Question

For a spacecraft, the following orbital parameters are given: e-1.2; perigee altitude-200 km, i=50 degrees; Ω=75 degrees; ω-80 degrees. Calculate r and v at perigee relative to the perifocal reference frame

Answer 1

The radius at perigee is calculated by adding the perigee altitude to Earth's radius, resulting in 6571 km. However, the given eccentricity indicates a non-closed hyperbolic trajectory, making the concept of perigee irrelevant and traditional calculations for velocity at perigee inapplicable.

Step-by-step explanation:

To calculate the radius r and velocity v at perigee for a spacecraft with given orbital parameters, the eccentricity (e), perigee altitude, inclination (i), right ascension of ascending node (Ω), and argument of perigee (ω) are taken into account. Using the perigee altitude and Earth's radius, we can determine r, the distance from the center of the Earth to the spacecraft at perigee. Since the perigee is the closest point to Earth in the spacecraft's orbit, r is the sum of Earth's radius (6371 km) and the altitude of the perigee. In this case, it's 6371 km + 200 km = 6571 km.

However, the given eccentricity of 1.2 indicates a hyperbolic trajectory since any eccentricity greater than 1 results in a non-closed orbit, and the concept of perigee becomes irrelevant for hyperbolic trajectories. For a proper elliptical orbit (with e < 1), Kepler's laws and other orbital mechanics equations such as the vis-viva equation would be used to calculate the velocity at perigee.