Astronomy 2

Overview of the Universe

Winter 2006

2. Lectures on Copernicus to Modern Times.

Joe Miller

Nicholas Copernicus (1473-1543)

• Proposed heliocentric system.• Inferior planets, superior planets, planetary configurations.

• Seasons.• Sidereal, synodic periods.• Distances of planets from the sun, assuming circular orbits.

• Still had epicycles.• Parallax problem still there.

Siderial period: one revolution around the sun with respect to the stars.

Synodic period: the time between two successive identical geometrical configurations.

Geometrical configurations:

Opposition: a planet is in opposition when it is opposite the sun in the sky.

Conjunction: a planet is lined up with the sun.

A superior planet has its orbit outside the earth’s orbit, an inferior planet’s orbit is inside.

Planetary Configurations

The time between two oppositions is one synodic period- easy to observe. But how do you determine the siderial period of another planet? We know what the earth’s is- one siderial year.

Inferior planet case:

Let S = synodic period, E =period of the earth,and P = siderial period of planet.

In one day earth goes 360o/E degrees around the sun.In one day planet goes 360o/P degrees around the sun.

In one synodic period, earth goes S(360o/E), and planet goes S(360o/P) . But an inferior planet makesone extra trip around the sun (it “laps” the earth), so it goes an extra 360o. Therefore

S(360o/P) - 360o = S(360o/E),

or dividing through by S and 360o, we have

1/P - 1/S = 1/E. Since E = 1 year, we can write 1/P = 1 + 1/S Inferior planets.

In like manner, for superior planets it is the earth that travels the extra 360o, so 360o has to be subtracted from the distance the earth travels and we get 1/P = 1 - 1/S. Superior planets.

The general formula:

1/S = 1/P(faster) - 1/P(slower)

The earth is the faster for superior planets, while it is the slower for exterior planets.

Copernicus determined the relative distances of the planets from the sun. He made direct use of the heliocentric system, the known sidereal and synodic periods of the planets, and observations of the times of specific planetary configurations: greatest elongation for inferior planets, and quadrature and opposition for superior planets.

He made two basic assumptions:1) Orbits of planets are perfect circles.2) All planets travel at constant speed in their orbits.

He defined the distance of the earth to the sun to be 1.0, so that all other distance would be in terms of the earth-sun distance.

Inferior planets (Mercury and Venus)

Superior planets (Mars, Jupiter, and Saturn)

Results

The seasons: primarily the result of the tilt of the earth’s axis.

Tycho Brahe (1546-1601)

• A Danish nobleman a disagreeable snob.• A clever instrument builder and excellent observer.

• Supernova of 1572 was beyond the moon’s orbit.• Comet of 1577 was at least three times the distance to the moon- no detectable parallax- and probably in orbit around the sun.

• 30 years of very accurate data on planetary positions.

• Hired Johannes Kepler.• Invented the “Tychonic System.”

The “Tychonic” System

Johannes Kepler (1571-1630)

Kepler was a staunch believer in the Copernican System

• His assignment by Tycho was to go over the observations of Mars’ positions. He spent 25 years doing this.

• He was driven to find order and harmony in the heavens. His reasoning could flip rapidly between the mystical and the rigorously scientific.

• He fit the five regular solids of Euclid between the orbits of the planets, which he thought was demonstration of the harmony of the universe.

Kepler first investigated the shape of Mars’ orbit. The approach was to use pairs of observations of Mars separated by one sidereal period of Mars. These pair would be made when Mars was in the same position in its orbit, but the earth was at a different position in its orbit.

By this approach Kepler finally worked out that the orbit of Mars was not a circle, as had been believed for nearly 20 centuries, but a mathematical curve called an ellipse.

The ellipse is a conic section:

Another view of conic sections:

Mathematics of ellipses:

a=semi-major axis

b=semi-minor axis

c=distance of focus from center

ca

=e the ellipticity or eccentricity of the ellipse

ca

=0 is a circle.

C

a

Kepler’s First Law:

A planet travels around the sun in an orbit of elliptical shape with the sun at one focus.

Kepler’s Second Law:

The line from a planet to the sun sweeps over equal areas in equal times.

In modern terminology, this is a result of the conservation of angular momentum.

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Kepler's Third Law : The Harmonic Law

The squares of the periods of the planets

are proportional to the cubes of their mean

distances from the sun.

p2 ∝ a3 or p2 = Ka3, where K is a constant

that depends on the choice of units for time and distance.

If units of earth years and astronomical units are

used, then K =1.0 and

p2 = a3 or you could write p1

2

p22 =

a13

a23 when you are

comparing two objects in orbit around the same central object.

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Galileo Galilei (1564-1642)

• Contemporary of Kepler• Modern scientist- rejected Aristotelian approach.

• Excellent observer and experimenter.– Explained earthshine.– Discovered the law of the pendulum– Noticed speed of hailstones.

• Founded the science of mechanics.• Heard about a telescope and immediately built one. His observations of the sky profoundly confronted modern beliefs.

Galileo founded modern mechanics: the study of the motions of objects.• The Law of Falling Bodies:

– In a vacuum, all bodies fall with the same uniform acceleration, regardless of their size or mass.

– Acceleration is the rate of change of velocity. It has funny-looking units: km/sec/sec or ft/sec/sec, etc.

• This contrary to Aristotle, who held that the heavier a body was, the faster it would fall compared to a lighter body.

Uniformly accelerated motion

Velocity changes the same amount over each identical period of time. Consider dropping an object:

Acceleration of gravity is 9.8m/sec/sec=32 ft/sec/sec

After 1 sec falling 9.8 m/sec

2 sec 19.6 m/sec3 sec 29.4 m/sec

At any time t, the velocity v is given by

v=at, where a is the acceleration.

Distance fallen:

The distance d traveled by an object is given by

d=vt. If the velocity is changing during the

time period, then one must use the average velocity.

For uniform acceleration starting from rest, the average

velocity is the final velocity divided by 2:

The appropriate average velocity v is

v=v(final)

2=at2

Therefore d=vt=at2t=

12at2

d=12at2

Galileo’s Law of Inertia

A body set into uniform motion will remain in uniform motion until interfered with.

By “uniform motion” we “mean moving with a constant speed in a straight line.” So the object neither changes speed nor direction.

This also means that an object at rest will remain at rest until it is interfered with. This idea is fundamental to our understanding of physics. Why did it take us 2000 years to get to this? Aristotle taught it is “natural” for all objects to remain at rest. Motion is unnatural and would be resisted.

Some examples:

1. A weight dropped from the mast of a ship.

2. Playing catch in an airplane.

3. A billiard ball.

Galileo’s discoveries with a telescope

• Craters and mountains on the moons- a tremendous challenge to prevailing.thinking.

• Sunspots.• The Milky Way consists of many unresolved stars.

• Saturn was not always round, but sometimes had two blobs near it.

• Venus went through phases: a real challenge to Aristotelian views.

• Discovered four moons of Jupiter! Caused tremendous excitement.

The Beginnings of Modern Physics

C. Huygens.

Explaining it all--the laws of mechanics.

First, a brief digression:

There are two kinds of quantities:

Scalar quantities have just a size or magnitude

Vector quantities have both a size and directionExamples:

Temperature is a scalar. It is just an amount.

Speed is a scalar. It measure how fast something is going.

Velocity is a vector. It specifies both speed and direction.

Acceleration is a vector quantity. It is a measure of the rate of change of velocity, a vector quantity, with respect to time. A change in velocity can be the result of any of three things:• A change in speed, but not direction.• A change of direction, but not speed.• A change of both speed and direction.• Examples:

– Car speeding up or slowing down on a straight road.

– Car traveling at constant speed on a circular track.

Huygens’ formula for acceleration in circular motion

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a = acceleration, a vector quantity

r = radius of circular path

v = velocity, a vector quantity

The two triangles are similar, so

Δv

v=

vΔt

r, where Δ means change.

Therefore Δv

Δt= a =

v 2

r

Isaac Newton (1643-1727)

• In his early 20’s, most of it in two years, he– Invented calculus– Developed the laws of motion– Developed the Law of Gravity– Developed optics theory– Developed a theory of colors

• He performed a great synthesis of the results of Copernicus, Galileo, Kepler, and Huygens, ultimately published in Philosophae Naturalus Principia Mathematica

Among many other things, the “Principia” vastly simplified a great range of behavior associated with moving objects into three apparently “simple” laws.

But first Newton introduced a new concept, mass.

Mass is the quantity of matter, of stuff, that an object has. The mass of an object is the same on the moon as it is on earth.

Second, he made clear that another important physical quantity is force, which is defined as apush or pull. Weight is a force, the amount of force an object exerts on a scale. An object’s weight is different on the moon than on the earth.

Newton’s Three Laws of Motion.

First Law:

In the absence of forces, the motion of an object does not change.

By “motion” we mean the “velocity.” Therefore, without forces there can be no accelerations. An object at rest will remain at rest unless it is acted upon by a force.

An alternative look at the First Law:

Newton introduced another new quantity, momentum.

Momentum = mass times velocity= mv, a vector quantity.

The First Law can be stated as

The momentum of an object does not change when no forces act on it.

It takes the action of a force to change the momentum of an object.

Newton’s Second Law:

A force acts so as to accelerate an object. The amount of acceleration is directly proportional to the applied force, but is inversely proportional to the object’s mass.

In terms of simple formulae:

a=Fm

acceleration equals force divided by mass

F =ma force =mass times acceleration.

An alternative way of writing Newton’ Second Law

Force = rate of change of the momentum, or

F =Δ(mv)

Δt "Δ" means "change in"

Newton’s Third Law

For every action there is equal and opposite reaction.

or

Forces come in opposing pairs.

or

In any system of objects, the total momentum is a constant.

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Newton’s three laws provide a complete prescription for motion. They are very powerful, though they appear very simple.

Using these three laws and Kepler’s Laws, Newton was able to explain the motion of the planets in terms of forces operating and masses involved. This is called dynamics, as opposed to what Kepler did, kinematics, a description of motion.

Newton’s starting assumption: there is some kind of force of attraction between all things in the universe. Let’s call it gravity. It is responsible for holding the planets in their orbits. What kind of force do we need to explain what we observe?

The force required to maintain an object in circular motion:

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F = ma Newton's Second Law

a =v 2

r Huygens' Formula

∴ F =mv 2

r "∴" means "therefore"

A simple derivation of the Law of Gravity using

Newton's Laws and Kepler's Third Law:

Let r =the radius of the planet's orbit

mp = the planet's mass

ms = the sun's mass

v= the orbital speed of the planet

The acceleration a is given by Huygen's Formula:a=v2

r

Therefore the force needed to keep the planet in its orbit is

F =mpv

2

r

Now we have to figure out how to calculate v.

Speed is distance divided by time, or v=dt

Let us use for d the distance around the orbit, 2πr, and the

time t to get around the orbit P, the orbital period of the planet. Then

v=dt

=2πrP

and v2 =4π 2r2

P2 Therefore F =mp(

4π 2r2

P2 )

r=

4π 2mpr2

rP2

F =4π2mpr

P2 . Remember Kepler's Third Law: P2 =Ka3 =Kr3

∴ F =4π2mpr

Kr3 =Cmp

r2

Thus we see that there is a constant part and a variable part to this

force law:

Fp =Cmp

r2

The force on the planet depends only on the mass of the planet divided by

the square of its distance from the sun. This is an example of an

inverse square law.

Now Newton could apply his Third Law. There must be an equal and

opposite force on the sun to accelerate the sun and conserve momentum.

Newton showed this will only happen if this equal force on the sun obeys

a similar law

Fs =C2ms

r2 , where is C2 some constant.

Thus the force depends on both the mass of the sun and the mass of the planet:

F ∝msmp

r2 or F =Gmsmp

r2 , where G is called the constant of gravity

Newton then made one of the most extreme generalizations in science. He postulated that this gravitational force exists between any two masses:

This applies to any two masses at any separation!

F =Gm1m2

d2

Newton derived Kepler’s three laws using his three laws of motion and his law of gravity.• First Law: all orbits of one object traveling around another are conic sections: circles, ellipses, parabolas, or hyperbolas.

• Second Law: Equal Areas rule is just an example of the conservation of angular momentum.

• Third Law. Since how gravity works is directly involved, masses have to be introduced, and Newton modified Kepler’s Third Law:

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ms + mp( )P2 = Ka3 where K is a constant. If we use earth years and the

astronomical unit for distances, then

ms + mp( )P2 = a3 . If we let the mass of the sun = 1.0,

then the masses of most planets are essentially negligable and we get back to

Kepler's version.

How is G measured?

The Cavendish Balance:

The Law of Falling Bodies Revisited

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Remember that Newton's Second Law states

a =F

m1

, where we have made it clear we are talking about m1.

But the Law of Gravity gives

F = Gm1m2

d2. Therefore

a =G

m1m2

d2

m1

= Gm2

d2. Acceleration does not depend on m1!

But what is the acceleration near the surface of the earth caused by the earth?

The effect of the earth’s gravity at its surface

Newton showed that the combined effect of all the mass of the earth acted as if all the mass of the earth was in one point at its center.

Inside the earth, only the matter closer to the center mattered.

Tides were explained by Newton as a result of gravitational forces:

Thus the tidal force is a differential gravitational force. The effect of the moon is about three times that of the sun.

Newton’s greatest triumph- the discovery of Neptune• Herschel- discovered Uranus by accident on March 13, 1781. By 1790 it was realized that its orbit was not as expected. Conclusion: another unknown planet was “perturbing” its motion.

• 1845- Adams calculated a position for the unknown planet and sent it to the Astronomer Royal Airy. Airy gave him a test problem to check him out, but Adams refused to do it.

• 1845- Simultaneously Leverrier did the same calculation and sent the position to Airy. He did the problem, and Airy started looking for the planet. He botched the job!

Neptune (cont.)

• September 23, 1845- Leverrier got tired of waiting and sent a note to Galle in Berlin with the position. The note arrived on this date. That very night Galle checked it out and found what we now call Neptune only 52 minutes of arc from the predicted position!

• Thus Neptune was discovered because of its gravitational effects, not its light.

Are Newton’s Laws true?

Until the 19th century everything looked good, except for one “minor problem”: the rate that Mercury’s orbit was precessing:

Newton’s physics predicted this would change at 531 seconds of arc per century. The observation was 574, a discrepancy of 43. Why. Another planet? Newton’s physics not accurate. Einstein to the rescue!

Finally, the first parallax of a star was observed by Bessel in 1838. The parallax of the nearest star is only 0.74 seconds of arc!

Summary of historical developments in astronomy• Without telescope:

– Eclipses of the sun and moon predictable– The earth’s axis precesses.– The relative distances of the planets from the sun accurately worked out.

– The orbits of the planets are shown to be ellipses with the sun at one focus.

– The speed of the planets in their orbits varies according to a simple law

– The periods and distances of the planets from the sun follow a simple formula.

– A single type of force whose behavior is described by a simple formula controls the motions of the planets, the moon, and much more!

• With telescope– The moon has craters and mountains, Saturn has rings, Venus has phases, the Milky Way is vast numbers of stars, Jupiter has moons, etc.

– The parallax of stars can be detected, convincing demonstrating the heliocentric theory is in, the geocentric theory is out.