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True anomaly of binary orbit

  1. Feb 21, 2016 #1
    I have a real doosy that has got me stumped.
    I need to solve the following equation for v:
    tan(v + ω) = tan(θ + Ω)sec(i)

    The symbols stand for the following values in an elliptical orbit of one point source around another (on the celestial sphere):
    where v = true anomaly; ω = argument of periastron; θ = position angle; Ω = position angle of the ascending node; i = orbital inclination (to line of sight)

    The following obvious solutions don't actually work (at least not for every case):
    v = atan(tan(θ + Ω)sec(i)) - ω
    v = atan2(sin(θ + Ω)sec(i), cos(θ + Ω)) - ω

    I think the reason the first one doesn't work is to do with quadrant issues when taking the arctangent.
    I think the reason the second one doesn't work has something to do with the direction of orbital motion. I've tried:
    v = atan2(± sin(θ + Ω)sec(i), cos(θ + Ω)) - ω, but that doesn't work either in every case.

    There is a convoluted solution on page 643 of:
    but try as I might I cannot make this work (in Octave/Matlab) and it does seem unnecessarily complicated.

    Either way I've been working on this for weeks and just can't get it. Any help would be hugely appreciated.

  2. jcsd
  3. Feb 21, 2016 #2


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    Science Advisor
    Gold Member

    Here's a calculator I made that will do it for you.
    But the method is a bit different. It computes it from the R and V vectors and the Sun's mu (G*M), rather than directly from the other elements.
    You can view the source and read the javascript. Search for "var TA = arctan2(TAy, TAx);" and work backwards from there.
  4. Feb 22, 2016 #3
    Thanks for the reply, but I can't see how this is relevant to my problem - that could be my fault though. They both involve arctangents but the knowns are different. All I know are the variables listed (except v of course) and the direction of motion/orbit (clockwise or anti). Another issue I forgot to mention that may be relevant is that in many cases there is data for more than one orbit. I've tried just selecting data for one orbit, but still doesn't work.
  5. Feb 22, 2016 #4
    Oops, that should be: tan(v + ω) = tan(θ - Ω)sec(i)
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