O N T H E I N T E G R A T I O N O F I N T E N S I T Y M E A S U R E M E N T S O F

T H E S O L A R C E N T E R A N D L I M B N E A R 3 0 0 N A N O M E T E R S

(Research Note)

KENNETH MOE

Consulting Atmospheric Physicist, 1520 Sandcastle Drive, Corona del Mar, CA 92625, U.S.A.

(Received 6 June, 1983)

Abstract. Two methods of integrating limb-darkening data to calculate the spectral irradiance are compared. Near 300 nm Moe and Milone's (1978) quadratic approximation to the limb darkening yields an integral 3% larger on average than the two linear segments of Kohl et aL (1980) when the same data are inserted in both"formulas. As one would expect, the difference between the two calculations is directly proportional to Moe and Milone's quadratic coefficient. When Moe and Milone's data are inserted in the formula of Kohl et al., the calculated irradiance is 4% higher than when the data of Kohl et al. are used in the same formula. Possible reasons for this difference are suggested.

1. Introduction

A question has recently arisen regarding the precision with which local measurements

of the specific intensity (spectral radiance) of the quiet Sun can be integrated over the

solar disc. One answer to that question is obtained in this brief report by utilizing the

measurements of Kohl et aI. (1980) and those of Moe and Milone (1978). The data of

Kohl et al. were taken at # = 1 and/2 = 0.23, while those of Moe and Milone were taken

at/2 --- 0.729, 0.320, 0.228, 0.158, 0.092, and 0.065. Several methods of integrating these data are compared in the present paper.

2. Integration over the Solar Disc

The solar flux or spectral irradiance, F~, is obtained by integrating the following expres- sion over the solar disc:

1 / m

Fz = 2~z(Re/r) 214(1 ) t R~(#)# d/2, (1) Q /

0

where R e is the solar radius, r the solar distance, I2(1) the specific intensity of the disc

center at wavelength )., R~(#) is the ratio of the intensity at # = cos 0 to the intensity

at the center (# = 1), and 0 is the heliocentric angle. In this paper we are concerned only with the definite integral in Equation (1).

Because they had data at only two points on the solar disc, Kohl et al. approximated

Rz(/2) by two straight lines, one going from # = 1 to/2 = 0.23, and the second going from # = 0.23 to 0 at/2 = 0. This approximation is represented by the dashed lines in Figure 1 for the average over the wavelength interval, 2 = 297.5 _+ 0.5 nm. The quadratic fit of

Moe and Milone to their six data points on the solar disc is represented by the solid

Solar Physics 88 (1983) 9-12. 0038-0938/83/0881-0009500.60. @ 1983 by D. Reidel Publishing Company.

1 0 K E N N E T H M O E

0 1.0 ~ _ ~ =

>. I -

z i , i I.- z .4 II

|

1 . 0

Fig. 1.

/ M O E and

' I i l i I i I I i , i . i . i . ~ , , J l .8 .6 .4 .2 0

= C O S I N E ( 0 )

Two approximations to the limb darkening near 297.5 nm.

line in the same figure. This particular wavelength interval was chosen because the two curves are clearly separated. In some intervals the curves overlap except at the extreme limb (/~ ~ 0).

According to C. A. Zapata (private communication), the value of the definite integral in Equation (1) calculated by Kohl et al. is

1 t ~

I K = ~ Ra(#)~ d# L/ 0

= 0.29 + 0.195R~(0.23). (2)

The quadratic case of Moe and Milone is easily integrated to give

1

I M = f Rx(/,)# d# 0

1 = 5 - (B2)/6 - (C2)/4, (3)

where B 2 is Moe and Milone's linear coefficient and C 2 is their quadratic coefficient. The data averaged over 1.0 nm, and the corresponding integrals are compared in Table I: the central wavelength is in column 1. Rx(0) and Ra (0.23) from Moe and Milone's quadratic equation are given in columns 2 and 3. IK and IM, derived by inserting Moe and Milone's quadratic fits in Equations (2) and (3), respectively, are in columns 4 and 5. Column 6 contains Rx(0.23) measured by Kohl et aL and column 7 gives I K calculated by inserting R~ (0.23) from column 6 in Equation (2). The data are good to only two decimal places, but calculations are carried to three places to avoid introducing errors by rounding.

I N T E G R A T I O N O F L I M B - D A R K E N I N G D A T A

TABLE I

Integration of the limb-darkening functions

11

Wavelength Values computed from Moe and Milone (nanometers)

Values from Kohl's data

R~(0) R~(0.23) I K I M R~(0.23) ]K

295.5 0.131 0.311 0.351 0.346 0.260 0.341 296.5 0.133 0.339 0.356 0.359 0.283 0.345 297.5 0.116 0.350 0.358 0.367 0.300 0.349 298.5 0.114 0.413 0.371 0.397 0.312 0.351 299.5 0.096 0.432 0.374 0.410 0.308 0.350 300.5 0.132 0.404 0.369 0.389 0.315 0.351 301.5 0.114 0.412 0.370 0.397 0.283 0.345 302.5 0.113 0.321 0.353 0.354 0.324 0.353 303.5 0.098 0.277 0.344 0.336 0.299 0.348 304.5 0.134 0.388 0.366 0.381 0.284 0.345 305.5 0.142 0.344 0.357 0.358 0.320 0.352

Average 0.120 0.363 0.361 0.372 0.299 0.348

r

O

x

H !

I-4

fff ,,1

W I - Z m

tl O Ill O Z LU n- W

N

Fig. 2.

.5./. - . 7 - . 6 - -~4 - 3 *~1

- 2 0 . / : - 3 0 -

- 3 5 �9

Q U A D R A T I C COEFF IC IENT , C2

Difference of the solar-flux integrals as a function of Moe and Milone's quadratic coefficient.

12 KENNETH MOE

3. Discussion

The effect of the two different methods of integration can be seen by comparing columns 4 and 5 of Table I: on the average, the quadratic curve yields an integral 3~ larger than the two linear segments. As one would expect, the difference, IK - IM, is nearly proportional to Ca, the quadratic coefficient. This can be seen in Figure 2, where this relationship is graphed. The difference caused by the different data sets can be found by comparing column 4 with column 7: IK calculated from Moe and Milone's data is on average 4 ~o larger than I x from the measurements of Kohl et al. The immediate cause is the larger values of Rz in column 3. The rms difference between corresponding calculations of IK is 5 ~o.

It should be remarked that the values in columns 6 and 7 depend on heliocentric angles, 0, which were established by comparing limb-darkening ratios measured by rocket with ratios measured from the ground in narrow windows by Peyturaux (1955) and by Pierce and Slaughter (1977). Therefore it is possible that the differences between column 4 and column 7 resulted from an imperfect correction for stray light in the ground-based data, small errors in wavelength, or the spreading in wavelength produced by Raman and Brillouin scattering, or by some combination of these effects.

One other method of integration has been tried, because column 2 indicates that the limb-darkening ratio does not fall to zero at the extreme limb. This method is to extrapolate the linear segment of Kohl et al. which extended from R a (1) to R 4(0.23) until it intersects the line # = 0. The resulting integrals differ from the corresponding integrals of the two-segment case by 0 or 1 in the third decimal place, so they are not tabulated. The reason the difference is negligible is made apparent by reference to Equation (1): the factor # in the integrand causes values near/~ = 0 to contribute negligibly to the definite integral.

Acknowledgements

I would like to thank Olav Moe for data and informative discussions relating to the

Skylab experiment; and John Kohl, Carlos Zapata, and Robert Kurucz for sending data and calculations from the Harvard-Smithsonian rocket experiment before they all were published.

References

Kohl, J. L., Parkinson, W. H., and Zapata, C. A.: 1980, Astrophys. J. Suppl. Ser. 44, 295. Moe, O. K. and Milone, E. F.: 1978, Astrophys. J. 226, 301. Peyturaux, R.: 1955, Ann. Astrophys. 15, 302. Pierce, A. K. and Slaughter, C. D.: 1977, Solar Phys. 51, 25.