ASTRON-150: CH 1 POSTLEC

A vocabulary in context exercise in which students match words to definitions describing elliptical planetary orbits, applying ideas from Kepler's Laws of Planetary Motion.

1. Earth is located at one FOCUS of the Moon's orbit. 2. According to Kepler's second law, Jupiter will be traveling most slowly around the Sun when at APHELION. 3. Earth orbits in the shape of a/an ELLIPSE around the Sun. 4. The mathematical form of Kepler's third law measures the period in years and the SEMIMAJOR AXIS in astronomical units (AU). 5. According to Kepler's second law, Pluto will be traveling fastest around the Sun when at PERIHELION. 6. The extent to which Mars' orbit differs from a perfect circle is called its ECCENTRICITY.

Each of the four diagrams below represents the orbit of the same comet, but each one shows the comet passing through a different segment of its orbit around the Sun. During each segment, a line drawn from the Sun to the comet sweeps out a triangular-shaped, shaded area. Assume that all the shaded regions have exactly the same area. Rank the segments of the comet's orbit from left to right based on the length of time it takes the comet to move from Point 1 to Point 2.

Check Q 4 Response: Although Kepler wrote his laws specifically to describe the orbits of the planets around the Sun, they apply more generally. Kepler's second law tells us that as an object moves around its orbit, it sweeps out equal areas in equal times. Because all the areas shown here are equal, the time it takes the comet to travel each segment must also be the same.

Consider again the diagrams from Parts A and B, which are repeated here. Again, assume that all the shaded areas have exactly the same area. This time, rank the segments of the comet's orbit based on the speed with which the comet moves when traveling from Point 1 to Point 2.

Check Q 4 Response: From Parts A and B, you know that the comet takes the same time to cover each of the four segments shown, but that it travels greater distances in the segments that are closer to the Sun. Therefore, its speed must also be faster when it is closer to the Sun. In other words, the fact that that the comet sweeps out equal areas in equal times implies that its orbital speed is faster when it is nearer to the Sun and slower when it is farther away.

Consider again the diagrams from Parts D and E, which are repeated here. Again, imagine that you observed the asteroid as it traveled for one week, starting from each of the positions shown. This time, rank the positions (A-D) from left to right based on how fast the asteroid is moving at each position.

Check Q 4 Response: Just as you found for the comet in Parts A through C, the asteroid must be traveling at a higher speed during parts of its orbit in which it is closer to the Sun than during parts of its orbit in which it is farther away. You should now see the essence of Kepler's second law: Although the precise mathematical statement tells us that an object sweeps out equal areas in equal times, the key meaning lies in the idea that an object's orbital speed is faster when nearer to the Sun and slower when farther away. This idea explains why, for example, Earth moves faster in its orbit when it is near perihelion (its closest point to the Sun) in January than it does near aphelion (its farthest point from the Sun) in July.

Each of the four diagrams below represents the orbit of the same asteroid, but each one shows it in a different position along its orbit of the Sun. Imagine that you observed the asteroid as it traveled for one week, starting from each of the positions shown. Rank the positions based on the area that would be swept out by a line drawn between the Sun and the asteroid during the one-week period.

Check Q 4 Response: Kepler's second law tells us that the asteroid will sweep out equal areas in equal time intervals. Therefore, the area swept out in any one week period must always be the same, regardless of the asteroid's location in its orbit around the Sun.

Consider again the diagrams from Part A, which are repeated here. Again, assume that all the shaded areas have exactly the same area. This time, rank the segments of the comet's orbit from left to right based on the distance the comet travels when moving from Point 1 to Point 2.

Check Q 4 Response: Kepler's second law tells us that the comet sweeps out equal areas in equal times. Because the area triangle is shorter and squatter for the segments nearer to the Sun, the distance must be greater for these segments in order for all the areas to be the same.

Consider again the diagrams from Part D, which are repeated here. Again, imagine that you observed the asteroid as it traveled for one week, starting from each of the positions shown. This time, rank the positions from left to right based on the distance the asteroid will travel during a one-week period when passing through each location.

Check Q 4 Response: Notice the similarity between what you have found here and what you found for the comet in Part B. Kepler's second law tells us any object will sweep out equal areas in equal times as it orbits the Sun, which means the area triangles are shorter and squatter when the object is nearer to the Sun, so that the object covers a greater distance during any particular time period when it is closer to the Sun than when it is farther away.

Each of the following diagrams shows a planet orbiting a star. Each diagram is labeled with the planet's mass (in Earth masses) and its average orbital distance (in AU). Assume that all four stars are identical. Use Kepler's third law to rank the planets from left to right based on their orbital periods, from longest to shortest. If you think that two (or more) of the diagrams should be ranked as equal, drag one on top of the other(s) to show this equality. (Distances are to scale, but planet and star sizes are not.)

Check Q 5 Response: Kepler's third law tells us that the orbital period of the planet depends on its average distance from its star, but not on the planet's mass. As Newton later showed with his version of Kepler's third law, this is actually an approximation that works well whenver the planet's mass is small compared to the mass of the star.

The following diagrams all show the same star, but each shows a different planet orbiting the star. The diagrams are all scaled the same. (For example, you can think of the tick marks along the line that passes through the Sun and connects the nearest and farthest points in the orbit as representing distance in astronomical units (AU).) Rank the planets from left to right based on their average orbital distance from the star, from longest to shortest. (Distances are to scale, but planet and star sizes are not.)

Check Q 5 Response: Note that the line that passes through the star and connects the nearest and farthest points of the planet's orbit is called the major axis, and half this line is the semimajor axis — which we consider the planet's average distance from the star.

The following diagrams are the same as those from Part A. This time, rank the planets from left to right based on the amount of time it takes each to complete one orbit, from longest to shortest. If you think that two (or more) of the diagrams should be ranked as equal, drag one on top of the other(s) to show this equality. (Distances are to scale, but planet and star sizes are not.)

Check Q 5 Response: Recall that the time it takes a planet to complete an orbit is called its orbital period. The pattern found in this Part illutrates one of the ideas that are part of Kepler's third law: Planets with larger average orbital distances have longer orbital periods.

The following diagrams are the same as those from Parts A and B. This time, rank the planets from left to right based on their average orbital speed, from fastest to slowest. If you think that two (or more) of the diagrams should be ranked as equal, drag one on top of the other(s) to show this equality. (Distances are to scale, but planet and star sizes are not.)

Check Q 5 Response: This pattern illustrates another of the ideas that are part of Kepler's third law: Planets with larger average orbital distances have slower average speeds.

Each of the following diagrams shows a spaceship somewhere along the way between Earth and the Moon (not to scale); the midpoint of the distance is marked to make it easier to see how the locations compare. Assume the spaceship has the same mass throughout the trip (that is, it is not burning any fuel). Rank the five positions of the spaceship from left to right based on the strength of the gravitational force that Earth exerts on the spaceship, from strongest to weakest.

Check Q 6 Response: Gravity follows an inverse square law with distance, which means the force of gravity between Earth and the spaceship weakens as the spaceship gets farther from Earth.

The following diagrams are the same as those from Part A. This time, rank the five positions of the spaceship from left to right based on the strength of the gravitational force that the Moon exerts on the spaceship, from strongest to weakest.

Check Q 6 Response: Gravity follows an inverse square law with distance, which means the force of gravity between the Moon and the spaceship increases as the spaceship approaches the Moon. Now continue to Part C for activities that look at the effects of both distance and mass on gravity.

The following diagrams show five pairs of asteroids, labeled with their relative masses (M) and distances (d) between them. For example, an asteroid with M=2 has twice the mass of one with M=1 and a distance of d=2 is twice as large as a distance of d=1. Rank each pair from left to right based on the strength of the gravitational force attracting the asteroids to each other, from strongest to weakest.

Check Q 6 Response: You have correctly taken into account both the masses of the asteroids and the distances between them.

The geocentric model, in all of its complexity, survived scientific scrutiny for almost 1,400 years. However, in modern astronomy, scientists seek to explain the natural and physical world we live in as simply as possible. The complexity of Ptolemy's model was an indicator that his theory was inherently flawed. Why, then, was the geocentric model the leading theory for such a long time, even though the heliocentric model more simply explained the observed motions and brightness of the planets?

From Earth, all heavenly bodies appeared to circle around a stationary Earth. The heliocentric model did not make noticeably better predictions than the geocentric model. Ancient astronomers did not observe stellar parallax, which would have provided evidence in favor of the heliocentric model. The geocentric model conformed to both the philosophical and religious doctrines of the time. Response: The geocentric model was the leading model for hundreds of years because it conformed to the common beliefs from observations and religious doctrine of the time. However, the heliocentric model gained widespread acceptance when astronomers obtained new evidence after the time of Copernicus. These astronomers popularized his view and helped pave the way for our current understanding of the solar system.

Two competing models attempt to explain the motions and changing brightness of the planets: Ptolemy's geocentric model and Copernicus' heliocentric model. Sort the characteristics according to whether they are part of the geocentric model, the heliocentric model, or both solar system models.

Geocentric This model is Earth-centered. Retrograde motion is explained by epicycles. Heliocentric This model is Sun-centered. Retrograde motion is explained by the orbital speeds of planets. Both geocentric and heliocentric The brightness of a planet increases when the planet is closest to Earth. Epicycles and deferents help explain planetary motion. Planets move in circular orbits and with uniform motion. Response: Ptolemy's geocentric model was based on the idea that Earth is the center of the universe, while Copernicus's heliocentric model was developed around the idea that the Sun is at the center. While these two models were based on opposing ideas, they shared a common belief in uniform circular motion and the use of epicycles. However, Copernicus's heliocentric model does not use epicycles to explain retrograde motion like Ptolemy's geocentric model. In order to explain retrograde motion, Copernicus uses the different orbital speeds of the planets as an explanation to the seemingly backward motion of the planets in the sky.

Copernicus's heliocentric model and Ptolemy's geocentric model were each developed to provide a description of the solar system. Both models had advantages that made each an acceptable explanation for motions in the solar system during their time. Sort each statement according to whether it is an advantage of the heliocentric model, the geocentric model, or both.

Heliocentric Explained planetary motions and brightness changes most simply Geocentric Rooted in widely accepted religious beliefs regarding Earth's place in the universe Both geocentric and heliocentric Predicted planetary positions accurately over relatively short time periods Planetary orbits and motions based on Greek ideologies of perfect form and motion Response: The geocentric model was compelling because it adhered to religious beliefs about Earth's centrality in the universe. The heliocentric model was compelling because it provided a simpler explanation for observed motions in the solar system. Because both models adhered to the belief in perfect form and motion, they made inaccurate predictions of planetary motions over long periods of time. Since neither model made better predictions than the other, both remained viable.

Johannes Kepler used decades of Tycho Brahe's observational data to formulate an accurate description of planetary motion. Kepler spent almost 30 years of his life trying to develop a simple description of planetary motion based on a heliocentric model that fit Tycho's data. What conclusion did Kepler eventually come to that revolutionized the heliocentric model of the solar system?

Kepler determined that the planetary orbits are elliptical. Response: One of the most crucial conclusions that Kepler reached using Tycho's data was that the planets do not move in circular orbits, but rather in elliptical orbits. Kepler also concluded that the planets do not move with uniform motion. Applying these ideas to the Copernican model, the revised heliocentric model could then accurately predict planetary positions over long periods of time.

The following diagrams are the same as those from Part A. This time, rank the pairs from left to right based on the size of the acceleration the asteroid on the left would have due to the gravitational force exerted on it by the object on the right, from largest to smallest.

Strongest Force The diagram shows an asteroid and the Sun that are separated by the distance d. The diagram shows an asteroid and Earth that are separated by the distance d. The diagram shows an asteroid and the Moon that are separated by the distance d. The diagram shows two asteroids that are separated by the distance d. The diagram shows an asteroid and a hydrogen atom that are separated by the distance d. Weakest Force Response: According to Newton's second law, the asteroid with the largest acceleration will be the one that has the strongest gravitational force exerted on it by the object on the right. That is why the ranking here is the same as the ranking for Part A.

The following five diagrams show pairs of astronomical objects that are all separated by the same distance d. Assume the asteroids are all identical and relatively small, just a few kilometers across. Considering only the two objects shown in each pair, rank the strength, from strongest to weakest, of the gravitational force acting on the asteroid on the left.

Strongest Force The diagram shows an asteroid and the Sun that are separated by the distance d. The diagram shows an asteroid and Earth that are separated by the distance d. The diagram shows an asteroid and the Moon that are separated by the distance d. The diagram shows two asteroids that are separated by the distance d. The diagram shows an asteroid and a hydrogen atom that are separated by the distance d. Weakest Force Response: Because the distance is the same for all five cases, the gravitational force depends only on the product of the masses. And because the same asteroid is on the left in all five cases, the relative strength of gravitational force depends on the mass of the object on the right. Continue to Part B to explore what happens if we instead ask about the gravitational force acting on the object on the right.

The following diagrams are the same as those from Part A. Again considering only the two objects shown in each pair, this time rank the strength, from strongest to weakest, of the gravitational force acting on the object on the right.

Strongest Force The diagram shows an asteroid and the Sun that are separated by the distance d. The diagram shows an asteroid and Earth that are separated by the distance d. The diagram shows an asteroid and the Moon that are separated by the distance d. The diagram shows two asteroids that are separated by the distance d. The diagram shows an asteroid and a hydrogen atom that are separated by the distance d. Weakest Force Response: Newton's third law tells us that the gravitational force exerted on the asteroid on the left by the object on the right will be equal in magnitude, but opposite in direction to the gravitational force exerted on the object on the right by the asteroid on the left. That is why the ranking here is the same as the ranking for Part A.

As you found in Part A, your weight will be greater than normal when the elevator is moving upward with increasing speed. For what other motion would your weight also be greater than your normal weight?

The elevator moves downward while slowing in speed. Response: When the elevator is moving downward, a downward acceleration would mean an increasing downward speed. Therefore, as your answer correctly states, an upward acceleration would mean a decreasing downward speed

Astronomers have made many observations since the days of Galileo and Kepler to confirm that the Sun really is at the center of the solar system, and that the planets revolve around the Sun in elliptical orbits. Which observation(s) could you make today that Galileo and Kepler could not have made to confirm that the heliocentric model is correct?

Transit of an extrasolar planet Orbital periods of Jupiter's moons Doppler shifts in stellar spectra of nearby stars Response: When Galileo observed the changing phases of Venus, he showed that at least one planet must be orbiting the Sun. Today, astronomers are confident that all of the planets in our solar system orbit the Sun because of Earth-based observational evidence that supports a heliocentric model. Doppler shifts and stellar parallaxes show that Earth is in motion around the Sun and is thus not stationary. More recent observations of extrasolar planets show astronomers that planets in other planetary systems are orbiting stars.

Galileo Galilei was the first scientist to perform experiments in order to test his ideas. He was also the first astronomer to systematically observe the skies with a telescope. Galileo made four key observations that challenged the widely accepted philosophical beliefs on which the geocentric model was based, thus providing support for the heliocentric model. From the following list of observations, which are the key observations made by Galileo that challenged widespread philosophical beliefs about the solar system?

Venus goes through a full set of phases. The Moon has mountains, valleys, and craters. Jupiter has orbiting moons. The Sun has sunspots and rotates on its axis. Response: Galileo made four key observations that went against the geocentric model and the common beliefs about the universe at the time. Observing that the Sun and Moon had surface blemishes disproved the idea that celestial objects were perfect. Galileo's observations of Jupiter's orbiting moons showed that there were other centers of motion in the universe. Galileo's most crucial observation was the observation of Venus in different phases, which directly supported the idea that objects orbit the Sun rather than Earth.

Suppose you are in an elevator. As the elevator starts upward, its speed will increase. During this time when the elevator is moving upward with increasing speed, your weight will be

greater than your normal weight at rest Response: Increasing speed means acceleration, and when the elevator is accelerating upward you will feel a force pressing you to the floor, making your weight greater than your normal (at rest) weight.

Suppose you are in an elevator that is moving upward. As the elevator nears the floor at which you will get off, its speed slows down. During this time when the elevator is moving upward with decreasing speed, your weight will be

less than your normal weight at rest Response: Even though the elevator is still moving upward, the fact that its speed is slowing means that the acceleration is downward; the situation is rather like that of a ball that is still on its way up after you throw it, even though it is being pulled downward with the acceleration of gravity. Because the acceleration of the elevator is downward, your weight is lower than normal.

Consider Earth and the Moon. As you should now realize, the gravitational force that Earth exerts on the Moon is equal and opposite to that which the Moon exerts on Earth. Therefore, according to Newton's second law of motion

the Moon has a larger acceleration than Earth, because it has a smaller mass Response: Newton's second law of motion, F=ma, means that for a particular force F, the product mass x acceleration must always be the same. Therefore if mass is larger, acceleration must be smaller, and vice versa.

If you are standing on a scale in an elevator, what exactly does the scale measure?

the force you exert on the scale Response: You probably recognize that neither your mass nor the gravitational force exerted on you change when you are in an elevator. The scale measures the force that is exerted on it, which in an elevator is a combination of the force due to your normal weight and a force due to the elevator's acceleration.