Rate Of Angular Momentum Transport Research Articles

A Study of the Accretion State of Magnetically Arrested Disks across Black Hole Spins for Radiatively Inefficient Accretion Flows

The study of magnetically arrested disks (MAD) has attracted strong interest in recent years because these disk configurations were found to generate strong jets, as observed in many accreting systems. Here, we present the results of 14 general relativistic magnetohydrodynamic simulations of advection-dominated accretion flow in the MAD state across black hole (BH) spins, carried out with cuHARM. Our main findings are as follows. (i) The jets transport a significant amount of angular momentum to infinity in the form of Maxwell stresses. For positive, high spin, the rate of angular momentum transport is about five times higher than for negative spin. This contribution is nearly absent for a nonrotating BH. (ii) The mass accretion rate and the MAD parameter, both calculated at the horizon, are not correlated. However, their time derivatives are anticorrelated for every spin. (iii) For zero spin, the contribution of the toroidal component of the magnetic field to the magnetic pressure is negligible, while for a fast-spinning BH, it is on the same order as the contribution of the radial magnetic component. For high positive spin, the toroidal component even dominates. (iv) For negative spins, the jets are narrower than their positive-spin counterparts, while their fluctuations are stronger. The weak jet from the nonrotating BH is the widest with the weakest fluctuations. Our results highlight the complex nonlinear connection between the black hole spin and the resulting disk and jet properties in the MAD regime.

Effects of the Planetary Field on the Accretion Process of a Planet in the Final Stage of Giant Planet Formation

Recent studies suggest that the magnetic field may play an important role in the formation of giant planets during the final stage of the formation process. In this paper, we construct a circumplanetary system around a planet that is in the final formation stage to investigate the effects of the planetary magnetic field on the accretion process of the planet at this stage. We find that at the early times of our magneto-hydrodynamic (MHD) simulation, the magnetic loops connecting the planet and the circumplanetary material inflate upward due to the build-up of the toroidal field pressure with magnetic islands forming inside the loops, which changes the flow pattern above the planet in comparison with the hydrodynamic case. We find that a low-density gap, which is produced by the strong magnetic pressure, appears along the disk’s surface and expands as the system evolves. Accompanied by the accretion flow above the disk surface, the disk surface field lines can reconnect with the magnetic loops anchored on the planet. Thus, the material above the disk’s surface can permeate into the loops anchored on the planet via reconnection and can be accreted to the planet through the loops. Comparing the results in the hydrodynamic and MHD simulations, we find that the mass accretion rate and the angular momentum transport rate from the infalling gas to the planet do not change apparently when the effects of the magnetic field are included, but these rates from the circumplanetary disk to the planet increase significantly by an order of magnitude.

Strong disk winds traced throughout outbursts in black-hole X-ray binaries.

Recurring outbursts associated with matter flowing onto compact stellar remnants (such as black holes, neutron stars and white dwarfs) in close binary systems provide a way of constraining the poorly understood accretion process. The light curves of these outbursts are shaped by the efficiency of angular-momentum (and thus mass) transport in the accretion disks, which has traditionally been encoded in a viscosity parameter, α. Numerical simulations of the magneto-rotational instability that is believed to be the physical mechanism behind this transport yield values of α of roughly 0.1-0.2, consistent with values determined from observations of accreting white dwarfs. Equivalent viscosity parameters have hitherto not been estimated for disks around neutron stars or black holes. Here we report the results of an analysis of archival X-ray light curves of 21 outbursts in black-hole X-ray binaries. By applying a Bayesian approach to a model of accretion, we determine corresponding values of α of around 0.2-1.0. These high values may be interpreted as an indication either of a very high intrinsic rate of angular-momentum transport in the disk, which could be sustained by the magneto-rotational instability only if a large-scale magnetic field threads the disk, or that mass is being lost from the disk through substantial outflows, which strongly shape the outburst in the black-hole X-ray binary. The lack of correlation between our estimates of α and the accretion state of the binaries implies that such outflows can remove a substantial fraction of the disk mass in all accretion states and therefore suggests that the outflows correspond to magnetically driven disk winds rather than thermally driven ones, which require specific radiative conditions.

Quasi-two-dimensional nonlinear evolution of helical magnetorotational instability in a magnetized Taylor–Couette flow

Magnetorotational instability (MRI) is one of the fundamental processes in astrophysics, driving angular momentum transport and mass accretion in a wide variety of cosmic objects. Despite much theoretical/numerical and experimental efforts over the last decades, its saturation mechanism and amplitude, which sets the angular momentum transport rate, remains not well understood, especially in the limit of high resistivity, or small magnetic Prandtl numbers typical to interiors (dead zones) of protoplanetary disks, liquid cores of planets and liquid metals in laboratory. Using direct numerical simulations, in this paper we investigate the nonlinear development and saturation properties of the helical magnetorotational instability (HMRI)—a relative of the standard MRI—in a magnetized Taylor–Couette flow at very low magnetic Prandtl number (correspondingly at low magnetic Reynolds number) relevant to liquid metals. For simplicity, the ratio of azimuthal field to axial field is kept fixed. From the linear theory of HMRI, it is known that the Elsasser number, or interaction parameter determines its growth rate and plays a special role in the dynamics. We show that this parameter is also important in the nonlinear problem. By increasing its value, a sudden transition from weakly nonlinear, where the system is slightly above the linear stability threshold, to strongly nonlinear, or turbulent regime occurs. We calculate the azimuthal and axial energy spectra corresponding to these two regimes and show that they differ qualitatively. Remarkably, the nonlinear state remains in all cases nearly axisymmetric suggesting that this HMRI-driven turbulence is quasi two-dimensional in nature. Although the contribution of non-axisymmetric modes increases moderately with the Elsasser number, their total energy remains much smaller than that of the axisymmetric ones.

Gravitational Instabilities in Circumstellar Disks

Star and planet formation are the complex outcomes of gravitational collapse and angular momentum transport mediated by protostellar and protoplanetary disks. In this review, we focus on the role of gravitational instability in this process. We begin with a brief overview of the observational evidence for massive disks that might be subject to gravitational instability and then highlight the diverse ways in which the instability manifests itself in protostellar and protoplanetary disks: the generation of spiral arms, small-scale turbulence-like density fluctuations, and fragmentation of the disk itself. We present the analytic theory that describes the linear growth phase of the instability supplemented with a survey of numerical simulations that aim to capture the nonlinear evolution. We emphasize the role of thermodynamics and large-scale infall in controlling the outcome of the instability. Despite apparent controversies in the literature, we show a remarkable level of agreement between analytic predictions and numerical results. In the next part of our review, we focus on the astrophysical consequences of the instability. We show that the disks most likely to be gravitationally unstable are young and relatively massive compared with their host star, Md/M*≥0.1. They will develop quasi-stable spiral arms that process infall from the background cloud. Although instability is less likely at later times, once infall becomes less important, the manifestations of the instability are more varied. In this regime, the disk thermodynamics, often regulated by stellar irradiation, dictates the development and evolution of the instability. In some cases the instability may lead to fragmentation into bound companions. These companions are more likely to be brown dwarfs or stars than planetary mass objects. Finally, we highlight open questions related to the development of a turbulent cascade in thin disks and the role of mode-mode coupling in setting the maximum angular momentum transport rate in thick disks.

Dark matter trapping by stellar bars: the shadow bar

We investigate the complex interactions between the stellar disc and the dark-matter halo during bar formation and evolution using N-body simulations with fine temporal resolution and optimally chosen spatial resolution. We find that the forming stellar bar traps dark matter in the vicinity of the stellar bar into bar-supporting orbits. We call this feature the shadow bar. The shadow bar modifies both the location and magnitude of the angular momentum transfer between the disc and dark matter halo and adds 10 per cent to the mass of the stellar bar over 4 Gyr. The shadow bar is potentially observable by its density and velocity signature in spheroid stars and by direct dark matter detection experiments. Numerical tests demonstrate that the shadow bar can diminish the rate of angular momentum transport from the bar to the dark matter halo by more than a factor of three over the rate predicted by dynamical friction with an untrapped dark halo, and thus provides a possible physical explanation for the observed prevalence of fast bars in nature.

Spiral-driven accretion in protoplanetary discs

Accretion discs are ubiquitous in the universe and it is a crucial issue to understand how angular momentum and mass are being radially transported in these objects. Here, we study the role played by non-linear spiral patterns within hydrodynamical and non self-gravitating accretion disc assuming that external disturbances such as infall onto the disc may trigger them. To do so, we computed self-similar solutions that describe discs in which a spiral wave propagates. Such solutions present both shocks and critical sonic points that we carefully analyze. For all allowed temperatures and for several spiral shocks, we calculated the wave structure. In particular we inferred the angle of the spiral patern, the stress it exerts on the disc as well as the associated flux of mass and angular momentum as a function of temperature. We quantified the rate of angular momentum transport by means of the dimensionless $\alpha$ parameter. For the thickest disc we considered (corresponding to $h/r$ values of about 1/3), we found values of $\alpha$ as high as $0.1$, and scaling with the temperature $T$ such that $\alpha \propto T^{3/2} \propto (h/r)^3$. The spiral angle scales with the temperature as $\arctan(r/h)$. The existence of these solutions suggests that perturbations occurring at disc outer boundaries, such as for example perturbations due to infall motions, can propagate deep inside the disc and therefore should not be ignored, even when considering small radii.

MHD simulations of the magnetorotational instability in a shearing box with zero net flux: the case Pm = 4

<i>Aims. <i/>This letter investigates the transport properties of MHD turbulence induced by the magnetorotational instability at large Reynolds numbers <i>Re<i/> when the magnetic Prandtl number <i>Pm<i/> is larger than unity.<i>Methods. <i/>Three MHD simulations of the magnetorotational instability (MRI) in the unstratified shearing box with zero net flux are presented. These simulations are performed with the code Zeus and consider the evolution of the rate of angular momentum transport as <i>Re<i/> is gradually increased from 3125 to 12 500 while simultaneously keeping <i>Pm<i/> = 4. To ensure that the small scale features of the flow are well resolved, the resolution varies from 128 cells per disk scaleheight to 512 cells per scaleheight. The latter constitutes the highest resolution of an MRI turbulence simulation to date.<i>Results. <i/>The rate of angular momentum transport, measured using the <i>α<i/> parameter, depends only very weakly on the Reynolds number: <i>α<i/> is found to be about 7 × 10<sup>-3<sup/> with variations around this mean value bounded by 15% in all simulations. There is no systematic evolution with <i>Re<i/>. For the best resolved model, the kinetic energy power spectrum tentatively displays a power-law range with an exponent -3/2, while the magnetic energy is found to shift to smaller and smaller scales as the magnetic Reynolds number increases. A couple of different diagnostics both suggest a well-defined injection length of a fraction of a scaleheight.<i>Conclusions. <i/>The results presented in this letter are consistent with the MRI being able to transport angular momentum efficiently at large Reynolds numbers when <i>Pm<i/> = 4 in unstratified zero net flux shearing boxes.

MHD turbulence in accretion disks: the importance of the magnetic Prandtl number

The magnetorotational instability (MRI) is the most likely source of MHD turbulence in accretion disks. Recently, it has been realized that microscopic diffusion coefficients (viscosity and resistivity) are important in determining the saturated state of the turbulence and thereby the rate of angular momentum transport. In this paper, we use a set of numerical simulations performed with a variety of numerical methods to investigate the dependance of α , the rate of angular momentum transport, on these coefficients. We show that α is an increasing function of the magnetic Prandtl number Pm , the ratio of viscosity over resistivity. In the absence of a mean field, we also find that MRI–induced MHD turbulence decays at low Pm .

Applications of the Miller equilibrium to extend tokamak computational models

An analytical model for the equilibrium flux surface geometry, widely known as the Miller equilibrium model, has been exploited to improve a variety of simplified tokamak computation models that incorporate an approximate flux surface geometry. Also discussed are improved models for an effective unelongated toroidal plasma representation of elongated flux surfaces; mapping temperature (and density) gradients measured at one poloidal location to other poloidal locations and to an average gradient over the flux surface; interpreting experimental heat diffusivities from local temperature gradients and average conductive heat fluxes; calculating the poloidal distribution of radial conductive heat fluxes; and evaluating the gyroviscous angular momentum transport rate.

MHD simulations of the magnetorotational instability in a shearing box with zero net flux

We study the influence of the choice of transport coefficients (viscosity and resistivity) on MHD turbulence driven by the magnetorotational instability (MRI) in accretion disks. We follow the methodology described in paper I: we adopt an unstratified shearing box model and focus on the case where the net vertical magnetic flux threading the box vanishes. For the most part we use the finite difference code ZEUS, including explicit transport coefficients in the calculations. However, we also compare our results with those obtained using other algorithms (NIRVANA, the PENCIL code and a spectral code) to demonstrate both the convergence of our results and their independence of the numerical scheme. We find that small scale dissipation affects the saturated state of MHD turbulence. In agreement with recent similar numerical simulations done in the presence of a net vertical magnetic flux, we find that turbulent activity (measured by the rate of angular momentum transport) is an increasing function of the magnetic Prandtl number Pm for all values of the Reynolds number Re that we investigated. We also found that turbulence disappears when the Prandtl number falls below a critical value Pm_c that is apparently a decreasing function of Re. For the limited region of parameter space that can be probed with current computational resources, we always obtained Pm_c>1. We conclude that the magnitudes of the transport coefficients are important in determining the properties of MHD turbulence in numerical simulations in the shearing box with zero net flux, at least for Reynolds numbers and magnetic Prandtl numbers that are such that transport is not dominated by numerical effects and thus can be probed using current computational resources.

MHD simulations of the magnetorotational instability in a shearing box with zero net flux

Aims. We study the properties of MHD turbulence driven by the magnetorotational instability (MRI) in accretion disks. To do this we perform a series of numerical simulations for which the resolution is gradually increased. Methods. We adopt the local shearing box model and focus on the special case for which the initial magnetic flux threading the disk vanishes. We employ the finite difference code ZEUS to evolve the ideal MHD equations. Results. Performing a set of numerical simulations in a fixed computational domain with increasing resolution, we demonstrate that turbulent activity decreases as resolution increases. The highest resolution considered is 256 grid cells per scale height. We quantify the turbulent activity by measuring the rate of angular momentum transport through evaluating the standard a parameter. We find a = 0.004 when (N x , N Y , N z ) = (64,100,64), a = 0.002 when (N x , N y , N z ) = (128,200,128) and a = 0.001 when (N x , N y , N z ) = (256,400, 256). This steady decline is an indication that numerical dissipation, occurring at the grid scale is an important determinant of the saturated form of the MHD turbulence. Analysing the results in Fourier space, we demonstrate that this is due to the MRI forcing significant flow energy all the way down to the grid dissipation scale. We also use our results to study the properties of the numerical dissipation in ZEUS. Its amplitude is characterised by the magnitude of an effective magnetic Reynolds number Re M which increases from 10 4 to 10 5 as the number of grid points is increased from 64 to 256 per scale height. Conclusions. The simulations we have carried out do not produce results that are independent of the numerical dissipation scale, even at the highest resolution studied. Thus it is important to use physical dissipation, both viscous and resistive, and to quantify contributions from numerical effects, when performing numerical simulations of MHD turbulence with zero net flux in accretion disks at the resolutions normally considered.

TOWARDS A NEW STANDARD THEORY FOR ASTROPHYSICAL DISK ACCRETION

We briefly review recent developments in black hole accretion disk theory, placing new emphasis on the vital role played by magnetohydrodynamic (MHD) stresses in transporting angular momentum. The apparent universality of accretion-related outflow phenomena is a strong indicator that vertical transport of angular momentum by large-scale MHD torques is important and may even dominate radial transport by small-scale MHD turbulence. This leads to an enhanced overall rate of angular momentum transport and allows accretion of matter to proceed at an interesting rate. Furthermore, we argue that when vertical transport is important, the radial structure of the accretion disk is modified and this affects the disk emission spectrum. We present a simple model demonstrating that energetic, magnetically-driven outflows give rise to a disk spectrum that is dimmer and redder than a standard accretion disk accreting at the same rate. We briefly discuss the implications of this key result for accreting black holes in different astrophysical systems.

Towards a new standard model for black hole accretion

We briefly review recent developments in black hole accretion disk theory, emphasizing the vital role played by magnetohydrodynamic (MHD) stresses in transporting angular momentum. The apparent universality of accretion-related outflow phenomena is a strong indicator that large-scale MHD torques facilitate vertical transport of angular momentum. This leads to an enhanced overall rate of angular momentum transport and allows accretion of matter to proceed at an interesting rate. Furthermore, we argue that when vertical transport is important, the radial structure of the accretion disk is modified at small radii and this affects the disk emission spectrum. We present a simple model demonstrating how energetic, magnetically-driven outflows modify the emergent disk emission spectrum with respect to that predicted by standard accretion disk theory. A comparison of the predicted spectra against observations of quasar spectral energy distributions suggests that mass accretion rates inferred using the standard disk model may be severely underestimated.

Simulations of Magnetorotational Instability in a Magnetized Couette Flow

In preparation for an experimental study of magnetorotational instability (MRI) in liquid metal, we present non-ideal two-dimensional magnetohydrodynamic simulations of the nonlinear evolution of MRI in the experimental geometry. The simulations adopt initially uniform vertical magnetic fields, conducting radial boundaries, and periodic vertical boundary conditions. No-slip conditions are imposed at the cylinders. Our linear growth rates compare well with existing local and global linear analyses. The MRI saturates nonlinearly with horizontal magnetic fields comparable to the initial axial field. The rate of angular momentum transport increases modestly but significantly over the initial state. For modest fluid and magnetic Reynolds numbers Re, Rm between 100 and 1000, the final state is laminar reduced mean shear except near the radial boundaries, and with poloidal circulation scaling as the square root of resistivity, in partial agreement with the analysis of Knobloch and Julien. A sequence of simulations at Rm=20 and 100 < Re < 25,600 enables extrapolation to the experimental regime (Rm ~ 20, Re ~ 10^7), albeit with unrealistic boundary conditions. MRI should increase the experimentally measured torque substantially over its initial purely hydrodynamic value.

Shearing Box Simulations of the MRI in a Collisionless Plasma

We describe local shearing box simulations of turbulence driven by the magnetorotational instability (MRI) in a collisionless plasma. Collisionless effects may be important in radiatively inefficient accretion flows, such as near the black hole in the Galactic center. The MHD version of ZEUS is modified to evolve an anisotropic pressure tensor. A fluid closure approximation is used to calculate heat conduction along magnetic field lines. The anisotropic pressure tensor provides a qualitatively new mechanism for transporting angular momentum in accretion flows (in addition to the Maxwell and Reynolds stresses). We estimate limits on the pressure anisotropy due to pitch angle scattering by kinetic instabilities. Such instabilities provide an effective collision rate in a collisionless plasma and lead to more MHD-like dynamics. We find that the MRI leads to efficient growth of the magnetic field in a collisionless plasma, with saturation amplitudes comparable to those in MHD. In the saturated state, the anisotropic stress is comparable to the Maxwell stress, implying that the rate of angular momentum transport may be moderately enhanced in a collisionless plasma.

Jet rotation: Launching region, angular momentum balance and magnetic properties in the bipolar outflow from RW Aur

Using STIS on board the HST we have obtained a spectroscopic map of the bipolar jet from RW Aur with the slit parallel to the jet axis and moved across the jet in steps of . After applying a velocity correction due to uneven slit illumination we find signatures of rotation within the first 300 AU of the jet (15 at the distance of RW Aur). Both lobes rotate in the same direction (i.e. with different helicities), with toroidal velocities in the range 5–30 km s-1 at 20 and 30 AU from the symmetry axis in the blueshifted and redshifted lobes, respectively. The sense of rotation is anti-clockwise looking from the tip of the blue lobe (PA 130° north to east) down to the star. Rotation is more evident in the [OI] and [NII] lines and at the largest sampled distance from the axis. These results are consistent with other STIS observations carried out with the slit perpendicular to the jet axis, and with theoretical simulations. Using current magneto-hydrodynamic models for the launch of the jets, we find that the mass ejected in the observed part of the outflow is accelerated from a region in the disk within about 0.5 AU from the star for the blue lobe, and within 1.6 AU from the star for the red lobe. Using also previous results we estimate upper and lower limits for the angular momentum transport rate of the jet. We find that this can be a large fraction (two thirds or more) of the estimated rate transported through the relevant portion of the disk. The magnetic lever arm (defined as the ratio between the Alfvèn and footpoint radii) is in the range 3.5–4.6 (with an accuracy of 20–25%), or, alternatively, the ejection index is in the range 0.025–0.046 (with similar uncertainties). The derived values are in the range predicted by the models, but they also suggest that some heating must be provided at the base of the flow. Finally, using the general disk wind theory we derive the ratio of the toroidal and poloidal components of the magnetic field at the observed location (i.e. about 80–100 AU above the disk). We find this quantity to be at 30 AU from the axis in the red lobe and at 20 AU from the axis in the blue lobe (assuming cylindrical coordinates centred on the star and with positive z along the blue lobe). The toroidal component appears to be dominant, which would be consistent with magnetic collimation of the jet. The field appears to be more tightly wrapped on the blue side.

Nonlinear Evolution of the Magnetorotational Instability in Ion‐Neutral Disks

We carry out three-dimensional magnetohydrodynamical simulations of the magnetorotational (Balbus-Hawley) instability in weakly ionized plasmas. We adopt a formulation in which the ions and neutrals are treated as separate fluids coupled only through a collisional drag term. Ionization and recombination processes are not considered. The linear stability of the ion-neutral system has been previously considered by Blaes & Balbus. Here we extend their results to the nonlinear regime by computing the evolution of the Keplerian angular momentum distribution in the local shearing box approximation. We find significant turbulence and angular momentum transport when the collisional frequency is on the order of 100 times the orbital frequency Ω. At higher collision rates, the two-fluid system studied here behaves much like the fully ionized systems studied previously. At lower collision rates, the evolution of the instability is determined primarily by the properties of the ions, with the neutrals acting as a sink for the turbulence. Since in this regime saturation occurs when the magnetic field is superthermal with respect to the ion pressure, we find that the amplitude of the magnetic energy and the corresponding angular momentum transport rate is proportional to the ion density. Our calculations show that the ions and neutrals are essentially decoupled when the collision frequency is less than 0.01Ω; in this case, the ion fluid behaves as in the single-fluid simulations and the neutrals remain quiescent. We find that purely toroidal initial magnetic field configurations are unstable to the magnetorotational instability across the range of coupling frequencies.

Nonlinear Outcome of Gravitational Instability in Optically Thick Disks

AbstractIn the outer parts of disks in active galactic nuclei and around young stellar objects (YSOs) the internal sources of heat may be too weak to prevent gravitational instability. We are thus motivated to examine the nonlinear outcome of gravitational instability in an optically thick disk. We use height-integrated, two-dimensional numerical simulations that include pressure (the gas has a polytropic equation of state), self-gravity, and cooling. The cooling function is calculated using a one-zone model and is characterized by a single dimensionless parameter τcool, which is the ratio of the cooling time to the orbital period. The simulations are done in the context of a local model of the disk (the “shearing sheet”). We also include the possibility of an effective “viscosity” characterized by a dimensionless parameter α.We find that the gravitational instability in thin, Keplerian disks leads to sustained angular momentum transport that is essentially local in character. Instability drives Q toward 1.7, and provides an angular momentum flux αeff ≃ 0.4/τcool. The disk breaks up when the cooling time is shorter than the dynamical time, which is equivalent to saying that αeff &lt; 1. We conclude that gravitational instability can raise the angular momentum transport rate in the outer parts of AGN and YSO disks (although YSO disks may be too thick for our local approach to be strictly applicable). This can occur over a factor of a few in radius. The gravitationally dominated region is bounded on the inside by that radius where other heating processes (e.g. MHD turbulence driven by the Balbus-Hawley instability) make Q &gt; 1. It is bounded on the outside by that radius where the cooling time is shorter than the dynamical time, or the disk is optically thin so that the equation of state is soft. At larger radius the disk will be clumpy.

The Solar-stellar Connection: Magnetic Braking and Stellar Activity

AbstractThe theory of coronal structure and of the associated angular momentum transport by magneto-thermo-centrifugal winds is reviewed, with reference to recent observations of rapidly rotating late-type stars. In particular, the combined optical and X-ray studies of AB Doradus are consistent with the predicted picture of a hot, magnetically-controlled ‘dead zone’ in near corotation with the star, and of cooler wind-emitting regions (analogues of the solar coronal holes). The standard theory predicts a rate of angular momentum transport that depends on the extent of the wind zone, the relative importance of thermal and centrifugal driving, and the variation of the dynamo-built surface field with rotation. For stars like the Sun, with approximately one tenth of the mass in the convective envelope, detailed comparison of the theory with the observed rotations of stars in clusters of different ages is sensitive to the strength of the coupling between the envelope and the radiative core, and is constrained by helioseismological limitations on rotational shear within the present-day Sun. An inferred strong braking during the early epochs may hint at a non-standard braking model, with the Maxwell stresses dominant far from as well as near the stellar surface.