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Orbit of Comet 17P Holmes - equations

  1. Apr 19, 2008 #1
    Im trying to learn how to calculate orbits.

    Comet 17P Holmes
    e: 0.432668
    q: 2.053207
    Ascending Node: 326.8586
    w: 24.2856
    Inclination: 19.1134
    Epoch: 2454600.5 (2000)
    Tp: 20070504.5695


    I can calculate Semi Major Axis, Semi Minor Axis, semi-latus rectum, Period in years

    [tex]a = q/(1-e)[/tex]
    [tex]b = a\sqrt{1-e^2}[/tex]
    [tex]l = a(1-e^2)[/tex]
    [tex]T = \sqrt{a^3}[/tex]


    semi-major axis a 3.61905727158
    semi-minor axis b 3.2627731245014
    semi-latus l 2.94156396627603518632608
    Period T 6.8848294081932


    How do I calculate
    Perihelion date

    at any date in the orbit
    distance to the Sun, Earth in AU
    orbital speed


    thank you for you help.

    Tony
     
  2. jcsd
  3. Apr 21, 2008 #2
    Checking on wiki as Im trying to teach myself about orbits
    [tex]\mu = GM[/tex] Standard gravitational parameter
    On Wiki I also see that
    [tex]\mu = 4\pi^2a^3/T^2[/tex] Standard gravitational parameter

    but when I calc them they do not match
    using the above orbital elements
    I get [tex]\mu = 39.478417604357[/tex] when using [tex]\mu = 4\pi^2a^3/T^2[/tex]
    and
    [tex]\mu = 132712440018[/tex] [tex]\mu = GM[/tex] this number was taking right from the wiki page for the GM of the Sun.

    lol I still dont have enough posts to post a link.

    what is the correct Standard gravitational parameter? Am I missing a step or doing it out of order?
     
  4. Apr 21, 2008 #3
    I assume you are talking about this wiki page:

    http://en.wikipedia.org/wiki/Standard_gravitational_parameter ?

    From another source ( http://www.projectrho.com/rocket/Orbits.htm ) I get GM = 1.32715 x 10^11 in km^3 / s^2 which does not exactly match, but is pretty close to the wiki number you quote.

    I have seen that in your first post you state that you have calculated the period T in years. but surely for the formula above you have to use the value in seconds. could you just repeat all the parameters you use as input for the formula together with their units (s, km, whatever) ?
     
    Last edited: Apr 21, 2008
  5. Apr 21, 2008 #4
    Thanks for the reply

    I have less then 15 posts so I cant post the link yet, but the wiki page I am talking about is the same one you posted.

    This is right on the page. So I used period in years.
    for elliptic orbits: [tex]4 \pi^2 a^3/T^2 = \mu [/tex] (with [tex]a[/tex] expressed in AU and [tex]T[/tex] in years, and with M the total mass relative to that of the Sun, we get [tex]a^3 / T^2 = M[/tex])

    e: 0.432668
    q: 2.053207 AU
    Ascending Node: 326.8586 Deg
    w: 24.2856 Deg
    Inclination: 19.1134 Deg
    Epoch: 2454600.5
    Tp: 20070504.5695
    Orbital period: 6.88 Years

    semi-major axis a 3.61905727158 AU
    semi-minor axis b 3.2627731245014 AU
    semi-latus l 2.941563966276 AU
    Period T 6.8848294081932 Years

    I wish that I could post links but on the wiki page wiki/Periapsis_distance

    its does not give the units of speed. would it be km/s ?
    Periapsis: maximum speed [tex]v_\mathrm{per} = \sqrt{ \frac{(1+e)\mu}{(1-e)a} } \,[/tex] at minimum (periapsis) distance [tex]r_\mathrm{per}=(1-e)a\!\,[/tex]

    Apoapsis: minimum speed [tex]v_\mathrm{ap} = \sqrt{ \frac{(1-e)\mu}{(1+e)a} } \,[/tex] at maximum (apoapsis) distance [tex]r_\mathrm{ap}=(1+e)a\!\,[/tex]


    How do I calculate the orbital speed at times other than Perihelion and Aphelion
     
  6. Apr 21, 2008 #5

    Janus

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    The units of speed will depend on the units used. If you use meters, kilograms and seconds, your answer will be in meters/sec. If you use AUs, years and the [itex]\mu[/itex] you arrived at above, your answer will be in AU/yr.
    (The reason you got a different value for [itex]\mu[/itex] than wiki was that you were using different base units.)
    You can use:

    [tex]v=\sqrt{\mu \left ( \frac{2}{r}- \frac{1}{a} \right )}[/tex]

    where [itex]r[/itex] is the radial distance from the Sun at that part of the orbit.

    You can find [itex]r[/itex] for any angle [itex]\theta[/itex] as measured from perihelion by

    [tex]r = a \left ( \frac{1-e^2}{1+ e \cos{\theta}} \right )[/tex]
     
  7. Apr 22, 2008 #6
    Thank you, your answer made it very clear to me, now that I know about AU/yr vs km/s.
     
  8. Apr 23, 2008 #7

    I'm having a problem with the value for r. r should not be smaller than 2.053207 or larger than 5.1849075431599 right?

    I calculate
    2.053207 AU [tex] r_\mathrm{per}=(1-e)a\!\,[/tex]
    5.1849075431599 AU [tex] r_\mathrm{ap}=(1+e)a\!\,[/tex]

    also shouldn't [tex] r_\mathrm{ap}-r_\mathrm{per}= a[/tex]


    semi-major axis a 3.61905727158 AU [tex]a = q/(1-e)[/tex]
    e: 0.432668
    q: 2.053207 AU
    Ascending Node: 326.8586 Deg
    w: 24.2856 Deg
    Inclination: 19.1134 Deg
    Epoch: 2454600.5
    Tp: 20070504.5695
    Orbital period: 6.88 Years


    [tex]r = a \left ( \frac{1-e^2}{1+ e \cos{\theta}} \right )[/tex]



    Code (Text):

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    Last edited: Apr 23, 2008
  9. Apr 23, 2008 #8

    Janus

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    Staff Emeritus
    Science Advisor
    Gold Member

    You made a mistake somewhere with the equation. You should never get an negative answer. The limits for cos[itex] \theta[/itex] are -1 and 1. Thus the limits of your answer will come out between:

    [tex]r = a \left ( \frac{1-e^2}{1+ e }} \right )[/tex]

    and

    [tex]r = a \left ( \frac{1-e^2}{1- e }} \right )[/tex]
     
    Last edited: Apr 23, 2008
  10. Apr 23, 2008 #9

    Janus

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    Staff Emeritus
    Science Advisor
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    Okay, here's your problem. Instead of adding e times cos [itex]\theta[/itex] to 1 like you should, you are adding 1 to e and multiplying this sum by cos [itex]\theta[/itex].
     
  11. Apr 23, 2008 #10
    ohhhhhhhhhhhhhhh its a good thing my webcam is off I have a big red face.

    yes works great now I also had to deg2rad the angle.

    thank you for your help
     
  12. May 11, 2008 #11
    Ok I have been away for a while trying to get past the next step, but I'm not having much luck in finding out how to do it.

    I would like to be able to calculate velocity and distance and position at a given time. if you can point me in the right direction a website or a book/video that would be huge help.

    I am thinking of ordering Orbital Mechanics by John Prussing and Astronomical Algorithms by Jean Meeus. Does any recommend these books or some other book?
     
    Last edited: May 11, 2008
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