Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 19: Problem 39

A quasar has a luminosity of \(10^{41} \mathrm{W}\), or \(\mathrm{J} / \mathrm{s}\), and \(10^{8} \mathrm{M}_{\text {sun }}\) to feed it. Assuming constant luminosity and 20 percent conversion efficiency, what is your estimate of the quasar's lifetime?

Short Answer

Expert verified
The quasar's estimated lifetime is approximately \(1.135 \times 10^6\) years.

Step by step solution

01

Identify given values

Given: Luminosity (L) = \(10^{41} \text{ W}\), Mass available (M) = \(10^{8} M_{\text{sun}}\), Efficiency (\(\eta\)) = 20% = 0.2.
02

Calculate total energy available

Use Einstein's mass-energy equivalence formula. Total energy (E) = \(\eta M c^2\). Assuming \(M_{\text{sun}} = 1.989 \times 10^{30} \text{ kg}\), \[E = 0.2 \times 10^8 \times 1.989 \times 10^{30} \times (3 \times 10^8)^2\].
03

Simplify the energy calculation

Calculate as follows: \[E = 0.2 \times 10^8 \times 1.989 \times 10^{30} \times 9 \times 10^{16}\] \[E = 3.5802 \times 10^{54} \text{ J}\].
04

Calculate the quasar's lifetime

Lifetime (t) = \(\frac{E}{L}\). Thus, \[t = \frac{3.5802 \times 10^{54}}{10^{41}} = 3.5802 \times 10^{13} \text{ s}\].
05

Convert lifetime to years

Convert seconds to years, using \(1 \text{ year} = 3.154 \times 10^7 \text{ s}\): \[ t = \frac{3.5802 \times 10^{13}}{3.154 \times 10^7} \approx 1.135 \times 10^6 \text{ years} \].

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!

Key Concepts

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

Luminosity
Luminosity is a measure of the total amount of energy emitted by an astronomical object per unit time. For quasars, which are extremely bright and distant objects, luminosity helps us understand the power and energy output. This is generally measured in watts (W) or joules per second (J/s). In the given exercise, the quasar has a luminosity of \(10^{41} \text{ W}\), indicating a very high energy output. Luminosity is crucial in determining how long the quasar can shine brightly, given its available fuel or mass.
Mass-Energy Equivalence
The concept of mass-energy equivalence is derived from Einstein's theory of relativity, summarized by the famous equation \(E = mc^2\). This equation indicates that mass (m) can be converted into energy (E) and vice versa, with the speed of light (c) squared acting as the conversion factor. In the context of the quasar, we use this principle to calculate the total energy that its available mass can produce. Given \(10^8 M_{\text{sun}}\) of mass, we apply the equation to find out the total energy potentially available to sustain the quasar’s luminosity.
Conversion Efficiency
Conversion efficiency (\(\eta\)) refers to how effectively the mass of the quasar is transformed into energy. In our problem, the efficiency is given as 20%, or 0.2. This means that only 20% of the quasar’s mass is converted into usable energy, while the rest is either lost or remains unused. Efficiency is a crucial factor because it directly influences how much energy the quasar can emit over its lifetime. In our exercise, this factor is used in the calculation \(E = \eta M c^2\) to determine the total energy available.
Einstein's Equation
Einstein's equation \(E = mc^2\) is foundational in understanding energy and mass interchangeability. This formula is crucial in astrophysics, especially when calculating phenomena such as quasar luminosity and lifespan. Here, it allows us to compute the energy output from the quasar’s mass. By plugging in the known values of mass and efficiency, we can estimate the total energy the quasar has to sustain its luminosity. This explains how the initial mass of the quasar translates into a specific lifetime, given its constant energy output rate.

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

What accounts for the differences among various types of AGNs? a. the type of the host galaxy b. the size of the central black hole c. the amount of dark matter in the galaxy's halo d. our viewing angle

Describe how elliptical galaxies and spiral bulges are similar.

If you observed a galaxy with an \(\mathrm{H} \alpha\) emission line that had a wavelength of \(756.3 \mathrm{nm}\), what would be the galaxy's redshift? Note that the rest wavelength of the H \(\alpha\) emission line is \(656.3 \mathrm{nm}\). a. 0.01 b. 0.05 c. 0.10 d. 0.15

Which of the following contributes the largest percentage to the total mass of a spiral galaxy? a. dark matter b. central black hole c. stars d. dust and gas

For a galaxy, the term morphology refers to a. its shape. b. its evolution over time. c. the motion of its stars. d. its overall density.
See all solutions

Recommended explanations on Physics Textbooks

Geometrical and Physical Optics

Read Explanation

Quantum Physics

Read Explanation

Famous Physicists

Read Explanation

Wave Optics

Read Explanation

Magnetism and Electromagnetic Induction

Read Explanation

Classical Mechanics

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