[amsat-bb] Conventions for apogee and perigee altitudes?
Phil Karn
karn at ka9q.net
Sun Oct 1 16:47:17 UTC 2017
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
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