H O W D E E P CAN ONE SEE INTO T H E S U N ?

T H O M A S R. A Y R E S

Center for Astrophysics and Space Astronomy, University of Colorado, Boulder, CO 80309-0391, U.S.A.

(Received 10 February, 1989; in revised form 30 May, 1989)

Abstract. Conventional wisdom dictates that the 1.642 gm H 'opacity minimum' is the best window to the depths of the solar photosphere. However, the violet continuum near 0.4 gm exhibits a larger intensity response to small thermal perturbations at depth, and thus might offer an even better view of the subsurface roots of granulation ceils and magnetic flux tubes.

1. Introduction

Newly-available infrared detector arrays have inspired keen interest in probing the subsurface properties of solar granulation cells and small-scale magnetic flux ropes at the well-known 1.642 ~tm H - 'opacity minimum' (Foukal, Little, and Mooney, 1989). Less well-known, however, is the curious fact that intensity response functions for the violet continuum (2 < 0.4 gin) peak in the same or slightly deeper layers. All other factors being equal (which they rarely are), the viewing of the depths of the solar photosphere might be better at the shorter wavelengths. Here, I explore the proposition numerically.

2. The VAL C' Model Atmosphere

Figure 1 illustrates the thermal profile of a modem model of the deep photosphere: the VAL C' reference atmosphere of Maltby et al. (1986; their Appendix A). I smoothly interpolated the authors' tabulated T(m) profile onto a finer grid of 60 plane-parallel layers (in the interval 0.080-7.400 g cm- 2) to improve the accuracy of the radiative transfer solutions. I calculated density distributions by solving the equations of hydro- static equilibrium, chemical equilibrium, and ionization equilibrium jointly by means of a Newton-Raphson iteration scheme. I included turbulent pressure support as recom- mended by Vemazza, Avrett, and Loeser (1973, VAL), with the microturbulent velo- cities of Maltby et al. (1986). I allowed for the LTE formation of H -, H2 + , and diatomic molecules of cosmically-abundant elements. I used the molecular fomaation parameters of Kurucz (1970) and the solar abundances of Withbroe (1976), with the exception of helium where I adopted an abundance of 0.088 (by number relative to hydrogen): the latter yields a mass fraction of Y = 0.25 as favored in solar evolution models (cf. Flannery and Ayres, 1978). I treated departures from LTE in hydrogen, but in a simple parametric way (see Ayres, Testerman, and Brault, 1986).

I compared the calculated density distributions with those tabulated by Maltby et aL :

the agreement between 1//e and nH in the deeper hotter layers (dominated by partial ionization of hydrogen) was within a percent or so: the discrepancies can be attributed

Solar Physics 124: 15-22, 1989. �9 1989 Kluwer Academic Publishers. Printed in Belgium.

16 THOMAS R. AYRES

.01 .04 .1 I ' ' ' ' i '

~-, 9 0 0 0 -

~" - VAL C' / 8000

.~ 7000

6000

5000

7"1.6421 ~ , , , I , ,~,I ,,,,I �9 0 1 . 0 4 . I I 10

nil1

" t I 10 ~'I ~ ' " I TO'400

17 10

1016 ~E

15 0 10

1014 O0

1013 z L.d d:3

1012 I I I I l l l l l l l I I I I I I l l l l t l l l

.5 .7 1 2 5 4 6 8 C O L U M N MASS DENSITY ( g r c m - 2 )

Fig. l. Temperature profile and density distributions for the VAL C' reference atmosphere. The abscissa is the column mass density, a 'depth' scale preferred in spectrum synthesis work. The upper panel illustrates the run of temperature, and the calculated continuum optical depths in the violet (upper scale) and near-IR (bottom scale, upper panel). The lower panel illustrates the derived density distributions: the rapid rise of the electron density for T > 6000 K is caused by partial ionization of hydrogen; at cooler temperatures,

metal ionization dominates.

completely to the small difference in helium abundance. In the shallower cooler layers, there were disagreements up to 20% in the electron densities, owing partly to small abundance differences (of the important electron donors), and partly to the details of the treatment of the metal ionization (! assumed LTE ionization, but with temperature- dependent partition function ratios as given by Gray (1976, his Appendix D)).

I tested my version of the VAL C' model against the empirical criteria I developed in an earlier study (Ayres, 1978): namely, deblanketed disk-center continuum intensities and low-order polynomial fits (in in g) to limb-darkening coefficients. I adopted the observational data described in the earlier work, but extended the coverage of the deblanketed continuum intensities from the original long wavelength cutoff in the red (0.65 ~tm) to the mid-IR (2.50 ~tm) by means of the homogeneous data set of Pierce and Allen (1977).

I smoothly interpolated the measured continuum intensities and limb-darkening coefficients onto a grid of 17 frequencies evenly spaced in wavenumber from 0.4-2.0 ~t- 1 (i.e., 2.5-0.5 ~). I synthesized the corresponding quantities for the VAL C'

HOW DEEP CAN ONE SEE INTO THE SUN? 17

model using the approach outlined in the previous paper (Ayres, 1978). Again I used Auer's (1976) Hermitian differencing scheme in a Feautrier-type ray solution to obtain high accuracy with the approximately ten points per decade spacing of the optical depth grids. Averaged over the reference frequencies, I find essentially no difference between the calculated and measured disk-center continuum intensities, with an r.m.s, dispersion of the mean of 1.4~. The excellent agreement indicates that the deep-photosphere temperature scale of the VAL C' model is well-matched to the radiometric calibration of solar intensities. The calculated fits of the VAL C' model to the first-order and second-order limb-darkening coefficients (b(2) and c(2) in the notation of Ayres (1978)) were less good however: b(2)c~ o was systematically 6 ~ too high with a dispersion of 2~o, while [c(2)/b(2)]calo was systematically 12~o too high with a dispersion of 6~o. These differences indicate that the gradient of temperatures is somewhat too steep in the shallower layers above %.5 ~m ~ 1. While it is possible to design thermal structures that meet the limb-darkening criteria to higher precision than the VAL C' model, it is doubtful that such models would bring any sensible improvement in practical applica- tions: the single-component plane-parallel hydrostatic simulation is only a minimal approximation to the true highly structured dynamic photosphere of the Sun.

While the VAL C' model meets essentially identically the intensity criterion in the optical and near-IR, the intensities I calculate in the b - f continuum of H - are somewhat lower than those reported by Maltby et aI. In particular, at 0.40 gm I calculate a disk-center brightness temperature of 6540 K compared with 6600 K of Maltby et al. (their Figure 15); the differences decrease to only tens of K at the H - opacity maximum near 0.90 ~tm (T~, ca1 = 6070 K); and are essentially negligible in the f - f continuum of H - longward of 1.642 gm (T~, ~al = 6780 K). The small discrepancies in optical disk-center intensities very likely can be traced to differences in the b - f cross-sections adopted here (John, 1988) and those contained in the PANDORA code used to construct the VAL C' thermal model. The gradient of temperature with column mass is very steep near %,5 ~m ---- 1, and the optical depth scale itself is a steep function of the column mass density (H- is a 'pressure-squared' opacity source). Thus, a small change in the b - f cross-section can measurably affect the emergent intensity.

In short, I find that the VAL C' thermal profile meets a number of critical criteria that describe the line-free photospheric radiation field, and thus it is eminently suited to explore the question of how deep one can see into the Sun.

3. Opacities and Response Functions

Figures 2(a) and 2(b) depict the depth distributions of the total opacity and its con- stituents; the thermal emission function; and the intensity response function for the H - opacity minimum at 1.642 gm and for a wavelength in the violet continuum, 0.40 gin. The opacity at the near-IR wavelength is dominated by Hff (just beyond the bound-free threshhold for photodetachment of the negative ion), while the opacity in the violet is dominated by H ~ (the free-free contribution there is small owing to the ~ 22 depen- dence of the Hff opacity).

18 THOMAS R. AYRES

.01 .04 .I 1 I 0 I ' ' ' '1 ' ' " 1 ' " ' 1 7-C

2 E ~ 1.0 1.642 u,m / 1

oo 0.8 ~ ._,

. . 0.6! ~ -,] ~ o . 4 ~ -~ E C 0.2 -1

o.o t .

f f , ' / , f f -1

/(X / / / / 10-, ,..oo

/ / I l l l i l f l l l [ i I I t I l t l l l l l l l .5 .7 1 2 3 4 6 8

COLUMN MASS DENSITY ( g r c m -2 )

Fig. 2a. Calculated opacity, thermal emissivity, and intensity response function (g = 1) for the H - opacity minimum at 1.642 gin. The total opacity (V~roT: lower panel) is dominated by Hff , except in the deep photosphere (T > 7500 K) where the f - f and b - f continua of neutral hydrogen begin to compete. The upper panel illustrates the normalized Planck function (~/~ma• and intensity response function (off/Cgmax) for the near-IR wavelength. While the thermal emission rises rapidly in the deeper layers of the photosphere, the contrast between ~ (z = 0.1) and N(z = 10) is only about a factor of 2 (the continuum z scale is illustrated in the upper portion of the panel). Note also that the response function peaks between T = 1 and 2, at a

column mass of about 5.3 g cm -2.

The ratio of the monochromatic opacities is approximately:

~:b~- (0.40 ~tm)/xf~-(1.64 ~tm) ~ 0.6(T/5000)- 1.5 exp(8762/T). (1)

Note that the nen H density factors present in both opacity sources cancel in the ratio, and thus it is purely temperature dependent. While the violet b - f opacity exceeds the near-IR f - f opacity throughout most of the middle photosphere (T < 6000 K), the two opacities become equal in the deep photosphere at about 7600 K, and in practice equality of the total opacities occurs at a lower temperature (~ 7000 K) owing to the significant additional contribution of hydrogen f - f in the near-IR.

Thus, Zl.642/~m " = 1 occurs ~ 0.5 g cm -2 deeper than Zo.4o p.na = 1 (corresponding to 20 km or �89 but the near-IR opacity builds up faster at depth so that 1~1.642 Hm = 10

actually falls in shallower layers than "Co.4o ~m = 10. The figures also illustrate the sharp contrast in the violet Planck function

HOW DEEP CAN ONE SEE INTO THE SUN.9 19

1.0 E

eO ~ o.8 eo

0.6

x 0.4 E

0.2

o.o

.01 .04 .I .4 1 ' ' ' ' I ' " ' ' I

0.400 ffrn

i i i s 7

1o ' '"I g-c

(

l H b , 10+1 ']-~

H+ /.~...., 4 IO -~

.,o,o,= . . . . . . . . . . . _ / - - ' , , / _1 lO-= L:_ . . . . . . . . . . -: . . . , j (r~ot .... HI. ........ / .~_ ..../ I I' Hf, 1 0 -'3 n

O I , l , l , l , l , I I I I I I l l l l l l l l l

.5 .7 1 2 ,3 4 6 8

COLUMN MASS DENSITY ( g r c m - 2 )

Fig. 2b. Same as Figure 2(a), for the violet continuum (0.40 gm). Here, the total opacity is dominated by H~ throughout the photosphere, and thus is strongly temperature sensitive due to the low binding energy of the negative ion. At T ~ 7000 K, the 0.40 gm opacity is similar to that of the 1.642 gm continuum, and falls below it in the deeper hotter layers. Note, also, that the contrast of the 0.40 gm thermal emission between z = 0.1 and z = 10 is a factor of about 20. The exponential rise of the violet Planck function in the deeper layers partially compensates for the optical attenuation factor and pushes the peak of the intensity response function to ~ ~ 2, at essentially the same column mass as that of the 1.642 gm 'opacity minimum'.

(~z = Cl/[exp ( c2 /T ) - 1]) between the shallow and deep layers of the photosphere ,

compared with the much flatter distr ibution of the nea r - IR thermal emission. The

accompanying intensity response functions (cd) indicate where the major contr ibut ions

to the emergent radia t ion field arise:

~z = d J z / d log m = in 10 M~e- ~* ~cxm, (2)

where rx = S~ ~c~dm.

As can be seen in the figures, (~71.64 2 gin peaks between r~ = 1 and 2, while %.4o gm

peaks at zz = 2. Both response functions drop precipi tously beyond ~ ~ 2 owing to the

exponential a t tenuat ion of the emerging radiat ion. The violet response function peaks

at a larger monochromatic optical depth because the exponential rise of the violet thermal

emission part ial ly compensa tes for the opt ical a t tenuat ion factor. As a result, the two

response functions reach maximum at essentially the same column mass. Furthermore ,

20 THOMAS R. AYRES

% . 4 0 !am extends to slightly larger column masses than does ~91.642 g in , owing partly to the compensation effect and partly to the less rapid inward increase of %.40 ,m"

The preceeding simulations demonstrate that the criterion for viewing the deepest

layers of the solar photosphere should not be based solely on the wavelength for which

"cx = 1 occurs at the largest column mass, but should consider in addition the depth

dependence of the radiation production, particularly in light of the sharp inward rise in temperature at and below the visible surface.

4. Response to Thermal Perturbations at Depth

I tested the response functions more concretely as follows. I imposed a _+ 100 K

step-function temperature perturbation on the VAL C' commencing at a given depth

rncrit and extending uniformly to the deeper layers. Physically the perturbed atmospheres

represent thermal profiles that are in horizontal pressure equilibrium with the reference

atmosphere, but are slightly hotter or cooler than the standard model below a given

depth: these are the types of structures one might wish to detect in a monochromatic

imaging experiment. I averaged the absolute values of the calculated brightness tempera-

ture deviations for the + 100 K steps: the resulting [AT~I loo are depicted in Figure 3

as a function of mcrit. Note the curious fact that a + 100 K perturbation extending over the optically active

portion of the atmosphere (corresponding to merit ~< 2), shows essentially a negligible

,x/

0 0

l-- <l

.01 .04 .I .4 I 10 ' ' ' ' I ' ''I ' '"I '7-0'400

50

20

10

0

o l i t , ,I , ,,,I,,,,I 7-1'642. I .04 .I I 10

IKlillliltl , I i I ,l,l,l,lll

.5 .7 1 2 3 4 6 8 COLUMN MASS DENSITY (gr c m - 2 )

Fig. 3. The absolute brightness-temperature response of the emergent continuum radiation field in the violet (dots) and near-IR (squares) when the underlying atmosphere is subjected to a step-function increase or decrease in temperature of 100 K commencing at a critical column mass and extending uniformly to deeper layers. The response of the brightness temperature in the violet is larger in the shallow layers, and comparable at depth, to that of the near-IR. When cast in terms of intensity contrast (the experimentally

accessible quantity), the comparison strongly favors the violet for all merit.

HOW DEEP CAN ONE SEE INTO THE SUN? 21

brightness temperature contrast in the near-IR (a few K) and only a modest contrast in the violet (a few tens of K). That behavior results from the resiliency of a thermal profile cast as temperature versus column mass, when the temperature exhibits a positive gradient inward. For example, raising the temperatures uniformly over a grid of fixed pressures (P = gm) increases the ionization, thereby bodily shifting the H -dominated v-scale to somewhat smaller column masses. However, since the tem- perature scale has shifted upward at a given column density compared with the initial T(m) profile, the new T*(z*) should be much closer to the original T('c) profile than the gross AT(m) might suggest. The reverse compensation phenomenon occurs when the temperature perturbation is negative.

Figure 3 demonstrates the key result: even though the absolute brightness temperature of the continuum is 240 K lower at 0.40 gm compared with 1.642 lam, a step-function AT in layers deeper than z = 1 produces the same AT~ at the two wavelengths. Conse- quently, the intensity contrast A ~ J i s significantly larger at the shorter wavelength, owing to the stronger temperature response of the violet Planck function.

5. Discussion

While ~ = 1 occurs in the deepest layers of the solar atmosphere at the 1.642 gm opacity minimum, and the brightness temperature of the deblanketed continuum is largest there, the near-IR might not be the best window to view the deep roots of thermal structures like granulation cells and the interiors of magnetic flux tubes. Instead, the violet continuum at and shortward of 0.40 gm offers the key practical advantage that tempera- ture perturbations at depth are mapped onto significantly larger - and therefore easier to detect - inwnsity contrasts. Furthermore, detector technology is considerably more mature at the shorter wavelengths, and the violet continuum offers the prospect of 4 x higher theoretical spatial resolution for a solar telescope of given aperture.

The major practical drawback of the violet continuum is the presence of line veiling. The problem becomes progressively worse shortward of 0.40 gm, even though in prin- ciple the continuum itself becomes progressively more transparent down to the Balmer edge (0.365 ~tm). A few relatively clean continuum windows exist at and longward of 0.40 gm, but even these likely are affected to some degree by line blanketing.

However, the problem of line veiling is not restricted solely to the optical continuum: it also severely affects the near-IR as well. For example, owing to the low quantum efficiency of their PtSi CCD camera, Foukal, Little, and Mooney (1989) employed a relatively broad 40-4 (FWHM) interference filter. According to Hall's (1970) Infrared Atlas, the filter passband (1.6255-1.6295 lam) is contaminated by several moderate- strength lines of Ti, Fe, and C, as well as by weaker absorptions due to OH. Although the line depressions caused by the molecular features might be slight, they arise in cooler parts of the photosphere greatly removed from the continuum-forming layers: they might well confuse the interpretation of near-IR monochromatic pictures by impressing on the continuum brightness distribution a faint negative ghost of high-altitude thermal inhomogeneities (cf., Ayres, Testerman, and Brault, 1986).

22 THOMAS R. AYRES

A simple test of the influence of line-veiling in the violet and near-IR could be performed by detailed spectrum synthesis of selected continuum windows (viz., Kurucz and Avrett, 1981) in the two regions.

Finally, as the new infrared detector arrays come into play in the 1-5 gm region, it would be fair to ask of any striking discoveries made at the 1.642 gm opacity minimum whether the putative phenomena also are seen in the violet continuum, where the view into the depths of the solar photosphere should be as good if not better.

Acknowledgements

Support was provided by the National Science Foundation through grant AST-8507029 to the University of Colorado, and by the National Aeronautics and Space Adminis- tration through grant NGL 06-003-057.

References

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3". 306, 284, Pierce, A. K. and Allen, R. G.: 1977, in O. R. White (ed.), The Solar Output and Its Variation, Colorado

Assoc. Univ. Press, Boulder, p. 169. Vernazza, J. E., Avrett, E. H., and Loeser, R.: 1973, Astrophys. J. 184, 605 (VAL), Withbroe, G. L.: 1976, Center for Astrophysics preprint No. 524.