Any orbit requires 6 elements to specify the position and motion fully. Since we live in 3-D space, it's equivalent to 3 spatial dimensions and 3 velocities. You could use (x,y,z) for the position and (vx,vy,vz) for the velocities. You could use spherical coordinates, or Euler angles. All of those give you, at any instant, the full position and motion in 3D of the satellite at a specific instance in time.

A more clever approach still uses 6 elements-- the minimum regardless of what dimensional or grid layout you choose. However, it results in a set of elements that let you predict future positions. If you specify the (x,y,z) positions and speeds, that tells you nothing about where the satellite will be next because (x,y,z) space doesn't factor in gravity.

However, since gravity means orbits trace out ellipses (as per Kepler's 3rd Law), and knowing the specific ellipse of an orbit lets you know the full path, defining the orbit elements using an ellipse gives you both the current position and movement, and a way of predicting where it will be next.

In implementation, then, the 6 elements are:

1) a = Semi-major axis = size

2) e = Eccentricity = shape

3) i = inclination = tilt

4) ω = argument of perigee = twist

5) Ω = longitude of the ascending node = pin

6) v = mean anomaly = angle now

The first two, a&e, yield the 2-D shape of the orbit. a gives you the size, and e gives you the squishyness. As a nuance, you can also get the period (time to do 1 orbit) of an elliptical orbit if you have that semi-major axis 'a' (p

^{2}/a

^{3}= 4 π

^{2}/MG)

The 3rd and 4th elements, i & ω, give you the 3D orientation. i is the tilt, the angle with which the entire orbit is tilted relative to the ecliptic plane. We define the 'ascending node' as the point where the orbit intersects the equatorial plane. w (argument of perigee or argument of periapsis) is the twist, the rotation or skew of that ellipse from a straight up-down, given as the angle from that infamous ascending node to the semi-major axis 'longest length diameter' of the ellipse.

The 5th parameter, Ω, ties it to Earth. Called many things-- longitude of the ascending node, right ascension of the ascending node, it tells you what longitude in the Earth-reference position the orbit goes over. It is measured CCW from vernal equinox (aka intersection of Earth's equator and ecliptic), so it's an absolute measure, and using the date you can translate it to an Earth 'right now' longitude. Since the Earth is turning underneath the orbit, that's pretty important to calculate.

The final parameter, v, is the mean true anomaly (or alternately, q, the true anomaly, or T

_{p}, the time of periapsis passage). That says, given the orbit, where the satellite is along that path. It's an angular measure from the usual reference point of perigee, or orbit's closest approach to Earth.

The excellent YouTube channel by 'mrg3' titled "Animation for Physics and Astronomy" has a good presentation of each 'Orbital Elements'.

Calliope will have a low eccentricity (e) orbit at 300-350km up (a), polar (i = 90 degrees), with the ω value probably close to 0 due to launching near the equator, Ω depending on the day of launch, and of course a wildly changing (but predictable) value v at any given time.

Things we'll consider in future columns:

* How they determine it (lasers, radar, radio Doppler, inertial, etc)

* What throws it off (tides, drag, solar, et cetera)

* Keplerian or Two-Line Element Sets (TLEs)

Until next week,

Alex

Launching Project Calliope, sponsored by Science 2.0, in 2011

News every Tuesday at The Satellite Diaries, every Friday at the Daytime Astronomer

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