RADIO PHOTOSPHERES OF LONG-PERIOD VARIABLE STARS

Transcription

1 THE ASTROPHYSICAL JOURNAL, 476:327È346, 1997 February 10 No copyright is claimed for this article. Printed in U.S.A. RADIO PHOTOSPHERES OF LONG-PERIOD VARIABLE STARS MARK J. REID AND KARL M. MENTEN Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA Received 1996 June 13; accepted 1996 September 5 ABSTRACT We report the detection of centimeter-wavelength emission from a sample of nearby long-period (Mira and semiregular) variables using the VLA. Six of the eight stars in the sample were detected. We Ðnd the continuum emission in the radio band to have a spectral index near 2.0, as expected for optically thick blackbody emission. The Ñux densities are a factor of B2 above the level expected from the optical photospheres of the stars. We monitored three stars over a period of nearly 2 yr and Ðnd Ñux density variations of less than ^15%. We partially resolved the stellar disk of W Hya and Ðnd an average diameter of 0A.080 ^ 0A.015 and a brightness temperature of 1500 ^ 570 K. Our observations suggest that long-period variables have a radio photosphere ÏÏ near is the stellar radius (deðned by line-free regions of the optical spectrum). For the physical conditions expected in the radio photosphere, free electrons, obtained predominantly from the ionization of potassium and sodium, provide the dominant opacity through free-free interactions with neutral H and H. A simple model with a single set of physical parameters can approximate all of our centimeterwavelength data, as well as providing plausible sizes and brightness temperatures at far-ir wavelengths. 2 At centimeter wavelengths, unity optical depth is achieved at a radius of about 4.8 ] 1013 cm, where the density and temperature are B1.5 ] 1012 cm~3 and B1630 K, respectively. The lack of variability of the centimeter-wavelength Ñux density for stars like o Ceti, R Leo, and W Hya limits variations of the temperature and/or radius of the radio photosphere to less than ^150 K 2R, where R * * and ^4 ] 1012 cm. Also, any periodic shocks or disturbances near 2R probably propagate outward at * less than B5 kms~1 and/or are mostly damped. The radio photosphere lies just outside of a molecular photosphere,ïï seen optically in strong absorption lines of metallic oxides, and just inside of the SiO maser shell and the dust formation zone. Indeed, the inner boundary of the SiO maser emission region may be determined by continuum opacity in the radio photosphere. Our study suggests that the density and temperature in the SiO shell are B5 ] 1010 cm~3 and B1300 K, respectively. Extrapolating our model outward to Z1014 cm radius, where signiðcant dust is detected at 10 km wavelength and H O 2 masers are found, gives densities [3 ] 109 cm~3 and temperatures [1100 K. Based on the sample of stars we observed, it appears that the Ñux density at radio frequencies from long-period variables can be modeled with a simple formula. Given the excellent agreement between measured and modeled Ñux densities, it is possible that distances can be estimated from a Ñux density measurement with a precision of about 10%. Since the radio-frequency emission from long-period variables has a well-deðned spectrum, is very compact, and is relatively constant in time, we suggest these stars can be used to determine the absolute Ñux density scale for millimeter- and submillimeterwavelength interferometers. Subject headings: radio continuum: stars È stars: fundamental parameters È stars: variables: other (long-period variables) È techniques: interferometric 1. INTRODUCTION Long-period variables (e.g., Miras and semiregular variables) vary dramatically in optical light and are thought to change temperature and size by about 30% over periods of 100È1000 days. We have a good understanding of neither the stellar pulsation mechanism nor even the observational properties of the pulsations. Nonradial pulsations have been suggested, and signiðcant departures from spherical symmetry have been observed for o Ceti (e.g., Karovska, Nisenson, & Papaliolios 1991; Hani et al. 1992). At a distance of a few stellar radii and beyond, dust condenses (see Danchi et al. 1994) and mass begins to stream away from the star as a result of radiation pressure on the dust and subsequent drag on molecules. Since a signiðcant fraction of the mass returned to the interstellar medium comes from red giant stars, this is an important process that a ects star formation and galaxy evolution. However, fundamental questions relating to the movement of mass outward to a few stellar radii and the conditions under 327 which dust forms still remain. Thus, new approaches to the study of the region between the photosphere and the dust formation zone are extremely valuable. Long-period variables have photospheric temperatures of B2500 K and radii of B2 AU or more. The Rayleigh- Jeans tail of the blackbody emission from such a star at a distance of 100 pc would be close to 1 mjy at a wavelength of 1 cm. This level of emission is easily detectable, for example, with the Very Large Array (VLA), and motivated us to conduct a survey of Mira and semiregular variables for radio emission from the stellar photosphere. Previously, most attempts to Ðnd centimeter-wavelength emission from stars have been aimed at detecting nonthermal (e.g., synchrotron) emission or emission from ionized stellar winds (see 4.1). Nonthermal emission is characterized usually by a falling spectral index (Ñux density, S, decreasing with increasing frequency, l), and ionized winds have mildly rising spectra. Given receiver and antenna performance characteristics, searches for such

2 328 REID & MENTEN Vol. 476 emissions favored observations at frequencies near or below 5 GHz. Since emission associated with a stellar photosphere would have a sharply rising spectrum (S P l2), such emissions would be difficult or impossible to detect at low frequency. This may, in part, explain why photospheric ÏÏ emission from stars has often not been detected in previous studies. In this paper, we provide a comprehensive report of our broadband spectral and high-resolution spatial information on the radio emission from long-period variables. Early results from our studies have been reported elsewhere (Reid & Menten 1990, 1993). In 2, we describe observations made with the VLA of a number of Mira and semiregular variables. In 3, we show clear detections of these variables at centimeter wavelengths, the spectral dependence of this radio emission, a measurement of the size and brightness temperature of one star, and the nature of temporal variations of the radio emission over the cycle of optical variability of a few stars. Remarks on previous centimeterwavelength observations of such stars are found in 4, along with a discussion of the opacity mechanism. We also present a model of the radiative transfer in the radio photosphere, as well as constraints on the physical conditions in this region. Finally, in 5 we discuss the use of Mira and semiregular variables in determining the absolute Ñux density scale for telescopes operating at millimeter and submillimeter wavelengths. 2. OBSERVATIONS 2.1. Measurements of Stellar Spectra We observed eight stars between 1989 December 2 and 1990 February 27, using the NRAO1 VLA in its smallest (D) conðguration. Table 1 lists the stars observed, including their optical positions and spectral types from Baudry et al. (1990) and their estimated distances from Hani et al. (1992). In addition, we observed VY CMa, an extreme supergiant star located near a region of star formation, and will report these results in a later publication (Menten & Reid 1997). We used a standard continuum observing technique, employing four 50 MHz bands (right- and left- 1 The National Radio Astronomy Observatory is operated by Associated Universities, Inc., under cooperative agreement with the National Science Foundation. circular polarization at two frequencies) and observed at 8, 15, and 22 GHz for most stars. The stars were selected to be nearby (to maximize their angular size) and distributed in right ascension. The Ñux density scale was determined from observations of 3C 286 or 3C 48. Table 2 lists the Ñux densities assumed for these primary calibrators for the three observing frequencies. The Ñux densities measured for the secondary calibrators are also listed in Table 2. The secondary calibrators were used to calibrate the interferometric amplitudes and phases of the stars. The calibrators were chosen to be strong at all three observing frequencies, nearby in angle to the selected stars, and to have no signiðcant structure resolved with the D-conÐguration baseline lengths. At the highest observing frequency of 22 GHz, we found that the average telescope gain decreased by about 10% between elevation angles of greater than 40 and 25. This a ected the low-declination source W Hya that, although observed near transit, was observed at elevation angles of about 25. Since the secondary calibrator for this star was observed near 40 elevation, we corrected the measured 22 GHz Ñux density of W Hya upward by 10%. Except for the elevation-dependent gain correction discussed above, calibration was handled in the standard manner recommended for VLA data with the Astronomical Image Processing System (AIPS) developed by NRAO. The Ðelds around the stars were imaged with maps of 256 by 256 pixels. We used pixel sizes of 3A.0, 1A.5, and 1A.0 at 8, 15, and 22 GHz, respectively. For each star, the location and total Ñux density were determined by Ðtting a two-dimensional Gaussian proðle in a small region of the map containing the stellar emission. While most maps were consistent with point-source emission, some of the higher frequency maps displayed slightly extended brightness distributions. This is probably due to imperfect calibration of the interferometric phases and is almost certainly not a property of the stars. Thus, we used the brightness integrated over the Ðtted Gaussian proðle as the best estimate for Ñux density Measurements of Stellar Size We attempted to measure directly the angular size of W Hya and o Ceti when the VLA was in its largest (A) conðguration. At 22 GHz, the VLA has a synthesized beam of B0A.08, which is comparable to the angular size, measured optically, for both stars. Since radio-frequency seeing ÏÏ TABLE 1 OPTICAL DATA FOR STARS OBSERVED WITH THE VLA Distanceb Star R.A. (J2000) Decl. (J2000) Spectral Typea (pc) o Ceti ^ [ ^ 0.15c M5eÈ9e 110 ^ 9 U Ori ^ ] ^ 0.20 M6eÈ9.5e 265 ^ 21 R Leo ^ ] ^ 0.16 M6eÈ9e 110 ^ 9 R Crt ^ [ ^ 0.27 M7 121 W Hya ^ [ ^ 0.38 M7.5eÈ9e 95 ^ 8 R Aql ^ ] ^ 0.14 M5eÈ9e 220 ^ 18 s Cyg ^ ] ^ 0.17 S6.2eÈ10.4e 135 ^ 11 R Cas ^ ] ^ 0.15 M6eÈ10e 160 ^ 13 NOTE.ÈUnits of right ascension are hours, minutes, and seconds, and units of declination are degrees, arcminutes, and arcseconds. a Positions and spectral types from Baudry et al. (1990). b All distances are based on the PL relation of Feast et al. (1989); values are from Hani et al. (1995), except for R Crt, where P \ 160d and m \[1.23 mag were used to calculate distance. c o Ceti has a large declination proper motion k of [0A.23 yr~1; the observations were conducted between 1984 and 1988.

3 No. 1, 1997 LONG-PERIOD VARIABLE STARS 329 TABLE 2 CALIBRATION SOURCES FOR SPECTRAL MEASUREMENTS FLUX DENSITY (Jy) SOURCE R.A. (J2000) DECL. (J2000) 8.44 GHz 14.9 GHz 22.4 GHz 3C 48a ] C 286a ] ]478b ] ]152c ] ]135d ] [166e [ ]201e ] [189f [ [129g [ ]211h ] ]318i ] NOTE.ÈUnits of right ascension are hours, minutes, and seconds, and units of declination are degrees, arcminutes, and arcseconds. a Primary calibrator; see Baars et al. (1977) and VLA Calibration Manual. b Secondary calibrator for R Cas. c Secondary calibrator for o Ceti. d Secondary calibrator for U Ori. e Secondary calibrator for R Leo. f Secondary calibrator for R Crt. g Secondary calibrator for W Hya. h Secondary calibrator for R Aql. i Secondary calibrator for s Cyg. (determined mostly by Ñuctuations in water vapor in the atmosphere) is rarely better than 0A.1, we developed a novel observing procedure in which H O masers associated with 2 a star serve as a phase reference for the continuum data. This allows the calibration of the data that results in nearly perfect seeing ÏÏ to be achieved. In 1990 February and March and 1991 July, we observed W Hya with the VLA in the A conðguration. The observations to measure stellar sizes were conducted in 4-IF continuum mode. The A-C intermediate-frequency (IF) pair (50 MHz bandwidth in right- and left-circular polarization) was centered at MHz, B50 MHz above the H O 2 6 ] 5 line. The B-D IF pair was set at a MHz bandwidth centered on the water line at a local standard of rest (LSR) velocity of 42.4 km s~1, assuming a rest frequency of MHz. Initial calibration using observations of a strong continuum source removes the electronic phase di erences among the four bands, adjusts for the slowly varying amplitude response of individual antennas, and partially compensates for the highly variable atmospheric phase variations. Following the initial calibration, the A-C and B-D IF data are separated and time averaged to 10 or 30 s records, depending on the coherence time of the array. The B-D IF data, containing very intense H O maser emission, are selfcalibrated ÏÏ on a record-by-record 2 basis. The resulting antenna-dependent phase corrections are then applied to the A-C IF data, containing the broadband continuum emission from the star. This calibration procedure was Ðrst described by Reid & Menten (1990) and is described in detail in the Appendix. The calibration e ectively removes atmospheric variations from the data, yielding nearly perfectly calibrated interferometric phases. Maps of the maser emission in the B-D IFs and the stellar continuum emission in the A-C IFs can then be made. We observed o Ceti on 1990 March 3 with a somewhat di erent observing setup than used for W Hya. Since the H O maser in o Ceti is weak, [5 Jy, we observed in 2 spectral-line mode, providing a sufficiently narrow bandwidth to match the H O line width, in order to achieve high 2 enough signal-to-noise ratio for calibration. Unfortunately, at the VLA for observations in spectral-line mode, one must use bandwidths of 25 MHz or less if any one of the bands is set below 50 MHz. Thus, in spectral-line mode, one cannot, for example, simultaneously use a 50 MHz band o the H O line and a 3 MHz band (containing, say, 64 line 2 channels) on the H O line. Dropping the o -line bandwidth 2 to 25 MHz reduced the signal-to-noise ratio for the broadband continuum data by a factor of 21@2, compared with the 50 MHz bands used for W Hya. We did detect compact continuum emission from o Ceti. However, the reduced signal-to-noise ratio made size measurements for this star that are of insufficient accuracy for use in this paper Measurements of Radio-Frequency L ight ÏÏ Curves Following the detection of radio-frequency emission from most of the variable stars in our D-conÐguration survey, we initiated a program to monitor the Ñux density variations of three stars: o Ceti, R Leo, and W Hya. The monitoring program consisted of observations at 8.44 GHz at roughly monthly intervals. The observing frequency was chosen to maximize signal-to-noise ratios and minimize difficulties in calibrating instrumental and atmospheric variation. Typically, we obtained 1 hr of on-source integration time for each star at each session. The monitoring program spanned the period from 1990 June to 1991 May and started with the VLA near the end of the A conðguration and proceeded through smaller (B and C) conðgurations to the D conðguration. In addition, our earlier measurements, described in 2.1, extended the time span to nearly 2 yr. Standard continuum calibration techniques were applied, similar to those described in 2.1 for the D-conÐguration observations used to determine the radio-frequency spectra of the stars. However, di erent secondary calibration

4 330 REID & MENTEN Vol. 476 sources were needed, since most of the observations were conducted with markedly longer interferometer baselines than in the D conðguration. Unfortunately, the observations during the summer of 1990 in the larger VLA con- Ðgurations were not usable, owing to large atmospheric phase Ñuctuations, and we report only those results starting with the B-conÐguration observations in 1990 September. 3. RESULTS 3.1. Stellar Spectra Six of the eight stars observed were detected with the D-conÐguration observations in at least one of the observing bands. The measured Ñux densities are given in Table 3. Also listed in Table 3 are the positions of the continuum emission calibrated with respect to compact extragalactic radio sources. The positions are a weighted average of those measured at di erent frequencies and, in some cases, at different epochs. No corrections for proper motions have been made. The optical positions (see Table 1) and radio positions of all stars, except o Ceti, are coincident to within their joint uncertainties (typically [0A.5). The only statistically signiðcant di erence between the radio and the optical positions can be found in the declination position of o Ceti. The radio-minus-optical declination di erence for o Ceti is [0A.71^0.18 However, o Ceti has a large declination proper motion of [0A.23 yr~1. Since the optical observations of Baudry et al. (1990) were conducted between 1984 and 1988 and our radio observations were near 1990, the radio position should be between about 0A.3 and 1A.4 south of the optical position as observed. The Ñux densities listed in Table 3 are plotted in Figure 1 for all stars with detected centimeter-wavelength emissions. The solid lines in the Ðgure represent the best-ðt power laws with spectral indices of 2. We Ðnd that all stars detected had power-law radio spectra (from 8 to 22 GHz) consistent with a spectral index near 2.0, which is characteristic of an optically thick blackbody spectrum. In order to compare the centimeter-wavelength Ñux densities with those observed at shorter wavelengths, in Figure 2 we also plot millimeter-wavelength Ñux densities (from Sopka et al. 1985; Walmsley et al. 1991; Altenho, Thum, & Wendker 1994) and selected infrared Ñux densities from the compilation of Gezari et al. (1993). The solid gray lines in these panels display the maximum and minimum radio emission expected from the stellar photosphere based on a simple model, motived by the data of Pettit & Nicholson (1933), in which the temperature, T, and radius, R, vary with optical phase, /, as T(/)\2300 ] 300 cos (/) K, (1a) and R(/) \ 2.0 ] 0.4 sin (/) AU. (1b) The model Ñux densities were computed, assuming distances of Hani, Scholz, & Tuthill (1995) and based on the period-luminosity (PL) relation of Feast et al. (1989), which assumes a distance modulus for the LMC of mag. One can see that the radio emission expected from this simple photospheric model tends to fall below the measured values. The measured Ñux densities are close to, but somewhat above, those expected from optically determined temperatures and photospheric sizes. Finally, we show, as the dashed lines in Figure 2, the Ñux density versus frequency predicted by the model for the radio emission discussed in detail in 4.3. This model Ðts the radio-frequency data better than the pure photospheric model given by equations (1a) and (1b). Also, the Ñux densities extrapolated from radio frequencies either provide a reasonable Ðt to the infrared data or fall below it. This is as expected for radiofrequency emission from the photospheres of stars that have a range of infrared excesses, owing to extended circumstellar dust shells. We note that our centimeter-wavelength measurements are totally insensitive to any reasonable circumstellar dust emission, because of the low dust emissivity and the spatial Ðltering provided by the interferometer Stellar Sizes The broadband observations at 22 GHz of W Hya, calibrated with the narrowband H O line data, reveal a par- 2 tially resolved continuum source. Figure 3 shows the (real part of the) fringe visibility versus projected interferometer spacing for observations in 1990 and Modeling the TABLE 3 RADIO-FREQUENCY SPECTRA FLUX DENSITYb (mjy) STAR R.A. (J2000)a DECL. (J2000)a 8.44 GHz 14.9 GHz 22.4 GHz o Ceti ^ 0.01 [ cB ^ ^ ^ 0.2 U Ori d ] d \0.09 \ R Leo ^ 0.01 ] ^ ^ ^ ^ 0.2 R Crt d [ d \ W Hya ^ 0.02 [ ^ ^ ^ ^ 0.3 R Aql ^ 0.1 ] ^ 0.4 \ ^ ^ 0.4 s Cyg ^ 0.1 ] ^ ^ ^ ^ 0.3 R Cas ^ 0.1 ] ^ ^ 0.05 \ ^ 0.3 NOTE.ÈUnits of right ascension are hours, minutes, and seconds, and units of declination are degrees, arcminutes, and arcseconds. a Positions have no corrections for proper motions. b From VLA D-conÐguration observations only; upper limits are twice the rms level in the map. c o Ceti has a large declination proper motion of [0A.23 yr~1. Since the optical observations of Baudry et al. (1990) were conducted between 1984 and 1988 and our radio observations were near 1990, the position of the star at the epoch of the radio observatons should be between about 0A.5 and 1A.0 south of the optical position as observed. d Star not detected; coordinates are from optical catalogs.

5 No. 1, 1997 LONG-PERIOD VARIABLE STARS 331 FIG. 1.ÈRadio-frequency spectra of six long-period variable stars. The measured values and their 1 p uncertainties plotted here are also listed in Table 3. A best-ðtting line with Ñux density, S, proportional to frequency, l, squared is also shown. variation of fringe visibility with baseline length for the 1990 observations yields a (uniform disk) diameter of 0A.09 ^ 0A.02. The 1991 observations yield a size of 0A.07^ 0A.02. Although taken at di erent phases of the light curve of this variable star, the lack of strong variability in the radiofrequency light ÏÏ curves, discussed in 3.3, suggests that changes in the extent of the stellar emission are probably much less at radio than at optical wavelengths. Thus, we average the two size measurements together and, for all further discussion in this paper, assume an angular size of 0A.080 ^ 0A.015 for W Hya at 22 GHz. For a distance of 95 pc, a 0A.08 diameter corresponds about 1.1 ] 1014 cm (7.6 AU). The total Ñux density of W Hya estimated from the A- conðguration data was 2.2 mjy. It was difficult to determe the absolute Ñux density scale for this observation because of the combined e ects of the low elevation angles of the observations, the 15 di erence in the declinations of the secondary calibrator (1337È129) and W Hya, and the small interferometer u-v range of 0È180 kj available for the primary Ñux density calibrator (3C 286). We estimate a systematic uncertainty of about 20% in the A-conÐguration Ñux density scale. Averaging the Ñux density of 2.2 ^ 0.4 mjy measured in the A conðguration with 2.98 ^ 0.3 mjy measured in the D conðguration yields an average 22 GHz Ñux density of 2.70 ^ 0.24 mjy for W Hya. Adopting a uniform disk diameter of 0A.080 ^ 0A.015 and a total Ñux density of 2.70 ^ 0.24 mjy yields a radio brightness temperature of 1500 ^ 570 K. Thus, we Ðnd radio emission from a region approximately twice the diameter and two-thirds the temperature of the optical photosphere of this variable star (cf. Hani et al. 1995) Radio-Frequency L ight ÏÏ Curves The 8 GHz Ñux densities of o Ceti, R Leo, and W Hya, measured at roughly monthly intervals from 1990 Septem-

6 332 REID & MENTEN Vol. 476 FIG. 2.ÈRadio-to-infrared spectra of six long-period variable stars. The centimeter-wavelength data (near 1010 Hz) are from this paper; a millimeterwavelength point (near 2 ] 1011 Hz) and selected infrared points (between 3 ] 1012 and 1014 Hz) are taken from the literature cited in the text. The solid gray lines bound a range of Ñux densities expected from the optical photosphere over the light cycle of the stars. The dashed line is for a single model for a radio photosphere shown in Fig. 7 and described in the text. ber through 1991 May, are tabulated in Tables 4A, 4B, and 4C. These tables also list the Ñux densities derived for the (secondary) calibrators, obtained from comparisons with the primary Ñux density calibrators. The Ñux densities of the three secondary calibrators decrease slowly and smoothly in time; this almost certainly reñects real changes in the secondary calibrators. (The monitoring program also included observations of the extreme supergiant star VY CMa, which will be reported elsewhere. However, we note here that the secondary calibrator associated with VY CMa increased Ñux density over the monitoring period.) The measured 8 GHz Ñux densities as a function of stellar phase and date (Tables 4AÈ4C), plus the 8 GHz Ñux densities determined as part of the radio-frequency spectrum measurements ( 3.1, Table 3), are plotted in Figure 4. This Ðgure also displays the radio emission expected from the optical photosphere (solid gray lines), based on the simple model given by equations (1a) and (1b). One can see that the radio emission expected from this model falls below the measured values (as noted in 3.1) and varies by about ^50%, considerably more than the measured Ñux densities. The 8 GHz emission from R Leo and W Hya appears constant to within measurement uncertainties. The 8 GHz emission from o Ceti, while also fairly constant in time, is suggestive of a periodic variation at a low level and out of phase with that expected from the model based on the optical light curve (solid gray line). The dashed line in Figure 4 for o Ceti is based on a model, discussed in 4.3.2, of the e ects of a periodic disturbance in the radio photosphere. While this model Ðts well the data for o Ceti, a

7 No. 1, 1997 LONG-PERIOD VARIABLE STARS 333 FIG. 3.ÈFringe visibility vs. interferometer baseline length for W Hya at 22 GHz measured in 1990 (optical phase of 0.7) and 1991 (optical phase of 0.0). The dotted lines indicate a uniformly bright disk model. Best-Ðtting diameters are 0A.09^0A.02 and 0A.07^0A.02 for the 1990 and 1991 data, respectively. constant Ñux density would Ðt almost as well. Thus, pending further data, we conclude conservatively that the 8 GHz Ñux densities for all three stars are consistent with a constant value within about less than ^15%. 4. DISCUSSION 4.1. Comparisons with Previous Centimeter-W avelength Results Mira and semiregular variables, as well as M-type supergiants, have been targets of centimeter-wavelength observations in the past. Table 5 lists results published prior to 1996 (excluding early reports from our work), including claimed detections, tentative detections, and upper limits. As we will show below, few of the previously published results are detections of photospheric emission from longperiod variables at high conðdence levels (e.g., [5 p). Based upon the results presented in 2, the following general statements seem appropriate. Few, if any, of the tentative ÏÏ detections ([5 p) are likely to be real. For example, the 5 GHz measurement for o Ceti of 0.74 ^ 0.25 mjy (Spergel, Giuliani, & Knapp 1983) is approximately 5 times greater than expected by extrapolating with S P l2 from our maximum 8.4 GHz value of 0.49 mjy. The 15 GHz measurement of 0.76 ^ 0.17 mjy (Drake, Linsky, & Elitzur 1987) near R Cas is approximately twice that expected from the marginal detections or upper limits cited in Table 3. Since, as Drake et al. (1987) point out, their 15 GHz emission peaks about 7A from the stellar position, this emission is probably not from R Cas. (Note that we Ðnd no statistically signiðcant emission at any position in our maps at 8, 15, or 22 GHz.). Our monitoring observations of three long-period variables are consistent with nearly constant (\^15%) 8.4 GHz Ñux densities. In light of these Ðndings, the strong Ñares required to explain the 11 and 15 GHz Ñux densities reported by Woodsworth & Hughes (1973) and Bowers & Kundu (1979) for R Aql would be unusual occurrences. However, since both of these measurement were made with single-dish telescopes (not interferometers), the chance for spurious changes in the total power mimicking a weak variable source should be considered. Three of the Ðve stars in Table 5 with Ðrm detections are R Aqr, V Hya, and IRC ] R Aqr is a symbiotic system consisting of an M-type giant and a hotter companion. The centimeter-wavelength emission detected by Spergel et al. (1983) is far too strong to be photospheric emission from the M-type giant and is associated with material ionized by the companion. V Hya is a semiregular carbon star (spectral class N6). Luttermoser & Brown (1992) point out that, at its estimated distance of 380 pc, the detected emission is an order of magnitude above that expected from the photosphere. Thus, this emission, as well as that from IRC ]10216, is probably chromospheric in origin, as has been well established for the early M-type giant a Ori (e.g., Altenho, Oster, & Wendker 1979; Newell & Hjellming 1982). Recently, Knapp et al. (1995) reported Ðrm detections at a frequency of 8.4 GHz of the Mira variable R Leo and the extreme supergiant VY CMa, both at levels of 0.26 ^ 0.03 mjy, as well as from a variety of other types of evolved stars. Their observational results for R Leo and VY CMa are consistent with our Ðndings. Knapp et al. point out that the 8.4 GHz emission from R Leo exceeds that expected from the optical photosphere and suggest a chromospheric component. However, as we show below, this emission is most likely not chromospheric but extended ÏÏ photospheric emission Opacity Source The primary questions raised by our detection of continuum emission at radio frequencies toward Mira and semiregular variables concern the source of opacity and the physical conditions under which the emission originates. Spectral indices near 2 (S P l2), characteristic of the Rayleigh-Jeans side of a thermal blackbody source, and Ñux densities close to those expected from the stellar photospheres suggest a direct photospheric origin for the emis-

8 334 REID & MENTEN Vol. 476 TABLE 4A MONITORING DATA FOR oceti AT 8.4 GHZ Visual Flux Density VLA Calibrator Flux Densitya Julian Date Phase (mjy) ConÐguration (Jy) 2,448, ^ 0.1 AB ^ ,448, ^ 0.02 BC ^ ,448, ^ 0.02 C ^ ,448, ^ 0.03 C ^ ,448, ^ 0.03 C ^ ,448, ^ 0.04 CD ^ ,448, ^ 0.04 D 0.775b 2,448, ^ 0.04 D ^ a Calibrator was 0239[125 with Ñux density based on observations at similar elevations of 3C 48 assumed to be 3.30 Jy. Formal uncertainties are listed with no contribution from systematic sources of error. b No primary calibrator available; secondary calibrator Ñux density interpolated from neighboring measurements. TABLE 4B MONITORING DATA FOR RLEO AT 8.4 GHZ Visual Flux Density VLA Calibrator Flux Densitya Julian Date Phase (mjy) ConÐguration (Jy) 2,448, ^ 0.1 BC ^ ,448, ^ 0.1 BC ^ ,448, ^ 0.02 C ^ ,448, ^ 0.02 C ^ ,448, ^ 0.03 D ^ ,448, ^ 0.02 D ^ ,448, ^ 0.03 D ^ ,448, ^ 0.02 D ^ a Calibrator was 0954]177 with Ñux density based on observations at similar elevations of 3C 286 assumed to be 5.26 Jy. Formal uncertainties are listed with no contribution from systematic sources of error. TABLE 4C MONITORING DATA FOR WHYA AT 8.4 GHZ Visual Flux Density VLA Calibrator Flux Densitya Julian Date Phase (mjy) ConÐguration (Jy) 2,448, ^ 0.1 BC ^ ,448, ^ 0.1 BC ^ ,448, ^ 0.03 C ^ ,448, ^ 0.03 C ^ ,448, ^ 0.03 D ^ ,448, ^ 0.03 D ^ ,448, ^ 0.03 D ^ ,448, ^ 0.03 D ^ a Calibrator was 1339[263 with Ñux density based on observations at similar elevations of 3C 286 assumed to be 5.26 Jy. Formal uncertainties are listed with no contribution from systematic sources of error. sion. However, our observations contradict this hypothesis in three ways. First, the centimeter-wavelength Ñux densities measured for our sample exceed those predicted from the optical photospheres, based on the size and temperature measurements of Hani et al. (1995), on average by a factor of about 2. Second, the measured diameter of W Hya (B1.1 ] 1014 cm or 7.6 AU) is larger, and the brightness temperature (B1500 K) is lower, than most estimates for the optical photosphere. Third, the lack of variability of the radio Ñux density, at levels expected from a pulsating star, argue strongly against emission directly from the optical photosphere. The most likely explanation for the observed radio spectrum is that electrons of sufficient density form a radio photosphere ÏÏ at a radius of about twice that of the optical photosphere. Mechanisms that might result in a radio photosphere include Thomson scattering, thermal bremsstrahlung involving H` or metallic ions, and free-free emission from interactions with neutral H and/or H. In order to assess the relative contributions of these processes, 2 we need to estimate the electron density in the radio photosphere Ionization Under conditions of thermal equilibrium, the ionization of a mixture of gases is governed by the Saha equation (cf.

9 No. 1, 1997 LONG-PERIOD VARIABLE STARS 335 FIG. 4.ÈRadio-frequency light curves ÏÏ for three long-period variable stars. The 8 GHz Ñux densities from Tables 4AÈ4C and from Table 3 are plotted, folded on the optical period in the left-hand panels and as a function of Julian Date in the right-hand panels. The solid gray line is the Ñux density expected from a simple model of the optical photosphere. The dashed lines are from models of the radio photosphere: the static model shown in Fig. 7 was used for R Leo and W Hya, while the shock ÏÏ model shown in Fig. 11 was used for o Ceti. The model Ñux densities were scaled to Ðt the data by adjusting the distances to 102, 125, and 100 pc for o Ceti, R Leo, and W Hya, respectively. Alternatively, small changes in the model temperature and density proðles could accomplish the same scaling. Gray 1992). For each atomic element, n 1 P \ (2nm )3@2(kT )5@2 2u (T ) e 1 e~i@kt, (2) n e h3 u (T ) 0 0 where n and n are the densities of singly ionized and neutral states, 1 respectively, 0 I is the ionization potential, T is the temperature, u (T ) and u (T ) are the partition functions 1 0 for the ionized and neutral states, m is the electron mass, k e is BoltzmannÏs constant, h is PlanckÏs constant, and P \ e n kt is the total electron pressure, where n is the electron density e obtained by summing the electrons e from all ions. We evaluated the Saha equation for the 25 most abundant elements, using solar abundance, ionization potential, and partition function data tabulated by Gray. The hydrogen density was nominally set to 1012 cm~3, and the 25 element equations were solved iteratively, updating the total electron pressure at each iteration, until convergence was achieved. Figure 5 displays the total electron density, along with

10 336 REID & MENTEN Vol. 476 TABLE 5 PUBLISHED CENTIMETER-WAVELENGTH DATA Frequency Flux Density Star (GHz) (mjy) Reference Telescope o Ceti ^ 0.25 Spergel et al VLA-C 5 \0.23 Drake et al VLA-C 11 \5 Altenho et al MPI 100 m 15 \0.47 Drake et al VLA-C R Leo... 5 \0.56 Spergel et al VLA-C ^ 0.03 Knapp et al VLA-C s Cyg... 5 \0.30 Spergel et al VLA-C RS Cnc... 5 \0.39 Spergel et al VLA-C 15 \0.30 Drake et al VLA-A R Lya ^ 0.12 Drake et al VLA-A R Cas ^ 0.17 Drake et al VLA-C R Aql... 5 \0.14 Drake et al VLA-A 5 \0.2 Drake et al VLA-C 11 [24(var) Woodsworth & Hughes 1973 ARO 46 m ^ 0.17 Drake et al VLA-C ^ 2.0 Bowers & Kundu 1979 MPI 100 m VY CMa... 5 \0.30 Spergel et al VLA-C ^ 0.03 Knapp et al VLA-C 15 \0.36 Drake et al VLA-A NML Cyg \0.38 Drake et al VLA-A NML Tau... 5 \0.33 Spergel et al VLA-C R Aqr ^0.2 Spergel et al VLA-C V Hya ^ 0.02 Luttermoser & Brown 1992 VLA-BC g Her ^ 0.12 Drake et al VLA-A IRC] ^ 0.05 Drake et al VLA-A ^ 0.10 Spergel et al VLA-C ^ 0.12 Drake et al VLA-A ^ 0.05 Sahai, Claussen, & Masson 1989 VLA-B ^ 1.2 Sahai et al OVRO 40 m signiðcantly contributing ion densities, as a function of temperature from these calculations. At temperatures below about 1600 K, the ionization of potassium (K) supplies most of the free electrons. This occurs because potassium has a very low ionization potential (4.34 ev) and in spite of its low abundance (10~6.85 relative to hydrogen). Sodium has the second lowest ionization potential (5.14 ev) of the elements considered, but a higher abundance (10~5.69 relative to hydrogen) than potassium, and it is the dominant source of electrons for temperatures between about 1600 and 2300 K. At temperatures above 2300 K, contributions from calcium and aluminum become signiðcant, followed by silicon, magnesium, and iron. Finally, above about 4000 K, the ionization of hydrogen dominates the production of free electrons. These conclusions assume no signiðcant departures from thermodynamic equilibrium, and exclude external radiative and cosmic-ray ionization T homson Scattering Thomson scattering of radio-frequency photons generated in the optical photosphere could explain why the observed size at centimeter wavelengths exceeds that of the optical photosphere, provided the scattering depth were large. The scattering depth, q, of a region containing elec- trons of density n is given by sc e P q \ p n dl, (3) sc Th e where p is the Thomson cross section and the integral is over the Th line-of-sight path through the scattering region. Evaluating equation (3) for n D 105 cm~3 and a path e length of 1.5 ] 1013 cm, or about 20% of the radius of the radio photosphere, gives q D 10~6. Thus, Thomson scat- sc tering is not likely to be signiðcant in the radio photosphere Electron-Ion Free-Free Emission Direct emission due to the acceleration of free electrons in the presence of a Coulomb Ðeld from positively charged ions (thermal bremsstrahlung) might explain the observed radio photosphere. The absorption coefficient, a, for ff thermal bremsstrahlung is as follows (e.g., Rybicki & Lightman 1979): A a \ 4e6 2n B1@2 Z2n n g e i ff, (4) ff 3m kc 3km T 3@2 l2 e e e where Z is the charge per ion, g is the Gaunt factor, T is the electron temperature, m and ff e are the electron mass and e charge, and n and n are e the electron and ion densities, respectively. For e n \ i n \ 105 cm~3, T \ 1500 K, and l \ 1010 Hz, equation e (4) i implies an optical e depth less than 10~3 at 10 GHz for a uniform shell of thickness 1.5 ] 1013 cm (B20% of the shell radius). To achieve optically thick emission up to a frequency of at least 22 GHz, and probably considerably higher, electron and ion densities signiðcantly greater than 107 cm~3 would be required. Based on the ionization equilibrium calculation presented in 4.2.1, this would require a temperature well above 2000 K and/or a total density greater than 1014 cm~3. Such conditions are comparable to that expected in the optical photosphere, and are much hotter and denser than expected in the radio photosphere (at about twice the radius of the optical photosphere) Electron-Neutral Free-Free Emission For temperatures appropriate for the radio photosphere of a long-period variable (below 2000 K), hydrogen will be

11 No. 1, 1997 LONG-PERIOD VARIABLE STARS 337 portional to the inverse square of the frequency from essentially 0 to 1013 Hz (30 km), to better than 1%, and to 1014 Hz (3 km), with about a 10% error. Thus, the polynomial equations for H~ and H~ opacities can be easily extended to other frequencies. 2 We Ðnd that, while the electron-ion free-free (thermal bremsstrahlung) absorption per ion exceeds the electronneutral free-free absorption per neutral particle by a factor of about 103, the neutral hydrogen densities exceed the ion densities by a factor of about 107, for the low temperatures appropriate for the radio photosphere. Thus, H~ and H~ free-free interactions provide roughly 104 times higher 2 opacity than electron-ion interactions and are the dominant opacity source in the radio photosphere. Figure 6 displays the optical depth for H~ and H~ freefree absorption as a function of temperature for a 2 total density of 2 ] 1012 atoms cm~3 and path length of 3 ] 1012 cm (0.2 AU). (In 4.3, we show that these parameter values reproduce our observations well.) This calculation assumes the equilibrium abundances of atomic and molecular hydrogen, based on the dissociation constants of Tsuji (1964). (Note that two tabular entries of Tsuji appear to have been typeset in error by 1 order of magnitude, and FIG. 5.ÈIon density vs. temperature. The density of the four most important ions supplying free electrons are plotted with solid lines, and the total density of free electrons is plotted with a dotted line. The calculations assumed thermodynamic equilibrium using the Saha equation, a total hydrogen density of 1012 cm~3, and solar abundances. mostly atomic and/or molecular, and the LTE abundance of H` will be very low. Under these conditions, free electrons come mostly from the ionization of metals, as discussed in We now consider the interaction of free electrons with neutral hydrogen atoms and molecules, i.e., H~ and H~ free-free opacity. (Note that the terms H~ and H~ free-free 2 are used even though the interactions do not involve 2 bound states of a negative ion.) Dalgarno & Lane (1966) provide formulas for the absorption coefficients for these interactions, which should be very accurate at low temperatures (\3000 K) and long wavelengths ([3 km). We numerically evaluated equation (20) of Dalgarno & Lane for the absorption coefficients at a frequency of 10 GHz, i(10 GHz), for temperatures between 1000 and 3000 K, and Ðtted them with third-order polynomials in temperature, T : i(10 GHz) \ a ] a T ] a T 2]a T3. For H~ opacity, we Ðnd a \] ] 10~17, 2 a \[ ] 10~20, a \]6.646]10~24, 0 and a \[7.853]10~28. 1 For H~ opacity, 2 we Ðnd a \] ] 10~18, a \ [ ] 10~21, a \] ] 10~24, and a \[ ] 10~28. For T in 2 units of kelvins, the above polynomials 3 return i(10 GHz) in units of cm4 dyn~1. The polynomial expansions are accurate to better than 0.6% over the Ðtted temperature range. These absorption coefficients are pro- FIG. 6.ÈOpacity at 10 GHz vs. temperature. H~ and H~ free-free opacities, resulting from the interactions of free electrons with 2 neutral H atoms and H molecules, are indicated with a dotted line and a dashdotted line, respectively. 2 Opacities are for a uniform slab of total (H ] 2H ) density of 2 ] 1012 cm~3 and a thickness (path length) of 3 ] 1012 cm 2 (0.2 AU). The solid line is the total opacity.

12 338 REID & MENTEN Vol. 476 we adopt dissociation constants, log [K (dyn cm~2)], of 10~ and 10~4.854 for temperatures 10 of p 1120 and 1440 K, respectively.) The sharp increase in optical depth with temperature (note the logarithmic ordinate in Fig. 6) occurs primarily as a result of the increasing density of free electrons. For example, over most of the temperature range of interest, an increase of 100 K results in about a factor of 10 increase in electron density and optical depth. This sharp change in electron density with temperature, and hence radius, provides a hard opacity edge ÏÏ sufficient to explain a blackbody-like spectrum at centimeter wavelengths Model of the Radio Photosphere The Ñux densities measured at centimeter wavelengths in this study can be compared with those expected from the optical photosphere. For example, using the photospheric (angular) sizes measured by Hani et al. (1995) and their e ective temperatures, calculated from bolometric magnitudes, one can predict the expected centimeter-wavelength emission. Our measured values exceed those predicted by factors ranging from about 1.5 to 3.0. Such excess emission has led some researchers to suggest that the centimeterwavelength emission includes a chromospheric component, as has been well established for hotter giants such as a Ori. However, we now show that the excess ÏÏ emission is as expected from an extended photosphere. Since the extended photosphere is partially opaque throughout the visible and near-ir portions of the spectrum where the blackbody function peaks, the total luminosity emerging from any spherical shell would be nearly constant (i.e., T 4r2 B constant), and temperature as a function of radius, T (r), would decrease approximately as r~1@2. In the Rayleigh-Jeans tail of the blackbody function, the emergent Ñux density, S, is proportional to Tr2and hence would scale as r3@2. For W Hya, we measured a radius at 22 GHz, R, approximately twice that measured optically, radio R, and thus one would expect S(R ) B 23@2S(R ). This is * radio * close to what is observed. In order to quantify better the physical conditions in the radio photosphere, we constructed a simple model of the extended atmosphere of a long-period variable. We assumed spherical symmetry and parameterized the radial behavior of temperature and density. The model allowed for an average static ÏÏ structure, as well as for simple empirical characterizations of periodic disturbances that might result from pulsations or shocks. Given the temperature and density at any position in the atmosphere, we determined the degree of ionization and the distribution of hydrogen between atomic and molecular states. Then, assuming the dominant opacity source is from H~ and H~ free-free interactions, we numerically evaluated the transfer 2 of radiation through the atmosphere. In the Rayleigh-Jeans limit, the formal solution of the radiative transfer equation for a thermal source of temperature T at an optical depth q is given by 0 P T (q ) \ T (q )e~q0 ] T (q)eq~q0 dq, (5) b 0 0 0q0 where T is the observed brightness temperature. First, we numerically b evaluated the integral in equation (5) along the line of sight to the center of the star. We started at a distance of 7.5 ] 1013 cm (5 AU) from the center of the star and summed T (q)eq~q0 dq with a step size, dq, of1.5]1011 cm (0.01 AU), until either q \ q \ 20 was reached or steps were taken (and ending symmetrically 5 AU beyond the center of the star). At each step, the model temperature and density were determined, the Saha equations were solved to obtain the total electron density, the densities of atomic and molecular hydrogen were evaluated, the coefficients for H~ and H~ free-free were calculated, and Ðnally the incremental optical 2 depth was obtained. The result of this integration was a line-of-sight opacity proðle. The integrations were then repeated, starting toward the center of the star and moving perpendicular to the line of sight (p) with step sizes of *p \ 1.5 ] 1012 cm (0.1 AU), in order to determine a center-to-limb brightness temperature proðle. We estimated the apparent stellar radius, R, as the halfwidth at half-brightness of the T (p) proðle. We obtained the total Ñux density, S, assuming b azimuthal symmetry, from S \ ; p/r T (q, p)2np *p/d2, where D is the stellar distance. p/0 Finally, b 0 we calculated a disk-averaged brightness temperature, ST T, deðned by the equation ST T \ SD2/(nR2). b b As discussed in 4.2.1, the free electron density was determined by solving the ionization equilibrium using the Saha equation. In order to speed up the calculations, we restricted the elemental mix to include only the four most important elements: K, Na, Al, and Ca. Tests indicated that inclusion of other elements makes a negligible di erence for temperatures below 2500 K. As mentioned in 4.2.4, the densities of atomic and molecular hydrogen were calculated assuming equilibrium dissociation constants. Bowen (1988) noted that, based on the work of Palla, Salpeter, & Stahler (1983), three-body reactions should dominate the formation of molecular hydrogen and that the formation rate can be slow. Thus, the transition from atomic to molecular hydrogen, with decreasing temperature at increasing radius, might not have time to reach equilibrium. For mass densities [10~13 g cm~3, or number densities of [6 ] 1010 cm~3, Bowen points out that the time to reach equilibrium would be Z5 ] 109 s, about 2 orders of magnitude greater than the stellar period. However, our observations suggest number densities in the radio photosphere of Z1012 cm~3, which shortens the equilibrium time to [2 ] 107 s or less than a stellar period. Thus, our use of equilibrium densities is unlikely to lead to large errors. (Note that in the worst possible case, where all hydrogen remains atomic, but the equilibrium calculation indicates pure molecular hydrogen, we would underestimate the opacity by a factor of B5: a factor of B2.5 from the decreased absorption coefficient and a factor of 2 from the decreased number of targets ÏÏ for electron free-free interactions.) Static Model Our static model for the atmosphere assumes a radial temperature distribution appropriate for a gray atmosphere: T 4(r) \ T 4 (1 [ J1 [ R 2/r2). (6a) R* * The model provides for a radial dependence of density (deðned as a mass density per atomic hydrogen mass) governed by hydrostatic equilibrium, assuming T changes slowly with radius: n(r) B GM eff m = n(r@) H k T (r@)r@2 dr@. (6b) r In equation (6a), R is the optical photospheric radius, and * T is the photospheric temperature. Equation (6a) rapidly R* P