REVEALING δ CEPHEI'S SECRET COMPANION AND INTRIGUING PAST

Classical Cepheid variable stars (from hereon: Cepheids) are precise Galactic and extragalactic distance tracers and thus of crucial importance for cosmology. The prototype of this class of stars, δ Cephei (HD 213306, HIP 110991), has been extensively ⁵ studied ever since the discovery of its variability 230 yr ago by Goodricke (1786). Until Baade's proposed method (Baade 1926) to test Shapley's pulsation hypothesis (Shapley 1914) bore fruit in the early to mid-twentieth century, Cepheids were thought to be binary stars on eccentric orbits (see, e.g., Gautschy 1997), inspiring a great deal of research, not least that by Christian Doppler.

The first scientists to measure the variability of δ Cephei's spectral lines and to determine their velocities were Belopolsky (1894, 1895) and Moore (1913), before radial velocities (RVs) became available for many Cepheids thanks to the observations by Joy (1937). Two decades later, Shane (1958) conducted a detailed analysis of δ Cephei's RV curve and concluded that no evidence of long-period changes in the mean velocity could be seen, a result that was confirmed much later by Kovacs et al. (1990).

Nowadays, Cepheids are known to be pulsating variable stars, although many Cepheids are also known to be binaries. ⁶ The binary fraction of Cepheids is being studied intensively with most recent estimates of the total binary fraction ranging around 60%; see, e.g., Evans et al. (2013) and Szabados et al. (2013). However, Cepheids cannot reside in very close-in binary systems (e.g., Neilson et al. 2014) due to their nature as evolved (super-)giant stars, which results in observed minimum orbital periods of approximately 1 yr. For long periods ($\gt 10$ yr), practical constraints create significant observational bias against companion detection.

To our knowledge, δ Cephei has not been shown to be a spectroscopic binary prior to this work. δ Cephei is, however, a known visual binary (Fernie 1966), whose companion HD 213307 (= HIP 110988) is itself an astrometric binary and thought to be physically associated (see Benedict et al. 2002 and references therein). HD 213307 was proposed to also be a spectroscopic binary (private communication by G. H. Herbig mentioned by Fernie 1966). Finally, δ Cephei is usually considered to be a member of the loose association Cepheus OB6 (de Zeeuw et al. 1999; van Leeuwen et al. 2007; Majaess et al. 2012).

Precise trigonometric parallax of δ Cephei has been measured by ESA (1997) and van Leeuwen et al. (2007) using observations made by the Hipparcos space mission and by Benedict et al. (2002) using measurements obtained with FGS 3 on board the Hubble Space Telescope (HST). Notable previous distance estimates include those reported by Fernley et al. (1989), Gatewood et al. (1993), Gieren et al. (1993), and Mourard et al. (1997).

Mérand et al. (2005) employed infrared long-baseline interferometry to study the Baade–Wesselink projection factor of δ Cephei and were later (Mérand et al. 2006) able to show the presence of an extended circumstellar envelope. The presence of this circumstellar environment was confirmed independently (using different methodologies) by Marengo et al. (2010) and Matthews et al. (2012).

In this paper, we present the discovery of the spectroscopic binary nature of δ Cephei. This discovery is demonstrated using new observations that are presented in Section 2. Using these new data, we reveal the presence of a hidden companion in Section 3.1. After combining our new RVs with literature data in Section 3.2, we determine the orbital solution in Section 3.3. We re-analyze the Hipparcos intermediate astrometric data (IAD) from the van Leeuwen (2007) reduction in Sections 4.1.1 and 4.1.2 to measure parallax and to investigate whether the available astrometric measurements are sensitive to the motion due to binarity. In Section 4.2, we investigate Gaia's expected sensitivity to the binary motion. We discuss how our discovery helps to better interpret other observations and begins to draw a complex picture of δ Cephei's rich and dynamical history in Section 5 before concluding in Section 6.

Line-of-sight (radial) velocities were measured from 136 observations taken between 2011 September and 2014 using the fiber-fed high-resolution (R ∼ 85 000) spectrographHermes (Raskin et al. 2011) at the Flemish 1.2 m Mercator Telescope located at Roque de los Muchachos Observatory, La Palma, Canary Islands. We utilize the high-resolution fiber (HRF) mode for all observations, since this is the most commonly used, i.e., best-understood, observing mode available for Hermes. The HRF mode offers optimal efficiency and the highest available spectral resolution.

The reduction pipeline available for Hermes performs pre- and overscan bias correction, flat-fielding using halogen lamps, and background modelization, as well as cosmic-ray removal. ThAr lamps are used for the wavelength calibration. RVs are determined via the cross-correlation technique (Baranne et al. 1996; Pepe et al. 2002) using a numerical mask designed for solar-like stars (optimized for spectral type G2).

These observations were started as part of a search for Cepheids belonging to open clusters (Anderson et al. 2013) with the goal of quantifying the precision limit for a classical Cepheid using Hermes RVs and of having high-quality spectra available for further study. The average signal-to-noise ratio $({\rm S}/{\rm N})$ of the spectra is higher than 200 near 6000 Å, with some spectra reaching up to 400.

Table 1 shows a sample of our Hermes RV measurements that we are making publicly available together with our standard star observations.

Table 1. Sample of the New Hermes RV Data Shown Here for Guidance Regarding Their Form and Content

BJD−2,400,000	Phase	vr	$\sigma ({{v}_{r}})$
		(km s−1)	(km s−1)
55816.469627	0.82074	0.802	0.018
55816.470920	0.82098	0.813	0.018
55816.472221	0.82122	0.824	0.018
55817.510667	0.01474	−21.125	0.018
55817.511961	0.01498	−21.202	0.018
55818.494094	0.19800	−31.354	0.018
55818.495384	0.19824	−31.362	0.018
55818.496675	0.19848	−31.342	0.018
55819.497269	0.38494	−21.989	0.018
55819.498564	0.38518	−21.984	0.018

Note. Radial velocities have been shifted to the CORAVEL-ELODIE zero point. Pulsation phase is defined here as 0 when ${{v}_{\,r}}={{v}_{\,\gamma }}$ at the steep part of the RV curve. Uncertainties are fixed at 18 m s⁻¹ to account for uncertainty in the wavelength calibration and zero-point calibration; see text.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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2.1. Zero Point, Stability, and Precision of Hermes RVs

To determine the RV zero point of Hermes, we observed eight RV standard stars listed in Udry et al. (1999a, 1999b); ⁷ see Table 2. These standard stars were chosen to cover the range of spectral types that δ Cephei exhibits during its pulsation cycles. We thus determine a mean systematic offset of 55 m s⁻¹ with respect to the ELODIE and CORAVEL zero point. We estimate this zero-point offset to be accurate to approximately 10 m s⁻¹ and note that additional scatter in the difference between Hermes and literature RVs can exist for various reasons of astrophysical origin, including binarity and planetary companions. We therefore increase our error margin by adding 10 m s⁻¹ in quadrature to the Hermes RV uncertainties when correcting for zero-point offsets.

Table 2. List of RV Standard Stars with Spectral Types from SIMBAD, Hermes RVs of Standard Stars from New Observations, Udry et al. (1999b), Offsets between New and Reference RVs, and rms of New Observations without (Subscript nc) and with Pressure Corrections Applied (Subscript corr) Following Anderson (2013, Section 2.1.5)

HD	Sp. Type	$\langle {{v}_{r}}\rangle$	${{v}_{r,{\rm ref}}}$	${\Delta }{{v}_{r,{\rm ref}}}$	rms$_{{\rm nc}}$	rms$_{{\rm corr}}$
		(km s−1)	(km s−1)	(km s−1)	(m s−1)	(m s−1)
10780	K0V	2.771	2.70	0.071	38.7	17.1
32923	G4V	20.627	20.50	0.127	70.1	14.8
42807	G2V	6.068	6.00	0.068	115.2	29.7
82106	K3V	29.784	29.75	0.034	45.8	20.8
144579	G8V	−59.452	−59.45	−0.002	42.7	20.0
168009	G1V	−64.581	−64.65	0.069	36.4	20.0
197076	G5V	−35.413	−35.40	−0.013	78.3	10.9
221354	K0V	−25.111	−25.20	0.089	53.8	15.5

The machine-readable version of this table contains all of the individual measurements of the standard stars. These individual values were used to determine the average velocities in the third column.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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We investigate the long-term stability and precision of Hermes RVs using the RV standard star observations. Thanks to the very high S/N of our spectra, photon noise (Bouchy et al. 2001) contributes only marginally to the uncertainty of our Hermes RVs. Instead, the precision of our RVs is dominated by the intra-night stability of the wavelength calibration and the long-term stability of the instrument.

To improve RV precision and stability, we employ a method developed in Anderson (2013, Section 2.1.5) to correct for the RV drift induced by atmospheric pressure variations that occur during the night. Pressure variations are the leading cause for drifts in Hermes RVs, since the temperature of the instrument is stabilized to within 0.01 K (Raskin et al. 2011). The RV drift correction due to pressure variations is based on changes in the refractive index of air and has been shown to be precise to approximately 10 m s⁻¹ in the case of the Coralie spectrograph, ⁸ for which RV drift corrections are measured using interlaced simultaneous ThAr exposures. For Hermes, our method improves RV precision by a factor of approximately 2.5, as measured by the decrease in rms for all standard stars.

Figure 1 shows both the uncorrected (red dashed lines) and corrected RVs (blue solid line) for the standard stars HD 144579 and HD 168009 as a function of time for the different observing runs during which they were observed. The highest precision of 9 m s⁻¹ is achieved for HD 168009 during a 10-night observing run in 2013 July. We find no evidence for long-term variations over the 3 yr duration of the observations. Based on all standard star measurements, we estimate the long-term precision of pressure-corrected Hermes RVs to be approximately 15 m s⁻¹. Accounting for the additional uncertainty due to zero-point offsets (see above), we adopt 18 m s⁻¹ as our error budget for the investigations based purely on Hermes RVs. ⁹ Note that this adopted uncertainty of 18 m s⁻¹ is more than a factor of 2000 smaller than the pulsation-induced peak-to-peak RV amplitude of 38.6 km s⁻¹.

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In this section we describe our analysis of δ Cephei's RV curve, starting with the initial discovery of δ Cephei's nature as a spectroscopic binary (Section 3.1). After subtracting the RV drift due to orbital motion observed in Hermes data, we establish a high-precision reference model for the pulsations alone, which we subtract from a combined data set of RVs from Hermes and several literature sources (see Section 3.2). Using this combined data set (of residuals), we determine the orbit for the binary system in Section 3.3.

We also searched the individual high-quality Hermes spectra and the cross-correlation profiles for a signature identifying the companion, finding none. We note that previous analyses of IUE (Evans et al. 1993) and HST/Cosmic Origins Spectrograph (COS) (Engle et al. 2014) spectra did not report evidence of a companion's signature, effectively ruling out early-type companions. Given the high S/N of our spectra and the even better S/N of the cross-correlation profiles, we estimate that the companion must be at least a factor of 100 fainter than δ Cephei, at least in optical bandpasses. Based on Geneva stellar evolution models (Georgy et al. 2013) and assuming an approximately 5 ${{M}_{\odot }}$ Cepheid, this implies a companion mass below approximately 1.75 ${{M}_{\odot }}$. This is consistent with the upper limit on companion spectral type (A3, i.e., ${{M}_{2}}\lt 2$ ${{M}_{\odot }}$) set by non-detection in IUE spectra (Evans 1992).

3.1. Discovery of δ Cephei B using HERMES RVs

Figure 2 shows the Hermes RVs obtained for δ Cephei, phase-folded with the best-fitting pulsation period of ${{P}_{{\rm puls}}}=5.366274$ days. The 10-fold RV uncertainty is shown in the upper right corner. The observation date is traced by symbol color going from red (first observations in 2011 September) to yellow (2014 September). The color coding reveals the average RV to be time dependent, thus revealing orbital motion caused by a hidden companion.

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We model the RV variations due to pulsation using the technique described in Anderson et al. (2013). In a nutshell, we fit a Fourier series to model the phase-folded RV curve and increase the number of harmonics until an F-test indicates spurious fit improvement. This approach yields fit residuals that clearly exhibit a strong temporal correlation with an rms of 337 m s⁻¹; see Figure 3.

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By running the fitting algorithm while assuming a model composed of the sum of a Fourier series and linear, quadratic, and cubic trends, we find that the orbital drift seen in Hermes RVs is best described by a cubic polynomial and that the best-fit pulsation model has 14 harmonics. We retain this model as our pulsation reference model for the following steps.

Figure 4 shows the residuals from Hermes RVs after subtracting the model that accounts for pulsations, as well as the cubic drift due to orbital motion. The residuals are flat over the observational baseline. However, the phase-folded residuals do exhibit some structure, which is exposed by applying a color scale to trace the observation date. The rms of our Hermes residuals is 47 m s⁻¹, i.e., higher than the 18 m s⁻¹ estimated from RV standard stars in Section 2.1.

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This higher-than-expected residual may be explained by small stochastic variations in the pulsation period seen in photometry of other Cepheids obtained with the Kepler and Microvariability and Oscillations of Stars (MOST) satellites (so-called period-jitter; see Derekas et al. 2012; Evans et al. 2015), which has been explained as being due to surface convection and granulation (Neilson & Ignace 2014). We note also that Anderson (2014) recently discovered RV curve modulation in Cepheids, albeit at much lower amplitudes. Since these effects (period-jitter or modulation) limit our ability to precisely reproduce the pulsation curve, we adopt the rms value of 47 m s⁻¹ as our Hermes RV uncertainty for δ Cephei for the remainder of our analysis.

3.2. Combination with Literature Data

The cubic drift seen in the residuals in Figure 3 indicates orbital motion at a timescale longer than the observational baseline achieved by our Hermes observations. We therefore searched the literature for data suitable for determining the orbit of δ Cephei.

δ Cephei is one of the most-studied variable stars, and several authors have previously published RV data for it, including Shane (1958), Barnes et al. (1987, 2005), Wilson et al. (1989), Butler (1993), Bersier et al. (1994), Gorynya et al. (1996), Kiss (1998), and Storm et al. (2004). Historically, δ Cephei's RV curve has been thought to be well understood, which may in part explain why its spectroscopic binary nature has gone unnoticed for so long. The main reason, however, is that δ Cephei's orbital signature has a small RV amplitude, high eccentricity, and long orbital period and thus requires high-precision velocimetry over an observational baseline spanning at least two years. For comparison, the various RV data sets available in the literature have typical observational baselines on the order of 1 yr and do not have sufficient precision to detect binarity during this time frame.

Zero-point offsets must be corrected for when combining RV data from different instruments and the literature. δ Cephei represents a particularly difficult case, because zero-point differences can be on the same order of magnitude as the difference in ${{v}_{\gamma }}$ due to orbital motion between data sets. Therefore, a well-determined common zero point is the key to determining the orbit of δ Cephei accurately.To this end, we adopted the CORAVEL-ELODIE RV zero point (see Udry et al. 1999a, 1999b, and Table 2).

Of the available literature sources, we excluded the following from our analysis due to insufficient precision (${{\sigma }_{{{v}_{r}}}}\gt 1$ km s⁻¹): Shane (1958), Barnes et al. (1987), Wilson et al. (1989). We further discarded the measurements by Butler (1993) and Storm et al. (2004), since no information is available to determine the zero-point differences with CORAVEL. Finally, we excluded the single measurement published by Gorynya et al. (1996).

Conversely, the measurements published by Bersier et al. (1994) and Barnes et al. (2005), as well as publicly available RV data from the ELODIE archive ¹⁰ (Moultaka et al. 2004), are already on the common CORAVEL-ELODIE RV zero point. We found that an offset of −0.35 km s⁻¹ to the measurements by Kiss (1998) is appropriate to improve agreement with the (contemporaneous) data from the ELODIE archive and Barnes et al. (2005). This offset is similar to the precision stated by Kiss (1998, ∼0.3 km s⁻¹). The determination of the zero-point offset between Hermes and CORAVEL-ELODIE is discussed in Section 3.1 above. Having thus calibrated the zero-point differences based on measurements of standard stars, we can use the combined data set to determine the orbit with confidence.

Classical Cepheids are known to exhibit changing periods due to their secular evolution. However, Cepheids can also exhibit erratic changes of unknown origin in their pulsation periods (see, e.g., Berdnikov et al. 2000). When combining RV data from the literature, we noticed that variable periods have to be accounted for. We attempted to use the ephemerides and rate of (pulsation) period change by Berdnikov & Ignatova (2002) to obtain accurately phase-folded RV curves, but we were unable to obtain a satisfactory result.

Since obtaining good phase-folding is required in order to correctly subtract the pulsation reference model, we phase-folded the combined data set in the following way. First, we separated the data set into three parts with different pulsation periods. The motivation for separating the evolution of the pulsation period in this way is the observation that the period of δ Cephei changes very slowly (Eddington 1919; Berdnikov & Ignatova 2002). We then determined the best-fit pulsation periods for each of these three epochs by minimizing the scatter in the residuals after fitting for the pulsation alone. We thus adopt the following pulsation periods: 5.3657 ± 0.0013 days for data by Bersier et al. (1994, mean epoch JD 2,444,467.12); 5.36615 ± 0.0005 days for data from ELODIE, Kiss (1998) and Barnes et al. (2005, mean epoch JD 2, 450, 398.61); 5.366274 ± 0.00006 days for Hermes RVs (mean epoch JD $2,\;456,\;430.38$). While the center values of this sequence would imply a slowly increasing pulsation period ($dP/dt\sim (0.5-1.1)\times {{10}^{-5}}\;{\rm s}\;{\rm y}{{{\rm r}}^{-1}}$), these adopted periods agree to within their uncertainties. After applying our updated pulsation periods, we shifted all three epochs for $\phi \equiv 0$ to occur at minimum radius, i.e., when the velocity is equal to ${{v}_{\gamma }}$ on the steep part of the RV curve. This provides us with an accurately phase-folded pulsation curve from which we subtract the pulsation reference model to reveal the orbital motion of δ Cephei.

Figure 5 shows the RV curve based on the combined data set. The residuals shown in the bottom panels clearly demonstrate the presence of orbital motion.

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3.3. Orbit Determination using HERMES and Literature RVs

Figure 6 shows the orbital motion of δ Cephei exposed by subtracting our pulsation reference model from the phase-folded combined data set. As can be seen in the top right panel, the orbit is well sampled except for the ascending part of the orbital RV curve. Our observations did not sample this part of the RV curve, since it occurred shortly (a few months) before the start of our observations.

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Table 3 provides the orbital solution for δ Cephei determined from the orbit-only combined RV curve using a standard Keplerian model. We used the tool Yorbit (D. Ségransan et al. 2015, in preparation) to first determine an initial orbital estimate via a genetic algorithm and then characterize the parameter uncertainties using the marginal distributions of Markov Chain Monte Carlo simulations with 500,000 iterations; see Figure 7. From the orbital solution and assuming ${{M}_{\delta {\rm Cep}}}\sim 5.0-5.25$ ${{M}_{\odot }}$ (see Section 6), we determine the minimum mass of the companion to be 0.2 ± 0.02 ${{M}_{\odot }}$. While ${{a}_{{\rm rel}}}$ and ${{a}_{1}}{\rm sin} i$ are listed assuming ${{M}_{\delta {\rm Cep}}}=5.25$ ${{M}_{\odot }}$, the stated values remain within the stated uncertainties if ${{M}_{\delta {\rm Cep}}}=5.0$ ${{M}_{\odot }}$ is adopted.

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Table 3. Orbital Solution for δ Cephei Based on the Combined Hermes and Literature Radial Velocities

Parm	${{v}_{\gamma }}$	T0	${{P}_{{\rm orb}}}$	K	e	ω	${{a}_{{\rm rel}}}$	${{a}_{1}}\;{\rm sin} \;i$	fm
Unit	(km s−1)	(days)	(days)	(km s−1)		(deg)	(AU)	(10−3 AU)	(10−3 ${{M}_{\odot }}$)
Value	−16.787	55649.68	2201.87	1.509	0.674	246.77	5.82	226.3	0.784
${{\sigma }_{+}}$	0.026	${{24.68}^{\dagger }}$	5.73	0.239	0.038	2.37	0.18	${{21.9}^{\dagger }}$	${{0.249}^{\dagger }}$
${{\sigma }_{-}}$	0.049	${{19.86}^{\dagger }}$	6.31	0.080	0.021	4.90	0.19	${{7.9}^{\dagger }}$	${{0.083}^{\dagger }}$

Note. ${{\sigma }_{+}}$ and ${{\sigma }_{-}}$ denote the upper and lower standard errors derived from marginal distributions. Quantities with superscript dagger have been computed using Gaussian error propagation. Other uncertainties were estimated by the MCMC analysis.

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Comparing our results to other known Cepheid orbits listed in the Cepheid binary database by Szabados (2003), we find that the semi-amplitude K of this orbit is the second smallest among all known binary Cepheid orbits, albeit with much larger eccentricity and somewhat longer period than W Sgr's orbit (Groenewegen 2008). This also helps to explain why δ Cephei has not previously been identified as a spectroscopic binary.

Projecting the orbit derived in Section 3.3 to the distance determined by Benedict et al. (2007, ϖ = 3.66 ± 0.15 mas) yields an orbital relative semimajor axis of ${{a}_{{\rm rel}}}=21.2$ mas, with an orbital barycentric semimajor axis of the Cepheid of ${{a}_{1}}\gt 0.84$ mas, a value more than five times the uncertainty estimated by Benedict et al. (2007), and even seven times the uncertainty stated by van Leeuwen et al. (2007, Hipparcos, 3.71 ± 0.12 mas). This prompts the question whether the observations taken by Hipparcos or HST are sensitive to the orbital motion; see Figure 6 for information as to which ranges of orbital phase were observed by these missions. We therefore explore the sensitivity of Hipparcos and Gaia astrometry to δ Cephei's hidden companion in this section.

4.1. Hipparcos IAD

To test whether Hipparcos (ESA 1997) was sensitive to the orbital motion of δ Cephei, we analyze the Hipparcos IAD published on the DVD attached to the new reduction by van Leeuwen (2007).

4.1.1. Parallax from Intermediate Astrometry Data

A five-parameter fit to the 95 measurements given on the DVD yields the residuals shown in Figure 8, left panel. There are several outliers, and fit quality is poor ($\chi _{{\rm red}}^{2}=2.01$, rms(O-C) = 1.43 mas). We obtain a best-fit parallax of ${{\varpi }_{{\rm all}}}=4.37\pm 0.27$ mas.

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We therefore discarded a total of six data points on the basis of their excess residual from the following satellite orbits: 180, 252, 396, 759, 1786, and 2126. We then repeated the five-parameter fit to 89 measurements and obtained a better fit (Figure 8, right panel; $\chi _{{\rm red}}^{2}=0.65$, rms(O–C) = 0.81 mas), as well as smaller parallax: ${{\varpi }_{5{\rm parm}}}=4.09\pm 0.16$ mas. Note that there is a discrepancy of ${\Delta }\varpi =0.38$ mas (0.21 mag) between the parallax value published in the van Leeuwen (2007, ϖ = 3.77 ± 0.16 mas) reduction and our result based on the IAD. There may be several reasons for such a discrepancy, including:

• 1.  
  Our fitting routine. We excluded this possibility by processing other stars, in particular δ Cephei's visual companion HD 213307, for which we obtained a parallax of 3.69 ± 0.46 mas, in exact numerical agreement with the published value (3.69 ± 0.46 mas; van Leeuwen 2007). As an additional cross-check, we applied the same methodology to more than 20 other Cepheids using the same routine and obtained results that agree well with the published values of van Leeuwen (2007).
• 2.  
  Our selection of IAD. As noted in the IAD header, 3% of the IAD were discarded to obtain the solution of van Leeuwen (2007), although it is not specified which ones. We attempted to reproduce their result by discarding only three measurements but did not succeed.
• 3.  
  The cluster solution. It appears that van Leeuwen (2007) used a special procedure for cluster stars to better remove outlier measurements for individual stars, under the assumption that all cluster stars are at indistinguishable distance. While our parallax result is larger than the previously published value, it is well within the wide range of parallaxes (2.57–5.28 mas) reported for presumed members of Cep OB6.
• 4.  
  Other unknown procedures or modifications that may have been applied specifically to δ Cephei's parallax to achieve the solutions presented by van Leeuwen (2007) and van Leeuwen et al. (2007).

Our parallax estimate ($\varpi =4.09\pm 0.16$ mas) is also considerably ($2\sigma$) larger than the HST-based result by Benedict et al. (2002, ϖ = 3.66 ± 0.15 mas). Such an increase in parallax would increase δ Cephei's absolute magnitude by 0.24 mag, making it intrinsically fainter than previously thought (${{M}_{V}}=-3.23$ mag using the absolute magnitude published by Benedict et al. 2002). We note that the HST-based period–luminosity relations presented by Benedict et al. (2007) seem to agree better with such an increase in absolute magnitude for δ Cephei.

The HST observations were taken at an orbital phase at which a significant parallax bias due to orbital motion is not very likely; see Figure 6. However, the assumption of a physical association between HD 213307 and δ Cephei in the loose association Cep OB6 was required to "reduce (the HST) astrometric residuals to near-typical levels" (Benedict et al. 2002, Section 5.5).

Concerning δ Cephei's membership in Cep OB6, we note that the association's discovery did not take into account RV data, since only little such information was available at the time (de Zeeuw et al. 1999). Inspection of the SIMBAD database yields RV information for 10 of the 19 presumed member stars (in addition to δ Cephei), with values ranging from −7 to −38 km s⁻¹ and median uncertainty of 1.2 km s⁻¹ based on measurements by Fehrenbach et al. (1996), Grenier et al. (1999), Famaey et al. (2005), Gontcharov (2006), and Kharchenko et al. (2007). The RVs are distributed as follows: HIP 110807 and HIP 112998 have ${{v}_{r}}\sim -7\pm 2$ km s⁻¹, HIP 110497 has ${{v}_{r}}=-13.3\pm 0.5$ km s⁻¹, HIP 109492 and HIP 110988 (=HD 213307) are close to δ Cephei's ${{v}_{\gamma }}=-16.8$ km s⁻¹ (to within 1 km s⁻¹), HIP 113993 has ${{v}_{r}}=-20.9\pm 1.1$ km s⁻¹, HIP 113316 has ${{v}_{r}}=-25.6\pm 8$ km s⁻¹, and three stars (HIP 110266, HIP 110275, HIP 110356) have ${{v}_{r}}\lt -30$ km s⁻¹ with reported uncertainties between 3 and 15 km s⁻¹. This relatively wide range of RVs suggests that Cep OB6 is not gravitionally bound, ¹¹ i.e., that the assumption of a common distance for all presumed member stars is not valid. This interpretation is corroborated by the wide range of parallax values of the presumed member stars (see item 3 above). However, more homogeneous and high-quality RV measurements of the presumed members of Cep OB6 are required to further illuminate this issue.

Although the above suggests that not all of Cep OB6's presumed members have indistinguishable distance, we note that the observational evidence does support the physical association between HD 213307 and δ Cephei, since their parallaxes and RVs agree to within the uncertainties. A re-assessment of the HST astrometric data without the assumption of cluster membership (this affects, e.g., the spectrophotometric parallax for HD 213307, which is part of the HST reference frame), as well as taking into account δ Cephei's orbital motion, would be useful for testing whether the difference between the published HST parallax and our result can be reconciled.

4.1.2. Orbit Analysis

We search for an orbital signature in Hipparcos IAD using the methodology described in Sahlmann et al. (2011b), which has been shown to reliably detect such orbital signatures in Hipparcos IAD (Sahlmann et al. 2011a, 2011b; Sahlmann & Fekel 2013). We use the spectroscopic orbital parameters given in Table 3 to fit the IAD with a seven-parameter model, which has the free parameters inclination i, longitude of the ascending node Ω, parallax ϖ, and offsets to the coordinates $({\Delta }\alpha ,{\Delta }\delta )$ as well as offsets to the proper motions $({\Delta }\mu \alpha ,{\Delta }\mu \delta )$. We then search a two-dimensional grid in i and Ω for its global ${{\chi }^{2}}$-minimum with a nonlinear minimization procedure. We determine the statistical significance of the derived astrometric orbit via a permutation test (Zucker & Mazeh 2001), for which we employ 1000 pseudo-orbits and derive parameter uncertainties using Monte Carlo simulations that include propagation of RV parameter uncertainties.

Only 41% of the orbits are probed by Hipparcos measurements, whose average measurement uncertainty of 1.07 mas is furthermore greater than the barycentric minimum semimajor axis of the orbit (0.84 mas). It may therefore not surprise that we determine virtually identical results for five-parameter (single star) and seven-parameter (binary) models, with flat residuals even for the five-parameter model (Figure 8). From an F-test, we obtain a probability of 4.4% that the single-star model is true, and the permutation test yields an orbit detection significant at the 1.8σ level (93.7%). While the evidence for the orbital signature is only marginally significant, it has been determined with two independent methods that are in agreement. It thus appears that there is some temporal coherence present in the Hipparcos data, although we cannot claim orbit detection using Hipparcos astrometry. We can, however, use our orbital analysis to set (very) loose constraints on the inclination ($10{}^\circ \lesssim i\lesssim 170{}^\circ$) of the orbit (see Figure 9), allowing us to place an upper limit on the companion mass with 0.2 ${{M}_{\odot }}$ $\lesssim \;{{M}_{2}}\lesssim 1.2$ ${{M}_{\odot }}$, which is fully consistent with the lack of spectral features due to the companion in the Hermes data set; see Section 3.

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While we are unable to claim detection of the orbit from Hipparcos astrometric data, we caution that the previous estimate of δ Cephei's proper motion may have been affected by the companion.

4.2. Gaia

Now that δ Cephei's nature as a spectroscopic binary is revealed, it is worth revisiting other observed features of the prototype of classical Cepheids in this new light.

Engle et al. (2014) recently provided evidence that δ Cephei is a soft X-ray source with a luminosity of ${{L}_{{\rm X}}}(0.3-2{\rm keV})\approx (4.5-13)\;\times \;{{10}^{28}}\;{\rm erg}\;{{{\rm s}}^{-1}}$ and peak flux at $kT=0.6-0.9$ keV. As these authors discuss, young (∼120 Myr for a 5 ${{M}_{\odot }}$ Cepheid, according to Geneva evolution models by Ekström et al. 2012; Georgy et al. 2013), low-mass main-sequence companions can provide coronal X-ray emission. In Sections 3.3 and 4.1.2, we constrained the mass range for the unseen companion to be $0.2\;{{M}_{\odot }}\lt {{M}_{2}}\lt 1.2$ ${{M}_{\odot }}$ based on RVs and Hipparcos astrometric measurements. As Cepheids are rarely detected in X-rays, it appears likely that δ Cephei's young main-sequence companion is responsible for the detected variable X-ray activity.

Due to the high eccentricity of the orbit, δ Cephei and its companion have recurrent close encounters, with a pericenter distance of ${{r}_{{\rm per}}}=(1-e){{a}_{{\rm rel}}}=1.89$ AU = 409 ${{R}_{\odot }}$= 9.5 ${{R}_{\star }}$, where ${{R}_{\star }}=43.3$ ${{R}_{\odot }}$ (Turner 1988; Mérand et al. 2005; Natale et al. 2008). It is also important to bear in mind that δ Cephei is currently in the core He burning phase, most likely on the second crossing of the instability strip as shown by the decrease in pulsation period (measured, e.g., by E. Hertzsprung and reported by Eddington 1919; Berdnikov & Ignatova 2002; Engle et al. 2014). It has therefore previously occupied the red giant branch, where its radius was even larger, approximately ${{R}_{\star ,{\rm RG}}}\sim 80$ ${{R}_{\odot }}$. Assuming an unchanged orbit, pericenter passage would have brought the two stars to within $5\times {{R}_{\star ,{\rm RG}}}$.

Taking these considerations further, we consider the possibility of previous interactions between δ Cephei and its companion. Since δ Cephei is on a highly eccentric orbit and the Roche-lobe formalism is valid only for circular orbits, we adopt the Roche-lobe formalism in the quasi-static approximation as presented by Sepinsky et al. (2007a). Assuming ${{M}_{\delta {\rm Cep}}}=5.2$ ${{M}_{\odot }}$, ${{M}_{2}}=0.7$ ${{M}_{\odot }}$, and equatorial velocity ${{v}_{{\rm eq}}}=10$ km s⁻¹, we obtain the volume-equivalent Roche-lobe radius at pericenter of ${{R}_{{\rm Roche},{\rm corr}}}=1.05$ AU $\approx 5.2\;{{R}_{\star }}\approx 2.8\;{{R}_{\star ,{\rm RG}}}$. Hence, at pericenter passage δ Cephei fills 19% of its (quasi-static) Roche lobe, and the Roche-lobe radius at pericenter is approximately 55% of the distance at pericenter passage (${{r}_{{\rm per}}}=1.89$ AU). At apocenter, the distance between the two stars is a factor $(1+e)/(1-e)=5.1$ larger, and δ Cephei fills only about $3.7\%$ of its Roche lobe. δ Cephei may thus become noticeably deformed due to tidal interactions close to pericenter passage, while being spherical near apocenter passage. During the red giant phase, this situation would have been even more extreme, if the past orbit was similar to the present-day orbit. The above considerations suggest that the observed circumstellar material (Mérand et al. 2006; Marengo et al. 2010) and bow shock (Matthews et al. 2012) may originate from previous and ongoing binary interactions. A more detailed investigation of such interactions is required to discuss this scenario in terms of δ Cephei's mass-loss history (see the discussion in Matthews et al. 2012) but is considered out of scope for this paper.

If δ Cephei and its companion have a history of episodal interactions at pericenter passage, then the high eccentricity of the orbit may appear surprising. However, Sepinsky et al. (2007b) showed that rapid circularization is not expected for all close-in binaries with this type of interaction. Applying our results for δ Cephei to their formalism shows that δ Cephei's orbit is not expected to have been rapidly circularized. Furthermore, δ Cephei is a visual binary whose outer companion is understood to be physically associated and on a very long period orbit (Benedict et al. 2002). The high eccentricity of the inner binary (the one shown in the present work) could thus have been driven up by the outer companion by a Kozai–Lidov mechanism (Kozai 1962; Lidov 1962) and may have varied significantly over its evolutionary history.

If δ Cephei and its companion have undergone significant interactions, one might expect the evolutionary status of δ Cephei to vary significantly from that of a Cepheid with a single star progenitor. We therefore examine δ Cephei's current evolutionary status in Figure 10 to search for signs of non-standard evolution. We compare observed absolute V magnitude, ${{(B-V)}_{0}}$ color (both from Benedict et al. 2002), and rate of (pulsation) period change $\dot{{{P}_{{\rm puls}}}}=-0.1006\pm 0.0002$ (Engle et al. 2014) with predictions from Geneva stellar evolution models ¹⁵ of solar metallicity that include rotation (Ekström et al. 2012; Georgy et al. 2013) and have been studied specifically in the context of classical Cepheids by Anderson et al. (2014). Strong disagreement between the observations and these predictions could be considered evidence for binary interactions, since no binary interactions are accounted for in these model predictions. However, we find that δ Cephei's location in both diagrams is consistent with a 5.25 ${{M}_{\odot }}$ Cepheid whose progenitor had a slightly faster than average initial rotation (${\Omega }/{{{\Omega }}_{{\rm crit}}}\sim 0.7$). Assuming these parameters for the evolutionary models yields an age of 112 Myr. Adopting our new parallax from Section 4.1.1 yields a smaller initial mass of $M\approx 5.0$ ${{M}_{\odot }}$ and thus an older age of 127 Myr, while leaving the implications regarding the rotational history unchanged.

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At the present level of accuracy, we thus do not find any irreconcilable discrepancies between the predicted and observed evolutionary states of δ Cephei. This shows either that binary interactions, if present, have had a negligible effect on the evolutionary path of δ Cephei, or that the interactions are weak and slow enough for δ Cephei to reach an equilibrium state similar to a non-interacting star.

In summary, the discovery of δ Cephei's nature as a spectroscopic binary helps to complete the puzzle created by a plethora of observations. While there is no clear evidence for a non-standard evolutionary path, δ Cephei is a particularly interesting example of the limitations that this evolutionary phase is subject to for binary stars (see Neilson et al. 2014) and deserves detailed observational follow-up and dynamical modeling to further investigate its intriguing past that may have been marked by tidal interactions due to both the inner (discovered here) and outer (HD 213307) companions.

Approximately 230 yr after the discovery of its variability (Goodricke 1786), we discover the spectroscopic binary nature of δ Cephei, archetype of classical Cepheid variable stars and one of the most-studied variable stars.

Our discovery is demonstrated using new high-precision RVs measured from high-quality optical spectra obtained with the high-resolution spectrograph Hermes. Combining these new high-precision data with lower-precision RVs from the literature, we determine the orbital solution for the spectroscopic binary, which is an inner binary to the outer visual binary system discussed by Benedict et al. (2002). δ Cephei thus appears to be a pair of binary stars.

We re-analyze Hipparcos IAD (see Section 4.1.1) and obtain the parallax $\varpi =4.09\pm 0.16$ mas ($d=244\pm 10$ pc) using a five-parameter (single star) model. This result is larger than the estimates reported by van Leeuwen (2007, $\varpi =3.77\pm 0.16$ mas), van Leeuwen et al. (2007, $\varpi =3.71\pm 0.12$ mas), and Benedict et al. (2002, 3.66 ± 0.15 mas). While these previously published results based on Hipparcos and HST astrometry agree to within their stated uncertainties, they all shared a common assumption of δ Cephei's membership in the loose association Cep OB6, which our analysis does not and which we argue should be revisited using RVs. Relaxing this assumption and accounting for orbital motion in a re-analysis of the HST astrometric data would be useful to test whether the existing HST astrometry can be reconciled with our Hipparcos-based result.

We perform an orbital analysis of the Hipparcos IAD in Section 4.1.2 and find tentative evidence for an orbital signature, although no detection can be claimed. Based on detailed simulations, we show that Gaia is highly sensitive to the astrometric orbit of δ Cephei and will likely model the full set of Keplerian parameters with better than 10% accuracy. The orbit will have to be accounted for when determining proper motions from Hipparcos and Gaia data.

Using the constraints provided by the optical spectra, the orbit measured from RVs, the astrometric orbital analysis,and assuming a mass of 5.0–5.25 ${{M}_{\odot }}$, we constrain the mass range of the companion to be $0.2\;{{M}_{\odot }}\lt {{M}_{2}}\lt 1.2$ ${{M}_{\odot }}$. Adopting the lower mass for δ Cephei mainly affects the upper mass limit, which would become 1.1 ${{M}_{\odot }}$ in this case. Given that the spectroscopic companion is expected to be the same age as δ Cephei, i.e., approximately 100–130 Myr (depending on mass and initial rotation rate of the progenitor), the reported X-ray emission detected using XMM-Newton (Engle et al. 2014) could be explained by magnetorotational activity of a young main-sequence star.

The close pericenter approach of the two stars has potentially far-reaching consequences for the explanation of the observed circumstellar environment of δ Cephei. Detailed modeling of the orbital and stellar evolution of this complex system is desirable to further improve our understanding of the archetype of classical Cepheids and its intriguing past.

Many thanks are due to everyone who aided in securing the analyzed data sets and, in particular, to the Hermes and Mercator teams. Damien Ségransan is acknowledged for assistance regarding the use of Yorbit. R. I. A. acknowledges Miranda A. Gaanderse for her creative assistance in finding the title for this paper and Dr. Paul I. Anderson for his careful reading of the manuscript. We thank the anonymous referees for their constructive reports. This research is based on observations made with the Mercator Telescope, operated on the island of La Palma by the Flemish Community, at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofsica de Canarias. Hermes is supported by the Fund for Scientific Research of Flanders (FWO), Belgium; the Research Council of K.U. Leuven, Belgium; the Fonds National de la Recherche Scientifique (F.R.S.-FNRS), Belgium; the Royal Observatory of Belgium; the Observatoire de Genève, Switzerland; and the Thüringer Landessternwarte, Tautenburg, Germany. This study also employed spectral data retrieved from the ELODIE archive at Observatoire de Haute-Provence (OHP). This research has made use of NASA's ADS Bibliographic Services, the SIMBAD database, and the VizieR catalogue access tool provided by CDS, Strasbourg, and Astropy, a community-developed core Python package for Astronomy (Astropy Collaboration et al. 2013). R. I. A. acknowledges funding from the Swiss NSF. J. S. is supported by an ESA research fellowship in space science.

Facilities: Mercator1.2 m, HIPPARCOS