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We can use this to find the total energy of the orbit,

GMm r

Note that, if you are in an orbit of a given semimajor axis a, and fire a rocket such that your energy increases, you must go to a higher orbit, making your energy less negative. In the higher orbit, your speed will actually be less than it was in the lower orbit. Thus, firing the engine to "accelerate" the rocket has the interesting effect of reducing its orbital speed.

Another consequence of equation (22.3) is that the energy of an orbit depends only on the semi-major axis, and not on the eccentricity. If we launch a rocket, once its fuel is used up its energy is determined by its location and its speed. How, then, can we determine the eccentricity of the resulting orbit? The direction of motion when the fuel is used up determines the eccentricity. It

Fig 22.14.

Angular momentum and orbital eccentricity. (a) If a rocket stops its powered flight at a distance r from the Earth while moving with a velocity v, its energy is determined (as the sum of kinetic plus potential energies at that point), independent of the direction of motion.This means that the semi-major axis of the ellipse is determined. However, the direction of motion determines the eccentricity. (b) Ellipses with the same semi-major axis, but with different eccentricities.

Fig 22.14.

Angular momentum and orbital eccentricity. (a) If a rocket stops its powered flight at a distance r from the Earth while moving with a velocity v, its energy is determined (as the sum of kinetic plus potential energies at that point), independent of the direction of motion.This means that the semi-major axis of the ellipse is determined. However, the direction of motion determines the eccentricity. (b) Ellipses with the same semi-major axis, but with different eccentricities.

also determines the total angular momentum, L. Referring to Fig. 22.14, the angular momentum is given by

The least eccentric orbit has the highest angular momentum for a given semi-major axis. It v r r

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