Question

NASA has sent satellites to the planet Mars. The method is to put the satellite into a Keplerian orbit whose perihelion is at the position of the Earth and whose aphelion is at the position of Mars. The satellite does not travel to Mars under rocket power; it just moves under the influence of the Sun's gravity. Calculate the number of days the trip to Mars will take.

Answer 1

Period of earth around sun T = 1 year = 31557600 s

aphelion length of earth a = (GMT²/4² )^1/3 = (6.67x10^-11x1.972x10³⁰x31557600²/4² )^1/3 = 149151.28x10⁶ m

perihelion length of earth b = [a²(1 - 0.9²)]^1/2 = [(149151.28x10⁶)²x(1 - 0.9²)]^1/2 = 65013.54x10⁶ m

In the problem aphelion length of earth is mars aphelion length

If perihelion is at the position of the Earth and aphelion is at the position of Mars

then the distance from mars to earcth R = (a² + b²)^1/2 = [(149151.28x10⁶)² + (65013.54x10⁶)²]^1/2

                                                          R = 91943.03x10⁶ m

Time taken by satellites to the planet Mars Tp = (R³/GM)^1/2 = [(91943.03x10⁶)³/(6.67x10^-11x1.972x10³⁰]^1/2

                                                     Tp = 2430870.49 s = 28.13 days