THE BANANA PROJECT. III. SPIN–ORBIT ALIGNMENT IN THE LONG-PERIOD ECLIPSING BINARY NY CEPHEI^*

Close binaries and star–planet systems might be expected to have well-aligned orbital and spin angular momenta, since all of the components trace back to the same portion of a molecular cloud. However, good alignment is not guaranteed. If a cloud is highly elongated, with its long axis tilted with respect to its rotation axis, then the cloud may give birth to binary stars with a strong spin–orbit misalignment (Bonnell et al. 1992). Alternatively, disks around young stars might become warped during the last stage of accretion. This warp could torque the orbit by a large angle while maintaining the orientation of the spins (Tremaine 1991). More generally, star formation may be a chaotic process, with accretion from different directions at different times (Bate et al. 2010), and perhaps we should not expect the angular momenta of the star and the disk to be as well aligned as they apparently were in the solar system.

There are also processes that could alter the stellar and orbital spin directions after their formation. A third body orbiting a close pair on a highly inclined orbit can introduce large oscillations in the orbital inclination and eccentricity of the close pair (Kozai 1962). Tidal dissipation during the high-eccentricity phases can cause the system to free itself of these "Kozai oscillations" and become stuck in a high-obliquity state (Fabrycky & Tremaine 2007). However, if dissipation is sufficiently strong then the system will evolve into the double-synchronous state, characterized by spin–orbit alignment (e.g., Hut 1981). Therefore, whether a close binary or a star–planet system is well aligned or misaligned depends on its particular history of formation and evolution. Even though this issue is important for a complete understanding of star formation, there has been very little observational input.

Another motivation comes from exoplanetary science. Recently it was revealed that many of the close-in giant planets ("hot Jupiters") have orbits that are strongly misaligned with the rotation axes of their parent stars (see, e.g., Hébrard et al. 2008; Winn et al. 2009; Narita et al. 2009; Triaud et al. 2010). The high obliquities are especially common among stars with higher masses and effective temperatures (Winn et al. 2010; Schlaufman 2010). The results of such studies are commonly interpreted as constraints on theories of the "migration" processes that presumably brought the planets inward from their more distant birthplaces. However, some theories invoke processes that produce large stellar obliquities for reasons having nothing to do with the planets, such as chaotic accretion (Bate et al. 2010) and magnetic interactions with the inner edge of the accretion disk (Lai et al. 2010). Although those theories have not been fully developed, it would seem that the mechanisms they propose should also operate in the case of binary stars, and therefore the theories might be tested by measuring the obliquities of binary stars.

For example, if interactions with a distant companion are important for close double stars, then this should also be true for star–planet systems. Other effects like tidal realignment do depend on the mass and mass ratio of the close pair. For a direct comparison with the exoplanet hosts, one would want to survey main-sequence F and G binaries, while (as we will describe) most of the existing data, including the data presented in this paper, are for earlier type stars. We hope to rectify this situation with future observations.

Measuring the orientations of stellar spin axes is not straightforward. Telescopes cannot usually resolve stellar disks, causing the information on spatial orientation to be lost. An optical interferometer, in combination with a high-resolution spectrograph, might allow for spatial resolution of rotationally broadened stellar absorption lines (Petrov 1989; Chelli & Petrov 1995), but so far such measurements are possible for only the very nearest and brightest systems (Le Bouquin et al. 2009).

One would be able to determine the inclinations of the spin axes with respect to the sky plane, based on empirical estimates of the projected stellar rotation speed (vsin i), the stellar radius (R_⋆), and the stellar rotation period (P_rot), using the equation

$$\begin{equation} i = \sin ^{-1} \left[ \frac{v\sin i}{(2\pi R_\star /P_{\rm rot})} \right]. \end{equation} \tag{ 1 }$$

Many investigators have pursued this path and found it to be blocked, for reasons described clearly by Soderblom (1985). Namely, the quantities vsin i, R_⋆, and P_rot are difficult to measure with high enough accuracy, and the flattening of the sine function near 90° prevents the method from discriminating even modest inclinations from edge-on inclinations. Despite these difficulties, Glebocki & Stawikowski (1997) used this method to argue that active binaries that are observed to be asynchronously rotating have misaligned spin axes.

Given a large sample of vsin i measurements of stars that have a known distribution of rotation speeds, or that are presumed to have similar rotation speeds (e.g., main-sequence stars of a given mass and age), it is possible to test for departures from an isotropic distribution of spin directions. Using this idea, Guthrie (1985) and Abt (2001) searched for, and did not find, a tendency for stars to be preferentially aligned with the Galactic plane. When the orbital orientation is also known, as is the case for visual binaries or eclipsing binaries, then such tests give information about spin–orbit alignment. Such studies are worthwhile but they are hampered by the nonlinearity of the sine function (as mentioned above) as well as uncertainty in the underlying distribution of rotational speeds, and the heterogeneity in the techniques for measuring vsin i. This method has been used in various guises by Weis (1974), Hale (1994), and Howe & Clarke (2009), among others. Recently, Schlaufman (2010) used this method on stars with transiting planets, finding evidence that more massive stars have high obliquities.

Indirect evidence on the orientation of young stars comes from the orientation of their disks—assuming star–disk alignment. The projected rotation angle of the disk can be traced by the linear polarization vector of the starlight reflected from the dust grains in disks (Monin et al. 1998). These authors and others (e.g., Donar et al. 1999; Wolf et al. 2001; Jensen et al. 2004; Monin et al. 2006) find that in most binary systems the disks around the individual stars are aligned with each other. However they also mention exceptions to this rule. One of these exceptions is the hierarchical triple system T Tauri, for which Skemer et al. (2008) and Ratzka et al. (2009) found that the disk around the northern component, the original T Tauri system, is viewed face on, while the disk around one of the southern components (T Tau Sa), separated by ≳100 AU from T Tau N, is viewed edge-on. Thus this system has misaligned circumstellar disks.

The method that provides the most accurate information for individual systems takes advantage of eclipses. During an eclipse of one star by another star or a planet, part of the rotating stellar surface is hidden, causing a weakening of the corresponding velocity component of the stellar absorption lines. Modeling of this spectral distortion reveals the relative orientation of the spin and orbital axes on the sky: the projected obliquity. This "rotation anomaly" was first predicted by Holt (1893). A claim of its detection in the δ Librae system was made by Schlesinger (1910), but more definitive measurements were achieved by Rossiter (1924) and McLaughlin (1924) for the β Lyrae and Algol systems, respectively. The phenomenon is now known as the Rossiter–McLaughlin (RM) effect. Various aspects of the theory of the effect have been worked out by Struve & Elvey (1931), Hosokawa (1953), Kopal (1959), Sato (1974), Ohta et al. (2005), Giménez (2006), Hadrava (2009), and Hirano et al. (2010). The work described in this paper, as well as in the previous papers in this series, is based on the RM effect.

Although it has been more than 80 years since its discovery, there are relatively few quantitative analyses of the RM effect observed in stellar binaries. It is ironic that at present there are more such papers about exoplanetary systems than about stellar binaries. In the past, observing the RM effect was generally either avoided (as a hindrance to measuring accurate spectroscopic orbits) or used to estimate stellar rotation speeds. Almost all authors explicitly or implicitly assumed that the orbital and stellar spins were aligned.

Table 1 shows the results our literature search for analyses of the RM effect in stellar binaries. (This table also gives the results for NY Cep, the subject of this paper.) We have tried to be comprehensive, but cannot claim our list to be complete. The authors would appreciate being notified of any omissions. The systems are listed in order of decreasing r_p, the radius of the primary star in units of the orbital semimajor axis, and thus the ordering is approximately a progression from closely interacting systems to well-detached systems (although this is not true for some of the Algol systems, which have large faint secondaries). Column 6 summarizes the results for the stellar obliquities. In most cases we have written "aligned?" because the RM data appear visually to display the pattern of a well-aligned system—a redshift during the first half of the eclipse, followed by a blueshift of equal amplitude during the second half of the eclipse—with the question mark indicating that no quantitative analysis was undertaken, and hence the uncertainty is unknown. For systems with orbital inclinations very close to 90° the results are especially ambiguous because in such cases there is a strong degeneracy between the projected obliquities and rotation rates (Gaudi & Winn 2007). For DE Dra we wrote "misaligned?" because Hube & Couch (1982) found the RM effect to be asymmetric about the mid-eclipse time, but gave no quantitative result for the projected obliquity. The only cases where quantitative results for the projected obliquities are given are from the Banana project.

Table 1. Measurements of Spin–Orbit Angles in Eclipsing Binaries

System	Spectral Type	Period (days)	rp	Eccentricity (e)	β	Reference
β Lyr	Be+B6-8II	12.9	0.52	<0.01	Aligned?	1,2
V1010 Oph	A7IV-V+?	0.66	0.4511 ± 0.011	≡ 0	Aligned?	3,4
W Umi	A3V+G9IV	1.7	0.363 ± 0.001	≡ 0	Aligned?	5
δ Lib	A0V+K0IV	2.33	0.30 ± 0.06	≡ 0	Aligned?	6,7
V505 Sgr	A2V+GIV	1.2	0.288 ± 0.001	≡ 0	Aligned?	8,9
AI Dra	A0V+F9.5V	1.2	0.284 ± 0.004	≡ 0	Aligned?	8,10
X Tri	A3V+G3IV	0.97	0.272 ± 0.003	≡ 0	Aligned?	5
RZ Cas	A3V+KIV	1.20	0.233 ± 0.001	≡ 0	Aligned?	5
U Sge	B8.5V+G3III	3.38	0.219 ± 0.002	0.04	Aligned?	5
WW Cyg	B8V+G4III	3.32	0.215 ± 0.002	≡ 0	Aligned?	5
Y Leo	A3V+K3IV	1.69	0.213 ± 0.001	≡ 0	Aligned?	5
RX Hya	A8+K0IV	2.28	0.211 ± 0.013	≡ 0	Aligned?	5
Algol	B8+K2IV	2.87	0.206 ± 0.003	≡ 0	Aligned?	11,12,13
RW Gem	B5-B6V+F0III	2.87	0.198 ± 0.002	≡ 0	Aligned?	5
Y Psc	A3V+K2IV	3.77	0.193 ± 0.001	0.12	Aligned?	5
TV Cas	A2V+G1IV	1.81	0.188 ± 0.15	≡ 0	Aligned?	5
ST Per	A3V+KIV	2.65	0.184 ± 0.004	≡ 0	Aligned?	5
U Cep	B7V+G8III	2.49	0.177 ± 0.009	≡ 0	Aligned?	5
TX Uma	B8V+F7-F8III	3.06	0.164 ± 0.002	≡ 0	Aligned?	5
W Del	A0-B9.5V+K0IV	4.81	0.151 ± 0.004	0.20	Aligned?	5
AA Dor	sdOB+?	0.26	0.14 ± 0.01	≡ 0	Aligned?	14
DE Dra	B0V+B2V	5.3	0.14	0.018 ± 0.011	Misaligned?	15
SW Syg	A2V+K1IV	4.57	0.138 ± 0.002	0.30	Aligned?	5
RY Per	B4V+F0III	6.86	0.137 ± 0.003	0.21	Aligned?	5
RZ Sct	B2II+A0II-III	15.19	0.136 ± 0.007	≡ 0	Aligned?	5
AQ Peg	A2V+K1IV	5.55	0.123 ± 0.002	0.24	Aligned?	5
RY Gem	A2V+K0III-K1IV	9.30	0.097 ± 0.014	0.16	Aligned?	5
NY Cep	B0V+B2V	15.3	0.086 ± 0.015	0.445 ± 0.004	βp = 2° ± 4°	16,17
DI Her	B4V+B5V	10.55	0.0621 ± 0.001	0.489 ± 0.003	βp = 72° ± 4°, βs = −84° ± 8°	18,19
V1143 Cyg	F5V+F5V	7.64	0.059 ± 0.001	0.5378 ± 0.0003	βp = 03 ± 15, βs = −12 ± 16	20,21

Notes. In Column 4, r_p denotes the radius of the primary star divided by the length of the orbital semimajor axis. For DE Dra and β Lyr no uncertainties in the primary radii are given in the references. In Column 5, the entry "≡ 0" indicates that the eccentricity was assumed to be zero by the authors. In Column 6, β_p and β_s denote angles between the projections of stellar and orbital spin axes for the primary and secondary, respectively, using the coordinate system of Hosokawa (1953). References. (1) Rossiter 1924; (2) Harmanec 2002; (3) Worek et al. 1988 (4) Corcoran et al. 1991; (5) Twigg 1979; (6) Bakış et al. 2006; (7) Worek 1985(8) Worek 1996; (9) Lázaro et al. 2006; (10) Lázaro et al. 2004; (11) McLaughlin 1924; (12) Struve & Elvey 1931; (13) Soderhjelm 1980; (14) Rucinski 2009; (15) Hube & Couch 1982; (16) Holmgren et al. 1990; (17) This study; (18) Popper 1982; (19) Albrecht et al. 2009; (20) Andersen et al. 1987; (21) Albrecht et al. 2007.

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For our program, we have begun by concentrating on relatively young, detached systems with r_p < 0.15, in order to minimize the effects of tidal interaction, and thereby study a more "primordial" distribution of obliquities. We also omit Algol-type systems, to avoid the extra complexity of the spin evolution caused by mass transfer between the stars. These criteria exclude the Algol-type systems studied by Twigg (1979) as well as all the systems listed in Table 1 except for DE Dra, V1143 Cyg, DI Her, and NY Cep. The result of misalignment in the DE Dra system by Hube & Couch (1982) is intriguing but needs to be checked. For V1143 Cygni we showed that both spin axes are well aligned with the orbital axis (Paper I, Albrecht et al. 2007). In contrast, for our second target, DI Her, we measured spin axes that are drastically misaligned with the orbital axis (Paper II, Albrecht et al. 2009). This was the first clear demonstration of such a strong misalignment in a close binary. It resolved the longstanding problem of the system's anomalous apsidal motion.

In this paper, we focus on the NY Cephei system, whose properties are summarized in Table 2. A summary of the history of observations of this system was given by Ahn (1992). It harbors two early B-type stars on an eccentric 1527 orbit. One peculiar characteristic of this system is the lack of secondary eclipses. Inferior conjunction occurs when the stars are relatively far apart, and with an orbital inclination of ≲80° the sky-projected separation of the stars is too large for eclipses. The orientation of the orbit is illustrated in Figure 1.

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Table 2. General Data on NY Cephei

HIP	113461
HD	217312
R.A.J2000	22h58m40s a
Decl.J2000	63°04'38''a
Vmax	7.5 maga
Spectral type	B0.5V+ B2Vb
Period	1527c
Eccentricity	0.48(2)b
Inclination	78(1)°b
Rp	6.8(7)R☉b
Rs	5.4(5)R☉b
Mp	13(1)M☉b
Ms	9(1)M☉b
Ls/Lp	0.37(6)b

Notes. R_p denotes the radius of the primary component and R_s the radius of the secondary component. L_p/L_s denotes the luminosity ratio between the primary and secondary. ^aData from ESA (1997). ^bData from Holmgren et al. (1990). ^cData from Ahn (1992).

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Section 2 of this paper presents our observations, describes our model for the spectroscopic data, and our analysis procedure. The results for the orbital and stellar parameters, including the measurement of the stellar orientation, are presented in Section 3. The results are discussed in Section 4.

Our analysis focused on the 11th échelle order of the Sophie CCD, which encompasses the wavelength range from 4459 Å to 4486 Å, including the two best absorption lines available for this study, He i (4471 Å) and Mg ii (4481 Å). The helium line is strong, with a line width dominated by pressure broadening. The magnesium line is relatively weak but has the virtue of being chiefly broadened by rotation and therefore well suited to the analysis of the RM effect. There are also a number of other weaker lines in this order that must be modeled simultaneously with the stronger lines. The lines are illustrated in Figure 3, which shows the spectrum of each star individually, after disentangling them with the modeling procedure described in this section. The spectrum from that order was binned to a resolution of about 9 km s⁻¹, giving 213 pixels in the region of interest. The S/N was estimated from the scatter in the continuum on both sides of the He i (4471 Å) and Mg ii (4481 Å) lines.

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Our spectral model is similar to the models described in Papers I and II. Because of the light from the foreground star, we cannot treat the RM effect as a simple wavelength shift, as is commonly done for eclipses of stars by planets. Instead we modeled the line profiles of both stars simultaneously, taking into account stellar rotation, surface velocity fields, orbital motion, and partial blockage during eclipses. By adjusting the parameters of the model to fit the observed spectra, we derived estimates of the orbital and stellar parameters.

For each star, and for each phase of the eclipse, we created a discretized stellar disk with about 200,000 pixels in a Cartesian coordinate system. We assumed the disk to be circular, since the stars are well detached and have slow rotation speeds relative to the breakup velocity. Each pixel has its own emergent spectrum, weighted in intensity according to a linear limb-darkening law, and Doppler shifted due to the combined effects of orbital motion, rotation, and macroturbulence. The orbital motion is specified by a Keplerian model common to all pixels. The rotation is assumed to be uniform (no differential rotation). Following Gray (2005), the macroturbulent velocity field is assumed to obey Gaussian distributions for the tangential and radial components, with equal amplitudes, brightnesses, and surface fractions.

The sky-plane coordinates of the foreground and background stars are calculated, and if the background star is being eclipsed then the eclipsed pixels are assigned zero intensity. The emergent spectra from the uncovered pixels of both stars are summed to create a model absorption line kernel. For the He i line a further convolution with a Lorentzian function is applied to account for pressure broadening. Figure 4 illustrates the effects of rotation, macroturbulence, pressure broadening, and eclipse blockage on the model absorption line kernel.

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The kernel is then convolved with a line list in the wavelength region between 4460 Å and 4485 Å. The positions and line strengths are obtained from the Vienna Atomic Line Database (VALD)⁴ for the stellar parameters given by Holmgren et al. (1990) and are listed in Table 2. The He i, Mg ii, Al iii, and some of the O ii lines consist of multiple lines with energy levels spaced closely enough (⩽0.3 Å) that we model them as single lines. We also omitted one S iii line which has nearly the same wavelength as one of the O ii lines.

Since we did not obtain new photometric data, we relied on the data presented by Ahn (1992) (in their Table 2) to constrain the orbital period and time of primary minimum light. The lack of error estimates by Ahn (1992) presented a complication, which we dealt with as follows. We fitted a linear function of epoch to the reported times, assuming equal errors in all the measurements, determined by the requirement χ² = N_dof. The resulting error in each time was 0.012 days, and the resulting ephemeris are given in Table 3 (see also Figure 5). Our result for the period agrees with that given by Ahn (1992), although we found a slightly different time of primary minimum light, presumably due to different weighting of the measurements.

Then we used a Monte Carlo approach to calculate the predicted times of minimum light at the epochs of our observations, in a manner that respects the correlation between the parameters of the ephemeris. We created 10⁵ fake data sets by adding Gaussian perturbations to the times of minima of Ahn (1992), with a standard deviation of 0.012 days. We then fitted a line to each of these fake data sets and used the linear fit to calculate the expected mid-eclipse time on 2009 October 12/13, the last night of our eclipse observations. The mean and standard deviation among all 10⁵ results were taken to be the value and the error in the mid-eclipse time. The results are given in Table 3.

Table 3. Results of Eclipse Timings

Parameter	This Work	Ahn (1992)
Period	1527566 ± 0.00002	1527566
Tmin,I 1973 [HJD −2400000]	41903.819 ± 0.006	41903.8161
Tmin,I 2009 [HJD −2400000]	55117.265 ± 0.016

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We now describe the model parameters in detail. Seven parameters describe the Keplerian orbit: the time of minimum light (T_min,I), the eccentricity (e), the argument of periastron (ω), the velocity semiamplitudes of the primary and secondary (K_p and K_s), and the velocity offsets (γ_p and γ_s).⁵ We held the period (P) fixed, as its uncertainty is negligible over the 3 months spanned by our observations.

Another four free parameters specify the photometric aspects of the eclipse: the light ratio between the two stars at the wavelength of interest (L_s/L_p at 4500 Å), the fractional radii of the stars (r_p and r_s), and the orbital inclination (i_o). The linear limb-darkening parameter (u_i) was held fixed at 0.4 for both stars (Gray 2005).

The model of the velocity fields introduces six parameters: the projected equatorial rotation speeds (vsin i_p and vsin i_s), the Gaussian width of the macroturbulence for the primary star (ζ_RTP), the half-width at half-maximum of the Lorentizan function representing pressure broadening for each star (ξ_p and ξ_s), and, finally, the parameter of greatest interest for this study, the sky-projected spin–orbit angle (β_p). The angle is defined according to the convention of Hosokawa (1953), such that β = 0° when the axes are parallel, β = ±90° when they are perpendicular, and β>0 when the time-integrated RM effect is a redshift for an orbit with i_o < 90°.⁶ The macroturbulence parameter for the secondary was held fixed to 20 km s⁻¹, since the secondary is significantly fainter than the primary, causing the results to be insensitive to this parameter.⁷

Eight additional parameters are needed for each star to describe the relative depth of the spectral lines in the wavelength range being modeled. In addition, three free parameters are needed to describe the quadratic function used to normalize the continuum level of each of the 46 spectra. The total number of parameters is therefore 7 (orbital) + 4 (photometric) + 6 (velocity fields) + 8 × 2 (line depths) + 3 × 46 (normalization), for a total of 171 adjustable parameters. Although this may seem like a large number, the 138 normalization parameters are "trivial" in the sense that they can be optimized separately in subsets of 3, for a given choice of the other parameters. Also, several of the other parameters are subject to a priori constraints. More detail on these points is given below.

The fitting statistic was

$$\begin{eqnarray} \chi ^{2} & = & \sum _{i=1}^{46} \sum _{j=1}^{213} \left[\frac{f_{\lambda _{i,j}}({\rm obs})-f_{\lambda _{i,j}}({\rm calc})}{\sigma _{i}}\right]^2\nonumber \\\ms & & +\left(\frac{{T}_{\rm min, I} - 2455117.265}{0.016}\right)^2 \nonumber \\\ms & & +\left(\frac{i_{o} -78^{\circ }}{1^{\circ }}\right)^2 + \left(\frac{{L}_{{\rm s}}/{L}_{{\rm p}} - 0.37}{0.06}\right)^2 \nonumber \\\ms & & +\left(\frac{R_{\rm p}/R_\odot - 6.8}{0.7}\right)^2 + \left(\frac{R_{\rm s}/R_\odot - 5.4}{0.5}\right)^2, \nonumber \end{eqnarray}$$

where $f_{\lambda _{i,j}}({\rm obs})$ is the observed flux in pixel j in observation i and $f_{\lambda _{i,j}}({\rm calc})$ is the calculated flux based on a particular choice of the model parameters, and σ_i represents the uncertainty in the flux pixel for observation i. The latter was taken to be the reciprocal of the S/N, estimated as described in Section 2. In this equation, the first two terms represent the usual sum-of-squares, and the remaining terms represent Gaussian priors based on the results of Holmgren et al. (1990) (except for T_min,I, which was discussed in the previous section).

To optimize the parameters we used Levenberg–Marquardt (LM) least-squares minimization. The main optimization was conducted for the nontrivial parameters. However, whenever χ² was computed within that process, the values of the trivial (normalization) parameters were first optimized using a separate three-parameter minimization for each observation. This process is equivalent to the "Hyperplane Least Squares" method that was described and tested by Bakos et al. (2010). To estimate the parameter uncertainties, we used the bootstrap method described by Press et al. (1992) with 5 × 10³ realizations.

With 171 parameters, 5 Gaussian priors, and 46 × 213 data pixels, there are 9632 effective degrees of freedom. The best-fitting model has χ² = 9148 or ${\overline{\chi }}^2 = 0.95$. The low ${\overline{\chi }}^2$ is probably the result of underestimating the S/N of the relevant lines. The estimate was based on the continuum nearer to the edges of the order, where the S/N is lower (see Figure 2). We have not attempted to correct for this effect.

Our result for T_min,I is 1.2σ away from the value calculated from the older eclipse timings. Since the ephemeris is based on observations obtained more than 20 years ago, and we do not know the true uncertainty in each measurement, we regard this as good agreement. Likewise the eccentricity is within 2σ of the value given by Holmgren et al. (1990). The measured value for ω agrees with the previous value, but as it is expected to change over time due to apsidal motion a more careful comparison is needed, as discussed in Section 4.

Our results for K_p and γ_p agree with the values found by Holmgren et al. (1990). However, our results for the secondary star are different: we find a lower value for K_s, and we found Δγ ≡ γ_s − γ_p to be consistent with zero. In contrast, Holmgren et al. (1990) found the secondary to be redshifted by 26 km s⁻¹ relative to the primary, and Heard & Fernie (1968) found an even greater difference of 32 km s⁻¹. We speculate that the previous measurements were subject to bias because of the relative faintness of the secondary, its rapid rotation, and the need to use pressure-broadened (and asymmetric) lines. Based on the agreement in γ_p it seems that the systemic velocity has not changed significantly over the last decades.

We further find with our fit to all spectra a luminosity ratio (L_s/L_p) of 0.36 ± 0.05 at 4500 Å. This result agrees with the value 0.35 that is obtained from comparing the equivalent widths of the He i (4471 Å) line profiles shown in Figure 3.⁸ Both values are consistent with 0.37 ± 0.06 found by Holmgren et al. (1990), based on spectrophotometry.

Our model gives projected rotation speeds of 78 ± 1 km s⁻¹ for the primary and 155 ± 4 km s⁻¹ for the secondary. The width of the Gaussian function describing the macroturbulence in the primary is 23 ± 3 km s⁻¹, and the half-widths at half-maximum of the Lorentzians describing pressure broadening are 36 ± 1 km s⁻¹ and 54 ± 6 km s⁻¹. These error estimates must be understood as internal to our model, which we recognize may not be completely realistic, especially as it pertains to pressure broadening. We have assumed that absorption lines form at a specific value of pressure, which is not necessarily true. For this reason, the Lorentzian widths have no simple physical interpretation, and the results may be somewhat biased for all other parameters affecting line broadening (namely vsin i and ζ). For comparisons to other analyses obtained with different instruments and different analysis procedures, we recommend using the more conservative error estimates given in Table 4. For the projected rotation rates, we find a similar result for the primary as did Holmgren et al. (1990), but we find a significantly higher vsin i for the secondary, perhaps for the same reasons given in Section 3.1.

As for the projected spin–orbit angle β_p, the focus of this study, we find β_p = 2° ± 4°. There is no correlation between this parameter and the line width parameters, so the concerns raised above about the oversimplified pressure-broadening model do not apply here. A strong correlation does exist with the time of minimum light (see Figure 10). This implies that future photometric observations to refine the eclipse ephemeris would lead to smaller uncertainty in β_p, although the uncertainty is already quite small.

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The absolute radii of the stars can be calculated from the fractional radii and the other orbital parameters. We find radii consistent with the values given in the literature, R_p = 6.0 ± 0.5 R_☉ and R_s = 5.8 ± 0.6 R_☉, which is not surprising since our results were strongly influenced by the priors on those parameters. For the stellar masses, we find M_p = 10.7 ± 0.8 M_☉ and M_s = 8.8 ± 0.5 M_☉ for the primary and secondary, respectively. The value found for the primary is smaller than that reported previously due to the previously mentioned discrepancy in the velocity semiamplitude of the secondary.

We have measured the angle between the projections of the orbital and stellar angular momentum vectors for the primary star in NY Cep and find that these projections are aligned within the uncertainty of our measurement ($\beta _{\rm {p}}=2^{\circ }\pm 4$°). We believe this result to be robust, even if our model is simplified in many respects. We have neglected any changes in the limb-darkening law within the lines as compared to the continuum, as well as a number of other "second-order" effects, such as gravity brightening and differential rotation. We experimented with more complex models including these phenomena and found that they produced effects too small to affect our conclusions about spin–orbit alignment. Likewise we experimented with a "two-layer" model for the pressure-broadened lines (see Figure 11.5 of Gray 2005) and found that while it provided a slightly better fit to the data, it introduced several new free parameters and led to no significant changes in any other parameters.

Our result for β_p gives a lower bound on the true angle between the stellar and orbital spins, the obliquity (ψ). However we expect the obliquity to be not much larger than β unless the primary spin axis is highly inclined toward the line of sight, an unlikely scenario.

For our discussion, we adopt the effective temperatures derived by Holmgren et al. (1990), which are listed in Table 2. Because we do not have any new photometry, we do not attempt to derive new effective temperatures or a new age for the system. We adopt the previous classification of NY Cep as a young system (∼6 Myr) at a distance of 750 ± 90 pc, making it consistent with membership in the Cepheus OB III association. Our finding of a lower mass for the primary does not change the picture of NY Cep significantly given the uncertainties in other parameters.

If the spin periods were equal to the orbital period, then the stellar rotation velocities would be 20 km s⁻¹ and 19 km s⁻¹ for the primary and secondary, respectively. These numbers are smaller than our results for vsin i, which are 78 ± 3 km s⁻¹ and 154 ± 6 km s⁻¹. Assuming ψ = 0 for both stars, the implied rotation periods are 3.9 and 1.9 days.

Nor do the stars seem to be pseudosynchronized. The ratio of rotational to orbital frequencies is 3.9 and 8.0 for the primary and secondary, respectively. In the Hut (1981) model of pseudosynchronization, the ratio would be 2.3, which is lower than the observed ratios. If the stars had been significantly influenced by tidal evolution, then one would expect the rotational frequency to be smaller for the secondary star than for the primary star, because the less massive star should synchronize first. In contradiction with this expectation, we observe the secondary to have a higher rotation frequency than the primary.

All of these findings are in agreement with a picture in which the system is only a few million years old, and tidal forces have not had enough time to alter the rotational state of the stars after their arrival on the zero-age main sequence.

With the data at hand it is not possible to investigate the internal structures of the two stars in a stringent way via observations of apsidal motion. First, the uncertainties in the radii and ages are so high that the theoretical expected apsidal motion has a high uncertainty. Using the system parameters, together with internal structure constants of k₂ = −0.080 ± 0.35 ⁹ for both stars, we derive an expected apsidal motion rate of 19⁺⁴¹₋₉ arcsec per cycle. Second, the measurements of the argument of the periastron (ω) are also quite uncertain. Using the values from Heard & Fernie (1968), who derived an ω of 50.9 ± 3.7, from Holmgren et al. (1990), and from this work, we derive a measured apsidal motion rate of 6 ± 11 arcsec per cycle. Therefore, the expected and measured apsidal motions are consistent, but the bounds are too weak to give additional information. To make progress in this field both the expected and measured values for the apsidal motion have to have greater precision and accuracy.