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 (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.

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.

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[]