A little help with some formulae?

[george7378]

I'm working through the algorithm on this website for orbit propagation, and I've got to the section which defines some formulae for precession:

http://www.castor2.ca/04_Propagation/07_Precess/index.html

There are a few things which don't make sense on this page, and I'm hoping you might be able to help.

- At the top, it says RE = 1.0 Earth Radii, but I don't know if this is trying to say that it equals 6378.135km, or if it is just equal to 1 (surely not?)

- Same question for the formula which says that a1 = 1.221640413 Earth Radii - is it saying that I should put the number 1.22... into the next formulae, or should I put it in kilometres?

- In the formula for d1, it uses an undefined constant called aE - does anyone know what this is supposed to mean?

- The formulae for a0 and p0 also give their answers in terms of Earth radii - does that mean that I'm supposed to continue using them in terms of Earth radii or should I convert them to km before putting them into future formulae?

Thanks, and I appreciate it if someone can take the time to explain what it should be, if they know!

 

[Urwumpe]

It uses multiples of Earth radius as unit of distance... like you could use 1 km as unit of distance. What goes in as unit of distance, should also go out.

 

[george7378]

Oh, so if I put 6378.135km in, I will get the answers in km? Also, I guess it doesn't matter what I put into the last two formulae (for the precession of the RA of the ascending node and the argument of perigee), because they cancel anyway.

---------- Post added at 15:49 ---------- Previous post was at 15:30 ----------

I've also got another quick question please - the formula for true anomaly on this page:

http://www.castor2.ca/04_Propagation/04_True/index.html

...only gives answers between 0 and 180 - when the eccentric anomaly becomes greater than 180, it starts to count down from 180 to 0 again, i.e. in the case that:

eccentricity = 0.0001492, and eccentric anomaly = 245 degrees, the formula will give a true anomaly of 115 degrees - is this a convention or am I missing something?

Thanks again!

 

[Miner1]

When converting between true anomaly and eccentric anomaly, the best formula to use is

[math]tan \frac{E}{2} = \sqrt{\frac{1-e}{1+e}}tan \frac{\nu}{2}[/math]

Because it involves half-angle tangents, there is no need for quadrant checks for the result will always be in the proper quadrant. I would not recommend using the formula given on that page.

Hope that helps!

 

[george7378]

Oh, thanks. I tried that formula too - it gives me the same results up to 180 degrees, then it gives me an error (I suppose this is an asymptote) at E=180, and then after E=180, it gives me the same as the other formula, but negative - i.e. when E=270, v=-90, when E=300, v=-60, etc...

Is this how it's supposed to be?

 

[Miner1]

Yes, this is the behavior you should expect. The defined range for the arc tangent is [math](- \frac{\pi}{2}, \frac{\pi}{2})[/math]. Also, the tangent function goes to infinity at [math]\frac{\pi}{2}[/math], so if you implement this in software you'd need to do a check that if E is close enough to 180 degrees, set the tangent to a very large number.

 

[C3PO]

Oh, thanks. I tried that formula too - it gives me the same results up to 180 degrees, then it gives me an error (I suppose this is an asymptote) at E=180, and then after E=180, it gives me the same as the other formula, but negative - i.e. when E=270, v=-90, when E=300, v=-60, etc...

Is this how it's supposed to be?

You only have to look at the left side to see why the formula doesn't work for 180°.

[math]tan \frac{180}{2} = tan 90 = \infty[/math]