Energy changes of cosmic rays in the interplanetary region

A clarification and discussion of the energy changes experienced by cosmic rays in the interplanetary region is presented. It is shown that the mean time rate of change of momentum of cosmic rays reckoned for a fixed volume in a reference frame fixed in the solar system is 〈p〉 =p V·G/3 (p=momentum,V is the solar wind velocity andG=cosmic-ray density gradient). This result is obtained in three ways:

1. by a rearrangement and reinterpretation of the cosmic-ray continuity equation;
2. by using a scattering analysis based on that of Gleeson and Axford (1967);
3. by using a special scattering model in which cosmic-rays are trapped in ‘magnetic boxes’ moving with the solar wind.

The third method also gives the rate of change of momentum of particles within a moving ‘magnetic box’ as 〈p〉_ad = ?p ?·V/3, which is the adiabatic deceleration rate of Parker (1965). We conclude that ‘turnaround’ energy change effects previously considered separately are already included in the equation of transport for cosmic rays.     

A Nonlinear Force-Free Magnetic Field Approximation Suitable for Fast Forward-Fitting to Coronal Loops. I. Theory

We derive an analytical approximation of nonlinear force-free magnetic field solutions (NLFFF) that can efficiently be used for fast forward-fitting to solar magnetic data, constrained either by observed line-of-sight magnetograms and stereoscopically triangulated coronal loops, or by 3D vector-magnetograph data. The derived NLFFF solutions provide the magnetic field components B _x (x), B _y (x), B _z (x), the force-free parameter α(x), the electric current density j(x), and are accurate to second-order (of the nonlinear force-free α-parameter). The explicit expressions of a force-free field can easily be applied to modeling or forward-fitting of many coronal phenomena.     

Very Fast and Accurate Azimuth Disambiguation of Vector Magnetograms

We present a method for fast and accurate azimuth disambiguation of vector magnetogram data regardless of the location of the analyzed region on the solar disk. The direction of the transverse field is determined with the principle of minimum deviation of the field from the reference (potential) field. The new disambiguation (NDA) code is examined on the well-known models of Metcalf et al. (Solar Phys. 237, 267, 2006) and Leka et al. (Solar Phys. 260, 83, 2009), and on an artificial model based on the observed magnetic field of AR 10930 (Rudenko, Myshyakov, and Anfinogentov, Astron. Rep. 57, 622, 2013). We compare Hinode/SOT-SP vector magnetograms of AR 10930 disambiguated with three codes: the NDA code, the nonpotential magnetic-field calculation (NPFC: Georgoulis, Astrophys. J. Lett. 629, L69, 2005), and the spherical minimum-energy method (Rudenko, Myshyakov, and Anfinogentov, Astron. Rep. 57, 622, 2013). We then illustrate the performance of NDA on SDO/HMI full-disk magnetic-field observations. We show that our new algorithm is more than four times faster than the fastest algorithm that provides the disambiguation with a satisfactory accuracy (NPFC). At the same time, its accuracy is similar to that of the minimum-energy method (a very slow algorithm). In contrast to other codes, the NDA code maintains high accuracy when the region to be analyzed is very close to the limb.     

Modified Coronal Index of the Solar Activity

The original coronal index of the solar activity (CI) has been constructed on the basis of ground-based measurements of the intensities of the coronal line of 530.3 nm (Rybanský in Bull. Astron. Inst. Czechoslov., 28, 367, 1975; Rybanský et al. in J. Geophys. Res., 110, A08106, 2005). In this paper, CI is compared with the EUV measurements on the CELIAS/SEM equipment based on the same idea as the original idea of the coronal index. The correlation is very good for the period 1996?–?2005 (r=0.94 for daily values). The principal result of this paper is the introduction of the modified coronal index (MCI) which in all uses and contexts can replace the existing CI index. Daily MCI values extend over a time period of six solar activity cycles. Future MCI measurements will be derived from more reliable measurements made by space-based observatories that are not influenced by the weather. MCI measurements are and will continue to be archived at the web site of the Slovak Central Observatory in Hurbanovo ( http://www.suh.sk/obs/vysl/MCI.htm ).     

Drift theory of charged particles in electric and magnetic fields

In this paper we review the drift theory of charged particles in electric and magnetic fields. No new physical interpretations are added to this classical topic, but through an alternative, simplified derivation of the guiding centre velocity, several complexities are eliminated and possible misconceptions of the theory are clarified. It is shown that:

1. The curvature/gradient drift velocity in the magnetic field, averaged over a particle distribution function is to lowest order in the direction of?×B/B ², while the average particle velocity is in the direction ofB×? P withP the scalar particle pressure.
2. These drift directions are correct for first-order expansions of the particle distribution function, and only second-order or higher expansions change these directions.
3. The?×B/B ² drift, which is the standard gradient plus curvature drift, and which is usually considered as a ‘single particle’ drift, need not be ‘reconciled’ with theB×? P, or ‘macroscopic, collective’ drift, as is often asserted in the literature. They are in fact related per definition and we show how.
4. When viewed in fixed momentum intervals (p,p+dp), the so-called Compton-Getting factor enters into the electric field (E×B)/B ² drift term.
5. The results are independent of the scale length of variation ofE andB, in contrast to existing drift theory. We discuss the implications of this result for three important cases.

     

Alfvén waves in the solar atmosphere

We examine the propagation of Alfvén waves in the solar atmosphere. The principal theoretical virtues of this work are: (i) The full wave equation is solved without recourse to the small-wavelength eikonal approximation (ii) The background solar atmosphere is realistic, consisting of an HSRA/VAL representation of the photosphere and chromosphere, a 200 km thick transition region, a model for the upper transition region below a coronal hole (provided by R. Munro), and the Munro-Jackson model of a polar coronal hole. The principal results are:

1. If the wave source is taken to be near the top of the convection zone, where n _H = 5.2 × 10¹⁶ cm^?3, and if B _⊙ = 10.5 G, then the wave Poynting flux exhibits a series of strong resonant peaks at periods downwards from 1.6 hr. The resonant frequencies are in the ratios of the zeroes of J ₀, but depend on B _⊙, and on the density and scale height at the wave source. The longest period peaks may be the most important, because they are nearest to the supergranular periods and to the observed periods near 1 AU, and because they are the broadest in frequency.
2. The Poynting flux in the resonant peaks can be large enough, i.e. P _⊙ ≈ 10⁴–10⁵ erg cm^?2s^?1, to strongly affect the solar wind.
3. ¦δv¦ and ¦δB¦ also display resonant peaks.
4. In the chromosphere and low corona, ¦δv ≈ 7–25 kms^?1 and ¦δB¦ ≈0.3–1.0 G if P _⊙≈10⁴-10⁵ erg cm^?2s^?1.
5. The dependences of ¦δv¦ and ¦δB¦ on height are reduced by finite wavelength effects, except near the wave source where they are enhanced.
6. Near the base, ¦δB¦ ≈ 350–1200 G if P _⊙ ～- 10⁴–10⁵. This means that nonlinear effects may be important, and that some density and vertical velocity fluctuations may be associated with the Alfvén waves.
7. Below the low corona most wave energy is kinetic, except near the base where it becomes mostly magnetic at the resonances.
8. ?₀ < δv ² > v _A or < δB ² > v _A/4π are not good estimators of the energy flux.
9. The Alfvén wave pressure tensor will be important in the transition region only if the magnetic field diverges rapidly. But the Alfvén wave pressure can be important in the coronal hole.

     

A regularization of the three-body problem

Letr ₁,r ₂,r ₃ be arbitrary coordinates of the non-zero interacting mass-pointsm ₁,m ₂,m ₃ and define the distancesR ₁=|r ₁?r ₃|,R ₂=|r ₂?r ₃|,R=|r ₁?r ₂|. An eight-dimensional regularization of the general three-body problem is given which is based on Kustaanheimo-Stiefel regularization of a single binary and possesses the properties:

1. The equations of motion are regular for the two-body collisionsR ₁→0 orR ₂→0.
2. Provided thatR?R ₁ orR?R ₂, the equations of motion are numerically well behaved for close triple encounters.

Although the requirementR? min (R ₁,R ₂) may involve occasional transformations to physical variables in order to re-label the particles, all integrations are performed in regularized variables. Numerical comparisons with the standard Kustaanheimo-Stiefel regularization show that the new method gives improved accuracy per integration step at no extra computing time for a variety of examples. In addition, time reversal tests indicate that critical triple encounters may now be studied with confidence. The Hamiltonian formulation has been generalized to include the case of perturbed three-body motions and it is anticipated that this procedure will lead to further improvements ofN-body calculations.     

Long-Term Variations of Solar Differential Rotation and Sunspot Activity: Revisited

Long-term variations of solar differential rotation and sunspot activity are investigated through re-analyzing the data on parameters of the differential-rotation law obtained by Makarov, Tlatov, and Callebaut (Solar Phys. 170, 373, 1997), Javaraiah, Bertello, and Ulrich (Astrophys. J. 626, 579, 2005a; Solar Phys. 232, 25, 2005b), and Javaraiah et al. (Solar Phys. 257, 61, 2009). Our results indicate that the solar-surface-rotation rate at the Equator (indicated by the A-parameter of the standard solar-rotation law) shows a secular decrease since Cycle 12 onwards, given by about 1?–?1.5×10^?3 (deg?day^?1?year^?1). The B-parameter of the standard differential-rotation law seems to also show a secular decrease since Cycle 12 onwards, but of weak statistical significance. The rotation rate averaged over latitudes 0^°?–?40^° does not show a secular trend of statistical significance. Moreover, the average sunspot area shows a secular increase of statistical significance since Cycle 12 onwards, while a negative correlation is found between the level of sunspot activity (indicated by the average sunspot area) and the solar equatorial rotation on long-term scales.     

Implementation and Comparison of Acoustic Travel-Time Measurement Procedures for the Solar Dynamics Observatory/Helioseismic and Magnetic Imager Time?–?Distance Helioseismology Pipeline

The Helioseismic and Magnetic Imager (HMI) instrument onboard the Solar Dynamics Observatory (SDO) satellite is designed to produce high-resolution Doppler-velocity maps of oscillations at the solar surface with high temporal cadence. To take advantage of these high-quality oscillation data, a?time?–?distance helioseismology pipeline (Zhao et al., Solar Phys. submitted, 2010) has been implemented at the Joint Science Operations Center (JSOC) at Stanford University. The aim of this pipeline is to generate maps of acoustic travel times from oscillations on the solar surface, and to infer subsurface 3D flow velocities and sound-speed perturbations. The wave travel times are measured from cross-covariances of the observed solar oscillation signals. For implementation into the pipeline we have investigated three different travel-time definitions developed in time?–?distance helioseismology: a Gabor-wavelet fitting (Kosovichev and Duvall, SCORE’96: Solar Convection and Oscillations and Their Relationship, ASSL, Dordrecht, 241, 1997), a?minimization relative to a reference cross-covariance function (Gizon and Birch, Astrophys. J. 571, 966, 2002), and a linearized version of the minimization method (Gizon and Birch, Astrophys. J. 614, 472, 2004). Using Doppler-velocity data from the Michelson Doppler Imager (MDI) instrument onboard SOHO, we tested and compared these definitions for the mean and difference travel-time perturbations measured from reciprocal signals. Although all three procedures return similar travel times in a quiet-Sun region, the method of Gizon and Birch (Astrophys. J. 614, 472, 2004) gives travel times that are significantly different from the others in a magnetic (active) region. Thus, for the pipeline implementation we chose the procedures of Kosovichev and Duvall (SCORE’96: Solar Convection and Oscillations and Their Relationship, ASSL, Dordrecht, 241, 1997) and Gizon and Birch (Astrophys. J. 571, 966, 2002). We investigated the relationships among these three travel-time definitions, their sensitivities to fitting parameters, and estimated the random errors that they produce.     

Validation of the 3D AMR SIP–CESE Solar Wind Model for Four Carrington Rotations

We carry out the adaptive mesh refinement (AMR) implementation of our solar–interplanetary space-time conservation element and solution element (CESE) magnetohydrodynamic model (SIP–CESE MHD model) using a six-component grid system (Feng, Zhou, and Wu, Astrophys. J. 655, 1110, 2007; Feng et al., Astrophys. J. 723, 300, 2010). By transforming the governing MHD equations from the physical space (x,y,z) to the computational space (ξ,η,ζ) while retaining the form of conservation (Jiang et al., Solar Phys. 267, 463, 2010), the SIP–AMR–CESE MHD model is implemented in the reference coordinates with the aid of the parallel AMR package PARAMESH available at http://sourceforge.net/projects/paramesh/ . Meanwhile, the volumetric heating source terms derived from the topology of the magnetic-field expansion factor and the minimum angular separation (at the photosphere) between an open-field foot point and its nearest coronal-hole boundary are also included. We show the preliminary results of applying the SIP–AMR–CESE MHD model for simulating the solar-wind background of different solar-activity phases by comparison with SOHO observations and other spacecraft data from OMNI. Our numerical results show overall good agreements in the solar corona and in interplanetary space with these multiple-spacecraft observations.     

Some properties of finite energy constant-α force-free magnetic fields in a half-space

Some useful properties of a finite energy, constant-α, force-free magnetic field B _α occupying a half-space D are presented. In particular:

1. Fourier and Green representations of B _α are obtained and used to derive conditions for the existence and uniqueness of a B _α having a given normal component B _z on the boundary ?D.
2. The asymptotic behaviour of B _α at infinity as well as stability results against changes in the boundary condition on ?D and in the value of α are established.
3. The energy of B _α is shown to be smaller than the energy of the open field having the same B _z on ?D, thus confirming an earlier conjecture (Aly, 1984).
4. B _α is proved to not be a Taylor-Heyvaerts-Priest state, in spite of the fact that its relative helicity H is finite and that it is the only solution of the Lagrange-Euler equation associated with the problem of minimizing the energy among all the fields having the same value of H and the same B _z on ?D.

     

Estimating Flare-Related Photospheric Lorentz Force Vector Changes Within Active Regions

It is shown that expressions for the global Lorentz force associated with a flaring active region derived by Fisher et al. (Solar Phys. 277, 59, 2012) can be used to estimate the Lorentz-force changes for strong fields in large structures over photospheric subdomains within active regions. Gary’s (Solar Phys. 203, 71, 2001) model for the stratified solar atmosphere is used to demonstrate that in large-scale structures with typical horizontal magnetic length scale ??300 km and with strong magnetic fields (≥?1 kG at the τ=1 opacity layer at 5000 Å), the Lorentz force acting on the photosphere may be approximated by a surface integral based on photospheric boundary data alone. These conditions cover many of the sunspot fields and major neutral lines that have been studied using Fisher et al.’s (2012) expressions over the past few years. The method gives a reasonable estimate of flare-related Lorentz-force changes based on photospheric magnetogram observations provided that the Lorentz-force changes associated with the flare have a lasting effect on the observed fields, and they are not immediately erased by post-flare equilibration processes.     

A relation between Giorgilli-Galgani's and Deprit's algorithms

If \(T = \sum\nolimits_{i = 1}^\infty {\varepsilon ^i } T_i\) and \(W = \sum\nolimits_{n = 1}^\infty {n\varepsilon ^{n - 1} } W^{\left( n \right)}\) are respectively the generators of Giorgilli-Galgani's and Deprit's transformations, we show that the change of variables generated byT is the inverse of the one generated byW, ifT _i =W ⁽ⁱ⁾ for anyi. The method used is to show that the recurrence which defines the first algorithm can also be obtained with the second one.     

Can We Constrain Solar Interior Physics by Studying the Gravity-Mode Asymptotic Signature?

Gravity modes are the best probes to infer the properties of the solar radiative zone, which represents 98% of the Sun’s total mass. It is usually assumed that high-frequency g modes give information about the structure of the solar interior whereas low-frequency g modes are more sensitive to the solar dynamics (the internal rotation). In this work, we develop a new methodology, based on the analysis of the almost constant separation of the dipole gravity modes, to introduce new constraints on the solar models. To validate this analysis procedure, several solar models – including different physical processes and either old or new chemical abundances (from, respectively, Grevesse and Noels (Origin and Evolution of the Elements 199, Cambridge University Press, Cambridge, 15, 1993) and Asplund, Grevesse, and Sauval (Cosmic Abundances as Records of Stellar Evolution and Nucleosynthesis CS-336, Astron. Soc. Pac., San Francisco, 25?–?38, 2005)) – have been compared to another model used as a reference. The analysis clearly shows that this methodology has enough sensitivity to distinguish among some of the models, in particular, among those with different compositions. The comparison of the models with the g-mode asymptotic signature detected in GOLF data favors the ones with old abundances. Therefore, the physics of the core – obtained through the analysis of the g-mode properties – is in agreement with the results obtained in the previous studies based on the acoustic modes, which are mostly sensitive to more external layers of the Sun.     

The influence of fluctuation fields on the force-free field

In Paper I (Hu, 1982), we discussed the the influence of fluctuation fields on the force-free field for the case of conventional turbulence and demonstrated the general relationships. In the present paper, by using the approach of local expansion, the equation of average force-free field is obtained as (1+b)?×B ₀=(α#x002B;a)B ₀#x002B;a ⁽¹⁾×B ₀#x002B;K. The average coefficientsa,a ⁽¹⁾,b, andK show the influence of the fluctuation fields in small scale on the configurations of magnetic field in large scale. As the average magnetic field is no longer parallel to the average electric current, the average configurations of force-free fields are more general and complex than the usual ones. From the view point of physics, the energy and momentum of the turbulent structures should have influence on the equilibrium of the average fields. Several examples are discussed, and they show the basic features of the fluctuation fields and the influence of fluctuation fields on the average configurations of magnetic fields. The astrophysical environments are often in the turbulent state, the results of the present paper may be applied to the turbulent plasma where the magnetic field is strong.     

Extreme Ultraviolet Late-Phase Flares: Before and During the Solar Dynamics Observatory Mission

The solar extreme-ultraviolet (EUV) observations from the Solar Dynamics Observatory (SDO) have revealed interesting characteristics of warm coronal emissions, such as Fe xvi 335 Å emission, which peak soon after the hot coronal X-ray emissions peak during a flare and then sometimes peak for a second time hours after the X-ray flare peak. This flare type, with two warm coronal emission peaks but only one X-ray peak, has been named the EUV late phase (Woods et al., Astrophys. J. 739, 59, 2011). These flares have the distinct properties of i) having a complex magnetic-field structure with two initial sets of coronal loops, with one upper set overlaying a lower set, ii) having an eruptive flare initiated in the lower set and disturbing both loop sets, iii) having the hot coronal emissions emitted only from the lower set in conjunction with the X-ray peak, and iv) having the first peak of the warm coronal emissions associated with the lower set and its second peak emitted from the upper set many minutes to hours after the first peak and without a second X-ray enhancement. The disturbance of the coronal loops by the eruption is at about the same time, but the relaxation and cooling down of the heated coronal loops during the post-flare reconnections have different time scales with the longer, upper loops being significantly delayed from the lower loops. The difference in these cooling time scales is related to the difference between the two peak times of the warm coronal emission and is also apparent in the decay profile of the X-ray emissions having two distinct decays, with the first decay slope being steeper (faster) and the delayed decay slope being smaller (slower) during the time of the warm-coronal-emission second peak. The frequency and relationship of the EUV late-phase decay times between the Fe xvi 335 Å two flare peaks and X-ray decay slopes are examined using three years of SDO/EUV Variability Experiment (EVE) data, and the X-ray dual-decay character is then exploited to estimate the frequency of EUV late-phase flares during the past four solar cycles. This study indicates that the frequency of EUV late-phase flares peaks before and after each solar-cycle minimum.     

Theoretical and Experimental Studies of the Rigidity Spectrum of the 27-Day Variation of the Galactic Cosmic Ray Intensity in Different Epochs of Solar Activity

Recently Pop (Solar Phys. 276, 351, 2012) identified a Laplace (or double exponential) distribution in the number of days with a given absolute value in the change over a day, in sunspot number, for days on which the sunspot number does change. We show this phenomenological rule has a physical origin attributable to sunspot formation, evolution, and decay, rather than being due to the changes in sunspot number caused by groups rotating onto and off the visible disc. We also demonstrate a simple method to simulate daily sunspot numbers over a solar cycle using the Pop (Solar Phys. 276, 351, 2012) result, together with a model for the cycle variation in the mean sunspot number. The procedure is applied to three recent solar cycles. We check that the simulated sunspot numbers reproduce the observed distribution of daily changes over those cycles.     

Properties of the 15 February 2011 Flare Seismic Sources

The first near-side X-class flare of Solar Cycle 24 occurred in February 2011 (SOL2011-02-05T01:55) and produced a very strong seismic response in the photosphere. One sunquake was reported by Kosovichev (Astrophys. J. Lett. 734, L15, 2011), followed by the discovery of a second sunquake by Zharkov, Green, Matthews et al. (Astrophys. J. Lett. 741, L35, 2011). The flare had a two-ribbon structure and was associated with a flux-rope eruption and a halo coronal mass ejection (CME) as reported in the CACTus catalogue. Following the discovery of the second sunquake and the spatial association of both sources with the locations of the feet of the erupting flux rope (Zharkov, Green, Matthews et al., Astrophys. J. Lett. 741, L35, 2011), we present here a more detailed analysis of the observed photospheric changes in and around the seismic sources. These sunquakes are quite unusual, taking place early in the impulsive stage of the flare, with the seismic sources showing little hard X-ray (HXR) emission, and strongest X-ray emission sources located in the flare ribbons. We present a directional time–distance diagram computed for the second source, which clearly shows a ridge corresponding to the travelling acoustic-wave packet and find that the sunquake at the second source happened about 45 seconds to one minute earlier than the first source. Using acoustic holography we report different frequency responses of the two sources. We find strong downflows at both seismic locations and a supersonic horizontal motion at the second site of acoustic-wave excitation.     

Gaussian variational equations for osculating elements of an arbitrary separable reference orbit

If a satellite orbit is described by means of osculating Jacobi α's and β's of a separable problem, the paper shows that a perturbing forceF makes them vary according to $$\dot \alpha _\kappa = {\text{F}} \cdot \partial {\text{r/}}\partial \beta _k {\text{ }}\dot \beta _k = {\text{ - F}} \cdot \partial {\text{r/}}\partial \alpha _k ,{\text{ (}}k = 1,2,3).{\text{ (A1)}}$$ Herer is the position vector of the satellite andF is any perturbing force, conservative or non-conservative. There are two special cases of (A1) that have been previously derived rigorously. If the reference orbit is Keplerian, equations equivalent to (A1), withF arbitrary, were derived by Brouwer and Clemence (1961), by Danby (1962), and by Battin (1964). IfF=?gradV ₁(t), whereV ₁ may or may not depend explicitly on the time, Equations (A1) reduce to the well known forms (e.g. Garfinkel, 1966) $$\dot \alpha _\kappa = {\text{ - }}\partial V_1 {\text{/}}\partial \beta _k {\text{ }}\dot \beta _k = \partial V_1 {\text{/}}\partial \alpha _k ,{\text{ (}}k = 1,2,3).{\text{ (A2)}}$$ holding for all separable reference orbits. Equations (A1) can of course be guessed from Equations (A2), if one assumes that \(\dot \alpha _k (t)\) and \(\dot \beta _k (t)\) depend only onF(t) and thatF(t) can always be modeled instantaneously as a potential gradient. The main point of the present paper is the rigorous derivation of (A1), without resort to any such modeling procedure. Applications to the Keplerian and spheroidal reference orbits are indicated.