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Modern Astronomy
Lecture 3 Jan. 29, 2002
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
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.)
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 is the ‘morning’ or
‘evening’ star – the
brightest object in the sky.
 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.
Tycho’s 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).
Tycho’s observatory
Tycho’s observatory, Uraniborg
on the island of Hven
Galileo and Kepler – the foundations of modern science
Galileo Galilei 1564 – 1642
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 can’t see something experimentally,
you aren’t allowed to say it is true”
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
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 church’s geocentric doctrine.
Galileo spent the rest of his life in house arrest.
 Check out “Galileo’s 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 Tycho’s 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 planet’s orbits and was enamored of
the ancient Pythagorean philosophy.
 Read Arthur Koestler’s 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
The ratio of circumferences
of the planet’s 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!
Kepler’s model of the 5 perfect solids
Although the Harmony of the
Spheres, and the Perfect
Solids came close to
reproducing the orbits, Tycho’s
data was too good, and Kepler
was too honest, to ignore the
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?)
Kepler’s Laws
1. The planets move in ellipses, with the Sun at
one focus.
2. The line from the Sun to the moving planet
sweeps out equal areas in equal times.
3. The square of the planet’s 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.
Major axis
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
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.
2. 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
Definition: The point on the
orbit at nearest approach to
the Sun is perihelion. The
point furthest from the Sun
is aphelion.
Kepler’s 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 planet’s 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:
P2 ~ a3
“~“ means “proportional to” )
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)
The value of the constant of proportionality for our solar system
can be fixed using the Earths orbit:
Earth’s 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)
P2 = a3
; a3 = 1 (AU)3
so kss = 1 in these units
(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
P22 = k a23
Divide (b) by (a):
The k’s 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
= 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.
 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