# Dependence of areal velocity on distance between sun and planet

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?

• Here's a nice animated diagram showing the relationship between the radius & velocity vectors in an elliptical orbit, plus a few other goodies. gregegan.net/SCIENCE/LRL/LRL.html Apr 23, 2023 at 14:33
• Definately looks cool, it's apparant here now that r and v are really not perpendicular. Apr 23, 2023 at 14:59
• You may like this answer of mine on the connection between the magnitudes of the tangential velocity, radial velocity, and angular momentum. Also check out the eccentricity vector. Apr 23, 2023 at 15:09

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 $${\bf v}$$.