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Appendix A: Monochromatic amplitude, phase, and radial velocity of a pulsation mode

For a non-rotating star, linear theory leads to the following expression for the first-order perturbation to the emergent Eddington flux relative to the unperturbed flux in presence of a single pulsation mode, (A.1)where ϵ_r is the intrinsic dimensionless radial amplitude of the pulsation mode characterised by an angular frequency ω (=2π/P, P being the pulsation period), a degree index l, and an azimuthal order m, where is a real function representing the viewing aspect (i is the inclination angle between the line of sight and the symmetry axis of the pulsation), where is a measure of the monochromatic amplitude of the relative flux, and where is the monochromatic phase of the relative flux with respect to an arbitrary zero point in time symbolised by the phase factor mφ₀. This equation corresponds to Eq. (41) from Randall et al. (2005) using a slightly different notation, and the reader is referred to that publication for more details.

The term , which multiplies the cosine function in the previous equation, corresponds to the observable amplitude of the pulsation mode in the Fourier domain. However, given that the intrinsic radial amplitude ϵ_r and the viewing factor are unknown a priori, the Fourier amplitude does not reveal the details of the geometry of the mode. Hence, in order to exploit the l dependence of the amplitude as a function of wavelength (frequency), it is customary to compute the ratio of the amplitude at a given frequency ν with respect to the amplitude at some reference frequency ν′. By doing this, the monochromatic ratio becomes independent of the unknown factors and still bears a wavelength-dependent signature of the degree index l of the mode. The phase difference , which is independent of , also bears a similar, but distinct signature. We note that this signature is lost in the adiabatic approximation as no phase shift is then predicted. Given data of sufficiently high quality, amplitude ratios and phase differences can thus be exploited to infer or constrain the l index of a pulsation mode. This is a well-known method that we were fully able to make use of in this paper.

Mode discrimination between various l solutions can be improved, at least in principle, if radial velocity measurements can be obtained and combined with amplitude ratios and phase differences. This is described particularly well in Cugier & Daszyńska (2001), but we also refer the reader to the original paper of Dziembowski (1977) and to the review of Stamford & Watson (1981). According to these authors and others, the pulsational velocity projected along the line-of-sight and integrated over the visible disk of the star may be written as (A.2)\newpage

where is given by (A.3)In this expression, R is the (unperturbed) radius of the star, g is the (unperturbed) surface gravity, and u_lν and v_lν are l-dependent monochromatic quantities related to the disk-integrated quantity b_lν through the relations The disk-integrated quantity may be written as (see e.g. Eq. (25) of Randall et al. 2005), (A.6)where h_ν(μ) is the limb-darkening law, and P_l(μ) is a Legendre polynomial.

If one divides the term in front of the cosine term in Eq. (A.2) by its equivalent in Eq. (A.1), one eliminates the unknown factor , and thus obtains another diagnostic tool to discriminate between l values, namely the monochromatic velocity-to-amplitude ratio, . It should be pointed out that the diagnostics based on amplitude ratios, phases differences, velocity-to-amplitude ratios, or combinations of the three work best when applied to the continuum part of the spectrum. This is because the theory was developed in this simpler framework, without having to add complicated velocity fields which are needed for a detailed description of the spectral lines, particularly the strong, dominant spectral lines. A potential difficulty with this approach in the particular case of EC 01541−1409, and more generally for pulsating hot subdwarf B stars, is that readily available radial velocity measurements – based on the central wavelengths of strong Balmer H I lines and strong He I and He II lines – cannot be easily connected with the pulsational behaviour of the continuum, which is formed significantly deeper in the atmosphere than the cores of these strong spectral features.

There is yet another potentially useful diagnostic that can be exploited by combining flux measurements with radial velocity measurements, and that is the predicted phase difference, , between the flux and the velocity (see Eqs. (A.1) and (A.2)). This difference (to be measured in some suitable part of the continuum) is independent of the term , and is also independent of the depth in the atmospheric layers since all moving parts must go through their equilibrium position at the same time during a modal cycle. The latter characteristic gives the phase lag technique the edge over the velocity-to-amplitude ratio diagnostic. If there is a pulsation node in the atmospheric layers – which is not the case for the pulsation modes of interest here – the phase could be off by a factor of π radian. Otherwise, the phase difference bears a signature of the degree index ℓ for a given mode.

© ESO, 2014