Consider a geosynchronous communications satellite of mass `m` placed in an equitorial circular orbit of radius `r_(0)`. These satellite have an "apogee engine" which provides the thrusts needed to reach the final orbit. Once an error by the ground controllers causes the apogee engine to be fired. the thrust happens to be directed the Earth and, despite the quick reaction of the ground crew to shut the engine off, an unwanted velocity variation `Deltav` is imparted on the satellite. we characterize this boost by the parameter `beta=Deltav//v_(0)`. the duration of the engine burn is always negligible with respect to any other orbital times, so that it can be considered as instantaneous.[Hing : under the action of central forces obeying the inverse square law, bodies follow trajectories descirbed by ellipses, parabolas or hyperbolas. in the approzimation `mlt ltM` the gravitating mass `M` is at one of the focuses. where `l` is a positive constant named the semilatus rectum and `epsilon` is the eccentricity of the curve. in terms of constants of motion:
`l=(L^(2))/(GMm^(2))` and `epsilon=(1+(2EL^(2))/(G^(2)M^(2)m^(3)))^(1//2)` where `G` is the Newton constant, `L` is the modulus of the angular momentum of the orbiting mass, with respect to the origin, and `E` is its mechanical energy, with zero potential energy at infinity. suppose `betalt1`
Determine the distance `r_(min)` of the satellite to the earth centre
A. `(betar_(0))/2`
B. `(r_(0))/(1+beta^(2))`
C. `(r_(0))/(1+beta)`
D. `(r_(0))/(1+sqrt(2beta))`
