Lecture Notes

Arny Chapter 1, Sections 3, 4, 5

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Chapter 1: History of Astronomy

Section 1.3: Astronomy in the Renaissance

Nicolaus Copernicus (1473-1543)
In 1514, he created a heliocentric model for the solar system,
The Copernican Model.

Features of this model include:

• The Sun at the center of the universe (heliocentric).
• Planets (Earth included) revolve around the Sun.
• Planets closer to the Sun move faster.
• Stars are very far away.
  (Necessary to explain lack of stellar parallax.)
• Earth rotates on its axis daily.

Does this model really explain the variety of observed celestial motions?

Earth's rotation explains the daily motion of the stars.
Earth rotates once every 23h 56m relative to the stars.
[The sidereal day.]
At the same time, Earth moves around the Sun.

Earth moves around the Sun
in the same sense it rotates.
In 23h 56m, the same star will
appear in the same place in the sky.
But the Earth will have to rotate a
little more (an extra 4 minutes) before
the Sun returns to the same spot in our sky.
See figure E2.2 on page 181 of the text.
The solar day is 24 hours.

[Full Page Diagram]
See figures 1.5 and 1.29 in text.
As Earth moves around the Sun, the Sun will appear against different constellations of stars.
Thus the Sun's motion along the ecliptic is really the Earth's motion.
Earth orbits the Sun, which is why the Sun returns to the same place among the stars in a year.

The Copernican model also explains retrograde motion.
Positions of Sun, Earth, and Mars, after one month of time.
The constellation of the Sun (which one it appears to be "in") depends on where Earth is relative to the Sun, which constellation happens to be behind the Sun from our viewpoint.
In one month, the Sun "changes position" by about one constellation.
Not because the Sun moved, but because Earth moved.

Now consider Mars, this is more complicated because both Earth and Mars are moving around the Sun.
Usually Mars will appear to move through the constellations in the "usual" direction, the same direction as the Sun.

When a planet is in the right place, like Mars in opposition, Mars can appear to move through the stars in the opposite direction from usual, retrograde.
The faster Earth is overtaking the slower Mars and giving the illusion of backwards or retrograde motion.

Maybe another analogy will help.
You're on the freeway doing about 80 (shame on you!) and you're passing a slower car.
That car is going from in front of you to behind you, it's like it's moving backwards.
Even though both cars are going forward on the freeway, from your point of view it can look like the slower car is moving backwards.
Because we are viewing other planets from the moving Earth, we get similar illusions.

This is how the Copernican Model explains retrogrades.
Retrogrades are not actually true backwards motion.
Retrogrades are an illusion which occur when a faster inside planet passes a slower outer planet.
This works correctly at explaining details of retrograde motions!

One more thing, remember that while all this planet motion is going on, the month of time shown in my diagram, the Earth is spinning.
So while the planets seem to slowly change their position on the celestial sphere, all the planets and stars are whirling around us on Earth, rising in the east and setting in our west.

Copernicus spent 30 years making his model as complete as Ptolemy's model before finally publishing it.
Copernicus calculated and measured many complex quantities to make his model work.
Copernicus did his own tinkering to make his model work better, but he never considered anything other than circular orbits (and it was that which made his model the most incorrect).

Copernicus failed to address most of the objections to the heliocentric model that we discussed before;

why we don't feel the Earth's spin,
why the Moon could keep up with a moving Earth,
and so on.

Aristotle explained why the Sun and planets moved around Earth.
Copernicus offered no explanations why Earth moves around the Sun.

Two competing models Ptolemy's and Copernicus's.
Which, if any, was correct?

Which predicts planetary motion better?
Neither!
Both performed well, but not great.
Typically both would be in error by maybe 1 degree, sometimes much more.

Which was simpler?
[Often preference is given to a simpler theory, although in the long run definite evidence is needed, this is called "Occam's razor".]
Both models contained very many extra rules and features, not clear which was simpler.

Why did the Copernican model catch on?
It didn't, not until hard evidence showed it to be correct.
The church prohibited any discussion of the subject, the church had proclaimed Aristotle's ideas as correct and would not tolerate any other opinion.
Still, many intellectuals learned of the Copernican Model and favored it over Ptolemy's Model.
This was generally because the Copernican Model had an underlying elegance in the idea of turning Earth into just another planet obeying the same laws as other planets.


Tycho Brahe (1546-1601)
I've heard various pronunciations, most often: Tie-koh Braw-hee.
Last of the great naked-eye astronomers, star charts.
(We are close to the invention of the telescope).
[Mention his golden nose? His original nose was lost in a duel with another student over who was the better mathematician, I guess Tycho lost.]

He determined rough distances
to the Sun, Moon, and planets
using parallax.
(See figure 1.25 on page 45 or
figures 12.1 on page 351.)

He observed supernovae and comets,
and showed (using parallax) that these
were things occurring beyond the atmosphere.
Tycho proved that the heavens were not constant
and unchanging as had been long believed.

Tycho created a model of the solar system.
Geocentric, Earth at the center with the Sun orbiting Earth.
All other planets orbited the Sun.
A good and interesting compromise.
But really it's just the Copernican Model slightly changed.
This model did not play a major role in astronomy.

Tycho's detailed astronomical measurements would be of immense use to his young assistant:
Johannes Kepler.
Tycho was jealous of the brilliant Kepler.
Afraid Kepler would make brilliant discoveries that would overshadow him (rightly so!).
Tycho did not allow Kepler access to his measurements and data.

[*** Usual end of lecture #3 ***]

Johannes Kepler (1571-1630)
Kepler had a wild idea that certain geometric shapes dictated the size of planetary orbits.
He would later prove that theory incorrect.
But the mathematical talent he showed got him hired by Tycho.
Kepler joined Tycho in 1600, Tycho died 10 months later and Kepler took over his position for a while.

Kepler simplified and improved the Copernican/Heliocentric Model.
With access to Tycho's accurate data, Kepler was determined to find a model that would accurately predict planetary positions.
Kepler succeeded magnificently.
Kepler was able to characterize the basic planetary orbits using just three laws (which we today call Kepler's Laws).
He published the laws in 1609.
Kepler's Model of the universe was based on Copernicus's view,
Heliocentric, Earth moves and spins.

Kepler's Laws
1. Planets move in elliptical orbits with the Sun at one focus of the ellipse.
Ellipses are oval-shaped loops.
Geometrically, an ellipse is the set of points with equal total distance from two foci (foci = plural of focus). [See Figure 1.33 on page 50.]

A circle is an ellipse with the two
foci coinciding [picture of a perfect
circle with a single dot in the center].

Ellipses can be non-circular
[picture of a stretched-out loop with
the two dots - foci - far apart].

In the extreme case (orbits that are
often seen for comets), the orbit can
be extremely elongated
[picture of a path like a rubber band
with opposite sides pulled apart].

Kepler's first law allows for non-circular orbits!
Circular motion gone (all previous models had assumed perfectly circular motions).
This greatly bothered Kepler, an ellipse lacks the beauty of a circle.
Why did God choose to use ellipses?
But the data was clear, planets were moving in ellipses whether he liked it or not.


2. The orbital speed of a planet varies so that a line joining the Sun and the planet sweeps over equal areas in equal time intervals.


This law tells us how the speed of
a planet varies as it goes around the Sun.
When the planet is further from the Sun,
it moves slower.
[See figure 1.34 on page 51.]
Perihelion and aphelion.

This law can be used to compute planetary
speeds at any time although the calculations
can be complex.
This law tells us that planets will be moving
fastest when they are closest to the Sun, slowest
when furthest.

Kepler rejected circular motion with his first law.
Now he rejects uniform motion with his second law.
That all heavenly motion be "uniform and circular" had been the foundation of all previous models.


3. [Let me just read this law first . . .]
The amount of time a planet takes to orbit the Sun is related to its orbit's size, such that the period, P, squared is proportional to the semimajor axis, a, cubed.

What do these words mean?
(period)^2 proportional to (semimajor axis)^3
"Proportional" means that one is always the same factor larger than the other.
Call this factor k.
P^2 = k a^3
Or, all planets have the same value for the ratio of P^2/a^3.

Explain what the major axis of an ellipse is.
Explain what the semimajor axis is.
See figure 1.33 on page 50.
It turns out (although I won't prove it),
the semimajor axis for an orbit is the same
as the average distance of the planet from
the Sun.
We will use those two terms, semimajor axis and average distance, interchangeably.

P is the time to go around the Sun.
For Earth, P = 1 year or 365 days.
a is the average distance from the Sun.
For Earth, a = 93 million miles or 150 million kilometers or 1 AU

So for Earth, (1 yr)^2 = k (1 AU)^3, which means k = 1 (yr^2 / AU^3).
But the point of Kepler's third law is that it is the same k for all the planets.
If we assume the values for P and a to be years and AUs, then k = 1, and Kepler's third law takes the simpler form, P^2 = a^3

Does this really work, for all the planets?
That's easy enough to test:

Planet	P (years)	a (AU)	P^2 / a^3
Earth	1	1	1.0000
Saturn	29.4577	9.539	0.9997
Venus	.6152	.723	1.0014

They weren't perfectly equal, this does indicate some minor flaws in Kepler's law.
But the distances weren't known well enough in Kepler's time for him to have computed with this much accuracy.
As far as Kepler could tell, the theory worked perfectly!
We'll explain the slight inaccuracy in the next chapter.
Kepler's law is accurate enough for our purposes and we will use it to solve problems.

Kepler's Third Law: P^2 = a^3
where

P is measured in years.
a is measured in AU
and when dealing with orbits around the Sun.

If we know P or a, we can solve for the other.

Kepler's third law says planets closer to the Sun orbit faster - just like Copernicus had said.
Kepler has gone further and given the exact relationship.

Galileo Galilei (1564-1642)
Galileo was a remarkable man.
He deserves much of the credit for the modern scientific method.
He believed in the physical testing of theories, not just the contemplation and revelation of the Greeks.

Galileo made major contributions to physics.
He studied forces and motions on Earth with the plan of then applying this knowledge to the cosmos (it would be Isaac Newton who would actually complete this plan).
We'll discuss the physics of Galileo in the next chapter.

Galileo did not invent the telescope.
But he improved it and was the first to use it as a tool.
With the telescope, Galileo made a series of startling discoveries.

I. Moon Landscapes
Galileo found that the Moon was covered with mountains, valleys, and craters.
It was a real place where people could presumably live.
It was not a smooth surface like many had thought.

II. Stars
Galileo discovered huge numbers of stars, too faint to be seen with the naked eye.
Why were they there? Suggests universe maybe not created just for humans.
He discovered that the Milky Way Nebula (cloud) was not really a cloud but actually a vast group of stars.

III. Satellites (Moons) of Jupiter
Four satellites, each clearly orbiting Jupiter.
Today these four moons are often called "Galilean Satellites".
This discovery was important for a few reasons.
a. It proved Earth was not the center of all heavenly motion. Galileo considered this convincing evidence for the Copernican model and against the Ptolemaic model.
b. It showed that orbiting moons are not left behind by a moving planet. This removed one of the objections to the Copernican (heliocentric) model that a moving Earth would leave our Moon behind.
c. Kepler (who was a contemporary of Galileo) showed that the Galilean satellites obeyed his laws of planetary motion (appropriate for motion around Jupiter).

IV. Sunspots
Using a telescope in a special way (not looking directly at the Sun with it), Galileo discovered sunspots, dark spots or blemishes on the Sun.
Galileo wasn't careful enough and blinded himself.
The Sun itself was imperfect.
The sunspots also showed the Sun rotates.
If the Sun can rotate, why not Earth?
Further, this contradicted Aristotle's claim that everything in the universe except Earth was perfect and without flaws.

V. Phases of Venus
Through the telescope, Galileo saw that the planets were actual worlds.
The stars, though, remain as mere pinpoints of light through telescopes.

Galileo saw that not only was Venus a world in its own right, but it went through a sequence of phases (New, First Quarter, Full, Last Quarter, and back to New).
Same as the phases for our Moon except Venus takes 584 days to complete the sequence compared to 30 days for our Moon.
This observation was a clear contradiction
of the Ptolemaic model.

In the Ptolemaic model,
Venus is always between
the Earth and Sun.
The Sun lights the side of
Venus mostly away from
Earth.
From Earth, we would
never see Venus fully lit (Full).
[The text does not include
a figure showing this.]

In the Copernican model,
Venus can show a new phase,
quarter phase, or full phase.
[See figure 1.29 on page 52.]

Galileo saw the complete range
of phases for Venus.
This is consistent with the Copernican
model, it contradicts the Ptolemaic model.

[Why does it take 584 days for Venus to go through its sequence of phases when Venus orbits the Sun in about 225 days? This "synodic" period of Venus will come up a few times in computer labs and be explained there.]


Section 1.4 - Isaac Newton and the Birth of Astrophysics

Isaac Newton (1642-1727)
Newton continued the study of forces and motions begun by Galileo.
He then went on to apply this knowledge to the solar system and to verify and to even improve Kepler's model.
The details of his new physics are covered in the next chapter (chapter 2).

Section 1.5 - The Growth of Astrophysics

We end our history of astronomy at this point, although we could certainly continue.
[Herschel, Kelvin, Maxwell, Einstein, Hubble, and so many more.]
These names will come up later in the quarter, but our current goal of understanding the motions we see in the sky do not require their discussion now.
The textbook section 1.5 gives a quick overview of some of these future developments, read it.

We are still not done with the people we've already discussed, that will be finished in chapter 2.

End chapter 1 lectures.

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