Correlation between equations of elliptical orbits and pendulums The equation for the period of a pendulum is:
$$T=2π\sqrt{\frac{L}{g}}$$
Where 'g' is the acceleration due to the gravitational field and 'L' is the length. 
The equation for the period in of a body travelling along an elliptical orbit is:
$$T = 2π\sqrt{\frac{a^3}{GM}}$$
Where 'a' is the semi-major axis. I can see that this is derived from Kepler's 3rd law. 
Is there are similar equation to Kepler's 3rd law for pendulum periods?
If a pendulum string is of fixed length, does that essentially make the motion of a circular orbit? It seems like a very similar relationship between length, gravity and periods.
My question: "Is there a deeper, more fundamental relationship between these equations?
 A: The equation for the period of a pendulum $(T=2π\sqrt{\frac{L}{g}})$ is only an approximation. That equation assumes, among other things, that gravity doesn't change with height, and that $\sin(\theta) = \theta$. 
Even if there were a connection between that approximation and elliptical orbits, that would not imply any connection between the true period of a pendulum and the period of an elliptical orbit.
A: We derive the time period of a pendulum T=√(L/g) considering the angular displacement of the pendulum , θ<4° i.e. the motion of the pendulum is approximately linear.
And the equation of S.H.M. is F∞-x, i.e. force acting on the body, executing S.H.M. , must be in same line of the displacement of the body.
So when a body moving in a elliptical does not execute S.H.M. like a simple pendulum.
Hence there is no logic behind the comparison of the time period of a simple pendulum and body moving in a elliptical orbit.    
A: These equations correspond to the period of a harmonic motion, which generally have the form
$$T=2 \pi \sqrt{\frac{m}{k}}$$
