# NavList:

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

**Re: Parallax in azimuth**

**From:**Robin Stuart

**Date:**2020 May 28, 08:29 -0700

My compliments to Dave Walden for posting a number of interesting and thought provoking navigational puzzles recently. With regard to parallax in azimuth; the topic of corrections to parallax in altitude due to the ellipsoidal figure of the Earth was discussed a while ago here and here.

For the particular problem posed*e*^{2} 0.00669454

Mean equatorial radius of the Earth, *a* 6378.14 km

Moon's geocentric distance, *R* 395191.48 km

Topocentric altitude, *h* 32.05569°

Latitude, *L* 39°

Topocentric azimuth, *Z * 88.83136°

The geocentric horizontal parallax is, sin HP = (*a*/*R*)

The result expressed as a series expansion in *e*^{2} and (*a*/*R*) for the geocentric altitude *h*' is*h*' = *h* + (*a*/*R*) cos *h* - (*a*/2*R*) *e*^{2} sin^{2}*L* cos *h* + (*a*/2*R*) *e*^{2} sin 2*L* cos *Z* sin *h*

Assuming a mean HP of 57.7' and plugging in the numbers reproduces the formula found in the Nautical Almanac. At time I didn't derive the equivalent series result for the parallax in azimuth but, having set the problem up as described in the attached document here, it's a simple extension. The result for the geocentric azimuth, *Z*', is*Z*' = *Z* + (*a*/2*R*) *e*^{2} sin 2*L* sin *Z* / cos *h*

I haven't found this formula in Chauvenet (although it might be there somewhere) and it seems more convenient to use than the ones he gives.

Plugging the numbers gives *Z*' = *Z* + 12.858" in close agreement with Kermit.

Robin Stuart