How can a satellite's speed decrease without its angular momentum changing?

Angular momentum, L, of an object is a vector quantity representing how much rotational motion it has. In the case of a satellite, its angular momentum depends on its mass, speed, and distance from the center of the Earth. Mathematically, L = m * r * v, where m is the mass of the satellite, r is the radius of its orbit, and v is its orbital speed.

The law of conservation of angular momentum states that if there are no external torques acting on a system, then the total angular momentum of the system remains constant. In the case of a satellite orbiting the Earth, as long as there are no external forces (such as atmospheric drag, thrust from rocket propulsion, etc.), the angular momentum of the system will remain constant.

To maintain a stable orbit, a satellite must have a specific speed that depends on the distance from the center of the Earth. The relationship between orbital speed (v), orbital radius (r), and the standard gravitational parameter (GM) of the Earth is given by the equation v = \sqrt{GM/r}. This equation shows that the satellite's speed linearly depends on the square root of the reciprocal of the orbital radius.

To decrease the satellite's speed without changing its angular momentum, we need to change another variable in the angular momentum formula. Let's denote the initial values as r1 and v1, and the final values as r2 and v2. Since the satellite's mass will not change, we need to change the orbital radius while maintaining the angular momentum constant, i.e.,L1 = L2 or m * r1 * v1 = m * r2 * v2. Based on the equation in Step 3, we can derive the relationship between r1 and v1, and r2 and v2: r1 * v1 = r2 * v2. To decrease the satellite's speed (v2 < v1), the orbital radius must increase (r2 > r1). This change in orbital radius requires energy, either from an external source or by converting the satellite's existing kinetic energy into potential energy. In conclusion, a satellite's speed can decrease without changing its angular momentum by increasing its orbital radius. This would require energy input to change the satellite's orbit or conversion of its existing kinetic energy into potential energy.