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Chapter 13: Problem 22

Star A and star B have the same luminosity, but star A is hotter than star B. Which of the following statements is true? a. Star A is smaller than star B. b. Star A is farther away than star B. c. Star A is larger than star B. d. Star A is closer than star B.

Short Answer

Expert verified
a. Star A is smaller than star B.

Step by step solution

01

- Understand Luminosity

Luminosity is the total amount of energy a star emits per unit time. Both stars A and B have the same luminosity.
02

- Understand Temperature and Luminosity Relationship

The luminosity of a star is related to its radius and temperature by the equation: \[ L = 4 \pi R^2 \sigma T^4 \]where \(L\) is luminosity, \(R\) is radius, \(T\) is temperature, and \(\sigma\) is the Stefan-Boltzmann constant. Given that both stars have the same luminosity, if one star is hotter, the other must be larger to compensate.
03

- Compare the Sizes of the Stars

Given that star A is hotter than star B, and they have the same luminosity, star A must have a smaller radius compared to star B to maintain the same luminosity.
04

- Identify the Correct Statement

From the understanding that star A is hotter and hence must be smaller to have the same luminosity as star B, the correct statement is: a. Star A is smaller than star B.

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Key Concepts

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

Star Temperature
Stars emit light and other forms of energy because of the reactions happening in their cores. These reactions release an enormous amount of heat, which can be visualized as the star's temperature. The temperature of a star is one of the fundamental properties that define it. For instance, a hotter star emits more energy per unit area across all wavelengths. This is influenced by its surface temperature. When comparing two stars, if one is hotter, it typically means it has a higher surface temperature. This higher temperature impacts the star's color as well; hotter stars tend to appear blue or white, while cooler stars look redder.
Stellar Radius
The radius of a star is the distance from its core to its surface. It's another critical factor in determining the star's luminosity. If two stars have the same luminosity but different temperatures, the star with a higher temperature must have a smaller radius. This is derived from the relation given by the Stefan-Boltzmann law, which connects a star's luminosity to both its surface temperature and radius. So, in our example, since star A is hotter than star B, star A must have a smaller radius to ensure both stars maintain the same luminosity.
Stefan-Boltzmann Law
The Stefan-Boltzmann law is the principle that mathematically relates the luminosity of a star to its radius and temperature. The law is expressed as: To relate this to our problem, when both stars A and B have the identical luminosity but differ in temperature, the Stefan-Boltzmann law helps us understand why star A must be smaller than star B.

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Most popular questions from this chapter

Betelgeuse (in Orion) has a parallax of \(0.00763 \pm 0.00164\) arcsec, as measured by the Hipparcos satellite. What is the distance to Betelgeuse, and the uncertainty in that measurement?

Star A and star B are two nearby stars. Star A has a parallactic angle 4 times as large as star B's. Which of the following statements is true? a. \(\operatorname{Star} \mathrm{A}\) is \(^{1 / 4}\) as far away as star \(\mathrm{B}\) b. Star A is 4 times as far away as star \(B\). c. Star A has moved through space \(1 / 4\) as far as star \(B\). d. Star A has moved through space 4 times as far as star \(B\).

Other than the Sun, the only stars whose mass astronomers can measure directly are those in eclipsing or visual binary systems. Why?

Star A is more massive than star B. Both are main-sequence stars. Therefore, star A is than star B. (Choose all that apply.) a. brighter b. fainter c. hotter d. colder e. larger f. smaller

T/F: The mass of a star is the primary indicator of its observable properties.
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