Momentum Loss Rate Research Articles

Time evolution of the Von Neumann entropy for a Kerr–Taub–NUT black hole

In this work, we study the evolution of an evaporating black hole, described by the Kerr–Taub–NUT metric, which emits scalar particles. We found that allowing the black hole to radiate massless scalar particles increases the angular momentum loss rate while decreasing the loss rate of the NUT parameter and black hole mass. In fact, it means that angular momentum will disappear faster than the other black hole parameters (mass and NUT parameter) during the evaporation process. We also calculate the time evolution of the mass, angular momentum, and NUT parameter in order to get the evolution of the Von Neumann entropy of the black hole. We found that the entropy follows approximately the so-called Page curve, where the β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document} parameter, which quantifies the amount of radiation, affects the evaporation process. Implying that high β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document} values accelerate the evaporation process of a Kerr–Taub–NUT black hole.

Suggested magnetic braking prescription derived from field complexity fails to reproduce the cataclysmic variable orbital period gap

Magnetic wind braking drives the spin-down of low-mass stars and the evolution of most interacting binary stars. A magnetic braking prescription that was claimed to reproduce both the period distribution of cataclysmic variables (CVs) and the evolution of the rotation rates of low-mass stars is based on a relation between the angular momentum loss rate and magnetic field complexity. The magnetic braking model based on field complexity has been claimed to predict a detached phase that could explain the observed period gap in the period distribution of but has never been tested in detailed models of evolution. Here we fill this gap. We incorporated the suggested magnetic braking law in MESA and simulated the evolution of for different initial stellar masses and initial orbital periods. We find that the prescription for magnetic braking based on field complexity fails to reproduce observations of The predicted secondary star radii are smaller than measured, and an extended detached phase that is required to explain the observed period gap (a dearth of non-magnetic with periods sim 2$ and $ sim 3$ hours ) is not predicted. Proposed magnetic braking prescriptions based on a relation between the angular momentum loss rate and field complexity are too weak to reproduce the bloating of donor stars in derived from observations and, in contrast to previous claims, do not provide an explanation for the observed period gap. The suggested steep decrease in the angular momentum loss rate does not lead to detachment. Stronger magnetic braking prescriptions and a discontinuity at the fully convective boundary are needed to explain the evolution of close binary stars that contain compact objects. The tension between braking laws derived from the spin-down of single stars and those required to explain and other close binaries containing compact objects remains.

Simulated Coronal Mass Ejections on a Young Solar-type Star and the Associated Instantaneous Angular Momentum Loss

Coronal mass ejections (CMEs) on stars can change the stars’ magnetic field configurations and mass-loss rates during the eruption and propagation and therefore, may affect the stars’ rotation properties on long timescales. The dynamics of stellar CMEs and their influence on the stellar angular momentum loss rate are not yet well understood. In order to start investigating these CME-related aspects on other stars, we conducted a series of magnetohydrodynamic simulations of CMEs on a solar-type star of moderate activity levels. The propagation and evolution of the CMEs were traced in the three-dimensional outputs and the temporal evolution of their dynamic properties (such as masses, velocities, and kinetic energies) were determined. The simulated stellar CMEs are more massive and energetic than their solar analog, which is a result of the stronger magnetic field on the surface of the simulated star than that of the Sun. The simulated CMEs display masses ranging from ∼1016 to ∼1018 g and kinetic energies from ∼1031 to ∼1033 erg. We also investigated the instantaneous influence of the CMEs on the star’s angular momentum loss rate. Our results suggest that angular momentum can either be added to or removed from the star during the evolution of CME events. We found a positive correlation between the amplitude of the angular momentum loss rate variation and the CME’s kinetic energy as well as mass, suggesting that more energetic/massive CMEs have a higher possibility to add angular momentum to the star.

Evidence for saturated and disrupted magnetic braking from samples of detached close binaries with M and K dwarfs

Context. Recent observations of close detached eclipsing M and K dwarf binaries have provided substantial support for magnetic saturation when stars rotate sufficiently fast, leading to a magnetic braking (MB) torque proportional to the spin of the star. Aims. We investigated here how strong MB torques need to be to reproduce the observationally inferred relative numbers of white dwarf plus M dwarf post-common-envelope binaries under the assumption of magnetic saturation. Methods. We carried out binary population simulations with the BSE code adopting empirically derived inter-correlated main-sequence binary distributions as initial binary populations and compared the simulation outcomes with observations. Results. We found that the dearth of extreme mass ratio binaries in the inter-correlated initial distributions is key to reproduce the large fraction of post-common-envelope binaries hosting low-mass M dwarfs (∼0.1 − 0.2 M⊙). In addition, orbital angular momentum loss rates due to MB should be high for M dwarfs with radiative cores and orders of magnitude smaller for fully convective stars to explain the observed dramatic change of the fraction of short-period binaries at the fully convective boundary. Conclusions. We conclude that saturated but disrupted, that is, dropping drastically at the fully convective boundary, MB can explain the observations of both close main-sequence binaries containing M and K dwarfs and post-common-envelope binaries. Whether a similar prescription can explain the spin down rates of single stars and of binaries containing more massive stars needs to be tested.

Magnetized rotating isothermal winds

ABSTRACT We consider the general problem of a Parker-type non-relativistic isothermal wind from a rotating and magnetic star. Using the magnetohydrodynamics code athena++, we construct an array of simulations in the stellar rotation rate Ω* and the isothermal sound speed cT, and calculate the mass, angular momentum, and energy loss rates across this parameter space. We also briefly consider the 3D case, with misaligned magnetic and rotation axes. We discuss applications of our results to the spin-down of normal stars, highly irradiated exoplanets, and to nascent highly magnetic and rapidly rotating neutron stars born in massive star core-collapse.

Numerical quantification of the wind properties of cool main sequence stars

ABSTRACT As a cool star evolves, it loses mass and angular momentum due to magnetized stellar winds that affect its rotational evolution. This change has consequences that range from the alteration of its activity to influences over the atmosphere of any orbiting planet. Despite their importance, observations constraining the properties of stellar winds in cool stars are extremely limited. Therefore, numerical simulations provide a valuable way to understand the structure and properties of these winds. In this work, we simulate the magnetized winds of 21 cool main-sequence stars (F-type to M-dwarfs) using a state-of-the-art 3D MHD code driven by observed large-scale magnetic field distributions. We perform a qualitative and quantitative characterization of our solutions, analysing the dependencies between the driving conditions (e.g. spectral type, rotation, and magnetic field strength) and the resulting stellar wind parameters (e.g. Alfvén surface size, mass-loss rate, angular momentum loss rate, and stellar wind speeds). We compare our models with the current observational knowledge on stellar winds in cool stars and explore the behaviour of the mass-loss rate as a function of the Rossby number. Furthermore, our 3D models encompass the entire classical Habitable Zones (HZ) of all the stars in our sample. This allows us to provide the stellar wind dynamic pressure at both edges of the HZ and analyse the variations of this parameter across spectral type and orbital inclination. The results here presented could serve to inform future studies of stellar wind-magnetosphere interactions and stellar wind erosion of planetary atmospheres via ion escape processes.

The winds of young Solar-type stars in the Pleiades, AB Doradus, Columba, and β Pictoris

ABSTRACT Solar-type stars, which shed angular momentum via magnetized stellar winds, enter the main sequence with a wide range of rotational periods Prot. This initially wide range of rotational periods contracts and has mostly vanished by a stellar age $t\sim {0.6}\, {\rm Gyr}$, after which Solar-type stars spin according to the Skumanich relation $P_\text{rot}\propto \sqrt{t}$. Magnetohydrodynamic stellar wind models can improve our understanding of this convergence of rotation periods. We present wind models of 15 young Solar-type stars aged ∼24 Myr to ∼0.13 Gyr. With our previous wind models of stars aged ∼0.26 and ∼0.6 Gyr we obtain 30 consistent three-dimensional wind models of stars mapped with Zeeman–Doppler imaging – the largest such set to date. The models provide good cover of the pre-Skumanich phase of stellar spin-down in terms of rotation, magnetic field, and age. We find the mass-loss rate $\dot{M}\propto \Phi ^{{0.9\pm 0.1}}$ with a residual spread of ∼150 per cent and the wind angular momentum loss rate $\dot{J}\propto {}P_\text{rot}^{-1} \Phi ^{1.3\pm 0.2}$ with a residual spread of ∼500 per cent where Φ is the unsigned surface magnetic flux. When comparing different magnetic field scalings for each single star we find a gradual reduction in the power-law exponent with increasing magnetic field strength.

Accounting for differential rotation in calculations of the Sun’s angular momentum-loss rate

Context. Sun-like stars shed angular momentum due to the presence of magnetised stellar winds. Magnetohydrodynamic models have been successful in exploring the dependence of this ‘wind-braking torque’ on various stellar properties; however the influence of surface differential rotation is largely unexplored. As the wind-braking torque depends on the rotation rate of the escaping wind, the inclusion of differential rotation should effectively modulate the angular momentum-loss rate based on the latitudinal variation of wind source regions. Aims. Here we aim to quantify the influence of surface differential rotation on the angular momentum-loss rate of the Sun, in comparison to the typical assumption of solid-body rotation. Methods. To do this, we exploited the dependence of the wind-braking torque on the effective rotation rate of the coronal magnetic field, which is known to be vitally important in magnetohydrodynamic models. This quantity has been evaluated by tracing field lines through a potential field source surface (PFSS) model, driven by ADAPT-GONG magnetograms. The surface rotation rates of the open magnetic field lines were then used to construct an open-flux weighted rotation rate, from which the influence on the wind-braking torque could be estimated. Results. During solar minima, the rotation rate of the corona decreases with respect to the typical solid-body rate (the Carrington rotation period is 25.4 days), as the sources of the solar wind are confined towards the slowly rotating poles. With increasing activity, more solar wind emerges from the Sun’s active latitudes which enforces a Carrington-like rotation. Coronal rotation often displays a north-south asymmetry driven by differences in active region emergence rates (and consequently latitudinal connectivity) in each hemisphere. Conclusions. The effect of differential rotation on the Sun’s current wind-braking torque is limited. The solar wind-braking torque is ∼10 − 15% lower during solar minimum, (compared with the typical solid body rate), and a few percent larger during solar maximum (as some field lines connect to more rapidly rotating equatorial latitudes). For more rapidly rotating Sun-like stars, differential rotation may play a more significant role, depending on the configuration of the large-scale magnetic field.

A model of anisotropic winds from rotating stars for evolutionary calculations

Context. The surface properties of rotating stars can vary from pole to equator, resulting in anisotropic stellar winds which are not included in the currently available evolutionary models. Aims. We developed a formalism to describe the mass and angular momentum loss of rotating stars which takes into account both the varying surface properties and distortion due to rotation. Methods. Adopting the mass-loss recipe for non-rotating stars, we assigned to each point on the surface of a rotating star an equivalent non-rotating star, for which the surface mass flux is given by the recipe. The global mass-loss and angular momentum loss rates are then given by integrating over the deformed stellar surface as appropriate. Evolutionary models were computed and our prescription is compared to the currently used simple mass-loss enhancement recipes for rotating stars. Results. We find that mass-loss rates are largely insensitive to rotation for models not affected by the bi-stability jump. For those affected by the bi-stability jump, the increase in mass-loss rates with respect to time is smoothed. As our prescription considers the variation of physical conditions over the stellar surface, the region affected by the bi-stability jump is able to grow gradually instead of the whole star suddenly being affected. Conclusions. We have provided an easy to implement and flexible, yet physically meaningful prescription for calculating mass and angular momentum loss rates of rotating stars in a one-dimensional stellar evolution code which compares favourably to more physically comprehensive models

Gravitational waves by perturbation of a slowly rotating thin-shell wormhole

In this paper, the gravitational wave generation by a slowly rotating thin-shell wormhole is considered. Since the rotating thin-shell wormhole is assumed to be an axisymmetric rigid body, the rotation axis coincides with the largest principal axis which means there is no precession motion. However, if there is a perturbation in the angular velocity, the rotating wormhole can move with precession by perturbation which make arise the gravitational waves. We derive the gravitational wave spectrum, energy loss rate, and angular momentum loss rate.

Angular momentum loss rates in Be stars determined by the viscous decretion disc model

ABSTRACT Circumstellar discs around Be stars are formed by the material ejected by the central star. This process removes excess angular momentum from the star as viscosity facilitates the mass and angular momentum transfer within the disc and its growth. The angular momentum loss rates (AMLR) of Be stars is a subject of debate in the literature. Through the modelling of the disc formation and dissipation phases observed from Be stars, their average AMLR can be determined and this is the goal of this work. We use the viscous decretion disc (VDD) model to provide a range of the average AMLR for Be stars and compare these rates with predicted values from the literature. We explore the reasons for discrepancies between the predicted values of average AMLR, using the VDD and Geneva stellar evolution models that were previously reported in literature and find that the largest differences occur when Be stars are rotating below their critical speeds. We show that the time over which the mass reservoir builds up is inversely proportional to the average AMLR. Also, we determine a revised value of the average AMLR for the Galactic Be star ω CMa of $4.7\times 10^{36}\, \mathrm{g\, cm^2\, s^{-2}}$, which is in better agreement with the values expected for a typical B2-type star. Finally, the effect of disc truncation due to the presence of a companion star is investigated and we find that this has a minimal effect on the average AMLR.

Tidal excitation of auto-resonant oscillations in stars with close-by planets

Context. Close-by planets can excite various kinds of oscillations in their host stars through their time-varying tidal potential. Aims. Magnetostrophic oscillations with a frequency much smaller than the stellar rotation frequency have recently been proposed to account for the spin-orbit commensurability observed in several planet-hosting stars. In principle, they can be resonantly excited in an isolated slender magnetic flux tube by a Fourier component of the time-varying tidal potential with a very low frequency in the reference frame rotating with the host. However, due to the weakness of such high-order tidal components, a mechanism is required to lock the oscillations in phase with the forcing for long time intervals (103–107 yr) in order to allow the oscillation amplitude to grow. Methods. We propose that the locking mechanism is an auto-resonance produced by the nonlinear dependence of the oscillation frequency on its amplitude. We suggest that the angular momentum loss rate is remarkably reduced in hosts entering auto-resonance and this contributes to maintaining those systems in that regime for a long time. Results. We apply our model to a sample of ten systems that show spin-orbit commensurability and estimate the maximum drifts of the relevant tidal potential frequencies that allow them to enter the auto-resonant regime. Such drifts are compared with the drifts expected from the tidal evolution of the planetary orbits and the stellar angular momentum loss in the magnetized winds, and we find that auto-resonance is a viable mechanism in eight systems, at least in our idealized model. Conclusions. The duration of the auto-resonant regime and the associated spin-orbit commensurability may be comparable with the main-sequence lifetimes of the host stars, indicating that gyrochronology may not be applicable to those hosts.

Efficient search method of anomalous reflection by the central object in an extreme mass-ratio inspiral system by future space gravitational wave detectors

In our previous work we investigated the effect of the hypothetical reflecting boundary near the black hole event horizon on the waveform from extreme mass-ratio inspirals (EMRIs). Even if the reflection efficiency is not extremely high, we found that a significant modification of the waveform can be expected. Then, the question is how to implement the search for this signature in the actual data analysis of future space gravitational wave antennas, such as LISA. In this paper we propose a simple but efficient method to detect the signature of the reflecting boundary. The interesting feature of the effect of the reflecting boundary on the orbital evolution of EMRIs is that the energy and angular momentum loss rates periodically oscillate in the frequency domain. The oscillation period is corresponding to the inverse time scale for the round trip of gravitational waves between the hypothetical boundary and the angular momentum barrier. We will show that this peculiar feature allows to detect the signature of the reflecting boundary without much additional computational cost.

The Viscosity Parameter for Late-type Stable Be Stars

Abstract Using hydrodynamic principles we investigate the nature of the disk viscosity following the parameterization by Shakura & Sunyaev adopted for the viscous decretion model in classical Be stars. We consider a radial viscosity distribution including a constant value, a radially variable α assuming a power-law density distribution, and isothermal disks, for a late-B central star. We also extend our analysis by determining a self-consistent temperature disk distribution to model the late-type Be star 1 Delphini, which is thought to have a nonvariable, stable disk as evidenced by Hα emission profiles that have remained relatively unchanged for decades. Using standard angular momentum loss rates given by Granada et al., we find values of α of approximately 0.3. Adopting lower values of angular momentum loss rates, i.e., smaller mass loss rates, leads to smaller values of α. The values for α vary smoothly over the Hα emitting region and exhibit the biggest variations nearest the central star within about five stellar radii for the late-type, stable Be stars.

Destination exoplanet: Habitability conditions influenced by stellar winds properties

AbstractThe cumulative effect of the magnetized stellar winds on exoplanets dominates over other forms of star‐planet interactions. When combined with photoevaporation, these winds will lead to atmospheric erosion. This is directly connected with the concept of habitable zone (HZ) planets around late‐type stars. Our knowledge of these magnetized winds is limited, making numerical models useful tools to explore them. In this preliminary study, we focus on solar‐like stars exploring how different stellar wind properties scale with one another. We used one of the most detailed physics‐based models, the 3D Alfvén wave solar model part of the Space Weather Modeling Framework, and applied it to the stellar winds domain. Our simulations showed that the magnetic field topology on the star surface plays a fundamental role in shaping the different stellar wind properties (wind speed, mass loss rate, and angular momentum loss rate). We conclude that a characterization of the Alfvén surface is crucial when studying star‐planet interaction as it can serve as an inner‐boundary of the HZ.

The contribution of alpha particles to the solar wind angular momentum flux in the inner heliosphere

Context. An accurate assessment of the Sun’s angular momentum (AM) loss rate is an independent constraint for models that describe the rotation evolution of Sun-like stars. Aims. In situ measurements of the solar wind taken by Parker Solar Probe (PSP), at radial distances of ~28−55 R⊙, are used to constrain the solar wind AM-loss rate. For the first time with PSP, this includes a measurement of the alpha particle contribution. Methods. The mechanical AM flux in the solar wind protons (core and beam), and alpha particles, was determined as well as the transport of AM through stresses in the interplanetary magnetic field. The solar wind AM flux was averaged over three hour increments, so that our findings more accurately represent the bulk flow. Results. During the third and fourth perihelion passes of PSP, the alpha particles contain around a fifth of the mechanical AM flux in the solar wind (the rest is carried by the protons). The proton beam is found to contain ~10−50% of the proton AM flux. The sign of the alpha particle AM flux is observed to correlate with the proton core. The slow wind has a positive AM flux (removing AM from the Sun as expected), and the fast wind has a negative AM flux. As with previous works, the differential velocity between the alpha particles and the proton core tends to be aligned with the interplanetary magnetic field. Conclusions. In future, by utilising the trends in the alpha-proton differential velocity, it may be possible to estimate the alpha particle contribution when only measurements of the proton core are available. Based on the observations from this work, the alpha particles contribute an additional 10−20% to estimates of the solar wind AM-loss rate which consider only the proton and magnetic field contributions. Additionally, the AM flux of the proton beam can be just as significant as the alpha particles, and so neither should be neglected in future studies.

The Solar Wind Angular Momentum Flux as Observed by Parker Solar Probe

Abstract The long-term evolution of the Sun’s rotation period cannot be directly observed, and is instead inferred from trends in the measured rotation periods of other Sun-like stars. Assuming the Sun spins down as it ages, following rotation rate ∝ age−1/2, requires the current solar angular momentum (AM) loss rate to be around 6 × 1030 erg. Magnetohydrodynamic models, and previous observations of the solar wind (from the Helios and Wind spacecraft), generally predict a values closer to 1 × 1030 erg or 3 × 1030 erg, respectively. Recently, the Parker Solar Probe (PSP) observed tangential solar wind speeds as high as ∼50 km s−1 in a localized region of the inner heliosphere. If such rotational flows were prevalent throughout the corona, it would imply that the solar wind AM-loss rate is an order of magnitude larger than all of those previous estimations. In this Letter, we evaluate the AM flux in the solar wind, using data from the first two orbits of PSP. The solar wind is observed to contain both large positive (as seen during perihelion), and negative AM fluxes. We analyze two solar wind streams that were repeatedly traversed by PSP; the first is a slow wind stream whose average AM flux fluctuates between positive and negative values, and the second is an intermediate speed stream that contains a positive AM flux (more consistent with a constant flow of AM). When the data from PSP are evaluated holistically, the average equatorial AM flux implies a global AM-loss rate of around (2.6–4.2) × 1030 erg (which is more consistent with observations from previous spacecraft).

How Much Do Underestimated Field Strengths from Zeeman–Doppler Imaging Affect Spin-down Torque Estimates?

Abstract Numerous attempts to estimate the rate at which low-mass stars lose angular momentum over their lifetimes fill the literature. One approach is to use magnetic maps derived from Zeeman–Doppler imaging (ZDI) in conjunction with so-called “braking laws.” The use of ZDI maps has advantages over other methods because it allows information about the magnetic field geometry to be incorporated into the estimate. However, ZDI is known to underestimate photospheric field strengths due to flux cancellation effects. Recently, Lehmann et al. conducted synthetic ZDI reconstructions on a set of flux transport simulations to help quantify the amount by which ZDI underestimates the field strengths of relatively slowly rotating and weak activity solar-like stars. In this paper, we evaluate how underestimated angular momentum loss rate estimates based on ZDI maps may be. We find that they are relatively accurate for stars with strong magnetic fields but may be underestimated by a factor of up to ∼10 for stars with weak magnetic fields. Additionally, we re-evaluate our previous work that used ZDI maps to study the relative contributions of different magnetic field modes to angular momentum loss. We previously found that the dipole component dominates spin-down for most low-mass stars. This conclusion still holds true even in light of the work of Lehmann et al.

The solar wind from a stellar perspective

Context.Due to the effects that they can have on the atmospheres of exoplanets, stellar winds have recently received significant attention in the literature. Alfvén-wave-driven 3D magnetohydrodynamic models, which are increasingly used to predict stellar wind properties, contain unconstrained parameters and rely on low-resolution stellar magnetograms.Aims.In this paper, we explore the effects of the input Alfvén wave energy flux and the surface magnetogram on the wind properties predicted by the Alfvén Wave Solar Model (AWSoM) model for both the solar and stellar winds.Methods.We lowered the resolution of two solar magnetograms during solar cycle maximum and minimum using spherical harmonic decomposition. The Alfvén wave energy was altered based on non-thermal velocities determined from a far ultraviolet spectrum of the solar twin 18 Sco. Additionally, low-resolution magnetograms of three solar analogues, 18 Sco, HD 76151, and HN Peg, were obtained using Zeeman Doppler imaging and used as a proxy for the solar magnetogram. Finally, the simulated wind properties were compared to Advanced Composition Explorer (ACE) observations.Results.AWSoM simulations using well constrained input parameters taken from solar observations can reproduce the observed solar wind mass loss and angular momentum loss rates. The simulated wind velocity, proton density, and ram pressure differ from ACE observations by a factor of approximately two. The resolution of the magnetogram has a small impact on the wind properties and only during cycle maximum. However, variation in Alfvén wave energy influences the wind properties irrespective of the solar cycle activity level. Furthermore, solar wind simulations carried out using the low-resolution magnetogram of the three stars instead of the solar magnetogram could lead to an order of a magnitude difference in the simulated solar wind properties.Conclusions.The choice in Alfvén energy has a stronger influence on the wind output compared to the magnetogram resolution. The influence could be even stronger for stars whose input boundary conditions are not as well constrained as those of the Sun. Unsurprisingly, replacing the solar magnetogram with a stellar magnetogram could lead to completely inaccurate solar wind properties, and should be avoided in solar and stellar wind simulations. Further observational and theoretical work is needed to fully understand the complexity of solar and stellar winds.

Do Non-dipolar Magnetic Fields Contribute to Spin-down Torques?

Abstract Main-sequence low-mass stars are known to spin down as a consequence of their magnetized stellar winds. However, estimating the precise rate of this spin-down is an open problem. The mass-loss rate, angular momentum loss rate, and magnetic field properties of low-mass stars are fundamentally linked, making this a challenging task. Of particular interest is the stellar magnetic field geometry. In this work, we consider whether non-dipolar field modes contribute significantly to the spin-down of low-mass stars. We do this using a sample of stars that have all been previously mapped with Zeeman–Doppler imaging. For a given star, as long as its mass-loss rate is below some critical mass-loss rate, only the dipolar fields contribute to its spin-down torque. However, if it has a larger mass-loss rate, higher-order modes need to be considered. For each star, we calculate this critical mass-loss rate, which is a simple function of the field geometry. Additionally, we use two methods of estimating mass-loss rates for our sample of stars. In the majority of cases, we find that the estimated mass-loss rates do not exceed the critical mass-loss rate; hence, the dipolar magnetic field alone is sufficient to determine the spin-down torque. However, we find some evidence that, at large Rossby numbers, non-dipolar modes may start to contribute.