Studio Exercise Week 1

We'll do the exercise in MAPLE, but don't scream. I'll step you through it. Begin by defining the formula for the radiance, Eq.8-35b:

> R:=2*Pi*h*c^2/(lambda^5*(exp(h*c/(lambda*k*T))-1));

1. Integrate this expression over wavelength to verify the Stefan-Boltzmann law, Eq.8-40. MAPLE needs some help with this, so perform a change of variables yourself. Redefine R in terms of a new variable x, and call it Rx,

   > xx:=h*c/(lambda*k*T);

   > dxdl:=-diff(xx,lambda);

   > Rx:=subs(lambda=h*c/(x*k*T),R/dxdl);

   Then integrate Rx over x from zero to infinity

   >F:=int(Rx,x=0..infinity);

   and confirm Eq.8-40. You can express the constant of proportionality "sigma" in terms of h, c, and k. Does this have the correct dependence on T?
2. Now derive Wien's law, Eq.8-39, by finding the value of the wavelength lambda which maximizes the radiance R.

   > lmax:=solve(diff(R,lambda)=0,lambda);

   You may want to simplify this expression. Does it have the correct dependence on T?
3. Next put in some numbers for the constants

   > h:=6.626E-34; c:=2.998E8 ; k:=1.381E-23;

   and check to make sure the values of the proportionality constants in the Stefan-Boltzmann law and Wien's law are correct. (Note that I'm using SI units everywhere.) This is also a good time to make some plots of the distribution for different temperatures. One way to do this is to define different functional forms for the different values of T, for example

   > T:=6000; R6:=R; and then plot this using > plot(R6,lambda=0..2E-6);

   where you need to make sure you stick with SI units for wavelength. You can combine several distributions on one plot, for example, using

   > plot({R4,R6,R8,R10},lambda=0..2E-6);

4. Finally, let's begin to apply this to stars by defining something like the "color" and relating it to temperature. Astronomers can measure some "part" of the spectrum using wavelength filters. Let's try the simplest approximation to this by integrating just over separate parts of the spectrum and comparing the parts.

   First, define some wavelength limits representing, say, the "blue" and "red" parts of the spectrum

   > lblo:=400E-9; lbhi:=500E-9; lrlo:=600E-9; lrhi:=700E-9;

   and define a "color index" by taking the difference between the "blueness" and "redness" of the spectrum. You have to force MAPLE to do this numerically using the evalf function since the integral is not analytic. For example,

   > CI6:=evalf(int(R6,lambda=lblo..lbhi)-int(R6,lambda=lrlo..lrhi));

   defines the color index for a blackbody with T=6000K. Calculate the color indices for several different temperatures, and plot the temperature as a function of color index. You can plot the data values with MAPLE using a command like

   > plot([CI4,4000,CI6,6000,CI8,8000,CI10,10000],style=LINE);

   Where is this color index useful in determining the temperature of a star? Where is it not so useful? Can you think of a different definition that might do a better job, at least for some of the temperatures?