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Chapter 16: Problem 55

Recall from Chapter 4 that the luminosity of a spherical object at temperature \(T\) is given by \(L=4 \pi R^{2} \sigma T^{4},\) where \(R\) is the object's radius. If the Sun became a white dwarf with a radius of \(10^{7}\) meters, what would its luminosity be at the following temperatures: (a) \(10^{8} \mathrm{K} ;\) (b) \(10^{6} \mathrm{K} ;(\mathrm{c}) 10^{4} \mathrm{K} ;(\mathrm{d}) 10^{2} \mathrm{K} ?\)

Short Answer

Expert verified
Luminosities are: (a) \(7.13 \times 10^{39} \mathrm{W} \), (b) \(7.13 \times 10^{31} \mathrm{W} \), (c) \(7.13 \times 10^{23} \mathrm{W} \), (d) \(7.13 \times 10^{15} \mathrm{W} \).

Step by step solution

01

Understand the Given Formula

The luminosity formula is given by \(L = 4 \pi \cdot R^{2} \cdot \sigma \cdot T^{4}\)where:- \( L \) is the luminosity,- \( R \) is the radius of the object,- \( \sigma \) is the Stefan-Boltzmann constant (\( \sigma \approx 5.67 \times 10^{-8} \mathrm{W\cdot m^{-2}\cdot K^{-4}} \)),- \( T \) is the temperature in kelvins.
02

Insert Known Values

Substitute the known values into the formula for each case:- \( R = 10^{7} \) meters,- \( \sigma = 5.67 \times 10^{-8} \mathrm{W\cdot m^{-2}\cdot K^{-4}} \),and calculate for each temperature given below:(a) \( T = 10^{8} \) K,(b) \( T = 10^{6} \) K,(c) \( T = 10^{4} \) K,(d) \( T = 10^{2} \) K.
03

Calculate Luminosity for Temperature (a)

For \( T = 10^{8} \) K,\[ L = 4 \pi \cdot (10^{7} \ m)^{2} \cdot 5.67 \times 10^{-8} \cdot (10^{8} \ K)^{4} \]\[ L = 4 \pi \cdot 10^{14} \cdot 5.67 \times 10^{-8} \cdot 10^{32} \]\[ L = 4 \pi \cdot 5.67 \times 10^{38} \]\[ L \approx 7.13 \times 10^{39} \mathrm{W} \]
04

Calculate Luminosity for Temperature (b)

For \( T = 10^{6} \) K,\[ L = 4 \pi \cdot (10^{7} \ m)^{2} \cdot 5.67 \times 10^{-8} \cdot (10^{6} \ K)^{4} \]\[ L = 4 \pi \cdot 10^{14} \cdot 5.67 \times 10^{-8} \cdot 10^{24} \]\[ L = 4 \pi \cdot 5.67 \times 10^{30} \]\[ L \approx 7.13 \times 10^{31} \mathrm{W} \]
05

Calculate Luminosity for Temperature (c)

For \( T = 10^{4} \) K,\[ L = 4 \pi \cdot (10^{7} \ m)^{2} \cdot 5.67 \times 10^{-8} \cdot (10^{4} \ K)^{4} \]\[ L = 4 \pi \cdot 10^{14} \cdot 5.67 \times 10^{-8} \cdot 10^{16} \]\[ L = 4 \pi \cdot 5.67 \times 10^{22} \]\[ L \approx 7.13 \times 10^{23} \mathrm{W} \]
06

Calculate Luminosity for Temperature (d)

For \( T = 10^{2} \) K,\[ L = 4 \pi \cdot (10^{7} \ m)^{2} \cdot 5.67 \times 10^{-8} \cdot (10^{2} \ K)^{4} \]\[ L = 4 \pi \cdot 10^{14} \cdot 5.67 \times 10^{-8} \cdot 10^{8} \]\[ L = 4 \pi \cdot 5.67 \times 10^{14} \]\[ L \approx 7.13 \times 10^{15} \mathrm{W} \]

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Key Concepts

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

Stefan-Boltzmann Law
In the study of luminosity, the Stefan-Boltzmann Law plays a crucial role. This law states that the total energy radiated per unit surface area of a black body is directly proportional to the fourth power of the black body's temperature. The formula used for calculating luminosity, which is the total amount of energy a star emits per second, is derived from this principle. This equation is given by:

\ L = 4 \pi R^{2} \sigma T^{4} \

Where:
  • \ L \ is the luminosity
  • \ R \ is the radius of the object
  • \sigma \ is the Stefan-Boltzmann constant (approximately \(5.67 \times 10^{-8} \mathrm{W \cdot m^{-2} \cdot K^{-4}}\))
  • \ T \ is the temperature in kelvins.
Students should note that a small change in temperature can lead to a significant change in luminosity because of the power of four relationship.
White Dwarf
A white dwarf is a stellar remnant composed mostly of electron-degenerate matter. These are the remnants of medium-sized stars that have exhausted their nuclear fuel and expelled their outer layers. What remains is a very hot and dense core. Even though white dwarfs no longer sustain fusion reactions, they still emit light due to the residual thermal energy. The significance of a white dwarf in the exercise lies in its density and the application of the Stefan-Boltzmann Law to determine its luminosity at various temperatures.
Astrophysics
Astrophysics is the branch of astronomy that delves into the physical properties of celestial bodies. It combines principles from both physics and astronomy to understand how stars, planets, and other objects in the universe behave. In this exercise, astrophysics is employed to explore how changes in temperature impact the luminosity of a star-like object. Recognizing the interconnectedness of temperature, radius, and emitted radiation enables us to predict stellar behaviors and better comprehend cosmic phenomena.
Temperature
Temperature is a critical factor in the physics of stars. It measures the thermal energy of the particles within a star. In the case of a white dwarf, significant variations in temperature have profound effects on luminosity. Using the previously mentioned formula, we see how an increase in temperature leads to a dramatic increase in luminosity (\fn{ proportional to } \ T^{4} \fn{ }). Whether it's \(10^{2}\) K or \(10^{8}\) K, understanding temperature helps astrophysicists determine the star's energy output and its lifecycle. This principle is fundamental to grasping stellar evolution and simulations.

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Most popular questions from this chapter

Why are most nearby stars low-mass, low-luminosity stars? a. Those stars are easier to see. b. The more distant low-mass, low-luminosity stars are not visible from Earth. c. Most stars are low-mass, low-luminosity stars. d. Most nearby stars formed in the same region as the Sun, and thus are similar.

Roughly how large does a planetary nebula grow before it disperses? Use an expansion rate of \(20 \mathrm{km} / \mathrm{s}\) and a lifetime of 50,000 years.

When the Sun runs out of fuel in its core, the core will be a. empty. b. filled with hydrogen. c. filled with helium. d. filled with carbon.

When compressed, ordinary gas heats up but degenerate gas does not. Why, then, does a degenerate core heat up as the star continues shell burning around it? a. It is heated by the radiation from fusion. b. It is heated by the gravitational collapse of the shell. c. It is heated by the weight of helium ash falling on it. d. It is insulated by the shell.

T Coronae Borealis is a well-known recurrent nova. a. Is it a single star or a binary system? Explain. b. What mechanism causes a nova to flare up? c. How can a nova flare-up happen more than once?
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