Difference between revisions of "Tutorial:Satellite Coverage"

Revision as of 10:38, 8 April 2014

This tutorial is about how to calculate the satellite coverage of a celestial body. This is particularly useful for RemoteTech. Note: When placing multiple satellites in one orbit, it is more important that the semi-major axis is identical for all satellites. With an identical semi-major axis, nil eccentricity, and approximately identical inclination the exact positioning of the apsides is not critical. So long as the eccentricity is as close to nil as possible, variances in apside positioning and inclination will only result in a small oscillation (analemma).

Distance between satellites

Illustrating the calculation

The distance between two satellites (if their angular distance is known) can be calculated with following formula:

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

Where ${\displaystyle \alpha }$ is the angular distance, ${\displaystyle a}$ the altitude above mean sea level, and ${\displaystyle r}$ the radius of the orbited body - it is basically an isosceles triangle, where the base length is the distance between the satellites and the legs are from the center of mass of the orbited body to the satellites. For example: two satellites at GSO around Kerbin with a distance of 90° are 4.9 Mm away. This calculation does not work if the altitude of the two satellites is not equal.

Satellite Line-of-Sight communication

Satellites require line-of-sight for communications. Depending on the number of satellites and altitude, it is not always possible to communicate with the other satellites in the same orbit. To mitigate this, either a higher orbit is chosen or more satellites are required. With the following formulas the altitude is linked with the number of satellites to determine at which altitude (and in what quantity) the satellites are required to allow communication.

When all satellites in orbit have the same orbital period, the most efficient way to position them is in the corners of a regular convex polygon. Neighboring satellites should have an angular distance of ${\displaystyle \alpha }$. Each satellite is placed on the same circular orbit and their position is a multiple of the following angle:

${\displaystyle \alpha ={\frac {\tau }{n}}={\frac {360^{\circ }}{n}}={\frac {2\pi }{n}}}$

In other words, a satellite is positioned at every multiple of ${\displaystyle \alpha }$. The connection between two neighboring satellites and the center of the orbited body form a triangle where the distance from the satellites to the body's center is identical, forming an isoceles triangle. The height of the triangle is the shortest distance between the body's surface and the connection between the satellites. To calculate the height the following formula can be used:

${\displaystyle h=\cos \left({\frac {1}{2}}\alpha \right)\cdot (a+r)}$

Note that ${\displaystyle h}$ must be greater or equal to ${\displaystyle r}$ to allow a successful line-of-sight connection:

${\displaystyle a\geq \left({\frac {1}{\cos \left({\frac {\tau }{2n}}\right)}}-1\right)\cdot r}$

To determine how many satellites are needed for a specific altitude (as before, ${\displaystyle h}$ must be greater than ${\displaystyle r}$):

${\displaystyle n\geq {\frac {\tau }{2\cdot \arccos \left({\frac {r}{a+r}}\right)}}}$

Tau (${\displaystyle \tau }$) is equivalent to ${\displaystyle 2\cdot \pi }$ (or 360°). The unit needed depends on which unit is used by cosine and arccosine in the calculations. Note that the number of satellites, ${\displaystyle n}$, must be greater than or equal to 3.

Two satellite configuration

When using two satellites it is not possible to position them 180° apart, as the orbited body obstructs the line-of-sight they require. To provide near-100% coverage of an orbital body with only two satellites (at 180° separation) requires a very large ${\displaystyle a}$ to ${\displaystyle r}$ ratio and is not optimal. Keep in mind that the two satellites cannot communicate directly (via line-of-sight) and you will need to take measures to ensure connectivity is maintained.