Question

2. To get from Earth to Saturn as economically and quickly as possible, spacecraft make use of an elliptical transfer trajectory called the Hohmann transfer orbit. As shown in the diagram, below, Hohmann transfer facilitates a smooth transition between two planetary bodies in roughly circular orbits located on the same plane. For our solar system, this plane is known as the ecliptic. Using data you can find online for the orbits of Earth, Saturn and the Sun and assuming that the orbits of Earth and Saturn are effectively circular with radi equal to their semi-major axes: Determine the semi-major axis of the Hohmann transfer orbit of a spacecraft being sent to Saturn a. from Earth. Cite any data sources you consult (a simple url date accessed is sufficient.) (5 marks) mass M is the minimum velocity any comparatively small b. The escape velocity for any large planetary object must attain if it is to escape the gravitational pull of M. This velocity, Vescape, decreases with increasing radial distance, r, from M. The escape velocity for M at a given r is: 2GM Vescape r where G is the universal gravitational constant and has a value of 6.668 x 101 m3/kg'sec2. Once the spacecraft has reached Saturn, what minimum speed must it attain to escape the gravitational influence of the ringed planet? (4 marks)

Answer 1

a) From figure it is clear that we need to find a Saturn is situated at aphelion distance from sun and earth is situated partition distance from sun so, partition distance a (1 - e) = 1 AU (for earth) aphelion distance = a(1 + e) = 9.5 AV (for Saturn) Taking two equation a -a e = 1 a + ae = a middot s rightarrow 2a = 10 s semi major axis a = 5.25 Av b) Vescape squareroot 2 Gm/r m_saturn = 5.68 times 10^26 kg radius_saturn = 5.82 times 10^7 kg therefore v_escape squareroot 2 times 6.6 times 10^-11 times 5.68 times 10^26/5.82 times 10^7 tildetilde 35.5 km/s