Optical spectroscopy of Be stars: peak separation of Balmer emission lines

TL;DR

This study analyzes high-resolution spectra of 93 Be stars to establish new linear relations between the peak separations of Balmer emission lines, extending the velocity range of previous models.

Contribution

It provides new empirical equations linking peak separations of Balmer lines in Be stars, applicable over a broader velocity range than prior research.

Findings

Derived linear relations between $\Delta V_eta$ and $\Delta V_eta$

Established validity of equations for wider velocity ranges

Analyzed spectra from multiple archives and time periods

Abstract

The Be stars display variable optical emission lines originating in the circumstellar disc. Here we analyse high resolution spectroscopic observations of Be stars and the distance between the peaks of H-alpha, H-beta, and H-gamma emission lines ($\Delta V_\alpha$, $\Delta V_\beta$, and $\Delta V_\gamma$ respectively). Combining published data, spectra from the ELODIE archive (obtained in the period 1998 -- 2003) and Rozhen spectra (obtained 2015 -- 2023) of 93 Be stars, we find a set of relations connecting $\Delta V_\alpha$, $\Delta V_\beta$ and $\Delta V_\gamma$. They are effective for $30 \le \Delta V_\alpha \le 500$ km s$^{-1}$, $80 \le \Delta V_\beta \le 600$ km s$^{-1}$, and $40 \le \Delta V_\gamma \le 300$ km s$^{-1}$. The new equations are in the form $y=ax + b$ and are valid for a wider velocity range than in previous studies.

Tables

Table 1. Table 1: Rozhen spectra – Δ ​ V α Δ subscript 𝑉 𝛼 \Delta V_{\alpha} , Δ ​ V β Δ subscript 𝑉 𝛽 \Delta V_{\beta} and Δ ​ V γ Δ subscript 𝑉 𝛾 \Delta V_{\gamma} .

object		file	$\Delta V_{\alpha }$	$\Delta V_{\beta }$	$\Delta V_{\gamma }$
			[km s-1]	[km s-1]	[km s-1]
BD$-$00 3543	HD 173371	20220520.01.fit	223.4	278.7	—
BD$-$00 3543	HD 173371	20220520.02.fit	223.7	281.4	—
BD$-$00 3543	HD 173371	20220512.1800s.fit	218.2	269.3	—
BD$+$02 3815	HD 179343	20220512.1800s..fit	227.4	270.7	—
BD$+$02 3815	HD 179343	20220521.01.fit	230.2	277.1	—
BD$+$02 3815	HD 179343	20220521.02.fit	229.9	278.3
BD$+$05 3704	HD 168797	20220520.01.fit	549.6	651.1
BD$+$05 3704	HD 168797	20220520.02.fit	521.1	658.3
BD+30 0591	X Per	20151223.1.fit	102.9	143.2	168.4
BD+30 0591	X Per	20160130.fit	96.8	127.0	135.7
BD+30 0591	X Per	20161211.fit	59.2	132.7	154.3
BD+30 0591	X Per	20171230.fit	127.0	150.6	163.5
BD$+$37 0675	HD 18552	20220120.1.120s.fit	151.2	234.2
BD$+$37 0675	HD 18552	20220120.1.600s.fit	151.6	228.3	257.9
BD$+$37 0675	HD 18552	20220120.2.120s.fit	150.2	236.5
BD$+$37 0675	HD 18552	20220120.2.600s.fit	150.1	230.9	280.7
BD$+$47 0857	psi Per	20220120.1.120s.fit	133.4	—
BD$+$47 0857	psi Per	20220120.1.300s.fit	132.2	174.0	205.1
BD$+$47 0857	psi Per	20220120.2.120s.fit	132.9	174.8	201.2
BD$+$47 0857	psi Per	20220120.2.300s.fit	131.7	176.5	206.9
BD$+$50 0825	HD 23552	20220320.1.20min.fit	184.0	261.2
BD$+$50 0825	HD 23552	20220320.2.20min.fit	187.0	245.1
BD$+$40 1213	HD 33604	20220119.1.20min.fit	—	29.8	39.7
BD$+$40 1213	HD 33604	20220119.2.20min.fit	—	26.5	48.7
BD$+$42 1376	HD 37657	20220119.1.20min.fit	412.7	467.4	—
BD$+$42 1376	HD 37657	20220119.2.20min.fit	404.2	421.1	—
CSI+44-04374	LS V +44 17	20161211.fit	290.6	346.9	—
CSI+44-04374	LS V +44 17	60m.1.20230211.fit	250.8	338.9	—
BD+53 2790	4U 2206+54	20160620.fit	467.0	503.4	—
BD+53 2790	4U 2206+54	20151227.fit	493.1	582.7	—
BD+53 2790	4U 2206+54	20190819.fit	516.4	599.7	—
BD+53 2790	4U 2206+54	20200905.fit	493.0	525.4	—
CSI+59-01302	LS I +59 79	20160621.fit	177.4	265.1	—
CSI+59-01302	LS I +59 79	20160923.fit	179.5	249.5	—
BD+59 0144	gamma Cas	20160621.1.fit	—	138.7	176.2
BD+59 0144	gamma Cas	20160621.2.fit	—	132.0	172.3

Equations

$$\Delta V_{\beta }=1.01(\pm 0.01)\Delta V_{\alpha }+64.2(\pm 2.5)\mathrm{km}{\mathrm{s}}^{-1}$$

$$\Delta V_{\gamma }=1.034(\pm 0.031)\Delta V_{\beta }+14.9(\pm 4.2)\mathrm{km}{\mathrm{s}}^{-1}$$

$$\Delta V_{\gamma }=1.04\Delta V_{\alpha }+81.8\mathrm{km}{\mathrm{s}}^{-1}$$

$$\Delta V_{\gamma }=1.065(\pm 0.062)\Delta V_{\alpha }+74.8(\pm 6.2)\mathrm{km}{\mathrm{s}}^{-1}$$

$$R_{disc}=R_1{\left(\frac{2v\sin i}{(1-ϵ)\Delta V\alpha }\right)}^2,$$

$$R_{disc}=R_1{\left(\frac{1.01\ast 2v\sin i}{(1-ϵ)(\Delta V_{\beta }-64.2)}\right)}^2,$$

$$R_{disc}=R_1{\left(\frac{1.065\ast 2v\sin i}{(1-ϵ)(\Delta V_{\gamma }-74.8)}\right)}^2,$$

Full text

††articletype: Article Type

\fundingInfo

Bulgarian National Science Fund – project K$\Pi$-06-H28/2 08.12.2018 "Binary stars with compact object"

1 Introduction

The Be stars are fast-rotating B-type stars which, at some point in their lives, have shown spectral lines in emission (Porter \BBA Rivinius, \APACyear2003). They are main sequence or evolved stars, belonging to luminosity classes III – V and having masses and radii ranging between M $\sim$ 3.6 – 20 M⊙ and R $\sim$ 2.7 – 15 R⊙ (Cox \BBA Pilachowski, \APACyear2000). In the optical spectra, the most prominent observational characteristics of Be stars are variable emission lines of different chemical elements such as hydrogen, helium, iron, etc. The emission lines may change their intensity and even disappear during the star’s life. In addition, an infrared excess is present in their continuum (Gehrz \BOthers., \APACyear1974). The emission lines and the infrared excess indicate the presence of a geometrically thin, equatorial, gaseous, decretion disc which orbits the star in near-Keplerian rotation (Rivinius \BOthers., \APACyear2013). Interferometric observations confirm the disc-like structure of the circumstellar material (Meilland \BOthers., \APACyear2007; Cochetti \BOthers., \APACyear2019)). The emission lines can be symmetric or asymmetric, with double-peak or more complicated structure (Hummel \BBA Vrancken, \APACyear1995; Silaj \BOthers., \APACyear2010) and the peaks’ separation can be used for an estimation of the disc radius and to test theoretical models.

In this note, we analyse the separation of the peaks of the Balmer emission lines formed in the circumstellar disc and the relations between them.

2 Observations

We analyse 35 optical spectra of Be stars obtained with the ESpeRo Echelle spectrograph (Bonev \BOthers., \APACyear2017) on the 2.0 m telescope of the Rozhen National Astronomical Observatory, Bulgaria, and 27 spectra from the ELODIE archive (Ilovaisky \BOthers., \APACyear2008), obtained with the 1.93 m telescope of Observatoire de Haute-Provence, France. Both instruments are fiber-fed Echelle spectrographs providing high resolution spectra in the optical range. The ESpeRo spectrograph has a resolution of $\sim$ 30 000 and covers the range 3900 Å – 9000 Å. The ELODIE spectrograph has a resolution $\sim$ 42 000 and covers the range 3895 Å – 6815 Å. The spectra Rozhen spectra are obtained during the period 2015 – 2023, the ELODIE – 1997-2003. Some examples of the profiles of H$\alpha$, H$\beta$ and H$\gamma$ emission lines are shown on Fig. 1.

On the spectra, we measure the distance between the peaks of the Balmer emission lines H$\alpha$ ($\Delta V_{\alpha}$), H$\beta$ ($\Delta V_{\beta}$), H$\gamma$ ($\Delta V_{\gamma}$). The measurements are listed in Table 2 and Table 2. These parameters are identical to $(\Delta\lambda)_{s}$, as shown on Fig. 3 of Slettebak (\APACyear1976). To measure the position, we applied Gaussian fitting at the top of each peak, as shown in Fig. 5 in the Appendix. The typical errors are $\pm 5$ km s*-1* for $\Delta V_{\alpha}$, $\pm 7$ km s*-1* for $\Delta V_{\beta}$, and $\pm 10$ km s*-1* for $\Delta V_{\gamma}$.

3 Relations

For the Be stars, Hanuschik \BOthers. (\APACyear1988) find that the peak separations of H$\beta$ and H$\alpha$ emission lines follow approximately the relation $\Delta V_{\beta}\approx 1.8\Delta V_{\alpha}$. Dachs \BOthers. (\APACyear1992) find that average $\Delta V_{\beta}/\Delta V_{\alpha}=1.6-1.8$. Using more data, we have found that this relation is not valid above $\Delta V_{\alpha}\approx 200$ km s*-1*, and a linear fit of the type $y=a+bx$ is more appropriate. In Fig. 3 we plot $\Delta V_{\beta}$ versus $\Delta V_{\alpha}$ for 56 Be stars. In this figure the black circles are form Hanuschik \BOthers. (\APACyear1988), Dachs \BOthers. (\APACyear1992), and Catanzaro (\APACyear2013). The green squares are ELODIE spectra (Table 2). The red pluses are Rozhen data. The Rozhen data include the new spectra from Table 2, and the published spectra obtained with the same telescope setup (Zamanov \BOthers., \APACyear2016, \APACyear2022). We find the following relationship between $\Delta V_{\alpha}$ and $\Delta V_{\beta}$:

[TABLE]

This relation is valid for the range of $\Delta V_{\alpha}$ from 40 km s*-1* to 530 km s*-1*. It is very similar to our finding for a smaller sample and smaller range (Zamanov \BOthers., \APACyear2022). As expected, there is a very strong correlation between these two quantities, with Pearson correlation coefficient 0.95, Spearman’s rank correlation 0.93, and significance $10^{-20}$.

In Fig. 3 we plot $\Delta V_{\gamma}$ versus $\Delta V_{\beta}$. The black circles are data from Slettebak (\APACyear1976) and Hanuschik \BOthers. (\APACyear1988). The green squares are ELODIE data, the red pluses - Rozhen data. For the Be stars, Hanuschik \BOthers. (\APACyear1988) find that the peak separations of H$\beta$ and H$\gamma$ emission lines are connected as $\Delta V_{\gamma}=1.2\Delta V_{\beta}$. Here using more data, we find the following connection between $\Delta V_{\beta}$ and $\Delta V_{\gamma}$:

[TABLE]

This relation is valid for the range of $\Delta V_{\beta}$ from 20 km s*-1* to 340 km s*-1*. There is a very strong correlation between these two quantities, with Pearson correlation coefficient 0.97, Spearman’s rank correlation 0.96, and significance $10^{-18}$.

For 14 objects we have $\Delta V_{\gamma}$ and $\Delta V_{\alpha}$ on the same spectrum, in total 36 data points, because some objects are observed several times. In Fig. 4 we plot $\Delta V_{\gamma}$ versus $\Delta V_{\alpha}$. These data points refer to spectra on which both $\Delta V_{\alpha}$ and $\Delta V_{\gamma}$ can be measured. We are not able to measure $\Delta V_{\alpha}$ and $\Delta V_{\gamma}$ on each spectrum because (1) if the $H\alpha$ emission is strong, the H$\alpha$ peaks are very close one to another and can merge; (2) if the $H\alpha$ emission is weak, the emission peaks in H$\gamma$ are not visible; (3) the sensitivity of the ESpeRo spectrograph is lower at H$\gamma$. From Eq. 1 and Eq.2 we expect the following connection between $\Delta V_{\gamma}$ and $\Delta V_{\alpha}$:

[TABLE]

This equation is represented as a blue dashed line in Fig. 4.

Because there are some objects with one measurement and a few objects with 3-4 measurements, we performed the linear fit, using different subsamples (bootstrap technique, e.g. Efron (\APACyear1979)) in a way to have in the subsamples 1 or 2 measurements for each object and thus not to give too much weight to the objects with 4 measurements. A linear fit in the form $y=ax+b$ gives coefficients in the range $0.98<a<1.11$ and $71.3\leq b\leq 82.6$, with average values $a=1.065$ and $b=74.8$:

[TABLE]

This fit is represented as a black solid line in Fig. 4. The correlation is strong with Pearson correlation coefficient 0.90, Spearman’s rank correlation 0.91, and significance $10^{-9}$.

The black solid line (derived from the fit over the data) and the blue dashed line (based on the relationships $\Delta V_{\alpha}$ vs. $\Delta V_{\beta}$ and $\Delta V_{\beta}$ vs. $\Delta V_{\gamma}$) are similar, which confirms that the results are selfconsistent. In all three cases ($\Delta V_{\beta}$ versus $\Delta V_{\alpha}$, $\Delta V_{\gamma}$ versus $\Delta V_{\beta}$, and $\Delta V_{\gamma}$ versus $\Delta V_{\alpha}$, presented in Fig. 3, 3, and 4, respectively) we find a linear relationship of the type $y=ax+b$, with $a\approx 1.0$.

4 Disc size

In the Be stars, the distance between the peaks of H$\alpha$ emission line can be regarded as a measure of the outer radius ($R_{disc}$) of the emitting disc (e.g. Huang, \APACyear1972).

[TABLE]

where $G$ is the gravitational constant, $M_{1}$ is the mass of the Be star, $v\,\sin{i}$ is the projected rotational velocity, $\sin{i}$ is the inclination to the line of sight. The term $(1-\epsilon)$ represents the fact that the Be stars are rotating below the critical value (Porter \BBA Rivinius, \APACyear2003).

The relations between $\Delta V_{\gamma}$, $\Delta V_{\beta}$, and $\Delta V_{\alpha}$ give us the possibility to estimate $R_{disc}$, when two peaks are visible in $H\beta$,

[TABLE]

or in $H\gamma$ emission:

[TABLE]

where $\Delta V_{\beta}$, $\Delta V_{\gamma}$, and $v\sin{i}$ are in km s*-1*.

Radius estimation through the method of Huang (\APACyear1972) is a good approximation for symmetric profiles with double peaked $H\alpha$. If the two peaks of $H\alpha$ are not clearly visible, the equivalent width of H$\alpha$ emission can be used instead [e.g. Section 3 of Coe \BOthers. (\APACyear2006), and Eq. 5 in Zamanov \BOthers. (\APACyear2022)]. The relationships given above by Eq. 6 and Eq. 7 offer a third possibility.

5 Discussion

Recently, Wang \BOthers. (\APACyear2018) and Wang \BOthers. (\APACyear2021), using the UV spectra from HST and IUE, searched and found hot subdwarf companions of Be stars. Their results indicate that probably the rapid rotation of most Be stars is a result of mass transfer in a close binary system, as suggested by Pols \BOthers. (\APACyear1991). In this scenario, many Be stars are expected to have companions that are the remnants of the mass donor. The donor star might be stripped and become a hot subdwarf star in a Be+sdO binary, it might explode to create a neutron star or black hole in a Be/X-ray binary, or the binary might be disrupted by the supernova explosion (Postnov \BBA Yungelson, \APACyear2014).

Potentially, the deviations of individual objects from the average behaviour (as expressed in Eq. 1, Eq. 2, and Eq. 4) can be used to investigate the influence of the secondary on the Be disc structure, disc truncation, (because the truncation occurs only at certain radii; Okazaki \BBA Negueruela (\APACyear2001)), warping of the disc (Martin \BOthers., \APACyear2011), etc. The correlations can also be useful to test the theoretical models for formation of the emission lines in Be discs like those presented in Fig. 14 of Iwamatsu \BBA Hirata (\APACyear2008).

The Be stars show spectral variability both on short time-scales – hours to months (Porter \BBA Rivinius, \APACyear2003; Paul \BOthers., \APACyear2017) and long time-scales – up to years and decades (Mennickent \BOthers., \APACyear1994). One major observational aspect of Be stars is that some of them change phase from Be $-$ Be-shell $-$ normal B star $-$ Be. It will be interesting to follow the evolution of Be stars on these diagrams during the phase changes.

6 Conclusions

We analysed high resolution spectra of Be stars. We find the relationships between separations of the peaks of the H$\alpha$, H$\beta$, H$\gamma$ emission lines. The correlations found should be useful for future theoretical modeling and better understanding of the circumstellar discs of the Be stars in general.

**Acknowledgments: ** This work was supported by the \fundingAgencyBulgarian National Science Fund \fundingNumberproject KP-06-H28/2 08.12.2018 "Binary stars with compact object".

**Conflict of interest: ** The authors declare no potential conflict of interests.

Appendix A Supporting information