# Computation slant range d between a satellite and a ground terminal

I am computing the free space path loss by using the following equation:

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.

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

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 $$d$$?

• "Satellite altitude and Earth radius" is insufficient information to calculate slant range. You need the position of the ground station and the satellite. Commented Aug 7, 2023 at 11:42
• @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 Commented Aug 14, 2023 at 11:53
• If you know the positions, you can calculate the elevation angle, and anything else you need. Commented Aug 14, 2023 at 12:03

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.

• The OP states they don't know the elevation angle, and that they do know the Earth radius. Commented Aug 7, 2023 at 14:14
• 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. Commented Aug 7, 2023 at 14:35
• Yes, that is what I commented on the question. From what they say, they don't have enough information to solve the problem. Commented Aug 7, 2023 at 14:55

Not sure you can get a mathematically precise answer, but then again we are implicitly assuming that Re is constant (i.e. Earth is a perfect sphere)

But let's analyze the geometry. Confusingly your phi is the same as the example's alpha and your alpha is the look angle from the satellite. Let's use your notation.

We know that in the case where phi = 0, alpha can be found as asin(Re/(Re+ho)). Let's call that Lambda. This is the angle TO THE HORIZON from the satellite.

We also know that in the case where phi = 90, alpha = 0. And when phi = 90, 6.6-3 reduces to: d = ho, which confirms the equation.

So do a linear interpolation between Lambda to 0 ~ 0 to 90: phi = 90(Lambda - Alpha)/Lambda

Now, in order to calculate the slant range, we use Alpha and ho (altitude) to calculate phi, and plug phi into 6.3-3.

This will be, as we say, close enough for government work. I use an approach like this when I am calculating the slant range of a beam from a satellite in my simulations.