Plasma effects on electromagnetic wave propagation

Transcription

1 Plasma effets on eletromagneti wave propagation & Aeleration mehanisms Plasma effets on eletromagneti wave propagation Free eletrons and magneti field (magnetized plasma) may alter the properties of radiation rossing the volume oupied by the plasma. Suh modifiation is frequeny dependent and an be measured, being indiret measurements of both ne and HII. Referenes: Fanti & Fanti, ap. 7 Longair,.3.3 &.3.4 Rybiky & Lightman, ap. 8 Daniele Dallaasa Radiative Proesses and MHD Setion -6 : Plasma effets & aeleration mehanisms

2 E-m wave propagation in a plasma Let's onsider an astrophysial plasma, omposed by ionized gas whih is, however, neutral as a whole. Maxwell equations are defined for vauum, but an be adapted to a plasma if we onsider harge and urrent densities ρe, j Let's onsider the dieletri onstant of an e-m wave with pulsation ω rossing a medium: 4πe ϵr = me ( ne ω ω o + i Ni ω ω i ) ω = pulsation inoming radiation ne = free eletrons number density ω o= free eletrons pulsation (=0) Ni =bound eletrons number density ω i= bound eletron pulsation (in the radio domain, and, in general, when ω ω i then an be negleted)

3 Plasma frequeny We define the refration index: nr nr εr 4πe ne me ω where the plasma frequeny νp has been defined as νp = e ne π me = n = e νp (ν ) (Hz) only waves with ν > νp an travel aross the region those with ν < νp are refeted (nr beomes imaginary) Below the plasma utoff frequeny there is no propagation of e-m waves 6 3 In the ionosphere: n e 0 m 7 implies νp 0 Hz 3 3 In the interstellar medium: ne 0 0 m 3 implies νp Hz

4 Wave propagation at ν > νp the e-m wave travels with group veloity v g ω = nr = k [ νp ( ) ( )] ν νp for ν νp then... v g ν A L B The time neessary to travel from A to B at a given frequeny is T A, B( ν) = (next page) L dl 0 v g L 0 [ ( )] νp ν dl = L 0 [ ( )] + νp ν dl =

5 Wave propagation T A,B (ν) = [...] = = = L dl + 0 L L + + L ν p 0 ( ν ) dl = [ ] [ ] e π me ν e π me ν where DM = A L 0 n dl = e DM L 0 n dl e is termed the Dispersion Measure L B

6 Dispersion Measure Observing at two different frequenies, the arrival time will be different! A delay is present Δ Tν ν [ ( )] L = T A, B (ν ) T A, B( ν ) = + Δ Tν ν = DM e π me DM e π me ν ( ν ν [ L + ( )] DM e πme ν ) In ase it is possible to detet this effet, namely measure Δ T ν ν in a given partiular ase, then it beomes feasible todiretly determine n e along a given LOS (to the objet).

7 Dispersion Measure What is a pulsar? Rapidly spinning netron star (many turns per seon Stong Magneti field (misaligned wrt the rotation axis) Narrow radiation one (interepting the LoS to the Earth) in pulsars

8 Dispersion Measure a diret measurement The slope of the pulse arrival time.vs. frequeny provides a measure of D.M. = L 0 n e dl Distanes, however, are diffiult to determine, exept in a few luky ases, like globular lusters

9 Faraday Rotation Mihael Faraday (79-867) Propagation effet arising from an external magneti field H whih auses an anisotropi transmission. Let's onsider what happens along the field diretion: m ωl d v dt = = ( v e E + x H eh ) me Let's us assume that the propagating e-m wave is polarized and sinusoidal as a superposition between a LCP and a RCP omponents. E (t) = E e i ω t ( ϵ ± ϵ ) o where + is for RCP and is for LCP. The dieletri onstant is no longer a salar and beomes a tensor: the two modes have different refration index (nr )R,L = νp (ν ) ±(ν / ν) os θ L where θ is the angle between the diretion of e-m wave propagation and H

10 Faraday Rotation Along the field diretion B o = Bo ϵ 3 and then in equation ** we get as a solution v (t) = ie E (t) me (ω ± ω H) εr, L = whih provides a dieletri onstant ωp ω(ω ± ω H) and therefore the propagation speeds of the two orthogonal modes are different, originating a shift in their relative phase, whih implies a rotation of the polarization vetor Δn = The differene between the refration indies is νp νl ν 3 os θ After a length dl there is a phase differene between the two waves ( ) πdl dϕ = Δn = λ π νδ n dl whih must be integrated!

11 In ase the Faraday sreen is spatially resolved, the net effet is just a rotation of the linear polarization vetor in a given diretion (radians) The rotation measure determines the magneti field along the LOS weighted on the eletron density ne and if also the D.M. is available H R.M. D.M. ne H dl ne dl

12 How to measure RM Polarization sensitive observations provide the measurement of m and χ at various disrete frequenies (wavelengths) Plot λ and χ get the slope of the best fit line

13 How to measure RM () Polarization sensitive observations provide the measurement of m and χ at various disrete frequenies (wavelengths) Plot λ and χ (±n π) get the slope of the best fit line obtain RM & the absolute orientation of χ blue points represents the observations red points are for the ±n π ambiguities vi o r p e p slo d M R es

14 Distribution of known pulsars in our galaxy: polarized emitters

15 Examples of RM and FR RM needs various frequenies to be measured Depolarization may take plae RM may be very different in various loations of the same radio soure -> loal to the r-soure

16 Examples of RM and FR () Stratified RM struture (impliations on H and ne)

17 Aeleration mehanisms Stohasti: - ollisions among partiles/louds (seond order Fermi proess) Systemati: - H field ompression + sattering/diffusion - shoks (first order Fermi proess) - e-m proesses (e.g. Low frequeny-large amplitude waves in pulsar magnetosphere) Requires ollisionless plasma (otherwise energy gain would be redistributed) Referenes: Fanti & Fanti, ap. 9 Longair, 7. & 7.3 & 7.4

18 Fermi's Aeleration basi proess Proposed by E. Fermi in 949 and refined in 954 Ingredients a harged partile moving at v a magnetized loud moving at u E fields annot survive given the enormous onduibility H field *only* in the loud When the harge moves into the moving loud (seen from the observer's frame) it ''feels'' an eletri field as well = e F 'e = e E' The harge ''feels'' a fore = e ( v E u H ( u v H + H E + ) v H dv = me dt ) u

19 Fermi's Aeleration: Let's elaborate on me basi proess -dv e dt [ u v = e H + H ] if we onsider a salar produt with v it beomes d e ( me v ) ( ) v ( u H) = e β ( u H) dt This means that the energy of the eletron hanges in ase the Lorentz fore is ative, and this requires that the magnetized loud is in motion However, the value of u depends on the referene frame...and ould be 0, as well

20 Fermi's Aeleration: basi proess -3- Elasti ollision (energy and momentum onservation): loud partile with v u, (m M)v ± Mu v' = v ± u given that m M m+ M u' u (Type I) (Type II) In terms of partile energy before and after the interation(εi,εf ) εf = = me (v ') me ( v Energy variation u Δε 4 v Δε 4 = me ( v ± 4uv + 4 u ) = εi ± u) = ( u ± 4 v type I ollision u v type II ollision +4 u v ) v u u

21 Fermi's Aeleration: basi proess -4- Type I interations happen more often than Type II v +u v u fi = f II = therefore l l Δε ε u u 8v u 8u = f I Δ εi + f II Δ εii = 4 ε = ε = ε = Δt F v l l v l v τf () A fator is more appropriate than 8 (valid for head on ollisions only) ε 8u = ε = Δt F l v τf Δε If we integrate in time we obtain: ε(t) = εo e t τf where τf lv u one the partile is aelerated at the required veloity, then should be able to leave the region where aeleration takes plae. Namely the onfining time τ should be of the order of (or slightly larger than) the aeleration time τ F

22 Fermi's Aeleration: τf basi proess -5lv u For initial e- veloities of ~ 0 km/s and given the typial loud size and number density in the ISM (distane 0-00 p) relativisti veloities are ahieved at In SNR the proess may be more effiient: ``louds'' may have higher veloities (03 km / s) ``l'' is small (0. p) and then 5 τ F 0 yr τ F 0 0 yr

23 Shok waves and Fermi's ollisions() s = γkt μ mh sound speed for an ideal gas (no H field) if a perturbation moves at a speed exeeding s, a disontinuity is reated region of partiles to be aelerated is moving at strong shok is moving at if v =(3 / 4) v partiles an ross several times the shok front before they gain enough energy to leave the aeleration region aelerating partile (v) v v v unperturbed v v=0

24 Shok waves and Fermi's ollisions () The ombination of the two veloities allow a partile to have Fermi I type ollisions in a row (and rebounds with unperturbed louds at rest). the ourrene between ollisions is and the energy gain in time is dε dt v =(3 / 4)v Δ ε =0 aelerating partile (v) 3 v ε Δ ε v = ε v v l τ onfining time v τf δ = + τ ( ) v f l 3 v v ε v l ( ) 3 v ε 4l = ετf v v unperturbed v =0

25 Fermi's Aeleration: Spetrum of Fermi's aeleration proesses: β = statistial energy inrease per ollision i.e. after a ollision ε = βεo k = # of ollisions p = probability to remain within the aeleration region Then, in a given time, after k ollisions εk = εo βk and the # of partiles with k ollisions is Nk = No pk Crab nebula (054) ln(nk / No) ln(p) = = m ln(εk / εo ) ln(β) Nk = No εk m ( ) εo +m N(ε)dε = ost ε i.e. power law energy distribution dε

26 Fermi's Aeleration: τf basi proess -5lv u For initial e- veloities of ~ 0 km/s and given the typial loud size and number density in the ISM (distane 0-00 p) relativisti veloities are ahieved at τ F 0 0 yr 3 In SNR the proess may be more effiient: ``louds'' may have higher veloities (0 km / s) and l is small (0. p) and 5 τf 0 yr The injetion problem: In the environment of a shok, only partiles with energies that exeed the thermal energy by muh (a fator of a few at least) an ross the shok and 'enter the game' of aeleration. It is presently unlear what mehanism auses the partiles to initially have energies suffiiently high.

27 Observations of radio supernovae The observations shown aside (images on the same sale!) require that effiient partile aeleration takes plae on time sales as short as a few weeks! Bietenholz + al. 00

28 Constraints from radio supernovae: SN993J in M8 Messier 8: spiral galaxy at ~3.7 Mp Apparent size : ~ 7 ' x 4 ' (NED) Bietenholz + al. 00

29 Shok waves and Fermi's ollisions (3): an example. Cas A 70 Jy at GHz. The SN ourred at a distane of approximately,000 ly away. The expanding loud of material left over from the supernova is now approximately 0 ly aross. Despite its radio brilliane, however, it is extremely faint optially, and is only visible on long-exposure photographs. It is believed that first light from the stellar explosion reahed Earth approximately 300 years ago but there are no historial reords of any sightings of the progenitor supernova, probably due to interstellar dust absorbing optial wavelength radiation before it reahed Earth (although it is possible that it was reorded as a sixthmagnitude star by John Flamsteed on August 6, 680 It is known that the expansion shell has a temperature of around 50 million degrees Fahrenheit (30 megakelvins), and is travelling at more than ten million miles per hour (4 Mm/s). A false olor image omposited of data from three soures. Red is infrared data from the Spitzer Spae Telesope, orange is visible data from the Hubble SpaeT, and blue and green are data from the Chandra X-ray Observatory