Observations Of Quiet Sun Research Articles

Persistent Homology Analysis for Solar Magnetograms

Understanding the magnetic fields of the Sun is essential for unraveling the underlying mechanisms driving solar activity. Integrating topological data analysis techniques into these investigations can provide valuable insights into the intricate structures of magnetic fields, enhancing our comprehension of solar activity and its implications. In this study, we explore what persistent homology can offer in the analysis of solar magnetograms, with the objective of introducing a novel tool that will serve as the foundation for further studies of magnetic structures at the solar surface. By combining various filtration methods of the persistent homology analysis, we conduct an analysis of solar magnetograms that captures the broad magnetic scene, involving a mixture of positive and negative polarities. This analysis is applied to observations of both quiet-Sun and active regions, taken with the Hinode/Solar Optical Telescope and SDO/Helioseismic and Magnetic Imager, respectively. Our primary focus is on analyzing the properties of the spatial structures and features of the magnetic fields identified through these techniques. The results show that persistent diagrams can encode the spatial structural complexity of the magnetic flux of active regions by identifying the isolated, connected, and interacting features. They facilitate the classification of active regions based on their morphology and the detection and quantification of interacting structures of opposing polarities, such as δ spots. The small-scale events in the quiet Sun, such as magnetic flux cancellation and emergence, are also revealed in persistent diagrams and can be studied by observing the evolution of the plots and tracking the relevant features.

Long-lived particles and the quiet Sun

The nuclear reaction network within the interior of the Sun is an efficient MeV physics factory and can produce long-lived particles generic to dark sector models. In this work we consider the sensitivity of satellite instruments, primarily the RHESSI spectrometer, that observe the quiet Sun in the MeV regime where backgrounds are low. We find that quiet Sun observations offer a powerful and complementary probe in regions of parameter space, where the long-lived particle decay length is longer than the radius of the Sun and shorter than the distance between the Sun and Earth. We comment on connections to recent model-building work on heavy neutral leptons coupled to neutrinos and high-quality axions from mirror symmetries. Published by the American Physical Society 2024

Formation of Hε in the solar atmosphere

Context. In the solar spectrum, the Balmer series line Hε is a weak blend on the wing of Ca II H. Recent high-resolution Hε spectroheliograms reveal a reversed granulation pattern and in some cases, even unique structures. It is apparent that Hε could potentially be a useful diagnostic tool for the lower solar atmosphere. Aims. Our aim is to understand how Hε is formed in the quiet Sun. In particular, we consider the particular physical mechanism that sets its source function and extinction, how it is formed in different solar structures, and why it is sometimes observed in emission. Methods. We used a 3D radiative magnetohydrodynamic (MHD) simulation that accounts for non-equilibrium hydrogen ionization, run with the Bifrost code. To synthesize Hε and Ca II H spectra, we made use of the RH code, which was modified to take into account the non-equilibrium hydrogen ionization. To determine the dominant terms in the Hε source function, we adopted a multi-level description of the source function. Making use of the synthetic spectra and simulation, we studied the contribution function to the relative line absorption or emission and compared it with atmospheric quantities at different locations. Results. Our multi-level source function description suggests that the Hε source function is dominated by interlocking, with the dominant interlocking transition being through the ground level, populating the upper level of Hε via the Lyman series. This makes the Hε source function partly sensitive to temperature. The Hε extinction is set by Lyman-α. In some cases, this temperature dependence gives rise to Hε emission, indicating heating. The typical absorption profiles show reversed granulation and the Hε line core reflects mostly the Ca II H background radiation. Conclusions. Synthetic Hε spectra can reproduce quiet Sun observations quite well. High-resolution observations reveal that Hε is not just a weak absorption line. Regions with Hε in emission are especially interesting to detect small-scale heating events in the lower solar atmosphere, such as Ellerman bombs. Thus, Hε can be an important new diagnostic tool for studies of heating in the solar atmosphere, augmenting the diagnostic potential of Ca II H when observed simultaneously.

The IAG spectral atlas of the spatially resolved Sun: Centre-to-limb observations

Context. Solar surface magneto-convection appears as granulation pattern that impacts spectral lines in terms of both shape and wavelength. Such induced effects also tend to vary over the observed solar disc because of the changing observation angle and, thus, the changing observation height as well. Centre-to-limb observations of the resolved Sun offer an insight into the variable spectral behaviour across different heliocentric observing positions, providing crucial information about limb darkening, convective velocities, and line profile variability relevant to radial velocity (RV) calculations. Thus, RV measurements and exoplanet transit spectroscopy depend on precise reference templates. Aims. We want to provide a spectroscopic centre-to-limb solar atlas at high spectral resolution and high-frequency accuracy. The atlas shall help improve the understanding of the solar atmosphere and convection processes. Methods. We performed high-resolution observations of the resolved quiet Sun with a Fourier transform spectrograph at the Institut für Astrophysik und Geophysik in Göttingen. Our dataset contains a wavelength range from 4200 Å to 8000 Å. We obtained 165 spectra in total, with a spectral resolution of Δν = 0.024 cm−1, corresponding to a resolving power R of 700 000 at ∼6000 Å. Results. We present a centre-to-limb solar atlas containing 14 heliocentric positions. To check for consistency, we investigated the Fe I 6175 Å absorption line and compared our line profiles with previous centre-to-limb observations and also with simulations. The line profile and also the bisector profiles are generally consistent with previous observations, but we have identified differences to model line profiles, especially close to the solar limb.

A benchmark study of atomic models for the transition region against quiet Sun observations

ABSTRACT The use of the coronal approximation to model line emission from the solar transition region has led to discrepancies with observations over many years, particularly for Li- and Na-like ions. Studies have shown that a number of atomic processes are required to improve the modelling for this region, including the effects of high densities, solar radiation, and charge transfer on ion formation. Other non-equilibrium processes, such as time-dependent ionization and radiative transfer, are also expected to play a role. A set of models which include the three relevant atomic processes listed above in ionization equilibrium has recently been built. These new results cover the main elements observed in the transition region. To assess the effectiveness of the results, this work predicts spectral line intensities using differential emission measure modelling. Although limited in some respects, this differential emission measure modelling does give a good indication of the impact of the new atomic calculations. The results are compared to predictions of the coronal approximation and to observations of the average, quiet Sun from published literature. Significant improvements are seen for the line emission from Li- and Na-like ions, intercombination lines, and many other lines. From this study, an assessment is made of how far down into the solar atmosphere the coronal approximation can be applied, and the range over which the new atomic models are valid.

Tracing the Magnetic Field Topology of the Quiet Corona Using Propagating Disturbances

The motion of faint propagating disturbances (PDs) in the solar corona reveals an intricate structure that must be defined by the magnetic field. Applied to quiet Sun observations by the Atmospheric Imaging Assembly (AIA)/Solar Dynamics Observatory (SDO), a novel method reveals a cellular network, with cells of typical diameters 50″ in the cool 304 Å channel and 100″ in the coronal 193 Å channel. The 193 Å cells can overlie several 304 Å cells, although both channels share common source and sink regions. The sources are points, or narrow corridors, of divergence that occupy the centers of cells. They are significantly aligned with photospheric network features and enhanced magnetic elements. This shows that the bright network is important to the production of PDs and confirms that the network is host to the source footpoint of quiet coronal loops. The other footpoint, or the sinks of the PDs, form the boundaries of the coronal cells. These are not significantly aligned with the photospheric network—they are generally situated above the dark internetwork photosphere. They form compact points or corridors, often without an obvious signature in the underlying photosphere. We argue that these sink points can either be concentrations of closed field footpoints associated with minor magnetic elements in the internetwork or concentrations of an upward-aligned open field. The link between the coronal velocity and magnetic fields is strengthened by comparison with a magnetic extrapolation, which shows several general and specific similarities, thus the velocity maps offer a valuable additional constraint on models.

Imaging of the Quiet Sun in the Frequency Range of 20–80 MHz

Radio emission of the quiet Sun is considered to be due to thermal bremsstrahlung emission of the hot solar atmosphere. The properties of the quiet Sun in the microwave band have been well studied, and they can be well described by the spectrum of bremsstrahlung emission. In the meter-wave and decameter-wave bands, properties of the quiet Sun have rarely been studied due to the instrumental limitations. In this work, we use the LOw Frequency ARray telescope to perform high quality interferometric imaging spectroscopy observations of quiet Sun coronal emission at frequencies below 90 MHz. We present the brightness temperature spectrum and the size of the Sun in the frequency range of 20–80 MHz. We report on dark coronal regions with low brightness temperatures that persist with frequency. The brightness temperature spectrum of the quiet Sun is discussed and compared with the bremsstrahlung emission of a coronal model and previous quiet Sun observations.

Disambiguation of Vector Magnetograms by Stereoscopic Observations from the Solar Orbiter (SO)/Polarimetric and Helioseismic Imager (PHI) and the Solar Dynamic Observatory (SDO)/Helioseismic and Magnetic Imager (HMI)

Spectropolarimetric reconstructions of the photospheric vector magnetic field are intrinsically limited by the 180^{circ} ambiguity in the orientation of the transverse component. The successful launch and operation of Solar Orbiter have made the removal of the 180∘ ambiguity possible using solely observations obtained from two different vantage points. While the exploitation of such a possibility is straightforward in principle, it is less so in practice, and it is therefore important to assess the accuracy and limitations as a function of both the spacecrafts’ orbits and measurement principles. In this work, we present a stereoscopic disambiguation method (SDM) and discuss thorough testing of its accuracy in applications to modeled active regions and quiet-Sun observations. In the first series of tests, we employ magnetograms extracted from three different numerical simulations as test fields and model observations of the magnetograms from different angles and distances. In these more idealized tests, SDM is proven to reach a 100% disambiguation accuracy when applied to moderately-to-well resolved fields. In such favorable conditions, the accuracy is almost independent of the relative position of the spacecraft with the obvious exceptions of configurations where the spacecraft are within a few degrees of co-alignment or quadrature. Even in the case of disambiguation of quiet-Sun magnetograms with significant under-resolved spatial scales, SDM provides an accuracy between 82% and 98%, depending on the field strength. The accuracy of SDM is found to be mostly sensitive to the variable spatial resolution of Solar Orbiter in its highly elliptic orbit, as well as to the intrinsic spatial scale of the observed field. Additionally, we provide an example of the expected accuracy as a function of time that can be used to optimally place remote-sensing observing windows during Solar Orbiter observation planning. Finally, as a more realistic test, we consider magnetograms that are obtained using a radiative-transfer inversion code and the SO/PHI Software siMulator (SOPHISM) applied to a 3D-simulation of a pore, and we present a preliminary discussion of the effect of the viewing angle on the observed field. In this more realistic test of the application of SDM, the method is able to successfully remove the ambiguity in strong-field areas.

Wavelength Dependence of Image Quality Metrics and Seeing Parameters and Their Relation to Adaptive Optics Performance

Ground-based solar observations are severely affected by Earth’s turbulent atmosphere. As a consequence, observed image quality and prevailing seeing conditions are closely related. Partial correction of image degradation is nowadays provided in real time by adaptive optics (AO) systems. In this study, different metrics of image quality are compared with parameters characterizing the prevailing seeing conditions, i.e. Median Filter Gradient Similarity (MFGS), Median Filter Laplacian Similarity (MFLS), Helmli–Scherer mean, granular rms-contrast, differential image motion, and Fried-parameter \(r_{0}\). The quiet-Sun observations at disk center were carried out at the Vacuum Tower Telescope (VTT), Observatorio del Teide (OT), Izaña, Tenerife, Spain. In July and August 2016, time series of short-exposure images were recorded with the High-resolution Fast Imager (HiFI) at various wavelengths in the visible and near-infrared parts of the spectrum. Correlation analysis yields the wavelength dependence of the image quality metrics and seeing parameters, and Uniform Manifold Approximation and Projection (UMAP) is employed to characterize the seeing on a particular observing day. In addition, the image quality metrics and seeing parameters are used to determine the field dependence of the correction provided by the AO system. Management of high-resolution imaging data from large-aperture, ground-based telescopes demands reliable image quality metrics and meaningful characterization of prevailing seeing conditions and AO performance. The present study offers guidance on how retrieving such information ex post facto.

Power spectrum of turbulent convection in the solar photosphere

The solar photosphere provides us with a laboratory for understanding turbulence in a layer where the fundamental processes of transport vary rapidly and a strongly superadiabatic region lies very closely to a subadiabatic layer. Our tools for probing the turbulence are high-resolution spectropolarimetric observations such as have recently been obtained with the two balloon-borne SUNRISE missions, and numerical simulations. Our aim is to study photospheric turbulence with the help of Fourier power spectra that we compute from observations and simulations. We also attempt to explain some properties of the photospheric overshooting flow with the help of its governing equations and simulations. We find that quiet-Sun observations and smeared simulations are consistent with each other and exhibit a power-law behavior in the subgranular range of their Doppler velocity power spectra with a power-law index of ≈ − 2. The unsmeared simulations exhibit a power law that extends over the full range between the integral and Taylor scales with a power-law index of ≈ − 2.25. The smearing, reminiscent of observational conditions, considerably reduces the extent of the power-law-like portion of the power spectra. This suggests that the limited spatial resolution in some observations might eventually result in larger uncertainties in the estimation of the power-law indices. The simulated vertical velocity power spectra as a function of height show a rapid change in the power-law index (at the subgranular range) from roughly the optical depth unity layer, that is, the solar surface, to 300 km above it. We propose that the cause of the steepening of the power-law index is the transition from a super- to a subadiabatic region, in which the dominant source of motions is overshooting convection. A scale-dependent transport of the vertical momentum occurs. At smaller scales, the vertical momentum is more efficiently transported sideways than at larger scales. This results in less vertical velocity power transported upward at small scales than at larger scales and produces a progressively steeper vertical velocity power law below 180 km. Above this height, the gravity work progressively gains importance at all relevant scales, making the atmosphere progressively more hydrostatic and resulting in a gradually less steep power law. Radiative heating and cooling of the plasma is shown to play a dominant role in the plasma energetics in this region, which is important in terms of nonadiabatic damping of the convective motions.

Propagation Effects in Quiet Sun Observations at Meter Wavelengths

Abstract Quiet Sun meterwave emission arises from thermal bremsstrahlung in the MK corona, and can potentially be a rich source of coronal diagnostics. On its way to the observer, it gets modified substantially due to propagation effects—primarily refraction and scattering—through the magnetized and turbulent coronal medium, leading to the redistribution of the intensity in the image plane. By comparing the full-disk meterwave solar maps during a quiet solar period and the modeled thermal bremsstrahlung emission, we characterize these propagation effects. The solar radio maps between 100 MHz and 240 MHz come from the Murchison Widefield Array. The FORWARD package is used to simulate thermal bremsstrahlung images using the self-consistent Magnetohydrodynamic Algorithm outside a Sphere coronal model. The FORWARD model does not include propagation effects. The differences between the observed and modeled maps are interpreted to arise due to scattering and refraction. There is a good general correspondence between the predicted and observed brightness distributions, though significant differences are also observed. We find clear evidence for the presence of significant propagation effects, including anisotropic scattering. The observed radio size of the Sun is 25–30% larger in area. The emission peak corresponding to the only visible active region shifts by 8–11ʹ and its size increases by 35–40%. Our simple models suggest that the fraction of scattered flux density is always larger than a few tens of percent, and varies significantly between different regions. We estimate density inhomogeneities to be in the range 1–10%.

First high-resolution look at the quiet Sun with ALMA at 3mm

We present an overview of high-resolution quiet Sun observations, from disk center to the limb, obtained with the Atacama Large millimeter and sub-millimeter Array (ALMA) at 3 mm. Seven quiet-Sun regions were observed at a resolution of up to 2.5″ by 4.5″. We produced both average and snapshot images by self-calibrating the ALMA visibilities and combining the interferometric images with full-disk solar images. The images show well the chromospheric network, which, based on the unique segregation method we used, is brighter than the average over the fields of view of the observed regions by ∼305 K while the intranetwork is less bright by ∼280 K, with a slight decrease of the network/intranetwork contrast toward the limb. At 3 mm the network is very similar to the 1600 Å images, with somewhat larger size. We detect, for the first time, spicular structures, rising up to 15″ above the limb with a width down to the image resolution and brightness temperature of ∼1800 K above the local background. No trace of spicules, either in emission or absorption, is found on the disk. Our results highlight the potential of ALMA for the study of the quiet chromosphere.

Signatures of quiet Sun reconnection events in Ca ii, Hα, and Fe i

We use observations of quiet Sun (QS) regions in the H$\alpha$ 6563 \AA, Ca~{\sc ii} 8542 \AA\, and Fe~{\sc i} 6302 \AA\ lines. We observe brightenings in the wings of the H$\alpha$ and Ca~{\sc ii} combined with observations of the interacting magnetic concentrations observed in the Stokes signals of Fe~{\sc i}. These brightenings are similar to Ellerman bombs (EBs), i.e. impulsive bursts in the wings of the Balmer lines which leave the line cores unaffected. Such enhancements suggest that these events have similar formation mechanisms to the classical EBs found in active regions, with the reduced intensity enhancements found in the QS regions due to a weaker feeding magnetic flux. The observations also show that the quiet Sun Ellerman bombs (QSEBs) are formed at a higher height in the upper photosphere than the photospheric continuum level. Using simulations, we investigate the formation mechanism associated with the events and suggest that these events are driven by the interaction of magnetic field-lines in the upper photospheric regions. The results of the simulation are in agreement with observations when comparing the light-curves, and in most cases we found that the peak in the Ca~{\sc ii} 8542 \AA\ wing occurred before the peak in H$\alpha$ wing. Moreover, in some cases, the line profiles observed in Ca~{\sc ii} are asymmetrical with a raised core profile. The source of heating in these events is shown by the MURaM simulations and is suggested to occur 430 km above the photosphere.

Manifestation of the Identity of Brightness Temperatures of the Polar Coronal Hole and Low-Latitude Coronal Holes in the Centimeter Wave Range

This paper is devoted to an analysis of brightness temperatures of the polar and low-latitude coronal holes on the Sun in the cm-wave range during periods of minimum solar activity. Data from observations of the polar coronal hole received by the RATAN-600 radio telescope during the solar eclipse of March 29, 2006, and low-latitude observations of coronal holes and quiet Sun received earlier with the RATAN-600 and BPR radio telescopes in the period of minimum solar activity have been used in the paper. The obtained good agreement between the brightness temperatures of cm-wave emission of the polar coronal hole above the North Pole of the Sun and of the low-latitude coronal holes against the background of the quiet Sun reveals the identity of the temperature properties of large coronal holes, irrespective of the mode of their arrangement and location on the Sun during the periods of minimum solar activity.

Solar H-alpha features with hot onsets

In H-alpha most of the solar surface is covered by dense canopies of long opaque fibrils, but predictions for quiet-Sun observations with ALMA have ignored this fact. Comparison with Ly-alpha suggests that the large opacity of H-alpha fibrils is caused by hot precursor events. Application of a recipe that assumes momentary Saha-Boltzmann extinction during their hot onset to millimeter wavelengths suggests that ALMA will observe H-alpha-like fibril canopies, not acoustic shocks underneath, and will yield data more interesting than if these canopies were transparent.

THE FORMATION OFIRISDIAGNOSTICS. VIII.IRISOBSERVATIONS IN THE C ii 133.5 nm MULTIPLET

The C II 133.5 nm multiplet has been observed by NASA's Interface Region Imaging Spectrograph (IRIS) in unprecedented spatial resolution. The aims of this work are to characterize these new observations of the C II lines, place them in context with previous work, and to identify any additional value the C II lines bring when compared with other spectral lines. We make use of wide, long exposure IRIS rasters covering the quiet Sun and an active region. Line properties such as velocity shift and width are extracted from individual spectra and analyzed. The lines have a variety of shapes (mostly single-peak or double-peak), are strongest in active regions and weaker in the quiet Sun. The ratio between the 133.4 nm and 133.5 nm components is always less than 1.8, indicating that their radiation is optically thick in all locations. Maps of the C II line widths are a powerful new diagnostic of chromospheric structures, and their line shifts are a robust velocity diagnostic. Compared with earlier quiet Sun observations, we find similar absolute intensities and mean line widths, but smaller red shifts; this difference can perhaps be attributed to differences in spectral resolution and spatial coverage. The C II intensity maps are somewhat similar to those of transition region lines, but also share some features with chromospheric maps such as those from the Mg II k line, indicating that they are formed between the upper chromosphere and transition region. C II intensity, width, and velocity maps can therefore be used to gather additional information about the upper chromosphere.

IRIS observations of the Mg ii h and k lines during a solar flare

The bulk of the radiative output of a solar flare is emitted from the chromosphere, which produces enhancements in the optical and UV continuum, and in many lines, both optically thick and thin. We have, until very recently, lacked observations of two of the strongest of these lines: the Mg II h & k resonance lines. We present a detailed study of the response of these lines to a solar flare. The spatial and temporal behaviour of the integrated intensities, k/h line ratios, line of sight velocities, line widths and line asymmetries were investigated during an M class flare (SOL2014-02-13T01:40). Very intense, spatially localised energy input at the outer edge of the ribbon is observed, resulting in redshifts equivalent to velocities of ~15-26km/s, line broadenings, and a blue asymmetry in the most intense sources. The characteristic central reversal feature that is ubiquitous in quiet Sun observations is absent in flaring profiles, indicating that the source function increases with height during the flare. Despite the absence of the central reversal feature, the k/h line ratio indicates that the lines remain optically thick during the flare. Subordinate lines in the Mg II passband are observed to be in emission in flaring sources, brightening and cooling with similar timescales to the resonance lines. This work represents a first analysis of potential diagnostic information of the flaring atmosphere using these lines, and provides observations to which synthetic spectra from advanced radiative transfer codes can be compared.

Spatial deconvolution of spectropolarimetric data: an application to quiet Sun magnetic elements

Observations of the Sun from the Earth are always limited by the presence of the atmosphere, which strongly disturbs the images. A solution to this problem is to place the telescopes in space satellites, which produce observations without any (or limited) atmospheric aberrations. However, even though the images from space are not affected by atmospheric seeing, the optical properties of the instruments still limit the observations. In the case of diffraction limited observations, the PSF establishes the maximum allowed spatial resolution, defined as the distance between two nearby structures that can be properly distinguished. In addition, the shape of the PSF induce a dispersion of the light from different parts of the image, leading to what is commonly termed as stray light or dispersed light. This effect produces that light observed in a spatial location at the focal plane is a combination of the light emitted in the object at relatively distant spatial locations. We aim to correct the effect produced by the telescope's PSF using a deconvolution method, and we decided to apply the code on Hinode/SP quiet Sun observations. We analyze the validity of the deconvolution process with noisy data and we infer the physical properties of quiet Sun magnetic elements after the deconvolution process.

Automatic Method for Identifying Photospheric Bright Points and Granules Observed by Sunrise

In this study, we propose methods for the automatic detection of photospheric features (bright points and granules) from ultra-violet (UV) radiation, using a feature-based classifier. The methods use quiet-Sun observations at 214 nm and 525 nm images taken by Sunrise on 9 June 2009. The function of region growing and mean shift procedure are applied to segment the bright points (BPs) and granules, respectively. Zernike moments of each region are computed. The Zernike moments of BPs, granules, and other features are distinctive enough to be separated using a support vector machine (SVM) classifier. The size distribution of BPs can be fitted with a power-law slope -1.5. The peak value of granule sizes is found to be about 0.5 arcsec^2. The mean value of the filling factor of BPs is 0.01, and for granules it is 0.51. There is a critical scale for granules so that small granules with sizes smaller than 2.5 arcsec^2 cover a wide range of brightness, while the brightness of large granules approaches unity. The mean value of BP brightness fluctuations is estimated to be 1.2, while for granules it is 0.22. Mean values of the horizontal velocities of an individual BP and an individual BP within the network were found to be 1.6 km/s and 0.9 km/s, respectively. We conclude that the effect of individual BPs in releasing energy to the photosphere and maybe the upper layers is stronger than what the individual BPs release into the network.

CALCULATED RESONANCE LINE PROFILES OF [Mg II], [C II], AND [Si IV] IN THE SOLAR ATMOSPHERE

NASA's Interface Region Imaging Spectrograph space mission, launched 2013 June 27, is intended to study the structure of the solar chromosphere and the transition region between the chromosphere and corona. The spectral lines to be observed include the Mg II k line at 2796.5 A, the C II 1334.5 A line, and the Si IV line at 1393.8 A, which are formed in the middle chromosphere, the upper chromosphere, and the lower transition region, respectively. Here we calculate the profiles of these lines from four models of the solar atmosphere, intended to represent the faint and mean internetwork, a network lane, and bright network. We show how the profiles change from the center of the solar disk toward the limb of the Sun and in response to outflows and inflows. These results are intended to cover the range of expected quiet-Sun observations and assist in their interpretation. We expect that the observations will lead to improvements in the models, which can then be used to estimate the required non-radiative heating in the different regions.