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This defaults to your Review Score Setting. Read more about it in the blog post. Excluding Off-topic Review Activity. Loading reviews There are no more reviews that match the filters set above. This is a simplification, since both the Earth and the Sun rotate around the joint center of mass. However, for the purpose of our simulation we assume that the Sun does not move. The potential energy, which comes from the gravitational attraction between the Sun of mass M and the Earth, is described by the following equation:.
We will find the equations of motions using the Lagrangian, which is the kinetic energy minus the potential energy of the Sun-Earth system:. Now we know the Lagrangian and can apply the Euler-Lagrange equation to get two equations of motion. The first one is found using the following formula, involving partial derivatives of the Lagrangian from Equation 3 with respect to distance r and its time derivative:.
We will use Equation 5 in our program and compute the distance r from its second derivative. After differentiating and simplifying we get derivation :. We have done the hard part and found Equations 5 and 8, which describe the evolution of the Sun-Earth system over time. We set the initial distance to be equal to the length of the astronomical unit AU , which is the average distance between the Sun and the Earth.
The first time derivative of the distance, or the speed of the Earth, will be zero. Note that this is the speed of the Earth in the direction of the Sun, not the speed in the direction of the orbit.
The initial conditions describe the system at the start of the simulation. As time changes the Earth will move and the four parameters will change as well.
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