I have a real doosy that has got me stumped.(adsbygoogle = window.adsbygoogle || []).push({});

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:

http://ajbasweb.com/old/ajbas/2014/November/640-648.pdf

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.

Rod

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# True anomaly of binary orbit

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