The current peak wavelength of the CMB corresponds to a blackbody of temperature 2.74K. What was the peak wavelength of the CMB when it was emitted? The current redshift of the CMB is z=1100. Give your answer in microns.
Which waveband regime is this?
1.09
The waveband regime is Infra Red (IR)
Oh, the peak wavelength of the CMB when it was emitted? Let me consult my cosmic comedy catalog for this one. Ahem, well, it turns out that back then, the CMB was so hip and trendy that it emitted its peak wavelength at a wavelength of 1.06 millimeters. That's right, it was totally grooving to the "millimeter wave" regime. It likes to keep things chill, you know?
But remember, wavelengths can be sneaky and mischievous, so they decided to do a little cosmic fashion show over time. Now, at a redshift of z=1100, the peak wavelength has shifted to about 0.266 microns. It's like the CMB has gone from rocking millimeter waves to flaunting the ultra-cool "infrared" regime. Talk about a style makeover!
So, in summary, the peak wavelength of the CMB when it was emitted was around 1.06 millimeters, but now it's strutting its stuff in the 0.266-micron range in the infrared regime.
To calculate the peak wavelength of the cosmic microwave background (CMB) when it was emitted, we can use Wien's displacement law, which states:
λpeak = (2.898 × 10^(-3) m · K) / T
where λpeak is the peak wavelength, T is the temperature in Kelvin, and 2.898 × 10^(-3) m · K is the Wien's constant.
Given that the current temperature of the CMB is 2.74K, we can plug this value into the equation:
λpeak = (2.898 × 10^(-3) m · K) / 2.74K
≈ 1.058 × 10^(-3) m
To convert this to microns, we can multiply by 10^6, since there are 10^6 microns in a meter:
λpeak ≈ 1.058 × 10^(-3) m ×10^6 μm/m
≈ 1058 μm
Therefore, the peak wavelength of the CMB when it was emitted is approximately 1058 microns.
As for the waveband regime, this corresponds to the microwave region of the electromagnetic spectrum, which typically ranges from 1mm to 1m in wavelength.
To find the peak wavelength of the Cosmic Microwave Background (CMB) when it was emitted, we need to take into account the effects of redshifting due to the expansion of the universe.
The redshift, denoted by z, is defined as the observed change in wavelength compared to the emitted wavelength. In the case of the CMB, it is currently redshifted to a value of z=1100.
The formula relating the redshift to the change in wavelength is given by:
1 + z = λ_observed / λ_emitted
Here, λ_observed is the wavelength of the CMB that we observe today, and λ_emitted is the wavelength at the time of emission (which we want to find).
Rearranging the formula, we get:
λ_emitted = λ_observed / (1 + z)
The current peak wavelength of the CMB is related to its temperature through Wien's displacement law, which states that the peak wavelength λ_max is inversely proportional to the temperature T:
λ_max = (2.898 × 10^-3 m·K) / T
Given that the current temperature of the CMB is 2.74K, we can calculate the current peak wavelength as:
λ_observed = (2.898 × 10^-3 m·K) / 2.74K
To convert this wavelength to microns, we divide by 10^-6 (since 1 meter = 10^6 microns).
Now, substituting the values into the formula, we have:
λ_emitted = λ_observed / (1 + z)
λ_emitted = [(2.898 × 10^-3 m·K) / 2.74K] / (1 + 1100)
Finally, to convert the wavelength from meters to microns, we divide by 10^-6:
λ_emitted_in_microns = λ_emitted / 10^-6
By plugging in the values and evaluating the expression, we can find the peak wavelength of the CMB when it was emitted and determine which waveband regime it belongs to.