First Detection of Orbital Motion for HD 106906 b: A Wide-separation Exoplanet on a Planet Nine–like Orbit

All of the reduced observations were obtained from the Mikulski Archive for Space Telescopes.⁵ The data had been processed automatically by the observatory pipeline, including the typical corrections for the dark current, flat field, and, most crucially, the geometric distortion to account for anamorphism in the detector of each instrument. For the ACS and WFC3 frames we used the combined "drizzled" image ("_drz"), whereas for STIS we used the individual reduced frames after the distortion correction had been applied ("_sx2").

We performed additional postprocessing to subtract the PSF of HD 106906. For the ACS observations we used the PSF of the reference star HR 4570 as a model for the PSF of HD 106906. Although this reference star was also observed with STIS, a background star near HR 4570 was in close proximity to HD 106906 b in several images when the two data sets were aligned, potentially biasing the companion astrometry. We instead used observations of HD 106906 taken at different telescope orientations as a reference PSF. This was also done for the WFC3 observations as no reference star was observed. For each image we optimized the PSF subtraction by shifting and scaling the reference PSF to minimize the residuals within an annulus surrounding HD 106906. The inner and outer radii were set at 50 and 100 pixels for ACS, 30 and 80 pixels for STIS (excluding regions of the annulus containing the occulting wedge or the bright diffraction spikes of the star), and 15 and 55 pixels for WFC3.

We queried the Gaia DR2 catalog for all sources within $120^{\prime\prime}$ of HD 106906 that had measurements of their position (α, δ), proper motion⁶ (${\mu }_{{\alpha }^{\star }}$, ${\mu }_{\delta }$), and parallax (π), including HD 106906 itself. We applied a small correction to the astrometry of HD 106906 necessary due to the measured offset between the ICRS and the Gaia DR2 bright star reference frames (Lindegren 2019, 2020), and a small correction to the astrometric uncertainties (Arenou et al. 2018). We excluded Gaia sources that either fell outside of the instrument field of view, were obscured by the coronagraphic optical elements, were contaminated by the diffraction spikes of HD 106906, or were too close to a bad pixel that could not be reliably corrected. Sources with count rates exceeding the linearity limit were also excluded. One source (Gaia DR2 6072902994265468544) was rejected for being a close binary that may have biased the fitting procedure.

After applying these cuts we were left with either two or three useful sources for ACS, between 15 and 21 useful sources for STIS, and between five and seven useful sources for WFC3. For each background star, we extracted an 11-by-11 pixel (approximately $5\lambda /D$ for ACS and $10\lambda /D$ for WFC3 and STIS) data stamp centered on the predicted position of the star and iteratively fit a 2D symmetric Gaussian using a Levenberg–Marquardt least-squares fitting algorithm. A constant offset was not fit as the background was subtracted by the PSF subtraction procedure described previously. We used the same algorithm to measure the coordinates of HD 106906 b within each image, the additional sources not within Gaia used to align the ACS images (see Section 4), and HD 106906 in the first acquisition image taken with ACS where the star is not saturated (see Section 5). The measured pixel positions for each background star in each image, and their propagated tangent plane offsets relative to HD 106906 using the model described in Section 4, are given in Table C1.

We used an injection and recovery framework to assess the uncertainty of the measurements of each background star and HD 106906 b in each image. We used TinyTim (Krist et al. 2011) to generate synthetic PSFs for each instrument configuration. The objective here was not to perfectly match the PSF of the stars within each image, rather it was to use a reasonably good approximation of the PSF to estimate the precision and accuracy of our Gaussian model. Oversampled PSFs were generated using the spectrum of a K7V star without any focus despace or jitter applied. The resulting PSFs were a reasonable approximation of the observations (Figure 5).

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Five thousand injections were performed per source at random locations within an annulus surrounding the source. The width of the annulus was five pixels, with a central radius of 20 pixels for ACS and STIS and 15 pixels for WFC3. The oversampled PSF was shifted by cubic interpolation to the subpixel position of the injection, pixelated to match the sampling of the detector, normalized to have a flux within 10% of the source being tested, and added into the image. The injected PSF was fit using the Gaussian model described previously.

This analysis revealed three sources of error; a random error associated with the signal-to-noise ratio of the injected source and two systematic errors, one associated with the asymmetry of the PSF and the other with the subpixel position of the injected source. The asymmetry of the TinyTim PSF was accounted for by adding a small correction to the recovered pixel position, estimated from a Gaussian fit to a 10 times oversampled TinyTim PSF generated for each instrument configuration. This offset (${\rm{\Delta }}x={x}_{\mathrm{injected}}-{x}_{\mathrm{recovered}}$) was estimated to be ${\rm{\Delta }}x=-0.008$ px and ${\rm{\Delta }}y=-0.012$ px for the unocculted ACS PSF, ${\rm{\Delta }}x=-0.006$ px and ${\rm{\Delta }}y=-0.031$ px for the occulted ACS PSF, ${\rm{\Delta }}x=-0.054$ px and ${\rm{\Delta }}y=-0.047$ px for the STIS PSF, and ${\rm{\Delta }}x=-0.013$ px and ${\rm{\Delta }}y=0.0$ px for the WFC3/IR PSF. The systematic error associated with the subpixel position of the injected source is demonstrated in Figure 6, after applying the correction due to the asymmetry of the instrument PSF. This effect was not seen at a significant level for the ACS PSF as this instrument is not undersampled. These results were not sensitive to the choice of spectral type used to construct the TinyTim PSF. We repeated the experiment for spectral types between K4 and M3, a probable range of spectral types for the distant background objects based on the small subset with effective temperatures reported in the Gaia catalog, and the values of the derived systematic and random errors were not significantly different.

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We accounted for the random error attributed to the signal-to-noise ratio and this remaining systematic error by conservatively adopting the 95th percentile of the absolute value of the difference between the injected and recovered position as our 1σ uncertainty. These uncertainties (${\sigma }_{x}$, ${\sigma }_{y}$) were calculated separately in the x and y directions. Our preliminary fits of the positions of the STIS background stars yielded ${\chi }_{\nu }^{2}\sim \,2$, indicating that our uncertainties were slightly underestimated. We combined in quadrature the uncertainties estimated for each source with an additional uncertainty of 0.05 px, leading to ${\chi }_{\nu }^{2}\sim \,1$ for these data.

We propagated the astrometric measurements for each source from the Gaia reference epoch (2015.5) to the epoch of the observation using a Monte Carlo approach. We started by generating 10⁵ random draws for each of the five astrometric parameters from a multivariate Gaussian distribution created from the catalog values, uncertainties, and covariances. These random draws were then propagated from 2015.5 to the appropriate epoch using a rigorous coordinate transformation accounting for perspective effects (Butkevich & Lindegren 2014). We adopted a radial velocity of 12.18 ± 0.15 km s⁻¹ for HD 106906, while assuming zero radial velocity for the background stars. These stars are likely so distant that their radial velocity has negligible effect on the coordinate propagation. The propagated spherical coordinates of each source (α, δ) were then transformed into tangent plane offsets relative to the propagated position of HD 106906 (${\alpha }_{0}$, ${\delta }_{0}$) following Bedin & Fontanive (2018) as

$$\begin{eqnarray}\begin{array}{rcl}{\xi }_{\mathrm{sky}} & = & \displaystyle \frac{\cos \delta \sin (\alpha -{\alpha }_{0})}{\sin {\delta }_{0}\sin \delta +\cos {\delta }_{0}\cos \delta \cos (\alpha -{\alpha }_{0})}\\ & & +\,\left(\pi {P}_{\alpha }-{\pi }_{0}{P}_{\alpha ,0}\right),\\ {\eta }_{\mathrm{sky}} & = & \displaystyle \frac{\cos {\delta }_{0}\sin \delta -\sin {\delta }_{0}\cos \delta \cos (\alpha -{\alpha }_{0})}{\sin {\delta }_{0}\sin \delta +\cos {\delta }_{0}\cos \delta \cos (\alpha -{\alpha }_{0})}\\ & & +\,\left(\pi {P}_{\delta }-{\pi }_{0}{P}_{\delta ,0}\right).\end{array}\end{eqnarray} \tag{ 1 }$$

The additional terms for both ${\xi }_{\mathrm{sky}}$ and ${\eta }_{\mathrm{sky}}$ account for the nonzero parallax of both HD 106906 (${\pi }_{0}$) and the background stars (π). The parallax factors in the ${\alpha }^{\star }$ and δ directions (${P}_{\alpha ,0}$, ${P}_{\delta ,0}$ for HD 106906 and P_α, P_δ for each source) were calculated from the Euclidean coordinates of the Earth relative to the solar system barycenter (${X}_{\oplus }$, ${Y}_{\oplus }$, ${Z}_{\oplus }$) at the epoch of the observation,

$$\begin{eqnarray}&&\begin{array}{l}{P}_{\alpha }={X}_{\oplus }\sin \alpha -{Y}_{\oplus }\cos \alpha ,\\ {P}_{\delta }={X}_{\oplus }\cos \alpha \sin \delta +{Y}_{\oplus }\sin \alpha \sin \delta -{Z}_{\oplus }\cos \delta .\end{array}\end{eqnarray} \tag{ 2 }$$

This process resulted in 10⁵ values for both ${\xi }_{\mathrm{sky}}$ and ${\eta }_{\mathrm{sky}}$ for each of the background stars at each epoch. We adopted the mean of these draws as the tangent plane offset, and calculated the covariance between them to account for correlated uncertainties within the Gaia catalog. The tangent plane offset and associated correlation coefficient for each source at each epoch is given in Table C1.

The tangent plane offsets (${\xi }_{\mathrm{sky}}$, ${\eta }_{\mathrm{sky}}$) were converted to detector coordinates (ξ, η) using the five parameters within our model—the pixel position of HD 106906 (x₀, y₀), and the pixel scale (p_x, p_y) and the position angle of north (${\theta }_{{\rm{N}}}$) on the detector—as

$$\begin{eqnarray}&&\left[\begin{array}{c}\xi \\ \eta \end{array}\right]=\left({\boldsymbol{R}}\left[\begin{array}{c}{\xi }_{\mathrm{sky}}\\ {\eta }_{\mathrm{sky}}\end{array}\right]\right)\left[\begin{array}{c}1/{p}_{x}\\ 1/{p}_{y}\end{array}\right]+\left[\begin{array}{c}{x}_{0}\\ {y}_{0}\end{array}\right],\end{eqnarray} \tag{ 3 }$$

and ${\boldsymbol{R}}$ the rotation matrix

$$\begin{eqnarray}{\boldsymbol{R}}=\left[\begin{array}{cc}-\cos {\theta }_{{\rm{N}}} & \sin {\theta }_{{\rm{N}}}\\ \sin {\theta }_{{\rm{N}}} & \cos {\theta }_{{\rm{N}}}\end{array}\right],\end{eqnarray} \tag{ 4 }$$

to rotate and scale the tangent plane offsets to match the orientation and pixel scale of the detector. The rotation matrix has a nonstandard form to account for the flip in the x-axis when transforming between detector and sky coordinates. We also rotated and scaled the covariance matrix

$$\begin{eqnarray}&&{{\boldsymbol{C}}}_{\xi \eta }=\left({\boldsymbol{R}}{{\boldsymbol{C}}}_{\xi \eta ,{\rm{sky}}}{{\boldsymbol{R}}}^{T}\right)\circ {\boldsymbol{S}},\end{eqnarray} \tag{ 5 }$$

where ◦ represents an element-wise multiplication. The scaling matrix ${\boldsymbol{S}}$ is defined as

$$\begin{eqnarray}{\boldsymbol{S}}=\left[\begin{array}{cc}1/{p}_{x}^{2} & 1/{p}_{x}{p}_{y}\\ 1/{p}_{x}{p}_{y} & 1/{p}_{y}^{2}\end{array}\right].\end{eqnarray} \tag{ 6 }$$

The predicted position of a background star (ξ, η) was compared to the measured pixel position (x, y) and adopted uncertainty (${\sigma }_{x}$, ${\sigma }_{y}$) to determine a goodness of fit ${\chi }^{2}$ for a given set of model parameters as

$$\begin{eqnarray}&&{\chi }^{2}={{\boldsymbol{r}}}^{T}{{\boldsymbol{C}}}^{-1}{\boldsymbol{r}},\end{eqnarray} \tag{ 7 }$$

where r is the residual vector

$$\begin{eqnarray}&&r=\left[\begin{array}{c}\xi -x\\ \eta -y\end{array}\right],\end{eqnarray} \tag{ 8 }$$

and ${\boldsymbol{C}}$ is the covariance matrix created from the combination of ${{\boldsymbol{C}}}_{\xi \eta }$ and a covariance matrix describing the uncertainty on the measured pixel position of the background star, assumed to be completely uncorrelated

$$\begin{eqnarray}{\boldsymbol{C}}={{\boldsymbol{C}}}_{\xi \eta }+\left[\begin{array}{cc}{\sigma }_{x}^{2} & 0\\ 0 & {\sigma }_{y}^{2}\end{array}\right].\end{eqnarray} \tag{ 9 }$$

For the STIS measurements, ${\sigma }_{x}$ and ${\sigma }_{y}$ here are the quadratic sum of the measurement uncertainty and the 0.05 px error inflation term described in Section 3.2.

There are only two Gaia sources within the coronagraphic ACS image, severely limiting our ability to constrain the location of HD 106906. Instead, we used the three Gaia sources within the second acquisition image to determine the star location in this image (${x}_{0}^{\mathrm{acq}}$, ${y}_{0}^{\mathrm{acq}}$) and measured the offset between the acquisition and coronagraphic image using the 11 sources common to both images, yielding the location of the star in the coronagraphic image (${x}_{0}^{\mathrm{coro}}$, ${y}_{0}^{\mathrm{coro}}$). The star position in the acquisition image was fit using the procedure described previously. The offset between the two images was determined by adding three additional parameters to our model; a counterclockwise rotation (${\rm{\Delta }}\theta$) about $(0,0)$, and a subsequent translation (${\rm{\Delta }}x$, ${\rm{\Delta }}y$) to align the 11 stars common to both images. The goodness of fit for this alignment step was calculated as

$$\begin{eqnarray}&&{\chi }_{\mathrm{align}}^{2}={{\boldsymbol{r}}}^{T}{{\boldsymbol{C}}}^{-1}{\boldsymbol{r}},\end{eqnarray} \tag{ 10 }$$

where

$$\begin{eqnarray}&&\begin{array}{l}r={\boldsymbol{R}}\left[\begin{array}{c}{x}^{\mathrm{acq}}\\ {y}^{\mathrm{acq}}\end{array}\right]+\left[\begin{array}{c}{\rm{\Delta }}x-{x}^{\mathrm{coro}}\\ {\rm{\Delta }}y-{y}^{\mathrm{coro}}\end{array}\right],\\ {\boldsymbol{C}}={\boldsymbol{R}}{{\boldsymbol{C}}}^{\mathrm{acq}}{{\boldsymbol{R}}}^{T}+{{\boldsymbol{C}}}^{\mathrm{coro}},\\ {\boldsymbol{R}}=\left[\begin{array}{cc}\cos {\rm{\Delta }}\theta & -\sin {\rm{\Delta }}\theta \\ \sin {\rm{\Delta }}\theta & \cos {\rm{\Delta }}\theta \end{array}\right],\end{array}\end{eqnarray} \tag{ 11 }$$

where ${x}^{\mathrm{acq}}$, ${y}^{\mathrm{acq}}$ and ${x}^{\mathrm{coro}}$, ${y}^{\mathrm{coro}}$ are the pixel position of the background star in the acquisition and coronagraphic image, and ${{\boldsymbol{C}}}^{\mathrm{acq}}$ and ${{\boldsymbol{C}}}^{\mathrm{coro}}$ are the corresponding covariance matrices. As previously, the measurements were assumed to be uncorrelated and these were used here for completeness. The goodness of fit for a given set of parameters was calculated from the ${\chi }^{2}$ of the fit of the Gaia sources in the acquisition image and ${\chi }_{\mathrm{align}}^{2}$ as

$$\begin{eqnarray}&&{\chi }^{2}=\displaystyle \sum _{i}^{{n}_{\mathrm{gaia}}}{\chi }_{\mathrm{Gaia},i}^{2}+\displaystyle \sum _{j}^{{n}_{\mathrm{align}}}{\chi }_{\mathrm{align},j}^{2}\end{eqnarray} \tag{ 12 }$$

for the ${n}_{\mathrm{gaia}}=3$ Gaia sources in the acquisition image and the ${n}_{\mathrm{align}}=11$ sources common to both images. With either the best-fit parameters, or their posterior distributions, the pixel position of HD 106906 in the coronagraphic image can be calculated as

$$\begin{eqnarray}&&\left[\begin{array}{c}{x}_{0}^{\mathrm{coro}}\\ {y}_{0}^{\mathrm{coro}}\end{array}\right]=\left[\begin{array}{cc}\cos {\rm{\Delta }}\theta & -\sin {\rm{\Delta }}\theta \\ \sin {\rm{\Delta }}\theta & \cos {\rm{\Delta }}\theta \end{array}\right]\left[\begin{array}{c}{x}_{0}^{\mathrm{acq}}\\ {y}_{0}^{\mathrm{acq}}\end{array}\right]+\left[\begin{array}{c}{\rm{\Delta }}x\\ {\rm{\Delta }}y\end{array}\right].\end{eqnarray} \tag{ 13 }$$

We used a maximum likelihood estimation to find the best-fit parameters for each of the four variants of our model when applied to each image. The complexity of the model adopted for each epoch was justified by comparing the average value of the Bayesian information criterion (BIC) for each variant of the model (Figure A1). The BIC is a metric that reduces overfitting by penalizing the excessive use of free parameters in a model. For both ACS and WFC3 we found small values of ΔBIC ranging from 0.5 to 6.6 when comparing the more complex models to the two-parameter model, suggesting that the improvement in ${\chi }^{2}$ was not sufficient to justify the additional parameters. For the STIS observations we found a large ΔBIC of 1352 when comparing the two-parameter and five-parameter models, indicating that the significant improvement in the goodness of fit justifies the use of the additional parameters for this model. This large improvement in the quality of the fit was due to a systematic error of the position angle of north reported in the FITS header for the STIS observations (incorrect by $0\buildrel{\circ}\over{.} 077$ for these data; see Appendix B).

We used the affine-invariant Markov Chain Monte Carlo ensemble sampler emcee (Foreman-Mackey et al. 2013) to sample the posterior distributions of the fitted parameters. We used uniform bounded priors on all free parameters, excluding nonphysical plate scales and limiting the orientation between 0 and 2π. We initialized 50 walkers near the maximum likelihood and advanced them for 1500 steps. The first 500 steps of each chain was discarded as a "burn-in" where the walker positions are still a function of their initial positions. The values and corresponding uncertainties for the fitted parameters for each image of each epoch are given in Tables C2, C3, C4, and C5 of the Appendix.

Relative astrometry between the star and planet was calculated by combining the posterior distributions for the pixel coordinate of the star with the measured pixel position and uncertainty of the planet. This was converted into a sky separation and position angle using the detector plate scales and orientation. The final astrometry for each epoch (and filter for WFC3) was calculated using a weighted mean of the measurements (${x}_{i}\pm {\sigma }_{i}$) from each image. For each offset (${\rm{\Delta }}{\alpha }^{\star }$, ${\rm{\Delta }}\delta$) the weighted mean $\bar{x}$ and corresponding uncertainty $\bar{\sigma }$ were calculated as $\bar{x}={\sum }_{i}{w}_{i}{x}_{i}/{\sum }_{i}{w}_{i}$ and $\bar{\sigma }=\sqrt{{\sum }_{i}{\sigma }_{i}^{2}{w}_{i}/{\sum }_{i}{w}_{i}}$ where ${w}_{i}=1/{\sigma }_{i}^{2}$, under the assumption that the measurements were not independent.

For the ACS observations we used the measurement of the star position in the first acquisition image as a prior for the location in the second under the assumption that there was no telescope motion between these images (Figure 1). We calculated a maximum offset between the two acquisition images of 0.2 px; larger offsets produced significant residuals when the two images were subtracted. We fit the ACS data with and without this prior and find consistent relative astrometry (Table C2), and we adopted the measurements derived from the fit using this prior as our final relative astrometry for the ACS epoch. The final astrometric measurements from each epoch are given in Table 2, and plotted in Figure 7. Our relative astrometric measurements derived from the ACS and WFC3 observations are consistent with previous measurements (Bailey et al. 2014; Zhou et al. 2020), with reduced uncertainties.

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Table 2.  Relative Astrometry between HD 106906 and HD 106906 b

UT Date	MJD	Instrument	${\rm{\Delta }}{\alpha }^{\star }$ (arcsec)	Δδ (arcsec)	ρ (arcsec)	θ (deg)
2004-12-01	53340.7	ACS	−5.6903 ± 0.0054	4.2997 ± 0.0054	7.1321 ± 0.0054	307.076 ± 0.043
2016-01-29	57416.9	WFC3 (f127m)	−5.6563 ± 0.0196	4.3128 ± 0.0159	7.1130 ± 0.0183	307.325 ± 0.140
2016-01-29	57416.9	WFC3 (f139m)	−5.6581 ± 0.0172	4.3152 ± 0.0145	7.1158 ± 0.0163	307.331 ± 0.125
2016-01-29	57416.9	WFC3 (f153m)	−5.6539 ± 0.0127	4.3096 ± 0.0114	7.1091 ± 0.0123	307.316 ± 0.096
2017-02-24	57808.8	STIS	−5.6585 ± 0.0039	4.3058 ± 0.0037	7.1104 ± 0.0038	307.270 ± 0.030
2018-06-07	58276.1	WFC3 (f127m)	−5.6565 ± 0.0175	4.3086 ± 0.0179	7.1106 ± 0.0176	307.296 ± 0.143
2018-06-07	58276.1	WFC3 (f139m)	−5.6573 ± 0.0157	4.3091 ± 0.0159	7.1115 ± 0.0158	307.296 ± 0.128
2018-06-07	58276.1	WFC3 (f153m)	−5.6568 ± 0.0120	4.3126 ± 0.0120	7.1133 ± 0.0120	307.321 ± 0.097

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We performed several experiments to gauge the consistency of our astrometry and search for potential biases, specifically testing to determine if the brightness of HD 106906 is biasing the measured Gaia astrometry of the nearby faint sources. The experiments we performed were all variations on using subsets of the available Gaia sources to obtain our final astrometry. In one experiment using the STIS frames, we limited our analysis to the same three Gaia background sources visible in the ACS observation to see if there was a small sample selection bias affecting the ACS astrometry. We did not uncover any significant bias in limiting our analysis to just these three stars. The measurement remained within 1σ of the STIS astrometry that used all Gaia sources, albeit with an increased uncertainty due to the smaller number of stars used.

In another experiment, we iteratively fit our model for the location of HD 106906, removing the closest star with each iteration until there were only three Gaia stars left. This experiment showed no obvious systematics, with the measured positions remaining around the same value (Figure 8, top row). The uncertainty on each measurement scaled as a function of the decreasing number of sources used. We also tried this same experiment but in the reverse direction, iteratively fitting our model by removing the farthest star from HD 106906 with each iteration. This analysis yielded the same result, with no significant change in the relative astrometry as a function of exclusion radius (Figure 8, bottom row).

We also conducted experiments to test the effect of a poor PSF subtraction on the ACS epoch where the background stars used to cross-register with the Gaia catalog are within the wings of the PSF of HD 106906, significantly closer than for the STIS epoch. We found that the final relative astrometry was insensitive to errors of the scaling of the reference star of 10%, and of relative shifts between the two stars of 1 pixel, both relative to the best-fit scaling factor and offsets found using the method described in Section 2.1. Errors of this magnitude produced PSF-subtracted images with significant residuals relative to the best-fit solution suggesting that the scaling and offsets were determined to a far greater accuracy.

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We fitted the measured astrometry with a Keplerian orbit using the Orbits for the Impatient (OFTI; Blunt et al. 2017) algorithm that is part of the orbitize software package (Blunt et al. 2020). OFTI uses a rejection-sampling algorithm to efficiently generate Bayesian posterior distributions of the orbital parameters. We fitted the following parameters: (1) semimajor axis, (2) eccentricity, (3) inclination, (4) argument of periastron, (5) position angle of the ascending node (where the companion has a positive radial velocity moving away from the observer), (6) epoch of periastron passage in units of the orbital period relative to MJD 58849 (2020), (7) parallax, and (8) system mass. The corresponding Bayesian priors placed on each parameter were (1) log uniform, (2) uniform, (3) sine, (4) uniform, (5) uniform, (6) uniform, (7) Gaussian (9.6774 ± 0.0429 mas), and (8) Gaussian (2.71 ± 0.14 M_⊙).

A random selection of orbits drawn from the posterior distributions is plotted in Figure 9, showing the plausible orbital trajectories relative to the resolved debris disk. The median and 1σ credible intervals for the orbital elements derived from the 10⁷ OFTI samples are given in Table 3. The marginalized distributions and their covariances for a subset of these parameters are shown in Figure 10. We apply algebraic transformations to the OFTI samples to additionally derive marginalized distributions for the period using Kepler's third law, the periastron distance using the relationship ${r}_{\mathrm{peri}}=a(1-e)$, and the mutual inclination, i_m, of the plane of the orbit of the companion relative to the debris disk using the formula:

$$\begin{eqnarray}&&\cos {i}_{{\rm{m}}}=\cos {i}_{\mathrm{disk}}\cos {i}_{{\rm{b}}}+\sin {i}_{\mathrm{disk}}\sin {i}_{{\rm{b}}}\cos ({{\rm{\Omega }}}_{\mathrm{disk}}-{{\rm{\Omega }}}_{{\rm{b}}}).\end{eqnarray} \tag{ 14 }$$

Because of the ambiguity of Ω_disk we calculated i_m for each orbit for the two possible pairs of (i_disk, Ω_disk) that are consistent with the southern part of the disk being behind the star (i_disk = 85 deg, Ω_disk = 104 deg and i_disk = 95 deg, Ω_disk = 284 deg; Kalas et al. 2015), selecting the smallest value to exclude retrograde orbits.

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Table 3.  Orbital Elements and Derived Parameters for HD 106906 b

Parameter	Unit	Median (±1σ)
Period (P)	yr	${15000}_{-6400}^{+17000}$
Semimajor axis (a)	au	${850}_{-260}^{+560}$
Periastron distance (${r}_{\mathrm{peri}}$)	au	${510}_{-320}^{+480}$
Eccentricity (e)	⋯	${0.44}_{-0.31}^{+0.28}$
Inclination (i)	deg	${56}_{-21}^{+12}$
Epoch of periastrona (τ)	⋯	${0.26}_{-0.15}^{+0.28}$
Epoch of periastron (t0)	yr	${3200}_{-1200}^{+7800}$
Parallax (π)	mas	9.677 ± 0.043
System mass (Mtotal)	M⊙	2.71 ± 0.14
		Family 1b	Family 2c
PA of ascending node (Ω)	deg	${99}_{-28}^{+26}$	${279}_{-29}^{+25}$
Argument of periastron (ω)	deg	${356}_{-86}^{+61}$	${176}_{-87}^{+61}$
Mutual inclination (im)	deg	${36}_{-14}^{+27}$	${44}_{-14}^{+27}$

Notes.

^a $\tau =({t}_{0}-2020)\mathrm{mod}P$. ^b $10\leqslant {\rm{\Omega }}\lt 190$ deg. ^c $190\leqslant {\rm{\Omega }}\lt 360$ deg or $0\leqslant {\rm{\Omega }}\lt 10$ deg.

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If HD 106906 b is bound to HD 106906, then the three-dimensional velocity of the companion must be lower than the gravitational escape velocity of the system. Radial velocity measurements of HD 106906 b relative to the host star have not yet been made, so we can only calculate a lower limit on the three-dimensional velocity. We estimated the projected linear velocity of the planet by calculating the instantaneous velocity of the companion at the midpoint of the orbit between the STIS and ACS epochs using the orbital element posterior distributions generated from OFTI. We found a projected velocity of the companion of 1.2 ± 0.2 km s⁻¹, which is 48% ± 8% of escape velocity of the system (2.6 km s⁻¹) estimated using a separation of 737 au and a mass of 2.71 M_⊙ (Figure 11). While we cannot use this measurement to definitively conclude that the planet is gravitationally bound to HD 106906, a lower limit greater than the escape velocity would have been definitive evidence that the companion is unbound. Future radial velocity measurements of the planet will be necessary to constrain the three-dimensional relative velocity to demonstrate conclusively that HD 106906 b is gravitationally bound to HD 106906.

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The difference between the parallax of the star and planet should be negligible if the planet is gravitationally bound, less than 1 μas given a 700 au orbit. We used the four astrometric measurements to fit a 10 parameter model for the relative astrometry of HD 106906 b following the procedure outlined by Nguyen et al. (2020). These 10 parameters included the standard five parameters (α, δ, π, ${\mu }_{{\alpha }^{\star }}$, and μ_δ) describing the astrometry of HD 106906 and five parameters (Δα, Δδ, Δπ, ${\rm{\Delta }}{\mu }_{{\alpha }^{\star }}$, and ${\rm{\Delta }}{\mu }_{\delta }$) describing the relative position and motion of HD 106906 b. We used the same MCMC package as previously to sample the posterior distributions of the 10 parameters. We adopted Gaussian priors on the five parameters describing the astrometry of HD 106906 from the Gaia catalog values and uncertainties, and uniform priors for the others. We initialized 256 walkers near the maximum likelihood estimate and advanced them for 2000 steps, discarding the first 800 steps. We measured a differential parallax of Δπ = 3.66 ± 5.73 mas, and a relative proper motion of Δμ_α = 2.78 ± 0.59 mas yr⁻¹, Δμ_δ = 0.77 ± 0.61 mas yr⁻¹. Our measurement was not particularly constraining; the differential parallax is consistent with zero but with a large uncertainty. Future astrometric monitoring of the system should greatly improve the uncertainties on this measurement.

Combining the orbital elements of HD 106906 b presented in Section 6.1 with the orientation and inclination of the debris disk from Kalas et al. (2015) yielded a mutual inclination between the plane of the orbit of the planet and the inner system of either ${36}_{-14}^{+27}$ deg or ${44}_{-14}^{+27}$ deg (3σ lower limit of 6 deg or 15 deg), depending on the true orientation of the orbital plane of the planet (Figure 9). We can confidently exclude a coplanar orbit (3σ lower limit is 13.8 deg or 5.1 deg depending on the value of Ω) as well as a radial trajectory for the companion. The periastron distance and the mutual inclination between the orbital plane and the inner system are highly correlated (Figure 12). Eccentric orbits with periastron distances interior to the sensitivity-limited outer radius of the eastern side of the debris disk (370 au) are highly misaligned with respect to the disk (i_m ≳ 40 deg), while more circular orbits with larger periastron distances are less misaligned (i_m ∼ 30 deg).

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Previous studies have simulated the dynamical influence of an eccentric and misaligned planetary orbit on the debris disk (Nesvold et al. 2017), assuming a small mutual inclination (∼10 deg) and a large eccentricity (e ∼ 0.7), corresponding to a small periastron distance (∼200 au). These simulations were able to qualitatively reproduce much of the observed morphology of the disk; however, a thorough exploration of parameter space for the orbit of the companion is needed in future work. Our results suggest that an eccentricity as large as e ∼ 0.7 would require a very large mutual inclination of ∼60 deg. This is larger than the maximum mutual inclination allowed based on the simulation and the lack of an observed vertical extent of the inner debris ring, potentially excluding some of the most eccentric solutions consistent with the astrometry presented in this work. The dynamical effect of a planet on an orbit with a lower eccentricity (e ∼ 0.4) but a larger mutual inclination (i_m ∼ 30 deg), also consistent with our astrometric measurements, has not yet been explored.

Table 4 compares our orbit constraints for HD 106906 b against the hypothetical Planet Nine (Batygin et al. 2019) and observations of 15 detached KBOs (Trujillo 2020) demonstrating an approximate commensurability. With age 15 ± 3 Myr, HD 106906 b shows that a Planet Nine-like orbital architecture can be established early, though identifying the most likely pathway requires future work. One scenario is that the massive planet initially formed in a disk around the eccentric binary star, migrated into an unstable resonance with the stars, pumping the eccentricities and inclinations of the planet, and then the planet's periastron was raised beyond the central planetary region by passing stars (Rodet et al. 2017; De Rosa & Kalas 2019). If future observations can show that the binary stars, the disk, and HD 106906 b are all moving in a prograde direction, then this pathway would be favored as opposed to an alternate mechanism whereby HD 106906 b was captured from another star or as a free-floating planet. These same pathways are debated for the origin of a hypothetical Planet Nine (Li & Adams 2016). Also compelling is to investigate the theory of how the entire HD 106906 system will evolve in the future. Though there are differences between HD 106906 and the solar system, HD 106906 provides an empirical initial configuration for analytical and numerical experiments to determine how it may resemble the solar system after 4.6 Gyr of evolution.

Table 4.  Comparison of a Selection of Orbital Elements for HD 106906 and Other Solar System Bodies

Param.	Unit	HD 106906	Planet 9	Detached KBOs
		Median (±1σ)	Range	Range
P	yr	${15000}_{-6400}^{+17000}$	8000–23000	1900–34000
a	au	${850}_{-260}^{+560}$	400–800	160–1000
${r}_{\mathrm{peri}}$	au	${510}_{-320}^{+480}$	200–640	40–80
e	⋯	${0.44}_{-0.31}^{+0.28}$	0.2–0.5	0.68–0.95
${i}_{{\rm{m}}}$	deg	${36}_{-14}^{+27}$, ${44}_{-14}^{+27}$	15–25	4.2–33.5

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