Since kinetic energy is proportional to the square of the velocity, any decrease in velocity (ΔV) would consequently lead to a decrease in kinetic energy, which could impact the stability of the orbit over time, as mentioned in the context of a week-long mission requiring additional fuel to maintain speed.
The question involves calculating the magnitude of velocity change required to transition between a circular and an elliptical orbit around the Earth. In orbital mechanics, this change in velocity is referred to as the ΔV (Delta-V). Using conservation of angular momentum and the vis-viva equation (which relates the speed of an orbiting body to its position in the orbit and the mass of the central body), one can calculate the velocity at any point in the orbit.
To find the velocity change ΔV1 at apogee required to enter the red elliptical orbit, we use the vis-viva equation given as:
V = √(μ(2/r - 1/a))
where μ is the standard gravitational parameter for Earth, r is the distance from Earth to the satellite at apogee, and a is the semi-major axis of the orbit. With the given apogee (ra = 26.371EE3 km) and perigee (rp = 25.873EE3 km), one can calculate the semi-major axis (a = (ra + rp) / 2) and subsequently find the satellite's velocity at the apogee of the elliptical orbit. The difference between this velocity and the orbital velocity of the initial green circular orbit at the same altitude gives the required ΔV1.
Since kinetic energy is proportional to the square of the velocity, any decrease in velocity (ΔV) would consequently lead to a decrease in kinetic energy, which could impact the stability of the orbit over time, as mentioned in the context of a week-long mission requiring additional fuel to maintain speed.