# Keplerian Elements

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To calculate a satellite's orbit, there are 8 elements needed. This page is a brief overview of what those elements are and their possible range.

Two excellent resources are this tutorial from AMSAT and this tutorial from Rutgers.

# Keplerian Elements

## Epoch Time

An epoch is a reference to a specific start date and time and then the total number of seconds can be calculated from that epoch. For example, Unix Epoch is based on January 1st, 1970 at UTC, so October 31st, 2019 12:00:00 would be calculated as 1572571116 seconds since Jan 01 1970.

Orbital Inclination

This is the angle between the equator and the orbit when looking from the centre of the earth. If the orbit was exactly over the equator, the orbital inclination would be 0 degrees. The orbital inclination ranges from 0-180.

## Right Ascension of Ascending Node (RAAN)

This one can be explained better from the resources but I will do my best to try and summarize. The Right Ascension of Ascending Node is an angle measured at the centre of the earth from a point in the sky to the north-south equatorial plane. Now, since the earth spins the latitude and longitude coordinate system does not work and so we need a reference system that is fixed in space, so astronomers use the Vernal Equinox. The Vernal Equinox is defined as the point at which the sun crosses the equatorial plane from south to north, which is what defines the first day of spring. Basically, when you draw a line from the centre of the earth to your satellite in orbit you can measure the Right Ascension of Ascending Node. If that point in the sky is the Vernal Equinox, then RAAN is 0. The range of RAAN is from 0 to 360 degrees.

## Eccentricity

This element defines the shape of the orbit and ranges from 0 to 1. If the eccentricity is 0 then the orbit is a perfect circle, while if the eccentricity is 1 then the orbit is a very long and skinny ellipse.

## Argument of Perigee

The Apogee is the point furthest from the earth and the Perigee is the point closest to the earth. The argument of Perigee is the angle between the perigee and the ascending node and ranges from 0 to 360 degrees.

## Mean Motion

According to Kepler’s law, as the satellite gets closer to the Earth it moves faster and when it is further away it moves slower. By determining how fast the satellite is traveling then we can calculate the altitude. Kepler’s law is $v = \frac{GM}{r}$ where

v is the velocity of the satellite in m/s

M is the mass of the earth at a constant 5.98 x 10^24

G is the gravitational constant 6.672 x 10^-11

r is the distance between the centre of the earth and the satellite in m

## Mean Anomaly

This determines at which position along the orbit the satellite is with the perigee as the reference. The range is from 0 to 360 degrees.

## Drag

If the satellite is at a very low altitude then atmospheric particles will cause the satellite to drag and slow down its velocity. Another drag force includes radiation pressure, which is the pressure exerted by solar radiation due to the exchange in momentum from the radiation particles.