Absorption Line Profiles for Absorption Line Profiles for Differentially Rotating 2 M � Differentially Rotating 2 M � Stellar Models Stellar Models Will Flanagan Will Flanagan HAO/NCAR HAO/NCAR Advisor: Dr. Keith MacGregor Advisor: Dr. Keith MacGregor + Roberto Casini and Andy Skumanich!
Objective Objective � Stellar rotation has been an important Stellar rotation has been an important � subject in astrophysics for over 400 years. subject in astrophysics for over 400 years. Galileo was the first to discover differential Galileo was the first to discover differential rotation. rotation. � Our project strives to diagnose differential Our project strives to diagnose differential � rotation in distant stars by analyzing line rotation in distant stars by analyzing line profiles. profiles. � A greater understanding of differential A greater understanding of differential � rotation in other stars can unveil clues to rotation in other stars can unveil clues to stellar formation and evolution! stellar formation and evolution!
Self-Consistent Field Method (SCF) Self-Consistent Field Method (SCF) � SCF is a method of treating rapid, SCF is a method of treating rapid, � differential rotation in stellar models. differential rotation in stellar models. � Though many others have used SCF for Though many others have used SCF for � stellar modeling (ex. Ostriker et al. 1968), stellar modeling (ex. Ostriker et al. 1968), Jackson, MacGregor, and Skumanich Jackson, MacGregor, and Skumanich (HAO) have had the most success. (HAO) have had the most success.
SCF of Jackson, MacGregor, SCF of Jackson, MacGregor, Skumanich Skumanich � Unlike their predecessors, Jackson, Unlike their predecessors, Jackson, � MacGregor, and Skumanich were able to MacGregor, and Skumanich were able to obtain converged models for all main- obtain converged models for all main- sequence masses. sequence masses. � These are the same models that I am These are the same models that I am � using to study the absorption line using to study the absorption line morphology of 2 M � stars due to morphology of 2 M � stars due to differential rotation. differential rotation.
So what do these SCF models give us? So what do these SCF models give us? � In the SCF models, angular momentum is a function In the SCF models, angular momentum is a function � = � (rsin rsin 2 ). As of distance from the axis of rotation � � = � ( 2 ). As of distance from the axis of rotation such, centrifugal force is also a function of such, centrifugal force is also a function of ’ distance from the axis of rotation � � ’ ( � )= � (rsin rsin 2 ). distance from the axis of rotation cent ( � )= � ( 2 ). cent � The SCF models use an effective potential The SCF models use an effective potential � , which � , which � is the gravitational and centrifugal potentials. is the gravitational and centrifugal potentials. = � + � ’ cent where � is gravitational potential � = grav + � ’ cent where grav is gravitational potential � � grav � grav and � ’ cent is centrifugal potential. and � ’ cent is centrifugal potential. = � (r), that is SCF assumes that the effective SCF assumes that the effective � � � = � (r), that is � potential � , increases monotonically monotonically with with spherical spherical potential � , increases radius . . radius Furthermore… … Furthermore
So what do these SCF models give us? So what do these SCF models give us? (continued) (continued) � Pressure P, density Pressure P, density � , and Temperature T � , and Temperature T � are all functions of these equal potential equal potential are all functions of these surfaces surfaces � P = P( P = P( � = � ( � (r)) � (r)) T = T( � (r)) � (r)) � = � ( � (r)) T = T( � (r)) � P = P(r ) = � (r ) T = T(r) so, P = P(r ) � � = � (r ) T = T(r) so, � The SCF models give us a stellar bodies The SCF models give us a stellar bodies � with concentric level surfaces level surfaces for effective for effective with concentric potential, pressure, density, and potential, pressure, density, and temperature. temperature.
Concentric level surfaces… … Concentric level surfaces
Important implication… … Important implication � The model photosphere is a level surface The model photosphere is a level surface � with constant � , P, � , and T. with constant � , and T. � , P, � But we know that rapidly rotating stars But we know that rapidly rotating stars � have a large temperature gradient have a large temperature gradient between the equators and the poles… … between the equators and the poles
How do we impose a Temperature How do we impose a Temperature gradient? gradient? 1/4 (gravity � von Zeipel 1924, T von Zeipel 1924, T eff ~ ( ~ ( � ) 1/4 (gravity eff � grav grav ) � darkening) darkening) � o / � e = 1 + � 2 0 = � o / � cr
What line are we looking at and What line are we looking at and ? why ? why II We are using the Mg II � We are using the Mg 8 4481 doublet. 8 4481 doublet. � 2 D 4f 2 F transition. It exists as the 3d 2 D 4f 2 � It exists as the 3d F transition. � � It is a prominent absorption line for a It is a prominent absorption line for a vast vast � range of temperatures with an oscillator range of temperatures with an oscillator strength of 0.95. strength of 0.95. 8 4481 (Kurucz 1979 ApJ)
II and MgI-III in Abundances of H II and MgI-III in Abundances of H our models our models
So what do these absorption So what do these absorption profiles look like? profiles look like?
What accounts for the shape of What accounts for the shape of these profiles… … these profiles
Modifications to the profile code… … Modifications to the profile code � I optimized the code for generating profiles I optimized the code for generating profiles � by using a more sophisticated algorithm by using a more sophisticated algorithm for calculating the Voigt function H(a, � ) for calculating the Voigt function H(a, � ) � ek 1981) and by exploiting í � (Humlí ek 1981) and by exploiting (Huml inherent symmetries in the profile inherent symmetries in the profile calculations. calculations.
Profile code modifications Profile code modifications (continued) (continued) � I also calculated the line to continuum I also calculated the line to continuum � opacity ratio as a function of latitude. opacity ratio as a function of latitude. l ( l / c ( ) c ( 2 ) = 6 ( 2 / 6 ( 2 ) � o o ( 2 ) = 6 � 2 ) 6 � 2 ) � � � � The continuum opacity was calculated The continuum opacity was calculated � using Rosseland mean opacities from both using Rosseland mean opacities from both OPAL 1995 and Kurucz 1993 OPAL 1995 and Kurucz 1993
Continuum Opacity Interpolations Continuum Opacity Interpolations � The Kurucz mean opacities are The Kurucz mean opacities are � interpolated from a table of values for interpolated from a table of values for temperature and pressure. 6 (T,P) temperature and pressure. 6 Ross Ross (T,P) � The OPAL mean opacities are interpolated The OPAL mean opacities are interpolated � from a table of values for temperature and from a table of values for temperature and density. 6 (T, � ) density. 6 Ross Ross (T, � ) � For the OPAL table, we used a routine to For the OPAL table, we used a routine to � convert � to P given H and He convert � to P given H and He abundances as well as temperature. abundances as well as temperature.
Comparison of Continuum Opacity Comparison of Continuum Opacity Models Models
Line Opacity… … Line Opacity
is relatively constant! � o o is relatively constant! �
Now to Principal Component Analysis…
Principal Component Analysis Principal Component Analysis (PCA) (PCA) � We are currently using Principal Component We are currently using Principal Component � Analysis (PCA) to analyze the morphology of a Analysis (PCA) to analyze the morphology of vast range of profiles with varying inclination profiles with varying inclination angles i, degrees of absolute and differential angles i, degrees of absolute and differential rotation � and 0 , and varying degrees of rotation � and 0 , and varying degrees of microturbulence > . microturbulence > . � PCA is a pattern recognition technique whose PCA is a pattern recognition technique whose � eigenprofiles we are using to look at the eigenprofiles we are using to look at the “principal components principal components” ” of our stellar spectra. of our stellar spectra. “
The Math Behind PCA The Math Behind PCA � PCA involves performing singular value PCA involves performing singular value � decomposition on a covariance matrix C decomposition on a covariance matrix C from N profiles � (observation matrix X). from N profiles � n n (observation matrix X). N � 11 � n1 � � 11 � n1 � N1 N1 C=XX T =U � V T X= � � � 1n � nn � 1n � nn � 1N � � 1N � Where U � V T is the singular value � Where U � NN NN decomposition (SVD) of the covariance matrix C.
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