Photospheric Conditions Research Articles

Resonant Absorption of Magnetohydrodynamic Body Modes under Photospheric Conditions

Observations reveal that surface and body modes exist in solar pores under photospheric conditions. While the effects of resonant absorption on photospheric surface modes are well established, its effect on body modes is not known yet. In this paper, we investigate resonant absorption for the body modes under photospheric conditions. We numerically solve the dispersion relation induced by thin boundary approximation and obtain the wave dispersion curves and damping rates of three arbitrarily chosen body modes for sausage and kink waves, respectively. The results show that resonant absorption for the body modes is weaker than for the slow surface modes in both cusp and Alfvén continua. The damping behavior of body modes is similar to slow surface modes while the higher body mode has stronger resonant absorption.

Slow body magnetohydrodynamic waves in solar photospheric flux tubes with density inhomogeneity

ABSTRACT Pores and sunspots are ideal environments for the propagation of guided magnetohydrodynamic (MHD) waves. However, modelling such photospheric waveguides with varying background quantities such as plasma density and magnetic field has thus far been very limited. Such modelling is required to correctly interpret MHD waves observed in pores and sunspots with resolved inhomogeneities such as light bridges and umbral dots. This study will investigate the propagation characteristics and the spatial structure of slow body MHD modes in a magnetic flux tube with a circular cross-section with inhomogeneous equilibrium density distribution under solar photospheric conditions in the short wavelength limit. For simplicity, the equilibrium density profile is taken to have a circular density enhancement or depletion. The advantage of this is that the strength, size, and position of the density inhomogeneity can be easily changed. Calculating the eigenfrequencies and eigenfunctions of the slow body modes is addressed numerically with use of the Fourier–Chebyshev Spectral method. The radial and azimuthal variation of eigenfunctions is obtained by solving a Helmholtz-type partial differential equation with Dirichlet boundary conditions. The inhomogeneous equilibrium density profile results in modified eigenvalues and eigenvectors. It was found that a localized density inhomogeneity leads to a decrease in the eigenvalues and the spatial structure of modes ceases to be a global harmonic oscillation, as the modes migrate towards regions of lower density. Comparing the homogeneous case and the cases corresponding to depleted density enhancement, the dimensionless phase speed undergoes a significant drop in its value (at least 40 per cent).

Magnetic field fluctuations in the shocked umbral chromosphere

Context. Umbral chromospheric observations show the presence of magnetoacoustic shocks. Several recent studies have reported magnetic field fluctuations associated with these shock waves. The mechanism behind these periodic magnetic field changes is still an unresolved question. Aims. We aim to study the properties and origin of magnetic field fluctuations in the umbral chromosphere. Methods. Temporal series of spectropolarimetric observations were acquired with the GREGOR telescope on 2017 June 18. The chromospheric and photospheric conditions, including the temporal evolution of the magnetic field, were derived from simultaneous inversions of the He I 10 830 Å triplet and the Si I 10 827 Å line using the HAZEL2 code. The oscillations were interpreted using wavelet analysis and context information from ultraviolet observations acquired with the Atmospheric Imaging Assembly on board the Solar Dynamics Observatory (SDO/AIA) and the Interface Region Imaging Spectrograph (IRIS). Results. The chromospheric magnetic field shows strong fluctuations in the sunspot umbra, with peak field strengths of up to 2900 G. This inferred field strength is comparable to the magnetic field strength in the upper photosphere. Magnetic field and velocity umbral oscillations exhibit a strong coherence, with the magnetic field lagging the shock fronts detected in the velocity fluctuations. This points to a common origin of the fluctuations in both parameters, and the analysis of the phase shift between photospheric and chromospheric velocity is consistent with upward wave propagation. These results suggest that the strong inferred magnetic field fluctuations are caused by changes in the response height of the He I 10 830 Å line to the magnetic field, which is sensitive to the high photosphere during and right after the shock. The analysis of extreme ultraviolet data shows a weak brightening in a coronal loop rooted in the umbra around the time of the measured magnetic field fluctuations. This coronal activity could possibly have some impact on the inferred fluctuations, but it is not the main driver of the magnetic field oscillations since they are found before the extreme ultraviolet event takes place. Conclusions. Chromospheric magnetic field fluctuations measured with the He I 10 830 Å triplet arise due to variations in the opacity of the line. After strong shocks are produced by the propagation of slow magnetoacoustic waves, the response of the line to the magnetic field can be shifted down to the upper photosphere. This is seen as remarkably large fluctuations in the line-of-sight magnetic field strength.

Modeling CME encounters at Parker Solar Probe with OSPREI: Dependence on photospheric and coronal conditions

Context.Coronal mass ejections (CMEs) are eruptions of plasma from the Sun that travel through interplanetary space and may encounter Earth. CMEs often enclose a magnetic flux rope (MFR), the orientation of which largely determines the CMEs’ geoeffectiveness. Current operational CME models do not model MFRs, but a number of research ones do, including the Open Solar Physics Rapid Ensemble Information (OSPREI) model.Aims.We report the sensitivity of OSPREI to a range of user-selected photospheric and coronal conditions.Methods.We modeled four separate CMEs observed in situ by Parker Solar Probe (PSP). We varied the input photospheric conditions using four input magnetograms (HMI Synchronic, HMI Synoptic, GONG Synoptic Zero-Point Corrected, and GONG ADAPT). To vary the coronal field reconstruction, we employed the Potential Field Source Surface (PFSS) model and varied its source-surface height in the range 1.5–3.0R⊙with 0.1R⊙increments.Results.We find that both the input magnetogram and PFSS source surface often affect the evolution of the CME as it propagates through the Sun’s corona into interplanetary space, and therefore the accuracy of the MFR prediction compared to in situ data at PSP. There is no obvious best combination of input magnetogram and PFSS source surface height.Conclusions.The OSPREI model is moderately sensitive to the input photospheric and coronal conditions. Based on where the source region of the CME is located on the Sun, there may be best practices when selecting an input magnetogram to use.

Comparison of Exact and Approximate MHD Slow Body Mode Solutions in Photospheric Waveguides

In this study, we explore the possibility of simplifying the modeling of magnetohydrodynamic slow body modes observed in photospheric magnetic structures such as the umbrae of sunspots and pores. The simplifying approach assumes that the variation of the eigenvalues of slow body waves can be derived by imposing that the longitudinal component of velocity with respect to the tube axis is zero at the boundary of the magnetic flux tube, which is in good agreement with observations. To justify our approach, we compare the results of our simplified model for slow body modes in cylindrical flux tubes with the model prediction obtained by imposing the continuity of the radial component of the velocity and total pressure at the boundary of the flux tube. Our results show that, to a high accuracy (less than 1% for the considered model), the conditions of continuity of the component of transversal velocity and pressure at the boundary can be neglected when modeling slow body modes under photospheric conditions.

Quasimodes in the cusp continuum in nonuniform magnetic flux tubes

Context. The study of magnetohydrodynamic (MHD) waves is important both for understanding heating in the solar atmosphere (and in particular the corona) and for solar atmospheric seismology. The analytical investigation of wave mode properties in a cylinder is of particular interest in this domain because many atmospheric structures can be modeled as such in a first approximation. Aims. The aim of this study is to use linearized ideal MHD to investigate quasimodes (global modes that are damped through resonant absorption) with a frequency in the cusp continuum, in a straight cylinder with a circular base and an inhomogeneous layer at its boundary that separates two homogeneous plasma regions inside and outside. We are particularly interested in the damping of these modes, and therefore try to determine their frequency as a function of background parameters. Methods. After linearizing the ideal MHD equations, we found solutions to the second-order differential equation for the perturbed total pressure in the inhomogeneous layer in the form of (1) Frobenius series around the regular singular points that are the Alfvén and cusp resonant positions, and (2) power series around regular points. By connecting these solutions appropriately through the inhomogeneous layer and with the solutions of the homogeneous regions inside and outside the cylinder, we derive a dispersion relation for the frequency of the eigenmodes of the system. Results. From the dispersion relation, it is also possible to find the frequency of quasimodes, even though they are not eigenmodes. As an example, we find the frequency of the slow surface sausage quasimode as a function of the width of the inhomogeneous layer for values of the longitudinal wavenumber relevant for photospheric conditions. The results closely match findings by other authors who studied the resistive slow surface sausage eigenmode. We also discuss the perturbation profiles of the quasimode and the eigenfunctions of continuum modes.

Stability of solar atmospheric structures harboring standing slow waves

Context.In the context of the solar coronal heating problem, one possible explanation for the high coronal temperature is the release of energy by magnetohydrodynamic (MHD) waves. The energy transfer is believed to be possible, among others, by the development of the Kelvin-Helmholtz instability (KHI) in coronal loops.Aims.Our aim is to determine if standing slow waves in solar atmospheric structures such as coronal loops, and also prominence threads, sunspots, and pores, can trigger the KHI due to the oscillating shear flow at the structure’s boundary.Methods.We used linearized nonstationary MHD to work out an analytical model in a cartesian reference frame. The model describes a compressible plasma near a discontinuous interface separating two regions of homogeneous plasma, each harboring an oscillating velocity field with a constant amplitude which is parallel to the background magnetic field and aligned with the interface. The obtained analytical results were then used to determine the stability of said interface, both in coronal and photospheric conditions.Results.We find that the stability of the interface is determined by a Mathieu equation. In function of the parameters of this equation, the interface can either be stable or unstable. For coronal as well as photospheric conditions, we find that the interface is stable with respect to the KHI. Theoretically, it can, however, be unstable with respect to a parametric resonance instability, although it seems physically unlikely. We conclude that, in this simplified setup, a standing slow wave does not trigger the KHI without the involvement of additional physical processes.

Magnetohydrodynamic Wave Modes of Solar Magnetic Flux Tubes with an Elliptical Cross Section

Abstract The purpose of this paper is to study the behavior of magnetohydrodynamic (MHD) wave modes that propagate in compressible magnetic flux tubes with an elliptical cross section embedded in a magnetic environment. The dispersion relation that describes the behavior of MHD wave modes permitted in an elliptical magnetic flux tube is solved numerically. Distortion of the spatial structure of the purely real eigenmodes from the well-known circular flux tube model has been considered. It has been studied under both photospheric and coronal conditions. It has been shown that (i) solutions in the form of even Mathieu functions are more sensitive to the value of eccentricity than solutions with the form of odd Mathieu functions; (ii) if the ellipticity of the cross section of the magnetic flux tube increases, a sausage mode (m = 0) cannot be easily identified; (iii) even solutions that correspond to the fluting mode (m = 3) can be misinterpreted as a kink mode (m = 1) due to their similarities. In contrast to the fluting modes that are polarized along the major axis and strongly depend on the ellipticity of the magnetic flux tube, the kink and sausage surface modes are practically unaffected by ellipticity. Several examples of the spatial structure of the eigenmodes permitted in the pores and sunspots have been visualized. The solutions obtained in the approximation of cylindrical symmetry are in agreement with previous studies.

Effect of Electrical Resistivity on the Damping of Slow Sausage Modes

Abstract Recent observations of slow surface sausage modes in a photospheric magnetic pore have been shown to be heavily damped. Numerical calculations have shown that electrical resistivity plays a significant role in this damping process. The aim of the present paper is to make an independent analytical derivation that would confirm the importance of electrical resistivity in the damping of these modes. An analytical dispersion relation in the framework of resistive magnetohydrodynamics is derived for sausage modes in a straight cylinder with a circular cross section and a discontinuous boundary. The effect of electrical resistivity on the damping of slow sausage modes in photospheric pore conditions is then studied, by solving the obtained dispersion relation numerically. The obtained results agree with those from the numerical calculations.

The Effect of Weak Magnetic Twist on Resonant Absorption of Slow Sausage Waves in Magnetic Flux Tubes

Abstract Observations show that twisted magnetic flux tubes are present throughout the Sun’s atmosphere. The main aim of this work is to obtain the damping rate of sausage modes in the presence of weak magnetic twist. Using the connection formulae obtained by Sakurai et al., we investigate resonant absorption of the sausage modes in the slow continuum under photospheric conditions. We derive the dispersion relation and solve it numerically, and consequently obtain the frequencies and damping rates of the slow surface sausage modes. We conclude that the magnetic twist can result in strong damping in comparison with the untwisted case.

The Wootton center for astrophysical plasma properties: First results for helium

AbstractThe Wootton Center for Astrophysical Plasma Properties (WCAPP) is a new center focusing on the spectroscopic properties of stars and accretion disks using “at-parameter” experiments. Currently, these experiments use the X-ray output of the Z machine at Sandia National Laboratories — the largest X-ray source in the world — to heat plasmas to the same conditions (temperature, density, and radiation environment) as those observed in astronomical objects. The experiments include measuring (1) density-dependent opacities of iron-peak elements at solar interior conditions, (2) spectral lines of low-Z elements at white dwarf photospheric conditions, (3) atomic population kinetics of neon in a radiation-dominated environment, and (4) resonant Auger destruction (RAD) of silicon at conditions found in accretion disks around supermassive black holes. In particular, we report on recent results of our experiments involving helium at white dwarf photospheric conditions. We present results showing disagreement between inferred electron densities using the Hβ line and the He I 5876 Å line, most likely indicating incompleteness in our modeling of this helium line.

Helical and rotating plasma structures in the solar atmosphere

Aims. We model helical or rotating signatures in the solar atmosphere to further understand the efficiency of the equilibrium conditions, for example magnetic twist, rotation, plasma-β, and viscous effects on the life of solar helical structures. Methods. Solar rotating structures, such as tornadoes, spirals, and whirls are modelled by considering a rotating and twisted magnetic cylinder residing in an environment with a straight magnetic field. A macroscopic approach proves adequate for working on the phase speed and damping of waves in solar atmospheric structures; as such, the magnetohydrodynamic theory is implemented. In this way the second order thin flux tube approximation is used for obtaining expressions for the frequency, deceleration, and damping of torsional waves in solar plasma structures in the presence of equilibrium rotation, magnetic twist, viscosity, and gravity. Results. The dependency of the dissipation effects regarding the torsional wave in the linear regime is highlighted. The dispersion relation for axisymmetric oscillations propagating along a rotating and twisted solar cylindrical plasma structure in the presence of plasma viscosity and gravity is obtained. In this way we present explicit expressions for the oscillation and damping of torsional waves. The explicit expressions shed light on the influence of the equilibrium and environmental conditions on the speed deceleration, frequency, and damping of the torsional wave that exists in various layers of the solar atmosphere. The dispersion of the torsional wave is highly controlled by the combined effects of the rotation and the plasma-β, where when both are zero, the magnetic twist becomes significant only when the plasma resistivity comes into play. Regarding damping, the dominant actor for coronal conditions is the magnetic twist. However, since the damping time is highly dependent on the plasma-β, for photospheric conditions, the rotation becomes very significant. The damping of torsional waves is inversely proportional to the elevation of the rotating structure. This means that if the torsional wave survives through the photosphere and chromosphere, the chance for it to extend through the corona and solar wind is very high by gradually dissipating energy, which gives more opportunity for it to be observed.

Resonant Absorption of Surface Sausage and Surface Kink Modes under Photospheric Conditions

Abstract We study the effect of resonant absorption of surface sausage and surface kink modes under photospheric conditions where the slow surface sausage modes undergo resonant damping in the slow continuum and the surface kink modes in the slow and Alfvén continua at the transitional layers. We use recently derived analytical formulas to obtain the damping rate (time). By considering linear density and linear pressure profiles for the transitional layers, we show that resonant absorption in the slow continuum could be an efficient mechanism for the wave damping of the slow surface sausage and slow surface kink modes while the damping rate of the slow surface kink mode in the Alfvén continuum is weak. It is also found that the resonant damping of the fast surface kink mode is much stronger than that of the slow surface kink mode, showing a similar efficiency as under coronal conditions. It is worth noting that the slow body sausage and kink modes can also resonantly damp in the slow continuum for those linear profiles.

Influence of Photospheric Magnetic Conditions on the Catastrophic Behaviors of Flux Ropes in Solar Active Regions

Since only the magnetic conditions at the photosphere can be routinely observed in current observations, it is of great significance to find out the influences of photospheric magnetic conditions on solar eruptive activities. Previous studies about catastrophe indicated that the magnetic system consisting of a flux rope in a partially open bipolar field is subject to catastrophe, but not if the bipolar field is completely closed under the same specified photospheric conditions. In order to investigate the influence of the photospheric magnetic conditions on the catastrophic behavior of this system, we expand upon the 2.5 dimensional ideal magnetohydrodynamic (MHD) model in Cartesian coordinates to simulate the evolution of the equilibrium states of the system under different photospheric flux distributions. Our simulation results reveal that a catastrophe occurs only when the photospheric flux is not concentrated too much toward the polarity inversion line and the source regions of the bipolar field are not too weak; otherwise no catastrophe occurs. As a result, under certain photospheric conditions, a catastrophe could take place in a completely closed configuration whereas it ceases to exist in a partially open configuration. This indicates that whether the background field is completely closed or partially open is not the only necessary condition for the existence of catastrophe, and that the photospheric conditions also play a crucial role in the catastrophic behavior of the flux rope system.

LABORATORY MEASUREMENTS OF WHITE DWARF PHOTOSPHERIC SPECTRAL LINES: Hβ

We spectroscopically measure multiple hydrogen Balmer line profiles from laboratory plasmas to investigate the theoretical line profiles used in white dwarf atmosphere models. X-ray radiation produced at the Z Pulsed Power Facility at Sandia National Laboratories initiates plasma formation in a hydrogen-filled gas cell, replicating white dwarf photospheric conditions. Here we present time-resolved measurements of H$\beta$ and fit this line using different theoretical line profiles to diagnose electron density, $n_{\rm e}$, and $n=2$ level population, $n_2$. Aided by synthetic tests, we characterize the validity of our diagnostic method for this experimental platform. During a single experiment, we infer a continuous range of electron densities increasing from $n_{\rm e}\sim4$ to $\sim30\times10^{16}\,$cm$^{-3}$ throughout a 120-ns evolution of our plasma. Also, we observe $n_2$ to be initially elevated with respect to local thermodynamic equilibrium (LTE); it then equilibrates within $\sim55\,$ns to become consistent with LTE. This supports our electron-temperature determination of $T_{\rm e}\sim1.3\,$eV ($\sim15,000\,$K) after this time. At $n_{\rm e}\gtrsim10^{17}\,$cm$^{-3}$, we find that computer-simulation-based line-profile calculations provide better fits (lower reduced $\chi^2$) than the line profiles currently used in the white dwarf astronomy community. The inferred conditions, however, are in good quantitative agreement. This work establishes an experimental foundation for the future investigation of relative shapes and strengths between different hydrogen Balmer lines.

An experimental platform for creating white dwarf photospheres in the laboratory: Preliminary results

We present the current status of the White Dwarf Photosphere Experiment at the Z Pulsed Power Facility at Sandia National Laboratories. This experiment has evolved into a unique platform for simultaneously measuring emission, absorption, and back-lighter continua spectra of plasmas with white dwarf (WD) photospheric compositions and conditions (Te∼1 eV, ne∼1016–1018e/cm3); our current experiments involve line profile measurements of hydrogen—corresponding to the most common surface composition in white dwarf stars, with future experiments planned for helium, carbon, and oxygen. These profiles will test line broadening theories used in white dwarf model atmospheres to infer the fundamental parameters (e.g., effective temperature and mass) of thousands of WDs. This experiment uses the large amount of x-rays generated from a z-pinch dynamic hohlraum to radiatively drive plasma formation in a gas cell. We reach significantly higher densities than the landmark study of Wiese et al. (1972), thereby putting competing line broadening theories to the test in a regime where their predictions strongly diverge. The simultaneous measurement of emission, absorption, and back-lighter continua in macroscopic plasmas represents a significant advance relative to hydrogen line profile experiments of the past.

TEMPORAL EVOLUTION OF THE SCATTERING POLARIZATION OF THE Ca II IR TRIPLET IN HYDRODYNAMICAL MODELS OF THE SOLAR CHROMOSPHERE

Velocity gradients in a stellar atmospheric plasma have an impact on the anisotropy of the radiation field that illuminates each point within the medium, and this may in principle influence the scattering line polarization that results from the induced atomic level polarization. Here we analyze the emergent linear polarization profiles of the Ca II infrared triplet after solving the radiative transfer problem of scattering polarization in time-dependent hydrodynamical models of the solar chromosphere, taking into account the impact of the plasma macroscopic velocity on the atomic level polarization. We discuss the influence that the velocity and temperature shocks in the considered chromospheric models have on the temporal evolution of the scattering polarization signals of the Ca II infrared lines, as well as on the temporally averaged profiles. Our results indicate that the increase of the linear polarization amplitudes caused by macroscopic velocity gradients may be significant in realistic situations. We also study the effect of the integration time, the microturbulent velocity and the photospheric dynamical conditions, and discuss the feasibility of observing with large-aperture telescopes the temporal variation of the scattering polarization profiles. Finally, we explore the possibility of using the differential Hanle effect in the IR triplet of Ca II with the intention of avoiding the characterization of the zero-field polarization to infer magnetic fields in dynamic situations.

An experimental platform for creating white dwarf photospheres in the laboratory

We present an experimental platform for measuring hydrogen Balmer emission and absorption line profiles for plasmas with white dwarf (WD) photospheric conditions (Te ∼1 eV, ne ∼1017 cm−3). These profiles will be used to benchmark WD atmosphere models, which, used with the spectroscopic method, are responsible for determining fundamental parameters (e.g., effective temperature, mass) for tens of thousands of WDs. Our experiment, performed at the Z Pulsed Power Facility at Sandia National Laboratories, uses the large amount of X-rays generated from a z-pinch dynamic hohlraum to drive plasma formation in a gas cell. The platform is unique compared to past hydrogen line profile experiments in that the plasma is radiation-driven. This decouples the heating source from the plasma to be studied in the sense that the radiation temperature causing the photoionization is independent of the initial conditions of the gas. For the first time we measure hydrogen Balmer lines in absorption at these conditions in the laboratory for the purpose of benchmarking Stark-broadened line shapes. The platform can be used to study other plasma species and to explore non-LTE, time-dependent collisional-radiative atomic kinetics.

The Wings of Ca ii H and K as Photospheric Diagnostics and the Reliability of One-Dimensional Photosphere Modeling

The extended wings of the Ca ii H and K lines provide excellent diagnostics of the temperature stratification of the photosphere of the Sun and of other cool stars, thanks to their LTE opacities and source functions and their large span in formation height. The aim of this study is to calibrate the usage of the H and K wings in a one-dimensional interpretation of spatially averaged spectra and in deriving per-pixel stratifications from resolved spectra. I use multi-dimensional simulations of solar convection to synthesize the H and K wings, derive one-dimensional models from these wings as if they were observed, and compare the resulting models to the actual simulation input. I find that spatially averaged models constructed from the synthesized wings generally match the simulation averages well, except for the deepest layers of the photosphere where large thermal inhomogeneities and Planck-function non-linearity gives large errors. The larger the inhomogeneity, the larger the error. The presence of strong network fields increases such inhomogeneity. For quiet photospheric conditions the temperature excesses reach about 200 K. One-dimensional stratification fits of discrete structures such as granulation and small-scale magnetic concentrations give satisfactory results with errors that are primarily due to steep temperature gradients and abrupt changes of temperature with depth. I conclude that stratification modeling using the H and K wings is a useful technique for the interpretation of solar high-resolution observations.

ANALYSIS OF A FRAGMENTING SUNSPOT USINGHINODEOBSERVATIONS

We employ high resolution filtergrams and polarimetric measurements from Hinode to follow the evolution of a sunspot for eight days starting on June 28, 2007. The imaging data were corrected for intensity gradients, projection effects, and instrumental stray light prior to the analysis. The observations show the formation of a light bridge at one corner of the sunspot by a slow intrusion of neighbouring penumbral filaments. This divided the umbra into two individual umbral cores. During the light bridge formation, there was a steep increase in its intensity from 0.28 to 0.7 I_QS in nearly 4 hr, followed by a gradual increase to quiet Sun (QS) values in 13 hr. This increase in intensity was accompanied by a large reduction in the field strength from 1800 G to 300 G. The smaller umbral core gradually broke away from the parent sunspot nearly 2 days after the formation of the light bridge rendering the parent spot without a penumbra at the location of fragmentation. The penumbra in the fragment disappeared first within 34 hr, followed by the fragment whose area decayed exponentially with a time constant of 22 hr. The depleted penumbra in the parent sunspot regenerated when the inclination of the magnetic field at the penumbra-QS boundary became within 40 deg. from being completely horizontal and this occurred near the end of the fragment's lifetime. After the disappearance of the fragment, another light bridge formed in the parent which had similar properties as the fragmenting one, but did not divide the sunspot. The significant weakening in field strength in the light bridge along with the presence of granulation is suggestive of strong convection in the sunspot which might have triggered the expulsion and fragmentation of the smaller spot. Although the presence of QS photospheric conditions in sunspot umbrae could be a necessary condition for fragmentation, it is not a sufficient one.