Processi di Radiazione e MHD

Transcription

1 Proessi di Radiazione e MHD Setion 0. Overview of elestial bodies and sky at various frequenies. Definition of main astrophysial observables. Radiative transfer 3. Blak body radiation 4. basi theory of radiation field 5. Liénard-Wiehert potentials and Larmor's formula daniele.dallaasa@unibo.it Daniele Dallaasa

2 Proessi di Radiazione e MHD Referenes: Radiative Transfer: Fanti & Fanti.3 Padmanaban 6.8 Radiative Transfer & Blak Body Radiation Rybiky Lightman Chap (pp -7) Blak Body Radiation Any textbook of Statistial Physis Eletromagneti (retarded, Liénard Wiehert) Potentials Longair 9.7, Rybiky Lightman.5 Padmanaban (pp 50-64) Setion

3 Emission proesses Observations: the eletromagneti spetrum ER M A H ER M N -T N O SYNCHROTRON [radio to optial / X-rays / gamma-rays] INVERSE COMPTON [X-rays, (gamma-rays)] A L TH BLACK BODY (radio + optial + (soft)x-rays) BREMSSTRAHLUNG (radio to gamma-rays) L CONTINUUM proesses LINE proesses (emission / absorption) Partiular (eletroni) transitions: ''allowed and prohibited'' e.g. m line (radio) a wealth of transitions from the radio to the X- (and gamma) rays

4 The Sky at various wavelengthss 408 MHz m -rays

5 The Galati plane

6 Individual objets: images ()

7 Individual objets: images ()

8 Individual objets: images (3)

9 Individual objets: images (4)

10 Individual objets: images (5) The Galati Centre (inlusive of Sgr A*) in the X-rays

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

12 Individual objets () Energy output: νfν Other examples

13 Windows on the Universe On the Earth, we get a small fration of the radiation oming from the Outskirts of the Universe

14 Glossary: observatonal quantities Spetral/Monoromati Luminosity L 4 D S [W Hz ] / [erg s Hz ] Power (Bolometri / Absolute Luminosity) L 0 L d [W ] / [erg s ] Flux Density L 6 S [W Hz m ] / [erg s Hz m ] [Jy 0 W Hz m ] 4 D (surfae) Brightness S B [W Hz m ster ] / [erg s Hz m ster ] d (speifi) Emissivity L J [W Hz m 3 ] / dv [erg s Hz m 3 ] Magnitude: m.5logsν + A

15 Problem: Radiative transfer In astrophysis, radiation may be originated anywhere in the Universe. There is a long journey before getting revealed. Speial attention must be paid to:. 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)

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

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

18 Radiative transfer: emission & absorption within a loud -D dl dv dω *. Soure Observer B(ν) l0 A loud is opaque to its own radiation: it is self-absorbed dbe (ν) db(ν) [ ] J(ν) 4π dl μ (ν)l dbe (ν)e o and integrating over lo we get B(ν, lo) [ ] [ ] J(ν) 4π emission μ (ν)l J(ν) o ( e ) 4 π μ(ν) μ (ν)l e o dl absorption μ (ν)l B(ν)loud ( e o )

19 Radiative transfer: emission & absorption within a loud -- [ ] μ (ν)l J(ν) o ( e ) 4 πμ(ν) B(ν, lo) μ(ν)lo τ(ν) optially THICK regime B(ν, lo) μ(ν)lo μ (ν)l Bloud ( e o ) J(ν) 4 π μ(ν) τ(ν) optially THIN regime B(ν, lo) Bloud μ(ν)lo J(ν) μ(ν)lo 4 π μ(ν) in priniple B(ν, lo) an grow as large as lo inreases J(ν) lo 4π Bloud ( e τ(ν))

20 Radiative transfer: general ase D dl dv D dw *. Soure Observer B0(n) l0 Absorption db(ν) db(ν) dl B(ν, lo) Emission [ ] [ ] [ ] μ(ν)b(ν)dl + μ(ν)b(ν) + μ ( ν) lo Bo e + J(ν) 4π J(ν) 4π J(ν) 4 π μ(ν) dl Leibniz's Diff. Eq. μ( ν) lo ( e ) τ (ν) Bo e + [ J(ν)lo 4 π τ(ν) ] ( e τ( ν))

21 Speial regimes τ(ν) B(ν, lo) Bo (ν)e + τ (ν) then e τ (ν) 0 B(ν, lo) [ J(ν) 4 π τ (ν) [ J(ν)lo 4 π τ(ν) ] ( e τ (ν)) ] Opt. THICK regime: bakground soure is fully absorbed. Only sattered photons may esape from the loud τ (ν) then e τ (ν) τ(ν)+ o(τ (ν)) B(ν, lo) Bo (ν) + [ ] ( e τ (ν)) τ (ν) J(ν)lo 4π Opt. THIN regime: unattenuated bakground soure + photons from the loud DEF.: Mean Free Path L mfp μ ( ν)

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

23 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 ν If I(ν) I'(ν) 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 (ν,t) is a universal funtion related to ν and T, but independent of enlosure properties: orollary: I(ν) is isotropi; in partiular I(ν) B (ν,t) is alled the Plank funtion I'(v)

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

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

26 Blak Body Radiation (5) Let's derive.vs. V and T respetively: ( ) S T V du T dt du 3 T dt 4u 3 T 4u 3T + 4 du 3 T dt du dt 4 u T log u 4 log T +log a providing the well known Stefan Boltzmann law u(t) a T4 I For an isotropi radiator and then u J 4 B,T d it is then possible to get the integrated Plank funtion B T 4 B T a 4 B,T d 4 T

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

28 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 (π/ λ)d (πν/ )d the box have sizes Lx,Ly,Lz λ, hv standing wave within the box # of nodes along a given Lx: nx kx Lx / π the wave hanged in ase Δnx Δkx Lx / π as a whole the number of states is ΔN ΔnxΔnyΔnz LxLyLz d3k/ (π)3 V d3k/ (π)3 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

29 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,,,... En total energy: n h from statistial mehanis, the probability of a given state of energy En : P E n Average energy: E n 0 E n e nh kt nh n 0 e but E n 0 e h e h e kt h kt e / kt nh kt kt h e h kt e nh h ln kt n 0 e nh kt then kt energy of a photon of frequeny v oupation number nv (Bose- Einstein statistis in limitless # of partiles with 0 hemial potential)

30 Blak Body Radiation (9) the Plank spetrum energy per solid angle, volume, frequeny: h E s u dv d d 3 h dv d d e kt h 3 I B u 3 h e kt then we have the expression(s) for the Plank law B(ν,T ) B(λ,T ) h ν3 e hν kt h 5 λ e h λ kt the two distributions peak at different plaes [ λmax vmax ] d d B, T d B, T d

31 B(ν, T) hν 3 hν kt e Visible light Blak Body Radiation (0) the Plank spetrum

32 Blak Body Radiation () B(λ, T) h λ 5 h λ kt e the Plank spetrum

33 Blak Body Radiation () B(ν, T) hν 3 the Plank spetrum hν kt e B(λ, T) h λ Remarkable features: A given urve is determined by T No rossing point among different B (ν,t) or different B (λ,t) Given a point in the plot it is possible to determine T 5 h λ kt e

34 Blak Body Radiation (3) Spetrum & approximations

35 Blak Body Radiation (4) Spetrum & approximations The ''Plank'' funtion: B(ν, T) h ν 3 hν e kt High photon energies, the Wien approximation holds : hν kt B(ν, T) h ν 3 e hν kt Low photon energies, the Rayleigh Jeans approximation holds : hν kt B(ν, T) 3 h ν kt hν kt ν ( )

36 Blak Body Radiation (5) find the peak frequeny: Wien displaement law B, T T x that beomes equivalent to solve 0 max 3 e x 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 e y 0; y.8 k.8 T h T h max kt max T 0.9 m deg Hz deg

37 Blak Body Radiation (6) Examples Wien displaement law

38 Blak Body Radiation (7) Examples big small

39 Blak Body Radiation (8) CMB spetrum

40 Total (bolometri) brightness Stefan-Boltzman's law 3 h kt Brightness Temperature B RJ,T k T h (R-J approximation) T B B,T B,T k k N.B. In some astrophyial ases, although it is given, it is NOT a real temperature In fat, it is also provided for non-thermal emission

41 TB(ν) never exeeds Tk (kineti temperature) absorption mehanisms play a role wait for radiative proesses... TB(ν) TBBB(ν) Energy (heat) TkBB Tk TBBB(ν) TkBB TB(ν) >Tk TBBB(n) TkBB are hosen to be between TB(n) 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)!!!!!!!!!!!!!!!!

42 Basi theory of radiation fields (0) In vauum: E 0 E B Maxwell equations B 0 B E t t let's onsider the url of third eq. and ombine it with fourth E ( E ) t E ( E ) ( E ) using the vetor identity E E 0 t 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(Eo Bo ) 4π

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

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

45 Basi theory of radiation fields (3) Maxwell equations the two potentials an be simplified (Lorentz gauge): A t A t 4 4 j solutions have the following expression (see Jakson for details): r,t e r,t A 3 d r ' r r ' 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 See Rybiky-Lightman pp 69-90

46 Basi theory of radiation fields (4) Liénard Wiehert potentials r o (t) with veloity u r o (t) ρe ( r, t) q δ[ r r o (t)] j ( r, t) q u (t)δ[ r r (t )] o A harge q is moving along a given trajetory r at a given position r we get: and q u q 3 r ρ ( r, t)d e 3 r j ( r, t )d the salar potential is ϕ( r, t) d r ' dt ' 3 ρe ( r ', t ') r r ' ( r r ' δ t ' t + ) Alfréd-Marie Liénard and inserting the above definition of harge density and integrating over distane ϕ( r, t) if we introdue ( r r o (t ') q δ t ' t + R (t ') r r (t ') o ) dt ' r r o (t ') R(t ') R (t ') and onsider the vetor potential as well we get Emil Wiehert 86-98

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

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

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

50 Radiation from moving harges () 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 n n q n q 3 [ n ] 3 R R /R, aeleration field perpendiular to ň it is the radiation field /R, veloity field generalization of Coulomb's law n E r,t. the Coulomb's law holds for and no aeleration; the E field point to the urrent position of the harge. In ase of aeleration the radiation field is then E rad r,t B rad r,t q n [ n ] 3 R E rad n r,t

51 / R /R see Erad - Brad - n right-hand system of perpendiular vetors, with Erad Brad and are onsistent with radiation solutions of Maxwell equations

52 Radiation from moving harges () omparison between the two omponents of the E [B] field [if β ] 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:. near zone, where R λ, and the Evel > Erad by a fator / u ; Coulomb effets dominate. far zone or wave zone, at large R (λ /u), where Erad dominates and whose dominane progressively inreases with R ; radiation is the only thing that matters relevant to astronomy!

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

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

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

56 Relativisti harge: (γ >>) 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 d ] d p d dw d d p d Vetor ] dp d Salar

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

58 Relativisti harge: (γ >>) 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 d ] d p d P dp d dw q dt 3m 3 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 y i H rg e en high ssion i m e

59 [entripetal aeleration (ont'd)] 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 06 more power than protons at the same energy (g)

60 Free eletrons are aelerated (deelerated) in the Coulomb field of atomi (ionized) nulei (free-free) and radiate energy Eletrons and nulei are in thermal equilibrium at the temperature T a) ions move at high veloities wrt the eletrons that are aelerated (Coulomb interation) and may gain higher energies while ions are deelerated (but P goes with m- and then p do radiate little energy) b) eletrons move at veloity u wrt the ions and during ''ollisions'' are defleted. The eletrostati interation deelerates the eletron whih emits a photon bringing away part of its kineti energy NB e and p have also homologous interations (e with e; p with p) but they do not emit radiation sine the eletri dipole is zero. Astrophysial examples: HII regions (04 ok) galati hot-oronae (07 ok) intergalati gas (07-8 ok)

61 Radiative proesses: foreword Continuum from radiative proesses from free-free transitions Classial physis holds when λdb( ħmkt)/ smaller that the typial sale d of the interation and quantum mehanis ( x p ħ ) an be avoided ħ/mv << d h Plank's onstant m mass of the (lightest) partile v veloity of the (lightest) partile d harateristi sale (of the interation or l of the eletromagneti wave if it is involved) ldb de Broglie wavelength impliations: Partiles at like points Luis (7th duke of) de Broglie hν Epartile (Nobel 99)