Problem 2 - An Introduction to Modern Astrophysics

Question

Suppose that all of the matter in the universe were converted into energy in the form of blackbody radiation. Take the average density of matter to be the WMAP value for the density of baryonic matter, \(\rho_{b, 0} .\) Use the below equation for the energy density of blackbody radiation to find the temperature of the universe in this situation. At what wavelength would the blackbody spectrum peak? In what region of the electromagnetic spectrum is this wavelength found? Explain how your result may be applied to Olbers's paradox. \\[ u=\frac{4 \pi}{c} \int_{0}^{\infty} B_{\lambda}(T) d \lambda=\frac{4 \sigma T^{4}}{c}=a T^{4} \\]

Step-by-Step Solution

Answer

The universe's temperature would be low. The peak wavelength would likely be in the microwave region, which explains the dark sky (Olbers's paradox).

Step 1: Understand the Energy Density Equation

The energy density of blackbody radiation is given by the equation \( u = rac{4 \sigma T^4}{c} = a T^4 \), where \( \sigma \) is the Stefan-Boltzmann constant, \( c \) is the speed of light, and \( a \) is the radiation constant. Our goal is to find the temperature \( T \) of the universe if all matter were converted to energy in the form of blackbody radiation.

Step 2: Relate Matter Density to Energy Density

We assume that the average density of baryonic matter, \( \rho_{b, 0}\), is entirely converted to radiation. Using Einstein's mass-energy equivalence \( E = mc^2 \), the energy density \( u \) of our universe would then be \( u = \rho_{b, 0} c^2 \).

Step 3: Equate and Solve for Temperature

Set the two expressions for energy density equal: \( u = \rho_{b, 0} c^2 = aT^4 \). Solve for \( T \) by rearranging the equation: \( T = \left( \frac{\rho_{b, 0} c^2}{a} \right)^{1/4} \). Substitute the known values for \( \rho_{b, 0} \), \( c \), and \( a \) to find the numerical value of \( T \).

Step 4: Find Wavelength of Peak Spectrum

Use Wien's Displacement Law, \( \lambda_{max} = \frac{b}{T} \), where \( b \) is Wien's displacement constant. Substitute the temperature \( T \) calculated in Step 3 to find \( \lambda_{max} \).

Step 5: Determine Spectrum Region

Evaluate the calculated \( \lambda_{max} \) and determine in which region of the electromagnetic spectrum it lies (e.g., infrared, visible, ultraviolet, etc.).

Step 6: Apply to Olbers's Paradox

Explain how the concept of blackbody radiation and the derived temperature relate to Olbers's paradox. The high density of radiation suggests a universe glowing brightly in a dark sky, explaining why the sky is not uniformly bright due to the universe's finiteness and expansion.

Key Concepts

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is essential in understanding the relationship between the temperature of an object and the energy it emits as blackbody radiation. According to this law, the energy emitted per unit area of a blackbody is proportional to the fourth power of the temperature of that body. The equation representing this relationship is given by \[ E = \sigma T^4 \] where

• \(E\) is the total energy emitted per unit area,
• \(\sigma\) is the Stefan-Boltzmann constant, approximately \(5.67 x 10^{-8} \, \text{W m}^{-2} \text{K}^{-4}\), and
• \(T\) is the temperature of the blackbody in Kelvin.

This principle applies universally, governing how stars, planets, and even the universe itself radiate energy.
In the original exercise, applying this law allows us to connect matter's energy density to temperature. This connection is crucial for solving problems where we convert matter into energy, as the universe does according to the blackbody radiation model.
Understanding the Stefan-Boltzmann Law helps us grasp how seemingly invisible cosmic processes influence large-scale observations, like Olbers's Paradox.

Wien's Displacement Law

Wien’s Displacement Law is another principle in blackbody radiation physics that describes how the peak wavelength of emission from a blackbody shifts with temperature. The law is mathematically defined as: \[ \lambda_{max} = \frac{b}{T} \] where

• \(\lambda_{max}\) is the wavelength at which the emission is highest,
• \(b\) is the Wien's displacement constant, approximately \(2.897 \times 10^{-3} \, \text{m K}\), and
• \(T\) is the temperature in Kelvin.

As the temperature of the blackbody increases, the peak wavelength shifts towards the shorter wavelengths of the spectrum, such as from infrared to visible or ultraviolet light.
Using Wien's Displacement Law in the exercise allows us to calculate the peak wavelength of the universe, considering it as a blackbody. With the temperature derived from the Stefan-Boltzmann Law, we can determine this peak wavelength and thereby assess which part of the electromagnetic spectrum it occupies.
This knowledge links deeply with our understanding of cosmic microwave background radiation and other astrophysical observations.

Olbers's Paradox

Olbers's Paradox is an intriguing phenomenon pointing out a discrepancy in our observations of the night sky. It argues that if the universe were infinite and filled uniformly with luminous stars, the night sky should be as bright as the surface of a star due to the collective light of countless stars. However, our night sky is mostly dark.
The resolution of Olbers's Paradox lies in the understanding of the universe's dynamics and evolution. The universe is not only finite but also expanding, meaning the light from many distant stars has not reached us yet or is redshifted beyond visible light.
In the context of the blackbody radiation problem from the exercise, applying these principles shows why the high-energy radiation we calculate doesn't lead to a bright night sky. Instead, it emphasizes the cosmic microwave background radiation that is a relic of the Big Bang and still permeates the universe. This faint glow is uniform across the sky and supports the notion of an evolving, finite universe, addressing Olbers's Paradox directly.