Period of a satellite in an elliptical orbit compared to circular orbit

[bullroar_86]

not sure what to do here..

Im being asked to compare the periods of 2 different satellites in orbit around a planet.


the first one is a circular orbit of radius = r

the second one orbits 1r to the left and 3r to the right around the planet.

I'll attempt to draw it here

0 is the planet




(---------0---------)------------------------)
<---r----><---r----><---------2r------------>



I understand how the period works in the first circular orbit.. but not the second one. any ideas?

 

[Doc Al]

Start by reviewing what Kepler's 3rd law says.

 

[bullroar_86]

it states that r^3/T^2 = K

is it just the average radius of the second satellite? (in this case 2r ?)

 

[BobG]

it states that r^3/T^2 = K

is it just the average radius of the second satellite? (in this case 2r ?)

Yes, the mean distance, average radius, or, more commonly, the semi-major axis.

The shape of the orbit doesn't matter (your second orbit has an eccentricity of .5)

 

[DaveC426913]

So, a satellite in an eccentric orbit will have a period that is equivalent to a circular orbit whose radius is equal to (aphelion minus perihelion) of the eccentric orbit?

So, if an asteroid happened to be on an orbit that went out as far as Jupiter, and in as far as Mercury, its orbital period would be equivalent to a circular orbit whose radius is (Jupiter's - Mercury's) orbit?

I'd always wondered that.

 

[SpaceTiger]

So, a satellite in an eccentric orbit will have a period that is equivalent to a circular orbit whose radius is equal to (aphelion minus perihelion) of the eccentric orbit?

A circular orbit whose radius is equal to the semimajor axis of the ellipse. Perihelion is

$$r_{peri}=a(1-e)$$

where a is the semimajor axis and e is the eccentricity. Aphelion is

$$r_{ap}=a(1+e)$$

So, the semimajor axis is given not by r_ap - r_peri, but rather:

$$a=\frac{1}{2}(r_{ap}+r_{peri})$$

This is what scales with period in Kepler's 3rd law:

$$P^2 \propto a^3$$

 

[James R]

So, a satellite in an eccentric orbit will have a period that is equivalent to a circular orbit whose radius is equal to (aphelion minus perihelion) of the eccentric orbit?[/quote[

No. A satellite in an eccentric orbit will have a period equal to that of a circular orbit of radius equal to the semi-major axis of the elliptical orbit. (The semi-major axis is the half-length of the longest axis of the ellipse.)