2
$\begingroup$

Suppose we want to build a number of space stations, all orbiting Earth at the same altitude, equally distanced from one another, and with the same inclination. How many of those stations are needed so that each of them is capable of communicating with any other station all times, assuming that each station can relay a signal to another station it is connected to?

$\endgroup$
9
  • 1
    $\begingroup$ All you need is 3 in an equilateral triangle if the orbital distance is at least equal to the radius of Earth. $\endgroup$
    – WarpPrime
    Apr 24, 2022 at 0:04
  • 4
    $\begingroup$ You would get more answers on Space Exploration Questions of satellite operations are more relevant to space exploration than astronomy $\endgroup$
    – James K
    Apr 24, 2022 at 5:20
  • 4
    $\begingroup$ I’m voting to close this question because it belongs on SE Space Exploration. $\endgroup$
    – Fred
    Apr 24, 2022 at 9:23
  • 1
    $\begingroup$ @LifeInTheTrees The OP asked for the satellite network to be operating at the same inclination and altitude, just different mean anomalies, so you really only need regular polygons and the formula for their in/excircles. $\endgroup$
    – WarpPrime
    Apr 24, 2022 at 13:57
  • 1
    $\begingroup$ @Fred "because belongs on" is never a close reason (except when there's an explicit close reason like for Earth Science here in Astronomy). You can close as off-topic, but the OP decides where to ask and moderators decide if and where to migrate. For more about why this particular comment wording is problematic see "I’m voting to close this question because it belongs on Physics SE" is bad and here's why $\endgroup$
    – uhoh
    Apr 24, 2022 at 22:26

1 Answer 1

2
$\begingroup$

3.

The absolute minimum number of satellites needed is 3, like I said in a comment, if the orbital distance is greater than or equal to the Earth's radius (at least 6371 km). However, if the orbital distance is less than 6371 km, you need more. For instance, if the radius is more than $6371(\sqrt2-1)\approx 2639$ km, then you will only need 4. Continuing, using the formula for the incircle and excircle of bicentric, regular polygons, where the incircle is the Earth and the excircle is the satellite network orbit, (see the Wikipedia article), the orbital radius is equal to $R=\frac{6371}{\cos{\frac{\pi}{n}}} - 6371$, where $n$ is the number of satellites in orbit.

Then given some orbital distance, $R$, from the Earth (relative to the surface), the minimum number of satellites needed is $n=\bigg\lceil\frac{\pi}{\cos^{-1}\big(\frac{6371}{R+6371}\big)}\bigg\rceil$, where $\lceil x \rceil$ is the smallest integer greater than or equal to a number, $x$. Hopefully this clarifies things.

$\endgroup$

Not the answer you're looking for? Browse other questions tagged .