Helioseismology of Sunspots: Confronting Observations with Three-Dimensional MHD Simulations of Wave Propagation

The propagation of solar waves through the sunspot of AR?9787 is observed by using temporal cross-correlations of SOHO/MDI Dopplergrams. We then use three-dimensional MHD numerical simulations to compute the propagation of wave packets through self-similar magnetohydrostatic sunspot models. The simulations are set up in such a way as to allow a comparison with observed cross-covariances (except in the immediate vicinity of the sunspot). We find that the simulation and the f-mode observations are in good agreement when the model sunspot has a peak field strength of 3 kG at the photosphere and less so for lower field strengths. Constraining the sunspot model with helioseismology is only possible because the direct effect of the magnetic field on the waves has been fully taken into account. Our work shows that the full-waveform modeling of sunspots is feasible.     

Intensity and Magnetic Field Distribution of Sunspots

We study the relationship between the brightness (I) and magnetic field (B) distributions of sunspots using 272 samples observed at the San Fernando Observatory and the National Solar Observatory, Kitt Peak, whose characteristics varied widely. We find that the I – B relationship has a quadratic form for the spots with magnetic field less than about 2000 G. The slope of the linear part of the I – B curve varies by about a factor of three for different types of spots. In general the slope increases as the spot approaches disk center. The I – B slope does not have a clear dependency on the spot size but the lower limit appears to increase as a function of the ratio of umbra and penumbra area. The I – B slope changes as a function of age of the sunspots. We discuss various sunspot models using these results.     

Sunspots and large-scale magnetic fields at various heights

Height variation of the magnetic field structure over groups of sunspots for heights ranging from the photosphere to the source surface (R = 2.5 Ro, where Ro is the radius of the Sun) is examined. For all heights, starting from the photospheric level, groups of sunspot are shown as being independent of long-lived boundaries of large-scale structures rotating with a period shorter than the Carrington period. At heights of 1–1.5 Ro, there is a clear relation between sunspot groups and boundaries separating the head and tail sunspots in the groups (the Hale boundaries). The rotation periods of these structures are close to the Carrington period, their lifespan being less than three to five rotations. The maximal intensity of the solar magnetic field drops by two orders when height increases from H = 1 to H = 1.1 Ro. Further decrease in intensity proceeds gradually (dropping by one order from H = 1.1 to 2.5 Ro). The results obtained can be considered as evidence that large-scale magnetic field structures and long-lived boundries between them (the lines dividing polarities of the magnetic field or zero lines) all exist irrespective of sunspot fields being generated by other sources than sunspots. At the photospheric level, active regions fields are superimposed on these structures.     

Turbulent effects of sunspot magnetic field reconstruction

We study the effects of two-dimensional turbulence generated in sunspot umbra due to strong magnetic fields and Alfven oscillations excited in sunspots due to relatively weak magnetic fields on the evolution of sunspots. Two phases of sunspot magnetic field decaying are shown to exist. The initial rapid phase of magnetic field dissipation is due to two-dimensional turbulence. The subsequent slow phase of magnetic field decaying is associated with Alfven oscillations. Our results correspond to observed data that provide evidence for two types of sunspot evolution. The effect of macroscopic diamagnetic expulsion of magnetic field from the convective zone or photosphere toward sunspots is essential in supporting the long-term stability and equilibrium of vertical magnetic flux tubes in sunspots.     

Horizontal Flows in the Photosphere and Subphotosphere of Two Active Regions

We compare horizontal flow fields in the photosphere and in the subphotosphere (a layer 0.5 Mm below the photosphere) in two solar active regions: AR?11084 and AR?11158. AR?11084 is a mature, simple active region without significant flaring activity, and AR?11158 is a multipolar, complex active region with magnetic flux emerging during the period studied. Flows in the photosphere are derived by applying the Differential Affine Velocity Estimator for Vector Magnetograms (DAVE4VM) on HMI-observed vector magnetic fields, and the subphotospheric flows are inferred by time–distance helioseismology using HMI-observed Dopplergrams. Similar flow patterns are found for both layers for AR?11084: inward flows in the sunspot umbra and outward flows surrounding the sunspot. The boundary between the inward and outward flows, which is slightly different in the photosphere and the subphotosphere, is within the sunspot penumbra. The area having inward flows in the subphotosphere is larger than that in the photosphere. For AR?11158, flows in these two layers show great similarities in some areas and significant differences in other areas. Both layers exhibit consistent outward flows in the areas surrounding sunspots. On the other hand, most well-documented flux-emergence-related flow features seen in the photosphere do not have counterparts in the subphotosphere. This implies that the horizontal flows caused by flux emergence do not extend deeply into the subsurface.     

Time – Distance Modelling in a Simulated Sunspot Atmosphere

In time?–?distance helioseismology, wave travel times are measured from the cross-correlation between Doppler velocities recorded at any two locations on the solar surface. However, one of the main uncertainties associated with such measurements is how to interpret observations made in regions of strong magnetic field. Isolating the effects of the magnetic field from thermal or sound-speed perturbations has proved to be quite complex and has yet to yield reliable results when extracting travel times from the cross-correlation function. One possible way to decouple these effects is by using a 3D sunspot model based on observed surface magnetic-field profiles, with a surrounding stratified, quiet-Sun atmosphere to model the magneto-acoustic ray propagation, and analyse the resulting ray travel-time perturbations that will directly account for wave-speed variations produced by the magnetic field. These artificial travel-time perturbation profiles provide us with several related but distinct observations: i) that strong surface magnetic fields have a dual effect on helioseismic rays?–?increasing their skip distance while at the same time speeding them up considerably compared to their quiet-Sun counterparts, ii) there is a clear and significant frequency dependence of both skip-distance and travel-time perturbations across the simulated sunspot radius, iii) the negative sign and magnitude of these perturbations appears to be directly related to the sunspot magnetic-field strength and inclination, iv) by “switching off” the magnetic field inside the sunspot, we are able to completely isolate the thermal component of the travel-time perturbations observed, which is seen to be both opposite in sign and much smaller in magnitude than those measured when the magnetic field is present. These results tend to suggest that purely thermal perturbations are unlikely to be the main effect seen in travel times through sunspots, and that strong, near-surface magnetic fields may be directly and significantly altering the magnitude and lateral extent of sound-speed inversions of sunspots made by time?–?distance helioseismology.     

Bimodal Distribution of Magnetic Fields and Areas of Sunspots

We applied automatic identification of sunspot umbrae and penumbrae to daily observations from the Helioseismic Magnetic Imager (HMI) on board the Solar Dynamics Observatory (SDO) to study their magnetic flux density (B) and area (A). The results confirm an already known logarithmic relationship between the area of sunspots and their maximum flux density. In addition, we find that the relation between average magnetic flux density ( $B_{\rm avg}$ ) and sunspot area shows a bimodal distribution: for small sunspots and pores (A≤20 millionth of solar hemisphere, MSH), $B_{\rm avg} \approx 800~\mbox{G}$ (gauss), and for large sunspots (A≥100 MSH), $B_{\rm avg}$ is about 600 G. For intermediate sunspots, average flux density linearly decreases from about 800 G to 600 G. A similar bimodal distribution was found in several other integral parameters of sunspots. We show that this bimodality can be related to different stages of sunspot penumbra formation and can be explained by the difference in average inclination of magnetic fields at the periphery of small and large sunspots.     

The cooling of a sunspot

A mechanism is proposed to explain the cooling of a sunspot in terms of the detailed interactions between the magnetic field and the convective motions. The mechanism provides that an axially symmetric concentration of magnetic field deforms the normal supergranule cell pattern below the sunspot into a radial outflow of plasma over a region of diameter 60 Mm.The flow occurs at depths where the magnetic and kinetic energy densities are approximately equal ( 5 Mm) and is described in terms of a Carnot refrigeration cycle. Application of the hydromagnetic equations to a very simple model shows that, because the magnetic field concentration causes the outflow, the field will itself decay in a time short compared with the lifetime of a spot. However, a slightly more sophisticated model does suggest conditions under which this decay is considerably reduced.Observations of the outward drift of magnetic knots around sunspots and of supergranule-type surface motions extending radially outwards from the penumbra of a spot to the nearest faculae are discussed in relation to the mechanism.     

Axial tilt angles of sunspot groups

Digitized Mount Wilson sunspot data from 1917 to 1985 are analyzed to examine tilt angles determined from the area-weighted positions of leading and following sunspots. These spot group tilt angles are examined in relation to other group characteristics to give information which may relate to the formation and evolution of sunspot groups and the magnetic connection of groups to subsurface magnetic flux tubes. The average tilt angle of all 24816 (multiple-spot) group observations in this study is found to be + 4.2 ± 0.2 deg, where the positive sign signifies that the leading spots lie equatorward of the following spots. Sunspot group areas are significantly larger on average for groups nearer the average tilt angle, which is similar to a result found earlier for active region plages. Average tilt angles are found to be larger at higher latitudes, confirming earlier results. There is a strong negative correlation between average daily latitudinal motion (plus to poles) and group tilt angle. That is, for groups within about 40 deg of the average tilt angle, smaller tilt angles are associated with more positive (poleward) daily drift. Groups nearest the average tilt angle rotate the fastest, on average, the amplitude differences being between about +0.1 and – 0.1 deg day^–1 for groups near and far from the average tilt angle, respectively. Groups with tilt angles near the average show a negative daily separation change between leading and following spots of close to 4 Mm day^–1 on average. Groups on either side of the average tilt angle show spot separations that are on average more positive. A similar effect is not seen for the daily variations of group areas. These results are discussed in relation to analogous recent results for active region magnetic fields. More evidence is found for a qualitative difference between the magnetic fields of sunspots and of plages, relating, perhaps, to a difference in subsurface connection of the field lines or to different physical mechanisms that may play a role for fields of different field strengths.Operated by the Association of Universities for Research in Astronomy, Inc., under Cooperative Agreement with the National Science Foundation.     

Microwave emission from a sunspot

From the gyroresonance brightness temperature spectrum of a sunspot, one can determine the magnetic field strength by using the property that microwave brightness is limited above a frequency given by an integer-multiple of the gyrofrequency. In this paper, we use this idea to find the radial distribution of magnetic field at the coronal base of a sunspot in the active region, NOAA 4741. The gyroresonance brightness temperature spectra of this sunspot are obtained from multi-frequency interferometric observations made at the Owens Valley Radio Observatory at 24 frequencies in the range of 4.0–12.4 GHz with spatial resolution 2.2″–6.8″. The main results of present study are summarized as follows: first, by comparison of the coronal magnetic flux deduced from our microwave observation with the photospheric magnetic flux measured by KPNO magnetograms, we show that theo-mode emission must arise predominantly from the second harmonic of the gyrofrequency, while thex-mode arises from the third harmonic. Second, the radial distribution of magnetic fieldsB(r) at the coronal base of this spot (say, 2000–4000 km above the photosphere) can be adequately fitted by $$B(r) = 1420(1 \pm 0.080)\exp \left[ { - \left( {\frac{r}{{11.05''(1 \pm 0.014)}}} \right)^2 } \right]G,$$ wherer is the radial distance from the spot center at coronal base. Third, it is found that coronal magnetic fields originate mostly from the photospheric umbral region. Fourth, although the derived vertical variation of magnetic fields can be approximated roughly by a dipole model with dipole moment 1.6 × 10³⁰ erg G^?1 buried at 11000 km below the photosphere, the radial field distribution at coronal heights is found to be more confined than predicted by the dipole model.     

Sunspot models with twisted magnetic field

The return-flux sunspot model is generalized by including azimuthal magnetic field B _Φ . The basic equation is obtained and numerical solutions are compared with the analogous solutions for the Schlüter-Temesvary sunspot theory for two cases: B _Φ ～ B _r and B _Φ ～ rB _r . The solutions demonstrate that the twisting of the sunspot magnetic field decreases with height. Our models confirm Yun's early statement: the azimuthal field only slightly influences sunspot structure.     

Naked sunspots

Naked sunspots are spots seen in Hα to be devoid of associated plage. In magnetograms and K-line little if any opposite polarity field is found, and in soft X-ray images a blank appears in the region of the spot. In almost all cases studied in which naked spots resulted the spot groups had emerged in unipolar regions of the same polarity as the naked spot. At least half of the naked spots are associated with coronal holes. The naked spots are long-lived and show rotation rates close to the Newton-Nunn curve. Most of the naked spots had bright rims in Hα, and the one spot observed to disappear left no trace in the background magnetic field. These spots may be a means by which separation of p from f magnetic polarity occurs.     

Penumbra filamentary structure as a purely higgs-field-induced phenomenon

Regarding areal-valued part of the U(1)-symmetric Higgs field, satisfactorily describing sunspots within an Abelian Higgs model, as being ofkink-type one demonstrates that the penumbra of a cylindrically-symmetric sunspot consists ofradial filaments whose number is twice of that representing magnetic flux quanta carried by the spot.     

Parameters of the Magnetic Flux inside Coronal Holes

The parameters of the magnetic flux distribution inside low-latitude coronal holes (CHs) were analyzed. A statistical study of 44 CHs based on Solar and Heliospheric Observatory (SOHO)/MDI full disk magnetograms and SOHO/EIT 284?Å images showed that the density of the net magnetic flux, B _net, does not correlate with the associated solar wind speeds, V _x . Both the area and net flux of CHs correlate with the solar wind speed and the corresponding spatial Pearson correlation coefficients are 0.75 and 0.71, respectively. A possible explanation for the low correlation between B _net and V _x is proposed. The observed non-correlation might be rooted in the structural complexity of the magnetic field. As a measure of the complexity of the magnetic field, the filling factor, f(r), was calculated as a function of spatial scales. In CHs, f(r) was found to be nearly constant at scales above 2 Mm, which indicates a monofractal structural organization and smooth temporal evolution. The magnitude of the filling factor is 0.04 from the Hinode SOT/SP data and 0.07 from the MDI/HR data. The Hinode data show that at scales smaller than 2 Mm, the filling factor decreases rapidly, which means a multifractal structure and highly intermittent, burst-like energy release regime. The absence of the necessary complexity in CH magnetic fields at scales above 2 Mm seems to be the most plausible reason why the net magnetic flux density does not seem to be related to the solar wind speed: the energy release dynamics, needed for solar wind acceleration, appears to occur at small scales below 1 Mm.     

Complex structure of the magnetic field of the CP star HD32633

The method of “virtual magnetic charges” is used to analyze the structure of the magnetic field of the CP star HD32633. The phase relation of its magnetic field differs strongly from a sine wave. The structure of the star’s field can be described fairly well by two dipoles located in the opposite regions of the star near its rotation equator. Each of these dipoles produces two pairs of magnetic spots of opposite polarity similar to sunspots. The dipoles are located at a distance of Δa=0.6 R from the center, where R is the radius of the star. The field strength at the poles is equal to ±42 and ±19 kG.     

Asymmetric flux loops in active regions,I

We investigate asymmetries of bipolar sunspot groups. We find that the magnetic field distribution of simple bipolar sunspot groups is significantly asymmetrical: the polarity inversion line is usually nearer to the main following polarity spot than to the main preceding one. This asymmetry grows with the age of the sunspot group. We suggest that this asymmetry has a causal link with two long-established asymmetries- the one in the proper motions of young sunspots, the other in the relative stability of p and f spots.In our view, these asymmetries together indicate that emerging flux loops, making sunspot groups, are not symmetrical but tilted eastward. The tilt is presumably caused by drag forces due to radial differential rotation in subphotospheric layers. In this paper we present observational indications supporting this hypothesis.     

Constructing Semi-Empirical Sunspot Models for Helioseismology

One goal of helioseismology is to determine the subsurface structure of sunspots. In order to do so, it is important to understand first the near-surface effects of sunspots on solar waves, which are dominant. Here we construct simplified, cylindrically-symmetric sunspot models that are designed to capture the magnetic and thermodynamics effects coming from about 500 km below the quiet-Sun τ ₅₀₀₀=1 level to the lower chromosphere. We use a combination of existing semi-empirical models of sunspot thermodynamic structure (density, temperature, pressure): the umbral model of Maltby et al. (1986, Astrophys. J. 306, 284) and the penumbral model of Ding and Fang (1989, Astron. Astrophys. 225, 204). The OPAL equation-of-state tables are used to derive the sound-speed profile. We smoothly merge the near-surface properties to the quiet-Sun values about 1 Mm below the surface. The umbral and penumbral radii are free parameters. The magnetic field is added to the thermodynamic structure, without requiring magnetostatic equilibrium. The vertical component of the magnetic field is assumed to have a Gaussian horizontal profile, with a maximum surface field strength fixed by surface observations. The full magnetic-field vector is solenoidal and determined by the on-axis vertical field, which, at the surface, is chosen such that the field inclination is 45° at the umbral – penumbral boundary. We construct a particular sunspot model based on SOHO/MDI observations of the sunspot in active region NOAA 9787. The helioseismic signature of the model sunspot is studied using numerical simulations of the propagation of f, p ₁, and p ₂ wave packets. These simulations are compared against cross-covariances of the observed wave field. We find that the sunspot model gives a helioseismic signature that is similar to the observations.     

Thin magnetic hydrogen atmospheres and the neutron star RX J1856.5–3754

RX J1856.5–3754 is one of the brightest nearby isolated neutron stars, and considerable observational resources have been devoted to it. However, current models are unable to satisfactorily explain the data. We show that our latest models of a thin, magnetic, partially ionized hydrogen atmosphere on top of a condensed surface can fit the entire spectrum, from X-rays to optical, of RX J1856.5–3754, within the uncertainties. In our simplest model, the best-fit parameters are an interstellar column density N _H≈1×10²⁰ cm^?2 and an emitting area with R ^∞≈17 km (assuming a distance to RX J1856.5–3754 of 140 pc), temperature T ^∞≈4.3×10⁵ K, gravitational redshift z _g ～0.22, atmospheric hydrogen column y _H≈1 g cm^?2, and magnetic field B≈(3–4)×10¹² G; the values for the temperature and magnetic field indicate an effective average over the surface.     

The velocity field associated with the birth of sunspots

A velocity field is found to occur prior to the birth of sunspots or during the rapidly developing phase of new spots. Fraunhofer lines are always shifted redwards in the observed active regions which are situated at various distances from the disk center. The velocity amplitude derived from Na i D1-line, 5895.940, amounts to, at maximum, 1.5 km s^–1 which is always a little larger than that derived from the weaker line, NI i 5892.883. The velocity field disappears when the spot ceases to grow. The lifetime of the velocity is, at least, 1 hr. The velocity field is interpreted in terms of the continuous downward flow in the process of formation of sunspots.Bray and Loughhead (1964) regard the disturbance in the granulation pattern accompanying the birth and growth of sunspot pores as an evidence of the existence of rising loops of magnetic flux. In view of the similarity of the phase of development of active regions and the lifetime in the observations by Bray and Loughhead and by us, we suggest that the velocity field may be a spectroscopic feature accompanying the rising magnetic loops in the photosphere and the chromosphere. We briefly discuss the observed mode of penetration of the magnetic flux to the solar surface before and after the appearance of AFS's.