© The Authors 2024

1. Introduction

One of the most important challenges in our understanding of the evolution of cataclysmic variables (CVs, Warner 2003) is period bouncers. These are CVs that, according to the theory of CV evolution, widen their orbital separation after reaching a minimum in their orbital period, P_orb, owing to an interplay between their mass-loss timescale, thermal timescale, and degeneracy (Paczynski & Sienkiewicz 1981). However, the predicted fraction of period bouncers (70% by Kolb 1993, 40% by Goliasch & Nelson 2015) is much greater than that inferred observationally (14% by Pala et al. 2020, a few percent by Inight et al. 2023a).

Pala et al. (2020) point out that current models of CV evolution (e.g. Knigge et al. 2011) possibly do not correctly describe their evolution¹. An important ingredient governing the evolution of such CVs is magnetic braking (MB). It is well established now that there may be a mechanism of angular momentum loss (AML) operating below the period gap (P_gap, 2 ≲ P_orb/h ≲ 3) in addition to AML by gravitational wave radiation (GWR). This is because the period minimum, P_min ≈ 70 min, by a system evolved solely with AML_GWR (Kalomeni et al. 2016) disagrees with observations, which find that P_min ≈ 80 min (Gänsicke et al. 2009). Knigge et al. (2011) suggested that the existence of an additional AML below the period gap that is stronger by a factor of 1.47 than the AML_GWR can reproduce the P_min of CV correctly. However, there is no evidence that AML in short-period CVs can be simply described by a scaling factor applied to AML_GWR or that this scaling factor remains constant throughout the evolution of these CVs. Changing the AML strength of CVs at short periods has strong implications not only on their evolution but also on their detectability. The latter strongly depends on the inferred mass-transfer rate of the system (Appendix C). The essence of our proposed solution to the dearth of observed period bouncers is that although period bouncers exist, they are either difficult to distinguish from pre-bouncers, or simply not detectable.

In Sect. 2, we describe our weakening MB paradigm and illustrate its results. In Sect. 3, we discuss the physical processes that can weaken MB in short-period CVs. We conclude in Sect. 4.

2. The weakening magnetic braking paradigm

Here, we describe our approach to study the implications of an MB strength that weakens around the CV period minimum.

2.1. Binary evolution calculation

We computed the evolution of CVs starting from a detached system with a fully convective donor of mass M_2,  i = 0.2 M_⊙, a WD accretor of mass M_1,  i = M_WD,  i = 0.8 M_⊙, and an initial period of P_orb,  i = 3.18 h using version r23.05.1 of MESA (Paxton et al. 2011, 2013, 2015, 2018, 2019; Jermyn et al. 2023). The system parameters were chosen such that P_orb,  i is the upper limit of P_gap and M_2,  i is the donor mass at P_gap reported by Knigge et al. (2011). All our results were obtained by modifying jdot_multiplier in project_inlist in the MESA code, which multiplies AML_GWR by the factor jdot_multiplier. We call this factor 1 + κ. We defined κ as a parametrised estimate of the strength of AML_MB. This approach is similar to that of Knigge et al. (2011). However, here κ may also depend on stellar parameters (Sect. 2.2). We assume fully conservative mass transfer so that the only mechanisms of AML are GWR and MB.

We note importantly that no unique M_2, gap exists for all CVs, as is known from observations and theoretical computations (e.g. Knigge et al. 2011; Sarkar & Tout 2022). This is because for un-evolved CV donors, M_2, gap and the lower end of the period gap, at which mass transfer resumes (P_gap, −), depend on M₁ and the MB strength above the period gap. If we assume that the initial strength of MB below the period gap depends on stellar parameters, M_2, gap and P_gap, − set the initial κ of our systems. In the next section, we choose how κ behaves as the CV evolves.

2.2. The method

We considered two qualitatively different sets of models, one in which κ is a constant throughout and another in which κ varies with stellar parameters. In the first set, we computed models evolved with κ = 0, 4, and 15 (Fig. 1). The latter two κs are purely ad hoc and were chosen to aid understanding of an MB that depends on stellar parameters, described later. The system with κ = 0 evolves solely with GWR and represents the minimum P_orb of a given M₂. Systems such as polars (AM Her systems, Li et al. 1994), in which there is no MB, follow this track. In zero-age CVs with some degree of MB, κ > 0 initially. We plot in Fig. 1 CVs with donor masses and periods, estimated from superhump periods, assuming M₁ = 0.75 M_⊙ by Knigge (2006, their Table 1)². There is quite a bit of scatter among these points and they do not seem to converge on a unique evolutionary track. This illustrates that varying strengths of MB likely operate below the period gap³. However, studying the significance of this effect is beyond the scope of this Letter. The track with κ = 15 matches with the systems with the biggest P_orb for a given M₂ in the catalogue of Knigge (2006). So, hereinafter, we assume that the tracks with κ = 15 and κ = 0 exhibit, respectively, the upper and lower limits of P_orb for a given M₂.

Fig. 1. Evolution of CVs below the period gap. The solid lines show the tracks on the M2 − Porb plane (lower x- and left y-axis). The dashed tracks in the M2 − B plane (lower x- and right y-axis) show the evolution of B (Eq. (1)) for the models in which κ depends on stellar parameters. The ticks on each solid track denote timesteps of 300 Myr. The different colors correspond to different κs. The grey points are CVs reported by Knigge (2006, their Table 1). We also plot observed period bouncer candidates from Table B.1. Eclipsing systems are plotted as circles, while non-eclipsing systems are plotted as triangles. The horizontal shaded region is the observed period minimum spike (80 ≲ Porb/min ≲ 86) reported by Gänsicke et al. (2009). The squares of different colors show when the system has Ṁ2 = 10−11 M⊙ yr−1. During further evolution, the system is presumably undetectable (see text).

The other set of tracks, in which κ varies with stellar parameters, illustrates the behaviour of the system when the strength of MB depends on stellar structure and changes as the donor evolves. We used the result of the strong- and weak-field dynamo for fully convective low-mass stars proposed by Morin et al. (2011) to model such an MB strength. They argued, based on spectropolarimetric observations by Morin et al. (2010), that two different magnetic field profiles exist in isolated fully convective stars with similar rotation rates and masses. The first is a strong and steady axial dipole field and the second is a weak, multi-polar non-axisymmetric field that is changing rapidly. Because donors in short-period CVs are fully convective, it is possible that a strong-field dynamo also operates in such CV donors in which it drives MB. So, we used the formula for the magnetic field given by Morin et al. (2011, their Eq. (2)) to define κ (for details on how they derive their magnetic field expression, we urge the reader to refer to their Sect. 4.2). Other physical mechanisms that may lead to a stellar-dependent κ are discussed in Sect. 3.

We defined a dimensionless quantity, B, as a proxy for the magnetic field as⁴

$$\begin{array}{c}\hfill B=\frac{6\mathrm{kG}}{19.5\mathrm{kG}}{\left(\frac{M_2}{M_{\odot }}\right)}^{1/2}{\left(\frac{R_2}{R_{\odot }}\right)}^{-1}{\left(\frac{L_2}{L_{\odot }}\right)}^{1/6}{\left(\frac{P_{\mathrm{orb}}}{\mathrm{d}}\right)}^{-1/2},\end{array}$$(1)

where R₂ and L₂ are the radius and the luminosity of the donor, respectively. We computed these using MESA. The last term in Eq. (2) of Morin et al. (2011) is (P_spin/d)^−1/2, where P_spin is the spin period of the M-dwarf. This becomes (P_orb/d)^−1/2 in our Eq. (1) because of tidal locking. The denominator 19.5 kG is the dipolar field at the time of the commencement of Roche lobe overflow (RLOF). This ensures that B < 1 throughout the evolution. We plot two tracks in which κ = 15 B² and κ = 15 B⁴. The exponents are ad hoc but highlight the varying degrees of the dependence of MB strength on the magnetic field, and hence the stellar structure. They also lead to the system attaining P_min at 86 and 80 min, respectively (Fig. 1), which are the upper and lower limits of the observed period minimum spike reported by Gänsicke et al. (2009). The behaviour of B can be understood as follows. Because of RLOF and the fact that the donors are close to thermal equilibrium, R₂, L₂, and P_orb are functions of M₂, and so B ≡ B(M₂) and $P_{\text{orb}}\propto R_2^{3/2}{M}_2^{-1/2}$. For our donors, $L_2\propto M_2^{\beta }$, in which 2 ≲ β ≲ 4 depending on the mass-transfer rate. If we define $R_2\propto M_2^{\alpha }$, we get $B\propto M_2^{3/4+\beta /6-7\alpha /4}$. We have α > 0 pre-bounce and α ≲ 0 post-bounce. Choosing β = 3, α = 0.6 pre-bounce and α = 0.3 post bounce (similar to Eq. (16) of Knigge et al. 2011), we get $B\propto M_2^{0.2}$ pre-bounce and $B\propto M_2^{0.725}$ post-bounce. So, post-bounce B decreases strongly because of a change in the M₂ − R₂ relation of the donor. The evolution of B is shown in Fig. 1.

2.3. Results

We follow the evolution of the models with κ = 15B² and κ = 15B⁴ in the M₂ − P_orb plane. At M₂ ≈ 0.2 M_⊙, these systems are driven by strong MB so they follow the track with κ = 15. However, B starts decreasing gradually at M₂ ≈ 0.125 M_⊙ and substantially when M₂ ≲ 0.05 M_⊙. This leads to the weakening of the MB strength. We note, importantly, that for all our models, the absolute value of AML decreases as the CV evolves (see Appendix A). So, the “weakening” of MB is the additional weakening of the MB strength caused by B (Fig. A.1). The weakening of MB is such that the donor star always adjusts to it on its thermal timescale. The extent of the weakening depends on the exponent of B. Close to their respective P_min, MB becomes negligible. This can be understood with Eq. (1) – further evolution decreases M₂ and increases R₂ and, as a consequence, P_orb. These systems, now only driven by GWR, evolve further to converge to the κ = 0 track. This causes their evolution timescale to drastically increase around and beyond their P_min. Owing to their long evolutionary timescales, these systems stall in the period minimum spike and spend a lot of time there compared to systems evolved with a constant κ. Because the systems are clustered around the period minimum spike, here it is very difficult to distinguish between pre-bounce and post-bounce systems observationally (Pala et al. 2018). We highlight that the weakening MB models also reproduce the period minimum reported by Knigge et al. (2011) but that the M₂ at which P_min is attained is much smaller than the 0.069 M_⊙ reported by Knigge et al. (2011). So, if MB weakens in near-P_min CVs, our models suggest that most of the period bouncer candidates in Fig. 1 are pre-bounce CVs.

At this stage the problem is far from over. Any MB strength below the period gap will only accelerate the evolution of short-period CVs towards their period minimum and drive more CVs to form period bouncers. This will lead to more period bouncers than are predicted solely using GWR (e.g. Kolb 1993; Goliasch & Nelson 2015), and thus exacerbate the classical problem of the dearth of observed period bouncers. For a detectable period bouncer we not only need P_orb ≥ P_min and M₂ ≤ M₂(P_min), but also Ṁ₂ > Ṁ_2, detect, where Ṁ_2, detect is the detection threshold in the mass-transfer rate, Ṁ₂.

All of our candidate bouncer CVs (Table B.1) have Ṁ₂ (estimated by Eq. (C.1) using WD properties) about a few times 10⁻¹¹ M_⊙ yr⁻¹ (also see Pala et al. 2022). So, we assume an optimistic detection threshold of Ṁ₂ = 10⁻¹¹ M_⊙ yr⁻¹ such that any system below this limit is undetectable. The impact of such observational selection effects have been explored in the past (e.g. Pretorius et al. 2007; Inight et al. 2023b). This limit is likely to change with emerging data from optical and X-ray surveys, such as SDSS-V (Kollmeier et al. 2017) and SRG/eROSITA (Predehl et al. 2021; Sunyaev et al. 2021), respectively. The former has already led to the discovery of new period bouncer candidates, which are optically fainter than much of the population (Inight et al. 2023a). The latter is five to 15 times deeper than the last all-sky X-ray survey, potentially revealing systems with lower accretion rates; for instance, the bouncer candidate reported by Galiullin et al. (2024). In Appendix C, we discuss the effects of using Ṁ₂ derived from X-ray luminosity and its limitations. Later in this section we also discuss the effect of changing Ṁ_2, detect.

A complete picture of how the weakening of MB can reduce the number of detectable period bouncers is shown in Fig. 2. We chose Ṁ_2, detect = 10⁻¹¹ M_⊙ yr⁻¹ (circles in each track), beyond which the system becomes undetectable. Although Ṁ_2, detect is model-independent, how a variable MB strength drives a system to reach this Ṁ_2, detect is model-dependent. In the t − Ṁ₂ plot, it can be seen that the Ṁ₂ of the k = 15B⁴ model attains Ṁ_2, detect earlier than that of the other two models. The time evolution of P_orb and M₂ is affected because of this. In the t − P_orb plot, the κ = 4 model bounces at 80 min and becomes undetectable at 105 min at t = 0.94 Gyr. The κ = 0 model bounces at 65 min and becomes undetectable at 78 min at t = 1.15 Gyr. The k = 15B⁴ model bounces at 80 min and becomes undetectable at 83 min at t = 0.48 Gyr. So the weakening MB system becomes undetectable earlier and without much change in P_orb post-bounce (there is only about a 3 min difference between bounce and non-detection). The κ = 4 and κ = 0 models have a large difference between the P_orb in which they bounce and in which they become undetectable. The same is true in the t − M₂ plot. In the model with weakening MB, there is little change in its M₂ between bounce and non-detection, while the change is more significant for the κ = 0 and κ = 4 models. The results do not change qualitatively if we take Ṁ_2, detect = 5 × 10⁻¹² M_⊙ yr⁻¹ (pluses in each track), the limit we expect from SRG/eROSITA (Galiullin et al. 2024). In other words, we conclude that the weakening of MB greatly slows down the evolution of the system, with lesser change in M₂ and P_orb, from the time it bounces to the time it becomes undetectable. However, we note that all our tracks asymptotically tend to Ṁ₂ = 10⁻¹² M_⊙ yr⁻¹. So, our idea makes a strong testable prediction for finding many low-luminosity bouncers aggregated at Ṁ₂ ≈ 10⁻¹² M_⊙ yr⁻¹ in the upcoming surveys (e.g. Galiullin et al. 2024). A dearth of CVs at these Ṁ₂ will easily falsify this theory, although finding such low Ṁ₂ might be difficult (see Appendix C). The evolution of the models in the Ṁ₂ − P_orb plane is shown in Fig. C.1 and discussed in detail in Appendix C.

Fig. 2. Time evolution of Porb, M2, and Ṁ2 for three choices of κ. The vertical black line in each subplot is the time at which the system attains its period minimum (tbounce). The circles and pluses in each track mark detection thresholds of Ṁ2 = 10−11 M⊙ yr−1 and 5 × 10−12 M⊙ yr−1 (see text). The shaded region in the top subplot is the observed period minimum spike.

For Ṁ_2, detect = 10⁻¹¹ M_⊙ yr⁻¹, our weakening MB model (κ = 15B⁴) predicts a reduction in the time spent by a system as a detectable bouncer by a factor of 2.35 compared to the GWR model. Simply reducing the fraction of bouncers predicted by solely using GWR in Kolb (1993, about 70%) and Goliasch & Nelson (2015, about 40%) by 2.35, we get equivalent fractions of detectable period bouncers of about 30% and 17%, respectively. We note that the estimates of Kolb (1993) and Goliasch & Nelson (2015) are the fraction of all period bouncers. The detectable ones are a small subset of it. Additionally, a further reduction to match the observationally inferred estimates (e.g. Inight et al. 2023a) is possible because we show that detectable bouncers populate the period minimum spike that is also populated by pre-bounce CVs. Here, they are difficult to distinguish observationally (Pala et al. 2018).

We note that although Inight et al. (2023a,b) show that bouncers make up only a few percent of the total CV population, this population also consists of magnetic CVs, which follow a different evolution than non-magnetic CVs (Li et al. 1994). If we assume that bouncers remain non-magnetic throughout (although see Sect. 3.2), a better estimate of the fraction of bouncers amongst all non-magnetic short-period CVs can be obtained from the top panel of Fig. 33 of Inight et al. (2023b). Here, we designate SU UMa systems as pre-bounce CVs and assume that most WZ Sge are bouncers. These assumptions, although crude, allow us to approximate that period bouncers make up roughly 30% of all non-magnetic CVs below the period gap in the SDSS I to IV catalogue.

We perform a direct comparison of our results with that of Inight et al. (2023b) using the distribution of the evolution time spent by our systems as a function of P_orb and Ṁ₂ in Fig. 3. Because we make no claims on the nature of MB above the period gap, we assume that all our models follow identical evolution till the lower end of the period gap. So, the time spent by a system at a given P_orb and Ṁ₂ interval below the gap is proportional to the number of systems in that interval. As was expected, it is seen that the intrinsic distribution of short-period non-magnetic CVs is dominated by bouncers. The intrinsic fraction of bouncers is about 74% for κ = 0, 82% for κ = 4, and 87% for κ = 15B⁴. This was also expected, because in the κ = 15B⁴ model we introduce a substantial MB at the beginning, which drives more CVs towards becoming bouncers.

Fig. 3. Distributions of the evolution time spent at each increment of Porb and Ṁ2 for three choices of κ. The stars denote pre-bouncers (M2 > M2(Porb = Pmin)) and triangles denote post-bouncers. Top panel: systems evolving from right to left as pre-bouncers (star symbols) and from left to right (triangles) post-bounce. Bottom panel: systems evolving from right to left.

Because the κ = 4 and κ = 15B⁴ models have similar P_min, we analysed how their features may compare with observations. We calculated the number of bouncers that would have to become invisible below a certain Ṁ₂ cut-off⁵ for our fraction of visible period bouncers to corroborate the 30% that we obtained from Fig. 33 of Inight et al. (2023b). For κ = 15B⁴, the limit is Ṁ₂ ≈ 1.2 × 10⁻¹¹ M_⊙ yr⁻¹, so that about 94% of the intrinsic bouncers become invisible. For κ = 4, the cut-off is Ṁ₂ ≈ 3.2 × 10⁻¹¹ M_⊙ yr⁻¹, so that about 90% of the intrinsic bouncers become invisible. The factor of 2.7 between the Ṁ₂ cut-off in the two models may not seem dramatic, so its significance needs to be emphasised. For an Ṁ₂ cut-off of ≈1.2 × 10⁻¹¹ M_⊙ yr⁻¹, the fraction of bouncers in the κ = 4 model increases to about 50%, meaning that 20% of bouncers exist for 1.2 × 10⁻¹¹ ≲ Ṁ₂/M_⊙ yr⁻¹ ≲ 3.2 × 10⁻¹¹. In other words, for any detection cut-off in Ṁ₂, our weakening MB models will be more affected than the constant MB ones and corroborate observations better.

We note that there is no reason to suggest that our analyses with an ad hoc power of B in an expression of MB derived from the prescription of Morin et al. (2011) would provide a robust comparison with the detailed observational work of Inight et al. (2023a,b). In this work, we just show that for a given observational cut-off, the number of detectable bouncers reduces strongly if MB weakens post-period minimum. The strength of this reduction, in turn, depends on how strongly MB weakens. An extreme case of this is if post-minimum, MB weakens on a dynamical timescale. In such a case, the CV detaches, as is suggested by Inight et al. (2023b), and remains so for about a Gyr (see Fig. 2) till GWR resumes RLOF.

It is important to discuss our results in the context of certain population synthesis studies; for instance, Pretorius et al. (2007) and Belloni et al. (2020). They find, using MB from Verbunt & Zwaan (1981) and Rappaport et al. (1983) above the period gap, respectively, and no MB below the gap, that the intrinsic CV population cannot contain as large a fraction of short-period systems, specifically period bouncers, as is predicted by theory and that selection effects (such as our Ṁ_2, detect cut-off) cannot reconcile the predictions of CV evolution theory with observations. At first glance, these results seem to defeat the main idea in this Letter, in which our claim that there is an MB strength about 15 times as strong as AML_GWR below the period gap exacerbates this apparent discrepancy. However, we note that the intrinsic populations constructed by Pretorius et al. (2007) and Belloni et al. (2020) entirely depend on their assumed MB strength above the period gap. There is evidence to suggest that a weaker MB possibly operates above the gap (Knigge et al. 2011). A weaker MB above the period gap lowers the birthrate of all CVs, which agrees with Pala et al. (2020), who find a lower space density of CVs. Figure 2 of Knigge et al. (2011) illustrates that the AML strengths via MB in Verbunt & Zwaan (1981) and Rappaport et al. (1983) are about two orders of magnitude stronger than GWR above the period gap. So, no matter what strength of MB we choose below the gap, too many CVs have already been driven below the period gap by the strong MB above the gap and so a discrepancy between theory and observations is bound to arise. What causes the discrepancy between theoretical predictions and observations is not what one chooses as MB below the period gap but what one chooses as MB above the period gap. In other words, a modest initial MB strength of about 15 AML_GWR below the period gap cannot undo the accumulation of CVs already dumped there by a strong MB above the gap. Because we make no claims about MB above the gap in this work, there is no reason for our idea to corroborate studies in which the theoretical CV population is entirely shaped by uncertain estimates of a strong MB above the period gap.

3. Physical processes driving the weakening of magnetic braking

We highlight a few physical processes that may cause the weakening of MB in short-period CVs. We note that this list is not exhaustive and that there can be additional mechanisms driving such a weakening.

3.1. Dynamo action in cool stars

In Sect. 2.2, we showed that if the strong-field dynamo proposed by Morin et al. (2011) operates in short-period CV donors, Eq. (1) causes B to reduce significantly for M₂ ≲ 0.07 M_⊙. There is observational evidence to suggest that stars with T_eff ≲ 2200 K such as L-dwarfs have significantly lower chromospheric activity compared to M-dwarfs despite being rapid rotators (Mohanty & Basri 2003). This means that the magnetic field strength drops from fully convective M-dwarfs to brown dwarfs. In Sect. 2.2, we showed that this drop may be due to the change in the mass-radius relation of the star. The results of the α² dynamo model proposed by Chabrier & Küker (2006) have also shown, similarly to Morin et al. (2011), that there is a transition in the magnetic field structure from a steady, large-scale field in late M-dwarfs to a toroidal, oscillatory one in brown dwarfs. In addition, the conductivity of the atmosphere of cool objects such as brown dwarfs decreases greatly, thereby hampering the formation of a hot corona that would drive stellar winds. The combined effect of weaker stellar winds and reduced magnetic field strength drives a weaker MB in brown dwarfs (Mohanty & Basri 2003; Chabrier & Küker 2006). In other words, if such a dynamo operates in short-period CV donors, MB reduces significantly as the donor enters the brown dwarf regime (M₂ ≲ 0.07 M_⊙).

3.2. White dwarf magnetism

Isern et al. (2017) suggest that cool WDs generate strong magnetic fields by a crystallization-driven dynamo. Schreiber et al. (2021) show that magnetic CVs can be explained by the rapid rotation and crystallization of the WD accretors, which can generate fields of several MG (Ginzburg et al. 2022). Schreiber et al. (2023) have recently proposed that such fields are generated in the accretor of short-period CVs post-period minimum. This field connects with that of the donor star, resulting in the detachment of period bouncers for several gigayears. They argue that this can lead to a reduction of about 60% in the number of observed bouncers.

We illustrate a variation in their analysis in which the CV may remain semi-detached. Schreiber et al. (2023) assume that the diffusion timescale of the magnetic field to the WD surface is 100 Myr (Fig. 3 of Ginzburg et al. 2022). However, recently Blatman & Ginzburg (2024) showed that the magnetic field on the WD surface gradually emerges on a gigayear timescale (their bottom right subplot in Fig. 1). By consistently taking into account phase separation, they find that the magnetic diffusion time is about 1 Gyr at the time of breakout and shortly afterwards (this also depends on the WD mass).The donor has a thermal timescale of around a few gigayears, depending on the mass-transfer rate. Since the thermal timescale of the donor is comparable to the diffusion timescale of the WD magnetic field, there is a possibility that the donor adjusts to the reduction in MB because of magnetic reconnection post-period minimum, while continually filling its Roche lobe. In such a case, the evolution will be similar to that presented in Sect. 2. However, such a weakening depends on the properties of the WD accretor, such as its mass and temperature, but is independent of the donor star transitioning from an M-dwarf to a brown dwarf. So, such systems would not necessarily experience an MB weakening at M₂ ≈ 0.07 M_⊙ but when the WD becomes magnetic (Schreiber et al. 2023).

3.3. Dead zones

The dead zone is the region around a spinning magnetised star in which the stellar wind is captured and forced to co-rotate along its magnetic field lines (Mestel & Spruit 1987). This leads to a reduction in wind mass loss and, as a consequence, the strength of MB. Dead zones were first studied by Mestel & Spruit (1987) who gave a simple description for isolated solar-like stars with different rotation rates. Subsequently, several groups have implemented the effects of dead zones in their calculations of MB torque in stellar spin-down (Réville et al. 2015; Garraffo et al. 2015). Because dead zones arise through the interplay of gravity, centrifugal force, and magnetism in the star, they should be at play in every system undergoing MB. This includes the donor stars in CVs. The only difference here is that, owing to tidal locking, P_orb governs the behaviour of the dead zone. We calculated the evolution of dead zones using the simple treatment of Mestel & Spruit (1987, their Eqs. (8) and (9)), adopting solar parameters for the coronal temperature and mean molecular weight. The choice of these parameters does not alter the qualitative behaviour of our dead zone calculations.

For the expression of the ratio of the magnetic pressure and the thermal pressure at the base of the dead zone, ζ_d, we can study the behaviour of two cases: ζ_d = 60(Ω/Ω_⊙) and ζ_d = 60(Ω/Ω_⊙)², from Table 1 of Mestel & Spruit (1987). Here, Ω is the orbital angular velocity of the CV. The evolution of the dead zone of the donor star for the models with constant κ in Fig. 1 is shown in Fig. 4. Here, f_DZ = R₂/R_DZ is the fraction of field lines contributing to MB in the system, where R_DZ is the equatorial radius of the dead zone. With no dead zones, f_DZ = 1. The value f_DZ,  i denotes the contribution of dead zones at the time of commencement of RLOF⁶. These tracks demonstrate how the dead zones would behave in a short-period CV. It is seen that when ζ_d ∝ Ω, f_DZ changes very little throughout the evolution. However, the dead zones grow (f_DZ becomes smaller) with decreasing M₂ when ζ_d ∝ Ω², with the drop becoming steep at M₂ ≈ 0.05 M_⊙. A stronger dependence of ζ_d on Ω yields a steeper drop in f_DZ. One way in which MB affects dead zones is through the generated magnetic field in the donor (say, by a strong-field dynamo or an α² dynamo) that governs the magnetic pressure outside the star (through ζ_d). Dead zones work as an additional mechanism of MB alteration that is always at play regardless of the physical mechanism that drives MB. It can further weaken MB if d ln ζ_d/d ln Ω ≳ 2 (Fig. 4).

Fig. 4. Evolution of the dead zone relative to that at the beginning of RLOF, fDZ/fDZ,  i, with M2 for the models with constant κ. The colours denote the same models as in Figs. 1 and C.1, with κ = 0 shown in blue, κ = 4 in green, and κ = 15 in black. The line styles denote the choice of ζd. The dead zone for each track was calculated post-evolution using the method of Mestel & Spruit (1987).

4. Conclusion

In this Letter, we have shown that the weakening of MB in short-period CVs can help explain the dearth of observed period bouncers. The main idea behind this stems from the fact that to reproduce the correct CV period minimum there ought to be some additional AML mechanism below the period gap. This need not necessarily be a constant scaling to GWR, as was employed by Knigge et al. (2011). We introduce an MB at the lower end of the period gap that decreases as the CV approaches its period minimum and find that such a prescription also correctly reproduces the period minimum at about 80 min. In contrast to the systems with constant scaling, these systems spend considerable time around the observed period minimum spike between 80 and 86 min even after they have passed their minimum period. There, they become difficult to distinguish from pre-bounce systems. The mass-transfer rate decreases below the current detection threshold during further evolution driven by GWR and weakening MB.

A direct comparison of our results with observations of the relative fraction of bouncers is difficult because of our ad hoc prescription of MB weakening and an uncertain detection threshold. So, we compare the constant MB models with the weakening MB ones for a range of detection thresholds and find that the latter shows a stronger reduction in the number of observable bouncers. Our models predict that the undetectable bouncers accumulate around Ṁ₂ ≈ 10⁻¹² M_⊙ yr⁻¹, which can be tested with upcoming surveys that will hopefully reveal low-luminosity bouncers.

The weakening of MB can be caused by physical processes such as a change in the dynamo action in the donor that drives weaker chromospheric activity, the emergence of magnetism in the white dwarf accretor that connects with that of the donor, and thus restricts the outflow of stellar winds, and dead zones in the donor trapping stellar winds. Nevertheless, we admit that the paucity of observed period bouncers may be caused by other selection effects, too, or as-yet-unrecognised physical effects.

Acknowledgments

We thank the referee for a detailed review which greatly improved the paper. A.S. thanks the Gates Cambridge Trust for his scholarship. A.S. also thanks Ken Shen and Elmé Breedt for discussions on the nature of short-period CVs. A.C.R. acknowledges support from an NSF Graduate Research Fellowship. A.S. and A.C.R. are grateful to Franco Giovanelli and the Golden Age of Cataclysmic Variables and Related Objects VI Workshop for facilitating fruitful conversations. S.G. acknowledges support from the Israel Ministry of Innovation, Science, and Technology (grant No. 1001572596), and from the US – Israel Binational Science Foundation (BSF; grant No. 2022175). A.S. and S.G. thank Daniel Blatman for the discussion on the emergence of magnetic fields in white dwarfs. C.A.T. thanks Churchill College for his fellowship.

References

Appendix A: Evolution of angular momentum loss

In Fig. A.1, we plot the evolution of the total AML rate, $\dot{J}\equiv {\mathrm{AML}}_{\mathrm{GWR}}+{\mathrm{AML}}_{\mathrm{MB}}$, with the donor mass. We note that although $-\dot{J}$ decreases throughout the evolution for each model, at M₂ ≈ 0.07 M_⊙ it decreases more steeply for the models in which κ depends on stellar parameters.

Fig. A.1. Evolution of $\dot{J}$ with M2 for the models in Fig. 1.

Appendix B: A catalogue of period bouncers

We present a catalogue of all known period bouncers in Table B.1. While there are more such candidate systems in the literature (some 25 to 30 in total), we require there to be precise estimates of 1) an orbital period, 2) donor mass, 3) WD mass (and radius), and 4) WD temperature for us to include one in our sample. If a donor mass is not available, we ensure that WD properties are well measured and that there is spectroscopic evidence for a brown dwarf donor. We also indicate the few systems known to be eclipsing because those have, on the whole, more precisely measured donor star parameters.

Appendix C: Computation of Ṁ₂ and evolution in the Ṁ₂ − P_orb plane

We computed Ṁ₂ using the relation derived given in Pala et al. (2022),

$$\begin{array}{c}\hfill L_{\mathrm{WD}}=6\times {10}^{-3}{L}_{\odot }\left(\frac{{\dot{M}}_2}{{10}^{-10}{M}_{\odot }{\mathrm{yr}}^{-1}}\right){\left(\frac{M_{\mathrm{WD}}}{0.9M_{\odot }}\right)}^{0.4}\end{array}$$(C.1)

which relates the WD mass, radius, and temperature to the accretion rate (Table B.1). It is also possible to estimate accretion rates from the X-ray luminosity or disc luminosity. The former requires a model of the X-ray emission mechanism, and the latter a model of disc geometry. Both require an estimate of accretion efficiency, which is often parametrised as η in the following expression:

$$\begin{array}{c}\hfill L=\frac{\eta }{2}\frac{G{M}_{\mathrm{WD}}{\dot{M}}_2}{R_{\mathrm{WD}}},\end{array}$$(C.2)

where R_WD is the WD radius and L is the observed accretion luminosity (either from the disc or from the boundary layer, in X-rays). However, the range of η in CVs is a subject of current debate (see Sect. 6.1 of Mukai 2017 for a thorough explanation). As an example, one model of accretion is advective dominated accretion flow (ADAF), which was first applied to explain the hard X-ray spectra of CVs in Narayan & Popham (1993). It was later extended in Narayan et al. (1996) to X-ray binaries observed in a low accretion state. In this work, accretion efficiencies have been shown to be very low, with η between 10⁻³ to 10⁻⁴. From Eq. (C.2), it is clear how failing to incorporate low efficiencies could lead to an underestimate of the accretion rate, given an observed luminosity. More recently, Liu et al. (2008) applied the ADAF model to X-ray spectra of CVs and found good agreement. Nevertheless, Mukai (2017) warns that a complete analysis of accretion efficiency in CVs, which takes into account interactions between disc annuli, is still needed.

In Fig. C.1, we describe how each model attains Ṁ_2, detect in the Ṁ₂ − P_orb plane. The model evolution can be explained as follows. Variation in the MB strength changes the M₂ − R₂ relation, and consequently the M₂ − P_orb relation of a CV, and so a CV attains P_min at a larger P_orb for a stronger MB (Fig. 1). Now, if MB stays constant post-bounce (say κ = 4), then as CV bounces P_orb increases and Ṁ₂ decreases. However, for the weakening MB model (say κ = 15B⁴), around the time when the CV bounces, MB becomes negligible. The CV now has to adjust its mass-transfer rate according to the current weak AML strength, but it cannot increase its P_orb because the P_orb where the CV bounced is too large for its AML strength. Ṁ₂ has to reduce with little change in P_orb. So, the κ = 15B⁴ system essentially stays at approximately 80 min after bouncing before becoming undetectable (when Ṁ_2, detect = 10⁻¹¹ M_⊙ yr⁻¹). The system stays in the period minimum spike (which is also populated by pre-bounce systems) before becoming undetectable. This is also seen in Fig. 1 (the square in which Ṁ_2, detect is attained is in the period minimum spike). This track explains observed candidates clustered at the lower end of the period minimum spike in Fig. C.1. Similarly, the track with κ = 15B² bounces at about 86 min but becomes undetectable at about 90 min. This track explains observed candidates clustered at the upper end of the period minimum spike. The system with κ = 4 emerges from the period minimum spike with Ṁ₂ > 10⁻¹¹ M_⊙ yr⁻¹. So, if such a constant κ is at play post-bounce, there should be systems populating the region with 86 ≲ P_orb/min ≲ 105 and Ṁ₂ ≳ 10⁻¹² M_⊙ yr⁻¹. These are not observed, indicating further that MB weakens post-period minimum. If SRG/eROSITA unveils systems with Ṁ₂ ≈ 10⁻¹² M_⊙ yr⁻¹, the κ = 15B⁴ track indicates the existence of a population of systems up to P_orb ≈ 110 min and the κ = 15B² track up to P_orb ≈ 115 min. However, because η in Eq. (C.2) is very uncertain and can easily be lower than even 10⁻⁴, such low-Ṁ₂ systems would have very low luminosities. This can make finding them very difficult, even with the newer SRG/eROSITA surveys.

Fig. C.1. Evolution of CVs below the period gap. Solid lines show evolution in the Ṁ2 − Porb plane for the same tracks as in Fig. 1. Ticks on each track denote timesteps of 300 Myr. The dotted vertical line denotes Ṁ2 = 10−11 M⊙ yr−1. During further evolution, the system is presumably undetectable (see text). Observed period bouncer candidates from Table B.1 are also plotted. Eclipsing systems are plotted as circles, while non-eclipsing systems are plotted as triangles. The systems labelled and marked with stars have their Ṁ2 derived from X-ray luminosity (Eq. (C.2)), while the others have Ṁ2 derived from WD properties (Eq. (C.1)). The horizontal shaded region is the observed period minimum spike (80 ≲ Porb/min ≲ 86) reported by Gänsicke et al. (2009).

Finally, we note that our accretion rate estimates, based on WD properties (Eq. (C.1)), place the accretion rates of systems such as EZ Lyn (Amantayeva et al. 2021) and SRGeJ0411+6853 (Galiullin et al. 2024) nearly an order of magnitude higher than that reported by authors using X-ray or disc luminosities. Amantayeva et al. (2021) estimated the accretion rate based on the optical disc luminosity, and assumed that η = 1 in Eq. (C.2) to obtain Ṁ₂ ≈ 3 × 10⁻¹² M_⊙ yr⁻¹ (EZ Lyn^* in Fig. C.1). Galiullin et al. (2024) incorporated a bolometric correction to the X-ray luminosity, which assumed a thermal bremsstrahlung model for the emission, to obtain Ṁ₂ ≈ (1.7 − 7.8) × 10⁻¹² M_⊙ yr⁻¹ (SRGeJ0411+6853^* in Fig. C.1). However, they did not explore a range of radiative efficiencies. In both cases, the accretion rates could have been underestimated. Another reason why these may have been underestimated is that Ṁ₂ (lower end of SRGeJ0411+6853^*) is smaller than that for the κ = 0 model. Assuming that the CV remains semi-detached, the estimates of the κ = 0 model set the minimum accretion rate post-bounce. Regardless of these uncertainties, the Ṁ₂ of EZ Lyn^* is only a factor of two smaller than that predicted by our κ = 15B⁴ model. It will agree with our model if we choose η = 0.5 in Eq. (C.2) to calculate Ṁ₂. The Ṁ₂ of SRGeJ0411+6853^* is already in general agreement with both the κ = 15B² and κ = 15B⁴ models. Our model tracks agree well with several systems in Fig. C.1, but notably the κ = 15B⁴ model is in good agreement with all the estimates of SDSSJ1501 and SDSSJ1035 — namely P_orb, M₂, and Ṁ₂ — while the κ = 15B² model is in agreement with the P_orb and M₂ estimate of EZ Lyn and within a factor of two of its Ṁ₂ estimate.

All Tables

All Figures

Fig. 1. Evolution of CVs below the period gap. The solid lines show the tracks on the M2 − Porb plane (lower x- and left y-axis). The dashed tracks in the M2 − B plane (lower x- and right y-axis) show the evolution of B (Eq. (1)) for the models in which κ depends on stellar parameters. The ticks on each solid track denote timesteps of 300 Myr. The different colors correspond to different κs. The grey points are CVs reported by Knigge (2006, their Table 1). We also plot observed period bouncer candidates from Table B.1. Eclipsing systems are plotted as circles, while non-eclipsing systems are plotted as triangles. The horizontal shaded region is the observed period minimum spike (80 ≲ Porb/min ≲ 86) reported by Gänsicke et al. (2009). The squares of different colors show when the system has Ṁ2 = 10−11 M⊙ yr−1. During further evolution, the system is presumably undetectable (see text).

In the text

	Fig. 2. Time evolution of Porb, M2, and Ṁ2 for three choices of κ. The vertical black line in each subplot is the time at which the system attains its period minimum (tbounce). The circles and pluses in each track mark detection thresholds of Ṁ2 = 10−11 M⊙ yr−1 and 5 × 10−12 M⊙ yr−1 (see text). The shaded region in the top subplot is the observed period minimum spike.
In the text

	Fig. 3. Distributions of the evolution time spent at each increment of Porb and Ṁ2 for three choices of κ. The stars denote pre-bouncers (M2 > M2(Porb = Pmin)) and triangles denote post-bouncers. Top panel: systems evolving from right to left as pre-bouncers (star symbols) and from left to right (triangles) post-bounce. Bottom panel: systems evolving from right to left.
In the text

	Fig. 4. Evolution of the dead zone relative to that at the beginning of RLOF, fDZ/fDZ,  i, with M2 for the models with constant κ. The colours denote the same models as in Figs. 1 and C.1, with κ = 0 shown in blue, κ = 4 in green, and κ = 15 in black. The line styles denote the choice of ζd. The dead zone for each track was calculated post-evolution using the method of Mestel & Spruit (1987).
In the text

	Fig. A.1. Evolution of $\dot{J}$ with M2 for the models in Fig. 1.
In the text

Fig. C.1. Evolution of CVs below the period gap. Solid lines show evolution in the Ṁ2 − Porb plane for the same tracks as in Fig. 1. Ticks on each track denote timesteps of 300 Myr. The dotted vertical line denotes Ṁ2 = 10−11 M⊙ yr−1. During further evolution, the system is presumably undetectable (see text). Observed period bouncer candidates from Table B.1 are also plotted. Eclipsing systems are plotted as circles, while non-eclipsing systems are plotted as triangles. The systems labelled and marked with stars have their Ṁ2 derived from X-ray luminosity (Eq. (C.2)), while the others have Ṁ2 derived from WD properties (Eq. (C.1)). The horizontal shaded region is the observed period minimum spike (80 ≲ Porb/min ≲ 86) reported by Gänsicke et al. (2009).

In the text