Mean motion of Halley's comet: units for semi-major axis and angular frequency

[Deadstar]

Homework Statement

I'm trying to find out the mean motion of Halleys Comet using the equation.

$$n = \sqrt{\frac{GM}{a^3}}$$

where we have taken the mass of Halleys comet to be negligible compared to the sun, M is the mass of the sun and a is the semi major axis.

The Attempt at a Solution

My question is what unit do I take a to be (i.e AU? m?). Nothing seems to give me an answer that makes sense.

And what units should n actually have? I seem to think it should be in revolutions per day but I'm not quite sure what the above formula is actually giving me. It would seem to be $$ms^{-1}$$ (or square root of, whatever that means...) if we take a in meters but then I get an answer of around $$2.6^10^{-9}$$ which is clearly wrong!

P.S. I'm a mathematician which is why I have zero grasp of units...

 

[tiny-tim]

Hi Deadstar!

If you put SI units in, you always get SI units out.

As you know, GMm/r² is a force, ie mass times acceleration, and so has dimensions of ML/T²

So G has dimensions of L³/MT²,

and so √(GM/a³) has dimensions of 1/T, and will have units of s^-1.

 

[Deadstar]

Hi Deadstar!

If you put SI units in, you always get SI units out.

As you know, GMm/r² is a force, ie mass times acceleration, and so has dimensions of ML/T²

So G has dimensions of L³/MT²,

and so √(GM/a³) has dimensions of 1/T, and will have units of s^-1.

Thanks tiny-tim.

This is actually what I got as well (not sure where that ms^-1 came from, might have been a mistype...) from using a different formula to derive the units.

So, I'm now getting 2.6x10^(-9) as my answer.

What is this telling me exactly? The change in angle per second? I can't seem to find any info on mean motion except the very small wikipedia link.

 

[tiny-tim]

Hi Deadstar!

(try using the X² icon just above the Reply box )

I'm guessing it's either radians per second or revolutions per second.

I've not seen the formula before … where did you get it from?

 

[Deadstar]

Hi Deadstar!

(try using the X² icon just above the Reply box )

I'm guessing it's either radians per second or revolutions per second.

I've not seen the formula before … where did you get it from?

Wikipedia.

http://en.wikipedia.org/wiki/Mean_motion

Although note that, M + m was taken to be just M as the comets mass was negligible compared to the sun.

 

[Deadstar]

I should also say that I will then be using the formula in Keplers equation.

$$n(t-\tau) = E - e \sin(E)$$

Where $$\tau$$ is the time of pericentre passage
E is the eccentric anomaly
e is the eccentricity

I'm trying to find E given some time t which I have.

 

[tiny-tim]

Well, it says it's the "angular rate", ie radians per second, but it gives no references, and I have no idea whether that's correct.