Solving Kepler's Problem: Understanding Awkward Integration

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In summary, the conversation discusses a step in understanding Kepler's problem involving a substitution and elementary integration to simplify an equation. The suggested substitution is w=1/r and completing the square in the radical. The person is seeking further clarification on this method.
  • #1
Piano man
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Hi,
I'm reading up on Kepler's problem at the moment, and there's a step in the book that I don't understand.

Starting off with the equation of the path [tex]\phi=\int\frac{M dr/r^2}{\sqrt{2m[E-U(r)]-M^2/r^2}}+\mbox{constant}[/tex]

The step involves subbing in [tex]U=-\alpha/r[/tex], 'and effecting elementary integration' to get
[tex]\phi=\cos^{-1}\frac{(M/r)-(m\alpha/M)}{\sqrt{(2mE+\frac{m^2\alpha^2}{M^2}})}+\mbox{constant}[/tex]

But it doesn't look very elementary to me:confused:

Does anyone have any idea?

Thanks.
 
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  • #2
Try the substitution w=1/r and then complete the square in the radical.
 

FAQ: Solving Kepler's Problem: Understanding Awkward Integration

How does Kepler's problem relate to awkward integration?

Kepler's problem is a mathematical challenge that involves calculating the motion of a planet around the sun using the laws of gravity. The integration, or solving, of the equations involved in this problem can be quite difficult and awkward, leading to the name "awkward integration".

What makes solving Kepler's problem so challenging?

The main challenge in solving Kepler's problem is the need to accurately calculate the position and velocity of a planet over time, taking into account its constantly changing acceleration due to the gravitational pull of the sun. This requires complex mathematical calculations and can be difficult to solve analytically.

What are some approaches to solving Kepler's problem?

There are several approaches to solving Kepler's problem, including numerical methods such as Euler's method and Runge-Kutta methods, as well as analytical methods like Laplace's method and the use of elliptic integrals. Each approach has its own advantages and limitations.

How is solving Kepler's problem important in the field of astrophysics?

Solving Kepler's problem is crucial for understanding the motion of celestial bodies in our solar system and beyond. It allows us to accurately predict the positions of planets, moons, and other objects in space, and has played a significant role in the development of modern astrophysics and space exploration.

What are some real-world applications of solving Kepler's problem?

The solutions to Kepler's problem have been used in various fields, including space travel, satellite navigation, and astronomical observations. For example, the orbits of satellites and spacecraft are calculated using Kepler's laws of motion, and the positions of planets and stars can be predicted using solutions to Kepler's problem.

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