Let P(x,y) be the position of the planet after time t after starting from rest from perihelion position A , referred to S as origin. Then
x=acosθ−ae, y=bsinθ By Kepler's second law of motion, we have: the radius vector drawn from the centre of the sun to the planet sweeps out equal area in equal time. The rate of description of area is constant:
h=2(rate of description of area)
2h=21r2dtdν Hence
h=xdtdy−ydtdx
dtdx=−asinθdtdθ, dtdy=bcosθdtdθ
h=(acosθ−ae)bcosθdtdθ+bsinθ(asinθdtdθ)
h=ab(cos2θ+sin2θ−ecosθ)dtdθ
h=ab(1−ecosθ)dtdθ
hdt=ab(1−ecosθ)dθ Integrate
ht=ab(θ−esinθ)+c1 Initially when t=0, the planet is is at A whose eccentric angle θ=0.
Find c1
h(0)=ab(0−esin(0))+c1=>c1=0 Therefore
ht=ab(θ−esinθ)
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