4.1.26. Line: transmission model for a single spectral line

This model calculates the transmission T(E) for a single spectral emission or absorption model. It is used as a multiplicative component, and can be used both for absorption and emission lines. The line profile is a Voigt profile (the convolution between a Gaussian and a Lorentzian profile):

T(E) = e^{-\tau(E)},

with the optical depth \tau(E) given by:

\tau(E) = \tau_0 K(x,y),

where K(x,y) is the Voigt profile given by

K(x,y) = \frac{y}{\pi} \int_{-\infty}^{\infty} \frac{e^{\displaystyle{-t^2}}}{y^2+(x-t)^2} \mathrm{d}t,

with

x = \frac{2\sqrt{\ln{2}}}{F_{\mathrm g}} (E-E_0)

and

y = \sqrt{\ln{2}} \frac{F_{\mathrm l}}{F_{\mathrm d}}

where F_{\mathrm g} and F_{\mathrm l} are the Full Width at Half Maximum of the Gaussian and Lorentzian components, respectively. Further, E_0 is the line centroid.

The program also calculates the equivalent width of the line.

The model can be used both in energy (keV) mode or wavelength (Å) mode. In each case, the quantities for the other scaling (centroid, equivalent width, and FWHMs of the Gaussian and Lorentzian components) are calculated. In addition, like all SPEX absorption models, also the covering factor can be varied.

Warning

The equivalent width is calculated at the energy grid in use for SPEX. Line flux outside this range is not taken into account. That may be important in the case of strong Lorentzian wings.

Warning

The parameter \tau_0 only represents the optical depth at the line centre for a pure Gaussian case. When the Lorentzian component becomes important, the “true” depth at line center is smaller.

Warning

When changing from energy to wavelength units, take care about the frozen/thawn status of the line centroid and FWHM.

Warning

You need to do a calc or fit command to get an update of the wavelength (for type=0) or energy (type=1).

The parameters of the model are:

tau0:

Optical depth \tau_0 at line center. Default value: 1. Positive values correspond to absorption lines, negative values to emission lines.

el:

Line center energy E_0 (keV). Default value: 6.4 keV.

fwhg:

FWHM of the Gaussian component F_g in keV. Default value: 0.1 keV.

fwhl:

FWHM of the Lorentzian component F_l in keV. Default value: 0.1 keV.

type:

Type of calculation. Type=0: energy units; type=1: wavelength units.

wl:

Line center wavelength E_0 (Å). Default value: 20 Å.

awhg:

FWHM of the Gaussian component F_g in Å. Default value: 0.1 Å.

awhl:

FWHM of the Lorentzian component F_l in Å. Default value: 0.1 Å.

ewk:

Equivalent width of the line in keV. Calculated by the program, not an input variable.

ewa:

Equivalent width of the line in Å. Calculated by the program, not an input variable.

icov:

Type of the covering fraction. Default value: 2 (constant, set by fcov). If icov=1, full covering is applied. If icov=3, covering fraction follows a tangent function that increases with energy. If icov=4, covering fraction follows an inverse tangent function that decreases with energy. See description in pion.

fcov:

The covering factor of the absorber if icov=2. Default value: 1 (full covering). If icov=3 or 4, it sets the covering factor at the high energy end.

lcov:

The covering factor of the absorber at the low energy end. Default value: 1. lcov is applied only when icov=3 or 4. See description in pion.

ecov:

The energy when the covering factor changes from lcov to fcov. Only applied if icov=3 or 4.

acov:

The width of the transit on covering factor. Only applied if icov=3 or 4.