Properties of ZDI and testing the forward module of ZDoTS

ZDI attempts to recover the three different magnetic field components (radial, azimuthal and meridional) in order to map the flux density for each surface magnetic field vector. At any given phase we are only able to record the line-of-sight component of the magnetic field (B). The key to ZDI is that by sampling many rotational phases we are able to

observe how Stokes V signatures, produced by distinct locations on the surface of the star, change with time. We illustrate this in Fig. 5.5.

Unfortunately ZDI is only weakly sensitive to meridional field on stars with mid to high inclination angles. Given that the rotational broadening used in Doppler imaging strongly favours observation of stars at high inclination angles (vobs =vrotsini), maps of meridional field have rarely been recovered. The field that we do recover is mostly cross- talk with the radial and occasionally the azimuthal field components. For a more detailed discussion of the effects of field orientation on Stokes V signatures see Donati & Brown (1997) and Hussain (1999). We also discuss meridional field on HD 155555 in Chapter 6. For now, and during the tests of ZDoTS in the following sections, we consider only radial and azimuthal field.

In Fig. 5.5 we show a small simulation which uses our new ZDoTS code to simulate magnetic regions on the stellar surface and then generate synthetic spectra. The purpose of this is to compare the behaviour of the Stokes V signatures as magnetic regions of radial or azimuthal field rotate into and out of our line-of-sight. Three identical magnetic spots, with peak fluxes of 1000 G and radii of 10◦_{, are placed on to the surface of a star of} radius 1 R and rotating withvsini=91km s−1 andi= 90◦(therefore simulating the K0 dwarf AB Dor except for the increased inclination angle). All three spots are located at the stellar equator, one is on the stellar meridian (position B) while the other two are at longitudes of _±60◦ _{either side of this (positions A and C). The magnetic spots can either} be thought of as three unique regions seen on the stellar surface at a particular rotational phase or equivalently as a single magnetic region observed at three different phases.

First we consider that the magnetic regions shown in Fig. 5.5 are composed of radial field. In this case the amplitude of the Stokes V signature increases as the spot rotates further onto the stellar disc. The amplitude of modulation of Stokes V signatures is simply Vrad ∝cos(∆φ), where ∆φis the angular distance away from the meridian. At position B in Fig. 5.5 the magnetic spot is on the meridian (∆φ=0◦_{) so the radial field} lines are directly pointing into our line-of-sight and we observe the maximum amplitude of the Stokes V signature. At positions A and C (∆φ=60◦_{) and so the amplitude of the} Stokes V signature should be half that observed at B. A close examination of Fig. 5.5 reveals that the amplitude of the signatures at A and C are in fact less than half that at B. This is because the effect of limb darkening must also be taken into account.

Figure 5.5: An illustration of the basic principles of ZDI. A single magnetic region is shown at three rotational phases: A, B and C on a star of vsini=91km s−1 _{and i = 90}◦_{. The} observed amplitude of the Stokes V feature is dependent upon the nature of the field, the line-of-sight component of the magnetic fieldBand the local limb darkening (see text). At position B the azimuthal field should go to zero as the region crosses the stellar meridian. This is because the horizontally orientated field has no component in our line-of-sight. In the diagram however there is a small residual signature which arises purely from the finite

We now consider magnetic regions that are instead composed entirely of azimuthal field. Fig. 5.5 shows that quite different behaviour is seen to that of the radial field. The Stokes V signatures appear to have the largest amplitude away from the meridian and virtually no signature on the meridian. Furthermore, the Stokes V signature ‘reverses’ when it crosses the meridian (the shape is different at A and C). This is because the sign of the field changes when it crosses the meridian: field that was at A pointing towards the observer is then pointing away at C. The amplitude modulation of the azimuthal field is Vazi ∝ sin(∆φ); therefore when on the meridian at position B, Vazi will tend to zero. Fig. 5.5 shows some residual flux which is due to the fact that the magnetic region has a finite size. In theory the maximum amplitude would be observed when at a longitude 90◦ away from the stellar meridian when the azimuthal field is directly into our line-of-sight. When the effects of limb darkening and foreshortening are taken into account, ZDI is most sensitive to azimuthal field approximately 70◦ _{(or 0.2 in phase) away from the stellar} meridian. This is of course dependent upon the nature of the limb darkening and so the temperature of the star.

In summary, it is the characteristic behaviour described above and illustrated in Fig. 5.5 that allows ZDI to distinguish between radial and azimuthal field. This simulation just illustrates part of the properties of ZDI outlined in detail by previous authors (e.g. Donati & Brown 1997). However it has also been useful to show that ZDoTS is capable of performing the forward operation of generating Stokes V spectra from a surface magnetic field distribution. We now focus on tests of the maximum entropy reconstruction (inverse operation) of ZDoTS to recover magnetic maps from synthetic spectra.