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China-Phys. Mech. Astron. () Vol. No. -2 energy level populationsof hydrogenatoms in the solar atmo- increase drastically as hydrogen density Nh and den- sphere. To study the formation of the Hβ line in the magne- sity Ne decrease, and as the temperature increases at low opti- tized solar atmosphere, the non-local thermodynamicequilib- cal depths. The departure coefficients tend to decrease at very rium (NLTE) population departure coefficients bi are defined thin depths. as: A sunspot umbral model with the distributions of tempera- ture T, hydrogen density Nh and electron density Ne in Figure LT E LT E bl = Nl/Nl , bu = Nu/Nu , (1) 1b was provided by Ding & Fang (1991)[11]. Assuming that the same thermodynamic mechanism op- where N is the actual population and NLT E the Saha- (l,u) (l,u) erates both in the quiet and sunspot umbrae models, Boltzmann values for the lower (l) and upper (u) levels. The the departure coefficients b of the hydroden atomic levels treatment of NLTE is suitable for analyzing the formation of i in the latter atmosphere model can be phenomenologically spectral lines in the upper solar atmosphere due to the weak- provided by using the tendencies of the departure coefficients ening contribution of particle collisions, while the approx- b found for the quiet Sun as a proxy. Figure 1b shows the imation of local thermodynamic equilibrium (LTE) is nor- i population departure coefficient b ≈ 1 in the deep umbral mally acceptable in the lower solar atmosphere due to the i model atmosphere, which increases with height in the inter- high density of the particle populations, where the departure val lg τ = [−6, −7.5] and decreases gradually with height in coefficients are b ≈ 1ineq. (1). i the range lg τ = [−7.5, −9] . In thermodynamic equilibrium, are distributed over their bound levels according to the Boltzmann excitation equation [12]. The continuum normally forms in the lower solar atmosphere, thus the Planck function B(T, ν) can be used as its source function 2hν3 1 BC(T, ν) = , (2) c2 exp(hν/kT) − 1 and the source function of Hβ line is 2hν3 1 S L(H ) = , (3) β c2 b 2 exp(hν/kT) − 1 b4

where the values of b2 and b4 are shown in Figure 1. The self-consistency of these departure coefficients of the Hβ line with the observations can be found in the following sections.

3 Hydrogen Lines, Magnetic Fields and Ion- ized Solar Atmosphere

To analyze the contribution of the magnetic and micro- electric fields to the broadening of hydrogen lines in the solar atmosphere, the Hamiltonian can be written in the following

Figure 1 Distribution of T, Nh, Ne, and the departure coefficients (b2 and condensed form b4) of the hydrogen levels. The continuum optical depth is displayed in lg τc scale in the atmospheric model of the quiet Sun (a) by Vernazza, Avrett & H = H0 + HZ + HS , (4) Loeser (VAL, 1981)[10] and the sunspot umbra model (b) by Ding & Fang (DF, 1991)[11]. where H0 includes the Hamiltonian HNR, H f ine and Hhyper, and HZ = µ0 Be · (L + 2S) and HS = e0 Ee · r. We have pur- Figure 1a shows the atmospheric model of the quiet Sun posely chosen the Hamiltonian HZ and HS of the Zeeman with the distributions of the departure coefficients b2 and b4 and Stark effects to analyze the broadening of hydrogen lines of the hydrogen atomic levels n = 2 and n = 4 at differ- (such as, the Hβ line) in the solar atmosphere, as magnetic ent continuum optical depths (Vernazza, Avrett & Loeser, and micro-electric fields commonly contribute to these lines.

1981[10]). It is found that the departure coefficients bi in The general results for the fine structure of hydrogenlines can the lower solar atmosphere are of the order unity, and they be found in the monograph of Bethe & Salpeter (1957)[13]. Hongqi Zhang Sci. China-Phys. Mech. Astron. () Vol. No. -3

Next, we will consider the hydrogen atoms and perturbing, charged particles in external magnetic and electric fields (eq. 4). The perturbation equation is

′ H0 + HZ + HS | ψi = (E0 + E ) | ψi, (5)

′ where E0 and E are the energy’s eigenvalue and perturba- tion, respectively. One can calculate the shifts △λ j of the sub-componentsof the by means of eq. (5) when the magnetic and electric fields are considered. Figure 2 Hβ line profiles of the quiet Sun (solid line) and of the sunspot Left-multiply eq. (5) by hψ|, and use the identity for the umbra (red dotted line) observed by the National Solar Observatory of USA (L. Wallace, K. Hinkle, and W. C. Livingston, http://diglib.nso.edu/ftp.html). perturbing Hamiltonian The blue plus symbols mark the results of the numerical calculation.

′ ′ ′ hn, l, j, m |H + H | n, l , j , m i≡ K ′ , j Z S j q,q (6) The observed profiles of the Hβ λ4861.34 Å line in the quiet Sun and sunspot umbra with the overlap of some pho- ′ the suffixes q and q in K refer to various possible wave func- tospheric lines in the wings are shown in Figure 2. In the fol- tion states. Assume the perturbed wave function to be a linear lowing, we also neglect the asymmetry of the spectral line, combination of unperturbed wave functions: which is normally believed to be caused by the gradient of the line of sight velocity field in the solar atmosphere[15]. 2n2 | ψi = cq′ | ψq′ i, (7) qX′=1 4 Formationof the Hβ Line in Solar Magnetic Atmosphere the eq. (6) then becomes The numerical Stokes profiles (I, Q, U, V)oftheHβ line can 2n2 ′ be calculated by means of the Unno-Rachkovsky equations (Kq,q′ − E δqq′ )cq′ = 0, (8) qX′=1 of polarized radiative transfer of spectral lines (Unno (1956) [16] and Rachkovsky (1962a, b) [17,18]) where q = 1, 2, ...., 2n2. Normally, the shifts of the energy levels depend on the distribution of the magnetic and electric I η0 + ηI ηQ ηU ηV I − S       field nearby the hydrogen . d  Q   ηQ η0 + ηI ρV −ρU   Q  µ   =     (9)       Normally, the general solution of the eigenvalue problem dτc  U   ηU −ρV η0 + ηI ρQ   U        for hydrogen lines relates to the direction of the magnetic  V   η ρ −ρ η + η   V     V U Q 0 I    field and microscopic electric field. By means of eq. (8), we       can find the 2n2 possible splits of the upper and lower energy where the symbols have their usual meanings as given by 2 levels and the corresponding 2n probabilities of the wave Landi DeglInnocenti (1976) [19], dτc =−κcds, ηI , ηQ, ηU , ηV functions in various directions and intensities of the electric are the Stokes absorption coefficient parameters and ρQ, ρU field for each sub-level of the hydrogen line. Then, we can and ρV are related to magneto-optical effects, S represents calculate the shifts △λ j of the sub-components j of the spec- the source functions related to eq. (2) for the continuum and tral line (Zhang, 1987)[5]. The calculation of the Hermitian eq. (3)for the Hβ line. matrix of the perturbing Hamiltonian was presented also by By comparing with the formulae of the non-polarized hy- Casini and Landi Degl’Innocenti (1993)[14]. drogen line proposed by Zelenka (1975)[20], the contribu- As the microscopic electric field in the solar atmosphere tions of the statistically distributed microscopic electric field is statistically isotropic, we assumed here that the polarized with the Zeeman effectsineqs. (6-8) rising from the magnetic states of the sub-components of the Hβ line still rely on the field in the solar atmosphere are considered in the absorption magnetic field. Because the perturbed wave function is a lin- matrix of eq. (9) forthe Hβ line (see, [5,21]). ear combination of unperturbed wave functions, the polarized It is assumed here that, when the degeneracy of an energy states of the sub-componentsof the line are also related to the level disappears under the actions of magnetic and micro- magnetic quantum number m, e.g., △m = 0 for the π compo- scopic electric fields, each magnetic energy level keeps its nent and △m = ±1 for both σ components. original departure coefficient. Hongqi Zhang Sci. China-Phys. Mech. Astron. () Vol. No. -4

In Figure 3, it is noticed that the numerically calculated as defined by Jin (1981) [22] and Stenflo (1994) [23], for the residual intensity at the core of the Stokes I profile is of the Stokes vector I = (I, Q, U, V)[24] order 0.5 in the umbral model atmosphere, and the profile of ∞ ∞ Stokes I is roughly consistent with the observed umbral one I(0) = C I(τc)dτc = C I(x)dx, (10) Z0 Z−∞ in Figure 2, if the blended lines are ignored. (The calculated result is also marked by plus symbols in the blue wing of the where C I(x) = (ln10)τcC I(τc) and x = lgτc, τc is the contin- line in Figure 2.) Note that it is weaker than that of the quiet uum optical depth at 5000 Å. Equation (10) provides the con- Sun. There are no obvious differences in the Stokes I val- tribution to the emergent Stokes parameters from the differ- ues between the umbral magnetic field strengths of 1000 and ent layers of the solar atmosphere with the equivalent source −2 ⋆ 3000 G. The amplitude of the Stokes V is of the order 10 , functions S I (Zhang, 1986)[25]. and those of Stokes Q and U are 10−4. These values are sim- The contribution functions of the Hβ line in the magnetic ilar to those of the quiet Sun (see Figure 11 of Zhang, 2019 atmospheric model of the quiet Sun were presented by Zhang [21]). This means measuring the transverse components of and Zhang (2000)[26] and Zhang (2019)[21]. The emergent the magnetic fields of sunspots is still a technological chal- Stokes parameters at the Hβ line center in the quiet Sun al- lenge due to their very weak signals. most form in the middle chromosphere(1500 –1600 km), and those in the blue wing, -0.45Å from the Hβ line center, form 1.0 8 in the photosphere (300 km)[5]. a6 b 3000 0.8 4 2.0 3 1000 & 3000 2000 0.00 2 a0.06 b

0.6 ) 1000 0.15 - 4 0.25 ri 0 1.5 2 0.45 0.4 rq (10 -2 ) 4 1.0 1

-4 Cq

0.2 Ci (10 -6

0.0 -8 0.5 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 (0.1nm) Wavelength (0.1nm)

8 4 0.0 -1 3000 -8 -7 -6 -5 -4 -3 -2 -1 0 -8 -7 -6 -5 -4 -3 -2 -1 0 6 c d τ τ lg c (Optical depth) lg c (Optical depth) 3000 4 3 3 4 2000 2 c d ) 1000 ) - 4 - 2 3 0 2 2000 2 rv (10 ru (10 -2 2 ) 2 1 -4 1 1000 Cu

Cv (10 1 -6 500 -8 0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0 Wavelength (0.1nm) Wavelength (0.1nm)

-1 -1 Figure 3 Profiles of Stokes parameters r , r , r , and r (i.e. I/I , Q/I , -8 -7 -6 -5 -4 -3 -2 -1 0 -8 -7 -6 -5 -4 -3 -2 -1 0 i q u v c c τ τ lg c (Optical depth) lg c (Optical depth) U/Ic and V/Ic) of the Hβ line calculated for the umbral atmospheric model by Ding & Fang (DF, 1991)[11]. The homogeneous magnetic field inten- Figure 4 Contribution functions Ci, Cq, Cu, and Cv of Stokes parameters sity of 500, 1000, 2000, 3000G for an inclination of ψ=30◦, an azimuth of I, Q, U, and V of the numerically calculated Hβλ4861.34 Å line for the um- ◦ ϕ=22.5 and µ = 1. Ic is the continuum. bral atmospheric model by Ding & Fang (DF, 1991)[11] at of △λ=0.45 (pluses), 0.25 (asterisks), 0.15 (diamonds), 0.06 (triangles) and 0.0 (squares) Å from the Hβ line center. B=1000 G, ψ=30◦, ϕ=22.5◦ and µ = 1. Moreover, the peak values of Stokes V are located at 0.2 τc is continuum optical depth at 5000 Å. The horizontal coordinates are in Å from the line center as calculated using the umbral model logarithmic scale. atmosphere in Figure 3, and at 0.3 Å for the quiet Sun (see Figure 11 of Zhang, 2019 [21]). Similar tendencies for Stokes Figure 4 shows the contributions of the Stokes parameter I, Q and U can be found as well. This reflects that the contribu- Q, U, and V in the umbra model at 0.0, 0.06, 0.15, 0.25, and tion of the Holtsmark broadening in the umbral atmosphere, 0.45 Å from the center of the Hβ line. It is noticed that the which depends on the density of the charged particles, is rela- contributions of the emergent Stokes parameters in the higher tive weaker than that in the quiet Sun. This result is consistent solar atmosphere (the continuum optical depth lg τc ≈−7) are with the abundance of the Ne in both models. much weaker than that in the lower atmosphere (lg τc ≈−1), The key is to know where in the solar atmosphere a given even though there are some differences in the Stokes param- spectral line is formed, which can be used to estimate the pos- eters at different wavelengths from the center of the Hβ line. sible formation height of the observed . This in- This result implies that the upper umbral atmosphere is trans- formation is provided by the contribution function C I, such parent to the Hβ line. The signals of the emergent Stokes Hongqi Zhang Sci. China-Phys. Mech. Astron. () Vol. No. -5 parameters of the Hβ line in the umbrae formed mainly in the electric field to broadening the Hβ line in the solar umbral lower solar atmosphere, while those near the line center in the atmosphere than in the quiet Sun. quiet Sun form in the chromosphere (see Figure 12 of Zhang, It is worth noting that, because the sunspot umbra exhibits 2019 [21]). This result can also be found by comparing the lower temperature characteristics compared to the quiet Sun, residual intensities at the center of the Hβ line for the quiet the umbral chromosphere is generally transparent to the spec- Sun and the umbra in Figure 2 due to different absorptions of tral lines that require higher temperature excitation, such as the line. the Hβ line. Moreover, the ratio of the emergent Stokes pa- A similar observational result of the sunspots in the white rameters of spectral lines from different depths of the solar at- light has been named as the Wilson effect. Bray and Lough- mosphere depends on the the parameters of the magnetic and head (1965)[27] contended that the true explanation of the atmospheric models used in the radiative transfer equations. Wilson effect lies in the higher transparency of the spot ma- In order to verify it, accurate observations are still necessary terial compared to the photosphere. The morphological evi- and important. dencecan befoundin Figure13 of Zhang (2019)[21]. Similar patterns of a sunspot umbra occur both in the photospheric 6 Acknowledgements and the Hβ filtergrams, even when the bright flare ribbons in Hβ are attached to the umbrae in the Hβ filtergram. The The author would like to thank the referees for their com- blended lines from the deep layer cause the reversed signals ments and suggestions, which improved the paper. This in the umbral areas of the Hβ as indicated by study is supported by grants from the National Natural Sci- Zhang (2019)[21,28]. ence Foundation of China (NSFC 11673033, 11427803, 11427901) and by Huairou Solar Observing Station, Chinese Academy of Sciences. Chromosphere Chromosphere 1 Zeeman, P., 1897, On the influence of Magnetism on the Nature of the Light emitted by a Substance, Phil. Mag. 43, 226 2 Hale, G. E., 1908, On the Probable Existence of a in Photosphere Photosphere Sun-Spots, Astrophys. J. 28, 315 Umbra 3 Ai, G. X. & Hu, Y. F., 1986, Principles of a solar magnetic field tele- scope, AcASn, 27, 173-180 Figure 5 A schematic sketch of the estimated formation height of the mea- 4 Zirin, H., 1988, Astrophysics of the Sun, Cambridge University Press sured Stokes parameters of the Hβ line for sunspot regions. The red thick 5 Zhang, H. & Ai, G., 1987, Formation of the H-beta line in solar chro- dashed line marks the major contribution heights of the Stokes parameters of mosphere magnetic field, Chin. Astron. Astrophys., 11, 42-48. the Hβ line, while the orange thin dashed line shows the weak contribution 6 Severny, A. B. & Bumba, V., 1958, On the penetration of solar mag- of the line in the chromosphere. netic fields into the chromosphere, Obs., 78, 33-35. 7 Tsap, T., 1971, The Magnetic Fields at Different Levels in the Active A simple schematic sketch of the formation heights of the Regions of the Solar Atmosphere (presented by N. V. Steshenko), in Stokes parametersobserved by the Hβ line is proposed in Fig- Solar Magnetic field, Ed by R. Howard, IAU Symp. 43, 223-230. 8 Giovanelli, R. G., 1980, An exploratory two-dimensional study of the ure 5. The red thick dashed line shows the rough detectable coarse structure of network magnetic fields, Solar Physics, 68, 49-69. heights in the Hβ magnetograms. In the umbrae, the observed 9 Georgoulis, M. K., Nindos, A. & Zhang, H., 2019, The source and en- Stokes parameters mainly form in deep layers in Figure 4, al- gine of coronal mass ejections, Philosophical Transactions of the Royal though weak contributions also arise from the chromosphere Society A: Mathematical, Physical and Engineering Sciences, vol. 377, issue 2148, p. 20180094 in Figure 5. This means that the Hβ line in the sunspot um- 10 Vernazza, J.E., Avrett, E.H. & Loeser, R., 1981, Structure of the solar brae forms at the continuum optical depth lg τc ≈−1 for our chromosphere. III - Models of the EUV brightness components of the calculations, i.e., about 300 km in the photospheric layer. quiet-sun, Astrophys. J. Suppl. Ser., 45, 635-725. 11 Ding, M. D., & Fang, C., 1991, A semi-empirical model of sunspot umbra, ChA&A, 15, 28-36. 5 Discussions and Summary 12 Mihalas, D., 1978, Stellar atmospheres, San Francisco: W.H. Freeman 13 Bethe, H.A. & Salpeter, E.E., 1957, Quantum Mechanics of One-and In this paper, the departure coefficients of the hydrogen atom Two-electron Atoms, Springer-Verlag. 14 Casini, R. & Landi Degl’Innocenti, E., 1993, The polarized spectrum in the model umbral atmosphere under the NLTE hypothesis of hydrogen in the presence of electric and magnetic fields, Astron. were presented and compared with their distributions in the Astrophys., 276, 289-302. quiet Sun. The locations of the peak values of the Stokes V 15 Almeida, S. & Lites, B. W., 1992, Observation and Interpretation of the of the Hβ line are close to the line center in the model umbral Asymmetric Stokes Q, U, and V Line Profiles in Sunspots, Astrophys. J., 398, 359-374 atmosphere as compared to those of the quiet Sun. This re- 16 Unno, W., 1956, Line Formation of a Normal Zeeman Triplet, Publ. sult is consistent with the weaker contribution of the micro- Astron. Soc. Jap., 8, 108-125. Hongqi Zhang Sci. China-Phys. Mech. Astron. () Vol. No. -6

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