Tuesday September 2
The Earth is littered with prehistoric observatories. Stonehenge is the most famous of these. But there are others sites left by cultures all over the world:
All human cultures have some Astronomical tradition. This is partly because cultures tend to associate religious stories with astronomical patterns, objects, and events. But it is also because astronomy is of great practical utility for both timekeeping and for navigation in pre-technical societies.
Most of our astronomical tradition comes down from the Greeks. In fact, the Greeks were the inheritors of the astronomical traditions and learning of the Egyptians and Babylonians.
The thing that distinguished the Greeks from earlier cultures is that the Greeks did not use astronomy as a means of explaining their religious beliefs. Instead, they viewed the Universe as something to explain rationally.
This view was first explicitly stated by Thales of Miletus around 600 BC.
The notion that one could explain the Universe mathematically was introduced by Pythagoras around 500 BC. He also drew together mathematics with both geometry and music.
One of the Greek scholars that you are likely to have heard of and read about in other contexts is Plato. Plato's work continues to influence western aethetics to this day, but his scientific work laid the foundations for about 2000 years of errors and blind alleys. Plato asserted that observations were inherently fraught with error (he was correct about this), and that therefore pure intellect (thought without observation) would give a truer picture of the Universe than observation (he was VERY wrong about this!).
Plato is also the first to assert that heavenly bodies must move in perfect circular paths at a constant speed. This is a reflection of the Greek notion that the heavens were the realm of perfection, while the Earth was the realm of imperfection.
This notion of the "perfection of the Heavens" was given its fullest expression in the works of Aristotle around 350 BC. Aristotle also gave the clearest expression of the Greek notion that the Universe was Geocentric (Earth-centered), and SMALL. If the whole Universe is required to spin around the Earth once a day, it MUST be small. That was the Greek line of thinking, anyway.
Thursday September 4
The Geocentric picture was not universally held by all the Greek philosophers. The best known proponent of a Heliocentric (Sun-centered) Universe was Aristarchus of Samos (around 250 BC). His model was generally rejected at the time, both on aesthetic grounds, and because no one was able to observe the stellar parallax that should result if the Earth orbits the Sun.
Trigonometric parallax is how your brain generates depth perception. Hold your arm out straight in front of you, and stick up you thumb. Now close one eye, and notice what part of the wall you thumb is covering. Now, without moving your hand or your head, look at your thumb with your other eye. Notice that your thumb is now covering up a different part of the wall. That's parallax.
If the Earth orbits the Sun, then one should be able to observe parallaxes for nearby stars. None was observed. Notice that this could be because the stars are all VERY far away. This would make the parallax too small to measure. But the Greeks rejected the notion that the Universe could possibly be so vast. In fact, the stars are all very far away, and it wasn't until the 19th century that observing techniques and technology were good enough to measure the parallax of the nearest stars.
One of the real triumphs of Greek observational astronomy is the work of Eratosthenes (around 200 BC). Eratosthenes was told that the Sun cast no shadows in Syene (now Aswan, in Egypt) at noon on the date of the summer solstice. Eratosthenes lived in Alexandria, at the mouth of the Nile. He could measure the length of shadows cast there at noon on the date of the summer solstice. This allowed him to find the angular distance between Syene and Alexandria. Then, estimating the linear distance between the two cities, he estimated the size of the Earth. His calculation agrees with modern measurements to within 10% or so.
The large bodies (and many of the small bodies) of the Solar System all orbit in the same plane (the ecliptic plane), and in the same direction. This is a crucial piece of information for understanding the origin of the Solar System, and of planetary systems in general.
Five of the other planets in the Solar System are visible to the naked eye:
Because Venus and Mercury are closer to the Sun than the Earth, they always appear near the Sun in the sky Mercury is always within about 20 degrees of the Sun, and is thus very difficult to see. Venus can be up to about 45 degrees away from the Sun, and is often a very bright object in the early evening or early morning sky.
Mars, Jupiter, and Saturn are more distant from the Sun than is the Earth. As a result, they can be up in the sky at any time of day or night.
The motions of the planets through the sky are VERY complicated. It turned out to be extremely difficult to construct a system based on circular motion at a constant speed ("perfect circular motion") that actually fit the observations the Greeks had available. This led to the adoption of a set of cumbersome contrivances.
Hipparchus, who also came up with the magnitude system for measuring stellar brightness, contributed the first of these contrivences to the Greek model of the Universe. As noted above, the Greeks postulated that the planets must move around the Earth in circular orbits, and at constant speeds ("the perfection of the heavens"). The problem with this is that the planets do not co-operate. They move through the sky at varying speeds, and sometimes even turn around ("retrograde loops"). Hipparchus suggested a way of fudging things: Don't have the Earth at the center of the orbit. Such an orbit is called an "eccentric",
The person who really took this idea and ran with it was Claudius Ptolemy (around 140 AD). He constructed a Geocentric model that was actually quite accurate, given the state of observations at the time. His model had an off-center Earth (Hipparchus's eccentric), and also included epicycles. In Ptolemy's model, the planets did not orbit the Earth directly. Instead, their epicycles orbited the Earth, and the planets moved along the epicycles. Thus the motion of the planets was along circular paths, but the various circles fit together to make a planetary orbit look like a spirograph drawing.
Ptolemy's epicyclic model of the Solar System was accurate to within the observational uncertainties when he constructed it (around AD 150). But over the centuries it slowly diverged from the observations. This is similar to the case of a watch that gains 1 second per day. One does not notice the problem initially, but over the course of a year, one winds up with a watch that is about 6 minutes fast.
Rather than abandon the basic model, astronomers in the middle ages simply added more elaborate epicycles to the existing model in order to bring it back into agreement with observations.
This persisted for about 1500 years. Then, within a century, the old model was abandoned for a new one. And the basic methodology of modern science was created.
The story begins with the work of Nicolaus Copernicus around 1500 AD. Copernicus was the first modern astronomer to propose a heliocentric model for the Universe. This was a problem for him because the geocentric model of Ptolemy had been adopted as doctrine by the Catholic Church. Not only was the Church VERY powerful, Copernicus was a church canon. So he was scared. He was a very cautious man. So much so that he didn't officially publish his book on the heliocentric model until he was on his deathbed.
One huge appeal of Copernicus's model is that the retrograde loops of the outer planets are natural in a heliocentric system. Furthermore, although one can dispense with epicycles in a heliocentric model, Copernicus retained them. This is because he adopted the Platonic notion of perfect circular motion as the natural state of heavenly bodies. As a result, although his model was somewhat easier to visualize than the Ptolemaic model, it wasn't a lot more accurate. His tables of planetary position could still be off by up to two degrees. This was substantially larger than the observational uncertainties. Copernicus was, in some sense, the first modern astronomer. The first modern physicist was Galileo Galilei (born in 1564). Galileo did NOT invent the telescope. It was invented by a dutch lensmaker in 1608. Galileo was the first person to apply the telescope to astronomy, beginning in 1609.
Galileo made a number of telescopic observations that challenged the physical and philosophical underpinnings of the Ptolemaic/Catholic cosmology:
The observation that the Milky Way is actually composed of many stars too faint to resolve with the naked eye is a philosophical point: Why should God create all these stars that are too faint to see with the eye?
The imperfections of the Moon and Sun undermine the philosophical notion that the Heavens must be the realm of perfection.
The fact that Jupiter has moons means that not everything revolves around the Earth. So the Earth is clearly NOT the center of all rotation.
The fact that Venus exhibits the full range of phases from crescent through full was particulary compelling. There is no way for this to be so in a geocentric Universe. Only if Venus orbits the Sun can it show a full range of phases.
Now, in addition to be brilliant, Galileo was also abrasive and obnoxious. This led the Church to tell him to stop writing on astronomy by about 1616 (seven years after he began telescopic observations).
In 1623 the pope died, and was replaced by a new pope. The new pope was more sympathetic to Galileo's work, and allowed him to continue it. Galileo went right back to being abrasive and obnoxious. He published "Il Dialogo sopra i due massimi sistemi del mondo" ("The Dialog") in February 1632. In this work, he defended Copernicanism, and actively made fun of the Pope. The Church banned The Dialog in August 1632, tried Galileo, and place him under house arrest for the last decade of his life.
Galileo was not so much an astronomer as he was the first true mathematical physicist. He made an enormous contribution to astronomy by his application of a new tool (the telescope). But the person who collected the crucial dataset for the overthrow of the Ptolemaic model was a classical observational astronomer. A danish nobleman named Tycho Brahe.
Tycho began his astronomical career while still a University student. He observed a conjuction of Jupiter and Saturn, and noticed that the existing astronomical tables had the timing and the approach of the conjunction very wrong. In typical fashion, his reaction was "I can do better than THAT!". And so he did. Over the course of his life, he accumulated the largest and most accurate set of observations of planetary positions that had ever been compiled.
He also did crucial observational work on the so-called Nova Stella of 1572. In modern parlance, this was a supernova, and exploding star. It is now known as Tycho's supernova. But it isn't as if Tycho "discovered" it. It was bright enough to see in the daytime. The assumption was that, as it was clearly a transitory phenomenon, it could NOT be part of the heavens (the heavens were still assumed to be perfect and unchanging), but must instead be an atmospheric event.
Tycho reasoned that if it were in the atmosphere, it should have a measurable parallax. The fact that it did not convinced him that it must be in the heavens, changeable or not.
Although Tycho was a brilliant observer, he was not a brilliant mathematician or a brilliant theorist. Thus he collected a tremendous dataset, but did not have the skills to interpret it. This task fell to his assistant and successor, Johannes Kepler. Kepler inherited Tycho's observing notebooks when Tycho died. Given this, and about 20 years of diligent effort, he was able to solve the problem of planetary orbits. His work is captured in what are now called KEPLER'S LAWS
Tuesday September 9
Kepler discovered how the planets orbit the Sun. But not why. Did they orbit the Sun because of magnetism? Magic? Wind from the beating of angels wings? He didn't know.
Galileo worked on a lot of things other than just astronomy. One of these things was gravity. (You may know of the famous and possibly apocryphal story of Galileo dropping two cannonballs of the leaning tower of Pisa.)
In Galileo's time, mechanics was still based on the philosophy of Aristotle. And, while Aristotelian physics has a certain aesthetic appeal, it doesn't make any sense when confronted with the actual world. Rocks keep moving through the air after you throw them, even though you are no longer exerting any force on them.
Galileo was the first modern physicist in that he was the first person to conduct his own experiments. This was truely revolutionary.
Among the things he discovered was that objects do not fall at a constant rate. Instead, they accelerate as they fall to the ground.
He also discovered (and this is where the Pisa story comes in) that the rate at which an object fall DOES NOT DEPEND ON ITS MASS!!!!!. Heavy objects fall no more quickly than light objects.
One other crucial discovery Galileo made was that moving objects continue to move unless some force acts to stop them.
This is as far as Galileo got with the subject. It occupied those last ten years of his life, while he was under house arrest. The matter was taken up, and put in its modern form by Isaac Newton. Newton first examined inertial or unaccelerated motion. This led him to what are now called Newton's Laws of Motion:
And the second law can be written algebraically as
From here, Newton went on to consider the behavior of falling objects. As falling objects accelerate downward, they MUST be acted on by a force. But, as noted by Galileo, the rate of acceleration does not depend on the mass of the falling object. This is inconsistent with Newton's second law UNLESS the force of gravity depends on the mass of the falling object.
Newton realized that his third law of motion (the reaction law) meant that any two massive bodies must exert equal gravitational forces on each other. So the force of gravity between any two objects has to depend on the masses of both objects.
Newton reasoned that the force of Gravity should follow an Inverse- Square Law. This is just a reflection of the geometry of a sphere. The surface area of a sphere increases as the square of the radius. Take the example of a light source of some fixed luminosity. If you get twice as far away from it, it appears only a quarter as bright. Newton reasoned that gravity should behave in this way. This led him to the following form for the Law of Gravity:
The m1, M2 are the two masses involved, the R1,2 is the seperation between the masses. The G is a constant (the Gravitational Constant).
Newton knew this law was consistent with his Laws of Motion, and with the observed behavior of falling objects near the Earth's surface (local gravity). He then asked himself the question: "Is this force on the Moon the same as the force of gravity we feel on the surface of the Earth?"
Now, the Moon is in a roughly circular orbit around the Earth. Circular motion also requires a force. This is because any change in velocity requires an acceleration. And velocity isn't the same as speed. Two objects that are moving at the same speed (say two cars on the highway, both moving 50 mph), but in opposite directions (one car moving north, and the other south) do NOT have the same velocity (in this example the velocities differ by 100 mph). To change the direction that an object is moving in requires a force, even if the object's SPEED remains constant. Among the consequences of Newton's Laws of Motion are rules that govern objects in circular motion (and you'll need to take a physics class if you want to see the derivation of this). So he knew that any object in circular motion must have an acceleration that depends on its speed and the radius of the circle involved:
Newton knew the distance to the Moon (this was known from parallax measurements), and he knew the moon's orbital speed (from its distance and its orbital period). So he knew the acceleration the Moon HAS to be experiencing to be in the orbit it is in:
Now we return to his question: "Is the force that keeps the moon in a constantly accelerating (nearly circular) orbit around the Earth the same as the force of gravity we feel on the surface of the Earth?"
Newton knew the size of the Earth, and he knew that the distance between the Earth and the Moon was about 60 Earth radii. Given this, he calculated
Or about 1/3600 of the acceleration due to gravity at the Earth's surface. The acceleration of gravity on the Earth's surface is about 9.8 m/sec/sec. This means the acceleration on the Moon due to the Earths gravity should be about 2.7x10^-3 m/sec/sec. This is exactly the acceleration that the Moon must be experiencing in order to stay in a circular orbit around the Earth.
I can't stress enough how important this was. Kepler had figured out the motions of the planets, but not why they move as they do. Newton figured out why. His law of gravity is one of the most important tools we have for understanding the behavior of objects in the Universe.
Thursday September 11
This is worth playing around with some, as we will be using this relationship for the rest of the semester. First of all, it is consistent with the 3rd law of motion (action-reaction). Notice that this means you exert as strong a force on the Earth as the Earth does on you. But the Earth is much bigger than you.
But consider the result of two bodies with masses that are fairly close to the same. Or, by analogy, consider the problem of riding a see-saw with someone who is only 3/4 of your weight. How does that work?
The two masses orbit a common center of mass.
One can derive a few more useful relations from the gravitational force law. The first is an equation for a circular orbit:
The second is an equation for the escape velocity of an object:
Don't get caught up the algebra. Just think about this for a bit. If there are two satellites in circular orbits around the Earth, the one in a higher orbit will move more slowly than the one in the lower orbit (because the radius is larger for the one in the higher orbit). And if you have two satellites in circular orbits, at the same altitudes, but one is orbiting the Earth, and the other is orbiting the Moon, the one orbiting the Earth will move more rapidly than the one orbiting the Moon (because the Earth is more massive than the Moon).
Now, recall Kepler's first law. The planets orbit the Sun in ellipses. Not circles. Why? It isn't because circular orbits are not allowed. It's because a circular orbit is a special case. For any circular orbit, there are an infinite number of elliptical orbits with the same starting point. So if you pick one, at random, it will be elliptical.
Now I want to go back to a point I made when I was discussing the observed properties of the Moon: We always see the same face of the moon. This means that its period of rotation has to be the same as its orbital period around the Earth (a sidereal month). I noted that this behavior is an example of Tidal Phase Locking. Now it's time to look into the matter.
The Moon orbits the Earth due to gravity. This causes tides. The essential point about gravity is that it obeys an inverse-square law:
This means that the oceans on the side of the Earth closest to the Moon feel a larger force of gravity due to the Moon than does the center of the Earth. And the center of the Earth feels a larger force than does the ocean on the far side of the Earth from the Moon. The result of this is that two tidal bulges are raised on the Earth. One on the side facing the Moon, and one on the side facing away from the Moon.
The Sun also contributes to the tides raised on the surface of the Earth. But the Sun is much further away than the Moon is, so it has a relatively minor effect on the Earth's tides. Still the strongest tides (Spring Tides) are those that occur near New and Full Moon, when the Moon and Sun are aligned with the Earth. And the weakest tides (Neap Tides) are those that occur near First and Third Quarter Moon, when the Moon is perpendicular to the Earth--Sun line.
It's time to talk about the "phase locking" part of Tidal Phase Locking. Tides also occur in solid rock, but they are much less dramatic than those occuring in oceans. The solid body tides on the Earth only produce tidal bulges with amplitudes of about 1cm. Still, this has a dramatic long term effect. The Earth spins much more rapidly than the orbital period of the moon (one day versus 27.3 days), and thus the actual tidal bulge on the Earth is always ahead of the Earth - Moon line. This causes a tidal torque: The Moon is dragging the Earth, slowing its rotation rate down. This is born out by the fossil record: The day was only 22 hours long 400 Myr ago.
Eventually, the tidal torque from the Moon will cause the Earth's rotation rate to slow down to the orbital period of the moon. The Earth is much more massive than the Moon, and gravity is a symmetric force, so the Earth is much more effective at slowing down the Moon than the Moon is at slowing down the Earth. In fact, the Earth has already slowed down the rotation period of the Moon to match its orbital period. This is why we always see the same face of the moon. Tidal Phase Locking is a common phenomenon, both for moons around planets, and for stars in binary star systems.
