<h1>3.</h1>
<p>The <strong>absolute</strong> 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 <span class="math-container">$6371(\sqrt2-1)\approx 2639$</span> 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 <a href="https://en.wikipedia.org/wiki/Bicentric_polygon" rel="nofollow noreferrer">the Wikipedia article</a>), the orbital radius is equal to <span class="math-container">$R=\frac{6371}{\cos{\frac{\pi}{n}}} - 6371$</span>, where <span class="math-container">$n$</span> is the number of satellites in orbit.</p>
<p>Then given some orbital distance, <span class="math-container">$R$</span>, from the Earth (relative to the surface), the minimum number of satellites needed is <span class="math-container">$n=\bigg\lceil\frac{\pi}{\cos^{-1}\big(\frac{6371}{R+6371}\big)}\bigg\rceil$</span>, where <span class="math-container">$\lceil x \rceil$</span> is the smallest integer greater than or equal to a number, <span class="math-container">$x$</span>. Hopefully this clarifies things.</p>

CC BY-SA 4.0
2022-04-24T13:56:15.677Z
2022-04-24T13:56:15.677Z
WarpPrime
<p>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?</p>

CC BY-SA 4.0
2022-04-24T00:04:58.610Z
All you need is 3 in an equilateral triangle if the orbital distance is at least equal to the radius of Earth.
WarpPrime
CC BY-SA 4.0
2022-04-24T02:09:45.977Z
What exactly does "each is capable of communicating with any" mean exactly? Do you mean every satellite has a clear line of sight with every other satellite at all times? Well, the minimum number is of course 1. You can put a few dozen in the same orbit spaced at 1 kilometer intervals and that will work too. I don't think you've defined the problem well enough yet. If you can add a sketch that will help. Also, you'll need to nail down what range of altitudes. Do you want to specify 400 km? (where the ISS is for example)?
uhoh
CC BY-SA 4.0
2022-04-24T03:30:10.717Z
@fasterthanlight Yes, 400 km.
At 400 km, the circumference of their orbit is 2 (6371+400) π = 42,543 km.

If, for example, you have only 2 stations separated by half the distance (i.e., 21, 271 km), will they be able to communicate with each other? I guess not because Earth would be in the way.
If we have 3 stations separated by a third of the distance (i.e, 14 km), will they be able to communicate with each other?

In other words, what's the maximum distance between two satellites orbiting at 400 km that would still allow them to communicate with each other using, say, radio waves?
William
CC BY-SA 4.0
2022-04-24T04:59:33.303Z
There isn't really an equation which will give you you the geometry of points of equal distance on a sphere in three dimensions. Like the number pi, it's irregular, so you have to use the platonic solids, tetrahedron cube dodecahedron octahedron... And use their radius otherwise all the points are not equally spaced
bandybabboon
CC BY-SA 4.0
2022-04-24T05:20:36.340Z
You would get more answers on [space.se] Questions of satellite operations are more relevant to space exploration than astronomy
James K
CC BY-SA 4.0
2022-04-24T09:23:18.180Z
I’m voting to close this question because it belongs on SE Space Exploration.
Fred
CC BY-SA 4.0
2022-04-24T13:57:52.470Z
@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.
WarpPrime
CC BY-SA 4.0
2022-04-24T22:26:10.383Z
@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](https://space.meta.stackexchange.com/q/2866/12102)
uhoh
CC BY-SA 4.0
2022-04-25T08:36:38.593Z
@fasterthanlight Then it can be a 2D equation because the satallites can be on a 2d circle equal spacing and altitude?
bandybabboon
<orbit><orbital-mechanics><artificial-satellite>
How to calculate the minimum number of satellites needed to maintain constant link between themselves?
William
# How to calculate the minimum number of satellites needed to maintain constant link between themselves?

(Asked by: username_0 on 2022-04-23)

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?

**Tags:** orbit, orbital-mechanics, artificial-satellite
> **Comments:**
> **username_1**: All you need is 3 in an equilateral triangle if the orbital distance is at least equal to the radius of Earth.  
> **username_2**: What exactly does "each is capable of communicating with any" mean exactly? Do you mean every satellite has a clear line of sight with every other satellite at all times? Well, the minimum number is of course 1. You can put a few dozen in the same orbit spaced at 1 kilometer intervals and that will work too. I don't think you've defined the problem well enough yet. If you can add a sketch that will help. Also, you'll need to nail down what range of altitudes. Do you want to specify 400 km? (where the ISS is for example)?  
> **username_0**: @fasterthanlight Yes, 400 km. At 400 km, the circumference of their orbit is 2 (6371+400) π = 42,543 km.  If, for example, you have only 2 stations separated by half the distance (i.e., 21, 271 km), will they be able to communicate with each other? I guess not because Earth would be in the way. If we have 3 stations separated by a third of the distance (i.e, 14 km), will they be able to communicate with each other?  In other words, what's the maximum distance between two satellites orbiting at 400 km that would still allow them to communicate with each other using, say, radio waves?  
> **username_3**: There isn't really an equation which will give you you the geometry of points of equal distance on a sphere in three dimensions. Like the number pi, it's irregular, so you have to use the platonic solids, tetrahedron cube dodecahedron octahedron... And use their radius otherwise all the points are not equally spaced  
> **username_4**: You would get more answers on [space.se] Questions of satellite operations are more relevant to space exploration than astronomy  
> **username_5**: I’m voting to close this question because it belongs on SE Space Exploration.  
> **username_1**: @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.  
> **username_2**: @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](https://space.meta.stackexchange.com/q/2866/12102)  
> **username_3**: @fasterthanlight Then it can be a 2D equation because the satallites can be on a 2d circle equal spacing and altitude?  

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## 1 answers

### Answer by username_1 on 2022-04-24. Score: 2

# 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](https://en.wikipedia.org/wiki/Bicentric_polygon)), 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.


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