3 Stokes $\vec{V}$ Zeeman signatures and longitudinal magnetic field measurements

No significant circular polarisation is detected within spectral lines in the circular polarisation spectra. This result is illustrated in Fig. 1, and is further supported by a comparison with a numerical simulation. Using the polarised line synthesis code ZEEMAN2 (Landstreet 1988; Wade et al. 2001) we have synthesised the Stokes I and V spectrum of $\eta$ Aql in the approximate spectral region employed by Plachinda (2000) for his magnetic field measurements, assuming a dipole field configuration which produces a longitudinal magnetic field of approximately -70 G (the value measured by Plachinda 2000 at the depicted phase). Although the detailed shape of the computed Stokes V depends on the adopted model magnetic configuration (a dipole), its amplitude is representative of any field configuration producing a -70 G longitudinal field. Comparing this synthesis with both the individual and averaged observed spectrum at phase $\sim$0.8, we find that predicted Stokes V Zeeman signatures which should be easily detectable in many individual lines in this region are completely absent in our observed spectrum.

We have also applied the Least-Squares Deconvolution (LSD) multiline analysis technique (Donati et al. 1997) to further refine our attempt to detect a magnetic field. From each observed spectrum of $\eta$ Aql we extracted LSD mean Stokes I and V profiles using a line mask constructed using VALD (Piskunov et al. 1995) line lists corresponding to $T_{\rm eff}=6000$ K, $\log g=2.0$ and a solar abundance table. LSD profiles of $\alpha ^2$ CVn were also extracted using the procedure described by Wade et al. (2000a). This line mask includes all lines with unbroadened central depths above 10% predicted to appear in the MuSiCoS spectrum of $\eta$ Aql (more than 4000 lines). More information about the extraction of LSD profiles and the construction of LSD line masks is provided by Shorlin et al. (2002). None of the $\eta$ Aql LSD profiles show any significant circular polarisation, and our most conservative upper limit on circular polarisation in the mean spectral lines of $\eta$ Aql during 10, 11 and 12 July 2001 is 0.01% (3 $\sigma$, peak-to-peak). Representative mean Stokes I and V profiles of $\eta$ Aql are shown in Fig. 3.

Finally, the longitudinal magnetic field and its formal uncertainty  $\sigma_{\rm B}$ were inferred from each of the extracted LSD Stokes I/V profile sets by numerical integration, using the expression:

$$\ensuremath{B_{\rm z}} =-2.14\times 10^{11}\ \frac{{\displays... ...\displaystyle {\lambda z c\ \int [I_{\rm c}-I(v)]\ {\rm d}v}},$$ (1)

(Wade et al. 2000b) where the wavelength $\lambda$ is in nm and  is in Gauss. The Landé factor z is a mean value obtained from all lines used to compute the LSD profile. The accuracy of this technique for determining high-precision longitudinal field measurements has been demonstrated by, e.g. Wade et al. (2000b), Donati et al. (2001) and Shorlin et al. (2002).

The longitudinal field measurements of $\eta$ Aql represent some of the best ever obtained, with $1\sigma$ formal uncertainties of 3-5 G. The largest longitudinal field measured has  $\ensuremath{B_{\rm z}} =14\pm 5$ G, and the most significant measurement is $\ensuremath{B_{\rm z}} =13\pm 4$ G, (3.3 $\sigma$). These measurements (in particular the validity of the error bar) will be discussed further in Sect. 6. The  measurements of $\alpha ^2$ CVn are also very precise, have uncertainties in the range 35-70 G, and are completely consistent with the field variation of this star as determined by Wade et al. (2000b, see Fig. 2).

The inferred values of the longitudinal field of $\eta$ Aql are reported in Table 1.

Figure 2: Longitudinal magnetic field variation of the prototypical magnetic Ap star $\alpha ^2$ CVn, observed as a magnetic standard within the context of this study. Shown are observations reported by Wade et al. (2000b, open circles), and those obtained within the context of this study and an analogous study of RR Lyr (Chadid et al. 2002, filled circles). The solid curve is a least-squares fit.