Measuring the Wilson depression of sunspots using the divergence-free condition of the magnetic field vector

Context: The Wilson depression is the difference in geometric height of unit continuum optical depth between the sunspot umbra and the quiet Sun. Measuring the Wilson depression is important for understanding the geometry of sunspots. Current methods suffer from systematic effects or need to make assumptions on the geometry of the magnetic field. This leads to large systematic uncertainties of the derived Wilson depressions. Aims: We aim at developing a robust method for deriving the Wilson depression that only requires the information about the magnetic field that is accessible from spectropolarimetry, and that does not rely on assumptions on the geometry of sunspots or on their magnetic field. Methods: Our method is based on minimizing the divergence of the magnetic field vector derived from spectropolarimetric observations. We focus on large spatial scales only in order to reduce the number of free parameters. Results: We test the performance of our method using synthetic Hinode data derived from two sunspot simulations. We find that the maximum and the umbral averaged Wilson depression for both spots determined with our method typically lies within 100 km of the true value obtained from the simulations. In addition, we apply the method to Hinode observations of a sunspot. The derived Wilson depression (about 600 km) is consistent with results typically obtained from the Wilson effect. We also find that the Wilson depression obtained from using horizontal force balance gives 110 - 180 km smaller Wilson depressions than both, what we find and what we deduce directly from the simulations. This suggests that the magnetic pressure and the magnetic curvature force contribute to the Wilson depression by a similar amount.Comment: 12 pages, 8 figures. Accepted for publication in Astronomy & Astrophysic

No universal connection between the vertical magnetic field and the umbra-penumbra boundary in sunspots

Context. It has been reported that the boundary between the umbra and the penumbra of sunspots occurs at a canonical value of the strength of the vertical magnetic field, independently of the size of the spot. This critical field strength is interpreted as to be the threshold for the onset of magnetoconvection. Aims. Here we investigate the reasons why this criterion, also called the Jur\v{c}\'ak criterion in the literature, does not always identify the boundary between umbra and penumbra. Methods. We perform a statistical analysis of 23 sunspots observed with Hinode/SOT. We compare the properties of the continuum intensity and the vertical magnetic field between filaments and spines and how they vary between spots of different sizes. Results. We find that the inner boundary of the penumbra is not related to a universal value of the vertical magnetic field. The properties of spines and filaments vary between spots of different sizes. Both components are darker in larger spots and the spines exhibit stronger vertical magnetic field. These variations of the properties of filaments and spines with spot size are also the reason for the reported invariance of the averaged vertical magnetic field at 50% of the mean continuum intensity. Conclusions. The formation of filaments and the onset of magnetoconvection are not related to a canonical value of the strength of the vertical magnetic field. Such a seemingly unique magnetic field strength is rather an effect of the filling factor of spines and penumbral filaments.Comment: 15 pages, 11 figures. Accepted for publication in Astronomy and Astrophysic

Energiezuinige Teeltplanning voor Potplanten : een rekenplatform voor energie-efficiënte scenario's in de Ficus-teelt

In het kader van het project Energiezuinige Teeltplanning voor Potplanten is gewerkt aan een adviessysteem dat potplantentelers in staat stelt om zelfstandig de energie-efficiëntie van teeltscenario’s te evalueren op basis van actuele, bedrijfsspecifieke gegevens. Daarnaast maakt het systeem het mogelijk om via internet teeltscenario’s te delen met teeltadviseurs en binnen bedrijfsvergelijkingsgroepen. Het systeem is gebaseerd op dynamische gewasmodellen voor Hortensia, Poinsettia en Ficus, en maakt voor het berekenen van kasklimaat en energiestromen gebruik van het rekenmodel KASPRO. Dit rapport beschrijft de werking van het systeem, de resultaten op 4 bedrijven per gewas, de reacties van de betrokken telers, en een evaluatie van het project. De belangrijkste mogelijkheden om de energie-efficiëntie van een teelt te verbeteren werden gevonden in verbeterde wijderzetschema’s en in het toepassen van de principes van ‘Het Nieuwe Telen’, door met name de temperatuur meer te laten variëren met het licht en het ontwikkelingsstadium van het gewas

High-order aberration compensation with Multi-frame Blind Deconvolution and Phase Diversity image restoration techniques

Context. For accurately measuring intensities and determining magnetic field strengths of small-scale solar (magnetic) structure, knowledge of and compensation for the point spread function is crucial. For images recorded with the Swedish 1-meter Solar Telescope, restoration with Multi-Frame Blind Deconvolution and Joint Phase Diverse Speckle methods lead to remarkable improvements in image quality but granulation contrasts that are too low, indicating additional stray light. Aims. We propose a method to compensate for stray light from high-order atmospheric aberrations not included in MFBD and JPDS processing. Methods. To compensate for uncorrected aberrations, a reformulation of the image restoration process is proposed that allows the average effect of hundreds of high-order modes to be compensated for by relying on Kolmogorov statistics for these modes. The applicability of the method requires simultaneous measurements of Fried's parameter r0. The method is tested with simulations as well as real data and extended to include compensation for conventional stray light. Results. We find that only part of the reduction of granulation contrast in SST images is due to uncompensated high-order aberrations. The remainder is still unaccounted for and attributed to stray light from the atmosphere, the telescope with its re-imaging system and to various high-altitude seeing effects. Conclusions. We conclude that statistical compensation of high-order modes is a viable method to reduce the loss of contrast occurring when a limited number of aberrations is explicitly compensated for with MFBD and JPDS processing. We show that good such compensation is possible with only 10 recorded frames. The main limitation of the method is that already MFBD and JPDS processing introduces high-order compensation that, if not taken into account, can lead to over-compensation.Comment: in press in Astronomy & Astrophysic

Canonical Forms in Interactive Exercise Assistants

Interactive exercise assistants support students in practicing exercises, and acquiring procedural skills. Many mathematical topics can be practiced in such assistants. Ideally, an interactive exercise assistant not only validates final answers, but also comments on intermediate steps submitted by a student, provides hints on how to proceed, and presents worked-out examples. For these purposes, fine control over the symbolic simplification procedures of the underlying mathematical machinery is needed. In this paper, we introduce views for mathematical expressions. A view defines an equivalence relation by choosing a canonical form of mathematical expressions. We use views to track and recognize intermediate answers, to help in presenting expressions to a user, and to control the granularity of the steps in worked-out examples. We develop the concept of a view, discuss the laws it satisfies, and show how views are composed, which means that they can be used for multiple exercise classes.

CRISP Spectropolarimetric Imaging of Penumbral Fine Structure

We discuss penumbral fine structure in a small part of a pore, observed with the CRISP imaging spectropolarimeter at the Swedish 1-m Solar Telescope (SST), close to its diffraction limit of 0.16 arcsec. Milne-Eddington inversions applied to these Stokes data reveal large variations of field strength and inclination angle over dark-cored penumbral intrusions and a dark-cored light bridge. The mid-outer part of this penumbra structure shows 0.3 arcsec wide spines, separated by 1.6 arcsec (1200 km) and associated with 30 deg inclination variations. Between these spines, there are no small-scale magnetic structures that easily can be be identified with individual flux tubes. A structure with nearly 10 deg more vertical and weaker magnetic field is seen midways between two spines. This structure is co-spatial with the brightest penumbral filament, possibly indicating the location of a convective upflow from below.Comment: Accepted for publication in ApJL 17 Oct 2008. One Figure adde

Small-scale convection signatures associated with strong plage solar magnetic field

In this work, we study and quantify properties of strong-field small-scale convection and compare observed properties with those predicted by numerical simulations. We analyze spectropolarimetric 630.25 nm data from a unipolar ephemeral region near sun center. We use line-of-sight velocities and magnetic field measurements obtained with Milne-Eddington inversion techniques along with measured continuum intensities and Stokes V amplitude asymmetry at a spatial resolution of 0.15 arcseconds to establish statistical relations between the measured quantities. We also study these properties for different types of distinct magnetic features, such as micropores, bright points, ribbons, flowers and strings. We present the first direct observations of a small-scale granular magneto-convection pattern within extended regions of strong (more than 600 G average) magnetic field. Along the boundaries of the flux concentrations we see mostly downflows and asymmetric Stokes V profiles, consistent with synthetic line profiles calculated from MHD simulations. We note the frequent occurrence of bright downflows along these boundaries. In the interior of the flux concentrations, we observe an up/down flow pattern that we identify as small-scale magnetoconvection, appearing similar to that of field-free granulation but with scales 4 times smaller. Measured RMS velocities are 70% of those of nearby field-free granulation, even though the average radiative flux is not reduced. The interiors of these flux concentrations are dominated by upflows.Comment: Accepted for publication in Astronomy and Astrophysic

Chromospheric Velocities of a C-class Flare

We use high spatial and temporal resolution observations from the Swedish Solar Telescope to study the chromospheric velocities of a C-class flare originating from active region NOAA 10969. A time-distance analysis is employed to estimate directional velocity components in H-alpha and Ca II K image sequences. Also, imaging spectroscopy has allowed us to determine flare-induced line-of-sight velocities. A wavelet analysis is used to analyse the periodic nature of associated flare bursts. Time-distance analysis reveals velocities as high as 64 km/s along the flare ribbon and 15 km/s perpendicular to it. The velocities are very similar in both the H-alpha and Ca II K time series. Line-of-sight H-alpha velocities are red-shifted with values up to 17 km/s. The high spatial and temporal resolution of the observations have allowed us to detect velocities significantly higher than those found in earlier studies. Flare bursts with a periodicity of approximately 60 s are also detected. These bursts are similar to the quasi-periodic oscillations observed at hard X-ray and radio wavelength data. Some of the highest velocities detected in the solar atmosphere are presented. Line-of-sight velocity maps show considerable mixing of both the magnitude and direction of velocities along the flare path. A change in direction of the velocities at the flare kernel has also been detected which may be a signature of chromospheric evaporation.Comment: Accepted for publication in Astronomy and Astrophysics, 5 figure