1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Time-weighted average distance in an elliptical orbit

  1. Dec 8, 2016 #1
    1. The problem statement, all variables and given/known data

    Using the polar formula for an ellipse, and Kepler's second law, find the time-weighted average distance in an elliptical orbit.

    2. Relevant equations

    The polar formula for an ellipse:

    $$r = \frac { a(1-e^2)} {1 \pm e cos \theta},$$

    Area of an ellipse:

    $$ A = \pi a b $$

    $$ b = \sqrt{a(1 - e^2)} $$



    3. The attempt at a solution

    I don't know if you'd call this much of an attempt, but I understand I need to be taking some sort of integral with respect to time. I have genuinely been staring at this for hours with no idea where to start, though, and I need some idea of how to get going.

    I've messed with just about every algebraic combination of the three equations above, but I haven't found anything that pops out at me and says "oh, that's it."

    Sorry if this is too vague, this is the first time I've ever really needed to post here. Let me know if I can add any more information.

    Thanks!
     
  2. jcsd
  3. Dec 8, 2016 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Welcome to PF!

    The problem mentions Kepler's second law. That will probably be very helpful.

    Also, in general, if you have a function of time ##f(t)##, how would you set up an integral to represent the time average of the function over a time interval from ##t = 0## to ##t = T##?
     
  4. Dec 8, 2016 #3
    Thanks!

    Allegedly we have all of the information needed in those three equations, and does Kepler's Second Law have an actual mathematical form? If it does, I've gone through four years of undergrad, a year of research, and a year of grad school with misinformation, haha.

    For such an integral, you'd need to do something with ## \theta ## going from ##0## to ##2\pi##, no?
     
  5. Dec 8, 2016 #4

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Yes.
    https://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion#Second_law

    Yes. But start with the general idea of setting up an integral for the time average of ##r(t)## for an orbit. In calculus, you probably covered finding the average of a function ##f(x)## over some interval ##a < x < b##. If you need a review, try a web search for "average of a function integral". Then you can apply the general idea to this problem.
     
  6. Dec 9, 2016 #5
    Okay, is this what you're talking about? It does look vaguely familiar.

    http://tutorial.math.lamar.edu/Classes/CalcI/AvgFcnValue.aspx

    One thing, though---there's no time in the equations above. I might be too burnt out from a week of intense finals and missing something.
     
  7. Dec 9, 2016 #6

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Yes

    Once you set up the time integration that represents the time average of ##r##, ##\left<r\right>##, you can try a change of integration variable from ##t## to ##\theta##. This will allow you to express ##\left<r\right>## as some integral with respect to ##\theta##.
     
  8. Dec 9, 2016 #7
    I'm sorry, I guess I'm not fully picking up on what you're saying. How am I setting up a time integral with nothing that has time in it?
     
  9. Dec 9, 2016 #8

    TSny

    User Avatar
    Homework Helper
    Gold Member

    ##r## varies with time as the planet moves in its orbit. So, you can think of ##r## as a function of ##t##, ##r(t)##. If you knew the function ##r(t)##, how would you set up an integral that represents the time average of ##r(t)##?
     
  10. Dec 9, 2016 #9
    I'd do something like

    $$ \frac{1}{t - 0} \int_{0}^{t} r(t) dr$$

    But I don't know how to get an ##r(t)## with what I have right now.
     
  11. Dec 9, 2016 #10

    TSny

    User Avatar
    Homework Helper
    Gold Member

    OK. You want the average over one orbit. So, what specific time should you use for the upper limit of the integral?

    Also, you wrote the differential inside the integral as ##dr##. Did you mean ##dt##?

    You'll be able to do a change of variable of integration from ##t## to ##\theta##. This is where Kepler's 2nd law will be helpful.
     
  12. Dec 9, 2016 #11
    Yeah, I meant to write ## dt ##, my bad.

    For the time of one orbit, you'd want to integrate from 0 to ##P##? Am I intended to use an equation for ##P## as my upper-bound of integration?
     
  13. Dec 9, 2016 #12

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Yes, assuming ##P## is the period of the orbit. Or you could use ##P/2## since the shape of the orbit is symmetrical.
    You won't need an equation for ##P##.
     
    Last edited: Dec 9, 2016
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Time-weighted average distance in an elliptical orbit
Loading...