diff --git "a/1NAyT4oBgHgl3EQf1PnY/content/tmp_files/2301.00733v1.pdf.txt" "b/1NAyT4oBgHgl3EQf1PnY/content/tmp_files/2301.00733v1.pdf.txt" new file mode 100644--- /dev/null +++ "b/1NAyT4oBgHgl3EQf1PnY/content/tmp_files/2301.00733v1.pdf.txt" @@ -0,0 +1,2885 @@ +Astronomy & Astrophysics manuscript no. TEOs_in_massive_EEVs +©ESO 2023 +January 3, 2023 +Theoretical investigation of the occurrence of tidally excited +oscillations in massive eccentric binary systems +P. A. Kołaczek-Szyma´nski and T. Ró˙za´nski +Astronomical Institute, University of Wrocław, Kopernika 11, 51-622 Wrocław, Poland +e-mail: kolaczek@astro.uni.wroc.pl +Received 16.10.2022; Revised 01.12.2022; Re-revised 28.12.2022; Accepted 02.01.2023 +ABSTRACT +Context. Massive and intermediate-mass stars reside in binary systems much more frequently than low-mass stars. At the same +time, binaries containing massive main-sequence (MS) component(s) are often characterised by eccentric orbits, and can therefore be +observed as eccentric ellipsoidal variables (EEVs). The orbital phase-dependent tidal potential acting on the components of EEV can +induce tidally excited oscillations (TEOs), which can affect the evolution of the binary system. +Aims. We investigate how the history of resonances between the eigenmode spectra of the EEV components and the tidal forcing +frequencies depends on the initial parameters of the system, limiting our study to MS. Each resonance is a potential source of TEO. +We are particularly interested in the total number of resonances, their average rate of occurrence and their distribution in time. +Methods. We synthesised 20,000 evolutionary models of the EEVs across the MS using Modules for Experiments in Stellar Astro- +physics (MESA) software for stellar structure and evolution. We considered a range of masses of the primary component from 5 to +30 M⊙. Later, using the GYRE stellar non-adiabatic oscillations code, we calculated the eigenfrequencies for each model recorded by +MESA. We focused only on the l = 2, m = 0, +2 modes, which are suspected of being dominant TEOs. Knowing the temporal changes +in the orbital parameters of simulated EEVs and the changes of the eigenfrequency spectra for both components, we were able to +determine so-called ‘resonance curves’, which describe the overall chance of a resonance occurring and therefore of a TEO occurring. +We analysed the resonance curves by constructing basic statistics for them and analysing their morphology using machine learning +methods, including the Uniform Manifold Approximation and Projection (UMAP) tool. +Results. The EEV resonance curves from our sample are characterised by striking diversity, including the occurrence of exceptionally +long resonances or the absence of resonances for long evolutionary times. We found that the total number of resonances encountered +by components in the MS phase ranges from ∼102 to ∼103, mostly depending on the initial eccentricity. We also noticed that the +average rate of resonances is about an order of magnitude higher (∼102 Myr−1) for the most massive components in the assumed +range than for EEVs with intermediate-mass stars (∼101 Myr−1). The distribution of resonances over time is strongly inhomogeneous +and its shape depends mainly on whether the system is able to circularise its orbit before the primary component reaches the terminal- +age MS (TAMS). Both components may be subject to increased resonance rates as they approach the TAMS. Thanks to the low- +dimensional UMAP embeddings performed for the resonance curves, we argue that their morphology changes smoothly across the +resulting manifold for different initial EEV conditions. The structure of the embeddings allowed us to explore the whole space of +resonance curves in terms of their morphology and to isolate some extreme cases. +Conclusions. Resonances between tidal forcing frequencies and stellar eigenfrequencies cannot be considered rare events for EEVs +with massive and intermediate-mass MS stars. On average, we should observe TEOs more frequently in EEVs containing massive +components than intermediate-mass ones. TEOs will be particularly well-pronounced for EEVs with the component(s) close to the +TAMS, which begs for observational verification. Given the total number of resonances and their rates, TEOs may play an important +role in the transport of angular momentum within massive and intermediate-mass stars (mainly near TAMS). +Key words. binaries: close – stars: early-type – stars: massive – stars: oscillations – stars: evolution – methods: numerical +1. Introduction +For many reasons, massive stars (≳ 8 M⊙) are of particular in- +terest to modern astrophysics. Primarily, they are progenitors of +core-collapse supernovae (e.g., Janka et al. 2007; Smartt 2009) +and long γ-ray bursts (e.g., Fruchter et al. 2006; Yoon et al. +2006). For billions of years, they contributed to the chemical +evolution of the entire Universe and interacted mechanically +with the surrounding interstellar medium (e.g., Ouellette et al. +2007; Svirski et al. 2012), also through their intense line-driven +stellar winds (Vink 2021). Furthermore, most of their remnants +are compact objects, such as neutron stars (NSs, and among them +magnetars and pulsars) and black holes (BHs), which allow the +empirical study of effects of general relativity. Finally, massive +stars can be observed at cosmological distances due to their enor- +mous luminosities, hence they dominate in the spectra of distant +starburst galaxies (see Eldridge & Stanway 2022, for a recent re- +view). These features of massive stars (and many others) demon- +strate that understanding the structure and evolution of massive +stars is one of the key tasks of astronomy. +As is well known, massive and intermediate-mass (≳ 2 M⊙ +and ≲ 8 M⊙) stars reside in binary systems much more fre- +quently than their lower-mass counterparts (Duchêne & Kraus +2013; Sana et al. 2012). Moreover, as shown, for example, by +Sana et al. (2014) and Moe & Di Stefano (2017), O-type dwarfs +in particular are often found in multiple systems. This shows that +binarity is inherent in the evolution of massive stars and cannot +be ignored when studying these objects as well as their final out- +comes. Many interesting phenomena in the Universe are the re- +Article number, page 1 of 24 +arXiv:2301.00733v1 [astro-ph.SR] 2 Jan 2023 + +A&A proofs: manuscript no. TEOs_in_massive_EEVs +sult of binarity among massive and/or intermediate-mass stars. +These include Be stars (Kriz & Harmanec 1975; Bodensteiner +et al. 2020), so-called ‘stripped stars’ (Götberg et al. 2020; +Shenar et al. 2020; El-Badry & Burdge 2022), BH-BH/BH- +NS/NS-NS mergers (progenitors of gravitational-wave events, +Abbott et al. 2016, 2019), ‘early’ stellar mergers (Tokovinin & +Moe 2020; Sen et al. 2022; Li et al. 2022), Ib/c supernova pro- +genitors (Langer 2012; Yoon et al. 2012), ‘massive Algols’ (de +Mink et al. 2007; Skowron et al. 2017; Sen et al. 2022), and even +Wolf-Rayet stars (Shenar et al. 2019; Pauli et al. 2022). +At the same time, binary systems that contain massive main- +sequence (MS) component(s) are often characterised by eccen- +tric orbits due to their relatively young age and the presence of +radiative outer layers, which are less vulnerable to tidal dissi- +pation compared to convective envelopes (e.g., Van Eylen et al. +2016). Both observational studies of large samples of massive bi- +naries (e.g., Moe & Di Stefano 2017) and hydrodynamical sim- +ulations of their formation (see Oliva & Kuiper 2020, and ref- +erences therein) suggest significantly non-zero eccentricities at +their birth. +Assuming that the periastron distance between the compo- +nents is sufficiently small1, the combined proximity effects, such +as ellipsoidal distortion, irradiation/reflection effect and Doppler +beaming/boosting, make such a system an eccentric ellipsoidal +variable (hereafter EEV, e.g., Nicholls & Wood 2012). Due to +the characteristic shape of the light curve of EEV during the +periastron passage (which can resemble an electrocardiogram), +EEVs are sometimes referred to as ‘heartbeat stars’ (Welsh et al. +2011; Thompson et al. 2012; Beck et al. 2014; Kirk et al. 2016; +Kołaczek-Szyma´nski et al. 2021; Wrona et al. 2022b). +The orbital phase-dependent tidal potential acting on the +components of EEV can induce tidally excited oscillations +(TEOs) in their interiors (Zahn 1975; Kumar et al. 1995; Fuller +2017; Guo 2021), which in turn can affect the evolution of the +binary system. However, many details of TEOs in massive and +intermediate-mass stars are still poorly understood including the +total number of TEOs and their frequency of occurrence. In our +study, we aim to shed light on this issue based on theoretical +modelling combined with machine learning (ML) techniques. +The paper is organised as follows. Section 2 provides a con- +cise characterisation of TEOs and specifies the purpose of our +paper. In Sect. 3 we present a detailed description of the adopted +methodology, including the assumptions made and the software +used to generate the theoretical models. We then analyse the +obtained models and present our findings in Sect. 4. Finally, +we summarise the entire work and draw several conclusions in +Sect. 5. +2. Properties of TEOs and the purpose of the paper +TEOs are tidally forced eigenmodes of a star with frequencies, +σnlm (in the co-rotating frame of the star), coinciding with inte- +ger multiples, N, of the orbital frequency, forb2. The resonance +condition can be written as follows: +fNm ≡ N forb − m fs ≈ σnlm, +(1) +where fNm corresponds to the frequency of the tidal forcing in +the rotating frame, fs stands for the rotational frequency of the +star, while the subscripts n, l and m denote the radial order, de- +gree, and azimuthal order of the specific eigenmode, respec- +tively. This property of TEOs makes them relatively easy to dis- +1 That is, of the order of a few radii of the larger component. +2 We denote the corresponding orbital period as Porb = 1/forb. +tinguish from other types of pulsations (e.g., self-excited oscilla- +tions) in frequency spectra, provided the orbital period is known. +There are numerous examples of photometrically or spectro- +scopically detected TEOs (e.g., Handler et al. 2002; Welsh et al. +2011; Hambleton et al. 2013; Fuller et al. 2017; Guo et al. 2019, +2020; Wrona et al. 2022a), also in massive binary systems (e.g., +Willems & Aerts 2002; Pablo et al. 2017; Kołaczek-Szyma´nski +et al. 2021, 2022). Most TEOs are damped normal modes, mean- +ing that without constant tidal forcing they would not be ob- +served in the star. More importantly, because of their damped na- +ture, TEOs dissipate the total orbital energy making the system +tighter, more circular, and synchronised with time. On the other +hand, if the TEO is naturally an overstable mode it can transfer +thermal energy from the star to the orbit via so-called ‘inverse +tides’ (Fuller 2021). Regardless of the type of TEOs, they unde- +niably contribute to the evolution of the (massive) binary system, +and can therefore influence the characteristics of the phenomena +and objects mentioned above. The efficiency of energy transfer +between the stellar interior and the orbit due to TEOs strongly +depends on the temporal behaviour of the resonance condition +given by Eq. (1). It is to be expected that most TEOs are ‘chance +resonances’, i.e. resonances in which the aforementioned condi- +tion is satisfied for a relatively short time. Under such circum- +stances, TEOs do not have enough time to reach high ampli- +tudes, hence their ability to dissipate orbital energy is somewhat +limited. However, if, after reaching resonance, both frequencies +on the left and right sides of Eq. (1) evolve at the same rate and +direction, TEOs can ‘tidally lock’ for a longer time compared to +the chance resonance scenario (Fuller 2017). This unique vari- +ety of TEOs is suspected to be responsible for occasional peri- +ods of rapid evolution of the orbital parameters in binary systems +(Fuller et al. 2017). +TEOs are are not only restricted to MS stars, they can also +occur in binaries with pre-MS stars (Zanazzi & Wu 2021), some +compact objects (white dwarfs, Yu et al. 2021), planetary sys- +tems (Ma & Fuller 2021) and even planet-moon systems (e.g., in +the Saturn-Titan system, Lainey et al. 2020). +Although the literature on theoretical studies of TEOs is in- +deed extensive (see e.g., Fuller 2017; Guo 2021, for recent re- +views), the question of their rate of occurrence and the role they +play in the evolution of massive stars is still a matter of debate. +Unfortunately, only a small number of papers refer exclusively to +massive EEVs. Witte & Savonije (1999a,b) studied gravity- (g) +and Rossby-mode TEOs in an uniformly rotating 10 M⊙ MS star +using their own two-dimensional (2D) code for different stellar +rotation rates and several orbital configurations. They found that +dynamical tides can effectively circularise and tighten the orbits +of EEVs in just a few Myrs if resonance locking occurs. How- +ever, these and many other previous works on TEOs were done +under the assumption of a compact (point-like) secondary com- +panion that is not subject to tidal perturbations during each peri- +astron passage. This is obviously not the case in real binary sys- +tems, where both components are responsible for the tidal evo- +lution of the orbit. As theoretically shown by Witte & Savonije +(2001), for an eccentric binary system consisting of two 10 M⊙ +stars, tidal dissipation can be further enhanced due to the simul- +taneous excitation of tidally-locked TEOs in both components. +In spite of the advanced mathematical formalism, the aforemen- +tioned papers only dealt with a few assumed component masses +and sets of orbital parameters. Only Willems (2003) attempted +a qualitative analysis of the hyperspace of the orbital parame- +ters favouring excitation of TEOs in massive EEVs on MS. He +found that for a mass range of 2 – 20 M⊙, the favourable orbital +period interval lies between ∼5 and ∼12 d when both compo- +Article number, page 2 of 24 + +Kołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs +nents are zero-age MS (ZAMS) stars. This interval shifts towards +longer orbital periods (up to ∼70 d) for components approaching +terminal-age MS (TAMS). +Although, as argued above, the role of TEOs in the life of +massive binary systems is still not well understood, we are not +aware of any published work that develops the qualitative analy- +sis carried out by Willems (2003) based on state-of-the-art stel- +lar evolution and oscillations codes. We would like to fill this +gap by combining the Modules for Experiments in Stellar As- +trophysics3 (MESA, Paxton et al. 2011, 2013, 2015, 2018, 2019) +stellar structure and evolution code with the GYRE4 (Townsend & +Teitler 2013; Townsend et al. 2018) non-adiabatic stellar oscil- +lation code. Our study aims to answer the following three ques- +tions: +1. How many resonances (given by Eq. (1)) can EEVs experi- +ence during their lifetime between ZAMS and TAMS? How +does this picture change with different initial parameters of +the binary system? +2. Can we distinguish several distinct types of EEV resonance +histories that are statistically related to the initial physical +and orbital parameters of binary systems? +3. Does the resonance history correlate in any way with the +properties of EEV near TAMS? For instance, are systems +that undergo mass transfer before reaching TAMS also sys- +tems with a higher total number of resonances encountered? +In order to answer the last two questions, we use of ML tech- +niques by performing a Uniform Manifold Approximation and +Projection5 (UMAP, McInnes et al. 2018) dimension reduction +analysis of the resonance histories obtained for simulated binary +systems. +3. Methods +Assuming that the dynamical tide excited in the component is +dominated by a single TEO close to resonance with the orbital +harmonic N, one can express its amplitude of luminosity change +as proportional to (after Fuller 2017, his eq. 2) +AN ∝ q +�R +a +�l+1 +|Qnlm|FNmLN, +(2) +where q is the ratio of the masses of the two components, R +stands for the radius of the component in which the TEO is +excited, and a is the semi-major axis of the relative orbit. The +quantity denoted Qnlm is known as the so-called overlap integral, +which describes the spatial coupling between a given oscillation +mode and the actual geometry of the tidal potential (Fuller 2017, +his eq. 4). In general, the larger the value of |n|6, the smaller the +Qnlm, hence eigenmodes with a large number of nodes in the ra- +dial direction have a much lower probability of tidal excitation7. +In addition to Qnlm, the Hansen coefficient FNm (Fuller 2017, +his eq. 5) is responsible for the temporal coupling of the forced +normal mode and the Nth component in the Fourier expansion +3 https://docs.mesastar.org/en/latest/ +4 https://gyre.readthedocs.io/en/stable/ +5 https://umap-learn.readthedocs.io/en/latest/ +6 We use |n| instead of n because GYRE assigns negative values of n to +g modes and positive ones to p modes. +7 More precisely, Qnlm does not vary strictly monotonic with n and +can change significantly between consecutive modes for given l and m. +However, the overall trend of Qnlm peaks for low values of |n| and falls +sharply for |n| ≫ 0. For a more detailed discussion on the behaviour of +Qnlm see, e.g., Burkart et al. (2012). +of the orbital motion. Quantitatively, it expresses the intuitive +principle that for more eccentric orbits, TEOs with larger orbital +harmonic numbers will be excited. This is because, as the ec- +centricity increases, the periastron passage takes less time for a +given orbital period, so eigenmodes with higher frequencies bet- +ter ‘match’ rapidly changing gravitational field, in terms of time +scale. Nevertheless, for very high N, the FNm drops rapidly (al- +most exponentially). This particular property of FNm is respon- +sible for the lack of excitation of the TEOs with extremely high +N. It is clear here that the frequency range of TEOs in massive +and intermediate-mass MS stars is limited on two sides inde- +pendently by Qnlm and FNm. On the low-frequency side, Qnlm +prevents the excitation of g modes with very high |n|, while on +the high-frequency side FNm decreases sharply, strongly limit- +ing the possible excitation of pressure (p) modes characterised +by high radial orders. The last term in Eq. (2), i.e. LN, denotes +the detuning factor given by the following formula, +LN = +fNm +� +(σnlm − fNm)2 + γ2 +nlm +, +(3) +where γnlm stands for damping/growth rate of the normal mode. +This Lorentzian-like factor reflects the mismatch between fNm +and σnlm. Given the typical values of |γnlm| for g and p modes +in massive and intermediate-mass MS stars (of the order of +∼ 10−7 – 10−3 d−1), LN is extremely sensitive to the difference +(σnlm− fNm). Hence, among many other factors, the LN undoubt- +edly plays a key role in the excitation of TEOs. +While the precise prediction of TEO amplitude is a difficult +task8, we are interested in analysing the changes in resonance +conditions dictated by the sum of all the contributing detuning +factors with passing time, t. Let us define the following dimen- +sionless quantity, +L(t) ≡ +� +nlm +Nmax +� +N=1 +LN(t). +(4) +In contrast to LN, associated with a single orbital harmonic, L +reflects the overall chance of TEOs being excited in the EEV +component. However, we must stress at this point that it does +not carry direct information on the amplitude of potential TEOs. +The first summation in Eq. (4) applies to all the normal modes +we consider in the modelling (Sect. 3.3). Obviously, the second +summation in Eq. (4) should run from N = 1 to +∞, but due +to time and physical constraints one has to truncate the series at +some reasonably chosen Nmax. For a detailed description of the +selection of Nmax values see Sect. 3.3. +In order to try to answer the questions raised in Sect. 2, we +have synthesised 20,000 binary evolution models and calculated +L(t) for both components in each of them. The whole procedure +is described extensively in the next four subsections. +3.1. General assumptions +From a practical point of view, a fully consistent calculation +of the evolution of binary systems taking TEOs into account is +8 In order to reliably predict the photometric amplitude of a TEO, one +needs to: (1) determine the exact value of LN, which is almost impossi- +ble given the uncertainties in both observations and stellar models, (2) +calculate the corresponding Qnlm and (3) know the eigenfunction of lu- +minosity fluctuations at the photospheric level, which is a challenge for +radiation pressure-dominated atmospheres of early-type stars (with in- +tense stellar winds). In addition, the equilibrium amplitude of a linearly +driven TEO is determined by various non-linear effects, for instance by +multi-mode coupling (Guo 2020; Guo et al. 2022). +Article number, page 3 of 24 + +A&A proofs: manuscript no. TEOs_in_massive_EEVs +very time-consuming, as it requires time steps shorter than the +times at which the resonances occur (several orders of magni- +tude shorter than the nuclear time scale, cf. Fig. 3). It would take +an enormous amount of time to perform such consistent calcula- +tions for 20,000 binaries with hundreds of resonances occurring +in each of them. Therefore, to make our project both feasible +and still scientifically useful, the models were synthesised un- +der the general assumption that each resonance encountered by +the EEV components is a chance resonance. By sacrificing the +ability to track resonantly-locked TEOs, we are able to decou- +ple evolutionary and seismic calculations and run them indepen- +dently, greatly simplifying the whole problem. We believe that +we can to do this for three reasons: (1) we are only interested +in obtaining some general statistical information about the reso- +nance conditions in a large number of simulated binary systems, +(2) the phenomenon of resonance locking is rare compared to the +rate of chance-resonance events, and (3) the impact of chance- +resonance TEOs on the orbit is limited due to their relatively +short time of existence (e.g., Witte & Savonije 1999b). In con- +clusion, we focus on finding candidate binaries that may or may +not experience numerous TEOs, rather than precisely predicting +their actual evolutionary histories, which is beyond the scope of +this paper. We believe that our results will serve as a starting +point for more detailed calculations performed for the most in- +teresting cases of massive EEVs. +3.2. Synthesis of binary evolution models +Since we assumed that we could separate stellar and orbital evo- +lution from seismic calculations, we first generated a set of bi- +nary evolutionary tracks and only then performed seismic anal- +ysis on them to find L(t). +3.2.1. Initialisation of models +We used the latest open-source 1D stellar evolution code MESA +(release 15140) compiled with the MESA Software Development +Kit (version 21.4.1, Townsend 2021) to compute a set of 20,000 +binary evolution models. The MESAbinary module (Paxton et al. +2015) allows the simultaneous evolution of binary system com- +ponents and their orbits. Throughout this paper, we use the sub- +scripts ‘1’ and ‘2’ to denote the primary (initially more massive) +and secondary components, respectively. +We assumed that both components have the same chemical +composition with metallicity Z = 0.02 and a solar-scaled mix- +ture of elements taken from Grevesse & Sauval (1998). Since +we were only interested in massive and intermediate-mass MS +EEVs that can exhibit TEOs during their lifetime, the initial sys- +tems consisted of two stars lying on the ZAMS and were charac- +terised by parameters randomly drawn from the following uni- +form distributions, U[α,β], on specific intervals [α, β]. +– Mass of primary component, log(M1/M⊙) ∼ U[log 5,log 30]. A +uniform distribution on a logarithmic scale was used instead +of a linear scale to cover the Hertzsprung-Russell diagram +(HRD) with more evenly distributed evolutionary tracks. +– The mass ratio, q ≡ M2/M1 ∼ U[0.2,0.95], where M2 corre- +sponds to the mass of the secondary component. The lower +limit for q was introduced due to the fact that the efficiency +in driving TEOs scales with q (cf. Eq. (2)), so it is less likely +to observe TEOs in a binary system at a small value of the +mass ratio. Moreover, if the generated q corresponded to +M2 < 2M⊙, a redraw was performed. +– Eccentricity, e ∼ U[0.2,0.8]. Range typical of the observed +EEVs. +– Periastron distance, rperi, normalised to the sum of compo- +nents’ radii, �rperi ≡ rperi/(R1 + R2) ∼ U[1,5.5]. However, +if the generated system was initially Roche-lobe overflow- +ing (RLOF) at the periastron, a redraw was performed. We +also assumed an upper value of �rperi because the overall +strength of tidal forces decays as r−3 +peri and simulating widely- +separated systems would contradict the aim of this paper. +– Tidally-enhanced wind factor, Bwind ∼ U[32,896]. Introduced +by Tout & Eggleton (1988) for red giants residing in binary +systems, it accounts for the tidal enhancement of the stellar +wind mass-loss rate due to the presence of a nearby com- +panion. The ad hoc chosen range of Bwind corresponds to a +maximum amplification of the ‘nominal’ wind mass-loss rate +by a factor of 1.5 – 10 (cf. Tout & Eggleton 1988, their eq. 2). +– The angular rotational velocity divided by its critical value9, +Ω/Ωcrit ∼ U[0.1,0.5]. The assumed range of initial Ω/Ωcrit +translates into the linear equatorial velocities between +∼50 km/s and ∼320 km/s in our simulations and reflects the +significant non-zero rotation velocities observed in massive +young MS stars (e.g., Dufton et al. 2006; Hunter et al. 2008). +– The overshoot mixing parameter, fov ∼ U[0.015,0.025]. In our +calculations, the overshooting of the material above the con- +vective, hydrogen-burning core was treated in the exponen- +tial diffusion formalism developed by Herwig (2000). An ad- +justable parameter, fov, describes the spatial extent of the +overshoot layer in terms of the local pressure-scale height, +but its value for massive stars is still under debate. We +adopted the range of fov after Ostrowski et al. (2017). +The parameters presented above were generated independently +for each EEV system. Moreover, the last two parameters, Ω/Ωcrit +and fov, were drawn independently for each component, so the +final hyperspace of parameters explored in our simulations in- +cluded {M1, q, e,�rperi, Ω/Ωcrit,1, Ω/Ωcrit,2, fov,1, fov,2, Bwind}. Fig- +ure 1 shows our initial sample of generated EEVs in the orbital +period versus eccentricity diagram. As expected, they occupy +the upper envelope of the aforementioned plane with the upper +boundary dictated by the onset of periastron RLOF on ZAMS. +The rest of the necessary parameters and ‘physics switches’ were +identical for each simulated binary. We will now briefly describe +them below. +3.2.2. Integration of the evolution +Nuclear reaction rates were calculated using ‘basic.net’ op- +tion in MESA. We used a convective premixing scheme (Paxton +et al. 2019, their Sect. 5) in combination with the Ledoux cri- +terion to define the boundaries of convective instability. This +specific approach of treating convection agrees with the results +of modern 3D hydrodynamic simulations (Anders et al. 2022). +Convective mixing was incorporated into the models via mix- +ing length theory (MLT) in the ‘Cox’ formalism (Cox & Giuli +1968, their chap. 14) with the value of the solar-calibrated mix- +ing length parameter αMLT = 1.8210 (Choi et al. 2016). As +9 By critical rotational velocity we mean the situation when the effec- +tive gravity at the stellar equator is zero, i.e. the centrifugal force and the +Eddington factor, Γ, balance the true gravity. MESA estimates this quan- +tity as Ωcrit = +� +(1 − Γ)GM/R3, where G is the gravitational constant +and Γ ≡ Lrad/LEdd. The Lrad and LEdd denote the radiative luminosity +and Eddington luminosity of the star, respectively. +10 There is some evidence that αMLT may depend on global stellar pa- +rameters such as mass (Yıldız et al. 2006) or metallicity (Viani et al. +Article number, page 4 of 24 + +Kołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs +Fig. 1. Distribution of initial orbital period and eccentricity for a sample +of 20,000 binaries evolved in our project. The initial normalised sepa- +ration at the periastron is colour-coded. The upper left-hand corner cor- +responds to the ZAMS EEVs, which experience RLOF at the periastron +and should therefore rapidly circularise their orbits. The lower right- +hand corner, on the other hand, is where relatively widely-separated +binaries can be found. +mentioned earlier, exponential overshoot mixing above the con- +vective core was also included11, but we neglected overshoot- +ing in the non-burning convection zones. For stars with masses +⩾ 15 M⊙, we activated the treatment of convection as ‘MLT++’ +(Paxton et al. 2013, their Sect. 7.2) to reduce superadiabaticity +in convective zones dominated by radiation pressure. Since we +used Ledoux criterion, semiconvection could appear in our stars +with its efficiency parameter, αsc = 0.01 (Langer et al. 1985). +In our case, semiconvection sometimes occurred in chemically- +modified layers left by the shrinking core. +Upon initialisation at the ZAMS, we relaxed both compo- +nents in ∼100 steps so that they rotated rigidly. Later, we al- +lowed our stars to rotate differentially during their evolution, +according to the so-called shellular approximation of rotation +(Meynet & Maeder 1997). Throughout the entire evolution, we +assumed that the rotation axes of the stars are perpendicular to +the orbital plane. MESAstar uses the mathematical formalism +of Heger et al. (2000) and Heger et al. (2005) to apply struc- +tural corrections, perform different types of rotationally induced +mixing and „diffusion” of angular momentum between adjacent +shells. The following rotational mixing mechanisms were taken +into account in MESA: dynamic shear instability, secular shear +instability, Eddington-Sweet circulation, Solberg-Høiland insta- +bility, and Goldreich-Schubert-Fricke instability (all described +in detail by Heger et al. 2000). Even the combined mixing coef- +ficients of the aforementioned rotational instabilities can be zero +in some parts of the star. However, this is clearly unrealistic due +to the presence of a nearby companion that induces additional +mixing throughout the star. To at least approximately account for +this process, we did not allow the total mixing coefficient, Dmix, +to fall below 105 cm2/s. This particular arbitrarily-selected value +is related to the mixing time scale, τmix ∼ (∆r)2/Dmix ≈ 15 Myr +at radial distance, ∆r = 0.1 R⊙. We cannot conceal here that ro- +2018). It is also very likely that αMLT is sensitive to the evolutionary +stage of the star and the type of convection zone (e.g., Wu et al. 2015). +Here, we have assumed a constant value of αMLT for simplicity. +11 Similarly to the αMLT, fov also can depend on different stellar param- +eters (e.g., Castro et al. 2014). +tation and mixing profiles in MS stars are still poorly understood +(except in the solar case). There are no definitive conclusions +as to what mixing mechanisms and whether they actually occur +in massive and intermediate-mass MS stars (see e.g., Pedersen +et al. 2021; Pedersen 2022, for a discussion of this problem and +its asteroseismic inference from B-type dwarfs). +Mass losses due to the radiation-driven stellar wind were +calculated according to the prescription given by Vink et al. +(2001). Their formulae take into account the presence of a +so-called bi-stability jump around the effective temperature of +Teff ≈ 26,000 K, caused by ionization and recombination of some +Fe ions. Nevertheless, the presence of a bi-stability jump is still +questionable and there is some evidence that the associated al- +most instantaneous change in the mass-loss rate may not be real +(cf. Krtiˇcka et al. 2021; Björklund et al. 2022). ‘Nominal’ wind +mass-loss rates in our simulations were modified in two ways: +(1) the rate was amplified by the aforementioned tidal mech- +anism, parametrized by Bwind (Tout & Eggleton 1988) and (2) +the effect of fast rotation at the surface, which can amplify the +mass-loss rate, was accounted for by the simplified power-law +description given by Heger et al. (2000) (their Sect. 2.6). We +assumed that the mass loss through the wind is completely non- +conservative, i.e. there is no mechanism that could transfer some +material back to the star or to a companion. +As we already mentioned above, MESAbinary allows the si- +multaneous integration of some stellar and orbital parameters +that are coupled to each other in a binary system. We have +switched on the MESA controls responsible for changes in the to- +tal orbital angular momentum caused by: (1) gravitational wave +radiation, (2) wind mass loss and (3) tidal spin-orbit coupling. +For the first process, the rate of orbital momentum loss was cal- +culated assuming point masses. The mass loss through the stellar +wind was completely non-conservative, so the angular momen- +tum lost via this channel was equal to the angular momentum +carried by the wind. The phenomenon (3), contributing to the +evolution of eccentricity, orbital and spin angular momenta, was +modelled using the theory of tidal interactions for radiative en- +velopes developed by Zahn (1977), Hut (1981) and Hurley et al. +(2002), after being adapted to the shellular approximation of ro- +tation. For stars with radiative envelopes, tidal dissipation pro- +cesses are dominated by tidally excited gravity modes that prop- +agate to the stellar surface, where they gain relatively large am- +plitudes and experience effective radiative damping (due to the +short local thermal time scale) and nonlinear damping. Conse- +quently, they deposit their energy and angular momentum in the +outer layers of the envelope. Following earlier calculations of +Zahn (1977), Hurley et al. (2002) delivered convenient formu- +lae to describe the tidal synchronisation and circularisation time +scales associated with the aforementioned phenomenon. Using +these time scales combined with the formalism presented by Hut +(1981), MESAbinary integrates the evolution of the eccentricity +and updates spin angular frequency of each shell in the stellar +model. Therefore, our calculations in MESAbinary took into ac- +count the approximate influence of the dynamical tide on the +orbit, at least up to the lowest possible order. Of course, the tidal +evolution formalism implemented in MESAbinary does not in- +clude the effects of resonance locking. For explicit formulae de- +scribing tidal processes in MESAbinary, we refer to Paxton et al. +(2015) (their Sect. 2). +We have completely ignored the effects of magnetic fields, +while bearing in mind that they may mainly affect the actual stel- +lar wind mass-loss rates, the efficiency of internal mixing pro- +cesses and synchronisation/circularisation time scales (e.g. via +the magnetic braking mechanism). The impact of fossil mag- +Article number, page 5 of 24 + +0.8 +RLOF on ZAMS +5.0 +0.7 - +4.5 +0.6- +4.0 +e +0.5 +3.5 +3.0 +0.4 - +wide +even +2.5 +0.3 +2.0 +0.2 +1.5 +100 +101 +102 +Porb (d)A&A proofs: manuscript no. TEOs_in_massive_EEVs +netic fields on the evolution of massive and intermediate-mass +stars was recently described by Keszthelyi et al. (2022). +All details on the parameters of our models in MESA can be +found in Appendix A, where we present the contents of our MESA +inlists. A concise description of the micro- and macrophysics +data sources used by MESA is provided in Appendix C. +3.2.3. Termination conditions +The evolution of the binary system was carried out until at least +one of the following termination conditions was met for any of +the components: +1. The component reached TAMS, i.e. the central mass abun- +dance of hydrogen fell below Xc ⩽ 10−4. +2. The eccentricity was reduced to e ⩽ 0.01. +3. The rotation velocity reached Ω/Ωcrit = 0.75 at the stellar +surface. +4. Episodic mass transfer between components due to the +RLOF in the periastron began. +The reasons behind providing the termination conditions out- +lined above are as follows. Our study is exclusively dedicated to +the MS phase of the evolution of EEVs, hence the first condi- +tion has to be enforced. The second condition is self-explanatory, +since we are interested in non-zero eccentricities that allow for +TEO excitation12. The third condition is related to the conver- +gence problems that can occur in MESAbinary when one of +the components nearly approaches the break-up velocity of rota- +tion. Numerous assumptions and descriptions of rotation-related +phenomena reach the limits of their applicability in MESA for +Ω/Ωcrit ≈ 1. Since for Ω/Ωcrit ≳ 0.75 the deviation from +spherical symmetry becomes significant, a 1D treatment of the +problem is no longer adequate. For instance, the way in which +such a star loses mass becomes fundamentally different from the +isotropic case. We have therefore decided to stop integrations un- +der such circumstances. The last condition is related to the diffi- +culty in correctly describing an episodic (near-periastron) RLOF, +when a ‘blob’ of material could be ejected from the RL-filling +component during each periastron passage. However, this kind +of orbital phase-dependent RLOF is not expected to be observed +in a binary for a long time due to strong tidal forces. They should +effectively suppress the eccentricity, making the system circular +(and so the second condition can be quickly met). +3.3. Asteroseismic calculations +A consequence of the assumption of the aligned vectors of the or- +bital and spin angular momenta is a rule for selecting the geome- +try of modes that can be tidally excited. Under such conditions, a +normal mode can be tidally excited only if +mod (|l+m|, 2) = 0, +e.g. the l = 2 TEOs will be characterised only by m = −2, 0, +2. +Here we restrict our study to only l = 2, m = 0, +2 modes be- +cause of two reasons. First, l = 2 modes correspond to the dom- +inant component in the series expansion of the variable tidal po- +tential. Modes with l > 2 undergo much weaker excitation due +to the dependence on (R/a)l+1, which enters Eq. (2). Second, +the values of FN,−2 are very small compared to their m = 0, +2 +counterparts. This can be easily seen in Fig. 2a, where we have +12 In theory, components of circular systems (e = 0) can also exhibit +TEOs, provided they do not rotate synchronously. However, the number +of modes observable as TEOs in these systems is much smaller than the +number of potential TEOs in EEVs. +plotted the maximum values of FNm for m = −2, 0, +2 and dif- +ferent eccentricities. FN,−2 is approximately at least 2 – 3 orders +of magnitude smaller than FN,0 or FN,2. +For each model of the stellar internal structure that was saved +during the synthesis of binaries in MESA, we calculated the oscil- +lation spectrum using the GYRE code in the non-adiabatic regime +and the second-order Gauss-Legendre Magnus integrator. The +frequencies σn,2,0 and σn,2,+2 corresponding to the non-adiabatic +calculations were searched by GYRE based on the preliminary +adiabatic calculations. Rotational effects (Coriolis force) were +taken into account using the so-called traditional approximation +of rotation (e.g., Aerts et al. 2010, their Sect. 3.8). We searched +for eigenvalues in the family of gravito-acoustic solutions. We +assumed the necessary (differential) rotation profile inside the +star from the MESA model. +As we argued in Sect. 3, it is necessary to choose a rea- +sonable range of frequencies to scan for eigenvalues based on +Qnlm and FNm. Therefore, we only searched for modes with +|n| ⩽ 30 and frequencies, σn,2,0 ∈ (forb, Nm=0 +max forb) or σn,2,+2 ∈ +(max{0, forb − 2fs,core}, Nm=+2 +max +forb − 2 fs,core). In the ranges shown, +Nm=0 +max and Nm=+2 +max +refer to the limits of N due to the decrease in +FN,0 and FN,2, respectively. The fs,core is the core rotation fre- +quency. We defined Nm=0 +max and Nm=+2 +max +as N for which FN,0 or FN,2 +is equal to 10−8, i.e. FNm starts to effectively prevent excitation +of TEOs. In practice, we numerically calculated the FNm func- +tions13 for different eccentricities and obtained the log Nm +max(e) +relations as a fit of a fourth-degree polynomial to a set of its dis- +crete points. A summary of this process is shown in Fig. 2b. For +low-e orbits, the typical range of N favourable for the excitation +of TEOs reaches N ∼ 101, in contrast to highly eccentric orbits, +which may exhibit as much as N ≈ 100 – 200 TEOs. Figure 2b +also shows another feature of m = −2 modes that makes them in- +ferior candidates for TEOs compared to axisymmetric and pro- +grade modes – as potential TEOs they always span a narrower +range of orbital harmonic numbers. +Defining the frequency range for σn,2,0 is quite straightfor- +ward, as these are axisymmetric modes and their frequencies +do not change when switching between inertial and co-rotating +frames. The situation is quite different when it comes to the +m = +2 modes. This time, due to the differential rotation inside +the star, σnlm = σnlm(r) = σnlm − mfs(r), where r is the radial co- +ordinate in the stellar interior and σnlm is oscillation frequency +in the inertial frame. For some eigenmodes, σnlm may change its +sign somewhere in the star, depending on the shape of the rota- +tional profile. This location is known as the critical layer, where +σnlm(r) → 0, and such a mode experiences severe damping due +to its very short radial wavelength (e.g., Alvan et al. 2013). To +exclude these modes from our experiment, the maximum fre- +quency of σn,2,+2 was set to (Nm=+2 +max +forb − 2 fs,core)14. This is be- +cause during evolution the core rotates almost rigidly and faster +than the envelope, hence the difference (Nm=+2 +max +forb − 2 fs,core) ⩽ +(Nm=+2 +max +forb−2fs,env), where fs,env stands for the rotation frequency +of the outermost part of the envelope. +More details of our calculations performed in GYRE can be +found in Appendix B, where we present the explicit contents of +our GYRE input file. +13 Using eq. 5 presented by Fuller (2017). +14 We note that this frequency is expressed in a rest frame co-rotating +with the stellar core. +Article number, page 6 of 24 + +Kołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs +Fig. 2. (a) Maximum values of the Hansen coefficients FNm versus +eccentricity for l = 2 modes and three different azimuthal orders, +m = −2, 0, +2 denoted by blue, green, and red points, respectively. We +note the marginal contribution of the m = −2 modes; (b) Dependence of +log Nm +max on eccentricity with the same colour-coding as in panel a. The +colour solid lines represent the best fits of the fourth-degree polynomi- +als, which we used to determine frequency ranges in the asteroseismic +calculations. +3.4. Derivation of L(t) +The introduction of differential rotation also has consequences +when it comes to interpreting the resonance condition from +Eq. (1). The quantity fs is no longer a constant value, so one has +to decide which fs to choose. Theoretical studies imply the in- +duction of g-mode TEOs (especially of high radial order) primar- +ily near the convective core boundary (e.g., Goldreich & Nichol- +son 1989) for stars with radiative envelopes. However, the res- +onances in our simulations are also due to p or g modes with +small radial order. Therefore, we decided to apply our resonance +condition to the envelope15 (not to the interface region near the +core boundary), rewriting Eq. (1) more accurately as +fNm ≡ N forb − m fs,env ≈ σnlm, +(5) +and use it in the subsequent modelling of L(t). It is essential to +note at this point that the resonance condition given by Eq. (5) +refers to fNm and σnlm expressed in a frame co-rotating with +the outer stellar envelope. Although in principle the morphol- +ogy of L(t) depends on the choice of the specific resonance con- +dition, we note that it does not affect at all resonances due to +15 In the exact approach, different resonance conditions would have to +be used for different modes, depending on the radial coordinate inside +the star where a given TEO is dominantly excited. Here we assume a +single form of resonance condition for all modes. +m = 0 modes and should not significantly affect resonances cor- +responding to p modes or low-|n|, m = +2 g modes. +Having a set of eigenfrequencies calculated by GYRE and +knowing the history of the binary evolution from MESA, we per- +formed the summation shown in Eq. (4). However, this was not +a direct summation running over the models saved by MESA and +GYRE, as their temporal resolution was still too coarse compared +to the duration of a typical resonance. To circumvent this prob- +lem, we interpolated the temporal variations of each oscillation +frequency and all necessary parameters of the binary system us- +ing Akima cubic spline functions (Akima 1970). Then, we were +able to calculate the values of L(t) on a uniformly-spaced time +grid with a constant time step of 2,000 years, which we assumed +to be identical for all EEVs in our simulations. From here on, +we will use the term ‘resonance curve’ as a proxy for the L(t) +time series. Figure 3 shows a compilation of example resonance +curves, although we postpone discussion of these to Sect. 4. To- +gether with the initial parameters of binary systems, resonance +curves are particularly important to us in this study. +3.5. ML analysis of the resonance curves +Although in Sects. 4.3 and 4.4 we analyse the resonance curves +based on various statistics, due to their global nature we do +not distinguish many details that are ‘hidden’ in the resonance +curves. To characterise the morphology of all resonance curves +in more detail (without having to perform a visual classification, +which is almost impossible due to the number and complexity +of the data set), we applied dimensionality reduction methods. +With these, we were able to explore the topology spanned by +the morphological features of the resonance curves. We carried +out the entire analysis described here separately for the sets of +curves L1(t) and L2(t), corresponding to the primary and sec- +ondary components, respectively. +As a first step, we summarised each resonance curve with a +vector Q that described its morphological features. We focused +our attention on two particular features: (1) the distribution of +log(L) values and (2) the distribution of apparent maxima at a +normalised time, t/Tmax, where Tmax stands for the max{t}. In +practice, we calculated vectors Qx and Qy which contained sets +of 1,000 quantiles of normalised times corresponding to local +maxima of L(t) and 1,000 quantiles of log(L), respectively. The +levels of both calculated quantiles were spanned evenly from 0 +to 1. Qx describes the overall distribution of apparent maxima +in time, reporting changes in the rate of resonance occurrence. +We deliberately used normalisation by Tmax because we want +the results to be sensitive only to the relative distribution of the +resonance events over the lifetime of the EEV. Otherwise, its val- +ues would be strongly correlated with the length of the resonance +curve itself16, rather than with the distribution of resonances over +time. On the other hand, Qy encapsulated the combined informa- +tion about the mean level of log(L), any long-term trends in the +resonance curve and the distribution of the heights of the max- +ima. In contrast to the Qx, we did not apply any normalisation +to Qy as its absolute values carry valuable information about the +strength of the resonances and the average level of the entire +resonance curve. The final vector Q was constructed as the con- +catenation of Qx and Qy, which had previously been scaled using +the variance in the sets of all Qx and Qy. The resulting Q has a +total of 2,000 dimensions. +16 Which in turn is an almost a direct approximation for the mass of the +primary component. +Article number, page 7 of 24 + +0 +log (max[FNm )) +-6 +-8 +a) +2.5 +m=+2 +m=0 +2.0 +m=-2 +max +1.0 +0.5 +(b) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +eA&A proofs: manuscript no. TEOs_in_massive_EEVs +Fig. 3. Sample of resonance curves obtained as described in Sect. 3. Each panel corresponds to a different arbitrarily-chosen binary system with +the rounded values of their initial parameters given on the right. The dark red and blue curves reflect the behaviour of L(t) for the primary and +secondary components, respectively. For clarity, L2(t) has been shifted vertically by three orders of magnitude downwards. Time t = 0 indicates +ZAMS. A sudden break in L2(t) on the bottom panel (after about 5.5 Myr) indicates L2 = 0, i.e. the absence of any resonances. The differences in +the height of the peaks are due to different values of γnlm and min{|σnlm − fNm|} for excited TEOs. +Article number, page 8 of 24 + +14.6Mo +M1 +M2 +3.0M +105. +0.32Qcrit +21 +22 +0.31Qcrit +fov,1 +0.022 +fov, 2 +0.019 +primary +secondary +103- +0.52 +e +Tperi +5.23 +Bwind +583.9 +101- +2 +0 +4 +6 +8 +10 +12 +M1 +5.8Mo +M2 +4.2Mo +105. +0.11 2crit +21 +22 +0.162crit +fov, 1 +0.016 +fov,2 +0.015 +103- +0.29 +e +uody +1.53 +Bwind +828.3 +101. +C2 / 103 +10 +20 +25 +5 +15 +30 +L +23.5 Mo +M1 +106. +M2 +13.1 Mo +L +0.43 2crit +21 +105. +0.44 2crit +22 +fov,1 +0.022 +104 +fov,2 +0.019 +0.62 +103. +Tperi +3.32 +Bwind +102. +806.0 +101. +2 +3 +4 +5 +6 +0 +M1 +19.8Mo +M2 +5.5Mo +105. +21 +0.24 Qcrit +22 +0.18Qcrit +104. +0.018 +fov, 1 +0.015 +fov,2 +103- +0.31 +e +Tperi +2.68 +102 +Bwind +893.8 +101- +5 +6 +0 +3 +4 +8 +t (Myr)Kołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs +In the next step, we performed a preliminary dimensional- +ity reduction of Q by means of the Principal Component Analy- +sis (PCA, Pearson 1901), obtaining pre-processed ‘morphology’ +vectors, θPCA. PCA is a method that orthogonally projects the +data into a coordinate system in which successive vector com- +ponents explain a smaller and smaller part of the data variance. +The target number of its dimensions returned by PCA for each +Q was set to 10. This value was chosen experimentally by ex- +amining the percentage of the total variance of the data set ex- +plained by successive PCA components. For L1(t), the first ten +PCA components explained a total of 99.8% of variance (first +component – 79% and second component – 19%). For L2(t), the +corresponding value was 98.5% (in this case, the first PCA com- +ponent explained 60% of the total variance, while the second +explained 22%). +We then performed the final 2D embedding by applying +UMAP on the collection of θPCA vectors. UMAP is a multi- +purpose non-linear dimensionality reduction technique that con- +structs a low-dimensional projection that preserves as accurately +as possible the topological structure of the input data. For in- +stance, in this case, a pair of embeddings of resonance curves +with similar properties (in the sense of their summary statistics +described above) are expected to lie in mutual vicinity on the +2D UMAP plane. Let us denote the UMAP results as θUMAP. +The manifold spanned by θUMAP (Sect. 4.5) allowed us to ef- +fectively examine the differences in the morphology of the res- +onance curves and their dependence on the initial parameters of +the simulated EEVs. +Unlike PCA, UMAP is a complex method, with many free +parameters that need to be adjusted with care, as the resulting +embedding may depend heavily on their choice. Appendix D +provides all the ‘technical’ details of this process, including the +values of the most important UMAP parameters adopted in this +study. +4. Results +4.1. General properties of synthesised EEVs +Before going into a detailed analysis of the resonance curves, we +briefly characterise the general properties of the models we have +synthesised using the MESAbinary and GYRE codes. +4.1.1. Evolutionary tracks in HRD +Figure 4 shows a pair of HRDs with a compilation of all 20,000 +evolutionary tracks that we obtained in our simulations for the +primary (Fig. 4a) and secondary (Fig. 4b) components. Although +it is impossible to clearly present thousands of evolutionary +tracks on a single HRD, we have highlighted and colour-coded a +small fraction of them in order to describe some of their features. +First of all, only a fraction of the primaries reached TAMS +when the central mass abundance of hydrogen dropped be- +low 10−4 (according to the first of our termination conditions, +Sect. 3.2.3). Many evolutionary tracks were interrupted at MS +due to the fulfilment of one of the other termination conditions. +Secondly, a number of tracks clearly change their character after +crossing the line corresponding to the bi-stability jump (around +Teff = 26,000 K). This is due to the associated sharp increase in +the wind mass-loss rate, as it tries to keep the stellar luminosity +constant. In some circumstances, the mass-loss rate is so high +that the star loses a significant part of its envelope17. This effect +17 We recall that these high mass-loss rates are not exclusively derived +from the description of Vink et al. (2001). Rotational and tidal amplifi- +‘pushes’ the star back to the high effective temperature region +and is particularly pronounced for the most massive stars in our +sample (cf. Fig. 4a, evolutionary tracks that ‘turn around’ and +cross the bi-stability jump for a second time). +4.1.2. EEV groups in terms of the termination condition +Only four of the seven18 termination conditions actually oc- +curred in our simulations. The majority of our EEVs (∼67.1 %) +ended up as MS RLOF systems in which the primary compo- +nent filled its Roche lobe during the periastron passage. The +next most numerous group (∼22.6 %) were systems in which the +primary component successfully reached TAMS (Xc ⩽ 10−4). +About 10.2 % of the binaries managed to circularise their orbits +before any other termination condition was met. The last group +contains only about 0.04 % of the total sample. This is the group +where the primary’s rotation velocity exceeded the maximum al- +lowed angular velocity (Ω / Ωcrit ⩾ 0.75). +Figure 5 presents these four groups of EEVs on the Porb-e +plane and allows a comparison of the initial (Fig. 5a) and final +(Fig. 5b) states of the aforementioned distribution. As expected, +the EEVs with the shortest orbital periods and high eccentricities +tended to circularise their orbits before leaving the MS. Their +trajectories in the Porb-e diagram (Fig. 5c) follow smooth, almost +vertical lines due to the strong tidal damping of eccentricity. On +the other hand, the integration of the evolution of systems with +large distances between components at periastron (�rperi ≳ 3.5) +has been terminated mainly due to the exhaustion of hydrogen +in the primary’s core. Although the majority of EEVs belonging +to this group do not significantly change their orbital parameters +during evolution, there is a subgroup of them that behaves quite +differently. It can be recognised as the distinct ‘cloud’ of green +dots in Fig. 5b, represented by the mainly horizontal green tracks +in Fig. 5c. These are systems that were characterised by very +strong stellar winds at the end of the MS phase and have lost +much of their envelopes, so that their orbital period has increased +significantly (Kepler’s third law). +The most numerous group of EEVs, in which the primary +component has filled its Roche lobe in the MS phase, forms a +kind of ‘bridge’ between the two previously mentioned groups +and merges with them. The shapes of the corresponding trajec- +tories on the Porb-e plane may vary from system to system, de- +pending on the interplay between tidal forces and the intensity +of stellar winds, so no single ‘type’ of track can be assigned +to them. However, many of them resemble the inverted Greek +letter ‘Γ’ – initially, the system drifts horizontally (towards the +longer orbital period) as a result of the mass loss and/or spin- +orbit coupling, and then undergoes more or less rapid circulari- +sation (moves vertically downwards) under the influence of the +intense tides, which come to the fore when the primary compo- +nent almost fills its Roche lobe. +Only 8 out of 20,000 EEVs underwent efficient spin up of +both components due to the pseudo-synchronisation (when the +rotation period of the star ‘matches’ the rate of orbital motion +at periastron, so that there is no net torque over an orbital cycle, +e.g., Hut 1981). These few systems are located in the upper right +cation mechanisms can significantly intensify stellar winds in our sim- +ulations. These phenomena are particularly well-pronounced when the +component is close to TAMS, i.e., its radius approaches the Roche-lobe +radius. +18 In Sect. 3.2.3 we give four types of termination conditions, but three +of them apply independently to both the primary and secondary compo- +nents. +Article number, page 9 of 24 + +A&A proofs: manuscript no. TEOs_in_massive_EEVs +Fig. 4. (a) HRD with the evolutionary tracks of primary components. The grey area corresponds to the region occupied by the full set of 20,000 +tracks, while a subsample of 100 randomly-selected tracks is indicated with coloured points connected by black solid lines. Each point rep- +resents one saved MESA model. The colour coding reflects the central hydrogen abundance. The effective temperature of the bi-stability jump +(Teff ≈ 26,000 K) is marked with the vertical dashed line. The abrupt change in the behaviour of some evolutionary tracks after crossing the bi- +stability jump region is due to a significant change in the wind mass-loss rate; (b) The same as panel (a), but for a set of secondary components. +We note the difference in the ranges of the two axes in panels (a) and (b). More details are discussed in the main text. +corner of Figs. 5a and b. Their orbits were initially highly eccen- +tric yet relatively widely-separated at periastron (�rperi ≈ 4.5 – +5.0). Thus, in combination with the lower masses of the pri- +mary components (M1 ≈ 5 M⊙), there was no effective tidal +dissipation. However, the envelopes of these stars tended to ro- +tate pseudo-synchronously with the orbit (due to the relatively +long nuclear time scale of the evolution of a 5 M⊙, the primaries +had enough time to do so). Consequently, this led to a very fast +rotation of the primary component, exceeding the threshold of +Ω/Ωcrit = 0.75. +4.1.3. Internal structure and asteroseismic properties +The shape of the resonance curve depends not only on the global +properties of the components and the orbit, but also on the inter- +nal structure of the stars, which directly affects seismic proper- +ties (i.e. the spectrum of eigenmodes). Therefore, within the lim- +ited volume of this paper, we would like to show at least one rep- +resentative example of the evolution of the internal properties of +the primary component for an arbitrarily-chosen EEV. Figure 6 +shows the evolution of a primary with mass M1 ≈ 13.6 M⊙ in a +system with an initial eccentricity e ≈ 0.4 and an initial orbital +period Porb ≈ 4.0 d. In our simulations, this particular system +finished its evolution due to the circularisation of its orbit after +about 12 Myrs. The HRD in Fig. 6 reveals the ‘non-standard’ +evolutionary track of the primary due to the sharp change in the +mass-loss rate after crossing the bi-stability jump (right panel in +the top row of Fig. 6). The same panel also shows how the pri- +mary’s surface rotation rate varies over time – as the mass-loss +rate increases, it loses a lot of spin angular momentum and slows +down its rotation. The eccentricity and orbital period monotoni- +cally decrease with time (middle panel in the top row of Fig. 6), +except for a short episode of increase in Porb caused by the ir- +reversible loss of a large part of the envelope. We have selected +three epochs in the evolutionary history of this EEV (labelled A, +B, C on the HRD), for which we have presented the appearance +of the rotational profiles, mode propagation diagrams and oscil- +lation spectra of the primary component in the bottom part of +Fig. 6. Epoch A corresponds to the phase of evolution just af- +ter leaving the ZAMS, epoch B is characterised by Xc,1 ≈ 0.45, +and finally, epoch C marks the situation just before the complete +circularisation of the EEV. Let us briefly describe the changes +occurring in each of the three types of diagram below. +The internal rotation profile of the primary is almost constant +for epoch A, but by then a division between a faster-rotating core +and a slower-rotating envelope begins to emerge. The aforemen- +tioned division becomes particularly apparent in epoch B, when +the core has developed a rotation rate approximately 1.25 times +that of the surface layers. As can be seen, the contracting core ro- +tates as a rigid body throughout the MS lifetime due to efficient +angular momentum transport supported by convection. The outer +part of the envelope also rotates almost rigidly, but this time it is +due to large-scale Eddington-Sweet meridional flows. The angu- +lar velocity gradient in the primary starts to gradually decrease +as the star reaches epoch C. Various mixing processes in the +chemically-modified layer left by the core lead to the diffusion +of angular momentum from the core to the envelope. Moreover, +the rotational profile inside the star becomes a smooth function +of the radius (rather than a step-like function as for epoch B). +Article number, page 10 of 24 + +0.7 +5.5 +(a) +(b) +5 - +- 0.6 +5.0 - + 0.5 +4 - +4.5 - +-0.42 +X +1og ( +3.5 - +- 0.2 +2 - +3.0 - +- 0.1 + bi-stability jump +1- +0.0 +4.6 +4.5 +4.4 +4.3 +4.2 +4.1 +4.6 +4.5 +4.4 +4.3 +4.2 +4.1 +4.0 +3.9 +log (Teff, 1 / K) +log (Teff, 2 / K)Kołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs +Fig. 5. Orbital period-eccentricity distributions of 20,000 modelled +EEVs; (a) Initial distribution of e as a function of Porb. Colour-coding +corresponds to the termination conditions described in Sect 3.2.3, i.e. +the RLOF of the primary component during periastron passages before +reaching TAMS (black), exhaustion of hydrogen in the primary’s core +(primary at TAMS, green), almost complete circularization of the or- +bit (e = 0.01, magenta), and the maximum allowed rotation rate of the +primary component (Ω / Ωcrit = 0.75, orange). A pair of dashed hori- +zontal lines mark the boundary values of the initial eccentricity, e = 0.8 +and e = 0.2; (b) Same as in panel (a), but for the final state of each +modelled binary system; (c) Random selection of 400 orbital evolution +tracks with the same colour-coding as in panels (a) and (b). +The majority of TEOs in our simulations belong to the g- +mode family of oscillations, so it is very important to control the +behaviour of the Brunt-Väisälä buoyancy frequency, NBV, in our +models. Together with Lamb frequency for l = 2 modes, S l=2, +they carry information about g and p mode cavities and their +evanescence regions (e.g., Aerts et al. 2010, their Sect. 3.4). The +evolution of NBV and S l=2 is presented in the middle column +of Fig. 6. The blue and grey shaded regions denote the posi- +tion of the l = 2 p-mode and g-mode propagation cavities, re- +spectively. The white areas that lie between the Brunt-Väisälä +and Lamb frequencies correspond to the evanescence regions. +During evolution, the receding convective core builds up a large +g-mode trapping cavity, which is very important for their fre- +quency spectrum. Additionally, the behaviour of the NBV just be- +low the stellar photosphere reveals a pair of thin subsurface con- +vection zones, expected for this type of star (e.g., Jermyn et al. +2022). Comparing the mode propagation diagrams for epochs A +and C, it can be seen that also the p modes can penetrate deeper +and deeper into the primary as it gradually depletes the hydrogen +in its core. +The right column in Fig. 6 contains most of the information +that is directly used to obtain the resonance curve. The horizontal +bars at the top of each panel correspond to the frequency range in +which GYRE looked for potential TEOs (according to the criteria +adopted in Sect. 3.3). We recall that that their width depends on +the Nm +max(e) functions, so as the system evolves towards lower ec- +centricities, these bars are shorter and shorter (i.e. fewer harmon- +ics of the orbital frequency can effectively drive TEOs). With the +thick, short vertical lines we mark the location of the tidal forcing +frequencies. As can be seen, the separation between successive +values of fNm becomes larger with passing time due to the in- +crease in forb. The eigenfrequencies found by GYRE are marked +with the long thin vertical lines, while the linear damping rates +of these modes are shown as black solid and dotted lines. The +presented set of three synthetic oscillation spectra reveals a typ- +ical structure for g modes with their asymptotic behaviour for +large radial orders (which correspond to lower frequencies). It +may appear that the dense ‘forests’ of eigenfrequencies end too +early relative to the left limits of horizontal bars. However, this +is not a mistake, but a direct consequence of the maximum |n| +we allowed in the calculations – modes with lower frequencies +would have larger radial orders than thirty. During the evolution +of the EEV, both the forcing frequencies and the oscillation spec- +trum shift, so the intersection of these two vertical line patterns is +virtually inevitable in most cases. Each of these intersections is +the source of a single resonance that can give rise to a noticeable +TEO at the level of the photosphere. +4.2. The ‘visual inspection’ of resonance curves +The resonance curves are characterised by a striking diversity in +terms of morphology, which is already partly evident in Fig. 3. +The four examples of L1(t) and L2(t) shown in this figure show +that the components of the EEVs can experience, firstly, a very +different number of resonances and, secondly, their distribution +in time can take various forms. The heights of the maxima of +the resonance curves are mainly dictated by the γnlm of the mode +to which the smallest difference corresponds, (σnlm − fNm). Sta- +tistically speaking, modes with larger |n| are more strongly non- +adiabatic (have larger damping rates), hence the maxima they +induce in the resonance curves are lower (cf. Eq. (3)). Another +factor determines the extent of the resonant maximum in time. It +is determined by the relative ‘velocity’ with which the eigenfre- +quency spectrum crosses the fNm spectrum. By ‘velocity’ here +we mean the rate of change of these two independent frequency +spectra. +It should be emphasised that there are also numerous cases in +which L(t) drops sharply to zero at some point (cf. L2(t) curve +in the bottom panel of Fig. 3) or resonances do not occur at all +(see Sect. 4.3 and Fig. 8). Such a situation can occur, for exam- +Article number, page 11 of 24 + +0.8 +initial +distribution +0.7 +0.6 +0.5 - +e 0.4- +0.3 +0.2 +0.1 - +RLOF of the primary +minimum eccentricity +(a) +primary at TAMS +maximum rate of rotation +0.0 +0.8 +final +0.7 +distribution +0.6 - +0.5 - +e0.4l +0.3 - +0.2 +0.1 - +(b) +0.0 +0.8 - +sample +tracks +0.7 - +0.6 - +0.5 - +e0.4l +0.3 - +0.2 - +0.1 - +(c) +0.0 +100 +101 +102 +Porb (d)A&A proofs: manuscript no. TEOs_in_massive_EEVs +Fig. 6. Summary plot of the properties of the primary component of one of the arbitrarily selected binary systems from our simulations. The +approximate initial parameters of this particular system were as follows: M1 ≈ 13.6 M⊙, M2 ≈ 3.6 M⊙, e ≈ 0.4, and �rperi ≈ 2.3. The integration +of the system was terminated because of its circularisation. The top row of panels shows, from left to right, evolutionary track in the HRD, the +evolution of the orbital period and eccentricity, and the temporal changes of the wind mass-loss rate and surface rotation velocity. The vertical +dashed lines in the latter two diagrams correspond to epochs A, B, C in the HRD. The lower part of the figure shows the internal rotation profile +(left column), the mode-propagation diagram (middle column) and the synthetic oscillation spectrum (right column) for epochs A, B, C (shown +in consecutive rows labelled with these letters). The rotation frequency inside the primary is drawn as a function of fractional radius, r / R. The +range of rotation frequency is different in the three panels. The mode-propagation diagram shows the dependence of the Brunt-Väisälä frequency +(black line) and Lamb frequency for l = 2 modes (blue line) on the fractional radius. The grey and blue shaded areas correspond to the propagation +cavities of the g and l = 2 p modes, respectively. The synthetic oscillation spectrum diagrams contain several different pieces of information. The +light blue and light red horizontal bars delineate the range of frequencies allowed by the FNm values. In the background of each, the blue and red +short vertical lines indicate tidal-forcing frequencies lying within these ranges. The synthetic oscillation spectra calculated by GYRE are marked +with red (σn,2,0) and blue (σn,2,+2) long vertical lines. Their corresponding linear damping rates are plotted as solid (γn,2,0) and dashed (γn,2,2) black +lines. The frequency scale on the abscissa axis refers to the rest frame co-rotating with the primary’s core. +Article number, page 12 of 24 + +4.0 - + 0.40 +F 0.350 +-6.25 - +C +3.9 - + 0.35 +V! +B +4.35 - +6.50 - +F 0.325 +3.8 - +- 0.30 +-6.75 - +F 0.300 +B +yr- +3.7 - +- 0.25 +log (M / (Mo) +-7.00 - + 0.275 +C +e + 0.20 +-7.25 - +F 0.250 Ci +3.5 - +- 0.15 +-7.50 - +F 0.225 +3.4 + 0.10 +-7.75 - +4.20 - +F 0.200 +iA +IB +3.3 - + 0.05 +C! +-8.00 - +dA +F 0.175 +3.2 → + 0.00 +4.15 +4.46 4.44 4.42 4.40 + 4.38 +2 +4 +4.36 +0 +2 +6 +10 +6 +8 +10 +12 +4 +8 +t (Myr) +log (Teff /K) +t (Myr) +Rotational profile +Mode propagation +Oscillation spectra +140 - +10-8 - +0.487 +120 - +-10-7 , +0.486 - +）100- +9-01- +p +(d-l) +C +08 +Fs-0[- +S +60 +0.484 - +10-4 - +40 - +0.483 - +-10-3 - +20 - +-10-2 +0.482 +-0 +140 - +0.46 - +-10-8 - +120 - +-10-7 , +0.44 - +）100- +-10-6 _ + (d-1) +80 - +-10-5 +S +uul +C +-10-4 +0.40 - +40 - +ε-01- +B +20 - +0.38 - +-10-2, +-0 +0.310 : +140 - +F8-01 +NBV +l=2, m=0 modes +0.305 - +120 - +St=2 +l= 2, m= 2 modes +-10-7 +g-modes +0.300 - +100 +n,2,0 + propagation cavity +-10-6 - +→-. n,2,2 +p +I = 2 p-modes +）0.295 - +fn,0 +- 08 + propagation cavity +2 +p +10-5. +fn,2 +S +C 0.290 - + 09 +range between +-10-4 . +0.285 - +40 - +range between +fmi=? and fmax? +0.280 - +-10-3 _ +c +20 - +0.275 - +-10-2 - +-0 +0.2 +0.4 +0.8 +0.0 +0.2 +0.6 +80 +1.0 +0.0 +1.0 +0 +0.4 +2 +6 +r/R +r/R +Frequency (d-1)Kołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs +ple, when the oscillation spectrum lies completely outside the +frequency range allowed by the FNm coefficients or the nuclear +timescale of the secondary is much longer than the same time +scale for the primary. Under such circumstances, the secondary +component will remain close to the ZAMS until the termina- +tion condition is met. Thus, it will not significantly change its +internal structure and oscillation spectrum. This in turn means +that the oscillation spectrum will not move relative to the tidal +forcing frequencies, effectively reducing the number of possible +resonance events. +4.2.1. ‘Long’ resonances +Our sample of resonance curves includes a particular group of +L(t) curves that exhibit exceptionally long duration resonances +compared to typical ones (we will refer to them as ‘long res- +onances’). Figure 7 presents parts of three representative res- +onance curves belonging to this group. The shaded regions in +the figure mark the position of the long resonances. As can be +clearly seen, the typical resonance usually lasts for about 103 – +104 years, which is approximately 100 times shorter than the du- +ration of a long resonance (of the order of 105 – 106 years). They +originate from the intersection of one of the fNm frequencies with +the σnlm frequency at a very small angle, in terms of their tempo- +ral evolution. As a result, they remain for a relatively long time +in very close vicinity, leading to a broad resonance overlapping +with narrower ones (originating from other intersections of the +fNm and σnlm frequency spectra; cf. especially the middle panel +in Fig. 7). The long resonances are interesting for at least two +reasons. First of all, they are natural candidates for resonantly- +locked TEOs. However, based on our simulations, it is difficult +to say whether an extended resonance would persist when the +back-reaction of a TEO on the orbit is taken into account. Sec- +ondly, if the energy exchange between the eigenmode and the or- +bit that corresponds to a long resonance is not efficient (i.e. there +is a small chance that a long resonance will be lost), such a reso- +nance should lead to a high-amplitude TEO without the need for +resonance locking. This is simply because it has enough time to +reach its saturation level due to the non-linear effects. However, +we did not find any significant correlations between the occur- +rence of a long resonance in L(t) and the initial parameters of +our EEVs. +4.3. Total number of resonances and the average rate of +their occurrence +The first feature of the morphology of the resonance curves that +we investigated is the total number of resonances that occurred in +the primary and secondary components, Nres,1 and Nres,2. How- +ever, we did not calculate these statistics directly from L1(t) and +L2(t), because some of the apparent maxima may actually be a +blend of more than one resonance event. This is especially true +when the involved γnlm differ by orders of magnitude. Then one +of the resonances is characterised by a notably smaller maxi- +mum, which seems to ‘hide’ in the dominant one. Instead, we +used a different approach that did not underestimate the ac- +tual number of resonances. When post-processing the generated +models, we simply counted each intersection of the σn,2,0 and +σn,2,+2 frequency spectra with their counterparts fN,0 and fN,2, re- +spectively. The results of such an analysis are depicted in Fig. 8. +The most important thing about Fig. 8 is that it shows the +absolute number of resonances. EEVs can experience hundreds +or even thousands of resonances during their evolution on MS. +Fig. 7. Example resonance curves of the primary component for three +different EEVs from our simulations that exhibit long resonances (high- +lighted by the shaded areas in each panel). We note a substantial differ- +ence in the width of typical and long resonances. These long resonances +are good candidate for excitation of high-amplitude or resonantly- +locked TEOs. +It would therefore be wrong to claim that these phenomena are +rare in massive and intermediate-mass EEVs, although in gen- +eral, resonances are quite short-lived compared to the nuclear +time scale. The total number of resonances experienced by the +primary component (Fig. 8a) shows a correlation with both its +initial mass and the initial eccentricity of the system. The mildly +decreasing trend of Nres,1 towards higher M1 originates from the +fact that the mean lifetime of the star on MS shortens with in- +creasing mass. On the other hand, the wide range of Nres is +mainly due to differences in initial eccentricity. The closer the +system is to a circular geometry at the beginning of evolution, +the statistically lower the value of Nres, which is self-explanatory +and also applies to the secondary component (Fig. 8b). EEVs +that have managed to circularise their orbits in the MS phase +(magenta dots in Fig. 8c) have on average lower initial eccentric- +ities and thus fewer resonances. The opposite behaviour is exhib- +ited by EEVs in which the primary component has had a chance +to reach TAMS (green dots in Fig. 8c). The secondary compo- +nents experience a slightly fewer resonances compared to the +primaries (Fig. 8b) and there is no clear division of the Nres,2 dis- +tribution with respect to the termination criterion (Fig. 8d). The +noticeably smaller number of resonances for secondary compo- +nents with masses M2 < 5 M⊙ comes from the conditions of +our simulations, i.e. secondaries with these masses occur in sys- +Article number, page 13 of 24 + +107. +106. +L +105. +104- +22.0 +22.5 +23.0 +23.5 +24.0 +24.5 +25.0 +25.5 +106 - +L 105. +104→ +3 +4 +5 +6 +7 +8 +9 +105. +① +L +104- +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +t(Myr)A&A proofs: manuscript no. TEOs_in_massive_EEVs +Fig. 8. Total number of resonances in the pri- +mary (a, c) and secondary (b, d) components +that we detected in our simulations as a function +of the initial masses of the components. The +initial eccentricity of the EEV is colour-coded +in panels (a) and (b), while the corresponding +scale is shown at the top of the figure. Pan- +els (c) and (d) are analogous to their counter- +parts in the top row, but the colours used reflect +the termination criterion (same as in Fig. 5). +The ordinate scale is logarithmically-scaled for +Nres > 10. Below this value, a linear scale was +applied in order to present components without +resonances, i.e. Nres,1 or Nres,2 equal to zero. +tems with decreasing mass ratios19. Hence, the large difference +in nuclear time scales between the components means that the +secondary component does not significantly change its eigenfre- +quency spectrum, resulting in a smaller number of resonances. +As we already mentioned in Sect. 4.2, some of the L1(t) and +L2(t) curves do not reveal any resonances, which is why they +lie in Fig. 8 on the horizontal line Nres = 0. This behaviour oc- +curred for only 0.07% of our primaries. They are all EEVs with +highly eccentric orbits that quickly filled their Roche lobes at pe- +riastron. There was much more such behaviour for secondaries, +about 7%, mainly for the intermediate-mass companions of the +much more massive primaries. +From an observational point of view, even more important +than the total number of resonances is the rate at which they +occur. Knowing this rate, for a given population of MS EEVs, +we can approximately say where we have a statistically higher +chance of observing TEOs. After all, our observations only cor- +respond to one particular moment in time, not the entire evolu- +tion. Knowing the values of Nres for each component and the age +of each system at termination, Tmax, we can calculate the average +rate of resonances as Rres ≡ Nres/Tmax. We show the distribution +of Rres,1 and Rres,2 in Fig. 9. It is very difficult to predict what the +dependence of Rres on the mass of the component will look like, +as it is the result of a complex interplay between many related +factors. On the one hand, it can be said that massive stars should +have a smaller Rres because their lifetimes are shorter and they +fill their Roche lobes relatively easily (in the considered range of +orbital parameters). On the other hand, however, massive stars +quickly change their internal structure (i.e. asteroseismic prop- +erties), so that their eigenfrequency spectra evolve rapidly, in- +creasing the likelihood of interaction with the structure of the +19 We recall that the minimum mass of the primary component consid- +ered in our study was equal to 5 M⊙. +tidal forcing frequencies. The question is, which of these pro- +cesses prevails? As can be seen in Fig. 9, it is the more massive +stars that are more likely to undergo resonances. Both primary +and secondary components with masses around 30 M⊙ have on +average an order of magnitude higher Rres (∼102 Myr−1) than +components with masses around 5 M⊙ (∼101 Myr−1, Fig. 9a and +b). Moreover, the dependence of the distributions shown in Fig. 9 +on the initial eccentricity and termination condition is inherited +from Fig. 8. +At this point, we can venture the conclusion that in the case +of MS EEVs, TEOs should be observed mostly in the upper part +of the MS (among early B- and O-type dwarfs), which still re- +quires observational verification on a large sample of massive +EEVs. Although we cannot extrapolate the obtained distributions +of Rres towards lower masses, these stars have an increasingly +extended convective envelope, which in turn should effectively +limit the photometric detection of g-mode TEOs. On the con- +trary, the envelopes of massive stars are radiative, which should +not prevent g-mode TEOs from propagating up to the vicinity +of the photosphere. Thus, they can be more easily detected by +analysing the light curves, especially in the era of high-quality +space-borne photometry. +4.4. Distribution of resonances over time +Since the average rate of resonances we have studied so far has +effectively obliterated any differences in the corresponding tem- +poral distribution, we can ask another important question: Are +there any distinctive moments in the evolution of the simulated +EEVs during which the systems experienced temporally higher +resonance rates? After visually inspecting hundreds of resonance +curves, we noticed that the aforementioned rate changes dramat- +ically in many cases (cf. the top panel of Fig. 3 as an example). In +Article number, page 14 of 24 + +e +0.3 +0.4 +0.6 +0.5 +0.7 +(a) +103 +102 +101 +0 +103 +102 +101 +RLOF of the primary +minimum eccentricity +primary at TAMS +maximum rate of rotation +0 +5 +15 +20 +25 +30 +5 +10 +10 +15 +20 +25 +30 +Mi/Mo +M2/MoKołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs +Fig. 9. Summary plots analogous to Fig. 8, but +showing the average rate of resonances occur- +ring in the simulated EEVs (average number +of resonances per Myr). Components that did +not exhibit any resonances during the simula- +tion have been omitted here as their Rres value +would simply be zero. The colour-coding is the +same as in Fig. 8. +order to compare the temporal distribution of resonance events +for the various EEVs we are dealing with, we performed this +type of analysis on subgroups of systems divided according to +the termination condition. We also normalised the time variable +by dividing it by Tmax of each resonance curve. This allowed +us to present the whole evolution of components on a convenient +and uniform interval, [0, 1]. Figure 10 shows the results obtained +for the primary components that have managed to deplete hydro- +gen in their cores. +Figure 10 also demonstrates that the distribution discussed +here is not uniform over time. Specific areas in this diagram are +clearly distinguishable. Nevertheless, this figure still contains in- +formation on the total number of resonances, which makes it +somewhat problematic to compare the shapes of these distribu- +tions for different masses of the components. We have, therefore, +prepared histograms of the times of resonances for five inter- +vals of the primary’s initial mass (every 5 M⊙). Separate sets of +histograms were generated for the primary and secondary com- +ponents and the three main termination conditions20. All his- +tograms are shown in Fig. 11. +The most diverse structure of the temporal distribution of res- +onances is shown by systems in which the primary component +has completed its evolution in our simulations at TAMS (Fig. 11a +and b). In fact, for all mass ranges, the distribution has two dis- +tinct maxima. The smaller of the two is located near the ZAMS, +while the other is just before reaching the TAMS. Their presence +can be explained by the rate of change in the stellar eigenspec- +trum, which is the highest (after averaging over all modes) at the +aforementioned moments of evolution. In particular, the rapid +changes in the radius of the star when it is close to complete de- +20 We did not prepare separate histograms for the EEVs, whose calcu- +lations were terminated due to the maximum allowed rotation rate. The +size of this group (only eight systems) was insufficient for this task. +Fig. 10. Time distribution of the resonances of the primary component +that reached TAMS. The abscissa axis corresponds to the normalised +time and the ordinate shows the initial mass of the primary compo- +nent. In addition, the ordinate is logarithmically scaled, so the set of +resonance curves is almost uniformly distributed in the vertical direc- +tion. The total number of resonances contained in one hexagonal bin is +colour-coded according to the scale on the right. +pletion of hydrogen in its core cause a very high ‘concentration’ +of resonances in the final MS phase. The height of this domi- +nant maximum decreases towards higher masses, but at the same +time it becomes wider and wider. The distribution for the sec- +Article number, page 15 of 24 + +3×101 +2×101 - +104 +Mi /Mo +101. +103 +6× 100 - +0.2 +0.4 +0.6 +0.8 +1.0 +t / Tmaxe +0.3 +0.4 +0.5 +0.6 +0.7 +102 +101 +0 +(a) +10-2 +102 +101 +109 +C +RLOF of the primary +minimum eccentricity +10-2 +primary at TAMS +maximum rate of rotation +5 +10 +15 +20 +25 +30 +5 +10 +15 +20 +25 +30 +Mi/Mo +M2/MA&A proofs: manuscript no. TEOs_in_massive_EEVs +Fig. 11. Histograms of the normalised times of resonances occurring +in the primary (left column) and secondary (right column) components. +The consecutive rows (from top to bottom) correspond to EEVs satis- +fying different termination conditions, as labelled in panels (a), (c), and +(e). The colour of the histogram is related to the initial mass range of +the primary and is described in the legend in panel (b). We note that +the histograms on the right (corresponding to the secondaries) refer to +the different mass ranges of the primary component, not the secondary. +For example, the yellowish histogram in panel (b) summarises the be- +haviour of all secondaries of the systems with the primaries having mass +M1 > 25 M⊙, i.e. without distinguishing the mass ranges of M2. The +range of the ordinate axes is the same in each panel. +ondary components also reveals this kind of maximum near the +TAMS, which is particularly well pronounced for companions +of primaries with masses ≳ 25 M⊙. Given these facts, an inter- +esting conclusion can be drawn. Massive and intermediate-mass +EEVs with at least one component leaving the MS should expe- +rience an increased rate of encountered resonances. One might +therefore suspect that there is a statistically higher chance of ob- +serving TEOs in more evolved EEVs. The properties of the his- +tograms for EEVs in which RLOF eventually occurred at the +periastron (Fig. 11c and d) are very similar to the case described +above. The only evident difference between the two is the reduc- +tion in maximum of the distribution near TAMS. This is due to +the fact that the primary is likely to start the RLOF earlier than it +reaches the TAMS, preventing the occurrence of a large number +of resonances in a relatively short time, as mentioned earlier. +Eccentric systems that are subject to effective circularisation +(Fig. 11e and f) behave quite differently from the two previous +cases. They experience the vast majority of their resonance phe- +nomena at the beginning of evolution, and then reduce the num- +ber of resonances almost monotonically, as the orbital eccentric- +ity becomes smaller and smaller with time. Hence, the chance of +observing TEOs in initially relatively tight EEVs (cf. Fig. 5a) is +largest in the vicinity of ZAMS, which stays in contrast to the +systems described above. +4.5. Investigation of the morphology of resonance curves +using UMAP +All the analysis described above was based solely on the distri- +bution of resonances in time, i.e. neglecting the actual morphol- +ogy of the resonance curves, e.g. differences in the height and +width of resonance maxima, mean level of L(t), long-term trends +in L(t), etc. Using the dimensionality reduction techniques pre- +sented in Sect. 3.5, we constructed 2D UMAP embeddings of +the space of resonance curves in terms of their morphological +features. Figures 12 and 14 show the results obtained for the res- +onance curves of the primary and secondary component, respec- +tively. We recall that the idea of the low-dimensional embedding +performed here is to preserve the distances between two points in +the original space as accurately as possible, so that the distances +in the 2D plane reflect the distances in the full (original) space +of morphological features (the vector of 2,000 quantiles, Q). In +other words, a pair of distant points in Figs. 12 and 14 should cor- +respond to resonance curves with notably different morphologies +and ,vice versa, a pair of resonance curves with similar proper- +ties is expected to lie in mutual vicinity on the 2D UMAP plane. +Thanks to this key property of UMAP and many other dimen- +sionality reduction methods we can effectively explore the entire +space of resonance curve morphologies. +4.5.1. UMAP plane for primary components +We begin with a discussion of Fig. 12. Firstly, the presented 2D +embedding does not indicate the presence of any well-separated +groups among the resonance curves for the primary components. +This is an observation that is true over the entire range of differ- +ent values of the UMAP free parameters (Appendix D) as well as +for the different summary statistics of the resonance curves that +were considered during the preliminary experiments. The mor- +phology of the resonance curves changes smoothly depending +on to the initial parameters of the simulated EEVs. +Secondly, as can be seen in Figs. 12b and c, the initial +eccentricity and normalised periastron distance are parameters +strongly correlated with the overall morphology of the resonance +curves of the primary components. Moreover, their gradients in +the UMAP plane are approximately orthogonal. Therefore, the +pair of these parameters is the primary factor that determines +the shape of L1(t). The termination condition (Fig. 12e) gener- +ally follows the behaviour of �rperi except at small periastron dis- +tances, when the morphology remains similar but the simulations +were terminated due to hydrogen depletion in the primary’s core +or near-complete circularisation of the orbit. The initial mass of +the primary component (Fig. 12a) and its initial angular velocity +of rotation (Fig. 12d) are second-order factors shaping the mor- +phology of the L1(t) resonance curves. In the inner part of the +plane, M1 is distributed almost randomly. The clear exception is +Article number, page 16 of 24 + +(b) +Mi/Mo≤10 +(a) +5 +1025 +3 +2. +0 +(c) +(d) +5 +RLOF of the primary +0 +(e) +(f) +5 +minimum eccentricity +4 - +3 - +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +t/ TmaxKołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs +the boundary of the plane which can be roughly divided into two +parts of mostly high or low initial mass of the primary. A simi- +lar conclusion can be drawn for the initial Ω1/Ωcrit,1. This time, +however, the upper right part of Fig. 12d reveals a well-defined +group of high initial Ω1 / Ωcrit,1 and�rperi. +It is difficult to include here a complete presentation of the +changes in the morphology of L1(t) as a function of their posi- +tion on the UMAP plane. Therefore, we only focus on some ex- +treme points to present some boundary cases. Figure 13 shows +examples of L1(t) from different areas of the morphological +plane. Primary components with lower masses and high initial +eccentricities are generally characterised by resonance curves +with high mean levels and a rich set of resonances, as shown +in Fig. 13a. The resonance curve depicted in Fig. 13b repre- +sents intermediate-mass fast-rotating primary with large initial +�rperi and low initial eccentricity. Here, the base level of L1(t) +increases by an order of magnitude and then the system expe- +riences a large number of resonances, during the evolution near +TAMS. The increase in the mean value of L1(t) is characteris- +tic of stars with a high initial rotation rates. Primary components +lying between points (a) and (b) generally do not manifest this +characteristic. Moving along a straight line on the plane from (a) +to (b), the increase in the number of resonances during the evo- +lution near TAMS becomes more and more apparent. Figure 13c +shows an example of a system with a small initial eccentricity +and a short periastron distance at ZAMS that is rapidly circu- +larising. As expected, the L1(t) resonance curves for such ob- +jects have a small number of resonance maxima and a low base +level. The case corresponding to the larger initial eccentricity is +shown in Fig. 13e, where the total number of resonances is much +greater. In this case, the evolution is mainly distinguished by a +decrease in the frequency of resonances with time, related to the +efficient circularisation of the orbit, and therefore a decrease in +the mean level of L1(t). Finally, the resonance curve in Fig. 13d +is representative of the most EEVs between points (c) and (d). +They are characterised by an approximately uniform distribution +of resonances over time and an almost constant base level of the +resonance curve. +4.5.2. UMAP plane for the secondary components +The situation for the secondary component (Fig. 14) is quite dif- +ferent from the previous case. The UMAP manifold obtained for +the set of L2(t) reveals slightly more complex structure than the +shape of embedding in Fig. 12. Since the time span of L2(t) is +largely determined by the mass of the primary component, the +resonance curves for the secondary components were terminated +at times not necessarily related to their actual evolutionary sta- +tus and are statistically shorter than they could be for primaries +of the same mass. For this reason, secondary components ex- +perience, on average, fewer resonances, but, at the same time, +their resonance curves can take more diverse forms compared to +L1(t). Undoubtedly, the main factor shaping the morphology of +the L2(t) is the initial eccentricity (Fig. 14c), which with the ex- +ception for two small areas, varies smoothly across the UMAP +plane. The other parameters (Fig. 14a, b and d) play a secondary +role, showing the complex and fine structures on the plane. As +can easily be seen in Fig. 14a, the extreme cases of L2(t) in terms +of their morphology belong almost exclusively to the EEVs with +intermediate-mass secondary components hat gather at the pe- +riphery of the plane. Some of these objects even form slightly +better separated groups, isolating from the central part of the +area. +Five limiting examples of resonance curves for secondary +components are shown in Fig. 15. Panels (a), (c) and (e) to- +gether with the area they approximately enclose contain res- +onance curves morphology very similar to that described for +primary components. The resonance curves belonging to the +‘clouds’ of points labelled as (b) and (d) in Fig. 15 are com- +pletely different. They are distinguished by the complete absence +of resonances during a certain period of the evolution of the +system. The differentiating feature of these cases is the disap- +pearance of resonances from some time to the end of evolution +(Fig. 15b) or the presence of resonances only around the middle +of the considered evolution time (Fig. 15d). +5. Summary and conclusions +In our paper, we aimed to investigate the temporal variation of +conditions that favour excitation of TEOs in EEVs with massive +and intermediate-mass MS components (Sect. 2) and see how +their picture changes with different initial parameters of the sys- +tem. In order to achieve this goal, we simulated the evolution of +20,000 EEVs using the MESA software in combination with the +GYRE stellar oscillations code (Sect. 3). Our calculations started +at ZAMS and were terminated if one of the conditions presented +in Sect. 3.2.3 was met. We considered only modes with l = 2, +m = 0, +2 because they are expected to be dominant TEOs. +We also assumed that all TEOs are due to chance resonances, +i.e. we neglected the effect of TEO on the orbit. Knowing the +evolution of the orbital parameters of simulated EEVs and the +temporal changes in the eigenmode spectra of the components, +we were able to derive resonance curves L1(t) and L2(t) defined +by Eqs. (4) and (3). The equations reflect the overall resonance +conditions, and thus indirectly also the chance of TEOs, sepa- +rately for the primary and secondary components of our simu- +lated EEVs. +After visually inspecting the obtained resonance curves, cal- +culating basic statistics for them and applying ML-based meth- +ods to the entire data set, our main results can be summarised as +follows. +1. Resonance curves are characterised by striking diversity in +terms of their morphology (Sect. 4.2). EEV components can +experience a very different number of resonances, and their +distribution over time can take various forms, including the +lack of resonances over a long periods of time. We also +distinguished a group of resonance curves that exhibit pro- +longed resonances, about two orders of magnitude longer +than typical (Sect. 4.2.1, Fig. 7). These long resonances +are the potential sources of high-amplitude and resonantly- +locked TEOs. +2. Resonances between tidal forcing frequencies and the spec- +trum of stellar normal modes are not rare events among mas- +sive and intermediate-mass MS EEVs (Sect. 4.3). Although +the total number of resonances depends mostly on the initial +orbital parameters, it is typically of the order of 102 – 103 for +a given system during the entire MS phase (Fig. 8). Let us +emphasise at this point that these numbers are rather lower +limits for the actual Nres in EEVs because we considered +only l = 2 TEOs. Taking higher degree modes into account +will certainly increase the reported values of Nres. +3. On average, the more massive a star is, the higher the rate of +resonances it experiences (Sect. 4.3). For the most massive +stars in our sample (≈ 30 M⊙), the average rate of resonances +can reach ∼ 102 Myr−1, which is approximately an order of +magnitude higher than for intermediate-mass stars (Fig. 9). +Article number, page 17 of 24 + +A&A proofs: manuscript no. TEOs_in_massive_EEVs +Fig. 12. 2D UMAP embedding of the manifold spanned by the morphological features of the resonance curves of the primary components. For +details on how to obtain the presented embedding, see Sect. 3.5. Panels (a) – (d) are colour-coded with respect to the initial parameters of the +simulated EEVs, as shown on the corresponding colour bars. The other initial parameters were omitted as they were not significantly related to +the location of the points on the presented map. The different colours of points in panel (e) correspond to the termination condition, as shown in +the legend on the right. The values on the abscissa and ordinate axes were omitted as they have no physical meaning. For clarity, the colour-coded +features have been averaged within the small hexagonal areas in each panel. A discussion of the figure can be found in Sect. 4.5. +Fig. 13. Variations in the morphology of the resonance curve for the primary component across the 2D UMAP plane from Fig. 12. The middle +panel in the bottom row shows the plane with colour-coding identical to that in Fig. 12a (without hexagonal binning). Panels (a) – (e), which +surround the area, show example resonance curves that correspond to the locations on the area masked with large red dots and labelled according +to the associated panel. The positions of points (a) – (d) have been chosen in such a way as to correspond to different extreme positions in the plain, +while point (e) refers to one of the intermediate cases. A discussion of the figure can be found in Sect. 4.5. +Article number, page 18 of 24 + +(a) +(b) +(c) + 5.0 +- 0.7 +25 + 4.5 + 0.6 + 4.0 +20 +1(M) +rperi + 3.5 +10.5 e +- +- 3.0 + 0.4 + 2.5 +10 +0.3 +2.0 +1.5 +(d) +(e) + 0.45 + RLOF of the primary +- 0.40 +0.35 +0.25 +- minimum eccentricity +- 0.20 +0.15 +: maximum rate of rotation(a) +(c) +105 +106 - +L +104 +105 +0 +20 +40 +60 +0 +8. +10 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +2 +6 +(b) 106 +(p) +LLF +(e) +105 +(a) +(d) +(e) +(C +103 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +t (Myr) +t (Myr)Kołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs +Fig. 14. Same as Fig. 12, but for a set of resonance curves of the secondary components. +Fig. 15. Same as Fig. 13, but for a set of resonance curves of the secondary components. +4. The distribution of resonances over time is not homogeneous +and depends primarily on whether the system circularises be- +fore the primary reaches the TAMS or RLOF occurs at the +periastron (Sect. 4.4, Fig. 11). We noticed a particular mo- +ment in the evolution of our EEVs near the TAMS, when the +components undergo an increased number of resonances in a +relatively short time (Fig. 11a and b). +5. The low-dimensional representation of the morphology of +the resonance curves, summarised by quantile-based statis- +tics and subsequently processed by UMAP, shows that its +manifold forms a rather smooth distribution without well de- +fined (separated) groups (Sect. 4.5, Figs. 12 and 14). Less +differentiated, at least in terms of the adopted method, are +the resonance curves of the primary components, for which +the initial eccentricity and the normalised periastron dis- +tance largely determine their morphological features. Al- +though secondary components experience far fewer reso- +nances, their shapes are generally more complex due to the +Article number, page 19 of 24 + +(a) +25 +(b) +(c) + 5.0 +- 0.7 + 4.5 + 20 + 0.6 +M2(Mo) + 4.0 +15 + 3.5 , +10.5 e +3.0 + 0.4 +10 + 2.5 +- 0.3 +2.0 +a +(e) + 0.45 + RLOF of the primary +- 0.40 +0.35 +crit, 2 + 0.25 +- minimum eccentricity +- 0.20 + 0.15 +- maximum rate of rotation106 +(a) +(b) +(c) +105 +105, +L2( +104 +104 - +20 +40 +60 +80 +100 +0 +2 +6 +8 +10 +(b) +106 +(p) +(e) +106 +2(t) +(a) +L +104 +105 = +(p) +0 +10 +20 +15 +0 +2 +4. +6 +8 +t (Myr) +t (Myr)A&A proofs: manuscript no. TEOs_in_massive_EEVs +predominant influence of the primary component on evolu- +tion time. +In light of the results obtained in our study, we can draw sev- +eral interesting conclusions. Firstly, statistically speaking, TEOs +are more likely to be discovered in more massive EEVs, as their +components have a higher average rate of resonances. This does +not necessarily mean that a higher absolute number of more mas- +sive EEVs exhibiting TEOs than less massive ones will be ob- +served21. However, when comparing two particular systems, one +with intermediate-mass components and the other with much +higher masses of the components, it is for the latter that we have +a statistically higher chance that some resonance is currently +underway there. Secondly, it seems that TEOs should be espe- +cially well visible in EEVs that contain a component approach- +ing TAMS. Given these facts, the ‘extreme-amplitude’ massive +EEV, MACHO 80.7443.1718 (Jayasinghe et al. 2021; Kołaczek- +Szyma´nski et al. 2022) fits this picture almost perfectly. Its pri- +mary component is a B0.5 Ib-II supergiant leaving the MS and, +more importantly, many high-amplitude TEOs have now been +detected in this extreme system. It is possible that what the pri- +mary component of MACHO 80.7443.1718 is currently under- +going corresponds to the resonance curve shown in Fig. 13b, i.e. +it is in the phase of a high resonance rate caused by relatively fast +changes in its radius and orbital parameters. Moreover, the am- +plitudes of TEOs observed in MACHO 80.7443.1718 vary over +time notably, suggesting that we may witness rapid changes in +the resonance conditions for the primary component of this par- +ticular EEV. It would therefore be very valuable to carry out +an observational study for a large sample of EEVs to verify +whether TEOs are common in massive and intermediate-mass +EEVs whose components have already depleted most of the hy- +drogen in their cores. With high-quality space-borne photometric +observations, both operational, such as the Transiting Exoplanet +Survey Satellite (Ricker et al. 2015) or BRITE-Constellation +(Weiss et al. 2014), and planned missions (e.g. Planetary Tran- +sits and Oscillations of Stars, Rauer et al. 2014), it is definitely a +feasible task. +The excitation of g-mode TEOs, which propagate deep in- +side the star, may be an underestimated mechanism for angular +momentum (AM) transport inside the components of EEVs. It +has long been suspected that self-excited oscillations and inter- +nal gravity waves22 efficiently redistribute AM in the radial di- +rection of the star (e.g., Rogers et al. 2013; Rogers & McElwaine +2017). Although the majority of our resonances have relatively +short durations (of the order of 103 – 104 years), they can be quite +frequent (especially near the TAMS), hence the question of their +contribution to AM transport and mixing processes becomes ur- +gent for the components of EEVs. Performing calculations that +would treat the evolution of the orbit, components and TEOs in a +fully self-consistent way seems particularly interesting for mas- +sive eccentric systems leaving the MS. +We have already entered the era of observational stud- +ies of distant star-bursting galaxies and stellar populations in +low-metallicity environments, that shaped the Universe in its +early epochs. Recalling that the metal-poor stars were much +more massive than their current metal-rich counterparts (e.g., +Hosokawa et al. 2013; Susa et al. 2014), we can ask what ef- +fect metallicity has on the occurrence of TEOs in massive EEVs +21 Due to the rapid decrease of the mass function towards larger stellar +masses (e.g., Chabrier 2003). +22 Gravity waves which are stochastically driven by the turbulent con- +vective motions near the interface of the convective core and the enve- +lope (see Bowman et al. 2020, for a recent review). +and how the results we presented depend on metals content. +Therefore, future studies of the importance of TEOs in massive, +metal-poor EEVs seems worthy further investigation, especially +because of the ongoing James Webb Space Telescope mission23 +(JWST, Gardner et al. 2006), which is certain to bring many dis- +coveries in the stellar astrophysics of early stellar populations, +including stars in eccentric binary systems. +Acknowledgements. PKS is indebted to his brother, Adam Karol Kołaczek- +Szyma´nski, who oversaw the purchase and assembly of a dedicated PC +workstation to enable the efficient calculation of models in MESA and GYRE. +Without his generous help, this project would literally never have been +completed. +The authors are thankful to Prof. Andrzej Pigulski for many important sugges- +tions and fruitful discussions that made this manuscript more comprehensible, +and to the anonymous referee for many inspiring comments that helped to +improve the manuscript. +PKS +was +supported +by +the +Polish +National +Science +Center +grant +no. 2019/35/N/ST9/03805. TR was partly founded from budgetary funds +for science in 2018-2022 in a research project under the program „Diamentowy +Grant”, no. DI2018 024648. Much of this work was developed and written +during IAU Symposium 361, „Massive Stars Near & Far”, held in Ballyconnell, +Ireland, 8 – 13 May, 2022. 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However, we have intentionally omitted any controls re- +lated to the names of the files or directories where the results +should be stored. All values that changed in the inlists depending +on the simulated binary system were enclosed in square brackets +– [ · ]. The parameters adopted below resulted in a typical num- +ber of about 2,400 zones in the radial direction of the star. The +number of models calculated per single EEV was typically of the +order of several hundred, mainly depending on the termination +condition. +A.1. MESAstar inlist +&star_job +!OUTPUT +history_columns_file = "my_history_columns.list" +profile_columns_file = "my_profile_columns.list" +show_log_description_at_start = .false. +save_photo_when_terminate=.false. +!MODIFICATIONS TO MODEL +new_rotation_flag=.true. +change_rotation_flag=.true. +new_omega_div_omega_crit=[Ω/Ωcrit] +num_steps_to_relax_rotation=100 +relax_omega_max_yrs_dt = 1d4 +relax_omega_div_omega_crit=.true. +set_initial_cumulative_energy_error = .true. +new_cumulative_energy_error = 0d0 +/ ! end of star_job namelist +&eos +/ ! end of eos namelist +&kap +use_Type2_opacities = .true. +Zbase = 0.02 +/ ! end of kap namelist +&controls +!SPECIFICATIONS FOR STARTING MODEL +initial_z=0.02d0 +!CONTROLS FOR OUTPUT +terminal_interval=100 +write_header_frequency=1 +photo_interval=100000 +history_interval=5 +star_history_dbl_format = "(1pes40.6e3, 1x)" +profile_interval=10 +max_num_profile_models=5000 +write_pulse_data_with_profile=.true. +24 Details of each keyword in MESAstar v. r15140 inlist can be +found at https://docs.mesastar.org/en/r15140/reference/ +star_job.html and https://docs.mesastar.org/en/r15140/ +reference/controls.html +25 Details of each keyword in MESAbinary v. r15140 inlist can be +found at https://docs.mesastar.org/en/r15140/reference/ +binary_job.html +and +https://docs.mesastar.org/en/ +r15140/reference/binary_controls.html +pulse_data_format="GYRE" +add_double_points_to_pulse_data=.true. +!WHEN TO STOP +max_model_number = 5000 +xa_central_lower_limit_species(1)="h1" +xa_central_lower_limit(1)=1d-4 +omega_div_omega_crit_limit=0.75 +!MIXING PARAMETERS +mixing_length_alpha=1.82d0 +use_Ledoux_criterion=.true. +num_cells_for_smooth_gradL_composition_term = 0 +alpha_semiconvection=0.01d0 +okay_to_reduce_gradT_excess=.true. +mlt_make_surface_no_mixing = .true. +overshoot_scheme(1)="exponential" +overshoot_zone_type(1) = "burn_H" +overshoot_zone_loc(1) = "core" +overshoot_bdy_loc(1) = "top" +overshoot_f(1) = [fov] +overshoot_f0(1) = 0.005 +do_conv_premix=.true. +set_min_D_mix=.true. +min_D_mix=1d5 +!ROTATION CONTROLS +am_D_mix_factor=0.0333333d0 +D_DSI_factor = 1 +D_SH_factor = 1 +D_SSI_factor = 1 +D_ES_factor = 1 +D_GSF_factor = 1 +!ATMOSPHERE BOUNDARY CONDITION +atm_option="table" +atm_table="photosphere" +!MASS GAIN OR LOSS +hot_wind_scheme="Vink" +hot_wind_full_on_T=1.2d4 +cool_wind_full_on_T=0.9d3 +Vink_scaling_factor=1d0 +no_wind_if_no_rotation=.true. +mdot_omega_power=0.43d0 +max_mdot_jump_for_rotation=5d0 +rotational_mdot_kh_fac = 1.0d3 +!MESH ADJUSTMENT +max_delta_x_for_merge = 0.01d0 +max_dq=1d-3 +min_dq=1d-16 +min_dq_for_split=1d-16 +!ASTEROSEISMOLOGY CONTROLS +num_cells_for_smooth_brunt_B = 0 +!STRUCTURE EQUATIONS +use_dedt_form_of_energy_eqn = .true. +!TIMESTEP CONTROLS +min_timestep_factor=0.5d0 +max_timestep_factor=2.0d0 +dH_div_H_limit=0.5d0 +delta_lgL_phot_limit = 0.05d0 +/ ! end of controls namelist +A.2. MESAbinary inlist +&binary_job +!OUTPUT/INPUT FILES +show_binary_log_description_at_start = .false. +binary_history_columns_file = +Article number, page 22 of 24 + +Kołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs +"my_binary_history_columns.list" +!STARTING MODEL +evolve_both_stars=.true. +change_ignore_rlof_flag = .true. +new_ignore_rlof_flag = .true. +/ ! end of binary_job namelist +&binary_controls +!SPECIFICATIONS FOR STARTING MODEL +m1=[M1] +m2=[M2] +initial_eccentricity=[e] +initial_period_in_days=-1 +initial_separation_in_Rsuns=[a] +!CONTROLS FOR OUTPUT +history_interval=5 +photo_interval=100000 +terminal_interval=100 +write_header_frequency=1 +!TIMESTEP CONTROLS +fa=0.02d0 +fa_hard=0.03d0 +fr=0.10d0 +fj=0.001d0 +fj_hard=0.005d0 +fe=0.02d0 +fr_dt_limit = 1.0d2 +fdm = 1d-3 +fdm_hard = 5d-3 +dt_softening_factor = 0.3d0 +varcontrol_ms=5d-4 +varcontrol_post_ms=5d-4 +dt_reduction_factor_for_j=5d-2 +!MASS TRANSFER CONTROLS +do_enhance_wind_1=.true. +do_enhance_wind_2=.true. +tout_B_wind_1 = [Bwind] +tout_B_wind_2 = [Bwind] +!ORBITAL JDOT CONTROLS +do_jdot_gr=.true. +do_jdot_ls=.true. +do_jdot_ml=.true. +do_jdot_mb=.false. +!ROTATION AND SYNC CONTROLS +do_tidal_sync=.true. +sync_type_1="Hut_rad" +sync_type_2="Hut_rad" +!ECCENTRICITY CONTROLS +do_tidal_circ=.true. +circ_type_1="Hut_rad" +circ_type_2="Hut_rad" +anomaly_steps=300 +/ ! end of binary_controls namelist +Appendix B: GYRE input file +The GYRE stellar oscillations code requires a single input file that +collects all the user-specified parameters of the asteroseismic +calculations being performed26. Below is our example file +gyre.in. As in Appendix A, we have omitted any keywords +related to specific file names and have highlighted variables by +26 Details of each keyword in GYRE v. 6.0.1 input file can be found +at +https://gyre.readthedocs.io/en/v6.0.1/ref-guide/ +input-files.html +enclosing them in square brackets. +&constants +/ +&model +model_type = "EVOL" +file_format = "MESA" +/ +&mode +l = 2 +m = 0 +n_pg_min = -30 +n_pg_max = 30 +tag = "m0" +/ +&mode +l = 2 +m = 2 +n_pg_min = -30 +n_pg_max = 30 +tag = "m2" +/ +&osc +inner_bound = "REGULAR" +outer_bound = "VACUUM" +adiabatic = .true.’ +nonadiabatic = .true.’ +/ +&rot +coriolis_method = "TAR" +Omega_rot_source = "MODEL" +/ +&num +ad_search = "BRACKET" +nad_search = "AD" +diff_scheme = "MAGNUS_GL2" +/ +&scan +grid_type = "LINEAR" +freq_min = [ f m=0 +min ] +freq_max = [ f m=0 +max ] +n_freq = [Nm=0 +freq ] +freq_units = "CYC_PER_DAY" +grid_frame = "INERTIAL" +freq_frame = "INERTIAL" +tag_list = "m0" +/ +&scan +grid_type = "LINEAR" +freq_min = [ f m=+2 +min +] +freq_max = [ f m=+2 +max ] +n_freq = [Nm=+2 +freq ] +freq_units = "CYC_PER_DAY" +grid_frame = "COROT_I" +freq_frame = "COROT_I" +tag_list = "m2" +/ +&grid +/ +&ad_output +/ +&nad_output +summary_file_format = "TXT" +Article number, page 23 of 24 + +A&A proofs: manuscript no. TEOs_in_massive_EEVs +summary_item_list = "freq,l,m,n_p,n_g,n_pg" +freq_units = "CYC_PER_DAY" +freq_frame = "INERTIAL" +The frequency scan limits, f m=0 +min , f m=0 +max , f m=+2 +min +, and f m=+2 +max , +were calculated as described in Sect. 3.3. The total numbers of +discrete frequency points, Nm=0 +freq , and Nm=+2 +freq , were obtained as +follows, +Nm=0,+2 +freq += ⌈( f m=0,+2 +max +− f m=0,+2 +min +)/(0.005 d−1)⌉, +(B.1) +where ⌈ · ⌉ denotes the ceiling function. +Appendix C: Data used by MESA +Our work uses the MESA stellar evolution code, which incorpo- +rates a vast compilation of knowledge, mainly from micro- and +macrophysics, collected by many authors. The MESAeos mod- +ule is a mixture of OPAL (Rogers & Nayfonov 2002), SCVH +(Saumon et al. 1995), FreeEOS (Irwin 2004), HELM (Timmes +& Swesty 2000), PC (Potekhin & Chabrier 2010) and Skye +(Jermyn et al. 2021) equation of states. Radiative opacities are +taken primarily from OPAL (Iglesias & Rogers 1993, 1996), +with low-temperature data from Ferguson et al. (2005) and +the high-temperature, Compton-scattering dominated regime by +Poutanen (2017). Electron conduction opacities are from Cas- +sisi et al. (2007) and Blouin et al. (2020). Nuclear reaction rates +are from JINA REACLIB (Cyburt et al. 2010), NACRE (An- +gulo et al. 1999) and additional tabulated weak reaction rates +from Fuller et al. (1985), Oda et al. (1994) and Langanke & +Martínez-Pinedo (2000). Screening is included via the prescrip- +tion of Chugunov et al. (2007). Thermal neutrino loss rates are +taken from Itoh et al. (1996). Roche lobe radii in binary systems +are computed using the fit of Eggleton (1983). +Appendix D: Adjustable parameters of the UMAP +UMAP, as a highly flexible method, is prone to returning mis- +leading results in the case of inappropriately set free parame- +ters. On the one hand, they can lead to the appearance of spuri- +ous groups and, on the other hand, to the loss of finer topologi- +cal structure. The vital UMAP parameters that need adjustment +are n_neighbors, min_dist, n_components and metric. The +n_neighbors parameter is the most important, as it controls +the balance between the local and global structure present in +the data that will be mapped to the embedding. We experi- +mented with different values of this parameter ranging from 5 to +1,000 (the default is 15) and concluded that the resonance curves +(summarised by the proposed statistics) always form a single +group, almost independently of the choice of n_neighbors. +Later, min_dist sets the minimum distance between two dif- +ferent points on the embedding. We tested its values from 0.0 to +0.5 and do not observe any significant effect on manifold. We +took the last two parameters, namely n_components that spec- +ifies the number of dimensions of the embedding, and metric +specifying the metric used for similarity calculation, as default +values. Finally, we used the following set of free parameters: +n_neighbors = 500, min_dist = 0.1, n_components = 2 +and metric = ’euclidean’. +Article number, page 24 of 24 +