Convert between Intensity as f'n of wavelength and intencity as f'n of frequency

[theneedtoknow]

Homework Statement

Bλ(T) = β Bν(T) / λ2
v = frequency
λ = wavelength

find β , if Bλ(T) is in units Watt/m2/μm while Bν(T) is in units Jy (= 10−26 Watt/m2/Hz)

Homework Equations

The Attempt at a Solution

I know that first of all Bλ(T) = Bν(T) * |dv/dλ| = Bv(T) * c / λ^2
so at least one part of Beta should be c

Do i multiple Bv(T) Times 10^26 * 10^-6 to account for the different units?

so that β = c * 10^20?
Or is this wrong?

 

[anathema]

Hello, thank you for your question. Your approach is correct, but there are a few minor errors in your calculation. Let me guide you through it step by step:

1. First, let's rewrite the equation using the given units:

Bλ(T) = β Bν(T) / λ^2
where Bλ(T) is in units Watt/m^2/μm and Bν(T) is in units Jy (= 10^-26 Watt/m^2/Hz).

2. Next, let's use the given information to express Bν(T) in terms of Bλ(T):

Bν(T) = Bλ(T) * |dλ/dv| = Bλ(T) * c / λ^2

Note that you have a small typo in your attempt, where you wrote "dν/dλ" instead of "dλ/dν". Also, you should use the speed of light in meters, not in Jy.

3. Now, we can substitute this expression into the original equation:

Bλ(T) = β * (Bλ(T) * c / λ^2) / λ^2

4. Simplify the equation by dividing both sides by Bλ(T) and multiply by λ^4:

λ^4 = β * c / λ^2

5. Finally, solve for β:

β = λ^6 * c

Note that since λ is in units of μm, you should convert it to meters before plugging it into the equation. Also, make sure to use the correct value for the speed of light in meters (not Jy).

Therefore, the final answer is:

β = (λ * 10^-6)^6 * 3 * 10^8 = 3 * 10^14 * λ^6

I hope this helps. Good luck with your calculations!

 

[kerimek]

Your attempted solution is on the right track. To convert between intensity as a function of wavelength and intensity as a function of frequency, you can use the following equation:

Bλ(T) = Bν(T) * |dν/dλ| = Bν(T) * c/λ^2

Where c is the speed of light and λ is the wavelength. To find β, you can use the fact that Bλ(T) is in units of Watt/m2/μm and Bν(T) is in units of Jy (= 10^-26 Watt/m2/Hz).

So, first multiply Bν(T) by 10^26 to convert it to units of Watt/m2/μm:

Bν(T) * 10^26 = Bν(T) * 10^26 * c/λ^2

Next, you can use the fact that 1 Jy = 10^-26 Watt/m2/Hz, so 1 Watt/m2/μm = 10^26 Jy:

Bν(T) * 10^26 * c/λ^2 = Bν(T) * 10^26 * c/λ^2 * 10^-26

This simplifies to:

Bν(T) * c/λ^2 = Bν(T) * 10^-26

Finally, you can rearrange the equation to solve for β:

β = Bν(T) * c/λ^2 * 10^-26

So, your final answer for β would be:

β = Bν(T) * c/λ^2 * 10^-26

This shows that β is dependent on both Bν(T) and c, and that multiplying Bν(T) by c/λ^2 and then by 10^-26 will give you the value of β in the appropriate units.