A better presentation of Planck's radiation law Article in American - - PDF document

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

12

READS

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

(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

• f 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 Bλ (the Planck Function defined with units of wavelength in the denominator) but not with Bν (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

arXiv:1109.3822v2 [astro-ph.SR] 23 Sep 2011

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 Bλ (the Planck function in units

• f 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 Bλ 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 m1 − m2 = −2.5log(F1 F2 ), (1) where F1 and F2 are the fluxes from stars 1 and 2 and m1 and m2 are the “magnitudes”

• f 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 m(λ) − m(λpeak) = −2.5 × log( Bλ(λ) Bλ(λpeak)). (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 Bλ 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.

450 500 550 600 650 700 Λnm 0.1 0.2 0.3 0.4 0.5 0.6 mm500 nm

• 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

• magnitudes. The y-axis values, which are given by -2.5 log(

Bλ(λ) Bλ(500nm)) 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 Bλ to define a ‘peak color’ involves an assumption of the choice of units. An alternative expression for the Planck function, written as Bν, 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

• f the Planck function peak at different wavelengths. At the Sun’s surface temperature, for

example, Bν 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 Bλ 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 Bλ, rather than into equal steps in frequency, as in Bν. 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. ν2

ν1

Bνdν = λ1

λ2

Bλdλ (3) where ν1 = c/λ1 and ν2 = c/λ2. And, similarly, for an infinitesimal step in the spectrum, Bνdν = −Bλdλ. (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 Bλ vs. λ plot, the x-axis of the Bν 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 Bν and Bλ also contain these same factors and so the y-values are also skewed–the values of

• ne 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 Bν vs. ν plot are compressed and the values on the y-axis are amplified relative to those on the Bλ 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, Iν, that has a flat spectrum

when plotted vs. frequency i.e. Iν = I0. (6) In terms of Iλ though, the spectrum is Iλ = Iν

• dν

dλ

• = I0

c λ2, (7) 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

spectrum such as whether more, less, or equal amounts of energy are radiated at different ends of the spectrum. We have two correct expressions for the Planck function and two correct ways of pre- senting spectra, but because of the apparent inconsistencies between them one is generally presented with a dilemma. How should one, for example, describe the “peak color,” or spec- tral peak, of a blackbody source? A better approach, which we advocate for in this paper, is to use a function known as the “Spectral Energy Distribution,” a quantity familiar in a number of fields of astronomy.2,3

II. THE SPECTRAL ENERGY DISTRIBUTION (OR “SED”)

We must, unfortunately, digress a moment to point out a potential source of confusion regarding the term “spectral energy distribution.” This term is sometimes used in the liter- ature synonymously with “spectrum” (i.e. intensity vs. wavelength or vs. frequency). The definition which we use in the discussion below, though, is very specific and different. Per- haps, the physics and/or astronomical community can devise and agree upon a new term which is defined as the function we discuss here. The spectral energy distribution (or SED) is the quantity λFλ plotted as a function of ln λ, or νFν as a function of ln ν, where Fλ and Fν are flux densities, defined as power per area per wavelength interval or per frequency interval, respectively. It’s also implicit that a spectral energy distribution contains data that cover a broad enough spectral range to include most of the energy received. Consider, now, the reasons for the disagreement between the Bν vs. ν and Bλ vs. λ plots discussed in the Introduction. First, we now have x-axis steps of d(ln ν) and d(ln λ) which, according to Eq. 5 are related to each other by d(ln ν) = dν ν = −dλ λ = −d(ln λ). (8) So, we see that the x-axis steps now have equal magnitudes across in the entire electromag- netic spectrum. And, regarding the y-axis values, multiplying Bν by ν and Bλ by λ exactly cancels the difference due to the dν and dλ in the denominators, as indicated by Eq. 5. Note also that the y-axis units of the SED are, simply, flux. Below we show more rigorously that SED type plots of Bν and of Bλ are identical. First, though, we show that the SED is a 5

physically meaningful function. Consider a source with a flux density in units of flux per frequency equal to Fν and in units of flux per wavelength equal to Fλ. The integration over the whole spectrum yields the same total flux and therefore we know that Ftotal = ∞ Fνdν = ∞ Fλdλ (9) We can alter this expression mildly by multiplying the middle expression by ν

ν and the right

expression by λ

λ. Equation 9 then becomes

Ftotal = ∞ νFν dν ν = ∞ λFλ dλ λ (10)

• r

Ftotal = ∞ νFνd(ln ν) = ∞ λFλd(ln λ). (11) A “spectral energy distibution” plot, in general, is a graph of the integrand in Eq. 11; the y-axis contains λFλ or νFν and the x-axis contains ln ν or ln λ. To avoid the confusing issue

• f the logarithms of quantities with units, the parameters ν and λ can be divided by 1 Hz

and 1 nm (or whatever units are most convenient). In practice, since SEDs are generally used to describe the radiation over the entire electro- magnetic spectrum, in order to display the orders-of-magnitude range astronomers making such plots generally need to plot the y-axis logarithmically. And so, SED plots in the lit- erature are often shown as log(λFλ) vs. log(λ) or log(νFν) vs. log(ν). And, for sources of known distance, the Fλ can be multiplied by 4πd2 to define SEDs in luminosity units, e.g. log(λLλ) vs. log(λ). For convenience, we can also incorporate the unitless variable x, defined as x = hν kT = hc λkT (12) so that dν ν =

h kT dν h kT ν = d(ln x)

(13) and dλ λ =

kT hc dλ kT hc λ = d 1 x 1 x

= d(ln(x−1)) = −d(ln x). (14) Equation 11 then becomes Ftotal = ∞ νFνd(ln x) = ∞ λFλd(ln x). (15) 6

(There is no ‘−’ sign in the right expression, even though d(ln λ) = −d(ln x), because the limits of integration needed to be switched as well, since x = 0 when λ = ∞.) We can now show that the two functions in the integrands on either side of Eq. 15 are identical. For discussion sake, we’ll consider a blackbody radiation source and, since intensity relates to flux density merely by dividing by the solid angle of the source, we can substitute in for flux density in Eq. 15 with intensity (a.k.a. surface brightness). We can, therefore, insert the Planck functions, Bν and Bλ, respectively, for the flux densities. Let’s first consider νBν, which is given by νBν = 2hν4 c2 1 exp( hν

kT ) − 1.

(16) We substitute in x = ( hν

kT ) or ν = kTx h

yielding νBν = 2(kT)4x4 h3c2 1 exp(x) − 1. (17) Now, we do the same for λBλ, which is λBλ = 2hc2 λ4 1 exp( hc

λkT ) − 1,

(18) and we substitute in λ =

hc kTx, yielding

λBλ = 2(kT)4x4 h3c2 1 exp(x) − 1. (19) One can see that the expressions on the right hand sides in Eqs. 17 and 19 are identical. Therefore, we see that the SED plots obtained using Bν and those using Bλ have identical quantities on the y-axis and identical steps along the x-axis. These plots, therefore, are the

• same. Since the SEDs are identical whether one measures Bλ or Bν there is no ambiguity

about which one to consider and, therefore, the peak of the SED is a better representation

• f the “peak color” of a radiation source.

In Figure 2 we show an SED plot for a blackbody source at the temperature of the Sun’s

• surface. As mentioned earlier, SEDs are often, by necessity, presented with log(νBν) (or

log(λBλ)) plotted on the y-axis, which has the effect of compressing the y-axis non-linearly, but the qualitative inferences from the plot, such as the location of the peak, are still correct. 7

2 1 1 2 3 lnx lnhΥkT 2.0106 4.0106 6.0106 8.0106 1.0107 1.2107 1.4107 Υ BΥ

• FIG. 2: Spectral energy distribution plot of blackbody radiation at the Sun’s surface temperature.

A. Peak of Spectral Energy Distribution of Blackbody Radiation Source

For the purpose of determining the peak color of a blackbody source, in general, the SED, we argue, is the correct function to use. With that in mind, the “Planckian SED”, like that displayed in Figure 2 becomes a generally useful plot. Let’s now calculate the location of the peak of the Planckian SED. To calculate and set the derivative of the expression in Eq. 17 to zero, since the SED is a function of ln x, we must first substitute in ln x = a. The function

• n the y-axis, then, is expressed as

νBν = 2(kT)4e4a h3c2 1 exp(ea) − 1. (20) Then, setting the derivative with respect to ‘a’ to zero, we have 2(kT)4e4a h3c2 1 (exp(ea) − 1)2[4 exp(ea) − 4 − (ea) exp(ea)] = 0. (21) The left hand side equals zero only when the part in square brackets is zero. Therefore, we wish to know the value of ‘a’ when 4 exp(ea) − 4 − (ea) exp(ea) = 0. (22) Converting ‘a’ back to ‘x’ we get 4ex − 4 − xex = 0, (23) which occurs at x = 0 and 3.9207, the latter value, clearly, being the one of interest. So, we see that the peak of a blackbody SED occurs at xpeak = 3.9207. (24) 8

Note that the value of x at the peak of the Planckian SED is independent of temperature. This follows naturally from the definition of x, where x =

hν kT , and that with blackbody

radiation νpeak ∝ T. Now, converting x to frequency and lambda we find that νpeak = 8.1656 × 1010T Hz K (25) and λpeak = c νpeak = 0.36739 T cm K . (26) Equations 25 and 26 can be considered the Wien Law for Planckian SEDs. Note that these values for λpeak and νpeak fall between the Wien law values for Bλ and Bν (in which λpeak = 0.29 T −1cm K and 0.51 T −1 cm K, respectively).

B. Peak Wavelength of the Sun’s Radiation

We can now, as a demonstration, apply Eq. 26 to the Sun. We first remind the reader that because of the eye’s logarithmic sensitivity the difference in apparent brightness of the Sun across the visible spectrum is small enough that the Sun is effectively white. We use

• Eq. 26 with the Sun, here, just as an example of relating a blackbody’s temperature to

the peak in its SED. Using T = 5800 K, we find that the Sun’s Planckian SED peaks at a wavelength of 633 nm, which translates to red. This discussion might bring into question the perceived colors of stars, in general. Cer- tainly, most people do see stars of different colors. The cool stars are definitely red and the hottest stars are blue. The perception of these colors, though, does not conflict with our discussion of the Sun being white. The radiation from the coolest stars, with surface tem- peratures around 4000 K, and the hottest stars, which get above 30,000 K, have Planckian SED peaks at wavelengths of 918 nm and shorter than 122 nm, respectively. These peaks are in the infrared and the ultraviolet, respectively and so the visible-wavelength spectra of these stars are significantly sloped from one end to the other. For a visual demonstration, we display in Figure 3 the ‘relative-magnitude’ spectra of the Planckian SEDs, defined by m − m(λλref) = −2.5 × log( λBλ(λ) λBλ(λref)) (27) for stars at these temperatures. We have chosen λref’s which yield curves that stay at positive values through the visible spectrum. The Planckian SEDs of stars of 4000 K and 9

• f 30,000 K are both seen to differ by over 3.5 magnitudes from one end of the visible

spectrum to the other. The Planckian SED of a 5800 K star, on the other hand, is relatively flat, differing by less than half a magnitude over most of the spectrum, and dropping by a magnitude only close by the blue edge.

6.1 6.2 6.3 6.4 6.5 lnwavelength1 nm 1 2 3 4 mSEDmSEDΛref

• FIG. 3:

The relative-magnitude spectra of the Planckian SEDs at temperatures of 30,000 K (blue line), 4000 K (red line) and 5800 K (black line) are shown. The lines represent plots of −2.5 log(

λBλ(λ) λBλ(λref)) vs. ln( λ 1m), where λref is 400, 700, and 633 nm for the three lines, respectively.

We see, therefore, that the coolest and hottest stars do have significant colors while stars with moderate temperatures, producing Planckian SEDs that peak in the visible band, are, effectively, white. The apparent assorted colors of stars that people do see most surely has more to do with the complex factors involved with color vision. One should consider, for example, the fact that there appear to be no green stars, even though there are certainly stars whose spectra peak in the green.

C. How to Introduce the SED at an Introductory Level

We feel that it is misleading for textbook authors and instructors who cover blackbody radiation to introduce only Bλ as the Planck function and the corresponding Wien Law. When students in these courses later encounter Bν they most likely feel that they don’t understand the Planck function as well as they thought they had. And the reason for this, ultimately, is because the first class gave them an incomplete discussion. 10

For science majors classes, we think that, in the long run, the education of students will be enhanced by the introduction of both Bλ and Bν along with the Planckian SED. The coverage of the material should include an explanation about why Bλ and Bν are different, that in order to spread the light into a spectrum one must choose whether to spread it by frequency or by wavelength, and that this choice changes the overall shape. The Wien Law for both cases can be given. The instructor can then introduce the Planckian SED as a function designed for plotting the spectrum in a way that accurately depicts how the energy is distributed across the spectrum regardless of the method by which the spectrum is obtained. And, then, the Planckian SED can be displayed and the λpeak and νpeak of the Planckian SED introduced. The confusing part of the SED is the x-axis. In science-majors classes, though, the students should be able to grasp the derivation we present in Eqs. 9-17. The instructor may decide to continue the derivation all the way through Eq. 26 or just jump to Eq. 26, depending on the time and speed of the class. By the time the students see Eq. 26, the point of the Planckian SED should be clear; it doesn’t contain the ambiguity posed by the differing peaks in Bν and Bλ. In non-science majors classes, currently, the Planck function is generally introduced in concept only, with a justified avoidance of presenting the analytical expression. In the same vein, we recommend for non-science majors classes that instructors and textbook authors introduce the Planckian SED conceptually. Just as plots of logBλ vs. logλ are currently shown with little explanation of the axes, we feel that plots of the Planckian SED can be given instead. The non-science majors students will not be asked to comprehend the parameters on the axes any more than they are currently. And then, for the Wien law, the equation for λpeak for only the Planckian SED would be given.

III. CONCLUSIONS

Since a spectrum can be defined by spreading the light into into bins of either equal frequency or equal wavelength, and these will produce spectra with significantly different peaks and shapes, we argue that simple and direct inferences from analysis of such spectra can be misleading. A much better and certainly consistent approach is to use the spectral energy distribution, defined by λFλ vs. ln ν or νFν vs. ln λ. As an example, we show that 11

the calculation of the peak of the Sun’s spectrum is ambiguous when considering the peak

• f the Planck function, but we get a consistent and definite result when calculating the peak
• f the Planckian SED.

We recommend that instructors and textbook authors who introduce blackbody radiation, the Planck curve, and the Wien Law, consider it standard practice to introduce the Planckian spectral energy distribution so that by the end of the discussion of this material the students are left with a clear and unambiguous concept of the location of the peak of the spectrum for blackbody radiation. The Wien Law equivalent, as given by νpeak = 8.1656 × 1010T Hz K (28) and λpeak = 0.36739 T cm K, (29) can be provided to introductory astronomy students.

Acknowledgments

The preparation of this paper was aided significantly by the numerous very helpful dis- cussions with Leo Fleishman. We are also grateful for helpful feedback by Michel Fioc.

1 See, for example, Pasachoff, J. M. and Filippenko, A. The Cosmos (Thomson Higher Education,

Belmont, CA, 2007), 3rd Edition

2 See, for example, Adams, F. C., Lada, C. J. and Shu, F. H.,“Spectral Evolution of Young Stellar

Objects,” The Astrophysical Journal 312, 788–806 (1987)

3 See, for example, Elvis, M., Wilkes, B. J., McDowell, J. C., Green, R. F., Bechtold, J., Willner,

• S. P., Oey, M. S., Polomski, E., and Cutri, R., “Atlas of Quasar Energy Distributions” The

Astrophysical Journal Supplement Series 95, 1–68 (1994)

12

View publication stats View publication stats