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Title: The beginning of the modern age in Astronomy began with Nicholas Copernicus 1473 1543, a cleric with


1
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
2
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 ?)

3
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
4
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
5
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.

6
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?

7
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.)
8
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!
9
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?)
10
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.

11
  • 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!
12
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.

13
  • 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?

14
  • 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
15
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
16
  • 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

17
  • 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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