In astrodynamics

Astrodynamics

Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and Newton's law of universal gravitation. It...

 or celestial dynamics, mean longitude is the longitude

Longitude

Longitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....

 at which an orbiting body could be found if its orbit were circular

Circular orbit

A circular orbit is the orbit at a fixed distance around any point by an object rotating around a fixed axis.Below we consider a circular orbit in astrodynamics or celestial mechanics under standard assumptions...

, and free of perturbations

Perturbation (astronomy)

Perturbation is a term used in astronomy in connection with descriptions of the complex motion of a massive body which is subject to appreciable gravitational effects from more than one other massive body....

, and if its inclination

Inclination

Inclination in general is the angle between a reference plane and another plane or axis of direction.-Orbits:The inclination is one of the six orbital parameters describing the shape and orientation of a celestial orbit...

 were zero. Both the mean longitude and the true longitude

True longitude

In astrodynamics true longitude is the longitude at which an orbiting body could actually be found if its inclination were zero. Together with the inclination and the ascending node, the true longitude can tell us the precise direction from the central object at which the body would be located at...

 of the body in such an orbit would change at a constant rate over time. But if the orbit is eccentric and departs from circularity (and let it still be supposed free from any perturbations), then the orbit would become a Keplerian ellipse, and then the progress of the orbiting body in true longitude along this orbit would no longer change at a constant rate over time. The mean longitude then becomes an abstracted quantity, still proportional to the time, but now only indirectly related to the position of the orbiting body: the difference between the mean longitude and the true longitude is usually called the equation of the center

Equation of the center

For further closely related mathematical developments see also Two-body problem, also Gravitational two-body problem, also Kepler orbit, and Kepler problem...

. In such an elliptical orbit, the only times when the mean longitude is equal to the true longitude are the times when the orbiting body passes through periapsis (or pericenter) and apoapsis (or apocenter).

In an orbit that is undergoing perturbations, an osculating orbit

Osculating orbit

In astronomy, and in particular in astrodynamics, the osculating orbit of an object in space is the gravitational Kepler orbit In astronomy, and in particular in astrodynamics, the osculating orbit of an object in space (at a given moment of time) is the gravitational Kepler orbit In astronomy,...

 together with its (elliptical) osculating elements can still be defined for any point in time along the actual orbit. For each successive set of osculating elements, a mean longitude can be defined, as in the unperturbed case. But here, the changes in mean longitude over time will not only be those due to some constant rate over time; there will also be superimposed perturbations (and the rate itself is also perturbed). A set of mean elements can still be defined for such an orbit, after abstracting the perturbational variations with time. The term 'mean longitude' was already used for the unperturbed and osculating cases, and the corresponding mean longitude member in a set of mean elements, after abstraction of the periodic variations, is sometimes therefore called the 'mean mean longitude'. To arrive at a true longitude from a mean mean longitude, the perturbational terms must be applied as well as the equation of the center.

Calculation

The mean longitude can be calculated as follows:


where:

• is orbit's mean anomaly
  Mean anomaly
  In celestial mechanics, the mean anomaly is a parameter relating position and time for a body moving in a Kepler orbit. It is based on the fact that equal areas are swept at the focus in equal intervals of time....
  
  ,
• is longitude of the orbit's periapsis
  Longitude of the periapsis
  In astrodynamics, the longitude of the periapsis of an orbiting body is the longitude at which the periapsis would occur if the body's inclination were zero. For motion of a planet around the sun, this position could be called longitude of perihelion...
  
  ,
• is the longitude of the ascending node
  Longitude of the ascending node
  The longitude of the ascending node is one of the orbital elements used to specify the orbit of an object in space. It is the angle from a reference direction, called the origin of longitude, to the direction of the ascending node, measured in a reference plane...
  
   and
• is the argument of periapsis
  Argument of periapsis
  The argument of periapsis , symbolized as ω, is one of the orbital elements of an orbiting body...
  
  .