Goals:

1. % of day covered
2. Average coverage - factoring the number of satellites covering a location which would be more useful for approximating average bandwith available for an area

Original Implementation

1. TLE to simulate satellites
2. Altitude,x,y to conical section on sphere
   1. Adjustable minimum elevation
   2. Swap out for more accurate model of earth (WGS-84 ellipsoid)
3. Map conical section to equi-rectangular area
4. Translate to pixel coords with resolution of about 100sq mi
5. Normalize? Overlay

For the core there is this naive implementation:

for each time step:
    for each pixel:
        convert to lat/long
        for each sat:
            if visible
                increase counter for pixel

This seems like a pretty gross way to do it. We touch every pixel * numSats, when we could reduce to numSats * visible area. However, the math to calculate visible area for a satellite doesn't seem to be a part of any library I've found. This is even more relevant with VLEO satellites since they cover so much less area. I'm not sure how efficient it is to go from sphere to lat/long on an equi-rectangular projection.

Resolution

2 options for spatial resolution in my head originally:

• 2048 x 1024 = 2,097,152
• 4096 x 2048 = 8,388,608‬

Instead of trying to use a 2d array to represent the footprints and dealing with the spherical nature of the Earth, I am instead going to try and use S2 level 9 cells to track coverage. According to the statistics page of the S2 library, this is about 1537000 cells, so I would need about 12MB of data to track the whole Earth. Each cell covers 324.29km^2 on average.

For temporal resolution: These satellites move across a location quite quickly, using a website like https://findstarlink.com we can find that a whole "train" of Starlink satellites crosses a locations view in about 4 or 5 minutes so our temporal resolution needs to be a minute if not faster if we want accurate results.

S2 based approach

The original implementation has lots of gross properties and requires dealing with many projections. So I tried using a new approach that used ~ equal area tesselated grids. The one I knew about was S2 so I tried that first.

for each time step:
    for each sat:
        calculate footprint
        get cells to cover footprint
        for each cell:
            increment counter

This means we only have to iterate over ~2000 (+/-600?) cells per satellite instead of ~2 million points each. However, this approach seems too slow. Comptuing all the covering cells of each satellite takes about 90s on a single core in Python. While this could be parallelized, I think maybe there are even smarter approaches to consider. Could we maybe just store S2Cap objects and test points against them to count the amount of coverage? What I really like from S2 is that it handles the issues of latitude and longitude meaning different distances at the equator vs the poles.

In the end for plotting, if we used the cell vertices as plotting locations we would cover the whole globe and not oversample at the poles or undersample at the equator.

S2, more like 2Slow... An H3 approach

So S2 was just being way to slow to return cells for a given covering. (90s to simulate one timestep) Now I have to make some guesses about projections for H3 and but it is ~4.5 times faster to simulate a timestep than S2 (now takes 20 seconds on my machine). I'm gonna proceed with the H3 approach for now.

Used constants

• Earth Mean Equitorial Radius = 6,378.1km
• Minimum angle for user terminals = 35deg

(It has recently come to my attention that there may be newer data from SpaceX that the minimum user terminal angle would be 25 degrees, so my data may be more conservative. The data I used is from the FCC document linked below.)

Starlink details

From the second reference below I found that Starlink is targeting user terminal minimum angles of 35deg This means that each satellite covers around 600,000km^2 at an altitude of about 340km. However, most of the starlink satellites already in space are elevating or are at 550 km

Stralink TLE's https://celestrak.com/NORAD/elements/starlink.txt

References

• https://arxiv.org/pdf/1906.12318.pdf
• https://licensing.fcc.gov/myibfs/download.do?attachment_key=1190019