Boyajian's Star B: The Co-moving Companion to KIC 8462852 A

We obtained observations of the KIC 8462852 system using the near-infrared imaging camera NIRC2 coupled with the natural guide star adaptive optics system (Wizinowich et al. 2000) on the Keck II telescope in 2014 (28 images), 2016 (13 images), and 2019 (10 images). The observations in 2014 (PI M. Liu) were obtained in K, H, and J bands and were used by B16 to confirm the existence of KIC 8462852 B as a candidate companion. The observations in 2016 and 2019 were obtained by our team (PIs Mann and Huber) in the $K^{\prime}$-band filter, with the aim of testing whether KIC 8462852 B is co-moving. In both 2016 and 2019, we obtained observations with KIC 8462852 placed at two distinct orientations and dither positions. We deliberately duplicated the 2016 orientations and positions in 2019, with the goal of enabling cross-epoch measurements of motion that are independent of residual errors in the correction of the camera's static geometric distortion. In our subsequent analysis, we henceforth treat each of the two dither positions in 2016 and 2019 as an independent observation, denoted as 2016-1/2019-1 and 2016-2/2019-2, respectively.

All images used the narrow camera, with adaptive optics in the natural guide star mode, in the position angle tracking mode. We linearized each science and calibration frame in Python using the methodology of the IDL task linearize nirc2.pro ⁷ (Metchev & Hillenbrand 2009), then dark subtracted and flat fielded science frames in the standard manner. We adopted bad pixel identifications from Kraus et al. 2016 and replaced them with the median of surrounding pixels, then detrended spatially correlated read noise from the mirrored positions of each quadrant (A. L. Kraus et al., 2021 in preparation).

Figure 1 displays a NIRC2 image from the 2019 epoch, with the co-moving binary companion marked as B, located 1.95'' to the east, and the two candidate companions marked as cc1, 3.8'' southwest and cc2, 2.8'' southwest. KIC 8462852 B was visible at a sufficient signal-to-noise ratio for astrometric analysis in all data sets; cc1 was sufficiently visible only in the K and $K^{\prime}$ bands at all three epochs; and cc2 was sufficiently visible only in the $K^{\prime}$ band in the 2016 and 2019 data sets.

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We used the Gaussian point-spread function (PSF) fitting routine described in Pearce et al. (2019) to precisely measure the (x,y) pixel position and uncertainty of the four components in each image. Details of the modeling and acceptance criterion are described in Pearce et al. (2019) and applied in the same manner to this data set. Briefly, we modeled the PSF of the primary and candidate companion as the sum of two two-dimensional Gaussian functions and varied the model parameters through a custom Gibbs Sampler Markov Chain Monte Carlo (MCMC) routine. Figure 2 displays our model and residuals for all four objects in a 2019 epoch image. We performed astrometric calibrations for the primary and each of the three candidate companions for each step along the MCMC chains, including optical distortion and plate scale error (Yelda et al. 2010; Service et al. 2016), and differential aberration and atmospheric refraction, then computed the separation and position angle from the primary for each (x,y) position in the chains for each candidate companion. We took the mean and standard deviation of the separation and position angle chains as the final value and uncertainty in an image. Positions in each image are given in Figures 3, 5, and 6 as filled circles, with the median error for images in an epoch given by offset crosses. We computed a weighted mean of image positions as the mean position in an epoch, represented by small crosses in those figures.

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The geometric distortion solution for NIRC2 removes almost all of the distortion introduced by the telescope and instrument, but it does leave residual systematic uncertainties of σ_s,pos ∼1 mas in the position measurements of individual sources, or ${\sigma }_{s}\sim \sqrt{2}$ mas in the relative astrometry between the two sources. For a given detector location, any further residual error associated with that location (such as from temperature variations and non-repeatable positioning of the telescope/instrument optics) appears to be negligible across a timescale of years when compared to the empirical scatter seen within an individual epoch (≲0.2 mas; Dupuy et al. 2016; T. J. Dupuy et al. 2021, in preparation). It is therefore possible to achieve substantial further improvement in the precision of relative astrometric measurements if source positions and orientations can be duplicated in multiple epochs. Our observations in 2019 replicated the two position/orientation arrangements used in 2016, so we have encapsulated these systematic uncertainties in the covariance matrix for the five epochs (2014, 2016-1, 2019-1, 2016-2, 2019-2):

$$\begin{eqnarray*}C=\left[\begin{array}{ccccc}{\sigma }_{14}^{2}+{\sigma }_{s}^{2} & 0 & 0 & 0 & 0\\ 0 & {\sigma }_{16-1}^{2}+{\sigma }_{s}^{2} & {\sigma }_{s}^{2} & 0 & 0\\ 0 & {\sigma }_{s}^{2} & {\sigma }_{19-1}^{2}+{\sigma }_{s}^{2} & 0 & 0\\ 0 & 0 & 0 & {\sigma }_{16-2}^{2}+{\sigma }_{s}^{2} & {\sigma }_{s}^{2}\\ 0 & 0 & 0 & {\sigma }_{s}^{2} & {\sigma }_{19-2}^{2}+{\sigma }_{s}^{2}\end{array}\right],\end{eqnarray*}$$

where each diagonal term contains a contribution from the statistical variance ${\sigma }_{{NN}}^{2}$ (estimated from the rms of the individual image measurements at that epoch) as well as the systematic variance ${\sigma }_{s}^{2}$, and the systematic variance also contributes to the off-diagonal terms for epochs that were taken with the same source positions/orientations and hence share a common systematic error.

In subsequent measurements of the χ² goodness-of-fit statistic, we then use its modified definition:

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

where r is the vector of residuals for the observations about the model being tested, and C⁻¹ is the weight matrix corresponding to the inverse of the observational covariance matrix.

We computed a relative velocity using the scipy least squares fitting function curve_fit (Virtanen et al. 2020) by fitting a linear function to the mean positions, weighted with the weight matrix C⁻¹. We also use this formulation of the χ² goodness of fit in the orbit fitting calculations presented in Section 3.1.2. We computed the contrast in each filter as the mean and standard deviation of the flux ratio of our analytical models in each image.

Table 1 displays the results of our relative astrometry and photometry for the three candidate companions.

Table 1. Keck/NIRC2 NGS AO Astrometry for KIC 8462852 Candidate Companions

Epoch	MJD	Filter	Nimages	Ncoadds	tint	Separation a	Position Angle b	Δmc
					(s)	(mas)	(deg)	(mag)
B
2014.79	56946.30	J	9	10	0.726	1952.78 ± 0.77	96.064 ± 0.014	3.884 ± 0.057
2014.79	56946.30	H	9	10	0.726	1952.69 ± 0.36	96.059 ± 0.010	3.704 ± 0.034
2014.79	56946.30	K	10	10	0.726	1952.61 ± 0.40	96.058 ± 0.013	3.525 ± 0.020
2016.72 dither 1	57651.23	$K^{\prime}$	2	10	1.0	1950.64 ± 0.14	96.064 ± 0.004	3.640 ± 0.004
2016.72 dither 2	57651.23	$K^{\prime}$	11	10	1.0	1951.07 ± 0.07	96.063 ± 0.004	3.638 ± 0.012
2019.44 dither 1	58646.50	$K^{\prime}$	2	20	1.0	1951.63 ± 0.09	96.069 ± 0.004	3.632 ± 0.006
2019.44 dither 2	58646.50	$K^{\prime}$	8	20	1.0	1951.88 ± 0.06	96.062 ± 0.004	3.626 ± 0.009
cc1
2014.79	56946.30	K	10	10	0.726	3871.5 ± 2.6	256.377 ± 0.042	5.77 ± 0.38
2016.72 dither 1	57651.23	$K^{\prime}$	2	10	1.0	3863.2 ± 1.3	256.380 ± 0.015	6.11 ± 0.03
2016.72 dither 2	57651.23	$K^{\prime}$	11	10	1.0	3862.0 ± 0.7	256.401 ± 0.007	6.07 ± 0.05
2019.44 dither 1	58646.50	$K^{\prime}$	2	20	1.0	3850.1 ± 1.1	256.370 ± 0.012	6.23 ± 0.02
2019.44 dither 2	58646.50	$K^{\prime}$	8	20	1.0	3848.3 ± 0.5	256.367 ± 0.008	6.24 ± 0.04
cc2
2016.72 dither 1	57651.23	$K^{\prime}$	2	10	1.0	2785 ± 7	232.51 ± 0.02	7.15 ± 0.09
2016.72 dither 2	57651.23	$K^{\prime}$	5	10	1.0	2788 ± 6	232.46 ± 0.05	7.04 ± 0.11
2019.44 dither 1	58646.50	$K^{\prime}$	2	20	1.0	2761 ± 3	232.49 ± 0.03	7.26 ± 0.10
2019.44 dither 2	58646.50	$K^{\prime}$	7	20	1.0	2763 ± 2	232.52 ± 0.03	7.44 ± 0.13

Notes. 0.029 deg = 1 mas tangential angular distance at the projected separation of B. N_images is the number of images in each epoch. t_int is integration time per coadd. Some images were excluded from the astrometry of cc2 due to insufficient detection of the object in those images.

^a Errors shown are the statistical error, corresponding to the thick error bars in Figures 3, 5, and 6. A systematic error of 1.4 mas applies to all separation measurements, corresponding to thin error bars. ^b Errors shown are the statistical error. A systematic error applies to all position angle measurements, which corresponds to a 1.4 mas tangential angular distance at the object's separation: 0.042° for B, 0.021° for cc1, and 0.029° for cc2. ^c Errors shown are the statistical error. We adopt a conservative systematic error of 0.05 mag to account for detector systematics.

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In Table 2 we summarize relevant properties of KIC 8462852 AB. B16 found that the primary (F3V, M = 1.43 M_⊙; T_eff = 6750 ± 120 K) has a space velocity inconsistent with any moving groups and were unable to estimate an age.

Table 2. System and Stellar Properties for KIC 8462852 AB

Property		Ref.
Distance (pc)	${451.0}_{-4.8}^{+4.9}$	1
ρ (mas)	1951.48 ± 0.23	Section 3.1
ρ (au)	880 ± 10	2, Section 3.1
PA (°)	96.063 ± 0.004	Section 3.1
KIC 8462852 A
Proper motion	μα = −10.422 ± 0.040	2
(mas yr−1)	μδ = −10.288 ± 0.041	2
Luminosity (L⊙)	4.3 ± 0.3	Section 2.3
Mass (M⊙)	1.36 ± 0.05	Section 2.3
Radius (R⊙)	1.51 ± 0.04	Section 2.2
Teff (K)	6750 ± 120	3
SpT	F3V	3
Age (Gyr)	∼1.2	Section 2.3
J (mag)	10.763 ± 0.021	4
H (mag)	10.551 ± 0.019	4
K (mag)	10.499 ± 0.020	4
KIC 8462852 B
Mass (M⊙)	0.44 ± 0.02	5, Section 2.3
Radius (R⊙)	0.45 ± 0.02	Section 2.3
Teff (K)	3720 ± 70	Section 2.3
SpT	M2V	3

References (1) Bailer-Jones et al. (2018) (2) Gaia DR2 (Gaia Collaboration et al. 2016, 2018); (3) B16; (4) Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006); (5) Mann et al. (2019)

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The nearby companion, KIC 8462852 B, is co-moving with A (demonstrated in Section 3.1), so its stellar parameters are also relevant to establish. In the absence of spectroscopic follow-up, we adopt the spectral type of M2V estimated by B16.

To update the stellar parameters for both components we used isoclassify (Huber et al. 2017), with input constraints including T_eff for the primary from B16, the 2MASS K magnitude, Gaia DR2 parallax (corrected for the zero-point offset in the Kepler field, Zinn et al. 2019), a solar metallicity prior with a width of 0.1 dex, and the median measured $K^{\prime}$ contrast in Table 1 with a conservative uncertainty accounting for errors in the synthetic fluxes (${\rm{\Delta }}K^{\prime} =3.64\pm 0.05$ mag). This procedure is essentially the same as that used in Kraus et al. (2016), but with an improved stellar classification method and a newer grid of isochrones (Choi et al. 2016) supplemented with the empirical relations by Mann et al. (2015) and Mann et al. (2019) for low-mass stars, as described in Berger et al. (2020). The resulting classification yields a self-consistent classification of the primary and secondary assuming both components have the same age and metallicity. We estimate an isochronal age of ∼1.2 Gyr, and the resulting updated stellar parameters are listed in Table 2. Note that the uncertainties do not account for systematic errors between different model grids.

Our measured relative motion indicates that KIC 8462852 and its close, bright neighbor are a common proper motion pair. We determined a relative motion in the plane of the sky of Δμ = 0.14 ± 0.44 mas yr⁻¹ (Δμ = 0.3 ± 1.0 km s⁻¹) over the 5 yr span of observations, which is consistent with a bound companion. Figure 3 (left) displays the motion of B relative to the primary. As discussed above, the 2016 epoch was obtained in two dither positions; we matched observations in 2019 to the same two positions on the detector as 2016, to enable comparison. The change in position from 2016–2019 is consistent between images in the same dither position (i.e., 2016-1 to 2019-1 is consistent with 2016-2 to 2019-2). Our astrometry is sufficiently precise that the differences between 2016 and 2019 could be due to orbital motion. The 2014 epoch is offset from the others, most likely due to the NIRC2 realignment which occurred in 2015 (Service et al. 2016).

KIC 8462852 A and B are in Gaia EDR3 ⁸ (Gaia Collaboration et al. 2020, 2016), which was released to the public while this paper was under review. Gaia EDR3 reports a total relative proper motion within 1σ of our measured value (0.44 ± 0.34 mas yr⁻¹) and parallaxes for A and B that are consistent to within their uncertainty. We continued to use Gaia DR2 for some calculations such as stellar parameters due to independent validations and the Kepler field zero-point.

3.1.1. Tests for Binarity

To test the level of consistency with co-movement, we computed the χ² goodness of fit for the specific cases of the companion being a distant background object (zero absolute proper motion) or completely co-moving (zero relative proper motion). Figure 3 (right) displays the common proper motion of KIC 8462852 B with KIC 8462852 A. Our measurements reject the null hypothesis that the object is a non-moving background star with zero proper motion (${\chi }_{\mathrm{non}\mathrm{moving}}^{2}=1060$, ${\chi }_{\mathrm{co}\mathrm{moving}}^{2}=25.6$, for 4 degrees of freedom). We interpret the disagreement with the zero relative proper motion case as likely due to orbital motion.

We performed two additional statistical tests to assess the probability of observing a non-bound star at the position, velocity, magnitude, and parallax of the common proper motion candidate companion. First, we used a statistical approach similar that in Section 4.5 of Correia et al. (2006) to estimate the probability of chance alignment given the surface density of similar objects in the vicinity. We queried the Gaia EDR3 catalog for objects within a 30° radius of KIC 8462852 A with proper motion within 1σ of the Gaia proper motion of KIC 8462852 A (μ_α = 10.4 ± 0.6 mas yr⁻¹, μ_δ = 10.3 ± 0.6 mas yr⁻¹) and parallax within 1σ of KIC 8462852 A (±0.025 mas), in order to determine the most conservative comparison. This returned 140 objects, with a surface density of Σ = 3.8 × 10⁻⁹ arcsec⁻². The probability of observing a field object within θ = 2'' of KIC 8462852 A is given as

$$\begin{eqnarray}&&P({\rm{\Sigma }},\theta )=1-{e}^{-\pi {\rm{\Sigma }}{\theta }^{2}}=2.4\times {10}^{-8}.\end{eqnarray} \tag{ 1 }$$

Second, we factored in the known demographics of binary companions (i.e., the frequency, mass ratio distribution, and semimajor axis distribution) by modifying the method of Deacon et al. (2016), hereafter D16, in their Appendix A for distinguishing likely binary systems from chance alignments of field stars. Equation A2 in D16 gives the probability of a pair being a true binary pair rather than a coincident pair of field stars as

$$\begin{eqnarray}&&P=\displaystyle \frac{{\phi }_{c}}{{\phi }_{c}+{\phi }_{f}},\end{eqnarray} \tag{ 2 }$$

where ϕ_c and ϕ_f are densities for companion and field stars, respectively. While D16 considered the full range of binary population, here we are only concerned with the binary fraction that falls within the relevant parameter space, and so we modified Equation A3 in D16 to

$$\begin{eqnarray}&&{\phi }_{c}={f}_{\mathrm{bin}}\times \displaystyle \frac{1}{A}\times \displaystyle \frac{1}{{\rm{\Delta }}m}\times \left[\displaystyle \frac{{e}^{-{\rm{\Delta }}{\mu }^{2}/2{\sigma }_{\mu }^{2}}}{2\pi {\sigma }_{\mu }^{2}}\right]\times \left[\displaystyle \frac{{e}^{-{\rm{\Delta }}{\pi }^{2}/2{\sigma }_{\pi }^{2}}}{\sqrt{2\pi }{\sigma }_{\pi }}\right],\end{eqnarray} \tag{ 3 }$$

where Δμ is relative proper motion in R.A./decl., Δπ is the parallax difference, Δm is the size of the magnitude bin used for potential similar companions, A is the total area of the separation bin used, and f_bin is the binary companion fraction in those bins. Using the binary demographics of Raghavan et al. (2010), we determined the binary fraction to be f_bin = 0.01 in a bin of Δρ = ± 0.5 dex of log₁₀ projected separation centered on the value for KIC 8462852 B, and q = ± 0.05 in the mass ratio between KIC 8462852 AB. We then estimated the corresponding apparent magnitude range (δ m_G = ± 0.6 mag) using the relations of Pecaut & Mamajek (2016) ⁹ . For ϕ_f , we performed a Gaia EDR3 query for all objects within a radius from 50,000 au (to exclude potential companions) to 30° within the same magnitude bin, +/− 0.5 mas yr⁻¹ proper motion in R.A. and decl., and +/− 0.25 mas in parallax, returning 132 objects. We computed the field density as

$$\begin{eqnarray}&&{\phi }_{f}=\frac{\mathrm{objects}}{A\times {\rm{\Delta }}m\times {(\mathrm{\Delta \mu })}^{2}\times {\rm{\Delta }}\pi }\end{eqnarray} \tag{ 4 }$$

We determined a density ratio of

$$\begin{eqnarray*}&&P=\displaystyle \frac{{\phi }_{c}}{{\phi }_{c}+{\phi }_{f}}=0.999972.\end{eqnarray*}$$

Given the extremely high density ratio for a binary companion compared to a field star, and the extremely small probability of chance alignment, we conclude that KIC 8462852 AB is a binary system.

3.1.2. Test for Orbital Motion

Since KIC 8462852 B is a bound companion, we assume that it follows a Keplerian orbit around the center of mass of the system. An object on a circular, face-on orbit at the current 880 au separation and total system mass of 1.9 M_⊙ would have a velocity of v_circ = 1.4 km s⁻¹, and period P = 18,600 yr. Our time baseline and measurement precision is therefore marginally capable of measuring linear orbital motion, but is very unlikely to yield a measure of acceleration. Astrometric monitoring alone is unlikely to yield a fit with well-constrained posteriors on the orbit elements, though a refined measure of the linear motion might offer meaningful limits on the joint values of some elements.

To verify this conclusion, we performed a fit to our astrometry for Keplerian orbital elements using our custom implementation of the Orbits for the Impatient (OFTI) rejection sampling algorithm (Blunt et al. 2017). OFTI is well suited to fit poorly constrained astrometric orbits with only small orbit fractions observed for which a Markov Chain Monte Carlo algorithm might not converge (Blunt et al. 2020). OFTI is described in detail in Blunt et al. 2017, and our implementation in Pearce et al. 2019. In brief, OFTI generates a random set of orbital elements from prior probability distributions, scales the semimajor axis and longitude of periastron to match observations, computes a χ² probability for each scaled orbit, and accepts an orbit if the probability of the orbit exceeds a randomly chosen uniform number on the interval [0,1]. We used a total system mass of 1.9 ± 0.2 M_⊙, based on the the mass estimates in Table 2. We ran our OFTI fitter until 100,000 orbits were accepted.

Figure 4 displays the posterior distributions for orbital elements, and periastron and apastron distances in our 100,000 orbit sample. The posterior distributions of orbital elements are similar to priors and do not meaningfully constrain the orbit of KIC 8462852 B relative to A. We also note that the data did not rule out high-eccentricity orbits with extreme values of apastron and periastron, however, the apparent prominence of high-eccentricity values is likely a reflection of the uniform eccentricity prior. We do not interpret this fit to reveal anything physically meaningful about the orbital elements due to the poor constraint.

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Figure 5 (left) displays the astrometry results for KIC 8462852 cc1. We measure a relative proper motion in the plane of the sky of Δμ = 5.0 ± 0.7 mas yr⁻¹ (Δμ = 10.4 ± 1.5 km s⁻¹), which exceeds the circular velocity at that separation by 6σ (v_circ ≈1 km s⁻¹), and is not consistent with being a bound companion. Figure 5 (right) shows that its relative motion is not consistent with being bound, nor with a non-moving background star. In testing for co-movement, our measurements fail to reject the null hypothesis that the object is a non-moving background star, yet neither is it consistent with being co-moving (${\chi }_{\mathrm{non}\mathrm{moving}}^{2}=500$, ${\chi }_{\mathrm{co}\mathrm{moving}}^{2}=250$).

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Additionally, we performed a search of all objects in Gaia DR2 within 0.5° of KIC 8462852 A, displayed in Figure 7, with our three candidate companions. While the proper motion of KIC 8462852 AB is distinct from the majority, cc1 is similar to the other objects with chance alignment. We conclude it is most likely a star with similar space velocity and a chance alignment.

Figure 6 (left) displays the astrometry results for KIC 8462852 cc2. We measure a relative proper motion in the plane of the sky of Δμ = 11.9 ± 2.5 mas yr⁻¹ (Δμ = 25.2 ± 5.2 km s⁻¹), which exceeds the circular velocity at that separation by >5σ (v_circ ≈1.2 km s⁻¹), and is not consistent with being a bound companion. Figure 6 (right) shows that its relative motion is more consistent with zero proper motion than with zero relative motion. Our measurements more strongly support that it is a background object rather than co-moving, but are not consistent with being completely non-moving (${\chi }_{\mathrm{non}\mathrm{moving}}^{2}=150$, ${\chi }_{\mathrm{co}\mathrm{moving}}^{2}=1670$). The motion of cc2 is consistent with the majority of nearby objects, as shown in Figure 7. The Gaia DR2 objects have a mean proper motion of 4.8 mas yr⁻¹, within 1σ of cc2's absolute proper motion of 5.1 ± 2.4 mas yr⁻¹. We conclude that cc2 is an unassociated, distant object that is aligned by chance.

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