HELP: simple TRIGONOMETRY

[Rihanna]

Hello my friends, How Are yoU toDay I have a question for you guys and i hope you to guide me to the correct answer

The path from the satellite to the earth station can be found as

(hs-he)/sin(theta) if theta >or= 5 degrees

where "hs" is satellite height and "he" is the earth station height

while the path equals

2(hs-he)/[{sin^2(theta)+(2(hs-he)/Re}^1/2}+sin(theta)] if theta < 5 degrees


Why? i mean what is the difference between those tow condition ( Please take into account the earth curvature Re = 8500 Km)

 

[godfreyl]

I would guess that the second equation is more accurate, but the first (simpler) equation gives results that are "close enough" when theta >= about 5 degrees.

i.e. First equation is an approximation that's fairly accurate for theta >= about 5 degrees.

 

[Rihanna]

I would guess that the second equation is more accurate, but the first (simpler) equation gives results that are "close enough" when theta >= about 5 degrees.

i.e. First equation is an approximation that's fairly accurate for theta >= about 5 degrees.

Thanks a lot Mr.godfreyl for your answer, but could you please explain to me in more details please

Sincerely

 

[godfreyl]

Hi Rihanna

By "path", I assume you mean the distance between the satellite and the earth station. In that case the first equation would be perfectly accurate if the world was flat, as shown in the diagram below.



Unfortunately, the world isn't flat so that equation doesn't work very well. Playing around in a spreadsheet, it seems like it's not only very inaccurate for low values of theta, but also if the satellite is too high. e.g. For a geostationary satellite, hs= 35,786 kilometers, and that equation is nowhere near accurate even for theta = 30 degrees (assuming I managed to figure out the correct answer).

Unfortunately the second equation you gave is wrong too. Maybe you made a mistake copying it or maybe I made a mistake in the spreadsheet, but either way I don't get good answers with it.

As best as I can figure it out, the correct equation is:
Path = 0.5*(-B+(B^2-4*C)^0.5)

where:
B = 2*Re*SIN(theta)
C = -((H+Re)^2-Re^2)

where:
H = height of satellite above ground
Re = radius of earth = 64000km, not 85000km

I ignored he, assuming the earth station is at ground level.

Cheers - Godfrey

 

[Rihanna]

Hi Rihanna

By "path", I assume you mean the distance between the satellite and the earth station. In that case the first equation would be perfectly accurate if the world was flat, as shown in the diagram below.



Unfortunately, the world isn't flat so that equation doesn't work very well. Playing around in a spreadsheet, it seems like it's not only very inaccurate for low values of theta, but also if the satellite is too high. e.g. For a geostationary satellite, hs= 35,786 kilometers, and that equation is nowhere near accurate even for theta = 30 degrees (assuming I managed to figure out the correct answer).

Unfortunately the second equation you gave is wrong too. Maybe you made a mistake copying it or maybe I made a mistake in the spreadsheet, but either way I don't get good answers with it.

As best as I can figure it out, the correct equation is:
Path = 0.5*(-B+(B^2-4*C)^0.5)

where:
B = 2*Re*SIN(theta)
C = -((H+Re)^2-Re^2)

where:
H = height of satellite above ground
Re = radius of earth = 64000km, not 85000km

I ignored he, assuming the earth station is at ground level.

Cheers - Godfrey

Tnx a lot godfrey for your efforts to helping me. However can you please give me the spreadsheet calculation, i want to understand wt you have calculated and how did you get the formula you gave me,

regards, Rihanna

 

[godfreyl]

Re = radius of earth = 64000km, not 85000km

Correction, I meant:
Re = radius of earth = 6400km, not 8500km

how did you get the formula you gave me,

Sorry, I didn't save the drawing I made to work it out.