Types of Radio Emission Mechanisms

Hyperfine Splitting and the 21 cm line Another important transition is not a transition from one principle quantum state to another, but rather a spin flip of electrons in neutral hydrogen atoms. The ground state of hydrogen has slightly different energies for electrons with spin parallel and antiparallel to the proton, as shown below:

For a thermal (Maxwellian) distribution, the distribution of speeds of the atoms is determined uniquely by the temperature, and a concise derivation of the thermal doppler width can be found here . The result is given as

An important question is how broad are these recombination lines? The "natural" width of lines is the inverse of their lifetime (how long they stay at one level before transitioning to the other level). These lifetimes are rather long, so the intrinsic width of the lines is narrow (about 0.1 Hz!, see here ). However, their observed width is mainly due to "Doppler broadening" resulting from motions of the atoms. This could be due to a combination of thermal motions and turbulent motions. The level of turbulence is not generally known, but the thermal broadening can be calculated based on an estimate of the temperature of the population of emitting atoms and any excess broadening can be ascribed to turbulence.

Radio recombination lines arise in HII regions , which are regions of the interstellar medium where the ordinarily neutral hydrogen gas is ionized by a nearby hot star, whose light is rich in UV photons. The UV photons ionize the atoms, and the electrons recombine and cascade downward toward the ground state. Some fraction (given by the Boltzmann and Saha equations) make the transition from 110 to 109, and emit a 5 GHz photon. Although the density of the interstellar medium is low (perhaps 10 4 cm -3 ), the path length is long ( ~0.5 pc ), so the emission can be observed as a weak radio line. Note that transitions around H40 a are in the 100 GHz (millimeter) range, and around H600 a are around 30 MHz range, so the entire radio spectrum can provide recombination lines.

The emission types that we have discussed up to now are transitions between bound states, for which quantum mechanical rules are of the first importance. However, for astrophysical plasmas in a high-energy environment, such as in stars, or the tenuous atmosphere around stars, it is free particles that dominate the radio emission mechanisms. (Note that bound state transitions can still be important at UV and soft X-ray wavelengths in such high-energy plasmas.) This includes all radio emission from the Sun and in the interplanetary medium, including the Earth's magnetosphere and ionosphere.

Recall that a charge at rest or in uniform motion (in the non-relativistic limit) has an electrostatic electric field (cgs units):

E = q/r 2 r ,

which is a radial field falling off as 1/r

. The power radiated is the square of the electric field, so the power from the radial part of an electrostatic field vanishes quickly, as 1/r

. However, when a transverse "kink" appears in the field, that transverse part falls off only as 1/r , so the power falls off as 1/r

. So any phenomenon that produces a kink in the field will result in radiation. The typical way to produce a kink in the electrostatic field of a charge is to accelerate the charge.

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The Larmor Formula (see Jackson, 2nd edition, section 14.2) expresses the radiated power from an accelerated charge per unit solid angle as:

dP/d W = q 2 a 2 sin 2 q / 4 p c 2 (1)

where q is the charge, a is the acceleration, and q is the angle from the direction of the acceleration. Note that the "radiation pattern" is a dipole pattern, which looks like this:

and the pattern arises naturally from the geometry. A dipole antenna (where charges travel in simple harmonic motion up and down the antenna) has exactly this radiation pattern. In the case of relativistic motion of charged particles, the pattern is altered, as we will discuss shortly (see Jackson, 2nd edition, section14.3). To get the total power radiated, simply integrate (1) over all solid angle to get P = 2/3 q

a

/ c

.

What kind of charged particles are good at radiating? Note that the force required to produce an acceleration is F = ma , so the acceleration is a = F/m , and dP/d W ~ 1/m 2 . So electrons are better at radiating than protons by a factor (m p /m e ) 2 = 4 x 10 6 ! For radio emission, we can assume that the only particles important for the emission are electrons .

Cherenkov Emission

For completeness, we briefly mention that the velocity of a free charge moving in a medium can actually exceed the speed of light in the medium. This causes the light to radiate in a cone, exactly like sound waves from an aircraft that exceeds the speed of sound. This effect gives rise to Cherenkov emission, which is responsible for the blue glow given off by nuclear reactors that are surrounded by water, and for emission by cosmic rays in the Earth's atmosphere. This is one of the few mechanisms that can result in radiation without the need for the particle to accelerate. Cherenkov emission plays no role in radiation in plasmas, however, because the wave speed of light in an ionized medium is greater than c !

Bremsstrahlung

This is an important type of radiation in many astrophysical plasmas. The word is german for "braking radiation," as is due to accelerations caused by collisions between electrons and ions (Coulomb collisions). Recall that the Coulomb force is F = qE , so the force on an electron in the electrostatic field of a proton is an attractive force F = - e 2 /r 2 r . In head-on collisions, the electron can actually bounce off the proton, and lose most of its energy, or even reverse its direction. In this case, the energy of the photons emitted is typically in the X-ray range. The hard and soft X-rays from astrophysical plasmas, including the Sun, are a mixture of bound-bound atomic transitions (soft X-rays), nuclear line emission (hard X-rays/gamma rays), and electron bremsstrahlung. In the hard X-ray range (10-1000 MeV), the emission is purely electron bremsstrahlung.

Radio emission, in contrast, is due to relatively gentle, distant collisions of electrons with ions. Picture the plasma as a soup of electrons and ions, with the electrons zipping around at great speed through the relatively immobile ions. The path of an electron is continually wiggling due to slight deviations as it passes each ion, and each wiggle produces radiation in a broad spectrum (related to the energy lost or gained in each collision). Note that in the case of local thermodynamic equilibrium (LTE), the electron is also absorbing radiation in approximate equality with its energy loss. We will only discuss emission today, but will discuss absorption next time.



An electron moving through relatively immobile

ions experiences many small accelerations, each

one producing a radiation pattern along its direction

of motion (perpendicular to its acceleration).

Let us look a bit more closely at a single collision. As shown in the figure below, the electron experiences a deviation of its straight line path by an angle q , which depends on the speed of the electron and the distance of the encounter, which is called the impact parameter, b . Bremsstrahlung has the peculiarity that the amount of deviation (and hence the power radiated) is actually greater when the electron is moving more slowly, and smaller when v is high. We will see this when we get to the final form of the emissivity.

One can calculate the emission from this process, but it is rather involved (see Rybicki & Lightman, "Radiative Processes in Astrophysics," 1979; or the web discussion here.). The outline of the procedure is to use the small-angle-approximation, for which the electron deflection is negligible, and consider the motion along a straight line, where the separation between charges is r = (b 2 + v 2 t 2 ) 1/2 . One uses the dipole approximation to determine the net acceleration over the path, and thus the emission from a single electron for a single collision

dW(b)/d w = 8Z 2 e 6 /(3 p c 3 m e 2 v 2 b 2 ), for b << v/ w. (2)

Note that this depends on the impact parameter b , and we limit the solution to b << v/ w because collisions at a given b lead to emission only at w < v/b . To extend this to a volume flux of electrons, all with speed v , note that the rate of collisions on any one ion is just n

e

v

, and these electrons will fill the area with element of area 2 p b db . Then the total emission (energy) per unit time per unit volume per unit frequency range is

dW b max dW(b) = n e n i 2 p v b db (3) d w dVdt b min d w

where the limits of the integral over impact parameter depend on the case to be evaluated. For any emission, there is a reasonable upper limit, where b max = v/ w . For radio emission (where the collision energy should be relatively low), we can take a lower limit as for a 90 o deflection, which occurs for b min = 4Ze 2 / p m e v 2 . For higher-energy emission (i.e. X-rays, with head-on collisions), a smaller impact parameter should be taken. When (2) is inserted into (3), we obtain



dW b max db = 16e 6 n e n i Z 2 /(3c 3 m e 2 v) d w dVdt b min b

= 16e 6 n e n i Z 2 /(3c 3 m e 2 v) ln (b max /b min ).

It is conventional to write this as

dW / d w dVdt = 16 p e 6 n e n i Z 2 /(3 3/2 c 3 m e 2 v) G ff (T, w ), (4)

where G

ff

(T, w ) is the Gaunt factor:

G ff (T, w ) = (3 1/2 / p ) ln [ p (kT) 3/2 / 2 1/2 Ze 2 m e 1/2 w ]. (c.f. Dulk 1985).

where we have made the assumption that the electron velocities are the thermal velocities,

v = (2kT/m) 1/2 .

Gaunt factors for many other cases can be calculated exactly (see Figure 5.2 of Rybicki and Lightman 1979).

Thermal Bremsstrahlung

The last step is to integrate the emissivity over a population of electrons, and in most cases for radio emission this is done explicitly for a thermal distribution (that is, the velocities are distributed according to a Maxwellian distribution). In other words, integrate (4) over all velocities. Let us use the symbol h n in place of the unwieldy dW / d n dVdt, which we call the emissivity. Then after integrating (and converting from w to n ) we get:

h n = (2 6 p e 6 /3m e c 3 )(2 p /3m e kT ) 1/2 n e n i Z 2 G ff (T, n ).

Note that for the solar corona, n e n i Z 2 ~ n e 2 , so the emissivity depends on the square of the electron density, and is inversely proportional to temperature (emissivity goes down for higher temperatures!). Also note that there is almost no frequency dependence--the only dependence is the weak (logarithmic) dependence in the Gaunt factor. We will see next time, when we introduce the absorption coefficient, that this is only true when the emission is optically thin.

It is important to recognize that the mechanism of Bremsstrahlung is not defined solely for a thermal distribution of particles (it is a common mistake to assume so). Calculation of Bremsstrahlung for other types of distribution is still a research topic, e.g. the recent paper by Fleishman and Kuznetsov (2014), which discusses emission from the kappa distribution and the n-distribution.

Gyroemission

There is another way to accelerate free particles, by considering the effect of the magnetic field and looking at the magnetic part of the Lorentz force F = qE + (q/c) v x B . For a plasma, there is usually no macroscopic electric field (except perhaps in current sheets), but often there is a non-negligible magnetic field (in which case the plasma is termed a magnetized plasma ). In this case, an electron of speed v will be accelerated perpendicular to both v and B , with magnitude

a = ev perp B/m e c

in the right-hand-rule sense (because charge is negative), as shown in the following figure:



Put the thumb of your right hand in direction of B, and an electron (negatively charged)

will gyrate in the direction of your fingers. A proton gyrates in the opposite direction.

Again, one can calculate the emission from this process, and again it is beyond the scope of this course. However, from the Larmor formula you can see that the power radiated will be

P = (2e 2 /3c 3 )(e 2 B 2 /m e 2 c 2 ) v 2 perp ,

for the non-relativistic case. For the relativistic case, this formula should be multiplied by g

. Some of the properties of this emission, which will be very important for understanding solar emission from both active regions and flares, are:

Cyclotron emission: In the non-relativistic case (low electron velocity, i.e. low temperature), the electrons gyrate at a fixed frequency, independent of their speed, called the gyrofrequency w B = eB/m e c , which depends only on magnetic field strength. The emission from a single electron, when viewed from afar, has a radiated power that varies sinusoidally, which gives rise to a cyclotron line at the gyrofrequency, f B = eB/2 p m e c = 2.8 x 10 6 B Hz (B in gauss) .

Gyroresonance emission: At slightly higher electron velocity ( T ~ 10 5 -10 6 K ), a relativistic effect comes in that changes the sinusoidal dipole pattern to a slightly asymmetric shape,

so that the radiated power peaks more strongly. This gives rise to harmonics of the cyclotron line (so-called gyroresonance lines),

f = sf B = seB/2 p m e c = 2.8 x 10 6 sB Hz (s = 1, 2, 3, ...).

This type of emission is responsible for bright coronal emission from solar active regions.

Gyrosynchrotron emission: At mildly relativistic speeds (electron energies 100-300 keV), the effect becomes stronger, and the lines go up to harmonics 10-100. The lines also become broader (thermal broadening), so that they blend together into a continuum emission. This form of the emission is called gyrosynchrotron emission, and is the type of emission responsible for most radio emission from solar and stellar flares.

Synchrotron emission: At highly relativistic speeds, the forward lobe becomes a narrow beam of width 1/ g 3 , and the emission comes in narrow pulses at the cyclotron frequency, beamed along the direction of motion. The pulses contain many many harmonics. This kind of emission is important in extreme energy environments such as black holes, neutron stars, and some extragalactic sources (generally associated with black holes).

These characteristics are for a single electron, or by extension a mono-energetic population of electrons. To determine the expected emissivity from a plasma, one must integrate the contribution of emission over a particular velocity distribution of electrons. There is a lot of effort in trying to determine an appropriate distribution to use for a given situation, and this is an area of active research. Typically there are two types of population that we consider: a thermal population (in which case the emissivity is expressed in terms of plasma parameters T , n e , and magnetic field parameters B and q ), or a powerlaw distribution in energy (in which case the emissivity is expressed in terms of energy distribution parameters N , d , and perhaps E o , and magnetic field parameters B and q ).

In some cases, such as thermal gyroresonance emission, or synchrotron emission from an isotropic powerlaw distribution, the emissivity can be written down analytically. A good overview is given by Dulk (1985). Here is the emissivity in these two cases:

Gyroresonance emissivity for a thermal distribution of electrons :

h n (s, q ) = p 2 m e /4c [ m s d (wm s )/ d w] - 1 b 2 nn p 2 (s 2 /s!) [s 2 b 2 sin 2 q /2] s - 1 [ b cos q ] - 1

x exp[ -( 1 - s n B / n ) 2 /2 m s 2 b 2 cos 2 q )] ( 1 -s| cos q| ) 2

Synchrotron emissivity for a powerlaw distribution of electrons :

h n /BN = 1/2 ( d - 1) E o d - 1 g( d ) 3 1/2 e 3 /8 p m e c 2 sin q [(2m e 2 c 4 /3 sin q ) n / n B ] - ( d - 1)/2

Transition Radiation

To round-out our discussion of radio emission mechanisms from free particles, it is worthwhile to point out a relatively new mechanism proposed for astrophysical plasmas, although it is a well-known mechanism in devices. This mechanism shares one characteristic with Cerenkov emission, in the sense that no acceleration is required to produce the emission. Instead, the transverse component of the electric field is produced by an inhomogeneity in the refractive index of the medium. To understand how the mechanism works, recall that the electrostatic field of a charged particle radiates outward with the group speed of light in the medium. Slight irregularities in these radiated field lines will occur when the group speed is not uniform. The mechanism was originally considered for the case of a density transition, hence the name Transition Radiation , but for astrophysical plasmas, where such density transitions would be smoothed out by motions of the hot particles, it is more appropriate to consider the case of density fluctuations.