5 Profile variations of Nd III and Pr III spectral lines and mode analysis

   
5.1 Moments of line profiles

The variability of Nd III and Pr III spectral features can be conveniently quantified by considering time variations of a few low-order moments of a line profile. This observational material can then be used to identify frequencies and modes of non-radial stellar oscillations. The moment technique was first developed by Balona (1986a, 1986b), then Aerts et al. (1992) extended and applied this method to line profile variations of the monoperiodic $\beta$ Cep star $\delta$ Ceti. The moment method turned out to be the best mode identification technique for non-degenerate slowly rotating pulsators, and therefore it seems especially suitable for roAp stars.

To measure the first four moments we transformed the wavelength scale $\lambda$ into velocity units v relative to the center-of-mass velocity, which is the zero point of the variation of the first moment. Then the nth order moment was computed as

$$\begin{array}{l} \langle V^n \rangle^j = \sum\limits_i v^n_i ... ...ambda^j, \\ W_\lambda^j = \sum\limits_i y^j_i, \\ \end{array}$$ (5)

where the zeroth order moment $W_\lambda$ is just the equivalent width of the line and y_i^j is the flux in the ith pixel of the jth spectrum, normalized to the continuum and subtracted from unity ( y_i^j=1-f_i^j). In order to obtain complete information about line profile variations we carried out the summation over the whole variable part of Nd III 6145.07 Å and Pr III 6160.24 Å line profiles, as determined from the standard deviation plot in the lower panels of Fig. 1.

Formal errors of the moment measurements were derived by applying the error propagation laws to formula (5):

$$\sigma^2\langle V^n \rangle^j =\sum\limits_i \left [ \frac{\s... ..._i)}{\sum_k y^j_k} (v^n_i - \langle V^n \rangle^j) \right ]^2,$$ (6)

where $\sigma(y^j_i)$ is the uncertainty of the flux in the ith pixel of the jth spectrum.

Figure 4 illustrates the variations of the line profile moments of doubly ionized REE lines. Similar to the analysis of the centroid RV variations (Sect. 4), we used non-linear least-squares sinusoid fitting to find the amplitudes, phases and constant terms of the moment variations. Time-series of each moment were fitted with

$$\langle V^n \rangle^j = \overline{\langle V^n \rangle}+K_n \cos (2\pi t^j/P + \varphi_n),$$ (7)

where the moments are defined in such a way, that $\overline{\langle V^n \rangle}$ is non-zero only for the even moments. K_n and $\varphi _n$ determined for each moment are listed in Table 2. According to the results of Sect. 4 we fixed the period P in the least-squares fit at P₂=12.20 min for Nd III 6145.07 Å and at P₁=12.45 min for Pr III 6160.24 Å.

 

 

Table 2: Variation of the moments of doubly ionized REE lines parametrised by a cosine curve (7) (parameters K_n and $\varphi _n$) and by expressions (8) (parameters A_n,k). Amplitudes K_n (n>0) and A_n,k are given in units of (ms^-1)ⁿ, where n is the moment order, while $W_\lambda$ variations are given in mÅ units. Phases $\varphi _n$ are expressed in units of the period.

Amplitude	Nd III 6145.07 Å			Pr III 6160.24 Å
zeroth moment: $W_\lambda$
K0	1.90	$\pm$	0.16	0.89	$\pm$	0.22
$\varphi_0$	0.342	$\pm$	0.013	0.253	$\pm$	0.039
first moment: $\langle V \rangle$
K1	0.452	$\pm$	0.022	0.784	$\pm$	0.028
$\varphi_1$	0.187	$\pm$	0.007	0.143	$\pm$	0.006
A1,1	0.452	$\pm$	0.020	0.784	$\pm$	0.024
second moment: $\langle V^2 \rangle$
K2	3.36	$\pm$	0.19	2.99	$\pm$	0.22
$\varphi_2$	0.416	$\pm$	0.009	0.377	$\pm$	0.011
A2,0	47.59	$\pm$	0.15	30.12	$\pm$	0.16
A2,1	3.03	$\pm$	0.22	2.63	$\pm$	0.24
A2,2	0.52	$\pm$	0.23	0.30	$\pm$	0.24
third moment: $\langle V^3 \rangle$
K3	71.49	$\pm$	4.70	86.76	$\pm$	2.95
$\varphi_3$	0.197	$\pm$	0.009	0.146	$\pm$	0.006
A3,1	71.15	$\pm$	4.70	86.11	$\pm$	2.91
A3,2	13.15	$\pm$	4.53	1.77	$\pm$	3.12
A3,3	0.22	$\pm$	4.41	6.33	$\pm$	2.94

From the theoretical expressions for the variation of the moments of line profiles of non-radially oscillating stars (Aerts et al. 1992) we expect the oscillation frequencies 2/P and 3/P to be present in the variation of the second and third moments. For $\gamma$ Equ the rotation frequency $\Omega$ is negligible in comparison with the pulsation frequency $\omega$. In this case the expressions of Aerts et al. (1992) for the variation of the first three moments of a monoperiodic pulsation are given by:

$$\begin{array}{lcl} \langle V \rangle &= &A_{1,1} \sin (\omega... ...i+\frac{3\pi}{2})+A_{3,1}\sin (\omega t +\psi). \\ \end{array}$$ (8)

In these formulas $\psi$ is the phase constant and the parameters A_n,k depend on the type of the pulsation mode ($\ell$ and |m| numbers), projected rotation velocity $v_{\mathrm{e}}\sin i$, pulsation velocity v_p, the angle between pulsation axis and line of sight and intrinsic stellar profile. The functions A_n,k can be used to identify pulsation mode and are usually estimated by considering periodograms of the three moments (Aerts et al. 1992). The small number of $\gamma$ Equ observations precluded us from using this approach, instead we directly fitted the variations of all three moments with expressions (8) using the Marquardt method. Figure 4 compares this fit with the observations and a simple cosine approximation of the moment curves. The best-fit amplitudes A_n,k and their formal errors are summarised in Table 2. Apparently variations of all four first moments of Nd III and Pr III lines are dominated by the fundamental pulsation frequency and our data do not give unambiguous support for the presence of the first and second harmonics of the main pulsation frequency in the variations of $\langle V^2 \rangle$ and $\langle V^3 \rangle$. Thus, amplitudes A_2,2, A_3,2 and A_3,3, given in Table 2, should be considered as the upper limits of these parameters. Longer and preferentially multi-site spectroscopic time-series are required for improving the phase curves of the second and third moments.

With the exception of Nd III 6145.07 Å and Pr III 6160.24 Å there are no other lines in the 6140-6166 Å spectral region that are suitable for accurate moment measurements. The other REE lines are weak and blended, while relatively unblended strong lines of lighter elements (Ba II 6141.71 Å, Fe II 6147.74 Å, Ca I 6162.17 Å, and Ca I 6163.76 Å) do not exhibit profile changes. Equivalent width variations at the level above 2 mÅ are also absent for these spectral lines.

Figure 5: Behavior of amplitudes (left panels) and phases (middle panels) across the profiles of Nd III 6145.07 Å and Pr III 6160.24 Å spectral lines. The bars are formal $1\sigma$ errors derived in a least-squares analysis of the profile variation. Phases $\psi (\lambda )$ are given in degrees, while amplitudes $D(\lambda )$ are in units of the continuum flux. The right panels show an estimate of the average line profile.

   
5.2 Amplitudes and phases across line profiles

During an oscillation cycle the flux at every pixel of the line profile varies with the same period(s). Therefore one can apply the usual time-series analysis to the variations of the fluxes in individual pixels. In particular, Fourier frequency analysis with the CLEAN algorithm (e.g. De Mey et al. 1998) helps to extract periodicities present in the pulsation spectrum. For moderately and rapidly rotating non-radial pulsators such line profile analysis is also the only way to detect pulsation modes with high $\ell$ numbers, since these modes cancel out in disk-averaged observables such as centroid RV or brightness.

Limited time-series of profile variations (similar to out $\gamma$ Equ data) are not well suited for the period determination, but can be efficiently analysed with the least-squares algorithm described by Mantegazza (2000). In this method variation of the ith pixel is approximated by the superposition of n sinusoidal components, corresponding to n detected frequencies

$$f_i^j = F_i + \sum^n_{k=1} D_{i,k} \cos (2\pi t^j/P_k + \psi_{i,k}),$$ (9)

where F_i, D_i,k, and $\psi_{i,k}$ are free parameters, optimized by the least-squares method assuming fixed set of periods P_k.

In the analysis of line profile variations of Nd III 6145.07 Å and Pr III 6160.24 Å we assumed that variations with a single period (P₂ for Nd III and P₁ for Pr III) are present and derived the estimate of the average line profile $F(\lambda)$ together with the variation of the pulsation amplitude $D(\lambda )$ and phase $\psi (\lambda )$ across the line profiles. Figure 5 shows these functions and their formal errors.

5.3 Mode identification

Unfortunately elaborate modelling techniques developed in recent years for the spectroscopic analysis of non-radial oscillations of normal stars are not directly applicable and cannot be easily adapted for roAp pulsations. In most of the studies of line profile variations in non-radially pulsating stars the pulsation axis is assumed to be aligned with the rotation axis. Therefore an inclination angle is one of the important parameters. In roAp stars a general model describing stellar pulsations is the oblique pulsator model (Kurtz 1982). According to this model the pulsation axis is aligned with the magnetic rather than the rotation axis. Hence, we have to use an inclination of the magnetic axis to the line of sight (angle $\alpha$) instead of the usual inclination angle. As the star rotates the angle $\alpha$ varies and therefore the pulsation amplitudes are modulated with rotation period. This effect has to be properly taken into account in the analysis of rapidly rotating roAp stars, but with the extremely slow rotation of $\gamma$ Equ we do not expect to see any rotational modulation of the pulsations and can treat the star as if it were a $\delta$ Scuti-type variable observed at inclination angle $i=\alpha$. Besides variable orientation of the pulsation axis, a strong magnetic field, which is present in the atmospheres of many roAp stars, alters the pulsation velocity field (e.g. Bigot et al. 2000) and simultaneously introduces Zeeman broadening and splitting of spectral lines. The latter effect makes a Gaussian approximation of the local line profile (Aerts et al. 1992; Schrijvers et al. 1997) highly questionable. Despite all these ambiguities, in this section we try to identify the pulsation mode of $\gamma$ Equ applying methods developed for normal non-radial pulsators to the variation of Pr III and Nd III spectral lines. This semi-qualitative analysis should be regarded with caution and it by no means aspires to reach any definite conclusions about the pulsation velocity distribution in the atmosphere of $\gamma$ Equ.

In principle diagrams that show the behaviour of the phases of each mode across the line profiles (Fig. 5) are able to discriminate between pulsation modes with different $\ell$ and m. Telting & Schrijvers (1997) determined linear relations between the observed phase difference and mode parameters $\ell$ and |m| for non-radially pulsating rotating stars. It is not correct to apply directly these relations for $\gamma$ Equ because they were obtained for a rapidly rotating star, which is not fulfilled in our case. However we expect to get a lower limit for $\ell$ using the proposed method. From Telting & Schrijvers' Eq. (9) we get $\ell \approx 0.10 + 1.09\vert\Delta\psi_0\vert/\pi$, where $\Delta\psi_0$ is the blue-to-red phase difference for the variations with the main pulsation frequency (middle panels in Fig. 5). For both Pr III and Nd III lines $\Delta\psi_0\approx 300\hbox{$^\circ$ }$ or 1.67$\pi$, therefore .

A consideration of the shape of the amplitude and phase diagrams for individual spectral lines opens another possibility for mode identification. In particular, the similarity of our pixel-by-pixel amplitude and phase diagrams to those derived for 5.31 d^-1 mode of $\delta$ Scuti-type variable HD 2724 (Mantegazza & Poretti 1998), for which $\ell=2\pm1$ and $m=2\pm1$ was obtained, allows us to propose $\ell\approx2\pm 1$ and $\vert m\vert\approx2\pm 1$ for the mode identification of the $\gamma$ Equ RV pulsations. The sign of m depends on the angle $\alpha$ between line of sight and pulsation (magnetic) axis. This angle can be determined from the parameters of the $\gamma$ Equ magnetic model suggested by Leroy et al. (1994). They found that published longitudinal and broadband linear polarisation magnetic observations can be fitted with a dipolar field B_d = 5.5 kG and two combinations of inclination angle i and angle $\beta$ between the magnetic and rotation axes ( $i=150\hbox{$^\circ$ }$, $\beta=80$$^\circ$ and $i=80\hbox{$^\circ$ }$, $\beta=150$$^\circ$). These two models yield the same $\alpha$:

$$\cos \alpha=\cos i \cos \beta + \sin i \sin \beta \cos \Phi,$$ (10)

where $\Phi$ is rotational phase. Assuming that the epoch of our $\gamma$ Equ observations roughly corresponds to $\Phi=0.8\pm0.1$(see Leroy et al. 1994; Scholz et al. 1997), we find $\alpha=132\hbox{$^\circ$ }\pm 10\hbox{$^\circ$ }$. This value of $\alpha$ together with the positive slope of the $\psi (\lambda )$ curves allows us to conclude that $\gamma$ Equ pulsates in prograde mode (negative m).

An alternative possibility of mode identification comes from moment analysis through the comparison between the observed moment amplitudes and calculations for a set of parameters $\ell, m, i=\alpha$, v_p and variance $\sigma$ of the intrinsic Gaussian stellar profile (Aerts et al. 1992; Aerts 1996). The most probable combination of ( $l, m, \alpha, \sigma$, v_p) can be defined as the one for which the weighted squared differencies between observed and calculated moment amplitudes,

$$Q = \sum_{n,k} (A^{\mathrm{obs}}_{n,k}-A^{\mathrm{calc}}_{n,k})^2/\sigma^2(A^{\mathrm{obs}}_{n,k}),$$ (11)

reaches its minimal value. We calculated theoretical moment amplitudes A^calc_n,k using the code of Aerts (1996) and compared them with amplitudes A^obs_n,k of the first three moments determined for Nd III and Pr III lines in Sect. 5.1 (see Table 2). For the projected rotation velocity we adopted value $v_{\mathrm{e}}\sin i$=0.003 kms^-1, estimated from the rotational period of 77 yr assuming the rigid rotation of $\gamma$ Equ. Since angle $\alpha$ is relatively well constrained by the published model of the magnetic topology, we considered only the values of $\alpha=130\hbox{$^\circ$ }\pm30\hbox{$^\circ$ }$. The variance of the intrinsic Gaussian profile is another parameter required for the moment amplitudes calculation. The lower limit of this parameter (non-magnetic broadening effects) can be estimated from the width of the resolved Zeeman components of Fe II 6149.26 Å: both lines give $\sigma_{6149}=2.6$ kms^-1. On the other hand the magnetic contribution to $\sigma$ should be of the order of the separation of magnetic $\sigma$ components, which for Nd III and Pr III lines amounts to an additional $\sigma_{\mathrm{mag}}\simeq 3$ kms^-1. Thus we adopted $\sigma\simeq4\pm2$ kms^-1 for the calculation of the weighted sum (11). Table 3 summarises the parameters $\sigma$, v_p and $\alpha$ which give the lowest Q(Nd III)+Q(Pr III) within the errors of the observed amplitudes A^obs_n,k for the three best-fit modes. Despite the relatively large scatter of v_p and $\alpha$ these results generally support a pulsational velocity v_p$\approx10$ kms^-1 and are consistent with the mode identification suggested above. Thus we conclude that the most probable parameters of the p-mode in $\gamma$ Equ are $\ell =2$ or 3 and $m=-\ell$ or $-\ell+1$.

 

 

Table 3: Estimate of the width of Gaussian profile $\sigma$, pulsational velocity v_p (both are given in kms^-1) and angle $\alpha$ (in degrees) for the three best-fit pulsation modes of $\gamma$ Equ.

		Nd III 6145.07 Å			Pr III 6160.24 Å
$\ell$	|m|	$\sigma$	vp	$\alpha$	$\sigma$	vp	$\alpha$
3	3	5-6	8-14	130-160	4	7-8	120-130
2	2	6	10-11	160	3-5	5-14	140-160
3	2	6	7	100-160	3-4	7-9	110-150

Figure 6: The upper panel shows a comparison between observations (dots) and synthetic spectrum calculations for two macroturbulent velocities: 2 kms-1 (thin line) and 10 kms-1 (thick line). The lower panels illustrate the effect of pulsations with sectoral $\ell =2$ (thin line) and $\ell =4$ (thick line) modes as seen in the average profiles of doubly ionized REE lines. The observed spectrum is shown by dots, while the dashed line corresponds to the synthetic profiles in the absence of pulsations.