Mathematical modelling to predict a vital path of comets , meteors and other minor planets?

Question

Mathematical modelling to predict a vital path of comets , meteors and other minor planets?

Answer 1

There is a short answer and a long answer to this.

First the short answer: suppose you have a solar system with only the Sun and the other object, say a comet with a mass much smaller than the Sun. Newton showed back in the 17th century that the orbit of the planet is then always an ellipse, with the Sun in one of the focal points. The ellipse has only a few free parameters, like the average distance, the eccentricity,  the inclination, the direction of the major axis, etc. A small number of observations of the system are enough to fix the parameters. The known orbit can now be used to predict the position *any* time in the future, and the error is limited only by the precision of your initial observations.

Now add a big planet to your mini solar system.

Suddenly the comet will feel not only the gravity from the Sun but also from the planet. The observer can still play the same game, do a number of measurements to fix the parameters of the Newtonian ellipse. And this will be enough for predictions a few days in the future. Over the longer term the predictions will be further and further away from the ellipse.

There are several ways out. People use for instance numerical integration of the orbit. Knowing the position vectors and velocity vectors of comet and planet, the computer program calculates all forces on the comet and uses those to make a small step forward in time. Then it recalculates the forces and makes a new step.

You can imagine that this becomes quickly complicated if you have eight planets and a few dozen moons :)