EART 160: Planetary Science

First snapshot of Mercury taken by MESSENGER

Flyby on Monday, 14 January 2008Closest Approach: 200 km at 11:04:39 PST

http://messenger.jhuapl.edu

Announcements

Seminar at 3:30 PM“Martian Impact Craters”

Nadine BarlowE&MS B210

HW 1 due in one week

Most Readings, Notes, available on Website

Today

Johannes Kepler1571-1630

Isaac Newton1643-1727

•Paper Discussion

•Stevenson (2000)

•Soter (2007)

•Celestial Mechanics

• Kepler’s Laws

• Conservation Eqns

• Newton’s Laws

• Gravity

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.

r

va

2

a

r

v

This is why planets don’t fall into the Sun.

And why it’s so hard to get to Mercury!

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

d

r

v┴ = v sin v

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

Earth

Moon

r

•The Moon is moving away from the Earth!

•The day is getting longer!

•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

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 impactJust Gravity

Without this, solar system explortationwould be slow and expensive.

Saved 19 years off Voyager 2’s tripto 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

• 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.