Tycho Brahe: Readings: Galileo and Newton Gravity

Tycho Brahe (1580's) was astronomy's 1st true observer. He built the Danish Observatory (using sextant's since telescopes had not been invented yet) from which he measured positions of planets and stars to the highest degree of accuracy for that time period (1st modern database). He showed that the Sun was much farther than the Moon from the Earth, using simple trigonometry of the angle between the Moon and the Sun at 1st Quarter.

Tycho's measurements were used to show that there was no detectable parallax with the naked eye, in support of the geocentric theory. So, even though his observations were the best for his time, his result was wrong, a lesson in how science is done.

Despite the lack of parallax, Tycho was doubtful of a pure geocentric model and attempted to formulate two compromise models shown below.

Neither successfully predicts the motion of the planets. The solution will be discovered by a student of Tycho's, who finally resolves the heliocentric cosmology with the use of elliptical orbits.

Kepler:

Kepler (1600's) a student of Tycho who used Brahe's database to formulate the Laws of Planetary Motion which corrects the problems of epicycles in the heliocentric theory by using ellipses instead of circles for orbits of the planets.

This is a key mathematical formulation because the reason Copernicus' heliocentric model has to use epicycles is due to the fact that he assumed perfectly circular orbits. With the use of ellipses, the heliocentric model eliminates the need for epicycles and deferents. The orbital motion of a planet is completely described by six elements: the semi-major axis, the eccentricity, the inclination, the longitude of the ascending node, the argument of the perihelion and the time of the perihelion.

The formulation of a highly accurate system of determining the motions of all the planets marks the beginning of the clockwork Universe concept, and another paradigm shift in our philosophy of science.

Galileo:

Kepler's laws are a mathematical formulation of the solar system. But, is the solar system `really' composed of elliptical orbits, or is this just a computational trick and the `real' solar system is geocentric. Of course, the answer to questions of this nature is observation.

The pioneer of astronomical observation in a modern context is Galileo. Galileo (1620's) developed laws of motion (natural versus forced motion, rest versus uniform motion). Then, with a small refracting telescope (3-inches), destroyed the the idea of a "perfect", geocentric Universe with the following 5 discoveries:

spots on the Sun

mountains and "seas" (maria) on the Moon

Milky Way is made of lots of stars

These first three are more of an aesthetic nature. Plato requires a `perfect' Universe. Spots, craters and a broken Milky Way are all features of imperfection and at odds with Plato's ideas on purely philosophical grounds. However, the laws of motion are as pure as Plato's celestial sphere, but clearly are not easy to apply in the world of friction and air currents etc. So these observations, by themselves, are not fatal to the geocentric theory. The next two are fatal and can only be explained by a heliocentric model.

Venus has phases

Jupiter has moons (Galilean moons: Io, Europa, Callisto, Ganymede)

Notice that planets with phases are possible in a geocentric model. But for a planet to change in apparent size with its phases, like Venus is impossible if the planet orbits the same distance from the Earth. And, lastly, if all bodies orbit around the Earth, then the moons of Jupiter, which clearly orbit around that planet, are definitive proof that the geocentric model is wrong.

Today our Solar System looks like this.

Aristotle's Laws of Motion:

With Galileo's observations and Kepler's laws of Planetary Motion, the heliocentric models begins to dominant scientific thinking by the 17th century, despite its theology problems. However, this is a kinematic description of cosmology, a contains only information about positions and motion. What is missing from 17th century cosmology is a dynamical description of the Universe, i.e. one that includes the cause of motion. Rationalism requires a logical progress from cause to effect. Cosmology will also require some cause for the motion of planets and stars. But first, ancient cosmologists needed to get straight the idea of motion on the Earth. The first work in this area was done, of course, by Aristotle.

In the center of the `School of Athens' by Raphael are Aristotle and Plato, Aristotle's hand level to the Earth symbolizing his realism view of Nature; Plato's hand pointed towards the heaven symbolizing the mystical nature to his view of the Universe. This image symbols the sharp change in the meaning of how `natural philosophy' or physics will be done for the 2,200 years.

Aristotle stands in the Greek philosophical tradition which asserts that nature is understandable. This tradition, opposed to the idea that nature is under the control of capricious deities which are to be appeased rather than understood, is one of the roots of science.

Aristotle constructed his view of the Universe based on a intuitive feeling of holistic harmony. Central to this philosophy was the concept of teleology or final causation. He supposed that individual objects (e.g. a falling rock) and systems (e.g. the motion of the planets) subordinate their behavior to an overall plan or destiny. This was especially apparent in living systems where the component parts function in a cooperative way to achieve a final purpose or end product.

Aristotle also provides a good example of the way in which what one knows or believes influences the way one understands new information. His theory of motion flows from his understanding of matter as constituted of four elements: air, earth, fire, and water. Objects, being solid like earth, would tend to clump together with other solids (earth), so objects tend to fall to earth, their natural place. Thus, falling is a natural motion.

 The way Aristotle believed objects to fall on the Earth The way objects actually fall to Earth

The difficulty comes in thinking about horizontal motion. Making an object move usually has a pretty obvious cause. What's difficult is explaining why something continues in motion.

 The way Aristotle thought projectiles moved The way projectiles "really" move

Think of a spear being thrown. At first, it is not in motion, but then the thrower's arm provides an impetus which accelerates it (our vocabulary, not Aristotle's). But then, what keeps it going after it leaves the thrower's hand? It should fall to earth immediately since there's nothing obvious pushing it!

Aristotle's answer was that as the spear flies through the air, it leaves a vacuum behind it. Air rushing in pushes the spear forward until its natural motion (falling) eventually brings it to earth.

Aristotle also thought about the causes which start things moving. In the spear scenario, it's easy to say that the thrower's arm moves the spear, but what moves the thrower's arm? Aristotle said that another motion moved the arm (muscle contraction?) but he also realized that some earlier motion must cause the muscle to contract and that earlier motion must also have its own initiator.

To avoid the idea that there is an infinite chain of causes, Aristotle argued that there must be an "unmoved mover," something which can initiate motion without itself being set in motion. This view was preserved by medieval Church during the Dark Ages and became the ruling paradigm.

Galileo's laws of Motion:

Aside from his numerous inventions, Galileo also laid down the first accurate laws of motion for masses. Galileo realized that all bodies accelerate at the same rate regardless of their size or mass. Everyday experience tells you differently because a feather falls slower than a cannonball. Galileo's genius lay in spotting that the differences that occur in the everyday world are in incidental complication (in this case, air friction) and are irrelevant to the real underlying properties (that is, gravity). He was able to abstract from the complexity of real-life situations the simplicity of an idealized law of gravity.

Key among his investigations are:

• developed the concept of motion in terms of velocity (speed and direction) through the use of inclined planes.
• developed the idea of force, as a cause for motion.
• determined that the natural state of an object is rest or uniform motion, i.e. objects always have a velocity, sometimes that velocity has a magnitude of zero = rest.
• objects resist change in motion, which is called inertia.
Galileo also showed that objects fall with the same speed regardless of their mass. The fact that a feather falls slowly than a steel ball is due to amount of air resistance that a feather experiences (alot) versus the steel ball (very little).

Much of this thinking dealt with objects on the Earth. Galileo didn't extend his ideas to beyond the Earth's surface, that was for an astronomer named Kepler.

Hammer and Feather on Moon

Newton:

Newton expanded on the work of Galileo to better define the relationship between energy and motion. In particular, he developed the following concepts:

• change in velocity = acceleration -> caused by force
• inertia = resistance to change in velocity and is proportional to the mass of the object
• momentum = quantity of motion energy and is equal to mass times velocity
• law of conservation of momentum = total momentum (mass x velocity) of an interaction is conserved -> is the same before and after
Example: Cars and Trucks on Ice!

A corollary to Newton's ideas was the so called Clockwork Universe model. A concept that states that the total momentum of the Universe is conserved, interactions redistribute the momentum, but the total never changes. In this model, God only starts the clock (initial cause), then it runs by itself for the rest of time.

Newton's laws of motion:

• 1st law: a body remains at rest or moves in a straight line of constant velocity as long as no external forces acts on it

• 2nd law: a body acted on by a force will accelerate such that force equals mass times acceleration (F=ma)

• 3rd law: for every action there is an equal and opposite reaction
Example: from Newton's 1st law we know that an object travels in a straight line unless acted upon by an external force. A circular orbit is clearly not a straight line, what is the force? Newton showed that the planets are acted on by the force of gravity arising from the Sun. Each orbit is a constantly changing velocity where gravity adds a small ``delta-vee'' at each moment. This ``delta-vee'' is what produces the elliptical curvature that is the orbit.

Example: from Newton's 2nd law when a baseball player hits a ball he applies a force, F, to the ball of mass m. Let's say he hits a tennis ball of mass of one-tenth the mass of a regular baseball (1/10m). What is the resulting acceleration? Ten times the acceleration of a regular baseball and, therefore, ten times the final velocity and ten times the distance hit.

Example: You are trapped on a lake of ice with a sandbag. Remembering Newton's 3rd law, how do you escape?

Newton's Law of Universal Gravitation:

Galileo was the first to notice that objects are ``pulled'' towards the center of the Earth, but Newton showed that this same force (gravity) was responsible for the orbits of the planets in the Solar System.

Objects in the Universe attract each other with a force that varies directly as the product of their masses and inversely as the square of their distances

All masses, regardless of size, attract other masses with gravity. You don't notice the force from nearby objects because their mass is so small compared to the mass of the Earth. Consider the following example: