Tug of war (astronomy)

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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[edit]

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}


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
 \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[edit]

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[edit]

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]


  1. ^ a b Isaac Asimov: Asimov on Astronomy Coronet Books,1976, ISBN 0-340-20015-4 pp125-139
  2. ^ Thomas Arny: Explorations, Mc Graw Hill, ISBN 0-07-561112-0 pp.543-545