Problem 42 - Astronomy: At Play in the Cosmos

Question

What is the temperature, in kelvins, of a star with a peak wavelength of \(6.7 \times 10^{-7}\) meter?

Step-by-Step Solution

Answer

The temperature is approximately 4325.37 K.

Step 1: Understanding Wien's Law

Wien's Law helps us determine the temperature of a black body based on the peak wavelength of the emitted radiation. The formula is given by \ T = \frac{b}{\lambda_{max}} \, where \( b \) is Wien's constant \( \approx 2.897 \times 10^{-3} \) m K, and \( \lambda_{max} \) is the peak wavelength.

Step 2: Substitute Given Values

We've been given \( \lambda_{max} = 6.7 \times 10^{-7} \) meters. Substituting this into Wien's Law, we get: \ T = \frac{2.897 \times 10^{-3}}{6.7 \times 10^{-7}}

Key Concepts

Black Body Radiation

Black body radiation refers to the type of electromagnetic radiation that a perfect black body emits. A black body is an idealized object that absorbs all incident light without reflecting any.
When it is heated, it emits radiation at all wavelengths. This spectrum of emitted radiation depends solely on the temperature of the body.
The concept is critical in understanding celestial objects like stars, which often behave like black bodies in terms of radiation emission.

• A perfect black body absorbs all wavelengths perfectly.
• It emits radiation at a characteristic spectrum dependent on its temperature.
• The emitted spectrum provides insight into the temperature and properties of the object.

Temperature Calculation

The temperature of a black body can be calculated using a direct relationship from Wien's Law. Essentially, we can determine how hot a star or similar object is by observing the peak wavelength of the radiation it emits.
For instance, if we know the peak wavelength of a star's emitted light, we can use Wien's Law to find its temperature. This is especially useful in astrophysics for classifying stars.
The formula used is:\[ T = \frac{b}{\lambda_{max}} \]where:

• \( b \) is Wien's constant, approximately \( 2.897 \times 10^{-3} \) meters Kelvin.
• \( \lambda_{max} \) is the peak wavelength in meters.

Peak Wavelength

The peak wavelength is the specific wavelength at which the emission of a black body is maximized.
This is a key feature in the spectrum of emitted radiation from stars or any radiating body. By identifying this peak wavelength, scientists can gain essential information about the object's temperature.
It serves as a focal point for calculations under Wien's Law, establishing a bridge between observed light and temperature determination.

• Emitting maximum radiation at the peak wavelength provides optimal data for analysis.
• It helps classify stars into various types based on their temperature.
• Measurable through instruments that observe and analyze radiant energy output.

Wien's Constant

Wien's constant plays a crucial role in the calculation of temperature from peak wavelength. It is a proportional constant in Wien's Law and is valued at approximately \( 2.897 \times 10^{-3} \) meters Kelvin.
Using this constant, scientists can link the observable property of peak wavelength to the intrinsic property of temperature. This simplifies the process of studying objects in space and aids in astronomical discoveries.
Wien’s constant is central to determining how energy distribution over different wavelengths changes with temperature.

• Provides the means to convert peak wavelength to temperature.
• Essential for the practical use of Wien's Law.
• Empowers the study of celestial bodies, helping classify and analyze stars by temperature.