/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 40 The orbital periods of Comets En... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 40

The orbital periods of Comets Encke, Halley, and Hale-Bopp are 3.3 years, 76 years, and 2,530 years, respectively. Their orbital eccentricities are 0.847,0.967 , and 0.995 , respectively. a. What are the semimajor axes (in astronomical units) of the orbits of these comets? b. What are the minimum and maximum distances from the Sun (in astronomical units) reached by Comets Halley and Hale-Bopp in their respective orbits? c. Which region of the Solar System did each likely come from? d. Which would you guess is the most pristine comet among the three? Which is the least? Explain your reasoning.

Short Answer

Expert verified
a. Semimajor axes: Encke: ~2.2 AU, Halley: ~17.9 AU, Hale-Bopp: ~322 AU. b. Halley: min ~0.59 AU, max ~35.2 AU; Hale-Bopp: min ~1.61 AU, max ~642 AU. c. Encke: Kuiper Belt, Halley: Kuiper/Oort Cloud border, Hale-Bopp: Oort Cloud. d. Most pristine: Hale-Bopp; least pristine: Encke.

Step by step solution

01

Calculate the Semimajor Axis using Kepler's Third Law

Kepler's Third Law can be expressed as \[ T^2 = a^3 \] where \( T \) is the orbital period in years, and \( a \) is the semimajor axis in astronomical units (AU). For each comet, solve for \( a \) using the orbital period provided: For Comet Encke: \( T = 3.3 \) years For Comet Halley: \( T = 76 \) years For Comet Hale-Bopp: \( T = 2530 \) years.
02

Step 1(a): Calculate for Comet Encke

Use Kepler鈥檚 Third Law: \( T_E^2 = a_E^3 \), where \( T_E = 3.3 \). \[ 3.3^2 = a_E^3 \] Solve for \( a_E \): \[ a_E = \sqrt[3]{3.3^2} = \sqrt[3]{10.89} \approx 2.2 \text{ AU} \]
03

Step 1(b): Calculate for Comet Halley

Use Kepler鈥檚 Third Law: \( T_H^2 = a_H^3 \), where \( T_H = 76 \). \[ 76^2 = a_H^3 \] Solve for \( a_H \): \[ a_H = \sqrt[3]{76^2} = \sqrt[3]{5776} \approx 17.9 \text{ AU} \]
04

Step 1(c): Calculate for Comet Hale-Bopp

Use Kepler鈥檚 Third Law: \( T_{HB}^2 = a_{HB}^3 \), where \( T_{HB} = 2530 \). \[ 2530^2 = a_{HB}^3 \] Solve for \( a_{HB} \): \[ a_{HB} = \sqrt[3]{2530^2} = \sqrt[3]{6,400,900} \approx 322 \text{ AU} \]
05

Calculate the Minimum and Maximum Distances

The minimum distance (perihelion) and maximum distance (aphelion) from the Sun can be calculated using the formulas: \[ r_{min} = a(1 - e) \] \[ r_{max} = a(1 + e) \] where \( a \) is the semimajor axis and \( e \) is the eccentricity.
06

Step 2(a): Calculate for Comet Halley

For Comet Halley: \( a_H \approx 17.9 \text{ AU} \), \( e_H = 0.967 \) Minimum distance: \[ r_{min,H} = 17.9(1 - 0.967) \approx 0.59 \text{ AU} \] Maximum distance: \[ r_{max,H} = 17.9(1 + 0.967) \approx 35.2 \text{ AU} \]
07

Step 2(b): Calculate for Comet Hale-Bopp

For Comet Hale-Bopp: \( a_{HB} \approx 322 \text{ AU} \), \( e_{HB} = 0.995 \) Minimum distance: \[ r_{min,HB} = 322(1 - 0.995) \approx 1.61 \text{ AU} \] Maximum distance: \[ r_{max,HB} = 322(1 + 0.995) \approx 642 \text{ AU} \]
08

Identify the Likely Origin Regions

Comets with semimajor axes less than 10 AU usually come from the Kuiper Belt, whereas those with greater semimajor axes (hundreds to thousands of AU) likely originate from the Oort Cloud. Based on this: Comet Encke with \( a \approx 2.2 \text{ AU} \) likely comes from the Kuiper Belt. Comet Halley with \( a \approx 17.9 \text{ AU} \) might come from the borderline region between the Kuiper Belt and the Oort Cloud. Comet Hale-Bopp with \( a \approx 322 \text{ AU} \) likely comes from the Oort Cloud.
09

Pristine Status of the Comets

Comets that have spent most of their time in the outer solar system (like those in the Oort Cloud) tend to be more pristine because they've had less interaction with the Sun. Therefore, Comet Hale-Bopp is likely the most pristine. Comet Encke, due to its short period and close approach to the Sun, is likely the least pristine.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

orbital period
The orbital period of a comet is the time it takes to complete one full orbit around the Sun. It's essentially the 'year' length for the comet. For instance, Comet Encke has an orbital period of 3.3 years, while Halley's Comet takes 76 years. Long-period comets like Hale-Bopp can take thousands of years to complete one orbit, in this case, 2530 years. This period is crucial in determining the distance and the path of the comet's journey through space, as explored next.
semimajor axis
The semimajor axis is half of the longest diameter of an elliptical orbit. It represents the average distance between the comet and the Sun. Using Kepler's Third Law, we can identify the semimajor axis with the formula \ \[ T^2 = a^3 \ \], where T is the orbital period and a is the semimajor axis in astronomical units (AU). For Comet Encke, its semimajor axis is approximately 2.2 AU, while Halley鈥檚 Comet鈥檚 semimajor axis is around 17.9 AU, and for Hale-Bopp, it is about 322 AU.
eccentricity
Eccentricity measures how much an orbit deviates from being circular. It ranges between 0 (perfect circle) and 1 (parabola). Bounded orbits like those of comets have eccentricity values close to but less than 1. Comet Encke, with an eccentricity of 0.847, has a more circular orbit compared to Halley's Comet (0.967) and Hale-Bopp (0.995), which have highly elongated orbits. High eccentricity values indicate that a comet spends a significant portion of its orbit far from the Sun.
Kuiper Belt
The Kuiper Belt is a region beyond Neptune, approximately 30-55 AU from the Sun. It's home to many icy bodies and dwarf planets. Short-period comets, like Comet Encke with a semimajor axis of 2.2 AU, are thought to originate from this region. The Kuiper Belt serves as a reservoir for these comets, and gravitational interactions can alter their orbits to send them into the inner solar system.
Oort Cloud
The Oort Cloud is a distant spherical shell of icy objects surrounding the solar system. It extends from about 2,000 to as far as 100,000 AU from the Sun. Long-period comets like Hale-Bopp, with a semimajor axis of 322 AU, typically come from this realm. Comets from the Oort Cloud are considered more pristine as they have spent most of their existence far from the Sun, experiencing minimal solar radiation and influence.
perihelion
Perihelion is the point in a comet's orbit where it is closest to the Sun. For instance, Halley's Comet has a perihelion distance of approximately 0.59 AU, while Hale-Bopp's perihelion is around 1.61 AU. These close approaches to the Sun often result in magnificent displays of comets' tails due to the increased sublimation of ices and release of gas and dust.
aphelion
Aphelion is the point where a comet is farthest from the Sun. For Comet Halley, this distance is about 35.2 AU, and for Hale-Bopp, it's around 642 AU. The enormous distances at aphelion explain why these comets can take so long to complete an orbit, as their travel through the outer regions of the solar system is extensive and slow.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The three types of meteorites come from different parts of their parent bodies. Stony-iron meteorites are rare because a. they are hard to find. b. the volume of a differentiated body that has both stone and iron is small. c. there is very little iron in the Solar System. d. the magnetic field of the Sun attracts the iron.

One recent estimate concludes that nearly 800 meteorites with mass greater than 100 grams (massive enough to cause personal injury strike the surface of Earth each day. Assuming you present a target of 0.25 square meter \(\left(\mathrm{m}^{2}\right)\) to a falling meteorite, what is the probability that you will be struck by a meteorite during your 100-year lifetime? (Note that the surface area of Earth is approximately \(5 \times 10^{14} \mathrm{m}^{2}\).)

Congress tasked NASA with searching for near-Earth objects because a. they might impact Earth, as others have in the past. b. they are close by and easy to study. c. they are moving fast. d. they are scientifically interesting.

Ceres has a diameter of \(975 \mathrm{km}\) and a period of about 9 hours. What is the rotational speed of a point on the surface of this dwarf planet?

The estimated amount of zodiacal dust in the Solar System remains constant at approximately \(10^{16}\) kg. Yet zodiacal dust is constantly being swept up by planets or removed by the pressure of sunlight. a. If all the dust disappeared (at a constant rate) over a span of 30,000 years, what would the average production rate, in kilograms per second, have to be to maintain the current content? b. Is this an example of static or dynamic equilibrium? Explain your answer.
See all solutions

Recommended explanations on Physics Textbooks

Measurements

Read Explanation

Scientific Method Physics

Read Explanation

Mechanics and Materials

Read Explanation

Fluids

Read Explanation

Electric Charge Field and Potential

Read Explanation

Electrostatics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.