A GROUND-BASED ALBEDO UPPER LIMIT FOR HD 189733b FROM POLARIMETRY

In this section, we briefly describe POLISH2 (POlarimeter at Lick for Inclination Studies of Hot jupiters 2), and we refer the reader to Wiktorowicz & Nofi (2015) for further inquiry. Rather than using a conventional half waveplate to convert incident polarization into an intensity modulation that may be measured by conventional imaging detectors, POLISH2 employs two photoelastic modulators (PEMs). The stress birefringent property of the fused silica PEMs preferentially retards the electric field component oriented ±45° from the stress axis. When coupled with a Wollaston prism, the PEMs impart a nearly sinusoidal intensity modulation on the photomultiplier tube detectors. The resonant frequencies of the PEMs are 40 and 50 kHz, which causes linear and circular polarization to modulate at linear combinations of these frequencies. After high speed digitization of the detector outputs, software demodulation simultaneously measures Stokes q = Q/I, u = U/I (fractional linear polarization), and v = V/I (fractional circular polarization), which describe the fractional polarization of incident light.

The SNR of each measured Stokes parameter is proportional to the modulation efficiency of the PEMs. For example, a modulation efficiency of 100% imparts the theoretical maximum amplitude of photometric modulation onto the detectors, while a modulation efficiency of 50% reduces the amplitude of modulation to half this value. Mueller matrix modeling of POLISH2 shows that the modulation efficiencies of Stokes q, u, and v are 86%, 81%, and 56%, respectively. Therefore, it is expected that uncertainties in circular polarization measurements will be ∼50% larger than in linear polarization.

While spatially resolved circular polarization of Jupiter has been detected at the ∼100 ppm level from multiple scattering (Kemp & Swedlund 1971; Kemp & Wolstencroft 1971; Michalsky & Stokes 1974), the sign of circular polarization is observed to reverse between northern and southern hemispheres. For exoplanets, it is expected that the dilution of circular polarization by direct light from the host star, as well as from integrating over the planetary disk, will cause exoplanet circular polarization to be more difficult to measure than linear polarization.

The two major improvements in the use of PEMs over waveplates are as follows: (1) simultaneous Stokes q and u measurements obviate systematic effects from waveplate rotation (heterogeneity in retardance across the optic itself), from atmospheric or astrophysical changes on short timescales, and potentially doubles the throughput of the measurement; and (2) high speed modulation enables photon-limited sensitivity via $1/\sqrt{{n}_{\mathrm{modulations}}},$ where "modulation" is defined by sequential measurement of Stokes ±q, for example. In contrast to PEMs, with a modulation timescale of order 10 μs, typical waveplate modulations have a duration of order minutes to minimize overhead due to waveplate rotation.

To directly detect scattered light from spatially unresolvable exoplanets, the ability to measure nightly changes in the polarization of starlight at the 10 ppm level or below is required. In this regime, accuracy (the ability to calibrate non-astrophysical change to some level) is far more important than sensitivity (the ability to measure a change to that level). While photon noise defines the fundamental limit to the accuracy of a measurement, instrumental and atmospheric systematic effects tend to dominate for many exoplanet investigations. Therefore, it is crucial that these systematic effects be removed or calibrated to the 1 ppm level to enable confident detection of scattered light from exoplanets at the 10 ppm level. We identify and partially correct for two dominant systematics: polarized sky foreground and non-zero telescope polarization.

As the POLISH2 field of view is $\sim 15^{\prime\prime}$ in diameter, a detectable quantity of sky photons is present even in the B band. This foreground tends to be polarized, especially when the moon lies ∼90° from the target. To mitigate this systematic effect, we perform one 30-s integration on a sky field $30^{\prime\prime}$ N of the target for every two integrations on-target. Therefore, one-third of all telescope time is devoted to monitoring sky polarization with a cadence of one minute.

At the focus of a telescope, detectable polarization is measured even when observing an unpolarized star. This is because reflectivity variations across the telescope mirrors cause a discrepancy in the intensities of pairs of light rays with equal but opposite incidence angles, which causes the polarization from certain patches of the mirror to dominate. Even at Cassegrain focus of a variety of telescopes, this so-called "telescope polarization" is found by many authors to be 10–100 ppm in amplitude (Hough et al. 2006; Wiktorowicz & Matthews 2008; Lucas et al. 2009; Wiktorowicz 2009; Berdyugina et al. 2011; Wiktorowicz & Nofi 2015).

The standard approach for measuring telescope polarization is to observe a handful of nearly unpolarized calibrator stars. In practice, however, the details of this process may impart additional systematics that may be misinterpreted as arising from the science target. For instance, Lucas et al. (2009) observe that 12 of their 75 telescope polarization measurements are inconsistent with the mean value of telescope polarization at the >3σ level, and they consequently reject these measurements. However, the probability of this occurrence in a normally distributed population is ∼10⁻⁹. Lucas et al. (2009) therefore note that the measurements must have a non-Gaussian distribution and advocate for repeated measurements if possible. We concur with this request, and we also caution that rejecting calibrator measurements that lie >3σ from the mean will introduce a bias into the science results, because measurements may not be drawn from a Gaussian distribution.

In addition, we advocate that the observing cadence on calibrator stars follow that of the science targets, lest biases be introduced. Wiktorowicz (2009) observes the same calibrator star during each night in the study, but the use of only one calibrator limits the accuracy of the zero point of polarization measurements. However, accurate measurement of the time-averaged polarization of an exoplanet host star is necessary only to describe the state of the intervening ISM dust particles and has no relevance to the scattering properties of an exoplanet atmosphere. Wiktorowicz (2009) found the square root of the weighted variance of nightly telescope polarization observations to be ${\sigma }_{q}=7.5\;{\rm{ppm}}$ and ${\sigma }_{u}=5.3\;{\rm{ppm}}$ (Wiktorowicz 2009, Table 4), while the upper limit to exoplanet variation was found to be 79 ppm. Therefore, observation of calibrator and science targets with the same nightly cadence enabled calibrations to be an order of magnitude more accurate than science observations. Observing the same five calibrator stars nearly every night, Lucas et al. (2009) expand this approach to accurately measure telescope polarization each night. While Berdyugina et al. (2011) observe 26 calibrator stars over 15 nights, the dates of observation for each star are not presented. Indeed, the long duration of each calibrator star observation, one to two hours, implies that very few calibrators were observed on successive nights. In an investigation where systematic effects are severe, it is imperative that the same control and science targets be observed during each night.

Therefore, we observe 3 to 12 telescope polarization calibrator stars during each night of a given run, and the nightly target list is reproduced for each night of the run. Depending on the time of year, the makeup of this list will of course vary based on observability. These stars are identified from both the PlanetPol group (Hough et al. 2006; Lucas et al. 2009; Bailey et al. 2010) and our own unpublished survey of nearby, bright stars with POLISH2 at the Lick Observatory Nickel 1-m telescope. Given that interstellar polarization scales roughly linearly with heliocentric distance (e.g., Hall 1949; Hiltner 1949; Fosalba et al. 2002), nearly unpolarized stars are expected to lie in the vicinity of the Sun. Due to this proximity, such suitable stars tend to be quite bright. Therefore, sufficient measurement sensitivity is typically obtained after nine minutes for each telescope polarization calibrator star.

Strongly polarized stars ($p=\sqrt{{q}^{2}+{u}^{2}}\sim 1\%$) represent the second type of calibrator. These stars are typically observed to determine the rotational position of the instrument; that is, the relationship between instrumental Stokes (q', u') and celestial (q, u). We typically observe the same two strongly polarized stars for nine minutes during each night of a run. Wiktorowicz & Nofi (2015) discuss absolute calibration of POLISH2 via sequential, laboratory injection of 100% Stokes q, u, and v. POLISH2 has recently demonstrated the ability to measure linear polarimetric variations at the 10⁻⁵ level over a single, 3.8-hr observation on an object only two magnitudes brighter than HD 189733. Spatial inhomogeneity in the surface albedo of asteroid (4) Vesta causes rotational modulation in its disk-integrated linear polarization. The amplitude of modulation varies with Sun-asteroid-observer phase angle, and POLISH2 has measured a peak-to-peak value of 294 ± 35 ppm (Wiktorowicz & Nofi 2015). Given the comparable amplitudes of polarization variations between (4) Vesta and HD 189733, as well as the comparable quantity of photons detected per period ($\propto {10}^{-0.4\;m}\times T$ for apparent magnitude m and period T), these observations show that POLISH2 is uniquely positioned to measure the polarimetric variability of spatially unresolvable exoplanet systems.

The 50 nights of HD 189733 observations were obtained over nine observing runs between 2011 August and 2014 July. An average of eight nearly unpolarized calibrator stars were observed during each of these runs for a total of 20 individual stars (Table 1). A total of 367 observations of such calibrator stars were performed, which represents a wealth of information on the B-band linear/circular polarization and stability of nearby stars. Indeed, high accuracy observations of nearby stars are essential in understanding the galactic magnetic field in the solar vicinity (Bailey et al. 2010; Frisch et al. 2010, 2012, 2015). We determine telescope polarization on a nightly basis from the weighted mean value of each Stokes parameter (Table 2). Figure 1 compares linear polarization of these stars in a q–u diagram, where measurements are subtracted by telescope polarization, while linear and circular polarization are compared in Figure 2. Both a histogram and the cumulative distribution function of values of p are shown in Figure 3. The square root of the weighted variances of all 367 observations are ${\sigma }_{q}=22,\;$ ${\sigma }_{u}=12,$ and ${\sigma }_{v}=32\;{\rm{ppm}}.$ We find that 75% of our B-band measurements have p < 31 ppm, while this value is 20 ppm for the optical red (590–1000 nm) observations of Bailey et al. (2010). Thus, our measurements are broadly consistent with PlanetPol observations of nearby stars (Bailey et al. 2010). We also present a large sample of B-band circular polarimetric observations of nearby stars, where 75% of measurements have $| v| \lt 47\;{\rm{ppm}}$ (Figure 4). This is consistent with the smaller subset of observations published in Wiktorowicz & Nofi (2015).

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Figure 5 shows nightly Lick 3-m/POLISH2 observations in the B band, while Figure 6 presents binned Stokes q, u, and p observations of the HD 189733 system, both from this study (Lick 3-m/POLISH2, B band) and from Wiktorowicz (2009; Palomar 5-m/POLISH, unfiltered). Both POLISH and POLISH2 measurements are subtracted by their time averages to highlight the time-variable component, and degree of linear polarization Δp is recalculated from Δq and Δu. Binned values of Δp are calculated from binned values of Δq and Δu. Also shown in Figure 6 are Rayleigh scattering models with geometric albedos of 0.231, 0.434, and 0.604. A detailed description of these models is given in Kopparla et al. (2015). In addition, we show the model fitting the reported detection from Berdyugina et al. (2011), which represents a geometric albedo of 0.61 ± 0.12 and is calculated using only single scattering. While it is clear that our model cannot be varied to explain our observations, the POLISH and POLISH2 data are intriguingly consistent. In particular, a high degree of linear polarization is observed near orbital phase 0.1, which is inconsistent with our Rayleigh scattering model. We note that while the data sets of Berdyugina et al. (2008; KVA 0.6-m/DIPol) and Berdyugina et al. (2011; NOT 2.5-m/TurPol) appear similar, and are also taken with different instrument and telescope combinations, this POLISH2 study dramatically improves the combination of telescope aperture and nights of observation on HD 189733.

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On average, each bin is composed of repeated observations on five nights at the same orbital phase (Figure 6), and the bin uncertainty is given by either the square root of the weighted variance or the standard error of the measurements. This choice is determined by whether the measurements in each bin are inconsistent with each other at the 3σ level from a χ² test. This tests whether measurements in each bin are drawn from a normally distributed population, because standard error decreases as $1/\sqrt{n}$ only for normally distributed measurements. Indeed, it can be seen that the uncertainties near orbital phase 0.5 and 0.85 are large due to relative disagreement between successive observations at these orbital phases (Figure 5). Therefore, polarization accuracy is hampered by systematic effects arising from telescope polarization variations. While is it possible that the significant variations between bins in Figure 6 arise from complicated scattering mechanisms in the HD 189733 system, we conservatively assume that these are caused by the known presence of telescope systematic effects.

Even if all variations are solely caused by telescope polarization, the amplitude of the scatter in degree of polarization p is only ∼50 ppm (Figure 6). This is clearly an interesting accuracy regime, as the scattered light amplitude of HD 189733b has been measured to ∼100 ppm from HST STIS (E13) in total intensity. To determine an upper limit to the polarimetric amplitude due to the planet, and thereby constrain the albedo of the exoplanet, we perform bootstrap resampling of our measurements. For the binned Stokes q and u data sets in Figure 6, we perform 10⁴ iterations of 10⁴ resampled data sets with the following procedure: (1) the orbital phase value of each bin is randomly assigned from the list of 11 bins, (2) each observable quantity (Stokes q and u) is resampled from its nightly mean and uncertainty, and the uncertainty is retained, and (3) degree of polarization p is calculated for each bin from the generalized MAS estimator (Section 3).

Since bootstrap resampling randomly assigns an orbital phase value for each synthetic data point, but exoplanet polarization is strongly correlated with orbital phase, bootstrap resampling cannot directly test whether observational data are consistent with scattering models. Indeed, bootstrap resampling is an investigation into the noise properties of the observational data, and it allows the maximum polarization amplitude of those models to be constrained.

Measurement of a linear polarization vector requires two sets of observations in an orthogonal basis. Stokes q represents the fractional polarization component in the north-south (+q) and east-west (−q) directions in the plane of the sky, while Stokes u represents the component in the northeast–southwest (+u) and northwest–southeast (−u) directions. If the orientation of the exoplanet orbital plane seen by the observer were known a priori (the longitude of the ascending node of the system, Ω), the observer may simply rotate the telescope or instrument about the optical axis such that instrumental Stokes q' (i.e., "up" on the detector as opposed to the direction toward celestial north) were aligned with the orbital plane. In the case that any interstellar polarization, inherent polarization of the host star, and polarization from an exozodiacal disk were all zero, Stokes $u^{\prime}$ would then be constant and Stokes $q^{\prime}$ would equal the degree of linear polarization $p^{\prime} .$ However, since Ω is unknown, the astrophysically interesting quantity p must be calculated from the variations in Stokes q and u in order to present a conclusive detection of scattered light from an exoplanet.

For this investigation, with a goal of albedo constraint as opposed to detection, we perform a χ² analysis between the values of p from bootstrap resampling and the modeled polarization phase curves. While it is possible to perform a bootstrap analysis on Stokes q and u data separately, the uncertain geometry of the system in the plane of the sky (Ω) introduces an additional free parameter that would significantly complicate the analysis. In the extreme case, where Ω = 0°, 45°, 90°, or 135°, no variations would be detected in one of Stokes q or u. One might then place an upper limit to the polarization of the planet that was erroneously low. Performing the bootstrap analysis on p obviates such concerns, because p is invariant through rotation ($p=p^{\prime}$). However, care must be taken in this analysis because the purpose is to reject model amplitudes rather than the models themselves. This is because our observations appear to be limited by systematic effects and prohibit conclusive detection at this time.

Even with systematic effects present, however, large-amplitude polarization variations from the exoplanet may be rejected with confidence by synthetic data whose polarization values are lower than model values. This is because the χ² residuals between low polarization data and the models directly test whether high-amplitude variations would have been detectable. While synthetic data having polarization higher than the model values also generate large χ² residuals, these data clearly should not be able to reject high-amplitude polarization variations from the exoplanet. Thus, a simple χ² test between all synthetic data and models cannot be used to constrain the polarization amplitude due to the planet. Instead, since synthetic data with polarization higher than model values cannot constrain the amplitude of planet polarization, we remove the contribution of these data to the χ² residuals. Rather than simply deleting these data, which would reduce the degrees of freedom in the χ² test and bias the result toward lower polarization amplitudes, we set their χ² residuals to zero but retain their degrees of freedom. Thus, our bootstrap analysis provides a conservative upper limit to the polarization amplitude of the planet.

The albedo constraint from a single bootstrap iteration out of 10⁴, which itself contains 10⁴ resampled data sets, is shown in Figure 7. Figure 8 shows the distribution of these χ² analyses, where Rayleigh scattering models with geometric albedos larger than 0.40 may be rejected by our observations with 99.7% confidence. By interpolating the amplitude of modeled degree of polarization versus geometric albedo, we find a 99.7% confidence upper limit to the linear polarimetric amplitude of HD 189733b to be 60 ppm in the B band. Table 4 shows that both the time averaged circular polarization, and any variability phase-locked to the orbital period of the planet, are consistent with zero.

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