Polarization measurements of the polluted white dwarf G29-38

We have observed G29-38 (ZZ Psc, WD 2326+049) with (two copies of) the HIPPI-2 polarimeter (Bailey et al., 2020b) during two observing runs. The first run was made at Mauna Kea in Hawaii with the Alt-Az 8.1-m Gemini North telescope during early 2018 July (denoted GN). The second run was made at Siding Spring Observatory in Australia with the equatorial 3.9-m Anglo-Australian Telescope (AAT) during 2018 August. HIPPI-2 is designed for an f/16 focus appropriate for the Gemini North observations. For the AAT run, the f/8 Cassegrain focus was used with the addition of a $\times 2$ negative achromatic (Barlow) lens. In combination with the selected instrument apertures (4 mm GN, 3.6 mm AAT) this gives on-sky aperture sizes of 6.4 arcsec (GN) and 11.9 arcsec (AAT).

HIPPI-2 is a high-precision optical linear aperture polarimeter based on the proven HIPPI (Bailey et al., 2015) and Mini-HIPPI (Bailey et al., 2017) instruments, and is described in full in Bailey et al. (2020b). It has recently been used to study polarization in the binary star system Spica (Bailey et al., 2019), the rapidly rotating star $\alpha$ Oph (Bailey et al., 2020a), and the red supergiant Betelgeuse (Cotton et al., 2020). To summarise its performance, in the SDSS $g^{\prime}$ band, in which our observations were made, the instrument has demonstrated precision of better than 4 parts-per-million (ppm). It achieves this through two stages of modulation. The first stage is rapid (500 Hz) and electrically driven in order to beat seeing noise; it is accomplished by a commercial Ferro-electric Liquid Crystal (FLC) modulator. The second stage of modulation is slower, and involves utilising an instrument rotator to achieve four instrument position angles (PA) of 0^∘, 45^∘, 90^∘ and 135^∘ per observation. The redundant angles (90^∘ and 135^∘) are used to cancel out instrumental effects. For the AAT run the FLC used was the MS series polarization rotator from Boulder Non-Linear Systems (BNS). At Gemini North we used an FLC from Meadowlark Optics (ML) with a design wavelength of 500 nm. The performance of each unit has been characterised by both laboratory and on-sky measurements, and is described in detail in Bailey et al. (2020b).

HIPPI-2 uses a Wollaston prism as an analyser, whence the light is fed into twin compact photomultiplier tubes (PMTs) used as detectors. For these observations we utilised two different pairs of Hamamatsu H10720-210 modules which have ultrabialkali photocathodes (Nakamura et al., 2010) providing a quantum efficiency of 43 per cent at 400 nm. In combination with the atmosphere, telescope and instrument optics this results in a mean effective wavelength for G29-38 observations of 451.7 nm (AAT) and 463.6 nm (Gemini North), as calculated by a bandpass model – the difference is mainly down to the reflectance profile of Gemini North’s silver coated mirror compared to the AAT’s aluminium mirror. The bandpass model also takes account of modulation efficiency of the instrument, the inverse of which must be multiplied by the raw Stokes parameter measurements to determine the true polarization. The mean modulation efficiency was significantly impacted by the performance characteristics of the different modulators; for AAT observations it was 63.5 per cent, on Gemini the figure was 93.4 per cent. The AAT run was the final one for the BNS modulator, the performance of which drifted over time, accounting for the comparatively low modulation efficiency. The bandpass model requires a source spectrum. For this purpose we chose to use the synthetic white dwarf spectra of Koester (2010). Koester et al. (2009) gives $T_{\rm eff}=$11 485 K and $\log(g)=$8.071 for G29-38; the two nearest models have $T_{\rm eff}=$11 000 K and 12 000 K. We therefore ran the bandpass calculations with both models separately and took the mean.

The telescope polarization (TP) and position angle (PA) calibrations rely respectively on the observation of low and high polarization standard stars. These observations and the associated calibrations for the two runs have both been reported previously (Bailey et al., 2020b). On the AAT the TP was 13.6 $\pm$ 1.1 ppm at a position angle of 80.9 $\pm$ 2.2^∘. On an Alt-Az telescope the TP varies with parallactic angle for any given Stokes parameter. The TP at Gemini North is very large (around 2,050 ppm in the SDSS $g^{\prime}$ band), and highly wavelength dependant, meaning that it also varies significantly with airmass. A model describing the TP at Gemini North was developed, and is presented elsewhere (Bailey et al., 2020b). The uncertainties surrounding the TP limit the achievable precision (in $g^{\prime}$) at Gemini North to 24.9 ppm.

It is normal procedure with HIPPI-2 to make a measurement of the sky at each PA immediately before or after each measurement of the object. At Gemini North this was done in the normal way with alternating trailing and leading skies. On the AAT for observations of G29-38 we took skies that bracketed every measurement and averaged them before making the subtraction. This precaution against changing sky conditions was taken because, at $m_{B}=13.17$ mag, G29-38 is one of the faintest objects so far observed with a HIPPI-class polarimeter.

The individual HIPPI-2 observations are given in Table 1 and presented in the form of a Q-U diagram in Figure 1, along with nightly means, run means, and the mean of both runs¹¹1We use $Q$ and $U$ to denote the linear polarization Stokes parameters, and $q$ and $u$ the normalised forms, i.e. $q=Q/I$, $u=U/I$.. In all cases the means are error weighted. The errors on the individual observations are fairly large, but these appear fairly consistent. This can be seen most clearly in the left-hand panel of Figure 1, where only one of the individual Gemini North observations deviates by more than 2$\sigma$ from the mean. More meaningfully, the nightly means are in good agreement within the formal error, as are the run means. The latter, in particular, gives confidence in the result, and implies stability in the system polarization on the scale of days to weeks. Combining all the data gives $p=$274.3$\pm$31.8 ppm at a PA of 92.5 $\pm$ 3.3^∘, a clear detection at a significance at 8.6$\sigma$.

Subsequent to the data acquisition it was decided to interrogate the data for evidence of effects resulting from pulsation. As a first step, variability was investigated by calculating the moments of the AAT and GN runs using the methodology of Brooks et al. (1994). The results are presented in Table 2. In each instance the calculated kurtosis and skewness (defined so 3 is normal skewness) are not inconsistent with the null hypothesis (of no variability) to a significant level (i.e. 95 per cent). The error variance, which is a measure of whether the standard deviation in the data is consistent with the associated errors, also indicates no variability for the AAT data. This is not surpising as the pulsations have periods much shorter than the length of typical AAT observations, so could be expected to average out, and our errors are of similar magnitude to the mean measurement. However, with the smaller errors associated with the GN data and slightly shorter dwell times, significant non-zero error variances are recorded, which warrants further investigation.

G29-38 has prominent pulsation periods around 800 to 900 s. The total exposure for an observation is usually 1000 s at Gemini North or 1280 s at the AAT, however each observation consists of four measurements at different instrument PAs, with one quarter the exposure. A set of four measurements is necessary to properly remove any instrumental polarization. As a result of triggering overheads, the dwell time for each measurement is about 10 to 20 s longer, meaning that each measurement corresponds to roughly a third of a pulsation cycle. The measurements are separated by about 50 to 100 s owing to Sky measurements. The net effect is that observations at Gemini North and the AAT have dwell times of around 19 min and 25 min respectively (see Table 1).

HIPPI-2 is not photometrically calibrated, and rotating the instrument can produce small jumps in intensity as a result of manufacturing/fabrication tolerances and alignment effects, but intensity (Stokes $I$) is recorded along with Stokes $Q$ and $U$. HIPPI-2’s software retains a record of every 2 s integration (which for convenience are grouped together in sets of 10 – called a repeat in Bailey et al., 2015). While the arrangement of the observations is far from ideal for this purpose, it is possible to investigate pulsation effects using the intensity and polarization recorded intra-measurement.

Each measurement has been broken down into its integrations, and a fifth-order polynomial fit to the intensity in that sequence to take out obvious dips from thin clouds or short-duration seeing variability. The fit minimum was then set to zero and the polynomial normalised. This gave an intensity range between 0 and 1 for each integration in each measurement which could be mapped to the polarization measurements. To allow for the calculation of statistical errors in the polarization Stokes parameters the integrations were grouped into their sets of 10. This normalised intensity scale was averaged over an integration set, and then that value used to make a cut of half the data, selecting either the high intensity half (Upper, $I_{Poly}>0.5$), middle (Middle, $0.25>I_{Poly}>0.75$) or the low intensity half (Lower, $I_{Poly}<0.5$). As an example for a single measurement, Figure 2 shows the Stokes $I$ parameters calculated in each channel in the middle panel in units of detector voltage; the polynomial to fit the average of the detectors is shown in the upper panel, and the corresponding (instrumental) Stokes $q$ determinations in the lower panel. The vertical grey lines divide up the integration sets. All four PA measurements of the same cut (Upper/Middle/Lower) are then used to calculate polarization values in the standard way. Because each measurement only corresponds to about a third of a pulsation period the upper and lower cuts of the data do not correspond directly to magnitude, but should be indicative of any polarigenic mechanism proportional to magnitude changes.

There are more significant differences between the various cuts of the data given in Table 3 – and depicted graphically in Figure 3 – than there are in the nightly means given in Table 1. This suggests that there are polarigenic effects resulting from pulsation. Yet, the evidence is inconsistent. The difference between the Upper and Lower cuts of the AAT data are striking in both $q$ and $u$ amounting to 100s of ppm significant to around 2$\sigma$, however the trends are not mirrored in the Gemini data. In the Gemini data it is the Middle and Upper cuts that are most different in $q$, whilst the trend in $u$ is reversed and is not significant.

If one assumes that pulsation effects are present, then the difference between the two runs could either be methodological or phenomenological in origin. The most obvious examples of the latter relate to the pulsations themselves. There are a number of pulsation modes in G29-38 with similar frequencies and beating between those frequencies could result in less prominent effects in one run than another. Although difficult to quantify, the pulsations do appear more prominent in the AAT data. Secondly, G29-38 is an unstable pulsator with modes that come and go, often on timescales of a few months (Kleinman et al., 1998) and so the star may be behaving differently in one run from the other. Other intrinsic changes in the system could affect the polarization, we examine these in Section 4.

Alternatively, the difference between the two runs could be due to how the data were taken. The shorter duration measurements in the GN run could be inadequately sampling the pulsation cycles for our normalised intensity assignment to be effective – the smaller the fraction of the pulsation sampled at any given PA, the less appropriate splitting based on intensity normalised over the exposure is. An observing program designed to measure distinctly the polarization of the pulsation peaks and troughs is needed to resolve the problem.

The measured polarization of G29-38 is a combination of intrinsic and interstellar polarization. G29-38 is within the Local Hot Bubble, very near to the Sun, so its interstellar polarization is expected to be low. The Local Hot Bubble is a region of space carved out by ancient supernovae with a very low gas and dust density that extends from 75 to 150 pc from the Sun (Liu et al., 2016). The interstellar medium within this region is not very polarizing, with typical values being around 0.2 to 2 ppm/pc (Bailey et al., 2010; Cotton et al., 2016a), which is an order of magnitude less than the region beyond the Local Hot Bubble (Behr, 1959). Nonetheless, in consideration of the small polarization magnitude measured for G29-38 in Section 2.1, the interstellar component may be significant, and here we make an attempt to measure and subtract it.

A common way of gauging the magnitude and orientation of interstellar polarization is to observe nearby intrinsically unpolarized control stars (Clarke, 2010). We have previously found stars with spectral types ranging from A to early K to be the least intrinsically polarized (Cotton et al., 2016a, b). Such stars are a good probe of the nearby interstellar medium so long as particularly active stars (Cotton et al., 2017, 2019a) and those that host prominent debris discs (Cotton et al., 2017) are avoided. The dust distribution within the Local Hot Bubble is patchy, making it difficult to measure the true level of interstellar polarization with a control that is not a wide companion. However the ISM is not as patchy within $\sim$25 pc, and we have previously devised a model for the nearest region of space (Cotton et al., 2017). The model relies on a weighted average of nearby control stars with angular separations of less than 35^∘ to determine the position angle, and simple piece-wise linear relationships that vary with Galactic latitude for the magnitude of interstellar polarization. For stars with Galactic co-ordinates $b<+30\degr$, the magnitude of interstellar polarization is given as

which for G29-38 at $d=$ 17.5 pc gives $p_{i}=16.6~{}\pm~{}2.6$ ppm.

A number of interstellar control stars are to be found in the Interstellar List in the appendix of Cotton et al. (2017). There are two stars in the Interstellar List that are just within 35^∘ of G29-38. So to be able to apply the method from Cotton et al. (2017) we have supplemented the Interstellar List with two new observations reported here in Table 4 for the first time. The interstellar control stars were observed with HIPPI-2 in the standard way, using the SDSS $g^{\prime}$ filter. The first of these control stars, HD 216385 was observed during the same AAT run as G29-38, and the same TP and PA calibrations apply. The second star was observed some time later in 2019 December at the AAT (denoted AAT-2). During this later run the TP was 20.1 $\pm$ 0.7 ppm at a PA of 104.8 $\pm$ 1.0^∘ determined from observations of HD 2151, 2$\times$ HD 10700 and 6$\times$ HD 48915; the PA was calibrated from observations of HD 84810 and HD 80558 with a standard deviation of 0.2^∘.

In nearby space it is often difficult to find flawless calibrator stars, this is the case here. Both HD 216385 and HD 222368 have spectral types associated with low intrinsic polarization, but HD 222368 is known to have both hot and cold infrared excesses. The (hot) excess in the K-band is 1.6 per cent (Nuñez et al., 2017), however we know that such hot dust is likely made up of nanoscale grains and not very polarizing (Marshall et al., 2016; Kirchschlager et al., 2018). The cold component has a fractional excess of just 1.1$\times 10^{-6}$ (Sibthorpe et al., 2018), which would only result in a polarization greater than 1 ppm should the disc have an asymmetric geometry. Thus we conclude that HD 222368 is a good interstellar calibrator.

In Figure 4 it can be seen that the nearest two stars to G29-38 have a similar position angle to each other; this bodes well for the reliability of the interstellar subtraction mechanism. According to Cotton et al. (2017) the position angle is then found by weighting the four stars within 35^∘ as

where $s_{a}$ is the separation to G29-38 in degrees. This gives ${\rm PA}_{i}=138.5^{\circ}$; which in turn gives $q_{i}=2.0~{}\pm~{}1.6$ ppm and $u_{i}=-16.5~{}\pm~{}1.6$ ppm.

Subtracting this interstellar polarization from that measured for G29-38 gives us a determination for its intrinsic component as $q_{\star}=-275.2\pm 31.8$ ppm and $u_{\star}=-7.8\pm 32.0$ ppm or alternatively $p_{\star}=275.3\pm 31.9$ ppm (hereafter referred to as $P_{\rm obs}$) at ${\rm PA}_{\star}=90.8~{}\pm~{}3.8^{\circ}$.

Ignoring for the moment geometrical effects, and other details (such as dust properties etc.), the simple overview of interpreting the observed polarization in terms of scattering from circumstellar dust is as follows:

1. 1.

   From the ratio $f$ of infrared flux $L_{\rm IR}$ to stellar flux $L_{\star}$ (Farihi et al., 2014),

   $$f=L_{\rm IR}/L_{\star}=0.039,$$

   (3)

   we know that the fraction of light from the white dwarf that is intercepted by the dust is $f^{\prime}=f/(1-A_{B})$, where $A_{B}$ is the Bond albedo – defined as the fraction of energy incident on the dust that is not absorbed by the dust²²2It is important to note that the word “albedo” is used in the literature to mean two different things – the Bond albedo, as defined here in the text, and the “geometric albedo”, or “backscattering albedo” which is the fraction of incoming radiation that is scattered back in the direction it came from. These two terms are often confused (see the discussion in the review article by Hughes et al., 2018).. We shall find that it is likely that $A_{B}$ is small and hence that $f$ does not differ greatly from $f^{\prime}$.

2. 2.

   We conclude that a fraction $f^{\prime}$ of all photons of a given frequency, which we shall call “optical” (e.g. SDSS $g^{\prime}$ band) strike the dust. Of these photons, a fraction $1-A_{B}$ are absorbed, and a fraction $A_{B}$ are scattered. Therefore the fraction of scattered star light is $A_{B}f^{\prime}$.

3. 3.

   If the typical polarization of the scattered light is $P_{\rm scat}$ then the observed polarization in the “optical” is

   $$P_{\rm obs}=P_{\rm scat}\times A_{B}\times f/(1-A_{B}).$$

   (4)

4. 4.

   For G29-38 we have $f\approx 0.04$. We have found $P_{\rm obs}\approx 0.0003$. From the models below we find a typical value of around $P_{\rm scat}\approx 0.3$. From this we see that we expect the approximate value of the Bond albedo to be

   $$A_{B}\approx 0.025.$$

   (5)

Thus, to summarise, the most direct outcome of our analysis will be an estimate of the Bond albedo $A_{B}$. However, in order to obtain an accurate estimate we would require both knowledge of the scattering properties of the dust, and of its geometric distribution, both with respect to the star and with respect to the observer; none of this do we have.

Since we do not know what the geometry of the dust is, we shall have to make some plausible assumptions and make do with simple models. It is evident that we shall not be able to elucidate the details of the dust geometry. The main thing that we shall be able to do is to shed light on the dust Bond albedo, and to then compare that with dust elsewhere. Recall that we are interested in the scattering properties at optical wavelengths.

To make a model we shall need some handle on the properties of the dust. In particular we are interested in $A_{B}$, the scattering phase function $I(\theta)$ which denotes what fraction of the scattered light is scattered through an angle $\theta$, and the polarization function $P_{s}(\theta)$ which denotes the polarization fraction of the light scattered through an angle $\theta$.

A number of the dust scattering models in the literature apply to objects (standard debris discs) in which the geometry can be resolved. For this reason, many of the models are quite sophisticated, and use dust models which take into account a range of dust particle sizes and properties.

But in our case, although the dust around G29-38 is often called a “debris disc”, there is no evidence that the dust is even in the form of an annular disc. The dusty material is assumed to have come from some ‘planetesimal’ or ‘comet’ which formed part of an original debris disc and which has been scattered inwards somehow to get closer to the white dwarf.

Since we have no knowledge of the actual dust distribution, we consider below some simple ideas that are discussed in the literature (with some slight modifications). Reach et al. (2005, 2009) suggest some idealised models of where the dust around G29-38 might be situated, as well as its mineralogy, in order to fit the infrared spectrum (from 1 to 35 $\mu$m). In the absence of any other information, we use their dust distributions as a starting point.

For the dust, the only parameters we need are the H-G parameter $g$, $A_{B}$ and $P_{\rm max}$. Everything scales linearly with $A_{B}$ (provided $A_{B}\ll 1$) and $P_{\rm max}$. Thus the main variable of interest is $g$.

Speculatively, if the dust distribution is clumpy, the orbits will complicate an investigation of the pulsations, since the orbital periods may be similar to the pulsation cycles. While typically an asymmetric dust distribution yields a higher polarization than otherwise expected, the observed polarization will be lower if individual exposures are long enough to integrate over a substantial portion of an orbit. Uneven dust distributions are observed; for WD 1145+017 (Vanderburg et al., 2015), the transits are interpreted as dust orbiting on a timescale of 4.5 to 4.9 hours. This is on the long side for observed dust around white dwarfs. Most estimates of the innermost orbital period of the dust are derived by fitting a series of blackbody profiles to the infrared excess, which for G29-38 and the assumption of a disc morphology (von Hippel et al., 2007) yields a ring with inner temperature 1150 K and outer temperature 725 K. This in turn yields $R_{\rm dust}$ = 0.15 to 0.28 $R_{\astrosun}$.

Extending to other dust morphologies and dust properties, Reach et al. (2009) show that for a spherical dust shell the distance of the innermost dust is weakly dependent on the optical depth, with $R_{\rm dust}$ = 1.3 to 2.6 $R_{\astrosun}$. Taking the mass of the white dwarf as 0.6 $M_{\astrosun}$, the Keplerian orbital period is 752 to 1917 s for the physically thin disc case, which is indeed a similar time scale to the pulsation cycles. For the spherical dust shell, the Keplerian orbital period of the hottest dust is instead 5 to 15 hr. In the former case, if measured instantaneously, the polarization would be higher than the mean polarization over a whole innermost orbit, and phase curve sampling would have to be considered. For this it would be advantageous to better determine the range of orbital periods, which would ideally require knowledge of both the radial thickness of the dust band and the dust particle size distribution (in particular the minimum dust grain size); the latter is something that might be furnished by multi-band polarimetry.