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Math Help - Find equation for Pluto's orbit.

  1. #1
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    Find equation for Pluto's orbit.

    According to Keplerís Laws, planets have elliptical orbits, with the sun at one of the foci. The farthest Pluto gets from the sun is 7.4 billion kilometers. The closest it gets to the sun is 4.4 billion kilometers. Find the equation for Plutoís orbit.

    My work:

    Using the perihelion, a - c = 4.4 billion km

    Using the aphelion, a + c = 7.4 billion km

    Adding 7.4 billion km + 4.4 billion km, I have 2a = 11.8 billion km

    Dividing by 2, a = 5.9 billion km

    b^2 = a^2 - c^2

    = ((59 x 10^9)^2) - (15 x 10^9)^2 = 3.256 x 10^21

    Equation of orbit = ((x^2)/(4.481 x 10^21)) + ((y^2)/(3.256 x 10^21)) = 1


    Correct? Thanks in advance.


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  2. #2
    MHF Contributor ebaines's Avatar
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    Re: Find equation for Pluto's orbit.

    You were good up to here:

    Quote Originally Posted by FatimaA View Post
    Dividing by 2, a = 5.9 billion km

    b^2 = a^2 - c^2
    But then things go awry:

    Quote Originally Posted by FatimaA View Post
    = ((59 x 10^9)^2) - (15 x 10^9)^2 = 3.256 x 10^21
    Check your powers of ten - 5.9 billion Km is 5.9 x 10^9 Km, not 59 x 10^9. Similarly 1.5 billion Km is 1.15 x 10^9 Km. This error ripples through the subsequent lines.

    Quote Originally Posted by FatimaA View Post
    Equation of orbit = ((x^2)/(4.481 x 10^21)) + ((y^2)/(3.256 x 10^21)) = 1
    How did you get the 4.481 x 10^21 term? It's probably just a typo, as 59^2 is 3481, not 4481.
    Thanks from FatimaA and topsquark
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  3. #3
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    Re: Find equation for Pluto's orbit.

    Hi ebaines, thanks for pointing that error out.

    I redid the problem and in the last few steps I determined:

    b^2 = a^2 - c^2

    b^2 = (5.9 x 10^9)^2 - (1.5 x 10^9)^2

    b^2 = 3.256 x 10^19

    Final answer in the form ((x^2)/(a^2)) - ((y^2)/(b^2)):

    ((x^2)/(3.481 x 10^19)) - ((y^2)/(3.256 x 10^19)) = 1
    Last edited by FatimaA; June 13th 2013 at 08:41 AM.
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  4. #4
    MHF Contributor ebaines's Avatar
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    Re: Find equation for Pluto's orbit.

    Looks good, except you left out the "= 1" for the right hand side of the equation. The only caveat is that the problem doesn't specify where the origin is, and you've arbitrarily centered the ellipse at the origin - that's OK, just want to point out that you could have centered the sun at the origin.
    Thanks from FatimaA and topsquark
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