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I want to estimate how many satellites (and how many number of planes) are needed for achieving a 4-fold global coverage on earth (global coverage with at least 4 satellites in view in each location). To do this I basically compute the area of coverage for a single satellite (A_Cap), and I divide the total earth surface by the coverage of a single satellite. The equations I use are (based on Figure 1):

Fig1. Geometry

Acap = 2*pi*Re^2*(1-cosd(Phi))

Where Phi is the cap angle and is computed as:

Phi = acosd(Re/(Re+h)*cosd(Alpha))-Alpha;

Where Re is the radius of the Earth, h is the orbit altitude and Alpha is the minimum elevation angle for consider the satellite in view.

To compute the numbe rof satellites I need I do:

Nsv = N_fold*A_Earth/Acap

Where N_fold is the needed number of satellites in view at any location (in my case 4). Doing this I belive I'm overestimating the number of satellites (since I'm just multiplicing the coverage of 1-fold by N). Do you have any other idea how I could solve this problem?

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  • $\begingroup$ "at least four satellites in view" sounds like the recipe for a GPS-like system. Answers to GPS constellation for Mars are in the ballpark of 15 to 18 minimum. Different but potentially insightful are answers to Minimum number of satellites to image the entirety of Earth's surface at all times $\endgroup$
    – uhoh
    Oct 15 at 11:41
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    $\begingroup$ Yes, I'm looking for something similar. My goal is to compute the number of satellites needed for covering the earth with 4 satellites in view as a function of the orbit altitude. $\endgroup$
    – Papadopoul
    Oct 15 at 12:25
  • $\begingroup$ Great! I've adjusted your title to match that. Please feel free to adjust further. $\endgroup$
    – uhoh
    Oct 15 at 14:17
  • $\begingroup$ You appear to be on the right path (assuming the tiling of coverage works out nicely) to cover a perfect sphere. However, the Earth is not perfect, and there are plenty of places where you can't get enough good GPS signals because of the surrounding terrain. So are you solving an ideal problem or the real problem? $\endgroup$
    – Jon Custer
    Oct 15 at 14:22
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    $\begingroup$ It is not clear whether you are looking for the theoretical minimum bound for the number of satellites (as a function of the cap angle "Phi"), or you are looking for a construction method. In the second case, Walker's method described in Circular orbit patterns providing whole Earth coverage, 1970 would be good start. Galileo and Iridium are based on such construction Wiki $\endgroup$
    – Ng Ph
    Oct 16 at 18:38
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Perhaps you are looking for a relationship in the form of (or wondering whether such a relationship exists):

Nmin(A_cap,n) = μ(n) * (A_Earth/A_cap)

A_cap: area of an instantaneous coverage by a single satellite, constant in time (circular orbits), modelled as a spherical cap.

Nmin: smallest number of satellites in a practical constellation that can provide seamless n-fold coverage.

μ(n): a fitting constant, independent of A_cap, function of n.

Fortunately, such approximate relationships exist, and μ(4)~7.2, whereas μ(1)~2.


However, I don’t know of any mathematical derivation of μ(n), even for μ(1). Mostly, μ(n) is derived from testing different constellations constructed using heuristic reasonings, such as the so-called Walker constellations (Wiki).

  • Case n=1

Let’s start with n=1 to get familiar with some published results.

Yuri Ulybyshev wrote a nice review in 2008 titled Satellite Design for continuous coverage: short historical survey

Figure 1, reproduced here for convenience, gives a plot for n=1 and Elevation =10° (that he called α).

enter image description here

As you have noted yourself, if you call Phi the Earth-centered half-cone angle of the spherical cap representing the individual coverage, then A_cap=2 π RE2 (1-cos(Phi)) So that (A_Earth/A_cap) = 2/(1-cos(Phi))

What I am claiming here is that the tight lower bound for N displayed in Figure 1 of Ulybyshev follows the trend of the relationship:

   Nmin= 4/(1-cos(Phi))

In other terms, μ(1) ~ 2.

Here is a check point so that we are on the same page with the detailed calculation:

H=1000Km (and El=10°) => Phi =21.6° => 2/(1-cos(Phi))= 28.4 => Nmin=56.8

This result (μ(1) ~ 2) was obtained independently by Beste in Design of satellite constellation for optimal continuous coverage. It is paywalled but Figure 3 is available (reproduced here, ψ is the half-cone angle that we called Phi).

enter image description here

  • Case n=4,

Take GPS. Since we know H (20200 Km) we can compute their Phi (66.3°), assuming that their design Elevation is 10° (a reasonable assumption for Satnav). We also know GPS requires 24 satellites. From this we can make the informed guess that μ(4) ~ 7.2, assuming that the designers of GPS did optimize their constellation for minimum number of satellites.

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  • $\begingroup$ Highly interesting! I dream of getting answers like this, so have an orb-mech bounty $\endgroup$ Oct 17 at 16:51
  • $\begingroup$ Thanks a lot to all of you for your answers, but especially to @Ng Ph for this extensive and didactic response! $\endgroup$
    – Papadopoul
    Oct 18 at 6:46
  • $\begingroup$ @Papadopoul (and se-stop-firing-the-good-guys), thanks for the compliments (well appreciated). In fact, a good question contains 80% of the answer, and yours is good. The paper from Beste gives some indication how to select the number of planes. It describes a procedure of construction close to the path se-stop-firing-the-good-guys has started working on. $\endgroup$
    – Ng Ph
    Oct 18 at 10:03
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Some notes on your approach, which I believe is a good one as a first order approximation.

Doing this I belive I'm overestimating the number of satellites

I would rather say it has to be an underestimate.

  1. Circles do not perfectly tile. To achieve a 1-fold coverage, for instance, there must be some overlap, requiring $\frac{2\pi}{3\sqrt{3}} \approx 1.21$ times more circle area.

For higher order coverings, the cover efficiency will asymptotically approach $1$, but for a 4-fold covering it's not clear that you can do better than just layering four 1-fold coverings (and even if some more clever solution exists, it's only going to be marginally better). So 20% more satellites just from the static geometry of the problem seems reasonable.

1-fold covering

  1. Satellites are not static. Even if you find a geometric solution that covers the whole planet 4-fold, it's likely to instantly break when their required relative motion is taken into account. The coverage pattern has to change as a function of time, and you are likely going to need extra satellites to make sure the 4-fold coverage holds all the time instead of just a specific instant.

For something that's clearly an overestimate, but will work, you could place satellites in longitude planes, tight enough to provide 4-fold coverage.

Some initial number juggling gives me an efficiency maximum where the planes are 1.55 single-satellite coverage radius apart, with the satellites in a plane spaced 0.36 at radii.

This should give a provable upper bound, but is inefficient because it gives unnecessary amounts of coverage near the poles.

longitude tiling

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