. mean distance of Callisto from the Sun, ds) is 778.3 · 106 km.
• The solar mass (Ms) 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 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.[1]

## References

1. ^ a b Asimov, Isaac (1976). Asimov on Astronomy. Coronet Books. pp. 125–139. ISBN 0-340-20015-4.
2. ^ Arny, Thomas. Explorations. Mc Graw Hill. pp. 543–545. ISBN 0-07-561112-0.