1.1

COURSE ON HIGH-ENERGY ASTROPHYSICS

Ac. Yr. 2017/18

1.2

SUMMARY

0. INTRODUCTION TO THE COURSE. RELEVANCE OF THE HIGH-ENERGY ASTROPHYSICS IN THE GENERAL CONTEXT (2h)

1. FUNDAMENTALS OF CLASSICAL ELECTRODYNAMICS (4h)

1. Basic formulae of electro-magnetism in the classical limit (from very first principles)

2. Electromagnetic waves 3. The relationship of electric charges and the radiation fields

(radiations from moving charges, Lienard-Wiekert potentials, the fundamental equation of electrodynamics, Larmor equation, dipole emission, multi-polar expansion, the radiation spectrum).

2. BREHMSSTRALUNG RADIATION (4h)

1. The classical limit. Electric dipole contribution. The Gaunt factor 2. Thermal Bremsstrahlung 3. Plasma cooling through free-free emission 4. Radiative transfer and Bremsstrahlung self-absorption 5. Relativistic Bremsstrahlung, non-thermal Bremsstrahlung 6. Applications to free-free emissions by astrophysical plasmas

3. GAS DYNAMICS AND PLASMA EFFECTS (4h)

1. Fundamentals of hydrodynamics. General Equations and Conservation Laws

2. Adiabatic and isothermal stationary flows 3. Sound waves 4. A generalized equation of state 5. Plasma effects. Plasma Oscillations and Plasma Frequency, Debye

length. 6. Particle collisions in plasmas 7. Momentum transfer among plasma particles: viscosity 8. Energy transfer among plasma particles: heat conduction 9. Shock waves 10. Effects of magnetic field. Limits on the heat conduction in plasmas.

1.3

4. HOT PLASMAS IN GALAXIES AND CLUSTERS OF GALAXIES (4h)

1. Fundamental physical parameters 2. Thermalization timescales. Heat conduction. 3. Magnetic field effects. 4. Ionization mechanisms. Collisional ionization. 5. Line emissions. Metal abundances in the plasma. 6. Cooling and heating mechanisms 7. Distribution and structure of the Intra-Cluster (IC) Plasma. 8. Origin, astrophysical and cosmological significance of the IC

plasma. 5. SYNCHROTRON RADIATION (3h)

1. Charges in static magnetic fields 2. Total synchrotron emitted power 3. Aberration, beaming, angular distribution of radiation 4. Synchrotron spectrum of a single pulse and its spectrum 5. The transition from the cyclotron to the synchrotron spectrum 6. Emission by a non-thermal electron distribution 7. The full treatment 8. Synchrotron self-absorption and spectral cutoffs 9. Synchrotron polarization 10. Limits of validity of our treatment 11. Electron energy losses and synchrotron spectral evolution

6. INVERSE COMPTON EMISSION. COMPTONIZATION EFFECTS (3h)

1. Electron scattering in the classical limit 2. Quantum effects: the Klein-Nishina cross-section 3. Compton scattering and Inverse Compton (IC) 4. Emitted power from single scattering and its spectrum 5. Emission by many particles 6. Effects of multiple IC scatterings (Compton parameter, spectral

distortions, Bose-Einstein distributions, the thermal and kinetic Sunyaev – Zeldovich effect)

7. Comptonization effects of radiation by hot plasmas

1.4

7. COSMIC RAYS AND PARTICLE ACCELERATION MECHANISMS (3h)

1. Observational properties of cosmic rays 2. Fermi second order and first-order mechanisms 3. Particle energy spectra for diffusive shock acceleration 4. The unipolar inductor 5. Limitations to the maximum energy

8. ACTIVE GALACTIC NUCLEI . RADIO GALAXIES AND QUASARS (3h)

1. Radio-galaxies and their synchrotron emission 2. Energetics of the synchrotron emission by radio galaxies 3. Evidence for non-stellar activity and energy production 4. Radio-Quiet Active Galactic Nuclei: Quasars and Seyfert galaxies

9. HIGH ENERGY EMISSION FROM ACTIVE GALACTIC NUCLEI (5h)

1. The Effects of multiple Inverse-Compton scatterings. Origin of X-ray emission of radio-quiet (and radio-loud) AGNs

2. Continuum and line emission. The photoelectric absorption. Origin of X-ray emission of radio-quiet (and radio-loud) AGNs by thermal Comptonization. Observational evidences.

3. Compton reflection 4. Main structural components in AGNs. Dust tori, accretion disks,

hot plasma coronae. 5. The AGN Unification Scheme and the X-ray Background. 6. BLAZARs and their phenomenology 7. Production of gamma-ray emission: Synchrotron Self Compton

and External Compton 8. Doppler boosting 9. Super-luminal motions in compact radio sources.

10. ACCRETION POWER IN ASTROPHYSICS (6h)

1. The compactness parameter 2. The Eddington limit. Critical accretion regimes 3. Bondi and spherically-symmetric accretion 4. Accretion with angular momentum. The analogue: accretion in

binary systems. 5. Plasma viscosity

1.5

6. (Thin) accretion discs. Observational tests. 7. Accretion in quasars and AGNs.

11. PROPAGATION OF RADIATION AND PARTICLES THROUGH PLASMAS (4h)

1. Propagation of electromagnetic waves through plasmas 2. Propagation along magnetic fields: the Faraday rotation 3. Cherenkov radiation 4. Electron-Positron Pair Production (in thermal and non-thermal

plasmas) 5. The photon-photon and particle-photon opacity 6. Hadronic interaction processes

12. CHERENKOV ASTRONOMY (2h)

1. The detection technique 2. Atmospheric showers 3. Imaging the shower 4. Existing facilities 5. The Very High Energy (VHE) extragalactic sky

13. THE COSMIC PHOTON-PHOTON AND PHOTON-PARTICLE OPACITIES (2h)

1. Extragalactic background radiations and their measurements 2. Statistical observables of extragalactic sources, background energy

density 3. The photon opacity, applications to current observations. 4. Constraints on background radiations and on VHE BLAZAR

spectra.

14. NEW FRONTIERS. NEUTRINO ASTROPHYSICS AND GRAVITATIONAL RADIATION (2h)

1. Neutrino astrophysics 2. Gravitational radiation and its astrophysical sources

1.6

MAIN PROPOSED READING A. Franceschini: These Lecture Notes G. Ribicki and A. Lightman: Radiation Processes in Astrophysics J. Frenk, A. King, D. Raine: Accretion Power in Astrophysics M. Longair: High Energy Astrophysics M. Vietri: Astrofisica delle Alte Energie C. Sarazin: Clusters of Galaxies G. Ghisellini: Radiative Processes in High Energy Astrophysics A. De Angelis, M. Pimenta: Introduction to particle and astroparticle physics

1.7

THE ELECTROMAGNETIC SPECTRUM AND HEA

1.8

1. CLASSICAL ELECTRODYNAMICS.

1.1 The classical (non-quantistic) limit

With reference to the more general quantum electro-dynamical treatment, we can use the classical limit (i.e. quantum-mechanical effects are negligible) when the De Broglie wavelength of the particle interacting with the photon field is small w.r.t. the typical size of the physical problem under investigation: h/p<<r (1.1) (h: Planck constant, p: particle moment, r: effective radius of the interaction, or radiation wavelength). The classical limit can also be referred to the energetic balance of the interaction: hν<<W (1.2) with W particle energy and ν photon frequency, which means that the particle does not transfer a too large fraction of its energy to the radiation field. Eqs. (1.1) and (1.2) express the same thing if we consider that W≈pv and ν≈v/r, with v the particle velocity. If we consider instead the relativistic theory, our treatment will fully consider relativistic effects (although we will avoid using tensorial formalism).

1.2 Basic formulae in the classical limit

1.2.1 Maxwell and Lorentz A first basic information is in the Lorentz force

vF q E Bc

= + ×

[1.3]

which provides an operative definition of the electric and magnetic fields. The second fundamental set of equations are the Maxwell:

4 0

1 4 1D B

B DE H jc t c c t

πρ

π

∇⋅ = ∇⋅ =

∂ ∂∇ × = − ∇ × = +

∂ ∂

[1.4]

where 0lim iV i

q

Vρ

∆ →≡

∆

∑ and 0

lim vi iV iq

jV

∆ →≡

∆

∑

, where ˆ ˆ ˆx y zx y z

∂ ∂ ∂∇ ≡ + +

∂ ∂ ∂

1.9

is the spatial differentiation operator and where : electric induction vector; : electric field vector; ε: dielectric constant (<1, depends on the polarization of the dielectric); µ: magnetic permeability (either <1 or >1, depending on the polarization status of the magnetic dipoles). We have and .

: magnetic field vector; : magnetic induction vector. and are microscopic fields (if we would have considered all the microscopic charges and currents, we would have identified the fields with and with ).

An immediate consequence of [1.4] is charge conservation : taking the divergence of the eq. in we get (after exchanging the temporal and spatial derivatives)

[1.5]

1.2.2 Radiative energy density, energy flux, Poynting theorem From Lorentz [1.3], taking the work per unit time made by the fields, we have

1.10

v vF q E j E⋅ = ⋅ = ⋅

which shows that the magnetic fields do not make work. For a large number of charges we have

1 vi ii

q E j EV

⋅ = ⋅∆ ∑

and from H∇ ×

by taking the internal product with E

:

1 ( )4

and using the vector identity ( ) ( ) ( ) :

1 ( ) .4

Dj E c H E Et

E H H E E H

B Dj E H c E H Et t

π

π

∂⋅ = ∇× ⋅ − ⋅ ∂

⋅ ∇× = ⋅ ∇× − ∇ ⋅ ×

∂ ∂⋅ = − ⋅ − ∇ ⋅ × − ⋅ ∂ ∂

Now assuming ε, µ constant in time, we have

2218 4

cBj E E E Ht

ε µπ π∂ ⋅ + + = −∇ ⋅ × ∂

[1.6]

which is the formal expression for the Poynting theorem, stating the conservation of energy of the electro-magnetic field in its various forms:

221 : energy density of the fields8

: flux of radiant energy (per unit solid angle) 4

: density of mechanical power from fields

field

mech

BU E

cS E H

dUj Edt

ε µπ

π

= +

= ×

⋅ =

As usual, there is ambiguity in associating the energy forms to the fields or the currents. In the following we adopt the CGS units in our calculations. In this system, physical units and constants are summarized in Appendix 1B. 1.2.3 Plane electromagnetic waves In the vacuum, the Maxwell become ( D

= E

, H

= B

)

1.11

0 0

1 1

E B

B EE Bc t c t

∇ ⋅ = ∇ ⋅ =

∂ ∂∇× = − ∇× =

∂ ∂

[1.7]

Taking the rotation of the third and time derivative of the fourth and after exchanging as usual the spatial and temporal derivatives, we get

2

2 2

2

1 1( ) ( )

and with the vector identity ( ) ( )

EE Bc t c t

E E E

∂ ∂∇× ∇× = − ∇× = −

∂ ∂∇× ∇× = ∇ ∇ ⋅ − ∇

2

22 2

1 0EEc t

∂∇ − =

∂

which is the D’Alembert equation. Analogously for the magnetic field, which appears to be entirely symmetric to the electric one in [1.7]. As well known, solutions are found in terms of plane waves

1 0

2 0

ˆ exp( [ ])

ˆ exp( [ ])

E a E i k r t

B a B i k r t

ω

ω

= ⋅ −

= ⋅ −

[1.8]

and k ω

being the wavenumber vector and wave pulsation. It is immediate to see the properties of these solutions, by substituting them into the Maxwell [1.7]

1 0 2 0

1 0 2 0 2 0 1 0

ˆ ˆ0 0

ˆ ˆ ˆ ˆ

ik a E ik a Bi iik a E a B ik a B a Ec cω ω

⋅ = ⋅ =

× = × = −

[1.9]

which implies that 1a and 2a are orthogonal to the propagation direction k and orthogonal one to the other. If we take the moduli of the third and fourth, we get the conditions

0 0 0 0

2 2 20 0

, orE B B Ekc kck c E B

ω ω

ω

= =

= =

[1.10]

By taking the solutions with positive values for ω and k, as implied by the discussion in the section 1.5 (electric and magnetic fields, and space and time are all real), we have the condition about the velocity of propagation of the wave. We define as the phase velocity of a wave the rate at which the phase of the wave propagates in space. This is the velocity at which the phase of any one frequency component of the wave travels). From [1.10], the phase velocity is

1.12

v phase ckω

= = [1.11]

that is the waves, as expected, travel at the speed of light in the vacuum. A more relevant quantity is the group velocity, i.e. the velocity with which variations in frequency of the wave propagate in space: this is the case of a wave packet, for example, and this is the relevant quantity to consider, because it is more directly related to an observable quantity. Since each amplitude envelope contains a group of internal waves, this speed is usually called the group velocity. In the vacuum the

group velocity, vgroup ckω∂

= =∂

, equals c too.

As it will be discussed in Sect. 11, for radiation travelling in a medium, there is a further quantity relating the phase speed, vphase and speed of light, c, that is known as refractive index, n = c/vphase = ck/ω. Taking the derivative of ω = ck/n, we get the group speed,

[1.12] Noting that c/n = vphase, this shows that the group speed is equal to phase speed only when the refractive index is a constant: dn/dk = 0. Otherwise, when the phase velocity varies with frequency, velocities differ and the medium is called dispersive. The phase velocity of electromagnetic radiation may – under certain circumstances (for example anomalous dispersion) – exceed the speed of light in a vacuum, but this does not indicate any superluminal transfer of information or energy.

1.13

1.3 The relationship of electric charges and the radiation fields Once we have discussed the fundamental observables, let us discuss now the fundamental relations of electro-dynamics, those linking the sources of the electro-magnetism (the charges) to the fields. 1.3.1 The electromegnetic potentials Electromegnetic potentials are very useful for various reasons: a) one scalar field φ and a vector potential A

are simpler to treat than the two vector fields E

and B

; b) the equations relating the potentials φ and A

to the sources are simpler than the Maxwell equations for E

and B

; c) the relativistic formulation of the electro-magnetism is much simpler in terms of the potentials than for the fields E

and B

.

Then from the second of the Maxwell 0B∇ ⋅ =

, it follows that B

can be expressed as the rotation of some vector field A

B A= ∇×

[1.13]

From [1.13] and the third Maxwell 1E c B t∇× = − ∂ ∂

, we can write:

1 0AEc t

∂∇× + = ∂

[1.13b]

such that the argument, which is irrotational, can be expressed as the gradient of some scalar field φ

1 or

1

AEc t

AEc t

φ

φ

∂+ = −∇

∂∂

= −∇ −∂

[1.14]

We can now insert the [1.13] and [1.14] into the Maxwell to establish the relation of the potentials with the e.m. field sources ρ and j

. The first Maxwell 4E πρ∇ ⋅ =

and [1.14] give 2

2

2 2

1 1 1 4Ac t c t c t

φ φφ πρ∂ ∂ ∂ ∇ − + ∇ ⋅ + = − ∂ ∂ ∂

1.14

having added here and subtracted the term 2

22

1tc

φ∂∂ . From the fourth Maxwell

H∇×

we have in a similar way: 2

2

2 2

1 1 4AA A jc t c t

φ π∂ ∂ ∇ − + ∇ ∇ ⋅ + = − ∂ ∂

.

Of course, there is some degree of freedom in the choice of the potentials. The so-called gauge transformations allow us to optimize these choices and simplify their definition. We can for example add to the vector potential the gradient of an arbitrary scalar field ψ keeping invariant the magnetic field:

such thatA A B Bψ→ + ∇ →

and to the scalar potential the time derivative of ψ keeping invariant the electric field

1 such that E Ec t

ψφ φ ∂→ + →

∂

.

Based on such a freedom to manipulate the potentials, we can choose one set of

transformations allowing us to eliminate the term 1Ac t

φ∂∇ ⋅ +

∂

. This gauge is

called the Lorentz gauge. In such a way, we get the following relations between potentials and sources:

22

2 2

22

2 2

1 4

1 4c t

AA jc t c

φφ πρ

π

∂∇ − = −

∂∂

∇ − = −∂

[1.15]

Eq. [1.15a] reduces to the Poisson equation for constant in timeφ = , while [1.15b] reduces to D’Alambert eq. for 0j =

in the absence of currents.

From now on we will use the Lorentz gauge exclusively, and the whole of classical electrodynamics reduces to the problem of solving the inhomogeneous wave equations for specified sources. In the static case (non varying potentials) eqs. [1.15] reduce to Poisson’s equation

2 2 44 ; A jcπφ πρ∇ = − ∇ = −

with familiar solutions

1.15

[1.16]

In the more general dynamical case, electromagnetic “news” travel at the speed of light. In the non-static case, therefore, it’s not the status of the source right now that matters, but rather its condition at some earlier time (called the retarded time) when the “message” left the source. Since this message must travel a distance

, the delay is , along the particle trajectory , such that the retarded time is

,

1.16

The natural generalization of Eqs. [1.16] to the non-static case is therefore

[1.17]

where the notation of everything included in the squared parentheses corresponds to the quantity computed at the retarded time (hence the name of equation of retarded potentials):

.

That all sounds reasonable – and surprisingly simple. But so far we did not prove that these solutions are correct. The more general proof that [1.17] is the general solution of the inhomogeneous wave equations [1.15], is reported in Jackson (1975) and Landau & Lifchiz (Theorie des champs). We defer to those textbooks for details. Through [1.17] it is easy to calculate the potentials and then the fields.

1.17

1.3.2 Radiation from moving charges. The Lienard-Wiekert potentials As an (important) application, let us consider a single charge q moving along a given trajectory . The particle velocity will be . So the charge and current densities will be

[1.18]

(the Dirac function localizes the charge along the trajectory). Let us calculate now the potentials produced by this charge, which is immediate using the delta function to set the condition about the retarded times:

The integral on time locates the charge in his position as a function of time. After substituting the ρ and integrating over the volume, we get, by passing from a spatial integral to a time-integral

.

If we now make the transformation of variables defined as:

1.18

0 ( )r r t

t t tc

′−′′ ′= − +

[1.19]

we can express the Jacobian of the transformation as 1

1

1

1

0

ˆ( ) ( ) ˆ1 ( ) ; ( ) 1 ( ) ( ) then

ˆ( ) ( ) ( )( , ) 1( )

dt n t u t K t K t n t tdt c

t dt n t u tr t qr r t c

β

δφ

−

−

−

′ ′ ′⋅ ′ ′ ′ ′= − = ≡ − ⋅ ′′ ′′ ′′ ′ ′⋅ = −∫ ′−

which is equivalent to the argument calculated at 0t′′ = or equivalently at the time

rett t′ = :

0

0

( , )

( )

qr tK r r

quA rK r r

φ

= ′−

= ′−

[1.20]

This simple expression for the potentials is known with the name of Lienard-Wiekert potentials. They differ from those of the static e.m. theory on two respects. First, because of the factor K at the denominator, which becomes important of course for relativistic speeds. This factor makes the potentials being concentrated in a narrow cone in the direction of the particle velocity, that is when u is close to the radial vector r. This is the so-called beaming effect, emerging when considering the effects of the Lorentz transformations on the direction of propagation of photons.

1 Let us calculate the Jacobian of the transformation 0r rt t t

c′−

′′ ′= − + , by considering that

0

( )( ); ; ( ) ;dR tR r r t R R R t dt′′ ′ ′≡ − ≡ =

so( )R t dtdt dt

c′ ′

′′ ′= +

. By differentiating 2 2( ) ( )R t R t′ ′=

we get 2 ( ) 2 ( ) ( )RR t R t u t′ ′ ′= − ⋅

because ( )R u t′= −

and from this ˆ( ) ( )R t n u t′ ′= − ⋅

where n is the versor of

0 ( )r r t′− . And, finally, ( )ˆ1 ( )dt u tn t

dt c′′ ′

′= − ⋅′

.

1.19

The second difference comes from the fact that all quantities should be calculated at the retarded time (see the squared parentheses), and we have already seen the meaning of this.

1.3.3 The fundamental equation of electro-magnetism The major consequence of retardation is that it makes possible for a particle to radiate. The potentials roughly fall off as 1/r so that differentiation to find the fields would give a 1/r2 decrease if this differentiation acted solely on the 1/r factor. As we show in this section, retardation allows an implicit dependence on position to occur via the definition of retarded time, and differentiation with respect to this dependence carries the 1/r behavior of the potentials into the fields themselves. We have seen that this allows radiation energy to flow to infinite distances. The problem is now to differentiate the potentials [1.20] through [1.13] and [1.14] to get the fields generated by a given particle. The extensive calculation is straightforward but very long and tedious (all details can be found in the classical book by Jackson - Classical Electrodynamics), we omit it and report only the results. If we want the fields at the point r and time t, we first need to calculate the position of the particle at , such that by emitting radiation from that position, that radiation will be seen by the observer at r and t. Of course, it may well be that there is no solution to this, and no radiation could be observed at all. Let us assume that the particle velocity and acceleration at the retarded time are and

, and let us further define . Then the

complete expressions for the fields are:

1.20

2

3 2 3

ˆ ˆ( )(1 ) ˆ( , ) ( )

ˆ( , ) ( , )

n q nE r t q nK R c K R

B r t n E r t

β β β β − − = + × − ×

= ×

[1.21]

The total field is composed of two terms, one 1R −∝ and one 2R −∝ . The latter is named the velocity field, as it depends only on the particle velocity β, and it is the generalization of the Coulomb law for moving charges (indeed, for β<<1 it becomes exactly the Coulomb law). The second term is named the acceleration field (or radiation field) as it depends on β , it scales 1R −∝ , and the direction of the E field is always orthogonal to the direction n . We name as far zone that dominated by the radiation field, near zone the other. We further note that the E and B fields and the direction of propagation n form a triad of mutually perpendicular vectors, and the moduli of E and B are identical, all properties that make [1.21] consistent with the Maxwell solutions in the absence of sources.

1.4 Radiation from non-relativistic particles

We consider here the case of radiation fields generated by charges moving with non-

relativistic speeds, 1ucβ = << . In such situation, we can first compare the two

radiation terms in [1.21] and get

2

rad

vel

E R RuE c c

β=

If we consider now a given Fourier frequency ν (ν is also the frequency of oscillation of the particle), we get for the ratio of the velocity variation and the required time u uν , so that

2

rad

vel

E Ru RE c

ν βl

= .

Then in the "near zone", R<λ, the velocity field dominates by a factor >1/β. Instead, in the "far zone", R>> λc/u, the radiation field dominates. Of course, the acceleration field is the only term of interest for astronomy.

1.21

1.4.1 The Larmor formula When β<<1, eqs. [1.21] do strongly simplify (of course we confine our analysis to the radiation fields only):

2ˆ ˆ( , )

ˆ( , ) ( , )

rad

rad rad

qE r t n n uRc

B r t n E r t

= × ×

= ×

[1.22]

So the calculation of the field is quite simple if we refer to the above figure. Note that this figure’s plane contains the acceleration and the versor n, so, due to the 2 external product operations, it also contains the E field. As for the moduli, we have

2sinrad rad

quE BRc

q= =

[1.23]

The radiant energy flux is then given by the Poynting vector in the direction n and has a modulus given by

2 2

2 2

2 4sin

4 4 4c c c q uS S E B E

R cq

π π π= = × = =

. [1.24]

The energy flow within a given differential area 2dA R d= Ω at the distance R from the source, 2 2 2 2 2 4sin 4cq u R d R cq πΩ , so that the power transported per unit solid angle, is then

2 2

2

3sin

4dW q u

dt d cq

π=

Ω

[1.25]

1.22

The total power is immediately obtained by integration on the solid angle ( 2 sind dπ q qΩ = ):

2 2 2 2

2

3 34

2sin4 3

dW q u q uP ddt c cπ

qπ

= = Ω =∫

[1.26]

(the integral amounting to the usual 8π/3). This is the well known Larmor formula. There are various points to note.

1. The radiation power is proportional to the square of the electric charge and to the square of the acceleration. The consequence is that it is much stronger when electrons are involved instead of protons, for a given force applied to the charges.

2. There is a characteristic dipolar behavior of the emitted power as a function of direction, 2sinP q∝

3. The electric field E and its direction are determined by the acceleration vector u and the versor n. If the motion of the charge is along a line, the radiation will be completely linearly polarized in the plane defined by n and u .

1.4.2 Radiation from for a set of charges: the electric dipole approximation So, everything is simple in the case of single isolated charges. When we have to deal with a set of non-relativistic charges with positions ir

and acceleration iu , the e.m. fields at large distances are the linear addition of all individual charge contributions. The obvious complication is that the retarded times will be different from one charge

1.23

to the other, and this impacts on the phases of the summed fields, and hence on the result of the sum of the fields. Let us imagine that we have completely ordered motions of the charges (i.e. a unique external force acting of them): even in such limiting situation, we have to account for the different phases of the fields coming from the different particles. And even worse, of course, if the charge motions are disordered. There is however a limiting situation for which we can neglect this complication: if we define L to be the size of the charge system and τ the timescale during which we may have a significant variation in the charge distribution, then the problem is reduced when the light travel time L/c by the radiation is

Lc

τ<< [1.27]

The approximation in eq. [1.27] is called the dipole approximation for a system of non-relativistic charges. Clearly, the inverse of the timescale τ corresponds to the characteristic frequency of the emitted radiation

1 1, so that charchar

L Lc

ν lτ ν

<< ⇒ >>

that is the differences in the retarded times become negligible when the size of the charge packet is small compared to the typical wavelength of the emitted radiation. It is equally obvious that these conditions can be expressed in terms of the charge velocity: assuming that the orbits of the charges have the same size of the charge distribution L, we have

1charL u cc

ν<< ⇒ <<

This is then equivalent to the approximation of non-relativistic motion. In such case we have

1.24

2

1 ˆ ˆ( , ) ( )irad i i i

ii

qE r t n n uc R

= × ×∑

If we then have the condition that the packet size L is much smaller than the radiation wavelength and hence the average distance to the observer 0R ,

0L R<< , [1.28]

then 0 ˆ ˆ( 1, );i iR R i N n n= and

2

0

22

2

2

2

the charge electric dipole moment

and from Poynting:

differential emitted power

total emitted power

ˆ ˆ( )( , )

sin4

23

rad

i ii

n n dE r tc R

d q r

dP dd c

dPc

qπ

× ×=

≡ ∑

=Ω

=

[1.29]

The charge bundle then behaves like a single emitting charge. 1.4.3 General Multipole Expansion The dipole approximation gives us the expression of the intensity (or the potentials, or the charges) only limited to the first term in the development of these quantities as a function of the ratio L/λ<<1. Let us see now briefly how the successive terms in this development emerge and produce contributions to the intensity much lower than the electric dipole term. These however may become significant whenever the dipole term vanishes. Let us start from the consideration of the retarded potentials, and consider the vector potential only (note that the scalar potential φ is negligible at large distances from the charges)

3( )1( )

j rA r d r

c r r

′ ′=′−∫

. [1.17]

1.25

Let us choose the origin in a point internal to the charge distribution. R0 is the radial position of the observer in the direction n. With reference to the figure below, and under the condition that the charge packet size L is sufficiently smaller than the observer distance L<<R0 , we can express the distance from every source to the observer as

0 0 ˆi i i iR r r R r R n r′ ′ ′= − = − − ⋅

We define 0 0ˆ /n R R≡

. Then, because 0r r R′−

, [1.17] becomes

3 30

0 0

ˆ ˆ1 1( ) R n r n rA r j t d r j t d rcR c c cR c

′ ′⋅ ⋅ ′ ′ ′= − + = +

∫ ∫

where 0' Rt tc

≡ − is the average retarded time of the charge distribution. By

developing the current j

in series around t' with respect to the small incremental time n r

c′⋅

we get, to the first order in this expansion

ˆ0

ˆ( ) ( ) ...

' n rc

j n rj t j tt c′⋅

=

′∂ ⋅′ + +∂

such that

( ) ( )2

0 0

1 1 ˆ( ) ( )A t j t dV n r j t dVcR c R t

∂′ ′ ′+ ⋅∫ ∫′∂

,

(note that temporal and spatial operators are interchanged here, as usual) or equivalently

2

0 0

1 1 ˆ( ) ( )i i i i ii i

A t q u q u n rcR c R t

∂ ′+ ⋅∑ ∑∂

1.26

It is clear that the time derivative in the second addendum will bring two additional terms to that of dipole. With some algebra it is then possible to obtain that, considering the first term in the development of the potential as a function of n r c′⋅

brings a total of three additive terms in the expression for the potential:

2

0 0 0

2 (electric quadrupole moment tensor)

(magnetic dipole moment vector)

1 1 ˆ( )6

, 1,3

(3 )i i i ii

i i ii

dA t D m ncR c R cR

D D n

D q x x r

m q r u

a aβ ββ

aβ a β aβ

a β

δ

+ + ×

= =∑

≡ −∑

≡ ×∑

[1.30]

Now, let us compute the e.m. fields in the far zone, where we can assume the electric potential to be negligible 0φ = (this can be seen to be consistent with the Lorentz gauge in the far zone). Then from

1 we haveAE c t∂= − ∂

1 ˆ ˆ= AB n E nc t

∂× = − ×

∂

and by differentiating [1.30] we get

2

0

1 1ˆ ˆ ˆ ˆ( )6

B d n D n m n nc R c

= × + × + × ×

[1.31]

while for the radiation intensity we have:

2 204

cdI B R dπ

= Ω

.

Note that when rising the B field to the square, the mixed terms in the product vanish and remain only the 3 main squared terms. Averaging over the full sky solid angle

2

2 2

3 5 3

2 1 23 180 3dI D mc c c

= + +

[1.32]

1.27

There are then three independent terms in the radiation field: the electric dipole term, the electric quadrupole and the magnetic dipole terms coming from the first order expansion. Let us consider them separately. A) The electric dipole term. This is null when the ratio of the charge to mass of the particles is constant. This is immediately seen by considering that

ii i i i i i

i i ii

q qd q r m r m rm m

= = =∑ ∑ ∑

such that its second derivative is proportional to the acceleration of the center of mass of the charge distribution, which is null for an isolated system. This is the case when dealing with a set of identical particles. B) The magnetic dipole term. Even in this case this is null when the ratio of the charge to mass of the particles is constant. In this case the magnetic moment is proportional to the angular momentum of the particle set, whose second derivative is still null (the kinetic moment being constant in time). It is also easy to verify that the same happens when we consider a couple of isolated particles of any given charge and mass (see exercise in Appendix 1A). C) The electric quadrupole term. This term becomes important when the others are null, as discussed above. However, this contribution is very small with respect to the dipoles (and becomes null when the third derivative of the position gets to zero,

0r =

). Comparing it to the dipole term we have

2

2 2

1270

quad

dip

I DI c d

=

[1.33]

which is typically quite small because of the large factor at the denominator. Note that this term is the only one contributing radiation when the particles have the same charge. This is typically the case for the gravitational radiation ("charges" exist of only one – positive – sign, the gravitational mass) and [1.33] explains why this carries only tiny amounts of energy and are so difficult to detect. Then given the same radial dependence of the fields in the electromagnetic and gravitational cases, our multi-pole expansion analysis can be applied as well to the gravitational one, as done above.

1.28

1.5 THE RADIATION SPECTRUM (taken from Ribicki & Lightman)

1.29

1.30

1.31

1.6 The spectrum of pulsed radiation for dipole emission Now, putting together [1.29] and [2.33], we can get a useful expression of the radiation spectrum in terms of the dipole emission in the case in which the radiation comes through a pulse. In such case we can define the Fourier transform of the dipole d as

From Larmor we need to take the second time derivative of the latter to get the electric field, so that we can express it in terms of the Fourier transform

[1.34]

such that the transform of E(t) is the transform of , or from [1.34]

[1.35]

and

such that the emitted power-spectrum is

1.32

24 23

2243 3

1 ( ) sin , and

8 8( ) ( )3 3

dW dd d cdW d dd c c

ω ω qω

π πω ω ωω

=Ω

= =

[1.36]

This expression will become useful when dealing with radiation coming through a set of pulses (see later Sections).

1.33

Appendix 1A

Magnetic dipole moment of two isolated charges

As an exercise, let us now consider the interesting case of the magnetic dipole emission by two isolated charges.

1.34

1.35

Appendix 1B

121 eV = 1.6 e erg−