SIMULTANEOUS LINEAR AND CIRCULAR OPTICAL POLARIMETRY OF ASTEROID (4) VESTA

POLISH2 is a dual photoelastic modulator (PEM) polarimeter mounted at the straight Cassegrain focus of the Lick Observatory Shane 3 m telescope. In the optical path downstream of the telescope secondary, two Hinds Instruments fused silica PEMs (I/FS40 and I/FS50) are located to minimize instrumental polarization. The compression/extension axis of the first PEM is oriented parallel to celestial north (+Q), while the second PEM is oriented northwest (−U). Acoustic resonance in the PEMs is induced, and their stress birefringence simultaneously modulates incident linear and circular polarization. Therefore, each Stokes parameter (Q, U, and V) is modulated at specific frequencies.

Downstream of the PEMs is a filter wheel with Bessell UBV filters. Instrumental polarization due to the filters is discussed later in this section. Next, a two-wedge Wollaston prism bifurcates the beam into opposite linear polarization states, where the polarization incident on the PEMs is AC coupled into an intensity modulation. Both modeling and telescope flat-field lamp observations show that polarization is modulated at sum and difference linear combinations of the PEM fundamental frequencies (f₁ = 40 kHz, f₂ = 50 kHz). Indeed, Stokes Q is modulated primarily at 100 kHz, U at 10 and 90 kHz, and V at 30, 50, and 130 kHz.

Both Wollaston output beams are focused at 15'' diameter field stops and field lenses and light is detected by Hamamatsu H10721-110 SEL photomultiplier tubes (PMTs). A pair of Stanford Research SR570 current preamplifiers converts the current output of the PMTs into a voltage that is digitized by a National Instruments PXIe-6124 analog to digital converter. Square wave voltage outputs from both PEM controllers are simultaneously digitized by this device, and the fundamental frequencies and phases of both PEMs are precisely measured in each 5 s data file. This allows frequent, precise knowledge of the modulation frequencies.

Since POLISH2 is an AC-coupled polarimeter, a lock-in amplifier approach is taken in software (Hough et al. 2006; Wiktorowicz & Matthews 2008). Briefly, both a sine and cosine are constructed for each modulation frequency, and the products of these sinusoids with mean-subtracted raw data from each PMT are calculated over 0.1 s data segments. The time average of each product is then taken, which removes the contribution of raw data varying at frequencies other than the modulation frequency. These time-averaged values, the so-called "in-phase" (X_i) and "quadrature" (Y_i) components of the power modulated at each frequency i, are the observable quantities:

$$\begin{eqnarray}&&{{X}_{i}}=2\langle {\rm sin} (2\pi {{f}_{i}}t)[S(t)-DC]\rangle ,\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{{Y}_{i}}=2\langle {\rm cos} (2\pi {{f}_{i}}t)[S(t)-DC]\rangle .\end{eqnarray} \tag{ 2 }$$

Here, S(t) and $DC=\langle S(t)\rangle$ are raw and time-averaged voltages from each PMT.

Wiktorowicz & Matthews (2008) show that polarized intensity (Stokes Q, U, or V) scales with ${{R}_{i}}=\sqrt{X_{i}^{2}+Y_{i}^{2}}$, while total intensity (Stokes I) scales with DC. Since fractional polarization is the quantity of astrophysical interest, we define normalized Stokes parameters $q\equiv Q/I$, $u\equiv U/I$, and $v\equiv V/I$ for clarity. Each of these is composed of ${{x}_{i}}\equiv {{X}_{i}}/DC$ and ${{y}_{i}}\equiv {{Y}_{i}}/DC$, which are measured at the appropriate modulation frequency i. Thus, fractional polarizations are given by the following:

$$\begin{eqnarray}&&{{q}_{0}}=\pm \sqrt{x_{q}^{2}+y_{q}^{2}},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{{u}_{0}}=\pm \sqrt{x_{u}^{2}+y_{u}^{2}},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{{v}_{0}}=\pm \sqrt{x_{v}^{2}+y_{v}^{2}},\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{{p}_{0}}=\sqrt{q_{0}^{2}+u_{0}^{2}},\end{eqnarray} \tag{ 6 }$$

where Equation (6) describes the degree of linear polarization.

It has long been known that the magnitude of a vector is biased toward larger values due to the uncertainty in vector components, especially for low S/N measurements. Thus, the subscript 0 in Equations (3)--(6) indicates the naive, biased estimator. To minimize this bias, we use the generalized MAS estimator (Plaszczynski et al. 2014):

$$\begin{eqnarray}&&q=\pm \left[ {{q}_{0}}-{{b}^{2}}\left( \frac{1-{\rm exp} (-q_{0}^{2}/{{b}^{2}})}{2{{q}_{0}}} \right) \right],\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{{b}^{2}}=\frac{{{x}^{2}}\sigma _{y}^{2}+{{y}^{2}}\sigma _{x}^{2}}{{{x}^{2}}+{{y}^{2}}},\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&\sigma _{q}^{2}=\frac{{{x}^{2}}\sigma _{x}^{2}+{{y}^{2}}\sigma _{y}^{2}}{{{x}^{2}}+{{y}^{2}}}.\end{eqnarray} \tag{ 9 }$$

Here, q is an estimate of the true, normalized Stokes parameter and $\sigma _{q}^{2}$ is an estimate of its variance. Similar equations exist for Stokes u and v. The position angle of linear polarization is then given by ${\Theta }=\frac{1}{2}{\rm arctan} (u/q)$.

To calibrate instrumental efficiency factors in each UBV filter, a linear polarizer and quarter-wave Fresnel rhomb are used to polarize and inject lamp light into POLISH2 with ∼100% Stokes q, u, and v sequentially. Instrumental polarization, caused by the filter between the PEMs and Wollaston, is assessed by rotating the filter wheel to a clear aperture. With ∼100% q (or u) injected, the removal of the B filter causes a difference of $\lt 200$ ppm in u (or q). Therefore, we expect astrophysical sources, where linear polarization is a maximum on the order of 1%, to experience a negligible ∼1 ppm crosstalk between q and u.

For sky subtraction, the telescope is nodded by 30'' between target and sky fields, where two 30 s target integrations are taken for each 30 s sky field integration to maximize S/N. We observed the asteroid (4) Vesta for 3.82 hr on the night of 2014 March 19 UT, which was composed of 14 minute integrations per band in a repeated BVU filter sequence. Since POLISH2 is an aperture-integrated polarimeter with a 15'' diameter field of view, the telescope tracked Vesta using JPL HORIZONS non-sidereal rates. For Vesta, sky-subtracted values of x_i and y_i are binned during each 14 minute integration. Non-zero telescope polarization (TP), caused by reflectivity variations across the telescope mirrors, is subtracted via measurements of seven bright, unpolarized stellar calibrators. Each calibrator star is observed for nine minutes. These stars are identified both from Bailey et al. (2010) and our unpublished survey of bright stars with POLISH2 on the Lick Observatory 1 m telescope.

Instrumental +Q is nominally aligned with celestial north. Comparison of the position angles of five strongly polarized stars with the catalogs of Heiles (2000; Aur, ζ Oph, 53 Per, 55 Cyg, and HD 154445), observed during two runs between 2014 March 19 and April 21 UT, indicates ${{{\Theta }}_{{\rm POLISH}2}}-{{{\Theta }}_{{\rm catalogs}}}=-1\buildrel{\circ}\over{.} 08\pm 0\buildrel{\circ}\over{.} 34$. Thus, instrumental +Q differs from catalogs at the 1° level with 3.2σ confidence, indicating that the absolute accuracy in position angle is ∼1°. The positioning accuracy of the Lick 3 m Cassegrain rotator is limited to $\sim 0\buildrel{\circ}\over{.} 3$, which may be the cause of the slight offset.

Fortuitously, Vesta was observed in the range ${\Phi }=90\buildrel{\circ}\over{.} 571\pm 0\buildrel{\circ}\over{.} 067$, where Φ is the angle between the Vesta-celestial north and Vesta–Sun directions. Thus, the rotation of observed Stokes q and u (in the celestial frame) to the scattering plane (denoted with primes, q' and u') involves a rotation of only $\mid {\Theta }-{\Theta }^{\prime} \mid \approx 0\buildrel{\circ}\over{.} 5$, which is accomplished by well-known relations (e.g., Bagnulo et al. 2006).

We measure the B band TP from observations of seven unpolarized stars taken during the same night as the Vesta observations (Table 1). The quantities in parenthesis indicate 1σ uncertainties in the last two digits. TP is calculated from the weighted mean of the x and y observables across all stars for each Stokes parameter. We detect significant telescope linear (and circular) polarization with 28σ (12σ) confidence and 3 ppm (7 ppm) accuracy, illustrating the simultaneous linear and circular polarimetric sensitivity of POLISH2.

Table 1 also lists the TP-subtracted, B band polarization of these unpolarized calibrator stars. For no star is linear polarization detected with ≥3σ confidence, which suggests all stars to be viable calibrators for TP. However, circular polarization of the peculiar A star HR 4295 is detected with 3.3σ confidence, and its lack of significant linear polarization suggests circular polarization to be intrinsic to the system as opposed to linear-to-circular conversion by the interstellar medium (Martin 1972). As it is a marginal detection, we do not attempt to further explain this result.

TP was only measured in the B band on the night of the Vesta observations, but Vesta data were taken in the UBV bands. To measure and correct for the wavelength dependence of TP, we utilize the UBV measurements of HR 4295, 4534, 6212, and 6220 obtained on 2014 April 20 and 21 UT (Table 2). Since q and u vary by $\lt 3\sigma$ between March and April, we use the difference between the April UBV measurements to estimate the U and V band TP in March. Inexplicably, the B band circular polarization measurements differ significantly between these runs, but, as Section 3.4 shows, this discrepancy lies within the uncertainty in Vesta's circular polarization.

Figure 1 shows a continuous 2.85 hr observation of the unpolarized calibrator star HR 4295 taken in the B band on 2014 March 12 UT. Measurement sensitivity, the ability to measure change, is given by median q, u, and v uncertainties in 13 minute bins (7.8, 6.7, and 17 ppm, respectively). Measurement accuracy, the ability to reject non-astrophysical change, is given by the square root of the weighted variance from bin to bin (6.4, 7.0, and 13 ppm in Stokes q, u, and v, respectively). Since measurement sensitivity and accuracy are essentially identical, a constant nature may only be rejected with 0.3σ, 1.1σ, and 0.2σ confidence from a χ² test. Therefore, over the timescale of the Vesta observations, we demonstrate that POLISH2 is stable to 7.0 ppm for linear polarization and to 13 ppm for circular polarization.

Zoom In Zoom Out Reset image size

Table 3 and Figure 2 (left column) detail our TP-subtracted, simultaneous Stokes q', u', and v measurements of Vesta in the B and V bands. Note that circular polarization v is invariant under rotation to the scattering plane. We confirm that Vesta's time-averaged linear polarization is parallel to the scattering plane (Θ' ≈ 90°), which is consistent with the phase angle of the observations. We discover that the magnitude of Vesta's time-averaged degree of negative branch polarization decreases with increasing wavelength with 14σ confidence (−6640 ± 110, −5204 ± 22, and −4990 ± 25 ppm in the UBV bands, respectively). The wavelength dependence of asteroidal linear polarization depends on the taxonomic class and phase angle of the observation (e.g., Cellino et al. 2015).

Zoom In Zoom Out Reset image size

Figure 2 (right column) shows the mean-subtracted polarization in each band to illustrate the amplitude of rotational modulation, which is immediately apparent in Stokes q' (Δq' panel). Here, a constant nature is rejected with 4.7σ confidence from a χ² test of q_x and q_y data (χ² = 55.8, ν = 16). We therefore fit sinusoids with Vesta's rotational period of 5.342 hr to our mean-subtracted BV band data (due to faintness in the U band), which provides a detection of rotational modulation with 12σ confidence. We find a peak-to-peak modulation of ΔP = 294 ± 35 ppm and a time-averaged polarization degree of $\langle p\rangle =-5105\pm 17$ ppm. Therefore, the ratio of peak-to-peak modulation to the time average is ${\Delta }p/\langle p\rangle =0.0577\pm 0.0069$.

After rotation to the scattering plane, Stokes u' is expected to be zero for single scattering events. Figure 2 (u' panel) shows that the weighted mean position angle of Vesta observations (Θ' panel) differs from 90° by $-1\buildrel{\circ}\over{.} 161\pm 0\buildrel{\circ}\over{.} 082$. This is consistent with the magnitude and sense of the $-1\buildrel{\circ}\over{.} 08\pm 0\buildrel{\circ}\over{.} 34$ rotational offset of POLISH2 in the plane of the sky, which is likely due to inaccuracy in the telescope Cassegrain rotator (Section 2.2).

Since time-averaged $\langle v\rangle =-28\pm 50$ ppm in the B band, we detect no significant broadband, circular polarization of Vesta and set a 1σ upper limit of $\mid v\mid \lt 78$ ppm. In addition, we detect no significant rotational modulation of circular polarization.