Reference no: EM133329266

BINNING IN FREQUENCY VS. BINNING IN WAVELENGTH

Spectra look different depending on whether they are computed per wavelength or per frequency. The spectrum of a blackbody when binned in frequency space (Bν) does not equal 1 the spectrum of the same blackbody when binned in wavelength space (Bλ). The often-used (and much abused) Wien displacement law (a.k.a. Wien peak law) for blackbodies actually refers to the peak of Bλ, not of Bν (a form of physics chauvinism, totally unjustifiable). The frequency where Bν peaks is not merely the speed of light c divided by the wavelength where Bλ peaks. The entire shapes of Bλ and Bν differ.

These facts may be a little confusing at first (I remember being confused myself), but they are just simple consequences of how we choose to bin photons when plotting spectra. We can either choose to bin the photons in bins of equal frequency width, or we can choose to bin the photons in bins of equal wavelength width. The spectrum will, in general, look different depending on our choice.

1. Suppose Astronomer A measures the spectrum of a point source (NOT a blackbody). This Astronomer collects photons from the object in N = 10 equally spaced bins in wavelength between λmin = 1000 and λmax = 10000 Angstroms. Each bin has a wavelength width of ?λ = (λmax -λmin)/N = 900 Angstroms. So the astronomer collects photons in the first bin having wavelengths between 1000 and 1900 Angstroms; photons in the second bin having wavelengths between 1900 and 2800 Angstroms; and so on. Photons are collected from the source for a total of ?t = 50 seconds. The telescope has a collecting area of ?A = 20000 square cm. When Astronomer A tallies up the ENERGY contained in each bin, it is discovered that each bin contains the same amount of energy ?E = 90 erg. From this result, plot the flux density in wavelength space Fλ between λmin and λmax. Label your axes. Give your answer in cgs units.

2. Astronomer B performs almost exactly the same experiment. The only difference is that Astronomer B collects photons in N = 10 equally spaced bins in FREQUENCY space, from νmin = c/λmax and νmax = c/λmin, where c is the speed of light. The bins for Astronomer B have equal frequency widths ?ν. Given (a), plot the flux density in frequency space Fν between νmin and νmax. Label your axes. Give your answer in cgs units. Hint: ?ν 6= c/?λ. Ask yourself why.

3. Does Fλ look different from Fν (remember it is one and the same object, observed under identical conditions. The only difference is how Astronomers A and B divide up the photons when plotting their results). Does Fλ have a peak? If so, specify λpeak. Does Fν have a peak? If so, specify νpeak.

4. How much TOTAL energy is collected by Astronomer A? Astronomer B?

5. For the Planck function expressed per unit frequency, what is the most probable frequency νp at a given temperature T? For the Planck function expressed per unit wavelength, what is the most probable wavelength λp at a given temperature T? For what range of temperatures does λp fall in the visible range of the electromagnetic spectrum?

By "most probable," we mean "where the spectrum peaks (is maximized)." This problem is really just an exercise in calculus and algebra. Your answer for λp should match the Wien displacement law given in standard textbooks. But given your results in parts (a)-(c), you should better appreciate why νp 6= c/λp. And when someone shows you a spectrum in the future, you will appreciate why it is important to ask whether the spectrum is binned per wavelength or per frequency.