Johannes Kepler discovered in the 1600s that planet orbits form ellipses, not circles.
Satellites (natural or human-made) orbit Earth as an ellipse.
Elliptical orbits remain fixed in space, and Earth spins under a fixed satellite orbit.
An orbit is a closed path around which a planet or satellite travels.
What Is an Ellipse?
An ellipse is the two-dimensional shape that is produced by a plane fully intersecting a cone.
A plane intersecting the cone at an angle perpendicular to the cone’s center line will form a special ellipse called a circle.
An ellipse has two foci instead of a center.
The sum of the distances from the foci is constant: A + B = constant
A circle is simply an ellipse with both foci located at the same spot.
A circle is a set of points fixed (constant distance) from a center point (focus): A = constant
Satellites orbit Earth with one focus at Earth’s center.
The other focus is an empty point, which may or may not be within Earth’s boundaries.
'a' defines ½ the major axis length.
'b' defines ½ the minor axis length.
'c' is the distance from the center of the ellipse to either focal point.
For a circle, a and b are equal to the radius, and both focal points are co-located at the center of the ellipse.
Diverse Orbits
Common orbit types:
LEO (Low Earth Orbit)
MEO (Medium Earth Orbit)
HEO (Highly Elliptical Orbit)
GEO (Geostationary Orbit)
How Are Orbits Described?
Orbits are described by a set of parameters called orbital elements (i.e., Keplerian elements).
The Keplerian element set consists of 6 parameters (plus a timestamp):
Two describe the size and shape of an orbit.
Three describe the orientation of the orbit in space.
One describes the location of the satellite within the orbit.
Eccentricity (e)
Eccentricity describes the roundness of an orbit, or the shape of the ellipse in terms of how wide it is.
The formula to calculate eccentricity is: e = \sqrt{1 - \frac{b^2}{a^2}}, where 'a' is the semi-major axis and 'b' is the semi-minor axis.
Eccentricity can vary from 0 to 1 (for “closed” orbits).
An eccentricity of 0 means the orbit is circular.
An eccentricity of 1 or greater means the orbit is not closed, which is used for interplanetary missions. Satellites in these types of orbits do not return to their starting point.
Values between 0 and 1 mean the orbit is elliptical.
Semi-Major Axis
The semi-major axis, denoted as 'a', describes the size of the ellipse.
It is half of the largest diameter (the major axis) of the orbit.
The semi-major axis originates from the center of the orbit, which makes it difficult to visualize from our reference point on Earth.
Important Points on the Orbit
Apogee defines the point in an orbit that is farthest from Earth.
Perigee describes the point in an orbit that is closest to Earth.
The suffix “gee” refers to Earth (e.g., apoapsis and periapsis).
Apogee altitude is the distance between the surface of the Earth and apogee.
Perigee altitude is the distance between the surface of the Earth and perigee.
In a circular orbit, apogee altitude and perigee altitude are the same.
A perfectly circular orbit has neither an apogee nor a perigee and is undefined.
Perfectly circular orbits cannot be achieved.
Generally, circular orbits are described by their altitude.
The semi-major axis is rarely used to describe circular orbits.
Semi-Major Axis (Altitude for circular orbits)
The semi-major axis is the only orbital parameter that determines the orbital period.
This is related to Kepler’s 3rd Law: The square of the period of a planet is proportional to the cube of its mean distance from the Sun.
Formula: T^2 \propto a^3
Where \mu = GM , G = Universal Gravitation Constant (6.67 x 10^{-11} m^3/kg*s^2) and M = Mass of the central body.
T = 2\pi\sqrt{\frac{a^3}{\mu}}
Describing the Orientation of the Orbit in Space
Orbits may have identical sizes and shapes (a and e), yet they can vary in their orientation in space.
Three additional Keplerian elements define this orientation:
Inclination
Right ascension of the ascending node
Argument of perigee
Inclination (i)
Inclination is the angle between the Earth’s equatorial plane and the plane of the orbit.