The orbit of the SOHO spacecraft permits acquisition of helioseismic data in long time series. The MDI scans the Ni 1.

Transcription

1 THE ASTROPHYSICAL JOURNAL, 515:832È841, 1999 April 20 ( The American Astronomical Society. All rights reserved. Printed in U.S.A. ABSORPTION/EMISSION OF SOLAR p-modes: MICHELSON DOPPLER INTERFEROMETER/SOHO OBSERVATIONS DEBORAH HABER, REKHA JAIN,1 AND ELLEN G. ZWEIBEL JILA, University of Colorado at Boulder, Boulder CO ; rjain=solarz.colorado.edu, dhaber=solarz.colorado.edu, zweibel=solarz.colorado.edu Received 1998 May 26; accepted 1998 November 24 ABSTRACT We search for sources and sinks of solar p-mode waves by creating absorption maps from Dopplergrams taken by the Michelson Doppler Interferometer instrument on board SOHO. Although the maps are noisy, we present evidence for sources and sinks, of duration D0.5È1 hour or less, based on an autocorrelation analysis of maps made from sequential intervals of data. Emission and absorption nearly balance each other in the data. This may imply that emission and absorption are inverse processes of each other rather than fundamentally di erent in nature. Subject headings: Sun: atmosphere È Sun: oscillations 1. INTRODUCTION The solar photosphere is observed to oscillate at the resonant acoustic frequencies of the solar envelope. Accurate measurements of the frequencies of these resonant acoustic, or p-mode oscillations, some 107 of which have now been detected, form the basis for helioseismic probes of solar structure. Helioseismology has led to a fairly well developed picture of the composition, thermodynamics, and largescale hydrodynamics of the solar interior (Christensen- Dalsgaard et al. 1996; Gough et al. 1996; Thompson et al. 1996). The statistical properties of solar oscillations appear to be steady with time, except for slow variations associated with the solar activity cycle. This suggests that the rates of mode excitation and mode damping are balanced. Stochastic excitation by turbulent convection slightly below the photosphere is now considered the most likely driver (Goldreich & Keeley 1977; Goldreich & Kumar 1988; Kumar & Lu 1991; Rast & Toomre 1993; Kumar 1994; Rast 1998b). Damping could occur either through interaction with turbulence (Goldreich & Keeley 1977; Goldreich & Kumar 1991), by radiative losses (Christensen- Dalsgaard & Frandsen 1983; Goldreich & Kumar 1991; Goldreich & Murray 1994), or by escape along magnetic Ðeld lines that thread the p-mode cavity (Spruit 1991; Bogdan et al. 1996). These theories of excitation and damping have signatures that can at least in principle be observed. In the classical formula for the emission of acoustic radiation by turbulence (Lighthill 1952), the power radiated by a source with mean velocity v is proportional to v8. This suggests that most of the p-mode energy could be emitted by localized, highvelocity events. This conclusion is supported by detailed hydrodynamical models of downward-directed plumes (Rast & Toomre 1993; Rast 1998a, 1998b). The more recent work highlights the role of thermal perturbations rather than Reynolds stresses in mode excitation. However, the acoustic emissivity is still expected to be doated by discrete events, namely, plumes. The work of Rimmele et al. (1995) provides observational support for the role of down- Ñows in mode excitation. Motivated by the arguments for localized, discrete acoustic sources, Brown (1991) used the WKB approximation to 1 Now at Department of Physics, UMIST, Manchester, England-M60 1QD. 832 work out the appearance of a point source of acoustic radiation near the top of the convection zone. His model shows that the signal spreads out rapidly and should be directly observable as an enhancement of p-mode power over only a small area. Such enhancements may have been detected at frequencies above the acoustic cuto frequency, i.e., for waves that are not trapped in the photospheric cavity (Brown et al. 1992). Goode, Gough, & Kosovichev (1992) calculated the appearance of a source by solving the linearized Ñuid equations, without the WKB approximation, in a simple model atmosphere. The seismic events ÏÏ described by their model have been reported by Rimmele et al. (1995) and Goode et al. (1998) in oscillations data. This paper reports on an observational study of p-mode excitation and damping from a di erent viewpoint. We apply an analysis method originally developed to study acoustic absorption in sunspots (Braun, Duvall, & LaBonte 1987) to produce maps of emission and absorption from time series of Dopplergrams obtained by the Michelson Doppler Interferometer (MDI) on board SOHO. Maps made from half-hourèlong data sets taken within an hour of each other are signiðcantly correlated with one another, while maps made from data taken further apart in time are uncorrelated. This suggests that we have detected acoustic sources and sinks that persist over 0.5È1 hour. Moreover, we Ðnd that for each day of data, the histogram of emission/ absorption is well Ðtted by a Gaussian, centered on zero. Although the width of the histograms appears to be inñuenced by noise, the signiðcance of the mean is high. This suggests that emission and absorption are inverse processes of each other, so that if the modes are convectively excited they are also convectively damped. However, it is known from line width measurements (Korzennik 1990; Chen et al. 1996) that the total damping of the modes is weak. In 4.3 we show that only a slight excess of emission over absorption would be required if an additional absorption process, with magnitude consistent with the line widths, were at work. In 2 of the paper we brieñy describe the data sets, and in 3 we describe the method of analysis. In 4 we present the main results, including an error analysis. Section 5 is a summary of the paper and our main conclusions. 2. THE DATA SETS AND PRELIMINARY ANALYSIS The orbit of the SOHO spacecraft permits acquisition of helioseismic data in long time series. The MDI scans the Ni

2 TABLE 1 DIFFERENT DATA SETS USED FOR ANALYSIS Parameter 1996 Jun 6È Jan Jul 5 Region... Sunspot Quiet-Sun Quiet-Sun Start and end times (TAI)... 09:56:30 and 11:31:30 00:01 and 12:00 09:00 and 12:59 Duration (hours) Field of view @@ ] 512@@ 300@@ ] 300@@ 384@@ ] 384@@ Center location (region)... (3.04, 291 ) ([3.0, 213 ) (18.3, 133 ) Resolution (pixel~1)... 2A 0A.6 0A.6 Number of pixels ] ] ] 640 Frequency spacing, *l (khz) Nyquist wavenumber (Mm~1) Maximum B (^) G and 1.2 kg 247 G and 285 G 100 G and 143 G Mean B ^ p... B [1.5 ^ 37 G [1.52 ^ 10.7 G ^ 9.1 G FIG. 1.ÈThe p-mode power as a function of frequency l and horizontal wavenumber k for the 25.6 hour, 1996 June 6 full-disk data set (azimuthally averaged).

3 834 HABER, JAIN, & ZWEIBEL Vol. 515 TABLE 2 PARAMETERS USED IN p-mode ABSORPTION MAPS l (mhz) k (Mm~1) l n \ 0 n \ 1 n \ 2 n \ 3 n \ A photospheric absorption line at a cadence of 12È60 s, yielding line-of-sight velocities in the solar atmosphere at about 200 km above the temperature imum. MDI operates in full-disk mode, which images the entire solar disk with 2A pixel~1 (4A resolution), and in a high-resolution mode with Ðeld of view restricted to near disk center, but with 0A.6 pixel~1 (1A.2 resolution). Recall that 1@@ B 750 km on the Sun. The instrument and types of data acquired are discussed by Scherrer et al. (1995). We use a solar based coordinate system rather than plane-of-the-sky coordinates to carry out our local analysis. We selected PostelÏs projection, which preserves great circles, or distances along which waves travel (for an extended discussion of projection methods see Bogart et al or Richardus & Adler 1972). In addition to being remapped, the data must be tracked to compensate for solar rotation, which would otherwise introduce unacceptable distortions over even a few hours. We tracked the quiet- Sun data using the Snodgrass coefficients for 1982/1984 (Snodgrass 1984), which give the surface rotation rate as a function of latitude (we applied the formula at the center of the tracked region). The 1996 June 6 data were tracked faster than the photospheric rate (Howard et al. 1984) to keep the sunspot region stationary during 25.6 hours. The properties of the data sets we usedètwo in which the Sun was magnetically quiet and one with small sunspots presentèare summarized in Table 1. In Figure 1 we show a diagram of mode power as a function of frequency l and local wavenumber k D l/r for the June 6 data. At the low orders we use in this work _ (see Table 2), the ridges are clearly deðned in the range of interest 0.42 Mm~1 ¹ k ¹ 0.85 Mm~1. 3. EMISSION AND ABSORPTION MAPS Braun et al. (1987, 1988, hereafter BDL) developed a technique for measuring the rate at which solar oscillations are emitted or absorbed by a patch on the solar surface. Their analysis led to the discovery that sunspots are strong absorbers of p-mode waves, a phenomenon that is still unexplained. We have used their method, with little modiðcation, to construct emission and absorption maps of patches of quiet and magnetized Sun. We brieñy summarize the procedure for readers who are unfamiliar with it. In order to measure emission or absorption from the neighborhood of a single point, we deðne an annulus with inner and outer radii R, R, centered on that point. We then interpolate the Doppler max velocity within the annulus, which is measured on a Cartesian grid, onto a polar grid with radial spacing of the gridlines dr equal to a pixel, and angular spacing dh equal to dr/r. We chose max the cubic interpolation method discussed by Keys (1981).2 The natural incog and outgoing radial waveforms in polar coordinates are Hankel functions. We therefore consider waves of the form ei(ut`mh)h(1)(kr), ei(ut`mh)h(2)(kr), m m where H(1) and H(2) are the incog and outgoing Hankel m m functions, respectively, and m is an integer. In order to project the p-mode power onto functions of this form, it is necessary to choose the k such that the Hankel functions at each m form an orthogonal set (since we do not need to represent all the p-mode power by these waveforms, it is not necessary that the set be complete). Following BDL, we have simpliðed the search for an orthogonal set by choosing only values of k such that kr [ o m o, in which case the Hankel functions are well approximated by the asymptotic forms A H(1),(2)(x) D 2 ebi(x~k*(2m`1)n+@4l), (1) m nxb1@2 where the ^ signs refer to the Hankel function superscripts 1 and 2, respectively. When equation (1) applies, the appropriate k are of the form k \ k \ 2nn n *R, (2) where *R 4 R [ R. Equations (1) and (2) together imply the orthogonality max relations P Rmax rdrhm (1)(k n r)h m (1)(k j r) R \ P R Rmax rdrhm (2)(k n r)h m (2)(k j r) \ 0 2 This method becomes cumbersome to implement for mapping the absorption and emission from a large array of points, so we use instead the cross-correlation technique described by Braun, LaBonte, & Duvall (1990): We take the FFT of the appropriate Hankel functions and multiply them by the two-dimensional FFTs of the data. Perforg the inverse FFT gives the cross-correlation.

4 No. 2, 1999 SOLAR p-modes 835 FIG. 3.ÈScan of absorption coefficient a (dashed line, left ordinate scale) and measured magnetic Ðeld B (solid line, right ordinate scale) across the sunspot pair observed on 1996 June 6. The absorption coefficient was computed for R \ 30 pixels, R \ 98 pixels. Absorption is well corre- max lated with the peak magnetic Ðeld strength. for all n and j, while P Rmax 2*R rdrhm (2)(k r)h(1)(k r) \ d n m j nj k n. (3) R n Note that while the same set of k can be used for all the azimuthal mode numbers m, this is valid only within the asymptotic approximation. The incog p-mode power at frequency u \ 2nl, azimuthal wavenumber m, and radial wavenumber k, in a n data set of duration t to t ] *T is 0 0 P (u, m, k ) \ o V (u, m, k ) o2, (4) in n in n where P V (u, m, k ) 4 k t0`*t n dte~iut in n 4*R*T t0 2n dhe~imh P Rmax rdrhm (2)(k r)v(r, h, t). (5) n FIG. 2.ÈAbsorption maps for modes n \ 1È4. The maps shown here are all for 30 ute time strings with a boxcar smoothing of 11 pixels. (a) Full-disk data from 1996 June 6, covering 488@@ ] 488@@. (b) Highresolution data from 1997 January 17 covering 293@@ ] 293@@. (c) Highresolution data from 1997 July 5, covering 384@@ ] 384@@. Contour lines corresponding to MDI magnetogram Ðeld strengths of ^125 and ^250 G are superposed on the absorption maps. P ] 0 R A similar deðnition holds for the outgoing power P, with the Hankel function H(1) replacing H(2). Once the out lowfrequency power associated m with solar granulation m is eliated by Ðltering, the power spectrum has strong peaks at p-mode frequencies u \ u(k ), independent of m, so we restrict the analysis to the n neighborhood of these frequencies. The noise associated with P and P for a single mode can be large. In making the emission/absorption in out maps, we have reduced the noise by sumg over groups of modes, all with m \ 0. The modes we selected from the p band are listed in Table 2. The frequencies were obtained by Ðtting azimuthally averaged power spectra for the 1996 June 6 data (see Haber et al for a description of the Ðtting technique). According to equation (2), Ðxing these modes constrains the choice of R and R. We work with the max summed incog and outgoing power in the modes listed in Table 2 P in 4 ; P in (u,0,l); P out 4 ; P out (u,0,l). (6)

5 836 HABER, JAIN, & ZWEIBEL Vol. 515 We deðne the absorption coefficient a a 4 P in [ P out P in ] P out. (7) Equation (7) di ers from the deðnition of the absorption coefficient introduced by BDL and used by subsequent authors a BDL 4 1 [ P out P in. (8) The latter deðnition is more appropriate for the study of absorption; a is typically 0.3È0.6 for sunspots. For the purpose of mapping BDL absorption and emission, we prefer the deðnition in equation (7) because a is then antisymmetric with respect to P and P and ranges from [1 to 1, with positive and negative in values out for absorption and emission, respectively. To summarize, the quantity a measures the mean absorption or emission of the p-modes listed in Table 2 by the region interior to R over the time *T covered by the data set. We map emission and absorption by treating each point in the Ðeld of view (excluding a margin near the edge) successively as the center of the coordinate system and computing a according to the procedure outlined above. The resulting maps are fuzzy ÏÏ in the sense that our method measures emission/absorption for the whole region interior to R, not just at the center of the annulus. In order to further reduce the noise, we smooth the maps by boxcar averaging a over a block of 11 neighboring pixels, all of which lie within a distance R of the central pixel. We defer an analysis of the noise to the following section. Sample maps are shown in Figure 2. In order to compare our analysis with previous results, we computed the absorption associated with the small sunspot pair in the MDI Ðeld of view on 1996 June 6. For R \ 60A, R \ 196A and 20 of the modes listed in Table max 2 with l º 418, n \ 0, 1, 2, 3 we Ðnd a maximum absorption of This corresponds to a D 0.44, which is in the BDL range reported by other investigators (BDL; Bogdan et al. 1993; Braun 1995; Braun et al. 1990). However, our annulus is smaller than the annuli chosen in previous work and does not enclose the sunspot group completely. This enables us to map the absorption within the spots and establish the correspondence between absorption and magnetic Ðeld strength. Figure 3 is a scan of the absorption coefficient together with the average magnetic Ðeld strength measured by MDI on that day. The enhancement of absorption with magnetic Ðeld strength appears clearly, although there is some noise near the edges. Note, however, that the Ðeld within this pair of sunspots is fairly weak, about 800 G at the maximum (which is not shown on this particular scan) and the region of enhanced absorption is quite small. This means that the sunspot regions appear rather weakly on absorption maps. As noted by Braun (1988) and Braun et al. (1990), there is also a question of whether there are absorption features that are artifacts of the data reduction procedure itself. To answer this question Braun looked at the autocorrelation of the Hankel functions (masked by the observing window). The autocorrelation represents the response of the Hankel transform to a single Hankel component, and the absorption maps are created by correlating the data with a set of Hankel functions. Although in principle this autocorrelation should give a d-function, he found that, due to the Ðnite observing window, there was an elongation of the autocorrelation function especially at the edges and corners, but which also showed up as side lobes for points near the center. Braun also found what appeared to be leakage of power in both wavenumber and azimuthal order around the edges of the frame due to incompleteness in the data coverage showing up as spurious artifacts in the active region maps of Although we have a di erent observing geometry and remapping strategy, our maps also su er from similar problems of elongation of the autocorrelation of the Hankel functions and leakage of power showing up as artifacts along the outer edges of the maps and the lower right-hand corner. 4. RESULTS 4.1. Absorption Maps and Correlation Analysis The absorption maps shown in Figure 2 appear randomly patchy to the eye. In Figure 2a, which shows absorption in a region containing two small sunspots, contours of magnetic Ðeld strength are overlain on the absorption map. The maps are relatively insensitive to the outer boundary R of the annulus. This is shown in Figure 4. In most of max the analysis we chose R \ 589, 982, and 1260 pixels, max respectively, with R \ 30 pixels in all cases. We developed a procedure for detering the degree of correlation between absorption maps of the same region at di erent times. This tests whether the maps are doated by noise and, if they are not, leads to estimates of the lifetimes of sources and sinks. We subdivided the data for each day into sets 30 utes in length and made an absorption map for each of these 30 ute data sets. This is about as close to an instantaneous map of absorption as can be obtained, because a data slice at least 30 utes long is necessary to isolate the individual modes, which have periods in the 5 ute range. We index these individual maps by n, so that a (i, j) is the value of a n for the nth map at location (i, j) on one of the days listed in Table 1. We computed the correlation function at spatial lag (I, J), C (I, J), between pairs of maps n and m by taking the fast n,m Fourier transform (FFT) of each map and making use of the cross-correlation theorem. The function is normalized in the same way as a coefficient of linear correlation (Taylor 1997). After carrying out the boxcar smoothing described at the beginning of 3, we shifted each array relative to its mean absorption a6 a8 n (i, j) 4 a n (i, j) [ a6 (i, j), n where a6, is typically very small; see 4.2 and Figure 7. Then, if F [a8 is the Fourier component of at the two- k n ] a8 dimensional wavenumber k, then the Fourier transform of C is n,m F k [C n,m ] \ F k *[a8 n ] É F k [a8 m ], where * denotes the complex conjugate and C (I, J) is found from the inverse Fourier transformation. We n,m normalize F [C ] by dividing by p p, where the p represent the dispersions k n,m of each map about n its m mean. This normalization makes the autocorrelation function at zero lag equal to unity [C (0,0) \ 1]. n,n Figure 5 shows scans of C (I,0) for two pairs of con- n,n`1 secutive time slices from the 1997 July 5 data set. There are

6 No. 2, 1999 SOLAR p-modes 837 FIG. 5.ÈCenter scans of the cross-correlation between absorption maps made from neighboring 30 ute time strings for the 1997 July 5 data. These single line graphs show the lag in the east-west direction at the zero lag position in the north-south direction, for the boxcar-smoothed data. The sharpness of the peaks at zero spatial lag in both directions indicates that we are seeing a real signal. Panel a shows times 09:00È09:29 cross-correlated with 09:30È09:59; panel b shows times 10:30È10:59 cross-correlated with 11:00È11:29. FIG. 4.ÈAbsorption maps for the 1996 June 6 data using wavenumbers that correspond to n \ 0È1 and di erent values of R and corresponding *k (deðned by eq. [2] with n \ 1). (a) R \ max pixels, *k \ Mm~1; (b) R \ pixels, *k \ max Mm~1; (c) R \ 98 pixels, *k \ Mm~1. max In all cases, R \ 30 pixels. max clear peaks at zero spatial lag and no other peaks of comparable height. We interpret this to mean that there is a high degree of spatial correlation between these maps, which are made from data centered on times 30 utes apart. The results of all the correlation analyses are presented in compact form in Figure 6. The abscissa is the di erence between the starting times of each pair of maps, and the ordinate is the correlation at zero lag, i.e., the peak value in plots such as Figure 5. Each day of data is represented by a di erent symbol. A rectangle containing the sunspot pair was removed from the absorption maps made from the June 6 data, because the sunspots are obviously stable features that boost the degree of correlation. The points at 45 utes were obtained by removing the Ðrst 15 utes of each day of data and then subdividing into 30 ute strings. Interleaving the absorption maps made from these staggered ÏÏ strings with the other maps makes it possible to calculate correlations at intermediate times; for example, the correlation of absorption at times 0È30 utes with absorption at times 45È75 utes gives a point on Figure 6 at 45 utes. Although Figure 6 shows considerable scatter, the trend is clear: The correlation decreases steadily in all cases as the

7 838 HABER, JAIN, & ZWEIBEL Vol. 515 FIG. 6.ÈScatter plot of peak cross-correlations for absorption maps (ordinate) made over time intervals of 30 utes (abscissa). A rectangular region covering the sunspots in the June 6 data was removed before any cross-correlations were carried out. A peak cross-correlation of ¹0.2 is at the noise level, and in these cases the peak value was not always at zero lag. The lines are drawn through the average peak cross-correlation for each time interval with di erent lines used for each day of data. time lag increases from 30 to 90 utes. After this time, the correlation levels o at an amplitude of about 0.2, which we associate with the level of noise. Figure 6 therefore demonstrates that absorption maps made close together in time (within an hour of each other) resemble each other more than absorption maps made from data several hours apart. If the maps were merely noise, we would not see this e ect. Therefore, we believe that we are seeing a real pattern of acoustic sources and sinks and that individual features in this pattern persist for not much more than an hour and not much less than half an hourèif the pattern changed completely in a short interval, we would not see much correlation even between temporally adjacent data sets. The fact that the correlation between maps made 30 utes apart is substantially below unity suggests that shorter duration events must play some role, which is consistent with current theories of acoustic excitation by small-scale surface convection. However, the correlation is probably also reduced by noise Statistics of Emission and Absorption In order to quantify the statistics of emission and absorption, we created histograms of a for each map. These histograms are generally well Ðtted by Gaussians with dispersion p and mean k ; some examples are shown in Figure 7. The large a absorption a by the sunspot in the June 6 data does not register strongly in the histograms, primarily because less than 10% of the total area of the map is covered by the sunspot, a in the sunspots rarely exceeds 0.2 (see Fig. 3), and a substantial part of the dispersion is probably due to noise. Removing a rectangle covering the sunspots equal to 10% of the total area typically reduces the mean absorption k to about half its (already small) value, while changing the a dispersion p by less than 3%. Whether the Ðeld of view is only quiet Sun a or contains an active region, the histograms suggest that emission and absorption are closely balanced, at least in the data sets we have exaed. Previous studies have reported a small net emissivity for the quiet Sun (Bogdan et al. 1993; Braun 1995). Both of these papers (see, especially, Fig. 10 of Braun, bearing in d that our deðnition of a yields a value about half the size of his; see eqs. [7] and [8]) suggest that the modes that are most strongly emitted by the quiet Sun are also most strongly absorbed by sunspots. The quiet-sun data in these two papers were taken at a time when sunspots were present on the disk, 2 days later, in the Bogdan et al. study and, contemporaneously, in the Braun study. Our quiet- Sun data were taken near solar imum, at a time when there was less activity on the disk. Thus, if absorption by sunspots is compensated by emission from the quiet Sun, we would not see it in our quiet-sun data. This does not explain the lack of pronounced emission seen in the quiet parts of the June 6 data. It may be that our use of modes with n \ 1È4 (see Table 2) has washed out emission of the p mode, which tends to be more strongly absorbed than the 1 modes of higher radial order. A more worrisome possibility is that there is some systematic error that o sets the mean absorption coefficient measured from the MDI data relative to the absorption coefficient measured in other spectral lines, with other instruments (each of the data sets discussed here uses a di erent spectral line: Fe I 5576 [Bogdan et al.], the Ca II K feature at 3933 [Braun], and Ni I 6768 for the present MDI observations). As D. C. Braun (1998, personal communication) has independently conðrmed the absence of net emission in the quiet-sun data sets from MDI, we regard the possibility of an error in our analysis as unlikely. We, like previous investigators, have failed to identify any systematic e ect that would di erentiate between incog and outgoing waves, as would be necessary to produce a persistent o set. For example, in order to see whether projection e ects might somehow be at work, we plotted histograms of a versus distance from disk center and found them to be Ñat. We also carried out an analysis of the formal random errors, using the method suggested by Braun (1988). If the probability of observing power P in some mode i is given by a s2 distribution with 2 dof, f (P) \ e~p@p0i, P 0i then the best estimate for the mean of the distribution P is 0i the observed value P itself and the best estimate for the variance of P, p2, is P i 2/2. If all the errors i Pi from i mode to mode and point to point are independent of one another, then the dispersion in the a is given by (e.g., Taylor 1997) A LaB2 p2\; ppi 2, (9) a LP i i where the sum is over both the incog and outgoing waves and the di erentiation is carried out using equation (7). If there are N modes in the sum and the p2 do not di er greatly from m each other, then it can be seen Pi from equation (9) that p2 P N~1. In our case, N \ 88 (cf. Table 2). a m m Let the true mean value of a be a6. The standard deviation of a6, p 2, expected from N measurements of a, is (e.g., Taylor 1997) } a p } 2\ p a 2 N a. (10)

8 No. 2, 1999 SOLAR p-modes 839 FIG. 7.ÈHistograms of absorption coefficient a for the three data sets, compared for di erent time intervals, always with the same bin size. The time string is 30 utes in (a) 1996 June 6, 2562 pixels, (b) 1997 January 17, 5002 pixels, (c) 1997 July 5, 6402 pixels. Panel d shows 1996 June 6 data of 25.6 hours; panel e shows 1997 January 17 data for 12 hours; and panel f shows 1997 July 5 data for 4 hours. The curves are Gaussian Ðts to the data, with dispersion and mean shown on each Ðgure. The dispersion is insensitive to duration, suggesting that errors contribute signiðcantly to the widths. Therefore, if random errors doate, then a histogram of a large number of values of a should have approximate width p and should be centered on a6 to an accuracy of p /JN. a We computed the formal variances p2 and p, deðned a a in equations (9) and (10), for all our data a sets, including } the ones used to make the histograms shown in Figure 6. We Ðnd that p is typically within 20% of 0.1, i.e., that it is comparable a to the p computed for the Gaussian Ðts to the histograms. The number of points at which a is computed, namely, N, is typically of order 2562È6402, so the dispersion of the a mean according to this formal analysis is of order 1.5È4 ] 10~4. The similarity of the observed width of the histograms to that predicted from formal error analysis suggests that the true shape of the a distribution function is quite possibly obscured by error. Support for this conclusion comes from the similar widths of a histograms computed for data sets of di erent durations. Since we argue in 4.1 that the mean duration of an acoustic source or sink is an hour or less, we would expect histograms made from data sets much longer than an hour to narrow in proportion to their duration. We do see a 10% narrowing of the histograms for longer duration sets, but it appears to be constant and not proportional to the length of the set. Although the errors in the dispersion of a are signiðcant, the errors in a6 are extremely small. Therefore, we have con- Ðdence in the balance between emission and absorption that we obtained, barring discovery of some systematic source of error Interpretation Finally, we attempt to assess the physical signiðcance of the near-zero mean emission. The widths! of the p-mode peaks in the power spectrum have been measured (Korzennik 1990). If we assume that the widths are due to damping, we can crudely estimate the mean value of a, a6,

9 840 HABER, JAIN, & ZWEIBEL Vol. 515 such that emission balances dissipation. Since the line widths can also be produced by nondissipative e ects, this gives an upper limit on the a6 required for a steady state. We assume that the propagation of waves away from their sources can be described by WKB ray theory, as calculated by Brown (1991). We assume that the solar envelope is a polytrope of index m and that the rays propagate between the surface, z \ 0, and a lower turning point z \ z. The lower turning point is approximately deðned by t c (z ) \ 2nlR /l (11) s t _ for a mode with degree l? 1 and frequency l. The sound speed c in a polytrope is given by s A c \ cgz, (12) s m ] 1B1@2 where c is the ratio of speciðc heats. Integration of the ray equations shows that the distance r between successive reñections of the ray from the surface z s \ 0is r \ nz, (13) s t and the time interval t between successive reñections is s t \ r s s c (z ). (14) s t Brown (1991) has shown that the decay of wave amplitude with horizontal distance from the source is rapid; Âv D r~2. Therefore, the outgoing power from a source is detectable only within a fairly small horizontal range r D r. Assug s that the rays generated by the source occupy a depth z, t writing the di erence in energy per unit mass between outgoing and incog power as *(v2) and taking the mean horizontal speed of the outgoing waves to be V, the w average observable source luosity L can be written as s L \ 2no*(v2)V z r, (15) s w t s where o is the average mass density. On the other hand, the dissipation rate E0 within the source volume is E0 \!nov2z r2. (16) t s Equating L and E0, as is required for a steady state, yields s *(v2) v2 \!r s. (17) 2V w The left-hand side of equation (17) can be identiðed with a6. On the right-hand side, V can be approximated by r /t. Equation (17) then becomes w s s a6 \!t s 2, (18) where the factor of 1 comes from the assumption that the source volume is cylindrical. 2 We now proceed to estimate a6 from simple ray theory and measured line widths. According to equations (11)È(15), a6 \ n2(m ] 1)!lR _ D 6 ] 107!l, (19) cgl l where in the last equality we have put m \ 3, c \ 5/3. Based on mean values from Table 2, we put l \ 3 ] 10~3 Hz, l \ 500. The mode lifetimes!~1 at the moderately large l we are considering have been measured by Korzennik (1990) and estimated by Chen et al. (1996) to be of order 2 hours;! D 10~4 s~1. Substituting these values into equation (19) yields the estimate a6 D 0.036, or a few percent. This is substantially larger than the mean values of a that we compute from our histograms, suggesting that the only signiðcant absorption is the localized absorption displayed on the maps. Nevertheless, the uncertainties in our calculation, the measured uncertainty in!, and theoretical uncertainty in the interpretation of! suggest that we can draw only provisional conclusions at this time. 5. SUMMARY AND CONCLUSIONS We have used Dopplergrams obtained from the MDI instrument on board SOHO to search for sources and sinks of p-mode wave emission located slightly below the solar photosphere. Theoretical studies suggest that the primary source of waves is acoustic emission from turbulent convection. Modeling has shown that, whether the acoustic emissivity is due to turbulent Reynolds stresses (as in the Lighthill process) or to thermal perturbations, the emissivity should be concentrated in small regions. The wave amplitude decreases rapidly with distance from these sources, due to geometrical and ray e ects. These two things together suggest that it might be possible to observe discrete sources. We produced absorption maps by applying the Hankel wave decomposition technique originally introduced by Braun et al. (1987) to study acoustic absorption by sunspots. This involves projecting the p-mode power observed within an annulus surrounding a particular point onto a set of incog and outgoing Hankel functions and comparing the power found in each. In this way we covered, point by point, large sections of the MDI Ðeld of view, on three di erent days of observing (see Tables 1 and 2 for a summary of the data sets and the modes we analyzed). Absorption maps are shown in Figures 2 and 4. We showed by means of a correlation analysis that individual absorption features persist for periods up to about half an hour to an hour and subsequently decay (Figs. 5 and 6). Histograms of the absorption coefficients are well Ðtted by Gaussians centered near zero mean absorption. Although the Gaussian shape itself is likely to be blurred by random errors, and hence not necessarily a reliable Ðt to the distribution function of emission and absorption, the formal signiðcance of the mean a6 itself is high. Therefore, our data appear to show that point sources of emission are balanced by point sources of absorption. This is a nontrivial result. It implies that the primary p-mode damping process is the inverse of the emission process and not a physically distinct mechanism such as radiative damping. However, two di erent factors mitigate the strength of this conclusion. One is that there may be some source of error that systematically shifts all our measured values of the emission and absorption coefficient. Some previous studies, using di erent instruments to observe di erent spectral lines, have in fact reported a small net emissivity for the quiet Sun. The second issue is that since p-modes are observed to have rather sharp line widths, and therefore are weakly damped, only a small net emissivity is required to

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