What is the "sunrise equation" for non-constant declination component?

Question

The standard Sunrise Equation (refraction neglected) is this:

$$ \cos(\text{hour angle}) = \tan(\text{declination})\tan(\text{latitude}) $$

This equation - which really has nothing to do with the Sun specifically (and hence my question trigger) - does seem to assume a constant declination. which is somewhat circular: since the declination of the Sun is not constant and given that we don't know the exact sunrise time we also don't know the exact declination. In the case of the Sun this error seems small and for practical proposes (especially with refraction and twilights) is probably quite good within seconds.

But, what about other bodies that move faster like the Moon, or if we pursue the theoretical accurate values? Maybe I'm lacking the correct terminology here, but I could not find any equation or a process if there is no analytical solution.

Answer 1

This equation - which really has nothing to do with the Sun specifically (and hence my question trigger) - does seem to assume a constant declination.

It might seem so but it doesn't. We can say that the equation is a necessary condition but it's not sufficient to solve the problem.

In practice you would get a list of hour angles and declinations for your object that bracket the expected time of rise or set either "writing with feathers using light from burning animal fat" or in a spreadsheet, and for each you'd calculate how far off you are from this equation being correct. In other words:

$$\Delta = \tan(\text{declination})\tan(\text{latitude}) \pm \cos(\text{hour angle})$$

where the $\pm$ distinguishes between rising and setting and $\Delta$ is some indicator of how far away the time of this line in the spreadsheet is from the correct time.

It's not linear but it's at least monotonic near the correct time.

You'd then find the two lines where the sign of $\Delta$ changes and start trying to converge on the time where it is close enough to zero (based on how accurately you need to know the rise/set time) using something simple like a binary search or if you can take derivatives Newton's method or if in Excel there is even a goal seek function.

So:

1. Get an ephemeris for declinations and hour angles for times spaced before and after the expected timeframe
2. Iterate and converge on a solution by either finding the zero crossing of $\Delta$ and asking for more ephemeris points, or using some kind of interpolation, either linear or otherwise.