Processi di Radiazione e MHD

Transcription

1 Proessi di Radiazione e MHD 0. Overview of elestial bodies and sky at various frequenies 1. Definition of main astrophysial observables. Radiative transfer 3. Blak body radiation 4. basi theory of radiation field 5. Lienard Wiehart potentials and Larmor's formula daniele.dallaasa@unibo.it...reeving students: Daniele Dallaasa Tuesday & Thursday 15:30 17:00 offie nd floor Astronomy Department

2 Emission proesses Observations: the eletromagneti spetrum TH ER M AL CONTINUUM proesses NO N TH ER M AL BLACK BODY (radio + optial + (soft)x rays) BREMSSTRAHLUNG (radio to gamma rays) SYNCHROTRON [radio to optial / X rays / gamma rays] INVERSE COMPTON [X rays, (gamma rays)] LINE proesses (emission / absorption) Partiular (eletroni) transitions: ''allowed and prohibited'' e.g. 1m line (radio) a wealth of transitions from the radio to the X and gamma rays

3 The Sky at various wavelengthss 408 MHz 1 m rays

4 The Galati plane

5 Individual objets: images (1)

6 Individual objets: images ()

7 Individual objets: images (3)

8 Individual objets: images (4)

9 Individual objets: images (5) Galati entre in the X rays (Sgr A*)

10 Radio IR O Individual objets: spetra (1) SEDs [Fν vs. ν ] X Radio IR O X

11 Individual objets () Energy output: νfν Other examples

12 On the Earth, we get a small fration of the radiation oming from the outskirts of the Universe

13 Glossary: observatonal quantities Spetral/Monoromati Luminosity L 4 D S [W Hz 1 ] / [erg s 1 Hz 1 ] Power (Bolometri / Absolute Luminosity) L 0 L d [ W ] / [erg s 1 ] Flux Density L S [W Hz m ] / [erg s Hz m ] [Jy 10 W Hz m ] 4 D (surfae) Brightness S B [W Hz 1 m ster 1 ] / [erg s 1 Hz 1 m ster 1 ] d (speifi) Emissivity L J [W Hz 1 m 3 ] / dv [erg s 1 Hz 1 m 3 ] Magnitude: m.5logsν + A

14 Problem: Radiative transfer In astrophysis, radiation may be originated anywhere in the Universe. There is a long travel before getting revealed. Speial attention must be paid to: 1. Photons may reah the Earth undisturbed after their origin. Photons may interat with matter before eventually reahing the Earth 3. Along a given Line of Sight (LoS) photons may ome from many ontributors 4. How photons are originated? 4a. Intrinsi prodution 4b. Sattering (hange of diretion/energy) 4. subtration (absorption) 4d. addition (emission) Magnitude: m.5logsν + A

15 Radiative transfer: emission D dl dv D dω. B(ν) * Observer Soure L l0 dv de d dl d d [ds ] J J dl dt 4 4 D the emission from the whole homogeneous & transparent loud is d ds J l0 from whih the brightness is 4 ds J J db dl B lo d 4 4 [ ]

16 Radiative transfer: absorption D dl D dω. B(ν) * Observer Soure L l0 Absorption: db B dl if onstant aross the homogeneous loud db dl then after integrating and going from log to exp B B,l o B e l o l o in a slightly different form S,l o S e is known as opaity l o

17 Radiative transfer: emission & absorption within a loud D dl dv D dω. B(ν) * Observer Soure lo A loud is opaque to its own radiation: it is self absorbed J db E dl 4 [ ] [ ] J l e dl 4 and integrating over l o we get emission absorption db db E e B,l o [ l o J 4 ] 1 e l o o B loud 1 e l o

18 Radiative transfer: emission & absorption within a loud B,l o [ ] J l l 1 e B loud 1 e B loud 1 e 4 o o l o 1 optially THICK regime B, l o J 4 l o 1 optially THIN regime B, l o B loud l o J J l o lo 4 4 in priniple B,l o an grow as large as l o inreases

19 Radiative transfer: general ase D dl dv D dω Soure B0(ν). * Observer l0 General ase Absorption Emission J db B dl dl 4 [ [ db B dl B,l o B o e l o B,l o B o e [ [ J 4 ] ] eq. diff. di Leibniz ] J 4 J l o 4 ] 1 e 1 e l o

20 speial regimes J l o B, l o B o e 1 e 4 [ ] then e 0 implies J B, l o i.e. in the optially THICK regime 4 the bakground soure is fully absorbed, and only sattered photons may esape from the loud 1 then e 1 o implies 1 e and then J l o B, l o B o i.e. in the optially THIN regime 4 unattenuated bakground soure with the addition of photons from the loud 1 [ ] [ Definition: Mean Free Path: optial depth (p. 14 Rybiky L) ] Lmfp 1 probability for a photon to travel a distane equal to

21 Blak Body Radiation (1) Thermodynamis: thermal radiation is partially responsible for heat exhange between bodies spetral absorption spetral refletion 1 spetral transmission at thermodynami equilibrium 1 spetral emissivity Kirhhoff's law of thermal radiation: Namely, at thermal equilibrium the emissivity of a body ( ) equals its absorbane ( ) impliation: It is not possible to thermally radiate more energy than a Blak Body (unless thermal equilibrium breaks down) j B, T

22 Blak Body Radiation () An enlosure at T do not let radiation in /out until thermal equilibrium is reahed. Then a small hole is made to measure radiation inside w/o disturbing equilibrium. massless photons do not onserve number and self interation is negligible the number of photons adjusts itself in equilibrium at T Let's onsider open enlosures at the same T separated by a filter allowing a single v If I( ) I'( ) energy will flow between the two enlosures, but they are at the same T and this violates the II priniple of TD I(v) filter (v) radiation field I (,T) is a universal funtion related to and T, but independent of enlosure properties: I'(v) orollary: I( ) is isotropi; in partiular I( ) B (,T) is alled the Plank funtion

23 Blak Body Radiation (3) A body at a given temperature T in an enlosure; in this ase: S B, T j B, T I( ) T), ( B [ Kirhhoff ' s law ] the transfer funtion for thermal radiation is: d I ds d I d I I B,T namely B,T S(v) B(v,T) within BB enlosure, throughout I(v) B(v,T) Distintion between thermal radiation: S(v) B(v,T) BB radiation: I(v) B(v,T) thermal radiation beomes BB radiation for optially thik media only.

24 Blak Body Radiation (4) thermodynamis u, P, T, V Let's onsider a BB enlosure with a piston: work (pdv) an be done/extrated (aphys ex: CMB) I law: du dq dq T II law: ds U ds uv Q heat,u total energy, S entropy ; P V du T u 3 ; u dv T ds is a perfet differential S T V wrong in old slides P dv V du T dt u 4 J d 1u dv 3T S V T V du T 4u 3T 4 B,T d 4u dv 3T

25 Blak Body Radiation (5) Let's derive.vs. V and T respetively: S T V 1 du T dt 4u 3T 1 1 du 3 T dt 4u 3 T log u 4 log T log a 4 1 du 3 T dt du dt 4 u T providing the well known Stefan Boltzmann law u T at4 For an isotropi radiator and then I u J 4 B,T d it is then possible to get the integrated Plank funtion B T 4 B T a B,T d 4 T 4

26 Blak Body Radiation (6) the flux oming out from an isotropially emitting surfae is the brightness: F where F d B,T d a 4 summary: erg m s 1 deg 4 a 4 B T T erg m 3 deg 4 Any (elestial) body with ( ) 1 an be approximated as a BB (thermal equilibrium) For a given temperature, the BB is the body with the highest possible emissivity

27 Blak Body Radiation (7) the Plank spetrum photon states in a BB avity: let's onsider a photon hv moving along d and its wave vetor k ( )d ( )d the box have sizes Lx,Ly,Lz, hv standing wave within the box # of nodes along a given Lx: nx kx Lx / the wave hanged in ase nx kx Lx / 1 as a whole the number of states is nx ny nz LxLyLz d k/ ( V d k/ ( 3 d k k dk d 3 d d 3 the density of states [must be x to take into aount polarization!] (i.e. number of states per solid angle, volume and frequeny) s N d dv d 3

28 Blak Body Radiation (8) the Plank spetrum Let's express the average energy for eah state: eah state of energy hv ontains n photons with n 0,1,,... total energy: En n h from statistial mehanis, the probability of a given state of energy En : P E n Average energy: n 0 E n e n 0 e but E n 0 e kt kt h 1 e kt h kt ln 1 / kt nh kt nh h e e nh E e h kt 1 1 e h nh kt n 0 e nh kt then kt h energy of a photon of frequeny v 1 oupation number nv (Bose Einstein statistis in limitless # of partiles with 0 hemial potential)

29 Blak Body Radiation (9) the Plank spetrum energy per solid angle, volume, frequeny: s E u u dv d d h e 1 h 3 dv d d e 1 h kt h kt I then we have the expression(s) for the Plank law B,T B, T 3 h 1 h e kt 1 h 1 5 h e kt 1 the two distributions peak at different plaes [ max vmax ] d d B, T d B, T d B

30 Blak Body Radiation (10) B,T B, T 3 h 1 h e kt 1 h 1 5 h e kt 1 B(λ,T) the Plank spetrum

31 Blak Body Radiation (11) the Plank spetrum visible light B,T B, T 3 h 1 h e kt 1 h 1 5 h e kt 1

32 Blak Body Radiation (1) B,T the Plank spetrum 3 h 1 h e kt 1 B,T h 1 h 5 e kt 1 B,T : BB spetral radiane W m sr 1 Hz 1 B, T : BB spetral radiane W m sr 1 m 1 T h k : : : : : : o absolute temperature of BB K frequeny Hz wavelength m 8 1 speed of light m s Plank ' s onstant J s o Boltzmann ' s onstant J K 1

33 Blak Body Radiation (13) Remarkable features: A given urve is determined by T Given a point in the plot it is possible to determine T Spetrum & approximations

34 Blak Body Radiation (14) Spetrum & approximations Plank funtion: B,T 3 h 1 h e kt 1 Low photon energies, the Rayleigh Jeans approximation: h kt B,T 3 h kt h High photon energies, the Wien approximation: h kt B,T 3 h h kt e kt

35 Blak Body Radiation (15) find the peak frequeny: Wien displaement law B, T T x that beomes equivalent to solve 0 max x 3 1 e given that x h max kt approximate root x.8 i.e. h max kt max and for wavelengths: B, T T max y 5 1 e y 0; y.8 k.8 T h T h max kt max T 0.9 m o K Hz o K 1

36 Blak Body Radiation (16) Examples Wien displaement law

37 Blak Body Radiation (17) Examples big small

38 Blak Body Radiation (18) Total (bolometri) brightness Stefan Boltzman's law Brightness Temperature (R J approximation) 3 h kt B RJ,T k T h T B 1 1 B,T B,T k k N.B. It is NOT a real temperature; it is also given for non thermal emission TB(ν) never exeeds Tk (kineti temperature) absorption mehanisms play a role wait for radiative proesses...

39 TB(ν) never exeeds Tk (kineti temperature) absorption mehanisms play a role wait for radiative proesses... TB(ν) Energy (heat) TBBB(ν) TkBB Tk TBBB(ν) TkBB TB(ν) >Tk TBBB(ν) TkBB are hosen to be between TB(ν) and Tk then, let's ouple the two bodies: TB(ν) > TBBB(ν) TkBB > Tk means that the BB heats the other body, and then TB(ν) would be heated by a older TBBB(ν)!!!!!!!!!!!!!!!!

40 Basi theory of radiation fields (0) In vauum: Maxwell equations E B 0 1 B E 0 1 E B t t let's onsider the url of third eq. and ombine it with fourth E 1 E t E using the vetor identity... E 1 E t E E 0 an idential equation holds for B given that Maxwell equations. are invariant for E B, B E solutions are waves whose time averaged Poynting vetor is S ℜ E o B o 4

41 Basi theory of radiation fields (1) In general: [1] [3] from []: B E E 4 e [] 1 B t A then [3] an be written as: Maxwell equations [ 4] B B 0 4 j 1 E t r,t vetor potential A E 1 A t 0 and the argument an be written as a gradient of a salar field 1 A t 1 E 1 equation [1] beomes E equation [4] is A A t A t 1 t 4 e 1 A t 4 j

42 Basi theory of radiation fields () using the vetor identity A A 1 t 1 A t A 1 A t A Maxwell equations A 4 j 1 t A beomes 4 j are not uniquely determined given that The potentials and A A A 1 t implies B B implies E E are known as Gauge transformations; is a salar funtion modifiations of and A The Lorentz gauge satisfies the Lorentz ondition: A 1 t whih greatly simplifies the above equations 0

43 Basi theory of radiation fields (3) Maxwell equations the two potentials an be simplified (Lorentz gauge): A 1 1 t A t 4 4 j solutions have the following expression: r,t e r,t A 3 d r' r r' 1 d 3 r' j r r' the integrals are omputed over the soure (volume) defining e and j. They are known as retarded potentials given they onsider a time delay required for light to travel from the origin of the potential (harge and urrent) to the observer

44 Basi theory of radiation fields (4) Liénard Wiehert potentials a harge q is moving along a given trajetory r r we get: at a given position ro t with veloity u r o t e r, t q [ r ro t ] j r,t q u t [ r ro t ] and e r, t d 3 r j r, t d 3 qu r q the potentials are r,t d 3 r ' dt ' e r ',t ' r r ' r r ' t ' t and inserting the above definition of harge density and integrating over distane r,t r ro t ' q t ' t dt ' r ro t ' t ' r r t ' R t ' R t ' if we introdue R o and onsider the vetor potential as well we get

45 Basi theory of radiation fields (5) Liénard Wiehert potentials() the potentials are r,t q R r,t A q 1 u t ' R in partiular if we onsider t ' t ret and hange variable to t '' R t ' t ' t ' t 1 dt ' R t ' t ' t ' t so that dt ' R t ret t t ret t ' t R t ' / 1 dt ' ' dt ' R t ' dt ' t ' R t ' if we onsider R t ' R and take the time derivative whose inrement beomes t ' R t ' R t ' R t ' R t ' R t ' R t ' R t ' R R t ' n namely R t ' u ; finally we introdue R

46 Basi theory of radiation fields (6) the inrement an then be written as dt '' Liénard Wiehert potentials(3) [1 1 n t ' u t ' ]dt ' 1 1 '' [1 n t ' u t ' ] dt dt ' and the potentials beome r,t r,t A 1 1 t ' u t ' ] t ' ' dt ' ' q R t ' [1 n 1 q 1 1 t ' u t ' ] t ' ' dt ' ' u t ' R t ' [1 n 1 the delta funtion an be integrated if t'' 0 (or equivalently t ' t ret ) yelding the Lienard Wiehart potentials where t ' r,t r,t A 1 1 n t ' u t ' q R t ret t ret qu R t ret t ret [t ' t ret ] differene wrt stati eletromagneti theory, relevant when u ~, onentrates the potentials about the partile veloity (beaming)

47 Radiation from moving harges (1) The differentiation of the Liénard Wiehert potentials (easy but lengthy) produes the radiation fields at a position r and a time t (omputed at the retarded time t ret and orresponding position r ret ): let u and then r,t E r,t B 1 n n 1 q n q 3 [ n ] 3 R R 1/R, aeleration field perpendiular to ň it is the radiation field 1/R, veloity field generalization of Coulomb's law r,t E n 1. the Coulomb's law holds for 1 and no aeleration; the E field point to the urrent position of the harge. In ase of aeleration the radiation field is then Erad r,t Brad r,t q n [ n ] 3 R Erad n r,t

48 Radiation from moving (non relativisti) harges [Rybiky Lightman] R(r,t). β Partile position at n β Partile position at tret t

49 effet of aeleration 1/R see Erad Brad n right hand system of perpendiular vetors, with Erad Brad and are onsistent with radiation solutions of Maxwell equations

50 Radiation from moving harges () omparison between the two omponents of the E [B] field [if 1 ] E rad E vel ~ R u and for a given harateristi osillation v then u E rad E vel ~ Ru u ur Two zones an be highlighted: 1. near zone, where R, and the Evel > Erad by a fator /u ; Coulomb effets dominate. far zone or wave zone, at large R /u), where Erad dominates and whose dominane progressively inreases with R ; radiation is the only thing that matters relevant to astronomy!

51 Radiation from moving harges (3) Larmor's formula if 1 fields are: Erad r,t Brad r,t q n ] [ n 3 R Erad n r,t q ] n u [n R at a given point the maximum field amplitudes an be omputed Erad Brad q u sin R Brad n ů the Poynting vetor providing the emitted power is then Erad 4 q u 4 3 q u sin 4 R sin dw dt d ad Er S

52 Radiation from moving harges (4) Larmor's formula () integrating over the angles we get the total emitted power Brad n Erad. u θ Larmor's formula

53 Dipole approximation of Larmor's formula P dw dt d 3 3 q a 3 3 dω Angular distribution: Brad q p 3 3m n N θ Erad. u θ u u Antiipare queste figure?

54 Relativisti harge: (γ >>1) Must write the Larmor's formula in an invariant form: salars are not affeted, let's write dt d. i W] 4 p m pi [ p, W..... [ dp i dp i dw q P 3 dt 3m d d [ dp i dp i ] d p d d d 1 dw d d p d Vetor ] dp d Salar

55 Relativisti harge: (γ >>1) salar & vetor variations Linear aeleration (p does not hange diretion, no entripetal aeleration) dp d dp d d p d dp d dp d P dw q 1 3 dt 3m dp d dp d The same as the non relativisti ase! 1 q 3m 3 dp d dp dt

56 Relativisti harge: (γ >>1) salar & vetor variations Centripetal aeleration (the hange in diretion largely exeeds the hange in veloity; the energy does not hange substantially during the interation) dp d [ dp i dp i ] d p d d d 1 P dp d dw q dt 3m 3 1 dw d dw d d p d dp d q 3m 3 lig ha ht rges em high iss ion dp d dp dt gh Hi gy r ene igh h sion is em

57 entripetal aeleration (ont'd) P q dw 3 dt 3m dp d q 3m 3 d p dt Important note(s): P ~ γ the most energeti harges are the most effetive emitters P ~ m the ''lightest'' harges are the most effetive emitters: eletrons (positrons) radiate 3 4 x 106 more power than protons at the same energy (γ)