Conundrum of Solving True Anomaly in Binary Orbits: A Scientist's Dilemma

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Discussion Overview

The discussion revolves around solving the equation for true anomaly (v) in the context of binary orbits. Participants explore the mathematical challenges associated with the equation and the implications of various orbital parameters, including the argument of periastron, position angles, and orbital inclination. The focus is primarily on theoretical and mathematical reasoning related to orbital mechanics.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Rod presents the equation tan(v + ω) = tan(θ + Ω)sec(i) and expresses difficulty in finding a valid solution for v, noting that common solutions do not work in all cases.
  • Rod suggests that quadrant issues may affect the first proposed solution, while the second solution's failure might relate to the direction of orbital motion.
  • A convoluted solution referenced from a paper is mentioned, but Rod finds it overly complicated and unworkable in his attempts using Octave/Matlab.
  • Another participant shares a link to an orbital elements calculator that computes true anomaly from R and V vectors, suggesting a different method that does not directly use the same elements as Rod's problem.
  • Rod questions the relevance of the calculator to his specific issue, emphasizing the differences in known variables and the direction of motion, and mentions complications arising from having data for multiple orbits.
  • Rod corrects the original equation to tan(v + ω) = tan(θ - Ω)sec(i), indicating an adjustment in the problem statement.

Areas of Agreement / Disagreement

Participants do not reach a consensus on a solution to the problem. There are multiple competing approaches and unresolved issues regarding the application of the equations and the implications of orbital parameters.

Contextual Notes

Rod's attempts to solve the equation are complicated by quadrant issues, the direction of orbital motion, and the presence of data for multiple orbits. The discussion highlights the dependency on specific definitions and the challenges of applying different methods to the same problem.

Rod_123
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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:
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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http://orbitsimulator.com/formulas/OrbitalElements.html
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.
 
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.
 
Oops, that should be: tan(v + ω) = tan(θ - Ω)sec(i)
 

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