1. Areal velocity definition

For a particle moving in a plane, the areal velocity is

$\frac{dA}{dt} = \frac{1}{2} r^2 \dot{\theta}$

where $r$ is the distance from the center and $\dot{\theta}$ is the angular speed.

2. Angular momentum of the planet

For a planet of mass $m$ moving in a circular orbit,

$L = m r^2 \dot{\theta}$

3. Relate areal velocity to angular momentum

From the above expression,

$r^2 \dot{\theta} = \frac{L}{m}$

Substitute into the areal velocity formula:

$\frac{dA}{dt} = \frac{1}{2} \left(\frac{L}{m}\right) = \frac{L}{2m}$

4. Match with options

$\frac{L}{2m}$ corresponds to Option C.

Final Answer

The areal velocity is

$\boxed{\frac{L}{2m}}$

So the correct option is C.