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## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: venus**

**From:**George Huxtable

**Date:**2004 Oct 11, 17:12 +0100

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. ================================================================ contact George Huxtable by email at george---.u-net.com, by phone at 01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. ================================================================