The optimal way (from an energy standpoint) to transfer from one circular orbit about a pr...

The optimal way (from an energy standpoint) to transfer from one circular orbit about a primary body B to another circular orbit is via the so-called Hohmann transfer, which involves transferring from one circular orbit to another using an elliptical orbit that is tangent to both at the periapsis and apoapsis of the ellipse. The ellipse is uniquely defined because we know rP (the radius of the inner circular orbit) and rA (the radius of the outer circular orbit), and therefore we know the semimajor axis a via Eq. (5.117) and the eccentricity e via Eq. (5.114) or Eqs. (5.119). Performing a Hohmann transfer requires two maneuvers, the first to leave the inner (outer) circular orbit and enter the transfer ellipse and the second to leave the transfer ellipse and enter the outer (inner) circular orbit.

Figure P5.107

A spacecraft S1 is transferring from circular low Earth parking orbit with altitude 100 mi to a circular orbit with radius rA. Plot, as a function of rA for rP ≤ rA ≤ 100rP, the change in speed ΔvP required at perigee of the elliptical transfer orbit as well as the change in speed ΔvA required at apogee. In addition, plot the time as a function of rA, again for rP ≤ rA ≤ 100rP, required for the orbital transfer. Assume that the changes in speed are impulsive; that is, they occur instantaneously.