How to calculate parallactic angle from a fixed alt-az position?

Question

Say we have an observer at a specified latitude and longitude on Earth who would like to observer a specific fixed alt-az sky position (rather than RA and dec), for example to look at geosynchronous satellites. Since the alt-az position of the observation is not changing, it seems to me that parallactic angle also would not be changing and so should be derivable from just alt-az and lat-long, but I haven't been able to figure out how to do it.

All the calculations I've seen online use RA or hour angles in calculating parallactic angle, which intermediary values I would like to avoid if mathematically possible.

Answer 1

I found an equation on this website,

$\require{cancel} \cancel{\sin{\theta_p}=\sin{\theta_{az}} \cos{\theta_{lat}}/ \cos{\delta}}$,

with $\theta_p$ being the parllactic angle, $\theta_{az}$ being the azimuth, $\theta_{lat}$ being the latitude of the observing stations, and $\delta$ being the declination. The website cites this equation as being from "Spherical astronomy, Small pg. 49." I am unable to find such a book (based on my searches, I'm wondering if the correct reference might actually be "Textbook on Spherical Astronomy, Smart", which I do not have ready access to), but in my experience so far this equation has been correct.

The above equation, my original answer, seems to provide something similar to the parallactic angle (something like the complement in each quadrant, with an extra negative sign, but I haven't investigated in detail). However, when testing against values gotten from astropy, it did not match the astropy values.

Instead, I found the following, taken from Astronomical Algorithms by Jean Meeus, which does match the astropy values:

$\theta_p = \text{atan2}( \sin{\theta_H}, \tan{\theta_{lat}}\cos{\delta} - \sin{\delta}\cos{\theta_H}),$

with $\theta_H$ being the hour angle, calculated via

$\theta_H = \text{atan2}( \sin{\theta_{az,mod}}, \cos{\theta_{az,mod}} \sin{\theta_{lat}} + \tan{\theta_{el}} \cos{ \theta_{lat} } ),$

Where $\theta_{el}$ is the elevation and $\theta_{az,mod}$ is the azimuth, but defined differently from $\theta_{az}$ as used originally and elsewhere in my answer. The zero-point $\theta_{az}$ is due north, increasing clockwise, and the zero-point $\theta_{az,mod}$ is due south, increasing clockwise.

Declination can be gotten from latitude, elevation, and azimuth:

$ \sin{\delta} = \sin{\theta_{el}} \sin{ \theta_{lat} } + \cos{ \theta_{el} } \cos{ \theta_{lat} } \cos{\theta_{az}} $,

Thus indeed, given an alt-az (or az-el depending on your preferred terminology) position and the latitude of the observing station, it is unnecessary to use RA or an independently determined hour angle in a calculation of parallactic angle. I say "independently determined hour angle" meaning that although hour angle is used in the calculation, it can be determined entirely from az/el and latitude, so one does not need to provide an hour angle for this calculation to work.