Question

Suppose we decide to send a spacecraft to Saturn, using one transfer orbit that connects the earth to Saturn. Craft starts in a circular orbit (centering the Sun) close to Earth (radius 1 AU) and is to end up in another orbit near Saturn (radius 10 AU). Ignore the effects of other planets in the solar system.

a) Find the eccentricity of the transfer orbit that gives the shortest time of travel.

b) Use Kepler's third law to estimate the time of transfer (i.e. how long it will take to travel from earth to saturn).

c) How much energy boost, , is required to launch the spacecraft of mass m=100,000 kg from the initial orbit to the transfer orbit?

d) Once you reach Saturn, how much energy boost, , is required to go from the transfer orbit to the final circular orbit of Saturn? Calculate the ratio of the final velocity (of the spacecraft) on Saturn's orbit to the initial velocity on Earth's orbit?

Answer 1

Orbital period is some of side real forced p = 21.3 days Spin period is same of cynodic period G = 5.1 days rightarrow period of return to make one cyclic is Hyperion T = ? 1/g = 1/s = 1/9 (or) 1/T = 1/G + 1/s substituting the shown values G, T = 4.115 days Perihelion: p = 1A.v (A) phelion A = 5.2 A.V formula rho = Q - e and A = a^t Qe, where a semi major axis, sigma e^iy = eccentricity. Solving both a = 3.1 A. V and e = 0.0674 A.V Velocity of the transfer orbit v^2 = mu((alpha + r) - (1/a)] mu = Gm = 1.2664 times 10617 f (or) Jupiter: x = 5.2 Ay \r\n rightarrow substituting shown values, v = 3922 times 1064 and orbital velocity, v_arb = 2.9785 times 10^4 Orbital velocity of Jupiter V_orb = 13.07 times 10^3 and v.f of object = 3.252 times 10^-4 (or) 0.52521 times 10^3 By Kepler's 3rd law T^2 = (4 times (3.14)^2/4x) times e^3 = 3.11246^k. 10^-19 k (31Fv)^-3 = 3.104