# Calculate time (or fraction of orb. period) of sat. in umbra

Greetings!

I would like to know how to calculate the time, or fraction of orbital period, that a satellite spends in umbra? (total solar eclipse for satellite when they aren't affected by Solar radiation pressure and solar panels don't generate power).

Information given would be satellite's orbit radius from the center of Earth, and orbit would be assumed circular.

Thank you and kind regards,
T

BobG
Homework Helper
Your satellite is in a circular orbit, so r (the radius) never changes. The Earth's shadow will project out nearly in straight lines. A straight line from the start of umbra to the end of umbra will equal the Earth's diameter. That, plus the radii for each of those points forms an iscoceles triangle. Some simple trig can get you the angle.

Once you have the angle (in radians), multiply it by the radius to get the length of the portion of the orbit that's in darkness. (optional since there's multiple ways to get what you want)

Divide your angle by your mean motion (average angular velocity) and you have the amount of time spent in darkness.

Comparing the time in darkness to the orbital period, dividing your angle by 2pi, or dividing your distance spent in darkness by the circumference will get you your percentage of the orbit in darkness.

Actually, since the Sun is larger than the Earth, the Earth's shadow is conical, but it's a cone that converges very slowly. If you want more accuracy, you really need to take into account that convergence. In practice, close is good enough - at least from a satellite operations planning point of view. I imagine there could be some scenarios (experiments, etc) where you'd actually need that extra accuracy.

Thank you BobG, I believe this will do!