HIGH ENERGY ASTROPHYSICS - Lecture 5
PD Frank Rieger ITA & MPIK Heidelberg Wed
1 Synchrotron Emission I
1 Overview
• Radiation of charged particles gyrating in magnetic fields. • Total emitted power for single particle (Larmor formula), spectral distribu- tion. • Characteristic Emission at ν ∝ γ2B, nearly linearly polarised. • For a non-thermal, power law electron distribution, n(γ) ∝ γ−p, emitted −(p−1)/2 spectrum a power law jν ∝ ν too. • Believed to be the fundamental radiation process at the lower end of the HE spectrum.
”Magnetobremsstrahlung” = Radiation emitted by relativistic charged particles (mostly electrons) due to acceleration (”gyro-motion”) in a static magnetic field. Note: In the sub-relativistic regime this process is called ”Cyclotron emission”
Approach: Semi-quantitative analysis of spectral features as detailed calcula- tions are lengthy (see, e.g., Rybicki & Lightman, Chap. 6).
Structure: • Motion of electrons in magnetic fields • Look at emission from a single electron • Consider electron distribution and opacity effects to obtain final spectrum
3 3 Motion of a Charged Particle in Static Magnetic Field
Remember: Particle gyrates around magnetic field with angular frequency (Lar- mor frequency) qB Ω = L γmc 2 and associated radius of gyration orbit (Larmor radius), γmv⊥/rL = (q/c) v×B,
2 v⊥ γmcv sin θ γmc rL = = ' ΩL qB qB Numerically: B m 1 Ω ' 1.8 × 107 e Hz L 1 G m γ c 3 1 G m rL ' ' 1.7 × 10 γ cm ΩL B me 10−6 G m ' 1.7 × 109 γ cm B me −6 ⇒ rL is small on cosmic scales (B ∼ 10 G in interstellar medium, B ∼ 1 G in center of AGN [but away from BH]).
4 4 Total Emitted Power for a Single Particle
• Remember: Larmor’s Formula for relativistically moving particle (lecture 4):
2 4 dE 2 q γ 2 2 2 = 3 (a⊥ + γ ak) dt K 3 c
with a⊥, ak = acceleration components perpendicular and parallel to direction of motion (not to magnetic field!).
2 • For gyro-motion, ak = 0 and a⊥ = v⊥/rL = ΩLv⊥ (centrifugal), with v⊥ = v sin θ, so 2 e2γ4 v2 e2B2 2 β2e4B2 P = ⊥ = γ2 sin2 θ syn 3 c3 γ2m2c2 3 m2c3 • For isotropic distribution of particles, average pitch angle 1 Z 4π 1 Z 2π Z π < sin2 θ >= sin2 θdΩ = dφ sin2 θ sin θdθ 4π 0 4π 0 0 1 2 = [cos(3θ) − 9 cos(θ)]π/12 = 2 0 3
5 • Average single particle synchrotron power [erg/s] for β → 1 4 e4B2 4 m 2 hP i = γ2 = cσ e γ2 u syn 9 m2c3 3 T m B with magnetic energy density B2 u := , B 8π and Thomson cross-section 2 2 8π e −25 2 σT := 2 = 6.65 × 10 cm 3 mec • Characteristic electron cooling timescale: 2 E γmec 8 1 tcool(γ) := = = 7.8 × 10 2 sec |dE/dt| < Psyn > B γ Note: 2 1. Psyn ∝ 1/m ⇒ Synchrotron radiation from charged particles with larger mass (e.g. protons) is usually negligible. 2 2 2. E = γmc ⇒ Psyn ∝ E uB, so more energetic particles radiate more.
6 5 Energy Evolution for Electrons due to Synchrotron Losses
Start at t0 with initial energy of electron = E(t0): 2 • From Psyn = −dE/dt and energy E = γmec 2 dE 2 4 1 B −3 2 = −c1E with c1 := cσT 2 2 = 1.6 × 10 B [1/erg/s] dt 3 (mec ) 8π
2 • Integrate (dE/dt)/E = −c1 from t0 to t (t > t0) to give 1 t 1 1 − = −c (t − t ) viz. − = c (t − t ) E 1 0 E(t) E(t ) 1 0 t0 0
E(t0) − E(t) = c1(t − t0) E(t)E(t0) so finally E(t ) 1 E(t) = 0 ∝ for large times 1 + c1(t − t0)E(t0) t
• Note: Timescale tcool for cooling down to 1/2 of initial energy E0 at t0 = 0: E0 E0 1 E(tcool) = = to give tcool(E0) = . 2 1+c1tcoolE0 c1E0
7 6 Single Electron Spectrum
Remember: Radiated power as a function of time, P (t) ∝ |~a(t)|2, reflects time- dependence of acceleration a(t). Distribution of this radiation over frequency, 2 Pν is called spectrum and reflects power (Fourier) spectrum of a (t).
For synchrotron emission, observed radiation varies with time due to relativistic beaming (in addition to fundamental gyro-motion), so expect observed spectrum to contain power at frequencies much higher than ΩL/2π.
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Figure 1: A charged particle in slow gyro-motion around a magnetic field will emit an approximate dipole pattern with maximum in the direction of motion v. For higher speeds, aberration (beaming) changes pattern to an asymmetric shape, with most emission contained in forward cone of half opening angle ∼ 1/γ.
8 • Aberration (Lecture 4): Radiation of relativistic particle is strongly beamed in forward direction, focused into cone of opening angle∆ α ' 2/γ [rad].
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1 2 Observer
• Observer only sees radiation once per obit when particle velocity is within 2/γ of line of sight. ∆tobs:=duration for which beam stays within line of sight
⇒ Observer sees narrow pulse of duration ∆tobs once per gyro-period
⇒ Most of radiation power must appear at frequency ωc ∼ 1/∆tobs.
(Basic motion is periodic, t ∝ 1/ΩL (discrete spectrum), but frequency spacing ΩL between successive harmonics becomes so narrow as to be negligible, so can treat as continuous spectrum)
9 Estimate Characteristic Frequency νc or ωc = 2πνc:
• Path length ∆s for which we see radiation: ∆s = r ∆α = r (2/γ).
• Radius of curvature r of path given by equation of motion: ∆~v q γm = ~v × B~ ∆t c With |∆~v| ' v ∆α and ∆s ' v∆t, i.e. ∆t = ∆s/v, this gives: ∆α qB sin θ ' ∆s γmcv so that ∆s γmcv r = ' ∆α qB sin θ
(Note: 3-dim r is different from projected/perpendicular Larmor radius)
2vmc • Path length ∆s ' r ∆α = r (2/γ) = qB sin θ.
10 • Times t1 and t2 at which particles passes point 1 and 2 related by ∆s = v(t2 − t1), thus 2mc t − t ' 2 1 qB sin θ
• Travel time effects: Arrival time difference is less than (t2 − t1) by ∆s/c: v ∆s h vi 2mc h vi 1 + c mc ∆tobs = (t2−t1)− = (t2−t1) 1 − = 1 − v ' 2 c c qB sin θ c 1 + c γ qB sin θ ⇒ Observed pulse has duration shortened by1 /γ2
Note: Converting from rad to Hz, observed duration would be = 2π∆tobs
• Spectrum (in terms of frequencies) will be broad with characteristic angular frequency (ωch = 2πνch) at which significant part of power is radiated:
1 2 qB 2 3 ωch ' = γ sin θ = γ Ω0 sin θ = γ ΩL sin θ tobs mc
with cyclotron frequency Ω0 := qB/(mc).
11 7 Resulting Observed Electric Field y t i s n e t n i
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The observed time-dependent E-Field, E(t), from an electron is a sequence of 1 1 pulses of width∆ tobs ' 2 = 3 , separated in time by ∆t ∼ 2π/ΩL. Note: γ Ω0 γ ΩL Larmor frequency ΩL = eB/γmc =: Ω0/γ.
12 8 Single Particle Synchrotron Spectrum Pν(γ)
Approach: Notation alert: Note E here refers to electric field, not particle energy!
• Want spectrum=power per unit frequency, Pν(γ) [erg/s/Hz], R ∞ 2 β2e4B2 2 2 satisfying 0 Pν(γ)dν = Psyn = 3 m2c3 γ sin θ.
⇒ Find dimensionless frequency distribution function F˜(ν/νc) with 1 Pν(γ) = Psyn F˜(ν/νc) νc satisfying Z ∞ Z ∞ 1 νc ! F˜(ν/νc)dν = F˜(x)dx = 1. νc 0 νc 0
• Start by Fourier-transforming E(t) ∝ g(ωct) (get from full L-W potentials), 1 Z ∞ Eˆ(ω) = E(t)eiωtdt , 2π −∞ R ∞ ˆ −iωt where inverse transform E(t) = −∞ E(ω)e dω.
13 • Remember Parseval’ theorem: Z ∞ Z ∞ E2(t)dt = 2π |Eˆ(ω)|2dω −∞ −∞
• Total energy W per unit area in pulse (Poynting theorem): dW c Z ∞ Z ∞ = E2(t)dt = c |Eˆ(ω)|2dω dA 4π −∞ 0 so total energy per area per unit time per unit frequency: dW 1 dW c ≡ = |Eˆ(ω)|2 dAdtdω T dAdω T
• Spectrum Pν(γ) = 2πPω(γ) from 1 dW c Z P (γ) ≡ = r2|Eˆ(ω)|2dΩ ω T dω T 2 where r distance, T = 2π/ΩL period and solid angle element dΩ = dA/r , using ∆θ ∼ 1/γ.
14 ⇒ Result: Total synchrotron power per unit frequency [erg/s/Hz] for a single electron with Lorentz factor γ (relativistic case β ' 1) √ e3B sin θ ν Pν(γ) = 3 2 F mc νc R ∞ 0 0 0.3 −x with F (x) := x x K5/3(x )dx ' 1.8 x e , K5/3 modified Bessel function of order 5/3, and γ entering via 3 eB 3 ν = γ2 sin θ = γ2 Ω sin θ c 4π mc 4π 0 ∞ Note: By convention, F is normalized somewhat differently compared to F˜, i.e. R F (x)dx = 8√π , introducing the 0 9 3 slightly different numerical value in Pν.
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!/!c !/!c
Figure 2: Single Particle Spectrum F (ν/νc) as function of normalized frequency ν/νc. Left: Log-log repre- sentation. Right: Normal representation. The spectrum is continuous and has a maximum at νmax = 0.29νc. 15 9 Example: Synchrotron Emission from the Crab Nebula
Figure 3: Crab Nebula (d ∼ 2 kpc) caused by SN explosion in 1054 A.D. - composite image: Chandra X-ray [blue], HST optical [red and yellow], Spitzer infrared [purple]. X-ray image is smaller than others as extremely energetic electrons emitting X-rays radiate away their energy more quickly than lower-energy electrons emitting in optical and infrared [Credits: NASA].
16 Emission as synchrotron radiation of relativistic electrons, characteristic frequency: 1 2 1 2 eB B 2 νc ' γ Ω0 = γ ' 280 −4 γ Hz 2π 2π mec 10 G for average B ∼ 10−4 G in Crab Nebula. • Optical Emission: Optical emission (HST) at ν ∼ 5 × 1014 Hz requires electrons with γ ∼ 106. 106 > ⇒ Cooling time scale tcool ∼ 2500 γ yr ∼ tage age of Nebula. • X-ray Emission: Chandra (ACIS, 0.2-10 keV) X-ray emission ν ∼ 1017 Hz requires γ ∼ 107, electrons cool quicker by factor ∼ 10. ⇒ X-ray emission spatially less extended. ⇒ tcool < tage ∼ 950 yr of Nebula, need continuous supply of fresh electrons. • Radio Emission: Crab Nebula also bright in radio (NRAO, ν ∼ 5 × 109 Hz), less energetic electrons needed, γ ∼ 5 × 103, size constrained by age of Nebula. ⇒ See synchrotron emission from broad distribution of electrons in energy space.
17 10 Synchrotron Spectrum for Power-law Electron Distribution
In many case, synchrotron radiation is emitted from relativistic electrons with power-law density distribution in energy, i.e.
−p −3 n(γ)dγ = n0γ dγ [cm] for γmin < γ < γmax, typically p ∼ 2−3 ⇒ ”Nonthermal synchrotron radiation”.
Resultant emission spectrum = frequency distribution of emitted power [erg/s/Hz/cm−3]: Z ∞ jν = < Pν(γ) > n(γ) dγ . 1
Hard to do analytically using exact < Pν(γ) > ⇒ Approximation: Photons are 2 only emitted at frequency νc ' γ νL, νL = Ω0/2π = eB/(2πmec),
2 Pν(γ) '< Psyn > δ(ν − νc) =< Psyn > δ(ν − γ νL) So Z γmax 2 4 γ 2 jν ' cσT uBn0 p δ(ν − γ νL) dγ 3 γmin γ
18 0 2 0 0 Substituting ν = γ νL, i.e., dν = 2νLγdγ ↔ dγ = dν /(2γνL) 2 2 Z γmaxνL Z γmaxνL (p−1)/2 1 1 0 0 1 νL 0 0 jν ∝ p−1 δ(ν − ν ) dν = 0 δ(ν − ν )dν 2νL 2 γ 2νL 2 ν γminνL γminνL 0 1/2 since γ = (ν /νL) . Thus p−1 − 2 2 uB ν jν = cσT n0 3 νL νL 2 2 for γminνL < ν < γmaxνL. Synchrotron spectrum of a power-law electron distribution of in- −p −α dex p, n(γ) ∝ γ , is a power-law jν ∝ ν with spectral index p − 1 α = . 2
Note: Outside limits, emissivity will be dominated by that of particles at γmin 2 and γmax, respectively. Thus for ν < νmin := γminνL (using asymptotic for F (x)) 1/3 jν ∝ ν , 2 and for ν > νmax := γmaxνL −ν/νmax jν ∝ e .
19 log flux spectra electron individual power-law superposition power-law 20 log frequency log 11 Polarization
• Electron acceleration vector changes only slightly during a pulse ⇒ quasi-constant polarization state.
• Synchrotron radiation turns out to be highly elliptically polarized, i.e. very nearly linearly.
• Polarization allows to measure magnetic field direction (⊥ to observed electric field vector; but note caveat: Faraday rotation and B-field inhomogeneities).
• For a power-law energy distribution of electrons with index p, maximum degree of linear polarization (polarized flux/total flux): (3p + 3) π(p) = ' 0.7 (3p + 7) for typical p = 2 − 3. ⇒ Observation of highly polarised emission as argument for synchrotron origin.
21 Circular Polarization
Linear Polarization
Circular Polarization
Cyclotron Radiation Synchrotron Radiation
Figure 4: Left: Non-relativistic cyclotron motion: When viewed in orbital plane, radiation is 100% linearly polarized with electric vector oscillating perpendicular to magnetic field B. Viewed from along B, emission is 100% circularly polarized. Right: For relativistic motion, radiation is beamed into direction of motion. The two components of circular polarization effectively cancel almost, whereas linear polarization largely survives.
22 Figure 5: Decomposition of synchrotron polarisation vectors on the plane of the sky. The radiation is almost nearly linearly polarised, and dominated by the component perpendicular to the projected magnetic field direction. From Rybicki and Lightman, Fig. 6.7.
23 12 Example: B-Field Direction Inferred from Observations
Figure 6: B-field vectors inferred from degree of radio (2 × 109 Hz) synchrotron polarization for the spiral galaxy M51 [distance ∼ 10 Mpc, B ∼ O(10µG)] by rotation of the observed electric field vector by 90◦. There is a tendency for B to run parallel to the spiral arms. 24 13 Exact Calculations
Evaluating spectrum for single electron, radiation can be split into two compo- nents linearly polarised across and along magnetic field as projected on the sky √ 3 e3B sin θ ν ν P (ν, γ) = F + G ⊥ 2 mc2 ν ν √ c c 3 e3B sin θ ν ν Pk(ν, γ) = 2 F − G 2 mc νc νc with P⊥ > Pk, where Z ∞ 0 0 G(x) := x K2/3(x) and F (x) = x K5/3(x )dx , x with Ki modified Bessel function of ith order. Total power for single electron:
Pν(γ) = P⊥(ν, γ) + Pk(ν, γ) ∝ F (ν/νc) . Fractional degree of linear polarization: P (ν, γ) − P (ν, γ) π(ν) := ⊥ k P⊥(ν, γ) + Pk(ν, γ)
For single particle π(ν) = G(ν/νc)/F (ν/νc). For a power law of index p, P⊥ and Pk must be integrated over particle energy. Result gives π(p) = (3p+3)/(3p+7).
25 1 F G
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Figure 7: Comparison of the synchrotron distribution functions F and G, noting that we have P⊥(ν, γ) ∝ F ν + G ν and P (ν, γ) ∝ F ν − G ν νc νc k νc νc
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