VLADIMIR ZEKOVIĆ, BOJAN ARBUTINA, ALEKSANDRA DOBARDŽIĆ
and MARKO PAVLOVIĆ
Abstract

By applying a method of virtual quanta we derive formulae for relativistic non-thermal bremsstrahlung radiation from relativistic electrons as well as from protons and heavier particles with power-law momentum distribution We show that emission which originates from an electron scattering on an ion, represents the most significant component of relativistic non-thermal bremsstrahlung. Radiation from an ion scattering on electron, known as inverse bremsstrahlung, is shown to be negligible in overall non-thermal bremsstrahlung emission. These results arise from theory refinement, where we introduce the dependence of relativistic kinetic energy of an incident particle, upon the energy of scattered photon. In part, it is also a consequence of a different mass of particles and relativistic effects.

Keywords: Bremsstrahlung; Cosmic rays; X-ray sources

Accepted 14 October 2013

PACS numbers: 03.50.-z, 41.60.-m, 78.70.Ck, 96.50.S-, 13.85.Tp, 97.80.Jp, 98.70.Qy

1 Introduction

Bremsstrahlung or deceleration radiation (form German bremsen ”to brake” and Strahlung ”radiation”) is important radiation mechanism in laboratory and astrophysical plasmas. Since protons are generally less motile than electrons, bremsstrahlung or free-free emission is often considered as a radiation (photon emission) of decelerating electron in the Coulomb field of an ion. Even synchrotron radiation of an ultra-relativistic electron gyrating about field lines in the magnetic field is sometimes referred as magneto-bremsstrahlung.

Thermal bremsstrahlung is often observed in astrophysical sources such as Hii or ionized hydrogen regions (e.g. Orion nebula) in radio domain or in clusters of galaxies i.e. hot intercluster medium, in the X-rays. It is produced by electrons with thermal (Maxwell-Boltzmann) distribution.

Accelerated electrons often have power law distribution and as a result produce non-thermal bremsstrahlung radiation. Power law distribution of particles can be produced in shock waves, so non-thermal bremsstrahlung radiation could be observed in objects such as supernova remnants . Also hard X-ray emission from clusters of galaxies in which accretion or merger shocks are present could be produced in part by non-thermal bremsstrahlung radiation .

In paper , the role of inverse bremsstrahlung radiation is considered in shocked astrophysical plasmas, defined to be the emission of a single photon when a high-speed ion collides with an electron that is effectively at rest. Their conclusion was that inverse bremsstrahlung can be neglected in most models of shock acceleration in supernova remnants and similar sources. However, problem of inverse bremsstrahlung detection on scales 100 pc, distant from these discrete sources, remains open.

Although electron-electron bremsstrahlung is normally ignored in comparison to normal electron-ion bremsstrahlung, it was pointed out that it can contribute to the hard X-ray emission from solar flares . These authors recognized the growing importance of electron-electron bremsstrahlung at electron (and photon) energies above 300 keV.

In this paper we consider the relativistic non-thermal bremsstrahlung. In the following section, by applying a method of virtual quanta (see also Ref. 8), we shall derive formulae for relativistic non-thermal bremsstrahlung radiation from ultra-relativistic electrons as well as from protons and heavier particles with power-law momentum distribution .

2 Analysis and Results

2.1 Relativistic Electron Bremsstrahlung

Let us first consider bremsstrahlung radiation of a relativistic electron. In the rest frame of the electron it appears that a proton, or in general an ion with charge , moves rapidly towards the electron. Electrostatic field of this ion is transformed into a transverse pulse, which to electron appears as a pulse of electromagnetic radiation – a virtual quanta. The quanta or photon scatters of electron and produces detectable bremsstrahlung radiation. In the primed (electron’s) frame, the spectrum of the pulse of virtual quanta has the form (see Ref. 9)

 dW′dS′dω′=Z2e2cπ2b′2v2(b′ω′γv)2K21(b′ω′γv), (1)

where is energy, surface element, circular frequency, impact parameter, speed of light, the Lorentz factor, , is velocity and the modified Bessel function of order one.

Assuming elastic scattering in the low-energy limit () we have

 dW′dω′=σTdW′dS′dω′, (2)

where is Thomson cross section. Since energy and frequency transform identically under Lorentz transformations , transverse lengths are unchanged , and on average, if the scattering is forward-backward symmetric, in the laboratory frame we have

 dWdω=8Z2e63πb2m2ec3v2(bωγ2v)2K21(bωγ2v). (3)

Emitted power per unit frequency of the single electron is

 dWdtdω=2πcni∫∞bmindWdωbdb, (4)

where is ion number density and we shall set . For a power-law distribution of electrons , where is energy index:

 dWdtdωdV=∫∞0dWdtdωN(E)dE (5)

where, as usual, the integration limits are set from 0 to in approximation of ultra-relativistic particles. We derive our solution in a more realistic way, by taking into account cosmic rays of lower energies and by not making the assumption that the particle’s speed is always close to , which leads to a more complex derivation. We use momentum instead of the energy distribution in (5) and we also introduce finite integration limits:

 dWdtdωdV = ∫pmaxpmindWdtdωN(p)dp, dWdtdωdV = 16Z2e63m2ec3nike∫pmaxpmin1vp−qdp∫∞xyK21(y)dy, (6)

where is constant of the momentum distribution of electrons and ; the momentum boundaries and will be discussed in the following text; collision parameter is ; and . If we express in terms of

 pc = √mec2ℏωx, (pc)d(pc) = −12mec2ℏωx2dx

and change it in (6), we derive

 dWdtdωdV = 16Z2e63m2ec4ni(kecq−1)(mec2ℏω)(1−q)/2⋅∫xmaxxmin12x(q−3)/2√1+mec2ℏωx (7) ⋅ [xK0(x)K1(x)−12x2(K21(x)−K20(x))]dx,

with integration limits and In the above derivation we used identity

 ∫∞xyK21(y)dy=xK0(x)K1(x)−12x2(K21(x)−K20(x)). (8)

We now derive analytical solution to (7), but first let us discuss the integration boundaries. Varying the momentum upper limit from to will always give values of very close to zero, so we use and The lower limit of momentum however small it is, will always give for photon energies in the scope of Thomson scattering. This enables us to use approximation of relation (8) (see Ref.9)

 xK0(x)K1(x)−12x2(K21(x)−K20(x))≈ln(0.684x), (9)

where and is Euler’s gamma constant, which then results in analytically more solvable expression than (7)

 dWdtdωdV = 16Z2e63m2ec4niKe(mec2ℏω)(1−q)/2 (10) ⋅ ∫xmax012x(q−3)/2√1+mec2ℏωx⋅ln(0.684x)dx.

Here we introduce the new method of solving complicated integrals in a way of getting very precise solutions analytically, only if limiting conditions can be defined so that integrals can take much simpler form and become solvable. As a first step, we make use of the integral mean value theorem which states that

 ¯¯¯¯¯¯¯¯¯¯f(x)=1b−a⋅∫baf(x)dx (11)

and we define some mean value so that Applying the theorem to relation (10) leads to

 dWdtdωdV = 16Z2e63m2ec4niKe(mec2ℏω)(1−q)/2⋅12⋅xmax⋅f(¯¯¯x) (12) f(¯¯¯x) = ¯¯¯x(q−3)/2√1+mec2ℏω¯¯¯x⋅ln(0.684¯¯¯x).

As a second step, we want to make relation between and so we define the two boundary cases of (10). First, when so that This is valid for low energy photons and ideally when In this limit relation (10) reduces to

 = 16Z2e63m2ec4niKe(mec2)1−q/2(ℏω)−q/2 ⋅ ∫xmax012xq/2−1ln(0.684x)dx.

Second case is defined over domain of high energy photons (), when and which leads to the high energy limit

 = 16Z2e63m2ec4niKe(mec2ℏω)(1−q)/2 ⋅ ∫xmax012x(q−3)/2ln(0.684x)dx.

Solutions to integrals (13) and (14) are now straightforward and are given by (15) and (16) respectively:

 = 16Z2e63m2ec4niKe(mec2)1−q/2(ℏω)−q/2⋅xq/2max ⋅ 1q[ln(0.684xmax)+2q], = 16Z2e63m2ec4niKe(mec2)(1−q)/2(ℏω)(1−q)/2⋅x(q−1)/2max ⋅ 1q−1[ln(0.684xmax)+2q−1].

We now make use of the first boundary solution (15) and equate it with relation (12) in the limit of very low photon energies

 ¯¯¯xq/2−1ln0.684¯¯¯x≡xq/2−1max2q[ln(0.684xmax)+2q]=xq/2−1max2qln(0.684xmax⋅missinge−2/q). (17)

Because in most cases spectral index lies in the range we can expand the series

 (missinge−2/q)q/2−1=missinge2/q−1≈2q

and substitute it in (17) to get

 ¯¯¯xq/2−1⋅ln0.684¯¯¯x=(xmax⋅missinge−2/q)q/2−1⋅ln(0.684xmax⋅missinge−2/q),

which then results in relation for low energy photons

 ¯¯¯x=xmax⋅missinge−2/q. (18)

We now equate the second boundary solution (16) with relation (12) written in the limit of very high photon energies

 ¯¯¯x(q−3)/2⋅ln0.684¯¯¯x≡x(q−3)/2max⋅2q−1ln(0.684xmax⋅missinge−2/(q−1)). (19)

As previously, we can represent with a series expansion

 (missinge−2/(q−1))(q−3)/2=missinge2/(q−1)−1≈2q−1

and change it in (19) to get relation which holds in the range of high energy photons

 ¯¯¯x=xmax⋅missinge−2/(q−1). (20)

As a balance between low and high energy solutions, we make use of the geometric mean of (18) and (20) to get our analytical solution to fit the whole range of photon energies

 ¯¯¯x=√x2max⋅missinge−2/q⋅missinge−2/(q−1)=xmax⋅missinge(1−2q)/(q(q−1)). (21)

We use this relation and after substituting it in (12), we derive

 dWdtdωdV = 16Z2e63m2ec4niKe(mec2ℏω)(1−q)/2⋅12x(q−1)/2max⋅missingeQ(q−3)/2 ⋅ √1+mec2ℏωxmax⋅missingeQ⋅[ln(0.684xmax)+1q+1q−1],

where

If we now express the lowest value of momentum , that we take into account in in terms of electron’s kinetic energy

 xmax=ℏωmep2min=ℏω⋅mec2T2e+2mec2Te (23)

and if we define that kinetic energy of a particle is related to the energy of a photon emitted by that particle as

 Te=ζeℏω, (24)

so that

 xmax=ℏω⋅mec2(ζeℏω)2+2ζeℏωmec2=12ζe⋅11+12ζeℏωmec2, (25)

where is generally not constant and can be a function of then we can rewrite expression (22) completely as a function of photon energy

 dWdtdωdV = 16Z2e63m2ec4niKe(ζeℏω)1−q⋅(1+2mec2ζeℏω)1−q/212⋅missingeQ(q−3)/2 ⋅    ⎷1+missingeQ−2(2+ζeℏωmec2)(2+ζeℏωmec2)2⋅[ln(0.684⋅ζe(2+ζeℏωmec2))+1q+1q−1].

This relation represents an approximate analytical solution to electron non-thermal bremsstrahlung emission, which holds for all photon energies in the scope of Thomson scattering.

There is a break in the spectrum of electron non-thermal bremsstrahlung emission around . As an approximation of (26), the two power laws are then given for low and high energy parts of the spectrum by (27) and (28) respectively

 dWdtdωdV = 16Z2e63m2ec4niKe(2mec2)1−q/2(ζeℏω)−q/2 ⋅ 14missingeQ(q−2)/2[ln(0.684⋅2ζe)+1q+1q−1], dWdtdωdV = 16Z2e63m2ec4niKe(ζeℏω)1−q ⋅ 12missingeQ(q−3)/2[ln(0.684⋅2ζe)+ln(1+12ζeℏωmec2)+1q+1q−1]

and the break point is found as

 ℏωb≈2mec2ζe. (29)

2.2 Relativistic Ion Bremsstrahlung

In the case of relativistic proton or heavier ion with charge , mass aaa is the mass number; for hydrogen i.e. proton it is 1, for helium i.e. -particle it is 4 (assuming nearly equal mass of proton and neutron), etc. and energy we are dealing with the scattering of quanta or photon over electron in the laboratory frame, assuming that during the interaction the electron thermal velocity is small enough, so that it is practically at rest and that the photon energy is small compared to electron’s rest energy, i.e. . If this is not the case the scattering would not be Thomson’s but Compton’s and Inverse Compton’s.

Since now we do not have to transform from primed to laboratory frame, we have

 dWdω=8Z2e63πb2m2ec3v2(bωγv)2K21(bωγv). (30)

If again , and () we have

 dWdtdωdV=16Z2e63m2ec3neki∫pmaxpmin1vp−qdp∫∞xyK21(y)dy, (31)

where is now constant of the momentum distribution of ions and If we now express in terms of

 (pc)4 (meAmpℏω)2x2−(pc)2−(Ampc2)2=0, (pc)2 =1+√1+(2mec2ℏωx)22(meAmpℏωx)2, d(pc)2 =⎡⎢ ⎢ ⎢ ⎢⎣−1+√1+(2mec2ℏωx)2(meAmpℏω)2x3+2(Ampc2)2x√1+(2mec2ℏωx)2⎤⎥ ⎥ ⎥ ⎥⎦dx=2c2pdp

and change it in (31), we get

 dWdtdωdV = 16Z2e63m2ec4ne(kicq−1)(Ampℏωme)1−q∫xmaxxmin⎛⎜ ⎜ ⎜ ⎜⎝2x21+√1+(2mec2ℏωx)2⎞⎟ ⎟ ⎟ ⎟⎠q/2−1 ⋅ ⋅ [xK0(x)K1(x)−12x2(K21(x)−K20(x))]dx.

As in previous section, we use approximation (9) and discussion about the integration boundaries to derive a more analytically solvable expression

 dWdtdωdV = 16Z2e63m2ec4neKi(Ampℏωme)1−q∫xmax0⎛⎝2x21+√1+(2mec2ℏωx)2⎞⎠q/2−1 (33) ⋅

We again make use of the integral mean value theorem and as in the case of electrons we define some mean value so that Applying this to relation (33) leads to

 dWdtdωdV = 16Z2e63m2ec4neKi(Ampℏωme)1−q⋅xmax⋅f(¯¯¯x), (34) f(¯¯¯x) = ⎛⎜ ⎜ ⎜ ⎜⎝2¯¯¯x21+√1+(2mec2ℏω¯¯¯x)2⎞⎟ ⎟ ⎟ ⎟⎠q/2−1 ⋅ ⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣1−12⋅(2mec2ℏω¯¯¯x)2√1+(2mec2ℏω¯¯¯x)2⋅(1+√1+(2mec2ℏω¯¯¯x)2)⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦⋅ln0.684¯¯¯x.

To make a connection to we introduce the two boundary cases of (33), first when and (33) then simplifies to an approximate relation

 dWdtdωdV ≈ 16Z2e63m2ec4neKi(Ampℏωme)1−q∫xmax0(2x21+2mec2ℏωx)q/2−1 ⋅ ⎡⎢⎣1−12⋅2mec2ℏωx1+2mec2ℏωx⎤⎥⎦⋅ln0.684x dx.

We can make the further simplification, which is the subcase when or the low energy limit when :

 = 16Z2e63m2ec4neKi(Ampc2)1−q/2(Ampℏωme)−q/2 (35) ⋅ ∫xmax012 xq/2−1ln0.684x dx.

In the second case when , (33) becomes

 dWdtdωdV∣∣∣ℏω→∞=16Z2e63m2ec4neKi(Ampℏωme)1−q⋅∫xmax0xq−2ln0.684x dx. (36)

As in the case of relativistic electron bremsstrahlung, solutions to (35) and (36) are straightforward and are given by (37) and (38) respectively:

 = 16Z2e63m2ec4neKi(Ampc2)1−q/2(Ampℏωme)−q/2⋅xq/2max ⋅ 1q[ln(0.684xmax)+2q], = 16Z2e63m2ec4neKi(Ampℏωme)1−q⋅xq−1max ⋅ 1q−1[ln(0.684xmax)+1q−1].

We first make use of boundary solution (37) and equate it with relation (34) in the limit of very low photon energies when

 1+√1+(2mec2ℏω¯¯¯x)2≈√1+(2mec2ℏω¯¯¯x)2≈2mec2ℏω¯¯¯x,

to get relation between and

 ¯¯¯xq/2−1ln0.684¯¯¯x≡xq/2−1max2q[ln(0.684xmax)+2q]=xq/2−1max2qln(0.684xmax⋅missinge−2/q). (39)

It is argued in previous section that we can expand series

 (missinge−2/q)q/2−1=missinge2/q−1≈2q

and change it in (39) to derive

 ¯¯¯xq/2−1⋅ln0.684¯¯¯x=(xmax⋅missinge−2/q)q/2−1⋅ln(0.684xmax⋅missinge−2/q),

which then results in a connection

 ¯¯¯x=xmax⋅missinge−2/q. (40)

We then equate (38) with relation (34) written in the limit of very high photon energies when and we get relation between and

 ¯¯¯xq−2ln0.684¯¯¯x≡xq−2maxq−1[ln(0.684xmax)+1q−1]=xq−2maxq−1ln(0.684xmaxmissinge−1/(q−1)). (41)

As previously, we can represent as series expansion

 (missinge−1/(q−1))q−2=missinge1/(q−1)−1≈1q−1

and change it in (41) to get relation which holds for high energy photons

 ¯¯¯x=xmax⋅missinge−1/(q−1). (42)

To get our analytical solution to fit the whole range of photon energies, we make compromise again by using the geometric mean of (40) and (42)

 ¯¯¯x=√x2max⋅missinge−2/q⋅missinge−1/(q−1)=xmax⋅missinge(2−3q)/(2q(q−1)). (43)

We now use this relation and after including it in (34), we derive an approximate analytical solution to relativistic ion (or inverse) non-thermal bremsstrahlung emission, which holds for all photon energies in the scope of Thomson scattering:

 dWdtd