Integrating the Blackbody Curve, Part 2


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In Part 1, the series approximation for the integrated blackbody spectrum were

F ( v ) = 15 π 4 ∑ m = 1 ∞ e − mv [ v 3 m + 3 v 2 m 2 + 6 v m 3 + 6 m 4 ] for large  v = 1 + 15 π 4 ∑ m = 1 ∞ e − mv [ v 3 m + 3 v 2 m 2 + 6 v m 3 + 6 1 − e mv m 4 ] for small  v . matrix { alignr F(v) # {}={} # alignl 15 over { %pi^4 } SUM from m=1 to infinity { e^-mv left [ v^3 over {m^{}} + {3 v^2 } over {m^2} + {6 v} over {m^3} + 6 over {m^4} right ] } # "for large " v ## {} # {} # {} # {} ## alignr {} # {}={} # alignl 1 + 15 over { %pi^4 } SUM from m=1 to infinity { e^-mv left [ v^3 over {m^{}} + {3 v^2 } over {m^2} + {6 v} over {m^3} + 6 {1 - e^mv} over {m^4} right ] } # "for small " v "." }

(2.1)

The first summation term dominates at large v. This term can be summed up into a closed-form expression as

∑ m = 1 ∞ q m m = ∑ m = 0 ∞ q m + 1 m + 1 = ∑ m = 0 ∞ ∫ 0 q x m dx = ∫ 0 q ∑ m = 0 ∞ x m dx = ∫ 0 q dx 1 − x = ∫ 0 1 − q dy y , y = 1 − x = − ln y | y = 1 1 − q = − ln ( 1 − q ) . matrix { alignr {} SUM from m=1 to infinity { alignc q^m over m } # {}={} # alignl {} SUM from m=0 to infinity { alignc { q^{m+1} } over { m+1} } ## alignr {} # {}={} # alignl {} SUM from m=0 to infinity { INT from 0 to q { x^m dx } } ## alignr {} # {}={} # alignl {} INT from 0 to q { SUM from m=0 to infinity { x^m dx } } ## alignr {} # {}={} # alignl {} INT from 0 to q { alignc dx over {1-x} } ## alignr {} # {}={} # alignl {} INT from 0 to {1-q} { alignc dy over y }, y=1-x ## alignr {} # {}={} # alignl {} left none -ln y mline {} sub {y=1} sup {1-q} right none ## alignr {} # {}={} # alignl {} -ln(1-q) "." }

(2.2)

The formula for F(v) for large v now becomes

F ( v ) = 15 π 4 ∑ m = 1 ∞ e − mv [ 3 v 2 m 2 + 6 v m 3 + 6 m 4 ] − 15 π 4 v 3 ln ( 1 − e − v ) . F(v) = 15 over { %pi^4 } SUM from m=1 to infinity { e^-mv left [ { 3 v^2 } over {m^2} + {6 v} over {m^3} + 6 over {m^4} right ] } - 15 over { %pi^4 } v^3 ln( 1 - e^{-v} ) "."

(2.3)

Applying this to the formula for large v gives an improvement of about half a magnitude, as shown in Figure 1.

Figure 1: Errors in series approximation to Blackbody curve after summation of first term

For small v, the fit becomes somewhat worse. See Figure 2.

Figure 2: Errors in series approximation to Blackbody curve after summation of first term for small v

The improvement for small v must be offset by the infinite series already added in Part 1.

Unfortunately the remaining terms do not appear to have convenient closed-form summations. One approach would be to use the logarithmic series accelerations such as methods by Herbert H. H. Homeier. Another is to subtract off similar summations, pushing the error terms off to higher orders.

The story continues in Integrating the Blackbody Curve, Part 3.


Categories Physics, Optics

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