University:
California State University, Northridge
Course:
MATH 360 | Abstract Algebra I
Academic year:

2024
Views:

55

Pages:

3
Author:

Lauren Larson

Orbital energy in terms of semi-major axis In order to compute the length of energy. where , we need to prove the following formula for the is the semi-major axis. In particular, this provides a simple way to compute semi-major axis. We use polar coordinates we have that in the plane of the orbit. As before Now, consider the coordinate transformation It follows that which can be written as This is the formula for rotation by the angle , which preserves the length of vectors. Hence, the length of the left hand side is the same as the length on the right, i.e. We return now to dealing with energy. We learned that the absolute value of the angular momentum momentum can be expressed as and that where is the parameter of the ellipse. We also have From these, we get Now we compute the energy in polar coordinates. Since the energy is a constant, we compute its value at any time. We choose the time of pericenter passage for convenience. Then , which can be proved by differentiating the polar equation for the orbit, and we also have from the same equation (with ) It follows that and Hence and the desired formula for the energy is shown. Now we return to the computation of the length of above, we have that . From the energy relationship and we also know that at pericenter passage Hence Finally, Therefore, the length of is . Notice that this result can be useful when computing orbital elements. both the direction of pericenter and the eccentricity. provides