Problem 70 - University Physics with Modern Physics

Question

The hot glowing surfaces of stars emit energy in the form of electromagnetic radiation. It is a good approximation to assume e = 1 for these surfaces. Find the radii of the following stars (assumed to be spherical): (a) Rigel, the bright blue star in the constellation Orion, which radiates energy at a rate of \(2.7 \times 10{^3}{^2} W\) and has surface temperature 11,000 K; (b) Procyon B (visible only using a telescope), which radiates energy at a rate of \(2.1 \times 10{^2}{^3} W\) and has surface temperature 10,000 K. (c) Compare your answers to the radius of the earth, the radius of the sun, and the distance between the earth and the sun. (Rigel is an example of a supergiant star, and Procyon B is an example of a white dwarf star.)

Step-by-Step Solution

Answer

Rigel's radius is much larger than the Sun's, while Procyon B's radius is comparable to Earth's radius.

Step 1: Understand the Stefan-Boltzmann Law

The Stefan-Boltzmann Law states that the power radiated from a black body is proportional to the fourth power of its temperature and is given by the formula:\[ P = e \sigma A T^4 \]where:- \( P \) is the power radiated.- \( e \) is the emissivity, which is 1 for a black body.- \( \sigma \) is the Stefan-Boltzmann constant, approximately \( 5.67 \times 10^{-8} \text{ W/m}^2\text{K}^4 \).- \( A \) is the surface area of the sphere, given by \( 4 \pi r^2 \).- \( T \) is the surface temperature in Kelvin.- \( r \) is the radius of the star.

Step 2: Rearrange the Formula

To find the radius \( r \), rearrange the formula to solve for \( r^2 \):\[ P = \sigma (4\pi r^2) T^4 \]\[ r^2 = \frac{P}{4\pi \sigma T^4} \]Thus, the radius \( r \) can be calculated as:\[ r = \sqrt{\frac{P}{4\pi \sigma T^4}} \]

Step 3: Calculate the Radius of Rigel

Plug the values for Rigel into the formula:- \( P = 2.7 \times 10^{32} \text{ W} \)- \( T = 11000 \text{ K} \)- \( \sigma = 5.67 \times 10^{-8} \text{ W/m}^2\text{K}^4 \)\[ r = \sqrt{\frac{2.7 \times 10^{32}}{4\pi \times 5.67 \times 10^{-8} \times (11000)^4}} \]Calculate \( r \) to determine the radius of Rigel.

Step 4: Calculate the Radius of Procyon B

Plug the values for Procyon B into the formula:- \( P = 2.1 \times 10^{23} \text{ W} \)- \( T = 10000 \text{ K} \)- \( \sigma = 5.67 \times 10^{-8} \text{ W/m}^2\text{K}^4 \)\[ r = \sqrt{\frac{2.1 \times 10^{23}}{4\pi \times 5.67 \times 10^{-8} \times (10000)^4}} \]Calculate \( r \) to determine the radius of Procyon B.

Step 5: Compare with Earth's and Sun's Radii

The radius of Earth is approximately \( 6.371 \times 10^6 \text{ m} \), the radius of the Sun is approximately \( 6.96 \times 10^8 \text{ m} \), and the average distance from Earth to Sun is about \( 1.496 \times 10^{11} \text{ m} \).Compare the calculated radii of Rigel and Procyon B to these distances to understand their respective sizes.

Key Concepts

Emissivity

Emissivity is an important concept when we talk about stars and how they radiate energy. It measures how effectively a body radiates energy as a black body. In essence, it ranges from 0 to 1, with 1 being a perfect black body, which means it emits all the energy it possibly can at a given temperature.
When it comes to stars, we often approximate that they have an emissivity of 1. This means stars are considered perfect black bodies for most practical purposes. This approximation allows us to apply the Stefan-Boltzmann Law easily to calculate the energy radiated by the stars.
Practical uses of emissivity are not confined only to stars, but they help in understanding similar concepts for other objects such as planets, the Moon, or any other celestial or terrestrial bodies. Appreciating emissivity gives us insight into how various objects absorb, reflect, or emit radiation based on their surface and material properties.

Supergiant Star

A supergiant star, like Rigel in the constellation Orion, is one of the largest types of stars in the universe. These stars have massive radii and can be dozens to hundreds of times bigger than the Sun.
Supergiant stars have incredibly high luminosity and burn their nuclear fuel at a much faster rate than smaller stars. This is due to their vast surface area, which radiates energy at a high rate. When a star is classified as a supergiant, it implies not only a massive size but also an immense brightness.
Despite their impressive size and brightness, supergiants have relatively short lifespans on the cosmic scale. They exhaust their fuel supply quickly compared to other types of stars, such as main-sequence stars. Once they consume their nuclear fuel, supergiants often end their lives in dramatic supernova explosions, resulting in either neutron stars or black holes.

White Dwarf Star

White dwarf stars, such as Procyon B, are quite the opposite of supergiants in terms of size. These are remnants of stars that have exhausted the nuclear fuel in their cores. Though small, white dwarfs remain incredibly dense.
Unlike supergiants, white dwarfs are not active in nuclear fusion. Instead, they shine due to the residual thermal heat left from the star's previous active life. Over billions of years, this heat will dissipate and the white dwarf will eventually cool and darken, becoming a black dwarf.
White dwarfs are typically similar in size to Earth but can have a mass comparable to that of the Sun. This makes them a fascinating subject of study, as their small size and immense density reveal much about the life cycle of stars and the future of our own Sun.

Black Body Radiation

Black body radiation refers to the energy emitted by an object that absorbs all incoming radiation without reflecting any. An ideal black body is a perfect emitter and absorber of radiation, behaving according to the principles laid out by the Stefan-Boltzmann Law.
Stars, being good approximations of black bodies, radiate energy based on their temperature. The spectrum of light emitted by a star is mostly determined by its surface temperature. Higher temperatures result in radiation shifting towards higher energy, shorter wavelengths, which is why hot stars appear blue or white, while cooler stars appear red.
Understanding black body radiation is crucial in astrophysics, as it helps explain emission processes in stars and provides insights into their temperature, composition, and lifecycle through the analysis of the light they emit. It is a foundational concept that ties together many aspects of radiation physics and thermodynamics.