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AST101 Lecture 3 Jan. 29, 2002

Modern Astronomy

The beginning of the modern age in Astronomy

began with Nicholas Copernicus (1473 1543), a

cleric with independent fortune.

Copernicus suggested that the Sun is at the

center of the universe (solar system), and that

the Earth rotates on its axis once a day to give

the apparent daily turn of the stars and the Sun.

He however kept the notion of epicycles and

deferents and the insistence on the primacy of

circles. The Copernican Revolution removes man

(and Earth) from the center of the Universe.

Copernicus objected to the ugliness of the

Ptolemaic theory. (Aesthetic arguments often

play a role in science a correct theory should

have some beauty. This notion continues today

beauty guides our models.)

Copernicus

Besides a being more sensible picture, are

there observational advantages of the new

ideas? Copernicus did give a more plausible

explanation for the maximum angle between Venus

(or Mercury) and Sun

Since Venus is closer to Sun on a smaller circle,

it never deviates from the Sun by more than angle

q. Can see full disk of Venus bright (when on

opposite side of Sun)

Venus

Venus is the morning or evening star the

brightest object in the sky.

Earth

Sun

q

- Which of the two positions of Venus in the

diagram is morning star and which is evening

star? (Hint In what sense does Earth rotate

relative to its orbital motion shown by ?)

Tycho Brahe (1546 1601) was an autocratic

Danish nobleman who devoted years of his life to

observing the positions of the planets in the

sky. He developed observational tools and

methods (with a grant from the Danish king that

would now be worth 1.5M to build an

observatory). Tycho observed a Nova stella

new star in the heavens in 1572 that we would

now call a supernova. The appearance of

something not previously present countered the

old idea of the unchanging heavens. Tychos main

accomplishment was the body of accurate

measurements of planets location in the sky over

20 years,. This proved invaluable to the next

generation of astronomers in understanding the

planets orbits (and the laws of Physics).

Tychos observatory

Tychos observatory, Uraniborg on the island of

Hven

Galileo and Kepler the foundations of modern

science

Galileo was an Italian mathematician and

philosopher who pioneered the use of experiments

and observations to understand the world. He

heard of the invention of the telescope in

Holland, and built a rudimentary telescope that

he turned on the heavens. Galileo also

pioneered experiments in physics, demonstrating

the rules that govern falling bodies. With

Galileo came the beginning of the notion that

Science is based on experiment If you cant see

something experimentally, you arent allowed to

say it is true

Galileo Galilei 1564 1642

Telescope observations of Galileo

- There are many more stars in the sky than can be

seen with the naked eye. If this is so, how can

we hold the opinion, as in the Middle Ages, that

the . heavens are provided for the sole benefit

of mankind? - Jupiter has four moons not observable to the

naked eye. (actually now see at least 28 moons!)

This is a shock to a geocentric view of the

world there are bodies that do not revolve

around Earth! - Venus shows phases from full to crescent. In

the geocentric model, there are only crescent

phases. Copernican system predicts all phases. - The moon has craters and mountains. The sun

shows blemishes called sunspots that come and go.

The sunspots reveal that the sun rotates on

its axis. The heavenly bodies are not perfect

orbs and have their own motions!

Galileo arrogantly published his findings

supporting the Copernican view and belittling the

Catholic church. His book Dialogues featured a

character Simplicio (a simpleton) who tries to

defend the churchs geocentric doctrine. Galileo

spent the rest of his life in house arrest.

- Check out Galileos Daughter, a recent best

seller by Dava Sobel based on the letters between

Galileo and his daughter.

Johannes Kepler 1571 1630

Kepler was one of most interesting characters in

scientific history with one leg in Middle Ages

and one in the Renaissance. As Tychos assistant

in court of Rudolf, Holy Roman Emporer in Prague,

Kepler inherited the extensive data collected by

Tycho to guide his calculations. He believed in

the Copernican model, and wanted to find the

underlying cause or model of the motions of the

planets. However, he was inclined to seek

mystical explanations for the planets orbits and

was enamored of the ancient Pythagorean

philosophy.

- Read Arthur Koestlers book The Sleepwalkers

how did Kepler span the divide between the Middle

Ages and the Renaissance?

The Music of the Spheres Kepler likened the

orbits of planets to strings that could be

plucked, sounding the Greek and Medieval

pentatonic scale (the black keys of the piano)

Saturn

Jupiter

Mars

Mercury

Venus

The ratio of circumferences of the planets

orbits were about right to give the pentatonic

scale. (Kepler had to invent the math to allow

him to calculate the tones.)

Kepler also thought the Five Perfect Solids of

Pythagoras and Plato could be the basis for the

planetary orbits He tried to inscribe and

circumscribe the spheres containing the orbits in

nested Platonic solids. The size of the spheres

that allowed the nesting were about right for the

known planets.

The perfect Platonic Solids

8 triangles

6 squares

4 triangles

12 pentagons

20 triangles

Calculating this model was a tour de force in

solid geometry!

Keplers model of the 5 perfect solids

Although the Harmony of the Spheres, and the

Perfect Solids came close to reproducing the

orbits, Tychos data was too good, and Kepler was

too honest, to ignore the discrepancies. He

then set out to find a more complete and accurate

representation of the known planet orbits using

painstaking calculations of the orbits found by

Tycho. After about 30 years, he wrote his

conclusions in the form of 3 Laws (buried in a

mass of mystic mumbo jumbo how did Newton find

the pearls of truth?)

Keplers Laws

- The planets move in ellipses, with the Sun at one

focus. - The line from the Sun to the moving planet sweeps

out equal areas in equal times. - The square of the planets orbital period (P) is

proportional to the cube of the semi-major axis

of the ellipse.

- The planets move in ellipses, with the Sun at one

focus.

Planet

Focii

Major axis

Sun

center

Minor axis

Semimajor axis a

Elliptical motion is a major departure from the

Ptolemic model based on circles!

Drawing an ellipse

Ellipse is the set of points for which the sum of

distances to 2 fixed points (the focii) is held

constant.

Eccentricity e is ratio of distance between focii

and length of major axis. An ellipse with e 0

is a circle (the two focii coincide).

- Draw your own ellipses, varying the separation

between focii from zero to ½ the major axis.

- The line from the Sun to the moving planet sweeps

out equal areas in equal times.

The light blue shaded areas represent the motion

of planet in the same fixed time interval. The

areas of all the light shaded sectors are the

same. Definition The point on the orbit at

nearest approach to the Sun is perihelion. The

point furthest from the Sun is aphelion.

Keplers 2nd law tells us that the planet does

not move with uniform speed another major

departure from the Ptolemaic model.

- Does planet move faster at perihelion or

aphelion?

- The square of the planets orbital period (P) is

proportional to the cube of the semi-major axis

of the ellipse.

In symbols If P period (time to revolve one

full turn) and a semi-major axis

means proportional to ) Equivalently P2 k

a3 The constant of proportionality k depends

(mainly) on the mass of the sun, so the relation

holds for all planets in a given solar system.

Same relation for another solar system, but with

a different value for k (see update by Newton on

what the constant k means)

P2 a3

The value of the constant of proportionality for

our solar system can be fixed using the Earths

orbit Earths period is 1 year (the

definition of year)

and semi-major axis is 1 AU (the definition of

AU). Thus for our solar system P2 kSS

a3 P2 1 (yr2) a3 1

(AU)3 so kss 1 in these units

P2 a3 (if P is in years and a is in AU)

for our solar system

Ratio relation For any two planets in the same

system, with periods P1 and P2 and semi-major

axes a1 and a2 P12 k a13 (a) P22

k a23 (b) Divide (b) by (a) The ks

cancel and we get (P2/P1)2 (a2/a1)3

- Example
- Mars orbits the Sun every 1.881 years. Predict

the size of its orbit (that is, find the

semi-major axis of Mars orbit). - P2(Mars) 1.8812 3.538 yr2
- Using P2 a3 3.528, a ? 3.538

1.524 AU - (Doing the cube root requires a good calculator!

You can try it in reverse to show that 1.5243

3.538 ) - Direct observation of Mars orbit gives a 1.524

AU, so prediction and observation agree.

3

- Check this calculation for another planet using

the data in Table 2.1 of the text

- Example
- In some other other planetary system, we see two

planets. The first planet revolves around its

star every 2 years and has a semi-major axis of 3

AU. The second revolves around the star every 16

years. What is the size of the orbit of the

second planet? - Let P1 period of planet 1 2 yr
- P2 period of planet 2 16 yr
- a1 semi-major axis for planet 1 3 AU
- a2 semi-major axis for planet 2 (unknown)
- Since this is a different planetary system, the

constant of proportionality is different from our

solar system. However, we can still use - (P2/P1)2 (a2/a1)3
- Thus (P2/P1)2 (16/2)2 82 64 (a2/a1)3 .
- Then ?64 4 (a2/a1), giving a2 4 a1 4x3

12 AU

3

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