Question about the Kepler equation

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In summary, the conversation discusses manipulating an equation to derive the value 7.27. Different attempts were made, including substituting variables and rearranging the equation, until the desired result was achieved.
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Homework Statement


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Homework Equations


1/r = a/b2 * (1+e cosθ)

The Attempt at a Solution


How do you derive 7.27?
I tried to substitute r in the orbit equation by (acosψ - ae) / cosθ and got
(cos ψ - e)/cosθ*(1+e cosθ)=b^2/a^2 I don't know what to do next. Please help, thank you.
 

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Clara Chung said:
(cos ψ - e)/cosθ*(1+e cosθ)=b^2/a^2
I believe this is OK. You just need to manipulate it to get (7.27). I don't know the quickest way, but I found something that works. I don't want to give too much away.
Rearrange your equation as (cos ψ - e)*(1+e cosθ)=(b^2/a^2)*cosθ. Substitute for b^2/a^2 in terms of e. Multiply everything out and see if you can get to the result.
 
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I tried hard by substituting (b^2/a^2) but still can't figure out the equation... Then I changed my mind to substitute for e. Thank you so much.
(cos ψ - e)*(1+e cosθ)=(b^2/a^2)*cosθ

(ecos ψ - 1+b^2/a^2)*(1+e cosθ)=(b^2/a^2)*cosθ*e

-(1-ecosψ)(1+ecosθ)+(b^2/a^2)e cosθ+(b^2/a^2)=(b^2/a^2)*cosθ*e

(1-ecosψ)(1+ecosθ)=(b^2/a^2)
 
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Nice.
 

What is the Kepler equation?

The Kepler equation is a mathematical expression that relates the orbital period and semi-major axis of a planet to its mass and the mass of the central body in a two-body system. It is used to calculate the position of a planet in its orbit at a given time.

Who discovered the Kepler equation?

The Kepler equation was derived by German astronomer and mathematician Johannes Kepler in the early 17th century. It is named after him in recognition of his contributions to the field of celestial mechanics.

What is the significance of the Kepler equation?

The Kepler equation is significant because it allows scientists to accurately predict the position of a planet in its orbit at any given time. It also provides insight into the dynamics of planetary motion and has been crucial in the development of modern theories of gravity and orbital mechanics.

How is the Kepler equation used in space exploration?

The Kepler equation is used in space exploration to calculate the trajectories of spacecraft and to plan and execute missions to other planets and celestial bodies. It is also used to analyze and interpret data from spacecraft observations of other planets and their moons.

Are there any limitations to the Kepler equation?

Yes, the Kepler equation is based on simplified assumptions and does not take into account the influence of other bodies or factors such as relativity and atmospheric drag. It is most accurate for predicting the positions of planets in our solar system and may not be as effective for more complex systems.

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