Planar Kinematics: Circular orbit?

Question

Assuming that the Earth's orbit around the Sun is circular, determine the mass of the Sun. It is also assumed that the mass of the sun is much greater than the mass of Earth. It is known the Earth completes a full orbit in 1 year.

Attempted Solution:

I want to do this problem without assuming a value for the radius of Earth's orbit.

Starting point: Kepler's Third Law: $$T^2 = \left(\frac{4\pi^2}{GM}\right)r^3$$ unknowns: $M$ and $r$

Since the orbit is circular, the speed $v$ is constant. More so, the acceleration is: $$\mathbf{a}_e = -r \dot{\theta}^2\hat{\mathbf{e}}_r$$ from planar kinematics of a circular orbit where $\hat{\mathbf{e}}_r$ is the unit vector that points in the direction of radius to Earth.

Here is where I am stuck unfortunately, I want to use the 2D planar kinematics equations of motion to solve for $r$ which can then be used in Kepler's Third Law to get $M$. Any hints would be greatly appreciated!

Answer 1

Kepler's second law for circular motion can be obtained by the following equations

• $\omega=\frac{2\pi }T\quad a_c=\omega^2r$
• $ma_c=\frac{GMm}{r^2}$

therefore to obtain $M$ from the second law we need some more information to obtain $r$.