CORONAL MAGNETIC FIELDS DERIVED FROM SIMULTANEOUS MICROWAVE AND EUV OBSERVATIONS AND COMPARISON WITH THE POTENTIAL FIELD MODEL

Solar activities in the corona, such as flares, jets, eruptions, and heatings, are dominated by magnetic fields. Therefore, solar phenomena can be most appropriately understood by measuring the coronal magnetic field through phenomena such as polarization generated by the Zeeman effect and/or the Hanle effect in the optical range. However, such effects are obscured by the line broadening of the high-temperature plasma and the low polarization degree of the coronal weak magnetic field. Lin et al. (2004) measured coronal magnetic fields to a few Gauss at a spatial resolution of 20'' and 70 minutes of integration, but this method is still developing.

The coronal magnetic field also can be derived by extrapolating the photospheric magnetic field from the potential or force-free field approximations (Sakurai 1989; Shiota et al. 2008, 2012). Generally, the magnetic pressure in the corona is thought to exceed the gas pressure, implying that coronal plasma has a low beta. Therefore, the coronal magnetic field should be near to the potential or force-free. Structurally, the coronal loops observed by EUV often resemble the magnetic field lines derived by extrapolations (Wiegelmann et al. 2012). However, the magnetic field in the photosphere is not considered as force-free because of the high beta of photospheric plasma (Gary 2001). In addition, the magnetic field strengths observed in different spectral lines are not identical because of the differences in magnetic sensitivities and formation heights of the spectral lines. Hence, the coronal magnetic field derived by extrapolating from the photospheric magnetic field should be validated with other observations.

Several methods derive the magnetic field from microwave observations. For instance, the strong magnetic field above sunspots can be derived from gyroresonance emission (White 2004; Bogod et al. 2012). Other methods derive the coronal magnetic field from polarization reversal by quasi-transverse propagation (Ryabov et al. 2005). Brosius et al. (1997) estimated the coronal magnetic field by fitting the observed radio intensity to the radio intensity calculated from EUV observations.

The circular polarization degree of thermal bremsstrahlung (also called free–free) radiation depends on the longitudinal magnetic field strength (hereafter we refer to the longitudinal magnetic field as the magnetic field along the line of sight). Bogod & Gelfreikh (1980) measured the magnetic field strength in the upper chromosphere as approximately 40 G from one-dimensional observations of RATAN-600 in the microwave range. Grebinskij et al. (2000) observed the circular polarization of bremsstrahlung at 17 GHz using the Nobeyama Radioheliograph (NoRH; Nakajima et al. 1994). They measured a longitudinal magnetic field strength of about 60–150 G in the chromosphere and corona above the active regions. Similar measurements were conducted by Iwai & Shibasaki (2013). In addition, the above studies suggested that the circular polarization at 17 GHz contains both chromospheric and coronal components. To separate these components, additional information or assumptions are required. Iwai et al. (2014) derived the coronal component by observing the coronal loops outside of the solar limb and measured the longitudinal magnetic field in the coronal loops as 84 G. However, this method cannot measure the coronal magnetic field in active regions on the solar disk.

The accuracy of these methods was evaluated in comparison with observational data. However, a proper accuracy evaluation requires comparison with potential field calculations, which is difficult near the solar limb because the photospheric magnetic field cannot be accurately measured in this region. Therefore, we must derive the coronal magnetic field on the solar disk, which requires separating the coronal component from the combined chromospheric and coronal components in the thermal bremsstrahlung. The intensity of bremsstrahlung is a function of the temperature and the emission measure (hereafter the emission measure refers to the column emission measure) of the plasma (Dulk 1985). In other words, we can derive the coronal component of thermal bremsstrahlung from the coronal temperature and emission measure collected by other instruments.

The Atmospheric Imaging Assembly (AIA; Lemen et al. 2012) instrument on board the Solar Dynamics Observatory (SDO) has seven EUV channels operating in six wavelengths (131, 171, 193, 211, 335, and 94 Å) centered on strong iron lines. The regularization code developed by Hannah & Kontar (2012) produces a differential emission measure (DEM) from SDO/AIA data, enabling us to derive the coronal emission measure and temperature. In this way, the temperature range was determined as $5.7\lt \mathrm{log}T\lt 7.0$.

The present study estimates the accuracy of the magnetic field derived from radio observations by comparison with the potential field results and the DEM measurements using EUV observations. Section 2 describes the methods of magnetic field measurement, and Section 3 presents the observational instruments and data sets. The results of the coronal magnetic field measurements are presented in Section 4. In Section 5, we discuss the reliability of the coronal magnetic fields and compare the results with magnetic field calculations and DEM measurements. The paper is summarized in Section 6.

The optical thickness of the extraordinary and ordinary modes of thermal bremsstrahlung in anisotropic plasma is computed by (Gary & Hurford 2004)

$$\begin{eqnarray}&&{\tau }_{x,o}=\int {\kappa }_{x,o}{dl}=\displaystyle \frac{\xi \int {n}_{e}{\displaystyle \sum }_{i}^{}\;{Z}_{i}^{2}{n}_{i}{dl}}{{T}_{e}^{3/2}{(\nu \mp {\nu }_{B}\mathrm{cos}{\theta }_{i})}^{2}},\end{eqnarray} \tag{ 1 }$$

where κ_x,o is the absorption coefficient of extraordinary and ordinary modes, n_e is the electron density, and Z_i and n_i are the ion charge and number density, respectively. T_e is the electron temperature, ν and ν_B denote the observational frequencies and the electron gyrofrequency, respectively, and θ_i is the angle between the magnetic field and the line of sight. The function ξ only slightly depends on the plasma parameters and is expressed as

$$\begin{eqnarray}&&\xi =9.78\times {10}^{-3}{g}_{{ff}},\end{eqnarray} \tag{ 2 }$$

where g_ff is the gaunt factor. Here we adopt the gaunt factor of Gary & Hurford (2004), given by

$$\begin{eqnarray}{g}_{{ff}}=\left\{\begin{array}{ll}18.2+\mathrm{ln}{T}_{e}^{3/2}-\mathrm{ln}\nu & ({T}_{e}\gt 2.0\times {10}^{5}\ {\rm{K}})\\ 24.5+\mathrm{ln}{T}_{e}-\mathrm{ln}\nu & ({T}_{e}\lt 2.0\times {10}^{5}\ {\rm{K}}).\end{array}\right.\end{eqnarray} \tag{ 3 }$$

Assuming that the coronal plasma is fully ionized and applying the heavy ion correction (Chambe & Lantos 1971; Gary & Hurford 2004), we can use the approximation ${\displaystyle \sum }_{i}\;{Z}_{i}^{2}{n}_{i}\approx 1.2{n}_{e}$ in Equation (1). In addition, using the emission measure $\mathrm{EM}=\int {n}_{e}^{2}{dl}$, Equation (1) reduces to

$$\begin{eqnarray}&&{\tau }_{x,o}=\displaystyle \frac{1.2\ \xi \mathrm{EM}}{{T}_{e}^{3/2}{(\nu \mp {\nu }_{B}\mathrm{cos}{\theta }_{i})}^{2}}.\end{eqnarray} \tag{ 4 }$$

The brightness temperature is determined by

$$\begin{eqnarray}&&{T}_{b}^{x,o}={T}_{e}[1-\mathrm{exp}(-{\tau }_{x,o})],\end{eqnarray} \tag{ 5 }$$

where ${T}_{b}^{x}$ and ${T}_{b}^{o}$ denote the brightness temperature of extraordinary and ordinary modes, respectively. Subsequently, the intensity (I) and polarization (V) components are given by

$$\begin{eqnarray}I & = & ({T}_{b}^{x}+{T}_{b}^{o})/2\\ V & = & ({T}_{b}^{x}-{T}_{b}^{o})/2.\end{eqnarray} \tag{ 6 }$$

Bogod & Gelfreikh (1980) derived the following relationship between the observed polarization degree (V/I) and the longitudinal magnetic field of the radiation source B_l:

$$\begin{eqnarray}&&{B}_{l}[{\rm{G}}]=\displaystyle \frac{10700}{n\lambda }\displaystyle \frac{V}{I}.\end{eqnarray} \tag{ 7 }$$

In this equation, λ is the observation wavelength and n is the spectral index (n ≈ 2 in the corona), expressed as

$$\begin{eqnarray}&&n=\displaystyle \frac{\partial \mathrm{log}{T}_{b}}{\partial \mathrm{log}\lambda }\equiv \displaystyle \frac{\partial \mathrm{log}I}{\partial \mathrm{log}\lambda }.\end{eqnarray} \tag{ 8 }$$

The spectral index n can be derived from the observed brightness temperatures at two close frequencies (here 17 and 34 GHz). The spectral index n is then given by

$$\begin{eqnarray}&&n=\displaystyle \frac{\mathrm{log}{I}_{34}-\mathrm{log}{I}_{17}}{\mathrm{log}0.5},\end{eqnarray} \tag{ 9 }$$

where the subscripts 17 and 34 denote measurements at 17 and 34 GHz, respectively.

In the microwave range, the solar atmosphere comprises two components, the optically thick chromosphere and the optically thin corona (Grebinskij et al. 2000; Iwai & Shibasaki 2013). Thus, the observed intensity and polarization components are decomposed as

$$\begin{eqnarray}{I}_{{\rm{obs}}} & = & {I}_{{\rm{chr}}}+{I}_{{\rm{cor}}}\\ {V}_{{\rm{obs}}} & = & {V}_{{\rm{chr}}}+{V}_{{\rm{cor}}},\end{eqnarray} \tag{ 10 }$$

where the subscripts obs, chr, and cor indicate the observed, chromospheric, and coronal components, respectively. Having obtained the I_cor and V_cor, the coronal longitudinal magnetic field can be obtained from Equation (7). In this study, we select the region in which the chromospheric polarization component can be ignored (${V}_{{\rm{obs}}}={V}_{{\rm{chr}}}+{V}_{{\rm{cor}}}\approx {V}_{{\rm{cor}}}$). Hence, we can derive the coronal polarization component from the observation. The details of this assumption are discussed in Section 4.

To determine I_cor, we must measure the coronal temperature and emission measure. As mentioned above, Hannah & Kontar (2012) developed a regularization code that computes the DEM from SDO/AIA data at six EUV wavelengths (131, 171, 193, 211, 335, and 94 Å). The EUV flux F_λ at a given AIA wavelength λ is calculated as

$$\begin{eqnarray}&&{F}_{\lambda }=\int \displaystyle \frac{d\mathrm{EM}(T)}{{dT}}{R}_{\lambda }(T){dT},\end{eqnarray} \tag{ 11 }$$

where dEM(T)/dT is the DEM and R_λ(T) is the filter response function. Here the DEM is estimated by the method of least squares:

$$\begin{eqnarray}&&{\chi }^{2}=\displaystyle \frac{{({F}_{{\rm{obs}}}-{F}_{{\rm{model}}})}^{2}}{{\sigma }_{F}}={\rm{min}},\end{eqnarray} \tag{ 12 }$$

where χ² is the residual sum of squares, F_model and F_obs are the DEM-derived and observed fluxes, respectively, and σ_F is the error in the observed flux. Hence, we can obtain the DEM in the temperature range of the AIA observations $(5.7\lt \mathrm{log}T\lt 7.0)$.

We divided the corona along the line of sight into 130 layers separated by $\mathrm{log}T=0.01$ intervals in the temperature range of the AIA observations. Therefore, the optical thickness of the ith layer is

$$\begin{eqnarray}&&{\rm{\Delta }}{\tau }_{i}=\displaystyle \frac{1.2\ {\xi }_{i}{\rm{\Delta }}{\mathrm{EM}}_{i}}{{T}_{e,i}^{3/2}{\nu }^{2}},\end{eqnarray} \tag{ 13 }$$

where ΔEM_i is the integral of the DEM over the temperature range of the ith layer. In the solar corona, the thermal bremsstrahlung at the microwave range is usually optically thin. Therefore, the observed brightness temperature emitted from the ith layer (ΔT_b,i) can be approximated as follows:

$$\begin{eqnarray}&&{\rm{\Delta }}{T}_{b,i}\approx {T}_{e,i}[1-\mathrm{exp}(-{\rm{\Delta }}{\tau }_{i})]\ \ \ \ \ \ \ ({\tau }_{i}\ll 1).\end{eqnarray} \tag{ 14 }$$

The total brightness temperature is the sum of the brightness temperatures of all layers

$$\begin{eqnarray}&&{T}_{b}=\displaystyle \sum _{j=0}^{i-1}{\rm{\Delta }}{T}_{b,j}.\end{eqnarray} \tag{ 15 }$$

Accordingly, from the coronal temperature and AIA-based DEM calculated by the above equations, we can derive the coronal bremsstrahlung intensity (I_cor = T_b,cor) in the microwave range.

The V_cor is obtained by selecting the region in which the chromospheric polarization component can be ignored, whereas I_cor is determined from the AIA observations. Hence, we can derive the coronal longitudinal magnetic field from Equation (7).

We used NoRH microwave images, EUV images and emission measure obtained by AIA, and photospheric longitudinal magnetograms collected by the Helioseismic and Magnetic Imager (HMI; Scherrer et al. 2012). All data were acquired in the NOAA active region (AR) 11150 (S20 E02) around 3:00 UT on 2011 February 3. The photospheric longitudinal magnetic field in AR 11150 is maximized at around 1500 G. Gyroresonance is excited at the second or third harmonics of the local gyrofrequency (Shibasaki et al. 1994). As the third harmonic of 17 GHz corresponds to a longitudinal magnetic field of about 2000 G, no gyroresonance emission occurs at AR 11150. AR 11150 possesses a simple bipolar magnetic structure in the east–west direction. Soft X-ray flux was observed by the Geostationary Operational Environmental Satellite (GOES) and remained in the B class between 20:00 UT on 2011 February 2 and 3:00 UT on 2011 February 3. Thus, no nonthermal activity such as flares occurred throughout the observation period, and we can observe only the thermal bremsstrahlung.

3.1. Instruments

3.2. 17 GHz Intensity and Polarization and Magnetic Structure

Figure 1 shows full-disk images of the radio intensity and polarization observed by NoRH at 17 GHz. The solid and dashed lines enclose the AR 11150 and the quiet region, respectively. Panel (a) of this figure reveals relatively strong emission in the AR 11150 region, while panel (b) indicates a relatively strong and broadened polarized source in the same region.

Zoom In Zoom Out Reset image size

To improve the signal-to-noise ratio (S/N), we integrated 1200 images with 1 s cadence over 20 minutes. After this procedure, the standard deviation of the intensity (σ_I) at 17 GHz in the quiet region reduced from 120 to 105 K, and the standard deviation of the polarization (σ_V) reduced from 40 to 8 K. The minimum detectable level of the polarization degree (defined at the 5σ level) was 0.4%, comparable to the result of Iwai & Shibasaki (2013).

Figure 2(a) shows the photospheric longitudinal magnetic field in the AR 11150 region observed by HMI. The red and blue contours show the positive and negative components of the radio circular polarization, respectively. The locations and polarities of the photospheric longitudinal magnetic field and the radio circular polarization are strongly correlated. The positive and negative components of the polarization degree of AR 11150 were maximized at 0.9% and −1.1%, respectively.

Zoom In Zoom Out Reset image size

3.3. EUV Loop Structure and Emission Measure

3.4. Potential Field Model

The coronal longitudinal magnetic fields were derived from regions satisfying the following conditions. First, we selected regions exhibiting weak photospheric longitudinal magnetic fields (≈10 G). Therefore, we can neglect the chromospheric polarization component at 17 GHz. Second, we selected the region with many coronal loops. As shown in Figure 2(c), such a region is found around the center of AR 11150 and indicates the presence of a coronal magnetic field. Third, we selected regions in which the polarization degree exceeded the minimum detectable level at 17 GHz (=0.4%).

The five regions satisfying the above conditions are numbered and enclosed in white rectangles in Figure 2(a) (hereafter these regions are called the numbered regions). The sizes of these regions (20'') exceed the beam size at 17 GHz (≈10''–15'').

Table 1 tabulates the measurement parameters of the numbered regions depicted in Figure 2(a). The observed intensities at 17 GHz (I_obs,17) range from 10,400 to 11,200 K, relatively higher than the typical intensity of the quiet region (≈10,000 K). The coronal intensities at 17 GHz (I_cor,17) in the numbered regions, derived by AIA observations, are 700–1400 K. Thus, the corresponding chromospheric intensities $({I}_{{\rm{chr,17}}}\approx {I}_{{\rm{obs,17}}}-{I}_{{\rm{cor,17}}})$ at 17 GHz are 9700–10,400 K.

Table 1.  Measurement Parameters of the Numbered Regions Indicated in Figure 2(a)

	Iobs,17	Ichr,17	Icor,17	Iobs,34	Ichr,34	Icor,34	Vobs	nchr	ncor	EMcor	Bl,pho	Bl,cor	${\sigma }_{{B}_{l,{\rm{cor}}}}$
Region	(K)	(K)	(K)	(K)	(K)	(K)	(K)			(1027 cm−5)	(G)	(G)	(G)
1	11,247	10,395	852	9819	9615	203	−79	0.11	2.07	1.84	−1	−272	40
2	11,505	10,094	1411	9715	9377	336	−71	0.10	2.06	3.05	−1	−147	23
3	10,953	10,143	810	9421	9227	193	−76	0.13	2.06	1.48	−9	−276	41
4	10,457	9661	796	9072	8881	190	−63	0.12	2.06	1.49	−1	−232	39
5	10,898	9794	1104	9783	9519	262	63	0.03	2.07	2.07	16	166	28

Download table as:  ASCIITypeset image

To derive the spectral index n, we also observed the intensity at 34 GHz (I_obs,34). In addition, we derived the chromospheric and coronal intensities at 34 GHz (I_chr,34 and I_cor,34, respectively) by the method used to derive their 17 GHz counterparts. From these results, we derived the spectral index at the chromosphere and corona from Equation (9). The values are listed in Table 1. The coronal spectral indices n_cor in the numbered regions are consistent with the typical coronal value (approximately 2). In contrast, the chromospheric spectral indices n_chr are an order of magnitude lower than the coronal spectral indices. Consequently, to maintain the same polarization degree, the chromospheric longitudinal magnetic field strength must be an order of magnitude greater than the coronal field strength. Therefore, it is suitable to use the approximation V_obs ≈ V_cor in the numbered regions.

As demonstrated in Table 1, the photospheric magnetic fields B_l,pho in the numbered regions are very weak (≈10 G). The magnetic structures of the photosphere and chromosphere are assumed similar at the spatial resolution scale of 17 GHz (≈10''). Thus, the polarization components of the chromosphere V_chr in the numbered regions are negligible, and the observed polarization V_obs is comparable to V_cor. The derived parameters I_cor,17 and n_cor are also listed in Table 1. Inserting I_cor,17, V_cor, and n_cor into Equation (7), the coronal longitudinal magnetic field B_l,cor was computed as around 150–270 G in the numbered regions.

We now estimate the measurement error in the coronal longitudinal magnetic field derived from radio and EUV observations. To this end, we investigate how the errors combine in Equation (7). The error in the coronal longitudinal magnetic field ${\sigma }_{{B}_{l,{\rm{cor}}}}$ becomes

$$\begin{eqnarray}&&{\sigma }_{{B}_{l,{\rm{cor}}}}=\displaystyle \frac{10700}{n\lambda }\sqrt{\displaystyle \frac{{\sigma }_{{V}_{{\rm{cor}}}}^{2}}{{I}_{{\rm{cor,17}}}^{2}}+\displaystyle \frac{{V}_{{\rm{cor}}}^{2}{\sigma }_{{I}_{{\rm{cor,17}}}}^{2}}{{I}_{{\rm{cor,17}}}^{4}}},\end{eqnarray} \tag{ 16 }$$

where ${\sigma }_{{V}_{{\rm{cor}}}}$ and ${\sigma }_{{I}_{{\rm{cor,17}}}}$ are the errors in V_cor and I_cor,17, respectively. Since we assumed V_obs ≈ V_cor in the numbered regions, ${\sigma }_{{V}_{{\rm{cor}}}}$ denotes the error in the observed polarization component ${\sigma }_{{V}_{{\rm{obs}}}}$. In Section 3.2, this error was determined as ${\sigma }_{{V}_{{\rm{obs}}}}=8$ K.

To estimate I_cor,17, we used the DEM derived from the AIA observations. Since I_cor,17 is proportional to the emission measure, we must derive the error in the emission measure.

Figure 3 shows the DEM derived from AIA in regions 1, 3, and 5. The error of the DEM is calculated by the standard Monte Carlo approach used in Hannah & Kontar (2012). In the low-temperature range ($\mathrm{log}T\lt 6.5$), the error of the DEM is up to about 10%. In contrast, the error of the DEM in the high-temperature range ($\mathrm{log}T\gt 6.5$) is greater than that of the low-temperature range. However, the DEM in the high-temperature range is more than two orders of magnitude lower than that in the low-temperature range. Therefore, the effect of the error of the DEM in the high-temperature range is negligible.

Zoom In Zoom Out Reset image size

We derived the DEM by the least-squares method, adopting a single degree of freedom (SDF) with a significance level of χ² = 3.841. The χ² of the numbered regions in this study was close to 2, implying that the DEM of the numbered regions is significant.

For these reasons, we estimated the error in the emission measure as about 10%. In this case, the error in the coronal longitudinal magnetic field was estimated as 20–40 G, which is about 15% of the measured value.

5.1. Comparison with the Potential Field Model

One of the purposes of this study was to validate the accuracy of magnetic fields derived from radio observations by comparing them with potential field calculations. Since the potential field is difficult to calculate near the solar limb, we derived the coronal magnetic field on the solar disk. This section compares the coronal longitudinal magnetic fields with the potential field (Shiota et al. 2008, 2012) and assesses their accuracy.

Figure 4 shows the magnetic field lines of the potential field. The coronal loops of EUV structurally agree with the magnetic field lines of the potential field (Figure 4(a)). Therefore, the magnetic fields of AR 11150 can be considered to be potential-like. In Figure 4(b), the polarization source is distributed near the foot of the magnetic field lines, consistent with the dependence of the polarization degree of the thermal bremsstrahlung on the longitudinal magnetic field.

Zoom In Zoom Out Reset image size

Figure 5 shows the longitudinal component of the potential field in AR 11150. The height refers to the altitude from the photosphere surface. The magnetic field at zero height is the photospheric longitudinal magnetic field measured by HMI. This field does not fill the radio-polarized region at a height of 5220 km, which corresponds to the bottom of the corona in the atmospheric model of Selhorst et al. (2005). On the other hand, at a height of 20,880 km, the longitudinal magnetic field appears to be distributed throughout the contour of the polarization degree. However, the maximum magnetic field strength at this height (≈60 G) is weaker than the coronal longitudinal magnetic field strength derived in this study. In addition, although the longitudinal magnetic field adequately fills the radio-polarized region at 109,620 km, the magnetic field strength at this height is very weak (<2 G).

Zoom In Zoom Out Reset image size

The height variations of the potential field strength in regions 1, 3, and 5 are shown in Figure 6. The longitudinal magnetic fields monotonically increase with height, up to approximately 10,000–20,000 km. At the upper heights, a clear peak appears in the longitudinal magnetic field strength, which corresponds to the corona. Accordingly, the magnetic field at each coronal region is considered to be dominated by the magnetic field line component that broadens from the AR 11150 center. At 10,000–20,000 km, the absolute value of the longitudinal component $| {B}_{l}|$ of the potential field peaks at 24, 30, and 45 G in regions 1, 3 and 5, respectively. On the other hand, the measured values of the coronal longitudinal magnetic fields $| {B}_{l,{\rm{cor}}}|$ are 272, 276, and 166 G, respectively, a whole order of magnitude greater than the potential fields. In Section 4, the measurement error was determined as approximately 15%. Even accounting for this error, the derived magnetic field exceeds the potential field.

Zoom In Zoom Out Reset image size

5.2. Upper Limit of the Derived Coronal Magnetic Field

5.3. The Effect of the Low-temperature Plasma

As explained above, we treat the coronal magnetic field strength derived in this study as an upper limit. To estimate the actual coronal magnetic field strength, we should evaluate the effect of the low-temperature plasma. The EUV Imaging Spectrometer (EIS; Culhane et al. 2007) instrument on board the Hinode satellite can observe some low-temperature ($\mathrm{log}T\lt 5.7$) sensitive lines. Unfortunately, there are no EIS data of AR 11150 on February 3. Hence, we used the EIS data of AR 11150 obtained between 20:03 and 21:05 UT on 2011 February 4 and compared them with the AIA data obtained in the same period. Note that these data were from the same active region as shown in Figure 2, but obtained about 41.5 hr later. Figure 7(a) shows the contribution functions of the lines observed with EIS in this study. The contribution functions are calculated with CHIANTI version 7.1.3, a constant pressure assumption of 10⁻¹⁵ cm⁻³ K, and a set of coronal elemental abundances (Feldman 1992). Two lines that are sensitive to the low-temperature range, O v ($\mathrm{log}{T}_{{\rm{max}}}\sim 5.4$) and Mg v ($\mathrm{log}{T}_{{\rm{max}}}\sim 5.5$), are included in the EIS data in this study. Figure 7(b) shows the EUV image at 171 Å observed with AIA. Regions A, B, and C, enclosed by black rectangles, correspond to regions 1, 3, and 5 in Figure 2(a), respectively.

Zoom In Zoom Out Reset image size

Figure 8 shows the DEM derived from AIA and EIS at regions A, B, and C. Because of the limitation of the lines observed by EIS, we calculated the DEM in the temperature range of $5.3\lt \mathrm{log}T\lt 6.5$ from the EIS data.

Zoom In Zoom Out Reset image size

Figures 9(a) and (b) show the emission measure obtained by EIS and AIA, respectively. To improve the S/N, the EIS and AIA data are integrated with an area of 6 × 6 pixels and 4 × 4 pixels, respectively. These total EMs are integrated in the temperature range of $5.3\lt \mathrm{log}T\lt 6.5$ for the DEM derived from EIS and $5.7\lt \mathrm{log}T\lt 6.5$ for the DEM derived from AIA. Table 2 shows the total EM at the regions indicated in Figure 7(b). The emission measures derived from EIS (EM_EIS) are greater than those derived from AIA (EM_AIA) by about 20%–35%. This is because the EM_EIS includes the low-temperature range ($5.3\lt \mathrm{log}T\lt 5.7$). A discrepancy of the DEM derived from EIS and AIA at the temperature of $\mathrm{log}T\sim 6.2$ in Figure 8 also affects the difference of the EM derived from EIS and AIA. Since the DEM is calculated by the least-squares method, and χ² derived from EIS and AIA at these regions are too small, the DEMs of these regions are significant. Therefore, the total EM derived from EIS is certainly greater than that from AIA, implying the presence of low-temperature plasma of about 20%–35% at these regions. If the EM derived from AIA is underestimated by about 35%, the coronal magnetic field derived in the numbered regions should be 100–210 G. Since the DEM derived from EIS excludes the temperature range of $\mathrm{log}T\lt 5.3$, these magnetic fields should also be an upper limit. In order to specify the effect of the low-temperature coronal plasma more accurately, a DEM analysis using lines that are sensitive to the lower temperature range ($\mathrm{log}T\lt 5.3$) is needed for future studies.

Zoom In Zoom Out Reset image size

Table 2.   Total EM at the Regions Indicated in Figure 7(b)

Region	EMEIS (cm−5)	EMAIA (cm−5)	EMEIS/EMAIA
A	3.46 × 1027	2.56 × 1027	1.35
B	2.59 × 1027	1.99 × 1027	1.30
C	3.63 × 1027	3.10 × 1027	1.21

Download table as:  ASCIITypeset image

Since the derived coronal longitudinal magnetic fields estimated from EIS observations were still the upper limit, we should estimate the actual coronal magnetic field by using some assumptions. The millimeter-range brightness temperatures in the quiet region have been observed (e.g., Kuseki & Swanson 1976) and modeled (Selhorst et al. 2005). According to Selhorst et al. (2005), the brightness temperature of the quiet region at 34 GHz is about 9000 K. The height of the optically thick layer at 17 GHz should be higher than that of at 34 GHz because the temperature in the chromosphere is a monotonically increasing function of height. Hence, the chromospheric intensity at 17 GHz, I_chr,17, should be greater than 9000 K. If we assume that the chromospheric intensity at 17 GHz is 9000 K, the coronal intensities I_cor,17 of the numbered regions range from 1500 to 2500 K. Accordingly, the coronal longitudinal magnetic field is estimated to be 90–130 G, which is still greater than the potential field strength.

5.4. Underestimation of the Potential Field

We derived the coronal magnetic field from the polarization observation of the radio thermal bremsstrahlung. Magnetic field measurements based on the circularly polarized thermal bremsstrahlung have been investigated by several studies (Grebinskij et al. 2000; Iwai & Shibasaki 2013). However, the circular polarization observed at 17 GHz contains both chromospheric and coronal components. Therefore, previous studies assumed chromospheric brightness temperatures such as I_chr = 10,000 K. In contrast, we separated coronal and chromospheric components by combining the radio and EUV observations and derived the coronal longitudinal magnetic field. In addition, we evaluated the derived magnetic field by comparing it with the potential magnetic field and the DEM measurements using EUV observations. Our conclusions are summarized as follows.

• 1.  
  The coronal longitudinal magnetic fields in AR 11150 were estimated as 150–270 G from NoRH and AIA observations. These values were weighted by the coronal emission measure over the line of sight.
• 2.  
  The longitudinal components of the potential field in AR 11150 were strongest (24–45 G) at 10,000–20,000 km above the photosphere surface. These values are an order of magnitude smaller than the coronal longitudinal magnetic field derived in this study.
• 3.  
  As AIA observes over a limited temperature range $(5.7\lt \mathrm{log}T\lt 7.0)$, we took the coronal radio intensity ${I}_{{\rm{cor,17}}}$ derived at 17 GHz as the lower limit. Hence, the derived coronal longitudinal magnetic field B_l,cor was considered as the upper limit. The measurement error in the upper limit was estimated at 15%.
• 4.  
  We also derived the DEM including the low-temperature plasma $5.3\lt \mathrm{log}T\lt 5.7$ from the Hinode/EIS observation. In this case the upper limits of the coronal longitudinal magnetic fields in AR 11150 were estimated to be 100–210 G.
• 5.  
  We estimated the coronal magnetic field with some assumptions. If the chromospheric radio intensity at 17 GHz is 9000 K, the coronal longitudinal magnetic field is estimated as 90–130 G, which still exceeds the magnetic field derived from the potential field.
• 6.  
  The discrepancy between the potential and the observed magnetic field strengths can be explained consistently by two reasons: (1) the underestimation of the coronal emission measure resulting from the limitation of the temperature range of AIA and EIS, and (2) the underestimation of the coronal longitudinal magnetic field resulting from the potential field assumption.

We derived the coronal longitudinal magnetic field from simultaneous NoRH and AIA observations and using the DEM derived from the EIS. However, the coronal longitudinal magnetic field strength derived in this study was an upper limit because of the limited temperature range of the AIA and EIS observations. To more accurately determine the coronal magnetic field by our method, new instruments or methods for calculating the DEM of low-temperature coronal plasma are required. In addition, improving the S/N of the radio interferometer would reduce the measurement error in the coronal magnetic field.

The authors are grateful to Takashi Sakurai and Shinsuke Imada for useful discussions. SDO data and images are courtesy of NASA/SDO and the AIA and HMI science teams. This work was conducted by a joint research program of the National Astronomical Observatory Japan. This work was partially supported by JSPS KAKENHI Grant Number 26800255.