Problem 75 - 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 54.2 million km, and Procyon B's radius is about 9,800 km, indicating Rigel is a supergiant and Procyon B is a white dwarf.

Step 1: Understanding the Stefan-Boltzmann Law

To find the radius of a star given its luminosity and temperature, we can use the Stefan-Boltzmann Law. This law states that the power emitted by a star is proportional to its surface area and the fourth power of its temperature. It is expressed as \( L = 4\pi R^2 \sigma T^4 \), where \( L \) is the luminosity (or power radiated), \( R \) is the radius, \( \sigma \) is the Stefan-Boltzmann constant \( (5.67 \times 10^{-8} \, \text{W/m}^2\cdot\text{K}^4) \), and \( T \) is the temperature.

Step 2: Solving for Radius

Rearranging the equation for radius, we have: \( R = \sqrt{\frac{L}{4\pi \sigma T^4}} \). This formula will allow us to compute the radius of each star using their given luminosity and surface temperature.

Step 3: Calculating Radius for Rigel

For Rigel, the luminosity \( L = 2.7 \times 10^{32} \, \text{W} \) and temperature \( T = 11,000 \, \text{K} \). Substitute these into the formula: \[ R = \sqrt{\frac{2.7 \times 10^{32}}{4\pi \times 5.67 \times 10^{-8} \times (11,000)^4}} \]. Calculating this gives \( R \approx 54.2 \times 10^6 \text{ km} \).

Step 4: Calculating Radius for Procyon B

For Procyon B, the luminosity \( L = 2.1 \times 10^{23} \, \text{W} \) and temperature \( T = 10,000 \, \text{K} \). Substitute these into the formula: \[ R = \sqrt{\frac{2.1 \times 10^{23}}{4\pi \times 5.67 \times 10^{-8} \times (10,000)^4}} \]. Calculating this gives \( R \approx 9.8 \times 10^{3} \text{ km} \).

Step 5: Comparing Results with Known Values

The radius of the Earth is approximately 6,371 km, the radius of the Sun is about 696,340 km, and the average distance from the Earth to the Sun is about 149.6 million km. Rigel's radius of \(54.2 \times 10^6 \text{ km} \) indicates it is much larger than our Sun, qualifying it as a supergiant. Procyon B's radius of \(9.8 \times 10^{3} \text{ km} \) makes it relatively close in size to Earth, typical of a white dwarf.

Key Concepts

Electromagnetic Radiation

Stars, like Rigel and Procyon B, emit energy in the form of electromagnetic radiation. This type of energy travels through space in the form of waves. Electromagnetic radiation includes a broad range of wavelengths such as gamma rays, X-rays, ultraviolet light, visible light, infrared, microwaves, and radio waves. The energy output of stars is directly related to their surface temperature and radius. High-temperature stars emit more energetic waves, primarily in the visible and ultraviolet spectrum. Understanding electromagnetic radiation helps scientists determine a star's behavior, composition, and structure.
Electromagnetic radiation from stars provides insight into the universe as it carries crucial information about the star's internal processes. As this radiation reaches Earth, it is captured by telescopes and analyzed to understand phenomena such as the chemical makeup of stars and their movement in galaxies.
The Stefan-Boltzmann Law is particularly important here. It links the luminosity of a star to its electromagnetic output, and allows us to calculate the energy a star emits from its surface at given temperatures.

Star Luminosity

Luminosity of a star, like the ones in this exercise, refers to the total amount of energy a star radiates per second. It's comparable to the light intensity of a bulb but on a cosmic scale. Luminosity is measured in watts and can vary widely from one star to another depending on size and temperature.
For instance, Rigel emits a luminosity of approximately \(2.7 \times 10^{32}\) watts, making it a very bright star due to its large size and high temperature. On the other hand, Procyon B has a much lower luminosity of \(2.1 \times 10^{23}\) watts, because it is much smaller and cooler compared to Rigel.
*Factors influencing star luminosity include:*

• Radius of the Star: Larger stars have more surface area, which allows them to emit more light.
• Surface Temperature: Hotter stars emit more energy per unit of surface area, thus appearing more luminous.

Luminosity is crucial for astronomers as it helps determine a star's stage in its life cycle, from its formation, through its active main-sequence phase, and eventually to its death.

Stellar Radius Calculation

To find a star's radius when given its luminosity and temperature, astronomers use a formula derived from the Stefan-Boltzmann Law. The law states that the power radiated per unit area is proportional to the fourth power of the temperature. Thus, the formula for radius is:\[R = \sqrt{\frac{L}{4\pi \sigma T^4}}\]This equation allows us to find the radius, \(R\), using the luminosity, \(L\), and temperature, \(T\), where \(\sigma\) is the Stefan-Boltzmann constant \((5.67 \times 10^{-8} \text{W/m}^2\cdot\text{K}^4)\).
The step-by-step solution given involves plugging Luminosity and Temperature for stars like Rigel \( (L = 2.7 \times 10^{32} \, \text{W},\ T = 11,000 \, \text{K}) \) and Procyon B \( (L = 2.1 \times 10^{23} \, \text{W},\ T = 10,000 \, \text{K}) \) into this formula. The radius is essential not just for size comparison, but also for determining the star's type and expected lifecycle events.

Supergiant and White Dwarf Comparison

Stars like Rigel and Procyon B are excellent representatives of supergiants and white dwarfs respectively. They illustrate how the size of a star can vary dramatically based on its stage in the stellar life cycle.

**Supergiants:**

• Rigel is a supergiant, characterized by its massive radius and brightness. With a radius of approximately \(54.2 \times 10^6\,\text{km}\), Rigel forms an imposing presence in the cosmos, far exceeding the size of our Sun.
• Supergiants are among the largest stars in the universe. Their extended radius means they emit a tremendous amount of light, making them highly luminous.

**White Dwarfs:**

• In contrast, Procyon B is a white dwarf with a radius of about \(9.8 \times 10^{3}\,\text{km}\), reflecting a stage of life where the star has exhausted its nuclear fuel and collapsed into a much smaller size comparable only to planetary dimensions.
• White dwarfs are the final evolutionary state of stars like our Sun. They are dense, not as bright, yet incredibly hot.

This stark contrast highlights the dynamic and varied nature of stars as they progress through their cosmic lives.