AST101: Chapter 2

In Ptolemy's Earth-centered model for the solar system, Venus's phase is never full as viewed from Earth because it always lies between Earth and the Sun. In reality, as Galileo first recognized, Venus is :

Full whenever it is on the opposite side of the Sun from Earth, although we cannot see the full Venus because it is close to the Sun in the sky.

In what way did Newton improve Kepler's laws?

He discovered the dependence on mass in the third law.

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? Check all that apply: Neptune has orbiting moons. The Sun has sunspots and rotates on its axis. Uranus has a ring system. Venus goes through a full set of phases. Jupiter has orbiting moons. Venus is only seen in a crescent phase. The Moon has a smooth, featureless surface. The Moon has mountains, valleys, and craters.

The Sun has sunspots and rotates on its axis. Venus goes through a full set of phases. Jupiter has orbiting moons. The Moon has mountains, valleys, and craters.

It took two centuries for the Copernican model to replace the Ptolemaic model because:

There was no scientific evidence to support either model until Galileo made his observations.

A circular orbit would have an eccentricity of

0

Which of the following is a contribution to astronomy made by Galileo? Venus appears almost fully lit when it lies on the far side of the Sun. The Moon has craters, mountain, valleys, and dark flat areas on its surface. Jupiter has four moons orbiting it. The astronomical telescope can show us far more detail than the eye can. All of the above.

All of the above.

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.

The planet with the most eccentric orbit is

Mercury

A fatal flaw with Ptolemy's model is its inability to predict the observed phases of

Mercury and Venus.

When would you expect to see Venus high in the sky at midnight?

Never.

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.

See image for answer.

PART E 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. Rank 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.

See image for answer. 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.

PART C 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. Rank 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.

See image for answer. 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.

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? Check all that apply : The heliocentric model did not make noticeably better predictions than the geocentric model. The complexity of the geocentric model was appealing to most ancient astronomers. Ancient astronomers did not observe stellar parallax, which would have provided evidence in favor of the heliocentric model. From Earth, all heavenly bodies appeared to circle around a stationary Earth. The geocentric model conformed to both the philosophical and religious doctrines of the time.

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. From Earth, all heavenly bodies appeared to circle around a stationary Earth. The geocentric model conformed to both the philosophical and religious doctrines of the time.

Scientists today do not accept the Ptolemaic model because:

The work of Tycho and Kepler showed the heliocentric model was more accurate.

T OR F: A planet (or comet) will speed up as it approaches the Sun.

True

T or F: Galileo's observations of sunspots proved the Sun was rotating, like the Earth.

True

Imagine that Venus is in its full phase today. If we could see it, at what time would the full Venus be highest in the sky?

At noon.

When would a new Venus be highest in the sky?

At noon.

The Ptolemaic model probably persisted for all these reasons except: It was consistent with the doctrines of the Catholic Church. It had the authority of Aristotle behind it. It accounted well for Galileo's observations of the phase cycle of Venus. It explains why stellar parallax was not observed by the Greeks. It used perfect circles, which appealed to geometry.

It accounted well for Galileo's observations of the phase cycle of Venus.

As shown in Figure 2.12 in the textbook ("Venus Phases"), Galileo's observations of Venus demonstrated that Venus must be

Orbiting the Sun.

The place in a planet's orbit that is closest to the Sun is called

Perihelion.

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.

See image for answer.

PART C 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.)

See image for answer. 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.

Which of the statements below is part of both the Ptolemaic and Copernican models?

The Moon orbits the Earth once a month.

Which of these observations of Galileo refuted Ptolemy's epicycles?

The complete cycle of Venus' phases

Which of these was not a part of the original Copernican model? Mercury speeds up at perihelion, and slows down at aphelion. Mercury must move faster in its orbit than any other planet. Venus can go all the way around the Sun. The Sun lies at the center of the solar system. The Earth rotates on its axis once a day.

Mercury speeds up at perihelion, and slows down at aphelion.

According to Kepler's third law, the square of the planet's period in years is

Proportional to the cube of its semimajor axis in A.U.

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.

See image for answer.

PART B 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.)

See image for answer. 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.

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. Drag the appropriate items to their respective bins.

See image for answer. 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.

In Ptolemy's Earth-centered model for the solar system, Venus always stays close to the Sun in the sky and, because it always stays between Earth and the Sun, its phases range only between new and crescent. The following statements are all true and were all observed by Galileo. Which one provides evidence that Venus orbits the Sun and not Earth?

We sometimes see gibbous (nearly but not quite full) Venus.

Kepler's second law implies what about planetary motion?

A planet moves faster when it is closer to the Sun.

The heliocentric model was actually first proposed by:

Aristarchus.

Which was a contribution to astronomy made by Copernicus?

He laid out the order and relative motion of the known solar system.

Tycho Brahe's contribution to Kepler's Laws of Planetary Motion was

His detailed and accurate observations of the planet's position.

PART B 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. Rank 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.

See image for answer. 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.

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? Check all that apply: Transit of an extrasolar planet. Stellar parallax in nearby stars. Orbital periods of Jupiter's moons. Doppler shifts in stellar spectra of nearby stars.

Transit of an extrasolar planet. Stellar parallax in nearby stars. Doppler shifts in stellar spectra of nearby stars.

PART D 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.)

See image for answer. 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.

PART A 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.)

See image for answer. 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.

PART A 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. Rank 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.

All will be ranked equally. 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.

PART F 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. Rank 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.

See image for answer. 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.

PART D 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. Rank from largest to smallest. 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.

See image for answer. 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.

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. Drag the appropriate items to their respective bins.

See image for answer. 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.