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 (in this case, the Sun) to another circular orbit is via the 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. This ellipse is uniquely defined because we know the perihelion radius re (the radius of the inner circular orbit) and the aphelion radius rj (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. Assume that the orbits of Earth and Jupiter are circular, use 150 × 106 km for the radius of Earth’s orbit, use 779 × 106 km for the radius of Jupiter’s orbit, and note that the mass of the Sun is 333,000 times that of the Earth.

Figure P5.146

A space probe S1 is launched from Earth to Jupiter via a Hohmann transfer orbit. Determine the change in speed Δve required at the radius of Earth’s orbit of the elliptical transfer orbit (perihelion) and the change in speed Δvj required at the radius of Jupiter’s orbit (aphelion). In addition, compute the time required for the orbital transfer. Assume that the changes in speed are impulsive; that is, they occur instantaneously.