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 can use Wien's displacement law, which states that the peak wavelength (λ_max) of the blackbody radiation is inversely proportional to the temperature (T) of the object emitting the radiation.
The formula for Wien's displacement law is:
λ_max = (b/T),
where b is a constant known as Wien's displacement constant (approximately equal to 2.898 × 10^(-3) m·K).
So, plugging in the given temperature of 2.72 K, we can calculate the peak wavelength as follows:
λ_max = (2.898 × 10^(-3) m·K) / (2.72 K)
λ_max ≈ 1.065 × 10^(-3) m or 1.065 mm
Therefore, the peak wavelength in the given direction is approximately 1.065 mm.
Now, to determine whether we are moving toward or away from the region of space, we need to consider the concept of redshift. Redshift refers to the stretching of light waves as objects move away from us. If an object is moving away, its light gets shifted towards longer wavelengths, which means a decrease in frequency.
In this case, since the CMBR appears to be at a temperature of 2.72 K (which corresponds to the expected temperature of CMBR), we can conclude that there is no redshift or blueshift in the observed radiation. Therefore, we are not moving towards or away from the region of space in question.
Finally, the velocity of our motion with respect to the CMBR is known as the peculiar velocity, and it represents our velocity relative to the average motion of matter in the universe.
The precise calculation of this velocity requires detailed observations and data analysis. However, it is estimated that the peculiar velocity of the Earth with respect to the CMBR is around 600 km/s in the direction of the constellation Leo.
Please note that these values are approximate, as the exact velocity may vary depending on different measurements and models.