Assuming that their orbit is circular with diameter of 8,000 miles how far did they travel?

Question

suppose the space shuttle astronuts orbit the earth 15.3 times. Assuming that their orbit is circular with diameter of 8,000 miles how far did they travel?

Answer 1

The diameter of the orbit is 8000 miles. So we'll times it by pi to get the circumference. This will be 25,132.74 miles. This distance was done 15.3 times, which will give us 384,530.94 miles.

Answer 2

We use Kepler's third law here. It states that

(T^2/ a^3) = const.

where 'T' is the period for 1 revolution, 'a' is the average distance of the orbit from center.

We need to know the const. first.

We know that Geo stationary (a subset of Geo-synchronous) satellite is almost 40,000km from the center of earth. So

a = 40 Mega meters (I use this term Mm).

T = 24 hrs.

const = 576/64000.

Since 576= 9 times 64,

const = 9/1000

Now your shuttle is at 8000 miles above earth. Add 4000 miles of earth's radius to get the distance from center (I have approximated; earth's radius is 3960miles; an error of 1/3 %).

That brings the distance to 12000miles = 19200km= 19.2Mm

T^2 = (19.2^3)*(9/1000) = 63.7

T = 7.98129 hrs. (7h 58' 53")

For 15.3 times orbiting

time taken = 15.3 X 7.98129 = 122.113737 hrs (122h 6' 50")

5d 2h 6' 50".

I don't claim accuracy here, but to arrive at a quick, but fairly good approximation (Engineers do that all the time).

P.S.: I feel the term 'circular orbit' is loosely used and is strictly not correct. Any body moving under a central (point) force  must describe a conic with eccentricity >0. If eccentricity = 0, it becomes circular. So it is almost circular.