I considered posting this to astronomy.stackexchange.com, but I've bugged them enough for today...
Let $p(t)$ be a parametric function that traverses an ellipse such that it sweeps out equal areas in equal time, as per the diagram above. In other words, $p(t)$ is the ideal elliptical orbit (no 3rd party peturbations). Additionally:
The ellipse's semimajor axis has length $m$.
The ellipse's eccentricity is $e$.
$p(t)$'s period is $T$. In other words, $p(T) = p(0)$
Question: what circle best fits this ellipse? More specifically, if we parametrize a circle as:
$$c(t) = \big(x_0+r\cos(bt-c), y_0+r\sin(bt-c)\big) $$
what values of $x_0$, $y_0$, $b$, and $c$ would minimize:
$$\int_0^T d(c(t),p(t)) \, dt$$
where $d$ is the linear distance between $c(t)$ and $p(t)$?
By introducing $x_0$, $y_0$, $b$, and $c$, I'm allowing for the possibility that:
The best fit circle's center is different from the ellipse's center.
The best fit circle is "out of phase" with the ellipse.
The best fit circle's period is not $T$, the ellipse's period.
These all seem unlikely (especially the last 2), but I want to allow for the most general parametric circle.
This ultimately answers the question: if we ARE going to pretend a planet's orbit is circular, what's the best circle to use?
(as a humorous note, stackexchange suggested my question was subjective and would most likely be closed, possibly because I used the word "best". Of course, in mathematics "best fit" is a perfectly valid, non-subjective concept)