Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 14: Problem 10

(For all of these problems, where necessary assume the normal ratio of total- to-selective extinction.) Suppose the albedo of a dust grain is constant in the infrared (where it will emit most of its energy, at a value \(a(\mathrm{IR}),\) and is constant over the visible and ultraviolet (where it will absorb most of its energy, with a value \(a(\mathrm{V})\). It is near a star whose spectrum is that of a blackbody at temperature \(T\), and whose radius is \(R\) : The grain is a distance \(d\) from the star. Derive an expression for the dust temperature.

Short Answer

Expert verified
The dust temperature is given by \[ T_d = \left( \frac{a(V) R^2 T^4}{4 d^2 a(IR)} \right)^{1/4} \]

Step by step solution

01

Determine the absorbed energy

The dust grain absorbs energy primarily in the visible and ultraviolet wavelengths. Assuming it acts like a blackbody at temperature T, the energy absorbed by the grain from the star per unit time is given by the star's luminosity multiplied by the absorption efficiency in the visible, which we denote as \( \frac{{a(V) L}}{{4 \pi d^2}}.\)
02

Express the luminosity of the star

The luminosity (L) of the star is given by the Stefan-Boltzmann law: \[ L = 4 \pi R^2 \sigma T^4 \] where \( \sigma\) is the Stefan-Boltzmann constant.
03

Write the absorbed power by the dust grain

Substituting the star's luminosity into the absorbed energy expression: \[ P_{\text{abs}} = \frac{{a(V) 4 \pi R^2 \sigma T^4}}{{4 \pi d^2}} = a(V) \sigma T^4 \left( \frac{R}{d} \right)^2 \]
04

Determine the emitted energy

The dust grain re-emits energy as a blackbody in the infrared. The power emitted by the grain can be expressed as: \[ P_{\text{emit}} = 4 \pi a^2 \sigma T_d^4 a(IR) \] where \( T_d \) is the temperature of the dust and \( a \) is the grain's radius.
05

Equate absorbed and emitted power

For thermal equilibrium, the power absorbed must equal the power emitted: \[ P_{\text{abs}} = P_{\text{emit}} \] \[ a(V) \sigma T^4 \left( \frac{R}{d} \right)^2 = 4 \pi a^2 \sigma T_d^4 a(IR) \]
06

Solve for the dust temperature \(T_d\)

Cancel out common terms and solve for the dust temperature \( T_d \): \[ T_d^4 = \frac{{a(V) T^4 \left( \frac{R}{d} \right)^2}}{4 \pi a^2 a(IR) \sigma} \] \[ T_d = \left( \frac{a(V) R^2 T^4}{4 d^2 a(IR)} \right)^{1/4} \]

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.

Albedo
Albedo refers to the reflectivity of a surface, meaning how much light is reflected by it rather than absorbed. In this exercise, we consider the albedo of a dust grain in two different spectral regions:
  • Infrared (IR): denoted as \(a(\text{IR})\), which accounts for the grain's behavior in the infrared wavelengths where it emits most of its energy.
  • Visible (V): denoted as \(a(\text{V})\), which accounts for the grain's absorption of energy in the visible and ultraviolet wavelengths.
A higher albedo in a specific range means more light in that range is reflected rather than absorbed, affecting the temperature of the dust grain.
Blackbody Radiation
Blackbody radiation refers to the type of electromagnetic radiation emitted by an idealized object which perfectly absorbs all incident radiation regardless of wavelength or angle. The star in the exercise is described as emitting blackbody radiation at a temperature \(T\).
  • A blackbody radiates energy at all wavelengths, with a peak that depends on its temperature due to Wien's Law.
  • The star's blackbody spectrum influences the dust grain, forcing it to absorb energy mostly in the visible and ultraviolet wavelengths and emit in the infrared.
This is essential for understanding how the energy absorbed and emitted by the grain leads to its temperature.
Thermal Equilibrium
Thermal equilibrium is the state in which the rate of energy absorption equals the rate of energy emission. In this problem, the dust grain reaches thermal equilibrium when:
  • The absorbed energy from the star (mainly in the visible and ultraviolet) equals the emitted energy in the infrared.
This balance can be expressed mathematically as:
\[ a(V) \frac{4 \text{π} R^2 \text{σ} T^4}{4 \text{π} d^2} = 4 \text{π} \text{a}^2 \text{σ} T_d^4 a(IR) \text{π} \text{a}^2\text{σ} T_d^4 a(IR) \]
By solving this equation, we can determine the dust grain's temperature.
Stefan-Boltzmann Law
The Stefan-Boltzmann Law links the total energy radiated per unit surface area of a blackbody to the fourth power of its temperature. It is given by:
  • \[L = 4 \text{π} R^2 \text{σ} T^4 \]
This law is critical in the problem because it allows us to express the luminosity of the star as a function of its temperature and radius.
In the context of the dust grain, we apply the Stefan-Boltzmann Law to both the star and the grain to determine the energy absorbed and emitted, ensuring we find the temperature of the dust grain in equilibrium.

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

(For all of these problems, where necessary assume the normal ratio of total- to-selective extinction.) Show that, in a gas, the number of particles hitting a surface per second per unit surface area is \(n v,\) where \(n\) is the number density of particles per volume and \(v\) is the speed of the particles.

(For all of these problems, where necessary assume the normal ratio of total- to-selective extinction.) Suppose we have an electric dipole made up of two charges, \(+q\) and \(-q,\) a distance \(d\) apart. The dipole is placed a distance \(r\) from a charge \(+Q\). (a) Find an expression for the net force on the dipole. (b) How is this related to chemical reactions in the interstellar medium?

(For all of these problems, where necessary assume the normal ratio of total- to-selective extinction.) Calculate the mass of an HI cloud with a circular appearance, with a radius of \(10 \mathrm{pc}\) and an average H column density of \(10^{21} \mathrm{cm}^{-2}\).

(For all of these problems, where necessary assume the normal ratio of total- to-selective extinction.) Calculate the virial mass for a molecular cloud with \(\left(v_{\mathrm{r}}\right)=5 \mathrm{km} / \mathrm{s}\) and \(R=20 \mathrm{pc}.\)

(For all of these problems, where necessary assume the normal ratio of total- to-selective extinction.) For an excitation temperature of \(100 \mathrm{K},\) what is the ratio of populations for the two levels in the \(21 \mathrm{cm}\) transition? (Take the statistical weight of the lower level to be 1 and the upper level to be \(3 .\))
See all solutions

Recommended explanations on Physics Textbooks

Magnetism

Read Explanation

Solid State Physics

Read Explanation

Torque and Rotational Motion

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Atoms and Radioactivity

Read Explanation

Work Energy and Power

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