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We know velocity of a planet in an elliptical orbit is given by: $$v^2 = GM * (\frac{2}{r} - \frac{1}{a})$$ in an elliptical orbit. [Here r is distance between particle and sun] source

We also know, areal velocity in an elliptical orbit is given by $$\frac{dA}{dt} = \frac{1}{2}vr$$

By putting value of velocity in this equation we find that areal velocity is dependant on r, and thus ever changing(since distance between sun and a planet is also changing)

But keplers second law states that areal velocity of a particle is always constant.

How do I resolve this contradiction? What am I doing wrong? I assumed r would cancel out leaving only contants behind. If I am putting the value of velocity wrong what is the correct one?

Qmechanic
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abhitruechamp
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1 Answers1

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The areal velocity is $$ \frac {dA}{dt}= \frac 12 |{\bf r}\times {\bf v}| $$ and the vector product ${\bf r}\times {\bf v}$ has magnitude $rv$ only if ${\bf r}$ is at right-angles to ${\bf v}$.

mike stone
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