## Monday, April 08, 2019

### [rjkjfjug] Eccentricity and the inverse square law

We consider a body in an elliptical orbit around the sun.  The following table gives eccentricity, the ratio of distances between aphelion and perihelion, and the ratio of sunlight received between perihelion and aphelion.

Eccentricity Distance ratio Power ratio
0.000 1.000 1.000
0.001 1.002 1.004
0.002 1.004 1.008
0.005 1.010 1.020
0.010 1.020 1.041
0.020 1.041 1.083
0.050 1.105 1.222
0.100 1.222 1.494
0.200 1.500 2.250
0.300 1.857 3.449
0.400 2.333 5.444
0.500 3.000 9.000
0.600 4.000 16.000
0.700 5.667 32.111
0.800 9.000 81.000
0.900 19.000 361.000
0.950 39.000 1521.000

An eccentricity of (sqrt(1.1)-1)/(sqrt(1.1+1)) = 0.0238 is the eccentricity that sees 10% additional power from the sun at perihelion.  Perhaps that is the threshold eccentricity beyond which an orbit ought not be called "roughly circular", because 10% is (subjectively) a significant difference in power.  Currently, only Venus, Neptune, and Earth have roughly circular orbits around the sun.  (Eccentricities change over time, because Jupiter.)

Mercury's eccentricity is 0.21.  Other planets.

At e = 0.172, the power ratio is 2.

At e = 1/3, the distance ratio is 2.