Description of an orbit: the orbital elements  

To know the state of motion of a body, it would be necessary to know its position and its speed. Furthermore, the values of these six parameters (the three components of the position and the three components of the speed) should be known at every instant. But, during a general motion, these values change constantly and knowing them at every moment is impossible.
In the reality, constant, stable orbits (as in the 2-body problem), can be described by 6 constant parameters, called orbital elements. In other words, to know the state of motion of the body, it is not necessary to know the position and the speed of the object at every instant. The knowledge of the numerical values of the 6 constant parameters that characterize the orbit is sufficient.

2 dimensional orbit 
Let's first consider the orbit solution of a 2-body problem: the orbit lies completely on a plane, called orbital plane. If the orbit is elliptic (click here to know more about ellipse ) and periodic, only two of the 6 orbital parameters are needed to completely define the orbit:


The orbit is totally defined by two constant parameters:

M = the semimajor axis

e = eccentricity defined by the formula

For what concerns the other conics, these two parameters have different definitions: in the case of a parabola, M, the major semiaxis is considered infinite, while for an hyperbola, it is considered negative. For what concerns the eccentricity, in the case of an ellipse, it will assume values ranging from 0 (for the circle) to 1. In the case of a parabola, it will assume value =1 and will be superior to 1 in the case of an hyperbola.
The values of eccentricity and the major semiaxis, together, define completely the shape and the other characteristics of the orbit. In fact, the value of the major axis is inversely proportional and of the opposite sign of the binding energy of the orbit. This means that a very little semiaxis corresponds to a strongly negative energy and therefore to a very bound orbit.
On the other hand, eccentricity is a clear indicator sign of the shape of the orbit giving, for example in the case of the ellipse, an information on how much the orbit is crushed.
If this planar orbit is considered in a 3dimensioal volume, a third parameter is needed to indicate the inclination of the orbital plane.

3 dimensional orbit: definition of the orbital parameters

If instead of a planar orbit, we consider a tridimensional orbit (for example, the orbit of an asteroid not considered on its plane) 6 parameters will be needed to define completely the orbit. These parameters are called the orbital elements.
On the following drawing, the coordinates axes are chosen to have a reference plane xy that is different from the orbital plane.

In the drawing, the orbital plane is the plane where the orbit lies. The intersection between the orbital plane and the reference plane is called nodal line. This nodal line passes through the ascending and descending node.
The first 3 orbital elements needed to define the kind, the shape and the inclination of the orbit are:
- a, the major semiaxis. If the orbit is elliptical the definition corresponds to the planar case, while the case of parabola corresponds to an infinite value of a;
- e, the eccentricity (as defined in the planar case);
- i the inclination of the orbital plane, the angle it forms with the plane xy.

Other three parameters are needed to describe the orientation of the orbit on the orbital plane:
- argument of the perihelion
- nodal longitude
- T period of the perihelion


Image courtesy of Andrea Carusi

 

How to confront the orbit of an asteroid with the orbit of the Earth

When considering the chance of an impact between two bodies, it is very important to compare the two orbits. For example, let's examine the orbit of an asteroid in comparison with the orbit of the Earth. To do so, it is possible to choose the orbital plane of the Earth as the plane xy. In this case we will talk of mutual nodes.