Transcript EART 160: Planetary Sciences

EART 160: Planetary Science
First snapshot of Mercury taken by MESSENGER
Flyby on Monday, 14 January 2008
Closest Approach: 200 km at 11:04:39 PST
http://messenger.jhuapl.edu
Announcements
Seminar at 3:30 PM
“Martian Impact Craters”
Nadine Barlow
E&MS B210
HW 1 due in one week
Most Readings, Notes, available on Website
Today
•Paper Discussion
•Stevenson (2000)
•Soter (2007)
•Celestial Mechanics
• Kepler’s Laws
Johannes Kepler
1571-1630
• Conservation Eqns
• Newton’s Laws
• Gravity
Isaac Newton
1643-1727
Kepler’s Laws
1. Each planet moves in an ellipse with the sun at
one focus.
2. The line between the sun and the planet
sweeps out equal areas in equal amounts of
time.
3. The ratio of the cube of the semimajor axis to
the square of the period is the same for each
planet.
Empirical laws – based solely on observation.
Kepler had no understanding of why this
occurs. YOU WILL!
Some Terms
• a – semimajor axis: long axis of the ellipse
• e – eccentricity: elongation of ellipse
– e = 0  circular
– e = 1  parabolic (unbound orbit)
• i – inclination: angle between orbital plane
and Earth’s orbital plane (ecliptic)
• P – period: time to complete one orbit
• Periapsis – closest approach of secondary object to
primary
– Perigee if primary is Earth
– Perihelion in primary is Sun
• Apoapsis – farthest point on orbit from primary
.
Newton’s Laws of Motion
1. A body at rest remains at rest and a body in
motion at a constant speed remains in motion
along a straight line unless acted on by a
force.
2. The rate of change of velocity of a body is
directly proportional to the force and inversely
proportional to the mass of the body.
3. The actions of two bodies are always equal in
magnitude and opposite in direction.
Newton’s Law of
Universal Gravitation
• Motions of planets around Sun are caused
by gravity – that is the force in the first two
laws.
• Force of gravity between any two objects
is proportional to the masses of both
objects
• Force of gravity between any two objects
drops off as the square of the distance
between them.
Explanation of Kepler’s Laws
• Kepler observed orbital periods and
distances, but didn’t know what caused it.
– Third Law only works for the Sun, using Earth
as a reference.
• Newton finds force of gravity is what
moves planets toward the sun.
• Can extend Kepler’s Third Law for any
object.
– Let’s do that now!
Kepler’s Third Law
• Compare orbital velocity to period
• I’ll show this for a circular orbit
• Works for elliptical orbit as well, but the
derivation is unpleasant and not very
informative.
• Should recover Kepler’s version if we stick
in the Sun’s Mass, keep times in years,
and distances in AU.
Circular Velocity
• Gravity imposes a centripetal acceleration
to an orbiting object.
v
a
v
a  
r
2



This is why planets don’t fall into the
Sun.
And why it’s so hard to get to Mercury!
r
Conservation Laws
• Momentum
– If the vector sum of the external forces on a system is
zero, the total momentum of the system is constant.
– Momenta of individual objects can change.
• Angular Momentum
– When the net external torque on a system is zero, the
total angular momentum of the system is constant.
– Angular Momenta of individual objects can change.
• Energy
– Cannot be created or destroyed
– Can be converted from one form to another
(e.g. from potential to kinetic)
Escape Velocity
• How fast does an object have to go to
escape the gravitational pull of a planet?
• Conservation of Energy
• Balance the Potential Energy due to
gravity against the Kinetic Energy due to
motion
• Collapse of solar nebula  lots of
potential energy lost. Where does it go?
Kepler’s Second Law
v
v┴ = v sin j
j
r
dq
Law of Areas
• Conservation of Angular Momentum
• Object moves fast near periapse (short
lever arm), slow near apoapse (long lever
arm.
• Energy shifts from kinetic to potential and
back.
• Conservation of Energy again!
Earth-Moon System
•The Moon is moving
away from the Earth!
Moon
•The day is getting longer!
r
•Earth’s spin angular
momentum turns into Moon’s
orbital angular momentum.
•This will continue until the spins
and orbits match (syncrhonous
rotation)
•Common for nearly all satellites
Earth
Kepler’s First Law
•
•
•
•
•
Derivation is unpleasant
Requires Differential Equations
Pure mathematics, no science involved
Shall we skip it?
Bound orbits are ellipses (or circles)
– Not enough KE to escape, keep orbiting
– Negative total energy! KE < -U  KE + U < 0
• Unbound orbits are hyperbolae (or parabolae)
– One pass and gone for good (e.g. many comets)
– Positive total energy. KE +U > 0.
Collisions
• Conservation of Momentum
• Inelastic collison: Kinetic energy not
conserved
– But total energy is! Some goes into heating
or deformation
– Objects may stick together (completely
inelastic)
• Elastic collision: Kinetic energy is
conserved
Inelastic Collisions
Dust Grains colliding during
solar system formation
Impacts
Elastic Collisions
“Collision” with no impact
Just Gravity
Without this, solar system explortation
would be slow and expensive.
Saved 19 years off Voyager 2’s trip
to Neptune!
Two-body problem
• All this is derived for two bodies, as if nothing
else exists in the universe.
• Good approximation if one body is very large.
• Third body causes perturbations
• Three-body problem is analytically unsolvable in
general.
• Good treatment of restricted three-body problem
in Murray and Dermott (1999) Solar System
Dynamics.
Next Time
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Formation of the Solar System
Distribution of solar system materials
Planet formation, composition, structure
Conservation of Energy, Ang. Momentum
Mars Crater talk today at 3:30
MESSENGER flyby Monday at 11.