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Diagnostic of Spectral Lines in Magnetized Solar Atmosphere SCIENCE CHINA Physics, Mechanics & Astronomy Print-CrossMark . Article . Vol. No. : doi: Diagnostic of Spectral Lines in Magnetized Solar Atmosphere: Formation of the Hβ Line in Sunspots Hongqi Zhang* Key Laboratory of Solar Activity, National Astronomical Observatories of the Chinese Academy of Sciences, 100101, Beijing China Received ; accepted Formation of the Hβλ4861.34 Å line is an important topic related to the diagnosis of the basic configuration of magnetic fields in the solar and stellar chromospheres. Specifically, broadening of the Hβλ4861.34 Å line occurs due to the magnetic and micro- electric fields in the solar atmosphere. The formation of Hβ in the model umbral atmosphere is presented based on the assumption of non-local thermodynamic equilibrium. It is found that the model umbral chromosphere is transparent to the Stokes parameters of the Hβ line, which implies that the observed signals of magnetic fields at sunspot umbrae via the Hβ line originate from the deep solar atmosphere, where lg τc ≈−1 (about 300 km in the photospheric layer for our calculations). This is in contrast to the observed Stokes signals from non-sunspot areas, which are thought to primarily form in the solar chromosphere. Solar spectrum, Polarization - Stokes parameters, Magnetic fields PACS number(s): Citation: Hongqi Zhang, Formation of Hβ Line in Solar Magnetic Atmospheres, Sci. China-Phys. Mech. Astron. , (), doi: 1 Introduction of Sciences for the measurements of the solar chromospheric magnetic fields (Zhang & Ai, 1987)[5]. Similar magne- With the discovery of the Zeeman effect by P. Zee- tograms were obtained at other observatories, such as Crimea man (1897)[1], scientists can therefore extract informa- and Kitt Peak [6-8]. These observationshave proved useful to tion on magnetic fields in remote objects through spectro- enhance our understanding of the different properties of solar polarimetry. Hale (1908)[2] was the first to make use of this active phenomena caused by magnetic fields [9]. principle in astrophysics, which led him to find pronounced Diagnostic on the formation layers of spectral lines in so- lar or stellar atmospheres is a basic topic, especially for the arXiv:2009.03573v1 [astro-ph.SR] 8 Sep 2020 Zeeman splitting in sunspots. Quantitative measurements of solar magnetic fields were made possible. Since the 1980s, measurements of the magnetic fields with Stokes parameters a modern solar vector magnetograph (Solar Magnetic Field of spectral lines. In the following, we present the analysis Telescope) was developed in China (Ai and Hu, 1986)[3]. It on the formation of the Hβ λ4861.34 Å line in the magnetic was the inauguration of the systematic observations and the fields of solar atmosphere, especially in the sunspot umbral corresponding theoretical research of solar magnetic fields in atmosphere. China. Chromosphere is a very important layer in the diagnostic 2 Solar Model Atmospheres of solar and stellar activities[4]. TheHβ λ4861.34 Å line has been used at Huairou Solar Observing Station (HSOS) of Na- The Hβ line forms in a relatively wide solar atmosphere. This tional Astronomical Observatories of the Chinese Academy is related to the basic states on the statistical distribution of c Science China Press and Springer-Verlag Berlin Heidelberg 2017 phys.scichina.com link.springer.com Hongqi Zhang Sci. China-Phys. Mech. Astron. () Vol. No. -2 energy level populationsof hydrogenatoms in the solar atmo- increase drastically as hydrogen density Nh and electron 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 Sun 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, atoms 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 spectral line 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 atom. 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.
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