University:
California State University, Northridge
Course:
ASTR 300 | Introductory Astrophysics
Academic year:

2024
Views:

393

Pages:

4
Author:

Scarlet Buchanan

Elliptic Orbits In this section we will prove that the solutions to the equation of motion are conic sections in general, and ellipses in particular. For the sake of simplicity, we consider only the simplified version for an earth orbiting satellite (13). This equation gives as a function of the time . From the previous section we know that the motion is restricted to a plane. To continue we introduce polar coordinates in this plane and write: and hence where is a unit vector perpendicular to , which forms a right hand system together with . Using this we can rewrite the angular momentum as In particular, we have where is the area swept over by . The conservation of angular momentum therefore directly implies Kepler's second law. To continue observe that and hence is perpendicular to itself and therefore parallel to . It turns out that Combining all these properties we can write: and the entire equation of motion becomes Using the expression for above we get Since the angular momentum is conserved we get (16) We can eliminate by using . Doing this reduces the equation of motion to a single non-linear second order ODE. Observe that we started out with a system of three second order non-linear ODE. What follows is a number of little tricks to simplify (16) even further. First observe that Differentiating this again with respect to we get Plugging this into (16) we get after some algebra (17) All this did so far is change the independent variable from to . (16) was a nonlinear second order equation and as such difficult to solve.. To continue we make a change of dependent variable, namely let Then and We observe that this introduces another non-linear term of the form However, after substituting this we see that this new nonlinear term cancels with the existing nonlinear term and we get (18) This new equation is now a linear second order equation with constant coefficients, the best of all possible worlds. This equation is easy to solve and its solutions are of the form (19) where the constants and introducing we get: depend on the initial conditions. After re(20) where and (20) is of course the well known equation of an ellipse in polar coordinates with eccentricity and focus at the origin. We have thus established Kepler's first law, namely that the orbits are elliptical.

Elliptic Orbits Lesson Notes

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