On 10/1/17 04:48, Alan wrote:
I expect for amateur purposes, a simple sphere is fine.
I'm interested in what others are doing so I can be at least somewhat consistent with their results.
My current method for computing apogee and perigee seems to give results that match many of the online tracking programs. That is, find the geometric apogee and perigee and subtract the radius of the earth under those points. I can't easily tell if they make the correction for geodetic latitude because the effect is small and the results are usually not given with a lot of precision. But you do end up with weird oddities like seeing the ISS at an altitude that exceeds calculated apogee. As long as everybody is happy with that little anomaly, I'm happy too.
If you want to compute the energy in a (decaying) orbit, there's a simple and straightforward formula:
specific orbital energy = -GM/(2*a)
where 'GM' (also known as mu) is the earth's gravitational parameter = 3.986004418e14 m^3/s^2 and 'a' is the orbit's semi-major axis in meters.
The specific energy is the sum in joules of the satellite's kinetic and potential energy per kilogram of its mass. Because of energy conservation, it will remain constant for any satellite in any trajectory unless it is gaining or losing energy from thrust, drag, or gravitational exchanges with a third body. It is always negative for a closed orbit because potential energy is zero at infinity and increasingly negative as you get closer. A positive specific energy means a hyperbolic (escape) trajectory.
For the ISS I calculate a specific energy of about -29.38 MJ/kg.
Problem is, you can't recover the semimajor axis from the apogee and perigee unless you know how they're computed. And if they're done the way I described, you also need the latitude of apogee and perigee so you can add the right earth radius, and that in turn usually requires knowing the inclination and argument of perigee, the latter changing steadily with time.
But if you have the mean motion or period you can compute the semi major axis from it using Kepler's third law:
a = (GM*P^2)^1/3
where the period P is expressed in seconds/radian = seconds/revolution / (2*pi), or
a = (GM/N^2)^1/3
where the mean motion N is expressed in radians/sec = rev/day * 2 * pi/86400.
So all you really need to compute specific energy and track decay is the satellite's period. That's it! Forget apogee and perigee...
You can also track the specific angular momentum of the satellite, which must also remain constant absent external forces besides simple 2-body gravity:
h = sqrt(GM * a(1-e^2))
The units of specific angular momentum are m^2/s, or area per time. This is where Kepler's second law comes from: the satellite sweeps out equal areas in equal time because that represents the satellite's (conserved) specific angular momentum.
In fact, we could describe an orbit with the angular momentum vector, which points normal to the orbit plane and defines RAAN and inclination, and the eccentricity vector, which points at perigee and defines the argument of perigee and the eccentricity. The two together define the mean motion, and if the epoch corresponds to a specific point on the orbit that sets the mean anomaly. That's all 6 Keplerian elements.
--Phil