Spiral Arm Pattern Motion in the SAO 206462 Protoplanetary Disk

We measure the spiral locations after transforming the deprojected r²-scaled ${{ \mathcal Q }}_{\phi }$ images into polar coordinates, where the horizontal axis is the counterclockwise angular deviation θ from the northwest semiminor axis of the disk, and the vertical axis is the stellocentric distance r. For each θ, we fit a Gaussian profile to its radial profile with scipy.optimize.curve_fit to obtain the peak location r with an error⁶ of δr.

When we inspect the radial profiles for the θ values, we notice multiple peaks or flat plateau in some regions in the S2 spiral, which is indicative of resolved and unresolved sub-spirals; we ignore these points to minimize their potential impact for location and subsequent speed measurement. See Figure 2 for the (θ, r ± δr) measurements used for subsequent analysis.

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To constrain the motion pattern for the spirals, we ignore features that can bias our results. For example, the spirals joining the edge of the coronagraph (S1: θ ≳ 360°; S2: θ ≳ 180°), and the branching feature at the S2 tip (e.g., Figure 1 inset), may bias our spiral arm location measurement. The r²-scaled surface brightness measurement of S3 is a factor of ∼ 5 lower than that of S2 in Figure 1, and thus the influence from the merging of S2 and S3 should be less than ∼ 20%. Therefore, we focus on 220° ≤ θ ≤ 360° for S1, and 100° ≤ θ ≤ 180° for S2. In our measurement, we additionally ignore the S1 blob (specifically, 237° ≤ θ ≤ 269°) for S1, see Section 3.2.1 for the justification. In Figure 2, we present the selected data points with 1° step for θ; we also present the S1 blob for the purpose of illustration only. We denote these chosen angles in Figure 1 by projecting a three-dimensional setup of a disk to the sky plane (Appendix A of Ren et al. 2019).

We constrain the morphology and the angular movement between the two epochs for each arm under two hypotheses. A (θ, r) pair will advance to a location of (θ + Δθ, r) between the observations. In the GI-induction scenario, each part of the arm moves at the local Keplerian speed, Δθ ∝ r^−3/2.⁷ In the companion-driven scenario, the entire arm corotates with its driver planet as a rigid body, thus ${\rm{\Delta }}\theta =\mathrm{Constant}$ and traces the motion of the driver.

We fit p-degree polynomials to the (θ, r ± δr) pairs in both epochs. Following Ren et al. (2020), we use dummy variables as proxies to simultaneously obtain morphological parameters for a spiral and speed for pattern motion. Noticing that the two arms might be moving under different rates, we begin with independent fitting for them. In this Letter, positive pattern speed corresponds to counterclockwise rotation.

3.2.1. Independent Motion

Spiral arms are expected to be trailing features in protoplanetary disks. The orientation of the spirals in SAO 206462 indicates that the disk is rotating counterclockwise. Nevertheless, we first obtain an S1 pattern speed of ∼ 1° yr⁻¹ clockwise under both the GI-induction and the planet-driven scenarios. Such a motion is in the opposite direction of the expected disk rotation inferred from spiral morphology. We notice that the result originates from the points at 237° ≤ θ ≤ 269°, see Figure 2. Such a region has been identified as a kink in Stolker et al. (2016a) and the S1 blob in Maire et al. (2017), which matches the location of a hypothesized forming planet based on arm morphology fitting (Muto et al. 2012). We thus exclude the S1 points at 237° ≤ θ ≤ 269°, which have ≳ 1 au residuals when we do not ignore them in our fitting, to minimize the impact from unresolved spirals around a forming planet (e.g., the twist in Boccaletti et al. 2020) on our pattern motion measurement.

The S1 pattern speed is 132 ± 013 yr⁻¹ in the planet-driven scenario. Taking into account the 008 instrumental north uncertainty of SPHERE (Maire et al. 2016), we obtain a propagated uncertainty⁸ of $\sqrt{{[1.00\mathrm{yr}\times (0\buildrel{\circ}\over{.} 13{\mathrm{yr}}^{-1})]}^{2}\,+\,2\,\times \,0\buildrel{\circ}\over{.} {08}^{2}}/(1.00\,\mathrm{yr})=0\buildrel{\circ}\over{.} 18$ yr⁻¹. For a ${1.6}_{-0.1}^{+0.1}\,{M}_{\odot }$ central star (Garufi et al. 2018), this corresponds to a driver located at ${49}_{-5}^{+6}$ au assuming a circular orbit.

The S2 pattern speed is 038 ± 006 yr⁻¹ in the planet-driven scenario. Taking into account the instrumental uncertainty, the motion rate is 038 ± 013 yr⁻¹. For a ${1.6}_{-0.1}^{+0.1}\,{M}_{\odot }$ central star, this corresponds to a driver located at ${120}_{-30}^{+30}$ au assuming a circular orbit.

We summarize the planet-driven motion rates in Table 1. We do not further calculate the individual motion under GI given such a treatment is less physically motivated; instead, we perform a joint GI motion in Section 3.2.2.

Table 1.  Pattern Motion Measurement for SAO 206462 Spirals

Mechanism	Parameter	Independent Fit		Joint Fit
		S1	S2	S1 and S2
Planet-Driven	Rotation Rate (yr−1)	132 ± 018	038 ± 013	057 ± 013
	Driver Locationa (au)	${49}_{-5}^{+6}$	${120}_{-30}^{+30}$	${86}_{-13}^{+18}$
	Driver Orbital Perioda (yr)	${270}_{-30}^{+50}$	${950}_{-240}^{+490}$	${630}_{-120}^{+190}$
GI-Induction	Rotation Rateb (yr−1)	⋯	⋯	046 ± 014
	Enclosed Massc (M⊙)	⋯	⋯	${0.10}_{-0.05}^{+0.08}$

Notes.

^aThe driver has a circular orbit along the midplane of the disk, its location is calculated for a ${1.6}_{-0.1}^{+0.1}\,{M}_{\odot }$ central star. ^bThe rate for GI-induction is calculated for a location of 40 au, multiply the rate by ${\left(\tfrac{r}{40\mathrm{au}}\right)}^{-3/2}$ to obtain that for other locations. ^cEnclosed mass within 39 au, which is inferred from Keplerian motion.

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3.2.2. Comotion

We obtain the best-fit p-degree polynomial description, $r(\theta )={\sum }_{j\,=\,0}^{p}{c}_{j}{\theta }^{j}$, where $p\in {\mathbb{N}}$ and ${c}_{j}\in {\mathbb{R}}$ is the coefficient for the jth term, of the spirals by minimizing the Schwarz information criterion (Schwarz 1978) that penalizes excessive use of parameters. For the S1 arm, the best-fit p = 8; the S2 arm, p = 5. These best-fit p parameters apply to both the GI-induction and the planet-driven scenarios.

We compare our motion measurement with a cross-correlation analysis of disk images (e.g., Ren et al. 2018). For the selected regions in Figure 1, the best-fit motion rate based on cross-correlation is − 12 ± 579 yr⁻¹ for S1, 11 ± 438 yr⁻¹ for S2, and − 22 ± 567 for both. These rates, which report the motion in the planet-driven scenario, are dominated by shadowing effects since most of the disk in 2016 is ∼ 0.7 × the brightness in 2015 (with the exception of the northwest S1 arm: ∼ 1.3 × ) in Figure 1. In addition, the uncertainty from cross-correlation analysis traces the broadening of the signals (Tonry & Davis 1979), thus the width of the spirals along the radial direction here, which is less informative on the real motion of the spirals. We therefore do not adopt the results from cross-correlation analysis. In this Letter, instead, we approximate the dust distribution for each angle with a Gaussian profile to locate the spines for the spiral arms, since we do not expect shadows to affect the radial distribution of dust particles. We note that an eccentric driver in the Calcino et al. (2020) simulation may drive the spiral arm motion differently; however, the corresponding arm motion has not been characterized.

We have assumed that the disk is infinitely thin in our deprojection procedure. Nevertheless, Andrews et al. (2011) report for SAO 206462 an aspect ratio, which is defined as the ratio between vertical scale height and radial separation (i.e., h/r), of $0.096{\left(\tfrac{r}{100\mathrm{au}}\right)}^{0.15}$ using the Submillimeter Array at 880 μm. We use diskmap (Stolker et al. 2016b) for the deprojection of the system to address such effects, and find that in the planet-driven scenario, the S1 motion is 112 ± 012 yr⁻¹, the S2 motion 038 ± 007 yr⁻¹, and the comotion 057 ± 007 yr⁻¹; all of these are consistent with our previous results within 1σ. In addition, although millimeter observation traces a different layer of dust from the scattered light observations, we expect that the impact from such a difference is not important given the low inclination of this system (e.g., see Ren et al. 2020 for their experiment on the impact of disk flaring).

We have used all data points in Figure 2 except the S1 blob in our motion analysis. To address possible bias from individual data pairs, we randomly discard 20% of the pairs and repeat the motion analysis procedure 10⁴ times. We find that for S1, the best-fit rotation rate for the driver is 132 ± 007 yr⁻¹, 038 ± 006 yr⁻¹ for S2, and 057 ± 004 yr⁻¹ for comotion; all within 1σ from our initial measurements.

We have only used the two observations that have the longest exposure times for motion analysis. To address the contribution from the three shorter observations, we repeat the measurement using all five epochs. We exclude the third epoch due to its compromised data quality with a 17 minute exposure and a seeing larger than 2''. For S1, the four-epoch result under the planet-driven scenario is 133 ± 012 yr⁻¹, which is consistent with the previous two-epoch result within 1σ. For S2, we obtain large uncertainties in the Gaussian fit for spiral arm location measurement. Furthermore, we could not properly approximate the data points at 150° ≲ θ ≲ 180° using Gaussian profiles in the two epochs with 34 minute exposures. We notice that the S2 arm is ∼ 2 times fainter than S1 in Figure 1, we thus conclude that a 34 minute exposure is not sufficient in capturing the S2 arm with high data quality. In addition, we note that these shorter exposures can only establish a 40 day timeline, which is a factor of 9 less than the longer exposures in Section 3. Therefore, we do not include the three short exposures in our analysis.

We investigate the robustness of our measurements using the two 34 minute observations on 2016 June 22 and 30 that establish a 7.9 day separation. After propagating the instrumental north uncertainty, we obtain that the angular motion rates under all planet-driven scenarios are 0° ± 6° yr⁻¹, or 000 ± 013 between the observations. The planet-driven arm motion in Table 1 during this 7.9 day period is expected to range from 001 to 003, which is within the 1σ interval of the estimate using the 2016 June data. In addition to instrumental north and statistical uncertainties, the uncertainty in our measurements may originate from effects including (but not limited to) compromised data quality with 34 minute observations, random noise, and shadows from inner disks. Specifically, shadows with a 1 week period can trace down inner disks at 0.1 au. Nonetheless, we cannot properly decompose these effects until time series monitoring of the system is available.

For the S1 arm, its motion is consistent with being driven by a planet when we exclude the S1 blob region. The driver is located at ${49}_{-5}^{+6}$ au assuming a circular orbit, which coincides with the stellocentric radius of the S1 blob. In comparison, on the one hand, Muto et al. (2012) fit the spiral morphology and report a theorized driver at 53 au, while Stolker et al. (2016a) obtain 24 au assuming the driver is located within the cavity: our morphological fit uses an identical observation and method, but, without such a requirement, returns loose constraints for the S1 driver—the position angle is 40° ± 100° and the stellocentric separation is 19 ± 15 au. On the other hand, Maire et al. (2017) identify the Muto et al. (2012) driver region as a bright S1 blob, and report a redder spectrum for the S1 blob than the spirals. Despite these results, we cannot constrain the position angle for the dynamically measured driver in this Letter, and thus we do not try to over-interpret the findings here. Nevertheless, such a location overlaps with the ALMA millimeter ring (see Figure 3, e.g., van der Marel et al. 2016; Cazzoletti et al. 2018). This overlapping could be explained by planet-disk interaction when two conditions—low mass planet and low disk viscosity—are met (e.g., Facchini et al. 2020).

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For the S2 arm, its motion is consistent with being driven by a planet at ${120}_{-30}^{+30}$ au assuming a circular orbit. This agrees with predictions based on morphological estimates (e.g., Muto et al. 2012; Bae et al. 2016; Dong & Fung 2017). Our morphological analysis based on static images for S2 returns a position angle of 20° ± 2° and location of 75 ± 2 au for the driver; yet we caution that the ignored regions, as well as the possibility of two interacting planets and thus spirals, may change the results.

Assuming S1 and S2 are driven by individual planets with circular orbits, we present the semimajor axes for the two hypothesized arm-driving planets in Figure 3. Comparing the motion rates for the two arms under this scenario, we obtain a 3.0σ difference, which offers tentative evidence that the two spirals are moving independently. Nonetheless, the colocation of the ALMA ring and the S1 driver from motion measurement requires a less massive planet (e.g., Facchini et al. 2020), which is in tension with the expectation that spiral arms with high arm-to-disk contrast should be excited by massive planets (e.g., Dong & Fung 2017). To further evaluate the difference between the spiral motion rates, we expect that a re-observation of the system after year 2020, which will establish a >5 yr timeline for motion measurement, is necessary (e.g., Ren et al. 2020).

We obtain the direct imaging constraints on the mass of the S2 driver with hot-start evolutionary models (i.e., Sonora, Bobcat; M. Marley et al. 2020, in preparation) using 2400 s of Keck/NIRC2 $L^{\prime}$-band archival observation on 2016 May 27 (Program ID: C264N2, PI: D. Mawet). We preprocess the data following Xuan et al. (2018), and subtract the stellar point-spread function using a principal-component-analysis-based speckle subtraction method with a matched filter (FMMF; Ruffio et al. 2017). We adopt the W1 magnitude of SAO 206462 (5.41 ± 0.05; CatWISE2020: Marocco et al. 2020) as its $L^{\prime}$-band magnitude, then follow Ruffio et al. (2018) to transform the Gaussian-distributed S2 pattern speed to planetary mass limit (with speed boundaries being the 2σ lower limit of S2 motion and the best fit of S1 motion). We obtain a planet-to-star flux ratio upper limit of 1.5 × 10⁻⁴ in $L^{\prime}$ band (i.e., ${\rm{\Delta }}L^{\prime} =9.6$) at 97% confidence level. Adopting an age of ${12}_{-6}^{+4}$ Myr (Garufi et al. 2018), we calculate a mass upper limit of 13 M_Jupiter. This upper limit is consistent with the driver mass in theoretical predictions (5–15 M_Jupiter: Bae et al. 2016; Dong & Fung 2017).

A driver on a highly eccentric orbit may excite spiral arms (e.g., eccentricity e = 0.4 for the MWC 758 system: Calcino et al. 2020). Nevertheless, the corresponding pattern motion characteristics have not been characterized yet. We therefore only discuss the impact of less eccentric drivers that do not trigger wiggles or bifurcations (e ≲ 0.2: Li et al. 2019; Muley et al. 2019). When e = 0.2, the S2 driver has a possible range of 102 au to 118 au, which is consistent with the estimated 1σ uncertainty in Table 1.

The motion rates of the two spirals differ by 3.0σ, yet the symmetry of the two spirals calls for a single-planet driver (e.g., Bae et al. 2016; Dong & Fung 2017). In comparison with the double-planet scenario where we obtain a χ² value of 1617 assuming independent Gaussian noise, we obtain χ² = 1662 in the single-planet scenario. The χ² difference between the two scenarios is Δχ² = 1662−1617 = 45. We adopt the Akaike information criterion (AIC) and Schwarz information criterion (SIC), both of which penalize excessive use of free parameters, to compare the two scenarios. We obtain a difference of ΔAIC = 43 and ΔSIC = 39, both are larger than the classical threshold of 10 (e.g., Kass & Raftery 1995). However, correlated noise could decrease the difference in the χ² (consequently the ΔAIC and ΔSIC values), we thus do not distinguish the two scenarios here.

In addition to statistical noise, alternative spiral formation mechanisms could bias our motion measurement. On one hand, shadows, which trace the motion of the inner disk that is under the influence of local dust dynamics in SAO 206462 (Stolker et al. 2017), can affect the formation of spirals (Montesinos & Cuello 2018) and thus impact the motion of spirals. On the other hand, an eccentric driver (e.g., Calcino et al. 2020) could change the spiral motion pattern and fitting results. We thus do not attempt to conclude on the number of planetary drivers, and instead present the possible orbits for both planet-driven scenarios in Figure 3.

We query and reduce available observations of the SAO 206462 system using HST/NICMOS in 1998 (F160W filter, PropID: 7857, PI: A.-M. Lagrange) and 2005 (F110W filter, PropID: 10177, PI: G. Schneider) in Grady et al. (2009). The apparent motion in Figure 4 is possibly consistent with independent motion. Nevertheless, we note that the NICMOS observations can be dominated by speckle noise, the NICMOS pixel size of 75.65 mas is ∼6 times that of the SPHERE pixel, and that our principal-component-analysis-based data reduction method (Soummer et al. 2012) can alter the morphology of the spirals in reference differential imaging. Furthermore, and most importantly, moving shadows may manifest spurious motion between the two observations—specifically, the shadow at ∼3 o'clock in 1998 may have moved to ∼1 o'clock in 2005, thus causing spurious motion (e.g., Debes et al. 2017). Therefore, we do not perform motion measurement by combining these observations as in Ren et al. (2018), nor do we distinguish between the independent motion and comotion mechanisms in this Letter.

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