# Predicting Satellite Flyover Calculations by Hand

Recently I used transit-finder.com to get a really cool picture of the ISS transiting the sun. The prediction from the website was spot-on. Since then I'm absolutely fascinated by how the prediction works. I have a background in engineering and math but not really sure where to start in order to make my own flyover or transit predictions. I see that there's plenty of freeware and websites that make these predictions but I can't find anything that really points me to the math behind what's happening. Does anyone know where I can find some sample problems/calculations or maybe something that walks through this process?

I guess more formally my question is: given the TLE data from a given satellite, how can I predict future flyovers of my location on my own?

There are approximate hand calculations you can do following Kepler's law but one quickly runs into problems that need to be solved numerically. Half-way through this answer I realised this is almost certainly a duplicate of some prior questions so you may find it gets closed by moderators, but I started so I'll finish.

If you want TLEs to be your starting point then the usual assumption is that you need an SGP propagator (a numerical model) to get anywhere. However in the interests of understanding why, here is a bit more explanation:

• Let us start by throwing away some of the information implied by the TLE and say that the orbit is just an ellipse with an eccentricity and a semi-major axis. We can then solve for position and time anywhere around that ellipse using geometry. Some bits of this are hand calcs, depending on which way you approach it.
• The ellipse is itself in motion relative to the Earth turning underneath it. This is still a hand calc and for some orbits this is still ok to predict roughly where a satellite will be on the next time around.
• The ellipse also does not remain fixed both with respect to either the inertial Earth, the Sun or the background of fixed stars. There are some relations for the Earth oblateness (flattening of sphere), tri-axiality (lumps corresponding to major continents), the gravitational effects of any number of other solar system bodies starting with the moon and the sun, solar radiation pressure and the Earth's atmosphere. Some of these change the orientation of the ellipse or its shape so that to still call it an ellipse is just a convenient handle. There are hand calc approximations for each of these but by the time you put it all together you will have made your own numerical propagator.
• You can do a numerical solution considering i) each orbit at a time, so that each update is an average for that orbit, ii) a full force model that runs at a smaller time step, or iii) somewhere between the two. The SGP propagator that works with TLEs follows approach iii) and is well known to be inacccurate for some purposes but ok for predicting most passes.

Things you can look up:

Accuracy of TLEs

Doing sums

Terminology