Question 1

If the size of the Earth's orbit were doubled,its period of revolution about the Sun would

Accepted Answer

According to Kepler's Third Law, the square of a planet's period of revolution is proportional to the cube of its semi-major axis (orbital radius). Doubling the size of Earth's orbit would mean doubling its semi-major axis, and therefore the period of revolution would increase by a factor of ∛(2^3) ≈ 1.59, or about 3 times.

Question 2

Bodes's law "predicts" that there is a planet with an orbit having a semimajor axis of 77 AU.What would be the orbital period of the "predicted" planet?

Accepted Answer

Using Kepler's third law, $P^2 = a^3$, where $P$ is the orbital period in years and $a$ is the semimajor axis in astronomical units (AU), we find that for $a = 77$, $P^2 = 77^3$. Solving for $P$, we get $P \approx 676$ years.

Question 3

An asteroid has an orbital period of 5 years.Approximately what is its average distance from the sun?

Accepted Answer

According to Kepler's Third Law, the square of the orbital period of a planet (or asteroid) is directly proportional to the cube of the semi-major axis of its orbit. This law can be simplified to $P^2 = a^3$, where $P$ is the orbital period in Earth years and $a$ is the average distance from the Sun in astronomical units (AU). For an orbital period of 5 years, solving for $a$ gives $a^3 = 5^2 = 25$, so $a \approx 2.92$, which rounds to approximately 3 AU.

Question 4

Why did Tycho Brahe reject Copernicus's model of the solar system?

Accepted Answer

Tycho Brahe rejected Copernicus's model primarily because he could not detect stellar parallax, which he expected to observe if the Earth was indeed orbiting the Sun. This absence of observable parallax was a significant argument against the heliocentric model at the time, as it suggested that either the stars were unimaginably distant (to make the parallax too small to detect with the naked eye) or that the Earth did not move.

Question 5

The Sun is located at one focus of the orbit of a planet.What is located at the other focus?

Accepted Answer

In the elliptical orbits of planets around the Sun, as described by Kepler's First Law of Planetary Motion, the Sun is located at one of the foci, and nothing is specifically located at the other focus. The shape of the orbit is determined by the gravitational forces and the initial velocity of the planet, not by the presence of an object at the other focus.

Question 6

The point of closest approach of a planet to the Sun is called the

Accepted Answer

The point of closest approach of a planet to the Sun is called the Perihelion.

Question 7

What is the orbital period of an asteroid that has a semimajor axis of 2.8 AU?

Accepted Answer

The orbital period of an asteroid (or any object orbiting the Sun) can be calculated using Kepler's Third Law, which states that the square of the orbital period (P, in years) is directly proportional to the cube of the semimajor axis of its orbit (a, in astronomical units, AU). The formula is P^2 = a^3. For an asteroid with a semimajor axis of 2.8 AU, the calculation is P^2 = 2.8^3 = 21.952, and taking the square root gives P ≈ 4.7 years.

Question 8

Suppose a planet has an elliptical orbit.The speed of the planet is 20 km/s when it is at its average distance from the Sun.Which of the following is most likely to be the planet's speed when it is farthest from the Sun?

Accepted Answer

According to Kepler's laws of planetary motion, a planet moves faster when it is closer to the Sun and slower when it is farther away. Therefore, when the planet is at its farthest point from the Sun (which is also called the aphelion), it will be moving at a slower speed than its average speed. Since the planet's average speed is given as 20 km/s, it is most likely that its speed when it is farthest from the Sun will be less than 20 km/s. Among the given choices, the only option that is less than 20 km/s is 15 km/s. Therefore, the best choice is A)15 km/s.

Question 9

What object is located at one focus of the orbit of the planet Mars?

Accepted Answer

According to Kepler's laws of planetary motion, each planet orbiting around the sun follows an elliptical path, with the sun located at one of the two foci of the ellipse. Therefore, one focus of the orbit of Mars would be the Sun.

Question 10

Why was Tycho unwilling to accept that the stars are too far away for him to be able to detect their parallaxes?

Accepted Answer

Tycho believed that he could measure the angular diameters of stars and therefore, he should be able to determine their parallaxes. However, due to the limited technology of his time, he was unable to accurately measure these quantities.

Answer: B
 Tycho assumed that the Earth was stationary, which meant that he did not consider the possibility of observing stellar parallaxes. It was only after Galileo discovered the principle of parallax that astronomers began to realize the importance of taking the Earth's motion around the Sun into account.

Answer: D
 This is a trick question, as Tycho was actually able to measure some stellar parallaxes during his lifetime. Although his measurements were not very accurate, he was able to observe parallax shifts from several stars and use them to estimate the distances to those stars. His observations provided important data for later astronomers to refine our understanding of parallax and the distances to stars.

Question 11

Suppose the distance between the two foci of an ellipse is much smaller than the semimajor axis of the ellipse.What can be said about the ellipse?

Accepted Answer

When the distance between the two foci of an ellipse is much smaller than the semimajor axis of the ellipse, the eccentricity of the ellipse is nearly zero. This means that the ellipse is almost circular, with the two foci located close to the center. As a result, a planet orbiting on that ellipse would have a relatively long orbital period, not a short one. The ellipse is also not necessarily long and skinny, and the foci are not necessarily located far from the semimajor axis.

Question 12

Kepler's law of equal areas in equal times predicts that

Accepted Answer

Kepler's law of equal areas in equal times states that a planet will sweep out equal areas in its orbit in equal times. This means that a planet will move faster when it is closer to the sun (at perihelion) and slower when it is farther away (at aphelion).

Question 13

Which of the following is one of Kepler's Laws of Planetary Motion?

Accepted Answer

Kepler's Third Law states that the square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit. In other words, the planet's distance cubed is proportional to its distance squared.

Question 14

A planet has a semimajor axis of 10 astronomical units.The orbital period would be

Accepted Answer

According to Kepler's third law, the square of the orbital period (in years) is proportional to the cube of the semimajor axis (in astronomical units). Using this formula, we get: orbital period^2 = (semimajor axis)^3. Plugging in the values for semimajor axis (10 AU) and solving for the orbital period, we get: orbital period^2 = 10^3, so the orbital period is 32 years (approximately). Therefore, the correct choice is D.

Question 15

Callisto is 1.76 times as far from Jupiter as is Ganymede.The orbital period of Callisto is 16.7 days.What is the orbital period of Ganymede?

Accepted Answer

According to Kepler's Third Law of Planetary Motion, the square of the orbital period of a planet (or moon in this case) is directly proportional to the cube of the semi-major axis of its orbit. This law can be simplified to $T_2^2/T_1^2 = R_2^3/R_1^3$, where $T$ is the orbital period and $R$ is the distance from the central body. Given that Callisto is 1.76 times as far from Jupiter as Ganymede, and the orbital period of Callisto is 16.7 days, we can set up the ratio for Ganymede (let's call its period $T_g$) as follows: $16.7^2/T_g^2 = 1.76^3/1^3$. Solving for $T_g$ gives an orbital period of about 7 days for Ganymede.

Question 16

Which of the following best describes Tycho's model of the solar system?

Accepted Answer

Tycho's model of the solar system is known as the geocentric model, in which the Sun orbits around the Earth, and the planets orbit around the Sun.

Question 17

A hypothetical planet orbits the Sun a distance of 4 AU.What is its orbital period?

Accepted Answer

According to Kepler's Third Law, the square of the orbital period (in years) is equal to the cube of the semi-major axis (in astronomical units). Therefore, $ T^2 = a^3 $, so $ T^2 = 4^3 = 64 $, and $ T = \sqrt{64} = 8 $ years.

Question 18

Suppose there were very remote planets in the solar system,with periods of about 100,000 years.Approximately how far from the Sun would such a planet be?

Accepted Answer

According to 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. This law can be represented as $P^2 \propto a^3$, where $P$ is the orbital period in years and $a$ is the semi-major axis in astronomical units (AU). For a planet with a period of about 100,000 years, we can rearrange Kepler's law to solve for $a$, knowing that Earth's period is 1 year and its distance from the Sun is 1 AU. The calculation shows that the planet would be approximately 2000 astronomical units from the Sun to have a 100,000-year orbital period.

Question 19

Kepler's Second Law states that

Accepted Answer

Kepler's Second Law states that a planet moves more rapidly when near the Sun. This is also known as the Law of Equal Areas, which states that a planet will sweep out equal areas in equal times as it orbits around the Sun. This means that when a planet is closer to the Sun, it will move faster in order to cover the same amount of area in the same amount of time as when it is further away from the Sun.

Question 20

According to Kepler's laws,a planet moves fastest in its orbit when it is

Accepted Answer

According to Kepler's second law, a planet moves faster when it is closer to the Sun in its elliptical orbit. This is because the gravitational force between the planet and the Sun is stronger when the planet is closer, resulting in a higher speed. Therefore, choice B, when the planet is nearest the Sun, is the best answer.

Quiz 4: Renaissance Astronomy