:

Use the Law of Sines:

r+z sin 90° =  r sin y sin y =  r r+z y = 63.° 464

Find range of limb. Use the Pythagorean Theorum:

(r+z)2 = L2 + r2 L2 = (r+z)2 - r2 = 2xr + z2 L = Ö   2zr + z2   L = 1696.536  km

Calculate the angle q, the number of degrees around planet to limb:

y+ q = 90° q = 90° - y q = 26.° 536

: When do we lose 100% coverage of Olympus Mons?

This occurs when the far slope becomes co-linear with the line-of-sight from the spacecraft. We assume, for Olympus Mons, a slope s = 20^°, a height h = 20 km, and that the basal diameter (110 km) is small compared to the radius of Mars, r = 3397 km. The nominal spacecraft altitude is z = 400 km.

Using the Law of Sines:

r+z sin (90° + s) =  r+h sin n sin n = æ è  r+h r+z ö ø cos s n = sin-1  é ë æ è  r+h r+z ö ø cos s ù û n = 57.° 742,     (which is  < y, as it should be.)

Calculate the angle f, the number of degrees around planet to the summit of Olympus. The sum of the angles of a triangle is 180^°:

180° = 90° + s + n + f f = 90° - s - n f = 12.° 258     and    q-f = 14.° 278

Find range of the summit from spacecraft. Again use the Law of Sines:

l sin f =  r+h sin n l = æ è  r+h sin n ö ø sin f l = 857.925  km

: When does Olympus Mons peek above the limb?

This occurs when the summit becomes co-linear with the line-of-sight from the spacecraft, it .e., when the nadir angle of the summit equals y. This calculation is easy, since all our triangles are right triangles:

l2 = (r+h)2 - r2 l = Ö   2rh+h2   l = 369.161  km

So the total range is L₂:

L2 = L + l L2 = 1696.349 + 369.161  km L2 = 2069.510  km

For the angle around the planet when this occurs, use Law of Sines:

l sin q¢ =  r+h sin 90° sin q¢ =  l r+h q¢ = 6.° 202 and the total   q+q¢ = 37.° 738

End.

-RLM 12 December 2001

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