Received 1999 April 27; accepted 1999 May 14; published 1999 June 17. ABSTRACT. Maps of the whole Sun at 48 GHz made during the period 1991â1993, ...
The Astrophysical Journal, 520:L63–L66, 1999 July 20 q 1999. The American Astronomical Society. All rights reserved. Printed in U.S.A.
SOLAR RADIUS VARIATIONS AT 48 GHz CORRELATED WITH SOLAR IRRADIANCE J. E. R. Costa,1 A. V R. Silva,1 V. S. Makhmutov,2 E. Rolli,3 P. Kaufmann,1 and A. Magun3 Received 1999 April 27; accepted 1999 May 14; published 1999 June 17
ABSTRACT Maps of the whole Sun at 48 GHz made during the period 1991–1993, using the large dish of Itapetinga Radio Observatory with the multibeam system, allowed solar limb determinations with unprecedented precision. From a large sample of maps, the solar radius at 48 GHz is found to vary in time. The observations showed that there is an apparent decrease of the measured radius as the solar cycle declines. We estimated the decrease to be 80 for half of the solar cycle. The 48 GHz solar radius variations are found to be very well correlated with the solar irradiance. Subject headings: Sun: atmosphere — Sun: corona — Sun: fundamental parameters — Sun: radio radiation However, the solar radius at other wavelengths such as microwave are important tools to determine the boundary conditions for the energy transport/balance from the subphotospheric to the atmospheric levels. We have measured the solar radius at 48 GHz from maps made during the observing campaigns from 1991 to 1993. The observations and method of determining the radius are described in the next section. Section 3 presents the results and describes the variations of the solar radius. Finally, the implications of the results and conclusions are discussed in § 4.
1. INTRODUCTION
The accurate determination of the solar photospheric radius has been an important problem in astronomy for many decades. A precise value of the solar radius is needed for calibration of solar models of stellar structure and evolution. Moreover, any changes in the optical radius of the Sun will affect the solar irradiance, which may in turn produce climate changes on Earth. Variations in the solar irradiance have been measured by spacecraft (Hickey et al. 1980; Willson et al. 1981; Laclare et al. 1996). However, the fraction due to the solar activity cycle needs to be deducted from these variations before any secular changes can be accessed. Variations in the measurement of the solar radius have been studied at optical wavelengths. Some recent works (Froehlich & Eddy 1984; Laclare et al. 1996; Delache, Laclare, & Sadsaoud 1985; Ribes et al. 1989; Ulrich & Bertello 1995; Rozelot 1998) find the radius to vary in accordance with some index of the solar activity cycle (for example, the envelope of sunspot number), whereas other authors find the variations in radius to be in opposition to the observed solar cycle (Gilliland 1981; Wittman, Alge, & Bianda 1993; Delache et al. 1993; Noeel 1997). Furthermore, there are reports of no observed variations (Neckel 1995; Antia 1998; Brown & Christensen-Dalsgaard 1998). The Sun’s diameter is also found to vary on other timescales: for example, 1000 days (Laclare 1983; Rozelot 1998) and 76 yr (Gilliland 1981). Variations of the solar radius have also been measured in radio at 8 and 13 GHz by Bachurin (1983), who reports an approximate increase in the mean radio radius of 0.010 R, at 13 GHz and 0.015 R, at 8 GHz from 1976 to 1981. Costa, Homor, & Kaufmann (1985) measured the solar radius at millimeter wavelengths and found it to be 1.023 R, at 22 GHz and 1.019 R, at 44 GHz, where R, is the optical radius. Laclare et al. (1996) observed bulges in high heliographic latitudes of the Sun. A similar result was found from 22 GHz observations (Costa et al. 1985). The solar emission of the quiet component of the solar disk, at microwave wavelengths, is mainly the thermal emission of chromospheric levels and above. Variation of the size of the optical solar disk has direct implications for total irradiance, which is mainly composed of optical emission.
2. OBSERVATIONS
A large number (over 500) of maps of the whole Sun were made at 48 GHz during the 3 yr period after the last solar maximum (1991–1993). Unfortunately, this data sample is too short to establish a dependence of the solar radius with the full solar cycle. Nevertheless, short-term variations have been observed in solar activity indices, and thus our analysis here will be limited by this. The multibeam system used at the 13.7 m dish of the Itapetinga radio telescope consists of five independent receivers operating simultaneously at 48 GHz (Hermann et al. 1992; Costa et al. 1995). The goal is to determine the position of the flaring sources with a precision of a few arcseconds (Hermann et al. 1992; Costa et al. 1995; Gime´nez de Castro et al. 1998; Raulin et al. 1998). From this data set, we have selected 279 solar maps during which the sky conditions were good. In this sample we had at least three receivers working simultaneously, resulting in 1074 independent solar maps for 279 periods. Each map takes 6.3 minutes of time and is made by scanning the Sun along the right ascension (R.A.) direction with readings taken every 50. 78 along 609. For every map, a total of 19 scans are obtained with a separation of 29 in the declination direction. Thus, a map is an array of 623 # 19 pixels. An example of a solar scan is shown in Figure 1. The limb of the Sun can be determined from the R.A. scans at the point where the map antenna temperature is half of the quiet-Sun temperature. Therefore, a total of about 34 coordinates of positions along the limb are determined from the maps for each beam. Since usually good data are available for at least three beams (receivers), the limb of the Sun is determined from more than 100 coordinates. These points are then fitted by a circle in order to determine the solar radius. Figure 1 also shows the halfintensity points (asterisks) and the best fit to a circle. The coordinates of every map have been rotated by the polar angle inclination, such that solar north points upward in every
1 CRAAE, Instituto Presbiteriano Mackenzie, Rua da Consolac¸a˜o 896, 2o. andar, 01302-000, Sa˜o Paulo, SP, Brazil. 2 P. N. Lebedev Physical Institute, Leninsky Prospect 53, 117924 Moscow, Russia. 3 Bern University, Sidlestrasse 5, CH-3012 Bern, Switzerland.
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Fig. 1.—Measurement of solar limb coordinates by drift scans at 48 GHz. The points in the parallel lines are the encoder measurements every 32 ms. The asterisks represent the solar limb or the half-intensity points of the quietSun temperature. The plotted circle is the best fit to these points.
map. We have also corrected for the Earth-Sun distance, which causes changes in the angular size of the Sun, dividing by the optical radius tabulated in the Nautical Almanac. In order to interpolate the solar limb coordinates where the antenna temperature is half of the quiet-Sun temperature, we have used the measured encoder positions. The alt-azimuth encoder resolutions at Itapetinga is 07. 001, and the tracking accuracy of the radome enclosed antenna is 10. 4 rms (Costa et al. 1995). The solar disk scans in the R.A. direction, made every 29 in declination, produce an oversampling of the equatorial limb in comparison to the polar limb. Two parallel scans cross the limb at two different angles, J 2 c/2 and J 1 c/2, where J is the angle measured from the solar equator with J 5 07 pointing east and 1907 pointing north, and c is the resulting angle between the two limb crossings. To overcome this problem of oversampling, we used a weighting function, q, inversely proportional to the angle c:
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Fig. 2.—Error distribution of the solar radius measurements at 48 GHz. (a) Normal distribution of errors. (b) Errors distributed throughout the year.
scope at 48 GHz is about 29, and we believe that nearby atmospheric effects are responsible for the limb irregularities rather than spicules (for example) since we cannot resolve them. We would like to point out that the independent receivers are offset in space by about 29, so they do not cross the limb at the same coordinates. The solar radius measurements follow a normal distribution (Fig. 2a), and no seasonal effects were observed in our sample (Fig. 2b). Solar disk deviations from a perfect circle were calculated as function of an angle which measures the solar radius from equatorial to polar directions. No deviations were observed down to the accuracy of 00. 1 in the solar radius measured in the present work with about 22,680 coordinate pairs of the 1074 maps. This confirms the result of Costa et al. (1985) for 44 GHz obtained in a smaller data sample. 3. VARIATIONS OF THE SOLAR RADIUS
(1)
From the selected sample of 279 maps during the 12 observing campaigns along 3 yr, we calculated the radius as described in the previous section. The value of the ratio of 48 GHz radius to the tabulated optical radius (R, ) as a function of the year of observation is plotted in Figure 3. The mean
where R 48 is the solar radius in arcminutes. The maximum J 1 c/2 occurs at 907 (for a tangent scan at the pole); since the scan separation is 29, the maximum J is about 757 which yields q 5 29. In order to estimate the polar radius and minimize the influence of limb active regions in the solar radius determination, we also made a second run using coordinates with heliographic latitudes higher than 407 only. Because of the high encoders and tracking accuracies of Itapetinga, the radio seeing is the most important limitation in the limb determination. However, it has been measured in 48 GHz to be negligible at least for short timescales (Costa et al. 1995). Nevertheless, for the timescale of a map in our sample (6.3 minutes) we may have a limb blurring effect; however, it would show up in the solar radius determination. The root mean square of the limb fit for our sample was about 50 for each receiver, whereas the dispersion of the radius measurements among the receivers was only 20. 5, as can be seen in Figure 2. This difference may be due to irregularities in the radio seeing or to randomly distributed structures in the solar limb. The half-power beamwidth of the Itapetinga tele-
Fig. 3.—Cycle 22 sunspot number. The thin line is the Solar Geophysical Data mean monthly sunspot number. The thick line is the 31 day averaged soft X-ray fluxes from GOES. The circles represent the solar radius at 48 GHz; their size is of the order of the rms errors of the measurements.
q5
{
(
4 arccos 1 2 2 2R48 cos 2 J 29
)
if 757 ≤ FJF ≤ 1057, elsewhere,
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sumption of a constant antenna velocity during the whole scan, which may have caused errors of about 0.6%–1% in position. However, the increment in the radius might also be explained by specific changes between both cycles similar to those found for Cycle 22, presented here. In Figure 3, we overplot the solar radius with the monthly mean sunspot number taken from Solar Geophysical Data Bulletins (NOAA, Solar Geophysical Data Prompt Reports 1991, 1992, and 1993) to show the good correlation with the solar cycle decay. The correlation of the mean monthly sunspot number with the 48 GHz solar radius for the period of our data sample results in a linear correlation coefficient of 0.57 for our set of 12 data points (number of campaigns), which implies a 95% confidence level (Bevington 1969). The linear fit found was
Fig. 4.—Solar constant. The dot-dashed line is the ERB solar irradiance normalized to match ACRIM II shown by the thin solid line. The thick solid line is the average between both experiments during common periods of observation. The circles are the irradiance values inferred from the solar radius measurements at 48 GHz.
radio radius is found to be R 48 5 1.025 5 0.002 R,.
(2)
The uncertainty reflects only the observed radius variation during the 3 yr of observation. The height of the solar radius at 48 GHz corresponds to 17,000 km above the photosphere or the transition region/lower corona. In Figure 3, the circles represent the mean value for every campaign, where the size of the circle represents the rms of each series. A decreasing trend in the radius value is clearly seen, the radius being larger in mid-1991 (near solar maximum activity) than in mid-1993. The variation in the solar radius, as claimed by Froehlich & Eddy (1984) for the optical radius, is seen to vary in phase with the solar cycle. The short period of available data here yield the following relation: R 48 5 1.029 2 0.0015(year 2 1990). R,
(3)
We used the above linear fit to find the total decrease of 80 if we extrapolate the fit to a period of 5.5 yr (half of the activity cycle). Bachurin (1983) found an increase of approximately 100 at 13 GHz and 140 at 8 GHz by comparing 1976 observations with 1980–1981 data. In this case, the radius was measured during the rise of activity (the maximum of Cycle 21 occurred in 1980). If we extrapolate Bachurin’s (1983) results to 48 GHz, assuming a power-law spectra for the solar radius variations, we should have a decrease of the 48 GHz solar radius of about 50 in 5.5 yr. The results seem to be in agreement if we consider that Bachurin’s measurements were made not only in the rise of Cycle 21 but near the peak. Costa et al. (1985) measured the solar radius in the 1978–1982 period. At that time, the mean solar radius at 44 GHz was found to be 1.019 of the optical radius, and it differs from the present result by about 60. We claim that in the present analysis for Cycle 22 we measured the solar radius using antenna encoder data precise to an accuracy of 30. 6 that was not digitally acquired in Cycle 21. Costa et al. (1985) used instead a data sample integrated in a matrix file of 19 rows by 21 columns in a grid of 2 # 2 0 and based their radius measurements on the as-
R 48 5 1.021 1 3.422 # 1025Savr , R,
(4)
where Savr is the mean monthly sunspot number. Also plotted in Figure 3, is the GOES soft X-ray daily flux,4 averaged by a running mean for 31 days, which shows an even better correlation with the solar radius variations. The most important result from our analysis was the correlation found between the solar radius variation and the solar constant measured from space. In Figure 4, we present the solar irradiance as measured by the experiments Earth Radiation Budget (ERB) and the Active Cavity Radiometer Irradiance Monitor II (ACRIM II).5 In order to compare the data set of both satellites, we normalize the ERB data with the ACRIM II in the period of observation overlap from 1991.8 to 1993.1, as shown in Figure 4. The data of the ACRIM II and ERB were adjusted to each other through a simple intercomparison with ERB. A similar procedure has already been used by other authors (e.g., Pap et al. 1994), and further details are given by Willson (1994). The main change of our normalization was the offset imposed to the measurements of ERB to coincide with the measurements of ACRIM II, as can be seen in Figure 4. This procedure should not affect the relative variation of the irradiance that we are interested in, as it modifies only the absolute calibration. Also plotted on Figure 4 is the solar irradiance (Irr) as inferred from the solar radius at 48 GHz showing a very good fit with the following empirical equation: Irr 5
[(R 48 /R, ) 1 8.082] watts m22. 150
(5)
The correlation coefficient for the solar radius and the solar constant variations during the period analyzed was 0.71, and this is surprisingly good (confidence level of 99%) for a period of 3 yr with 12 data samples. 4. DISCUSSION AND CONCLUSIONS
An increment of the photospheric radius of 00. 1 or in the effective temperature of the photosphere of about 800 K can account for the measured variation of 0.04% of the solar constant during a solar cycle found by Willson & Hudson (1988). However, measurements of the photospheric radius variation with the cycle have been controversial. Laclare et al. (1996), from long-period observations of the optical radius, found variations of about 00. 2 which were out of phase with the sunspot 4 5
See the National Geophysical Data Center at http://spidr.ngdc.noaa.gov. See the Distributed Active Archive Center at http://daac.gsfc.nasa.gov.
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number. Ulrich & Bertello (1995) found a total variation of 00. 35 for a whole cycle, with measurements of a temperature minimum line, i.e., in a higher level of the photosphere. The variation of the 48 GHz radius of about 80 (for a half-cycle) is then not a common expansion valid for all atmospheric levels as we can see from photospheric measurements. The atmospheric level of the 48 GHz emission measured here is about 17,000 km above the photosphere. This height corresponds to the base of the corona or the transition region according to usual atmospheric models of temperature and plasma pressure. However, in the corona, the structure of the magnetic field plays an important role in the shape and size of the atmosphere, from the minimum to the maximum of the solar cycle. Complications among the involved energy sources and losses (as reviewed by Stix 1991 and references therein) make the balance in the corona a complex problem; nevertheless, its heating is by far the most interesting open question. We present here a very strong correlation between the 48 GHz emitting level and the solar irradiance. Moreover, the excess of the soft X-ray emission from 1992.1 to 1993.7 correlates very well with the solar radius expansion (see Fig. 3) and strongly suggests that coronal heating occurs through the magnetic field by its explosive or non-
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explosive dissipation in active regions over sunspots or other quiescent regions. Recently, SOHO observations (Schrijver et al. 1997; Day 1998) have shown the existence of a “magnetic carpet,” composed of short-lived small magnetic loops throughout the quiet Sun. It is the reconnection of these loops which accounts for the heating of the corona. The authors wish to thank John R. Hickey and the Nimbus7 ERB Experiment Team, Richard C. Willson (ACRIM I and II), Robert B. Lee and the ERB Science Team for their solar data, and the Distributed Active Archive Center (sponsored by NASA’s Earth Science enterprise) at the Goddard Space Flight Center for putting these data in their present format and distributing them. The authors also thank the National Oceanic and Atmospheric Administration for distributing the X-ray solar data of the GOES satellite in the National Geophysical Data Center. This research was partially supported by the Brazilian agency FAPESP. We are grateful to P. Lagrotta for helping to organize the 48 GHz Itapetinga database. CRAAE is a joint consortium of the Brazilian Institutions Mackenzie, INPE, USP, and UNICAMP.
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