Astronomy Assignment #4 Solutions

Text Problems:

Unit Number / Questions for Review / Problems
12 / 5, 6, 7, 8, 9 / 10, 11, 12, 13

Unit 12

Problem 10: All objects orbiting the Sun obey Kepler’s Laws. Suppose a comet is found with an orbital period of 64 years. If its orbit is circular, what is its distance from the Sun?

This is a Kepler’s 3rd Law () problem where the orbital period P is given as 64 years. There are several ways to mathematically solve for the value of the semimajor axis a. I’ve listed the method using the simplest conceptual steps below.

The comet’s orbit has a semimajor axis of 16 AU. Since the orbit is circular, the comet will always be 16 AU from the Sun.

Problem 11: In 2003, astronomers discovered Sedna, an object in the outer Solar System with a semimajor axis of 526 AU. What is its orbital period?

This is a Kepler’s 3rd Law () problem where the semimajor axis a is given as 526 AU. There are several ways to mathematically solve for the value of the orbital period P. I’ve listed the method using the simplest conceptual steps below.

Sedna requires 12,060 years to complete one orbit around the Sun.

Problem 12: Suppose an asteroid is discovered with an orbit that brings it as close as 0.5 AU from the Sun and as far away as 1.5 AU. What is the semimajor axis of its orbit? What is its orbital period? What is the eccentricity of its orbit?

This problem requires that you understand the relationship between perihelion distance, aphelion distance and the semimajor axis of an orbit. See the figure to the right that is similar to figure 12.3 in the text. The figure indicates that the sum of the perihelion distance and the aphelion distance is equal to twice the semimajor axis:

The problem tells us that . Thus a = 1 AU. The semimajor axis of the asteroid is 1 AU.

Next is a Kepler’s 3rd Law () problem where the semimajor axis a is derived as 1 AU. There are several ways to mathematically solve for the value of the orbital period P. I’ve listed the method using the simplest conceptual steps below.

The asteroid requires 1year to complete one orbit around the Sun.

Finally, the eccentricity of the comet can be determined from the figure above (similar to Fig 12.3 in the text) by noticing that . Thus

The eccentricity of the asteroids orbit is 0.5 – pretty non-circular in shape.

Problem 13: If a comet orbits the Sun and reaches 1 AU at its closest approach to the Sun, and its orbital period is 27 years, what is its maximum distance from the Sun?

First use Kepler’s 3rd Law ()to find the semimajoraxis a where the orbital period P is given as 27 years. There are several ways to mathematically solve for the value of the semimajor axis a. I’ve listed the method using the simplest conceptual steps below.

The comet’s orbit has a semimajor axis of 9 AU.

Next, if you understand the relationship between perihelion distance, aphelion distance and the semimajor axis of an orbit ( See the figure to the right that is similar to figure 12.3 in the text), you can find the aphelion distance. The figure indicates that the sum of the perihelion distance and the aphelion distance is equal to twice the semimajor axis:

The problem tells us that. Thus the asteroid has an aphelion distance of 17 AU from the Sun.

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