The most efficient way to send a spacecraft from the earth to another planet is by using a Hohmann transfer orbit (Fig. P13.87). If the orbits of the departure and destination planets are circular, the Hohmann transfer orbit is an elliptical orbit whose perihelion and aphelion are tangent to the orbits of the two planets. The rockets are fired briefly at the departure planet to put the spacecraft into the transfer orbit; the spacecraft then coasts until it reaches the destination planet. The rockets are then fired again to put the spacecraft into the same orbit about the sun as the destination planet.

(a) For a flight from earth to Mars, in what direction must the rockets be fired at the earth and at Mars: in the direction of motion, or opposite the direction of motion? What about for a flight from Mars to the earth?

(b) How long does a one way trip from the earth to Mars take, between the firings of the rockets?

(c) To reach Mars from the earth, the launch must be timed so that Mars will be at the right spot when the spacecraft reaches Mars€™s orbit around the sun. At launch, what must the angle between a sun€“Mars line and a sun€“earth line be? Use data from Appendix F.

Figure P13.87:

Appendix F:

Fundamental Physical Constants*

Other Useful Constants*

Astronomical Data^€ 

Prefixes for Powers of 10

Orbit of Mars Hohmann transfer orbit Sun Orbit of earth Name Symbol Value 2.99792458 x 10 m/s 1.602176487(40) 10-1 c 6.67428(67) 10-1! N m/kg 6.62606896(33) 10-34 J .s 1.3806504(24) 1023 J/K 6.02214179(30) 1023 molecules/mol 8.314472(15) J/mol K 9.10938215(45) 10-31 kg 1.672621637(83) 10-27 kg 1.674927211(84) 10-27 kg 4m x 10-7 Wb/Am 8.854187817... 10-12 C/N m? 8.987551787... 10 N m/C Speed of light in vacuum Magnitude of charge of electron Gravitational constant G Planck's constant Boltzmann constant k NA Avogadro's number Gas constant Mass of electron me Mass of proton Mass of neutron Permeability of free space Permittivity of free space 1/Hc2 1/4