For the elliptical orbit of a spacecraft around the earth,...

For the elliptical orbit of a spacecraft around the earth, determine the speed $v_{A}$ at point $A$ which results in a perigee altitude at $B$ of $200 \mathrm{km} .$ What is the eccentricity $e$ of the orbit?

Key Concepts

Elliptical Orbits

Elliptical orbits describe the paths of objects under the influence of a central gravitational force where the path is an ellipse with the central body at one focus. These orbits are characterized by varying distances from the central body, with distinct perigee (closest point) and apogee (farthest point), which are essential for determining orbital parameters and mission profiles.

Eccentricity

Eccentricity is a dimensionless parameter that quantifies how much an orbit deviates from being circular. A value of zero corresponds to a perfect circle, while values between zero and one indicate the degree of elongation of the ellipse. It is a critical factor in assessing orbital shape and is related to the distances at perigee and apogee.

Vis-viva Equation

The vis-viva equation provides a direct relationship between the orbital speed of an object, its distance from the focal body, and the semi-major axis of the ellipse. This fundamental relation is widely used in orbital mechanics to determine the velocity at any point in the orbit, given the body's instantaneous radius and the overall orbital energy.

Conservation of Angular Momentum

Conservation of angular momentum in orbital dynamics states that in the absence of external torques, the angular momentum of the spacecraft remains constant. This principle explains why a spacecraft travels faster when it is closer to the central body (at perigee) and slower when it is farther away (at apogee), linking orbital speed to the radial distance.

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