NavList:

Frank,

I finally got around to thinking about your “Lunars: altitude accuracy” post. Now that I see you’re talking about the effect of error in the moon’s altitude measurement, I derived the same expression you gave,

dLD = (cos h/ 60 tan LD) dH

where I’ve taken the moon’s average HP = 57.3’ to get your round number of “60” (which BTW is dimensionless). This only includes the effect of the moon’s parallax -- refraction and augmentation being small in comparison. Under these approximations and using 14-place precision, I compared the errors computed from both the exact equation and the above approximation. For all cases I tried, of LD, Hmoon and Hsun, the above expression was correct to within about 0.002’ of arc – plenty good enough.

Now from your last post, presumably your expression for error from the sun’s altitude would be

dLD = (cos h/ 60 sin LD) dH

But I don’t see how this same factor of “60” would arise because parallax is not an issue here – or do I misunderstand something again??

Finnally for the P.S.
Yes of course you’re right about the approximate clearing expression. I only showed the effect of one altitude change on the lunar distance. I find it easier to expand the cos LD instead of LD, and when I include the effects of the altitude changes of both bodies, by expanding the exact result in terms of dh and dH to 2nd order, I get :

cos LDo = cos LD +
cos AH cos LD dH + cos Ah cos LD dh +
cos D dH^2/2 + cosLD dh^2/2 +
{cos(h + H) cos(h – H)/cos h cos H + tan h tan H cos LD} dh dH

where h and H are the altitudes of the two bodies. And yes, “I misspoke” about the RBA; cos Ah & cos AH are indeed the “corner cosines” between the LD arc and the adjacent sides, coh and coH, respectively (the co-altitudes). And dh and dH are the small changes in altitudes due to the altitude’s refraction and parallax adjustments. The altitudes h and H are the altitudes of the bodies corrected for everything else, augmentation, semi-diameter, and instrument error.

Loony in Oshkosh,
JK

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