Orbital elements

To know the state of motion of a body, it would be necessary to know the values of the six parameters corresponding to its position and its speed. But, during a general motion, these values change constantly and knowing them at every moment is a very difficult task.
n the reality, for what concerns constant, stable orbits, the motion of a body can be completely described knowing the values of 6 constant parameters, called orbital elements that characterize the orbit. In this hyothesis, knowing the position and the speed of the object at every instant becomes useless.
First of all, let's consider the orbit of an object of the Solar System (a planet or a minor body). This orbit can be an ellipse (as the first of Kepler's laws states), or an opened orbit such as parabola or hyperbola. Anyhow, this orbit will always be a conic and it will always lie on a plane called the orbital plane. The intersection between this plane and a reference plane (which can be chosen as the plane where the Earth's orbit lies) is called nodal line. This nodal line passes through the ascending and descending node.

The first 2 orbital elements, needed to define size and shape of the orbit on this plane, are the semimajor axis a and the eccentricity e.

These two parameters can be defined on the orbital plane; for an
elliptical orbit:
- 2a is the length of the major axis (M in the image on the left);
- e, the eccentricity, gives an indication of how much the ellipse is elongated: it is 0 for a circle, and tends to 1 for more and more elongated orbits. It can be calculated with the formula (where m and M are defined in the image):
Three more parameters are needed to describe where is the orbit, and how it is oriented:
- i the inclination of the orbital plane, the angle that this plane forms with a reference plane;
- the longitude of node, the angle between a reference direction and the nodal line, that goes from the Sun to the ascending node;
- the argument of perihelion, the angle from the nodal line to the line joining the Sun and the perihelion, the point on the orbit closest to the Sun.

Finally, the position on the orbit of the body of interest can be specified by giving the sixth parameter:
- T, the time of passage at the perihelion.



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