I am computing the free space path loss by using the following equation: enter image description here

enter image description here

My goal is to compute d (distance between satellite and user) = SE ( see the picture below). Parameters I know: satellite altitude and radios of Earth.

enter image description here

$$ \psi = 90^{\circ} + \phi $$

Using the Law of Sines:

$$ \frac{r_s}{sin (90^{\circ} + \phi)} = \frac{d}{sin(\gamma ) } $$

$$ cos(\phi) = \frac{r_s}{d} sin(\gamma) $$

$\gamma$ yield the coverage area on the surface of the earth assuming the satellite has symmetrical coverage about nadir (see the first picture)

So I don't know $d$ and $\phi$.

I am confused how I can compute $d$ without $\phi$ or $\phi$ without $d$?

  • $\begingroup$ "Satellite altitude and Earth radius" is insufficient information to calculate slant range. You need the position of the ground station and the satellite. $\endgroup$ Aug 7, 2023 at 11:42
  • $\begingroup$ @OrganicMarble thank you for your reply. I can probably set the position of the ground station and/or satellite fixes or random in my simulation. ok, then I don't need to know the elevation angle. right? I can use cartesian coordinate to compute the distance between satellite and user $\endgroup$
    – Aid22
    Aug 14, 2023 at 11:53
  • $\begingroup$ If you know the positions, you can calculate the elevation angle, and anything else you need. $\endgroup$ Aug 14, 2023 at 12:03

1 Answer 1


You do know two more key pieces of information. The radius of the earth and the angle above the elevation. Then including the third part of the law of sines.

$$ \frac{r_s}{sin (90^{\circ} + \phi)} = \frac{d}{sin(\gamma ) } = \frac{r_e}{sin(\alpha ) } $$

focusing on the part where you know 3 of the 4 variables.

$$ \frac{r_s}{sin (90^{\circ} + \phi)} = \frac{r_e}{sin(\alpha ) } $$

solve for alpha.

then you will know two of the three angles, geting the 3rd angle is simple becasue they have to equal 180 degrees.

then go back to the law of sines

$$ \frac{r_s}{sin (90^{\circ} + \phi)} = \frac{d}{sin(\gamma ) } = \frac{r_e}{sin(\alpha ) } $$

the only thing you don't have is d the slant range. use the third part and either of the other two parts to solve for the slant range.

  • $\begingroup$ The OP states they don't know the elevation angle, and that they do know the Earth radius. $\endgroup$ Aug 7, 2023 at 14:14
  • $\begingroup$ If they don't know the angle, then nothing can be calculated. They can calculate the angle from the position of the satellite and the position of the ground equipment. Otherwise they can use every angle from 0 to 90 degrees to get a range of values. $\endgroup$ Aug 7, 2023 at 14:35
  • $\begingroup$ Yes, that is what I commented on the question. From what they say, they don't have enough information to solve the problem. $\endgroup$ Aug 7, 2023 at 14:55

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.