1.3 Photospheric magnetic field measurements
1.3.1 Doppler and Zeeman e ff ect
Practically all the observed solar radiation originates in the photosphere and produces a spectrum that resembles that of a black body at a temperature of about 5800 K (Fig. 1.4).
Most of this emitted electromagnetic energy corresponds to the visible and near-infrared parts of the spectrum.
In the solar photosphere, the temperature of the plasma is sufficiently low so that re-combination occurs and therefore, in addition to protons and electrons, the gas is also composed of ions, neutral atoms, and some molecules, which have multiple energy lev-els. Consequently, the photons can be absorbed and emitted at characteristic wavelengths (with those wavelengths corresponding to energies equal to the energy difference between two electron energy levels in an atom or molecule), thus producing a solar spectrum that contains a huge number of well-defined spectral lines. Since the plasma temperature in the photosphere decreases upwards (and the conditions for local thermodynamic equilib-rium are approximately satisfied - see below), its spectrum is dominated by absorption lines.
Because of the ubiquitous temperature gradients in the photosphere global thermo-dynamic equilibrium is not possible. However, the solar photosphere generally satisfies the conditions oflocal thermodynamic equilibrium(LTE), i.e., the thermodynamic state of a given atomic/ionic/molecular specie at a certain place can be well described with a single temperature, which at the same time is sufficient to represent the velocity distri-bution function of the gas particles at such place as a Maxwellian, the population of the atomic states with the Saha and Boltzmann equations, and the local radiation field with the homogeneous and isotropic black-body form given by the Kirchhoff-Planck function (see e.g., Stix 2002).
Thus, for spectral lines that are formed under LTE conditions, the velocity distribution function of the gas particles (for a given specie) produces the broadening of the spectral lines such that the root-mean-square width or Doppler width, vD, can be related to the temperature,T, as:
1 General Introduction
Energy
Ju=1
Mu
Ml +1
-1 0
0
σb π σr
ΔM= +1 0 -1
B=0 B>0
Normal Zeeman effect
Jl=0
Figure 1.6: Normal Zeeman effect: Example of an atomic transition occurring between two energy levels, in which the lower level has Jl = 0 and the upper level has Ju = 1, in the absence of an external magnetic field (left) and in the presence of an external field (right) which produces the splitting of the upper level into 2Ju+1 different sublevels.
vD=
r2kT
m +v2MIC = c∆λD
λ0 (1.1)
wherekis the Boltzmann constant,mis the mass (at rest) of the atom,vMIC is a parameter that accounts for motions on smaller scales than the mean free path of the photons and is called themicroturbulence velocity,λ0is the central wavelength of the spectral line,cis the speed of light, and∆λDis the Doppler width in terms of wavelength.
On top of that, the gas emitting the photons in the photosphere is in constant motion, mainly due to convection. Such convective motions produce a non-relativistic Doppler effect which can be seen as an additional broadening of the spectral lines when the wave-length shifts produced by up- and downflows are superposed, or as asymmetric spectral lines when velocity gradients exist.
Spectral line broadenings can also be produced by the presence of magnetic fields in the solar surface. This is a result of the Zeeman effect, which is the splitting of the spectral lines due to modifications in atomic structure produced by an external magnetic field. If such an external magnetic field is uniform at the atomic scale, then the Hamiltonian of the atomic system is given by (Baym 1969):
H = H0+HB (1.2)
whereH0is the Hamiltonian of the unperturbed atomic system in the absence of an exter-nal magnetic field and the additioexter-nal termHB is themagnetic Hamiltonian, which in the weak field regime of the linear Zeeman effect (spin-orbit coupling) is given by:
1.3 Photospheric magnetic field measurements
Energy
Ju=3
Mu
Ml +1
-2 0
0
σb π σr
ΔM= +1 0 -1
B=0 B>0
Anomalous Zeeman effect
Jl=2
+3 +2
-1 -3
+1 -2 +2 -1
Figure 1.7: Anomalous Zeeman effect: Example of an atomic transition occurring be-tween two energy levels, in which the lower level has Jl = 2 and the upper level has Ju = 3, in the absence of an external magnetic field (left) and in the presence of an exter-nal field (right) which produces the splitting of the lower and upper levels into 2Jl+1 and 2Ju+1 different sublevels, respectively.
Figure 1.8: Polarization of the different Zeeman components in an emission line for lines-of-sight parallel and antiparallel to the magnetic field vector, as well as in the plane per-pendicular toB~(x-yplane). Adapted from Landi Degl’innocenti and Landolfi (2004).
1 General Introduction
HB = e~
2mec(~L+2S~)·B~ =µB(~L+2S~)·B~ (1.3) with e being the absolute value of the electron charge, ~ = h/2π the reduced Planck constant,methe electron mass,cthe speed of light,~LandS~ the total orbital angular mo-mentum and the total spin angular momo-mentum respectively, andB~ the external magnetic field vector. µB= 9.27×10−21erg G−1is the so-calledBohr magneton.
Nonetheless, if the external magnetic field is strong enough so that the splitting of the energy levels is of the same order as the energy separation between two degenerate states, then the orbital angular momentum,L, and the spin angular momentum,S, couple more strongly to the external magnetic field than to each other, and therefore precess independently around the external magnetic field (LS-decoupling). The splitting of the energy levels in the strong-field scheme can be well described by the Paschen-back effect where, unlike in the Zeeman effect, the J quantum number is no longer a constant of motion (a detailed discussion is provided by, e.g., Landi Degl’innocenti and Landolfi 2004).
The eigenvalues of the total Hamiltonian in equation 1.2 can be found by applying first order time-independent perturbation theory (e.g., Landi Degl’innocenti and Landolfi 2004, and references therein), which leads to the splitting of the energy levels into 2J+1 magnetic sublevels of slightly different energies as follows:
EJ,M =EJ +µBgMB. (M =−J, ...,0, ...,J) (1.4) Here,EJ is an eigenvalue ofH0,M is the magnetic quantum number which is the projec-tion of the total angular momentumJalong the direction of the magnetic field (J≡ L+S), andgis called the Landé factor which inLS−coupling can be written as:
gLS = 1+ J(J+1)+S(S +1)−L(L+1)
2J(J+1) (1.5)
and must be set to zero whenJ =0 since no splitting occurs in this case.
The so-callednormal Zeeman effect(Fig. 1.6) occurs in a transition where the Landé factors of the two levels are equal or if the energy levels haveJ = 0 (M = 0) and J = 1 (M = −1,0,1) respectively, meaning that the splitting results in exactly three different sublevels forming a Lorentz triplet. In a transition different to the particular case described above, the more generalanomalous Zeeman effectoccurs (Fig. 1.7), in which the splitting of an energy level results in more than three different sublevels and the Landé factors of the two transition levels are different. Thus, in the latter case, an effective Landé factor is defined as follows for a given transition (Shenstone and Blair 1929):
ge f f = 1
2(gu+gl)+ 1
4(gu−gl)[Ju(Ju+1)−Jl(Jl+1)] (1.6) wherelandurefer to the lower and upper energy levels of the transition respectively. The effective Landé factor provides information on the magnetic sensitivity associated to the spectral line resulting from such transition.
In a quantum mechanical treatment, due to conservation of the angular momentum, the selection rule for electric-dipole transitions (which are the simplest kind of interaction between atoms and radiation field, see e.g., del Toro Iniesta 2003, Landi Degl’innocenti
1.3 Photospheric magnetic field measurements and Landolfi 2004) establishes that only those transitions that satisfy∆M = Mu −Ml = 0,±1 are allowed, with the particular case of transitions from Mu = 0 to Ml = 0 being forbidden for∆J = 0 (see Figs. 1.6 and 1.7 for an example of the allowed transitions in a normal and an anomalous Zeeman effect, respectively). Generally, transitions with
∆M = 0 are calledπ components. Transitions with ∆M = −1 correspond to red-shifted wavelengths with respect to the unperturbed line and are called σr components, while transitions with∆M = +1 have blue-shifted wavelengths with respect to the unperturbed line and are calledσbcomponents.
The resultant wavelengths associated to all possible transitions between the sublevels of two energy levels can be derived from equation 1.4. By considering the particular case of a transition that occurs in the visible or infrared part of the spectrum (where the Larmor frequency, which is the angular frequency of the circular motion of a charged particle under the presence of a uniform magnetic field, is much smaller than the frequency of the unperturbed line) between two levels that have each angular momentum quantum numbers given byJuandJlfor the upper and lower energy levels respectively, and Landé factorsgu
andgl respectively, then the allowed transitions produce the splitting of the spectral lines into components with different wavelengths which are given by:
λJMuJl
uMl =λ0− λ20eB
4πmec2(guMu−glMl), (1.7) where λ0 is the wavelength of the unperturbed line, and Mu and Ml are the magnetic quantum numbers of the upper and lower sublevels, respectively. Equation 1.7 implies that larger magnetic field strengths as well as larger wavelengthsλ0 produce larger line splittings and thus, due to the quadratic dependence of the wavelength separation onλ0, the Zeeman effect is a handy diagnostic tool to infer the strength of photospheric magnetic fields particularly in the visible and infrared parts of the solar spectrum.
In practice, for the relatively small splittings seen outside sunspots, the splitting of a spectral line is generally calculated by considering the wavelength separation between the center of gravity of theσ components (those that result from∆M = ±1 transitions), λσ, and the unperturbed wavelength,λ0, as follows:
∆λ=|λ0−λσ|= λ20eB
4πmecge f f, (1.8)
so that large effective Landé factors (implying high magnetic line sensitivities) make pos-sible the measurement of relatively weak magnetic fields in the solar photosphere. How-ever, the effects of weak photospheric magnetic fields on the spectral lines can be very similar to those produced by plasma motions (line broadening) when the wavelength sep-aration in the line splitting is so small that the Zeeman components appear superimposed to each other. Hence, the identification of the nature of a spectral line broadening is, at first, not straightforward. However, velocity and temperature effects on the photons do not involve polarization, whilst a stream of photons emitted in the presence of a magnetic field show a net polarization (e.g., Herzberg 1945).
For the allowed transitions mentioned above, the so-calledπcomponents correspond to linearly polarized photons in the direction of the magnetic field and thus, they can be observed only when the line-of-sight is perpendicular toB. The so-called~ σrcomponents display right-handed circular polarization when observed in the direction of the magnetic
1 General Introduction
field (for emission lines), while theσbcomponents have left-handed circular polarization when the line-of-sight is along the direction of the magnetic field. However, the polariza-tion of theσcomponents is generally elliptical for arbitrary directions, with the particular cases of being circular along the direction ofB~ and linear in a direction perpendicular to B~ (Fig. 1.8), with their handedness mentioned above being reversed for a line-of-sight in the opposite direction to B~ and also for an absorption line. Thus, the analysis of the polarization of the light provides us information on the orientation of the magnetic field with respect to the observer.