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

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?

• The article is paywalled for some readers (me at least) I can try again from a more academic location but if your question is about the analytical method I think you should write it out here for others to try it and propose a way to use it. Also have a look at Laplace Method for Orbit Estimation which includes some potentially helpful links – uhoh Dec 16 '20 at 1:35
• Thanks @uhoh, I added more details to the equations. Describing everything might be a bit tedious though, and I trust people will find ways to download papers given their DOI which do not necessarily involve paying a certain amount of money to a scholarly publisher or enrolling a research institution. – astrojuanlu Dec 16 '20 at 17:56
• Thanks for the edit! Yes this is a challenge; you have a perfectly reasonable and interesting question about a technique/procedure and it requires an extensive description, but that lengthy description seems to be paywalled. I would love to know some of those ways, is this one? Are there any others? I'd love to read this paper! – uhoh Dec 16 '20 at 21:46
• I didn't want to mention it explicitly, but yes :) (that's how I obtained the paper myself, by the way!) – astrojuanlu Dec 17 '20 at 22:10