Question

All solutions are to be given in SI units. Problem 1: You are observing a satellite in an inclined orbit (Fig. ). The inclination is defined as the angle between the orbital plane and the inertial x-y plane. You can assume you are observing from the center of the earth. Fig 1: Satellite S in the orbit around the Earth, x,y,z inertial space-fixed system. Left: 3D view; right: view of the system in the x,y-plane (viewing along the negative z-axis). a) Compute the inertial velocity and acceleration in the most general way expressed in the frame that is rotating with the satellite. b) Assume that the satellite is in a circular geostationary orbit (42'000km distance from the earth center, one exact revolution around the earth in one day) at zero inclination. Compute the values for the inertial (space-fixed) velocity and acceleration and the rotating (earth-fixed) velocity and AAE 340 Homework acceleration, when the satellite is passing the x-axis (position (42000,0,0)km) c) Compute the expression again for the same circular geostationary orbit but now with an inclination of 15 degrees. What are the inertial velocity and inertial acceleration compared to velocity and acceleration in the rotating frame (non- inertia), assuming the satellite is crossing the xy plane at the location of the positive x-axis (position (42000,0,0)km, see Fig.

Answer 1

Initial velocity v = sigma^3_i = 1 v^0_i logicaland e^0_i v^0_i = v middot e^0_i (dv/dt)_inertial = sigma^3_i = 1 dv^0_i/dt e^0_i Initial body rotating frame. V = sigma^3_i = 1 v_i logicaland e_i (dv/dt)_inertial = sigma^3_i = 1 dv/dt e^^_i + sigma^3_i = 1 v_i d e^^_i/dt (dv/dt)_inertial = (dv/dt)_body + sigma^3_i = 1 v_i d e^^i/dt Acceleration: - (d^2 r/dt^2)_inertial = [(d/dt)_body + omega times] [(dr/dt)_body + omega times r] = (d^2 r/dt^3)_body + 2 omega times (dr/dt)_body + d omega/dt times r + omega times (omega times r)