Appendix A: Equations for Blackbody Radiator

            This appendix analyzes the standard spectral data for a blackbody radiator and derives several convenient relationships from this, including an expression for photon rate. The results are applied to calculate the photon rate for the cosmic microwave radiation received by the COBE satellite, and the photon rate that is radiated by the sun.

A.1 Handbook Data for Blackbody

            Reference [5], page 6-153, gives the following equations describing an ideal blackbody radiator, where Kx is an unspecified constant, P is power, A is surface area, T is temperature in degrees Kelvin, and l is wavelength:

            dP/dl   =   K_x/l⁵(exp[K₁/lT] - 1)                                    (A-1)

            K₁  =  1.438  cm-°K                                                       (A-2)

            P/A  =  5.679x10^-12 T⁴  (watt/cm²)/°K⁴                         (A-3)

Max[d(P/A)/dl]  =  1.290x10^-11 T⁵ (watt/cm²)/cm-°K⁵              (A-4)

In Eq. A-4, the unit mm (which means 10^-6 meter) was replaced by 10^-4 cm. Dividing Eq. A-4 by Eq. A-3 gives

            Max[dP/dl]  =  2.272 T P  (cm-°K)^-1                            (A-5)

From Ref [5], page 6-154, the following were derived by interpolating between the data in the table:

            l_mT  =  0.290  cm-°K                                                    (A-6)

            l_hT  =  0.4107  cm-°K                                                  (A-7)

            [dP/dl] at l_h =  0.772 Max[dP/dl]                              (A-8)

where l_m is the wavelength of maximum dP/dl and l_h is the wavelength for half of the power integral. Half of the integral of the power spectrum lies below the wavelength l_h and half lies above it. Substituting Eq. A-5 into Eq. A-8 gives

            [dP/dl] at l_h =  1.754  T P (cm-°K)^-1                          (A-9)

A.2 Spectrum in Terms of Normalized Frequency

            By combining the above information, the spectrum of Eq. A-1 can be expressed as

dP/dl   =   [56.39 T P]/{(l/l_h)⁵(exp[3.501(l_h/l)] - 1) }                  (A-10 

This is equal to the value of Eq. A-9 when l = lh. 

            We need the power spectrum of a blackbody expressed in terms of frequency f, which is equal to c/l.  Rather than show the spectrum directly in terms of frequency f, it is often desirable to use the variable (1/l) as a normalized frequency, which is proportional to f. Since d(1/ l) = - dl/l2, the spectrum relative to (1/l) is related as follows to the spectrum relative to l

             dP’/[d(1/ l)]  =   - l² [dP’/dl]  =   l² [dP/dl]                    (A-11)

The variable P’ (for the 1/l spectrum) is zero at zero frequency or infinite wavelength, whereas the variable P (for the l spectrum) is zero for zero wavelength or infinite frequency. Consequently, dP’ is equal to –dP. The prime on dP’ is henceforth dropped. Applying Eq. A-11 to Eq. A-10 gives the power spectrum in terms of (1/l); 

 dP/d(1/ l)  =  [56.39l_h²TP]/{(l/l_h)³(exp[3.501(l_h/l)] - 1)}  (cm-°K)^-1

                                                                                                (A-12)

By Eq. A-7, l_hT is equal to 0.4107 cm-°K, and so this can be expressed as

 dP/d(1/ l)  =  [23.16 l_h P]/{(l/l_h)³(exp[3.501(l_h/l)] - 1)}   (A-13)

                  =  [ 9.512(P/T) cm-°K]/{(l/l_h)³(exp[3.501(l_h/l)] - 1)} 

This can also be expressed as follows in terms of the frequency ratio f/f_h, where f_h is the frequency at the wavelength l_h:

dP/d(1/l)  =  [9.512 (P/T)(f/f_h)³]/{exp[3.501(f/f_h)] - 1)}  cm-°K    (A-14)

Setting the derivative of this equal to zero shows that its maximum (peak) value occurs at the frequency

            f_p  =  0.8059 f_h                                   (A-15)

The maximum value of Eq. A-14 (at this frequency f_p) is

            Max[dP/d(1/l)]  =  0.3151 (P/T) cm-°K                          (A-16)

Comparing Eqs. A-7, A-15 shows that this maximum ("peak") value occurs at a wavelength lp given by

            lpT  =  0.5096  cm-°K                                                  (A-17)

            Equation A-14 can be approximated quite accurately, except at low frequencies, by ignoring the -1 term in the denominator. At the frequency f_h , the exact denominator is 32.15, and the denominator in the approximation is 33.15. The numerator is multiplied by the ratio (33.15/32.15) to obtain the following approximation, which matches the original equation exactly at the frequency f_h:

dP/d(1/ l)  =  [9.81 (P/T)(f/f_h)³]/{exp[3.501(f/f_h)]}  cm-°K  (A-18)

A.3 Spectrum in Terms of Photon Rate

            Let us denote the photon rate in photons per second as N*. The photon rate over a small frequency band is denoted DN*. The energy of a photon is denoted u_p and is equal to

            u_p  =  hf  =  hc/l                                 (A-19)

where  Planck's constant h is given in Ref [17], page 7-3, as

            h  =  6.6251x10^-27  erg-sec                                     (A-20) 

The photon rate within a small interval D(1/l) is equal to

DN*  =  [DP/u_p]  =  (1/u_p)[dP/d(1/ l)]D(1/l)            (A-21)

Energy u_p is equal to u_p(h)(f/f_h), where u_p(h) is the photon energy at the frequency f_h; and the increment D(1/l) is equal to (1/l_h)D(f/f_h). Hence Eq. A-21 can be expressed as

            u_p(h)DN*  =    (f_h/f)[dP/d(1/l)](1/l_h) D(f/f_h)           (A-22)

Substitute into Eq. A-22 the approximation given in  Eq. A-18 to obtain 

u_p(h)DN*  »   [9.81(P/l_hT)(f/f_h)2D(f/f_h)]/{exp[3.501(f/f_h)]}   cm-°K

                                                                                       (A-23)
Integrate this to obtain the total photon rate N*

u_p(h)DN*  »   [9.81(P/l_hT)cm-°K] ¦ {(f/f_h)²]/exp[3.501(f/f_h)]}d(f/f_h)

                                             (A-24)  
The integral is performed over limits from zero to infinity. In accordance with Eq. A-7, replace lhT by (0.4107 cm-°K).  Define the exponent 3.501(f/f_h) as the variable x. In terms of the variable x, Eq. A-24 becomes 

            u_p(h)N*  »    0.5566  P ¦   x² e^-x dx                              (A-25)

It can be shown that the integral (from zero to infinity) is equal to 2, and so Eq. A-24 becomes

            N*  »   2(0.5566) P/u_p(h)  =   1.113 P/u_p(h)                (A-26)

  Define u_p(h)/1.113 as the parameter u_bb . Equation A-26 becomes

            N*  =  P/u_bb                                                               (A-27)

This appendix treats this as an exact relation, but the analysis using the blackbody spectrum from which our value of u_bb was derived was not exact. Nevertheless the error in the analysis should not be important. The wavelength l_bb is defined in terms of u_bb by

             ubb  =  hc/lbb                                                              (A-28)

  Apply the value for lhT in Eq. A-7, noting that u_p(h) is equal to 1.113 u_bb. The corresponding value for l_bb is

            l_bbT  =  0.457  cm-°K                                                    (A-29)

  Thus, one can compute from Eq. A-29 the effective photon wavelength l_bb for a blackbody radiator of temperature T, and from Eq. A-28 one can compute the effective photon energy ubb for that blackbody radiator. Equation A-27 shows that the photon rate from the blackbody is equal to the power P radiated from the blackbody divided by u_bb.

A.4 Intensity of COBE Cosmic Microwave Radiation

             In 1989, the Cosmic Background Explorer (COBE) satellite made accurate measurements of the cosmic background radiation at wavelengths from 0.05 cm to 1 cm. Strong signals were detected, coming almost uniformly from all directions, with a spectrum that accurately matches a blackbody at a temperature of 2.73 °K. Narlikar [6] shows on page 328 a plot of this spectrum versus reciprocal wavelength (1/l). The following maximum value was read from the plot:

Max{dP/d(1/l)}  =  1.16x10^-4AW [(erg/sec)/cm²]/(steradian-cm^-1)

                                                                                                (A-30)   
where P is power (erg/second), A is the capture area of the receiver (cm²), and W is the acceptance solid angle of the receiver (steradians). Equation A-16 showed that the maximum value of a blackbody power spectrum versus (1/l) is equal to 0.3151 P/T cm-°K, which is (0.1154 P cm) for a blackbody temperature T of 2.735 °K. Setting Eq. A-30 equal to (0.1154 P cm) gives the total power P of the COBE spectrum:

            P  =  1.005x10^-3 AW [(erg/sec)/cm²]/steradian        (A-31)

This is the power received within a small acceptance solid angle W. When this solid angle W is set equal to p steradians, the equation gives the total power falling on a surface in space of area A. The power per surface area is  

P/A  =  1.005x10^-3 p [(erg/sec)/cm²]  =  3.157x10^-10 watt/cm²  (A-32)

This is expressed in watts, where 1 watt = 10⁷ erg/sec. Equation A-3 gave a general expression for the power per unit area radiated from the surface of an ideal blackbody of temperature T. Setting this temperature T equal to 2.73 ° K gives

   For ideal blackbody at 2.735 ° K:

            P/A  =  3.154x10¹⁰ watt/cm²                                                (A-33)

The values of Eqs. A-32, A-33 are practically identical. Therefore, the blackbody microwave radiation level measured by the COBE satellite is the same as the level that would be radiated from the surface of an ideal blackbody of the measured blackbody temperature 2.73 ° K.

A.5 Photon Rate for Solar Radiation

            Let us consider the radiation from the sun. Table 6-3 in Chapter 6 gives the following for the luminosity of the sun, which is the power radiated from the sun:

L_sun  =  P_sun  =  3.826x10³³ erg/sec  =  3.826x10²⁶ watt      (A-34)

This was denoted L_sun in Universe [1], but here is denoted P_sun. The diameter of the sun, which is denoted D_sun, is 1,392,530 kilometers (km). Hence, the surface area of the sun is

Asun  =  p D_sun2  =  6.092x10¹² km²  =   6.092x10¹⁸ meter²  (A-35)

Dividing Eq. A-34 by Eq. A-35 gives the power per unit surface area radiated from the sun:

P_sun/A_sun  =  62.8x10⁶ watt/meter²  =  6280 watt/cm²            (A-36)

The sun radiation can be approximated as an ideal blackbody. Equation A-3 gave the following for power per unit-area radiated from a blackbody of temperature T

            P/A  =  5.679x10^-12 T⁴  (watt/cm²)/°K⁴             (A-37)

Setting Eq. A-37 equal to A-36 gives the following for the equivalent blackbody temperature of solar radiation, which we denote T_sun

            T_sun  =  [6280/(5.679x10^-12)]^1/4  =  5770 °K         (A-38)

Reference [7], page 43, states that the surface temperature of the sun is 10,430 ° F, which is 5777 ° C or 6050 ° K. Since this is only 5 percent greater than the ideal blackbody temperature (5770 ° K), the solar radiation closely approximates the radiation from an ideal blackbody. 

            Equations A-27 to A-29 showed how to calculate the photon rate of the radiation from a blackbody when the total power P and blackbody temperature T are known. Applying these principles gives the following for the effective photon wavelength lbb of solar blackbody radiation

            (l_bb)_sun  =  (0.458 cm-°K)/T  =  7.94x10^-5 cm                 (A-39)

The blackbody temperature T of the sun was set equal to 5770 °K. The energy of a photon at this wavelength, which is denoted u_bb, is computed from                                                                                               

            (u_bb)_sun  =  hc/(l_bb)_sun  =  2.50x10^-12 erg                          (A-40)

where h is Planck's constant (6.6251x10-27 erg-sec) and c is the speed of light (3x10¹⁰ cm/sec). As was shown in Eq. A-27, the photon rate N* of the power radiated from a blackbody is equal to P/u_bb . Divide the solar radiated power Psun of Eq. A-34 by the solar blackbody photon energy (u_bb)_sun of Eq. A-40 gives the following photon rate N* for radiation from the sun

            N*_sun =  P_sun/(u_bb)_sun  =  1.530x10⁴⁵  photon/sec            (A-41)

A.6 Photon Density for Blackbody Radiator

            Equation A-3 gave the following for the power per unit area radiated by a blackbody:

            P/A  =  5.679x10^-5 T⁴  (erg/sec)/cm²-°K⁴                  (A-42)

This has been converted from watts to erg/sec by noting that 1 watt is equal to 10⁷ erg/sec. From Eqs. A-27, A-28, the photon rate per unit area is equal to

N*/A   =   (P/A)/u_bb   =  [(P/A) l_bb]/(hc)                                  (A-43)

Substitute into this the expression for l_bbT in Eq. A-29:

N*/A   =  [(P/A) l_bb]/(hc)  =  [(P/A)(0.457 cm-°K)]/(hcT)            (A-44)

Combine Eqs. A-42, A-44

N*/A   =   [2.595x10^-5 T³]/(hc)            (erg/sec)/(cm-°K³)        (A-45)

Substitute into this the speed of light c (3x10¹⁰ cm/sec) and the value of Planck’s constant h given in Eq. A-20:

            N*/A  =  1.3056x10¹¹ T³  (photon/sec)/(cm²- ° K³)            (A-46) 

This is a general expression for the intensity of the radiation from an ideal  blackbody, expressed in terms of photon rate.

            An ideal blackbody radiator operates under conditions of thermal equilibrium, and under this condition the photons are packed as closely as is physically possible. Let us define the parameter leff such that the radiation from an ideal blackbody has one photon within a cube of dimensions leff on a side. The number of photons within a cube of thickness leff is equal to

            N  =  (N*/A)(l_eff)² (l_eff/c)  =  (N*/A)(l_eff)³/c      (A-47)

Since (l_eff)² is the cross section area of the cube, the expression (N*/A)(l_eff)² is the rate at which photons enter the cube. This rate is multiplied by (l_eff/c), which is the time for light to travel the thickness of the cube, to obtain the number of photons N within the cube. Set this number N equal to unity, and solve for l_eff:

            (l_eff)³  =  c/(N*/A)                                  (A-48)         

Substitute into this the expression for N*/A in Eq. A-46, and the value for the speed of light c.

            (l_eff)³  =  0.2297/T³   (cm- ° K)³                      (A-49)         

Taking the cube root of this gives

            l_effT  =  0.612   cm- ° K                                 (A-50) 

Equation A-7 showed that l_hT is equal to 0.4017 cm-°K. Hence l_eff is related as follows to l_h:

            l_eff  =  1.49 l_h                                                  (A-51)

Remember that l_h is the half power wavelength of the blackbody spectrum: half of the power lies below l_h and half lies above it. This result can be summarized by

A blackbody radiates the maximum possible photon rate for thermal equilibrium at its wavelength band. This photon rate is equivalent to one photon within a cube having a thickness of l_eff , which is  approximately equal to 1.50 l_h (where  l_h is the half-power wavelength of the spectrum}. About ¾ of the power lies below the wavelength l_eff  .

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