Problem 59 - College Physics

Question

How large is the sun? By measuring the spectrum of wavelengths of light from our sun, we know that its surface temperature is \(5800 \mathrm{~K}\). By measuring the rate at which we receive its energy on carth, we know that it is radiating a total of \(3.92 \times 10^{26} \mathrm{~J} / \mathrm{s}\) and behaves nearly like an ideal blackbody. Use this information to calculate the diameter of our sun.

Step-by-Step Solution

Answer

The sun's diameter is approximately 1,390,000 km.

Step 1: Understand the Stefan-Boltzmann Law

To find the sun's diameter, we use the Stefan-Boltzmann Law which relates the power radiated by a blackbody to its temperature and surface area: \[ P = \sigma A T^4 \]where \(P\) is the total power radiated, \(\sigma\) 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.

Step 2: Rearrange the Equation to Solve for Surface Area

We know the total power radiated \(P = 3.92 \times 10^{26}\) J/s and the temperature \(T = 5800\) K, so we solve for the area \(A\): \[ A = \frac{P}{\sigma T^4} \]

Step 3: Calculate the Surface Area

Substitute the values into the equation:\[ A = \frac{3.92 \times 10^{26}}{5.67 \times 10^{-8} \times (5800)^4} \]Compute \(A\) to find the surface area of the sun.

Step 4: Surface Area of a Sphere Relation

For a sphere, the surface area \(A\) is given by \(A = 4\pi R^2\), where \(R\) is the sun's radius.Rearrange this to solve for radius: \[ R = \sqrt{\frac{A}{4\pi}} \]

Step 5: Calculate the Radius of the Sun

Substitute the surface area from Step 3 into the sphere equation:\[ R = \sqrt{\frac{3.83 \times 10^{18}}{4\pi}} \]Compute \(R\) to find the radius of the sun.

Step 6: Convert Radius to Diameter

The diameter \(D\) is twice the radius:\[ D = 2R \]Use the radius found in Step 5 to find the sun's diameter.

Key Concepts

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is foundational in understanding how stars, like our Sun, emit energy. This law states that the total energy radiated per unit surface area of a blackbody is directly proportional to the fourth power of its absolute temperature. Mathematically, it is expressed as:\[ P = \sigma A T^4 \]where:

• \( P \) is the power output or energy radiated by the blackbody,
• \( \sigma \) 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 blackbody, and
• \( T \) is the absolute temperature in Kelvin (K).

This law is crucial for determining how much energy a star like the Sun emits, based on its temperature and size. It allows scientists to estimate the Sun's diameter when combined with its known surface temperature and emitted power.

Blackbody Radiation

A blackbody is an idealized physical object that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. It is a perfect emitter, radiating energy at maximum efficiency depending on its temperature. This concept is central in stellar physics, as many stars, including our Sun, approximate blackbody behavior. The spectral distribution of radiation from a blackbody helps determine its temperature. For the Sun, its light spectrum provides a surface temperature of approximately 5800 K. By assuming the Sun behaves as a near-ideal blackbody, we can use the Stefan-Boltzmann Law to calculate characteristics like energy output and surface area, and consequently, its diameter.

Surface Temperature

The surface temperature of a celestial body such as the Sun is critical for understanding its physical properties. It determines how brightly the object shines and directly influences the amount of energy it radiates. In the context of the Sun, its surface temperature is measured to be around 5800 Kelvin. This temperature is derived from analyzing the Sun's emitted spectrum of light and is a key component in applying the Stefan-Boltzmann Law to calculate the Sun's emitted power and size. Knowing the surface temperature allows astronomers to further study and comprehend the processes happening in a star's outer layers and to make broader assumptions about its internal structure and life cycle.

Solar Radius

The solar radius is an essential measurement in astrophysics, representing the Sun's size. Once the surface area is calculated using the Stefan-Boltzmann Law, the solar radius can be found using the formula for the surface area of a sphere:\[ A = 4\pi R^2 \]where \( R \) is the radius. By rearranging this formula, we can solve for the radius:\[ R = \sqrt{\frac{A}{4\pi}} \]Understanding the solar radius is crucial because it helps scientists compare the Sun’s size with other celestial objects. The radius also aids in calculating other significant parameters, such as the Sun's density and gravitational field, enhancing our understanding of stellar structures and dynamics.