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This is about some minor details in correcting the parallax of Venus, in
view of the small discrepancies that have arisen in tackling Michael Dorl's
Venus lunar.

We are neglecting all the other corrections here, considering the globe to
be spherical, not spheroidal, and considering only parallax, the biggest
correction, which can approach a whole degree.

There's no problem about calculating the Horizontal parallax, HP, the value
for parallax when the Moon is on the horizon. The problem comes in
calculating the way parallax varies with altitude of the Moon.

1. Correcting the Observed Altitude OA to get the True Altitude TA.

I've always happily taken-
parallax P = HP cos OA,

which is then added to OA to obtain TA.  This is in accord with W M Smart's
Spherical Astronomy, section 117, except that he gives it in terms of the
observed zenith angle, which he calls z' so that it becomes

P = HP sin z'

Actually, it's given by Smart as sin P = HP sin z', but for such small
angles (less than 1 degree) sin P is almost exactly the same as P (when
expressed in radians).

This is the way I have always dealt with parallax.

However, it seems to differ from what Meeus offers, in his Astronomical
Algorithms. In chapter 40, under "Parallax in horizontal coordinates", he
says-

"Due to the parallax, the apparent altitude of a celestial body is smaller
than its "geocentric" altitude h. Except when high accuracy is needed, the
parallax P in altitude may be calculated from

sin P = sin HP cos h.

Except in the case of the Moon, the parallax is so small that we can
consider P and HP to be proportional to their sines, and then we have

P= HP cos h."

To the accuracy that concerns us, that approximation of sines by their
angles is perfectly good, even in the case of the Moon.

But note the difference between Smart and Meeus. Smart multiplies HP by the
cos of the observed alt OA; Meeus multiplies it by the cos of the
"geocentric", or true angle TA, as I understand it. They can't both be
right.

Why does Meeus say that his sine formula is valid "except when high
accuracy is needed"? Usually, he is very helpful to the reader, but here,
he does not go on to explain what should be done if higher accuracy is
required. Is the reason for his inaccuracy that the cos multiplier should
really be cos OA, not cos h?

What I read in Meeus is somewhat disturbing, but I still think the correct
procedure is what I have outlined above, multiplying by cos (observed
altitude). Any comments from our geometers / trigonometers?

2. Correcting the True Altitude TA to compute what the Observed Altitude OA
should be (as if it had been observed).

This is the converse problem, which arises when the Moon's altitude has not
been observed. It arose in last week's question from Michael Dorl.

You might think that all you have to do is to subtract an equivalent
correction from TA to get OA. The correction, remember (but not Meeus'
correction) was-

OA = TA - P , or OA = TA - HP cos OA

That would be OK, but you don't yet know OA to find its cosine.

If instead we approximated that by the expression OA = TA - HP cos TA, we
will get significantly wrong answers

Try some numbers to see.

For correcting OA to find TA, take HP = 55', OA = 45deg. Then parallax P is
55' cos 45deg or 38.8909 minutes. So TA is 45deg 38.89', or 45.6482
degrees. Fair enough.

For the reverse operation, we are faced with a TA of 45.6482 and we might
calculate OA by subtracting off a correction 55' cos 45.6482. The
correction is 38.4484 minutes, or 0.6408 degrees, to be subtracted from our
value of TA of 45.6482, which will give a value for OA of 45.0074 deg.

So this operation hasn't (quite) returned us to the original value. The two
corrections differ by .0074 degrees, or 0.44 arc-minutes. Not a lot, but
significant in terms of a lunar distance "clearance" calculation, which is
expected to be particularly exact. Interestingly, it seems to be very
similar in magnitude to the discrepancy between Frank Reed's solution to
Michael Dorl's problem, and my own.

To increase accuracy, we need to find a better approximation, to bring us
closer to-
OA = TA - HP cos OA, when we don't yet know what OA is.

My own routine, for reverse correction for parallax of the Moon's TA, takes
the Moon's distance D from Earth's centre in Astronomical Units (which has
been computed elsewhere) to determine HP.

The angle to be subtracted from TA to obtain OA is, in this routine-

Arc-tan ( cos TA / ((23455 * D) - sin TA))

The amount of this correction now corresponds very closely (with a change
of sign) to the correction that's added to OA to obtain TA.

To simplify matters, I have avoided referring to the correction- "Reduction
in the Moon's Horizontal parallax" which allows for the spheroidal figure
of the Earth, but such a correction needs to be made (and is).

George.

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contact George Huxtable by email at george@huxtable.u-net.com, by phone at
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