This is Kepler's Problem, and this is what, one of the reasons offered for why people resisted the area rule. The other reason, of course, is, why should the planets sweep out equal areas at equal times? Said crudely, how can they know, how could they be calculating the areas, et cetera?
But the problems in working with this are by themselves enough. So, here's the thing. Start at the top of the line, this is my writing. Fraction of the period. It is the mean speed, mean angular motion. That times the time from when you started at aphelion at over 2 pi, is a fraction of the total time.
Right? That's how we're gonna specify time. As how far away you are from the aphelion after a certain number of mean increments. So, we're gonna do it in terms of the mean anomaly, m, which is m over 2 pi. So far, so good. That's the fraction of the circle.
We now look at the area of the sector, asp in the ellipse, over the area of the ellipse. And we notice that the ratio of those two areas is identical to the ratio of the area of the sector of the circle to the whole circle. Why? Because when you drop those perpendiculars, q, p, down, the ratio of the height removed is always a constant all the way around of the ellipse to the circle.
That may not make sense, but in other words, looked at that way, each little slice of the ellipse is the same proportion of the slice of the circle above it. Fair enough? So, we can write out the solution for the circle and it's simply a square over 2 times e times a square e2 times two the sine of e.
You can do that trigonometry yourself. You cancel all the terms out and you get e times e times the sine of e. That became Kepler's equation, the thing at the bottom. That's a transcendental equation. Given the time, how do you solve for e? And if you think about it just a moment, sine of e is an infinite series, e is not an infinite series.
This is an infinite equation. It's a transcendental equation. They didn't know enough to know what a transcendental equation was. That's Linus's term I'm pretty sure. But, the point is that meant you could not simply calculate given time, where you are in a planet except iteratively. I took a course in my course in celestial mechanics under Brower was my sophomore year, handheld calculators didn't exist yet.
And all the problems you constantly regret being asked to find where a planet was at some time. And you sat there doing that bloody iterative calculation to figure out how to do it to whatever level of accuracy he was demanding. Kepler supplied tables. But then he makes, this is from the Epitome.
He makes the following remark, here there is no direct way, but one who wishes to compute this without tables must employ the rule of assumptions. What's the rule of assumptions? Namely as shown in the following figure. Assuming the eccentric anomaly arc ap prime has such and such an amount and for the eccentric anomaly so assumed, computing its mean anomaly, asp prime.
And if the result is the amount of the mean anomaly that was proposed to the eccentric anomaly, will have been assumed correctly. But if the result is not such an amount, the assumption will have to be corrected by means of the result and the work repeated. And that's what you'd do.
And I will pass it around right after break. There's a whole book here, Solving Kepler's Equation Over Three Centuries. It's still with us. We do it all the time now. It's very quick, but clever solution, fast converging solution after fast converging solution was found. That was the trick.