See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/51939342 A better presentation of Planck's radiation law Article in American Journal of Physics · September 2011 DOI: 10.1119/1.3696974 · Source: arXiv CITATIONS READS 12 865 2 authors , including: F. P. Wilkin Union College 41 PUBLICATIONS 1,785 CITATIONS SEE PROFILE All content following this page was uploaded by F. P. Wilkin on 17 March 2015. The user has requested enhancement of the downloaded file.
Why Using the Peak of the Planck Function to Explain a Star’s Color is Incorrect; A Better Approach Uses the Spectral Energy Distribution Jonathan M. Marr < marrj@union.edu > & Francis P. Wilkin < wilkinf@union.edu > Union College, Schenectady, NY arXiv:1109.3822v2 [astro-ph.SR] 23 Sep 2011 (Dated: September 27, 2011) Abstract A common discussion in Astronomy 101 courses involves using the Wien Law to explain why hotter stars are bluer and cooler stars redder. Sometimes this is used to address the yellow color of the Sun. We argue here that this is an incorrect approach and is potentially deceiving for astronomy students. The main problem with this approach is that it only appears to work when using B λ (the Planck Function defined with units of wavelength in the denominator) but not with B ν (in which the units of wavelength are replaced by units of frequency). We discuss these flaws and argue that converting the Planck function to a “spectral energy distribution” is a better function for determining the ‘peak color’ of a blackbody radiation source. We discuss how the spectral energy distribution can be explained at introductory levels and propose that it become a commonly discussed concept in introductory classes. 1
I. INTRODUCTION The concept of blackbody radiation, along with the associated Stefan-Boltzmann Law and Wien Law, is so fundamentally important to understanding stars that this otherwise complex topic is viewed as a crucial pillar of physics to be introduced in all Astronomy 101 courses. A commonly presented example of the Wien law uses it to explain the Sun’s appearance. By simple application of the Wien Law, one finds that at the Sun’s surface temperature of 5800 K, the wavelength of the peak of B λ (the Planck function in units of power per area per steradian per wavelength interval) occurs at approximately 500 nm, which is in the green near the center of the visible band. An unfortunate mistake that is sometimes made then is to say that this indicates that the Sun is actually green and that it appears yellow due to the greater scattering of shorter wavelengths by the atmosphere. A better answer, as given in many texts, 1 is to explain that the Sun is white but looks yellow because of the atmospheric scattering. The sensitivity of the human eye to light is a logarithmic dependence and so the relative amounts of the different colors in the intrinsic spectrum of the Sun’s radiation is, actually, barely noticeable across the visible spectrum. To show this quantitatively we have converted B λ into a function similar to relative stellar magnitudes, a familiar scale regarding naked-eye visual measurements. Stellar magnitudes are logarithmically related to flux and are defined by m 1 − m 2 = − 2 . 5 log ( F 1 ) , (1) F 2 where F 1 and F 2 are the fluxes from stars 1 and 2 and m 1 and m 2 are the “magnitudes” of stars 1 and 2. (Note that a brighter star has smaller magnitude.) The brightest star in the sky, Sirius, has a magnitude -1.4 and the faintest stars visible with the naked eye on a moonless night at a location with no artificial lights are about magnitude 6. We have similarly devised a ‘relative-magnitude’ spectrum by calculating a wavelength-dependent logarithmic brightness given by B λ ( λ ) m ( λ ) − m ( λ peak ) = − 2 . 5 × log( B λ ( λ peak )) . (2) Figure 1 shows the ‘relative-magnitude’ spectrum for Planck curve at the Sun’s surface temperature. As can be seen in Figure 1, across most of the visible spectrum, from 400 nm to 630 nm, the Sun’s B λ varies by less than a third of a magnitude and the greatest 2
difference, between the peak and the red edge, is about 0.6 magnitudes. These differences are too small to cause the Sun to appear to be of a specific color. Therefore, using the Wien law to assign a specific color of the Sun is, actually, a gross overinterpretation of its power. In reality, the peak is broad and the total wavelength range of the visible band is narrow. m � m � 500 nm �� 0.6 0.5 0.4 0.3 0.2 0.1 Λ � nm � 450 500 550 600 650 700 FIG. 1: The relative-magnitude spectrum of the Sun’s Planck spectrum. The logarithmic bright- ness of the Sun at different wavelengths is shown on a scale similar to that of relative stellar B λ ( λ ) magnitudes. The y-axis values, which are given by -2.5 log( B λ (500 nm ) ) for a temperature of 5800 K, represent the magnitudes at each wavelength relative to the magnitude at 500 nm. Since a larger magnitude represents a fainter brightness, the peak of the Sun’s spectrum appears as a minimum in this plot. A more significant problem with using the peak of B λ to define a ‘peak color’ involves an assumption of the choice of units. An alternative expression for the Planck function, written as B ν , is defined with units of power per area per steradian per frequency interval. A simple fact that commonly causes great consternation when first encountered is that these two forms of the Planck function peak at different wavelengths. At the Sun’s surface temperature, for example, B ν peaks at a wavelength of 880 nm, which is in the infrared – not green or even in the visible band. How could this be? Isn’t the Planck function defined well enough that regardless of which units we use we should come to the same qualitative conclusions about the source? The difference really revolves around the method by which the spectrum is obtained or defined. In calculating the peak of the blackbody spectrum in terms of B λ one assumes that the detector (whether it be a human eye or a spectrograph) infers the “color” of the radiation 3
by binning the detected radiation into equal steps in wavelength, as in the definition of B λ , rather than into equal steps in frequency, as in B ν . In principle, both functions are correct physical descriptions of the radiation. For example, the integral of these functions over any specific range of the spectrum, should agree, i.e. � ν 2 � λ 1 B ν dν = B λ dλ (3) ν 1 λ 2 where ν 1 = c/λ 1 and ν 2 = c/λ 2 . And, similarly, for an infinitesimal step in the spectrum, B ν dν = − B λ dλ. (4) The difference in the shapes of the plots is due to the non-linear relation between dν and dλ , dν = − ν λdλ, (5) which affects the relative shapes of the plots in two ways. First, the x-axis steps, when comparing the two plots, are skewed. Relative to the B λ vs. λ plot, the x-axis of the B ν vs. ν plot is stretched at the higher-frequency (and smaller-wavelength) end and compressed at the lower frequencies (and larger wavelengths). And, similarly, the denominators of B ν and B λ also contain these same factors and so the y-values are also skewed–the values of one plot are magnified relative to the other plot at one end and vice versa at the other end. At the low-frequency end, for example, the steps along the x-axis on the B ν vs. ν plot are compressed and the values on the y-axis are amplified relative to those on the B λ vs. λ plot. This apparent disagreement in the plots, actually, has nothing to do with the Planck function. Consider, for example, a source with an intensity, I ν , that has a flat spectrum when plotted vs. frequency i.e. I ν = I 0 . (6) In terms of I λ though, the spectrum is � dν � c � � I λ = I ν � = I 0 λ 2 , (7) � � dλ � and so appears to decrease with wavelength with a power-law index of -2. They do not disagree, really, because when integrated correctly one finds they give the same amount of energy radiated in any particular segment. However, the visual representation is certainly misleading. Many young scientists could easily be deceived about a basic aspect of the 4
Recommend
More recommend
