Black Body Radiation (Awkward integral)

[CanIExplore]

Homework Statement

What percentage of the Sun’s blackbody radiation spectrum falls into the visible light spectrum (400-700 nm). Where T=5000K
Hint: Integrate over frequencies

Homework Equations

B=2h$$\nu$$³c^-2 (e^h$$\nu$$/kT-1)^-1

Where $$\nu$$ is the frequency of the light.

The Attempt at a Solution

Ok so the problem is very straightforward, I'm just having trouble evaluating the integral. I need to integrate the equation for the brightness (B) over the frequency $$\nu$$ where the limits are given by the span of wavelengths in the visible part of the spectrum. The integral then just looks something like this:
$$\int B$$=$$\int \nu$$³(e^$$\nu$$-1)^-1 , where i excluded the constants.

I tried integration by parts but it didn't work. I also plugged it into mathematica and got a very weird, long answer that didn't make sense. When I asked my TA about it he told me I had to solve it numerically. What does it mean to solve an integral numerically? Am I just supposed to plug in the lower limit and then subtract that from what I get when I plug in the upper limit? I am thinking of the fundamental theorem of calculus here.

Thanks

 

[Cyosis]

This integral doesn't have a primitive in terms of elementary functions. To evaluate this integral you can use the Raleigh-Jeans approximation by looking it up in your book or by using the Taylor series of the exponent up to first order.

 

[phyzguy]

When you integrate a function numerically, you basically divide the area up into a bunch of narrow rectangles or trapezoids and add them all up. There are a lot of ways to do this, try reading this:

http://en.wikipedia.org/wiki/Numerical_integration

To accomplish this, you could write a program to do it, or use a canned program. Mathematica has a function NIntegrate which will do numerical integration and, given the function and the endpoints, will just return a number.

 

[CanIExplore]

Thanks guys

 

[paranormal]

for any help!

I understand your frustration with the integral not working out as you had hoped. It is important to remember that not all integrals can be solved analytically, and sometimes numerical methods are necessary. In this case, it means using a computer or calculator to evaluate the integral at specific values of frequency and then summing those values to approximate the total area under the curve.

To answer the question, we need to find the total area under the curve for the visible light spectrum (400-700 nm). This can be done by breaking the integral into smaller intervals and evaluating each one numerically. For example, we can divide the spectrum into 10 intervals of 30 nm each (400-430 nm, 430-460 nm, etc.) and use a computer or calculator to evaluate the integral at the midpoint of each interval. The sum of these values will give us an approximation of the total area under the curve for the visible spectrum.

Another method is to use a numerical integration technique, such as the trapezoidal rule or Simpson's rule, which can give more accurate results than simply evaluating the integral at a few points.

In conclusion, while it may be frustrating that the integral cannot be solved analytically, numerical methods provide a way to still find an approximation of the desired result. This is a common approach in many scientific fields, and it is important to be familiar with these methods in order to solve complex problems.