A C O M P A R I S O N OF S Y N T H E T I C A N D M E A S U R E D

S O L A R C O N T I N U U M I N T E N S I T I E S

A N D LIMB D A R K E N I N G C O E F F I C I E N T S

T. R. AYRES

Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards,

Boulder, Colo. 80309, U.S.A.

(Received 1 November, 1977)

Abstract. Absolute continuum intensities and wavelength-dependent low-order polynomial fits to optical and infrared continuum limb darkening provide useful discriminants among single-component models of the solar photosphere. The thermal structure in best quantitative agreement with the recent center-limb measurements by Pierce and Slaughter (1977) and by Pierce et. al. (1977) is the semi- empirical model by Vernazza, Avrett and Loeser (VAL). However, the VAL model M temperatures must be scaled upward by a factor of 1.0154-0.005 to be consistent with the Labs and Neckel absolute calibration of continuum high points in the optical region 0.40-0.65 t~m.

1. Introduction

The most straightforward, semiempirical models of the solar photosphere are those constructed to match limb darkening behavior of continuum high points (see e.g. Avrett, 1977). Although ground-based center-limb measurements can probe only a relatively thin zone of the photospheric thermal structure ( - 1.5 ~< log %.5 , ~ ~< 0.5; Mihalas, 1970), an understanding of this region of the solar atmosphere is crucial in chemical abundance determinations (Withbroe, 1978; Ross and Aller, 1976); in empirical estimates of pressure broadening parameters (Holweger, 1972; Shine, 1973; Ayres, 1977); and in studies of convective energy transport in the deep photosphere (Spruit, 1974).

Owing to the importance of the continuum forming region of the solar atmos- phere, I will compare the recent center-limb studies of optical and infrared continuum intensities by Pierce and Slaughter (1977) and by Pierce et al. (1977) with predictions based on several current photospheric thermal structure models, using calibrated specific intensity measurements by Labs and Neckel (1968, 1970) to set an absolute temperature scale.

2. Observations

Pierce and Slaughter (1977) and Pierce et al. (1977) have published extensive measurements of continuum center-limb behavior over the wavelength range of 0.3-2.4/zm. These data are tabulated in the form of second- and fifth-order polynomial fits (in tz and In/~) t o empirical limb darkening curves. The obser- vations were obtained at Kitt Peak by repeated drift scans along activity-free

Solar Physics 57 (1978) 19-26. All Rights Reserved Copyright (~) 1978 by D. Reidel Publishing Company, Dordrecht, Holland

20 T . R . AYRES

diameters on the large solar image of the 1.5 m McMath telescope. The authors have carefully corrected their measurements to account for scattered light and seeing distortions. Full details concerning the observational methods and data reduction can be found in the original references.

Instead of attempting to match the apparent limb darkening curves at a representative sample of wavelengths (e.g. Vernazza et al., 1976), I decided for economy's sake to model the wavelength dependence of the second-order (In 12) polynomial coefficients themselves. A somewhat restricted wavelength range of 0.4-2.4 12m was chosen for comparison with synthetic limb darkening behavior, owing to the increasing influence of line blanketing on the continuum 'windows' shortward of - 0 . 4 12m (Labs and Neckel, 1970), and because the low-order polynomial fits become progressively less reliable with decreasing wavelength below about 0.6 12m (Pierce and Slaughter, 1977). Longward of 0.6 12m, where the low-order polynomial representation is more accurate, the intrinsic scatter among neighboring limb darkening curves is comparable to the accuracy of the second- order relative to the fifth-order fits (~< 1% ). Hence, little additional information can be gained by going to the higher order representation.

The second-order In 12 representation of the limb darkening data is expressed as

Ix (In 12______) = 1 + bx (2) In 12 + cA (2)[ln/2] 2 . (1) Ix(0)

Since the wavelength dependences of the bx (2) and cx (2) coefficients have similar shapes, I decided to fit the quantity [c(2)/b(2)]x instead of cx(2) itself.

The center-limb behavior data provide information on the temperature gradients in the region of continuum formation (Mihalas, 1970). An absolute temperature scale can be determined only by fitting calibrated continuum intensities. I have adopted the photometry of Labs and Neckel (1968,1970) for this purpose, includ- ing those disk center high points which coincide with continuum windows measured by Pierce and Slaughter (0.40-0.65 12m). By smoothly interpolating among several of the calibrated high points, I estimate a reference central intensity at 0.5 12m of /0.5(0)= 4.15 +0.1 • 10 l~ ergs cm -2 s -1 sr -1 12m -1. The uncertainty in Io.5(0) cor- responds to the • absolute accuracy cited by Labs and Neckel (1968). However, this uncertainty may be somewhat optimistic owing to the larger syste- matic differences obtained in other, independent calibrations covering the same region of the solar energy distribution (Pierce and Allen, 1977).

3. Continuum Synthesis

The synthesis of b(2)x and [c(2)/b(2)]x limb darkening coefficients and disk center continuum intensities to compare with the measured data is straightforward. A solar model is initially specified as a run of temperature T(K) with reference continuum optical depth ~'05,r~. The pressure stratification of the TOO model is determined by solving the hydrostatic equilibrium condition in planar geometry.

A C O M P A R I S O N O F S Y N T H E T I C A N D M E A S U R E D S O L A R C O N T I N U U M INTENSITIES 21

Densities are obtained from pressures assuming the perfect gas law and an equation of state based on LTE for molecular hydrogen, H~- and the important electron donor metals, and non-LTE for neutral hydrogen and H-.

Hydrogen departure coefficients for the Vernazza et al. (1976; VAL) model M were kindly provided by E. H. Avrett. Departure coefficients for the other models considered here (Section 4 below) were interpolated from the VAL model accord- ing to temperature.

Chemical abundances used in the ionization equilibrium are those of Withbroe (1978), with the exception of helium which is taken to be 0.088 (by number relative to hydrogen) in order to agree with solar evolution models (e.g. Flannery and Ayres, 1978).

Background opacities adopted in the ~" -+ P conversion and in the monochromatic intensity synthesis are: The bound-free and free-free continua of H- and neutral hydrogen; H I Rayleigh and electron scattering; H~-; and Peach's (1970) Mg i, Si I. and A1 t photoionization continua.

The most important of these background opacities is H-. A photodetachment cross section for H- was constructed from the recent multi-channel J-matrix calculations of Broad and Reinhardt (1976), which imply a cross section maximum of 0-a = 3.98 x 10 -17 cm 2 at 0.96/zm, and a value (interpolated) at 0.5/zm of 0-0.5 = 2.85 • 10 -17 cm 2. Using the VAL H- cross section, which differs somewhat from the cross section adopted here (see e.g. Vernazza et al., 1976), produces no significant differences in either the pressure stratification of the continuum models or in the synthetic limb darkening coefficients. (Indeed, the differences between LTE and non-LTE representations of a given T(r ) model also have a negligible effect on the computed center-limb behavior, as do moderate changes in the T(r) structure above to.5 ~ 10 -2 and below r0.5 - 10.)

Limb darkening coefficients at a particular wavelength are obtained by synthesizing monochromatic intensities at the 18 'normal'/z positions adopted by Pierce and Slaughter (1977) and by Pierce et al. (1977), and then fitting a second- order least-squares polynomial in In/x to the calculated center-limb behavior. Auer's (1976) Hermitian differencing scheme is used in a Feautrier-type ray solution in order to obtain reasonably accurate intensities for the ten-points-per- decade optical depth spacing chosen here (ro.s = 10 - ~ ~ 10 l"s in steps of 0.1 in the log). The continuum synthesis for each photospheric model was carried out at a sufficiently large number of wavelength points ( - 4 0 ) to accurately determine the wavelength dependence of the b(2) and c(2) coefficients.

4. Results and Conclusions

The T(r ) models chosen for the center-limb and absolute intensity comparisons are illustrated in Figure 1. These are: the Vernazza et aL (1976) model M, which is based on an extensive analysis of ultraviolet, optical and infrared continuum data; a preliminary photospheric model by Allen (1977; see Avrett, 1977) derived to fit

22 T.R. AYRES

9000

8000

VA L

~7000

<:[ 6 d

I M

6000

KURUCZ (i974

ALLEN (197ro) //,4 :/'

~~" VERNAZZA e~_ (i976)

5000 --

" L . . . . . - ~ "

4 0 0 0 ~ L ' - I I ' ~ [ ] [ l I 1 1 1 -~ - I I 1 1 1 l I I I [ I I I I I I I I I I 1 I

LOG To,sff m

Fig. 1. Adopted thermal structure models of solar photosphere.

measurements of the center-limb behavior of infrared window intensities, especi- ally in the vicinity of the H- opacity minimum ( - 1.6 t~m); and the elaborate LTE line-blanketed radiative equilibrium model by Kurucz (1974), including mixing- length convection. These three models are representative of the range of thermal structures currently advocated for the solar photosphere (Avrett, 1977). The remaining model, VAL 1.015, is a modification of the Vernazza et al. (1976) model M, and will be described in subsequent discussion.

The b(2)a and [c(2)/b(2)]~ coefficients calculated using these thermal structures are illustrated in Figures 2 and 3, respectively. The triangles in these figures refer to the measured data described in Section 2, while the synthetic limb darkening coefficient curves are labeled to correspond with the thermal models of Figure 1. It is clear from these comparisons that the two empirical solar models - those of VAL and of Allen - reproduce the measured center-limb behavior of optical and infrared continuum intensities reasonably well, whereas the purely theoretical model

A C O M P A R I S O N OF SYNTHETIC AND M E A S U R E D SOLA R C O N T I N U U M INTENSITIES 23

I

t- 08 I

0 6

0.4

x~0 .5

0.2

J

~- V,& L 1.015 i /

0.1 ~ ~ I 1 k ~ ~ k I ~ ~ ~ ~ I ~ ~ t ~ L

0.5 1.0 1.5 2.0 2.5 k~lr~

Fig. 2. Comparison of synthetic b(2) limb darkening coefficients with measured values. Theoretical curves are labeled to correspond to thermal models of Figure 1. 'VAL 1.000' refers to the unscaled VAL

model M. Note inverse wavelength scale.

evidently has too steep a temperature gradient near ~'0.5 = 1 and too much 'curva- ture' to fit the limb-darkening data adequately. The failure of the radiative equili- brium model to fit the empirical center-limb behavior may result from an unrealis-

tic treatmer~t of convection in the deeper layers of the photosphere (e.g. Relyea and

Kurucz, 1978). In Figure 4, disk center intensities synthesized using the model photospheres of

Figure 1 are compared with the absolute photometry of Labs and Neckel (1968, 1970). The solid dots represent calibrated specific intensity estimates for window points in the optical region (0.40-0.65/zm) as described in Section 2. Unlike the previous comparison, here we find that the synthetic continuum of Kurucz's model (and that of Allen's model) adequately fits the calibrated intensity information, while the VAL model intensities fall somewhat below. The differences in this

comparison between the Kurucz and Allen thermal structures on the one hand, and the VAL model on the other, are the absolute temperature scales. In particular, the VAL model is somewhat cooler near ~'0.5 = 1 than either the Kurucz or Alien thermal structures (see e.g. Figure 1).

The somewhat cooler temperatures of the VAL model M near z0.5---1 can be attributed to a compromise adopted by Vernazza et al. (1976) because of what they felt was an inconsistency between absolute intensity measurements near the 1.6/zm opacity minimum and continuum center-limb data in the 0 .8-2 .0/zm region. In

24

0.50

0.40

0.30

0.20

0.10

0.08

0.06

T. R. A Y R E S

r r , , f , , , , I ' ~ ' ' I ' ' ~ ' I

ALLEN

KURUCZ /

r ,.. I . . ,~ ,~E-~ % ~\~

"-VAL 1.000, VAL 1.015

J

~ f

, l~t: , , , I , , , , I , , I , I , , , I I

0.5 1.0 1.5 2.0 2.5 X2 I Fm

Fig. 3. Same as Figure 2 for [c(2)/b(2)].

T E

~L T 5 4

Tt/~

m

o - 0 5

.-.~ .,, ' . . . . . ]~ ' I

~.,/--KU~UCZ "%. .---<.. ,%, ~ ~ , o"~"~ VAL 1.015

_ \ ~ " "';;' �9 ;" ....

a

VAL 1000 " ~

I ~ I

0.4 0.5 0.6 ,kffm

Fig. 4. Comparison of computed and measured disk center continuum intensities 0.40-0.65 t~m.

A COMPARISON OF SYNTHETIC AND MEASURED SOLAR CONTINUUM INTENSITIES 25

particular, the maximum measured central brightness temperature at 1.6 ~m cited Vernazza et al. (Tb~6600 K; e.g. their Figures 9 and 16) is roughly 150 K lower than would be expected on the basis of homogeneous photospheric models which match calibrated high points in the optical region and reproduce infrared continuum center-limb behavior (Avrett, 1977). However, the recent critical review of solar irradiances by Pierce and Allen (1977; their Table 2) removes this discrepancy. The equivalent brightness temperatures for the infrared continuum energy distribution adopted by Pierce and Allen (e.g. Tb ~ 6760 K at 1.6 fzm) is entirely consistent with homogeneous models based on the Labs and Neckel calibration of spectral high points in the optical region.

Although the VAL model M fits the continuum limb darkening data more satisfactorily than the other two photospheric models considered here, the less- than-optimum agreement with absolute central intensities in the optical and infrared regions might cause problems in certain applications, for example in modeling the damping wings of a strong, temperature sensitive spectral line using profile data calibrated in absolute units based on Labs and Neckel (or Pierce and Allen). It is, however, possible to preserve the excellent fit of the VAL model to the

T A B L E I

V A L 1 . 0 1 5 M o d e l

Te~ = 5770 K; g = 2 .74x 104 c r n s - 2

l o g ~'o.5 T(K) bvi- b l (H I) log to.5 T(K) bH- b l (H I)

- 2 . 5 4470 0.990 0.88 - 0 . 5 5745 1.001 1.23 - 2 . 4 4505 0.991 0.92 - 0 . 4 5875 1.001 1.21 - 2 . 3 4545 0.992 0.97 - 0 . 3 6020 1.001 1.18 - 2 . 2 4590 0.994 1.03 - 0 . 2 6175 1.001 1.15 - 2 . 1 4630 0.995 1.09 - 0 . l 6340 1.001 1.13

- 2 . 0 4670 0.996 1.15 O. 6515 1.001 1.10 - 1.9 4715 0.998 1.22 0.1 6710 1.001 1.08 - 1.8 4755 0.999 1.28 0.2 6920 1.001 1.06 - 1.7 4800 0.999 1.28 0.3 7145 1.001 1.05 - 1.6 4845 1.000 1.28 0.4 7370 1.001 1.04

- 1.5 4895 1.000 1.28 0.5 7595 1.001 1.02 - 1.4 4950 1.000 1.28 0.6 7810 1.001 1.02 - 1.3 5005 1.000 1.28 0,7 8025 1.000 1.02 - 1.2 5060 1.001 1.28 0,8 8220 1.000 1.01 - 1.1 5120 1.001 1.28 0.9 8425 1.000 1.01

- 1.0 5195 1.001 1.28 1.0 8625 1~000 1.01 - 0 . 9 5290 1.001 1.27 1.1 8830 1.000 1.01 - 0 . 8 5385 1.001 1.27 1.2 9035 1.000 1.01 - 0 . 7 5500 1.001 1.26 1.3 9235 1.000 1.01 - 0 . 6 5620 1.001 1.24 1.4 9440 1.000 1.00

1.5 9640 1.000 1.00

26 T.R. AYRES

center-limb measurements, while at the same time improving the fit to the absolute window intensities, simply by multiplying the photospheric temperatures by a constant factor. For the adopted reference continuum intensity Io.5(0)= 4.15 + 0.1 x 101~ ergs cm -2 s -1 sr -1/xm -~, the appropriate scaling factor is 1.015 + 0.005. Center-limb coefficients and absolute ~intensities have been computed for such a model - denoted VAL 1.015 - and are illustrated in Figures 2--4. The VAL 1.015 thermal structure itself is depicted in Figure 1, and a tabulation for reference purposes is provided in Table I.

The VAL 1.015 model is a good candidate for single-component studies of the continuum forming layers of the solar photosphere, and for those stellar appli- cations where scaled solar models are useful. However, if future absolute cali- brations of solar irradiances reveal systematic differences compared to the work of Labs and Neckel, the VAL temperature structure could be rescaled accordingly.

Acknowledgements

I would like to thank E. H. Avrett for informative discussions and for providing hydrogen ionization data for the VAL model M, and J. L. Linsky for his critical reading of the manuscript and helpful comments. This work was supported by the Harvard-Smithsonian Center for Astrophysics Fellows program and by the National Aeronautics and Space Administration under grants NGR-06-003-057 and NAS5-23274 to the University of Colorado.

References

Alien, R. G.: 1977, Ph.D. thesis, Univ. of Arizona, in preparation. Auer, L. H.: 1976, J. Quant. Spectr. Radiative Transfer 16, 931. Avrett, E. H.: 1977, in O. R. White (ed.), The Solar Output and Its Variations, Colorado Assoc. Univ.

Press, Boulder, p. 327. Ayres, T. R.: 1977, Astrophys. J. 213, 296. Broad, J. T. and Reinhardt, W. P.: 1976, Phys. Rev.. A14, 2159. Flannery, B. P. and Ayres, T. R.: 1978, Astrophys. J., (in press). Holweger, H.: 1972, Solar Phys. 25, 14. Kurucz, R. L.: 1974, SolarPhys. 34, 17. Labs, D. and Neckel, H.: 1968, Z. Astrophys. 69, 1. Labs, D. and Neckel, H.: 1970, SolarPhys. 15, 79. Mihalas, D.: 1970, Stellar Atmospheres, W. H. Freeman, San Francisco. Peach, G.: 1970, Mere. Roy. Astron. Soc. 73, 1. Pierce, A. K. and Alien, R. G.: 1977 in O. R. White (ed.), The Solar Output and Its Variations, Colorado

Assoc. Univ. Press, Boulder, p. 169. Pierce, A. K. and Slaughter, C. D.: 1977, Solar Phys. 51, 25. Pierce, A. K., Slaughter, C. D., and Weinberger, D.: 1977, Solar Phys. 52, 179. Relyea, L. J. and Kurucz, R. L.: 1978, Astrophys. J. Suppl., (in press). Ross, J. E. and Aller, L. H.: 1976, Science 191, 1223. Shine, R. A.: 1973, unpublished Ph.D. Thesis, University of Colorado. Spruit, H.: 1974, SolarPhys. 34, 277. Vernazza, J. E., Avrett, E. H., and Loeser, R.: 1976, Astrophys. J. Suppl. 30, 1. Withbroe, G. L.: 1978, Solar Phys., (in press).