Resolving the Circumstellar Environment of the Galactic B[e] Supergiant Star MWC 137.II. Nebular Kinematics and Stellar Variability^*

Four long-slit optical spectra were obtained on 2018 November 8 with the Nordic Optical Telescope (NOT) using the Andalucia Faint Object Spectrograph and Camera (ALFOSC). ALFOSC's field of view (FOV) is $6\buildrel{\,\prime}\over{.} 4\times 6\buildrel{\,\prime}\over{.} 4$ and the pixel scale is 021 pix⁻¹.

We utilized Grism #17 with a slit width of 05, covering a spectral range of 6315–6760 Å with a spectral reciprocal dispersion of 0.29 Å pix⁻¹ (R ∼ 10 000). This choice is the same as for our previous observations (Kraus et al. 2017), covering Hα and the nebular lines [N ii] λ λ 6548,6583 and [S ii] λ λ 6716,6731.

The slits were centered on the star and had P.A.s of 0°, 35°, 295°, and 345° (see Figure 1) following the convention of measuring P.A.s from north to east. ⁸ For each slit position the exposure time was 30 minutes and the observations were carried out with a seeing of ∼1''.

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The spectra were reduced (bias, flat-fielding, wavelength calibration) using standard IRAF ⁹ routines, and were corrected for both the heliocentric and the systemic radial velocity (RV). For the latter we used the value of 42 km s⁻¹ determined by Kraus et al. (2017).

Additional observations have been taken on 2019 November 19 and 21 with the Spectral Camera with Optical Reducer for Photometric and Interferometric Observations No. 2 (SCORPIO-2, Afanasiev & Moiseev 2011), mounted at the Russian 6 m telescope (BTA). We used SCORPIO-2 in the Scanning Fabry–Perot mode in the order 751, operating with 40 spectral channels and providing a spectral resolution of 0.4 Å (R ∼ 15, 400). The total field of view of SCORPIO-2 is $6\buildrel{\,\prime}\over{.} 1\times 6\buildrel{\,\prime}\over{.} 1$. The CCD detector, E2V 42-90, has a size of 4600 × 2048 pixels with a pixel size of 13.5 × 13.5 μm and a pixel scale of 018 per pixel. During the observations with the FPI only the central part of the detector was used in 4 × 4 read-out binning mode providing the scale 071 per pixel.

The observations were centered on the [S ii] λ λ 6716,6731 lines. We refrained from using the filter covering Hα and the adjacent [N ii] λ 6583 line because of the brightness of the central star in Hα leading to strong saturation and blending effects when scanning the central regions. Instead, we gave preference to the two nebular lines [S ii]. Their lower intensities allowed us to take longer exposures, hence deeper images, without the risk of saturation.

During the first observing night, the observations were obtained in a single orientation, whereas two different orientations were used during the second night. The different orientations help to correct for parasitic ghosts. For each set of observations, the exposure times were 30 sec per channel. Data reduction was performed following the recipes of Moiseev (2002, 2015) and Moiseev & Egorov (2008). The reduced data cubes for the two orientations from the second night have been co-added. Unfortunately, the observing conditions were poor, with seeing changing from 27–35 to 31–38 between the two observations. To combine the two data cubes, we had to degrade the one taken with the slightly better conditions, resulting in a final spatial resolution (measured full-width at half-maximum of the image of the star) of just 38. The data cube from the first night was taken under significantly better seeing conditions of 12–15, resulting in a final spatial resolution of 15.

The data have been corrected for heliocentric and systemic RV and continuum-subtracted. Moreover, for each [S ii] line the data has been converted into RV, and the emission in both lines has been co-added in velocity space for a better signal-to-noise ratio (SNR) of the nebular structure. The final SNR has values spreading from 5–10 in the faintest to 55–60 in the brightest nebular regions.

A new Hα image of the MWC 137 nebula was acquired with the 1.54 m Danish Telescope at La Silla on 2019 October 27. The Danish Faint Object Spectrograph and Camera (DFOSC) was used with an FOV of $13\buildrel{\,\prime}\over{.} 7\times 13\buildrel{\,\prime}\over{.} 7$ and a pixel scale of 039 pix⁻¹. The narrow-band Hα filter, ESO #693, centered at 656.23 nm and with a width of 6.21 nm, was used. The total exposure time was 35 minutes, achieved with 7 dithered single exposures of 5 min each. The spatial resolution of the combined frame was 15.

In addition, we have at our disposal the raw images of MWC 137 presented by Marston & McCollum (2008). A single, 20 minutes long exposure was acquired on 2001 September 28 with the 60 inch telescope on Mt. Palomar with a spatial resolution of 21. The used FOV, pixel scale, and narrow-band Hα filter are all comparable to our observations with the Danish Telescope. For more details on these observations see Marston & McCollum (2008).

For both facilities, standard routines in IRAF were used for data reduction and for combining the individual frames.

To date, MWC 137 (TIC 437994564) has been observed twice with high cadence (120 s) by the Transiting Exoplanet Survey Satellite (TESS). The first observation period was between 2018 December 15 and 2019 January 6 (BJD = 2458468.27–2458490.05) in Sector 6 with camera 1 and CCD 3. The second visit took place about two years later, between 2020 December 18 and 2021 January 13 (BJD = 2459201.74–2459227.57) in Sector 33 with camera 1 and CCD 4. We accessed the calibrated light curves; ¹⁰ (Feinstein et al. 2019) through the Mikulski Archive for Space Telescopes (MAST). Calibrated means that the flux series has been corrected for common instrumental systematics for the amount captured by the photometric aperture and for crowding from known nearby stars. From these calibrated TESS light curves we used only the highest quality measurements (with quality flag 0000000000000000) for the analysis.

Our new Hα image of MWC 137 from 2019 was used in comparison to the Hα image from 2001 to explore any morphological changes and/or expansion of the large-scale nebula in the plane of the sky over the total time span of 18.1 years. We utilized more than 40 stars in the FOV to first match both images, pixel by pixel, using the IRAF tasks geomap and geotran. The rms of the matching was 0.1 pixels. At first glance, no obvious morphological evolution or expansion of the nebula was seen from a simple by-eye inspection (blinking) of the images.

Next, we used the so-called magnification method (see, e.g., Reed et al. 1999; Liimets et al. 2018) which uses the residual images to detect displacements in the nebula down to 0.1 pixels. For this, further image processing was required such as matching the spatial resolution of the 2019 image to the slightly worse one of the 2001 image and matching the flux. This matching was performed as described in Section 3 and 4.1 in Liimets et al. (2018).

After that, the 2001 frame was magnified, using the IRAF task geotran with a steadily increasing magnification factor, and then subtracted from the 2019 image. If the nebula would have expanded in the plane of the sky, the resulting residual images should reveal a distinct pattern based on which the magnification could be determined (see Reed et al. 1999). However, we detected neither morphological changes nor an expansion in the plane of the sky.

This means that, considering the time span of 18.1 yr, a possible expansion in the plane of the sky is proceeding with a tangential velocity of less than 52.7 km s⁻¹ for a pixel scale of 039 and the distance of 5.15 kpc toward MWC 137. In fact, the maximum RV component measured in the nebula is only about 40 km s⁻¹ (see Kraus et al. 2017, and Section 3.3 below) and hence too small to create detectable changes in the nebular morphology to date. For the idealized case of a ballistically expanding sphere with a tangential velocity of 40 km s⁻¹, we can hence expect to detect an indication of expansion earliest after a time span of about 24 yr, that is in the year 2025. However, the lack of significant RV of the nebula around the position of the star questions a ballistic expansion so that the real expansion of the nebula in the plane of the sky might proceed (much) slower.

Although the data from the second night had longer exposure times, we present the data from the first night because of their significantly higher spatial resolution. The lack of a second orientation observation resulted in several artifacts that appear on the final cube, such as false emission at both the stellar position and the position of several stars in the vicinity. These could not be properly corrected but were identified based on the deeper data cube. Within the nebula structure, only at the central star's position, a ghost artifact was identified and masked and is excluded from our discussion. This masked area around MWC 137 has a radius of 25, corresponding to a sphere with a radius of 0.062 pc.

Slicing the FPI data cube with respect to RV, we find that the nebula is detected from about −51 km s⁻¹ to about +62 km s⁻¹ with respect to the nebular systemic velocity as shown in Figure 2. For a better comparison, the intensity in all velocity bins is displayed on a linear scale and in all panels the intensity range is the same.

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Starting from the very blue, the first features that become visible are two structures in the northern nebular regions. The northeastern ring or arm-like structure brightens until it reaches maximum intensity at +0.5 km s⁻¹ and then starts to fade and disappear for velocities redder than +52 km s⁻¹. The second, most blueshifted emission is related to a smaller ring or arm-like structure northwest of MWC 137. The northern domain of this feature reaches maximum intensity at −20 km s⁻¹ and disappears for velocities redder than +11 km s⁻¹. Starting from about −30 km s⁻¹, a southwestern arm starts to appear. Its maximum intensity is reached around +11 km s⁻¹ and it remains visible beyond redshifted velocities of +52 km s⁻¹. Around +11 km s⁻¹, this feature seems to connect to the large northeastern arm structure. A further, high-intensity region appears west of the central object in the velocity interval −10 km s⁻¹ to +20 km s⁻¹. In the velocity bins at + 0.5 km s⁻¹ and +11 km s⁻¹, this emission seems to spread around the central object in east–west direction. Finally, the southwestern arm-like structure embraces some diffuse emission in the southern region with velocities from +31 km s⁻¹ to approximately +52 km s⁻¹.

Furthermore, the large FOV of the FPI reveals emission features at greater distances from the star, in regions that have not been covered by the MUSE mosaic of Mehner et al. (2016). These are (i) diffuse emission apparently emanating from the northwestern nebular regime and (ii) a bow structure toward the southwest at a P.A. of 225°–245°. Both emission structures are visible on our Hα images taken with ALFOSC and with the Danish telescope and are also present on the image of Marston & McCollum (2008). But Marston & McCollum (2008) mentioned only diffuse emission. The presence of the bow structure on all images taken between 2001 and 2019 and with different instruments implies that it is real and not an artifact.

These two emission features appear in the velocity interval from −10 to +21 km s⁻¹. For better visibility of these features, we co-added the images in this velocity range and show the combined image in Figure 3. The vertex of this bow or arch has a distance of about 80'' from the central star and an extension of about 36'' from northwest to southeast. These measurements translate into physical scales, projected to the plane of the sky, of 2 pc and 0.9 pc. But whether its origin is linked to MWC 137 is difficult to tell. At the least this arch or bow seems not to be related to the jet discovered by Mehner et al. (2016), which emerges at a P.A. of 18°–20°.

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Finally, we use the FPI data cube to measure the radial velocities and velocity dispersions of the combined [S ii] lines over the full nebula. Visual inspection of the profiles reveals that the lines are single-peaked everywhere in the nebula. The emission lines are fitted with a Voigt profile and the velocity dispersion has been corrected for instrumental broadening as described in Moiseev & Egorov (2008). The results are shown in Figure 4. The general trend is that the intense nebula arm structures have lower red- or blueshifted velocity compared to the gas inside and outside (in projection onto the plane of the sky) of these structures. In addition, we observe that the regions with highest radial velocities have the tendency to also display high velocity dispersion, but with no strict coincidence.

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The choice of filter setting for these observations was optimized such that both [S ii] lines were simultaneously observed. However, as we can see in Figure 4, the emission can become rather broad in some nebular regions, so that not always the full profile of either the blue or the red [S ii] line was covered by the narrow filter, hampering a more detailed analysis of the physical conditions across the entire nebula.

The long-slit spectra taken with ALFOSC cover, besides Hα, the forbidden emission lines [N ii] λ λ 6548,6583 and [S ii] λ λ 6716,6731. The slightly better observing conditions under which these spectra were taken compared to the FPI images and, in particular, the longer spectral coverage allow for more precise measurements of the kinematics as well as a proper determination of the physical parameters of the nebula.

For both doublets we focused on their more intense line to perform the RV measurements. These are [N ii] λ 6583 and, over large regions of the nebula, [S ii] λ 6716. For each line, we fit the emission profiles in the 2D spectra, line by line, to determine RV as a function of distance from the star. The lines appear single-peaked and are not blended so that we fit them with a single Gaussian profile. These measurements are presented for all four P.A.s in Figure 5 and are provided in an online table following the layout of Table 1. Profiles were excluded from the measurements if they were (i) too weak, (ii) contaminated by a cosmic ray that could not be removed, or (iii) contaminated with a neighboring line. The latter happens especially for the [N ii] λ 6583 line in the vicinity of the central star where Hα is saturated, preventing any precise measurements of that line. The uncertainties of the velocity measurements are rather small, with an average value of 0.95 km s⁻¹, and are included in Table 1. They have been determined from the rms of the wavelength calibration. Contributions from the fitting to the total error are negligible.

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From Figure 5 we can see that in general both nebular lines display the same trend in the gas dynamics. However, some deviations are noticeable. First, in the close vicinity of the hot central star the [S ii] lines disappear and the [N ii] lines become very weak on all our spectra. This points toward a higher electron density favoring collisional depopulation of the levels along with a hotter gas and, therefore, preferentially higher ionized material. In fact, the higher density is obvious from the much more intense Hα emission toward the center, whereas the slightly higher ionization state follows from high-resolution spectra of the innermost 2'' which display considerable emission of [S iii] λ 6312 and only very weak emission in [S ii] λ 6716 (Kraus et al. 2017, their Figure 3). These hindrances in measurements result in a gap in the RV values in all P.A.s within a radius of about 15–20 from the central star.

Second, we notice considerable differences in the RVs of the two lines in the regions with highest blueshifted RV values toward toward the north and northwest for P.A.s at 0°, 325°, and 295° (the same is seen for P.A. 345°, Kraus et al. 2017) where the [N ii] displays significantly higher velocities than the [S ii]. We see the same trend in the redshifted RV maximum for P.A. = 295° in the southeast. The deviations can reach values of Δv_rad ≈ 10–15 km s⁻¹. Because these differences are significantly larger than the measurement errors, they are considered real. Taking into account that the ionization potential of N i (I.P. = 14.5 eV) is higher than that of S i (I.P. = 10.4 eV) and that the ionization of the elements is governed by the radiation flux from the hot central star, the emission in [N ii] should be formed closer to the star than the emission in [S ii]. A relation between RV and the ionization potentials of the elements has been found in H ii regions (see, e.g., Esteban & Peimbert 1999; Caicedo-Ortiz & Fernandez 2019) where it has been used to trace the velocity stratification along the line of sight. Application to the nebula of MWC 137 suggests that within these specific regions we see deceleration of the material from inside out.

In agreement with the FPI data, the long-slit spectra reveal that the northern nebular region is predominantly blueshifted with respect to the nebular systemic velocity, reaching values up to −36.2 km s⁻¹, whereas the southern part is mostly redshifted with velocities up to +34.6 km s⁻¹. Despite this general trend, we also note strong variability of the RV values along most P.A.s, with multiple maxima and minima, in agreement with the inhomogeneous velocity pattern across the nebula seen in Figure 4 and with the wiggly structures detected along the P.A.s presented by Kraus et al. (2017).

To visualize the RVs over the measured nebular regions, we utilize the RV values of the [N ii] λ 6583 line and combine them with those of our previous long-slit spectra for the three P.A.s at 189, 62°, and 325° (respectively, at 3411, 298°, and 35° according to the notation in Figure 4 of Kraus et al. 2017). In Figure 6 we present the color-coded RVs along all seven slits. For a comparison with the nebular structures, these velocities are overlaid on the Hα emission nebula.

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From this image we can deduce that the largest region displaying highly blueshifted RVs with values exceeding −40 km s⁻¹ resides in the northwestern area of the nebula, in particular along P.A. = 325°. This area corresponds to the region in which neither cold molecular gas nor significant emission from warm dust is seen (Kraus et al. 2017). A further, though smaller blueshifted domain occurs toward the northeast, beyond the outer rim (maximum intensity in the Hα image) of the nebula at P.A. = 35°. Here, the velocities reach values around −36 km s⁻¹ whereas they tend toward zero within the region of maximum intensity. A similar behavior is found for the most northern region along P.A. = 0°. Also here the RV is around zero on top of the maximum intensity and reaches highest blueshifted values beyond that region. For the other P.A.s with blueshifted emission (295°, 325°, and 345°) the situation is different. Here, the emission with maximum blueshift either coincides with the intensity peaks or occurs at distances closer to the central star.

Turning to the southern nebular region we find the highest redshifted velocities in the southeastern portion, with a narrow peak of about +35 km s⁻¹ reached within an apparent gap in the Hα double-ring structure along P.A. 295°. For the P.A.s at 325°, 345°, and 0°, the highest redshifts occur (much) closer to the central star and are not correlated with the regions of maximum emission intensity. The only exception is P.A. = 62°. Here the redshifted emission coincides with the maximum intensity. But the redshift in that region is with values smaller than +10 km s⁻¹, much less prominent than along other P.A.s.

Our measurements also provide the full-width at half-maximum (FWHM) values of the emission lines. These are shown in Figure 7 for [N ii] 6583 Å and [S ii] 6716 Å along all seven P.A.s. As for the RV values, we find that both forbidden lines display the same behavior of their FWHM values, with noticeable differences only in regions in which the lines are very faint and thus have a higher measurement uncertainty. For most P.A.s, a significant portion of the nebula displays some average FWHM value of 30–40 km s⁻¹ (whereby 30 km s⁻¹ correspond to the velocity resolution of the long-slit spectra), but in certain regions it can reach values of 70–80 km s⁻¹. Exceptions are the P.A.s at 325° and 295°, which exhibit considerably higher FWHM values over large nebular portions. We also find that the nebular areas with high FWHM values correlate more or less with the regions of high RV, in agreement with what is seen in Figure 4.

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From their low-resolution MUSE data Mehner et al. (2016) estimated average values of T_e ∼ 10, 000 K and n_e ∼ 300 cm⁻³ for the electron temperature and density across the nebula. Our spectra have three times higher resolution, but are short in wavelength coverage and thus contain just two strategic forbidden lines, [S ii] λ λ 6716,6731, which allow us to investigate the electron density distribution along the P.A.s, but we cannot estimate the electron temperature.

From our spectra, we measure, line by line, the line fluxes of both [S ii] lines and then average the values every three lines before computing their ratio. This binning reduces the measurement errors, particularly in the very faint nebular regions, and also better represents the real values considering a spatial resolution of ∼1'' during the observations compared to the pixel size of 021. A conservative uncertainty estimate of the flux measurements is 5%, resulting in an uncertainty on the observed line ratios of about 7%.

From theory, the intensity ratio of these two [S ii] lines is given by

$$\begin{eqnarray}&&\displaystyle \frac{I(6716)}{I(6731)}=\displaystyle \frac{N(6716)A(6716)\lambda (6731)}{N(6731)A(6731)\lambda (6716)},\end{eqnarray} \tag{ 1 }$$

where the parameters N, A, and λ denote the level population, Einstein transition probability, and laboratory wavelength of the corresponding transition. This ratio is an implicit function of the electron density for a given temperature and vice versa.

To compute the level population we consider a five-level atom in which the levels, from which the forbidden lines arise, are excited by collisions with free electrons. Collision parameters and Einstein transition probabilities are taken from Tayal & Zatsarinny (2010). The level population is calculated for electron temperatures between 5000 and 20,000 K and for electron densities ranging from 1 to 15,000 cm⁻³. The line ratios as a function of electron density are shown in the top panel of Figure 8 for five different electron temperature values, whereas the observed line ratios from north to south across the nebula are plotted in the bottom left panel, and the resulting electron densities, considering a constant electron temperature of the nebula of 10,000 K, are in the bottom right panel.

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The densities display high local fluctuations with the majority of measurements spreading from 200 to 500 cm⁻³. But we also find a clear increase in electron density toward the position of the central star. This can be seen in the color-coded image shown in Figure 9, where the electron density distribution along each P.A. is displayed and overlaid on the Hα emission nebula. From this image it also becomes also evident that higher densities occur in the regions of high intensities. Moreover, in the northwestern region where the RVs are highest, the density rapidly decreases. The gaps, for example, in the northern part of the P.A. at 0° and in the inner region of the P.A. at 295° are caused by either an extremely faint, hence not measurable [S ii] λ 6731 line, or a contamination of that line by a strong cosmic ray, or a combination of both.

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A few outliers appear with flux ratio values greater than the theoretically predicted one. We checked these flux measurements by eye, and they seem to be appropriate. Therefore, we believe that in these extremely low-density regions our assumption of pure collisional excitation of the levels might not hold anymore.

The TESS time series of the central object MWC 137 taken with high cadence (120 s) in sector 6 (between 2018 December 15 and 2019 January 6) and sector 33 (between 2020 December 18 and 2021 January 13) are displayed in the top left and right panels of Figure 10, respectively. A slightly higher mean flux value was noted for sector 33, corresponding to a difference of 0.022 mag. Considering the 2 yr time gap between the observations, this brightening (if real and not just caused by very slight changes in the aperture and centering between sectors or due to contamination from nearby sources) might be related to a possible long-term variability of 332.4 d. This period has been suggested by Maravelias et al. (2018) based on their analysis of the ground-based light curve provided by the ASAS survey (Pojmanski 1997), covering a total of about 2500 days but with very sparse coverage. Such a long period cannot be recovered by the current set of TESS data. Therefore, we restrict our analysis to the light variability for each sector independently, considering their individual mean magnitudes.

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The two pieces of the light curve of MWC 137 display clear but complex variability. To search for periodicities in these data sets, we perform a frequency analysis utilizing Period04 (Lenz & Breger 2005) in a standard pre-whitening procedure. The amplitude spectra of the original data are shown in the lower parts of Figure 10. They have been computed up to the Nyquist frequency (∼360 d⁻¹ in both cases) and they display clear differences in the signal strengths, especially in the low-frequency (<0.5 d⁻¹) domain. No significant peaks are detected beyond 2 d⁻¹. We also include the amplitude spectrum of the combined data set (bottom panel).

The short observational baselines within the TESS sectors lead to several constraints for the identification of significant frequencies. First, it does not allow one to measure signals with periods greater than ∼10 days, so that we ignore any detections in the low-frequency (≤0.1 d⁻¹) range. Second, while the Rayleigh resolution, defined as 1/T where T is the time span of the observations, amounts to 0.046 d⁻¹ and 0.039 d⁻¹ for the data in sectors 6 and 33, respectively, high-amplitude side lobes appear around dominant peaks in the amplitude spectra (see Figure 11). Therefore, we follow the conservative approach proposed by Loumos & Deeming (1978) and discard frequencies with a separation of less than 2.5 times the Rayleigh resolution to previously identified frequencies with higher amplitude. Third, a higher threshold (S/N ≥ 5) for the significance of identified peaks is recommended for an observational baseline of less than a few months of space photometry (Baran et al. 2015). Finally, a densely populated amplitude spectrum in the low-frequency domain requires a wider window size of 5 d⁻¹ instead of the conventional 1 d⁻¹ for the computation of the underlying noise spectrum (Burssens et al. 2020).

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Applying all these constraints during the frequency analysis, we end up with only a small number of frequencies that we consider reliable. These frequencies are shown in the amplitude spectra of Figure 10 and listed in Table 2. We clearly identify one dominant signal in both sectors with a period of 1.93 days. The frequencies of its first two harmonics (2F1 and 3F1) are included in Figure 10. They are both seen in the amplitude spectrum of sector 33, whereas in sector 6 only the first harmonics are evident. Their signals fall slightly below our detection criterion (4 < S/N < 5) so that we include their frequencies only at the bottom of Table 2. The closeness of frequencies F2 in sector 6 and F4 in sector 33 suggests that they might be the same. If true, then this frequency could be another persistent one, while the other two frequencies (F3, F5) that show high amplitudes in sector 33, are not evident in sector 6 and thus seem to be stochastic.

In a previous study, Kraus et al. (2017) proposed a toy model for the nebula of MWC 137, consisting of two interwoven double cones offset by 55 in the north–south direction. The northern one was situated and sheared such that it fit the large-scale putative double-ring structure, whereas the southern one was oriented in such a way that its symmetry axis was aligned with the jet axis.

Our new observations provide no complementary evidence that would support this model. Instead, the deeper image in the lines of [S ii] imply that the large-scale nebula consists of helical or spiral-arm-like structures rather than a double-ring (see Figure 2). The RV measurements suggest that these helical structures are blueshifted in the northern nebular regions and redshifted in the southern ones, whereas the diffuse gas displays complex velocity variations. The lack of information about the velocity of the gas in the plane of the sky clearly hampers our ability to draw conclusions about the real motion of the nebular structures in three-dimensional space. For the detection of the nebular expansion in the plane of the sky, however, our current baseline of 18.1 yr is still too short.

The significantly improved distance estimate toward MWC 137 based on Gaia, which agrees very well with the value derived by Mehner et al. (2016), allows us to finally settle the luminosity of the star and to better constrain its initial and current mass. The huge range of previously proposed distances resulted in luminosity estimates spreading from $\mathrm{log}(L/{L}_{\odot })=4.18$ (Alonso-Albi et al. 2009) up to at least 5.37 (Esteban & Fernandez 1998), and thus in masses ranging from 12 to at least 32 M_⊙ when compared to stellar evolutionary tracks of rotating and non-rotating stars (see Muratore et al. 2015). The most recent mass estimate has been performed by Mehner et al. (2016), who fit the location of MWC 137 in a color–magnitude diagram with GENEVA isochrones for the cluster and evolutionary tracks from Ekström et al. (2012) based on which they claim best agreement for an age of ∼10 Myr and a mass of ∼15 M_☉. At the same time, they estimate the luminosity of MWC 137 as $\mathrm{log}(L/{L}_{\odot })\sim 6$, which clearly contradicts the low mass assigned to the object.

We believe that this mismatch between mass and luminosity arises from the fact that Mehner et al. (2016) only matched the models with absolute visual magnitude, M_V , and neglected the effective temperature of the star. Temperature estimates in the literature all point toward a hot central object of T_eff = 25,000–31,000 K (see, e.g., Hillenbrand et al. 1992; Esteban & Fernandez 1998; Alonso-Albi et al. 2009). Using the values of Mehner et al. (2016) for the visual magnitude V and the extinction A_V , along with the Gaia distance (Table 3), the absolute visual magnitude of MWC 137 is M_V = − 7.2 ± 0.3 mag, in excellent agreement with the value of Mehner et al. (2016).

Table 3. Stellar Parameters of MWC 137

V	E(B − V)	AV	d	MV	$\mathrm{log}{T}_{\mathrm{eff}}$	$\mathrm{log}L/{L}_{\odot }$	Min	M	R	Ωin/Ωcrit	Age
(mag)	(mag)	(mag)	(kpc)	(mag)	(K)		(M⊙)	(M⊙)	(R⊙)		(Myr)
11.95 a	1.8 a	5.6 a	${5.15}_{-0.60}^{+0.79}$	−7.2 ± 0.3	4.40–4.49 b	5.84 ± 0.13	49 ± 11	${37}_{-5}^{+9}$	${25}_{-9}^{+20}$	0.425 ± 0.025	4.7 ± 0.8

References.

^a Mehner et al. (2016). ^b Hillenbrand et al. (1992); Esteban & Fernandez (1998); Alonso-Albi et al. (2009).

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In Figure 12 we display evolutionary tracks for non-rotating (left panel) and rotating stellar models with Ω_in/Ω_crit = 0.568 (v/v_crit = 0.4, middle panel) from Ekström et al. (2012) for solar metallicity. If MWC 137 had an initial mass of 15 M_☉ as proposed by Mehner et al. (2016), a value of M_V = − 7.2 mag is achieved only when the star has reached an effective temperature $\mathrm{log}{T}_{\mathrm{eff}}\leqslant 4$ according to the models of both rotating and non-rotating stars. Considering the proper effective temperature range for MWC 137 of $\mathrm{log}{T}_{\mathrm{eff}}=4.40\mbox{--}4.49$, the real initial mass is considerably higher than 15 M_☉, spreading from about 35 to 65 M_⊙.

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To further narrow down the mass range, in Figure 12 we included information about the surface abundance ratio of ¹³C/¹²C. This ratio is very sensitive to stellar rotation and may help to make inferences about the initial rotation speed of MWC 137. While we cannot measure the carbon isotope abundance ratio directly on the stellar surface, Kraus (2009) has shown that the enrichment of the nearby circumstellar environment in the isotopic molecule ¹³CO provides a reasonable tracer of the stellar surface enrichment in ¹³C for stars with strong winds or mass ejections. The circumstellar matter of MWC 137 indicates a carbon isotope abundance ratio of ¹³C/¹²C = 25 ± 2 (Oksala et al. 2013; Muratore et al. 2015). While the non-rotating tracks do not display significant surface enrichment at early stellar evolutionary phases, the models with rotation predict that the amount observed in MWC 137 is reached already during the main sequence and is significantly lower around the position of MWC 137 in the HR diagram. Therefore, to achieve the observed value, the initial rotation rate of MWC 137 must be clearly non-zero but also lower than Ω_in/Ω_crit = 0.568.

We interpolate the pre-calculated evolution tracks of Ekström et al. (2012) using their tool SYCLIST ¹¹ to obtain stellar evolution tracks for a dense mass grid and covering rotation rates between Ω_in/Ω_crit = 0 and 0.568. The evolution of the carbon isotope ratio is followed along these tracks to select the best fitting models for the parameters of MWC 137. We find, that the most likely initial mass of MWC 137 is 49 ± 11 M_☉ with an initial rotation rate of Ω_in/Ω_crit = 0.40–0.43. Slightly higher rotation rate values (up to 0.45) are still possible for models at the upper mass limit. Representative tracks are shown in the right panel of Figure 12. From these tracks we deduce a current stellar mass of ${37}_{-5}^{+9}\,{M}_{\odot }$ and a luminosity of $\mathrm{log}(L/{L}_{\odot })=5.84\pm 0.13$ for MWC 137. The latter is in good agreement with the luminosity of $\mathrm{log}(L/{L}_{\odot })\sim 6$ derived by Mehner et al. (2016). The final set of stellar parameters of MWC 137 is presented in Table 3. The table also includes the plausible age of MWC 137, which is found to range from 3.9 to 5.5 Myr, in excellent agreement with the estimated ages of the other stars in that cluster (see Mehner et al. 2016).

The TESS time series, taken in sectors 6 and 33 with a separation of two years, contains at least one stable period of 1.93 days (see Sect. 3.5). Its putative decrease in amplitude between the two observing runs (see Figure 10) might not be real but caused by the simultaneous weakening or disappearance of possible, unresolved stochastic signals in the vicinity of the dominant frequency. Considering the stellar parameters of MWC 137 (see Table 3), it is obvious that a period of 1.93 days is way too short to be interpreted as the rotation velocity of the star. The eligible stellar models predict an equatorial rotation velocity of <10 km s⁻¹ for the current evolutionary state of MWC 137, resulting in a minimum stellar rotation period of ∼100 days. It is also unlikely that the 1.93 days period could be caused by the orbital motion of a hidden companion. Considering the parameter range for MWC 137, the corresponding orbital radius of the companion would be 21 ± 1 R_⊙ and thus smaller than the radius of MWC 137. Therefore, a different alternative for the appearance of the 1.93 day period seems to be more likely, and this is stellar pulsation. Low-amplitude variability with periods of 1–3 days were found in several massive early B-type supergiants and were ascribed to opacity-driven oscillations (see, e.g., Lefever et al. 2007), supporting such an interpretation.

The other stochastic or random signals seen in the amplitude spectrum of MWC 137 might be related to internal gravity waves. The appearance of such waves has been suggested for early-type stars, and they seem to be suitable for generating stochastic photometric variability (Rogers et al. 2013). The clearly variable amplitude and evanescence of these signals further support such an interpretation.

But other or additional mechanisms might be at work and are worth being mentioned. As a massive, early-type star MWC 137 has a line-driven wind. Recent investigations have shown that perturbations in the velocity field at the base of the wind, for example, by stochastic waves, can cause wind instabilities (Krtička & Feldmeier 2021). These instabilities can generate variability of the mass flux and hence wind inhomogeneities that might lead to measurable (quasi-periodic or stochastic) changes in photometry. A time-variable wind of MWC 137 is evident from moderate changes in its hydrogen Balmer line emission at optical wavelengths (Zickgraf 2003; Kraus et al. 2017) as well as from variable Pfund line emission in the near-infrared (Muratore et al. 2015). Furthermore, the central star is surrounded by a compact region of molecular and neutral gas, possibly arranged in a series of rings (Maravelias et al. 2018), and the absence of photometric lines in the spectra implies that the star is embedded in a dense envelope. A compact disk has also been proposed to be the source of the emanating jet and blobs discovered by Mehner et al. (2016). Dynamical dusty blobs and arc-like structures have been found in the close vicinity of MWC 137 as well (Kraus et al. 2017). Therefore, it is clear that any time-variable structures around the central star can easily imprint variabilities on the observed photometric signals.