# When Averaging Fails

## Earth's Orbit, and pertubations by Jupiter

A colleague claims that the Earth orbits the center of mass of the solar system.

This is incorrect ... but why? Let's explain.

Gravity is inverse square, so Jupiter's effect on the Earth's orbit, while nontrivial, is attenuated by the square of the distance Indeed, Jupiter pulls on both the Earth and the Sun. When Sun and Jupiter are in opposition (opposite sides of the Earth), Jupiter pulls a little stronger on the Earth than the Sun, and when they are in conjunction (same side of the Sun), Jupiter pulls a bit more strongly on the Sun than the Earth. In both cases, the net effect is to pull the Sun and Earth apart ... a tiny bit, about 30 parts per million of the Sun's gravitational pull.

Warning: Simplifications Follow:

Assume circular orbits, with the radius equal to the semimajor axis. In conjunction, the distance from the Earth to Jupiter is A+1, where A = 5.2044 AU is Jupiter's radius in astronomical units. The ratio of Jupiter's mass to the Sun's mass is M = 9.5456e-4 . So, Jupiter's gravitational acceleration the Earth, relative to the Sun's acceleration, is acc = M/(A+1)^2 . The effect on the Sun is acc = M/A^2 , and the difference between these accelerations is diff_c = M ( 1/A^2 - 1/(A+1)^2 ) .

Similarly, in opposition, the difference is diff_o = M ( 1/(A-1)^2 - 1/A^2 ) . Both tend to pull Sun and Earth apart, but the opposition effect is a 80% larger.

Jupiter orbits the Sun every 11.862 earth years. The synodic period is 398.88 earth days or 1.092 earth years; that is the average time between pairs of conjunctions and oppositions.

Here are the masses (normalized to the Sun) and average distances from the Sun (normalized to the 1AU Earth orbit) for the four large planets:

 Jupiter Saturn Uranus Neptune Mass in Suns 9.5456E-4 2.8581E-4 4.3655E-5 5.1510E-5 Distance in AU 5.2044 9.5826 19.2184 30.1104 Center of Mass in AU 4.9679E-3 2.7388E-3 8.3898E-4 1.5510E-3 tidal effect in opposition 1.8758E-5 7.6756E-7 1.3331E-8 3.9704E-9 tidal effect in conjunction 1.0445E-5 5.6044E-7 1.1403E-8 3.5937E-9 tidal effect average 1.4601E-5 6.6400E-7 1.2367E-8 3.7821E-9 tidal effect ratio 1.80 1.37 1.17 1.10

AverageFail (last edited 2017-10-20 05:10:09 by KeithLofstrom)