Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 4: Problem 39

A comet orbits the Sun with a sidereal period of \(64.0\) years. (a) Find the semimajor axis of the orbit. (b) At aphelion, the comet is \(31.5\) AU from the Sun. How far is it from the Sun at perihelion?

Short Answer

Expert verified
The semi-major axis of the orbit is approximately 52 AU. The comet is approximately 72.5 AU from the Sun at perihelion.

Step by step solution

01

Solving Part (a)

According to Kepler's Third Law, the square of the period of orbit \(T\) is proportional to the cube of the semi-major axis \(a\). In units where \(G = 4\pi^2 \), the average distance \(a\) from the Sun (in AU) and the period \(T\) (in Earth years) satisfy the equation \( T^2 = a^3 \). We can rearrange this equation to solve for \(a\), obtaining \( a = \sqrt[3]{T^2} \). Substituting \( T = 64.0 \) years, we find that the semi-major axis is approximately \( a = \sqrt[3]{(64.0)^2} \approx 52 \) AU.
02

Solving Part (b)

The distance of the comet from the Sun at aphelion is given as 31.5 AU, which we'll denote as \( r_a \). The semi-major axis that we found in Part (a) is the average of the distances at aphelion and perihelion, i.e., \( a = (r_a + r_p) / 2 \), where \( r_p \) is the distance at perihelion. If we solve this equation for \( r_p \), we get \( r_p = 2a - r_a \). Substituting the actual values \( a = 52 \) AU and \( r_a = 31.5 \) AU, we find that the comet is approximately \( r_p = 2*52 - 31.5 \approx 72.5 \) AU from the Sun at perihelion.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

Imagine a planet like the Earth orbiting a star with 4 times the mass of the Sun. If the semimajor axis of the planet's orbit is \(1 \mathrm{AU}\), what would be the planet's sidereal period? (Hint: Use Newton's form of Kepler's third law. Compared with the case of the Earth orbiting the Sun, by what factor has the quantity \(m_{1}+m_{2}\) changed? Has \(a\) changed? By what factor must \(P^{2}\) change?)

How did the models of Aristarchus and Copernicus explain the retrograde motion of the planets?

The mass of Saturn is approximately 100 times that of Earth, and the semimajor axis of Saturn's orbit is approximately \(10 \mathrm{AU}\). To this approximation, how does the gravitational force that the Sun exerts on Saturn compare to the gravitational force that the Sun exerts on the Earth? How do the accelerations of Saturn and the Earth compare?

Explain why the semimajor axis of a planet's orbit is equal to the average of the distance from the Sun to the planet at perihelion (the perihelion distance) and the distance from the Sun to the planet at aphelion (the aphelion distance).

Use the Starry Night Enthusiast \({ }^{\mathrm{TM}}\) program to observe the changing appearance of Mercury. Display the entire celestial sphere (select Guides > Atlas in the Favourites menu) and center on Mercury (double-click the entry for Mercury in the Find pane); then use the zoom controls at the right-hand end of the toolbar (at the top of the main window) to adjust your view so that you can clearly see details on the planet's surface. (Click on the + button to zoom in and on the - button to zoom out.) (a) Click on the Time Flow Rate control (immediately to the right of the date and time display) and set the discrete time step to 1 day. Using the Step Forward button, observe and record the changes in Mercury's phase and apparent size from one day to the next. Run time forward for some time to see these changes more graphically. (b) Explain why the phase and apparent size change in the way that you observe.
See all solutions

Recommended explanations on Physics Textbooks

Particle Model of Matter

Read Explanation

Solid State Physics

Read Explanation

Electric Charge Field and Potential

Read Explanation

Translational Dynamics

Read Explanation

Radiation

Read Explanation

Geometrical and Physical Optics

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.

Sign-up for free