large distances away from each other, the patched-conic approximation is much more accurate for interplanetary trajectories than for translunar trajectories. The patch point between the hyperbolic trajectory relative to the departure planet and the heliocentric transfer orbit occurs at the planet's sphere of influence radius relative to the Sun, as defined above in Orbital flight. Given the Sun's mass ratio of 333,432 times that of Earth and distance of , the Earth's sphere of influence radius is (roughly 1,000,000 kilometers). Heliocentric transfer orbit The transfer orbit required to carry the spacecraft from the departure planet's orbit to the destination planet is chosen among several options: A Hohmann transfer orbit requires the least possible propellant and delta-v; this is half of an elliptical orbit with aphelion and perihelion tangential to both planets' orbits, with the longest outbound flight time equal to half the period of the ellipse. This is known as a conjunction-class mission. There is no "free return" option, because if the spacecraft does not enter orbit around the destination planet and instead completes the transfer orbit, the departure planet will not be in its original position. Using another Hohmann transfer to return requires a significant loiter time at the destination planet, resulting in a very long total round-trip mission time. Science fiction writer Arthur C. Clarke wrote in his 1951 book The Exploration of Space that an Earth-to-Mars round trip would require 259 days outbound and another 259 days inbound, with a 425-day stay at Mars. Increasing the departure apsis speed (and thus the semi-major axis) results in a trajectory which crosses the destination planet's orbit non-tangentially before reaching the opposite apsis, increasing delta-v but cutting the outbound transit time below the maximum. A gravity assist maneuver, sometimes known as a "slingshot maneuver" or Crocco mission after its 1956 proposer Gaetano Crocco, results in an opposition-class mission with a much shorter dwell time at the destination. This is accomplished by swinging past another planet, using its gravity to alter the orbit. A round trip to Mars, for example, can be significantly shortened from the 943 days required for the conjunction mission, to under a year, by swinging past Venus on return to the Earth. Hyperbolic departure The required hyperbolic excess velocity v∞ (sometimes called characteristic velocity) is the difference between the transfer orbit's departure speed and the departure planet's heliocentric orbital speed. Once this is determined, the injection velocity