Tug of war (astronomy)

Ratio of gravitational forces on a satellite from the Sun and host planet

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

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According to Isaac Newton's law of universal gravitation

F = G ⋅ m 1 ⋅ m 2 d 2 {\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

F p = G ⋅ m ⋅ M p d p 2 {\displaystyle F_{p}={\frac {G\cdot m\cdot M_{p}}{d_{p}^{2}}}}
F s = G ⋅ m ⋅ M s d s 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

F p F s = M p ⋅ d s 2 M s ⋅ d p 2 {\displaystyle {\frac {F_{p}}{F_{s}}}={\frac {M_{p}\cdot d_{s}^{2}}{M_{s}\cdot d_{p}^{2}}}}

## Example

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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

F p F s = 1.9 ⋅ 10 27 ⋅ ( 778.3 ) 2 1.989 ⋅ 10 30 ⋅ ( 1.883 ) 2 ≈ 163 {\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.

## The table of planets

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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

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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

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1. ^ a b Asimov, Isaac (1976). Asimov on Astronomy. Coronet Books. pp. 125–139\. ISBN 0-340-20015-4.
2. ^ Arny, Thomas (August 1997). Explorations. Mc Graw Hill. pp. 543–545\. ISBN 0-07-561112-0.