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I'm currently working through some practice exercises and one is about how many minutes per orbit a spacecraft spends above a certain latitude. How can this be calculated? I can't find any equations and visualising the problem using GMAT hasn't helped me.

Here are the parameters:

  • Semimajor axis a = 35,000km
  • Eccentricity e = 0.7
  • Argument of Perigee = 320 degrees

The spacecraft is in a polar prograde orbit (assuming polar means dead on 90 degrees here) and the question is how many minutes per orbit are spent above the 70 degree latitude boundary.

I don't even really know where to start, and can't find any help in the textbook or other materials!

distressed_dave
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    Welcome to Space Exploration! If this a homework exercise then you have already been given some clues in your course. Sorry to be obtuse but this will go in much deeper the more you can contribute to it yourself. For example, what equations have you been given regarding orbits etc? You will need something that tells you about where the satellite is with respect to the perigee and its probably also worth doing a diagram. Just to get a conceptual handle on things ask yourself to start with how different the answer would be if the perigee were instead above the north pole or the south pole. – Puffin Jan 10 '21 at 17:27
  • I think above a certain latitude means the projected latitude is greater than a specified anount, thus between the soecified latitude and the pole. – Oscar Lanzi Jan 10 '21 at 22:56
  • @Puffin thanks for responding! The closest equation that we have covered is this one: https://prnt.sc/wkj31y and I believe it demonstrates the level of which we have covered. I've plotted it using GMAT, which produces a diagram of the orbit, and it's clear that the spacecraft spends a significant amount of time above the 70 degree mark: https://prnt.sc/wkj3wo. By with respect to the perigee, do you mean argument of perigee? That's given in the intial question as AoP - I'll edit that for clarity. Thank you! – distressed_dave Jan 10 '21 at 22:56
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    @Uwe I believe the implication of the question is what proportion of the orbit is spent above 70 degrees latitude, ie as it travels over the north pole and begins to journey southwards again to the 70 degree line. – distressed_dave Jan 10 '21 at 22:58
  • Kepler's third law is applicable here, but it would be much easier for a circular orbit than for this elliptical orbit. – Uwe Jan 10 '21 at 23:11
  • For a circular orbit you could use the math in this answer, for an elliptical orbit in a given plane you can first calculate what radius $r$ or angle $\theta$ of the ellipse will be above the desired latitude and then solve for $t(\theta)$ or $t(r)$ using this answer. This looks like a very fun problem to solve! If you do, and nobody has posted an answer, please feel free to post an answer here yourself. Welcome to Space! – uhoh Jan 11 '21 at 01:54
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    It may be that the purpose of your homework question is to get you to become familiar with the terms true anomaly, mean anomaly and eccentric anomaly. You could go to the section called "Position as a function of time" in the wiki link that @Uwe gave you and then make yourself a diagram like the one at figure 5 that applies to your situation. – Puffin Jan 11 '21 at 15:21

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