Trigonometry related to Kepler's Equation

In summary, the conversation is about deriving Kepler's Equation and the use of trigonometry to calculate the true anomaly, represented by nu. The question is raised about how the third equation is incorporated into the fourth equation. The source of the equations is discussed and it is suggested to use the first two equations instead.
  • #1
solarblast
152
2
In deriving Kepler's Equation, there's a little trig work that gets at nu (see figure), which is the true anomaly. I'm curious how the third equation makes it into the 4th (last) equation. It's been a very long time since I've played with trig. nu of course is the atan of the last equation.
 

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  • #2
solarblast said:
In deriving Kepler's Equation, there's a little trig work that gets at nu (see figure), which is the true anomaly. I'm curious how the third equation makes it into the 4th (last) equation. It's been a very long time since I've played with trig. nu of course is the atan of the last equation.


You should write down, with LaTeX, directly the equations here so that it'll be possible to check the notation and give, eventually, and answer, otherwise

it is way too cumbersome.

DonAntonio
 
  • #3
See <http://rip94550.wordpress.com/2011/05/02/elliptical-orbits-%E2%80%93-deriving-keplers-equation/>. Attached is the part I was looking for.
 

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  • #4
You're using the wrong equation as the source. Use the first two. From the second, compute [itex]1+\cos\nu[/itex]. Next compute [itex]sin\nu/(1+\cos\nu)[/itex]. This is just [itex]\tan(\nu/2)[/itex] per the half angle formula. Use the same formula to simplify the right hand side.
 
  • #5
What source are you referring to? The link or one of the equations in the attachment? The link is 10 pages long.
 
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  • #6
I'm referring to the attachment in your first post.
 
  • #7
Ah, that was produced by a CM "pdf book" on the web. I'm satisfied with the longer attachment. I may contact the author of the pdf.
 

FAQ: Trigonometry related to Kepler's Equation

What is Kepler's Equation?

Kepler's Equation is a mathematical expression that describes the relationship between the orbital period and distance of a planet or satellite in an elliptical orbit around a central body, such as the sun.

How is trigonometry related to Kepler's Equation?

Trigonometry plays a crucial role in solving Kepler's Equation, as it involves the use of trigonometric functions such as sine, cosine, and tangent to calculate the position and velocity of a planet or satellite at a given time in its orbit.

Can Kepler's Equation be used for all types of orbits?

Yes, Kepler's Equation can be applied to all types of orbits, including circular, elliptical, and parabolic orbits. However, it is most commonly used for elliptical orbits.

How is Kepler's Equation used in space exploration?

Kepler's Equation is used by scientists and engineers to accurately predict the position and velocity of spacecrafts in orbit around other planets or moons. It is also used to plan and execute spacecraft trajectories, such as the Hohmann transfer orbit.

Are there any limitations to Kepler's Equation?

Yes, Kepler's Equation assumes that the orbiting body is in a vacuum and is affected only by the gravitational pull of the central body. It also assumes that the orbits are stable and circular or elliptical in shape. In reality, there may be other external forces and perturbations that can affect the orbit, making the calculations less accurate.

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