Tutorial:Satellite Coverage

This tutorial is about how to calculate the coverage of a celestial body. This is for example useful for RemoteTech. When placing multiple satellites in one orbit, it is more important that the semi-major axis is identical for all satellites. As long as the eccentricity is not very high this is more precise, than trying to make both apsides equal.

Distance between satellites

To determine the distance between two satellites if their angular distance is known can be calculated with following formula:

$l=\sin \left({\frac {1}{2}}\alpha \right)\cdot (a+r)$

With $\alpha$ being the angular distance, $a$ the altitude above mean sea level and $r$ the radius of the orbited body. For example two satellites around Kerbin with a distance of 90° are 4.9 Mm away.

General coverage calculations

To determine what altitude is needed to cover a complete body only the radius of the orbited body and the number of used satellites is important.

When all satellites in orbit have the same orbital period, the most efficient way to place them is, is in the corners of a regular convex polygon. The interior angle of each point is then expressed with following formula:

$\alpha ={\frac {n-2}{n}}\cdot 180^{\circ }$

Each satellite is placed on the same circle with following distance:

$\delta ={\frac {360^{\circ }}{n}}$

The satellites with an angle of $\delta$ ahead and behind need to be at least $\alpha$ degrees away. The celestial body with a radius of $r$ has at an altitude of $a$ above mean sea level this angular size:

$\beta =2\cdot \sin ^{-1}\left({\frac {r}{r+a}}\right)$

To have the satellites in sight the celestial body's size $\beta$ has to be lower than $\alpha$:

${\frac {n-2}{n}}\cdot 180^{\circ }=\left(1-{\frac {2}{n}}\right)\cdot 180^{\circ }=\alpha \geq \beta =2\cdot \sin ^{-1}\left({\frac {r}{r+a}}\right)$

The lowest altitude is then, when $\alpha =\beta$:

${\begin{aligned}\left(1-{\frac {2}{n}}\right)\cdot 180^{\circ }&=2\cdot \sin ^{-1}\left({\frac {r}{r+a}}\right)\\\sin \left(\left(1-{\frac {2}{n}}\right)\cdot {\frac {180^{\circ }}{2}}\right)&={\frac {r}{r+a}}\\{\frac {1}{\sin \left(\left(1-{\frac {2}{n}}\right)\cdot 90^{\circ }\right)}}&={\frac {r+a}{r}}=1+{\frac {a}{r}}\\\left({\frac {1}{\sin \left(\left(1-{\frac {2}{n}}\right)\cdot 90^{\circ }\right)}}-1\right)\cdot r&=a\\\end{aligned}}$

This determines the height needed for n satellites, but can be reversed to get how many satellites are needed for a specific altitude:

${\begin{aligned}\left(1-{\frac {2}{n}}\right)\cdot 180^{\circ }&=2\cdot \sin ^{-1}\left({\frac {r}{r+a}}\right)\\1-{\frac {2}{n}}&={\frac {2}{180^{\circ }}}\cdot \sin ^{-1}\left({\frac {r}{r+a}}\right)\\-{\frac {2}{n}}&={\frac {1}{90^{\circ }}}\cdot \sin ^{-1}\left({\frac {r}{r+a}}\right)-1\\{\frac {1}{n}}&=-{\frac {{\frac {1}{90^{\circ }}}\cdot \sin ^{-1}\left({\frac {r}{r+a}}\right)-1}{2}}\\n&=-{\frac {2}{{\frac {1}{90^{\circ }}}\cdot \sin ^{-1}\left({\frac {r}{r+a}}\right)-1}}\\\end{aligned}}$

These calculations work only with at least three satellites.

Two satellite configuration

When using two satellites it is not possible to place them 180° apart, as they have no connection to each other than. So only two satellites can cover the complete celestial body. But when making the distance between both satellites as high as possible to 180°, it is possible to cover most of the body.