3 Convolution of profiles and microturbulence

Neither the BALMER9 code nor the SYNTHE code perform profile convolutions, but all the profiles are simply added. In the BALMER9 code, for separations larger than 0.2 Å from the line center, a Lorentz profile (representing the natural broadening and the resonance broadening) is added linearly to the Stark-thermal Doppler profile interpolated in the VCS tables. For separations smaller than 0.2 Å no Lorentz profile was considered.

In the SYNTHE code, the Doppler profile, the Stark profile, and the Lorentz profile (for natural broadening, resonance broadening, and van der Waals broadening from He I and H₂) are still summed together. The very inner core is that of the profile (Doppler, Stark, or Lorentz) with the largest full width at half maximum FWHM.

This method due to Peterson (1993), which we shall call the PK approximation, would be rigorously true for the wings of two Lorentzians. Since the wing-dependence of the Stark profile differs from that of a Lorentzian only by $\sqrt(\Delta\lambda)$, one might expect the approximation to be good, as we verified that it is.

 

 

Table 1: Models used for H$\alpha$ tests.

$T_{\rm e}$(K)	$\log~g$	$\xi_{\rm t}$(km s-1)	Comment
4500	1.5	3.0	solar abundances
4760	1.3	2.3	CS22892-052 (cf. Sneden et al. 1996)
5770	4.4	1.0	Sun
8000	3.5	2.0	like cool Ap or Am
8000	1.5	12.0	test of large $\xi_{\rm t}$
12 000	3.0	2.0	hot star

Figure 3: H$\alpha$ profiles for a model with $T_{\rm eff} = 8000~{\rm K}$, $\log~g$ = 1.5. The lower curve is for a CSII calculation with an assumed microturbulence $\xi _{\rm t} = 12$ km s-1. The upper curve, displaced upward for purposes of illustration, was made using BALMER9, the older interpolation scheme for VCS tables, and the PK approximation. There is no perceptible difference in the two profiles beyond the line core.

Replacing the sum of the Stark and Lorentz profile in BALMER9 by a convolution takes a large amount of computing time in that the $\Delta\lambda$ step of the convolution has to be very small (less than 0.001 Å) in order to account for the narrow full width at half maximum FWHM of the Lorentz profile. This problem can be overcome by including a microturbulent velocity $\xi_{\rm t}$ in the computations.

Both the VCS and CS tables include thermal Doppler, but not microturbulent broadening. The BALMER9 code makes no provision for the inclusion of microturbulence in the line profiles owing to the sum of the Stark-thermal-Doppler profile, interpolated in the VCS tables, with the Lorentz profile. The SYNTHE code does allow for a microturbulence in that it adds the Stark profile to a Doppler-microturbulence Gaussian profile.

The only way to rigorously include all broadening mechanisms is to do a convolution of the Stark-thermal Doppler profile, interpolated in the VCS or CS tables, with a profile which includes both the Lorentz broadening and turbulent motions. If we assume a Gaussian distribution of microturbulent velocities, the VCS or CS profiles need to be convolved with a Voigt profile.

To check BALMER9 and SYNTHE profiles we did calculations using the new CS profiles with improved interpolation, and a full convolution including a microturbulent velocity. We shall refer to such profiles and to the corresponding code with the abbreviation CSII (Convolution, Stehle, improved interpolation). Table 1 shows models parameters for which we made calculations of an H$\alpha$ profile in order to test the effects of the various approximations and improvements mentioned above. All models were generated with the ATLAS9 code (Kurucz 1993a). Solar abundances were assumed for all but CS22892-052, for which abundances were chosen to roughly match those of Sneden et al. (1996).

We find, with one exception, that the BALMER9 profiles computed with no convolutions and no microturbulent velocity are in excellent agreement with CSII calculations. The only exception occurs for the supersonic microturbulent velocity $\xi_{\rm t}=$ 12 km s^-1. In this case the line core of the profile computed for $\xi_{\rm t}=$ 12 km s^-1 is larger than that computed without microturbulence, as is shown in Fig. 3. However, the H$_{\alpha }$ profile computed by SYNTHE with no convolutions, but by assuming $\xi_{\rm t}=$ 12 km s^-1 agrees well with the CSII profile.

The effect of a microturbulent velocity $\xi_{\rm t}$ will be small until $\xi_{\rm t}$ approaches the sound speed. It is not surprising, therefore, that the only case we have found where plots of H$\alpha$ obtained using BALMER9 with the PK approximation and CSII differed significantly is that for $\xi_{\rm t}$ of the order of the sound speed. Even in this situation, only the deepest parts of the core were affected. The line wings still matched beautifully.

The calculations of Fuhrmann et al. (1993, 1994) included Lorentz broadening by a full convolution, while BPO used the PK approximation. The above comparisons led us to conclude that any differences between their results and other calculations (e.g. CGK or Gardiner et al. 1999) cannot be attributed to the PK approximation or to different Stark profiles (VCS or CS) - the immediate line core excepted.