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### lect18

Course: A 453, Fall 2009
School: Rochester
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Word Count: 355

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18: Lecture Solution of Equation of Radiative Transfer The equation looks fairly simple dI = (j - I )ds If we knew the state of matter, we should be able to calculate j and and solve this. Problem: to know state of matter (i.e. ionization, excitation, T), we need to know I, but to know I we need to know the state of matter. i.e. a real problem Typlically approached iteratively Guess one of the...

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18: Lecture Solution of Equation of Radiative Transfer The equation looks fairly simple dI = (j - I )ds If we knew the state of matter, we should be able to calculate j and and solve this. Problem: to know state of matter (i.e. ionization, excitation, T), we need to know I, but to know I we need to know the state of matter. i.e. a real problem Typlically approached iteratively Guess one of the quantities, solve for the other refine the guess etcetera In Thermodynamic Equilibrium divide by . Solutions giving: 1 dI j = I ds = S ds = d where is the "optical depth" j the "source function" In T.E., Kirchoff showed the emissivity of an object must equal its absorptivity, and therefore S = B (T), the Planck function i.e. j = B (T), Note that if S > I, dI/ds > 0, i.e. I increases and vice-versa until I = S Solution, given a known intensity I0 at s = 0 show on board, Take case S = const., consider emission from optically box thin and optically thick Formal Solution for intensity I Equation becomes, in terms of source function and optical depth dI + I = S d multiply both sides by e dI e + e I = S e d consider that d dI e I = e + e I d d ( ) so the equation of radiative transfer becomes d e I = S e d ( ) this can be formally integrated along the path starting at s = 0 where I = I0 and = 0 to the new position s = s I0 s, = 0 I s = s, = Solution given by integral of source function S attenuated by exp(-) along path The solution is d ' ( e d 0 ' ...

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Rochester - A - 453
Lecture 20 2nd Moment Equation of Transfer Multiply both sides of the equation of transfer by cos() and integrate over the unit sphere. dI 2 cos ( ) = cos( ) ( I S ) d 4 sterdagain d d but and and sod cos 2 ( ) = 0 so: d2 4 44 I
Rochester - A - 453
Lecture 24: Absorption lines in real stellar atmosphere The reversing layer model used to date was a convenient approximation Gives main features of real lines Line shape gaussian for weak lines Thermal doppler plus turbulent doppler give gaussia
Rochester - A - 241
Handout 19: Evolution of stars on the Hertzsprung-Russell DiagramFrom H.W. #7:Adiabatic T gradient is : cP cV so dln( T) dln( P) dln( T) dln( P) 1 cP cV cPIf the T-gradient required by radn xfer in order to transport the L outward is steeper t
Rochester - A - 241
Handout 20: Evolution and Nature of starsTo recapLow T stars are fully convectiveHigh Kramers opacity in the interior High heat capacity just below the photosphereH and He ionizationLuminosity determined by matching convective interior to radia
Rochester - A - 453
Uniform stellar model M := M sun R := Rsun := M ( r) := 4 3 r 3 := 0.62 M constant, the &quot;water sun&quot; 4 R3 3 G M ( r) dr P( r) = P( R) 2 r R 2rhydrostatic equilibriumP( r) :=2 3 G R r9(22)set P(R) = 0P( 0
Rochester - A - 241
Handout 15: Virial Theorem E = P.E. + K.E = (1/2)P.E. = -K.E.The virial theorem is crucial for an overview of the stellar interiorSee discussion Chap. 2 of C&amp;OHolds when P.E. is from a force ~ 1/r2i.e. gravity and electrostatic forcesConsider
Rochester - A - 453
Atomic absorption cross sections =2 e_2me cf= f ( ) f-value, line strength.line shape has property ( ) d = 1 0 FWHM = c 02hence 0 =( )1 FWHMapproximately, and FWHM2 e_2so, at line center 0 f , , 0 := f
Rochester - A - 241
Handout 3: Chap. 5: Light &amp; MatterIf everything radiated like a blackbody, spectra would be quite boring. In fact, deviations from a blackbody tell us a lot about stars (and the rest of the universe). Deviations occur for many reasons. Some objects
Rochester - A - 241
Handout 23: Star FormationReview collapse of a starConditions for collapse (Jeans)If gravity &gt; thermal pressure collapsei.e. not hydrostatic equilibriumFor a given cloud with (,T)M &gt; MJeans ~ T3/2-1/2 Or R &gt; RJeans ~ T1/2-1/2Note: such regi
Rochester - A - 241
Handout 7 Chap. 9 Stellar AtmospheresGoalsUnderstand why stellar spectra closely resemble black bodiesi.e. shape of F vs ~ B(Te) L = 4r2Te4Deviations from a blackbodyi.e. absorption linesGive Temperature Give abundances of nCa/nH, etc. Give L
Rochester - SCIENCEV - 323
Rochester - A - 241
Handout 9: Radiative transfer in a starSince Intensity is conserved in free space, we need only deal with interaction of light with matterTo properly account for these interactionsWe need the density of matter Also, the compositioni.e. H:He:C:N:O
Rochester - A - 453
Lectures 2-3 notes 1 2 g t = h 2 2 10 kmg := free-fall time at const. accel=45.16 secg=980.66 cmsec-3-2Msun4 3 Rsun 3 1= = 1.39 gmcmaverage solar densityG 54.7 min23 1 = 0.49 hr 32 G free-fall time scale7 1
Rochester - SCIENCEV - 323
Rochester - A - 453
More realistic line profiles, using Eddington Approximation for T structure Te :=5800 K4model the solar spectrumT ( ) := Te 3 2 + 4 3Eddington Approximation for T structure1.5 .104T(0) = 4877 KT 2 = 5800 K 3T 0 i( )
Rochester - A - 241
Nature of LightTwo aspects both importantWaveYoungs double slit destructive and constructive interference of monochromatic light. i.e.If (path length) = n, constructive If (path length) = (n+1/2), destructive Where = c, fundamental constantM
Rochester - A - 241
Handout 16: Nature of a StarR/Rsun 10 5 2 1 &lt;T&gt; 106 K 2 106 K 5 106 K 107 K t tK-H ~ 104 y tK-H ~ 105 y tK-H ~ 106 y Nucl. Reactions start Nucl. Reactions dominatetnucl ~ 1010 y Main SequenceThe contraction stops when d(Enucl)/dt in the interior
Rochester - A - 453
0 :=656.3 nm := 20.04 nm 0 :=.2Balmer alpha line , 0 := 0 e () 0 Gaussian profile continuum flux 0 proportional to column densityFc :=1FF , ,2 0 := Fc e i i :=( ) ) , 0()reversing layer model f
Rochester - A - 241
Handout 6 Chap. 8 Spectral ClassificationClass based on strength of absorption lines in spectrum (see v.g.)T increasing O B A F G K M H lines Ca II TiO molecules weak S weak weak S S weakweak S, strongVisible (Balmer) H linesArise from n=2 lev
Rochester - SCIENCEV - 323
Rochester - AST - 111
Astronomy 111 Lab ManualFall 2004LAB #8The Solar wind and the Earths magnetosphereI. Introduction Geomagnetic storms are a natural hazard, like hurricanes and tsunamis, which the National Oceanic and Atmospheric Administration (NOAA) Space Env
Rochester - AST - 142
Physical and astronomical constantsGravitational constant Boltzmann constant Stefan-Boltzmann constant Planck constant Speed of light Electron mass Proton mass Neutron mass Hydrogen mass Atomic mass unit Quantum of electric charge, cgs Quantum of el
Rochester - AST - 111
AST111 Lecture 4aTelescopes4m Mayall telescope of NOAO on Kitt PeakWhat a Telescope does Light gathering power, so we can see fainter objects. Telescopes can also be made to gather light at wavelengths that we can't see with our eyes. Provide
Rochester - AST - 111
Problem Set #1, AST111 Solutions1) Sidereal and solar periods. See PS (Planetary Sciences) problems 1.3-1.5. A planet which keeps the same hemisphere pointed toward the Sun must rotate once per orbit in the prograde direction (i.e., in the same rota
Rochester - AST - 142
Astronomy 142 Midterm Exam #1Thursday March 20, 2008 11:05AM-12:20PM, B+L407Exam rules: you may consult only one page of formulas and constants and a calculator while taking this test. You may not consult any books, nor each other. All of your wor
Rochester - AST - 111
Problem Set #1, AST111due Tuesday Sept 14, beginning of class1) Sidereal and solar periods. See PS (Planetary Sciences) problems 1.3-1.5. A planet which keeps the same hemisphere pointed toward the Sun must rotate once per orbit in the prograde di
Rochester - AST - 111
Problem Set #5, AST111Solutions 1) Calculate the equilibrium temperature of the Moon as a function of latitude. Assume that the moon is a rapid rotator with zero obliquity, has a bond albedo of Ab=0.07 and an emissivity of 1.For a small ribbon of
Rochester - AST - 142
Astronomy 142 Midterm Exam #1 SolutionsShort Problems (Choose 4 of 6 parts) In the absence of extinction an O5 star has colors U-B =-1.1 and B-V=-0.3. Its absolute magnitude is MV =-5.6. Interstellar extinction has extinction ratio A(B)/A(V) = 1.34
Rochester - AST - 111
Problem Set #7, AST111 Solutions1) The Voyager 1 spacecraft detected nine active volcanoes on Io. Assume that, on average, there are nine volcanoes active on Io, and that the average eruptive rate per volcano is 50km3/yr. The radius of Io is 1820 km
Rochester - AST - 111
Astronomy 111 Lab ManualFall 2004Scaling of Splash Craters and Ejecta TrajectoriesIntroductionThe surface of many bodies in our solar system exhibit evidence for impact craters. Some surfaces, like portions of the Moon appear saturated with cra
Rochester - AST - 111
Overview of the Solar systemThis is a composite image of all the planets in our solar system. From the top is Mercury (taken by the Mariner 10 spacecraft); Venus (taken by the Pioneer Venus Orbiter); a full disk of the Earth (taken by the Apollo 17
Rochester - AST - 142
Today in Astronomy 142: the Milky Way, continuedStellar relaxation time Virial theorem Differential rotation of the stars in the disk The local standard of rest Rotation curves and the distribution of mass The rotation curve of the GalaxyFigure: sp
Rochester - AST - 111
Problem Set #5, AST111due Tuesday Oct 12, beginning of class 1) Calculate the equilibrium temperature of the Moon as a function of latitude (see PS problems 3.5, 3.6). Assume that the moon is a rapid rotator with zero obliquity, has a bond albedo of
Rochester - AST - 111
AST111, Lecture 1bPlanetary properties (overview continued) The Celestial Sphere (our coordinate system).Planetary properties (continued): Measuring Mass The orbital period of a moon about a planet depends on the semi-major axis and on the plane
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Student projects for PHY103 (2008)The physics of/and music encompasses a wide range of topics, some that we will cover in labs and lectures in this class. However the diversity of subject matter related to this topic cannot be captured in a single s
Rochester - AST - 111
AST111 - Elementary Astronomy: Origins of the Solar System and BeyondLectures: TR 11:05am -12:20pm, Bausch and Lomb Hall (B+L) Room 270 Workshops: M7-9pm, B+L480, by tradition. Problem sets are due the next day at the beginning of class. Labs: R 6:
Rochester - AST - 111
Problem Set #4, AST111Solutions 1) On Spectral resolution and Absolute magnitude. When you observe with your eye you see light over a large wavelength range (about 4000-7000). We can think of the eye as having spectral resolution R ~ 5. Supposing yo
Rochester - AST - 111
Relected light from dust in AB Aurigas debris diskAST111, Lecture 3bAST111, Lecture 3bThe Dynamics of Small Bodies Dissipative and Radiation ForcesAdditional Forces on Small BodiesFor planets, gravity is the largest force. However small bodi
Rochester - AST - 111
Astronomy 111 Lab ManualFall 2004CCD cameras: basic operation, and the pinhole cameraIntroductionCharge-coupled devices (CCDs) were invented at AT&amp;T Bell Laboratories in the late 1960s. A CCD consists of a silicon wafer patterned in the manner
Rochester - AST - 142
Astronomy 142 Workshop #8: Problems1. The epicyclic frequency. Consider a star in a nearly circular orbit in a galaxy. We can think of the motion of the star as a radial oscillation about a circular orbit of radius R0 . The frequency of radial oscil
Rochester - BST - 466
BST 446 Homework # 3 (Due in one week from 2/13) 1. In the last homework assignment, you have constructed the variableEDULEVEL for theDOS data. Use these variables to answer the following questions. (a) Use the Pearson chi-square statistic to test
Rochester - BST - 466
data dd; set dosSurivivalBygender; nf=effsize-failed; array t[2] nf failed; do i=0 to 1; depression=i; count=t[i+1]; output; end; keep gender Uppertime depression count; run; proc freq data=dd; weight count; tables Uppertime*depression*gender/cmh NOC
Rochester - BST - 466
Rochester - BST - 466
3.3. The generalized linear models The term generalized linear model (GLM) was rst introduced in a landmark paper by Nelder and Wedderburn (1972), in which a wide range of seemingly disparate problems of statistical modelling and inference were set i
Rochester - BST - 466
BST 446 Homework # 3 (Due in one week from 2/8) 1. A categorical variable X with more than two possible levels is said to follow a multinomial distribution. The multinomial distribution is an extension of the binomial distribution for modeling two-le
Rochester - BST - 466
libname path &quot;C:\&quot;; symbol1 c=blue; proc freq data=path.survival; tables Year*majorminordepp/NOCOL NOCUM NOPERCENT NOROW; run; data one; set path.survival; if majorminordepp=1 and year=0 then delete; if majorminordepp=1 then year=year-1; if gender=.
Rochester - BST - 466
DATA ssd; INPUT gender dep count; DATALINES; 1 0 211 1 1 103 1 3 65 2 0 175 2 1 44 2 3 19 ; proc freq data=ssd; weight count; table gender*dep/all trend cmh; exact fisher chisq trend jt; run;31, 2008 428 The FREQ Procedure Table of gender by dep gen
Rochester - BST - 466
Chapter 1. Introduction This course focuses on modeling relationship among variables that contain discrete outcomes. Clarication of discrete outcomes: 1. binary outcome (yes, no) 2. ordinal (e.g. 5 point Likert scale: strongly disagree, disagree, neu
Rochester - BST - 466
BST 446 Midterm Exam 1. The DDPC data set contains information for those patients in the depression study ofseniors who identied an informant. (a) We are interested in the distribution of the depression status for patients based on the diagnosis, r
Rochester - BST - 466
Journal of Applied Statistics Vol. 33, No. 7, 679 690, August 2006Modifying the Exact Test for a Binomial Proportion and Comparisons with Other ApproachesALAN D. HUTSONUniversity at Buffalo, The State University of New York, New York, USAABSTRA
Rochester - BST - 466
Fisher Exact Test. s Why the distribution of the cell count n11 conditional on the mariginal counts follows the Hypergeometric distribution? Notation: Y Present (1) n11 n21 n+1X Present (1) Absent (0) TotalAbsent (0) n12 n22 n+2Total n1+ n2+ n
Rochester - BST - 466
BST 466 Midterm Exam 1. The PPW data set is from a study about depression in post partum women with N = 198. Each of the 198 women in the study has been screened for depression based on the Edinburgh postnatal depression (EPDS) scale, and undergone a
Rochester - BST - 466
BST 446 Homework # 1 (Due in class 2/04/08) 1. A sample of 32 severely agitated patients diagnosed as having schizophrenic spectrumdisorders was treated by two second-generation oral antipsychotic medications for 5 days. One of the outcomes of inte
Rochester - BST - 466
22 23 24DATA ms; INPUT ms gender count; DATALINES;NOTE: The data set WORK.MS has 4 observations and 3 variables. NOTE: DATA statement used (Total process time): real time 0.00 seconds cpu time 0.00 seconds29 30 31 32; PROC SORT DATA = ms; BY
Rochester - BST - 466
Chapter 5. Analyses of discrete survival time Survival data analysis is widely used in research studies and investigations when interest lies in the time to occurrence of some events of interest. In many applications, including studies involving seri
Rochester - BST - 466
4.7.2 Zero-inated Poisson model In addition to data clustering, another common cause of overdispersion is the excessive amount of zeros in the data. In many biomedical and psychosocial research, the distribution of zeros often exceeds the expected fr
Rochester - BST - 466
3). Goodman-Kruskal Gamma, Kendalls tau-b, Stuarts tau-c and Somers D All these measures consider whether the column variable Y tends to increase as the row variable X increases and vice versa. Each pair of subjects is classied as concordant or disco
Rochester - BST - 466
Rochester - BST - 466
Rochester - BST - 466
3.4.3. Models for ordinal responses Ordinal response occurs more frequently than nominal response in practice. For ordinal response, it is natural to model the cumulative response probabilities j = Pr (Y j) , j = Pr (Y = j) , j = 1, . . . , J 1, j