Efficient method to compute rise and set times of satellites (visibility problem)?

Question

Today I found the paper "Rise and Set Time of a Satellite about an Oblate Planet" (Pedro Ramón Escobal, 1963), which presents "a closed-form solution to the satellite visibility problem". It reduces to finding the roots to this trascendental equation, which only has the eccentric anomaly $E$ as independent variable:

$$ F \triangleq a (\cos{E} - e)\mathbf{P}\cdot\mathbf{Z} + a\sqrt{1 - e^2}\sin{E} \mathbf{Q}\cdot\mathbf{Z} - G = 0 $$

where $\mathbf{P}$ and $\mathbf{Q}$ are functions of $(\Omega, \omega, i)$, $\mathbf{Z}$ is the unit vector pointing towards the geodetic zenith as a function of $E$, and $G$ is a constant related to the site geodetic coordinates and the planet ellipsoid of choice.

I implemented all the equations, you can find a rough draft on this open pull request to orbit-predictor.

However, after reflecting on this a lot I don't see the big advantage of this method. Several times in the article, Escobal says that this equation gives a "better or improved estimate", but gives no hints about how to find the nearest root of the trascendental equation. Yes, there is an approximate method, but it doesn't work for mid inclinations or altitudes above LEO, so it's not useful for the general case.

I have read several articles that propose numerical methods, and they all cite Escobal "analytical method", without really clarifying whether it works or not for their use case. It seems to me that, even though everybody cites this seminal work, nobody has really tried to reproduce or implement it.

Am I missing something? Do people just implement some sort of "brute force" method with a resolution of tens of seconds to minutes and that's it?

Answer 1

From what I can tell, you're right -- every article I've read on the topic cites Escobal, because they all feel they have to acknowledge the existence of an analytical solution, but then they all describe their favorite numerical method, and give tables of its performance on various sets of simulated data, but none of them seem to actually compare their results to Escobal's method.

However, these methods aren't just brute force -- they represent many different attempts to construct a function in the same spirit as Escobal's, the roots of which give the value of some chosen parameter (not necessarily a classical anomaly) at which the rise/set occurs, and then solve that numerically (as indeed even Escobal's must be), by various approximation methods. For example:

• Lawton 1987 ("Numerical method for rapidly determining satellite-satellite and satellite-ground station in-view periods", JGCD 10.1, 32) uses Fourier series, but it only works well for low eccentricity.

• Alfano, Negron, and Moore 1992 ("Rapid Determination of Satellite Visibility Periods", J Ast Sci 40.2, 281-296) generalize this and require lots of fitting parabolas and solving cubics, but claim good results for many orbit types.

• Han, Gao, and Sun 2016 ("Rapid Satellite-to-Site Visibility Determination Based on Self-Adaptive Interpolation Technique", arxiv) is new to me, but looks very interesting.

That said, if you want really accurate determination of the visibility times, considering things like high order earth gravity models, atmospheric refraction and delay of the signal of interest, terrain obscuration, and other real-life complications, there's no substitute for brute force refinement of an approximate answer given by some such method. There simply is no analytic, or even sort of analytic, solution that attempts to include effects like these.

On the other hand, it's not clear why it would be useful to work this hard, so something easy to code that gets you to within a few tens of seconds is good enough for most practical purposes I can imagine. If you're not just doing a simulation, you're actually trying to get signals to and from a real satellite, you need to have an antenna that can track the object. You point it at the horizon where and a bit before your high-accuracy numerical propagator says it should appear, and you start trying to close the link. Once you do, your phased-array or monopulse or other antenna tracking hardware driver kicks in, and it and the numerical propagator do the rest. Seen this way, rise times are important but set times are trivial: you just keep trying to send or receive the signal until it doesn't get through anymore, and then you set up to begin the next pass.