Can you pls expain Keplers Laws and their derivation in a simple way ?
where the constant
has the dimension of length. Then
Differentiation with respect to time is transformed into differentiation with respect to angle:
Substitute into the radial equation of motion
Divide by the right hand side to get a simple non homogenous differntial equations for the orbit of the planet:
An obvious solution to this equation is the circular orbit
These solutions are
where and are arbitrary constants of integration. So the result is
Choosing the axis of the coordinate system such that , and inserting , gives:
If this is the equation of an ellipse and illustrates Kepler's first law.
Only the tangential acceleration equation is needed to derive Kepler's second law.
The magnitude of the specific angular momentum
where the expression in the last parentheses vanishes due to the tangential acceleration equation.
The area swept out from time t1 to time t2,
depends only on the duration t2 â t1. This is Kepler's second law.
which is Kepler's third law for the special case.
In the general case of elliptical orbits, the derivation is more complicated.
The area of the planetary orbit ellipse is
The areal speed of the radius vector sweeping the orbit area is
The period of the orbit is
implying Kepler's third law
- Hope this helps.