Answer to Question #201984 in Astronomy | Astrophysics for Krishna

Question

A space probe is about to launch with the objective to explore the planets Mars and Jupiter. To

use the lowest amount of energy, the rocket starts from the Earth’s orbit (A) and flies in an ellip-

tical orbit to Mars (B), such that the ellipse has its perihelion at Earth’s orbit and its aphelion at

Mars’ orbit. The space probe explores Mars for some time until Mars has completed 1/4 of its orbit

(C). Aer that, the space probe uses the same ellipse to get from Mars (C) to Jupiter (D). There the

mission is completed, and the space probe will stay around Jupiter.

The drawing below shows the trajectory of the space probe (not drawn to scale):

Sun

Earth

Mars

Jupiter

A

B

D C

Below you find the obrital period and the semi-major axis of the three planets:

Orbital period Semi-major axis

Earth 365 days 1.00 AU

Mars 687 days 1.52 AU

Jupiter 4333 days 5.20 AU

How many years aer its launch from the Earth (A) will the space probe arrive at Jupiter (D)?

Answer 1

The major axis of the first trajectory is $0.5(a_E+a_M) = 0.5(1.00+1.52) = 1.26$ AU.

The time of flight is a half of period on such an orbit, so $T_1 = 0.5 P_1 = 0.5\cdot(1.26)^{3/2},$ because of the third Kepler's law ($T^2 = a^3,$ when T is in years and a is in AU). So $T_1 = 0.71$ yr.

Next we should take into account one-fourth of the Martian year, or $0.25(1.52)^{3/2} = 0.47$ yr.

The second trajectory has the major axis $0.5(a_J+a_M) = 0.5(5.20+1.52) = 3.36$ AU.

The time of flight is a half of period on such an orbit, so $T_2 = 0.5 P_2 = 0.5\cdot(3.36)^{3/2} = 3.1$ yr.

Therefore, the total time is $0.71+0.47+3.1 = 4.28$ yr.