Atmospheric Radiation Fall 2008
|
|
- Egbert Page
- 5 years ago
- Views:
Transcription
1 MIT OpenCourseWare Atospherc Radaton Fall 8 For nforaton about ctng these aterals or our Ters of Use, vst:
2 .85, Atospherc Radaton Dr. Robert A. McClatchey and Prof. Sara Seager 3. Scatterng of Radaton by Molecules and Partcles a. Introducton Here, we ll deal wth wave aspects of lght rather than quantu aspects. Consder coponents of electrc feld n utually perpendcular drectons, parallel and perpendcular to the plane of propagaton and propagatng n the z drecton: ( tkz) Er are crcular frequency ( tkz) E ae k () The ntensty s gven by: I E E E E a a * * r r r () Let us frst consder Sngle Scatterng. We ay consder a sngle partcle or a sall volue of partcles such that tterng events wll all be sngle tterng events. Fg. p = phase atrx p = phase functon dv P I dv I k P I or I k 4R 4R k = tterng cross-secton per unt volue k, [k ] N length dv dv, [ ] length N = total no. of partcles Q, where G geoetrcal cross sec ton G.85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page of 6
3 , Q Q Scatterng effcency G G P s the phase atrx and provdes the angular dstrbuton and polarzaton of the ttered lght. For our purpose here, lets consder the total ntensty of the radaton whether polarzed or not. Then the ter,, s the phase functon or tterng dagra whch defnes the probablty for tterng of unpolarzed ncdent lght n any drecton. p s noralzed such that: p p d where d eleent of sold angle 4 (3) Let us defne <cos > = ). cos p d 4 where s the tterng angle (see Fg. <cos > = ansotropy paraeter (or asyetrc paraeter) and can vary between + and -. <cos > = for sotropc tterng. We defne analogous ters for partcle absorpton and we have: k k k ext abs ext abs Q Q Q ext abs Then, we defne: rq (4) k Q sngle tterng albedo k Q ext ext ext For practcal applcatons, k and can be taken as constants and P s a functon only of the tterng angle,. Ths specal case s vald for: () randoly orented partcles, each of whch has a plane of syetry () randoly orented asyetrc partcles, f half the partcles are rror ages of the others. (3) Raylegh tterng and Me tterng. Another portant defnton: a where a = partcle radus. nr n where nr = real part of the ndex of refracton and n = agnary part. n s responsble for absorpton and s responsble for tterng. n r.85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page of 6
4 For water, n r =.33 across the vsble and near nfrared. And n wll depend on the knds of aterals dssolved n the water drops. It wll therefore be uch ore of a functon of wavelength. Returnng to the analytcal expresson for the electrc feld as a bea of radaton passng through a sngle partcle: E(z, t) E e ( t kz) The ntensty of radaton vares as the square of the E-feld. So, we have: ( tkz) I E e k and So, we have : E(z, t) E e z (nrn ) t E e zn znr t e and I E e 4zn 4znr t e and usng the defnton of sze paraeter daeter of drop, we have: a where we wll take z = a = I E e e 4n ( 4n r zt) absorpton (5) b. Me tterng stll sngle tterng. E s Er exp( kr kz) S (, )S 4(, ) r s kr S E 3(, )S (, ) E (6) at dstance R n the far feld If we consder sotropc, hoogenous, spheres, we have: S( ) S I S ( ) k R And we have the transforaton atrx: F I The S values are n general coplex nubers and functons of tterng angle..85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 3 of 6
5 F * * * * SS SS SS SS * * * * SS SS SS SS SS SS SS SS SS SS SS SS * * * * * * * * whch s proportonal to the phase atrx: The noralzaton condton on P leads to: F CP (8) C 4 F d 4 and snce = effectve cross secton, we have: IR d and I 4 C 4 F Iod k 4 4 where we ve used I= F k R I o So we have: k p * * F (SS SS) 4 defnes phase functon F k p 4 * * (SS SS) (9) k p * * 43 k p * * F (SS SS ) F (SS SS) 4 4 For a sngle sphere, we have: S n ann nn(n ) bn n S n bnn nn(n ) an n ().85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 4 of 6
6 n and n are functon only of and relate to Legendre Polynoals. a n and b n functons. are functons of a x and nr n and nvolve Sphercal Bessel We also have: Q (n )(a a b b) * * n n n n x n Q (n) R (a b) () ext e n n x n 4 n(n) * * n * cos R e(ana n bnb n) R e(anb n) n n(n) x Q n All above s for Sngle Scatterng fro a Sngle Sphere. In general, f optcal thckness s not too large, sngle tterng can be appled to a dstrbuton of partcles assued to be ndependent. We then have: r r r r k (r)n(r)dr r Q (r)n(r)dr r r ext ext ext r r () k (r)n (r) dr r Q (r)n (r) dr where n(r) = sze dstrbuton, descrbng the nuber of the partcles havng rad between r and r+dr over the range r to r..85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 5 of 6
7 c. Geoetrc Optcs: When r >>, we can use ray theory of lght due to Fresnel (see Van de Hulst, Ch. 3, Lght Scatterng by Sall Partcles). Ternology s: dffracton external reflecton double refracton 3 frst ranbow 4 second ranbow and f P( ) P ( ), then P ( ) sn dd, s for non-absorbng spheres fro Fresnel theory. 4 real =.33 =..5.5 always true n geoetrc optcs often suffcent to consder just these For non-absorbng partcles, dffracton = \ of ttered lght. Thus, the geoetrc optcs ltng value of Q ext =...85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 6 of 6
8 d. Raylegh Scatterng: r and r where nr n Partcle can be consdered to be n hoogenous external electrc feld Radaton penetrated partcle quckly. partcle own feld s neglgble n the process. E k pencdent kn where angle between dpole sne R oent and drecton of tter (3) 3 ( cos ) 3 sn sn ( cos ) 4 p( ) 3 cos 3 cos (4) p 3 ( cos ) (5) 4 whch gves angular dstrbuton of ntensty ttered by sall partcles the Raylegh tterng. We also obtan: 8 4 Q x Qabs 4xI (6) 3 note that Q Q as x abs note 4 th power of x or -4 dependence Ths result s the sae as Me tterng as ltng case as x (r<<)..85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 7 of 6
9 .85 Lecture Notes (Atospherc Radaton) Multple Scatterng Refer back to Eq. fro the frst set of Atospherc Radaton lecture notes where we dscussed Case III whch arses due to the followng two condtons: F B (T) () I(,, )>>B ( ) () The resultng equaton of transfer s: di (,, ) I(,, ) J(,, ) d (3) where / o J (,, ) P(,, ', ') I ( ', ', )sn 'd 'd ' e P(, )F 4 4 and the foral soluton s: d ' ( ')/ I(,,,, ) J( ',,,, )e s ( ' )/ d ' I(,,,, ) J( ',,,, )e (4) Due to the coplextes of evaluatng the ntegrals n Eq. 4, a nuber of technques have been used to generate nuercal results:. Dscrete Ordnates. Doublng or Addng Method 3. Successve Orders of Scatterng 4. Iteraton of Foral Soluton 5. Invarant Ebeddng 6. Method of X and Y Functons 7. Sphercal Haroncs Method 8. Expanson n Egenfunctons 9. Monte Carlo Method We wll focus soe attenton on the Dscrete Ordnates Method and apply an avalable coputer progra to soe exercses..85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 8 of 6
10 Radatve Transfer n a Scatterng Atosphere. Coordnate syste n a plane parallel atosphere Here poston defned by z (or ) only. Recall that optcal depth related to alttude z by d = -dz where s the extncton coeffcent. cos = ; = nclnaton to upward outward noral Notaton cos o = o ; o = nclnaton to upward outward noral.85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 9 of 6
11 cos = ; = nclnaton to upward outward noral Fro sphercal geoetry, the cosne of the tterng angle, can be expressed n ters of the ncong and outgong drectons n the for: cos ' ' cos ' (5) Let us now dgress for a oent and exane the propertes of Legendre polynoals (whch coe to play n a varety of ways n radatve transfer probles). We ay consder wrtng the phase functon n ters of Legendre polynoals n the for: P(cos ) = N C P (cos ) (6) Legendre polynoals have the followng for, and orthogonal and recurrence propertes: d n!d n n P ( ) P n( ) ( ) (n,...) n n So, 3 P ( ) P ( )... P( )P( k )d = k k (7) P(h) P( ) P( ) Usng Eq. 5, the Phase Functon defned above ay be wrtten as follows: N P(,,, ) C P ( ) ( ) cos ( ) (8) Fro the orthogonalty condton, the expanson coeffcents are gven by: C P ( ) P ( )d,...n where we note that the phase functon s noralzed to unty:.85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page of 6
12 4 P( )dd There s an addton theore for Legendre polynoals whch allows us to wrte the Phase Functon as follows: N N P(,,, ) C P ( )P ( ) cos ( ) (9) where P denotes the Assocated Legendre polynoals and:! C (,)C l,,...n n! (),,, otherwse In vew of the expanson of the phase functon, the dffuse ntensty ay also be expanded n a cosne seres n the for: N I(,, ) I (, )cos( ) () Substtutng Eqs. 9 and nto Eq. 3, and usng the orthogonalty of the assocated Legendre polynoals, the equaton of transfer splts nto (N+) ndependent equatons, and ay be wrtten as: N di (, ) I (, ) (,) C P ( ) () d 4 x P ( )I (, )d C P ( )P ( )F e 4 N =,, N Let us rewrte these equatons as follows: di (, ) d I (, ) J (, ) (3).85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page of 6
13 wth the source functon gven by: J (, ) ( ) C P ( ) P ( ) I (, )d N, 4 4 N C P ( )P ( )Fe (4) To proceed wth the soluton of Eq. 3, we frst dscretze the equaton by replacng wth (= -n,., n, wth n=,,.) and the ntegral wth a su wth the weghts, a j n jn f( )d f( )a (5) j j The hoogeneous soluton for the set of frst-order dfferental equatons ay be wrtten: n k j j j (6) jn I (, ) L ( )e where and L j j ( ) and kj denote the egenvectors and egenvalues, respectvely, are coeffcents to be deterned fro approprate boundary condtons. On substtutng Eq. 6 nto the hoogeneous part of Eq. 3, the egenvectors ay be expressed by ( N n,) j q q j q 4( jk j ) qn ( ) C P ( ) a P ( ) ( ) The partcular soluton ay be wrtten n the for (7) I (, ) Z ( )e p (8) Fro Eq. 3, we have N Z ( ) C P ( ) j 4 (9) n F x aqp ( q)z ( q) P ( ) qn Equatons 7 and 9 are lnear equatons n j and z and ay be solved nuercally. The coplete soluton for Eq. 3 s the su of the general.85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page of 6
14 soluton for the assocated hoogeneous syste of the dfferental equatons and the partcular soluton. Thus, n kj j j jn I (, ) L ( )e Z ( )e () = -n,. +n In order to deterne the unknown coeffcents, L j, a q, boundary condtons ust be posed. In the dscrete-ordnates ethod for radatve transfer, analytcal solutons for the dffuse ntensty are explctly gven for any optcal depth. Thus the nternal radaton feld can be evaluated wthout addtonal coputatonal effort. And furtherore, useful approxatons can be developed fro ths ethod for flux calculatons. Advantages of Dscrete Ordnate Method a) In prncple - nuercal coputatons can be done for any order of approxaton. b) The nternal radaton feld s deterned - not just the Reflecton & Transsson. c) Accurate results (to about %) are achevable wth only a few streas (3-4) for ost cases. We wll utlze the Dscrete Ordnate coputer progra to do a few excercses..85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 3 of 6
15 Multple Scatterng Coputatonal Technques. Dscrete Ordnates (We ll dscuss n detal n a few nutes.). Doublng or Addng Prncple: If reflecton and transsson s known for each of two layers, the reflecton and transsson fro the cobned layer can be obtaned by coputng the successve reflectons back and forth between the two layers. If the two layers are chosen to be dentcal, the results for a thck hoogenous layer can be bult up rapdly n a geoetrc (doublng) anner. 3. Successve Orders of Scatterng Prncple: Intensty s coputed ndvdually for photons ttered once, twce, three tes, etc. wth the total ntensty obtaned as the su over all orders. If the ntensty s expanded n a Fourer seres, the hgh frequency ters arse fro photons ttered a sall nuber of tes. Therefore, ost Fourer ters can be obtaned wth soe accuracy by coputng a few orders of tterng. 4. Iteraton of Foral Soluton Drect soluton of ntegral over source functon by dvdng atosphere nto layers wth sall optcal thckness. 5. Invarant Ibeddng Dfferental Equatons are developed whch gve the change of reflecton and transsson atrces when an optcally thn layer s added to the atosphere. It s a specal case of the Doublng or Addng technque. 6. Method of X and Y Functons Involves the deternaton of ntegral equatons for functons whch depend upon only one angle and are drectly related to Reflecton and Transsson atrces. The ntegral equatons need to be solved nuercally. The ntegral equatons are copletely specfed by a character functon dependng on the partcular phase functon. Ths ethod s due to Chandrasekhar. 7. Sphercal Haronc Method Intensty s edately expanded nto a fnte nuber of sphercal haroncs and then the Phase Functon s expanded n Legendre polynoals slar to the Dscrete Ordnate ethod. 8. Expanson n Egenfunctons Standard technque for solvng dfferental equatons. Fnd hoogenous soluton and partcular soluton. Apply boundary condton. Drect applcaton to coplete RTE s ponderous. Dscrete Ordnates technque depends on ths approach for solvng dscretzed set of equatons. 9. Monte Carlo Method Scatterng of an ndvdual photon can be consdered to be a stochastc process, wth the Phase Functon beng the probablty densty functon for tterng at a gven angle. Photons are allowed to play a gae of chance n a coputer and by recordng the hstory of a suffcent nuber of photons, the radaton feld can n prncple be deterned to an arbtrary accuracy. The basc splcty of ths ethod allows great flexblty, and hence t can be appled to coplcated probles whch would be vrtually nsoluble by other ethods..85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 4 of 6
16 Isotropc Scatterng and Dscrete Ordnates Pertnent RTE: di(,, )/di(,, ) P(,,, ) I(,, )d d 4 e P(, )F 4 () For sotropc tterng, we have: P(,,, ) and I (, ) I(,, )d ().e. Intensty s azuthally ndependent. di(, ) I(, ) I(, )d F e d 4 (3) Applyng Gaussan Quadrature, and settng I = I (, ), we have: di I Ia Fe d 4 n j j jn n,..., n (4) Snce ths s lnear dfferental equaton, we need to seek the general soluton (soetes called the hoogenous soluton) and then the partcular soluton. Hoogenous soluton: k Try (guess) I g e where g and k are constants. di I I a (5) j j d j g( k) a g j j So, g ust be of the for L k where L s a constant. Substtutng ths back nto Eq. 5, we get the characterstc equaton for egenvalue k.85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 5 of 6
17 n a n j aj ( k) ( k ) jn j j j Note dfference n suaton (6) Ths Eq. has n roots, zero. General Soluton s: k =..n whch when = ncludes K values of n k L e I n,..., n (7) k Partcular Soluton: Try: I F he n,..., n 4 We have: n h h ah j j jn or h ah (8) n j j jn h ust be of the for = (9) wth n aj [ 4 j j () Addng the hoogenous and partcular solutons, we obtan: I n j k j Lj e F e kj 4 = -n,.. +n () The L j are deterned fro boundary condtons..85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 6 of 6
12.815, Atmospheric Radiation Dr. Robert A. McClatchey and Prof. Ronald Prinn
.85, Atospherc Radaton Dr. Robert A. McClatchey and Prof. Ronald Prnn 3. Scatterng of Radaton by Molecules and Partcles a. Introducton Here, we ll deal wth wave aspects of lght rather than quantu aspects.
More informationApplied Mathematics Letters
Appled Matheatcs Letters 2 (2) 46 5 Contents lsts avalable at ScenceDrect Appled Matheatcs Letters journal hoepage: wwwelseverco/locate/al Calculaton of coeffcents of a cardnal B-splne Gradr V Mlovanovć
More informationScattering by a perfectly conducting infinite cylinder
Scatterng by a perfectly conductng nfnte cylnder Reeber that ths s the full soluton everywhere. We are actually nterested n the scatterng n the far feld lt. We agan use the asyptotc relatonshp exp exp
More informationXII.3 The EM (Expectation-Maximization) Algorithm
XII.3 The EM (Expectaton-Maxzaton) Algorth Toshnor Munaata 3/7/06 The EM algorth s a technque to deal wth varous types of ncoplete data or hdden varables. It can be appled to a wde range of learnng probles
More informationFermi-Dirac statistics
UCC/Physcs/MK/EM/October 8, 205 Fer-Drac statstcs Fer-Drac dstrbuton Matter partcles that are eleentary ostly have a type of angular oentu called spn. hese partcles are known to have a agnetc oent whch
More informationLeast Squares Fitting of Data
Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 1998-2014. All Rghts Reserved. Created: July 15, 1999 Last Modfed: February 9, 2008 Contents 1 Lnear Fttng
More informationFinite Vector Space Representations Ross Bannister Data Assimilation Research Centre, Reading, UK Last updated: 2nd August 2003
Fnte Vector Space epresentatons oss Bannster Data Asslaton esearch Centre, eadng, UK ast updated: 2nd August 2003 Contents What s a lnear vector space?......... 1 About ths docuent............ 2 1. Orthogonal
More informationSystem in Weibull Distribution
Internatonal Matheatcal Foru 4 9 no. 9 94-95 Relablty Equvalence Factors of a Seres-Parallel Syste n Webull Dstrbuton M. A. El-Dacese Matheatcs Departent Faculty of Scence Tanta Unversty Tanta Egypt eldacese@yahoo.co
More information5.04, Principles of Inorganic Chemistry II MIT Department of Chemistry Lecture 32: Vibrational Spectroscopy and the IR
5.0, Prncples of Inorganc Chemstry II MIT Department of Chemstry Lecture 3: Vbratonal Spectroscopy and the IR Vbratonal spectroscopy s confned to the 00-5000 cm - spectral regon. The absorpton of a photon
More informationIntegral Transforms and Dual Integral Equations to Solve Heat Equation with Mixed Conditions
Int J Open Probles Copt Math, Vol 7, No 4, Deceber 214 ISSN 1998-6262; Copyrght ICSS Publcaton, 214 www-csrsorg Integral Transfors and Dual Integral Equatons to Solve Heat Equaton wth Mxed Condtons Naser
More informationOn Pfaff s solution of the Pfaff problem
Zur Pfaff scen Lösung des Pfaff scen Probles Mat. Ann. 7 (880) 53-530. On Pfaff s soluton of te Pfaff proble By A. MAYER n Lepzg Translated by D. H. Delpenc Te way tat Pfaff adopted for te ntegraton of
More informationSolutions for Homework #9
Solutons for Hoewor #9 PROBEM. (P. 3 on page 379 n the note) Consder a sprng ounted rgd bar of total ass and length, to whch an addtonal ass s luped at the rghtost end. he syste has no dapng. Fnd the natural
More informationSlobodan Lakić. Communicated by R. Van Keer
Serdca Math. J. 21 (1995), 335-344 AN ITERATIVE METHOD FOR THE MATRIX PRINCIPAL n-th ROOT Slobodan Lakć Councated by R. Van Keer In ths paper we gve an teratve ethod to copute the prncpal n-th root and
More informationLeast Squares Fitting of Data
Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 1998-2015. All Rghts Reserved. Created: July 15, 1999 Last Modfed: January 5, 2015 Contents 1 Lnear Fttng
More informationElastic Collisions. Definition: two point masses on which no external forces act collide without losing any energy.
Elastc Collsons Defnton: to pont asses on hch no external forces act collde thout losng any energy v Prerequstes: θ θ collsons n one denson conservaton of oentu and energy occurs frequently n everyday
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationThe Parity of the Number of Irreducible Factors for Some Pentanomials
The Party of the Nuber of Irreducble Factors for Soe Pentanoals Wolfra Koepf 1, Ryul K 1 Departent of Matheatcs Unversty of Kassel, Kassel, F. R. Gerany Faculty of Matheatcs and Mechancs K Il Sung Unversty,
More informationTitle: Radiative transitions and spectral broadening
Lecture 6 Ttle: Radatve transtons and spectral broadenng Objectves The spectral lnes emtted by atomc vapors at moderate temperature and pressure show the wavelength spread around the central frequency.
More informationC/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1
C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned
More informationExcess Error, Approximation Error, and Estimation Error
E0 370 Statstcal Learnng Theory Lecture 10 Sep 15, 011 Excess Error, Approxaton Error, and Estaton Error Lecturer: Shvan Agarwal Scrbe: Shvan Agarwal 1 Introducton So far, we have consdered the fnte saple
More informationCHAPTER 10 ROTATIONAL MOTION
CHAPTER 0 ROTATONAL MOTON 0. ANGULAR VELOCTY Consder argd body rotates about a fxed axs through pont O n x-y plane as shown. Any partcle at pont P n ths rgd body rotates n a crcle of radus r about O. The
More informationChapter 12 Lyes KADEM [Thermodynamics II] 2007
Chapter 2 Lyes KDEM [Therodynacs II] 2007 Gas Mxtures In ths chapter we wll develop ethods for deternng therodynac propertes of a xture n order to apply the frst law to systes nvolvng xtures. Ths wll be
More informationRate of Absorption and Stimulated Emission
MIT Department of Chemstry 5.74, Sprng 005: Introductory Quantum Mechancs II Instructor: Professor Andre Tokmakoff p. 81 Rate of Absorpton and Stmulated Emsson The rate of absorpton nduced by the feld
More informationSpecial Relativity and Riemannian Geometry. Department of Mathematical Sciences
Tutoral Letter 06//018 Specal Relatvty and Reannan Geoetry APM3713 Seester Departent of Matheatcal Scences IMPORTANT INFORMATION: Ths tutoral letter contans the solutons to Assgnent 06. BAR CODE Learn
More informationWhat is LP? LP is an optimization technique that allocates limited resources among competing activities in the best possible manner.
(C) 998 Gerald B Sheblé, all rghts reserved Lnear Prograng Introducton Contents I. What s LP? II. LP Theor III. The Splex Method IV. Refneents to the Splex Method What s LP? LP s an optzaton technque that
More informationFormal solvers of the RT equation
Formal solvers of the RT equaton Formal RT solvers Runge- Kutta (reference solver) Pskunov N.: 979, Master Thess Long characterstcs (Feautrer scheme) Cannon C.J.: 970, ApJ 6, 55 Short characterstcs (Hermtan
More informationChapter One Mixture of Ideal Gases
herodynacs II AA Chapter One Mxture of Ideal Gases. Coposton of a Gas Mxture: Mass and Mole Fractons o deterne the propertes of a xture, we need to now the coposton of the xture as well as the propertes
More informationApplied Nuclear Physics (Fall 2004) Lecture 23 (12/3/04) Nuclear Reactions: Energetics and Compound Nucleus
.101 Appled Nuclear Physcs (Fall 004) Lecture 3 (1/3/04) Nuclear Reactons: Energetcs and Compound Nucleus References: W. E. Meyerhof, Elements of Nuclear Physcs (McGraw-Hll, New York, 1967), Chap 5. Among
More information> To construct a potential representation of E and B, you need a vector potential A r, t scalar potential ϕ ( F,t).
MIT Departent of Chestry p. 54 5.74, Sprng 4: Introductory Quantu Mechancs II Instructor: Prof. Andre Tokakoff Interacton of Lght wth Matter We want to derve a Haltonan that we can use to descrbe the nteracton
More informationA DISCONTINUOUS LEAST-SQUARES SPATIAL DISCRETIZATION FOR THE S N EQUATIONS. A Thesis LEI ZHU
A DISCONTINUOUS LEAST-SQUARES SPATIAL DISCRETIZATION FOR THE S N EQUATIONS A Thess by LEI ZHU Subtted to the Offce of Graduate Studes of Texas A&M Unversty n partal fulfllent of the requreents for the
More informationFrequency dependence of the permittivity
Frequency dependence of the permttvty February 7, 016 In materals, the delectrc constant and permeablty are actually frequency dependent. Ths does not affect our results for sngle frequency modes, but
More informationChapter 1. Theory of Gravitation
Chapter 1 Theory of Gravtaton In ths chapter a theory of gravtaton n flat space-te s studed whch was consdered n several artcles by the author. Let us assue a flat space-te etrc. Denote by x the co-ordnates
More informationMultipoint Analysis for Sibling Pairs. Biostatistics 666 Lecture 18
Multpont Analyss for Sblng ars Bostatstcs 666 Lecture 8 revously Lnkage analyss wth pars of ndvduals Non-paraetrc BS Methods Maxu Lkelhood BD Based Method ossble Trangle Constrant AS Methods Covered So
More informationThe Impact of the Earth s Movement through the Space on Measuring the Velocity of Light
Journal of Appled Matheatcs and Physcs, 6, 4, 68-78 Publshed Onlne June 6 n ScRes http://wwwscrporg/journal/jap http://dxdoorg/436/jap646 The Ipact of the Earth s Moeent through the Space on Measurng the
More informationBAYESIAN CURVE FITTING USING PIECEWISE POLYNOMIALS. Dariusz Biskup
BAYESIAN CURVE FITTING USING PIECEWISE POLYNOMIALS Darusz Bskup 1. Introducton The paper presents a nonparaetrc procedure for estaton of an unknown functon f n the regresson odel y = f x + ε = N. (1) (
More informationSection 8.3 Polar Form of Complex Numbers
80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the
More informationEXAMPLES of THEORETICAL PROBLEMS in the COURSE MMV031 HEAT TRANSFER, version 2017
EXAMPLES of THEORETICAL PROBLEMS n the COURSE MMV03 HEAT TRANSFER, verson 207 a) What s eant by sotropc ateral? b) What s eant by hoogeneous ateral? 2 Defne the theral dffusvty and gve the unts for the
More informationPhysics 3A: Linear Momentum. Physics 3A: Linear Momentum. Physics 3A: Linear Momentum. Physics 3A: Linear Momentum
Recall that there was ore to oton than just spee A ore coplete escrpton of oton s the concept of lnear oentu: p v (8.) Beng a prouct of a scalar () an a vector (v), oentu s a vector: p v p y v y p z v
More information,..., k N. , k 2. ,..., k i. The derivative with respect to temperature T is calculated by using the chain rule: & ( (5) dj j dt = "J j. k i.
Suppleentary Materal Dervaton of Eq. 1a. Assue j s a functon of the rate constants for the N coponent reactons: j j (k 1,,..., k,..., k N ( The dervatve wth respect to teperature T s calculated by usng
More informationBoundaries, Near-field Optics
Boundares, Near-feld Optcs Fve boundary condtons at an nterface Fresnel Equatons : Transmsson and Reflecton Coeffcents Transmttance and Reflectance Brewster s condton a consequence of Impedance matchng
More informationHomework 4. 1 Electromagnetic surface waves (55 pts.) Nano Optics, Fall Semester 2015 Photonics Laboratory, ETH Zürich
Homework 4 Contact: frmmerm@ethz.ch Due date: December 04, 015 Nano Optcs, Fall Semester 015 Photoncs Laboratory, ETH Zürch www.photoncs.ethz.ch The goal of ths problem set s to understand how surface
More informationˆ A = A 0 e i (k r ωt) + c.c. ( ωt) e ikr. + c.c. k,j
p. Supp. 9- Suppleent to Rate of Absorpton and Stulated Esson Here are a ouple of ore detaled dervatons: Let s look a lttle ore arefully at the rate of absorpton w k ndued by an sotrop, broadband lght
More informationarxiv: v2 [math.co] 3 Sep 2017
On the Approxate Asyptotc Statstcal Independence of the Peranents of 0- Matrces arxv:705.0868v2 ath.co 3 Sep 207 Paul Federbush Departent of Matheatcs Unversty of Mchgan Ann Arbor, MI, 4809-043 Septeber
More information( ) + + REFLECTION FROM A METALLIC SURFACE
REFLECTION FROM A METALLIC SURFACE For a metallc medum the delectrc functon and the ndex of refracton are complex valued functons. Ths s also the case for semconductors and nsulators n certan frequency
More informationCHAPTER 14 GENERAL PERTURBATION THEORY
CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves
More informationModule 1 : The equation of continuity. Lecture 1: Equation of Continuity
1 Module 1 : The equaton of contnuty Lecture 1: Equaton of Contnuty 2 Advanced Heat and Mass Transfer: Modules 1. THE EQUATION OF CONTINUITY : Lectures 1-6 () () () (v) (v) Overall Mass Balance Momentum
More informationMoments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.
Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these
More informationSeveral generation methods of multinomial distributed random number Tian Lei 1, a,linxihe 1,b,Zhigang Zhang 1,c
Internatonal Conference on Appled Scence and Engneerng Innovaton (ASEI 205) Several generaton ethods of ultnoal dstrbuted rando nuber Tan Le, a,lnhe,b,zhgang Zhang,c School of Matheatcs and Physcs, USTB,
More informationAN ANALYSIS OF A FRACTAL KINETICS CURVE OF SAVAGEAU
AN ANALYI OF A FRACTAL KINETIC CURE OF AAGEAU by John Maloney and Jack Hedel Departent of Matheatcs Unversty of Nebraska at Oaha Oaha, Nebraska 688 Eal addresses: aloney@unoaha.edu, jhedel@unoaha.edu Runnng
More informationtotal If no external forces act, the total linear momentum of the system is conserved. This occurs in collisions and explosions.
Lesson 0: Collsons, Rotatonal netc Energy, Torque, Center o Graty (Sectons 7.8 Last te we used ewton s second law to deelop the pulse-oentu theore. In words, the theore states that the change n lnear oentu
More informationQuantum Particle Motion in Physical Space
Adv. Studes Theor. Phys., Vol. 8, 014, no. 1, 7-34 HIKARI Ltd, www.-hkar.co http://dx.do.org/10.1988/astp.014.311136 Quantu Partcle Moton n Physcal Space A. Yu. Saarn Dept. of Physcs, Saara State Techncal
More informationDenote the function derivatives f(x) in given points. x a b. Using relationships (1.2), polynomials (1.1) are written in the form
SET OF METHODS FO SOUTION THE AUHY POBEM FO STIFF SYSTEMS OF ODINAY DIFFEENTIA EUATIONS AF atypov and YuV Nulchev Insttute of Theoretcal and Appled Mechancs SB AS 639 Novosbrs ussa Introducton A constructon
More informationInductance Calculation for Conductors of Arbitrary Shape
CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationEPR Paradox and the Physical Meaning of an Experiment in Quantum Mechanics. Vesselin C. Noninski
EPR Paradox and the Physcal Meanng of an Experment n Quantum Mechancs Vesseln C Nonnsk vesselnnonnsk@verzonnet Abstract It s shown that there s one purely determnstc outcome when measurement s made on
More information4.5. QUANTIZED RADIATION FIELD
4-1 4.5. QUANTIZED RADIATION FIELD Baground Our treatent of the vetor potental has drawn on the onohroat plane-wave soluton to the wave-euaton for A. The uantu treatent of lght as a partle desrbes the
More informationOn the number of regions in an m-dimensional space cut by n hyperplanes
6 On the nuber of regons n an -densonal space cut by n hyperplanes Chungwu Ho and Seth Zeran Abstract In ths note we provde a unfor approach for the nuber of bounded regons cut by n hyperplanes n general
More informationImplicit Integration Henyey Method
Implct Integraton Henyey Method In realstc stellar evoluton codes nstead of a drect ntegraton usng for example the Runge-Kutta method one employs an teratve mplct technque. Ths s because the structure
More informationOur focus will be on linear systems. A system is linear if it obeys the principle of superposition and homogenity, i.e.
SSTEM MODELLIN In order to solve a control syste proble, the descrptons of the syste and ts coponents ust be put nto a for sutable for analyss and evaluaton. The followng ethods can be used to odel physcal
More informationPHYS 1443 Section 002 Lecture #20
PHYS 1443 Secton 002 Lecture #20 Dr. Jae Condtons for Equlbru & Mechancal Equlbru How to Solve Equlbru Probles? A ew Exaples of Mechancal Equlbru Elastc Propertes of Solds Densty and Specfc Gravty lud
More informationChapter 10 Sinusoidal Steady-State Power Calculations
Chapter 0 Snusodal Steady-State Power Calculatons n Chapter 9, we calculated the steady state oltages and currents n electrc crcuts dren by snusodal sources. We used phasor ethod to fnd the steady state
More informationRevision: December 13, E Main Suite D Pullman, WA (509) Voice and Fax
.9.1: AC power analyss Reson: Deceber 13, 010 15 E Man Sute D Pullan, WA 99163 (509 334 6306 Voce and Fax Oerew n chapter.9.0, we ntroduced soe basc quanttes relate to delery of power usng snusodal sgnals.
More informationPHYS 705: Classical Mechanics. Canonical Transformation II
1 PHYS 705: Classcal Mechancs Canoncal Transformaton II Example: Harmonc Oscllator f ( x) x m 0 x U( x) x mx x LT U m Defne or L p p mx x x m mx x H px L px p m p x m m H p 1 x m p m 1 m H x p m x m m
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More information1 Definition of Rademacher Complexity
COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture #9 Scrbe: Josh Chen March 5, 2013 We ve spent the past few classes provng bounds on the generalzaton error of PAClearnng algorths for the
More informationAtmospheric Radiation Fall 2008
MIT OpenCourseWare http://ocw.mt.edu.85 Atmospherc Radaton Fall 008 For nformaton about ctng these materals or our Terms of Use, vst: http://ocw.mt.edu/terms. .85, Atmospherc Radaton Dr. Robert A. McClatchey
More informationA new Approach for Solving Linear Ordinary Differential Equations
, ISSN 974-57X (Onlne), ISSN 974-5718 (Prnt), Vol. ; Issue No. 1; Year 14, Copyrght 13-14 by CESER PUBLICATIONS A new Approach for Solvng Lnear Ordnary Dfferental Equatons Fawz Abdelwahd Department of
More information1. Statement of the problem
Volue 14, 010 15 ON THE ITERATIVE SOUTION OF A SYSTEM OF DISCRETE TIMOSHENKO EQUATIONS Peradze J. and Tsklaur Z. I. Javakhshvl Tbls State Uversty,, Uversty St., Tbls 0186, Georga Georgan Techcal Uversty,
More informationOne-sided finite-difference approximations suitable for use with Richardson extrapolation
Journal of Computatonal Physcs 219 (2006) 13 20 Short note One-sded fnte-dfference approxmatons sutable for use wth Rchardson extrapolaton Kumar Rahul, S.N. Bhattacharyya * Department of Mechancal Engneerng,
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationBy M. O'Neill,* I. G. Sinclairf and Francis J. Smith
52 Polynoal curve fttng when abscssas and ordnates are both subject to error By M. O'Nell,* I. G. Snclarf and Francs J. Sth Departents of Coputer Scence and Appled Matheatcs, School of Physcs and Appled
More informationISSN: ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 3, Issue 4, October 2013
SSN: 77-3754 SO 9:8 Certfed nternatonal Journal of Engneerng nnovatve Technology (JET) Volue 3 ssue 4 October 3 nfluence of Kerr Effect on Tweeer Center Locaton n Nonear Medu Van Na Hoang Thanh Le Cao
More informationPHYS 705: Classical Mechanics. Hamilton-Jacobi Equation
1 PHYS 705: Classcal Mechancs Hamlton-Jacob Equaton Hamlton-Jacob Equaton There s also a very elegant relaton between the Hamltonan Formulaton of Mechancs and Quantum Mechancs. To do that, we need to derve
More informationThe Feynman path integral
The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space
More informationHigh-Contrast Gratings based Spoof Surface Plasmons
Suppleentary Inforaton Hgh-Contrast Gratngs base Spoof Surface Plasons Zhuo L 123*+ Langlang Lu 1+ ngzheng Xu 1 Pngpng Nng 1 Chen Chen 1 Ja Xu 1 Xnle Chen 1 Changqng Gu 1 & Quan Qng 3 1 Key Laboratory
More informationDetermination of the Confidence Level of PSD Estimation with Given D.O.F. Based on WELCH Algorithm
Internatonal Conference on Inforaton Technology and Manageent Innovaton (ICITMI 05) Deternaton of the Confdence Level of PSD Estaton wth Gven D.O.F. Based on WELCH Algorth Xue-wang Zhu, *, S-jan Zhang
More information16 Reflection and transmission, TE mode
16 Reflecton transmsson TE mode Last lecture we learned how to represent plane-tem waves propagatng n a drecton ˆ n terms of feld phasors such that η = Ẽ = E o e j r H = ˆ Ẽ η µ ɛ = ˆ = ω µɛ E o =0. Such
More informationWaveguides and resonant cavities
Wavegudes and resonant cavtes February 26, 2016 Essentally, a wavegude s a conductng tube of unform cross-secton and a cavty s a wavegude wth end caps. The dmensons of the gude or cavty are chosen to transmt,
More informationMatrix Mechanics Exercises Using Polarized Light
Matrx Mechancs Exercses Usng Polarzed Lght Frank Roux Egenstates and operators are provded for a seres of matrx mechancs exercses nvolvng polarzed lght. Egenstate for a -polarzed lght: Θ( θ) ( ) smplfy
More informationOn Syndrome Decoding of Punctured Reed-Solomon and Gabidulin Codes 1
Ffteenth Internatonal Workshop on Algebrac and Cobnatoral Codng Theory June 18-24, 2016, Albena, Bulgara pp. 35 40 On Syndroe Decodng of Punctured Reed-Soloon and Gabduln Codes 1 Hannes Bartz hannes.bartz@tu.de
More informationThe Geometry of Logit and Probit
The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.
More informationCOS 511: Theoretical Machine Learning
COS 5: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture #0 Scrbe: José Sões Ferrera March 06, 203 In the last lecture the concept of Radeacher coplexty was ntroduced, wth the goal of showng that
More informationCME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 13
CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 13 GENE H GOLUB 1 Iteratve Methods Very large problems (naturally sparse, from applcatons): teratve methods Structured matrces (even sometmes dense,
More informationNumerical Solution of Ordinary Differential Equations
Numercal Methods (CENG 00) CHAPTER-VI Numercal Soluton of Ordnar Dfferental Equatons 6 Introducton Dfferental equatons are equatons composed of an unknown functon and ts dervatves The followng are examples
More informationECE 107: Electromagnetism
ECE 107: Electromagnetsm Set 8: Plane waves Instructor: Prof. Vtaly Lomakn Department of Electrcal and Computer Engneerng Unversty of Calforna, San Dego, CA 92093 1 Wave equaton Source-free lossless Maxwell
More informationThermal-Fluids I. Chapter 18 Transient heat conduction. Dr. Primal Fernando Ph: (850)
hermal-fluds I Chapter 18 ransent heat conducton Dr. Prmal Fernando prmal@eng.fsu.edu Ph: (850) 410-6323 1 ransent heat conducton In general, he temperature of a body vares wth tme as well as poston. In
More informationProblem 1: To prove that under the assumptions at hand, the group velocity of an EM wave is less than c, I am going to show that
PHY 387 K. Solutons for problem set #7. Problem 1: To prove that under the assumptons at hand, the group velocty of an EM wave s less than c, I am gong to show that (a) v group < v phase, and (b) v group
More informationDescription of the Force Method Procedure. Indeterminate Analysis Force Method 1. Force Method con t. Force Method con t
Indeternate Analyss Force Method The force (flexblty) ethod expresses the relatonshps between dsplaceents and forces that exst n a structure. Prary objectve of the force ethod s to deterne the chosen set
More informationEXACT TRAVELLING WAVE SOLUTIONS FOR THREE NONLINEAR EVOLUTION EQUATIONS BY A BERNOULLI SUB-ODE METHOD
www.arpapress.co/volues/vol16issue/ijrras_16 10.pdf EXACT TRAVELLING WAVE SOLUTIONS FOR THREE NONLINEAR EVOLUTION EQUATIONS BY A BERNOULLI SUB-ODE METHOD Chengbo Tan & Qnghua Feng * School of Scence, Shandong
More information1.3 Hence, calculate a formula for the force required to break the bond (i.e. the maximum value of F)
EN40: Dynacs and Vbratons Hoework 4: Work, Energy and Lnear Moentu Due Frday March 6 th School of Engneerng Brown Unversty 1. The Rydberg potental s a sple odel of atoc nteractons. It specfes the potental
More informationTHEOREMS OF QUANTUM MECHANICS
THEOREMS OF QUANTUM MECHANICS In order to develop methods to treat many-electron systems (atoms & molecules), many of the theorems of quantum mechancs are useful. Useful Notaton The matrx element A mn
More informationUsing T.O.M to Estimate Parameter of distributions that have not Single Exponential Family
IOSR Journal of Mathematcs IOSR-JM) ISSN: 2278-5728. Volume 3, Issue 3 Sep-Oct. 202), PP 44-48 www.osrjournals.org Usng T.O.M to Estmate Parameter of dstrbutons that have not Sngle Exponental Famly Jubran
More informationITERATIVE ESTIMATION PROCEDURE FOR GEOSTATISTICAL REGRESSION AND GEOSTATISTICAL KRIGING
ESE 5 ITERATIVE ESTIMATION PROCEDURE FOR GEOSTATISTICAL REGRESSION AND GEOSTATISTICAL KRIGING Gven a geostatstcal regresson odel: k Y () s x () s () s x () s () s, s R wth () unknown () E[ ( s)], s R ()
More informationA Proof of a Conjecture for the Number of Ramified Coverings of the Sphere by the Torus
Journal of Cobnatoral Theory, Seres A 88, 4658 (999) Artcle ID jcta99999, avalable onlne at httpwwwdealbraryco on A Proof of a Conjecture for the Nuber of Rafed Coverngs of the Sphere by the Torus I P
More informationCHAPTER 5: Lie Differentiation and Angular Momentum
CHAPTER 5: Le Dfferentaton and Angular Momentum Jose G. Vargas 1 Le dfferentaton Kähler s theory of angular momentum s a specalzaton of hs approach to Le dfferentaton. We could deal wth the former drectly,
More informationIntroduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:
CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and
More informationDeparture Process from a M/M/m/ Queue
Dearture rocess fro a M/M// Queue Q - (-) Q Q3 Q4 (-) Knowledge of the nature of the dearture rocess fro a queue would be useful as we can then use t to analyze sle cases of queueng networs as shown. The
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationPoisson brackets and canonical transformations
rof O B Wrght Mechancs Notes osson brackets and canoncal transformatons osson Brackets Consder an arbtrary functon f f ( qp t) df f f f q p q p t But q p p where ( qp ) pq q df f f f p q q p t In order
More information