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Solar Limb Darkening and the Overshooting Approximation 848 F ���������������� ����������������������������� �������������������������������������� ����������� ����� ���������� ������� ���� � �������������� ������� ��� �������� ��� ����� ������� � ��� ������ ����� ����� ����� ��� ��� ��� ��� ��� ��� ��� ��� ���������������������������� ��������������������� ����������� � Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 Interferometry Collaborators S. T. Ridgway (NOAO, LESIA, Observatorie de Paris-Meduon) P. Kervella, A. Mérand, V. Coudé du Foresto (LESIA, Observatorie de Paris-Meduon) D. Mozurkewich (Seabrook Engineering) CHARA Team (Georgia State University) Model Atmosphere Collaborators H.-G. Ludwig (Lund Observatory) R. Kurucz (Smithsonian Astrophysical Observatory) P. Hauschildt, A. Schweitzer (Hamburger Sternwarte) program phoenix F. Allard (CRAL-ENS, Lyon) c --------------- c-- E. Baron (University of Oklahoma) c-- used modules: c-- use phoenix_variables T. Barman (Wichita State University) use phx_interfaces, dummy_takeband => takeband implicit none C. I. Short (St. Mary’s University) ************************************************************************ * the main program for phoenix * version 13.0.0 of 30/Jan/2002 by Peter H. Hauschildt et al. ************************************************************************ Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 What’s to Come.... Intro: All about limb darkening Part I: Procyon, Convection, and Limb Darkening Part II: Deneb, Stellar Winds, and Limb Darkening Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 Limb Darkening Basics �������������������������������� �� �� �� �� �� �� �� � � � � � � � � � � � �� � � � ����������� � � � � � � Center-to-Limb Intensity Profile red - weak limb darkening ������������������� blue - strong limb darkening Intensity RatioIntensity μ= Cosθ Gray (1992) Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 Early Multi-Wavelength Limb Darkening ��������������������������������������������������� ������ ������ ����������������������������������������������������������� Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 Early Model Limb Darkening Models Schwarzschild (1906) Models vs. Contemporary Data ������������������ ��������������������� ��������������������� 1906 - K. Schwarzschild Derived a center-to-limb profile for the Sun with a radiative equilibrium temperature structure. He showed this to be consistent with observations, ruling out an adiabatic equilibrium temperature structure. Assumptions: mass absorption coefficent is both wavelength and depth independent. Adapted from K. Schwarzschild (1906) “Über das Gleichgewicht der Sonnenatmosphäre” Angular-dependent intensity is replaced by Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen. Math.-phys. Kalsse, 295, 41 its mean. Translation in D. H. Menzel, Ed., Selected Papers on the Transfer of Radiation (1966) NY: Dover Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 New Models, New Physics, Hydrogen and the Origins of Convective Instability 1921 - E. A. Milne Replaced Schwarzchildʼs mean intensity by an angular average producing a radiative equilibrium temperature structure with better flux conservation, yielding a limb darkening coefficent of in better agreement with observations. 1925 - C. Payne and H. N. Russell Established hydrogen as the principal component of the solar atmosphere. 1930 - A. Unsöld Investigated the effects of hydrogen ionization on the stability of radiative equilibrium against convection. 1939 - R. Wildt Recognizes the importance of _ wavelength dependent H bound-free and free-free opacity. This opacity causes the solar atmosphere to be unstable to convection. Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 Wavelength-Dependent Opacity of the Solar Atmosphere � ����� � �� ������ � ���������� � �� ��������� ������������������ � ��� ��� ���� ���� ���� ���� ��������������� Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 1976ApJS...30....1V Reconstructing the Sun’s Temperature Structure by Inverting the Planck Function 1976ApJS...30....1V Vernezza, Avrett, & Loeser (1976) ApJS 30, 1 Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 18 CHAPTER 2. BASIC RADIATIVE TRANSFER 18 CHAPTER 2. BASIC RADIATIVE TRANSFER Eddington-Barbier approximation. The emergent intensity at the stellar surface (τν = 0, µ > 0) is given by: Eddington-Barbier approximation. The emergent intensity at the stellar surface (τν = 0, µ > 0) is given by: + ∞ tν /µ I (τ =0, µ) = S (t ) e− dt /µ. (2.43) ν ν + ∞ ν ν tν /µ ν Iν (τν =0, µ) =0 Sν(tν) e− dtν/µ. (2.43) � �0 Substitution of Substitution of ∞ n 2 n 18 Sν(τν) = anτνCHAPTER= a0 + a1τν + a2τ 2.ν + ... BASIC+ anτν RADIATIVE TRANSFER ∞ n=0n 2 n Sν(τν) = anτ�ν = a0 + a1τν + a2τν + ... + anτν n and use of 0∞ x exp( x) dx = n! gives n�=0− � I+(τ =0, µ) = a + a µ + 2a µ2 + ... + n! a µn. n ν ν o 1 2 n andEddington-Barbier use of ∞ xLinkingexp( Intensityx approximation.) dx = ton Depth:! gives The Eddington-BarbierThe emergent Approximation intensity at the stellar surface 0 Truncation− of both expansions after the first two terms produces the important Eddington- (τν = 0, µ > 0)Barbier is given approximation by: + 2 n � I+(τ =0, µ) S (τ = µ) (2.44) Iν (τν =0, µ) = aoν +ν a1µ +≈ 2aν 2νµ + ... + n! anµ . which is exact when+ Sν varies linearly with∞τν. Likewise fort theν /µ emergent flux: Iν (τν =0, µ) = Sν(tν) e− dtν/µ. (2.43) Truncation of both expansions after the+ first0 two terms produces the important Eddington- ν (0) � πSν(τν = 2/3). (2.45) Barbier approximation F ≈ Substitution of A formal derivation is given+ on page 85, a simple one in Exercise 2 on page 225. Figure 2.3 illustrates the Eddington-BarbierIν (τν =0 approximation, µ) S simplistically,ν(τν = µ) Figure 2.4 its application (2.44) to solar limb darkening, Figure 2.5 its application≈ to line formation at increasing sophis- tication. ∞ n 2 n which is exact whenSSν(ντνvaries) = linearlyanτν with= a0 τ+ν.a1 Likewiseτν + a2τν for+ the... emergent+ anτν flux: n=0 � + θ ν (0) πSν(τν = 2/3). Iν (2.45) n ∞ −Fτν ≈ 0 and use of 0 x exp( x) dx e= n! givesSν A formal derivation is given− on page 85, a simple one in Exercise 2 on page 225. Figure 2.3 � + −τν 2 n I (τν =0, µ) = aSoν+e a1µ + 2a2µ + ... + n! anµ . illustrates the Eddington-Barbierν approximation simplistically,1 Figure 2.4 its application to solar limb darkening, Figure0 2.5 its application to line formation at increasing sophis- Truncation of both expansions0 1 after 2 the 3 first 4 twoτ termsν produces the important Eddington- tication. τ ν Barbier approximation 2 + I (τν =0, µ) Sν(τν = µ) (2.44) Figure 2.3: The Eddington-Barbierν approximation. Left: the integrandFrom Rob Rutten’sSν exp(excellentτ lectureν ) measures notes: the contri- ≈ http://www.phys.uu.nl/~rutten/Astronomy_lecture.html− bution to the radially emergent intensity Iν (τν =0, µ=1) from layers with different optical depth τν . The Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitáθ di Pisa 13 April 2005 which is exact whenvalue of SνSatντν varies= 1 is a good linearly estimator of the with area underτν. the Likewise integrand curve, fori.e., the total emergent contribution.I flux: Right: for a slanted beam the characteristic Eddington-Barbier depth is shallower than for a radial beam;ν it lies at τν =−µτ. ν e Sν +(0) πS (τ =0 2/3). (2.45) Fν ≈ ν ν −τν A formal derivation2.3 is Line given transitions on pageSν e 85, a simple one in Exercise 2 on page 225. Figure 2.3 illustrates the Eddington-BarbierBound-bound transitions between approximation the lower l and upper simplistically,1 u energy levels Figure of a discrete 2.4 its application to solar limb0 darkening,electromagnetic Figure energy-storing 2.5 system its application such as an atom, to ion line or molecule formation may occur at as: increasing sophis- tication. 0– radiative 1 excitation; 2 3 4 τν τ ν 2 θ Iν Figure 2.3: The Eddington-Barbier approximation. Left: the integrand Sν exp( τν ) measures the contri- −τν 0 − bution to the radially emergente intensitySIνν (τν =0, µ=1) from layers with different optical depth τν . The value of Sν at τν = 1 is a good estimator of the area under the integrand curve, i.e., the total contribution. Right: for a slanted beam the characteristic Eddington-Barbier−τν depth is shallower than for a radial beam; Sν e it lies at τ = µ. ν 1 0 0 1 2 3 4 τν τ ν 2.3 Line transitions 2 Bound-bound transitions between the lower l and upper u energy levels of a discrete Figure 2.3: The Eddington-Barbier approximation. Left: the integrand Sν exp( τν ) measures the contri- − electromagneticbution to the radially energy-storing emergent
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