1. Continuum Emission

The X-ray spectra of HMXB between 1 and 20 keV may be described by a cut-off power law (1983). A spectrum of this form may be produced by Comptonization of soft X-rays in an optically thick medium (e.g. 1985).

The infalling matter will be heated up as its gravitational potential energy is converted to thermal energy. How this energy emerges as radiation depends on the geometry of the accretion column and the magnetic field dependent opacity of the infalling matter. If the matter is continuously decelerated by Coulomb interactions it will thermalize on the surface and the radiation will be a soft black body with temperature $\sim$ 10⁷K. However it is not clear how this can happen as the Coulomb stopping length is greater than the neutron star radius. If the infalling matter is stopped at a shock and then settles subsonically to the surface the resulting spectrum will be similar to that of thin thermal bremsstrahlung and a black body. If the electron density is high enough this spectrum will be Comptonized. This collisionless shock might be produced by radiation pressure.

Reasonable plasma parameters for the accretion column may be roughly estimated (e.g. 1992). We can estimate the temperature by assuming the polar cap radiates as a black body: $$\begin{eqnarray} T_{\rm bb}&\approx &\left({L_{\rm x}\over \sigma A}\right)^{1\o... ...1 over4\} &=&2\times10^5L_{37}^{1\over4}A_{10}^{-{1\over4}}\rm\ K \end{eqnarray}$$
where L₃₇ is the luminosity in units of 10³⁷ erg s^-1 and A₁₀ is the polar cap area in units of 10¹⁰ cm². If there is a shock above the surface the temperature will be higher (1985):

$$\approx {3\over8} {GM_{\rm x}m_{\rm p}\over k R_{\rm x}}$$ T (14)

$$\left(\vphantom{{M_{\rm x}\over M_\odot}}\right. {M_{\rm x}\over M_\odot} \left.\vphantom{{M_{\rm x}\over M_\odot}}\right)$$ = (15)

where m_p is the proton mass and R₆ is the neutron star radius in units of 10⁶ cm. For an optically thin plasma with T > 10⁸ K cooling is dominated by bremsstrahlung. Thus if the post-shock material is optically thin a thermal bremsstrahlung spectrum (1979) will be emitted. For an ion of atomic number Z the emission rate is

$$\varepsilon_{\nu}^{\rm ff} {1\over2} {h\nu\over kT} \bar{g}_{\rm ff}^{}$$ (16)

so that

$$\varepsilon_{\nu}^{\rm ff} \approx {n_{\rm He}\over n_{\rm H}} {1\over2} {h\nu\over kT} \bar{g}_{\rm ff}^{}$$ (17)

for an astrophysical plasma where $\bar{g}_{\rm ff}^{}$ $\sim$ 1 is the velocity-averaged Gaunt factor and n_H and n_He are the number densities of H and He. The total power emitted per unit volume will be

$$\varepsilon^{\rm ff}_{} {n_{\rm He}\over n_{\rm H}} \left(\vphantom{{M_{\rm x}\over M_\odot}}\right. {M_{\rm x}\over M_\odot} \left.\vphantom{{M_{\rm x}\over M_\odot}}\right)^{1\over2}_{} {1\over2} \bar{g}_{B}^{}$$ (18)

where $\bar{g}_{B}^{}$(T) $\sim$ 1.2 is the frequency average of $\bar{g}_{\rm ff}^{}$. The density of the accretion column should be greater than the spherical free-fall density

$$\rho {\dot M\over v_{\rm ff} 2 A} \sim$$ (19)

The optical depth will be somewhere between the Thomson depth of free-falling material across an accretion column width of $\sim$ 1 km and the stopping length for infalling protons on atmospheric protons: $$0.6L_{37}\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hb... ...mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}22.$$
Soft photons from the settling accretion mound will be Comptonized to produce a power-law for $E\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}kT$
and a turnover above E $\sim$ kT.

Diffusive Comptonization is described by the Kompaneets equation (Kompaneets (1957)) $${\partial D\over\partial y} = {1\over x^2}{\partial\over\partial x}\left[x^4\left(D+D^2+ {\partial D\over \partial x}\right)\right]$$
where D($\nu$) is the photon phase space density which is related to the total flux I($\nu$) by $$D(\nu) = {c^2 I(\nu)\over8\pi h \nu^3}.$$
Here

$${h\nu\over kT}$$ (20)

is the photon energy in units of kT and $$y = \sigma_{\rm T}nct\left({kT\over mc^2}\right).$$
For an infinite medium the Kompaneets equation may be solved to obtain a spectrum. At low frequencies where D $\gg$ 1, D = ${1\over x}$ which is the Rayleigh-Jeans spectrum. At high frequencies where D $\ll$ 1, D = exp(- x) which is the Wien spectrum.

For a finite non-relativistic thermal distribution of electrons the Compton y parameter is

$${4kT\over mc^2} \tau_{\rm es}^{} \tau^{2}_{\rm es}$$ (21)

where $\tau_{\rm es}^{}$ is the optical depth for electron scattering. The Compton y parameter is essentially the average fractional energy change per scattering times the mean number of scattering. It provides a measure of how much the input spectrum will be affected by Compton scattering. For y_NR $\gg$ 1 we have saturated Comptonization, resulting in a Wien spectrum. For y_NR $\ll$ 1 the resulting spectrum is a modified black body. The regime where $y_{\rm NR}\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$>$}}}1$
is known as unsaturated Comptonization. In this case the Kompaneets equation has to be solved.

Sunyaev and Titarchuk (1980) solved the Kompaneets equation for the upscattering of low-energy photons by a spherical distribution of hot electrons. For an initial phonon energy x₀ the emergent spectrum is F(x, x₀) = x³D. For x$\le$x₀

$${\alpha(\alpha+3)\over2\alpha+3} \left(\vphantom{{x\over x_0}}\right. {x\over x_0} \left.\vphantom{{x\over x_0}}\right)^{\alpha+3}_{}$$ (22)

and for x$\ge$x₀

$${\alpha(\alpha+3)x_0^\alpha\over\gamma(2\alpha+1)} \int_{0}^{\infty} \alpha \left(\vphantom{1+{t\over x}}\right. {t\over x} \left.\vphantom{1+{t\over x}}\right)^{\alpha + 3}_{}$$ (23)

where

$$\alpha \left[\vphantom{{4\over y_{\rm NR}}+{9\over4}}\right. {4\over y_{\rm NR}} {9\over4} \left.\vphantom{{4\over y_{\rm NR}}+{9\over4}}\right]^{1\over2}_{} {3\over2}$$ (24)

For $y_{\rm NR}\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$>$}}}1$
the form of the spectrum is a power-law with an exponential cut-off for photon energies above $\sim$ 4kT.