Why can't we use the semi-minor axis in Kepler's third law? Why can't we use the semi-minor axis in Kepler's third law? The formula for the third law is:
$$\dfrac{a^3}{T^2} = \text{const.} $$
Instead of semi-major axis (a) why can't we use the semi-minor axis (b)?
$$\dfrac{b^3}{T^2} \neq \text{const.} $$
I just checked it (I calculated the results for some planets using the semi-minor axis) and I really don't know why it's not constant.
The relationship between $a$, $b$ and the eccentricity of an orbit $e$ is
$$ a = b\left(\frac{1+e}{1-e}\right)\ .$$
It is clear therefore from a mathematical point of view that you cannot get a similar constant relationship by replacing $a^3$ by $b^3$ in Kepler's third law. You would have to use an expression involving both $b$ and $e$.
Another way of saying this is that you can have (infinitely) many orbits that have the same $a$ and $T$, but which have different $b, e$ combinations.
I guess from a fundamental point of view it boils down to being able to express the total energy of an orbit as
$$ E = -\frac{GMm}{2a}$$
which is independent of the shape of the ellipse.