Question

Use least squares to find Keplers 3rd Law

Planets Mercury & Neptune

Mercury: Semi-Major Axis 0.387098 AU Rotate Around Sun in Earth Years: 0.240846

Neptune: Semi-Major Axis 30.11 AU Rotate Around Sun in Earth Years: 164.8

Answer 1

Kepler's 3rd law states that

,

where P=period of revolution around the sun

X=length of semi-major axis.

In the problem , using the two data given , we have to derive the Kepler's 3rd law, i.e the relation between P and X.

using the two data points, given in the problem, we try to fit the data in a straight line, using least square fit mehtod. the slope and intercept of the line will give the required relation.

we will fit the data in the graph between log(P) and log(X).

here n=number of data points

_{subscript 1 represent for mercury, and 2 represent for neptune.}

putting these values in the above equation we get and .

Hence from the analysis , it is clear that using the least square fit , we arrived at the Kepler's 3rd law.

,

where P=period of revolution around the sun

X= length of semi-major axis.