Solar Physics 81 (1982), 207–215.
D. Reidel Publishing Co.,
Dordrecht, Holland, and Boston, U.S.A.
THE QUIET_{ } SUN BRIGHTNESS TEMPERATURE AT 127 MHz 
KAZIMIERZ_{ } M. BORKOWSKI
Toruñ Radio Astronomy Observatory, Nicolaus Copernicus University, Toruñ, Poland
(Received 15 May, 1980; in final form 22 April, 1982)
Abstract. A correction and analysis of routine total flux measurements made in Toruñ at 127 MHz allowed for the determination of the peak brightness temperature of the Sun (above 9×10^{5} K) in three periods groupped around the last minimum, and for estimation of the brightness temperature of the coronal holes (7.3 ± 1.4)×10^{5} K. 
The past decade has brought new information about the structure of the corona and the transition region of the Sun. In recent years, due to the controversy between observation and theory, a growing interest in the meter and decameter wavelengths range spectrum of an undisturbed Sun can be observed (e.g., ChiuderiDrago et al., 1977; Dulk et al., 1977; Kundu et al., 1977; Erickson et al., 1977; Lantos, 1978; Trottet and Lantos, 1978; Chambe, 1978; Alvares and Yurovsky, 1979; Meyer, 1979; ChiuderiDrago, 1980; Sheridan and Dulk, 1980). Here are reported some new results which are based on the long1asting (from 1958 up to the present) routine solar observations at 127 MHz carried out in Toruñ. The observations were made, as usual, with a low resolution (10l baseline) EW interferometer (Borkowski et al., 1975).
This work was also promoted by the fact that commonly available reduced data of this series are unsuitable for detailed analyses of quiet Sun emission. Firstly, the published results (e.g. in QBSA) are rounded off to the integers of solar flux units (1 s.f.u. = 10^{22} W m^{2} Hz^{1}) which, for quiet periods of solar activity at frequencies of the order of 100 MHz, means a reduction of accuracy of up to about 30%. Secondly, the Toruñ data have the drawback of having been calculated with the omission of a few important sources of systematic errors (Borkowski, 1981a, b).
For the present study, a few periods of a relatively quiet Sun have been chosen, for which observations were taken elsewhere at close frequencies. The already existing daily data of the flux density has been corrected for various effects — the potential sources of errors. Then, assuming the resulting values to be absolute, the peak brightness temperature of the Sun has been determined for periods closest to the last minimum of the 11year activity. Finally, an attempt is made to separate the hole and arch contributions to the overall temperature by correlation with Nançay (France) results at 169 MHz.
All uncorrected Toruñ data were taken from an observations diary and are listed in Table I. The observing period in 1975 corresponds to the one of Kundu et al. (1977). Those in 1976 were selected after Erickson et al. (1977). Here the flux density on 18 August 1976 is easily proved to represent an active Sun (a noise storm at 127 MHz was observed), thus it will be omitted in further analysis. The data set of 1978 refers to the period considered by Alvares and Yurovsky (1979). The day 12 May 1979 was chosen for a special study by the participants to the workshop held in Nançay in 1979 (Elgaröy et al., 1981).
Table I
Original daily mean flux densities at 127 MHz
Period  Daily data (s.f.u.) 
25 May – l June, 1975  3.1 2.7 3.6 2.9 3.3 3.3 3.3 3.4 
16 – 23 July, 1976  2.8 2.7 – – 2.8 3.0 3.0 2.6 
18 – 21 August, 1976  (11.2) 3.0 2.7 3.0 
17 – 25 Ju1y, 1978  3.5 3.5 3.8 3.5 3.8 3.4 3.5 3.8 3.8 
12 May, 1979  4.2 
In addition to the above data, the daily results have also been corrected for the remaining parts of the periods 10 May through 11 June, 1975 and 1 through 23 July, 1976, which were used — with similar data obtained in Nançay at 169 MHz (Lantos, 1981) — for an estimation of the hole brightness temperature at 127 MHz. Both the Toruñ and Nançay data were selected for days of relatively quiet Sun and reliable calibrations. Of the set of 51 dates available in these two periods three (13 and 23 May, and 8 June 1975) were rejected as representing outliers.
The correction of the Toruñ data included the following effects:
(1) A change of flux density scales from those previously used in Toruñ to the recent
one prepared by Baars et al. (1977) with respect to Cassiopeia A (Cas A).
(2) Ellipticity of the Earth's orbit.
(3) Average Earth's atmosphere attenuation (nighttime attenuation of the Cas A
signal was assumed to be negligible).
(4) Signal weakening due to the differential refraction.
(5) Quiet Sun visibility for the Toruñ interferometer with an assumption of 34' EW
half power diameter of the Sun at 127 MHz, and the brightness distribution taken as
an average of the distributions observed by Kundu et al. (1977) and Lantos
and Avignon (1975).
(6) Confusion of the Cas A and Cygnus A (Cyg A) emissions during calibration of the
observing system. The calculation was based on theoretical modeling (see Appendix).
(7) Discrete adjustments of the antenna elevation angle (in declination).
(8) Sectionally linear approximation of the receiver nonlinear gain
during data processing.
Detailed treatment of these (and a few other of lesser importance) factors is presented elsewhere (Borkowski, 1981a, b). Results of the calculations are summarized in Table II. The only correction introduced to the Nançay fluxes was the adjustment to the 1 AU distance.
Table II
Correction coefficients for the Toruñ 127 MHz solar flux measurements

Table III
Results of determinations of the flux density and peak brightness temperature

The total correction coefficients of Table II were applied to the corresponding mean flux densities for analysed periods. Table III contains these results along with the ones obtained elsewhere for the same periods. The errors quoted for Toruñ data in Table III are the rms deviations from the means.
Kundu et al. (1977) assumed the Cas A flux density to be 1.5 s.f.u., while Baars et al. (1977) scale gives the value reaching almost 1.6 s.f.u. at the epoch. Therefore, their appropriately corrected results are also included in Table III.
It is rather obvious that the results obtained for the period 25 May to 1 June, 1975 do not represent the minimum quiet Sun. The slope of the spectrum at metric waves as given by Erickson et al. (1977) with Gergely's (1978) completion implies that 4.2 s.f.u. at 121.5 MHz corresponds to 4.7 s.f.u. at 127 MHz (i.e., 15% higher than the value derived from Toruñ observations). The authors of the 121.5 MHz measurements estimate their accuracy to be about 15%, so that there is no need to discuss the discrepancy.
Similarly, the flux measurements of Erickson et al. for July 1976 are consistent with the author's determinations within a few percents. Those for August 1976, however, exceed the Toruñ data by more than 20% (when extrapolated to 127 MHz). One ofthe possible explanations is that the already noticed activity on 18 August, 1976 may have given a rise to Erickson et al.'s final result. This, in turn, may have resulted in too steep slope of their spectrum.
Alvares and Yurovsky (1979) determined the quiet Sun flux density at 245 MHz. They found their result in good agreement with the spectrum of Erickson et al. (1977):
 (1) 
where f is the frequency (MHz), and claim the spectrum to be correct up to 245 MHz (originally determined up to 121.5 MHz). There are some objections to this conclusion. Alvares and Yurovsky assumed (basing on Kundu, 1965) that the tota1 solar radio flux during quiet conditions within 2 years after the start of a new cycle (their observations refer to July 1978) may rise no more than 6% above the minimum level. The Toruñ data presented here show a rise of more than 20% .Even larger enhancements (more than 50%) may be found in currently reported data at higher frequencies, say 200–260 MHz as considered by the authors. Another source of misunderstanding may be the spectrum alone. The formal errors quoted in Equation (1) indicate the uncertainty factor of the flux extrapolated to 245 MHz (15.3 s.f.u.) as large as 2.7 (270%). Moreover, even a minute change in formula (1) may cause a large error in the final result. Actually, Alvares and Yurovsky's adopted 15.3 s.f.u. is a1ready 13% lower than that calculated using the more exact expression (Gergely, 1978)
 (la) 
It means that Alvares and Yurovsky's result coincided with the spectrum so accurately only by accident. The result, in fact, does not correspond to the minimum quiet Sun flux density. By the same token, the correction factors derived by them for eight other stations, seem incorrect themselves.
The Toruñ daily flux of 12 May 1979 falls inside the accuracy limits (13%) of the spectrum determined for this day and the frequency range of 150–450 MHz by Elgaröy et al. (1981):
lg S = 2.9355 + 1.765 lg f.  (2) 
The peak brightness temperature of the Sun may be computed with the help of the formula (Aubier et al., 1971)
 (3) 
where l is the wavelength (m), S — the flux density (s.f.u.), and q_{1} and q_{2} are the half power diameters of the radio Sun along EW and NS directions (arc min).
To calculate T_{p}, 34 and 33' respectively were adopted as the diameters at 127 MHz basing on the measurements of Kundu et al. (1977) made at 121.5 MHz. The resulting temperatures (Table III) agree well with the results of Kundu et al. (1977) and Erickson et al. (1977).
The quiet Sun was shown to be composed of at least two distinctive forms: holes and long distance arches or loops. Recent findings clearly revealed that these structures possess different physical properties and contribute in an unknown and variable proportion to the total radio flux.
The Toruñ interferometer has too short a baseline to resolve the Sun, and thus to make it possible to measure the hole brightness temperature directly. Still, the temperature may be inferred from the changes of the total flux by the use of correlation techniques. The inference requires the physical conditions in holes to remain constant in time, which seems to be already satisfactorily established (e.g. Trotted and Lantos, 1978).
The alreadymentioned sample of 48 data points — paired values consisting of the flux densities at 127 and 169 MHz — have been analysed with the technique of linear regression by the method of least squares. The data points and their best fitted line are plotted in Figure 1. The linear relationship of the fluxes describes the equation
S(127) = 2.272 + 0.173 S(169).  (4) 
Fig. 1.^{ } Plot of the Nançay 169 MHz daily flux densities against Toruñ 127 MHz ones for the periods 10 May through 11 June, 1975 and 1 through 23 July, 1976. The solid line is the best fit to the data points. The thickened section on the bottom scale denotes the 5% confidence interval at the 95% confidence level of the predicted 3.1 s.f.u. 127 MHz value. 
The correlation coefficient of this fit equals 0.44, what in this case implies that the value can be considered significant at the probability level of less than 1%.
Substituting the quiet Sun (understood here as completely covered by the holes) flux density at 169 MHz: S_{H}(169) = (4.75 ± 0.25) s.f.u. (Lantos and Avignon, 1975; Lantos, 1981) into (4) one gets the corresponding value at 127 MHz:
S_{H}(127) = (3.1 ± 0.4) s.f.u. 
Again, it is the flux density ascribed to a fictitious Sun entirely surrounded by the holes. The estimate of uncertainty of the above va1ue represents the ±5% confidence interval at the 95% confidence level (e.g. Bendat and Piersol, 1971).
From Jeans law it follows for the hole brightness temperature:
 (5) 
Taking the brightness temperature of the holes at 169 MHz as T_{H}(169) = (6.3 ± 0.7) ×10^{5} K (Trottet and Lantos, 1978; Lantos, 1980) and using the corresponding hole fluxes at the two frequencies the formula (5) yields
T_{H}(127) = (7.3 ± 1.4) ×10^{5} K, 
where the uncertainty (20%) resulted from already pointed inaccuracies of the quantities involved in Equation (5).
Other determinations of the hole temperature at metric frequencies range from 4.4 ×10^{5} K (the 'best' estimate at 160 MHz) to 8.2 ×10^{5} K (at 80 MHz) as found by Dulk et al. (1977). Considering still other determinations at 169 MHz, which enclose 6.3 ×10^{5} K and 7.9 ×10^{5} K (Lantos and Avignon, 1975, ChiuderiDrago et al., 1977), the derived value at 127 MHz of 7.3 ×10^{5} K seems reasonable.
As the hole temperature is believed to be constant throughout the solar activity cycle, it is easy to show that variations of the radio flux follow these of the loop mean brightness temperature T_{L} (averaged over the area A_{L} covered by the quiet region) according to the proportionality
 (6) 
Further, if the dependence of A_{L} on the frequency is negligible, then combining (6) and (4) yields the relation:
 (7) 
With T_{L}(169) = 11.5 ×10^{5} K, which is the loop brightness temperature at 169 MHz (Trottet and Lantos, 1978), this formula leads to the temperature of 8.9 ×10^{5} K at 127 MHz. Accuracy of this estimate is about 20%.
Similarly to the result of the hole temperature, the comparison with other authors determinations shows that the above loop temperature is safely acceptable.
In summary, basing on the Toruñ total flux measurements at 127 MHz taken in periods close to the last minimum of solar activity, it has been determined that the quiet Sun flux densities which, unlike the Toruñ up to the present results, fall well within the error limits of the spectra prepared by a few other authors. The respective peak brightness temperatures of the Sun were also inferred and found to be in accord with available results.
By comparison with the Nançay measurements at 169 MHz, it is calculated that the minimum quiet Sun (basic) component of the flux at 127 MHz is (3.1 ± 0.4) s.f.u., which corresponds to the brightness temperature of (7.3 ± 1.4) ×10^{5} K. This parameter is attributed to the coronal hole. These derivations were extended to arrive at the brightness temperature of the solar quiet (loop) region of 8.9 ×10^{5} K.
The author thanks S. Gorgolewski, R. Schreiber, and J. Usowicz for critical reading of the infant version of this paper. Grateful thanks are also given to T. Gergely and Yu. Yurovsky for further criticism on the paper, and special thanks to P. Lantos, whose ideas are widely incorporated in Section 5, and who kindly reduced the Nançay observations at 169 MHz used in this work.
AppendixThe detector output voltage of a simple total power radio interferometer in the onedimensional case may be expressed by (Borkowski, 1980b)
where A(f) is the filter transfer function, G(x,f) the antenna voltage radiation pattern with x being the direction cosine of a point on the sky, G(x,f) is the mutual coherence function in the frequency (f) domain (completely incoherent source is assumed), t is the time delay (made up of the geometrical and instrumental delays), n is the power of the (assumed) powerlaw detector (n = 1 for the linear detector, and n = 2 for the square law one; Borkowski, 1980a), Â denotes the real part of the complex quantity, and the subscript 0 stands to indicate the center of the bandwidth pattern (f_{o}) and of the antenna beamwidth (x_{o}). The formula (A1) presumes the two antennas being identical, and also insignificant variations of the source brightness distribution and antenna pattern over the passband and with time. For more complete discussion of the quantities involved in (A1) (except for the n) the reader is referred to the paper by Swenson and Mathur (1968), where a correlation (multiplying) interferometer is considered. In the case of a finite set of the point sources, each of the flux density S_{i} the coherence function is proportional to
where d(x) is the Dirac deltafunction. When assuming a narrow bandwidth of the system, the expression (A1) now conveniently reduces to
Straightforward extension of this expression to two dimensions allows to model the simultaneous observations of Cas A and Cyg A during the calibration of the broadbeam Toruñ interferometer. Relatively complex features normally seen on the records were compared with the computed pattern. The comparison speaks in favor of the models (Borkowski, 1981b), and thus justifies the application of by far the largest correction factor listed in Section 2. 
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