Problem 15 A manned mission to Mars uses a heliocentric Hohmann transfer from lo
ID: 3279673 • Letter: P
Question
Problem 15 A manned mission to Mars uses a heliocentric Hohmann transfer from low earth orbit. Determine (3 marks) a) The eccentricity of the orbit b)The total energy per unit mass of the transfer orbit c) The minimum and maximum velocities of the orbit and where these occur (3 marks) (4 marks) d) The time to complete the transfer from Earth to Mars (3 marks) On the way back from Mars, using the same Hohmann transfer, the crew is informed they Earth to achieve a desired course correction. The desired flyby deflection is 50 degrees and can return to Mars to save a previously thought dead crew member using a flyby around encounters earth tangential to earths orbit. Determine e) The eccentricity of the flyby passage f) How close the spacecraft will get to the earth's surface g) The maximum speed (relative to Earth) of the flyby and where this occurs (3 marks) (4 marks) (3 marks) h)The change in speed relative to the sun that the spacecraft obtains after the flyby ldentify and explain what could be changed to achieve a larger increase in the (4 marks) i)If the spacecraft can provide a maximum thrust of Av-5km/s, what is the critical velocity of the spacecraft relative to the sun (ignoring drag) on approach to Mars (but outside it's sphere of influence) above which it has to flyby Mars instead of (3 marks) change in speed orbiting to pickup the crew memberExplanation / Answer
A) Eccentricity could help people walk on the Red Planet one day. Mars, one of Earth's closest planetary neighbors, has one of the highest orbital eccentricities of all the planets. An eccentric orbit is one that looks more like an ellipse than a circle. Because Mars travels in an ellipse around the sun, there are times when it's close to Earth and times when it's farther away. Astronauts wishing to travel to Mars can get there quickly by choosing an arrival time when Mars is closest to Earth.
B) The minimum-energy transfer between circular orbits is an elliptical trajectory called the Hohmann trajectory. It is shown at right for the Earth-Mars case, where the minimum total delta-v expended is 5.6 km/s. The values of the energy per unit mass on the circular orbit and Hohmann trajectory are shown, along with the velocities at perihelion (closest to Sun) and aphelion (farthest from Sun) on the Hohmann trajectory and the circular velocity in Earth or Mars orbit. The differences between these velocities are the required delta-v values in the rocket equation.
C) The simplest high-exhaust-velocity analysis splits rocket masses into three categories..
where u measures the energy expended in a manner analogous to delta-v. After some messy but straightforward algebra, we get the high-exhaust-velocity rocket equation. Variable exhaust velocity and gravity considerably complicate the problem. When the exhaust velocity is varied during the flight, variational principles are needed to calculate the optimum . The key result is that it is necessary to minimize.
Even the simplest problem with gravity, the central-force problem, is difficult and requires advanced techniques, such as Lagrangian dynamics and Lagrange multipliers. In general, trajectories must be found numerically, and finding the ptimum in complex situations is an art.
D) This, by the way, is called a Hoeman Transfer Orbit, and is the main stay of interplanetary space travel. It depends on the details of the orbit you take between the Earth and Mars. The typical time during Mars's closest approach to the Earth every 1.6 years is about 260 days.