CH.2

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?

1. Stellar parallax in nearby stars 2. Transit of an extrasolar planet 3. Doppler shifts in stellar spectra of nearby 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? (four points)

1. The Moon has mountains, valleys, and craters. 2. The Sun has sunspots and rotates on its axis. 3. Venus goes through a full set of phases. 4. Jupiter has orbiting moons.

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.

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

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.

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

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.

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

Geocentric model

Epicycles, perfect circles & spheres

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.

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

Heliocentric model

Galileo knew about and had accepted Copernicus's heliocentric (Sun-centered) theory. It was Galileo's observations of Venus that proved the theory. Using his telescope, Galileo found that Venus went through phases, just like our Moon

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.

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.

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

Why the Galileo controversy?

It is commonly believed that the Catholic Church persecuted Galileo for abandoning the geocentric (earth-at-the-center) view of the solar system for the heliocentric (sun-at-the-center) view. The Galileo case, for many anti-Catholics, is thought to prove that the Church abhors science, refuses to abandon outdated teachings and is not infallible. For Catholics, the episode is often an embarrassment. It shouldn't be

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.

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

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

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.

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.

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

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.)

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 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.

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

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.)

Note that the line that passes through the Sun and connects the nearest and farthest points in the orbit is called the major axis, and half this line is the semimajor axis — which is the planet's average distance from the Sun

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.

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

Kepler's First law

Planetary orbits are ellipses, Sun at one focus, not circles!

Kepler's Second law

Planets sweep out equal areas in equal times

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.)

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

Epicycles

Secondary circular orbits around a deferent-used to explain retrograde motion

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

Kepler's Third law

The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit

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.)

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

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

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.

You have correctly taken into account both the masses of the asteroids and the distances between them

According to Kepler's second law, Jupiter will be traveling most slowly around the Sun when at _________.

aphelion

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

Orbital Speeds

be found using v = SQRT(G*M/R). The R value (radius of orbit) is the earth's radius plus the height above the earth - in this case, 6.59 x 106 m. Substituting and solving yields a speed of 7780 m/s

Figure 2.21 in the textbook ("Gravity"), showing the motion of a ball near Earth's surface, depicts how gravity...

causes the ball to accelerate downward

A major flaw in Copernicus's model was that it still had...

circular orbits

The extent to which Mars' orbit differs from a perfect circle is called its ________________.

eccentricity

Earth orbits in the shape of a/an _________ around the Sun.

ellipse

Newton's 1st law

every object will remain at rest or in uniform motion in a straight line unless compelled to change its state by the action of an external force. This is normally taken as the definition of inertia. The key point here is that if there is no net force acting on an object (if all the external forces cancel each other out) then the object will maintain a constant velocity. If that velocity is zero, then the object remains at rest. If an external force is applied, the velocity will change because of the force

If the Sun and its mass were suddenly to disappear, Earth would...

fly off into space

Earth is located at one ______ of the Moon's orbit.

focus

Newton's 3rd law

for every action (force) in nature there is an equal and opposite reaction. In other words, if object A exerts a force on object B, then object B also exerts an equal force on object A. Notice that the forces are exerted on different objects. The third law can be used to explain the generation of lift by a wing and the production of thrust by a jet engine

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

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

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

If Earth's orbit around the Sun were twice as large as it is now, the orbit would take...

more than two times longer to traverse

Figure 2.26(b) in the textbook ("Orbits") shows the orbits of two stars of unequal masses. If one star has twice the mass of the other, then the more massive star...

moves more slowly than the less massive star

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

never

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

orbiting the Sun

According to Kepler's second law, Pluto will be traveling fastest around the Sun when at ______________.

perihelion

The mathematical form of Kepler's third law measures the period in years and the ______________ in astronomical units (AU).

semimajor axis

How long would a radar signal take to complete a round-trip between Earth and Mars when the two planets are 1.2 AU apart?

t = 1200 seconds

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

Gravity

the force by which a planet or other body draws objects toward its center. The force of gravity keeps all of the planets in orbit around the sun

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

the force you exert on the scale

Newton's 2nd law

the velocity of an object changes when it is subjected to an external force. The law defines a force to be equal to change in momentum (mass times velocity) per change in time. Newton also developed the calculus of mathematics, and the "changes" expressed in the second law are most accurately defined in differential forms. (Calculus can also be used to determine the velocity and location variations experienced by an object subjected to an external force.) For an object with a constant mass m, the second law states that the force F is the product of an object's mass and its acceleration a: F = m * a

What is the speed of a spacecraft moving in a circular orbit just above the lunar surface?

v = 1700 m/s

What is the escape speed from the Moon?

v = 2400 m/s