© The Authors 2025

1. Introduction

Almost all massive stars (≳95%) have one companion or more than one (see e.g. Evans et al. 2011; Sana et al. 2013; Mahy et al. 2013; Kobulnicky et al. 2014; Moe & Di Stefano 2017; Almeida et al. 2017; Barbá et al. 2017; Villaseñor et al. 2021; Trigueros Páez et al. 2021; Banyard et al. 2022; Merle 2024). Roughly two-thirds of the massive stars in a binary are close enough to their companion that they will interact during their lifetime (Sana et al. 2012). This leads to many interesting phenomena. It is for instance possible for both massive stars in a binary to become a black hole (see for a review Mandel & Broekgaarden 2022). A binary with two black holes is an interesting object to study in relation to, for example, the large number of gravitational wave detections of double black holes (Abbott et al. 2023).

The formation path for black hole binaries and also high-mass X-ray binaries (e.g. Tauris & van den Heuvel 2003) typically involves a first mass-transfer phase that can have crucial consequences for the following evolution (see Dorozsmai & Toonen 2024). In this paper, we focus on binaries that contain a Wolf-Rayet (WR) star and an O-star that likely have experienced this phase (e.g. Paczyński 1967; Vanbeveren et al. 1998). WR stars are typically viewed as stars that lost (most of) their envelope and therefore contain little to no hydrogen on their surface, but have a high abundance of other elements, although some were found to be main-sequence stars (see e.g. Martins 2023). In general, the mass of WR stars ranges from 10 to 25 M_⊙, and an O-star is their progenitor (Crowther 2006). WR+O binaries can be the progenitors of interesting phenomena such as X-ray binaries and the aforementioned black hole-black hole binaries (e.g. Tauris & van den Heuvel 2003; van den Heuvel et al. 2017). However, in order to understand the formation of WR+O binaries and their potential to evolve into X-ray binaries and double black holes, the mass-transfer process must be understood. Two of the key ingredients of this process are the fraction of mass that is accreted by the accreting star and the fraction that is lost from the system (the mass-transfer efficiency), and the amount of angular momentum that this lost material that was removed from the system. The efficiency of the mass transfer can strongly affect the evolution of the stars within the binary and the binary as a whole (see e.g. Shao & Li 2016). We studied whether it is possible to determine the values of the mass-transfer efficiency and angular momentum loss of these systems and the evolutionary phase of the donor at the start of mass transfer. Because of the significant mass loss, WR+O stars can in principle also form because the WR progenitor loses its envelope via a stellar wind. However, the properties of the observed systems, such as the spins of the O stars and the orbital properties, suggest that the majority of WR+O binaries have experienced mass transfer (Shara et al. 2017; Cherepashchuk 2018; Vanbeveren et al. 2018).

Mass transfer is classified depending on the phase in the evolution of the donor the mass in which transfer starts. The first type of mass transfer is Case A. This is when the donor star is still on the main sequence (MS) at the moment it fills its Roche lobe (e.g. Pols 1994; Wellstein et al. 2001; Nelson & Eggleton 2001; de Mink et al. 2007; van Rensbergen et al. 2010; Sen et al. 2022). The phase when hydrogen burning in the core of the star has stopped but helium burning has not yet started at the onset of mass transfer is called Case B mass transfer (e.g. Paczyński 1967; van Rensbergen et al. 2011; Podsiadlowski et al. 1992; Klencki et al. 2022). There are also cases when Case A is followed by a second phase that is Case B. These cases are called Case AB mass transfer (see e.g. Sen et al. 2022). When mass transfer starts after helium ignition in the core, it is called Case C (e.g. Podsiadlowski et al. 1992). We focused on the first two cases of mass transfer because Case C mass transfer is expected to be unstable more often than Case A or Case B (e.g. Ge et al. 2015), and we studied the formation of WR+O binaries via stable mass transfer.

Petrovic et al. (2005) studied three WR+O binaries and concluded for all three that they must have undergone Case A mass transfer. Shao & Li (2016) also studied different WR+O systems based on models, but were unable to draw a clear conclusion on which type of system undergoes Case A or Case B mass transfer. Sen et al. (2022) studied the first mass-transfer phase in detail and compared their models to observations of systems that are currently transferring mass, but also showed results for the systems after the mass transfer. However, all these studies compared the observed systems to one or more specific binary models. We take a different approach and derive for each observed WR+O binary the type of evolution could have led to the current system. In this way, we constrained the fraction of the transferred mass that was lost from the system and the specific angular moment it took with it.

We used the VIIth catalogue of Galactic WR stars (van der Hucht 2001) to choose suitable candidates for our study (Sect. 3). This resulted in a sample of 21 WR+O binaries and allowed us to take a more statistical approach to study their mass transfer. This paper is structured as follows. In Sect. 2 we introduce the method with which we constrained the possible progenitors of the observed WR+O binaries and the resulting mass-transfer efficiency, angular momentum loss, and mass-transfer type. In Sect. 3 we introduce the properties of the binaries we studied. In Sect. 4 we present the results, which we discuss in Sect. 5 and summarise in Sect. 6.

2. Method

We inferred the mass-transfer type and (limits on) the mass-transfer efficiency and angular momentum loss that the observed systems experienced. In order to do this, we estimated the initial masses and the period of the progenitors to the WR+O binaries. In this section, we describe the methods we used to estimate these initial properties of WR+O binaries. We then discuss how we used these initial properties to determine the mass-transfer efficiency and angular momentum loss of the binary systems, and we determine whether the systems experienced Case A or Case B mass transfer.

2.1. Initial masses

The observed masses of the WR and O star can be used to estimate the initial mass of the WR star in the binary. The way in which this is done differs between Case A and Case B. For Case B, this can be done based on the relation between the core mass and the initial mass, as found by Wellstein & Langer (1999). As a WR star consists of just the core of the progenitor star, we can use this relation. Petrovic et al. (2005) made a linear fit of this relation, which resulted in Eq. (1),

$$\begin{array}{c}\hfill M_{\mathit{1},i,B}=\frac{M_{\mathit{WR}}+4.92}{0.53},\end{array}$$(1)

where M_WR is the mass of the WR star, and M_1, i, B is its progenitor mass for Case B.

For Case A, we used Figure 7 from Shao & Li (2016) to derive a similar relation. In this figure, they displayed the relation between the current WR mass and the initial mass of that star for different initial mass ratios and initial periods. We fit a straight line that traced the lower boundary of the majority of possible masses (i.e. those on the shortest orbital periods). This resulted in the relation between the current WR mass and the initial primary mass for Case A mass transfer that is found in Eq. (2). Since this relation is approximately a lower limit, the outcome for M_1, i, A based on a measured WR mass is an upper limit. This allowed us to find the (approximate) initial primary mass for Case A,

$$\begin{array}{c}\hfill M_{1,i,A}=\frac{M_{\mathit{WR}}+4.86}{0.41},\end{array}$$(2)

where M_1, i, A is the maximum progenitor mass for Case A.

With the initial mass of the WR star, we set some constraints on the parameter space of the secondary mass. First, the current WR star has to have been the donor star for it to have lost its envelope. Since the more massive star is the donor star, the initial mass of the current O star cannot be higher than the initial WR mass. Another way to derive an upper limit is by using the total mass of the system. Since we assumed that the O star has accreted mass, or at least not lost mass, we state that the initial O-star mass M_O, i ≤ M_O, f, the final O-star mass. From these two limits, the lower of the two becomes the final upper limit for M_2, i. We also set a lower limit to obtain a better restricted parameter space. To do this, we used the knowledge that the total mass of the system cannot have grown, but only decreased or stayed the same. Thus, the sum of the initial masses M_1, i + M_2, i ≥ M₁ + M₂ = M_T, the current total mass, which can be rewritten to M_2, i ≥ M_T − M_1, i. This gives the lower limit for the initial mass of the companion. If the difference is especially large, this might become negative, in which case, we used zero as a lower bound instead. Together with the upper limit, we thus determined the range of the initial secondary mass for each of our systems.

The change in both the donor mass and the accretor mass can be expressed in terms of the accretion efficiency β as ΔM_a = −βΔM_d. This can also be expressed in terms of the total mass M_T and the mass ratio q = M_a/M_d (Soberman et al. 1997) as follows:

$$\begin{array}{c}\hfill \frac{M_T}{M_{T,i}}=\left(\frac{1+q}{1+q_i}\right)\left(\frac{1+\beta q_i}{1+\beta q}\right).\end{array}$$(3)

2.2. Initial period and mass-transfer type

We cannot really estimate the initial period. We therefore considered many possible periods. Using the estimated initial masses for Case A and Case B, we then determined which periods lead to Case A and which to Case B mass transfer. To do this, we used the fact that mass transfer starts when a star fills its Roche lobe R_L. More specifically, we used the fact that a star needs to have exhausted the hydrogen in its core before the onset of mass transfer for the mass transfer to be considered Case B. When we knew the initial mass of the donor, we determined its radius at the end of the main sequence, also called the terminal age main sequence (TAMS). This gives the condition R_L ≥ R_d, TAMS for Case B mass transfer. When the Roche lobe is smaller than the donor radius at TAMS, this indicates that Roche-lobe overflow (RLOF) has already occurred before, and the mass transfer is Case A.

We related R_L to the period P and the mass ratio of the binary and the mass of the donor star M_d. To do this, we made use of the relation found by Eggleton (1983) for 0.1 < Q < 10, where Q again is the mass ratio, but inverted: M_d/M_a,

$$\begin{array}{c}\hfill \frac{R_L}{a}\approx \frac{0.44Q^{1/3}}{{(1+Q)}^{1/5}}·\end{array}$$(4)

With Kepler’s third law,

$$\begin{array}{c}\hfill P=9.859·{(1+Q)}^{-1/5}{M}_d^{-1/2}{R}_L^{3/2}{\left(\frac{G}{4{\pi }^2}\right)}^{-1/2}·{10}^{-7}\mathrm{d}.\end{array}$$(5)

The dependence on the mass ratio Q has become small enough here, and we set Q ≈ 1 to simplify the expression, resulting in

$$\begin{array}{c}\hfill P\propto M_d^{-1/2}{R}_L^{3/2}.\end{array}$$(6)

With this equation, we determined the period boundary between Case A and Case B for each initial mass. We set it to the period where R_L = R_d, TAMS.

To derive the donor mass and radius at the relevant points of the donor evolution, we ran a simulation of the star with the code called modules for experiments in stellar astrophysics (MESA) (Paxton et al. 2018, and previous publications). We used the inlists provided by Klencki et al. (2020). These inlists were designed to simulate single stars from the beginning of the MS to the point in their evolution at which carbon in the core was exhausted. Even though we studied binaries, it is a safe assumption that the stars behave roughly like single stars until one of them fills their Roche lobe and mass transfer starts. Since we only studied the star up until the point at which it is possible for Case B mass transfer to take place, we only ran the simulations until the hydrogen in the core was exhausted (defined here as a hydrogen mass fraction in the core lower than 10⁻⁴).

We only made slight alterations to the inlists provided by Klencki et al. (2020). We changed the initial mass and set the metallicity to Z = Z_⊙. We first created pre-zero age main-sequence (ZAMS) models for each Case A and Case B initial primary mass using one set of inlists. Their outcome was used to simulate the stars from the ZAMS until the helium in their core was exhausted. This gave us the parameters we searched for to determine the boundaries for the period for Case A and Case B mass transfer.

However, because solar metallicity might not be the correct metallicity for the systems, we repeated this process, but with Z = 0.5 Z_⊙. We did not study these outcomes in detail, but we briefly discuss some effects of this change in metallicity in Section 5.

For more massive stars, the radius expands strongly at the end of the main sequence and obfuscates the boundary between Case A and Case B. In order to take this into account, we included the end of the main sequence in the calculations assuming Case A and Case B mass transfer.

2.3. Accretion efficiency and angular momentum loss

We sampled the parameter space of the initial periods and using 200 data points within the limits for the secondary initial masses and the initial periods, with an upper boundary of 10³ days for the period. In the minimum period, a ZAMS star would fill its Roche lobe. The values for both parameters were spaced out evenly, linear for the masses and logarithmic for the periods. For each of the initial period and mass estimates, we derived the accretion efficiency β and the loss of angular momentum in terms of the specific angular momentum loss per unit mass of the binary,

$$\begin{array}{c}\hfill \gamma =\left(\frac{\mathrm{\Delta }{J}_{\mathrm{lost}}}{\mathrm{\Delta }{M}_{\mathrm{lost}}}\right)\left(\frac{M}{J}\right).\end{array}$$(7)

An equation for β is fairly straightforward using Eq. (3). After multiplying both sides by (1 + βq) and rearranging the terms, we obtained an equation where it is only dependent on the initial and current masses of the system,

$$\begin{array}{c}\hfill \beta =(\frac{1+q}{1+q_i}-\frac{M_T}{M_{T,i}}){(q\frac{M_T}{M_{T,i}}-q_i\frac{1+q}{1+q_i})}^{-1}.\end{array}$$(8)

We also related the angular momentum loss γ to the periods and masses of the WR+O binary and its possible progenitors. To do this, we used two equations, Kepler’s third law to relate P/P_i, and M_T/M_T, i to the change in semi-major axis a/a_i. This change is related to γ under the assumption that γ is constant over time (Pols & Marinus 1994),

$$\begin{array}{c}\hfill \frac{a}{a_i}={\left(\frac{M_d}{M_{d,i}}\frac{M_a}{M_{a,i}}\right)}^{-2}{\left(\frac{M_T}{M_{T,i}}\right)}^{2\gamma +1}.\end{array}$$(9)

This was then equated to the relation for a/a_i from Kepler’s third law and was rewritten to give an equation for γ that relates the period and masses of a WR+O binary and its prior,

$$\begin{array}{cc}\hfill {\left(\frac{M_d}{M_{d,i}}\frac{M_a}{M_{a,i}}\right)}^{-2}{\left(\frac{M_T}{M_{T,i}}\right)}^{2\gamma +1}& ={\left(\frac{M_T}{M_{T,i}}\right)}^{1/3}{\left(\frac{P}{P_i}\right)}^{2/3}\hfill \end{array}$$(10)

$$\begin{array}{cc}\hfill {\left(\frac{M_T}{M_{T,i}}\right)}^{2\gamma +\frac{2}{3}}& ={\left(\frac{P}{P_i}\right)}^{2/3}{\left(\frac{M_d}{M_{d,i}}\frac{M_a}{M_{a,i}}\right)}^2,\hfill \end{array}$$(11)

so that

$$\begin{array}{c}\hfill \gamma ={log}_{M_T/M_{T,i}}\left({\left(\frac{P}{P_i}\right)}^{1/3}\frac{M_1{M}_2}{M_{1,i}{M}_{2,i}}\right)-\frac{1}{3}·\end{array}$$(12)

2.4. Limiting possible solutions

The size of the grid is quite large, so that it can be useful to limit the part of the grid in which we are interested. We therefore studied this in more detail. We first disregarded all points where γ < 0 since this would mean that the system had gained angular momentum in some way. We also used an upper limit of γ < 5 since values higher than this limit are probably nonphysical. This is clear from considering mass loss through a circumbinary ring, as in Soberman et al. (1997),

$$\begin{array}{c}\hfill \gamma =\frac{M_T^2}{M_d{M}_a}·{\frac{a_r}{a}}^{1/2}.\end{array}$$(13)

Here, a_r is the radius of the ring. For γ = 3 and M₁ = 20 M_⊙ and M₂ = 10 M_⊙, we would obtaine a radius of the ring a_r that is four times as large as the radius of the binary. At some point, the question is whether this is still a reasonable radius for a circumbinary disk and if is it still circumbinary, or just a disk. For γ = 5, this radius becomes even larger: a_r ≈ 11.1 ⋅ a. A circumbinary disk with such a radius is not physically plausible, and we therefore used this value for γ as an upper limit. We only consider points where 0 < γ < 5 and deemed high values (γ > 3) unlikely.

The critical q values further limit the parameter space. For q > q_crit, mass transfer becomes unstable. When this occurs, we do not expect the binary to have become a WR+O binary. We therefore also disregarded the areas where M_1, i/M_2, i > q_crit. However, the exact value is hard to determine because it depends on many variables, such as stellar radius and mass, but also on properties, such as efficiency and the case of the mass transfer. Ge et al. (2015) determined q_crit values for a range of different scenarios for conservative mass transfer. They reported that the critical q value for a 16 M_⊙ primary star ranges from 1.793 to 8.646. We do not expect the systems we studied to have undergone conservative mass transfer, and we therefore did not use this range for q_crit. This shows, however, that there might not be a single correct value for this critical q, but instead a range for Case A and Case B. We disregarded results that lay outside the strictest boundary, but we took the part into account that was in between this and the most forgiving boundary.

From the literature, the following ranges are satisfactory for either Case A or Case B mass transfer (Gallegos-Garcia et al. 2022; Klencki et al. 2021),

$$\begin{array}{cc}\hfill 1.6& \le q_{\mathrm{crit},\mathrm{A}}\le 3\hfill \\ \hfill 4& \le q_{\mathrm{crit},\mathrm{B}}\le 10\hfill \end{array}$$

Finally, detailed studies (e.g. Sen et al. 2022) reported that very close binaries lead to a merger of the two stars, regardless of their mass ratio. The limiting period is highly uncertain and depends on assumptions, but the lower range of Case A periods is very likely ruled out as well.

2.5. Most likely progenitors

We also took prior information into account about the most common properties of binary systems similar to those of the WR+O star progenitors. The properties we considered are the mass ratio and the period. To obtain this information, we used the COSMIC code by Breivik et al. (2022) and the results of Moe & Di Stefano (2017) to create a sample of O+MS binaries with a size of 10⁵ and M_1, min = 16. This was to ensure that the primary would be an O star and that the sample was large enough. The random seed of the sample was 20. Since these types of binaries are expected to be the progenitors of the WR+O binaries we studied, we used this distribution to determine the likelihood of the different scenarios we might find for the progenitor evolution, such as the initial secondary mass. To do this, we studied the distribution of the O+MS mass ratios and periods. In Figure 1 we show a 2D histogram of the period and mass ratio of the sample. In the sample, we used a range for the primary mass (16 M_⊙ < M₁ < 150 M_⊙) such that the initial primary star was an O star and q > 1 because the O star was to be the donor and therefore had the higher initial stellar mass.

Fig. 1. Period and mass ratio distribution for a sample of 105 O+MS systems where 16 M⊙ < M1 < 150 M⊙ and q > 1, based on the distributions found by Moe & Di Stefano (2017). The colour bar gives the number of systems per cell.

The distribution shows that the closer to q = 1, the more binaries, in particular for the systems with periods shorter than several dozen days. For periods, there is a clear preference for lower periods of log P ≲ 4, and slightly so for log P ≲ 2. Following these results, we therefore expect the progenitors of the WR+O binaries to most likely have a mass ratio close to one and an orbital period that is not too high. These are the only possible values for the progenitors, but shows which values are more likely than others to have occurred. We used this result to determine the likelihood of certain scenarios we might find in our results. Scenarios with lower q are more likely based on this O+MS distribution.

Finally, we determined for this distribution which systems we expect to undergo Case A mass transfer and for which systems we do not expect this based on our calculated period boundary between Case A and B. About one-third of the O+MS systems (36,44%) is expected to undergo Case A mass transfer, and the initial periods of the other two-thirds are too long for them to undergo Case A mass transfer. If all O+MS binaries have an equal likelihood to become a WR+O binary and the observed WR+O binaries we studied are not too severely biased, we would expect most of them not to have undergone Case A mass transfer.

3. Objects

The systems we studied were all taken from the VIIth catalogue of Galactic Wolf-Rayet stars by van der Hucht (2001). This catalogue contains all the WR stars that are in the Milky Way. This iteration contains a total of 226 WR stars. We filtered the WR stars in the catalogue using the following criteria:

• The WR star is in a binary.

• Its companion is an O star.

• The orbital period of the system is known.

• The primary and secondary mass are known.

In addition to the WR star being in a binary, its companion was to be an O star since these systems are likely progenitors of double black hole binaries. To be able to usefully determine the remaining WR+O systems that are in the catalogue, some additional information was required: the masses of the two stars and their orbital period. However, the catalogue does not contain information on the stellar masses for all binaries. We therefore had to obtain these masses from other sources, and these are indicated in the table.

This resulted in a sample of 24 WR+O binaries in the Galaxy. Initially, we worked with this sample. However, during the simulations of the WR progenitor stars, the masses of some of the progenitors were found to be too high to do this properly. These were the progenitors of the three systems with M_WR ≳ 45 M_⊙, WR22, WR47, and WR141. The other 21 had M_WR ≲ 25 M_⊙. The high WR masses of the three systems resulted in initial primary masses of M_1, i ≳ 100 M_⊙. These initial masses proved to be too high to subsequently run proper simulations of the donor star from ZAMS to TAMS. Therefore, we discarded these three systems from our selection. WR35a was found to be a binary after the latest version of the van der Hucht (2001) by Gamen et al. (2014).

An overview of the final sample consisting of 21 WR+O binaries and some of their properties is listed in Table 1. When no source is specified, the data were taken from van der Hucht (2001). The mass ratio was determined using the masses, and the Roche-lobe radius of the WR star R_L was determined using Eq. (5).

4. Results

We present the results of our study in figures for each object. As an example, we use WR9, which is shown in Figure 2. The main graph is divided into two parts, in which the upper half shows Case B and the lower half shows Case A. The initial mass of the WR star as estimated is given in a box for each of the two cases. The x-axis at the bottom shows initial secondary masses covering the allowed ranges (see Sect. 2.1) for both of the cases. The y-axis shows the range of initial periods. The colour gradient within the parts for Case A and B displays the γ values we found for each point in the grid of initial secondary masses and periods for Case A and Case B, and the colour bar on the side shows the legend. Every integer value of γ is marked with a black line for clarity. Values of γ below zero (white) and above 5 (black) were ruled out. Some parts of the grid appear to be more shaded out than others. These shaded areas indicate the critical q values. The parts without shading always experience stable mass transfer based on the q_crit range used. The partly shaded out parts could have experienced either stable or unstable mass transfer, depending on the exact value for q_crit. The parts that are shaded out heavily always experience unstable mass transfer based on the critical q values. For the part of the graph that displays the grid for Case B, we show the lower limit of P_i for Case B mass transfer to occur as the dashed line. In the part below the dashed line, the shaded areas are different than for the rest of the graph. This represents late Case A mass transfer that may behave more like Case B due to the composition of the donor. It is unclear whether the critical q values would then also be those of Case B. Therefore, we (also) show the critical q values of Case A. These do not correspond to the same possible values for the initial secondary mass M_2, i of the lower, Case A, part of the graph because the initial mass of the primary M_i is different, and therefore, the mass ratio q is also different. Lastly, the smaller graph at the bottom displays the mass-transfer efficiency β for both cases of mass transfer, depending only on the initial secondary mass and not on the initial period. This graph uses the same values for the x-axis as the other two graphs. The y-axis displays the value for β for a certain mass ratio q.

Fig. 2. Resulting values for γ (colour) and β (lines in lowest graph) for the initial parameter space of WR9. Each integer value of γ is marked with a solid line. The shaded boxes indicate whether mass transfer will be stable. Parts without shading will be stable, lightly shaded parts can be either stable or unstable, depending on the exact value of qcrit The heaviest shaded parts will always experience unstable mass transfer. The dashed line indicates the period boundary for Case B. Dark grey shades indicate excluded mass ranges (see text).

For each observed system, we determined which of the two mass-transfer types is possible and the possible and likely values of β and γ. We present the results according to the possibilities.

4.1. Mass-transfer type

4.1.1. Case A is more likely

For many systems, we conclude that Case A is the most likely, and this is the largest part of our sample. Fourteen of the 20 systems we studied fall into this category based on our results: WR9, WR21, WR30, WR31, WR42, WR48, WR62a, WR68a, WR79, WR127, WR139, WR151, WR153, and WR155.

As an example, we consider WR9 (Figure 2). The results for the other systems are provided in the Appendix A. For this system, Case B is not fully excluded, but to have experienced Case B, it would take on extremely high values (γ ≳ 4) which while not impossible are not plausible. A similar conclusion holds for the other systems in this category. Instead, the Case A results are possible in the sense that values of γ in the expected range are found. For the late Case A results (top panel below the dashed line), solutions with plausible γ values are possible for many of the systems, while for others, the γ values are rather high again (γ ≳ 3)

Some limits are imposed on all systems by the critical q values concerning which progenitors would have undergone stable Case A mass transfer. The degree to which these limits affect the possibilities differ for each system, but all systems have options towards the upper limit of the secondary mass, depending on which value for the critical q is assumed. However, for about half of these systems stable Case A evolution is only possible if the higher critical q value of 3 is true. These systems, WR31, WR42, WR48, WR62a, WR68a, WR151, and WR155, have the most different initial mass ratios, and all have rather short periods (but they overlap with the other systems). The progenitor mass we used for Case A is an upper limit. Lower progenitor masses increase the initial mass ratio and thus allow for more stable mass transfer. The expected progenitor distribution peaks towards shorter periods and more equal masses and places the a priori most likely progenitors also in the region where stable Case A mass transfer is possible. In these scenarios, the mass-transfer efficiency most likely would have been very low. We discuss the mass-transfer efficiency for all systems in more detail in Section 4.2. We repeat here that towards the bottom of the Case A box, the systems likely merge (see Section 2.4).

In summary, based on the values for γ, we exclude Case B mass transfer as a plausible scenario for these 14 systems. Case A mass transfer with a low mass-transfer efficiency seems to be the most likely scenario for these binaries based on our results.

4.1.2. Case A and Case B are both possible

For some systems in our sample, we were unable to determine the most likely cases of mass transfer because Case A and Case B both seem possible. Three systems in our sample are unclear: WR11, WR113, and WR133. WR35a is also marginally in this class because a small range of Case A is allowed. However, Case B seems more likely for this system (see below). We examine the plot WR11 as an example (Figure 3). The plots containing the other results are shown in the Appendix A.

Fig. 3. Same as Figure 2, but for WR11.

For these three systems, for Case A and Case B there are many possibilities in which the mass transfer would have been stable, regardless of which value for q_crit is assumed. While this poses some limits for Case A, it still leaves enough possibilities. The values for γ are also within reason (≲3) for Case A and large parts of the Case B grid. Finally, one case of mass transfer might be found to be more likely than the other based on a comparison of the results to the progenitor distribution. This distribution shows that low initial mass ratios are more likely with initial periods that are also short, although for the Case B region, the mass ratios are more equally distributed (see Fig. 1). While this would shift the preference to a lower β for Case A, it does not impose any other limits for these three systems that would make either Case A or Case B very unlikely because even for low q and P, there are still plausible scenarios for both cases of mass transfer to be found.

Our results have no further options to rule out any of the two cases of mass transfer. Therefore, these systems might have undergone either Case A or Case B mass transfer based on our results.

4.1.3. Case A is not possible

Instead of Case A being more likely, we also have three systems where based on our results we deem Case B mass transfer to be the most likely scenario. These systems are WR97, WR35a, and WR140.

We followed the same procedure for γ and q_crit. For WR97 (Figure 4), all possible Case A progenitors are predicted to undergo unstable mass transfer even by the most least restraining q_crit value, and therefore, we excluded them as a possibility. For Case B, some limitations are also set by the critical q values for this system, but in a part, mass transfer is still certainly stable with reasonable γ values. WR97 is peculiar in the sense that the masses of the WR and the O star are quite low.

Fig. 4. Same as Figure 2, but for WR97.

For WR35a (see the Appendix A), there is a small range of possibilities for Case A, but Case B is more likely. Due to the high current mass ratio, only low values of β are possible.

For WR140 (Figure 5), there are also options for Case B, but the situation is different. For Case A, there are no solutions with positive γ. This is a consequence of the very wide period of the system, which means that it could even have been Case C or might have avoided mass transfer altogether.

Fig. 5. Same as Figure 2, but for WR140.

Finally, for WR 137 (Figure 6), neither Case A nor Case B are likely scenarios. There are no solutions because the period of the system is very wide. As in the case of WR 140, mass transfer may have been avoided.

Fig. 6. Same as Figure 2, but for WR137.

4.2. Mass-transfer efficiency and angular momentum loss

The possible mass-transfer efficiencies of a system can be determined by using the q_crit values for each case of mass transfer as an upper limit. This was done by determining the expected β for each system for the two cases of mass transfer as a function of the initial secondary mass. This was then combined with the critical q values for Case A and Case B to determine an upper boundary for each system. However, because there is a range of values for q_crit instead of one value, a range for β_max followed. For the lower and upper boundary of q_crit, we found the matching mass-transfer efficiency and used this as an upper limit. For Case A, this corresponds to 1.6 < q_crit, A < 3 and for Case B to 4 < q_crit, B < 10. An overview of the matching upper limits for β for each binary is given in Table 2. We only list the limits for plausible cases of mass transfer. More detail of this determination was described in previous sections.

For Case A, we expected to find a higher mass-transfer efficiency with values ranging from ∼0.1 − 0.7, while for Case B, we expected to find lower values of ≲0.25 for the efficiency (e.g. Sen et al. 2022; Shao & Li 2016; de Mink et al. 2007). For the Case A results, the efficiency is a combination of a low efficiency of the fast mass-transfer phase and a higher efficiency in the slow phase (see Sen et al. 2022). However, the values we found for β_max for the systems show something different. Firstly, the upper limits we found for Case B are often very high (≳0.9), regardless of which value for q_crit was taken. This seems to contradict the expectations we had of finding β ≲ 0.25 for Case B for one of the systems where we expected Case B to be the most likely scenario, WR140. However, a high maximum value does not automatically have to mean that this is also the efficiency that the mass transfer had. For WR97 and WR35a, we found low maximum values of β.

For Case A, we find very different values for β_max. The majority of the systems we studied has β_max < 0.5 even for the least restraining value of q_crit. The strictest boundary leaves no systems with β_max > 0.5 and roughly half of the systems with β_max ∼ 0, which would mean that Case A mass transfer is not the most likely scenario for these systems, combined with these critical q values. These results also disagree with the expectations we had, just as for Case B. We expected values of β_max ∼ 0.1 − 0.7. While these possibilities again are not excluded by our results, a lower maximum seems the more likely scenario based on our results. This is supported even more strongly when we again take the progenitor distribution and the higher likelihood of lower q into account, and with this, lower values for β.

Therefore, we find that at least for Case A mass transfer, the mass-transfer efficiency was most likely low due to the progenitor distributions. This is even more so because of the limitations posed by q_crit.

The most likely values for the parameter γ are often fixed to a certain value, or it is assumed that the mass that is lost takes away the angular momentum of the accretor. For binaries with initial mass ratios not far from unity, the latter implies values of γ around one. The transition from γ from below to above unity is visible in most of the figures. It implies that values around unity are indeed possible. However, the low γ values typically come from initially short periods that may very well lead to mergers, suggesting that higher γ values above one are more likely. The figures in the appendix show that some systems (WR31, WR35a if it was Case B, WR139, WR79, WR151, WR153, and WR155) only have solutions of γ above one. Our initial masses of the WR progenitors are upper limits (see Sect. 2.3), and lower initial masses lead to somewhat lower values of γ (Eq. (12)).

5. Discussion

We discuss the assumptions we made and their possible influence on the results and conclusions. Firstly, we assumed that the companion star did not or could not have lost mass with respect to its initial mass. However, this is not entirely correct. Stars can lose mass through stellar winds. While the amount of mass that is lost in this way is not expected to be high, it does mean that it is possible for the secondary star to slightly decrease in mass with respect to its initial mass M_2, i.

Another caveat that has to do with the mass is the way in which the masses of the WR progenitors were determined for Case A and Case B mass transfer. We used a linear relation for each, as explained in Section 2. These relations are an approximation. While the initial progenitor masses are close to the outcomes of these equations, they might also be slightly lower or higher than the masses we used for our results. However, by including part of the Case A parameter range in the top panels of the plots, we cover a significant fraction of this uncertainty. Another factor for this uncertainty is that some of the WR and O-star masses are uncertain, as we showed in Section 3. We did not systematically vary the masses, but the effect of the changes in the mass is as follows: Lower WR masses move the curves down in the figures, that is, they increase the values of γ. Increasing the O-star masses has the same effect. We therefore do not expect a specific bias in our results, except for the one mentioned above, which is due to our Case A progenitor masses being upper limits.

We assumed solar metallicity for the systems that we studied, although we are not fully certain that this is accurate. We therefore also examined the effect of a lower metallicity on the most likely scenario for a system. The outcomes for β and γ for a certain combination of initial parameters are not altered because they do not depend on metallicity. However, the period boundaries are dependent on the radius of the star at certain points in its evolution, which does change with metallicity. The resulting core masses also depend on the metallicity.

We ran the simulations to determine these boundaries again, but for Z = 0.5 Z_⊙. For these simulations, the radius of the star at the TAMS is smaller for a lower Z. This radius is indicated with the dot for each track. Because of this, the values for P_i also become lower.

This difference in evolution for our results is shown in a new figure for WR11 to illustrate the effect (Figure 7). There are less possible scenarios for Cases A and B. Maybe more interestingly, for Case B lower and more plausible values for γ become possible within these new boundaries, whereas the possibilities for the initial period decrease drastically for Case B. At the same time, the conclusion that γ > 1 most likely changes as there now is an upper limit on γ just above one. Based on these results, a different metallicity might therefore influence the likelihood of the four scenarios we studied and our conclusion on the most likely value of γ. It might be interesting to study this in more detail in future research.

Fig. 7. Same as Figure 3, but for a metallicity of Z = 0.5 Z⊙.

Lastly, there is also the uncertainty on the exact value for the critical q above which mass transfer becomes unstable. We used different values from the literature and all values in between as a possible range, but in reality, there is a single value of q_crit for each system that is able to determine whether the mass transfer will be unstable. Since it requires much information to accurately determine this, as reported for example by Ge et al. (2015), it was not possible for us to do so as we did not have all of the required information. Therefore, these regions provide an uncertainty on which progenitors may be possible for stable mass transfer for Case A and Case B mass transfer. We already mentioned this briefly in the results, but it is important recall this when interpreting the results.

6. Conclusion

The goal of this research was to study the type of mass transfer that WR+O binaries might have experienced in the past and determine the mass-transfer efficiency. To do this, we used the knowledge that their progenitors are O+MS binaries and that these binaries are most likely to have a low mass ratio q. Based on this and the other possible restrictions discussed in Section 2, we studied 21 WR+O binaries and their possible cases of mass transfer, the efficiency β, and the angular momentum loss.

To determine which case of mass transfer the WR+O binaries might have experienced, we distinguished between three different scenarios: most likely Case A, both Case A and Case B plausible, and Case A not plausible. The first scenario was the largest group in our sample. Fourteen out of 21 systems most likely experienced Case A mass transfer based on our results. The other two groups contain far fewer systems.

For the systems experiencing Case A (i.e. the majority of cases), we find the mass-transfer efficiency to have been low and no likely fixed value (as found by de Mink et al. 2007). For Case B, values up until one are possible for some systems, while for Case A, the highest possible value we found to be possible is β_max = 0.84.

This is not what we expected from analysing the distribution of O+MS binaries, as we only expected that about one-third of these systems undergo Case A mass transfer. Our sample may not be complete enough to include progenitors that are representative of O+MS binaries, but it may suggest that the products of Case B mass transfer do not contribute in the same way to the observed WR+O star population as post-case-A systems. Post-case-B binaries tend to have longer periods, which are harder to measure spectroscopically. However, some objects in our sample have long periods, so that the selection effect can be subtle. One effect that might hide post- Case-B systems may be that the stripped stars remain cool for longer (Dutta & Klencki 2024). Based on our results, we cannot conclude which of these phenomena is the cause of this discrepancy or whether it is caused by selection effects.

With these results, we conclude that the observed WR+O binaries are most likely to experience Case A mass transfer, but Case B is also a possibility for some. This is not representative of which case of mass transfer we expect their progenitors (O+MS binaries) to undergo.

The values of γ cannot be determined accurately, but values above one are implied for most cases. However, a lower metallicity would lower the estimates of γ.

Acknowledgments

We thank the referee for detailed comments that improved the paper. We would like to thank Alina Istrate for her help and advice on using MESA and for her advice in general. We also would like to thank Jakub Klencki for helpfully answering some of our questions. G.N. is supported by the Dutch science foundation NWO.

References

Appendix A: Additional system figures

In this section additional figures for the systems not discussed in section 4.1 can be found.

Fig. A.1. The same as figure 2 but for WR21.

Fig. A.2. The same as figure 2 but for WR30.

Fig. A.3. The same as figure 2 but for WR31.

Fig. A.4. The same as figure 2 but for WR35a.

Fig. A.5. The same as figure 2 but for WR42.

Fig. A.6. The same as figure 2 but for WR48.

Fig. A.7. The same as figure 2 but for WR62a.

Fig. A.8. The same as figure 2 but for WR68a.

Fig. A.9. The same as figure 2 but for WR79.

Fig. A.10. The same as figure 2 but for WR127.

Fig. A.11. The same as figure 2 but for WR139.

Fig. A.12. The same as figure 2 but for WR153.

Fig. A.13. The same as figure 2 but for WR113.

Fig. A.14. The same as figure 2 but for WR133.

Fig. A.15. The same as 2 but for WR151.

Fig. A.16. The same as 2 but for WR155.

All Tables

All Figures

	Fig. 1. Period and mass ratio distribution for a sample of 105 O+MS systems where 16 M⊙ < M1 < 150 M⊙ and q > 1, based on the distributions found by Moe & Di Stefano (2017). The colour bar gives the number of systems per cell.
In the text

Fig. 2. Resulting values for γ (colour) and β (lines in lowest graph) for the initial parameter space of WR9. Each integer value of γ is marked with a solid line. The shaded boxes indicate whether mass transfer will be stable. Parts without shading will be stable, lightly shaded parts can be either stable or unstable, depending on the exact value of qcrit The heaviest shaded parts will always experience unstable mass transfer. The dashed line indicates the period boundary for Case B. Dark grey shades indicate excluded mass ranges (see text).

In the text

	Fig. 3. Same as Figure 2, but for WR11.
In the text

	Fig. 4. Same as Figure 2, but for WR97.
In the text

	Fig. 5. Same as Figure 2, but for WR140.
In the text

	Fig. 6. Same as Figure 2, but for WR137.
In the text

	Fig. 7. Same as Figure 3, but for a metallicity of Z = 0.5 Z⊙.
In the text

	Fig. A.1. The same as figure 2 but for WR21.
In the text

	Fig. A.2. The same as figure 2 but for WR30.
In the text

	Fig. A.3. The same as figure 2 but for WR31.
In the text

	Fig. A.4. The same as figure 2 but for WR35a.
In the text

	Fig. A.5. The same as figure 2 but for WR42.
In the text

	Fig. A.6. The same as figure 2 but for WR48.
In the text

	Fig. A.7. The same as figure 2 but for WR62a.
In the text

	Fig. A.8. The same as figure 2 but for WR68a.
In the text

	Fig. A.9. The same as figure 2 but for WR79.
In the text

	Fig. A.10. The same as figure 2 but for WR127.
In the text

	Fig. A.11. The same as figure 2 but for WR139.
In the text

	Fig. A.12. The same as figure 2 but for WR153.
In the text

	Fig. A.13. The same as figure 2 but for WR113.
In the text

	Fig. A.14. The same as figure 2 but for WR133.
In the text

	Fig. A.15. The same as 2 but for WR151.
In the text

	Fig. A.16. The same as 2 but for WR155.
In the text