N I V E R U S E I T H Y T O H F G E R D I N B U High Energy Astrophysics, 2011–12 Bremsstrahlung and galaxy clusters Philip Best Room C21, Royal Observatory; [email protected] www.roe.ac.uk/∼pnb/teaching.html 1 Bremsstrahlung emission Having derived the Larmor formula for radiation from an accelerated charge, we can now ap- ply it to different physical situations and look at the astrophysical implications. We begin with Bremsstrahlung emission (or ‘braking radiation’) which is the acceleration of a charged particle due to the Coulomb field of another charged particle. e- Ze+ Figure 1: The process of Bremsstrahlung emission Here we will concentrate on electron-ion collisions, since electron-electron or ion-ion interactions involve equal and opposite accelerations, and the radiation field from each interferes destructively to give essentially no overall radiation. We will treat the ion as fixed and ignore its acceleration, due to its large inertia (relative to that of the electron). We will also ignore the change in the path of the electron; this is clearly a simplification which breaks down for very close encounters. 1.1 Parseval’s theorem; obtaining the spectral shape dE Larmor’s formula indicates that the energy radiation, dt , depends on the acceleration of the par- ticle, v˙ (t). Consider the Fourier transform of this acceleration: 1 v˙ (ω) ≡ dtv˙ (t)eiωt. (1) Z Here we have used the same symbol v˙ for the Fourier transform of the acceleration (the argument t or ω tells you which it is). We will use Parseval’s theorem (a proof of which can be found in any good textbook on Fourier transforms) 1 dt|v˙ (t)|2 = dω|v˙ (ω)|2 (2) Z 2π Z to work out the spectrum of radiation arising from the entire encounter. ∞ dE ∞ e2 v 2 Etot = dt − = dt 3 | ˙ (t)| Z−∞ dt Z−∞ 6πǫ0c e2 1 ∞ v 2 = 3 dω| ˙ (ω)| 6πǫ0c 2π Z−∞ e2 1 ∞ v 2 = 2 3 dω| ˙ (ω)| 6πǫ0c 2π Z0 ∞ ≡ dωI(ω). (3) Z0 Here, e2 v 2 I(ω)= 2 3 | ˙ (ω)| (4) 6π ǫ0c is the energy emitted per unit frequency interval; in other words, this quantity indicates the spec- trum of the radiation. For any acceleration mechanism, we can Fourier transform v˙ (t) to determine v˙ (ω), and hence derive I(ω). 1.2 Spectrum of Bremsstrahlung 1.2.1 Single interaction of an electron with an ion Let us consider an electron moving at velocity v, whose straight-line course would take it to a closest approach at distance b (known as the impact parameter) from a stationary ion. We will work in the rest-frame of the electron (and call this frame S). In this frame we place the electron at x = 0, y = b, z = 0, and the ion travels at speed v along the x axis. We will set t =0 to be the time when the interaction peaks (ie. the ion passes x = 0). y y=b e− + Ze v x Ion Figure 2: The frame S set-up for the Bremsstrahlung emission calculation. 2 In order to determine the acceleration of the electron, we need to determine the electric field due to the ion, in frame S (and then F = ma and F = Eq give the acceleration trivially). In S′, the fields are simple: Ze E′ r′ B′ = ′3 ; =0 (5) 4πǫ0r ′ ′µ ′ ′ ′ Similarly the 4-potential in frame S is simply given by A =(φ /c, A )=(Ze/(4πǫ0r c), 0). This can be transformed to S′ using the inverse Lorentz transformation matrix, and then by using E = −∇φ − ∂A/∂t (exercise for student: see Tutorial 2), the following result: γZe(x − vt) Ex = 2 2 2 2 3/2 4πǫ0 [γ (x − vt) + y + z ] γZey Ey = 2 2 2 2 3/2 4πǫ0 [γ (x − vt) + y + z ] γZez Ez = (6) 2 2 2 2 3/2 4πǫ0 [γ (x − vt) + y + z ] B need not concern us, as long as the change in electron speed is ≪ c. At the electron’s position, (x,y,z)=(0, b, 0), these equations evaluate to −γZevt Ex = 2 2 3/2 4πǫ0 [(γvt) + b ] γZeb Ey = 2 2 3/2 4πǫ0 [(γvt) + b ] Ez = 0 (7) The electron undergoes acceleration both along its direction of travel (x) and perpendicular to its direction of travel (the y direction). The accelerations are given simply by mev˙x = −Exe and mev˙y = −Eye. The ratiov ˙y/v˙x = Ey/Ex = b/vt gets very large as t → 0, the peak of the interaction (where the denominators in Equation 7 are smallest). This indicates that acceleration in the y-direction is dominant, and so we will consider only this term from now on. The y-acceleration is: γZe2b v˙y(t)= − (8) 2 2 3/2 4πǫ0me [(γvt) + b ] Fourier transforming this, γZe2b ∞ eiωt v˙ (ω)= − dt (9) y 3/2 4πǫ0me Z−∞ [(γvt)2 + b2] Changing variables to u = γvt/b then gives Ze2 1 ∞ eiωbu/(γv) v˙ (ω)= − du (10) y 3/2 4πǫ0me bv Z−∞ [1 + u2] 3 2ωb The integral is γv K1[ωb/(γv)], where K1 is a modified Bessel function of order unity. Hence, Ze2 1 2ωb v˙y(ω)= − K1[ωb/γv] (11) 4πǫ0me bv γv It is instructive to look at functional form of K1 at large and small values: for y ≪ 1, K1(y) → 1/y, 1/2 π while for y ≫ 1, K1(y) → 2y exp(−y). At low frequencies (ω ≪ γv/b), therefore, −Ze2 v˙y(ω) ≃ . (12) 2πǫ0mebv The intensity spectrum of the low frequency radiation is then given by (cf. Equation 4) e2 v 2 I(ω) = 2 3 | ˙ (ω)| 6π ǫ0c e2 Ze2 2 = 2 3 6π ǫ0c 2πǫ0mebv Z2e6 1 = 4 3 3 2 2 (13) 24π ǫ0c me (bv) Similarly, at high frequencies −Ze2 v˙ (ω) ≃ e−ωb/γv, (14) y 1/2 (8π) ǫ0mebv and hence 2 6 Z e ω −2ωb/γv I(ω)= 3 3 3 2 3 e . (15) 24π ǫ0c me γbv This exponential cut-off tells us that there is little power emitted at frequencies above ω ≈ γv/b. The origin of this exponential cut-off arises since the duration of the collision can be approximated as the time that the ion takes to travel from x = b to x = −b, ie τ ≈ 2b/γv. This timescale corresponds to a frequency ω ≈ 2π/τ ≈ πvγ/b, which is of the same order of magnitude as the exponential cut-off frequency. I(ω ) ω γ ω = v b Figure 3: The Bremsstrahlung spectrum from the interaction of an electron with an ion. 4 1.2.2 Integration across all collision parameters for one electron The calculations so far relate to the interaction of one electron with one ion, with impact parameter b. In order to work out the full spectrum of radiation, we need to consider all collisions, by integrating across all possible collision parameters. For simplicity, from now on we will consider only non-relativistic Bremsstrahlung, and therefore take γ = 1. The rate at which an electron undergoes collisions with ions separated by distance b to b + db is given by ni v 2πbdb, where ni is the number density of ions. Therefore, at low frequencies, bmax Z2e6 1 I(ω) = 4 3 3 2 2 2nivπbdb Zbmin 24π ǫ0c me (bv) 2 6 Z e ni 1 bmax = 3 3 3 2 ln (16) 12π ǫ0c me v bmin where we have set bmin and bmax as the minimum and maximum impact parameters that we will consider [note that by convention we write Λ = bmax/bmin]. Setting these limits is essential, since the logarithm diverges as either bmin → 0 or bmax → ∞. We therefore need to consider what values to take for the limiting b values. Fortunately, due to the logarithm involved, we don’t need to be too precise in this: e.g. the difference between ln(105) and ln(1010) is only a factor of two. Estimating bmax is the easier of the two. We showed above that for a single interaction there was an exponential cut-off in the spectrum above ω ≈ v/b. For a given ω, therefore, only collisions with b ∼< v/ω will make an important contribution: higher b collisions will be on the exponential tail of their spectrum. Hence, we take bmax = v/ω. (17) For bmin we can consider two different possibilities. One assumption that we have made is that the deflection angle is small; this will clearly break down if v is too slow. A reasonable approximation for this limit is that the kinetic energy of the electron is comparable to the electrostatic potential energy at closest approach, ie. 2 1 2 Ze mev ≈ (18) 2 4πǫ0b or Ze2 bmin = 2 (19) 2πǫ0mev An alternative possibility is to adopt a quantum approach, which becomes important if the change in speed is comparable to the speed itself. For example, if ∆v = 2v, then ∆p = 2mev, and Heisenberg’s uncertainty principle (∆x∆p ≈ ~) says that ∆x ≈ ~/(2mev). Thus we take ~ bmin = (20) 2mev Clearly the reality is that we should take whichever is the larger or these two values; which this is depends upon the properties of the system. For gases with low electron speeds, such as HII regions 5 at T ≃ 104K, the first estimate gives the higher value and so Equation 19 is the appropriate one to use. For hot gases in clusters of galaxies, with T ≃ 107K, the quantum limit is larger, and so Equation 20 gives the appropriate limit.