HIGH ENERGY ASTROPHYSICS - Lecture 10

PD Frank Rieger ITA & MPIK Heidelberg Wednesday

1 Pair Plasmas in Astrophysics

1 Overview

• Pair Production γ + γ → e+ + e− (threshold) • Compactness parameter • Example I: ”internal” γγ-absorption in AGN (Mkn 421) • Example II: EBL & limited transparency of the Universe to TeV photons • Pair Annihilation e+ + e− → γ + γ (no threshold) • Example: Annihilation-in-ﬂight (energetics)

2 2 Two-Photon Pair Production

Generation of e+e−-pairs in environments with very high radiation energy density:

γ + γ → e+ + e−

• Reaction threshold for pair production (see lecture 9):

2 2 e1e2 − p1p2 cos θ = m1m2c + δm c (m1 + m2 + 0.5 δm)

With mi = 0, ei := i/c = pi (photons) and δm = 2me:

2 4 2 4 12(1 − cos θ) = 0.5 (δm) c = 2mec

2 4 ⇒ 12 = mec for producing e+e− at rest in head-on collision (lab frame angle cos θ = −1).

12 • Example: for TeV photon 1 = 10 eV, interaction possible with soft photons 2 2 ≥ (0.511) eV = 0.26 eV (infrared photons).

3 • Cross-section for pair-production:

πr2 1 + β∗ σ (β∗) = 0 (1 − β∗2) 2β∗(β∗2 − 2) + (3 − β∗4) ln γγ 2 1 − β∗

with β∗ = v∗/c velocity of electron (positron) in centre of momentum frame, 2 and πr0 = 3σT /8.

In terms of photon energies and collision angle θ (cf. above):

∗2 ∗ ∗ 2 2 ∗2 −1/2 12(1 − cos θ) = 2Ee and Ee = γ mec = mec (1 − β ) ∗ where Ee =total energy of electron (positron) in CoM frame, so 1 (1 − cos θ) = 2γ∗2m2c4 = 2m2c4 1 2 e e (1 − β∗2)

s 2m2c4 ⇒ β∗ = 1 − e 12(1 − cos θ)

4 1 + - "+" --> e + e T !

/ 0.1 "" !

0.01 1 10 100

#1 #2 (1-cos$)

Figure 1: Cross-section for γγ-pair production in units of the Thomson cross-section σT as a function of 2 2 interacting photon energies (1/mec )(2/mec ) (1−cos θ). The cross-section rises sharply above the threshold 2 4 2 4 12(1 − cos θ) = 2mec and has a peak of ' σT /4 at roughly twice this value, i.e. at 12(1 − cos θ) = 4mec . 5 • At low energies, large annihilation probability of created pairs (cf. later). • Maximum of cross-section(isotropic radiation ﬁeld, average of cos θ → 0) 2 4 σγγ ' 0.25σT at 12 ' 4mec ⇒ TeV photons interact most eﬃciently with infrared photons

1 TeV t := 2 ' 1 eV 1 ⇒ produced e+e− will be highly relativistic and tend to move in direction of initial VHE γ-ray. • High-energy limit (β∗ → 1): πr2 1 + β∗ σ (β∗) = 0 (1 − β∗2) 2β∗(β∗2 − 2) + (3 − β∗4) ln γγ 2 1 − β∗ 3σ (1 + β∗)(1 + β∗) ⇒ σ (β∗) → T (1 − β∗2) 2(1 − 2) + (3 − 1) ln γγ 16 (1 − β∗)(1 + β∗) ∗ 3 σT ∗2 1 σγγ(β ) ' ∗2 ln(4γ ) − 1 ∝ 8 γ 1t

⇒ γ-rays with 1 can interact with all photons above threshold, but cross- section decreases.

6 • Delta-function approximation for cross-section in power-law soft photon ﬁeld (cf. Zdziarski & Lightman 1985):

−α1 (e.g., γ-ray with 1 interacting with power-law diﬀerential energy density n(2) ∝ 2 ; approximating interac- 2 4 tion by cross-section peak at 12 ' 2mec [head-on] and accounting for proper normalization) 2 δ σT 2 2 2mec σγγ(1, 2) ' 2 δ 2 − 3 mec mec 1

• Compare (noting δ(ax) = (1/|a|)δ(x)): 2 2 max 1 12 σT 2mec 2 2mec σγγ ∼ σT δ 2 4 − 1 = δ 2 − 4 2mec 4 1 mec 1 2 σT 2 2 2mec = 2 δ 2 − 4 mec mec 1 with 2 4 12 ' 2mec (from peak location for head − on)

7 • In general optical depth for γγ-absorption of a γ-ray photon in diﬀerential soft photon ﬁeld nph(2, Ω, x) including interaction probability ∝ (1 − cos θ): Z R Z Z ∞

τγγ(1) = dx dΩ (1 − cos θ) 2 4 d2 nph(2, Ω, x) σγγ(1, 2, cos θ) 0 4π 2mec 1(1−cos θ)

−α Example: For isotropic power-law soft photon ﬁeld (Fν ∝ ν ) with energy −α1 spectral index α = α1 − 1, i.e. nph(2) ∝ 2 using δ-approximation: Z ∞ 2 4 σT 2mec τγγ(1) ' R d2 nph(2) 2 δ 2 − 3 2 4 mec /1 1 2 4 2 4 σT 2mec 2mec 1 α1 α ' R nph ∝ 1 ∝ 1 3 1 1 1

⇒ τγγ increase with increasing 1 (there are more targets if α > 0).

8 • Estimating optical depth for homogeneous non-relativistically moving source:

L( > t) Ltσγγ τγγ(1) ' ntσγγR ' σγγ ' 3 4πRct 4πRmec

with σγγ ' σT /4 and nt number density of target photons, and where last 2 equality holds for typical photon energies < t >∼ mec .

⇒ source can be opaque for high target luminosities Lt and small sizes R

• If absorbing radiation ﬁeld is external to source (e.g., BLR in AGN, EBL interactions), γγ-absorption leads to exponential suppression (lect. 3):

obs int −τγγ() Fν () = Fν () e

9 • If absorption is internal, i.e. happens within source, from formal solution of

radiation transfer equation: (cf. lecture 3, no background source Iν(0) ' 0, and constant source

function Sν := jν/αν with τν = ανR):

−τν jνR −τν jνR −τν Iν(τν) ' Sν 1 − e = 1 − e = 1 − e ανR τν

−τγγ() int obs int 1 − e Fν () ⇒ Fν () ' Fν () ' for τγγ 1 τγγ() τγγ()

• Compactness parameter lc for measuring relevance of internal γγ-absorption:

Lγ σT lc := 3 4πR mec

(Note: Sometimes, lc is deﬁned without 4π in denominator) > > At MeV energies, τγγ ∼ 1 ⇔ lc ∼ 4. ⇒ if lc 1, then high-energy photons are likely to be absorbed.

10 3 Example I: ”Internal” γγ-absorption in AGN: Mkn 421 (d ∼ 140 Mpc)

Figure 2: Spectral Energy Distribution (SED) for the BL Lac object Markarian 421 as seen by diﬀerent instruments [Credits: Abdo, A. et al., ApJ 736 (2011)]. 11 • Application to Markarian 421 – assume ”characteristic” numbers: 14 44 Schwarzschild rS ∼ 10 cm, LIR ∼ 10 erg/s (cf. SED), IR ∼ 1 eV ∼ 10−12 erg

LIRσγγ 5 rs ⇒ τ(VHE) ' ∼ 3 × 10 1 4πRcIR R ⇒ IR ﬂux from extended region R rs and/or jetted-AGN emission (variability may help to distinguish)

• Note: Modiﬁcation for relativistically moving source (”blob”): (1) Apparent vs intrinsic luminosity (ﬂux enhancement by beaming, lect. 3): 4 0 Lt = D Lt (2) Short-term variability vs size of emitting region (Doppler formula): 0 0 ∆tobs = ∆t /D ⇒ R = cD∆tobs

(3) Photon energy boosting with D = 1/[γb(1 − βb cos i)]: 0 0 t = Dt ⇔ t = t/D ⇒ Optical depth: 0 0 0 Ltσγγ Lt Lt τ ' 0 0 ∝ 0 0 ∝ 4 4πR ct R t D

12 4 Example II: ”External” Absorption of TeV photons in the Ex- tragalactic Background Light

• Extragalactic background light (EBL) = accumulated light from all galaxies (optical/UV) reprocessed by dust (infrared) ⇒ TeV photons from a distant AGN traversing EBL will get absorbed.

Figure 3: Sketch illustrating absorption of VHE gamma-rays from distant AGN by interaction with the EBL [Credits: M. Raue].

13 Figure 4: EBL energy density in the local Universe (z = 0) with model curves to observations (data points) [Credits: J. Primack]. The EBL is usually diﬃcult to measure due to foreground emission from within the solar system and the Milky Way. Note: λ =1 Angstroem= 10−8 cm corresponds to ν = c/λ = 3 × 1018 Hz.

• TeV photons of energy 1 primarily interact with infrared photons of energy 1 TeV 2 ' 1 eV, corresponding to 1

h c 4 λ2 = = 1.2 × 10 (1/1 TeV) Angstrom = 1.2 (1/1 TeV) µ m 2

14 Figure 5: Gamma-ray ﬂux attenuation for γ-ray photons of energy Eγ from sources at diﬀerent redshifts z. The plateau between 1 and 10 TeV at low redshifts is a consequence of the mid-IR valley in the EBL spectrum [Credits: J. Primack]. • EBL spectrum and intensity depends on cosmological time (star formation history) and hence on redshift z. ”Knowing” intrinsic source spectra, γγ absorption imprint can be used to diagnose EBL evolution.

15 5 Pair Annihilation

Decay of e+e−-pairs into two photons (no threshold):

e+ + e− → γ + γ

• Annihilation cross-section in center of momentum (CoM) frame: " # 2 ∗2 ∗ ∗ ∗ πr0 γ + 4γ + 1 ∗ p ∗2 γ + 3 σe+e−(γ ) = ln γ + γ − 1 − γ∗ + 1 γ∗2 − 1 pγ∗2 − 1

∗ ∗ 2 2 2 with γ = E /mec lepton energy in CoM frame, r0 = e /mec (cgs) classi- 2 cal electron radius. Note: Thomson cross-section σT := 8πr0/3.

• Low-energy (β∗ << 1, thermal electrons) limit:

1 2 3σT σ + − ' πr = e e β∗ 0 8β∗ ⇒ Annihilation probability is high for leptons nearly at rest. 2 ⇒ Decay produces two γ’s very close to rest energy of electron: hν ∼ mec ⇒ 0.511 MeV ”annihilation line”

16 10 + - e + e --> "+"

1 T ! / - e + e !

0.1

1 * "

Figure 6: Cross-section σe+e− for pair annihilation in units of the Thomson cross-section σT as a function of CoM electron/positron Lorentz factor γ∗.

17 • High energy (γ∗ 1) limit: 2 πr0 ∗ σ + − ' (ln 2γ − 1) e e γ∗ Decay produces two photons, but photons have broader energy spread ⇒ ”annihilation spectrum”.

• If ambient density of electrons is ne, characteristic positron annihilation time scale (assuming σe+e− ∼ 3σT /8) is:

1 6 ne tann ' ∼ 4 × 10 −3 year cneσe+e− cm

+ + + + + + • Note: e are created in π -decay, i.e. π → µ +νµ with µ → e +ν¯µ+νe, the charged pion’s being created in pp-collisions (lect. 9), p+p → p+n+π+.... Since pp-collisions also produce π0 which decay into γ-rays, ﬂux of interstellar positrons created by this process can be estimated from γ-ray luminosity of interstellar gas.

18 6 Example: Positron-electron-annihilation in ﬂight

Maximum/minimum photon energy in case of relativistic e+ and stationary e−:

+ − 4-vectors Pe = γme(c,~v), Pe = mec(1, 0), Pγ1 = e1(1, −~n), Pγ2 = e2(1, ~n), with ~n = ~v/|~v|:

+ − Pe + Pe = Pγ1 + Pγ2

2 2 ⇒ P = (P + + P − − P ) γ1 e e γ2

2 2 2 ⇒ 0 = P + + P − + P +2 P +P − − 2P +P − 2P −P e e γ2 e e e γ2 e γ2

v ⇒ 0 = 2m2c2 +2 γm2c2 − 2γm ce 1 − − 2m ce e e e 2 c e 2

i 1 + γ 2 1 + γ ⇒ ei := = mec = mec v p 2 c γ(1 ∓ c ) + 1 γ ∓ γ − 1 + 1

(minus sign for e2; plus sign for e1 once solved for Pγ2)

19 Have i 2 1 + γ ⇒ ei := = mec c γ ∓ pγ2 − 1 + 1

+ 2 2 T In terms of kinetic e -energy: T = E − mec = (γ − 1)mec ⇔ γ = 2 + 1: mec 2 2 2 2 mec (T + 2mec ) mec mec i = √ = √ = 2 2 2 2 2 −1/2 T + 2mec ∓ 2mec T + T T 1+2mec /T 1 ∓ (1 + 2mec /T ) 1 ∓ 2 T (1+2mec /T ) 2 −1/2 x For T mec , expansion (1 + x) ' 1 − 2 , 2 mec i ' 2 1 ∓ (1 − mec /T ) Hence m c2 ' e (plus sign) 1 2

2 ' T ( for minus sign) ⇒ forward-going photon (maximum) takes almost all the kinetic energy of the positron, while energy of backward-moving photon (minimum) is only half the rest mass energy. ⇒ in lab. frame, photon spectrum by annihilation will spread over interval [1, 2].