How Do You Calculate the Intensity of a Star at Different Wavelengths?

[Krikri]

Homework Statement

We have a Star with known temperature T, Radius R and distance from us.
1) Say we know the effective temperature $T_{eff}$ and we want to calculate the intensity $I(λ.T) at λ=5500 A$.
2) Suppose that $I(λ,T)$ is constant over the range of visual band and $Δλ(visual)=900A$ then calculate $I(ν,T)$ of the star in the Visual band.
3)Measure the flux of the star above the Earth's atmosphere at the visual band (take into account that the black body intensity is independent of the viewing angle thus $\int_0^{π/2} sinθ cosθ \, dθ=1/2$)

Homework Equations

1) Planck's Law $I(λ,T)=\frac{2hc^2}{λ^5} \frac{1}{e^(\frac{hc}{kλt})-1}$ or $I(ν,T)=\frac{2hν^3}{c^2} \frac{1}{e^(\frac{hν}{kt})-1}$
2)$Ι(n,Τ)dν=-Ι(λ,T)dλ$

The Attempt at a Solution

1) For the first i just put the values in the Planck's formula and do the calculations
2) For the second I am confused. What does it mean $I(λ,T)$ is constant over the visual band and if so isn't $I(ν,T)$ also constant? Secondly I don't know how to measure the intensity over a range, only for a single value of $λ,ν$. And why do i need $Δλ$ because if there is an integral i need the limits not the range right?
3) For the third question i don't know nothing. If you can point me into a direction to look it would be great

Any help or hint is helpful.

 

[mfb]

2) For the second I am confused. What does it mean $I(λ,T)$ is constant over the visual band and if so isn't $I(ν,T)$ also constant?

The first one is (assumed to be) constant (it has the same value everywhere in the interval), the second one is not. This is related to the non-linear relationship between wavelength and frequency. A specific wavelength range (e.g. 1 A) corresponds to a small frequency range at large wavelengths, but it corresponds to a large frequency range at short wavelengths. The intensity cannot be constant in both expressions at the same time.

Secondly I don't know how to measure the intensity over a range, only for a single value of $λ,ν$. And why do i need $Δλ$ because if there is an integral i need the limits not the range right?

The integral of a constant function is just its value multiplied by the range.

3) For the third question i don't know nothing. If you can point me into a direction to look it would be great

You can calculate the surface area of the star and the surface of a spherical shell with distance [star-earth]. The ratio between the two will be relevant.