How do you calculate how long a point on earth will be visible by a satellite of a given altitude?

Question

I've calculated the distance of the arc of the earth the satellite can see, and I know the satellite velocity. but, I can't seem to figure out why my "visibility time" is 30 seconds off of the correct answer.

For example, this website says at 400km, the satellite visibility time is 10.1 minutes, but I get 9.6 minutes.

Interestingly enough, I have the same visibility half angle, velocity, and range-to-earth calculated as the site. I just can't figure out the satellite viewing time

http://www.csgnetwork.com/satorbdatacalc.html

Answer 1

The Earth is rotating while the satellite is orbiting (in the same direction as the Earth rotation). You have to take that into account. It will add more time to the 'viewing' time.

For a given arc of satellite motion from tangent to tangent, how many degrees of longitude are represented. In the time it takes for the satellite to move in that arc, how much time will have elapsed for Earth rotation at:
4 min/degree * cos(latitude).

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Answer 2

I always found that drawing these problems on a piece of paper helped me visualize them:

1.) draw a circle (earth)

2.) Draw a line straight down from the satellite to the earth (this is the height of it's orbit

3.) draw a line tangent to the earth that also intersects the satellite.

4.) the length of that arc is how far the satellite can see that point on earth to figure out how long that is you should figure that the earth makes one revolution in 1 day. so now you know how long.