Two Interesting (to me!) Topics Neither topic is in Goldstein. Taken from the undergraduate text by Marion & Thornton. Topic 1: Orbital or Space Dynamics –For those who have applied interests! Topic 2: Apsidal Angles & Precession –For those who have basic physics interests!

Orbital Dynamics (from Marion) Briefly look at objects launched the earth to other planets: If, in planet orbit problem, replace sun by earth & earth by satellite (including moon), the orbit is obviously still elliptical! –Earth satellites are also governed by Kepler-like Laws; earth at focus of ellipse. –The following mostly ignores this & considers space probe to be orbiting the sun. “Space Dynamics” or “Orbital Dynamics”: Dynamics of rocket flight to other planets. –The real problem: Very complex because of gravitational attraction to many objects & because of the orbital motion. –Ignore rocket-planet attraction: Assume: rocket is only attracted to sun & is in orbit around it!

Goal: Trip to Mars (or any planet) by a rocket. Search for the most economical method. Rocket Orbit around the sun: Changed by single or multiple thrusts of rocket engines. Simplest maneuver: Single thrust applied in Earth orbital plane, which doesn’t change the direction of orbital angular momentum (  plane), but changes eccentricity e & energy E. –Moves the rocket from the elliptic orbit of Earth into a new elliptic orbit, which intersects the elliptic orbit of Mars. –Assumes Earth, Mars, elliptic orbit of rocket are in the same plane. –Assumes relative positions of Earth & Mars are correct so the rocket arrives at Mars orbit when Mars is there “to meet it”! See figure next page.

Hohmann transfer  Rocket trip shown in figure. Rocket orbit about the sun, Mars & Earth orbits are all ellipses in the same plane! Simplest approx: Earth & Mars orbits are circular; rocket orbit is ellipse. (Earth: e = ; Mars: e = ). Can show path shown uses minimum total energy

Only 2 rocket engine burns are required: 1) First burn injects the rocket from circular Earth orbit to elliptical orbit which intersects Mars orbit. 2) Second burn transfers the rocket from elliptical orbit to Mars circular orbit. –Use energy conservation to analyze this problem: –Circles & ellipses, total energy is (combining previous eqtns) E = - (k)/(2a); k = GMm; 2a = major axis. Circular Earth orbit, radius r 1, velocity v 1 : E 1 = -(k)/(2r 1 ) = T + V = (½)m(v 1 ) 2 - (k)/(r 1 )  v 1 = [(k)/(mr 1 )] ½  Velocity of rocket in circular Earth orbit.

Transfer elliptical orbit (figure): Want major axis: a t = r 1 + r 2 r 1 = Earth circular orbit radius r 2 = Mars circular orbit radius Total energy of rocket in transfer elliptical orbit at perihelion = r 1 :E t = - (k)/(2a t ) = -(k)/(r 1 + r 2 ) New velocity = v t1 : E t = T + V = (½)m(v t1 ) 2 - (k)/(r 1 ) Solve for transfer velocity: v t1 = [(2k)/(mr 1 )] ½ [(r 2 )/(r 1 +r 2 )] ½  Transfer speed from Earth circular orbit to elliptical orbit is: Δv 1 = v t1 - v 1

Similarly, transfer speed from elliptical orbit to Mars circular orbit radius r 2, speed v 2 is Δv 2 = v 2 - v t2 with v 2 = [(k)/(mr 2 )] ½ and v t2 = [(2k)/(mr 2 )] ½ [(r 1 )/(r 1 +r 2 )] ½ –Direction of v t2 is along v 2 in figure Total speed increment is Δv = Δv 2 + Δv 1 Time required to make transfer = one half period of the transfer elliptical orbit: T t = (½)τ t Period of elliptic orbit is (μ  m): (τ t ) 2 = [(4π 2 m)/(k)](a t ) 3 ; a t = r 1 + r 2  T t = π[(m)/(k)] ½ (r 1 + r 2 ) 3/2

Calculate time needed for a spacecraft to make a Hohmann transfer from Earth to Mars & the heliocentric transfer speed required, assuming both planets are in coplanar, circular orbits. Answers: T t = 259 days v t1 = 3.27  10 4 m/s v 1 = 2.98  10 4 m/s (Orbital speed of earth) Δv 1 = 2.9  10 3 m/s

Transfers to outer planets:  Should launch rocket in direction of Earth’s orbit, in order to gain the Earth’s orbital velocity. Transfers to inner planets:  Should launch rocket in opposite direction of Earth’s orbit. In each case: the velocity relative to the Earth, Δv 1 is what is important.

Hohmann transfer path gives the least energy expenditure. It does not give the shortest time. –Round trip from Earth to Mars to Earth: Requires Earth & Mars relative positions to be right  Need to remain on Mars 460 days, until they are again in same relative positions! –Total trip time: Time to get there = 259 days Time to return = 259 days Time to remain there = 460 days Total time = 978 days = 2.7 years!!

Figure shows other schemes, including using gravity of Venus as a “slingshot effect” to shorten the time.

Consider spacecraft (probes) sent to outer reaches of solar system. Several of these Interplanetary Transfers since the late 1970’s. Divide the journey into 3 segments: 1. Escape from Earth 2. Heliocentric transfer orbit (just discussed) to region of interest in solar system. 3. Encounter with another body (planet, comet, moon of planet,..) Fuel required is huge! A “trick” has been designed to “steal” energy from bodies which the spaceship gets near! (M planet >> m spacecraft )  Small energy loss to the planet!

Simple flyby = “slingshot” effect: Uses gravity to assist, saving fuel. Qualitative discussion! Spacecraft in from r =  to near a planet (B in fig.). Path  hyperbola (in B frame of reference!). Initial & final velocities, with respect to B = v i, v f. Net effect of gravitational encounter with B = deflection angle δ, with respect to B.

Look at the spacecraft-planet system in an INERTIAL frame, in which B is also moving! Looks quite different because B is moving! See figure. Initial velocity = v i, velocity of B = v B, final velocity = v f. Vector addition: v i = v B + v i, v f = v B + v f

Vector addition: v i = v B + v i, v f = v B + v f Can have: |v f | > |v i |. Encounter with B can both increase speed & change direction! Detailed analysis: Speed increase occurs when craft passes BEHIND planet B! (Goes into a temporary, partial elliptical orbit about B). Similarly, decrease in speed occurs when craft passes in front of B.

1970’s: NASA flyby of 4 largest planets. + many of their 32 moons with a single probe! Planets were aligned to make this efficient. Slingshot effect on each flyby. Scaled back to Jupiter & Saturn only. Voyager 1 & 2. Later extended to Uranus & Neptune. Path of Voyager 2 is in the figure. 12 year mission!

Galileo satellite: Probed comet encounter with Jupiter. On its way, flybys of Earth & Venus (“boosted” each time!). 6 year mission. ISEE-3 = ICE: Earth, Moon vicinity & comet. See figure. 7 year journey.