Propagation Effects in Quiet Sun Observations at Meter Wavelengths

The data used for this study come from a period of very low solar activity. These data were obtained on 2015 December 03 as a part of the G0002 solar observing program. This day is characterized as a period of low solar activity by solarmonitor.org. EUV images from the Helioseismic and Magnetic Imager (HMI) instrument, on board the Solar Dynamic Observatory (SDO) show a smooth featureless disk across much of the photosphere, except for a few flux emerging footprints of active regions near the eastern and western limbs (Figure 1). Four weak bipolar regions were reported. Three of them were located very close to the limb at N11W78, N12W83, and S07W91, while one was located at N05W46. Only two GOES C-class X-ray flares were reported in the past two days, and on this day two C-class flares were reported from the active region 12458 at 04:25 UT (C2.2) and 06:01 UT (C1.2), respectively. The composite solar map from the Atmospheric Imaging Assembly (AIA) instrument on board the SDO shows the absence of on-disk active regions and the presence of a large coronal hole (Figure 1). Quantifying the extent of the coronal holes using the Coronal Hole Identification via Multi-thermal Emission Recognition Algorithm (CHIMERA; Garton et al. 2018) on AIA multithermal images led to a detection of three coronal holes covering a very substantial ∼23.8% of the total solar disk.

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Following the terminology introduced by Gibson et al. (2016), the coronal distribution of n_e, T_e, and B can be said to define the "physical state" of the system. For the date of our observations, the physical state is determined using the MAS model for solar corona. MAS self-consistently solves MHD equations in the 3D spherical coordinates of the solar corona/inner heliosphere.³ As input, MAS uses the synoptic HMI magnetogram in Carrington coordinates. The physical parameters n_e, T_e, and B are self-consistently evolved globally using resistive thermodynamic MHD equations. The simulated values of n_e, T_e, and B are normalized with photospheric values. Over the years, the results from this approach has been benchmarked with observations in various wave bands (e.g., Lionello et al. 2009; Linker et al. 2011, etc.). Once the physical state of the corona is available from this model, referred to as PSIMAS, the FORWARD package (Gibson et al. 2016) is used to synthesize the radio images. FORWARD is a comprehensive IDL package for coronal magnetometry that uses the physical state of the corona to synthesize a broad range of coronal observables, including radio images, using various "physical processes." The relevant physical processes for radio emissions are the thermal bremsstrahlung or the free–free emission and the gyroresonance emissions, though the contribution of the latter is significant only at much higher frequencies than are considered here. The synthetic observables obtained, radio maps in this case, can be compared directly with the observations.

For thermal bremsstrahlung, which is always included in a FORWARD radio emission calculation, opacity results from collisions between electrons and ions. The free–free emission optical depth (${\tau }_{{\rm{f}}{\rm{f}}}$) originating from collisions of coronal ions and electrons (Dulk 1985; Gelfreikh 2004), which is used by the FORWARD package (Gibson et al. 2016), is given by

$$\begin{eqnarray}&&{\tau }_{{\rm{f}}{\rm{f}}}=0.2\int \displaystyle \frac{{n}_{{\rm{e}}}^{2}}{{T}_{{\rm{e}}}^{3/2}{\left(f\pm {f}_{B,{\rm{L}}{\rm{o}}{\rm{S}}}\right)}^{2}}dS,\end{eqnarray} \tag{ 1 }$$

where S, f, and ${f}_{B,\mathrm{LoS}}$ are the coordinates along the LoS, the frequency of observation, and the LoS gyro-frequency, respectively, in cgs units. The plus and minus signs represent the $o-$ and the $x-$ modes, respectively. Here, ${f}_{B,\mathrm{LoS}}={f}_{B}| {\cos }(\theta )|$, where θ is the angle between the LoS and the local magnetic field B, and ${f}_{B}({Hz})=2.8\,\times \,{10}^{6}\ B$, with B expressed in Gauss. FORWARD does not incorporate any propagation effects, i.e., it does not take into account the scattering and refraction caused by the coronal density structures that are a part of the PSIMAS model or the refraction due to the large-scale density decrease with coronal height. The ${\tau }_{{\rm{f}}{\rm{f}}}$ is computed simply using rectilinear propagation of radio waves through the coronal volume. It also computes the T_B, the brightness temperature associated with a given ray, and provides 2D plane-of-sky (PoS) images of T_B in various Stokes parameters. The n_e, T_e, and $\vec{B}$ vary across the LoS, and it is not straightforward to visualize these quantities in a 3D volume surrounding the Sun. To provide a visual sense for the spatial variations of these parameters, Figure 2 shows the PoS maps of various LoS-averaged quantities determined using PSIMAS and extracted using FORWARD. The first three of these, i.e., n_e, T, and $| B|$, define the physical state of the corona, and the last one, ${\tau }_{{\rm{f}}{\rm{f}}}$ at 240 MHz, is obtained by using the physical processes of thermal bremsstrahlung. The ${\tau }_{{\rm{f}}{\rm{f}}}$ map has been convolved with the corresponding MWA point-spread function (PSF) to facilitate comparison with MWA images in Section 4. A distinct circular feature is found to be associated with the solar limb in all of these LoS-integrated quantities. As one proceeds from the center of the Sun toward the limb, the lengths of the LoS segment between any two coronal heights become increasingly longer. This leads to a tendency for the ${\tau }_{{\rm{f}}{\rm{f}}}$ to increase with increasing distance from the center. As the LoS goes past the limb, its length through the corona abruptly increases, as the part of corona "behind" the plane of the sky also becomes accessible to the LoS, giving rise to a pronounced limb "brightening." Its smeared out appearance in the rightmost panel of Figure 2 is due to the convolution with the MWA PSF. A comparison of Figure 2 with the Figure 1 (right panel), shows that the coronal hole region is associated with low n_e, low T, and a somewhat higher $| B|$. In the vicinity of the active regions on the western limb, an increase in T and $| B|$ are evident. Except for what has been noted above, the solar disk shows features with low contrast levels for n_e and $| B|$, while the T structure is more elaborate. The ${\tau }_{{\rm{f}}{\rm{f}}}$ map at 240 MHz clearly shows the expected limb brightening and a signature of the coronal holes. We divide the LoS into N segments with a given segment denoted by i, and i = 0 denoting the base of the corona and $i=N-1$ the edge of the simulated volume closest to the observer. The change in T_B as the radiation traverses the i^th segment of the LoS is given by

$$\begin{eqnarray}&&{T}_{{\rm{B}},i}={T}_{{\rm{B}},i-1}{e}^{-d{\tau }_{i}}+{T}_{{\rm{e}},i}(1-{e}^{-d{\tau }_{i}})\end{eqnarray} \tag{ 2 }$$

where $d{\tau }_{i}$ is the τ corresponding to the i^th segment of the LoS, and ${T}_{{\rm{e}},i}$ is its electron temperature. The PoS map is produced by integrating along the entire LoS using the above equation. Here, we only use Stokes I T_B maps at MWA frequency bands generated using FORWARD.

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The top row of Figures 4 and 5 show MWA T_B maps. Figure 4 shows radio maps for three representative observation frequencies—108 MHz, 162 MHz, and 217 MHz. Figure 5 shows a stack of T_B maps at all of the frequencies for which images were obtained, with the frequency of observation shown along the vertical axis. This figure takes advantage of the fact that the emission at higher radio frequencies is expected to arise at lower coronal heights to provide a convenient way to track the evolution of the radio morphology with coronal height. It is evident from these figures that the thermal bremsstrahlung from the disk dominates the emission. A gradual and systematic evolution of the single dominant compact emission feature in the western hemisphere, which splits into two features at higher coronal heights, is clearly seen. An increase in the size of the Sun with height is also evident. The 5σ threshold used to delimit the Sun is $\approx 0.2$ MK. The solar size at the equator is larger than that at the poles, and this is more prominent at lower frequencies. An overall drop in T_B of the compact source as well as of the extended emission features with increasing coronal height is also evident. However, the spatial T_B distribution is rather smooth, with the contrast between the on-disk features even at the highest frequency being only about a factor of two, which steadily decreases to about 25% by 108 MHz.

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Limb brightening is clearly evident in the FORWARD ${\tau }_{{\rm{f}}{\rm{f}}}$ map (Figure 2, rightmost panel, eastern limb), though it is less prominent in the FORWARD radio maps (Figure 4, middle row). As pointed out in Section 3.1, the ray paths used by FORWARD for computing the observed maps do not take propagation effects into account and are hence unrealistic. However, detailed studies taking coronal refraction into account also show limb brightening at frequencies ranging from 1200 MHz to below 200 MHz (Smerd 1950). As an impact of the large-scale refraction, the location of the peak of the brightness profile is seen to shift closer to the solar center with decreasing frequency. It is quite pronounced at decimeter wavelengths and has been observed (e.g., Swarup & Parthasarathy 1955). At meter wavelengths, however, the limb brightening is much less pronounced, and to the best of our knowledge, is yet to be reported observationally. In addition to scattering, which is expected to smear out the limb brightening, there are also other factors which have led to this. These include the intrinsic difficulties in imaging the quiet Sun with sufficient dynamic range, and the presence of contaminating features like coronal holes and active regions.

To look for evidence for limb brightening, we examine a position angle far from the contaminating effects of coronal features. The top panel of Figure 6 shows the 108 MHz T_B map in radial-azimuthal coordinates. The bottom panel shows the radial profiles at various frequencies along the azimuth of θ = −13.6° (lying in the southeastern quadrant) for both MWA and FORWARD maps. The FORWARD maps show the clear presence of limb brightening between 20ʹ and 25ʹ. Similar features are not seen in the MWA maps. Much weaker T_B enhancements are seen between 10ʹ and 15ʹ in the MWA maps. These could be the signatures of residual limb brightening much reduced by propagation effects. We note the tendency for the peak of the T_B profile to move to lower radii with increasing frequency. Some other expected tendencies, like the increase in radial size and the decrease in observed T_B with decreasing frequency, are also evident.

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The bottom row of Figure 4 shows the overlays of radio T_B contours over AIA EUV 193 Å images. At the higher frequency end of our observations, the large-scale correspondence between the EUV and radio emissions is clearly evident. The EUV bright regions also tend to have regions of higher T_B nearby. The large coronal hole, occupying much of the northern hemisphere, is clearly seen in the EUV with a thin tongue dipping into the southern hemisphere. At 240 MHz, the coronal hole region is associated with lower T_B, though by 108 MHz, the same region has higher T_B values than its neighboring regions. In fact, at 108 MHz, the enhanced T_B sits atop the thin coronal hole tongue.

This tendency for some coronal holes to transition from regions of relatively lower T_B to regions of relatively higher T_B as one goes from lower to higher coronal heights has been noticed earlier, as well (e.g., Mercier & Chambe 2015), and was the subject of a recent study by Rahman et al. (2019), who proposed an explanation based on refractive effects.

The significant impact of propagation effects has long been recognized for the active emissions arising from plasma emission mechanisms, where the generated radiation is at the local plasma frequency or its harmonic (Steinberg et al. 1971; Arzner & Magun 1999; Thejappa & MacDowall 2008; Kontar et al. 2019). These propagation effects are equally applicable to the quiet emissions, as well.

The top and middle rows of Figure 4 show the T_B maps observed by MWA and those computed by FORWARD, respectively. The FORWARD maps have been convolved with the MWA restoring beam to facilitate comparison. At 217 MHz, it seems reasonable that the impacts of scattering and refraction can morph the FORWARD map to look like what is observed by the MWA. The FORWARD maps show a larger contrast and comparatively compact emission features at the locations of active regions present close to the limb, as seen in the EUV maps. The large-scale refraction would tend to move features at the limb to smaller radial distances, and the scattering would smear them over a larger angular size. As one proceeds to lower frequencies, the variance between the MWA and FORWARD maps grows. While the FORWARD maps are all very similar, in the MWA maps, the emission from the active region on the northeastern limb disappears with decreasing frequency, a new compact source appears in the southeastern quadrant, and the coronal hole transitions from being radio faint to be being radio bright. While the peak T_B of the FORWARD maps is close to 1.6 MK at all frequencies, for the MWA maps it drops from 1.5 MK to 1.2 MK and 0.9 MK as one goes from 217 MHz to 162 MHz and 108 MHz, respectively. The disk-averaged T_B for FORWARD, T_B,FWD, and MWA, T_B,MWA, maps are listed in Table 1. T_B,MWA is consistently significantly less than T_B,FWD, except at the highest two frequencies, where the two match within the uncertainties.

Table 1.  The Brightness Temperature Calculated from MWA and FORWARD, along with their Optical Depth Estimates

Frequency	${{\rm{T}}}_{{\rm{B}},{\rm{M}}{\rm{W}}{\rm{A}}}$	${{\rm{T}}}_{{\rm{B}},{\rm{F}}{\rm{W}}{\rm{D}}}$	${\tau }_{\mathrm{MWA}}$	${\tau }_{\mathrm{FWD}}$	${{\rm{\Phi }}}_{\mathrm{MWA}}$	${{\rm{r}}}_{\mathrm{eff},\mathrm{MWA}}$	${{\rm{\Phi }}}_{\mathrm{FWD}}$	γ
(MHz)	(MK)	(MK)			(103 arcmin2)	(arcmin)	(103 arcmin2)
108	0.51 ± 0.06	0.86	0.901 ± 0.005	13.46	2.33 ± 0.05	27.20 ± 0.10	1.86	2.03
132	0.62 ± 0.13	0.86	1.279 ± 0.005	9.48	2.26 ± 0.05	26.85 ± 0.10	1.81	1.57
145	0.63 ± 0.04	0.85	1.369 ± 0.011	7.98	2.24 ± 0.05	26.71 ± 0.09	1.78	1.52
162	0.66 ± 0.05	0.85	1.471 ± 0.009	6.79	2.1 ± 0.05	25.87 ± 0.09	1.72	1.50
179	0.65 ± 0.06	0.83	1.558 ± 0.009	5.54	2.11 ± 0.05	25.93 ± 0.09	1.63	1.44
196	0.75 ± 0.07	0.81	2.625 ± 0.036	4.72	2.07 ± 0.05	25.65 ± 0.09	1.60	1.21
217	0.86 ± 0.10	0.78	$\gt 1$	3.94	2.01 ± 0.04	25.29 ± 0.09	1.60	1.03
240	0.75 ± 0.10	0.72	$\gt 1$	3.15	1.97 ± 0.04	25.02 ± 0.09	1.51	1.02

Note. The ${\tau }_{\mathrm{MWA}}$ is calculated using Equation (1), while ${\tau }_{\mathrm{FWD}}$ is the free–free optical depth obtained from the FORWARD code. ${{\rm{\Phi }}}_{\mathrm{MWA}}$ and ${{\rm{\Phi }}}_{\mathrm{FWD}}$ are the areas of the radio solar disk calculated from the MWA and FORWARD images, respectively. r${}^{\mathrm{eff},\mathrm{MWA}}$ is the effective radius corresponding to ${{\rm{\Phi }}}_{\mathrm{MWA}}$. We use 5σ cutoff to define the radio Sun, where σ is the rms computed over a region of the map far from the Sun. The last column lists the ratio of disk-integrated flux densities from FORWARD and MWA maps, $\gamma ={S}_{\nu ,{\rm{F}}{\rm{W}}{\rm{D}}}/{S}_{\nu ,{\rm{M}}{\rm{W}}{\rm{A}}}$.

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4.4.1. Source Sizes

A comparison of the observed and predicted angular sizes of the Sun can be a potentially useful way to quantify the impact of scattering. Table 1 lists the solar disk area estimated from the MWA, ${{\rm{\Phi }}}_{\mathrm{MWA}}$, and FORWARD, ${{\rm{\Phi }}}_{\mathrm{FWD}}$, maps. ${{\rm{\Phi }}}_{\mathrm{MWA}}\gt {{\rm{\Phi }}}_{\mathrm{FWD}}$ at all frequencies by 25–30%, with no significant trend with frequency. For the MWA maps, a 5σ cutoff is used to define the extent of the radio Sun, where σ is the rms in the map in a region far away from the Sun. For the FORWARD map, the radio Sun is delimited using a T_B contour corresponding to the 5σ value used for the MWA maps.

We also explore the use of the observed compact source to estimate the impact of scattering. For this, we define a region of size comparable to the PSF around the brightest source seen in the FORWARD maps, and then estimate the size of the corresponding region in the MWA maps which encloses the same radio flux density. Though simple, and really valid only for an isolated or a very bright compact source, this exercise can still provide interesting information about the impact of propagation effects. Figure 7 shows the regions in FORWARD maps marked by a contour at 1.5 MK for frequency bands by solid lines. The area of this region varies between 1.73 and 2.65 times the PSF area for 240 MHz to 108 MHz (Table 2). The regions marked by dashed contours are the regions from MWA maps around the peaks and enclosing the same flux density as the FORWARD contour. The locations of the flux density peaks are also indicated. The area enclosed by the MWA contours is always larger that enclosed by the FORWARD contours, by a factor between 1.26 and 2.08, and is listed in the column titled "Ratio" in Table 2. We note that as the FORWARD maps are convolved with MWA PSF, the true size of the source can be smaller and, hence, these numbers represent a lower limit. At higher frequencies, the corresponding region in the MWA maps is a single unbroken contour. As one proceeds to frequencies below 196 MHz, another contour emerges in the MWA maps. In all likelihood, this is an unrelated emission component and only illustrates the limitations of this exercise. While the FORWARD maps are very similar across frequencies in their broad features, the MWA maps are not, and some of these differences arise due to reasons other than propagation effects.

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Table 2.  Source Sizes of Regions Emitting Equal Flux Density in FORWARD and MWA Maps, as Shown in Figure 7

ν	FORWARD	MWA	Ratio	Shift	Type III
(MHz)	(PSFs)	(PSFs)		(arcmin)	(PSFs)
108	2.65	5.5	2.08	10.44 ± 3.63	1.91
132	2.44	4.13	1.69	10.17 ± 3.27	1.72
145	2.34	3.92	1.67	8.58 ± 3.28	1.61
162	2.55	4.28	1.68	8.58 ± 3.09	1.48
179	2.29	4.02	1.76	10.87 ± 3.09	1.36
196	2.04	3.06	1.50	10.44 ± 3.11	1.27
217	1.83	2.39	1.31	8.33 ± 3.10	1.15
240	1.73	2.19	1.26	8.86 ± 3.37	0.94

Note. The source sizes are given in units of the area of the PSF. The ratio is simply the size seen by MWA divided by that estimated from FORWARD maps. The second-to-last column shows the distance between the locations of the peaks in FORWARD and MWA maps. The size of the major axes of the PSFs are listed as the error bars. The last column shows the expected FWHM of the type III burst sources in units of MWA PSFs for comparison.

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The impact of scattering on the type III source sizes (Kontar et al. 2019) can be modeled as,

$$\begin{eqnarray}&&{\rm{\Phi }}(\{^\circ \})=(11.78\pm 0.06)\times {\nu }^{(-0.98\pm 0.05)},\end{eqnarray} \tag{ 3 }$$

where Φ and ν are in degrees and MHz, respectively. Type III bursts are believed to arise from compact sources, so it is useful to use this as a point of comparison. The last column of Table 2 lists the expected sizes of type III sources based on Equation (3) in units of the MWA PSFs for the corresponding frequency. The expected linear size of a type III source is found to increase by ∼39% from 240 MHz to 196 MHz, while an increase of ∼35% is observed in the linear size of the compact source seen in MWA maps. The observed fractional change in the MWA source size is consistent with the observed type III source size relation.

Another interesting finding from this exercise is that the location of the peak of the emission has substantial offsets between the two maps. These offsets are tabulated in Table 2 and lie between ∼8ʹ and 11ʹ, much larger than the ∼3ʹ PSF. The displacement vector has essentially the same orientation at all frequencies, and is nearly perpendicular to the local radial direction. Recent work by Kontar et al. (2019) suggests that shifts of this magnitude can be produced by propagation effects through the turbulent and anisotropic corona, and tend to be larger for sources closer to the limb. Additionally the nonradial nature of the offset provides evidence for the presence of significant nonradial anisotropy in the corona.

We compare the disk-integrated flux density, ${S}_{\nu }$, which is not sensitive to the distribution of flux density, obtained from MWA and FORWARD maps. A good match implies a remarkable success on multiple fronts. On the observational front, it attests to the ability of MWA images to capture the bulk of solar emission and provides an independent verification of the techniques used for flux calibration (Oberoi et al. 2017) and making T_B maps (Mohan & Oberoi 2017). On the modeling front, it provides an independent verification of the veracity of both the physical state of the corona as determined by the MAS model, and the ability of FORWARD maps to predict a key radio observable based on the physical processes of thermal bremsstrahlung.

Figure 8 shows a comparison of ${S}_{\nu }$ from FORWARD (S_ν,FWD) and MWA (S_ν,MWA) maps. For a given frequency, S_ν,FWD was computed using the following equation,

$$\begin{eqnarray}&&{S}_{\nu ,{\rm{F}}{\rm{W}}{\rm{D}}}=\displaystyle \frac{2k}{{\lambda }^{2}}\,{\int }_{Sun}{T}_{\nu ,{\rm{F}}{\rm{W}}{\rm{D}}}\,d{\rm{\Omega }},\end{eqnarray} \tag{ 4 }$$

where k is the Boltzmann constant and λ the wavelength of observation, and the integration is carried out over a closed contour bounding the Sun.

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S_ν,MWA is the sum of all of the flux density enclosed within the 5σ contour in the radio map. The error bars on S_ν,MWA represent 1σ of the flux density variation across the six baselines used for flux calibration and over 4 min of time. S_ν,FWD and S_ν,MWA are remarkably consistent at the highest two frequencies. At lower frequencies, S_ν,MWA is always significantly lower than S_ν,FWD and seems to settle at a value 2–4 SFU lower than S_ν,FWD at frequencies below 180 MHz. This is very curious and implies the presence of a systematic effect that is leading to either an overestimate of S_ν,FWD or an underestimate of S_ν,MWA. The excellent match between S_ν,FWD and S_ν,MWA at the two highest frequencies suggests that they could be used to explore the impacts of propagation effects.

As preparation for understanding the differences between S_ν,FWD and S_ν,MWA below 200 MHz, we first examine the 3D data FORWARD cube used for estimating S_ν,FWD. Just as in the photosphere, the coronal plasma density and temperature also show a significant variation. The top panel of Figure 9 shows the radial electron density profiles extracted from the 3D model used by FORWARD over regions illustrative of the active region (AR), a quiet Sun (QS) region, and the coronal hole (CH). The density profile predicted by the Newkirk model (Newkirk 1967) is also shown for comparison. At the base of the corona, the electron densities are very similar for all of these regions and the Newkirk model, but they diverge quickly with increasing radial distance. We note that for all of the regions considered, the Newkirk model suggests significantly larger electron densities all through the corona. By 1.25 R_⊙, the FORWARD coronal hole electron density is about an order of magnitude smaller than that suggested by the Newkirk model. The distribution of plasma frequencies in thin radial shells at a few selected coronal heights spanning the 1–2 R_⊙ range are shown in the bottom panel of Figure 9. A variation of a factor of few in plasma frequency is seen at all heights. The striking departures between the 3D FORWARD data and the expectations from the Newkirk model drives home the limitations of spherically, or at least azimuthally, symmetric coronal electron density models. The use of these models is quite common, e.g., to estimate the coronal height of radio sources using the frequency of plasma emission. It is, however, useful to keep in mind that this is driven solely by lack of better options; its results are only indicative in nature and should not be interpreted in great quantitative detail.

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To make the task of comparing S_ν,MWA and S_ν,FWD more tractable, we first compute an average coronal model and then do radiative transfer through it. The average LoS is constructed by first dividing the 3D coronal model into thin radial shells (about 200 km in thickness) and computing the average physical properties for each shell. In Section 3.1, we have simulated T_B via radiative transfer from the base of the corona along an LoS (Equation (2)) all the way to the observer, by regarding the LoS to be composed of numerous small segments. Knowing the angular size of the source allows us to compute ${S}_{\nu }$ given ${T}_{\nu }$ (Equation (4)). Let S_i denote the flux density measured from the i^th line segment all the way to the observer. Then, ${{dS}}_{i}={S}_{i}-{S}_{i+1}$ is the flux density contributed by the i^th segment of the LoS. ${\sum }_{j}^{N}{{dS}}_{i}$, where N is the location of the observer, then provides an estimate of the flux density contributed by the part of the LoS beyond the jth radial segment. This quantity is shown in Figure 10 for each of the MWA frequencies. At any given radial distance, the plotted flux density represents the flux density contributed by the part of the LoS beyond that radial distance, all the way to the observer. The filled box indicates the radial height at which the integrated ${S}_{\nu }$ along the average LoS equals the observed S_ν,MWA. This can be used as indicator of the effective coronal depth from which the LoS at a given frequency receives a contribution. The ⋆ indicates the radial height at which the average plasma frequency of the coronal model equals the radio frequency of observation. As shown in Figure 9, there is a large spread in the value of the plasma frequency in a given radial shell. This spread is shown by the error bar associated with the ⋆. This suggests that the thermal radiation at low frequencies comes not only from higher coronal heights, but also from a much wider height range depending on coronal structures.

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That at low radio frequencies the observed T_B for the Sun is much lower than the MK coronal T_e has been known for a while. The commonly accepted explanation argues that scattering arising due to the presence of inhomogeneities in the turbulent coronal medium effectively pushes the height of the critical layer, below which the LoS cannot penetrate, to much higher heights than that corresponding to the plasma frequency (e.g., Thejappa & MacDowall 2008). As the FORWARD model does not incorporate any propagation effects, including scattering, the observed value of S_ν,MWA being less S_ν,FWD is consistent with this explanation. However, the rather sharp change from the two flux densities being consistent above 200 MHz to S_ν,MWA falling below S_ν,FWD suddenly at lower frequencies does seem somewhat surprising.

The MAS model is based on HMI extrapolated magnetic fields, and it loses accuracy as one proceeds to higher coronal heights. However, lower radio frequencies pick up a larger fraction of their flux density contributions from exactly these higher coronal heights. This suggests the possibility that perhaps the coronal model in use is not well suited for modeling higher coronal heights and is giving rise to systematic discrepancies between S_ν,MWA and S_ν,FWD.

We start with the assumption that the FORWARD maps represent the true solar radio maps in absence of any propagation effects. For 218 MHz and 240 MHz, S_ν,FWD and S_ν,MWA are consistent, and this makes a quantitative comparison interesting.

The physical picture we model is shown in Figure 11. For a region R1, some fraction of the radiation emanating from it will be scattered out of the LoS to the observer, and similarly some radiation from neighboring regions will get scattered into the nominal LoS originating from R1. ${T}_{{\rm{B}},{\rm{R}}1,0}$ represents the T_B of radiation emanating from region R1 in the FORWARD maps (represented by 0 in the subscript). To keep the problem tractable, we represent the temperature of the neighboring region from which the radiation is scattered into the LoS by ${T}_{{\rm{B}},{\rm{N}},0}$, and it is arrived at by computing the mean T_B over a somewhat arbitrarily defined region in the vicinity of R1. This region was chosen to be the one bounded by the 60% contour in the FORWARD maps. This includes all of the bright regions on the solar disk. It is reasonable to expect that the flux density scattered into the line of sight will be dominated by the contribution from the brighter regions in the map. We parameterize the strength of scattering by a parameter α. We assume that the nature of inhomogeneities in the coronal regions is the same everywhere, and hence it is parameterized by the same α. The T_B observed in the MWA maps, ${T}_{{\rm{B}},{\rm{R}}1,{\rm{M}}{\rm{W}}{\rm{A}}}$, depends on ${T}_{{\rm{B}},{\rm{R}}1,0}$, ${T}_{{\rm{B}},{\rm{N}},0}$, and α, and can be modeled as

$$\begin{eqnarray}&&{T}_{{\rm{B}},{\rm{R}}1,{\rm{M}}{\rm{W}}{\rm{A}}}=(1-\alpha )\,{T}_{{\rm{B}},{\rm{R}}1,0}+\alpha \,{T}_{{\rm{B}},{\rm{N}},0}.\end{eqnarray} \tag{ 5 }$$

Equation (5) can be rearranged to yield α

$$\begin{eqnarray}&&\alpha =\displaystyle \frac{{T}_{{\rm{B}},{\rm{R}}1,{\rm{M}}{\rm{W}}{\rm{A}}}-{T}_{{\rm{B}},{\rm{R}}1,0}}{{T}_{{\rm{B}},{\rm{N}},0}-{T}_{{\rm{B}},{\rm{R}}1,0}}.\end{eqnarray} \tag{ 6 }$$

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The case α = 0 implies no scattering; the strength of scattering increases with increasing values of α, and $\alpha \gt 1$ is unphysical.

In the MWA and FORWARD maps, three distinct kinds of regions are easy to identify—bright active regions (AR), quiet Sun regions (QS), and coronal hole regions (CH). For the quantitative analysis, three representative regions on the Sun, shown in Figure 12, are chosen for each of these categories. The regions corresponding to AR and CH are chosen from the FORWARD map, as they are easier to identify there, and the QS region is chosen to be near the center of the disk. The size of the selected regions is in the range 3–4 × PSF size. The mean brightness temperature within the regions and the estimated values of α are listed in Table 3. The variation in T_B within these regions is found to be in the range of 10–15% (Table 3). For T_B,FWD, this variation is used to compute an uncertainty on it. Uncertainties in the values of α are computed by propagating the uncertainties in estimates of T_B,MWA and T_B,FWD.

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For 217 MHz and 240 MHz, ${\alpha }_{{\rm{A}}{\rm{R}}}$ is ∼0.5. However, ${\alpha }_{{\rm{Q}}{\rm{S}}}$ and ${\alpha }_{{\rm{C}}{\rm{H}}}$ are larger than unity, though with large uncertainties. This implies that, for the QS and CH regions, the model proposed in Equation (6) is insufficient for quantitatively modeling the strength of scattering, or some of the assumptions made are being violated, or both.

Assuming FORWARD models to be true representations of solar emission, we attempt to obtain a lower limit on the fraction of scattered emission, ${f}_{{\rm{s}}{\rm{c}},{\rm{R}}1}$ from a region R1 using

$$\begin{eqnarray}&&{f}_{{\rm{s}}{\rm{c}},{\rm{R}}1}=\displaystyle \frac{| {T}_{{\rm{B}},{\rm{R}}1,{\rm{M}}{\rm{W}}{\rm{A}}}-{T}_{{\rm{B}},{\rm{R}}1,0}| }{{T}_{{\rm{B}},{\rm{R}}1,0}}.\end{eqnarray} \tag{ 7 }$$

A small value of f_sc does not necessarily imply a weak or low strength of scattering, but a larger value of f_sc does imply an increasing strength of scattering. We expect the observed T_B for the AR region to decrease, and that for the QS and CH regions to increase. The values of f_sc obtained are presented in Table 3. The lowest estimated value of f_sc is ∼0.14 for AR at 217 MHz, and the largest is ∼3 at the same frequency for CH. Such large values imply that scattering plays an important role in redistributing the flux density.

The strength of scattering is well known to depend on the level of density inhomogeneities, parameterized by $\epsilon =\tfrac{{\rm{\Delta }}N}{N}$ , where N is the local average electron density and ΔN is a measure of the departure from it. Following earlier works, we assume that the inhomogeneities have a Gaussian distribution with a characteristic correlation length scale h (Chandrasekhar 1952; Hollweg 1968; Steinberg et al. 1971). For such a medium and for small-angle scattering, the scattering optical depth, ${\tau }_{{\rm{S}}{\rm{C}}}$, along an LoS at a frequency f is given by

$$\begin{eqnarray}&&{\tau }_{{\rm{s}}{\rm{c}}}(r)={\int }_{r}^{{\rm{\infty }}}\displaystyle \frac{\sqrt{\pi }{f}_{{\rm{p}}{\rm{e}}}^{4}(r)}{2{\left({f}^{2}-{f}_{{\rm{p}}{\rm{e}}}^{2}\left(r\right)\right)}^{2}}\displaystyle \frac{{\epsilon }^{2}}{h}dr,\end{eqnarray} \tag{ 8 }$$

where f_pe is the electron plasma frequency and r the radial coordinate. To take the effect of scattering into account, we modify the radiative transfer equation (Equation (2)) as follows:

$$\begin{eqnarray}\begin{array}{r}\begin{array}{c}\begin{array}{ccc}{T}_{{\rm{B}},i} & = & {T}_{{\rm{B}},i-1}{e}^{\left.-(d{\tau }_{i}+d{\tau }_{{\rm{s}}{\rm{c}},i}\right)}\,+\,{T}_{{\rm{e}},i}(1-{e}^{-d{\tau }_{i}})\\ & & +\,\displaystyle \frac{1}{\gamma }\,{T}_{{\rm{B}},{\rm{m}}{\rm{e}}{\rm{a}}{\rm{n}},i}(1-{e}^{-d{\tau }_{{\rm{s}}{\rm{c}},i}}).\end{array}\end{array}\end{array}\end{eqnarray} \tag{ 9 }$$

Here ${T}_{{\rm{B}},i}$ represents the radiation temperature at the i^th segment of the LoS, ${T}_{{\rm{B}},{\rm{m}}{\rm{e}}{\rm{a}}{\rm{n}},i}$ corresponds to the mean brightness temperature in the neighboring regions, and $d{\tau }_{{\rm{s}}{\rm{c}},i}$ is the scattering optical depth for the i^th LoS segment. The ratio of S_ν,FWD to S_ν,MWA is γ and is listed in Table 1. The first term now also includes a parameterization of the fraction of incident brightness temperature removed from the LoS due to scattering. The second term is as before, and the third term represents the contribution scattered into the LoS from the neighboring regions. Though it is not expected to be the case, for lack of a better option, and h are assumed to remain constant along the LoS.

Refraction can have a considerable impact on the propagation of rays out through the corona. Even for the simplest case of an isothermal spherically symmetric corona, the apparent shift of sources toward the disk center due to refraction can lead to a change in the observed T_B. This effect is the smallest for an LoS close to the center of the solar disk and is neglected here for simplicity. We assume that the ${T}_{{\rm{B}},{\rm{F}}{\rm{W}}{\rm{D}}}$ has been modified only due to scattering to result in ${T}_{{\rm{B}},{\rm{M}}{\rm{W}}{\rm{A}}}$. We also need to choose a region over which ${T}_{{\rm{B}},{\rm{m}}{\rm{e}}{\rm{a}}{\rm{n}},i}$, needed in Equation (9), can be computed. Somewhat arbitrarily, we use a region of diameter of 11ʹ around the central LoS (shifted toward the brighter region) to compute ${T}_{{\rm{B}},{\rm{m}}{\rm{e}}{\rm{a}}{\rm{n}},i}$ (Figure 12). This is justified because the flux scattered in will be dominated by the brighter regions in the neighborhood. Figure 13 shows the variation in ${T}_{{\rm{B}},{\rm{m}}{\rm{e}}{\rm{a}}{\rm{n}}}$ along the LoS for the frequencies of interest.

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Using the stated assumptions, one can relate T_B,MWA to T_B,FWD for the central LoS with the impact of scattering parameterized by a single parameter, ${\epsilon }^{2}/h$. For each of the MWA frequencies, the left panel of Figure 14 shows the expected value of the observed T_B for the central LoS as a function of ${\epsilon }^{2}/h$. The observed values of T_B,MWA for each of the observing frequencies are marked, along with their uncertainties, and for all of them ${\epsilon }^{2}/h$ lies in the range 1–10 $\times \,{10}^{-5}\,{\mathrm{km}}^{-1}$. For the lower frequencies, with increasing ${\epsilon }^{2}/h$, T_B tends to decrease and settle down to low values, while at higher frequencies, it tends to continue to increase to larger values. The uncertainty in T_B has been translated to that in ${\epsilon }^{2}/h$ in the middle panel of Figure 14. Averaging across frequency yields ${\epsilon }^{2}/h\approx (4.9\pm 2.4)\times {10}^{-5}\,{\mathrm{km}}^{-1}$, and the rms across the frequencies is used as the uncertainty. The right panel of Figure 14 shows the values of arrived at by assuming h = 40 km (Steinberg et al. 1971), with an average value of  = (4.28 ± 1.09)%.

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In the current analysis, the major sources of uncertainties can be categorized into those arising from observations and those from limitations of the modeling formalism.

5.3.1. Observational Uncertainties

5.3.2. Limitations of Models