Suppose when we look in one half of the sky, the CMBR appears to be at a temperature of 2.72K. What is the peak wavelength in that direction? Are we moving toward or away from the region of space? What is our velocity with respect to the CMBR?
To find the peak wavelength in the given direction, we need to use Wien's displacement law.
Step 1: Recall Wien's Displacement Law, which states that the peak wavelength (λ_peak) of blackbody radiation is inversely proportional to its temperature (T):
λ_peak = constant / T
Step 2: Use the given temperature of the CMBR (2.72K) to find the peak wavelength:
λ_peak = constant / 2.72K
Step 3: The constant in Wien's Displacement Law is known as Wien's constant and it is equal to approximately 2.898 × 10^-3 meters-kelvin (m-K).
λ_peak = (2.898 × 10^-3 m-K) / 2.72K
Step 4: Calculate the peak wavelength in meters:
λ_peak ≈ 1.066 × 10^-3 meters or 1.066 mm
Regarding the motion of our region of space relative to the CMBR:
Step 5: Our motion can be determined using the Doppler effect. If the CMBR appears at a specific temperature (2.72K), we can assume that we are moving relative to that region of space.
Step 6: If the observed temperature is higher, it suggests that we are moving towards the source of radiation, and if it is lower, we are moving away.
Step 7: Since the given temperature is the same as the standard value for the CMBR (2.72K), it implies that we are not moving directly towards or away from the CMBR region of space. So, our velocity with respect to the CMBR can be considered negligible.
To calculate the peak wavelength of the Cosmic Microwave Background Radiation (CMBR), we can use Wien's displacement law, which states that the wavelength corresponding to the peak intensity of radiation is inversely proportional to its temperature.
Wien's displacement law formula is given by:
λ_max = (b / T)
Where:
λ_max is the peak wavelength
b is Wien's displacement constant (2.89777 x 10^-3 m·K)
T is the temperature in Kelvin
In this case, the CMBR temperature is given as 2.72K. Plugging this value into the formula, we get:
λ_max = (2.89777 x 10^-3 m·K / 2.72K)
Calculating this gives us the peak wavelength in that direction.
Regarding our motion with respect to the region of space, the CMBR provides us with information about our velocity. Due to the Doppler effect, the CMBR appears blue-shifted in the direction towards which we are moving and red-shifted in the opposite direction.
If the CMBR appears at a temperature of 2.72K when we look in one half of the sky, it means that we are moving towards that region of space. The blue-shift indicates that we are moving towards the CMBR source.
To determine our velocity with respect to the CMBR, we can use the concept of the CMBR dipole. As we move through space, our velocity causes a dipole pattern in the CMBR, resulting in a hotter side and a cooler side.
The magnitude of our velocity can be calculated using the following formula:
v = ΔT_CMBR * c / T_CMBR
Where:
v is our velocity
ΔT_CMBR is the temperature difference between the hotter and cooler sides of the CMBR dipole
c is the speed of light (approximately 3 x 10^8 m/s)
T_CMBR is the temperature of the CMBR (2.72K)
By substituting the given values into the formula, we can calculate our velocity with respect to the CMBR.