6/16 Eclipses: We don t have eclipses every month because the plane of the Moon s orbit about the Earth is different from the plane the ecliptic, the Earth s orbital plane about the Sun. The planes of the two orbits are tilted about 5 degrees. Most of the time during full and new phases, the Moon lies above or below the Sun in the sky. For an eclipse to occur, two conditions must be met: 1. The phase of the Moon must be either full (gives a lunar eclipse) or new (gives a solar eclipse). 2. The Moon must be in or near the plane of the ecliptic. Terminology: If two planes are not parallel, they will intersect in a line. The line of intersection between the ecliptic plane and the plane of the Moon s orbit about the Earth is called the line of nodes. The nodes are the two points where the Moon s orbit intersects the ecliptic plane. For an eclipse to occur, the Moon must be in a new or full phase and located at or near a node. Lunar Eclipse: 1. Lines are drawn from the top and bottom of the Sun to the top and bottom of the Earth. These lines show limits where light from the Sun can reach various locations behind the Earth. 2. Outside the shadows, the Moon is completely in sunlight. In the penumbral shadow, sunlight is partly blocked. In the umbral shadow, sunlight is completely blocked. 3. If, in a given eclipse, the Moon does not make it into the umbral shadow, the eclipse will be partial. The eclipse will be total if the Moon makes it into the umbral shadow. Notes: 1. The Moon does not disappear during a total eclipse. Red light is scattered in the Earth s atmosphere into the shadow of the Earth and this red light illuminates the Moon, giving it a blood red appearance.
2. More solar eclipses occur in a given amount of time than lunar eclipses but everyone on the dark side of the Earth can see a lunar eclipse, while one must be in specific locations to see a total solar eclipse. More people have seen lunar eclipses than have seen solar eclipses. Solar Eclipse: 1. Note that there are umbral and penumbral shadows produced by the Moon, but they are much smaller than those produced by the Earth because the Moon is smaller than the Earth. 2. Not everyone on the day side of the Earth can see the eclipse. Most of those who can will only see a partial eclipse because they will be in the penumbral shadow of the Moon. Note that the umbral shadow on the surface of the Earth is tiny. 3. If the Moon is at a point in its orbit where it is farther from the Earth, it will not completely block the Sun at anytime during the eclipse. The eclipse will be an annular eclipse, one where the limn of the Sun is visible around the edge of the Moon. Predicting Eclipses: 1. Modern way use celestial mechanics: Newton s laws of motion and his law of gravity. 2. Ancient peoples didn t have celestial mechanics how did they predict eclipses? 3. They did have writing and they kept records of eclipses. 4. The found that if a certain eclipse occurred on a given day, the same eclipse would occur 18 years 11 1/3 days later. This is called the Saros cycle. Chapter 2 Gravitation and the Motion of the Planets Historical Perspective Recall the CS. The CS rotated about the Earth every 24 hours, carrying the stars with it from east toward west. The Greeks noted that seven objects moved or wandered on the CS called planets after the Greek word for wanderer. Wander is a poor choice of words these objects followed along definite paths with regularity. There were seven ancient planets: In order of their distance from the Earth Moon, Mercury, Venus, Sun, Mars, Jupiter, and Saturn.
All these ancient planets exhibited an apparent motion against the background of the fixed stars on the CS from west toward east. What this means is the following: Observe a planet when it is west of a particular star in the sky. Some time later a few days, weeks, or months depending on the planet it will be to the east of the star. In the figure at right, the left side shows the orientation of planet and star at one time and the right side shows the orientation at a later time. Note that both planet and star rise in the east and set in the west on any given day. This motion is called prograde motion. But all the ancient planets except for the Sun and Moon occasionally exhibit motion in the opposite direction from east toward west. This is called retrograde motion. (Aside: Modern Meanings of Prograde and Retrograde If we were to view the solar system from above the Earth s north pole, we would see that almost all objects orbit in a counterclockwise direction and spin in the that same direction. This is called prograde motion. Some objects orbit about their primary in a clockwise direction (some of the outer moons of Jupiter) and some spin about their axis of rotation in a clockwise direction (Venus). This is called retrograde motion.) First Model of the Universe Ptolemaic Earth-centered or geocentric model. A few other ancient ideas: 1. The Earth was imperfect, but the heavens were perfect. We can see imperfections on the Moon with the naked eye it is slightly imperfect due to its proximity to the Earth. 2. The Greeks believed the circle to be the perfect shape so felt that the planets should move on circles. Ptolemy first century AD put planets on circles. He put planets on circles he called epicycles. The epicycles in turn orbited about other circles called deferents. The Earth is located near but not at the center of the deferent. The purpose of the epicycle was to account for retrograde motion the Sun and Moon had no
epicycles because they showed no retrograde motion. In the model, the speed of the planet on the epicycle is greater than the speed of the epicycle on the deferent. When the planet is farthest from the Earth, the two velocities add and we have prograde motion. When the planet is closest to the Earth, the velocities subtract. Since the velocity of the planet on the epicycle is greater than the velocity of the epicycle on the deferent, the planet now appears to move in the opposite direction retrograde motion. The model worked fairly well in making predictions of planetary positions. Ptolemy had to fit 7 planets in his system but he had 7 parameters: velocity of the planet on the epicycle, the velocity of the epicycle on the deferent, radius of the epicycle, radius of the deferent, how far the Earth was off center from the deferent, how far off the deferent the center of the epicycle is located, and the equant (the point in space where the center of a planet s epicycle appears to move with constant speed). It is always possible to use a model with seven parameters to fit a system with seven things to fit. Ptolemy s model did a pretty good job at predicting planetary positions. But, over time, it fell out of synchronization with the actual planetary positions, and the parameters would have to be remeasured. Predictions of Ptolemy s Geocentric Model: 1. We observe that Mercury and Venus never get far from the Sun. 2. This means that the epicycles of these two planets must orbit about the Earth at the same rate as the Sun. 3. This means that Mercury and Venus are always closer to the Earth than the Sun. 4. This means that Mercury and Venus can only show crescent phases. Copernicus The geocentric model lasted for about 1500 years, when Copernicus (a Polish monk) pointed out that putting the Sun at the center of the solar system made things easier. Copernicus developed the sun-centered or heliocentric model of the solar system. In this model, only the Moon was left in orbit about the Earth. The Earth was placed in orbit about the Sun. The planets then were in order of their
distance from the Sun were: Mercury, Venus, Earth, Mars, Jupiter, and Saturn. This model has a very natural way of explaining retrograde motion in this model, the closer a planet is to the Sun, the faster it moves. Thus, the retrograde motion of the Mars is due to the Earth catching and passing Mars in its orbit. It explains in a natural way why Venus and Mercury don t get far from the Sun. Note also that, in this model, Venus can be farther from the Earth than the Sun and will show gibbous and full phases. In this model, calculations were simple; it was easier to calculate the locations of the planets. Galileo His Observations: 1. He looked at the Moon saw the craters, saw mountains, and large planes that he interpreted as seas. He saw that the Moon was even more imperfect than the ancients had thought. 2. He looked at the Sun and saw sunspots the Sun was also imperfect. 3. He observed the Milky Way and found it to be composed of thousands and thousands of stars. 4. He looked at Saturn and saw the rings more imperfection. 5. He looked at Jupiter and saw four little pinpoints of light moving back and forth across Jupiter from night to night. He correctly concluded that these were moons in orbit about Jupiter The Galilean moons of Jupiter: Io, Callisto, Ganymede, and Europa. Proof that there were objects in the solar system that did not orbit the Earth. One of the arguments against the heliocentric model was that the moving Earth would leave the Moon behind. The moons of Jupiter were not left behind as Jupiter moved through the sky. 6. He looked at Venus and saw Venus go through a complete cycle of phases the Ptolemaic model was wrong! Note that this result does not mean that all geocentric models are wrong. Tycho Brahe was the greatest observational astronomer of all time who never used a telescope. He could not detect the parallax of the stars he felt that some sort of geocentric model was necessary. Tycho put the Earth at the center and at rest. He put the Sun and Moon in orbit about the Earth. He put the remaining planets in orbit about the Sun. Kepler s Three Laws of Planetary Motion:
1. The orbit of a planet is an ellipse with the Sun at one focus. About ellipses: 1. How to draw an ellipse a. You need a piece of paper, a pencil, a length of string, and two thumbtacks. b. Use the thumbtacks to secure the ends of the length of string with the string loose. c. Push the pencil against the string, making it taut. d. The ellipse is produced by keeping the string taut as you move the pencil over the paper. e. Each thumbtack is a focus of the ellipse. f. Since the length of the string constant, the total distance from a focus to the ellipse and back to the other focus is constant. g. If the two foci are brought on top of one another, the ellipse becomes a circle. h. In the figure, a is the semimajor axis of the ellipse; b is the semiminor axis of the ellipse. 2. An imaginary line from the Sun to a planet sweeps out equal areas in equal times. Means that the speed of planet is greater when its closer to the Sun. Example of conservation of angular momentum applied to planetary motion. The angular momentum of a spinning body depends on three things: the rate spin, the mass of the object, and how the mass is distributed. For a given rate of spin, the angular momentum is larger if the mass is far from the axis of rotation and small if it is near the axis of rotation. Increasing the spin increases the angular momentum. Under many conditions, angular momentum is constant. Think of an ice-skater spinning slowly with extremities extended. Bringing extremities close to the body would decrease the angular momentum if the spin didn t change. To keep angular
momentum constant, the spin must increase to compensate. 3. The square of the period of a planet is proportional to the cube of the semimajor axis of its orbit. Notes: A. The period of a planet is the time for the planet to complete one orbit about the Sun. B. The semimajor axis of a planet s orbit is the average distance of the planet from the Sun. C. For a circular orbit, the semimajor axis becomes the radius of the orbit. If we agree to measure the period in years and if we agree to measure the semimajor axis of a planet s orbit in AU (astronomical units), the Kepler s third law as written by Kepler can be written: Example: Given that Jupiter s distance from the Sun is roughly 5 AU, find its period. Newton s Contributions: Newton s Three Laws of Motion: 1. In the absence of external influences, a body at rest will stay at rest and a body moving with constant speed in a straight line will continue to do so. 2. If a body suffers an external influence, we say it has a force applied to it. The total force on a body will cause its motion to change; it will accelerate. The total force on a body is proportional to the acceleration it causes with the constant of proportionality being defined to be the mass of the body: 3. It f two bodies A and B interact, the force that A applies to B is equal in size but opposite in direction to the force that B applies to A. Discussion of the first law. The first law appears to be contained within the second and it is. Why state it?
Newton s laws are not always valid they are not valid in, say, an accelerating automobile. We use Newton s first law to tell if we are in a proper frame of reference. Note it they ancients had known Newton s first law, the retrograde motion of the planets would have told them that the Earth was moving. Discussion of the second law. It does four things: 1. It operationally defines force; that is, it gives a method that, in principle, force can be measured. 2. It operationally defines mass. Apply a known force to a body, measure its acceleration, and the ratio of force to acceleration gives the mass. 3. Given a desired path for a body to follow, it allows us to calculate the forces needed to produce that path. 4. Given the forces on a body, it allows us to calculate the path of the body. Note: If we are in an accelerated frame of reference in which Newton s laws are not valid, we can still use Newton s second law with the introduction of fictitious forces artifacts of the motion of the reference frame. The centrifugal force is a fictitious force in a rotating frame of reference. Discussion of the third law. Usually hear it stated: To every action, there is an equal and opposite reaction. Note that if the action is the force of A on B, then the reaction is the force of B on A. The third law holds even in accelerated frames of reference. Unit of force: US Customary Unit pound or lb The SI unit of force is the newton = N. If we use Newton s second law F = ma. 2 2 1 N = (1kg)(1m/s ) = 1 kg m/s Note that 1 N is roughly 1/5 of a lb. Newton s Law of Gravity: The gravitational force between two spherical bodies is proportional to the product of their masses and inversely proportional to the square of the distance between their centers:
11 2 2 where G = 6.67 10 N m /kg. We can use Newton s Law of Gravity to explain the tides: The tides are caused by the gravity of the Moon and the Sun concentrate on the Moon. Without the Moon or Sun, the water would be evenly distributed around the Earth (we are ignoring the presence of land masses). Because the Moon s gravity gets weaker with distance, it pulls harder on the water on the near side of the Earth than it does on the Earth pulls water away from the Earth forming a tidal bulge.