Every few hundred years most of the planets line up on the s

Every few hundred years most of the planets line up on the same side of the Sun. Calculate the total force on the Earth due to Venus. Jupiter, and Saturn, assuming all four planets are in a line. The masses are mV=0.8l5mE, mJ=3l8mE, mSat = 95.1mE, and the mean distances of the four planets from the Sun are 108. 150, 778. and 1430 million km. What fraction of the Sun\'s force on the Earth is this?

Solution

Because we are interested in the ratio of the forces, we can simplify the maths by using units where:
- mass of earth = 1unit (1Me)
- the unit of distance is 1 million km
- G = 1 unit
(conversion factors to SI units will cancel when we take the ratio)

Mass of sun = 1.99 x 10^30 /5.98 x 10^24 = 3.328x10^5 units

Sun-earth force, F = GMm/d^2 = 1 x 3.328x10^5 x 1 / 150^2 = 14.8 units

Venus-earth force, FV = 1 x 0.815 x 1 /(150-108)^2 = 4.62x10^-4 units

Jupiter=earth force, FJ = 1 x 318 x 1 /(778-150)^2 = 8.06x10^-4 units

Saturn-earth force, FS= 1 x 95.1 x 1 /(1430-150)^2 = 5.80x10^-5 units

Total force, FT, from the 3 planets on earth (remembering Venus pulls in the OPPOSITE direction to Jupiter and Saturn) is:

FT = -FV + FJ + FS = -4.62x10^-4 + 8.06x10^-4 + 5.80x10^-5 = 4.02x10^-4 units

The ratio FT/F = 4.02x10^-4 / 14.8 = 2.72x10^-5

Since 1/2.72x10^-5 = 36800, as a fraction the ratio is 1/36800