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A&A
Volume 690, October 2024
Article Number L1
Number of page(s) 6
Section Letters to the Editor
DOI https://doi.org/10.1051/0004-6361/202451829
Published online 27 September 2024

© The Authors 2024

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1. Introduction

Magnetic fields of low-mass stars force the mass lost in winds to co-rotate with the star up to the Alfvén radius. This causes the terminal specific angular momentum of the wind to be higher than the specific angular momentum of the stellar surface. The resulting angular momentum loss is called magnetic wind braking and represents a fundamental ingredient of stellar astrophysics. Magnetic braking drives the spin-down of single stars (Schatzman 1962; Mestel 1968) and the secular evolution of virtually all close binary stars (see Belloni & Schreiber 2023a, for a recent review). Despite this importance, magnetic braking remains poorly understood. In particular, different prescriptions are used for single stars and stars in binaries.

Decades ago, Skumanich (1972) found that the rotational periods (Prot) of Sun-like stars are proportional to the square root of their age, which translates to an angular momentum loss of (with M and R representing the mass and the radius of the star). However, observations of chromospheric activity, coronal X-ray emission, flare activity, and magnetic field strengths in low-mass main-sequence stars reveal that these observables are all correlated and increase with rotation only up to a mass-dependent critical rotation rate. For shorter periods, the relation between activity and rotation saturates (e.g. Stauffer et al. 1994; Reiners et al. 2009; Medina et al. 2020). The assumption that these observables also relate with magnetic braking led to saturated magnetic braking prescriptions being postulated, in which the dependence of the magnetic braking torque on the spin period becomes shallower above a given rotation rate (e.g. Sills et al. 2000; Andronov et al. 2003).

More recent measurements of low-mass main-sequence stars in young clusters revealed a bimodal distribution of fast and slower rotation rates that is difficult to explain with Skumanich-like or saturated magnetic braking prescriptions (e.g. Meibom et al. 2011; Newton et al. 2016). A relation between the strength of magnetic braking and the field complexity has been suggested by Garraffo et al. (2016) to potentially solve this issue (Garraffo et al. 2018a, hereafter, CG18a).

In close binary stars with orbital periods shorter than ∼10 days, tidal forces cause the stellar rotation to be synchronised with the orbit (e.g. Levato 1974; Meibom et al. 2006; Fleming et al. 2019). A Skumanich-like magnetic braking prescription therefore predicts very strong orbital angular momentum loss in these close binaries (Rappaport et al. 1983). These high angular momentum loss rates for close binaries represent a key ingredient for the standard evolution theory of cataclysmic variables (CVs). According to this scenario, CVs with donor stars that still contain a radiative core experience strong angular momentum loss due to magnetic braking, which causes the donor stars to be bloated and the mass transfer rates to be high. When the secondary star becomes fully convective, at an orbital period of ∼3 hours, magnetic braking becomes much weaker, which allows the donor star to shrink and detach from its Roche lobe. The binary evolves as a detached system towards shorter periods until the mass transfer rate resumes at a much lower rate at a period of ∼2 hours.

In other words, to produce a detached phase that covers a period range from ∼2 to ∼3 hours, two conditions need to be fulfilled. First, above the gap, the donor star needs to be significantly oversized compared to its equilibrium radius. Second, at the fully convective boundary, the mass transfer timescale needs to become longer than the radius adjustment timescale of the donor star. In the standard scenario for CV evolution, the first condition requires strong magnetic braking above the gap, while the second condition is met by assuming a drastic decrease in magnetic braking at the fully convective boundary (see e.g. Belloni & Schreiber 2023a, for more details).

Recently, Schreiber et al. (2021a) showed that the late appearance of white dwarf magnetic fields (Bagnulo & Landstreet 2021), possibly related to a crystallisation-driven dynamo (Isern et al. 2017; Schreiber et al. 2021b, 2022, 2023; Ginzburg et al. 2022; Belloni et al. 2024a), affects the evolution of CVs and should cause a reduction in magnetic braking if the field is strong enough. The evolution described above might thus fully apply only to CVs with weakly or non-magnetic white dwarfs. For CVs containing strongly magnetic white dwarfs, so-called polars, the white dwarf magnetic field can reduce the wind zones of the secondary star, thereby significantly reducing magnetic braking. This reduction has been predicted to cause a much less pronounced period gap, or none at all (see also Webbink & Wickramasinghe 2002; Belloni et al. 2020).

The evolution of CVs is observationally constrained by the measured orbital period distribution (Knigge et al. 2011; Inight et al. 2023a,b; Schreiber et al. 2024) and the mass transfer rate, which can be determined from the mass-radius relation of the donor stars (Knigge et al. 2011; McAllister et al. 2019) or the temperature of the white dwarf (Pala et al. 2017, 2022). The orbital period distribution of CVs containing weakly or non-magnetic white dwarfs indeed shows a dearth of systems with periods between ∼2 and ∼3 hours. This period gap seems to be at least much less pronounced (maybe even absent) for CVs containing strongly magnetic white dwarfs (Schreiber et al. 2024). The mass transfer rates of CVs measured from donor star radii are significantly higher at periods longer than three hours (Knigge et al. 2011; McAllister et al. 2019). These observations are roughly consistent with the mass transfer rates determined from white dwarf temperatures (Pala et al. 2022).

These observational constraints support the idea that the field of a strongly magnetic white dwarf reduces magnetic braking. For non-magnetic CVs, the observations agree reasonably well with predictions made assuming a Skumanich-like magnetic braking for donor stars that still contain a radiative core, and significantly weaker magnetic braking for fully convective stars (Knigge et al. 2011; Belloni et al. 2018).

However, CV evolution has turned out to be difficult to explain when applying more recent prescriptions derived for single stars. In an attempt to unify magnetic braking prescription for single stars and close binaries, Garraffo et al. (2018b, hereafter, CG18b) applied their magnetic braking prescription based on field complexity to the evolution of CVs and claimed that it is possible to reproduce the observed orbital period distribution. If true, this result would greatly reduce the tension between magnetic braking prescriptions derived for single stars and those used in binary evolution studies.

Here we use detailed binary calculations to show that the magnetic braking prescription suggested by CG18b can explain neither the existence of the period gap for CVs containing weakly or non-magnetic white dwarfs nor the large radii derived from observations of such CVs above the gap. We then discuss possible avenues towards a unified magnetic braking prescription.

2. Binary star simulations

We used the MESA code (Paxton et al. 2011, 2013, 2015, 2018, 2019; Jermyn et al. 2023), version r-23.03.1, to compute the evolution of CVs1. The MESA equation of state is a blend of the OPAL (Rogers & Nayfonov 2002), SCVH (Saumon et al. 1995), FreeEOS (Irwin 2004), HELM (Timmes & Swesty 2000), PC (Potekhin & Chabrier 2010), and Skye (Jermyn et al. 2021) equations of state. Radiative opacities are primarily from OPAL (Iglesias & Rogers 1993, 1996), with low-temperature data from Ferguson et al. (2005) and the high-temperature, Compton-scattering dominated regime from Poutanen (2017). Electron conduction opacities are from Cassisi et al. (2007) and Blouin et al. (2020). Nuclear reaction rates are from JINA REACLIB (Cyburt et al. 2010), NACRE (Angulo et al. 1999), and additional tabulated weak reaction rates (Fuller et al. 1985; Oda et al. 1994; Langanke & Martínez-Pinedo 2000). Screening is included via the prescription of Chugunov et al. (2007). Thermal neutrino loss rates are from Itoh et al. (1996). Roche lobe radii in binary systems are computed using the fit of Eggleton (1983). Mass transfer rates in Roche lobe overflowing binary systems are determined following the prescription of Ritter (1988).

In our simulation we furthermore assumed the white dwarf mass to be constant, that is, that the same amount of mass that is accreted during a nova cycle is expelled during the eruption, in rough agreement with model predictions (Yaron et al. 2005). The mass expelled during nova eruptions was assumed to carry the specific angular momentum of the white dwarf. While this likely underestimates the true value for CVs containing lower-mass white dwarfs (Schreiber et al. 2016), using this form of consequential angular momentum loss should not lead to significantly different predictions for the secular evolution of most CVs, which contain relatively massive white dwarfs (≳0.8 M⊙). Our simulations take into account angular momentum loss through gravitational radiation according to Paczyński (1967). The orbital angular momentum loss through magnetic braking suggested by CG18a is summarised in what follows.

3. Magnetic braking and field complexity relations revisited

The magnetic braking prescription we tested is based on a relation between angular momentum loss and the complexity of the magnetic field. This relation has been claimed to explain not only the spin-down rates of single stars (CG18a) but also the CV orbital period distribution (CG18b). This prescription for angular momentum loss through magnetic braking , developed initially by Garraffo et al. (2016), can be written as follows:

(1)

where

(2)

with B representing the field strength. The dipole angular momentum loss rate is given by

(3)

where Ω = 2π/Porb is the angular velocity, and c is a constant related to the wind efficiency and is assumed to be 1 × 1041 g cm2 (CG18a). Finally, the field complexity parameter, n, is defined as

(4)

The parameters a and b were set to 0.02 and 2 in CG18a, respectively. However, when applying their model to CVs, slightly different values (a = 0.01 and b = 1) were adopted (CG18b, their Sect. 2) with the justification that the parameter values were not well constrained in their previous study, which only dealt with stars with masses greater than 0.3 M⊙. As the aim of this Letter is to test the magnetic braking prescription that was suggested to reproduce the orbital period distribution of CVs, we used a = 0.01 and b = 1.

For close binary stars, the rotational period, Prot, is equal to the orbital period, Porb (i.e. the orbit is synchronised). The dependence of the convective turn-over time on the stellar mass is given by the empirical relation of Wright et al. (2011):

(5)

Admittedly, this frequently used empirical dependence might need to be updated using larger, recently established datasets (Jao et al. 2022). However, given that we want to test the prescription calibrated by CG18b for CVs, we followed them in adopting the above relation. We furthermore note that this empirical convective turn-over time is different from the calculated global convective turn-over time used in alternative magnetic braking prescriptions (e.g. Van & Ivanova 2019, their Eq. (16)).

Applying their magnetic braking algorithm to CVs, CG18b dropped the term that depends on the field strength in Eq. (2) as it is negligible as long as n remains below 7, which is the case for the parameter space relevant to the orbital period gap in CVs. Combining the above equations then results in a largely simplified angular momentum loss prescription through magnetic braking:

(6)

To simulate CV evolution, CG18b then assumed a mass-radius relation for the donor stars in CVs derived from observations (Knigge et al. 2011). This last step of their procedure is potentially problematic because it swaps cause and effect. In CVs, magnetic braking causes angular momentum loss, which drives the mass transfer, and it is this mass loss of the donor star that determines the mass-radius relation of the donor star. In other words, angular momentum loss (largely through magnetic braking) determines the mass-radius relation of CV donors, which therefore cannot be assumed a priori.

4. CV evolution driven by magnetic braking related to field complexity

We show in Fig. 1 the dependence of the field complexity parameter and the convective turn-over time on the orbital period assuming the CG18b magnetic braking prescription for a white dwarf mass of 0.83 M⊙ and an initial donor mass of 0.65 M⊙. The bottom panel confirms that n indeed remains below ∼7 beyond the period gap if the formula from Wright et al. (2011) is used for the turn-over time. However, CG18b state that they obtained qualitatively identical results by assuming a constant convective turn-over time of 100 days. This is not the case in our simulations. The assumption that n remains below 7, and that the dropped term in Eq. (2) is thus indeed negligible, cannot be justified.

thumbnail Fig. 1.

Comparison of the turn-over time, τ, evolution (upper panel) based on the empirical relation proposed by Wright et al. (2011, solid orange line) and a constant value of 100 days (dashed black line) and the corresponding magnetic field complexities, n (lower panel). The shaded region corresponds to the period gap for non-polar CVs according to Schreiber et al. (2024). The dashed grey vertical line indicates the lower period gap edge according to Knigge et al. (2011). The assumption of a constant turn-over time is not justified, and it is incompatible with the condition n < 7.

In Fig. 2 we show the relative angular momentum loss we obtain using the magnetic braking prescription of CG18b. The strength of angular momentum loss is similar to theirs if the dependence of the convective turn-over time on mass (Wright et al. 2011) is taken into account. For a constant value, a convective turn-over time of 100 days, the angular momentum loss rate becomes much lower, clearly inadequate to describe CV evolution.

thumbnail Fig. 2.

Relative angular momentum loss of CVs with M1 = 0.83 M⊙ and M2 = 0.65 M⊙, with a constant τ = 100 days (dashed black line) calculated using the empirical relation from Wright et al. (2011, solid orange line). The shaded region corresponds to the period gap for non-polar CVs according to Schreiber et al. (2024). The dashed grey vertical line indicates the lower period gap edge according to Knigge et al. (2011).

In Fig. 3 we show the evolution of the donor star radius and mass for a white dwarf mass of 0.9 M⊙, which corresponds roughly to the mean white dwarf mass in CVs (Zorotovic et al. 2011; Pala et al. 2022), and an initial donor star mass of 0.85 M⊙, assuming the magnetic braking prescription based on field complexity (CG18b). The initial orbital period was assumed to be 12 hours. Also shown is the mass-radius relation derived by Knigge et al. (2011) from observations of CVs. The predicted radii are significantly smaller than those derived from observations, and the donor star does not sufficiently shrink to detach from its Roche lobe. Instead, the donor star mass decreases continuously.

thumbnail Fig. 3.

Donor star radius (R2) as a function of its mass (M2) according to Knigge et al. (2011) (dashed red line) compared to the evolution predicted by MESA for a CV with initial M1 = 0.9 M⊙ and M2 = 0.85 M⊙ and assuming magnetic wind braking as in CG18b (solid black line). The radii and masses of CV donors derived from observations (McAllister et al. 2019) are shown as blue dots with their respective error bars. It is clear that the prescription based on the complexity of the field fails to reproduce the observations.

Figure 4 shows the evolution of the donor mass, the rate of angular momentum loss due to magnetic braking, and the mass transfer rate as a function of the orbital period for different white dwarf and donor star masses. The resulting accretion rates in the considered period range are roughly of the same order as the observed ones (Pala et al. 2022; Dubus et al. 2018), but a detached phase that could explain the orbital period gap (Knigge et al. 2011; Schreiber et al. 2024) is not predicted. In contrast to the claims by CG18b, our detailed simulations of CV evolution show that the steep decrease in angular momentum loss through magnetic braking predicted by their prescription does not cause the systems to detach, and the orbital period gap thus remains unexplained.

thumbnail Fig. 4.

Evolution of the donor mass (upper panel), angular momentum loss rate (central panel), and mass transfer rate (lower panel) for CVs with M1 = 0.9 M⊙, M2 = 0.42 M⊙ (dashed blue line), M1 = 0.83 M⊙, M2 = 0.65 M⊙ (solid orange line), and M1 = 0.9 M⊙, M2 = 0.85 M⊙ (dash-dotted pink line). The angular momentum loss rate is calculated as in CG18b. The shaded region corresponds to the period gap according to Schreiber et al. (2024). The dashed grey vertical line indicates the lower period gap edge according to Knigge et al. (2011). For the lowest-mass donor (blue track), the onset of mass transfer occurs close to the upper edge of the period gap, and the binary evolves through a well-known loop (often called the period flag) at the onset of mass transfer (e.g. Stehle et al. 1996, their Fig. 4). As the donor stars do not detach from their Roche lobe, the considered magnetic braking model cannot explain this crucial feature in the observed period distribution.

5. Concluding discussion

Assuming that the efficiency of magnetic braking is related to the magnetic field complexity is very reasonable because the rate of magnetic wind braking must be related to the number of open field lines. In particular, given that the field strength saturates in a similar fashion for fully convective stars and those that still contain a radiative core (Wright & Drake 2016), a dependence of angular momentum loss through magnetic braking on field complexity could represent an elegant way to reproduce the orbital gap. This idea goes back several decades (Taam & Spruit 1989). Finding a prescription for magnetic braking that depends on field complexity and that can reasonably well explain the spin-down of single low-mass stars and key observables of CVs would therefore represent a significant step forwards in our understanding of magnetic braking. This is exactly what has been attempted by CG18b, and their results seemed promising.

However, instead of modelling the detailed evolution of the donor star, CG18b assumed a mass-radius relation similar to those derived from observations of CVs (Knigge et al. 2011), which might not represent the mass-radius relation predicted by their angular momentum loss prescription. It is important to note that CG18b were well aware of the limitations of their simulations and stated that more detailed simulations were required.

Here we filled this gap by implementing their prescription into the stellar evolution code MESA and simulating evolutionary tracks of CVs. We find that the magnetic braking model from CG18b does not allow fundamental observables of CVs to be reproduced, such as the donor star radii and the famous orbital period gap. The differences between our results and those of CG18b are most likely related to the assumed mass-radius relation. Detailed stellar evolution calculations are required to obtain the mass-radius relation for a given prescription of angular momentum loss through magnetic braking since the expansion of the donor star is driven by mass loss, which in turn is driven by angular momentum loss. In our detailed simulations, the magnetic braking prescription proposed by Garraffo et al. (2015) and CG18a does not produce sufficiently bloated donor stars, and the steep decrease in their angular momentum loss rate is not sufficient to cause a detached phase. A more drastic reduction in magnetic braking is required to explain the observed orbital period distribution of CVs.

This result emphasises the previously mentioned tension (e.g. Knigge et al. 2011; Belloni et al. 2024b; Schreiber et al. 2024) between braking laws that describe the rotational evolution of single low-mass stars (Matt et al. 2015; Sills et al. 2000; Andronov et al. 2003; Garraffo et al. 2018a) and those used to reproduce the evolution of close compact binaries ranging from CVs to AM CVn stars (Belloni & Schreiber 2023b), low-mass X-ray binaries (Van & Ivanova 2019), or detached white dwarf plus M-dwarf post common envelope binaries (Schreiber et al. 2010). While relatively weak saturated magnetic braking seems to do relatively well in describing the evolution of single stars, much higher angular momentum loss rates are needed to explain the observations of close compact binaries. Apart from the different strengths of magnetic braking, a drastic decrease (often assumed to be a discontinuity) at the fully convective boundary is required to explain the CV orbital period gap, and such a discontinuity is usually not incorporated into saturated magnetic braking laws.

Given the recent evidence of saturated magnetic braking in close main-sequence binaries (El-Badry et al. 2022) and observations of detached white dwarf plus main-sequence binaries (Schreiber et al. 2010), which indicate a drastic change at the fully convective boundary similar to that required to explain the CV orbital period gap, Belloni et al. (2024b) developed a disrupted (at the fully convective boundary) and saturated prescription that could simultaneously explain observations of both types of detached binaries. A natural next step is to test the very same model for CVs.

While this disrupted and saturated magnetic braking prescription represents a promising candidate to describe binary star evolution, one important piece of the puzzle, an explanation for the dramatically different angular momentum loss rates through magnetic braking in single stars and close binaries, is obviously missing. Perhaps the possibility of tidally enhanced mass loss, which is required to explain a number of observed binary star systems (Tout & Eggleton 1988), should be investigated in more detail.

Data availability

Supplementary material is available at https://zenodo.org/doi/10.5281/zenodo.13632684

Acknowledgments

MRS and DB thank for support from FONDECYT (grant numbers 1221059 and 3220167). VOG and MRS thank C. Garraffo and J. Drake for their insightful feedback. We also thank O. Toloza for her helpful comments and suggestions.

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All Figures

thumbnail Fig. 1.

Comparison of the turn-over time, τ, evolution (upper panel) based on the empirical relation proposed by Wright et al. (2011, solid orange line) and a constant value of 100 days (dashed black line) and the corresponding magnetic field complexities, n (lower panel). The shaded region corresponds to the period gap for non-polar CVs according to Schreiber et al. (2024). The dashed grey vertical line indicates the lower period gap edge according to Knigge et al. (2011). The assumption of a constant turn-over time is not justified, and it is incompatible with the condition n < 7.
In the text
thumbnail Fig. 2.

Relative angular momentum loss of CVs with M1 = 0.83 M⊙ and M2 = 0.65 M⊙, with a constant τ = 100 days (dashed black line) calculated using the empirical relation from Wright et al. (2011, solid orange line). The shaded region corresponds to the period gap for non-polar CVs according to Schreiber et al. (2024). The dashed grey vertical line indicates the lower period gap edge according to Knigge et al. (2011).
In the text
thumbnail Fig. 3.

Donor star radius (R2) as a function of its mass (M2) according to Knigge et al. (2011) (dashed red line) compared to the evolution predicted by MESA for a CV with initial M1 = 0.9 M⊙ and M2 = 0.85 M⊙ and assuming magnetic wind braking as in CG18b (solid black line). The radii and masses of CV donors derived from observations (McAllister et al. 2019) are shown as blue dots with their respective error bars. It is clear that the prescription based on the complexity of the field fails to reproduce the observations.
In the text
thumbnail Fig. 4.

Evolution of the donor mass (upper panel), angular momentum loss rate (central panel), and mass transfer rate (lower panel) for CVs with M1 = 0.9 M⊙, M2 = 0.42 M⊙ (dashed blue line), M1 = 0.83 M⊙, M2 = 0.65 M⊙ (solid orange line), and M1 = 0.9 M⊙, M2 = 0.85 M⊙ (dash-dotted pink line). The angular momentum loss rate is calculated as in CG18b. The shaded region corresponds to the period gap according to Schreiber et al. (2024). The dashed grey vertical line indicates the lower period gap edge according to Knigge et al. (2011). For the lowest-mass donor (blue track), the onset of mass transfer occurs close to the upper edge of the period gap, and the binary evolves through a well-known loop (often called the period flag) at the onset of mass transfer (e.g. Stehle et al. 1996, their Fig. 4). As the donor stars do not detach from their Roche lobe, the considered magnetic braking model cannot explain this crucial feature in the observed period distribution.
In the text

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