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Stellar mergers as the origin of magnetic massive stars

Abstract.About ten per cent of ‘massive’ stars (those of more than 1.5 solar masses) have strong, large-scale surface magnetic fields1,2,3. It has been suggested that merging of main-sequence and pre-main-sequence stars could produce such strong fields4,5, and the predicted fraction of merged massive stars is also about ten per cent6,7. The merger hypothesis is further supported by a lack of magnetic stars in close binaries8,9, which is as expected if mergers produce magnetic stars. Here we report three-dimensional magnetohydrodynamical simulations of the coalescence of two massive stars and follow the evolution of the merged product. Strong magnetic fields are produced in the simulations, and the merged star rejuvenates such that it appears younger and bluer than other coeval stars. This can explain the properties of the magnetic ‘blue straggler’ star τ Sco in the Upper Scorpius association that has an observationally inferred, apparent age of less than five million years, which is less than half the age of its birth association10. Such massive blue straggler stars seem likely to be progenitors of magnetars, perhaps giving rise to some of the enigmatic fast radio bursts observed11, and their supernovae may be affected by their strong magnetic fields12. Access provided by Main.We conduct three-dimensional (3D) ideal magnetohydrodynamical (MHD) simulations of the merger of a 9-Myr-old binary consisting of a 9Mʘ and an 8Mʘ (where Mʘ is the solar mass) core-hydrogen burning star with the moving-mesh code AREPO13, which is ideally suited for such simulations (see Methods). The binary configuration and evolutionary stage are chosen such that the resulting merger product is expected to have a total mass similar to that of τ Sco (about 17Mʘ; ref. 10) and that the binary could have formed at the same time as did other stars in the Upper Scorpius association.Snapshots of the density ρ, a passive scalar indicating material from the primary star, and of the absolute magnetic-field strength B of the 3D MHD simulation are shown in Fig. 1. Upon contact of the binary, a dynamical phase of mass transfer with rates as high as 17Mʘ yr−1 sets in from the more massive (initially 9Mʘ) primary star onto the less massive (initially 8Mʘ) secondary star. Mass is lost through the outer Lagrangian points, draining angular momentum and thereby accelerating the coalescence. The accretion stream shears on the surface of the secondary star and it is in this accretion stream, of size 0.8 solar radii (Rʘ), that the magnetic field is amplified on an e-folding timescale of about 0.2–1 d (Fig. 1d, g). The maximum magnetic-field strength saturates at about 108 G, which corresponds to an amplification of the magnetic energy by a factor of 1018 (Extended Data Fig. 2). At saturation, the magnetic energy is comparable to the turbulent energy (about 5%–30%), which is the source of the magnetic-field amplification (Extended Data Fig. 3). In the final merger, the amplified field is advected throughout the merger product and is therefore also present in the core of the merger remnant. When the primary star is disrupted around the secondary and the cores of the two stars merge, vortices form at the interface of the two former cores (Fig. 1e) that further contribute to the magnetic-field amplification (see also Supplementary Video 1). The maximum ratio of magnetic to gas pressure reaches 30% in localized regions but is less than 1% in the phase leading up to the merger.Fig. 1: Dynamical evolution of the merger of two main-sequence stars.Panels a–c show density snapshots in the orbital plane; panels j–l are edge-on views of the density. The passive scalar (white colour; panels d–f) indicates material from the 9Mʘ primary and thus visualizes the mixing of the two progenitor stars during the merger. The passive scalar and the magnetic-field strengths (panels g–i) are shown in the orbital plane. The times given in each panel are relative to the time when the cores of the two stars coalesce (panels b, e, h, and k).Full size image The local conditions in the differentially rotating accretion stream (rotational frequency of Ω ≈ 10 d−1, Alfvén velocity of about 1 km s−1 and rotational shear of q = −d lnΩ/d lnr ≈ 0.4) indicate that the magneto-rotational instability14 is the key agent providing the turbulence needed to exponentially amplify the magnetic fields. In the shearing layer, the fastest-growing mode of the magneto-rotational instability has a characteristic size of 0.1Rʘ and growth timescale of 0.5 d (ref. 15), in agreement with the size of the accretion stream and the observed growth timescale of the magnetic fields in our simulation.Because of the large amount of angular momentum, a torus of 3Mʘ forms that surrounds the central, spherically symmetric 14Mʘ core of the merger product (Fig. 1c, l). The central merger remnant is in solid-body rotation while the centrifugally supported torus rotates at sub-Keplerian velocities. The innermost core of the merger remnant consists of material from the former secondary star while the torus is dominated by core material from the former primary star (Fig. 1f).We continue the 3D MHD simulation for 10 d after the actual merger, that is, about 5 d after the merger remnant has settled into its final core-torus structure. This corresponds to roughly 5 Alfvén crossing timescales through the 14Mʘ core and we do not observe large changes in the magnetic field structure and strength. The ratio of toroidal to total magnetic field energy is 80%–85%, which is in a regime where magnetic-field configurations are thought to be stable in stellar interiors16. Because of the high conductivity of stellar plasmas, Ohmic decay of the field occurs only on a timescale similar to or even longer than the stellar lifetime (see Methods). We therefore expect the magnetic field to be long-lived.Most of the torus is expected to be accreted rapidly onto the central merger remnant and to form an extended envelope (see Methods). For the long-term evolution of the merger remnant we therefore assume that the innermost 16.9Mʘ end up in the merger product and that less than 0.1Mʘ remain in a disk (Extended Data Fig. 1). Under these assumptions, we follow the further evolution of the merger product in the one-dimensional (1D) stellar evolution code Mesa17. As suggested by the 3D MHD simulations, we assume the formed remnant to rotate rigidly at a rate close to break-up. The magnetic flux at the end of our 3D simulation at a mass coordinate of 16.9Mʘ is 3.5 × 1023 G cm2, such that the surface magnetic-field strength of the merger remnant on the main sequence would be 9 kG for a radius of about 5Rʘ (assuming magnetic flux conservation). This is well within observed surface field strengths of magnetic stars1,2,3. Because it is impossible to follow the evolution of an inherently 3D magnetic field in a 1D stellar evolution code, we assume that the radial magnetic-field strength in our 1D model follows that of a magnetic dipole. It contributes to internal angular-momentum transport and additional angular-momentum loss from the surface through a magnetized stellar wind (magnetic braking) but has otherwise no influence on the structure and evolution (see Methods).Because of the coalescence, the stellar interior is heated and the star is out of thermal equilibrium. A thermal relaxation phase sets in, during which the star reaches a maximum radius of about 200Rʘ and a luminosity of 2.5 × 105Lʘ, (where Lʘ is the solar luminosity) before it contracts back to the main sequence, after which it continues its evolution in a manner similar to that of a genuine single star of initially 16.9Mʘ (Fig. 2). During the thermal expansion, the star reaches critical rotation, leading to additional mass (less than 0.01Mʘ) and angular-momentum loss (roughly 7% of the star’s total angular momentum). In the subsequent contraction phase, the surface spins down from critical rotation to about 50 km s−1 (about 10% of critical rotation). This is not driven by angular-momentum loss but by an internal restructuring of the star (see also Methods). The spin of the merger product on the main sequence is thus set by the angular momentum that remains in the merger product after the viscous accretion of the torus and the corresponding outward angular-momentum transport.Fig. 2: Long-term evolution of the merger product in the Hertzsprung–Russell diagram.After most of the torus is accreted, the merger remnant rotates rapidly and a thermal relaxation phase sets in, during which the star first expands before it contracts back to the main sequence (grey line). The direction of evolution is indicated by the black arrow at the beginning of this phase. The colour-coding shows the surface rotational velocity, vrot, in terms of the critical Keplerian velocity, vcrit. Once on the main sequence, the merger product continues its evolution similar to that of a genuine single star of the same mass of 16.9Mʘ (orange line). The black hatched rectangle indicates observations of τ Sco (31,000 K ≤ Teff ≤ 33,000 K, 4.3 ≤ logL/Lʘ ≤ 4.5; see Table 1). The small cartoons are artist’s impressions of key evolutionary phases: (1) the contact phase before the actual merger, (2) the merger product with its torus, (3) during the thermal relaxation as a critically rotating star shedding mass, (4) as a main-sequence star with a strong surface magnetic field and (5) after the terminal supernova explosion that may form a magnetar.Full size image Once back in thermal equilibrium, the merger product is a slow rotator with effective temperature, luminosity and surface gravity in agreement with τ Sco (Table 1 and Fig. 2). This outcome is independent of the magnetic field chosen in the 1D models. Consequently, our merger product also looks like a rejuvenated blue straggler compared to other, apparently older stars in the Upper Scorpius association, mainly because of the shorter lifetime associated with the now more-massive star. τ Sco is enriched in nitrogen on the surface but this is currently not reproduced by our model. However, on average, the envelope of our merger model is nitrogen-rich because it is made out of core material of the former primary star (Fig. 1f). These enriched layers could easily be exposed by further mass loss or could be mixed to the stellar surface. For example, we have not considered mixing induced by the magnetic fields or during the viscous evolution of the torus. In conclusion, our merger scenario is able to explain the magnetic nature, atmospheric parameters, slow rotation and blue-straggler status of τ Sco.Table 1 Comparison of our merger model with τ ScoFull size table Strong amplification of magnetic fields is also observed in the coalescence of white dwarfs18, the merger of neutron stars19, and the common-envelope phase of a star spiralling into the envelope of a giant companion20. Thus, mergers of stars in general seem to provide the right conditions to produce strong magnetic fields. The coalescence of other main-sequence stars (having, for example, lower masses) with stars in different evolutionary phases (such as pre-main-sequence stars) is also expected to generate strong magnetic fields. Merging is therefore also a promising mechanism with which to explain the origin of magnetic fields in Ap stars and their suggested remnants, white dwarfs with surface field strengths in excess of 106 G (ref. 5).The magnetic flux in the innermost 1.5Mʘ of our merger model at the end of the MHD simulation is about 4 × 1028 G cm2. If all of the magnetic flux is conserved until core collapse of the merger product occurs, a resulting neutron star of radius 10 km would have a surface magnetic-field strength of about 1016 G. Such strong fields are thought to affect the explosions of core-collapse supernovae12 and appear to be able to explain the strong magnetic fields inferred for magnetars (1013–1015 G)21, which may give rise to some of the enigmatic fast radio bursts11. The birthrate of magnetars in our Galaxy of about 0.3 per century22 and the rate of occurrence of Galactic core-collapse supernovae of about 2 per century23 suggest that 15% of all Galactic core-collapse supernovae have produced a magnetar, which is consistent with the 10% incidence of magnetic massive stars24. Super-luminous supernovae and long-duration gamma-ray bursts have been suggested to be powered by rapidly rotating and highly magnetized cores25,26. Because of its slow rotation, our merger model is not expected to result in such events, in line with the low rate of these transients (less than 0.1% of core-collapse supernovae)24. However, rarer merger cases such as the coalescence of stellar cores in a common-envelope event27 could form rapidly rotating and highly magnetized stellar cores that may then power long-duration gamma-ray bursts and super-luminous supernovae. Taken together, this enables our merger model to explain the strong magnetic fields observed in a subset of massive stars and potentially also the origin of magnetars.Methods.3D MHD merger simulations.AREPO’s MHD solver.The AREPO code uses a second-order finite-volume method to solve the ideal MHD equations on an unstructured grid13,31,32. The grid is generated in each timestep from a set of mesh-generating points that move along with the flow, thus ensuring a nearly Lagrangian behaviour while regularizing the mesh by adding an additional term. The fluxes over the cell boundaries are computed using the HLLD solver and the divergence of the magnetic field is effectively controlled by employing the Powell scheme33, as shown in refs 31, 32. We use ideal MHD here because the resistivity is very small in the highly conducting plasma of stellar interiors (see Ohmic dissipation below).Initialization of the binary progenitor.The binary progenitors have an initial helium mass fraction of Y = 0.2703 and solar metallicity Z = 0.0142 (ref. 34). The stellar structures are imported from 1D Mesa17,35,36,37 models (version 9793) that employ exponential convective-core overshooting with a parameter of fov = 0.019, which effectively corresponds to a step convective-core overshooting of about 0.16 pressure scale heights.Mapping the 1D stellar structures into a 3D hydrodynamics code leads to discretization errors in the hydrostatic equilibrium; thus, a relaxation method38 is employed to create stable stellar models. The 1D stellar models from MESA are mapped onto an unstructured grid consisting of HEALPix distributions on spherical shells39. In the ensuing AREPO simulations, spurious velocities are damped away, resulting in stable stellar models according to the criteria outlined in ref. 38. The initial seed magnetic field was set up in a dipole configuration with a polar surface field strength of 1 μG.The relaxed single-star models are subsequently used to set up the binary star merger. It is computationally not feasible to simulate the merger beginning from Roche-lobe overflow until the actual merger occurs. We therefore speed up the merging process by artificially decelerating each cell for a certain time (about 1.5 orbits) and starting the calculation from then. The duration of this deceleration phase influences the outcome of the merger only marginally (see resolution study below).Resolution study and initial conditions.We ran simulations for different resolutions and initial binary setups to ensure that the amplification of the magnetic field is robust against variations of these parameters. The evolution of the total magnetic field energy over time is shown in Extended Data Fig. 2. The standard run shown in the main text is Model 1. The evolution of the magnetic energy is slightly different for the various configurations but the overall behaviour and the final energy are very similar. Model 2 tests a lower resolution for otherwise identical initial conditions. Model 3 was started at an earlier time with a larger initial separation. The resolution was set up with roughly 4 × 106 cells for Model 1 and about 4 × 105 cells for the other models.Magnetic-field saturation.The magnetic-field amplification switches off if the necessary physical conditions of the amplification process are no longer met. For the magneto-rotational instability14, this could be the case if the magnetic field becomes so strong that the fastest-growing mode exceeds the spatial region of interest (for example, it becomes larger than the star), if the amplification timescale becomes excessively long or if there is no longer differential rotation. Such a situation may go along with an equipartition of the magnetic energy with the energy source (for example, differential or turbulent energy) that drives the magnetic-field amplification.In our models, the initially fast, exponential magnetic-field amplification is consistent with being driven by the magneto-rotational instability. After the merger, the central star is in solid-body rotation such that the magneto-rotational instability cannot operate within the central star any more. In the torus, however, the magneto-rotational instability is still active and the magnetic-field strength is indeed found to increase until the end of the simulation. The fastest-growing magneto-rotational instability mode always fits into the central merger remnant and the magneto-rotational instability amplification timescale stays short compared to the runtime of our simulation.Using the kinetic energy in the radial and z directions as a proxy for the turbulent energy, which is generally thought to power the magnetic-field amplification, we find that the magnetic energy reaches a level of about 5%–30% of the kinetic energy (Extended Data Fig. 3). This supports the idea that the magnetic-field amplification ceases when approaching energy equipartition.Ohmic dissipation of magnetic fields.Stable magnetic fields can diffuse out of the stellar interior by Ohmic resistivity and thereby dissipate16. However, because the hot stellar interior is highly conducting, the resistivity is low and the dissipation of magnetic fields is slow. Indeed, for Spitzer’s resistivity40:$$\eta =7\times 1{0}^{11}ln\varLambda {\left(T\right)}^{-3/2}{{\rm{cm}}}^{2}{{\rm{s}}}^{-1}$$ (1) with T the temperature in Kelvin and lnΛ the Coulomb logarithm, which is of the order of 10 for stellar interiors. The diffusion timescale of magnetic fields is τdiff = R2/η ≈ 108–1011 yr for temperatures of 105–107 K and a length scale of R = 1Rʘ. These estimates depend on the still-uncertain resistivity in stellar interiors but it appears that Ohmic dissipation of the amplified magnetic fields does not play a part in this merger because the lifetime of the merger product is instead of the order of 107 yr. It might, however, be relevant for some evolved stars16.1D long-term evolution of the merger product.The amplified magnetic fields are too weak to affect the stellar structure directly. However, they can contribute to the angular-momentum transport through the stellar interior and may lead to additional angular-momentum loss from the stellar surface (magnetic braking). Below, we describe the assumed magnetic field structure in our 1D stellar evolution models, and our implementation of the interior angular-momentum transport through the magnetic fields and magnetic braking. We then explain how our 1D models are set up, on the basis of the outcome of the 3D MHD simulations and provide more details on the spin-down of the merger product. As before, we use the Mesa stellar evolution code in version 979317,35,36,37.Assumed large-scale magnetic field in 1D computations.It is not possible to follow the evolution of a 3D magnetic field in a 1D stellar evolution code. Moreover, the final configuration of the magnetic field after the accretion of the torus is uncertain at present. Hence, we assume that the radial magnetic-field strength in our 1D model follows that of a magnetic dipole, B(r) = μBr−3, (where r is the radius of the product of the merged stars) with dipole moment μB = 2 × 1037 G cm3. This assumption is conservative in the sense that it results in a surface magnetic field of the merger product on the main sequence of a few hundred Gauss, which is lower than that expected from magnetic flux freezing of our 3D model but reminiscent of that of τ Sco41. Using larger or smaller magnetic-dipole moments does not affect our conclusions. The dipole field diverges for r → 0 and we therefore cap its field strength at 109 G.We further assume that the magnetic field is expelled from convective regions if the convective energy density uconv is larger than the magnetic energy density uB, that is, if:$${u}_{{\rm{conv}}}=\frac{1}{2}\rho {v}_{{\rm{conv}}}^{2} > {u}_{{\rm{B}}}=\frac{{B}^{2}}{8{\rm{\pi }}}$$ (2) Here, ρ is the gas density and vconv is the velocity of convective eddies as predicted by mixing-length theory. This treatment of the static magnetic field means that it contributes only to the angular-momentum transport in radiative regions.Angular-momentum transport in the stellar interior through a large-scale magnetic field.We treat the transport of angular momentum through the stellar interior as a diffusive process. Magnetic fields cause Maxwell stresses and can thus transport angular momentum. To obtain the effective diffusion coefficient of this process (which we call effective viscosity, νeff), we consider differentially rotating, spherical shells and assume that the stresses due to magnetic fields are effectively similar to the classical Newtonian dynamic shear, S:$$S=\frac{{\rm{d}}F}{{\rm{d}}A}={\nu }_{{\rm{eff}}}\rho \frac{\partial v}{\partial r}$$ (3) where dF is the force exerted by the shear on an area dA and ∂v/∂r is the radial gradient of the velocity v. In spherical coordinates (r, φ, θ), the torque dτ on a surface element dA = r2sinθdφdθ due to a shear force dF is:$${\rm{d}}\tau =r\sin \theta {\rm{d}}F=r\sin \theta {\nu }_{{\rm{eff}}}\rho \frac{\partial v}{\partial r}{\rm{d}}A$$ (4) Introducing the angular velocity Ω (v = rsinθΩ), we have ∂v/∂Ω = rsinθ and thus ∂v/∂r = rsinθ∂Ω/∂r. Integrating equation (4) over φ and θ, we obtain the overall torque on a shell at radius r as:$$\tau =\frac{8{\rm{\pi }}}{3}{\nu }_{{\rm{eff}}}{r}^{4}\rho \frac{\partial \varOmega }{\partial r}$$ (5) From a physical point of view, the shear exerted by magnetic fields reduces differential rotation and attempts to establish solid-body rotation (∂Ω/∂r = 0). The amount of angular momentum ΔJ that needs to be transported to achieve solid-body rotation in neighbouring, differentially rotating shells is ΔJ = IΔΩ, where I is the moment of inertia. The angular-momentum transport across a shell of thickness Δr occurs with an Alfvén velocity \({v}_{{\rm{A}}}=B/\sqrt{4{\rm{\pi }}\rho }\) , that is, on an Alfvén timescale of τA = Δr/vA, such that:$$\frac{{\rm{d}}J}{{\rm{d}}t}\approx \frac{\Delta J}{{\tau }_{{\rm{A}}}}=\frac{I\left(\partial \varOmega /\partial r\right)\Delta r}{{\tau }_{{\rm{A}}}}=I\left(\frac{\partial \varOmega }{\partial r}\right){v}_{A}$$ (6) Equating equation (6) and equation (5), we find the desired effective viscosity for angular-momentum transport in shells rotating differentially because of a large-scale magnetic field:$${\nu }_{{\rm{eff}}}=\frac{3I}{8{\rm{\pi }}{r}^{4}\rho }{v}_{{\rm{A}}}$$ (7) The moment of inertia of a single shell in a stellar evolution model depends on the spatial discretization. To make the effective diffusion coefficient independent of resolution of the spatial discretization of the stellar model, we define ‘shells’ to have a thickness of 20% of the local pressure scale height HP. We modulate the effective viscosity with a factor fA that is thought to adjust the timescale over which solid-body rotation is achieved in neighbouring shells. We set fA = 0.5 in our calculations and note that small variations in fA hardly change our results.In the above analysis, we have not made explicit assumptions about the magnetic-field geometry, but it will of course matter in reality. For example, if there is no radial magnetic-field component, the Maxwell stress is zero, such that there is no angular-momentum transport in the radial direction through the magnetic field. In our approach, the field geometry enters indirectly through the Alfvén velocity, which depends on the absolute magnetic-field strength, which itself is a function of radius r.Magnetic braking.Stellar winds can couple to large-scale magnetic fields and thereby enhance the loss of angular momentum, a process called magnetic braking. The torque on the stellar surface from magnetic braking is:$$\frac{{\rm{d}}{J}_{{\rm{mb}}}}{{\rm{d}}t}=\frac{2}{3}\dot{M}{\varOmega }_{\ast }{R}_{{\rm{A}}}^{2}$$ (8) where \(\dot{M}\) is the stellar wind mass loss rate, Ω∗ is the stellar surface angular velocity and the factor 2/3 accounts for the moment of inertia of thin spherical shells.In MHD simulations of magnetic braking of hot, massive stars42, the Alfvén radius in equation (8) is found to be:$$\frac{{R}_{{\rm{A}}}}{{R}_{\ast }}\approx 0.29+{\left({\eta }_{\ast }+0.25\right)}^{1/4}$$ (9) with R∗ the stellar radius and η∗ the wind magnetic confinement parameter:$${\eta }_{\ast }=\frac{{B}_{{\rm{eq}}}^{2}{R}_{\ast }^{2}}{\dot{M}{v}_{\infty }}$$ (10) where Beq is the equatorial, surface magnetic-field strength and v∞ is the terminal wind velocity42. For the terminal wind velocity, we use observational results for O to F stars43.From a technical point of view, stellar winds in stellar evolution codes take away the specific angular momentum of their former Lagrangian mass shells. In our models, the additional angular momentum lost through magnetic braking is then taken away from a thin surface layer after the mass shells that are lost in the wind have been removed.Import of the merger remnant into a 1D stellar evolution code.Immediately after the merger, the evolution is driven mainly by that of the torus and its interplay with the central star. Two timescales are most relevant: the accretion and cooling timescales, τacc and τcool, respectively. The accretion timescale sets the time over which the torus is accreted by the central remnant whereas the cooling timescale describes the time over which the torus loses the heat produced by the accretion.Matter in the rotationally supported torus can only be accreted onto the central star if its angular momentum is transported outwards; hence, the accretion timescale is given by the angular-momentum transport timescale. We assume that the matter and angular momentum flow in the torus can be described by an α-disk model with an effective viscosity α that, for example, might be provided by the magnetic fields or the magneto-rotational instability14. Here and throughout, the term ‘viscous’ is used in the phenomenological sense of an effective viscosity which acts on large scales owing to the presence of an enhanced turbulent transport. It should not be confused with the true microscopic particle viscosity, which is negligible for the problems of interest here. Using the mass accretion rate for such an α-disk model:$${\dot{M}}_{{\rm{acc}}}\approx 3\alpha {\left(\frac{h}{r}\right)}^{2}\varOmega {M}_{{\rm{disk}}}$$ (11) The accretion timescale of the torus is then:$${\tau }_{{\rm{acc}}}=\frac{{M}_{{\rm{disk}}}}{{\dot{M}}_{{\rm{acc}}}}=\frac{1}{3}\frac{{r}^{2}}{\alpha {h}^{2}\varOmega }=0.02\;{\rm{yr}}\left(\frac{1{0}^{-2}}{\alpha }\right){\left(\frac{r/h}{2}\right)}^{2}\left(\frac{{h}^{-1}}{\varOmega }\right)$$ (12) Here, h/r is the ratio of disk height and radius, Mdisk is the mass in the disk and Ω is the angular velocity of the disk, which generally depends on radius. In our case, the accretion timescale is equivalent to the viscous timescale τvisc.Mass accretion leads to (turbulent) heating through the release of gravitational potential energy, Egrav. On the one hand, if this energy can be lost efficiently from the system via fast cooling (τcool ≪ τacc), the torus becomes thinner or at least keeps its shape. On the other hand, if the cooling is inefficient (τcool ≫ τacc), the torus becomes thicker and evolves into a thermally supported extended envelope. Assuming that the star-torus structure radiates at a fraction fEdd of its Eddington luminosity, LEdd, and that photon cooling is the dominant cooling process, the cooling timescale can be approximated as:$$\begin{array}{c}{\tau }_{{\rm{cool}}}=\frac{{E}_{{\rm{grav}}}}{{f}_{{\rm{Edd}}}{L}_{{\rm{Edd}}}}=\frac{1}{{f}_{{\rm{Edd}}}}\frac{G{M}_{{\rm{core}}}{M}_{{\rm{disk}}}/{R}_{{\rm{core}}}}{4{\rm{\pi }}G({M}_{{\rm{core}}}+{M}_{{\rm{disk}}})c/\kappa }\\ \approx 0.8\times 1{0}^{3}{\rm{yr}}\left(\frac{1}{{f}_{{\rm{Edd}}}}\right)\left(\frac{1+X}{1.7}\right)\left(\frac{{M}_{{\rm{core}}}{M}_{{\rm{disk}}}/{M}_{\odot }}{{M}_{{\rm{core}}}+{M}_{{\rm{disk}}}}\right)\left(\frac{{R}_{\odot }}{{R}_{{\rm{core}}}}\right)\end{array}$$ (13) where Mcore and Rcore are the mass and radius of the central star, and κ is the opacity. In the last step, we assumed that the opacity is dominated by electron scattering, that is, κ = 0.2(1 + X) cm2 g−1 with the hydrogen mass fraction X. Even for fEdd = 1, the cooling time in our case is of the order of 500–700 yr (Mcore = 14Mʘ, Mdisk = 3Mʘ, 3 ≤ Rcore/Rʘ ≤ 4) and thus much longer than the accretion timescale in equation (12). The expectation therefore is that the torus is rapidly accreted onto the central star and evolves into a thermally supported, extended envelope before its thermal relaxation and cooling process sets in.These arguments are analogous to previous work on the merger remnant of two white dwarfs44. More detailed simulations of the viscous evolution of this double white-dwarf merger remnant45 support the (analytic) expectations44 and indeed show a rapid transformation of the torus into a thermally supported envelope. Given the similarity of the physical situation and the ratio of accretion and cooling timescales in our case, it seems reasonable that large fractions of our torus will also evolve into a thermally supported envelope on a viscous timescale.The accretion and cooling timescales (Eqs. (12) and (13)) depend on radius through the radially declining angular velocity Ω of the torus and the radius Rcore at which matter is accreted onto the central star. At a radius of about 54Rʘ, both timescales are comparable such that cooling is inefficient inside and efficient further out (Extended Data Fig. 1). In our standard model, we therefore assume that the mass interior of 54Rʘ—that is, the innermost 16.9Mʘ—transforms into a star with an extended envelope on a viscous timescale. The remaining outer part of the torus is assumed to cool efficiently and evolve into a thin disk. This configuration then forms the initial condition of our 1D stellar evolution computations.To import the outcome of the 3D simulation into the 1D stellar evolution code MESA, we model a star that has the same chemical and thermal structure as the 3D merger remnant. We first relax a star of given total mass to the chemical structure of the 3D merger remnant before imposing the thermal structure by matching the 3D entropy profile. A comparison of the chemical and entropy structure of the 1D 16.9Mʘ merger remnant with the 3D profiles is shown in Extended Data Fig. 4. Our 1D model closely matches the structure of the merger remnant of the 3D simulation.Setting the rotational profile of the merger remnant requires further consideration. We argued above that the fast viscous evolution of the star-torus structure converts most of the torus into an extended envelope by transporting angular momentum outwards. This angular-momentum transport sets the initial conditions for our 1D merger evolution. In the viscous evolution of the remnant of a double white-dwarf merger, efficient outward angular-momentum transport is found such that the rotational profile of the central star remains a solid-body rotator and smoothly transitions into a near-Keplerian profile at the boundary between the central star and outer disk45. The same evolution and outcome has been found by others who studied the aftermath of double-white-dwarf mergers within a prescribed viscosity model but also within more self-consistent MHD simulations44,46. In all cases, a large fraction of angular momentum is transported outwards, allowing for the rapid accretion of a large fraction of the torus.Our 3D merger simulation also shows that the central star reaches solid-body rotation with the angular velocity matching that of the layer between star and torus, which is approximately 80% of the Keplerian value. The surface of the central star does not reach 100% Keplerian rotation because the torus is not only centrifugally supported but also thermally supported. We therefore assume that our merger remnant is a solid-body rotator that rotates at 90% of the critical Keplerian velocity at the surface after the viscous evolution.Restructuring of the stellar interior during the thermal relaxation phase after the merger.During the thermal relaxation, the merger product first approaches critical surface rotation before the model spins down rapidly (Fig. 2 and Extended Data Fig. 5). This spin-down is not driven by angular-momentum loss but can be understood as follows. The internal magnetic field keeps the star close to solid-body rotation such that the total angular momentum J of the star is \(J={r}_{{\rm{g}}}^{2}M{R}_{\ast }^{2}{\varOmega }_{\ast }\) with the stellar radius R∗ , the moment of inertia factor \({r}_{{\rm{g}}}^{2}\) , the stellar mass M and the angular velocity Ω∗ . For constant angular momentum J and mass M, the surface rotational velocity evolves according to:$${v}_{{\rm{rot}}}=\frac{J}{{r}_{{\rm{g}}}^{2}M{R}_{\ast }}\propto {\left({r}_{{\rm{g}}}^{2}{R}_{\ast }\right)}^{-1}$$ (14) In the contraction phase when the star spins down (about 102–104 yr after the merger), the radius decreases by a factor of 4 while \({r}_{{\rm{g}}}^{2}\) increases by a factor of 20, fully explaining the observed spin-down of the merger product by a factor of about 5 (Extended Data Fig. 5). This change in \({r}_{{\rm{g}}}^{2}\) is because, after the coalescence, the core of the merger is hotter and denser than in full equilibrium, which leads to core expansion while the envelope contracts (Extended Data Fig. 5). Because magnetic braking is unimportant for the spin-down, our conclusions regarding the final spin and surface properties of the merger product are almost independent of the magnetic field structure and strength used in the 1D stellar models. 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The post-merger magnetized evolution of white dwarf binaries: the double-degenerate channel of sub-Chandrasekhar type Ia supernovae and the formation of magnetized white dwarfs. Astrophys. J. 773, 136 (2013).ADS.Article. Google Scholar. Download references .Acknowledgements.This work was supported by the Oxford Hintze Centre for Astrophysical Surveys, which is funded through generous support from the Hintze Family Charitable Foundation. S.T.O., F.K.R. and F.R.N.S. acknowledge funding from the Klaus Tschira foundation.Author information.Author notes.These authors contributed equally: Fabian R. N. Schneider, Sebastian T. Ohlmann.Affiliations.Zentrum für Astronomie der Universität Heidelberg, Astronomisches Rechen-Institut, Heidelberg, Germany.Fabian R. N. Schneider.Heidelberger Institut für Theoretische Studien, Heidelberg, Germany.Fabian R. N. Schneider., Sebastian T. Ohlmann. & Friedrich K. Röpke.Department of Physics, University of Oxford, Oxford, UK.Fabian R. N. Schneider., Philipp Podsiadlowski. & Steven A. Balbus.Max Planck Computing and Data Facility, Garching, Germany.Sebastian T. Ohlmann.Zentrum für Astronomie der Universität Heidelberg, Institut für Theoretische Astrophysik, Heidelberg, Germany.Friedrich K. Röpke.Max-Planck-Institut für Astrophysik, Garching, Germany.Rüdiger Pakmor. & Volker Springel.Authors.Search for Fabian R. N. Schneider in:.PubMed • . Google Scholar .Search for Sebastian T. Ohlmann in:.PubMed • . Google Scholar .Search for Philipp Podsiadlowski in:.PubMed • . Google Scholar .Search for Friedrich K. Röpke in:.PubMed • . Google Scholar .Search for Steven A. Balbus in:.PubMed • . Google Scholar .Search for Rüdiger Pakmor in:.PubMed • . Google Scholar .Search for Volker Springel in:.PubMed • . Google Scholar .Contributions.F.R.N.S. initiated the project and carried out the 1D MESA computations. S.T.O. carried out the 3D AREPO simulations. F.R.N.S. and S.T.O. wrote most of the manuscript. P.P. and F.K.R. assisted with the 1D and 3D computations, respectively. S.A.B. in particular helped to analyse and understand the magnetic-field amplification process. V.S. and R.P. wrote the AREPO code and supported S.T.O. with the 3D simulations. All authors contributed to the analysis, discussion and writing of the paper.Corresponding authors.Correspondence to Fabian R. N. Schneider or Sebastian T. Ohlmann.Ethics declarations. Competing interests. The authors declare no competing interests.Additional information.Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Peer review information Nature thanks Christopher Tout and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.Extended data figures and tables.Extended Data Fig. 1 Ratio of viscous and cooling timescales..The τvisc/τcool ratio is shown as a function of radius (and mass) of the merger after 6 d for different disk thicknesses h and viscosity parameters α.Extended Data Fig. 2 Evolution of total magnetic field energy for different simulation setups..Model 1 is the standard run shown in the main text. Models 2 and 3 have a lower resolution. Model 3 started with a larger initial separation. The times for all models are normalized with the time of merger set to 0.Extended Data Fig. 3 Ratio of magnetic and radial, kinetic (that is, turbulent) energy in our 3D MHD simulations..The magnetic energies, EB, of our models approach equipartition with the turbulent energy, for which we use the kinetic energy of motions in the radial and z directions, Ekin,r,z, as a proxy. Inset, the ratio of magnetic and kinetic energy on a linear scale, showing that our models reach values of about 5%–30%. The dashed horizontal line indicates equipartition of magnetic and kinetic energy. The small, periodic wiggles in the curves before coalescence are caused by the orbital motion of the binary.Extended Data Fig. 4 Comparison of the final 3D and initial 1D merger product..a, b, The entropy (a) and the hydrogen (H1), helium (He4), carbon (C12), nitrogen (N14) and oxygen (O16) mass fractions (b) of the 3D merger remnant (thick grey lines) and the 1D stellar model (dashed lines) are compared.Extended Data Fig. 5 Rotational velocity and internal mass readjustment of the 1D merger model..a, Equatorial surface rotational velocity vrot (blue solid line) and rotational velocity in terms of critical Keplerian velocity vcrit (red dashed line) as a function of time after the merger. b, Radial location of various mass coordinates in steps of 5% of the total mass (see percentage labels for the black dotted and dashed lines) and moment of inertia factor \({r}_{{\rm{g}}}^{2}\) (blue solid line) as a function of time. The black solid line indicates the stellar surface.Supplementary information. Video 1.Magnetic-field evolution in the orbital plane. Similar to Fig. 1g–i, in this video the evolution of the absolute magnetic-field strength is colour-coded while the geometry of the magnetic field is visualised using line-integral convolution. This illustrates that the magnetic field is initially amplified locally before it organises itself on larger scales.Rights and permissions.Reprints and Permissions.About this article.Received.08 April 2019Accepted.05 August 2019Published.09 October 2019Issue Date.10 October 2019DOI.https://doi.org/10.1038/s41586-019-1621-5 Comments.By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

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