The Gemini Planet Imager View of the HD 32297 Debris Disk

The HD 32297 disk is bright enough to be detected in raw individual frames, as illustrated in Figure 1(a). In the combined Stokes I image (Figure 1(b)), the disk is strongly detected above the background of the PSF halo outside of ≈03, although measuring accurate surface brightness still requires an additional step of PSF subtraction; this is performed in Section 3.2.

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Because light from the star is intrinsically unpolarized, there is no leftover halo in the Stokes Q_ϕ and U_ϕ images. In the former, the disk is strongly detected from just outside the edge of the coronagraphic mask (≈015) out to a sensitivity-limited distance of about 12 from the star. This data set provides the smallest stellocentric distance at which the disk is clearly detected to date. Under the assumption of single scattering as in the optically thin regime, U_ϕ should be null throughout the image (Canovas et al. 2015). This is true outside of 025, where we use the U_ϕ map to evaluate the noise associated with the Q_ϕ map by measuring the standard deviation in concentric annuli. In the inner region, however, a U_ϕ signal is observed at approximately the same location as the disk at a level of 5%–10% of the Q_ϕ signal. This could either be a consequence of multiple scattering, implying that the disk is not quite optically thin, or an indication of uncorrected polarization systematics. Because the strongest signal in the U_ϕ map is offset by about 2 pixels perpendicular to the disk major axis from the strongest Q_ϕ signal, we deem the latter interpretation as likely correct. Despite various attempts to improve data reduction, we could not find a satisfactory method to fully remove this artifact. We thus evaluate the uncertainty associated with the Q_ϕ map with the same method at these inner regions as at larger radii, noting that this may introduce a bias because the dispersion between pixels within annuli is not driven by random noise.

To more clearly reveal the HD 32297 disk in total intensity, it is necessary to subtract the residual starlight in the Stokes I image. As in previous GPIES disk analyses (e.g., Kalas et al. 2015; Draper et al. 2016), we implemented several independent methods, each with their own advantages and limitations. The resulting PSF-subtracted images are presented in the top row panels of Figure 2.

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First, we used a standard Angular Differential Imaging (ADI) approach with pyKLIP-ADI (Wang et al. 2015), a custom implementation of the KLIP algorithm (Soummer et al. 2012). This method is highly effective for point source discovery but results in systematic self-subtraction of extended objects such as disks. In the particular situation of edge-on disks, strong negative "wings" are imprinted on each side of the disk, especially when the total field rotation is modest as is the case here. To minimize self-subtraction, we adopted a conservative set of parameters, using only 5 KL modes and averaging images with 3–9 annuli. In the resulting image, the disk is traced all of the way to the coronagraphic mask, with an apparently smooth brightness profile.

To mitigate self-subtraction, we also used pyKLIP with Reference Differential Imaging (RDI). Here, we first assemble a library of nearly 25,000 H-band images of stars observed with GPI, from which frames with known astrophysical signals or instrumental issues were removed. We then select the 500 images that are most highly correlated with each individual frame of HD 32297. The PSF is then estimated by applying the same KLIP process as above to this set of reference images. While this approach prevents self-subtraction, pyKLIP-RDI can still suffer from over-subtraction, as any astrophysical signal can be misinterpreted as a PSF "feature" by the algorithm. This is particularly relevant in the case of a bright disk like HD 32297, where the ratio of disk-to-PSF signal approaches or even exceeds unity in some parts of the image. We thus employed a conservative set of parameters (5 KL modes, averaged over 3–9 annuli, 500 reference PSFs chosen). Despite significant, low-frequency background fluctuations in the resulting image, the disk is clearly detected at all radii outside of the coronagraphic mask.

To sidestep self- and over-subtraction in a different way, we also employed the mask-and-interpolate (MI) PSF subtraction at the single-frame level (Perrin et al. 2015). We first mask out a 15 pixel high box centered on the disk, as well as the four satellite spots. The masked pixels are then replaced with the result of interpolating through the neighboring unmasked pixels with a fourth-order polynomial function. The resulting image is then smoothed with a 13 pixel (≈018) running median box to only model the low spatial frequency component of the PSF, and it is subsequently subtracted from the original frame. Residual fluctuations in the background are significantly lower than in the RDI case, except close to the inner working angle where the interpolation scheme fails to reproduce the sharp intensity gradients of the PSF. The region interior of ≈025 from the star is too uncertain to consider in our subsequent analysis, but the disk is strongly detected outside of this radius.

Finally, we applied the Nonnegative Matrix Factorization (NMF) method as implemented within pyKLIP. NMF is an iterative method based on the decomposition of the PSF into separate components that only contain positive pixels (Ren et al. 2018). Similar to the RDI process, we selected the 500 most correlated frames in the library of GPI images and used the first five modes computed by NMF to subtract the PSF. The resulting total intensity image for HD 32297 reveals a smooth brightness profile, albeit with leftover background fluctuations that are intermediate in strength between the RDI and MI methods. Like the ADI and RDI methods, the NMF method yields a strong detection of the disk all of the way to the edge of the coronagraphic mask.

Apart from the bright disk, all four PSF-subtracted images are marked by a diagonal negative residual pattern (along position angles ∼15° and ∼190°). This likely is a consequence of the "butterfly" structure of the PSF visible in the raw total intensity images (see Figure 1) and that is imparted by winds in the atmosphere (Madurowicz et al. 2019). To improve the quality of the final images, we perform a fourth-order polynomial fit in concentric annuli after masking out a vertical box centered on the disk; to improve the fit, the process handles each side of the disk separately. Effectively, this performs a second mask-and-interpolate subtraction, on a single annuli basis. The resulting images are shown in Figure 2. In the case of the RDI and NMF methods, which are characterized by more structured residuals, the subtraction residuals and amplitude of the background fluctuations become too high to produce a clean image of the disk inside of 03 from the star. For these images, we do not attempt to measure the absolute brightness of the disk closer in.

In both total and polarized intensity, the HD 32297 disk is revealed as a sharp, almost linear feature on each side of the star along position angle (PA; measured in the usual east of north convention) of 4790 ± 017, as measured from the geometric fit presented below, where the uncertainty incorporates the astrometric calibration precision (De Rosa et al. 2020). As was found in past scattered light images of the system (e.g., Boccaletti et al. 2012), the GPI data reveal that the spine of the disk is not perfectly straight as would be the case for a perfectly edge-on viewing geometry. Instead, the spine is slightly curved and passes to the NW of the star (see Figure 3), indicating that this side is the front side of the disk under the assumption that scattering is preferentially in the forward direction. We find no conclusive evidence of the back side of the disk.

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Comparing the PSF-subtracted images of the disk to radiative transfer models can yield simultaneous constraints on both the disk geometry and dust scattering and, thus, physical properties. This is a computationally intensive task and results are often fraught with model-dependent biases and ambiguities, however. To take advantage of the high-fidelity GPI images, we instead adopt a two-step empirical approach. In the first step, we ignore the surface brightness profile along the disk, which is dictated by the surface density and scattering phase function, and focus on the spine morphology to assess the disk geometry. Having established the system geometry, we can then constrain the dust scattering properties. We defer to Section 4 for the interpretation in terms of the physical properties of the dust.

The marked curvature of the spine and the uniform vertical FWHM along the disk spine (see below) are best explained if the disk is radially narrow since a broad ring would yield a smeared appearance due to line-of-sight project effects. This allows us to employ a simple model consisting of a circular ring of radius R_d, whose center can be offset by δ_x from the star along the major axis, and observed with an inclination i. We do not explore the possibility of an offset along the minor axis of the disk as the nearly edge-on configuration of the system renders this effect negligible. To incorporate the halo of blown-out dust to this simple model, we assume that the spine extends horizontally outside the ring ansae, i.e., with no offset from the disk major axis. On the larger scale, the halo is markedly curved, but this effect is only significant outside of the GPI field of view. Implicitly, this model assumes that the disk is an intrinsically narrow ring whose eccentricity is small. For a given PA of the disk major axis and (x_⋆, y_⋆) position of the star, we measure the spine by rotating the image so that the disk major axis is horizontal, binning the image by a factor of three (i.e., a resolution element) along the horizontal axis, and fitting a Gaussian function to the intensity profile perpendicular to the disk. Uncertainties are assigned at the pixel level based on the standard deviation in concentric 1 pixel wide annuli and propagated through the Gaussian fit for both of the total intensity maps, thus neglecting residual correlated noise.

To explore the six-dimensional parameter space, we use a Metropolis–Hastings Markov Chain Monte Carlo (MCMC) algorithm. We first perform the fit using the Q_ϕ (hereafter, polarized intensity) image since (1) it provides a clear detection down to a smaller inner working angle, and (2) it is not subject to systematic biases introduced by PSF subtraction. As illustrated in Figure 3, the data are reasonably well fit by this simple model (${\chi }_{\mathrm{red}}^{2}=2.1$). The resulting model parameters are: i = 8821${}_{-0.08}^{+0.06}$, ${R}_{d}={101.7}_{-2.1}^{+1.5}$ au and ${\delta }_{x}=-{0.9}_{-1.5}^{+1.3}$ au. The corresponding posteriors are shown in Figure 4. We find no significant offset between the star and ring center, with a 3σ upper limit on the ring eccentricity of e < 0.05, yielding further support to our simple geometric model.

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We then applied the same fitting method to each of the four PSF-subtracted images. The resulting posteriors are also shown in Figure 4. For each data set, the posteriors are much narrower than the posteriors from the fit to the polarized intensity image. The total intensity posteriors are also inconsistent with one another. The narrow posteriors are a consequence of the fact that uncertainties are underestimated due to correlated residuals in the PSF-subtracted images. The offsets between the various posteriors are likely a consequence of subtle, but significant, modifications to the disk spine introduced by the PSF subtraction process. To illustrate this point, we show in Figure 3 the spine vertical offset observed in the RDI total intensity image assuming the exact same disk geometric parameters as the best fit to the polarized intensity image. Despite modest deviations from the spine location derived from the polarized intensity image, the fit is much worse (${\chi }_{\mathrm{red}}^{2}=9.9$), and marginal differences observed on both sides (especially around positions −160 and +70 au) conspire to push the fit toward significant eccentricity in the ring. Given this experience, we adopt the geometrical parameters obtained from fitting the polarized intensity image.

Overall, while our geometric modeling is in reasonable agreement with past scattered light studies (Boccaletti et al. 2012; Currie et al. 2012; Esposito et al. 2014; Bhowmik et al. 2019), we find a significantly smaller disk radius of ≈100 au instead of ≈130 au. Most of these studies used total intensity images to assess the ring geometry, thus possibly introducing a systematic bias compared to our analysis of the polarized intensity image of the disk. However, Bhowmik et al. (2019) also analyzed polarized observations and also favor a larger disk radius. Inspection of their Figure 3 reveals a similar shape for the disk spine as we find here but with a global vertical displacement that can significantly bias the model fitting. This highlights the difficulty in assessing the location of the disk ansae in the edge-on configuration. We defer a more thorough discussion of the disk's viewing geometry to Section 5.

From the same Gaussian fit as described above, we also measured the vertical FWHM of the disk. The results for the polarized intensity image are shown in Figure 5. After subtracting quadratically the instrumental FWHM from the weighted average over all positions along the disk, we estimate the true FWHM of the disk to be about 0063, or 8.3 au. While this is generally consistent with past studies (Boccaletti et al. 2012; Currie et al. 2012; Esposito et al. 2014), we differ from these studies in that we find no significant trend as a function of stellocentric distance. We believe that the trends suggested in past analyses were affected by significant PSF subtraction artifacts. This is further supported by the fact that the FWHM measured with the RDI, MI, and NMF total intensity images shows a significant decline inside of 05, well below the value measured in the polarized intensity image. The lack of a stellocentric dependency of the disk FWHM is consistent with the hypothesis of a radially narrow disk, as projection effects would result in an increase in FWHM close to the minor axis otherwise.

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Except for the ADI method, we have tuned our PSF subtraction methods with an eye toward preservation of the disk surface brightness profile. One of the main motivations to do this was to measure the polarization fraction in the disk. In the Appendix, we show that injecting a model disk into an empty data set and applying the RDI, MI, and NMF methods yields surface brightness profiles that match the injected one to within 10% or better when considering the peak surface brightness along the spine, where PSF subtraction artifacts are smallest. We then proceed to measure the surface brightness profile of the HD 32297 disk using the same Gaussian as used in our geometric analysis. We also note that, since the disk is indeed an optically thin, narrow ring seen almost perfectly edge-on, limb brightening will significantly affect the observed surface brightness close to the ansae. On the other hand, because both total and polarized intensity are affected in a similar way, we expect the polarization fraction map to be mostly free of this effect. Either way, we will take the effect into account in the radiative transfer modeling presented in Section 4.

Figure 6 presents the surface brightness profile in both polarized and total intensity. In total intensity, the profiles measured in the RDI-, MI-, and NMF-processed images agree within ≈10% of another, with the exception of a possible local maximum at ≈09 in the MI image (most noticeable on the SW side of the disk). Given the amplitude of differences between the various PSF subtraction methods (and in line with the surface brightness profile obtained by Bhowmik et al. 2019), we consider this feature, which could indicate the ring ansae, as marginally significant at best. The surface brightness profile from the ADI image has a similar shape overall but is ≈40% lower than in the other images. Overall, this is consistent with the results of our injection-recovery tests, and the match between the other three methods for the HD 32297 data set provides further confidence in the reliability of the surface brightness profiles derived here.

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Both the total and polarized intensity profiles are highly symmetrical about the star, with differences never exceeding 20% at any stellocentric distance. This is in contrast with past claims of significant asymmetries in the inner 1'' (e.g., Schneider et al. 2005; Currie et al. 2012). Subsequent analyses suggested that PSF subtraction artifacts could be misinterpreted as physical asymmetries (Esposito et al. 2014). In agreement with Bhowmik et al. (2019), we do not recover the local "gaps" observed at ≈07 in total intensity by Asensio-Torres et al. (2016). Instead, the polarized intensity profile plateaus at the location of these putative gaps, and we conclude that the PSF subtraction method employed by these authors amplified these features into apparent surface brightness deficits.

All profiles share a steep decline outside of ≈1'', i.e., in the disk halo. We performed power-law fits and found that the surface brightness profile follows approximately r⁻⁴ and r⁻⁵ in polarized and total intensity, respectively. These are in reasonable agreement with previous studies (Boccaletti et al. 2012; Currie et al. 2012; Esposito et al. 2014), although the limited field of view of our observations significantly reduces the precision of our estimates. Inside of a marked inflection point around the disk ansae, the total intensity brightness profile follows r^−1.5, with suggestive evidence for a gradual steepening toward the smallest projected separations. Again, this is in reasonable agreement with past studies of the system.

Contrary to the total intensity surface brightness profile, the polarized intensity profile displays a broad plateau over the 04–09 range. The outer edge of this plateau lies ≈15–20 au outside the ring radius inferred in Section 3.3. This may indicate that the ring has a nonnegligible radial extent, an issue that we will revisit in Section 4. Inside of this plateau, the polarized surface brightness profile follows r^−1.5, similar to the total intensity profile. While it could be tempting to interpret the break at 04 as an indication for a secondary ring (with a radius of ≈50 au), the absence of any "kink" in the disk spine at that location argues against this scenario. Instead, the central peak in polarized surface brightness must be due instead to sufficiently strong forward scattering to overwhelm the polarization decline inherent to the smallest scattering angles.

Combining the total and polarized intensity surface brightness profile, we compute the polarization fraction along the disk spine. The results are shown in Figure 7. We observe a steady rise, from about 7% at a projected distance of 035 from the star, to 15% at the ring ansae, and up to 20%–30% at 13. Our results match well with those obtained by Asensio-Torres et al. (2016). This degree of linear polarization is within the range of near-infrared observations of debris disks (Tamura et al. 2006; Perrin et al. 2015; Draper et al. 2016; Esposito et al. 2018).

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To constrain the properties of the dust grains in the HD 32297 disk, we need to extract the scattering phase function (SPF) and the polarizability curves, i.e., the dependency of the total intensity and degree of linear polarization as a function of scattering angle. Under the assumption of a narrow ring, there is a simple analytical transformation between the projected position of a point along the ring spine into a scattering angle. We therefore use the best-fit geometry derived above from the polarized intensity image to estimate the scattering angle for every point along the spine out to the location of the ring ansae. The resulting curves are shown in Figure 8. One caveat in this process is that close to the ansae, the back side of the disk can contribute significantly to the observed surface brightness since the difference in scattering angle between the front and back side is small, leading to limb brightening. Therefore, we expect that the true SPF of HD 32297 declines more steeply at the largest scattering angles than we measure here. On the other hand, if the polarizability curve is symmetric about 90° (as seen in cometary dust, e.g., Frattin et al. 2019), this effect would cancel out when we compute the polarization fraction, and we thus expect the polarizability curve we derive to be more robust.

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The H-band SPF we derive for HD 32297, which declines by a factor of about 2.5 between scattering angles 30° and 60°, where contribution from the back side should be minimal based on the disk's curved spine, is consistent with the nearly universal SPF observed for solar system, debris disks, and protoplanetary disks dust populations (Hughes et al. 2018). On the other hand, it clearly deviates from that observed in the HR 4796 debris disk (Perrin et al. 2015; Milli et al. 2017) as the latter shows a minimum at a scattering angle of ≈60°. There are too few polarizability curves published to date for debris disks to draw a definitive conclusion, but the curve we obtain for HD 32297 is much more consistent with that observed in the HD 35841 system (Esposito et al. 2018) than that in HR 4796 (Perrin et al. 2015).

4.1.1. Modeling Setup

Here, we wish to reproduce the SPF and polarizability curves derived from our observations of the HD 32297 disk. We adopt the Mie model, valid for compact, spherical dust grains of homogeneous composition. We note that observations of both laboratory and astrophysical dust populations suggest that this assumption is not optimal (e.g., Pollack & Cuzzi 1980; Hedman & Stark 2015; Milli et al. 2017). However, it is computationally tractable in the context of large dust grains, a problem not yet solved for grain aggregates that are likely to represent a better model of astrophysical dust (e.g., Arnold et al. 2019).

Two components are necessary to build a dust model: the grain size distribution and the dust composition. We follow standard approaches and assume a power-law size distribution, $N(a){da}\propto {a}^{-\eta }\,{da}$ ranging from a_min to a_max. Collisional cascade models predict a size distribution with η ≈ 3.5 (e.g., Dohnanyi 1969; Marshall et al. 2017), although deviations from a pure power law are likely (e.g., Thébault & Augereau 2007). On the other hand, the dust composition is a more challenging issue to handle. It is most often addressed either as a fixed, presupposed composition or as a mixture with variable proportions of several individual compositions (using effective medium theory). While easiest to implement, the first approach can lead to significantly biased results or, worse, a lack of a model that fits the data well if an incorrect composition is picked. The dust mixture suffers from increasing the number of free parameters and, in the worst case scenario, a critical component may be left unexplored. For instance, Rodigas et al. (2015) consider 19 different dust compositions, plus vacuum, to represent porosity, when modeling the HR 4796 debris disk. Even then, only a subset of the data is well fit by the resulting model. To circumvent these issues, we adopt a more direct approach, which consists of fitting for the material's complex refractive index m = n + i k, as this is the quantity from which Mie theory predicts the SPF and polarizability curve. A similar approach was adopted by Graham et al. (2007) in their modeling of the polarized scattered light imaging of the AU Mic debris disk and was instrumental in identifying the need for a large dust porosity in that system.

Because HD 32297 is nearly, but not quite, edge-on, we expect that the back side of the disk contributes to the signal close the ansae. To account for this effect in our models, and taking advantage of the absence of a lateral offset of the central star, we modify the Mie-computed SPF by adding the contributions of the front and back sides using supplementary scattering angles. Similarly, we compute the average of the front and back side polarized intensity signals to obtain the final version of the model polarizability curve. Approximating the disk as being exactly edge-on, we perform this correction at all scattering angles, noting that the correction is only significant close to the ansae. In addition, because monochromatic calculations can experience interference fringes in model SPF and polarizability curves, we compute the Mie models at nine wavelengths spanning the bandpass of the GPI H-band filter and average the resulting curves over the wavelength prior to computing the model likelihood. Finally, we normalize all SPFs to their average value in the 40°–60° range of scattering angle in order to focus on the shape of these curves.

We set up three independent parallel-tempered MCMC chains using the emcee package (Foreman-Mackey et al. 2013). The first one fits only the HD 32297 SPF, the second only the polarizability curve, and the third fits both curves simultaneously. In all cases, the model likelihood is based on a standard χ² test between the observed and modeled curve. Each of these runs includes 2 temperatures and 50 walkers. Walkers are initially distributed using uniform priors spanning the ranges indicated in Table 1. We remove the first 80% of each chain as a burn-in and use the final 20% to obtain values reported in Table 1, with 1440 iterations kept after burn-in for our SPF fit, 3284 iterations for our polarizability fit, and 3784 iterations for our joint fit. Inspection of the movements of walkers in the parameter space confirm that the chains are well converged.

Table 1.  Best-fitting Dust Properties Based on the SPF Alone, the Polarizability Curve Alone, and Both Curves Simultaneously

Parameter	Prior	Best-fitting Model			Median ±1σ
	Range	SPF	Polar.	Joint	SPF	Polar.	Joint
							Peak 1	Peak 2
$\mathrm{log}({a}_{\min }\,(\mu {\rm{m}}))$	[−1 .. 1]	−0.08	−0.10	−0.14	−0.09${}_{-0.01}^{+0.01}$	−0.11${}_{-0.58}^{+0.01}$	−0.143${}_{-0.004}^{+0.004}$	−0.564${}_{-0.002}^{+0.002}$
$\mathrm{log}({a}_{\max }\,(\mu {\rm{m}}))$	[1 .. 3]	2.84	0.341	2.99	${2.01}_{-0.67}^{+0.68}$	${2.07}_{-0.66}^{+0.61}$	≥2.50	≤1.05
η	[2 .. 5]	4.14	3.56	3.52	${4.21}_{-0.13}^{+0.13}$	${3.54}_{-0.20}^{+0.11}$	${3.516}_{-0.01}^{+0.008}$	3.52${}_{-0.02}^{+0.02}$
n	[1 .. 5]	3.31	2.64	3.78	${3.30}_{-0.10}^{+1.07}$	${4.13}_{-1.51}^{+0.67}$	${3.78}_{-0.03}^{+0.03}$	3.49${}_{-0.06}^{+0.06}$
log k	[−7 .. 1]	−6.16	−1.29	−1.44	≤−2.83	−1.37${}_{-0.09}^{+0.11}$	−1.44${}_{-0.01}^{+0.01}$	−0.77${}_{-0.02}^{+0.02}$

Note. The range explored for each quantity is indicated in the second column. The upper and lower limits are reported at the 95% confidence level.

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4.1.2. Results

Our final best-fitting model from each of these runs is displayed in Figure 9, with parameters described in Table 1. While both the observed SPF and polarizability curves are reasonably well reproduced when either quantity is fit separately, the corresponding reduced χ² values are 11.8 and 2.3, respectively. These imperfections are driven by the fact that the SPF (and to a lesser degree, the polarizability curve) measured on the NE and SW sides of the disk are formally inconsistent with one another, and the best-fit model is a compromise between both sides. As a result, the formal parameter uncertainties derived from the MCMC process are likely underestimated. Nonetheless, Figure 9 illustrates that our best-fitting models reproduce the overall shape of both the SPF and polarizability curves, suggesting that the values of the best-fitting parameters can be considered as reliable.

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Although the model parameters for all three fits ("SPF only," "polarizability only," and "joint") are significantly different, all three model SPFs are similar to the observed one (left panel in Figure 9). This suggests that the SPF of the HD 32297 dust disk is consistent with a large swath of the parameter space, indicating that this quantity has limited discriminatory power as far as dust properties are concerned. This is qualitatively consistent with the observations that many astrophysical dust populations share similar scattering SPFs (Hughes et al. 2018). On the other hand, the polarizability curve may be significantly more constraining, since the "SPF only" dust model is highly inconsistent with the observed polarizability curve. Specifically, due to its much steeper size distribution and very small value of the imaginary part of the refractive index, that model predicts a negative polarization at most relevant scattering angles, i.e., polarization vectors that are radially organized instead of ortho-radial. This is readily excluded by the fact that the Stokes Q_ϕ map shows only a positive signal along the disk. Unsurprisingly, the joint fit resembles the "polarizability only" fit much more than the "SPF only" fit.

Turning our attention to the best-fitting model parameters, we first note that the "joint" fit leads to two distinct families of models, as illustrated in Figure 10. The family characterized by a large value of a_max, which is referred to as "Peak 1" in Table 1, is consistent with both the "SPF only" and "polarizability" fits, and we thus consider it as the most plausible model. Besides this consistency, the other family of models is characterized by a very narrow grain size distribution (in particular, ${a}_{\max }\lesssim 11\,\mu {\rm{m}}$ at the 95% confidence level) that seems physically unlikely. In the remainder of the analysis, we focus on the first family of models.

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All three fits yield consistent minimum grain sizes, a_min ≈ 0.8 μm. Conversely, we find that the maximum grain size is constrained to be large, with a 95% confidence level lower limit of 440 μm. Finally, we note that, while the "SPF only" and "polarizability only" fits each constrain the size distribution power-law index well, they yield inconsistent values: η ≈ 4.2 and 3.5, respectively. The "joint" fit favors the latter value, which is consistent with collisional models.

In both the "SPF only" and "polarizability only" fits, we find multimodal posteriors spanning a large fraction of the explored range for the real part of the refractive index but with little overlap between one another, indicating ambiguities in the fit. Striving to achieve a compromise between the two observed quantities, the "joint" fit has a significantly narrower posterior, 3.5 ≲ n ≲ 3.8. The imaginary part of the refractive index also reveals significant tension between the "SPF only" and "polarizability only" fits: the former yields an upper limit on k, $\mathrm{log}k\lesssim -2.8$, while the latter has a well constrained posterior, $\mathrm{log}k\approx -1.4\pm 0.1$. The "joint fit" posterior prefers the latter solution with a secondary peak at $\mathrm{log}k\approx -0.8$.

We defer the interpretation of the results of our dust fitting to Section 5. For now, we note that, while the SPF and polarizability fit leave some unsolved ambiguities and tensions, the best-fitting model yields an acceptable fit to both quantities. In turn, this allows us to fix the dust properties and perform image fitting to assess the geometrical properties of the disk.

4.2.1. Modeling Setup

To model the GPI total intensity and Stokes Q_ϕ images, we use the best-fitting combined dust model derived in the previous section and explore the geometrical structure of the disk. We use the MCFOST radiative transfer code (Pinte et al. 2006) to produce synthetic scattered light images. We model the debris disk density structure with the widely used functional form

$$\begin{eqnarray*}&&\rho (r,z)\propto \displaystyle \frac{{e}^{-{\left(\tfrac{\left|z\right|}{h(r)}\right)}^{{\gamma }_{\mathrm{vert}}}}}{\sqrt{{\left(\tfrac{r}{{r}_{{\rm{c}}}}\right)}^{-2{\gamma }_{1}}+{\left(\tfrac{r}{{r}_{{\rm{c}}}}\right)}^{-2{\gamma }_{2}}}},\end{eqnarray*}$$

following Augereau et al. (1999). The critical radius, r_c, marks the transition between two power-law density regimes (with indices γ₁ > 0 and γ₂ < 0, respectively). We set γ_vert = 2 to yield a Gaussian vertical profile and a bow-tie shape for the disk, i.e., a constant h/r ratio. We further restrain the radial extent of the disk with inner and outer hard edges at radii r_in and r_out, mostly for computational purposes.

Given a disk geometry and a set of dust properties, MCFOST produces a full Stokes synthetic datacube with pixel scale, orientation, and field of view set to match our GPI observations. The Stokes Q and U maps are converted to a Stokes Q_ϕ image, while the star is masked out of the Stokes I image, before both are convolved by the instrumental PSF as estimated by the satellite spots. We then mask regions that are closer than the inner working angle of the Stokes I image to only consider the same pixels that were used in deriving the SPF and polarizability curve in the previous section. We also set an outer radius of 16, outside of which no trustworthy data are available. Finally, we mask out pixels that lie more than 035 from the disk spine to ensure that the fitted region include both disk-dominated and background-dominated pixels. A likelihood is then computed using a pixel-by-pixel χ² calculation. Exploration of the parameter space is conducted through three independent MCMC processes—one fit for Stokes I, a second for Stokes Q_ϕ, and a third joint fit—each using three temperatures and 100 walkers. Our final results again include only the final 40%–45% of each MCMC chain, when the chains had visually achieved convergence (in total, this includes 1660, 2420, and 1380 iterations for our Stokes I fit, Stokes Q fit, and joint fit, respectively). Consistent with the dust modeling conducted above, we adopt the MI PSF-subtracted total intensity image of the disk.

In this aspect of our modeling, the geometrical free parameters are the disk inclination (i), the critical radius (r_c), the volume density power-law indices (γ₁, γ₂), the reference scale height (h₀, defined at r₀ = 100 au), and the disk inner radius (r_in). Given the large halo that extends well beyond the GPI field of view, we cannot constrain the disk outer radius with our data and, thus, set r_out = 200 au. Finally, we set the total disk mass (M_d) as a free parameter that defines the total amount of dust in the system, based on a representative grain density of 3.5 g cm⁻³. So long as the disk remains optically thin, this acts as a simple multiplicative factor that serves to adjust the absolute surface brightness of the model to the observed one. We initialize γ₁ and γ₂ with uniform distributions, and all other free parameters are assigned a Gaussian prior, either based on our empirical geometrical analysis (i, r_c, h₀, from Section 3.3) or assuming a conservatively broad range (r_in, M_d). The explored ranges for each parameter are indicated in Table 2.

Table 2.  Best-fitting Geometrical Properties Based on the Stokes I Image, the Stokes Q_ϕ Image, and Both Images Simultaneously

Parameter	Prior Ranges		Best-fitting Model			Median ±1σ
	Initial	Full	I	Qϕ	Joint	I	Qϕ	Joint
i (°)	87 ± 1	[70 .. 90]	88.84	88.65	88.74	${88.88}_{-0.02}^{+0.01}$	${88.59}_{-0.05}^{+0.04}$	88.75 ± 0.02
h0 (au)	5 ± 2	[0.1 .. 10]	0.10	0.10	0.10	0.13 ± 0.02	${0.102}_{-0.001}^{+0.002}$	${0.11}_{-0.01}^{+0.04}$
rc (au)	100 ± 20	[50 .. 150]	99.79	95.81	98.35	${97.74}_{-0.79}^{+0.96}$	${93.87}_{-0.90}^{+1.21}$	${98.20}_{-0.56}^{+0.37}$
rin (au)	50 ± 20	[1 .. 100]	50.50	7.38	39.85	${51.8}_{-2.8}^{+2.2}$	${22.2}_{-3.0}^{+7.8}$	${41.8}_{-3.4}^{+2.6}$
γ1	[0 .. 5]	[0 .. 5]	4.01	3.08	3.42	${3.77}_{-0.20}^{+0.23}$	${3.44}_{-0.31}^{+0.32}$	${3.37}_{-0.12}^{+0.11}$
γ2	[−5 .. 0]	[−5 .. 0]	−4.79	−4.96	−4.96	$-{4.60}_{-0.12}^{+0.10}$	≤−4.84	≤−4.87
${\mathrm{log}}_{10}({M}_{{\rm{d}}}({M}_{\odot }))$	−9 ± 2	[−12 .. −4]	−7.60	−7.31	−7.57	−7.61 ± 0.01	$-{7.37}_{-0.02}^{+0.01}$	$-{7.60}_{-0.02}^{+0.01}$

Note. The full range explored for each quantity is indicated in the second column, whereas the initial range indicates the Gaussian prior that is used for most parameters.

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4.2.2. Results

The results of our MCMC chain are summarized in Table 2. Figure 11 displays the full posterior distribution for all parameters in the combined fit, and Figure 12 shows the model images for the overall best-fitting model. While the posteriors appear multimodal, particularly for H₀, we inspected the movement of the walkers in the MCMC chains to confirm that the chains had been decoupled from their initial state. The results of fitting separately the Stokes I and Q_ϕ images are mostly similar to those of the joint fit; however, the scatter in some of the parameter values exceeds the nominal uncertainties from the MCMC chains. For instance, fitting the polarized intensity image yields a 10% smaller disk radius and a 02 lower inclination. In the following, we consider all three separate fits holistically in our analysis.

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Overall, the best-fitting model reproduces well the observed images, at least within the region used in the likelihood function. However, given the high signal-to-noise of our data set, the residuals are statistically significant, indicating that the model has some shortcomings. In particular, both the Stokes I and Q_ϕ residual maps reveal a thin trace along the disk spine, suggesting that the model is slightly too extended vertically. Since we have allowed the vertical thickness of the disk to be very small (h/r as low as 0.1%), it is possible that this is due to our use of a slightly too broad instrumental PSF. We also find systematic residuals in the Stokes Q_ϕ image at the location of the ring ansae. This may be a consequence of imperfection in the dust scattering properties as derived in the previous section. Furthermore, we also note that the best-fitting model underpredicts the polarized intensity in the immediate vicinity of the inner working angle of that image, a region not included in the fit. Finally, there are marginally significant positive residuals in the total intensity image outside of the ring radius. This is likely due to the lack of treatment of the halo in our model; however, we stress that only the region within the ring radius is strongly detected in our data. Altogether, in spite of these limitations, we consider that the quality of the fit is sufficient to warrant a discussion of the main results from our exploration of the parameter space.

All geometric parameters are well constrained in the fit, except for r_in as a consequence of the steep inner surface density profile. Considering first the joint fit (to the Stokes I and Q_ϕ images simultaneously), we derive an inclination of i ≈ 887 ± 01, which is about 05 higher than the best-fit value obtained in the geometrical analysis in Section 3.3. This is an indication that either our model is imperfect, or some of the assumptions that we used in empirically deriving the ring geometry are incorrect. This is further supported by the fact that we find a rather broad ring, with an inner radius 2–4 times smaller than r_c ≈ 100 au. Nonetheless, the latter is consistent with the ring radius we had derived in Section 3.3. Although dust extends over a broad range of stellocentric distances, the power-law volume density profiles are relatively steep (γ₁ ≈ 3.5 and γ₂ ≲ −4.5). The surface density profile is characterized by an FWHM of about 40 au. This ≈40% radial width is uncomfortably high to fully validate the narrow ring approximation of our initial geometric fitting and derivation of the SPF and polarizability curves. Nonetheless, the effect of this width is to blur the location of the ring spine and the dependencies on scattering angle, not to systematically bias these. We therefore expect our prior estimates to be representative of the true quantities, which is supported by the good match in the mean ring radius, for instance.

The surprisingly small disk scale height (h/r ≈ 0.2%) appears to contradict our finding that the disk is marginally resolved along the vertical direction (see Section 3.3). Aside from the possibility that the PSF we used in the image modeling may not be a perfect match to the HD 32297 data set, this may be an indication that the vertical density distribution is not Gaussian. If the profile is more condensed in the center, e.g., following a Lorentzian or exponential profile, the assumption of a Gaussian profile in both our initial geometrical analysis and in radiative transfer modeling would overestimate slightly the vertical extent of the disk. Finally, the PSF subtraction process could have slightly attenuated the lower surface brightness regions away from the midplane in the total intensity image, thus leading to a similar effect. The fact that the fit to the polarized intensity image also favors a very small disk thickness rather points to other explanations, however.

Finally, the total dust mass, M_d ≈ 0.01M_⊕, should be considered with caution, as this quantity is strongly correlated with dust properties, particularly the minimum grain size and porosity. Since we have not attempted to fit for an actual composition, the true mean grain density is not a parameter of our model and is degenerate with the total mass. It is nonetheless interesting to note that this is much smaller than the dust mass derived from the millimeter emission of the system (≈0.6M_⊕; MacGregor et al. 2018).

The combination of high angular resolution and exquisite image fidelity enabled by GPI offers an opportunity to determine the ring geometry in a precise manner. This is further enhanced by the fact that the polarized intensity image does not require any PSF subtraction. It is thus interesting to note that the disk radius that we determined here, both from the direct geometric analysis and from the direct image fitting (Sections 3.3 and 4.2, respectively), is markedly smaller than has been found in past studies: around 100 au compared to 130 au. We emphasize that this difference is not a result of the updated distance to the system, as we have already accounted for it. In other words, the angular radius of the ring we find is about 40% smaller than those reported in previous studies. Notably, the two methods we employed rely on very different aspects of the data. Image fitting is inherently weighted by the signal-to-noise and, thus, by the local brightness of the disk, which places a different emphasis on different regions of the disk. The geometric approach, instead, is mostly independent of the surface brightness profile. Arguably, the latter is a more robust approach to determining the disk geometry. In particular, self- and over-subtraction effects have a much more direct influence on the surface brightness distribution, and inadequately taking them into account is more likely to introduce biases than focusing on the spine of a nearly edge-on disk like HD 32297. The latter is now precisely traced as close as 012 from the star (this study; see also Bhowmik et al. 2019), and the remaining dominant source of uncertainty may actually be the location of the star itself. In particular, Bhowmik et al. (2019), who derived a disk radius from their SPHERE images that is consistent with past studies, suggest an offset of ≈001 of the star in the direction perpendicular to the disk, whereas our analysis reveals no such offset. While the nominal precision in the position of the star with instruments such as GPI and SPHERE is significantly better, this suggests that systematic uncertainties are not fully understood in these complex instruments.

Despite these systematic errors associated with scattered light images, the submillimeter emission of the system supports an 80–120 au radial range for the ring (MacGregor et al. 2018). Even though the inferred surface density profile rises as roughly r² in their best-fitting model, the r⁻² illumination dependency of impinging starlight yields a flat surface brightness profile and, thus, a roughly 100 au radius for the scattered light ring. Similarly, the mid-infrared emission from the system suggests an inner disk radius of about 80–90 au (Fitzgerald et al. 2007; Moerchen et al. 2007). While the scattered light images of the system may probe physically distinct grains (and there is evidence for millimeter-emitting dust in the halo surrounding the parent body belt; MacGregor et al. 2018), it seems implausible that the scatterers would be located exclusively outside of the parent body belt. This is definitely not the case in the well-studied, lower inclination, HR 4796 system (Kennedy et al. 2018). We therefore conclude that the HD 32297 ring is indeed centered at about 100 au, as inferred from the modeling of our near-infrared image.

Outside of the parent body ring, the HD 32297 system is characterized by a large-scale halo structure that lies mostly outside of our field of view. Consistent with past imaging, our observations confirm that the disk extends radially beyond the ring ansae, smoothly connecting the parent body ring and the outer halo. This confirms that the dust located in the halo most likely represents small dust grains that originated in the parent body belt before being radiatively pushed on high-eccentricity orbits, where another mechanism then sweeps them out in the NW direction. The fact that the halo is undetected to the NW of the star in our total intensity image, despite this region being the brightest of the halo (e.g., Schneider et al. 2014), could simply be due to the use of PSF subtraction techniques that effectively cancel out extended, low-gradient surface density structures. On the other hand, our polarized intensity image is free of such an effect and, yet, we find no evidence of the presence of the halo. To assess the meaningfulness of this non-detection, we compare our Q_ϕ image with the Hubble Space Telescope (HST)/STIS broadband image from Schneider et al. (2014) in the following manner: we compute surface brightness profiles in 015 bands orthogonal to the disk midplane and located 075 on either side of the star, i.e., roughly at the disk ansae. Both profiles are averaged to improve signal-to-noise given the lack of marked asymmetry in the halo within the central arcsec. We further rebin the GPI data to roughly match the 005 pixel scale of the STIS image. The resulting surface brightness profiles are shown in Figure 13. The swept-back halo is clearly visible in the STIS surface bright profile, most prominently as an extended structure to the NW of the disk, but it is absent in the GPI polarized intensity image with a high degree of significance. Besides the NW extension of the profile, we also find the GPI surface brightness profile to be much narrower around the disk spine than the STIS one. This indicates that the halo also extends radially in front of the parent body ring.

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There are two main possible explanations for the lack of detection of the halo in the STIS Q_ϕ image despite the high signal-to-noise detection of the ring itself: either the halo is much bluer than the main ring, or it is characterized by a significantly lower polarization fraction. The latter is inconsistent with the observations that the polarization fraction keeps rising outside of the parent body ring (Figure 7). On the other hand, the large-scale structure of the system has long been known to be much bluer than the star itself (Kalas 2005), whereas the main ring itself is neutral or slightly red (e.g., Esposito et al. 2014; Rodigas et al. 2014). We therefore believe that the blue color of the halo is primarily responsible for the lack of detection in our data set. Additionally, we note that the halo is also not apparent to the NW of the star in the J-band Q_ϕ SPHERE image of the system (Bhowmik et al. 2019). Overall, the only evidence for the halo in near-infrared images of the system is the curved extension of the disk midplane beyond the ansae due to limb brightening.

One of the motivations to obtain high-fidelity scattered light images of debris disks is to constrain the properties of the dust grains they contain. The surface brightness and color of debris disks are the primary observables affected by dust composition in scattered light images. In addition, polarization measurements provide further information regarding the porosity of grains, since, all else equal, large, porous grains have similar properties to smaller, compact grains (Graham et al. 2007; Shen et al. 2009). Assuming that Mie theory accurately describes the scattering properties of dust grains, the minimum grain size and the power-law index for the grain size distribution should be tightly constrained by measurements of the scattering phase function and polarization fraction. Indeed, our modeling successfully reproduced both the SPF and the polarizability curve observed for HD 32297. The size distribution inferred from our modeling, with a minimum grain size of ≈1 μm that is commensurable with the blowout size and a slope consistent with collisional cascade models, is in good agreement both with past studies of the system (including through thermal emission; see for instance Donaldson et al. 2013) and with general theoretical expectations.

Despite these apparent successes, it is important to emphasize that the refractive index derived from our analysis lies in a region of the parameter space that is far from all standard dust species, as illustrated in Figure 14. Worse still, no combination of such species (including void to represent porosity) is consistent with the inferred refractive index. This casts serious doubt on the physical meaning of the other dust parameters that were considered in this analysis. In other words, while we did find a combination of parameters that well reproduces the observed SPF and polarizability curve, it may be the case that this is only a practical empirical model but not one to be trusted at the physical level. There is increasing evidence that dust grains in both the solar system and in debris disks are aggregates of smaller, sub-micron monomers (e.g., Bentley et al. 2016), in which case the Mie model is irrelevant. Unfortunately, despite significant strides toward characterizing the scattering properties of aggregates, it remains beyond the reach of current models to consider aggregates whose sizes exceed the blowout size by one or more order of magnitude (Arnold et al. 2019), which we know are present in debris disks. Furthermore, it may also be instructive to revisit the assumption that the grain size distribution follows a simple power law. Collisional models suggest a more complex underlying structure when factoring in the effects of stellar gravity and radiation pressure in addition to the collisional cascade replenishing the disk (e.g., Krivov et al. 2006; Thebault et al. 2014).

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Leaving aside the physical interpretation of the SPF and polarizability curves, our data provide a robust empirical characterization of the scattering properties of the HD 32297 dust ring. With the number of debris disks with estimated SPFs and/or polarizability curves slowly rising, it is now possible to perform model-independent comparisons between systems to identify commonalities and differences between systems. Hughes et al. (2018) pointed out that most solar system dust populations share a similar SPF and that the few debris disks with estimated SPFs also match that template. The SPF we have derived for HD 32297 is also in reasonable agreement with that "generic" SPF. On the other hand, the SPF determined by Milli et al. (2017) for the HR 4796 ring is markedly different. Combined with the unusual polarization fraction curve observed in that system (Perrin et al. 2015), this suggests that this latter disk is characterized by a markedly different dust population. While such comparisons are best performed by extracting the SPF from observations, this process suffers from possible ambiguities and possible biases, as we have already discussed.

To illustrate the effects of a different SPF on the appearance of a debris disk, we compared the modeled surface brightness profile along the spine of a nearly edge-on disk (using the geometric parameters indicated in Table 2) assuming three distinct SPFs: the best-fitting Mie model presented in Section 4.1, the "generic" SPF from Hughes et al. (2018), and the HR 4796 SPF from Milli et al. (2017). The latter SPF is not defined at all scattering angles due to our particular viewing geometry of the system, so we performed linear extrapolations of the SPF for scattering angles <15° and >165°. These regions are behind the coronagraphic mask once the disk is observed with the viewing geometry of HD 32297; therefore, the details of this extrapolation are not critical to the comparison. The results of this exercise are illustrated in Figure 15. All three models under-predict the surface brightness profile outside of the main ring radius, but both the best Mie model and the generic SPF match the data extremely well inside of that projected distance. This confirms that the total intensity scattering properties of the HD 32297 dust are consistent with most other astrophysical dust populations. Conversely, if this disk was characterized by an SPF that is similar to that observed for HR 4796, its surface brightness profile would be dramatically different, with a nearly flat surface brightness profile that is inconsistent with the observations. This is due to the combination of (1) the fact that the HR 4796 SPF has its minimum at a scattering angle of ≈50° with significant backscattering at angles ≳100°, and (2) limb brightening in the optically thin ring. This is further evidence that the scattering properties in the HD 32297 and HR 4796 debris disks are clearly distinct.

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An additional qualitative feature of the SPF in HD 32297 is the sharp peak observed in polarized intensity close to the inner working angle of our observations. Since the polarization fraction is expected by symmetry to drop to zero at 0° scattering angle, this indicates the SPF itself must be characterized by a very sharp forward scattering peak, reminiscent of those of HR 4796 and β Pic (Millar-Blanchaer et al. 2015; Perrin et al. 2015; P. Arriaga et al. 2020, in preparation). On the other hand, there are several edge-on debris disks that have been imaged in polarized intensity that do not show such a feature (Olofsson et al. 2016; Engler et al. 2017; Esposito et al. 2018, 2020, submitted). This further hints at the fact that the scattering properties of dust populations in debris disks are not all identical; however, interpreting them in terms of physical properties of the grains may still be out of reach.

Finally, another qualitative approach to constraining the SPF in the case of nearly edge-on disks like HD 32297 is to assess whether the back side of the ring contributes to the scattered light image. Bhowmik et al. (2019) presented a tentative detection in total intensity using ADI PSF subtraction, which would imply a strong backscattering peak. We do not confirm this feature in our observations of HD 32297. It is possible that differences between PSF subtraction introduce different artifacts, precluding a definitive conclusion. However, we point out that, given the derived disk radius and inclination, the projected separation along the minor axis between the front and back side of the ring is in the 004–007 range, which makes it extremely challenging to detect. In polarized intensity, which is unaffected by PSF subtraction, the lack of an increase in vertical extent of the disk just inside of the ansae indicates that the back side of the ring does not contribute significantly to the observed surface brightness, indicating that the SPF drops significantly around a scattering angle of 90°. This latter conclusion matches our conclusion that the SPF of HD 32297 is qualitatively different from that observed in HR 4796.

Debris disks are believed to be associated with planetary-mass objects, as readily illustrated in some well-known systems, such as β Pic and HR 8799. No point source is evident in our total intensity image of the system. We note, however, that observations with GPI's polarization mode are not optimal for detection of planets unless they are highly linearly polarized. Instead, integral field spectroscopy observations provide the deepest search for planets. Bhowmik et al. (2019) presented such observations of HD 32297, reaching a contrast limit of 10⁻⁵ or better outside of 05. At an assumed age of 30 Myr, this corresponds to an upper limit on any planetary-mass object of 4–5 M_Jup based on the COND models (Baraffe et al. 2003). An important caveat, however, is that the detection limit is much worse along the bright disk spine, so that the upper limit computed only applies outside of the plane of the disk. Specifically, a 10⁻⁵ contrast point source would have its peak pixel brightness equal to that of the disk and, therefore, would be marginally detectable at best, at a distance of ≈12 from the star if it lies in the plane of the disk. At closer separation, the contrast degrades proportionally to the disk peak surface brightness, reaching 10⁻⁴ (or a planet mass of ≈12 M_Jup) at ≈045.

An indirect probe of the presence of planetary-mass bodies in the system is through their dynamical interaction with the disk. The lack of significant lateral asymmetry and of local (photometric or morphological) perturbation in the parent body belt supports the picture of a dynamically cold, azimuthally symmetric system, seemingly ruling out strong planet–disk interactions. The scale height of the belt can be also related to its dynamical excitation, since the scale height is directly related to the velocity dispersion of the solid bodies. The disk scale height derived from our geometric analysis (h/r ≈ 0.04) is consistent with dynamical models that only consider collisions between grains and radiative forces (Thebault 2009). Therefore, the dynamical state of the HD 32297 main belt can be fully explained without invoking the presence of planetary-mass bodies stirring the system. A planet could, however, be responsible for the inner dust depletion (inside of 30–50 au) without introducing measurable local perturbation. In addition, Lee & Chiang (2016) proposed that an interior planet on an inclined orbit is responsible for the "double wing" in the extended outer halo (Schneider et al. 2014); however, this could also arise from interaction with secondary gas (Lin & Chiang 2019). Current observations of the HD 32297 system are thus inconclusive regarding the presence and structure of its planetary system.