Problem 78 - University Physics with Modern Physics

Question

The Sizes of Stars. 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 \(2.7 \times 10^{32} \mathrm{W}\) and has surface temperature \(11,000 \mathrm{K} ;\) (b) Procyon \(\mathrm{B}\) (visible only using a telescope), which radiates energy at a rate of \(2.1 \times 10^{23} \mathrm{W}\) and has surface temperature \(10,000 \mathrm{K}\) . (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 \(\mathrm{B}\) is an example of a white dwarf star.)

Step-by-Step Solution

Answer

Rigel's radius is about \( 5.9 \times 10^9 \text{ m} \) and Procyon B's is \( 7 \times 10^6 \text{ m} \).

Step 1: Understanding the Stefan-Boltzmann Law

The power emitted by a star can be calculated using the Stefan-Boltzmann Law, which is given by \( P = e \cdot \sigma \cdot A \cdot T^4 \), where \( P \) is the power radiated, \( e \) is the emissivity (given as 1), \( \sigma \) is the Stefan-Boltzmann constant \( 5.67 \times 10^{-8} \, \text{W/m}^2 \text{K}^4 \), \( A \) is the surface area, and \( T \) is the temperature in Kelvin. Since the star is spherical, the surface area \( A = 4 \pi R^2 \), where \( R \) is the radius.

Step 2: Calculating Rigel's Radius

For Rigel, rearrange the formula to find \( R \): \( R = \sqrt{\frac{P}{4 \pi \sigma T^4}} \). Substitute the known values: \( P = 2.7 \times 10^{32} \), \( T = 11,000 \, \text{K} \), and \( \sigma = 5.67 \times 10^{-8} \). Calculating gives \( R \approx \sqrt{\frac{2.7 \times 10^{32}}{4 \pi \cdot 5.67 \times 10^{-8} \cdot 11,000^4}} \).

Step 3: Solving Rigel's Equation

Upon solving, \( R \approx 5.9 \times 10^9 \, \text{m} \). This indicates that Rigel's radius is approximately 5.9 billion meters.

Step 4: Calculating Procyon B's Radius

For Procyon B, use the same rearranged formula: \( R = \sqrt{\frac{P}{4 \pi \sigma T^4}} \). Substitute: \( P = 2.1 \times 10^{23} \), \( T = 10,000 \, \text{K} \). Calculate \( R \approx \sqrt{\frac{2.1 \times 10^{23}}{4 \pi \cdot 5.67 \times 10^{-8} \cdot 10,000^4}} \).

Step 5: Solving Procyon B's Equation

On solving, \( R \approx 7 \times 10^6 \, \text{m} \). This indicates that Procyon B's radius is about 7 million meters.

Step 6: Comparison with Known Astronomical Radii

The radius of the 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 the Sun is about \( 1.496 \times 10^{11} \, \text{m} \). Rigel, therefore, is much larger than the Sun, while Procyon B is about the size of the Earth.

Key Concepts

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is crucial in understanding how stars emit energy. This law states that the power emitted by a perfect black body is proportional to the fourth power of its temperature. Mathematically, it's expressed as \( P = e \cdot \sigma \cdot A \cdot T^4 \). Here, \( P \) represents the power emitted, \( e \) is the emissivity of the surface (often taken as 1 for stars, assuming they are perfect emitters), \( \sigma \) is the Stefan-Boltzmann constant (\( 5.67 \times 10^{-8} \, \text{W/m}^2 \text{K}^4 \)), \( A \) is the surface area, and \( T \) is the temperature in Kelvin. The law helps in calculating the total energy emitted per second by the surface of a star. Since stars are generally spherical, their surface area can be calculated using \( A = 4 \pi R^2 \). This formula is key to determining the energy output of stars based on their size and temperature.

Stellar Radii Calculation

To determine a star's radius, we use the rearrangement of the Stefan-Boltzmann Law formula. By expressing the equation in terms of radius, \( R = \sqrt{\frac{P}{4 \pi \sigma T^4}} \), we find the stellar radius given its power output and temperature. This formula essentially tells us how the energy radiated from the star, its temperature, and the universal constant \( \sigma \) come together to define the radius. The derivation involves recognizing that the star's surface area (\( A = 4 \pi R^2 \)) impacts its total radiative output.
For stars like Rigel or Procyon B, knowing \( P \) and \( T \) allows us to precisely calculate their radii, offering insights into their physical size compared to other astronomical bodies like Earth or the Sun.

Emissivity and Radiation

In astrophysics, emissivity is a measure of an object’s ability to emit thermal radiation. For stars, we assume emissivity \( e = 1 \), implying they are perfect emitters, like ideal black bodies. This simplification allows us to apply the Stefan-Boltzmann Law directly.
Emissivity impacts how much energy a body radiates at a given temperature. A star with an emissivity less than 1 would radiate less energy than predicted for its surface temperature. This concept is vital when contrasting actual celestial objects with theoretical predictions. Understanding emissivity helps bridge the gap between theoretical models and real observations of stellar properties.

Supergiant Stars

Supergiant stars, like Rigel in Orion, are enormous and luminous stars with immense radii, surpassing even the Sun. They represent an advanced stage in stellar evolution wherein these stars have expanded after exhausting the hydrogen in their cores. Their vast energy outputs originate from their large volumes and high surface temperatures.
Rigel, for instance, radiates at an astounding rate of \( 2.7 \times 10^{32} \) W, with an impressive radius derived from the Stefan-Boltzmann formula. Its massive size and energy output classify it as a blue supergiant, characterized by high temperatures and significant brightness compared to other stars. Supergiants play crucial roles in enriching the cosmos with heavy elements due to their stellar winds and eventual supernova explosions.

White Dwarf Stars

White dwarfs, such as Procyon B, are stellar remnants left after a star has exhausted its nuclear fuel. Unlike supergiants, white dwarfs have very small radii and are extremely dense. They form from stars that were once similar to our Sun, casting off their outer layers and leaving the core behind to cool.
Procyon B, with a radius of about \( 7 \times 10^6 \) m, approximates the size of Earth. Despite their small size, white dwarfs can radiate considerable power due to their high temperatures. Typically, they shine dimly compared to their massive siblings like supergiants. Understanding white dwarfs aids in comprehending stellar evolution and the ultimate fate of stars not massive enough to become neutron stars or black holes.