He II λ4686 EMISSION FROM THE MASSIVE BINARY SYSTEM IN η CAR: CONSTRAINTS TO THE ORBITAL ELEMENTS AND THE NATURE OF THE PERIODIC MINIMA^{* † ‡ §}

We monitored the system with the ctio 1.5 m telescope and the fiber-fed chiron spectrograph (Tokovinin et al. 2013) from early 2012 through mid 2014 as an extension of the monitoring efforts presented by Richardson et al. (2010, 2015). The fiber is 2.7 arcsec on the sky, which is large enough so that we should not suffer from large spatial variations in the observed background nebulosity. The resulting spectra have a resolving power between 80,000 and 100,000 and exhibit a strong blaze function. In order to remove the blaze function and have a realistic normalized spectrum, we compared each spectrum to a spectrum obtained of HR 4468 (B9.5 Vn), which has very few spectral features except for Hα and Hβ in the spectral window 4500–7500 Å. The stability of chiron allowed us to achieve a good rectification of the continuum without the need of frequent observations of standard stars (in fact, we only needed one for each observation mode). The wavelength calibration of the data was performed using a ThAr lamp spectrum obtained on the same night as the observations of η Car. We also used the narrow line emission component (originating in the nebulosity around the central source whose velocity is relatively well known) to check the wavelength solution.

The spectra from OPD (Observatório do Pico dos Dias; operated by LNA/MCT-Brazil) were collected at the Zeiss 0.6 m telescope with the Lhires iii spectrograph equipped with a Atik 460EX CCD. A set of additional spectra, with higher spectral resolution, was taken at the Coudé focus of the 1.6 m telescope to check for continuum normalization and heliocentric transformations of the Lhires iii spectra.

Data reduction was done with iraf in the standard way. For the Lhires iii data set, the typical resolution element was about 95 km s⁻¹, whereas for the Coudé spectra it was about 50 km s⁻¹. This resolution element was enough to give information on the velocity field of the region forming the He ii λ4686 line (FWHM> 400 km s⁻¹). Both data sets presented a typical S/N of about 550.

In addition to η Car, a bright A-type star (in general a spectrophotometric standard) was also observed in order to aid the normalization process of the stellar continuum. Wavelengths were transformed to the heliocentric reference system and checked against the narrow line components reported by Damineli et al. (1998).

Spectral data of η Car were also obtained at the Complejo Astronómico El Leoncito (CASLEO), Argentina, from 2014 March through August. The frequency of observations was increased around the periastron passage during 2014 July/August.

The spectra were collected using the reosc spectrograph in its echelle mode, attached to the 2.15 m "J. Sahade" telescope. A Tek 1024 × 1024 pixel² CCD (24 μm pixel), was used as detector, providing a dispersion of 0.2 Å pixel⁻¹. The wavelength coverage ranges from 4200 through 6750 Å.

The normalization of η Car spectra was performed by dividing it by the continuum of a hot star, usually ω Car or θ Car. Additional residuals were minimized by fitting a low-order polynomial function and defining ad hoc spectral ranges to constrain the continuum to the region 4550–4750 Å.

High spatial sampling (0.05–0.1 arcsec pixel⁻¹) spectroscopic mapping of the region was recorded at critical binary orbital phases between 2009 June and 2014 November using the HST/STIS (see Table 1). The spectra of interest utilized the $52\times 0.1$ arcsec² aperture with the G430M grating centered at 4706 Å. The mappings were accomplished by a pattern of slit positions centered on η Car. While the first two mappings were done at a spacing of 0.1 arcsec, subsequent mappings were done at 0.05 arcsec spacing. Allowance for potential detector saturation was provided by a sub-array mapping directly centered on η Car. Due to solar panel orientation constraints, the aperture position angle changed between observations. As η Car is close to the HST orbital pole, visits were done during continuous viewing zone opportunities, thus increasing observing efficiency more than two-fold. Spatial mappings from these data indicate that the HST/STIS response to the central source has an FWHM = 0.12 arcsec.

A data cube of flux values was constructed for each spatial position in right ascension and declination at 0.05 arcsec spacing and in velocity relative to He ii $\lambda 4686$ at 25 km s⁻¹ intervals ranging from $-8,000$ to $+10,000$ km s⁻¹.

For the HST/STIS data, the EW of He ii $\lambda 4686$ was measured using the same procedure as for the ground-based observations. However, unlike the space-based observations, emission from the central source cannot be separated from the surrounding nebulosity in ground-based observations due to atmospheric seeing. Hence, direct comparison between the two data sets might be hampered by unwanted contaminations, not only from nebular emission but also from continuum scattered off fossil wind structures (Teodoro et al. 2013). For space-based observations, the contribution from such contaminations is directly proportional to the slit aperture, whereas for ground-based observations, they are always present.

Figure 2 shows that, as we increase the size of the aperture, the amount of He ii $\lambda 4686$ emission relative to the continuum decreases. This indicates that the He ii $\lambda 4686$ emission and adjacent continuum do not come from the same volume. Hence, in order to properly compare the HST/STIS measurements with those obtained by ground-based telescopes, we measured the He ii $\lambda 4686$ EW using the final spectrum obtained from summing up the spectra from the entire 2 × 2 arcsec² mapping region of the HST/STIS data cube.

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The data obtained with the Goodman spectrograph were processed and reduced using standard iraf tasks to correct them for bias and flat-field, as well as to perform the extraction and wavelength calibration of the spectra. For the latter, we used a CuAr lamp to determine a low-order (between 3 and 5) Chebyshev polynomial solution for the pixel-wavelength correlation. Observations of a hot standard star (HD 303308; O4 v) were obtained—either just before or following those of η Car—in order to correct the spectra by the low-frequency distortions caused by instrumental response. The final product was a data set of spectra with an S/N per resolution element typically in the range from 200 to 1000 (90% of the data) and spectral resolution element of about 85 km s⁻¹.

High spectral resolution data were also collected with the 1.0 m McLellan telescope at the Mt. John University Observatory (MJUO)²⁷ in Tekapo, New Zealand. Spectra were obtained with the fibre-fed High Efficiency and Resolution Canterbury University Large Echelle Spectrograph (Hearnshaw et al. 2002) using the 100 μm optical fiber (corresponding to a 2 arcsec seeing), which delivers spectra with spectral resolution of about 3.5 km s⁻¹. hercules operates in the spectral range from 3800 to 8000 Å. The spectrograph is thermally stabilized, with the optical components contained within an evacuated tank. This ensures excellent stability and high precision. The detector is a 4096 × 4096 pixel² Fairchild 486 CCD with 15 μm pixels that samples the entire free spectral range in a single exposure. Cooling down to the operating temperature ($\approx 170$ K) is done via a CryoTiger closed-circuit liquid refrigerant system.

Observations for the campaign were taken during 2014 May through 2014 October. The reduction software used was an in-house sequence of packages written using matlab. We typically obtained one to three spectra of η Car with exposure time varying between 600 and 1200 s, and one spectrum of the bright hot star θ Car, with exposure time between 300 and 600 s, depending on sky conditions. The hot star was observed in order to determine the continuum and telluric features for the normalization process. We took a Th–Ar lamp spectrum before and after each science exposure for precise wavelength calibration. Flat-fields were taken at the beginning or end of each night in order to correct the science data for variations on the detector response.

By tracing the orders in the Th–Ar spectral images along the axes defined by the flat fields and using the location of the spectral lines used for wavelength calibration, a full set of wavelength-calibrated axes is obtained along which the stellar orders can be traced. Median filtering was used to remove cosmic rays. The spectral orders of each science image were extracted and merged into a single one-dimensional spectrum.

All the data obtained by the members of the SASER²⁸ were fully processed and reduced by them using the standard basic procedure for long slit observations, namely, correction for the bias level and pixel-to-pixel variations, extraction of the spectrum, and wavelength calibration using a comparison lamp. Most of the data were delivered without any continuum normalization. However, they presented variations in intensity within the range 4600–4750 Å that was accounted for by using a linear fit to this region.

In this paper, we used nine additional high-resolution ($R\sim 50,000$) spectra covering the wavelength range from 3620 to 8530 Å to study the variations in the He ii $\lambda 4686$ strength during the recovery and fading phases after periastron passage. These additional data were obtained with the BESO (Fuhrmann et al. 2011) attached to the 1.5 m Hexapod-telescope at the Universitätssternwarte Bochum on a side-hill of Cerro Armazones in Chile. All spectra were reduced with a pipeline based on a MIDAS package adapted from FEROS, the similar ESO spectrograph on La Silla. The results of our measurements for these additional data are listed on Table 3.

To determine the period of the spectroscopic cycle for He ii $\lambda 4686$, we analyzed the EW measurements using two approaches. (1) We used data encompassing only the last three periastron passages, and then only the data from the interval of sharp decrease in EW prior to the deep minimum. (2) We used the entire EW curve and all data sets,  including the 1992.4, 2003.5, 2009.0, and 2014.6 events

First, we focused our attention on the decreasing phase of the EW²⁹ just before the onset of the minimum. During this phase, the EW rapidly decreases from its absolute maximum to a minimum level, which seems to be close to zero. One of the difficulties in assessing the real minimum intensity level is that we need data with both high resolving power and high S/N, which is not always available for a long-term dedicated monitoring. Also, small variations in the spectrum induced by data processing, reduction, and/or normalization of the spectra are the major contributors to stochastic fluctuations during the minimum intensity phase.

We noted that, as opposed to 2003.5 and 2009.0, the decreasing phase for the 2014.6 periastron passage did not occur at a linear rate. Therefore, instead of adopting the methodology used by Damineli et al. (2008a) for the disappearance of the narrow component of the He i $\lambda 6678$ line, we used the approach suggested by Mehner et al. (2011). This method consists in finding the minimum (EW_min) and maximum (EW_max) value for the EW during periastron passage, ignoring the time when they occur. Then, the mid-point is determined by ${\mathrm{EW}}_{{\rm{m}}}=({\mathrm{EW}}_{\mathrm{min}}+{\mathrm{EW}}_{\mathrm{max}})/2$. Next, the observed EW that is nearest to EW_m is found, for which the corresponding time, JD(EW_m), is taken as reference. Applying this procedure to consecutive periastron passages allows us to determine the period by calculating the difference between JD(EW_m)'s. Figure 4 illustrates this methodology and shows the results for data from the last three events (2003.5, 2009.0, and 2014.6). The mean period obtained by using this approach was 2022.9 ± 0.2 days, where the uncertainty is the standard deviation of the mean (the standard error is 0.14 days).

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We also used the entire curve (including all the observations from 1992.4 up to 2014.6) to determine the period for the He ii $\lambda 4686$. We folded the EW curve using trial periods to determine which period would result in the least dispersion of the data. This method, called phase dispersion minimization (PDM; Stellingwerf 1978), is frequently used to search for periodic signals in the light curve of eclipsing binary systems. In this work, we adopted the Plavchan algorithm (Plavchan et al. 2008), which is a variant of the PDM method. The Plavchan algorithm folds the light curve to trial periods and, for each period, computes the ${\chi }^{2}$ difference between the original and the box-car smoothed data, but only for a predefined number of worst-fit subset of the data. Thus, the best period is the one that produces the lowest ${\chi }^{2}$ value.

Using the Periodogram Service available at the NASA Exoplanet Archive³⁰ , we tested a sample of 300 trial periods, equally distributed between 2010 and 2040 days. The number of elements of the worst-fit subset was set to 50 and we adopted a smoothing boxcar size of 0.01 in phase. With these parameters, the result of the analysis using the Plavchan algorithm is shown in Figure 5(a).

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The maximum of the PDM power distribution, which corresponds to the minimum ${\chi }^{2}$, occurred for a period of 2022.8 days. Although the uncertainty associated with this result cannot be obtained directly from the PDM analysis, an estimate of the standard deviation of the mean can be determined by using a weighted bootstrap technique with resampling, where a random sample with a predefined size is drawn from the original period data set (in this case, within the range 2010–2040 days). The probability of drawing a given period was determined by the PDM distribution itself and we restricted the size of each drawn sample to be 75% the size of the original data set. Repeating this procedure N times, where $N\to \infty$, allows us to obtain an estimate of the true standard deviation of the mean. Figure 5(b) shows that this method rapidly converges: for $N\gtrsim 5000$ the dispersion of the results is about 1%, suggesting a standard deviation of the mean of about ±1.6 days.

The results from the two methods described in this section are consistent and can be used to obtain a mean period of 2022.9 ± 1.6 days.

As an update to the previous work by Damineli et al. (2008a), we used the new result from He ii λ4686 in combination with previous period determinations in order to obtain a mean period of the spectroscopic cycle. Table 4 lists some of the methods used to determine the period. That table is an adapted version of Table 2 from Damineli et al. (2008a), which now includes an updated value for the period determined from the X-ray light curves, including data from 1992–2014 (M. F. Corcoran et al. 2016, in preparation), and the new determination from the He ii $\lambda 4686$ EW (from 1992–2014).

Table 4.  Period of the Spectroscopic Cycle Determined by Different Methods (Adapted from Damineli et al. 2008a)

Method	Perioda (days)
V band	2021 ± 2
J band	2023 ± 1
H band	2023 ± 1
K band	2023 ± 1
L band	2023 ± 2
Fe ii $\lambda 6455$ P Cyg radial velocity	2022 ± 2
He i $\lambda 6678$ broad comp. radial velocity	2022 ± 1
Si ii $\lambda 6347$ EW	2022 ± 1
Fe ii $\lambda 6455$ P Cyg EW	2021 ± 2
He i $\lambda 6678$ narrow comp. EW	2026 ± 2
He i $\lambda 10830$ EW	2022 ± 1
X-ray light curve (1992–2014)	2023.5 ± 0.7
He ii $\lambda 4686$ EW (1992–2014)	2022.9 ± 1.6
Weighted mean ± standard error	2022.7 ± 0.3

Note.

^aThe error is the standard deviation of the mean.

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The weighted mean period was determined by a two-step approach. First, an unweighted mean of the 13 periods listed in Table 4 was determined. Then, a new mean was determined by weighting each period by its absolute difference regarding the unweighted mean. This approach has the benefit of not being so sensitive to discrepant measurements, which has a great impact on the uncertainties in the period determination.

The result suggests a weighted mean period for the spectroscopic cycle of 2022.7 ± 0.3 days, where the error is the standard error of the mean (the standard deviation is 0.9 days). Thus, the mean period has not changed with the new measurements, but its uncertainty has considerably been reduced. Evidently, this improvement was mainly due to the fact that we adopted a weighted approach to the period determination, and not because we included an updated measurement from X-rays and a new datum from He ii λ4686.

We tested the hypothesis that the overall distribution of the EW of the He ii $\lambda 4686$ around the event is recurrent, i.e., it comes from the same distribution. We limited our analysis to the past two events (2009.0 and 2014.6) because they were monitored over almost one year (centered on the event) at a relatively high time sampling.

The period obtained from the He ii $\lambda 4686$ EW was used to fold the observed curve of the 2009.0 event by 2022.9 days in order to compare it with the recent 2014.6 event. Three non-parametric statistical tests were employed to address the similarities between the EW curves from 2009.0 and 2014.6: the Anderson–Darling k-sample test (Scholz & Stephens 1987; Knuth 2011), the two-sample Wilcoxon test (also known as Mann-Whitney test; Hollander & Wolfe 1973; Bauer 2012), and the Kolmogorov–Smirnov test (Birnbaum & Tingey 1951; Conover 1971; Durbin 1973; Marsaglia et al. 2003). We used the R package (R. Core Team 2014) to perform the statistical analysis.

All of the statistical tests used here are non-parametric, which means that they make no assumption on the intrinsic distribution of the samples under comparison. The null hypothesis, H₀, for these tests is that the samples come from the same parent distribution. The tests return a parameter, called test statistic (${T}_{\mathrm{obs}}$), which is then compared with a critical value (${T}_{{\rm{c}}}$) that is calculated based on the size of the samples and on the chosen significance level, α. Thus, the null hypothesis is rejected at the significance level α if ${T}_{\mathrm{obs}}\geqslant {T}_{{\rm{c}}}$, which is known as the traditional method. Another equally valid approach is to calculate the probability, under the null hypothesis, of getting a T as large as ${T}_{\mathrm{obs}}$, which is called the p-value method. In this case, the p-value must be compared to the chosen significance level, α, and the null hypothesis is rejected when $p(T\geqslant {T}_{{\rm{c}}})\leqslant \alpha$. For the analysis presented here, we established a significance level of $\alpha =0.05$, and we always discard H₀ whenever one of the two methods rejects the null hypothesis.

Figure 6 shows the equivalent width curve (ranging from JD = 2454787.5 through 2454926.0 for 2009.0 and from JD = 2456810.96 through 2456966.37 for the 2014.6 curve) for which we performed these statistical analyses. Table 5 indicates that both conditions, ${T}_{\mathrm{obs}}\leqslant {T}_{{\rm{c}}}$ and $p(T\geqslant {T}_{{\rm{c}}})\geqslant \alpha$, are true for all tests performed on the entire curve, suggesting that we cannot reject the null hypothesis. Note that, under the assumption that the null hypothesis is valid, a value of $p(T\geqslant {T}_{{\rm{c}}})\geqslant 0.1$ is classified as weak evidence against the null hypothesis at the significance level $\alpha =0.05$, whereas $p(T\geqslant {T}_{{\rm{c}}})\geqslant 0.2$ is considered as no evidence against the null hypothesis (Fisher 1925, 1935). Thus, the results of the analysis for the entire curve suggest that there is only weak evidence of significant changes in the He ii λ4686 equivalent width from one cycle to another.

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Table 5.  Results of the Two-sample Statistical Analysis, Performed on the Equivalent Width Curve of 2009.0 and 2014.6 (Figure 6), to Test for the Null Hypothesis (H₀) that the Two Curves Have the Same Distribution

Hypothesis Test	$| {T}_{\mathrm{obs}}|$	${T}_{{\rm{c}}}$ a	$p(T\geqslant | {T}_{\mathrm{obs}}| )$	Reject H0?
Entire Curve (${N}_{1}=54$ and ${N}_{2}=122$)
ADb	0.156	1.960	0.300	No
Wrsc	0.626	1.960	0.532	No
KSd	0.127	0.222	0.583	No
P1 Only (${N}_{1}=11$ and ${N}_{2}=32$)
AD	1.041	1.960	0.120	No
Wrs	0.334	1.960	0.738	No
KS	0.364	0.475	0.229	No
P2 Only (${N}_{1}=20$ and ${N}_{2}=20$)
AD	6.195	1.960	0.001	Yes
Wrs	3.00	1.960	0.003	Yes
KS	0.600	0.430	0.002	Yes
P3 Only (${N}_{1}=23$ and ${N}_{2}=70$)
AD	−0.021	1.960	0.365	No
Wrs	0.045	1.960	0.964	No
KS	0.199	0.327	0.501	No

Notes. ${T}_{\mathrm{obs}}$ is the test statistic and $p(T\geqslant | {T}_{\mathrm{obs}}| )$ is the probability, assuming that H₀ is valid, of obtaining a test statistic as extreme as observed. N₁ and N₂ are the sample size for 2009.0 and 2014.6, respectively.

^aCritical value of the test for a significance level $\alpha =0.05$. ^bAD: Anderson–Darling test. ^cWrs: Wilcoxon rank sum test. ^dKS: Kolmogorov–Smirnov test.

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We also looked for changes in the intensity of each peak (P1, P2, and P3) separately. For this, we defined three data subsets, each one containing one peak, to be analyzed in the same way as the entire curve. The first subset, containing P1, is composed of data within the range 2454787.5–2454826.36 for 2009.0 and within 2456810.96–2456848.81 for 2014.6. The second subset, which contains P2, is composed of data within 2454827.36–2454846.29 for 2009.0 and within 2456850.81–2456869.74 for 2014.6. Finally, the third subset contains P3 and is composed of data within 2454847.29–2454926 for 2009.0 and within 2456870.73–2456966.37 for 2014.6.

The results shown in Table 5 suggest that P1 and P3 did not change significantly between 2009.0 and 2014.6. In contrast, P2 had a statistically significant increase in strength over this period; the mean difference between the two epochs is about 0.53 Å, which corresponds to a relative increase of about 26% from 2009.0 to 2014.6.

Caution is advised regarding the lack of variability of P1 and P3, since their different sample size (resulting from different frequencies of observations) might have some influence on the statistical analysis in the case where significant changes occurred during epochs not covered by the monitoring. Since P2 does not suffer from different sampling between 2009.0 and 2014.6, the result obtained for this peak is more reliable than that for P1 and P3. Nevertheless, even if we adopt a different time interval to be used for the statistical analyses of each peak (e.g., reducing the time interval for P3 to the range 2456880–2456935), the outcome of the tests remain unchanged. Therefore, our results suggest that P3 might be composed of a broad and a narrow component. The former has been detected in previous events, but the latter was only detected in 2014.6 due to the better time sampling of observations. Further discussion on this discrepancy in P3 between 2009.0 and 2014.6 is presented in Section 5.

3.3.1. Comparing EW Measurements from HST/STIS and Ground-based Telescopes

There has been a debate about discrepancies resulting from measuring the EW of the He ii $\lambda 4686$ using HST/STIS and ground-based observations. Recently, Davidson et al. (2015), based on five measurements using HST/STIS data, suggested that the He ii $\lambda 4686$ EW for the 2014.6 event was systematically different from the past cycles. Their conclusion relies on the direct comparison between HST/STIS and ground-based observations for the past two cycles (2009.0 and 2014.6).

In the case of η Car, HST/STIS observations have undoubtedly higher quality than the ground-based ones in the sense that they allow us to obtain the spectrum of the central source with relatively less contamination from the surrounding nebulosities. Nevertheless, space-based observations could not be scheduled so as to have the same time coverage possible from the ground, which were performed almost on a daily basis (at least around the event, for the past two cycles). Also, ground-based optical spectra frequently are obtained with a much higher resolving power than possible with HST/STIS, allowing us to have better understanding of the line morphologies.

Thus, it is a good practice to compare the data obtained with HST/STIS with those obtained from ground-based telescopes. However, for the reasons mentioned in Section 2.4, aperture corrections must be applied to the measurements obtained from space-based telescopes in order to be properly compared with the ground-based measurements. This correction must be performed by summing up the observed EW and the aperture correction factor obtained from Figure 2. Figure 7 shows the result of performing such corrections to the measurements published in Davidson et al. (2015). The amount of correction for each measurement was determined from our HST/STIS data using the closest observation. After the corrections were applied, no significant differences are observed (within the uncertainties of the measurements).

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Although we did not detect significant variations in the overall behavior of the EW curve of the He ii $\lambda 4686$ EW in a timeframe of 5.54 yr, we cannot discard the possibility of changes over longer timescales. However, this can only be adequately addressed by continuously monitoring η Car over several more cycles.

A comparison between the He ii $\lambda 4686$ line profile at phases around apastron and periastron is shown in Figure 8. That figure leaves no doubt about the positive detection of this line in emission at phases around apastron for the last cycle (between 2009.0 and 2014.6). Indeed, as can be seen from Figure 8(a), there is a noticeable hump in the range from 4675 to 4694 Å, where the He ii $\lambda 4686$ is located at phases around periastron. As an illustration of what a zero He ii $\lambda 4686$ emission would look like, Figure 8(b) shows two spectra taken at the onset of the He ii λ4686 deep minimum (2014 July 31), using CTIO/CHIRON and HST/STIS data. The region where the He ii $\lambda 4686$ line was previously detected is flat, showing no evident line profile.

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Due to the restricted wavelength interval adopted to calculate the He ii λ4686 EW (4675–4694 Å), contamination from the broad component of the iron emission lines on both sides of the integration region are always included in the measurements at all phases. This effect can be especially significant at phases around apastron, when the intensity of the He ii λ4686 line is relatively low and contaminations become stronger.

Although we cannot reliably determine the exact magnitude of the contaminations, we can estimate a minimum value for the He ii λ4686 EW at apastron by reducing the size of the integration region so that we keep only the observed line profile. As can be seen in Figure 8(a), the He ii λ4686 line profile is evident in the range 4679–4691 Å (shaded area in that figure). The EW measured within this reduced region, at apastron, was about −0.051 Å for SOAR and −0.048 Å for HST/STIS, resulting in a minimum of −0.05 ± 0.01 Å. For the sake of completeness, the EW measured within the wider wavelength interval (4675–4694 Å) was −0.083 Å for SOAR and −0.075 Å for HST/STIS, which implies an EW of −0.08 ± 0.01 Å. Hence, the minimum EW of the He ii λ4686 emission line at apastron is −0.07 ± 0.01 Å.

Assuming that the He ii $\lambda 4686$ emission is produced close to the apex of the shock cone, variations in the observed emission can, in principle, be explained by the increase in the total opacity along the line of sight to the emitting region, as the secondary moves deeper inside the dense primary wind. Intuitively, this mechanism would cause a gradual decrease (or increase, after periastron passage) of the observed flux that would depend on the extent and physical properties of the optically thick region in the extended primary wind, and also on the orbital orientation to the observer.

This same approach was used by Okazaki et al. (2008) to show that the overall behavior of the RXTE X-ray light curve can be reproduced by assuming that the X-ray emission comes from a point source located at the apex of the shock cone and prone to attenuation by the primary wind. Recently, Hamaguchi et al. (2014) suggested that the variations across the spectroscopic events are composed of a combination of (1) occultation of the X-ray emitting region by the extended primary wind and (2) decline of the X-ray emissivity at the apex. In any case, opacity effects (either attenuation or occultation) can play an important role on the observed intensity of the radiation. Since the X-ray light curve shares some similarities with the He ii $\lambda 4686$ EW curve (both rise to a maximum before falling to a minimum when there is no emission at all), we tested the hypothesis that the variations in the He ii λ4686 EW could also be the result of intrinsic emission attenuated by the extended primary wind.

We used 3D SPH simulations of η Car from Madura et al. (2013) to calculate the total optical depth in the line of sight to the apex at each phase by using

$$\begin{eqnarray}&&{\tau }_{{\rm{A}}}(\varphi )={\kappa }_{e}\;{\displaystyle \int }_{{z}_{0}(\varphi )}^{R}\rho \;{dz},\end{eqnarray} \tag{ 1 }$$

where ρ and ${\kappa }_{e}$ are, respectively, the mass density and the mass absorption coefficient of the material in the line of sight to the apex. The integration starts at the position of the apex at each phase ${z}_{0}(\varphi )$ and goes up to the boundaries of the 3D SPH simulations, which, in this case, is a sphere with radius $R=154.5\;{\rm{AU}}$. For the present work, we assumed that electron scattering is the dominant process for the attenuation of the radiation in the line of sight, which corresponds to ${\kappa }_{e}=0.34$ cm² g⁻¹. Therefore, under these circumstances, the synthetic equivalent width, ${\mathrm{EW}}_{{\rm{syn}}}$, at each orbital phase, was obtained using

$$\begin{eqnarray}&&{\mathrm{EW}}_{{\rm{syn}}}(\varphi )={\mathrm{EW}}_{0}(\varphi )\;{e}^{-{\tau }_{{\rm{A}}}(\varphi )},\end{eqnarray} \tag{ 2 }$$

where ${\mathrm{EW}}_{0}(\varphi )$ is the intrinsic equivalent width (corresponding to the unattenuated flux). In the present work, we included only two mechanisms responsible for EW₀: (i) a continuous production of He ii λ4686 photons that varies reciprocally with the square of the distance D between the stars (i.e., in radiative conditions; see Fahed et al. 2011) and (ii) an additional, temporary contribution from the "bore hole" effect that depends on how deep the apex of the shock cone penetrates inside the primary wind (see Madura & Owocki 2010).

We included the contribution from the "bore hole" effect because, near periastron, due to the highly eccentric orbit, the wind–wind interacting region penetrates into the inner regions of the primary wind, eventually exposing its He⁺ core.³¹ The contribution from this mechanism to the observed He ii λ4686 flux is proportional to how large the "bore hole" is (see Figure 10). This assumption relies on the fact that high-energy radiation produced in the shock cone inside the He⁺ region (the red line between the cone and the sphere in Figure 10) can create He⁺⁺ ions, whose recombination will produce He ii λ4686 photons. Some will eventually escape through the "bore hole" and be detected by the observer.

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The radius ${r}_{{\rm{bh}}}$ of the "bore hole" is wavelength dependent and varies with orbital phase. Considering the He⁺ region, ${r}_{{\rm{bh}}}$ as a function of the orbital phase is given by

$$\begin{eqnarray}&&{r}_{{\rm{bh}}}{(\varphi )=({R}_{{\mathrm{He}}^{+}}^{2}-{r}^{2}(\varphi ))}^{1/2},\end{eqnarray} \tag{ 3 }$$

where $r(\varphi )$ is the distance between the primary star and the plane formed by the aperture of the "bore hole" (see Figure 10), given by

$$\begin{eqnarray}&&r(\varphi )=\displaystyle \frac{{c}^{2}\;{z}_{0}(\varphi )+{({c}^{2}({R}_{{\mathrm{He}}^{+}}^{2}-{z}_{0}^{2}(\varphi ))+{R}_{{\mathrm{He}}^{+}}^{2})}^{1/2}}{{c}^{2}+1},\end{eqnarray} \tag{ 4 }$$

and $c=\mathrm{tan}\alpha$, where α is the half-opening angle of the cone formed by the wind–wind interacting region. In Equations (3) and (4), ${R}_{{\mathrm{He}}^{+}}$ is the radius of the He⁺ region in the primary wind and ${z}_{0}(\varphi )$ is the distance between the primary and the apex at a given orbital phase. Figure 11(a) shows the contribution from each mechanism to the total intrinsic equivalent width EW₀. The relative contribution was chosen so that the transition between the two regimes occurred smoothly, as required by the observations. Thus, by combining the intrinsic strength for the line emission with the total opacity in the line of sight, we were able to calculate a synthetic equivalent width curve for different orbit orientations. The results for selected orbital orientations are shown in Figures 11(b)–(e).

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4.2.1. The Direct View of the Central Source

Based on the comparison between the overall profile of the observed He ii λ4686 equivalent width curve from the past three cycles and those synthetic curves shown in Figure 11, one can readily discard models with $0^\circ \lesssim \omega \lesssim 180^\circ$. Orientations with $\omega =0^\circ$ produce results that have excessively high optical depths before periastron passage and way too little after it. This orientation cannot reproduce the observed rise of the equivalent width before periastron passage and also overestimates its strength after it. Orientations with $\omega =90^\circ$ produce symmetrical profiles, which do not correspond to the observations. Orientations with $\omega =180^\circ$ can reproduce fairly well the observations before periastron but fail to reproduce the observed equivalent width after periastron (they underestimate P3).

Regardless of the overall profile of the synthetic equivalent width curves, the crucial problem of models with $0^\circ \lesssim \omega \lesssim 180^\circ$ is that they cannot reproduce the observed phase of the deep minimum—the week-long phase where the observed equivalent width is zero.

Orientations with $230^\circ \lesssim \omega \lesssim 270^\circ$, on the other hand, seem to provide a good overall profile, as they predict an increase just before periastron passage followed by a rapid decrease to zero right after periastron passage, and the return to a lower (in modulo) equivalent width peak before fading away (P3). Thus, we focused our analysis on ω in this range.

The duration of the interval when the synthetic equivalent width remains near zero (and whether it is ever reached) is also regulated by the orbital inclination. Thus, we compared the observations with 16 synthetic equivalent width curves obtained from the permutation of four values of orbital inclination ($i\in \{126^\circ ,135^\circ ,144^\circ ,153^\circ \}$) and five values of longitude of periastron ($\omega \in \{225^\circ ,234^\circ ,243^\circ ,252^\circ ,261^\circ \}$). Then we calculated the rms error between each model and the observations. The values for i and ω were obtained from a predefined grid within the 3D SPH models. Also, note that the orbital plane is parallel to the plane of the sky for $i=0^\circ$ or $i=180^\circ$, whereas for $i=90^\circ$ or $i=270^\circ$ they are perpendicular to each other. Thus, the set of orbital inclinations that we investigated in this work was chosen based on the premise that the orbital axis is aligned with the Homunculus polar axis (see Madura et al. 2012).

For each model, we also searched for the time shift to be applied to the models that would result in the least root mean square value between the model and the observations. Examples of the results of this analysis are shown in Figure 12. The minimum rms error was reached for an orbit orientation with {$i=144^\circ$, $\omega =243^\circ$} and a time shift $\mathrm{JD}(\mathrm{obs})-\mathrm{JD}(\mathrm{syn})=-3.5\;{\rm{days}}$. A comparison between the best model (with the derived time shift applied) and the observations is shown in Figure 13.

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Regarding the mean value and uncertainty of these results, statistical analysis showed that there are no significant differences between a model with a combination of $i=144^\circ$ and $234^\circ \lesssim \omega \lesssim 252^\circ$. In fact, within the range of orbit orientations that we focused our analysis on, only these models resulted in rms error significantly lower than the others at the $2\sigma$ level. Therefore, the mean values we adopted for i and ω are, respectively, $144^\circ$ and $243^\circ$ (coincidently equal to the best match), whereas the uncertainty on both values is defined by the step in the number of lines of sight used to produce the synthetic equivalent width curves from the 3D SPH models, which was set to $9^\circ$. In order to estimate the mean and uncertainty on the time shift, we adopted a sample composed of the time shift that resulted in the least rms error for each one of the 16 models. The result was a mean time shift of −4.0 ± 2.0 days (the standard error is ±0.5 days). This means that periastron passage occurs four days after the onset of the He ii λ4686 deep minimum.

4.2.2. The Event as Viewed from High Stellar Latitudes

An independent way to verify the reliability of our results would be analyzing the He ii λ4686 equivalent width curve from different viewing angles and comparing them with the results from the direct view of the central source. Fortunately, in the case of η Car, this is possible due to the bipolar reflection nebula—the Homunculus nebula—that surrounds the binary system. Each position along the Homunculus nebula "sees" the central source along a different viewing angle (Smith et al. 2003a). An interesting position is the FOS4 (e.g., Davidson et al. 1995; Humphreys & HST-FOS eta Car Team 1999; Zethson et al. 1999; Rivinius et al. 2001; Stahl et al. 2005), a region about 1 arcsec² in area located approximately 4.5 arcsec from the central source along the major axis of the Homunculus³² that is believed to reflect the spectrum originating at stellar latitudes close to the polar region (for comparison, the spectrum from the direct view of the central source seems to arise from intermediate stellar latitudes; see e.g., Smith et al. 2003a).

We compared our results with those obtained at FOS4 by Mehner et al. (2015) using the same analysis that we did for the direct view of the central source. Note, however, that the time shift obtained from the observations at FOS4 needs to be corrected by the light travel time and the expansion of the reflecting nebula. We achieved this by using the Homunculus model derived by Smith (2006), to determine the necessary parameters of FOS4 (see Figure 14). In a coordinate system where a stellar latitude of $\theta =90^\circ$ corresponds to the pole of the receding NW lobe and $\theta =270^\circ$ corresponds to the pole of the approaching SE lobe, the spectrum reflected at FOS4 corresponds to a stellar latitude of $\theta ({\rm{FOS4}})=258^\circ$, whereas the central source is viewed at ${\theta }_{0}=229^\circ$.

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Figure 15 shows the search for the best match between the models and the observations. Since the orbital axis is closely aligned with the Homunculus polar axis, we investigated models corresponding to stellar latitudes $\theta \in \{252^\circ ,261^\circ ,270^\circ \}$ and $\omega \in \{225^\circ ,243^\circ ,252^\circ \}$. The best match was found for {$\theta =261^\circ$, $\omega =243^\circ$} and a time shift of −21.0 days. This model resulted in a rms error significantly lower (at the $2\sigma$ level) than any other one. Note that this time shift of 21 days, obtained empirically (Smith 2006), is consistent with previous estimates by Stahl et al. (2005) and Mehner et al. (2011) based on geometrical arguments.

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Since the observations at FOS4 were not as frequent as for the direct view, our analysis is subject to aliasing, which explains the high frequency oscillations observed in Figure 15. Despite that, we determined a mean time shift at FOS4 of −21.6 ± 1.3 days (the standard error is 0.4 days). Note that the smaller uncertainty of this result, when compared with the direct view of the central source, is not real. This is just an effect introduced by the fact that we used a smaller sample for the trial models for the polar region than for the direct view. Also, the variations between the trial models for the polar region are not as large as the ones for the direct view, which reduces the dispersion of the minimum rms error (all the models for the pole region have similar time shift).

The time shift we derived for FOS4 has to be corrected by light travel delay. Considering that the lobes are expanding at 650 km s⁻¹, the spectrum reflected at the FOS4 position would be delayed by ${\rm{\Delta }}t=17\;{\rm{days}}$ relative to the direct view of the central source. This means that 17 days, out of the observed −21.6 days, are due to light travel effect that must be taken into account in order to compare with the observations of the central source (again, the minus sign means periastron occurs after the onset of the He ii λ4686 deep minimum). Therefore, the best model for the observations at FOS4 results in an effective time shift of 17 days−21.6 days $=-4.6\pm 0.4\;{\rm{days}}$, which is in agreement with the results obtained from the direct view of the central source. Figure 16 shows the best model compared with the observations at FOS4 (corrected for light travel time). An interesting result is that when the "bore hole" effect is included, the models overestimate the equivalent width at FOS4, which suggests that the contributions due to this mechanism is small at high stellar latitudes. This makes sense, given the fact that the contribution from the "bore hole" effect will be significant toward where the cavity is pointing (i.e., low and intermediate latitudes).

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The net result is that the ephemeris equation derived from the reflected observations at FOS4 (high stellar latitude), after correction by the travel time delay, is the same as for the direct view (intermediate stellar latitude). Therefore, the variations across the event for the He ii λ4686 seem to be ultimately determined by the high opacity in the line of sight to the He ii λ4686 emitting region during periastron passage, and not by a decrease in the intrinsic emission.

The results presented in this paper allowed us to determine the time of the periastron passage, ${T}_{0}(2014.6)$, from two different line of sights. For the direct view, ${T}_{0}(2014.6)=2456874.1$ (standard deviation: ±2.8 days), whereas for the FOS4 ${T}_{0}(2014.6)=2456874.7$ (standard deviation: ±2.3 days). Therefore, the time of periastron passage is given by

$$\begin{eqnarray}&&\text{JD (periastron passage)}=2456874.4+2022.7\;E,\end{eqnarray} \tag{ 5 }$$

where $E=({\rm{\Phi }}-13)$, Φ is the mean anomaly plus cycle³³ , and 2022.7 corresponds to the mean period of the spectroscopic cycle. The time of periastron passage determined by Equation (5) has an uncertainty of ±2.0 days (standard error), which is the result of error propagation from all the parameters used to determine the mean values shown in that equation. Table 6 summarizes all the orbital elements derived in this work and Figure 17 illustrates the orientation of the orbit projected onto the sky using the mean value for the orbital elements.

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Table 6.  Orbital Elements Derived in the Present Work, Assuming Orbital Eccentricity $\epsilon =0.9$

Parameter	Symbol	Value or Range
Inclination	i	$135^\circ$–$153^\circ$
Longitude of periastron	ω	$234^\circ$–$252^\circ$
Period	P	2022.7 ± 0.3a
Time of periastron passage	T0(2014.6)	2456874.4 ± 1.3a

Note.

^aThe error is the standard error of the mean.

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The comparison between the He ii λ4686 emission and the X-rays is inevitable because both are tracers of high-energy processes that might be interconnected. For example, X-rays produced in the wind–wind shock might be used to doubly ionize neutral helium, in which case the He ii λ4686 emission should vary in a similar way to the X-rays.

In η Car, the onset of the deep minimum, as determined from the observed He ii λ4686 and 2–10 keV X-ray emission light curves, occurs almost at the same time (see e.g., Teodoro et al. 2012; Hamaguchi et al. 2014). Incidentally, no significant changes in the time of the start of the deep minimum were observed over the past three cycles (M. F. Corcoran et al. 2016, in preparation). However, the recovery of the He ii λ4686 emission and the 2–10 keV X-ray flux occurs in a completely different way. So far, the observations support a scenario where the He ii λ4686 emission presents a recovery phase (after the deep minimum) that is stable and repeatable, whereas for the X-rays, recovery cannot be predicted. Indeed, in 1998.0 and 2003.5, the full recovery of the 2–10 keV X-ray flux occurred about 90 days after the onset of the deep minimum, but in 2009.0 it happened 30 days earlier than that. Even more intriguing, in 2014.6, the observations indicate that the recovery was intermediate between the 1998.0/2003.5 and 2009.0. Thus, the question is why does the He ii λ4686 and the X-ray emission enter the deep minimum at the same time but recover from it so differently? Are they connected?

Hamaguchi et al. (2014) showed that the 2–10 keV X-ray flux can be split into two regimes that behave differently in time: the soft (2–4 keV) and the hard X-ray band (4–8 keV). Those authors showed that the hard X-ray flux (4–8 keV) drops to a minimum before the soft X-ray flux (2–4 keV). In fact, by the time the latter reaches the minimum, the former is already recovering from it. Interestingly, He ii λ4686 emission enters the minimum phase about 4 days after the hard X-rays and about 12 days before the soft X-rays. This fact not only corroborates the idea that the He ii λ4686 emitting region is located in the vicinity of the apex (likely close to the hard X-ray emitting region), but is also consistent with the scenario proposed in this paper, in which the onset of the deep minimum is regulated by the orbital orientation and opacity effects, as the secondary moves toward periastron. In this scenario, emission from the apex should disappear first, followed by emission produced in regions gradually farther away from the WWC apex (i.e., downstream of the WWC structure). Therefore, the time when the deep minimum starts should be approximately the same for all features arising from spatially adjacent regions, as might be the case for He ii λ4686 and the X-rays (especially the hard band flux).

Using the ephemeris from Hamaguchi et al. (2014), the deep minimum phase in the hard X-rays occurred, approximately, from $\mathrm{JD}=2454843$ through $\mathrm{JD}=2454859$, which suggests that superior conjunction would have occurred within this interval. According to Equation (5), the 2009.0 periastron passage occurred on $\mathrm{JD}=2454851.7$, which falls approximately in the middle of the phase of minimum hard X-ray flux. Moreover, note that, for $\omega =243^\circ$, superior conjunction occurs very close to periastron passage, only 3.6 days later. Therefore, the orbital orientation as derived from X-ray and He ii λ4686 emission are consistent. This is not surprising in the scenario we propose here. Indeed, close to periastron passages, we expect the hard X-ray and the He ii λ4686 to behave similarly (although they might not have a causal relation) because the apex of the shock cone is inside the primary He⁺ core where the He ii λ4686 emission is being produced at these phases. The soft X-rays, however, are produced in extended regions along the shock cone, and will eventually be blocked by the inner dense parts of the primary star at a later time.

After periastron passage, the He ii λ4686 emission behaves differently from the X-rays likely due to intrinsic physical processes that inhibit or even shut off the latter, but not the former. For example, if the shock cone structure would switch from adiabatic to radiative during periastron passage, the result would be a substantial decrease of the hardest X-ray emission because the wind–wind interacting region would be cooler. The subsequent recovery of the X-ray emission would then depend on stochastic processes in order to re-establish the emission from the hot plasma in the wind–wind interaction region. This is not, however, the case for the He ii λ4686 emission because, during periastron passage, the He⁺ region of the primary star is exposed by the wind–wind interacting region, and, therefore, He ii λ4686 emission can still be produced during this phase, even if there is very low X-ray emission from the WWC region.