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Distance between Mars and Earth

  1. Dec 13, 2014 #1

    MMS

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    Hi guys,

    There are many calculators and applications out there that can calculate you the distance between Earth and Mars at every time, but I'm trying to find the actual function that gives me so and draw the path of it.

    My objective is finding the distance R (and Phi of t), as shown below, as a function of time.
    I'm trying to sort out some geometry here but I keep getting functions that are implicit (with R and Phi).

    Any ideas how to find R and Phi and as a function of time each explicitly?

    Untitled.png
     
  2. jcsd
  3. Dec 13, 2014 #2

    Danger

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    Gold Member

    I don't do orbital mechanics myself, but I know that it's a lot more complex than you seem to think. To start with, both orbits are elliptical, not circular, and of course their speeds are different.
    Here's a starting point: http://en.wikipedia.org/wiki/Orbital_mechanics
     
  4. Dec 13, 2014 #3
    What is phi supposed to represent?

    You can resolve the polar coordinates into cartesian coordinates, and then easily determine the distance.

    Chet
     
  5. Dec 13, 2014 #4

    MMS

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    Approximately speaking, they're circular. I can tell you both their time periods aswell (Kepler's 3rd law).

    It's simply the angle that is defined in the image.
     
  6. Dec 13, 2014 #5

    MMS

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  7. Dec 13, 2014 #6

    Danger

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    Gold Member

    Okay, sorry. I thought that you meant something that you could use to programme a spacecraft trajectory.
     
  8. Dec 13, 2014 #7
    You have a line drawn from mars to earth that is also tangent to earth's orbit. I hope you realize that you can't generally do this.

    Chet
     
  9. Dec 13, 2014 #8

    MMS

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    In that specific time it is shown to be tangent (as if someone snapped a picture at that time). Of course, it isn't the general case which I'm looking for.
     
  10. Dec 13, 2014 #9

    MMS

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    I'm still far from orbital mechanics so no need to worry about that now. :D
     
  11. Dec 13, 2014 #10
    So, again, what does phi look like when it's not tangent, and why do you want to know phi?
     
  12. Dec 13, 2014 #11

    MMS

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    This could be phi at a different time:

    Untitled.png

    Knowing phi and R as a function of time helps me plot its path as shown below.
    image024.gif
     
  13. Dec 13, 2014 #12

    MMS

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    I believe I have reached the answers. I'd be happy if someone could give it a look:

    My final answer for R(t) using the law of cosine is:
    image.png

    And phi(t) using arctan(y/x) for R:
    Untitled.png
     
  14. Dec 13, 2014 #13
    Yes. That looks correct.

    Chet
     
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