Never Go To Space Alone®
Never Go To Space Alone™

Argument of Periapsis – how maths is responsible for getting us to the moon.

In the beginning...

Copernicus and Galileo assumed that orbits were perfectly circular but as we began to study the motion of planets, we discovered that planets don’t move in perfect circles, rather they move in more of an elongated elliptical shape.

17th century Kepler is often credited with developing the first plausible model of planetary orbit. He was a key figure in the scientific revolution and best known for his laws of planetary motion that was the foundation for Newton’s theory of gravity.

Newton developed general laws of celestial motion in the late 1680s which enabled Edmund Halley to accurately establish the orbits of various comets – one of which still famously bears his name today.

Halley's Comet - Photo Credit: Professor Barnard, Wisconsin.

A further milestone in orbit calculation was Gauss’ discovery and publishing of the six orbital elements that completely describe an orbit. These are: two elements that define the shape and size of the ellipse (Eccentricity and Semimajor axis); the tilt (inclination) and pin (longitude of ascending node) of the ellipsis, the twist (Argument of Periapsis) and the epoch. The epoch defines the position of the orbiting body along the ellipse at a specific time – we’ll discuss the Argument of Periapsis later.

The six orbital elements:

  • a = Semi-major axis = size.
  • e = Eccentricity = shape.
  • i = inclination = tilt.
  • ω = argument of perigee = twist.
  • Ω = longitude of the ascending node = pin.
  • v = mean anomaly = angle now.

Why is this important to space travel?

The theory of determining orbit is how we can now pin point people on GPS receivers, accurately track comets and planets through our solar system and given rise to the ability to operate satellites to accurately determine their future locations as necessary.

Dr. Robert Goddard at Clark University Photo Credit: NASA

Astrodynamics was born and developed by astronomer Samuel Herrick who specialised in celestial mechanics  in the 1930s. He consulted the rocket scientist Robert Goddard whose ultimate goal was space travel was encouraged to continue his work on space navigation techniques as Goddard believed they would be needed in the future. As early as 1936 Goddard developed a way to use maths and celestial mechanics to develop the navigation methods needed for space travel and satellite operations.

Astrodynamics numerical techniques were coupled with new powerful computers in the 1960s, and man was ready to travel to the moon and return.

Argument of Periapsis – what is it?  

The argument of periapsis is the angular distance between the ascending node and the point of periapsis (see Figure 1).

The prefixes "peri-" and "ap-" are commonly applied to the Greek or Roman names of the bodies which are being orbited. Going by a couple of different names (The Argument of Perifocus, or The Argument of Pericenter), the Argument of Periapsis (ω) is a way of talking about the orbit of a planet, asteroid or comet - it describes the angle of an orbiting body’s periapsis relative to its ascending node.

For sun-centred orbits the argument is entitled, ‘perihelion’; for earth-centred orbits we use: ‘perigee’ and for general use you may see ‘pericenter’ replace periapsis.

Argument of Periapsis diagram
Argument of Periapsis diagram. Credit: EVONA - Space sector recruitment

In order to understand the argument,  you need to understand a few key phrases.

• The periapsis is the point when the orbiting object comes the closest to the thing it is orbiting around.

• The ascending node is the point where the body crosses the plane of reference from North to South.

• The angle is measured in the orbital plane and in the direction of the motion.

Calculation of Argument of Periapsis

In astrodynamics the argument of periapsis ω can be calculated as follows: ω=arccosn⋅e|n||e|

(if ez<0 then ω=2π−ω)


n is the vector pointing towards the ascending node (i.e. the z-component of n is zero),

e is the eccentricity vector (the vector pointing towards the periapsis).

In the case of equatorial orbits, though the argument is strictly undefined, it is often assumed that: ω=arccosex|e|


ex is x-component of the eccentricity vector e.

In the case of circular orbits it is often assumed that the periapsis is placed at the ascending node and therefore ω=0.

Planets' Argument of Periapsis calculations in degrees

Planets' Argument of Periapsis calculations in degrees
Planets' Argument of Periapsis calculations

The value of the Argument of Periapsis to astrodynmics

Without understanding the six orbital elements including the Argument of Periapsis alongside astrodynamics and celestial mechanics, space travel and space exploration as we know it wouldn’t be possible. Navigation relies upon the understanding of orbits. There simply wouldn’t be any satellites that we could effectively manipulate in space because we couldn’t plan for their future positions or adequately adjust or predict to avoid other orbiting bodies.

This is why advancements in STEM will be responsible for getting us further in space exploration. It’s not just about technological, scientific or engineering advancements, but also the fundamental understanding of mathematics that will push us further than we have ever gone before. The future of space Industry recruitment is reliant upon attracting a new generation of workforce into the sector. Getting children interested in all aspects of STEM is key to the long-term sustainability of the sector.