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.     

Similarity considerations and conservation laws for magneto-static atmospheres

The equations of magnetohydrostatic equilibria for a plasma in a gravitational field are investigated analytically. For equilibria with one ignorable spatial coordinate, the equations reduce to a single nonlinear elliptic equation for the magnetic potential A. Similarity solutions of the elliptic equation are obtained for the case of an isothermal atmosphere in a uniform gravitational field. The solutions are obtained from a consideration of the invariance group of the elliptic equation. The importance of symmetries of the elliptic equation also appears in the determination of conservation laws. It turns out that the elliptic equation can be written as a variational principle, and the symmetries of the variational functional lead (via Noether's theorem) to conservation laws for the equation. As an example of the application of the similarity solutions, we construct a model magnetostatic atmosphere in which the current density J is proportional to the cube of the magnetic potential, and falls off exponentially with distance vertical to the base, with an e-folding distance equal to the gravitational scale height. The solutions show the interplay between the gravitational force, the J × B force (B, magnetic field induction) and the gas pressure gradient.     

The polarization of continuum radiation in sunspots

Expressions are derived for the Stokes parameters of light scattered by a layer of free electrons and hydrogen atoms in a sunspot. A physically reasonable sunspot model was found so that the direction of the calculated linear polarization agrees reasonably with observations. The magnitude of the calculated values of the linear polarization agrees generally with values observed in the continuum at 5830 Å. Circular polarization in the continuum also accompanies electron scattering in spot regions; however for commonly accepted values of the longitudinal magnetic field, the predicted circular polarization is much smaller than observed.     

Basic properties of sunspots: equilibrium,stability and long-term eigen oscillations

A number of fundamental questions as regards the physical nature of sunspots are formulated. In order to answer these questions, we apply the model of a round-shaped unipolar sunspot with a lower boundary consisting of cool plasma and with strong magnetic field at the depth of about 4 Mm beneath the photosphere, in accordance with the data of local helioseismology and with certain physically sound arguments (the shallow sunspot model). The magnetic configuration of a sunspot is assumed to be close to the observed one and similar to the magnetic field of a round solenoid of the appropriate size. The transverse (horizontal) and longitudinal (vertical) equilibria of a sunspot were calculated based on the thermodynamic approach and taking into account the magnetic, gravitational, and thermal energy of the spot and the pressure of the environment. The dependence of the magnetic field strength in the sunspot center, B ₀, on the radius of the sunspot umbra a is derived theoretically for the first time in the history of sunspot studies. It shows that the magnetic field strength in small spots is about 700 Gauss (G) and then increases monotonically with a, tending asymptotically to a limit value of about 4000 G. This dependence, B ₀(a) includes, as parameters, the gravity acceleration on the solar surface, the density of gas in the photosphere, and the ratio of the radius of the spot (including penumbra), a _p, to the radius of its umbra a. It is shown that large-scale subsurface flows of gas in the sunspot vicinity, being the consequence but not the cause of sunspot formation, are too weak to contribute significantly to the pressure balance of the sunspot. Stability of the sunspot is provided by cooling of the sunspot plasma and decreasing of its gravitational energy due to the vertical redistribution of the gas density when the geometric Wilson depression of the sunspot is formed. The depth of a depression grows linearly with B ₀, in contrast to the quadratic law for the magnetic energy. Therefore, the range of stable equilibria turns out to be limited: large spots, with radius a larger than some limit value (about 12–18 Mm, depending on the magnetic field configuration), are unstable. It explains the absence of very large spots on the Sun and the appearance of light bridges in big spots that divide the spot into a few parts. The sunspots with B ₀≈2.6÷2.7 kilogauss (kG) and a≈5 Mm are most stable. For these spots, taken as a single magnetic structure, the period of their vertical eigen oscillations is minimal and amounts, according to the model, to 10–12 hours. It corresponds well to the period derived from the study of long-term oscillations of sunspots using SOHO/MDI data.     

The circular polarization of sources of synchrotron radiation

The degree of circular polarizationp _c is calculated for two models of a source of synchrotron radiation:

1. A source with an inhomogeneous magnetic field and isotropic angular distribution of the electrons with respect to the magnetic field;
2. A source with a homogeneous magnetic field and anisotropic angular distribution of the electrons in which the anisotropy of angular distribution substantially increases with the electron energy.

The first model can be used to describe extended radio-sources; and the second, to describe compact radio-sources. For those sources, whose observed polarization properties correspond to the first model, we obtain an integral equation which connects the observed distribution of the sources with the extent of their linear and circular polarization (p _l andp _c ) and the unknown distribution of the sources over the strengthB and the degree of homogeneity ?=(B ₀/B)² of the magnetic field;B ₀ is a homogenous field,B ₀?B. A solution of the integral equation obtained is found for a particular case. This solution makes it possible to determine the distribution of different types of sources over ? if the distribution of these sources in the extent of linear polarization is known. The formulae obtained make it possible to indicate which sources with a known degree of linear polarization should be expected to exhibit highest circular polarization. In the discussion of the first model the question is raised as to the information one can get about the magnetic field by using observations of both linear and circular polarization for a separate source, and for a number of sources. It is shown that the determination of the most probable values ofB and ? in a separate source based on the known values ofp _l andp _c for the source, is possible only if one knows the distribution overB and ? of the sources of the type to which the source in question belongs. The observational data now available make it possible to find the distribution of the sources only over ?. Since the distribution overB and ? is at present unknown, even a very strong upper limit forp _c in the case of a separate source does not enable us to give an exact upper limit for the strength of the magnetic field in this source. In the first model the upper limit for the magnetic field can be obtained only if the upper limit ofp _c is known for a certain number of sourcesN, withN?1. This limit allows for much stronger fields than are usually admitted. This last fact should be taken into consideration when one deals with the results of observations of circular polarization in sources with strong magnetic fields. The first model presents some difficulties when we compare it with observations of some compact sources. The second model can explain why one observes in these sources a violation of the lawp _c ～v ^?1/2 and a change of sign inp _c when the frequency of the observationsv changes.     

A note on the values of the vertical gradient of the magnetic field in the return flux sunspot model

The Return Flux (RF) sunspot model (Osherovich, 1982) imposes a restriction on the value of the vertical gradient of the magnetic field, dB/dz, analogous to a restriction implied by the self-similar sunspot model of Schlüter and Temesvary (ST). The maximum value of the gradient, (dB/dz)_max, is shown to be 10% smaller in the RF model than in the ST model. The dependence of (dB/dz)_max on the sunspot radius is predicted.     

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.     

Polarization of the atmosphere as a foreground for cosmic microwave background polarization experiments

We quantify the level of polarization of the atmosphere due to Zeeman splitting of oxygen in the Earth’s magnetic field and compare it to the level of polarization expected from the polarization of the cosmic microwave background radiation. The analysis focuses on the effect at mid-latitudes and at large angular scales. We find that from stratospheric balloon borne platforms and for observations near 100 GHz the atmospheric linear and circular polarized intensities are about 10⁻¹² and 100 × 10⁻⁹ K, respectively, making the atmosphere a negligible source of foreground. From the ground the linear and circular polarized intensities are about 10⁻⁹ and 100 × 10⁻⁶ K, making the atmosphere a potential source of foreground for the CMB E (B) mode signal if there is even a 1% (0.01%) conversion of circular to linear polarization in the instrument.     

Sunspot as an isolated magnetic structure: Stability and oscillations

A simple energy model of a sunspot as a compact magnetic feature is described where the main energy contribution is provided by the coolest and most compressed part of the magnetic force tube of the spot at depths ranging from Wilson’s depression level (300–500 km) down to 2–3 thousand km. The equilibrium and stability conditions for such a system are analyzed using the variation principle, and oscillations of the system as a whole about the inferred equilibrium position are studied. The sunspot is shown to be stable in the magnetic field strength interval from 0.8–1 to 4–5 kG. The dependence of the eigenfrequency on magnetic field strength ω(B) is computed for the main oscillatory mode, where only the umbra of the sunspot takes part in oscillations, ω = ω ₁ (B). Lower subharmonics may appear in the case where penumbra too becomes involved in the oscillatory process: ω ₂ = ω ₁/2, ω ₃ = ω ₁/3. Theoretical curves agree well with the observational data obtained in Pulkovo using various independent methods: from temporal variations of sunspot magnetic field and from line-of-sight-velocity measurements. The periods of oscillations found range from 40 to 200 minutes.     

Model calculations of the degree of polarization of solar S-coniponent emission at millimetre waves

Model calculations of the S-component accounting for the emission from uniform sunspot and inhomogeneous plage regions are applied to a prediction of the spectral distribution of the degree of circular polarization at millimetre waves. The results are compared with observations adopted from the literature and a sufficient agreement between model computation and measurement can be slated. However, observations with higher spatial resolution than presently available are needed in order to check the predicted fine structures and to verify details of the applied model.     

Magnetic fields in flares and active prominences

We study the longitudinal magnetic field in a number of active limb prominences showing fields in excess of 30 G. The objects fall into three groups: surges, caps and active region prominences. There appears to be an upper limit of 150–200 G for the field strength in prominences.

A model of surges is presented in which a pre-surge axi-symmetric magnetic field is established by a line current in the corona. We observe particle acceleration in surges that indicates v×B≠0 in these objects during periods comparable to the Alfvén transit time.

The strong fields observed in caps seem to run between parts of active regions in accordance with Hale's law of sunspot group polarities.

     

An interpretation of the broad-band circular polarization of sunspots

The broad-band circular polarization of sunspots is discussed on the basis of the observations made in the Okayama Astrophysical Observatory. The observation with the spectrograph proves that it is the integrated polarization of spectral lines in the observed spectral range. A velocity gradient in the line-of-sight can produce this integrated polarization due to the differential saturation between Zeeman components of magnetically sensitive lines. The observed degree of polarization and its spatial distribution in sunspots is explained when we introduce a differentially twisted magnetic field in addition to the velocity gradient. The differential twist has the azimuth rotation of the magnetic field along the line-of-sight and generates the circular polarization from the linear polarization due to the magneto-optical effect. The required azimuth rotation is reasonable and amounts at most to 30°. The required velocity gradient is compatible with the line asymmetry and its spatial distribution observed in sunspots. The observed polarity rule leads to the conclusion that the sunspot magnetic field has the differential twist with the right-handed azimuth rotation relative to the direction of the main magnetic field, without regard to the magnetic polarity and to the solar cycle. The twist itself is left-handed under the photosphere, when the sunspot is assumed to be a unwinding emerging magnetic field.     

Observations of the longitudinal magnetic field in the transition region and photosphere of a sunspot

The Ultraviolet Spectrometer and Polarimeter on the Solar Maximum Mission spacecraft has observed for the first time the longitudinal component of the magnetic field by means of the Zeeman effect in the transition region above a sunspot. The data presented here were obtained on three days in one sunspot, have spatial resolutions of 10 arc sec and 3 arc sec, and yield maximum field strengths greater than 1000 G above the umbrae in the spot. The method of analysis, including a line-width calibration feature used during some of the observations, is described in some detail in an appendix; the line width is required for the determination of the longitudinal magnetic field from the observed circular polarization.The transition region data for one day are compared with photospheric magnetograms from the Marshall Space Flight Center. Vertical gradients of the magnetic field are computed from the two sets of data; the maximum gradients of 0.41 to 0.62 G km^–1 occur above the umbra and agree with or are smaller than values observed previously in the photosphere and low chromosphere.     

Linear and Circular Polarization Properties of Jets

I discuss the transfer of polarized synchrotron radiation in relativistic jets. I argue that the main mechanism responsible for the circular polarization properties of compact synchrotron sources is likely to be Faraday conversion and that, contrary to common expectation, a significant rate of Faraday rotation does not necessarily imply strong depolarization. The long-term persistence of the sign of circular polarization, observed in some sources, is most likely due to a small net magnetic flux generated in the central engine, carried along the jet axis and superimposed on a highly turbulent magnetic field. I show that the mean levels of circular and linear polarizations depend on the number of field reversals along the line of sight and that the gradient in Faraday rotation across turbulent regions can lead to`correlation depolarization'. The model is potentially applicable to a wide range of synchrotron sources. In particular, I demonstrate how the model can naturally explain the excess of circular over linear polarization in the Galactic Center (SgrA^*) and the low-luminosity AGN M81^*.     

A Note on Saturation Seen in the MDI/SOHO Magnetograms

A type of saturation is sometimes seen in sunspot umbrae in MDI/SOHO magnetograms. In this paper, we present the underlying cause of such saturation. By using a set of MDI circular polarization filtergrams taken during an MDI line profile campaign observation, we derive the MDI magnetograms using two different approaches: the on-board data processing and the ground data processing, respectively. The algorithms for processing the data are the same, but the former is limited by a 15-bit lookup table. Saturation is clearly seen in the magnetogram from the on-board processing simulation, which is comparable to an observed MDI magnetogram taken one and a half hours before the campaign data. We analyze the saturated pixels and examine the on-board numerical calculation method. We conclude that very low intensity in sunspot umbrae leads to a very low depth of the spectral line that becomes problematic when limited to the 15-bit on-board numerical treatment. This 15-bit on-board treatment of the values is the reason for the saturation seen in sunspot umbrae in the MDI magnetogram. Although it is possible for a different type of saturation to occur when the combination of a strong magnetic field and high velocity moves the spectral line out of the effective sampling range, this saturation is not observed.     

Limiting polarization effect—a key link in investigating the mean profiles of radio pulsars

Using a well-known method for calculating the propagation of waves in an inhomogeneous medium, we have managed to reduce the problem of wave propagation in pulsar magnetospheres to a system of two ordinary differential equations that allow the polarization characteristics of the radio emission to be quantitatively described for any magnetic field structure and an arbitrary density profile of the outflowing plasma. We confirm that for ordinary pulsars (period P ∼ 1 s, magnetic field B ₀ ∼ 10¹² G, particle production multiplicity parameter λ ∼ 10⁴), the polarization is formed inside the light cylinder at a distance of the order of a thousand neutron star radii. For reasonable magnetic field strengths and plasma densities on the emission propagation path, the degree of circular polarization is found to be ∼5–20%, in good agreement with observations.     

Two Regularities in the Coronal Green-Line Brightness – Magnetic Field Coupling and the Heating of the Corona

To study the quantitative relationship between the brightness of the coronal green line 530.5 nm Fe xiv and the strength of the magnetic field in the corona, we have calculated the cross-correlation of the corresponding synoptic maps for the period 1977 – 2001. The maps of distribution of the green-line brightness I were plotted using every-day monitoring data. The maps of the magnetic field strength B and the tangential B _t and radial B _r field components at the distance 1.1 R _⊙ were calculated under potential approximation from the Wilcox Solar Observatory (WSO) photospheric data. It is shown that the correlation I with the field and its components calculated separately for the sunspot formation zone ±30° and the zone 40 – 70° has a cyclic character, the corresponding correlation coefficients in these zones changing in anti-phase. In the sunspot formation zone, all three coefficients are positive and have the greatest values near the cycle minimum decreasing significantly by the maximum. Above 40°, the coefficients are alternating in sign and reach the greatest positive values at the maximum and the greatest negative values, at the minimum of the cycle. It is inferred that the green-line emission in the zone ±30° is mainly controlled by B _t, probably due to the existence of low arch systems. In the high-latitude zone, particularly at the minimum of the cycle, an essential influence is exerted by B _r, which may be a manifestation of the dominant role of large-scale magnetic fields. Near the activity minimum, when the magnetic field organization is relatively simple, the relation between I and B for the two latitudinal zones under consideration can be represented as a power-law function of the type I ∝ B ^q. In the sunspot formation zone, the power index q is positive and varies from 0.75 to 1.00. In the zone 40 – 70°, it is negative and varies from −0.6 to −0.8. It is found that there is a short time interval approximately at the middle of the ascending branch of the cycle, when the relationship between I and B vanishes. The results obtained are considered in relation to various mechanisms of the corona heating.     

Two-photon annihilation radiation in strong magnetic field: the case of small longitudinal velocities of electrons and positrons

Spectra, angular distributions, and polarization of two-photon annihilation radiation in a magnetic field are studied in detail in the case of small longitudinal velocities of annihilating electrons and positrons which occupy the ground Landau level. Magnetic field essentially affects the annihilation if its magnitudeB is not very low in comparison withB _cr=4.4×10¹³G, which may take place near the surface of a neutron star. The magnetic field broadens the spectra (the width depends on an angle betweenB and a wave vector) and leads to their asymmetry. The angular distribution may be highly anisotropic, being fan-like or pencillike for different photon energies . The total annihilation rate is suppressed by the magnetic field (B ^–3 forBB _cr).The radiation is linearly polarized; the degree and orientation of the polarization depend onB, and . The polarization may reach several tens percent even for comparatively small fieldsB 0.1B _crtypical for neutron stars. This means that the polarization may be detected, e.g., in the annihilation radiation from the gamma-ray bursts.     

Reconstruction of Wolf Sunspot Numbers on the Basis of Spectral Characteristics and Estimates of Associated Radio Flux and Solar Wind Parameters for the Last Millennium

A reconstruction of sunspot numbers for the last 1000 years was obtained using a sum of sine waves derived from spectral analysis of the time series of sunspot number R _z for the period 1700–1999. The time series was decomposed in frequency levels using the wavelet transform, and an iterative regression model (ARIST) was used to identify the amplitude and phase of the main periodicities. The 1000-year reconstructed sunspot number reproduces well the great maximums and minimums in solar activity, identified in cosmonuclides variation records, and, specifically, the epochs of the Oort, Wolf, Spörer, Maunder, and Dalton Minimums as well the Medieval and Modern Maximums. The average sunspot number activity in each anomalous period was used in linear equations to obtain estimates of the solar radio flux F _10.7, solar wind velocity, and the southward component of the interplanetary magnetic field.     

The polarization of optical radiation from the magnetic white dwarfs

It is pointed out that, because of the large Faraday rotation an outlet of linear polarization from the photosphere of a white dwarf is hampered. In accordance with this fact it is proposed to distinguish two types of magnetic white dwarfs. The first type (its representative is Grw 70°8247) has a linear polarization which is comparable in magnitude with the circular one. Polarization of radiation from the white dwarfs of the first type cannot arise in the photosphere. It arises in the corona of the star either as a result of cyclotron emission of hot electrons (T～10⁶ K) or as a result of scattering of slightly polarized emission from the photosphere in the corona. For the first type dwarfs such magnetic fields are required thatω _B ω_opt, i.e.B(1?3)×10⁸G. The white dwarfs of the second type (its representative is G 99-37) have their linear polarization much smaller than the circular one. Polarization of these white dwarfs can arise as a result of the transfer of radiation in the nonisothermal photosphere. Magnetic fields required for the second type can be much smaller:B cos γ=(1?10)×10⁶ G. It is shown that the photospheric model allows to obtain the quantitative accordance of the theory with all the observational data for G 99-37 and is not in accordance with the data for Grw 70°8247, at the same time the model with cyclotron emission from the corona explains the magnitude of both linear and circular polarization and their wavelength dependence for Grw 70°8247.