Final answer:
To find the areal velocity for a circular orbit, the orbital speed is used, which for a circular orbit is √(GM/r). Thus, the areal velocity is ½√(GMr), which matches Earth's actual areal velocity about the Sun when using Earth's orbital parameters.
Step-by-step explanation:
The areal velocity for a circular orbit of radius r about a mass M is a measure of the rate at which area is swept out by the line segment joining the center of mass M to the orbiting object. This rate is constant for circular orbits, as per Kepler's second law which states that a line joining a planet and the Sun sweeps out equal areas during equal intervals of time.
To show this, we can start by noting that the areal velocity is ½ times the base times height for the infinitesimal triangle swept out. The base of this triangle is the arc length, which is the orbital speed v times the time interval Δt, and the height is the radius r. The orbital speed v for a circular orbit can be found using the centripetal force which is provided by gravity. This gives v = √(GM/r), and therefore the areal velocity is ½rv = ½ r √(GM/r) = ½√(GMr).
For Earth's areal velocity about the Sun, given the mass of the Sun and Earth's orbital radius, the above formula gives a value that matches the actual Earth's areal velocity calculated using the area of its orbit divided by the orbital period (1 year).