Physics 55Friday, September 23, 2005

1. Kepler’s empirical laws of planetary motion2. Newton’s laws of motion and related concepts such as

mass, acceleration, and forces.

Need to Understand Some Physics of Gravity and Light

To make further progress in understanding astronomy, you need to know some basic physics concepts:

1. Newton’s laws: mass, force, and universal law of gravitation.2. Fundamental conservation laws of energy, momentum, angular

momentum.3. What we can learn from light: surface temperature, speed toward or

away, rotation rate, presence of atmosphere, atomic composition, presence and strength of electric or magnetic fields, …

• Planet’s motion in sky results from combination of true motion and Earth’s motion. Planet orbits Sun each sidereal period.

• What we see recurs every synodic period (relative configuration of Earth, Sun, planet).

• For an inferior planet, a synodic period has elapsed when the planet has “lapped” Earth.

• For a superior planet – when Earth has lapped it. • If sidereal period is P days, planet moves 360/P degrees a day.

Earth moves 360/E degrees a day where E=365 is sidereal period of Earth.

• Planet laps Earth (completing a synodic period of S days) when S*(360/P) = S*(360/E) + 360. This is same as 1/P = 1/E + 1/S.

• For a superior planet find similarly 1/P = 1/E – 1/S.

Calculating Periods

Galileo’s Smoking Scope

• Smoking gun evidence for Copernician model required new technology.

• Galileo (1610) turns new telescope up and finds phases of Mercury.

• The correlation between phase and position in sky agrees with heliocentric model, not with Ptolemaic model.

Cultural Issues

• Galileo also discovers that Jupiter is itself accompanied by moons that orbit the planet, much like our Moon orbits Earth. Nature repeats on different scales.

• Motion of Jupiter’s moons studied closely, forms first Nautical clock for longitude measurement.

• Despite all this, Galileo tried for heresy and sentenced to house arrest.

• Galileo made many other discoveries of importance. With his telescope, he discovered mountains on the moon, size and shape of planets, nature of Milky Way, moving spots on Sun, among others.

• He also studied mechanics, properties of motion in general. Formulated principle of inertia: object tends to remain in its state of motion unless disturbed externally. We know we need to work to move things. Galileo’s insight reminds us we also work to stop or turn them.

• Galileo almost got mechanics right. What stopped him was the fact that the mathematics needed to formulate the theory was not known. To make progress, Newton had to invent Calculus.

More on Galileo

Brahe and Kepler

• First steps to deeper insight were careful observations by Brahe (1580) of planetary motion to great precision.

• Using these, Kepler (1609) finds three laws of planetary motion.

2 2

2 2

2

2

x1

semimajor axis, semiminor axis

eccentricity 1

y

a ba b

be

a

Kepler’s Laws

• 1. Orbit of a planet is an ellipse with Sun at one focus.

• Ellipse is shape of all points such that sum of their distances from two points (foci) is constant.

• Eccentricity (e) measures how far the foci are relative to size. e=0 is a circle.

• “Other focus” is nothing (not even same for all planets).

• Typically e small, .017 for Earth, .2 for Mercury.

• 2. Line connecting planet to Sun sweeps out equal areas in equal time intervals as planet orbits.

• Planet moves faster at perihelion, slower at aphelion.

• This causes slight change in rate of Sun’s motion discussed earlier.

This effect much more dramatic for comets which follow highly eccentric orbits.

• 3. Square of sidereal period is proportional to cube of semimajor axis.

• This relates the orbital motions of different planets orbiting same Sun.

• Write this as P2 = a3. This is valid if P is measured in years and a in AU.

Recall 1AU = 1.496 × 108 km = 93 million miles is average Earth-Sun distance.

• Kepler’s laws are amazing progress. They give planetary motion with unprecedented accuracy. What’s more, they are universal: they apply to any orbital system, from an atom through Saturn’s moons to Galaxy clusters.

• In physics such universality means there are fundamental laws at work here.

• These were found by Newton (1670) who at first was not thinking at all about Astronomy.

PRS Question

An asteroid with an orbital period of 8 years lies at

an average distance from the Sun equal to1. 2 AU

2. 4 AU

3. 8 AU

4. 16 AU

5. Need to know the asteroid’s mass.

PRS Question

The period of revolution p of a point on a spinning CD is related to its distance r from the center of the CD (its axis of rotation) by the expression

1. p2 =c r3 for some constant c.2. p2 =c r for some constant c.3. p =c r for some constant c.4. p does not depend on r.

PRS Question: Prediction

If steel ball is being swung around in circle on a rope and if the rope breaks at point P, which path does ball follow next? (For PRS, A=1, B=2, etc.)

PRS Question: Prediction

PRS Question: Prediction

Newton’s Laws of Motion

1. An object upon which no forces act will move in a straight line with constant velocity. (inertia)

• Familiar when velocity is zero – object at rest will stay there.

• Velocity is a vector – has direction as well as magnitude. So constant velocity means no change in direction or speed.

• Need to remember – in our world two forces (at least) always get in the way: gravity pulls us down; friction slows all motion. To see Newton 1 need to minimize these or imagine them removed.

• Acceleration a is rate of change of velocity v. So measured in (m/sec)/sec or m/sec2.

• Like v it is a vector and has direction. Note that changing direction of v requires acceleration, just as does changing magnitude of v.

2. When a force acts on an object, it will change its velocity. The acceleration will be proportional to the force (and pointed in the same direction). The proportionality constant is called mass.

• F = ma

• m is mass. Measured in kg – total amount of stuff.

• F is force. Measured in kg × (m/sec2) = N(ewton) • Acceleration of gravity here is g = 9.8 m/sec2, so force of gravity on 1kg.

is 9.8 N.

My Van

• My van can go 0 to 60 mph in 12 sec.• This is an acceleration of

a = (60 mi/hr)/12 sec = 5 (mi/hr)/sec

= (5 ×1609 m/3600 sec)/sec = 8045 m/3600 sec2

= 2.34 m/sec2

• Its mass is 800 kg. Force required is F = ma = 800 kg×2.34 m/sec2

= 1788 kg×m/sec2 = 1788 N

a = (1) 5 mph/sec

(2) 5 m/sec2

(3) 5 miles/sec2

F = (1) 1743 m/sec2

(2) 1788 N

(3) 1967 N

Uniform Circular Motion

• Planet moves at uniform speed v around circle of radius R. Period is P=2R/v

• Is velocity constant?

• NO. Direction changes.

• Guess acceleration:

• Points inwards

• Grows with larger v (m/sec).

• Smaller with larger R (m).

• Measured in m/sec2. • a = v^2/R.

• So F = ma = m v^2/R

a = (1) v/ R2

(2) v2 R

(3) v2 / R

Numbers for Earth

• As Earth spins, we move at

Vspin = (2R)/P= 4×107 m/24 hr

= 463 m/sec = 1036 mph

• As Earth orbits, we move at

Vorbit = (2R)/P

= 6.28× 1.5× 1011 m/365×86400 sec.

= 29871 m/sec = 100,595 mph

• When rocks in space hit Earth the relative velocities are about 100,000 mph. That is why they burn in atmosphere as meteors!

• a = v2/R =

4632/6.38×106

= 0.034 m/sec2

• a = v2/R =

298712/1.5×1011

= 0.0059 m/sec2

• Let’s compute the forces required to keep in these circular motions a person of mass m=100 kg. He weighs 9.8 m/sec2× 100 kg = 980 N

• F = ma =

100kg.× 0.034m/sec2

= 3.4 N

• F = ma =

100kg.× 0.0059m/sec2

= 0.59 N

a = (1) 9.8 m/sec2

(2) 0.034 m/sec2

(3) 0.45 m/sec2

• 3. When one object applies a force to another, the latter applies a force to the former, equal in magnitude and opposite in direction. (action and reaction).

• This explains how we walk. I push Earth back, it pushes me forward!

Deductions by Newton1. Elliptical orbit suggested to Newton an inverse-square law for gravity.2. Kepler’s first law was almost but not exactly correct: ellipses are the true shape of an

orbit only for two isolated masses. Can deduce position, mass of unknown planets from tiny deviation of known planet from ellipse.

3. Kepler’s second law holds for any central force, is really a statement about conservation of angular momentum.

4. Kepler’s third law can be generalized to a more useful form that allows one to deduce the mass of the less massive object in orbit.

5. Two masses orbit around their center of mass, which is at a focus of their elliptical orbits. Important for binary stars, Pluto and its moon Charon.

6. Other conic sections such as parabolas and hyperbolas can describe unbounded orbits of one mass moving near a second mass.

7. Escape velocity, implication for black holes.8. Tidal stresses, origins of tides, Roche limit, black holes.

Brief Review at Whiteboard of Acceleration, Mass, Forces, Gravity

Logic:

Objects like balls and planets often have nonuniform motion called acceleration. Acceleration has physics units speed over time or m/s.

Experiments and thinking suggested to Newton that acceleration can only arise from something called a force, which acts on a body to change its speed or direction. Many kinds of forces: gravitational, electrical, magnetic, friction.

Acceleration a is related to force F by a positive quantity called the mass m of the object:

a = (1/m)F.

The bigger the mass, the smaller the acceleration for a given force. Note: mass is measured in units of kilograms kg; force is measured in units of mass x acceleration = kg m/s2 called a “newton” and abbreviated as N.

By brilliant mathematical and scientific thinking, Newton discovered a formula for the gravitational force F of one mass on another mass. Newton also realized that the formula is universal and applies to any two masses in space: apples, Moons, planets, stars.

Simplest Motion: Uniform Motion

Nonuniform Motion: AccelerationSpeed is not constant or direction of motion is not constant (but speed can be constant in case of circular motion).

Where are speeds large in this picture if stroboscope samples at equal times?

Where are speeds small in this picture?

A

B

Demo: Ball in Circular Motion

Ball would go in straight line (B) if a force didn’t act on the ball, which here is the string pulling the ball toward the person at the center of the circle.

PRS Question

Newton’s Great Insight:Nonuniform Motion Caused by Forces

Something from one object like Sun somehow influences motion of other object like Earth. That “something” is still not understood in any fundamental sense but Newton discovered could be described by an astonishingly simple and precise mathematical rule now known as the universal law of gravitation.

The gravitational force becomes weaker with distance but has an effect no matter how far one object is from other object.. Total force on object is sum of forces from all other objects so depends on relative positions of all the other objects. Mathematics can be hard, computers have helped to obtain insight.

1 2gravity 1 12

11 3 2

=

~ 6.7 10 m /(kg s )

M MF G M a

d

G

Orbit: “Falling Around the Earth”

Marvelous insight and calculation by Newton: Moon falls around Earth exactly as apple falls to the ground, gravity is quantitatively universal.

How to Fly“The Hitchhiker’s Guide to the Galaxy”

There is an art, it says, or rather, a knack to flying.

The knack lies in learning how to throw yourself at the ground and miss. Pick a nice day, it suggests, and try it.

The first part is easy. All it requires is simply the ability to throw yourself forward with all your weight, and willingness not to mind that it's going to hurt. That is, it's going to hurt if you fail to miss the ground. Most people fail to miss the ground, and if they are really trying properly, the likelihood is that they will fail to miss it fairly hard.