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Re: Moon Parallax, Math Trivia (was Re: venus)
From: Herbert Prinz
Date: 2004 Oct 22, 01:04 -0400

```Frank Reed wrote:

> Herbert P wrote:
> "I believe you are mistaken. There is no resemblance to Kepler's
> equation. The characteristic feature of the latter is that the
> variable
> E (eccentric anomaly) appears as a term E as well as a term sin E,
> making it impossible to express E as explicit function of the mean
> anomaly. "
>
> Here's the direct equation for the calculation of the true altitude
> from the observed altitude and the horizontal parallax:
>   TA = OA - HP cos TA .
> OK? You agree, I hope, that this bears a very STRONG resemblance to
> Kepler's equation. Now invert it. Solve for TA. Can't be done in
> "closed form", right? See what I mean now?? If you solve this
> iteratively for TA to, say, one part in a million and then compare
> with the Arctan solution, you will get slightly different results.
> They are not inverses. They solve slightly different mathematical
> problems. That's the "trivia" I was getting at.

Frank,

In your message of Oct 20, you said

"Now for a bit of trivia: the Arctan formula, which is plenty accurate
enough for practical puposes, is actually an approximation."

This is the message I was responding to. The Arctan formula is strict.

(I understand that you were arguing that the other formula has no closed
form inversion and can thus not be equivalent to the Arctan formula. But
the conclusion that you should have drawn is that the other formula is
an approximation. Which it is. But George said this already.)

Herbert Prinz

```
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