ASTRONOMY 161Introduction to Solar System Astronomy

Class 8

Applying Newton’s LawsFriday, January 26

Applying Newton’s Laws: Key Concepts

(1) Newton modified and expanded Kepler’s Laws ofPlanetary Motion.

(2) Kepler described how planets move; Newtonexplained why they move.

(3) Tides are caused by the difference between theMoon’s gravitational force on different sides of theEarth.

(4) Tidal forces are slowing the Earth’s rotation &enlarging the Moon’s orbit.

(1) Newton modified and expandedKepler’s Laws of Planetary Motion

Kepler’s First Law:

The orbits of the planets aroundthe Sun are ellipses with theSun at one focus.

Newton’s revision:

The orbits of any pair of objectsare conic sections with thecenter of mass at one focus.

As the Earth pulls on Moon, Moon pulls on Earth.

Both Earth and Moon orbit the center of mass of theEarth-Moon system:

Center of mass = balance point:closer to more massive object.

Artificial satellites as envisagedby Isaac Newton:

To put an object into orbit,launch it sideways with alarge enough speed.

How large is large enough?

The shape of the orbit depends on the speed of thesatellite at launch:

Low speed = closed orbit, a circle or ellipse.High speed = open orbit, a parabola or hyperbola.Circles, ellipses, parabolas, and hyperbolas are called

conic sections.

To remain in a circularorbit just above theEarth’s surface, asatellite must havev = 7.9 km/sec.

To attain an open orbit,a satellite mustreach at least 11.2km/sec.

Some extra math:

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Kepler’s Second Law:A line from a planet to the Sun

sweeps out equal areas inequal time intervals.

Newton’s revision:Angular momentum

is conserved.

The product of the orbital speed (v) and the distancefrom the center of mass (r) is constant:

v x r = const

As r increases, v must decrease.

Kepler’sThird Law:

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Kepler’s third law applies only to objectsorbiting the Sun.

Newton’s revision applies to all pairs ofobject orbiting each other.

Newton’s revision can be used to find massesof distant objects (e.g., binary stars).

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(2) Kepler described how planets move;Newton explained why they move that way.

Kepler’s laws result naturally from Newton’slaws of motion and Newton’s law of

gravity.

Kepler’s laws of planetary motion, asmodified by Newton, are

UNIVERSAL!

(3) Tides are caused by the differencebetween the Moon’s gravitational force

on different sides of the Earth.

Time between high tides= 12 hours, 25 min

Time between moonrises= 24 hours, 50 min

The gravitational force between two objectsdecreases as the distance between themincreases.

The Moon’s gravitational pull on an object will be7% greater on the closer side of the Earth thanon the further side.

If the Moon’s pull wereconstant, thenEarth would beundistorted.

After subtractingaverage pull, Earthis stretched inMoon’s direction.

Result: TWO tidalbulges, on oppositesides of Earth.

Why do we notice tidesat the seashore?

Rock is stiff: Tidalbulges in rock areonly 0.3 m high.

Water is fluid: Tidalbulges in water are1 meter high.

Water bulges riseabove rock bulges.

The Sun also createstides on Earth.

High tides are highestwhen Sun, Earthand Moon line up(called ‘springtide’).

High tides are lowestwhen Sun, Earth &Moon are at rightangles (‘neaptide’).

(4) Tides forces are slowing the Earth’srotation and enlarging the Moon’s orbit.

The ocean’s tidal bulges press down on the oceanfloor.

Friction robs energy from Earth’s rotation and usesit to heat the ocean.

This process is known as “tidal breaking”. (Thinkof the tidal bulges as brake pads!)

The length of the day is increasing by0.002 seconds per century.

Moon creates tidal bulges on Earth:Earth creates BIGGER bulges on Moon.

The Moon has already undergone tidal braking.The Moon’s rotation has slowed to the point where

rotation period equals the orbital period.This is why the Moon always keeps the same side

turned to us.

Google Moonhttp://moon.google.com/

Friction between the tidalbulges and ocean floordrags the bulges in thedirection of the Earth’srotation.

Bulges lead Moon by about 10degrees.

The leading bulge steadily tugsthe Moon into a largerorbit.

The average Earth-Moondistance is increasing by4 meters per century.

How do we know the distanceis increasing?

Measure the distance to themoon with great accuracyand watch it change!

Shoot the moon with a laser andwatch it bounce off

How do we know the distanceis increasing?

Measure the distance to themoon with great accuracyand watch it change!

Shoot the moon with a laser andwatch it bounce off

Few closing questions:

1) If the Sun was twice as massive, how long wouldbe the year on Earth?

2) Is the center of mass of the Solar System inside oroutside the Sun?

3) How long will it take for a day on Earth to doublein duration?

4) How long will it take for the Moon to double itsdistance from Earth?