Physics 117

Chapter 4

A brief history of astronomy

and

Gravity

Summary of the previous lecture

• Motions in space: decompose them along the three spatial dimension

• Circular motion at constant speed:

• non-zero acceleration due to change in direction

• acceleration points toward the center

• F=mv2/r=m2r

where =2/T=v/r=rotational speed T=period of revolution

• Projectile motion: typical example of decomposition

• vx=(vx)initial=const

• vy =(vy)initial-gt

• Extended object when launched rotates around their center of mass

meanwhile it translates in space

• Torque: =r·FExtra Ex. Ch.3: 13,17,25Pag. 76

Babylonian cosmology

In Mesopotamia developed the most ancient civilization (2300-1600 B.C.) and with it the most ancient cosmology.

The Earth was a flat disk surrounded by a river beyond which there were high mountains which were supporting the sky. This was made of heavy iron (siderurgy from Latin word siderus=iron sidera=sky). There was a tunnel inside the mountains linking a door in the east to one in the west. It was used by the sun in is cyclic motion.

Egyptian cosmology

For the ancient Egyptians the sky was a roof placed over the word and supported by four columns placed at the four cardinal points. The Earth was a rectangle, longer from north to south whose surface was bulgy and having at the center the river Nile (obviously!). The stars were suspended from the sky by strong cables.

An interesting creation myth is that the goddess of the nigh, Nuit, and her husband, Sibu, the earth god, where separated suddenly by the god Shu. She was drag up in the sky and he was frozen on earth. So the earth is contorted and the night sky lies above it.

Ancient Greek cosmology

Early physics

Eratosthenes276-197 B.C.

The geocentric system

All the ancient civilizations considered natural to put the Earth at the center of the universe. This vision was inherited by western Europe via the ideas of the

ancient Greeks.

Aristotle: the Earth was the center of the universe and the Sun and the other planets had to revolve around it in perfect geometrical figures, circles. Later on (circa 150 A.D.) problems with variable motion of planets induced Ptolemy to add epicyclical motion to planets.

Ptolemy

Copernicus and the heliocentric universe

Nicolaus Copernicus (1473-1543):Simplified explanation of planets’ motion via an heliocentric system and simple circular orbits of constant speed (uniform circular motion)

The crisis of the Copernican paradigm and the new astronomy of Kepler

Johannes Kepler (1571-1630) was a convinced Copernican but studying new observations on the motion of Mars he realized that they were not compatible with the Copernican system.After several years he realized that the only way to conciliate the Copernican system with observation was to abandon the idea of perfectly circular orbits.This lead him to discovers the three laws of planetary motions now called Kepler’s laws.

Kepler’s three laws

1. Planets moves along elliptical paths with the sun at one focus of the ellipse2. During a given interval of time a line drawn from the planet to the sun

sweeps out an equal area anywhere along its path3. If T is the sidereal period of a planet (the time for a complete orbital

revolution) and Rm is the mean radius of the orbit of the planet then

T2=K(Rm)3 with K equal for all the planets of the solar system

Graphical representation of Kepler’s first law Graphical representation of Kepler’s second law

Projectiles and Planets

The legend is that Newton saw an apple fall in his garden in Lincolnshire, thought of it in terms of an attractive gravitational force towards the earth, and realized the same force might extend as far as the moon. He was familiar with Galileo's work on projectiles, and suggested that the moon's motion in orbit could be understood as a natural extension of that theory.

To see what is meant by this, consider a gun shooting a projectile horizontally from a very high mountain, and imagine using more and more powder in successive shots to drive the projectile faster and faster. The parabolic paths would become flatter and flatter, and, if we imagine that the mountain is so high that air resistance can be ignored, and the gun is sufficiently powerful, eventually the point of landing is so far away that we must consider the curvature of the earth in finding where it lands.

In fact, the real situation is more dramatic---the earth's curvature may mean the projectile never lands at all. This was envisioned by Newton in the

Principia.

Projectiles and Planets: a never ending fall

The following diagram is from his later popularization, A Treatise of the System of the World, written in the 1680's:

The mountaintop at V is supposed to be above the earth's atmosphere, and for a suitable initial speed, the projectile orbits the earth in a circular path. In fact, the earth's curvature is such that the surface falls away below a truly flat horizontal line by about five meters in 8,000 meters (five miles). Recall that five meters is just the vertical distance an initially horizontally moving projectile will fall in the first second of motion. But this implies that if the (horizontal) muzzle velocity were 8,000 meters per second, the downward fall of the cannonball would be just matched by the earth's surface falling away, and it would never hit the ground!

[This is just the motion, familiar to us now, of a satellite in a low orbit, which travels at about 8,000 meters (five miles) a second, or 18,000 miles per hour. ]

Newton realized that the moon's circular path around the earth could be caused in this way by the same gravitational force that would hold such a cannonball in low orbit, in

other words, the same force that causes bodies to fall.

Tides

Mt. St. Michel on the north coast of France at low tide (up) and high tide (down).

The water surrounding this island is the Gulf of Saint-Malo.

The tides vary along the shores of the oceans of the earth with a period of about 12.5 hours.  Curiously, the circadian rhythm period of 25 hours, corresponding to two tidal periods, subconsciously affects a large variety of animal life, including humans. 

Tides are due to the gravitational attraction of moon and to a lesser extent, the sun on the Earth. Because the moon is closer to the Earth than the sun, it has a larger effect and causes the Earth to bulge toward the moon. At the same time, a bulge occurs on the opposite side of the Earth due to inertial forces

High and low tides

These bulges remain stationary while Earth rotates.  The tidal bulges result in a rhythmic rise and fall of ocean surface, which is not noticeable to someone on a boat at sea, but is magnified along the coasts.

Usually there are two high tides and two low tides each day, and thus a variation in sea level as the tidal bulge passes through each point on the Earth's surface.

Because the Sun also exerts a gravitational attraction on the Earth, there are also monthly tidal cycles that are controlled by the relative position of the sun and moon.

The highest high tides occur when the Sun and the moon are on the same side of the Earth (new moon) or on opposite sides of the Earth (full moon). The lowest high tides occur when the Sun and the moon are not opposed relative to the Earth (quarter moons).

Gravitational locking

The tidal bulges are not aligned with the Earth-Moon direction, due to the rotation of the Earth they precede the moon so the high tide do not occur when the moon is overhead but later (as much as 6 hours later)

The bulge nearest the Moon is actually a bit ahead of the Earth-Moon line. That bulge has mass; not a lot, but some. Since it has mass, it has gravity, and that pulls on the Moon. It pulls the Moon forward in its orbit a bit. This gives the Moon more orbital energy.

An orbit with higher energy has a larger radius, and so as the bulge pulls the Moon forward, the Moon gets farther away from the Earth. This has been measured and is something like a few centimeters a year.

Of course, the Moon is pulling on the bulge as well (Newton’s third law!). Since the Moon is "behind" the bulge (relative to the rotation of the Earth), it is pulling the bulge backwards, slowing it down. Because of friction with the rest of the Earth, this slowing of the bulge is actually slowing the rotation of the Earth! This is making the day get longer.

Gravitational locking

Does this sound familiar?

It should. Since Earth's gravity is much stronger than the Moon's, the tides from the Earth on the Moon are much stronger than the Moon's tides on the Earth. The Moon has tidal bulges just like the Earth, and so it too was slowed by the Earth's pull on its nearer bulge. Eventually, the Moon's rotation was locked so that it took the same time to spin once on its axis as it takes to go around the Earth. This is why we always see the same face of the Moon! And this happened to the Moon before the Earth because the Earth's tides are so much stronger.

Eventually, the Earth's rotation will slow down so much that the bulge will line up exactly between the centers of the Earth and the Moon. When this happens, the Moon will no longer be pulling the bulge back, and the Earth's spin will stop slowing. But when this happens, the time it takes for the Earth to rotate once will be slowed to exactly the same time it takes for the Moon to go around the Earth once! If you were to stand on the Moon and look at the Earth, you would always see the same face of the Earth.