Variable Dynamics in the Inner Disk of HD 135344B Revealed with Multi-epoch Scattered Light Imaging^∗

Imaging polarimetry data sets were obtained on 2016 May 04, May 12, June 22, and June 30 with the near-infrared imager (IRDIS; Dohlen et al. 2008) of SPHERE at the European Southern Observatory's Very Large Telescope (VLT). Observations were carried out with the broadband J-filter (BB_J, 1.245 μm) in DPI (Langlois et al. 2014) mode. The pixel scale of the detector is 12.26 mas pix⁻¹ (Maire et al. 2016). An apodized Lyot coronagraph was employed (N_ALC_YJH_S, 185 mas mask diameter), allowing for an integration time of 32 s. The four standard half-wave plate orientations were cycled with two or four subsequent integrations per half-wave plate orientation. The extreme adaptive optics system (SAXO; Fusco et al. 2006) provided a typical Strehl ratio of ∼75% in the H band (see Table 1).

Seeing conditions were mostly good (<07), except on 2016 May 12 when the observations were executed with an average seeing of 21 and the presence of strong winds. Nonetheless, the AO loop remained closed for a total integration time of 17 minutes. The integration time was 34 minutes or longer for the other observations. The point-spread function (PSF) of the sequence on 2016 May 04 was partly affected by low-order aberrations caused by low wind speeds (1.4 m s⁻¹ on average). The PSF quality was evaluated from the non-coronagraphic images recorded by the differential tip-tilt sensor after which 13 polarimetric cycles were removed, leaving a total of 11 cycles (47 minutes).

Standard calibration procedures were applied with the SPHERE data reduction and handling (DRH) pipeline (v0.18.0; Pavlov et al. 2008) which included sky subtraction, flat field correction, and bad pixel interpolation. The frames with horizontally and vertically polarized flux were separated and subsequently processed by a custom pipeline for DPI data. We obtained coronagraphic images with four symmetric satellite spots, induced by a periodic modulation applied to the deformable mirror, before and after the science sequence. These frames were used to determine the position of the star behind the coronagraph mask. To center the coronagraphic DPI data, we interpolated linearly between the start and end position of the star. Stokes Q and U images were obtained with the double-difference method (Hinkley et al. 2009) and subsequently collapsed with a mean stacking.

To correct for instrumental polarization, we used the method described by Canovas et al. (2011) which assumes that the central star is unpolarized. The U_ϕ signal, which provides an estimate of the noise level in the single-scattering limit, was minimized by stepwise changing the inner and outer radius of the annulus used to measure the (assumed to be unpolarized) signal close to the star. The azimuthal counterparts of the Stokes Q and U images, Q_ϕ and U_ϕ, were calculated with an additional minimization applied on the U_ϕ image by correcting for a minor rotational offset of the half-wave plate (Avenhaus et al. 2014) for which the optimized values ranged from −20 to −13. We note that the procedure of minimizing the U_ϕ signal is not strictly valid because part of the scattered light flux from the disk will be present in the U_ϕ image due to multiple scattering (Canovas et al. 2015). The effect will be small when the disk inclination is low; however, the high signal-to-noise ratio (S/N) of the disk detection around HD 135344B might reveal a real signal in the U_ϕ image (Stolker et al. 2016). Finally, the images were rotated by −18 toward the true north orientation (Maire et al. 2016).

Flux frames were obtained at the start of each sequence by shifting the star away from the coronagraph, with a shorter integration of 0.87 s and an additional neutral density filter (ND1.0) in the optical path to avoid saturation. The flux frames are used to determine the angular resolution of the images (see Table 1), as well as the scattered light contrast of the disk (see Section 3.1). The data reduction procedure included a dark-frame subtraction, flat-field correction, and bad-pixel interpolation. To remove any residual background and bias from the images, we calculated the mean pixel value (with the inner 15 masked) for each detector column and subtracted that value from all pixels in that column. The PSF of HD 135344B was fitted with a 2D Gaussian profile, which yielded a typical FWHM of 41 mas. More details on the observations and conditions are provided in Table 1.

HD 135344B was observed with the REM at La Silla, Chile, in June 2016. The visible camera, ROSS2, is a simultaneous multi-channel imaging camera that delivers the $g^{\prime} r^{\prime} i^{\prime} z^{\prime}$ bands onto four quadrants of the same CCD detector. A dichroic enables simultaneous observations with the infrared camera, REMIR, with a similar field of view of $10^{\prime} \times 10^{\prime}$. Observations were executed with 3 s exposures in the $g^{\prime} r^{\prime} i^{\prime} z^{\prime}$ bands and 1 s exposures in the JHK-bands. For the JHK photometry, a standard five-position dither pattern was used and additional sky frames were obtained. Data was acquired during a total of twelve nights, but the data of two nights were rejected due to thick clouds.

The photometry of HD 135344B was measured differentially with respect to HD 135344A (SAO 206463), an A0V star with a separation of $21^{\prime\prime}$ from HD 135344B, which appears to be photometrically stable (Sitko et al. 2012). Differential photometry allowed us to measure with high precision the absolute fluxes of HD 135344B, also in variable conditions or with the presence of thin clouds. The JHK magnitudes of HD 135344A were retrieved from the 2MASS catalog (Skrutskie et al. 2006) while the $g^{\prime} r^{\prime} i^{\prime} z^{\prime}$ magnitudes were calculated through a transformation of the BVRI magnitudes with the relations from Jordi et al. (2006). The BVRI magnitudes of HD 135344A were obtained from Sitko et al. (2012): $B=7.879\pm 0.003$ mag, $V=7.756\pm 0.003$ mag, $R=7.708\pm 0.006$ mag, and $I=7.662\pm 0.004$ mag.

We retrieved all of the available archival near-infrared interferometric data of HD 135344B from the Precision Integrated-Optics Near-infrared Imaging ExpeRiment (PIONIER) instrument (Le Bouquin et al. 2011) at the Very Large Telescope Interferometer (VLTI). The data were taken during multiple epochs from 2011 to 2013.¹³ The instrument recombines the four auxiliary telescopes that were positioned in the short (A1-B2-C1-D0), intermediate (D0-G1-H0-I1), and long (A1-G1-I1-K0) baseline configurations. The projected baseline, B, ranged from 7 to 135 m, enabling a maximum angular resolution of $\lambda /2B=1.2$ mas across the seven spectrally dispersed channels in the H-band (${ \mathcal R }\simeq 40$). Each observation of HD 134453B was preceded and followed by an observation of a calibration star to characterize the instrumental and atmospheric contribution to the visibilities and closure phases (i.e., the transfer function). Calibration stars were identified with the SearchCal tool (Bonneau et al. 2006, 2011). The data were reduced with the pndrs package, described in detail by Le Bouquin et al. (2011). The 27 calibrated OIFITS files (Pauls et al. 2005) are available in the Optical interferometry DataBase (OiDB) (Haubois et al. 2016).

The observing conditions were best during the two epochs in 2011 April, with a coherence time of ∼5 ms and a seeing of 05–07, while the conditions were significantly poorer when the other data sets were taken. We rejected low S/N measurements by selecting only the data points where the error estimates were in the first quartile of the total distribution of errors. The dominating factor in the error estimate is the stability of the transfer function during the night, which is determined by the temporal scatter of the calibrator visibilities. For each data set, the quality assessment was done independently for the six visibilities, four closure phases, and seven spectral channels. The selected measurements were retrieved from 11 out of 13 epochs, with 2011 May 26 and 2012 April 27 excluded.

Scattered light images are displayed in chronological order (top to bottom) in Figure 1. Besides the newly obtained data sets from 2016, we also show the 2015 J-band imagery from Stolker et al. (2016). The first column shows the unscaled Q_ϕ images, defined such that positive values correspond to azimuthally polarized flux. The images are normalized to the disk-integrated Q_ϕ flux, measured with an annulus aperture between 01 and 10, and shown with identical dynamical range. The second column shows the corresponding U_ϕ images, containing flux with a $\pm 45^\circ$ rotational offset of the direction of polarization with respect to the Q_ϕ flux. The third column contains the Q_ϕ images with a stellar irradiation correction applied (i.e., r²-scaling). The fourth column shows unsharp-masked images that were obtained by smoothing the r²-scaled Q_ϕ images with a Gaussian kernel ($\sigma =200$ mas) and subtracting the smoothed images from the original r²-scaled images. This procedure enhances the contrast of small-scale features by removing low spatial frequencies. The dynamical range of the unsharp-masked images is limited to positive values, and for each image separately normalized to the peak intensity.

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The SW direction ($\mathrm{PA}=180^\circ$–$270^\circ$) of the disk, located around the major axis (see Figure 1), appears relatively bright in all r²-scaled images. In contrast to the shadowing variations, the origin of that brightness enhancement is presumably intrinsic as the bright wedge in scattered light coincides with the (sub)millimeter emission peak of the crescent-shaped dust continuum (Pérez et al. 2014; Stolker et al. 2016). An enhancement of the surface density and/or midplane temperature will elevate the height of the scattering surface; therefore, a larger geometrical cross section of the disk surface is irradiated, which increases the scattered light flux. In this work, we focus on local brightness variations between the five epochs. We refer the reader to Muto et al. (2012), Garufi et al. (2013), and Stolker et al. (2016) for a detailed analysis and discussion of the scattered light detection of the spiral arms and cavity edge.

The U_ϕ images in Figure 1 contain contributions from noise, residuals of instrumental polarization, and possibly multiple scattered light from the disk. The residual signal could not be removed with the minimization steps explained in Section 2.1, but a more detailed analysis is required to determine if the remaining signal is an instrumental artifact. We distinguish between two different type of signals in the U_ϕ images that appear to be related to the total integration time and therefore the S/N. First, the relative contribution of noise is particularly well visible at separations ≳300 mas from the star in the last three epochs for which the total integration time was shortest, while the relative contribution is lower in the first two epochs. Second, the U_ϕ images show an enhanced signal within 300 mas of which the relative strength is larger in the first two epochs. The inner signal reaches only mildly above the background noise level in the remaining epochs. The inner U_ϕ signal appears to be variable between epochs, revealing a variety of brightness patterns. For example, the first epoch shows a complex pattern of multiple positive and negative lobes, whereas the second epoch displays an anti-symmetric signal which is bisected in a positive and negative side. The relative strength of the U_ϕ flux with respect to the Q_ϕ flux is quantified in Section 3.3.

Although we deem it likely that the remaining U_ϕ signal is an uncorrected instrumental artifact, we speculate that the increasing strength of the innermost U_ϕ signal with increasing S/N could be a result of multiple scattered light from the cavity edge. A minor fraction of the scattered light will be non-azimuthally polarized because of the small inclination of the outer disk ($16^\circ$–$20^\circ ;$ van der Marel et al. 2015, 2016). Furthermore, a small fraction of the stellar light will scatter in the extended inner disk atmosphere (see Section 4.3) before it scatters from the outer disk. The U_ϕ signal from those photons will get modulated by the temporal variations in the inner disk, such that each epoch may show a different U_ϕ image. However, more detailed analysis and modeling is required in order to determine if the U_ϕ signal is real and if the temporal variations could be caused by the inner disk.

A comparison of the r²-scaled Q_ϕ images in Figure 1 shows epoch-to-epoch brightness variations. Azimuthal brightness minima are visible in all images with variations in their location, shape, and strength. The locality of the brightness minima (e.g., the shadow lane at $\mathrm{PA}=169^\circ$ on 2015 May 03) points toward a shadowing effect, likely caused by dust in the (unresolved) inner disk (Stolker et al. 2016). Furthermore, the brightness variations occur on a timescale similar to the dynamical timescale of the inner disk (the finest temporal resolution is eight days) and the variability timescale of the near-infrared photometry. Minor brightness variations in the unscaled Q_ϕ images are also visible in the region between the cavity edge and the coronagraph, possibly related to the flow of gas and small dust grains from the outer disk.

Azimuthal brightness variations, and their changes between epochs, are more evidently revealed with polar projections of the scattered light images, which are displayed in Figure 2. For clarity, we choose the unsharp-masked images for the identification of the shadow features in the polar projections. We caution that applying an unsharp mask may introduce a bias in the identification of brightness variations. However, the shadow features, which we will discuss below and are marked with arrows in Figure 2, are also visible in the regular r²-scaled Q_ϕ images, but the contrast between shadowed and non-shadowed regions is smaller.

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Asymmetric illumination/shadowing variations are visible in the scattered light imagery of all five epochs. Here we list the main characteristics of the shadow features, in consonance with the locations that are pointed out in Figure 2.

• 1.  
  Epoch 1, 2015 May 03—Three narrow shadow lanes are present at position angles of $34^\circ$, $169^\circ$, and $304^\circ$, and an azimuthally broader dimming is visible in a north-northwest direction that is bound by two of the shadow lanes (Stolker et al. 2016).
• 2.  
  Epoch 2, 2016 May 04—The eastern half of the disk is mildly shadowed, approximately in the position angle range of $10^\circ$–$170^\circ$. Deeper shadows are superimposed near the edges of the global shadow. The deepening is particularly well visible in the north, extending at the cavity edge from $\mathrm{PA}=-50^\circ$ to $\mathrm{PA}=50^\circ$. Radially outward, the location of the shadow shows an azimuthal gradient. There is a hint of a localized shadow lane at $\mathrm{PA}=170^\circ$, approximately colocated with the southern shadow lane in the first epoch.
• 3.  
  Epoch 3, 2016 May 12—The bisection of the brightness distribution from the second epoch seems to have disappeared, although the poor observing conditions and the short total integration time (see Table 1) make the identification of the shadows challenging. The broad, northern shadow from the previous epoch is still present. There is also a hint of the broad, southern shadow, whereas the narrow southern shadow has disappeared.
• 4.  
  Epoch 4, 2016 June 22—A broad shadow is present between $\mathrm{PA}=-90^\circ$ and $\mathrm{PA}=30^\circ$ upon which finer shadow variations are superimposed, including narrow shadow lanes at the boundary with the non-shadowed region, similar to the northern shadow features in the first epoch. The shadow lane at $\mathrm{PA}=30^\circ$ possibly coincides with the location of the northeast shadow lane detected in first epoch.
• 5.  
  Epoch 5, 2016 June 30—The cavity edge is shadowed between $\mathrm{PA}=-45^\circ$ and $\mathrm{PA}=40^\circ$, while shadowing of the exterior spiral arm only occurs from $\mathrm{PA}=0^\circ$ onwards. The radial extent of the broad shadow increases with increasing position angle, similar to the broad shadows in the first and second epoch. The shadow covers the full radial extent of the disk between $\mathrm{PA}=0^\circ$ and $\mathrm{PA}=40^\circ$, similar to the shadow feature at the same location in the second and third epoch. The narrow shadow lanes from the fourth epoch seem to have disappeared.

The brightness depth of the shadow lanes appears typically larger than the broader shadows (e.g., epoch 1 in Figure 1), indicating larger density enhancements in the inner disk. Indeed, there is no evident radial dependence in the depth of the shadow lanes, even though the height of the outer disk increases with radius. The brightness depth of the broad shadows is typically deepest at the cavity edge but weakens toward larger radii (e.g., epoch 4 in Figure 2), presumably an effect of the increasing outer disk height. The broad shadow in north-northwest direction ($\mathrm{PA}\simeq -80^\circ$–30°) of all epochs shows an azimuthal gradient which is possibly a result of the asymmetric spiral arm perturbation of the disk surface.

The absence of strong azimuthal gradients, caused by a light travel time effect, in the location of the narrow shadows provides a lower limit on the radius from where the shadows are cast. For example, the position angle of a shadow will change by $10^\circ$ between the inner disk and 80 au (500 mas) if the responsible dust clump is located at 0.15 au (see also Kama et al. 2016). Arguably, some of the shadow lanes show very minor tilts in Figure 2, but the precision and angular resolution of the observations challenge the identification of light travel time effects. The best candidate is the shadow lane at $\mathrm{PA}=169^\circ$ in the 2015 epoch. Stolker et al. (2016) speculated that its azimuthal tilt could be caused by orbital motion in the inner disk from which an orbital radius of 0.06 au (at 140 pc) was estimated. For the other shadow lanes, we may conclude that the dust clumps responsible are presumably located at distances ≳0.15 au.

A quantification of the azimuthal brightness variations is shown in Figure 3. The mean Q_ϕ flux is measured in position angle bins of 10° wide across a radial separation of 01–07 and divided by the angular area of a pixel. The polarized surface brightness (in counts s⁻¹ arcsec⁻²) is normalized to the total Stokes I flux (in counts s⁻¹), which is measured with a circular aperture on the unsaturated, non-coronagraphic flux frames after a correction for the integration time and response of the neutral density filter. The optimal aperture size (15) was determined by measuring the photometric flux with a large range of aperture sizes (up to 30), from which it was established that the total encompassed flux flattened for apertures larger than ∼15. The mean error bar is calculated from the standard error on the individual contrast points.

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The polarized surface brightness contrast in Figure 3 shows typical values in the range of $(2\mbox{--}6)\times {10}^{-3}$, except in the southwest direction where the contrast goes up to $8\times {10}^{-3}$ at $\mathrm{PA}\simeq 240^\circ$. The integrated disk brightness consists mainly of signal from the cavity edge and the spiral arms (see unscaled Q_ϕ images in Figure 1), which are both asymmetrically shadowed resulting in dimming variations of 20%–45%. The contrast variation is minimal (10%) in the south-southwest direction, where the large-scale density and/or scale height enhancement affects the disk surface. The opening angle of the scattering surface is larger in that direction, such that shadowing by the inner disk requires dust to be located at higher altitude above the midplane. The contrast variations provide an upper limit on the local changes in optical depth through the inner disk atmosphere. The stellar radiation that is transmitted through the atmosphere will be attenuated by a factor ${e}^{-\tau }$; therefore, the relative change in optical depth, ${\rm{\Delta }}\tau$, can be calculated from the minimum and maximum contrast (see Figure 3). However, this only provides an upper limit on the optical depth variations because (i) the total flux does also affect the contrast and is correlated with the shadowing of the outer disk (see Section 4.3), (ii) part of the thermal emission from the inner disk may illuminate the outer disk without being affected by any local obscurations, and (iii) PSF smearing will lower the brightness contrast between shadowed and non-shadowed regions.

Multi-epoch photometry in the $g^{\prime} r^{\prime} i^{\prime} z^{\prime}$ and JHK-bands are displayed in Figure 4, covering 10 nights between 2016 June 07 and 2016 July 04 with nearly a daily sampling from June 16 till June 22. The error bars reflect the uncertainty on both the science and calibration star. The early June photometry in the $g^{\prime} r^{\prime} i^{\prime} z^{\prime}$-bands starts with a minor increase of 3%, remains approximately constant in mid-June, and increases with 2% by the end of June.

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The temporal course of the JHK photometry appears more irregular with variations up to 10% with respect to the mean (horizontally dashed lines in Figure 4) in the JHK-bands. During the first half of 2016 June, the fluxes increased with approximately 6%, 9%, and 10% in the J-, H-, and K-bands, respectively, following the trend of the $g^{\prime} r^{\prime} i^{\prime} z^{\prime}$ fluxes but with a larger fractional increase. In mid-June, the JHK fluxes also remained approximately constant but they increased further from June 22 onwards, in contrast to the $g^{\prime} r^{\prime} i^{\prime} z^{\prime}$ fluxes. The final epoch shows a decline in the J- and H-bands, while the K-band photometry remained constant. Although the JHK variability in the first part of June seems correlated with the $g^{\prime} r^{\prime} i^{\prime} z^{\prime}$ photometry, in the second half no correlation is apparent; that is, the JHK fluxes increased up to 10% with respect to the mean while the $g^{\prime} r^{\prime} i^{\prime} z^{\prime}$ photometry remained constant. The total temporal coverage of the photometry is too short to reveal any trends and the sampling is too sparse to resolve possible variations on timescales less than one day.

In addition to the absolute REM photometry, we measured the relative disk photometry of the SPHERE data, which is presented in Figure 5. The disk-integrated polarized flux was determined from the Q_ϕ and U_ϕ images with an annulus aperture (01–10) centered on the star. The Stokes I flux was measured with a circular aperture (15 radius) from the dedicated flux frames. Absolute pixel values were used for the U_ϕ photometry. The photometric contrast (top panel in Figure 5) is calculated as the ratio of the Q_ϕ and Stokes I flux after correcting for the difference in integration time and the response of the neutral density filter. Relative photometry allows for an epoch-to-epoch analysis without requiring an absolute flux calibration, assuming that both the coronagraphic sequence and the total flux data were obtained during similar observing conditions. The Q_ϕ and U_ϕ surface brightness errors are computed as the standard deviation within an aperture centered on each pixel with a radius of 62 mas (i.e., 1.5 resolution elements) and propagated accordingly to an integrated error (3%–5%). The uncertainty on the total flux (1%–2%) is computed in a background-limited region as the standard error of the sum, $\sigma \sqrt{{N}_{\mathrm{pix}}}$, with an annulus aperture equal in size to the aperture used for the Stokes I photometry.

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The integrated contrast in Figure 5 varies between $5.4\times {10}^{-3}$ and $7.2\times {10}^{-3}$. The photometry is not calibrated, so only relative variations of ${Q}_{\phi }/\mathrm{Stokes}\,I$ and ${U}_{\phi }/{Q}_{\phi }$ are meaningful. The second epoch shows a consistent decrease of both the Q_ϕ and Stokes I flux with respect to the first epoch. In the third epoch, the contrast decreased by 20% possibly due to the poor observing conditions (see Section 2.1), particularly affecting the Q_ϕ photometry. During the last two epochs, the relative increase of the total flux is large compared to the Q_ϕ flux, resulting in relatively low contrast. The increase of the total flux during the last epochs seems consistent with the REM photometry (see June 22 and 30 in Figure 4). The relative U_ϕ photometry shows an increase in the second epoch due to the residual within 200 mas (see Section 3.1), possibly caused by multiple scattered light from the inner and/or outer disk, while the relative U_ϕ photometry in the third epoch is larger due to the enhanced noise residual.

The normalized, squared visibilities, V², across the spectrally dispersed H-band channels of the multi-epoch PIONIER observations are displayed in the left panel of Figure 6. The visibilities decrease continuously with increasing spatial frequency (i.e., $B/\lambda$), indicating that the region from which the H-band flux originates is resolved. At the longest baselines, there appears no turnover point to an asymptotic value of the stellar flux, so the circumstellar emission is not over-resolved. The closure phases, ${\rm{\Delta }}\phi$, are consistent with zero within the error bars ($| {\rm{\Delta }}\phi | \lesssim 3^\circ$), therefore, the brightness distribution of the inner disk is point symmetric within the uncertainties and at the spatial resolution probed by the observations. There is no significant dispersion visible for visibility points obtained with baselines of similar lengths but different position angles, which implies that the inclination of the inner disk is small. The diagonal scatter of the visibilities is an effect of chromatic dispersion (Lazareff et al. 2017). Coverage of the (u,v)-plane is shown in the right panel of Figure 6.

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The orientation and characteristic radius of the inner disk H-band emission is determined by fitting, in Fourier space, a parametric model to the visibilities, following the procedure described in detail by Lazareff et al. (2017). This allowed us to apply a ${\chi }^{2}$ minimization and to assign formal error bars to the inferred parameter values. We parameterized the H-band emission with an elliptical brightness distribution that is radially parameterized by a weighted combination of a Gaussian and pseudo-Lorentzian profile. The inner rim is not fully resolved by the longest baselines, which justifies an ellipsoidal distribution instead of a broadened ring.

The best-fit model (${\chi }^{2}=1.10$) corresponds to an inclination and major axis position angle of $18\buildrel{\circ}\over{.} {2}_{-4.1}^{+3.4}$ and $57\buildrel{\circ}\over{.} 3\pm 6\buildrel{\circ}\over{.} 3$, respectively. The values are, within the uncertainties, very similar to those of the outer disk (see the discussion in Section 4.1.1). The best-fit position angle is shown in Figure 1 in comparison with the outer disk value from van der Marel et al. (2015). The half-flux semimajor axis is 0.71 ± 0.03 mas (=0.11 au), which implies that a significant fraction of the H-band emission originates from within the silicate sublimation radius (${R}_{\mathrm{sub}}=0.2\,\mathrm{au};$ Carmona et al. 2014). A detailed overview of the fitting results is provided in the Appendix, where the best-fit values of all parameters are listed with their dependence on the cutoff level of the selection criterion for the (u,v) points.

We caution that the visibilities were combined from multiple epochs, while the inner disk is variable on a timescale of days or less (see Section 3.3). The visibilities in Figure 6 depend on the absolute H-band flux and the relative contributions of the star and disk. Therefore, an additional uncertainty has been introduced by combining the visibilities from multiple epochs. While the absolute H-band flux is variable and not measured at the nights of the observations. Also, we made the assumption that the orientation of the inner disk did not change due to precession between those epochs.

4.1.1. Near-infrared and Submillimeter Interferometry

4.1.2. Quasi-stationary Shadowing by a Minor Disk Warp?

The location and width of the shadows provide further constraints on the orientation of the inner disk with respect to the outer disk. Three scenarios remain possible: (i) the inner disk and outer disk are aligned, such that shadowing occurs only through uplifting of dust in the inner disk atmosphere, (ii) a minor misalignment (${\rm{\Delta }}\theta \sim 1^\circ$–$2^\circ$) is present that might not cast a stationary shadow, but additional variations in the inner disk will cause preferential shadowing in the direction where elevation of the inner disk above the outer disk midplane is largest, (iii) an intermediate misalignment (${\rm{\Delta }}\theta \sim 2^\circ$–$10^\circ$) is present which casts a broad, stationary shadow when the misalignment of the inner disk is similar to the opening angle of the scattering surface of the outer disk. An example of the third scenario is the broad shadow detected with HST on the TW Hya disk (${\rm{\Delta }}\theta =8^\circ ;$ Rosenfeld et al. 2012).

The inclination of the inner disk could be either smaller or larger than the outer disk given the uncertainties on its orientation (see Section 2.3) that could result in a broad shadow in an approximately northwest or southeast direction, respectively. Interestingly, a broad shadow seems present in all of the images in a north-northwest direction (see Figure 2), although with small variations in its precise location, shape, and depth. This may imply that the inclination of the inner disk is slightly smaller (i.e., more face-on) than the outer disk. Variable fine structure is present in the north-northwest shadow, which requires additional optical depth variations through the atmosphere of the inner disk as will be discussed in more detail in Section 4.2.

A broad dimming was also present in the northwest direction of the Subaru/HiCIAO H-band imagery by Muto et al. (2012) and the VLT/NACO H and ${K}_{{\rm{s}}}$-bands imagery by Garufi et al. (2013). This was interpreted in both studies as a depolarization effect that can occur when the disk is inclined because the polarization efficiency peaks around the major axis where scattering angles are close to $90^\circ$ (e.g., Murakawa 2010; Min et al. 2012). The inclination of the outer disk is relatively small, so a strong depolarization effect might not be expected. We speculate that the broad dimming in the HiCIAO and NACO imagery might be the same quasi-stationary shadowing effect that is seen in the SPHERE imagery, possibly related to a minor disk warp.

To illustrate this scenario, we have adopted the DIsc ANAlysis (DIANA; Woitke et al. 2016) radiative transfer model of HD 135344B. The left image in Figure 7 displays the raytraced Q_ϕ image for a setup in which the inner disk is aligned with the outer disk ($i=20^\circ$, $\mathrm{PA}=63^\circ ;$ van der Marel et al. 2015). A mild depolarization effect is seen in both directions of the minor axis, but the effect is slightly stronger on the near side (southeast) due to the flaring geometry of the disk surface (Min et al. 2012). In the right image of Figure 7, we changed the orientation of the inner disk to the best-fit values from Section 3.4. The misalignment here is $2\buildrel{\circ}\over{.} 6$; that is, comparable to the disk warp seen in the debris disk around β Pic (${\rm{\Delta }}\theta =4\buildrel{\circ}\over{.} 6;$ Heap et al. 2000). The atmosphere of the inner disk casts a mild shadow on the outer disk in a northwest direction, reminiscent of the broad shadow on the cavity edge of the SPHERE images in Figures 1 and 2. In the opposite direction, the outer disk becomes more strongly irradiated, which introduces an azimuthal brightness modulation along the cavity edge (see also Rosenfeld et al. 2012), although intertwined with the modulation by the polarization efficiency of the outer disk.

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4.2.1. Observational Perspective and Variability Timescale

4.2.2. Inner Disk Processes Affecting the Dust Dynamics

4.2.3. Face-on Variant of the UX Orionis Phenomenon?

Shadow variations on the outer disk depend on changes in the vertical distribution of dust in the inner disk. More specifically, the strength of the shadowing is set by the inner disk radius, where the transition region from optically thin to thick occurs highest above the midplane. The height of this transition region is mainly determined by the pressure scale height and the surface density if the dust opacities are constant throughout the inner disk. For a flaring geometry, the thickest part will be close to the outer edge of the inner disk. Alternatively, a puffed-up inner rim may shadow the outer disk in case the exterior of the inner disk is fully shadowed by the rim (Dullemond & Dominik 2004; Dong 2015).

4.3.1. Parametric Model Setup

4.3.2. Polarized Scattered Light versus Thermal Emission

For each model, we computed the disk-integrated Q_ϕ intensity of the outer disk and the total J-band flux, which are displayed in the top panel of Figure 8. Several effects of the inner disk mass and the aspect ratio on the polarized intensity are evident in the colored map. Increasing the aspect ratio from very small values up to ∼0.05 results in a larger fraction of the stellar light being reprocessed by the inner disk and reemitted in the J-band, thereby increasing the scattered light flux from the outer disk. Similarly, the increase of the polarized intensity with increasing dust mass is also the result of a larger fraction of thermal radiation from the inner disk scattering from the outer disk. In this regime of the aspect ratio, the opening angle of the inner disk is too small to shadow the outer disk.

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Shadowing of the outer disk by the inner disk starts to have an effect for ${H}_{0}/{r}_{0}\gtrsim 0.05$, such that the opening angle of the inner disk atmosphere is comparable to or larger than the opening angle of the scattering surface of the outer disk. The polarized intensity decreases with increasing aspect ratio when the inner disk dust mass is smaller than ∼10⁻⁹ ${M}_{\odot }$ because the optical depth through the inner disk atmosphere toward the outer disk increases. Consequently, a larger fraction of the stellar light is attenuated by extinction in the inner disk. For inner disk masses larger than ∼10⁻⁹ ${M}_{\odot }$, the effect of enhanced reprocessing by the inner disk starts to dominate over the shadowing effect. More specifically, even though increasing the aspect ratio beyond 0.05 will increase the amount of shadowing of the outer disk, enhanced reprocessing by a larger amount of dust in the inner disk results in an increased irradiation of the outer disk in the J-band. Since the exponent is negative for both the surface density and the scale height profile, the inner disk is thickest at the location of the inner rim (i.e., the radius where the vertical $\tau =1$ surface is located highest above the midplane). As a result, reprocessed stellar light by the inner rim can directly irradiate the outer disk, in contrast to an inner rim which is partly shielded from the outer disk by a flaring scale height profile.

The J-band photometry of the radiative transfer models is superimposed on the polarized surface brightness in the top panel of Figure 1. The inner disk dust mass and aspect ratio have a similar effect on the J-band flux because both parameters affect the amount of reprocessing by the inner disk. While the stellar J-band flux remains constant, the thermal emission from the inner disk is positively correlated with both the inner disk dust mass and the reference aspect ratio. The SEDs of all models are shown in the bottom panel of Figure 1. As expected, the near- and mid-infrared excess varies greatly in the explored parameter space.

4.3.3. Observed and Simulated Scattered Light Contrast

4.3.4. Evidence for Extended Variations in the Inner Disk