5.1

5. SYNCHROTRON RADIATION1

5.1 Charge motions in a static magnetic field

Charged particles moving inside a static magnetic field continuously accelerate due to the Lorentz force and continuously emit radiation. In the case of non-relativistic motions, the emitted radiation is named cyclotron radiation and is due to the spiraling motion of particles and consequent acceleration. The radiation frequency is just that of the gyration qB/mc, where q and m are the electric charge and mass of the particle. This is a line emission, if we neglect broadening factors. The nature of the emission changes radically if the particle becomes relativistic, and the emission is then named synchrotron emission. The motion of a generic charge is ruled by the Lorentz dynamical equation of motion

[5.0]

and the energy conservation equation

1 This Section particularly benefited, in addition to the Rybicki & Lightman book, by lecture notes on High Energy Astrophysics by A. Cavaliere.

5.2

(the second term has been neglected because of the scalar product and because an assumption about the lack of significant large scale electric fields - that would be immediately erased by charge motions - was made). Then we use the second eq. into the former and get (see the figure below for an illustration of the relevant quantities)

[5.1]

The solution is then an helicoidal motion (see graph below), with the velocity along the direction of the B field keeping constant, and a uniform circular motion around that direction with rotational pulsation given by

5.3

BqBmc

ωγ

= . [5.2]

5.2 Total synchrotron emitted power

The total emitted power by the particle is easily studied in the instantaneous reference frame of the electron (K'). Here the particle will not remain at rest for long time, but just for a short time interval and the electron motion in that frame will in any case be non-relativistic during this time. So we can use here the Larmor formula for the emitted power. Alternatively, one might use the full relativistic eq. (1.21), but this would be a quite more complicated derivation (see Sect. 5.7 below). Because energy and time transform in the same way from K' to the system K of the observer (and of the B field), we have

22

3

'; '''

'2' '3

dW dW dt tdW dWP Pdt dt

qP P ac

γ γ= =

= = =

= =d

We need to consider how the vector acceleration transforms from one system (K) to the other (K'). It can be seen (e.g. Rybicki & Lightman) that the following relativistic rules apply if the particle is at rest in K', where parallel and orthogonal refer to the direction of the instantaneous velocity vector

3 2' ; 'a a a aγ γ⊥ ⊥= =

.

From this and [5.2] we have

2

2 2 24 2 2 2 2 2

03 2 2 2

222 2 2

0 2 2

v v

2 2v3 3

v; sin

B BqBa rmc

q q BP r c Bc m c

ermc c

ω ωγ

γ β γγ

β β a

⊥

⊥

⊥ ⊥ ⊥

⊥

⊥

= = =

= =

= = =

[5.3]

5.4

0r is the classical electron radius, q e= the electron charge. a is the pitch angle, that is the angle between the velocity and B field vectors, as shown in the figure:

For an isotropic distribution of charge velocities, we average over α as:

2 22 2

22 2 2 2

0

such that2sin ,

4 323

d

P r c B

β ββ aπ

β γ

⊥ = Ω =

=

∫

or, if we use the Thomson cross section 208 3T rσ π= and the energy density of

the magnetic field UB

2 2 24 with / 83 T B BP c U U Bσ β γ π= = [5.4]

For the cyclotron emission we have instead 24 v

3 T BP Uc

σ= .

5.3 Aberration, beaming, angular distribution of radiation

Recall that in the lab reference system, in which the electron is seen moving with relativistic velocity, the emission is strongly enhanced (beamed) in the direction of the velocity. According to the relativistic aberration law, we have

5.5

where α is in the lab frame (K) and α' in the electron frame (K'). This implies that

, except when . We can see this by a power series development

If this condition does not verify, then (see below figure for an illustration of effect)

[5.5]

Case of

Case of

5.6

5.4 The transition from the cyclotron to the synchrotron spectrum

Let us consider here the evolution of a synchrotron spectrum at increasing electron velocity and energy. This makes a transition from a cyclotron line emission to the synchrotron continuum one that is illustrated by the following figures.

5.7

For another graphical representation we refer to Ghisellini (High Energy Astrophysics):

From cyclo to synchro: if the emitting particle has a very small velocity, the observer sees a sinusoidal (in time) electric field E(t). Increasing the velocity the pattern becomes asymmetric, and the second harmonic appears. For 0 < β ≪ 1 the power in the second harmonic is a factor β2 less than the power in the first, see eq. (5.4). For relativistic particles, the pattern becomes strongly beamed, the emission is concentrated in the time ∆tA. As a consequence the Fourier transformation of E(t) must contain many harmonics, and the power is concentrated in the harmonics of frequencies ν ∼1/∆tA. Broadening of the

harmonics due to several effects ensures that the spectrum in this case becomes continuous. Note that the fundamental harmonic becomes smaller at increasing γ (since νB ∝ 1/γ).

The spectrum of emission of the first 20 harmonics of mildly relativistic cyclotron radiation for an electron with v = 0.4c (Bekefi, 1966).

5.8

Cyclotron lines are observed more frequently in absorption in the high-energy spectra of neutron stars, like in the above figure, where the first 4 harmonics are observed. From this, the value of the (strong) stellar magnetic field can be derived from

:

where the last factor accounts for the gravitational redshift . 5.5 Qualitative behavior of the synchrotron spectrum of a single pulse

We have seen that the spectral emission in the case of a cyclotron is concentrated essentially in a line of frequency . All the electron orbit is seen by the observer. Instead, in the synchrotron case, because of the relativistic beaming, the radiation field seen by the observer is concentrated inside a narrow cone of width around the electron velocity vector. As a consequence, the observer will see photon emission only from a small fraction of the orbit: therefore, the emitted spectrum will

Hard X-ray spectrum of the neutron star X0115+63 observed with Beppo-SAX, showing the first four cyclotron harmonics. (From Santangelo et al. 1999)

5.9

include a much wider range of photon frequencies. All this is easily understood in terms of the Fourier transformation.

Let us consider the above figure (from Ribicki & Lightman) representing the portion of the orbit seen by the observer. a is the radius of the orbit. The points 1 and 2 correspond to the extremes of the sector seen: in 1 the emission cone enters the line-of-sight, in 2 it exits it. The emission cone is shown in the two points.

The angle Δθ is that subtending the sector, and is given by . The distance Δs between positions 1 and 2 is then

[5.6]

Now the radius of the orbit can be obtained from

[5.7]

where is the gyration frequency of the relativistic electron from [5.2] and where α is the pitch angle (obviously, the radius a increases when α decreases, that is the velocity approaches the B direction). So, from [5.6] and [5.7]

[5.8]

5.10

In the observer frame K, the transit times t1 and t2 are such that 2 1v( )s t t∆ = −

2 12sinB

t t tγω a

∆ = − =

This time interval however does not correspond to the times of arrival to the observer of the emitted photons, because these will be shorter by an amount equal to /s c∆ , the time that radiation emitted from 1 takes to travel from 1 to 2. So the arrival times will be

32 v 11sin sin

arr

B B

tcγω a γ ω a

∆ = −

because 2(1-v/c)∼1/γ2. In conclusion, the observed pulse will have a temporal

duration 31γ times the period of the electron gyration.

From consideration of the Fourier transform, we then expect the presence of characteristic photon frequencies in the observed spectrum proportionally larger by a factor 3γ , that is

3

3 21 1 1sin sin2 2 2char B cycl

qBmc

γn ω γ ω a a γ nπ π π γ

= = = =

We then define a critical frequency for the synchrotron emission as:

23 sin2C cycln γ ω a= [5.9]

5.11

So we expect significant emission to happen at this frequency. We will see later that indeed the peak synchrotron emission happen at this critical frequency. Let us see now some further scaling relations concerning the spectrum of a single radiation pulse by a single orbiting electron. Due to the relativistic beaming and the enormously increased emitted power in the direction of the electron velocity vector, the dependence of the electric field intensity E on the angle θ (the polar angle between the instantaneous velocity vector and the observer line-of-sight) goes under the product γθ . So also concerning the temporal

dependence we have [ ]( ) ( )E t g tγθ∝ , where t is the time in the K observer

frame. This means that a given value of E will be obtained for large values of θ if γ is proportionally small, or vice-versa small θ if γ is large. Let us set the origin of times t and paths s when 0θ = (this corresponds to when the velocity vector is along the visual). We will have:

(arrival time of the signal)/ ; (1 v / )v vss ss a t ccθ − −− − − [5.10]

/ sinv Bss a γγθ γ ω a− − . Since 21 v/c 1/2γ− −

, we have from [5.10]

2 22 , so that 2 ( sin )v B cs t t tγ γθ γ γ ω a ω∝− −

So for the time dependences of the E field we can write

[ ]( ) cE t f tω∝ ,

where [ ]f x is an a-dimensional function, so that every dependence on t goes under

the product ctω . The Fourier transform of E will be

and by transforming( ) ( )exp( )

1( ) ( )exp

c c

c c

E f t i t dt t t

E f i d

ω ω ω x ω

ωω x x xω ω

+∞

−∞

+∞

−∞

∝ → =

∝

∫

∫

For the power spectrum of the pulse from the single charge we will have (after defining T as the electron rotation period):

1 ( )

c

dW dW P CFdt d T d

ωω ωω ω = = =

[5.11]

5.12

that is, every dependence of the spectrum on frequency will go under the ratio c

ωω .

F is an a-dimensional function and C is a quantity that does not depend on the frequency ω. We can easily determine the value of this constant and verify that it does not depend on γ either. Indeed if we write the total power

( )( ) ;cc c

P P d C F d C F x dx xω ωω ω ω ωω ω = = = ≡ ∫ ∫ ∫ [5.12]

if we now compare this relation with [5.3] and [5.4], in the ultra-relativistic limit we get

4 2 2 2 2

2 3

2

with the definition 2 sin

33 sin

2c

q BPm c

qBmc

γ β a

γ aω

=

=

From [5.12], ( )0cC P F x dxω∞ = ∫ and we finally obtain

3

23 sin( )

2 c

q BP Fmc

a ωω ωπ =

[5.13]

We need later to find the full correct expression for the function F(x), which requires a rather complex calculation (Sect. 5.7 below). However, what we have found already allows us to obtain the key results of the synchrotron emission without going into complex calculations, just using the general expression [5.13].

5.6 Emission by a non-thermal electron energy distribution

We have seen in Sect. 5.4 that the emission spectrum of a single charge can be expressed as a function of an a-dimensional function F of the ratio cω ω , that is the whole dependence of the individual spectrum on electron energy is completely contained in that ratio. We need now to define the distribution of energies for a set of emitting electrons. This choice will be such that later comparison of predicted spectra will bring them into agreement with observations. We anticipate that this requires to introduce a power-law energy distribution of non-thermal electrons:

5.13

[5.14]

Of course, the normalization constant C1 may or may not depend on direction. Then

We change the integration variable to and, remembering that

; ; , the Jacobian gets

, so

where the limits will depend on γ1 and γ2 but if these limits are sufficiently wide, we have x1 =0 and x2=∞, such that

[5.15]

Figure a) Emission spectrum for a power-law energy distribution for emitting particles. b) Observed radio spectrum of a radio-source, showing indeed the power-law behaviour, in addition to a low-frequency cutoff due to synchrotron self-absorption. The spectral index α of the emitted radiation is simply related to that of the energy distribution of electrons, and the synchrotron spectrum will be a simple power-law as in figure.

log S(n)

log n

-(p-1)/2

5.14

Typical values of the energy index p are typically between 2 and 3. Then the radio spectral index will assume values around 0.7, as we will see in some detail below. How does this compare with data on the flux spectrum of cosmic rays in our Galaxy will be discussed in Sect. 7.

As we see from the figures, cosmic rays in the Galaxy show indeed energy spectral slopes between 2 and 3 over an enormous interval of particle energies. This is completely consistent with the observed synchrotron spectra of radio-galaxies (similar acceleration mechanisms appear to operate on such different scales).

5.15

More information on the cosmic rays that we directly collected near the Earth, as well as those inferred in Galactic and extragalactic radio-sources are reported in the following Section 6, including their presumed origin and composition.

5.7 The complete analysis

So far we have developed an heuristic analysis of the synchrotron emission that allowed us to obtain in a simple way the main results. We need to mention here a more complete and rigorous treatment, that we take from Rybicki & Lightman. The geometry of the process that we adopt is illustrated in the figure below.

The time considered is the retarded time as usual. It is assumed to be t'=0 at the moment the particle transits the origin of the system. The instantaneous velocity v is taken to be along the x axis. is the line-of-sight versor, is a versor orthogonal to v at t'=0 within the plane (x,y) containing the electron orbit (with curvature radius a) at that time. We define . Then the calculation starts from the general equation of electrodynamics in the "wave zone", eq. [1.21]. We can show that the total emitted power by the particle can be expressed as the sum of two contributions, one with the electric field parallel to and the other parallel to . We then start from the famous general eq. of electrodynamics, eq. (1.21) of Sect.1 (considering the acceleration term only, of course):

5.16

5.17

5.18

5.19

5.8 Synchrotron self-absorption and spectral cutoffs

If there aren't cutoffs in the electron energy distribution that might affect the spectrum, it could happen in some circumstances - e.g. for high particle densities - that the photons emitted by synchrotron are re-absorbed by the synchrotron process itself before exiting the source. This implies a synchrotron optical depth τ>1. If τ>>1, we can make the argument extreme, by stating that the intensity of the photon field produced is I(ν)=flux of radiant energy ∼ (number of independent states of the photon field)×(average energy of each state ε). From the black-body theory

5.20

we borrow the first factor being given by 2 32 /sn cn= , while for calculating ε we can consider it to correspond to an equilibrium situation between the radiation field and the emitting particles. The appropriate value for ε is that of electrons emitting at frequency ν:

2 2c Bn n γ ω ε∝ ∝= ∝

such that the characteristic energy of the emitting photons is 1/2ε n∝ . Overall, in the self-absorption regime, τ>>1, the spectrum will be:

2 0.5 5 / 2( ) ( )S In n n n n∝ ∝ ⋅ ∝ . [5.20]

We immediately verify that the absorption coefficient aν for synchrotron [defined by the transfer equation dI(ν) = aν I(ν) ds], can be obtained from the relation of the source function S(ν)=jν/aν with S(ν)∼ν2.5 , jν ∼ν-(p-1)/2 , hence

( 4) / 2pna n − +∝ [5.21]

that, considering the typical values of p, shows a strong inverse dependence on frequency aν ∼ ν 3. The meaning of this is that high values of optical depth, or the self-absorption, happen at low frequencies (Figs. 6), 3

n nτ a n −= ⋅ ∝ .

5.9 Synchrotron polarization

Let us first consider the simplest case of a single non-relativistic particle radiating cyclotron emission. The geometrical situation is illustrated in the figure, assuming for simplicity the particle simply rotating in a plane orthogonal to B.

Figure 6. Overall spectrum of the synchrotron emission, including synchrotron self-absorption at low frequencies.

5.21

The emitted radiation is described as usual in terms of the Larmor equation and of the time variations of the electric dipole. In our case, the dipole d

d

makes a rotation in the x-y plane, a motion that can be represented as the sum of two orthogonal oscillating

dipoles 1dd

and 2dd

.

At the point A of the figure only the oscillating dipole 1dd

contributes radiation: here

the polarization is just linear. At point B both 1dd

and 2dd

dipoles contribute, hence the polarization is circular. At intermediate directions the polarization is intermediate, i.e. it is elliptical. Elliptical polarizations as observed above and below the orbit are opposite. All this is useful to understand the more complex synchrotron situation. In the case of the synchrotron emission, the flux is linearly polarized looking in the direction of maximal emission, that is of vd , and is elliptically polarized when looking outside this direction. If the line-of-sight is outside the emission cone of the particle, the emission is essentially null. The situation is well illustrated by the figure below. The direction of the B field and of the particle motion are indicated, as well as the path made on the sky by the velocity vector and the trace of the emission cone in shaded grey. In analogy with what happens for the cyclotron, the polarization inside the emission cone in the inner and outer shaded region, with respect to the trace of the velocity vector, is elliptical, in one sense and in the opposite, respectively. So, for a roughly uniform distribution of electrons, these two elliptical polarization components cancel out, and only the linear polarization survives. If we want to quantitatively estimate the degree of polarization, we need to refer to the analysis in Sect. 5.7, where we have seen that the emitted wave in the observer

5.22

frame can be split into two components, one with the electric field ε

d

parallel to the

projection of the B field on the celestial sphere, and the other component ε⊥d

orthogonal to it. The corresponding radiation powers are ( )P dW dω ω⊥ ⊥= and

( )P dW dω ω=

. The degree of polarization can then be immediately obtained from

( ) ( ) ( )( )( ) ( ) ( )

P P G xP P F x

ω ωω

ω ω⊥

⊥

−P = =

+

[5.24]

where G(x) and F(x) are the two functions defined in Sect. 5.7, and F(x) corresponds to the total emitted power.

Emission cone of the single particle, where the line-of-sight and direction of the magnetic field are indicated. Because of the geometry of the emission, the synchrotron light is polarized linearly along the direction of the velocity vector (i.e. along the cone). Polarization is almost frequency independent. Once integrated over all frequencies, we obtain an average (linear) polarization fraction of about P∼75%, a rather high value. For an entire population of particles with power-law distribution function as in [5.14], we get

1

7 / 3p

p+

P =+

[5.25]

5.23

which makes still a very large fraction assuming a typical value for the spectral index p=2.5, P∼70%. It has a slow dependence on the energy spectral index p, which is illustrated in the following graph 2.

Note however that all this assumes a completely ordered B field, which is not a realistic assumption; the field may vary even chaotically inside the source or on its surface, which diminishes the polarization. Values of about a few tens of percent are often observed in radio-galaxies. The direction of polarization is orthogonal to the direction of the projection of the B field on the celestial sphere: polarization studies then allow us to estimate the

orientation of the B field on the sky. The observational evidence for polarized light offers important indications about the physical origin of radiations in cases in which this might not be obvious. One such case is the X-ray emission by the Crab nebula (that is a very important astrophysical source and often used as a cosmic calibrator at high energies). This source will be further discussed in Sect. 5.11.3. Observations of the degree of linear polarization of these X-rays have demonstrated that the origin of this flux is synchrotron emission by a population of extremely energetic electrons. Note in any case that the presence of polarized light is not always evidence for non-thermal processes in operation: other phenomena can originate it, e.g. scattering and reflections, or even anisotropic absorption by polarized orientation of dust particles.

5.10 Validity limits of our treatment

Two basic assumptions have been made in treating synchrotron. From the quantum mechanical limit condition that hν<<W, since , , where W is the particle (electron) energy, we have that

2 The calculation is simply: , after change of variable

.

P

p

5.24

135 10B Gaussγ < [5.26]

as the condition such that quantum corrections are negligible. From eq. (5.4), and what we will discuss in the next Sect., we can immediately understand that the quantum effects will be in the sense of reducing the emitted power because of the decrease of the cross section with respect to Tσ .

Another limitation to our treatment comes from considering that the radiation reaction can become important and modify the helical orbit, hence the spectrum, if the particle in an orbit loses a too large fraction of its energy:

22 2 22(1 orbit)

4TBE P t c mc

eB mcπγσ β γ γ

π= ∆ = <<

or

2 162

0

5 10eB Gaussr

γ <<

Since 2 2B mcγ π n= ,

3

2 20

2 70 !mc e mhch MeVe r e

π n n<< ⇒ << [5.27]

(See implications of this when discussing about the Crab Nebula emission below.)

5.11 Energy losses of particles and synchrotron spectral evolution

5.11.1 Generalities We consider here how the synchrotron spectrum evolves as an effect of the energy losses of the emitting particles. Every electron produces photons from transformation of its kinetic energy according to (note that the B component considered in the following is just that orthogonal to the line-of-sight):

0

2 22 2

2

2

2 2 0

;4 4

4

TT

tT

dE B d Bcdt dt mc

d B dtmc

γ

γ

γ σσ γ γπ π

γ σγ π

= − = −

⇒ =−∫ ∫

5.25

whose solution details how the electron energy changes with time:

[5.28]

for electrons and sufficiently long times t . Indeed the classical treatment requires that the time-scale for evolution is sufficiently long (see chap.5.10). Vice versa, the characteristic time for the energy loss through radiation emission is:

. [5.29]

From all this we see that particles with higher energy loose energy faster than the others. These energy losses produce a progressive modification of the emitted spectrum: the electrons with the highest energies producing the photons of highest frequencies loose energy fast and move to lower energies in an energy-differential way, as shown in the figure.

Consequently, we expect a bending of the synchrotron spectrum at the highest frequencies, progressing towards the lower energies with time. From [5.28] we get the synchrotron frequency where the energy loss effect starts to manifest itself:

[5.30]

5.26

5.11.2 Synchrotron spectral evolution The exact solution of the problem of estimating the spectral evolution by synchrotron emission requires to solve the energy continuity equation for the electron population, where N is the phase-space density of electrons:

[5.31]

The detailed process then depends on the source function S, which details how the energetic electrons are injected into the volume, if this happens in a continuous way or impulsively at the start time. It can be shown (see details in Blumenthal & Gould 1974) that if the injection is continuous we have that the energy distribution spectrum bends by , hence the synchrotron spectrum bends by (see figure below, showing the evolution in the case of a continuous injection).

Instead, in the case of an impulsive injection of electrons at t=0, we have a qualitatively similar situation, in which the bending in the photon spectrum gets larger: instead of -0.5, it is now . This difference may be intuitively understood, considering that in this latter case there is no compensation to the energy loss by the continuous electron injection as in the previous case.

5.27

5.11.3 The case of the Crab Nebula

A very interesting illustration of the energy loss effect by synchrotron emission is offered by a particular galactic object, which may be taken as a prototypical synchrotron source: the Crab Nebula. We have already mentioned that the evidence for linear polarization demonstrates the synchrotron nature of this emission up to the X-rays at least. The source (in particular the diffuse component within the network of filaments that constitute the nebula) has been observed in any detail at all frequencies. The Crab total broad-band spectrum is shown in the figure.

70 MeV

5.28

Optical and X-ray images of the Crab nebula.

Note among other things that the strong spectral break at about 1022 Hz corresponds to the limit of validity of the synchrotron treatment of 70 MeV in [5.27]. For photons at higher energies the electrons lose their energy very rapidly during one single orbital round. So the portion above 1023 Hz a different emission mechanism should be invoked (see Sect. 9).

5.29

This continuum spectrum is composed of various sections, all with a power-law behavior. Let us concentrate for the moment on the region between radio and the optical-IR: above a power-law spectrum with a rigorously constant index 0.3a extending from 107 to 1012 Hz, we see a first bending at about 1013 Hz that can be interpreted as due to energy losses in the electrons. On the other hand, we know exactly the time of the birth of the nebula, A.D.1054 (from Chinese records), so from this we can calculate the magnetic field intensity permeating the nebula, from eq. [5.30]:

7 4 7 2/3 45 10 / 2 10 (950 3 10 ) 5 10B Gauss−≈ ⋅ ≈

(note that this value depends weakly on the cutoff frequency and its uncertainty, * 1/3B n −∝ ). The bending of the spectrum is approximately 0.5a∆ − , which is

consistent with a continuous injection 3. There is in any case a stronger argument in favor of a continuous injection since the formation of the remnant 960 years ago: if the B field is B∼5 10-4 Gauss, energetic electrons emitting in X-rays loose energy very quickly: to emit at ν=3 1018 Hz, they need to have

1/21/2 187

4

105 1010

mc HzeB B Gauss

n nγ −

×

but then they lose energy in a time 8

77 4 2

5 10 4 10 sec 1 yrs!5 10 (5 10 )

t −

And high-frequency photons even shorter times. Such energetic electrons need to be continuously refurbished to the nebula. This argument demonstrates that there is an engine in the source doing the job, and this is the pulsar (indeed, this argument was known even before the discovery of the pulsar in 1966). Electrons with 75 10γ have an energy of 2.5 107 MeV=25 TeV, or about 100 erg, the kinetic energy of a flying mosquito. Instead the energies of the most energetic cosmic rays appearing in Fig. pag.14 correspond to that of a fast tennis ball! None of these energies can be reached in terrestrial accelerators (not even the last-generation LHC!, in particular 25 TeV is 2 times more than the highest energy achieved in LHC in proton collisions – note it is much more difficult to accelerate electrons, see Sect. 6), the main reason being the energy losses of the accelerated electrons due to their synchrotron emission: to reach higher energies we would need much lower B fields to constrain them, which would mean however accelerating rings of much larger radii.

3 There is also a second change in the spectral index in the far-UV that is clearly related to the acceleration mechanism.

5.30

A synchrotron spectrum with 0.3( )P n n −∝ would mean a flow of electrons injected

with an energy distribution 1.61( )N Cε ε −

.