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 = 0.0167; Mars: e =
0.0934). 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 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.
• 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
• Similarly, transfer speed from elliptical orbit to Mars
circular orbit radius r2, speed v2 is
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 is Δv = 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
• 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
• 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.
• 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! (Mplanet >> mspacecraft)  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 = vi, vf. 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 = vi, velocity of B = vB, final velocity =
vf. Vector addition: vi = vB + vi, vf = vB + vf
• 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 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.