Problem 40 - Astronomy: At Play in the Cosmos

Question

What is the wavelength of peak radiation, in meters, for a white dwarf with a temperature of \(35,000 \mathrm{K}\) ? What kind of light is this?

Step-by-Step Solution

Answer

The peak wavelength is approximately 82.8 nm, which is in the ultraviolet range.

Step 1: Understand the Problem

We need to find the wavelength of peak radiation for a white dwarf star with a given temperature, specifically 35,000 K. This will help us determine the type of electromagnetic radiation it emits most intensely.

Step 2: Using Wien's Displacement Law

Wien's Displacement Law relates the temperature of a black body, like a star, to the wavelength of peak emission. The formula is: \[ \lambda_{max} = \frac{b}{T} \] where \( \lambda_{max} \) is the peak wavelength in meters, \( b \) is Wien's constant \( 2.897 \times 10^{-3} \) meters Kelvin, and \( T \) is the temperature in Kelvin.

Step 3: Substitute the Values

Input the given temperature into Wien's Displacement Law formula: \[ \lambda_{max} = \frac{2.897 \times 10^{-3}}{35,000} \].

Step 4: Calculate the Peak Wavelength

Perform the division: \[ \lambda_{max} = \frac{2.897 \times 10^{-3}}{35,000} \approx 8.28 \times 10^{-8} \text{ meters} \].

Step 5: Determine the Type of Light

The peak wavelength, \( 8.28 \times 10^{-8} \) meters, is approximately 82.8 nm. This falls within the ultraviolet (UV) range of electromagnetic radiation.

Key Concepts

White Dwarf Star

A white dwarf is the final evolutionary stage of certain stars. These stars have exhausted their nuclear fuel and have shed their outer layers, leaving behind a hot, dense core. Most of the light we see from a white dwarf comes from this leftover core. White dwarfs are fascinating because:

• They are very dense; a white dwarf's mass is similar to that of the Sun, but its volume is comparable to that of Earth.
• Despite their high temperatures, they gradually cool over billions of years as they lose their heat.
• White dwarfs do not undergo further nuclear reactions once they are formed, making them stable and long-lasting.

These stars are crucial for understanding stellar evolution and the future fate of our sun. When studying phenomena like electromagnetic radiation from white dwarfs, we often use principles of physics, such as Wien's Displacement Law, to analyze their properties.

Electromagnetic Radiation

Electromagnetic radiation refers to the waves of the electromagnetic field, propagating through space and carrying electromagnetic radiant energy. This phenomenon encompasses a range of wavelengths, including:

• Radio waves at the longest wavelength end.
• Microwaves, infrared, visible light, ultraviolet, X-rays, and gamma-rays, continuing towards shorter wavelengths.

In the context of stars such as white dwarfs, electromagnetic radiation is emitted across multiple wavelengths due to their high temperatures. To understand which wavelength is most intensely emitted, scientists utilize Wien's Displacement Law. By calculating the peak wavelength using this law, we can find out the primary type of light a star like a white dwarf emits most strongly. This is crucial for classifying the type of radiation we observe from celestial objects.

Ultraviolet Light

Ultraviolet light is a type of electromagnetic radiation with a wavelength shorter than that of visible light but longer than X-rays. It ranges from about 10 nm to 400 nm in wavelength. Ultraviolet light is divided into three categories:

• UV-A (320-400 nm) - This is the least harmful form and most reaches the Earth's surface.
• UV-B (280-320 nm) - A portion of these rays is absorbed by the ozone layer, but some reach the Earth and can cause skin damage.
• UV-C (100-280 nm) - These are the most harmful, but are completely absorbed by the Earth's atmosphere.

When a white dwarf star has a peak emission wavelength of around 82.8 nm, it falls into the ultraviolet part of the spectrum, specifically within the UV-C range. Such emission highlights the intense energy and heat these stars emit, which are not visible to the naked eye. Understanding the properties of ultraviolet light contributes to our knowledge about the energetic processes occurring in different parts of the universe.