Free-free absorption & emission

Free-free emission arises from the acceleration of a charged particle, most com- monly an electron, and hence is referred to as bremsstrahlung, braking radiation. This acceleration may be due to the electron’s interaction with matter, or its deflection by an electric or magnetic field.

It is possible that coherent radiation may be produced by resonance with wave-modes in the plasma; mechanisms known as plasma oscillation, and electron- cyclotron emission. Plasma oscillations, also known as Langmuir waves, are caused by the displacement of electrons in a plasma, which are then subject to a restoring Coulomb force. From the equation of simple harmonic motion the frequency of the oscillation, and hence the emission, can be shown to be (in the “cold” case):

νp =

s nee2 πme ≈

9000√ne [Hz] (2.1)

where ne is the electron density, e is the electron charge, and me is the elec-

tron mass. For a typical chromospheric electron density, ne ∼ 109cm−3, we get νp = 280 MHz. Electron-cyclotron emission arises from electrons spiralling around

magnetic field lines, and has a frequency, often known as the gyro-frequency, of

νgyro = eB

2πme

= 2.8B [MHz] (2.2)

whereBis the magnetic field density measured in Gauss. For field strengths typical of cool stars (Sennhauser & Berdyugina, 2011), we would expect νgyro ∼ 50 MHz.

Figure 2.1: Emission produced by the deflection of an electron by Coulomb inter- action with a nucleus.

Non-equilibrium electron distributions may produce similar emission by three-wave interaction bringing about plasma emission (Robinson, 1993).

Of greater relevance to this work is non-coherent, thermal free-free emission. This emission occurs a a result of the deflection of electrons by ions, as can be seen in Fig. 2.1. In the relatively rarefied environment of stellar atmospheres small- anlge deflections, with high impact parameters, dominate the emission, and as a result can be computed without recourse to quantum mechanics. The emission,

f f (dEdtdν), is (see Rybicki & Lightman (1979) and Hubeny & Mihalas (2014)): f f = 32π2_e6 3√3c3_m2 e r 2me πkB Z2neniT−1/2e−hν/kBTgf f [erg/cm3/s/Hz] (2.3)

Where Z is the ion charge, and gf f is the free-free Gaunt factor, a quantum

mechanical correction factor. This is a function of temperature and frequency, and is tabulated in Karzas & Latter (1961). In the notation of the transfer equation derived in Chapter 1, the traditional notation for astrophysics,ηf f =f f/4π.

Since free-free absorption, χf f, and emission are collisional processes they oc-

cur at LTE rates, and hence are related by Kirchoff’s law1 _{to the LTE intensity,}

Planck’s Law, Bν: Bν = ηf f χf f = 2hν 3 c2 1 ehν/kBT −1 (2.4) χf f = √ 32πe6 3√3chpkBm3e Z2_n enigf f √ T ν3 (1−e −hν/kBT_{) (2.5)}

With these values defined we can now use the equation of radiative transfer

∂Iν

∂s =ην −χνIν (2.6)

In thermodynamic equilibrium, which is a valid approximation for free-free emission in an optically thick slab, the intensity is constant, hence

Iν = ηf f χf f

=Sν =Bν (2.7)

Free-free emission falls-off exponentially at high frequencies, hence we are only interested in low frequency, radio wavelengths. As a result we can use the Rayleigh- Jeans law in place of Planck’s Law

Bν =

2ν2_k

BT

c2 (2.8)

In radio astronomy the temperature which produces a given intensity is referred to as the brightness temperature, TBr, and we will frequently use this as a direct proxy for intensity1_.

We can now solve the transfer equation in this relatively simple case.

∂Iν ∂τν

=Iν−Sν (2.9)

This equation has the solution

Iν(0) =Iν(z)e−τν+

0

Z

τ(z)

Sνe−τνdτ (2.10)

1_{There is no requirement thatTBrbe reflective of the thermodynamic temperature of an ob-}

ject. In the case of non-thermal emissionTBris still often used, despite there being no physically meaningful sense in which the brightness has a related temperature.

which has the physical interpretation of the first term being radiation shining through the medium, and the second being the radiation produced in the medium (this is for a single ray, passing through a semi-infinite slab). In the thermal equilibrium case, where the ηf f and χf f are constant through the medium, and

writing in terms of brightness temperature rather than intensity,

Tb =Te(1−e−τν) (2.11)

whereTe is the electron temperature throughout the medium. In the case that the

medium is optically thick TBr =Te, and in the case that the medium is optically

thin TBr =τνTe.

From this we can define a spectral index,

α = d log10(Fν)

d log10(ν)

(2.12) which is the slope of the flux-frequency power-law. For instance, in the optically thick case the flux will clearly follow the frequency dependence of the Rayleigh- Jeans law, hence α= 2. In the optically thin case, given no background illumina- tion, the flux will follow the frequency dependence of the Gaunt factor, α≈ −0.1. By measuring the flux at multiple frequencies it is possible to compute the spec- tral index, which has the power to differentiate between optically thin and thick emitting regions, and hence to help diagnose the properties of the medium.

In Chapter 3 of this work we will present a simple radio flux model, designed to predict the thermal, free-free emission from stellar chromospheres. In this chapter the concepts introduced here will be expanded upon and used practically in an attempt to determine the temperature structure of the chromosphere.