Ratio between Semi major and Velocity

Main Question or Discussion Point

Hi, has anybody shown that the ratio between two planets semi-major equals the ratio of their velocity. I've been studying the solar system using geometry and came up with formulas that show this relationship. I've checked my work several times and can't find anything wrong with it. I'll post my work up maybe later on today or tomorrow so you guys can check it.
 

Answers and Replies

Janus
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If you are talking about the ratio of the orbital velocities vs ratio of semi-major axi, then these two ratios are not directly related.

For a circular orbit, the orbital velocity can be found by:

[tex]V = \sqrt{\frac{GM}{a}}[/tex]

With G being the gravitational constant, M the mass of the Sun (Or body being orbited) and a the semi-major axis.

With two planets orbiting the same Sun the only thing that changes is a. Thus the orbital velocity changes inversely to the squareroot of the semi-major axis.
 
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It may also be helpful to note that one can calculate the orbit speed at apoapsis distance (a) based on arbitrary eccentricity (e):

[tex]v = \sqrt{\frac{GM}{a} \frac{1 - e}{1 + e} }[/tex]

Where e = 0, this works out to the same as the formula for circular orbit given in the post above.

Where e = 1, this works out to 0 velocity, which is equivalent to radial in-fall.

Likewise, at periapsis distance (p), the formula is:

[tex]v = \sqrt{\frac{GM}{p} \frac{1 + e}{1 - e} }[/tex]

Where e = 1, things obviously fall apart due to division by zero. In other words, it's nonsensical to talk about orbit and radial in-fall in the same breath, since they are mutually exclusive concepts.
 
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Janus
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Or you can use

[tex]v=\sqrt{GM \left ( \frac{2}{r}- \frac{1}{a} \right )}[/tex]

to find the velocity at a distance of r, for an orbit with a semi-major axis of a (this assumes that the eccentricity of the orbit allows for the given value of r for the given orbit.)
 

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