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Two Interesting (to me!) Topics

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!.

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Two Interesting (to me!) Topics

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  1. 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!

  2. 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!

  3. 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.

  4. 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 = 0.0167; Mars: e = 0.0934). Can show path shown uses minimum total energy

  5. 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 r1, velocity v1: E1 = -(k)/(2r1) = T + V = (½)m(v1)2 - (k)/(r1)  v1 = [(k)/(mr1)]½  Velocity of rocket in circular Earth orbit.

  6. Transfer elliptical orbit (figure): Want major axis: at = r1 + r2 r1 = Earth circular orbit radius r2 = Mars circular orbit radius • Total energy of rocket in transfer elliptical orbit at perihelion = r1: Et = - (k)/(2at) = -(k)/(r1 + r2) New velocity = vt1: Et = T + V = (½)m(vt1)2 - (k)/(r1) Solve for transfer velocity: vt1 = [(2k)/(mr1)]½[(r2)/(r1+r2)]½  Transfer speed from Earth circular orbit to elliptical orbit is: Δv1 = vt1 - v1

  7. Similarly, transfer speed from elliptical orbit to Mars circular orbit radius r2, speed v2is Δv2 = v2 - vt2 with v2 = [(k)/(mr2)]½ and vt2 = [(2k)/(mr2)]½[(r1)/(r1+r2)]½ • Direction of vt2 is along v2 in figure • Total speed increment isΔv= Δv2 + Δv1 • Time required to make transfer = one half period of the transfer elliptical orbit: Tt = (½)τt Period of elliptic orbit is (μ m): (τt)2 = [(4π2m)/(k)](at)3 ; at = r1 + r2 Tt = π[(m)/(k)]½(r1 + r2)3/2

  8. 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: Tt = 259 days vt1 = 3.27  104 m/s v1 = 2.98  104 m/s (Orbital speed of earth) Δv1 = 2.9  103 m/s

  9. 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,Δv1 is what is important.

  10. 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!!

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

  12. 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! (Mplanet >> mspacecraft)  Small energy loss to the planet!

  13. 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 = vi, vf. Net effect of gravitational encounter with B = deflection angle δ, with respect to B.

  14. Look at the spacecraft-planet system in an INERTIALframe, in which B is also moving! Looks quite different because B is moving! See figure. • Initial velocity = vi, velocity of B = vB, final velocity = vf. Vector addition: vi = vB + vi, vf = vB + vf

  15. Vector addition:vi = vB + vi, vf = vB + vf • Can have: |vf| > |vi|. • Encounter with B can both increase speed & change direction! • Detailed analysis: Speed increase occurs when craft passes BEHINDplanet B! (Goes into a temporary, partial elliptical orbit about B). Similarly, decrease in speed occurs when craft passes in front of B.

  16. 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!

  17. 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.

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