9/13/2005 PHY 101 Week 3 1

The solar system SunPlanetsAsteroidsComets

PlutoNeptuneUranusSaturnJupiterMarsEarthVenusMercurySun

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Historical figures in the Copernican Revolution

Ptolemy – the geocentric model, that the Earth is at rest at the center of the Universe.

Copernicus – published the heliocentric model.

Galileo – his observations by telescope verified theheliocentric model.

Kepler – deduced empirical laws of planetary motion from Tycho’s observations of planetary positions.

Newton – developed the full theory of planetary orbits.

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The Copernican Revolution

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Nicolaus CopernicusThe Earth moves, in two ways.• It rotates on an axis (period = 1 day).• It revolves around the sun (period = 1 year).

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Johannes Kepler (1571 – 1630)

… discovered three empirical laws of planetary motion in the heliocentric solar system

1. Each planet moves on an elliptical orbit.2. The radial vector sweeps out equal

areas in equal times. 3. The square of the period is proportional

to the cube of the radius.(needed for the CAPA)

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How did Kepler determine the planetary orbits?

Compare the heliocentric model to naked-eye astronomy

The most complete data had been collected over a period of many years by Kepler’s predecessor, Tycho Brahe of Denmark.

The inner planet is Earth; the outer one is Mars. Plot their positions every month. Mars lags behind the Earth so its appearance with respect to the Zodiac is shifting.

Earth

Mars

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Ellipse Geometry

To draw an ellipse: Take a string. Tack down the two ends. Put a pencil in the string and pull the string taut. Move the pencil around keeping the string taut.

An ellipse is the locus of points for which the sum of the distances to two fixed points is fixed.

The two fixed points are called the focal points of the ellipse.

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Parameters of an elliptical orbit (a,e)

► Semi-major axis = a = one half the largest diameter

► Eccentricity = e = ratio of the distance between the focal points to the major diameter

For example, this ellipse has a = 1 and e = 0.5.

Perihelion = r2 = 0.5 Aphelion = r1 = 1.5

► Perihelion and aphelion

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Isaac Newton

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The observed solar system at the time of Newton

SunMercuryVenusEarthMarsJupiterSaturn

(all except Earth are named after Roman gods, because astrology was practiced in ancient Rome)

Three outer planets discovered later…Uranus (1781, Wm Herschel)Neptune (1846 Adams; LeVerrier) Pluto (1930, Tombaugh)

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To explain the motion of the planets, Newton developed three ideas:

1. The laws of motion2. The theory of universal gravitation3. Calculus, a new branch of mathematics

Newton solved the premier scientific problem of his time --- to explain the motion of the planets.

Isaac Newton

“If I have been able to see farther than others it is because I stood on the shoulders of giants.”

--- Newton’s letter to Robert Hooke,probably referring to Galileo and Kepler

mF

a

221

r

mGmF

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Newton’s Theory of Universal GravitationNewton and the Apple

(The apple never fell on his head, but sometimes stupid people say that, trying to be funny.)

Newton asked good questions the key to his success.

Observing Earth’s gravity acting on an apple, and seeing the moon, Newton asked whether the Earth’s gravity extends as far as the moon.

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In fact there are nine planets. The center of mass of the solar system is fixed (). To a first approximation the center of mass is at the Sun.

The motion of the planets

Diagram of a planet revolving around the sun.The eccentricity e is grossly exaggerated ― real orbits are very close to circular.

() actually it moves around the center of the galaxy

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Centripetalacceleration

Answer : 790 N

Example. Determine the string tension if a mass of 5 kg is whirled around your head on the end of a string of length 1 m with period of revolution 0.5 s.

For an object in circular motion, the centripetal acceleration is a = v 2/r . (Christian Huygens)

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Three concepts --–• Centripetal acceleration• Centripetal force• Centrifugal force

rv

ar

2

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Circular Orbits(a pretty good approximation for all the planets because the eccentricities are much less than 1.)

Centripetal acceleration

2

rv

ar

Newton’s second law

2

2

rGMm

rmv

There is a subtle approximation here: we are approximating the center of mass position by the position of the sun. This is a good approximation.

(velocity)

(acceleration)

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Circular Orbits

The planetary mass m cancels out. The speed is then

r

GMv

Time = distance / speedi.e., Period = circumference / speed

or ,

GM

rT

GMr

rv

rT

322 4

22

Kepler’s third law: T 2 r 3

Period of revolution

2

2

rGMm

rmv

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Generalization to elliptical orbits(and the true center of mass!)

)( mMGa4

T32

2

where a is the semi-major axis of the ellipse

The calculation of elliptical orbits is difficult mathematics.

The story of Newton and Halley

Many applications ...

GMa4 32

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FIN

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Newton’s figure to explain planetary orbits

Newton’s theory has stood the test of time. We use the same theory today for planets, moons, satellites, etc.

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The Law ofEqual Areas

Newton: Angular momentum is conserved—in fact that’s true for any central force—so the areal rate is constant.

Perihelion : fastestAphelion : slowest

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Newton said of himself…

“ I know not what I appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier sea-shell, whilst the great ocean of truth lay all undiscovered before me.”

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Applications

• Earth and Sun• Other planets• Moon and Earth• Artificial satellites• Exploration of the solar system

Some of these are covered in the CAPA problems.

GM

amMG

aT

32322 44

)(

Here is the general formula. We can always neglect m (mass of the satellite) compared to M (mass of the center).