1. Areal velocity definition

For a particle moving in a plane, the areal velocity is the rate at which the radius vector sweeps area:

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

2. Angular momentum of the planet

The angular momentum magnitude for a planet of mass $M$ is

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

So,

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

3. Relate areal velocity and angular momentum

Substitute into the formula for areal velocity:

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

Hence,

$\frac{dA}{dt} = \frac{L}{2M}$

4. Match with the options

The correct option is:

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

So, Option B is correct.