Download 7 Radiation of relativistic particles 7.1 General properties of relativistic radiation sources

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7.1
Radiation of relativistic particles
General properties of relativistic radiation sources
The wave 4-vector is K α = (ω/c, k) = (ω/c)(1, n). The Lorentz transformation, K 0 = Λ̂K, to the
frame moving with the velocity v in the x direction is presented by the matrix


γ
−βγ 0 0
−βγ
γ
0 0

Λ̂ = 
(7.1)
 0
0
1 0
0
0
0 1
p
where β = v/c, γ = 1/ 1 − β 2 . The reverse matrix describing the transformation from the
moving to the lab frame,K = Λ̂−1 K 0 , is obtained by replacing v by −v in Λ.
Doppler effect. If light is emitted by a moving source, the frequency detected in the lab
frame is
ν0
,
(7.2)
ν=
γ(1 − β cos θ)
where θ is the angle between the ray and the direction of motion (in the lab frame). Specifically
if the source moves towards the observer, ν = γ(1 + β)ν 0 .
Aberration and relativistic beaming. If a ray is emitted at the angle θ0 in the source
frame, the emission angle in the lab frame is
tan θ =
k2
sin θ0
k20
=
.
=
k1
γ(k10 + βk00 )
γ(cos θ0 + β)
(7.3)
One sees that the radiation from a highly relativistic source, γ 1, is beamed in a forward cone
of opening angle θ ∼ 1/γ.
Transit time effect. If the source is moving near the speed of light along a direction which
lies close to the line of sight, then the source almost catches up with its own radiation but not
quite. Consider an object which moves from a point P1 to a point P2 in a time ∆t in the observers
frame. If the angle between the direction of motion and the line of sight is θ, the time difference
between the time of receptions of photons emitted at P1 and P2 is
∆trec = ∆t(1 − β cos θ).
(7.4)
Specifically for a highly relativistic source moving towards the observer one gets ∆trec = ∆t/2γ 2 .
This effect can give the illusion of apparent transverse motion which is greater than the speed
of light. The source shifts in the plane of the sky by L⊥ = v∆t sin θ. Therefore the apparent
transverse velocity,
β sin θ
L⊥
=
,
(7.5)
βapp =
c∆trec
1 − β cos θ
could be as large as γβ.
Lorenz invariancy of the radiation power. In most cases the emission is symmetric in
the sense that the energy emitted in the direction n is the same as in −n. Then the momentum
loss in the comoving frame of the source is zero. Let the source radiate the energy ∆E 0 during
the time ∆t0 in the comoving frame. The corresponding energy-momentum 4-vector is (∆E 0 , 0);
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the displacement 4-vector is (∆t, 0). Both these 4-vectors are transformed to the observer’s frame
according to the Lorentz transformation to yield ∆E = γ∆E 0 ; ∆t = γ∆t0 . Now one can see that
the radiation power is a relativistic invariant,
dE 0
dE
= 0.
dt
dt
7.2
(7.6)
Synchrotron radiation
A non-relativistic electron rotates in the magnetic field with the Larmor frequency
νB =
eB
= 2.8 · 106 B Hz
2πme c
(7.7)
where B is in Gauss. Non-relativistic electrons radiate predominantly at the Larmor frequency
and a few lower harmonics.
The gyration frequency of a relativistic electron is
ν̃B =
νB
.
γ
(7.8)
Radiation of a highly relativistic electron is beamed to within the angle ∼ 1/γ therefore only
the radiation emitted during the fraction 1/γ of the gyration period is received by an observer.
Because of the transit time effect, the observed duration of the pulse is even γ 2 times smaller so
that the observer sees pulses of the duration
∆t ∼
1
1 1
= 2 ,
2
γ ν̃B γ
γ νB
(7.9)
the interval between the pulses being 1/ν̃B . The maximum Fourier component of the spectral
decomposition of the observed pulse corresponds to the reciprocal of the pulse duration
νm = νB γ 2 .
(7.10)
The detailed calculation shows that a relativistic electron in the magnetic field radiates the continuum spectrum with the maximum close to 0.5νm sin θ, where θ is the electron pitch angle. At
ν νm , the spectrum is proportional to ν 1/3 . At ν νm , the intensity decreases exponentially.
Taking into account that the total radiation power is a relativistic invariant, one can write
P =
2 e2 a02
,
3 c3
(7.11)
where a0 is the electron acceleration in the comoving frame. In the lab frame, a = 2πν̃b c sin θ =
eB sin θ/me γ. In order to perform transformation, let us define the 4-vector of acceleration
dU
γ dγ dγv
W =
=
,
,
(7.12)
ds
c dt cdt
√
where U = dx/ds = γ(1, v/c) is the velocity 4-vector; ds = c2 dt2 − dx2 = cdt/γ the interval.
In our case, a · v = 0, therefore dγ/dt = 0; this implies W = (γ/c)2 (0, a). In the comoving
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frame, W 0 = (0, a0 /c2 ), therefore writing out the relativistic invariant W 2 in both frames yields
a02 = γ 4 a2 . Now one finally finds
P =
2 e4 B 2 2 2
γ sin θ = 2σT UB cγ 2 sin2 θ,
2
3
3 me c
(7.13)
where UB = B 2 /8π is the magnetic energy density.
The relativistic particles typically have a power-law distribution
N (γ)dγ = Kγ −p dγ.
(7.14)
In most cases, p = 2 ÷ 3. Each electron radiates mostly at ν ∼ νm . Since the radiation spectrum
is narrower than the breadth of the power-law electron energy spectrum, one can assume that an
electron of energy E = me c2 γ radiates away its energy at ν = νm sin θ = γ 2 νB sin θ. Assuming
isotropic angular distribution of electrons, one can write the energy radiated in the frequency
range ν to ν + dν in the unit solid angle in the form
jν dν =
where
γ=
ν
νB sin θ
P
N (γ)dγ,
4π
1/2
;
dγ =
(7.15)
dν
.
2(νB ν sin θ)1/2
Now θ is the angle between the magnetic field and the line of sight. Finally one gets
−α
σT cKUb sin θ
ν
p−1
jν =
.
;
α=
4πνB
νB sin θ
2
(7.16)
(7.17)
There is an absorption process corresponding to any emission process. In principle, one can
find the synchrotron absorption coefficient from the detailed balance. A simpler way to take into
account the synchrotron absorption within the source (often called self-absorption) is to notice that
the radiation intensity from any body cannot exceed the blackbody intensity, Bν = 2kT (ν/c)2 ,
whose temperature is determined by the electron energy; in our case kT ∼ γme c2 . Taking into
account the relation (7.16) between the electron energy and the emitted frequency, one gets the
maximal possible intensity of the synchrotron radiation
Iν =
8πm3e c 5
ν
eB
1/2
.
(7.18)
Note that this intensity depends only on the magnetic field within the source.
If a source of the size R is transparent, the intensity of the emitted radiation is just Iν ∼ jν R.
Typically jν grows with the decreasing frequency (see, e.g., eq. (7.17) for the power-law electron
spectra). In this case, the source becomes opaque at small frequencies and then the intensity is
described by eq. (7.18). The transition frequency, νabs , is determined by the condition
8πm3e c 5
ν
eB abs
1/2
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= jνabs R.
(7.19)
7.3
Compton scattering
Kinematics of the Compton scattering is described by the conservation of 4-momentum:
P + ~K = P1 + ~K1 .
(7.20)
Here P = me cU , P 2 = m2e c2 ; K 2 = 0. In order to find the final state of the photon, one has to
move the term with K1 to the lhs and square the obtained equation:
(P + ~K − ~K 0 )2 = m2e c2 .
(7.21)
If the electron was initially at rest, P = (me c, 0), one gets
ν1 =
ν
,
1 + (hν/me c2 )(1 − cos θ)
(7.22)
where θ is the scattering angle. At hν me c2 , the photon frequency does not change; this is the
Thomson scattering.
The cross-section is the Klein-Nishina formula. At hν me c2 , it is reduced to the Thomson
cross-section (3.24); in the opposite limit,
1
3 me c2
2hν
+
σ = σT
ln
.
(7.23)
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hν
m e c2 2
Let us now consider the scattering on a moving electron. The electron sees the photon frequency
ν 0 = νγ(1 − β cos θ),
(7.24)
where θ is the angle between the velocity of the electron and the photon. At γ 1, one gets
typically ν 0 ∼ γν. If
γhν me c2 ,
(7.25)
the photon frequency does not change in the scattering process, ν10 = ν 0 . Due to the relativistic
beaming, the scattered photon is directed in the lab frame to the direction close to the electron
velocity; then eq. (7.2) yields the frequency of the scattered photon in the lab frame:
ν1 =
ν10
2ν10
=
∼ γν10 .
γ(1 − β cos θ)
γ(θ2 + 1/γ 2 )
(7.26)
Now one sees that the photon frequency increases significantly as a result of scattering
ν1 ∼ γ 2 ν.
(7.27)
This process is called the inverse Compton scattering.
Now let us consider a highly relativistic electron moving through the isotropic radiation field
with the energy density Urad . Let the average photon energy be ε0 ; then the photon number
density is N = Urad /ε0 . Due to relativistic beaming, the electron sees the radiation beam directed
towards the velocity. The energy flux in the beam is F 0 = ε0 N 0 c. The photon number density is
the zeroth component of the 4-vector (N, j), where j is the photon flux. In the lab frame, j = 0;
therefore the Lorentz transformation yields N 0 = γN . Taking into account that ε0 ∼ γε, one
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finds F 0 ∼ γ 2 Urad c. Let us now assume that the typical radiation frequency satisfies the Thomson
scattering condition (7.25). Then the scattered photons have the characteristic energy ε ∼ γ 2 ε0 .
The power of the scattered radiation in the electron frame is P 0 = σT F 0 . The Thomson scattering
satisfies the forward-backward symmetry condition, which yields the Lorentz invariancy of the
radiated power. Therefore one finally finds the power of the scattered radiation as
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P = σcUrad γ 2 ,
3
(7.28)
where the factor 4/3 is obtained in rigorous calculation. Note the similarity between the Compton
power and the synchrotron power (7.13) from a source with the chaotic magnetic field when
sin2 θ = 2/3.
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