Orbital Mechanics Notes

A ground trace is the path a satellite traces over the ground, like a magic marker hanging from the satellite.
It's a projection of the satellite’s orbit onto the Earth.
Satellites appear to move westward due to the Earth's eastward rotation.
After a full day, the ground trace of a satellite with an approximate 90-minute orbital period would span all longitudes because the Earth is continually rotating below the orbit.

Inclination determines the northern and southern latitude limits of a satellite's orbit.
For example, a satellite with a 45° inclination will have a ground trace ranging from 45° north to 45° south.
The inclination of an orbit can be determined by examining its ground trace.

Equatorial Orbit: An orbit with an inclination of 0 degrees.
- A satellite in an equatorial orbit passes directly over the equator.
Polar Orbit: An orbit with an inclination of 90 degrees.
- A satellite in a polar orbit passes over the entire Earth.

Inclination is determined by noting the northern and southern latitude limits of the ground trace.
Orbital period can be determined using a simple calculation.

The orbit of a satellite remains fixed in space while the Earth rotates underneath it.
The westward regression of the ground trace is due to the Earth's rotation.
Calculate the time it takes for the Earth to rotate one degree: $1440 \text{ minutes} / 360 \text{ degrees} = 4 \text{ min/degree}$
Determine how many degrees per pass the satellite’s orbit regresses on consecutive orbits, using the equatorial crossing as a common reference point (e.g., 25 degrees).
Calculate the time it took the Earth to rotate that many degrees to find the satellite's orbital period: $25 \text{ degrees} * 4 \text{ min/degree} = 100 \text{ minutes}$

Satellites can have identical eccentricities, semi-major axes, and inclinations (e, a, and i) but be oriented differently in space.
Longitude can’t be used as a reference because the Earth rotates underneath the orbits.
RAAN is the angle measured along the equatorial plane between:
- A vector pointing to a fixed reference point in space (the first point of Aries, also known as the vernal equinox).
- The point on the orbit where the orbital motion is from south to north across the equator (ascending node).

Even with the same e, a, i, and W, orbits can have different orientations around the Earth.
Argument of perigee describes the orientation of the orbit within the orbital plane, indicating the location of apogee and perigee.

Describes the satellite’s position within an orbit at any instant.
It is the angle between the perigee point and the satellite’s location, measured in the direction of the satellite’s motion.
True anomaly is 0 degrees at perigee and 180 degrees at apogee.

The Keplerian element set consists of 6 parameters:
- Two describe the size and shape of an orbit:
  - Eccentricity (e)
  - Semi-major axis (a)
- Three describe the orientation of the orbit in space:
  - Inclination (i)
  - Right ascension of the ascending node (W)
  - Argument of perigee (w)
- One describes the location of the satellite within the orbit:
  - True anomaly (u)
A time stamp (epoch) must be included to indicate when the values were accurate.

Kepler’s 1st Law: Satellites travel around Earth in elliptical paths with the center of Earth at one of the foci.
- The speed of a satellite changes as the distance between it and Earth changes. It moves fastest at perigee and slowest at apogee.
Kepler’s 2nd Law: A line drawn between Earth and a satellite sweeps out equal areas during equal time periods anywhere along the orbit.
Kepler’s 3rd Law: The period of an orbit (T) is related to its semi-major axis (a) by: $T^2 = \frac{4\pi^2}{m} * a^3$