Approximation for the Planck Radiation Function Marcus Hatch Institute of Optics, University of Rochester, Rochester, New York 14627. Received 14 December 1972. This describes a polynominal approximation for the in- tegral of the Planck radiation function where x is the dimensionless parameter hv {kT) -1 . This is useful because the integral cannot be expressed in closed form and the polynomial approximation can be computed much more rapidly than by numerical integra- tion of the above integral. For x > 15 it is sufficiently ac- curate to use the Wien approximation x 3 exp(—x), which can be integrated in closed form. The region 0 ≤ x ≤ 15 March 1973 / Vol. 12, No. 3 / APPLIED OPTICS 617
Transcript
Approximation for the Planck Radiation Function Marcus Hatch
Institute of Optics, University of Rochester, Rochester, New York 14627. Received 14 December 1972. This describes a polynominal approximation for the in
tegral of the Planck radiation function
where x is the dimensionless parameter hv {kT)-1. This is useful because the integral cannot be expressed in closed form and the polynomial approximation can be computed much more rapidly than by numerical integration of the above integral. For x > 15 it is sufficiently accurate to use the Wien approximation x3 exp(—x), which can be integrated in closed form. The region 0 ≤ x ≤ 15
March 1973 / Vol. 12, No. 3 / APPLIED OPTICS 617
Table I. Constants that Appear in the Polynominal in Eq. (4)°
"The number following the D is the power of ten, e.g., 1.234D-05 = 1.234 X l0-5 . The interval denotes the range of x over which the polynomial is valid. A blank entry indicates that this term was not used.
618 APPLIED OPTICS / Vol. 12, No. 3 / March 1973
Table I I . A Comparison of the Polynominal Approximation given in Eq. (4) with the Numerical Integration of Eq. (1) by Weddle's Rulea
a Both are multiplied by 15Π-4 to normalize the area to 1.0.
was divided into fifteen equal parts and for each interval Chebyshev polynomials were fitted to the Planck function
In each interval the degree of the polynomial was chosen to keep the maximum fractional error less than 10 - 8 , which is single precision accuracy for many digital computers.
The fraction error is the absolute value of
where the subscript a denotes the approximation. The polynomial expression was then integrated and the constant of integration adjusted to obtain an approximation
Table I lists these constants. The data were incorporated into a computer subroutine that uses the appropriate polynomial for the interval in which x occurs. Table II compares the normalized function 15π-4P{x)a with numerical integration. Weddle's rule1 was used with 576 su-bintervals in all cases but the last two, which used 2880. The accuracy is better than that obtained with most ta-bles2 that contain only five or six significant figures.
References 1. R. A. Buckingham, Numerical Methods (Pitman, London,
1962), p. 80. 2. M. Czerny and A. Walther, Tables of the Ffactional Functions
for the Planck Radiation Law (Springer, Berlin, 1961).
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