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Issue
A&A
Volume 682, February 2024
Article Number L7
Number of page(s) 5
Section Letters to the Editor
DOI https://doi.org/10.1051/0004-6361/202348807
Published online 06 February 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.

This article is published in open access under the Subscribe to Open model. Subscribe to A&A to support open access publication.

1. Introduction

According to the standard scenario for the evolution of cataclysmic variables, angular momentum loss due to magnetic braking is orders of magnitude stronger than gravitational radiation if the secondary stars still contain a radiative core. The correspondingly large mass transfer rates drive the donor stars out of thermal equilibrium. As soon as the donor star becomes fully convective, at about an orbital period of ∼3 h, magnetic braking becomes much less efficient, the donor star has time to relax, and the system becomes a detached binary until reduced magnetic braking and gravitational radiation bring the stars close enough to restart mass transfer (at a much lower rate) at an orbital period of ∼2 h (e.g. Kolb 1993; Belloni & Schreiber 2023a). As this evolutionary scenario includes a drastic decrease in magnetic braking at the fully convective boundary, it is known as disrupted magnetic braking.

The main motivation for developing the previously described scenario has been the observed dearth of cataclysmic variables in the period range between ∼2 − 3 h, the so-called period gap. However, whether observed samples do indeed provide statistically significant evidence for a reduction in cataclysmic variables with orbital periods between ∼2 − 3 h and whether this perhaps depends on the considered sub-type of cataclysmic variables has been intensively discussed during the last decades.

Verbunt (1997) claimed that the gap is not statistically significant for cataclysmic variables with the highest mass transfer rates (so-called nova likes), which was refuted by Hellier & Naylor (1998) and Kolb et al. (1998). Hellier & Naylor (1998), as well as Webbink & Wickramasinghe (2002), argued that polars, that is, cataclysmic variables in which the white dwarf magnetic field connects with that of the donor star that synchronises the white dwarf spin and the orbit, should be less affected by magnetic braking and therefore should show a shorter and less pronounced period gap. While Wheatley (1995) found no evidence for a difference in the period distribution of polars and non-polars, more recent works show that the gap in the polar distribution of the Ritter & Kolb catalogue (Ritter & Kolb 2003) of cataclysmic variables is less pronounced than that for non-magnetic cataclysmic variables (Pretorius et al. 2013).

A recently established volume-limited sample of cataclysmic variables seems to show a period gap but is subject to low-number statistics (Pala et al. 2020). In contrast, the period distribution of the large and homogeneous sample of cataclysmic variables spectroscopically discovered by the Sloan Digital Sky Survey (SDSS) has been claimed to not show clear evidence for a period gap (Inight et al. 2023a).

As magnetic braking is not only of crucial importance for cataclysmic variables but also for other binary stars, and as it also drives the spin down of single low-mass stars, one might have hoped that independent observational or theoretical constraints could settle the discussion. Unfortunately, this has not been the case.

On the one hand, from cataclysmic variables and related objects, there seems to be strong support for disrupted magnetic braking from other observables than the period distribution of cataclysmic variables (Schreiber et al. 2010; Knigge et al. 2011; Zorotovic et al. 2016; McAllister et al. 2019). On the other hand, observations of single stars as well as main sequence binaries argue against the disrupted magnetic braking scenario (e.g. Gossage et al. 2023; El-Badry et al. 2022), and favour saturated magnetic braking prescriptions instead. According to these saturated magnetic braking prescriptions, the relation between angular momentum loss and the spin period of the mass losing star saturates for rotation periods below a critical value in a similar fashion as chromospheric activity, coronal X-ray emission, and flare activity saturate (e.g. Reiners et al. 2009; Newton et al. 2017; Johnstone et al. 2021).

Unfortunately, most saturated magnetic braking prescriptions proposed so far predict angular momentum loss rates that are too weak to explain the orbital period gap, that is, the resulting mass transfer rates are not high enough to sufficiently increase the radius of the donor star. This means that, if the period gap exists, we need much stronger magnetic braking in cataclysmic variables than what is usually assumed for single main sequence stars or main sequence star binaries. If, instead, the period gap does not exist, most evidence would point towards a universal saturated magnetic braking prescription and we would need different explanations for the above listed independent evidence for disrupted magnetic braking. It is thus important to determine whether the period gap is a real feature (and, if so, for which type of cataclysmic variable) or if it just appeared in early samples of cataclysmic variables due to observational biases and selection effects as suggested by Inight et al. (2023a).

2. The SDSS I–IV sample

In two extensive works, Inight et al. (2023a,b) recently established large samples of cataclysmic variables identified by SDSS and convincingly showed that the orbital period distributions of these more homogeneous samples are significantly different to that of the Ritter & Kolb catalogue. In this work, we mainly consider the SDSS I–IV sample as cataclysmic variables in this catalogue are all serendipitous identifications through low-resolution SDSS spectroscopy. This sample represents the largest homogeneous sample of cataclysmic variables currently available. SDSS spectroscopy has been crucial for cataclysmic variable studies as, for example, the large number of low-accretion rate systems below the gap and at the orbital period minimum, which had been predicted for a long time, was only discovered thanks to early SDSS samples of cataclysmic variables (Gänsicke et al. 2009).

Following the idea of Webbink & Wickramasinghe (2002), Belloni et al. (2020) showed that the period distributions predicted for polars and for the rest of the cataclysmic variable population (from now on non-polar cataclysmic variables) should be different with respect to the gap because in polars the white dwarf magnetic field reduces the wind zones of the donor star and thereby the efficiency of magnetic braking. According to this prediction, the mass transfer rates above the gap should be lower, and the donors should be less (if at all) driven out of thermal equilibrium. Without a significantly inflated donor star at long orbital periods (above the gap), a much shorter or no detached phase is expected to be generated for polars by disrupted magnetic braking.

We therefore show in Fig. 1 the cumulative distributions of polars and non-polar cataclysmic variables separately. It becomes immediately clear that in the period range where the period gap has been claimed to exist, the period distribution of polars differs from that of non-polars. Only the latter ones show clear evidence for a period gap. Looking at the cumulative distributions and using different binning for histograms, we identified periods between 147 and 191 min by eye as the region with the most obvious decrease in the number of non-polar cataclysmic variables per period interval. Only five of the 256 non-polar cataclysmic variables, which corresponds to 2.0%, have periods in this range. The lower boundary of the gap (147 min) is different to the value (129 min) previously found by Knigge et al. (2011). The period distribution of polars does not show an obvious gap, that is, a reduction in the number of systems in a certain period range. In the range of periods we defined as the gap for non-polar cataclysmic variables (147 − 191 min), we found nine out of 64 polars (14.1%). In what follows, we explain how we derived the statistical significance of the previously described observations.

thumbnail Fig. 1.

Period distribution of both polars (red line) and non-polar cataclysmic variables (black line and grey histogram) from the SDSS I–IV sample of cataclysmic variables recently published by Inight et al. (2023b). A dearth of systems between 147 and 191 min (light blue shaded region) in the sample of non-polars can clearly be detected. We claim that the reduced number of non-polar cataclysmic variables in this period range represents the (in)famous period gap of cataclysmic variables. The dashed vertical lines indicate the boundaries of the period gap as identified by Knigge et al. (2011), while the upper boundary perfectly fits with what we found here, the lower edge of the period gap in the SDSS I–IV sample seems to be located at longer periods. The period distribution of polars does not show evidence for a reduction in the number of systems in the gap.

3. The statistical significance of the period gap

To determine the statistical significance of the decrease in the number of non-polar cataclysmic variables in the period range between 147 and 191 min, we performed several unimodality and bimodality tests using only systems with periods between 103 and 235 min (as this range is sufficient to cover the period gap). If the observed decrease in the number of cataclysmic variables is statistically significant, the period distribution around the gap should be consistent with a bimodal distribution.

As in the previous section, we separated polars and non-polars. Given that even for the homogeneous SDSS I–IV sample of cataclysmic variables, observational biases cannot be excluded, we defined an additional sample excluding nova likes from the non-polars. Nova likes are cataclysmic variables with stable accretion disks, have the largest mass transfer rates, and are typically found just above the period gap. If observational biases were dominating, these systems would be overrepresented which could in principal explain the period gap in observed magnitude-limited samples. By eliminating nova likes from the non-polar sample, we therefore tested how strong observational biases towards the detection of nova likes could impact our results on the statistical significance of the period gap. All samples were established based on the classifications by Inight et al. (2023b).

For each of the three samples, we first applied unimodality tests and, in the case the null hypothesis (the true number of modes is one) can be rejected with more than 95% confidence, we performed bimodality tests. We used the tests suggested by Silverman (1981) and Ameijeiras-Alonso et al. (2019) as provided in the multimode package implemented by Ameijeiras-Alonso et al. (2021) in the numerical tool R (R Core Team 2022). For the Silverman (1981) test, we first determined the critical bandwidth and then generated resamples using the distribution associated with the corresponding kernel density estimation. In the case of the Ameijeiras-Alonso et al. (2019) test, we used the exact excess mass value to perturb the sample data. For each test, we chose three different values for the number of bootstrap replicates (100, 500, and 1000).

The results of our tests are as follows. Regarding the sub-sample of non-polars, we can reject the null hypothesis (the true number of modes is equal to one) with at least 98.0 and at least 99.6% confidence for the Silverman (1981) and the Ameijeiras-Alonso et al. (2019) test, respectively. This strong evidence for the existence of the period gap slightly decreases if the brightest cataclysmic variables (nova likes) are excluded, but it remains above 95 and 96.6%. Performing bimodality tests for both samples, we find that the null hypothesis (i.e. that the distribution is bimodal) cannot be rejected (p values exceeding 20%). Consequently, the period distribution of non-polars is bimodal and this statement remains true when nova likes are excluded.

In contrast, unimodal tests for polars provide p values largely exceeding five per cent, which means the null hypothesis (the true number of modes is equal to one) cannot be rejected. However, given the small sample size of polars (only 39 polars have periods between 103 and 235 min), we cannot exclude the existence of a less pronounced gap similar to the one found in the Ritter & Kolb catalogue (Pretorius et al. 2013; Schwope et al. 2020). From these test results, we conclude that (i) there is a statistically significant gap in the period distribution of non-polar cataclysmic variables from SDSS I–IV; (ii) this gap is not caused by observational biases favouring the detection of nova-like cataclysmic variables; and (iii) there is no statistically significant evidence for the existence of a similar period gap in the distribution of polars from SDSS I–IV.

We show in Fig. 2 the probability density functions derived with the kernel density estimation method adopting Gaussian kernels and the critical bandwidth determined according to Silverman (1981). In line with the test results described above, we show a bimodal distribution for the non-polar samples and a unimodal distribution for the polar sample. The locations of the modes and the corresponding densities are given in Table 1.

thumbnail Fig. 2.

Probability density functions based on kernel density estimations with the Gaussian kernel and adopting the critical bandwidth of Silverman (1981) of the period distribution of different sub-samples of cataclysmic variables. From the top to bottom panels, we excluded the polars, the polars plus the nova likes, and finally the non-polars. The vertical dashed lines indicate the location of the modes assuming bimodal distributions (top and middle panels) and a unimodal distribution (bottom panels).

Table 1.

Characteristics of the probability density derived using the kernel density estimation method adopting Gaussian kernels with the critical bandwidth calculated according to Silverman (1981).

4. Comparison with other samples

The SDSS I–IV sample presented by Inight et al. (2023b) represents the largest sample of cataclysmic variables identified in a homogeneous way (SDSS spectroscopy). In what follows we compare the period distribution from SDSS I–IV with those of other recently established catalogues of cataclysmic variables. For this exercise we only consider non-polars because the number of polars in the other samples is too small to draw any meaningful conclusions concerning the polar period distribution.

Figure 3 shows the cumulative distributions of non-polar cataclysmic variables from SDSS I–IV (Inight et al. 2023b), the plate survey that is part of SDSS V (Inight et al. 2023a), the incomplete but volume-limited (300 pc) Gold sample (Inight et al. 2021), as well as the largely complete volume-limited 150 pc sample (Pala et al. 2020). The period distributions from the two SDSS samples are very similar (the KS test provides a p value of 0.50). Also the two volume-limited samples seem to agree with each other, that is, show no evidence to be not drawn form the same parent sample (p value of 0.75). However, we clearly observe a difference between the SDSS and the volume-limited samples (p values below 0.09 and below 0.02 for comparison with SDSS V and SDSS I–IV, respectively).

thumbnail Fig. 3.

Period distributions of non-polar cataclysmic variables from SDSS I–IV (Inight et al. 2023b), the plate survey that is part of SDSS V (Inight et al. 2023a), the incomplete but volume-limited (300 pc) Gold sample (Inight et al. 2021), as well as the largely complete but small 150 pc sample (Pala et al. 2020). While both SDSS samples show a clear period gap in the range between 147 and 191 min, the volume-limited samples contain too few cataclysmic variables with periods exceeding ∼120 min to derive meaningful constraints concerning the period gap. This result illustrates that the SDSS samples are still significantly biased against cataclysmic variables with periods shorter than two hours.

The differences arise from the fact that the volume-limited samples are more dominated by short orbital period cataclysmic variables and contain few non-polar cataclysmic variables with periods longer than 120 min. Because of this, both volume-limited samples do not provide strong evidence in favour or against the existence of the period gap nor do they allow one to distinguish between the lower gap boundary as defined by Knigge et al. (2011) using the Ritter & Kolb catalogue (129 min) and the one we found in the SDSS samples (147 min). However, the relative number of non-polar systems in the period gap (defined as 147−191 min) is similarly small in all samples, that is, 0/31, 1/97, 1/55, and 5/256 for the 150 pc, Gold, SDSS V, and SDSS I–IV samples, respectively.

The comparison between the different samples shows that all currently known samples containing large numbers of non-polar cataclysmic variables with periods exceeding 120 min (i.e. SDSS I–IV, SDSS V, Ritter & Kolb) show a period gap. Current volume-limited samples do not contradict this finding but provide no additional constraints. The difference between the period distributions of volume-limited samples and that of the SDSS samples indicate the presence of observational biases affecting the SDSS samples.

5. Potential biases

First of all, we do not see any reason why non-polar cataclysmic variables should be more difficult to detect when they have periods between 147 and 191 min than non-polar cataclysmic variables below the gap. The only observational bias one could imagine would be a drastic decrease in the mass transfer rate, the extreme case of which is exactly what disrupted magnetic braking predicts (no mass transfer at all). As shown in Sect. 3, the gap remains to be statistically significant even if the brightest cataclysmic variables (nova likes) are excluded from the non-polar sample. In addition, the fraction of cataclysmic variables is similarly low in the SDSS and the volume-limited surveys. All this shows that the gap is extremely unlikely to be caused by observational biases.

We also do not see any reason why polars should be easier to detect in the period gap than non-polar cataclysmic variables, which implies that the difference between the two subgroups in the SDSS I–IV sample is real. This shows that the existence of the gap is related to magnetism which strongly supports the idea of disrupted magnetic braking.

However, compared to the volume-limited samples, the SDSS samples contain significantly fewer systems below the period gap, that is, the SDSS samples are still biased against short period low mass transfer systems. This bias might affect the location of the lower boundary of the gap. A large volume-limited sample of cataclysmic variables, which perhaps can be provided by combining eROSITA detections with suitable follow-up observations, is needed to finally determine the exact location and depth of the gap from a fully representative sample.

However, it is fundamentally important to note that, for the time being, the exact location of the period gap does not provide critical information. Individual evolutionary tracks of cataclysmic variables predict the detached phase and the restart of mass transfer at slightly different orbital periods depending on the initial donor mass and metallicity as well as the white dwarf mass. Some cataclysmic variables start mass transfer in the gap and are not affected by disrupted magnetic braking. Furthermore, the appearance of white dwarf magnetic fields during cataclysmic variable evolution can alter the evolution of individual systems (Schreiber et al. 2021). A razor-sharp orbital period gap is thus not expected and one should not search for it.

Once a large and volume-limited sample of cataclysmic variables has been established, one might be able to extract further information on magnetic braking by comparing the exact shape of the period distribution with theoretical predictions. In the absence of this large volume-limited sample, the key information we can retrieve from the currently available data is the following: all existing samples of cataclysmic variables are consistent with a deep period gap in the distribution of non-polar cataclysmic variables and a less pronounced (or no) gap in the period distribution of polars.

6. Conclusion

The Ritter & Kolb catalogue has provided evidence for a period gap in the cataclysmic variable orbital period distribution. The existence of this period gap formed the basis for developing the theory of disrupted magnetic braking. In contrast to recent claims, the orbital period distributions of cataclysmic variables from SDSS samples provide further evidence for disrupted magnetic braking. A clear period gap is present in the distribution of non-polar cataclysmic variables and a similarly profound and extended gap can be excluded for polars from SDSS. A consistent picture for cataclysmic variable evolution therefore remains to contain the following ingredients:

(i) Magnetic braking above the gap is stronger than predicted by the models of saturated magnetic braking.

(ii) Magnetic braking needs to decrease significantly around the fully convective boundary.

(iii) Magnetic braking is reduced in polars as the wind from the secondary star is trapped within the magnetosphere of the white dwarf (e.g. Belloni et al. 2020).

Despite these crucial results on magnetic braking (or actually rather because of them), it might be difficult to soon find a unified magnetic braking prescription. Such a universal magnetic braking prescription would need to convincingly explain observations of single stars, which seem to favour some kind of saturated magnetic braking (Gossage et al. 2023), reproduce the donor star radii in cataclysmic variables as well as disrupted magnetic braking (Knigge et al. 2011), and provide the strong angular momentum loss rates that are required to explain persistent low-mass X-ray binaries (Van & Ivanova 2019) and AM CVn binaries descending from cataclysmic variables with evolved donors (Belloni & Schreiber 2023b).

A good starting point might be a disrupted and saturated magnetic braking prescription, which could work for cataclysmic variables as well as detached main sequence and post common envelope binaries (Belloni et al. 2024). However, this alone would not solve the differences between the strength of magnetic braking in cataclysmic variables, the progenitors of low-mass X-ray binaries, and single stars. Potentially new dependencies on the age and evolutionary status of the stars need to be considered.

It appears unlikely that the large mass transfer rates derived for cataclysmic variables above the gap from the radii of their donors (McAllister et al. 2019), and partly also from the white dwarf temperatures (Pala et al. 2022), are caused by consequential angular momentum loss instead of magnetic braking as speculated by El-Badry et al. (2022). Consequential angular momentum loss might play a role in explaining the white dwarf mass distribution of cataclysmic variables (Schreiber et al. 2016), but it can hardly be the main angular momentum loss mechanism above the gap. This is because there is no obvious reason for consequential angular momentum loss to be less efficient in polars and/or to decrease at the fully convective boundary.

As a final remark, while all samples of the cataclysmic variables currently available agree with the main predictions of disrupted magnetic braking, important issues related to our understanding of cataclysmic variable evolution still need to be addressed. On the observational side, a large and complete volume-limited sample is not yet available and we therefore cannot fully exclude that observational biases play a role. Concerning theoretical predictions, recent evidence points towards the appearance of white dwarf magnetic fields during cataclysmic variable evolution, which can lead to significant deviations from the standard scenario of cataclysmic variable evolution (Schreiber et al. 2021). A binary population synthesis code including the predicted transition from non-polar cataclysmic variables to polars is not yet available. Last but not least, the fact that large numbers of cataclysmic variables that passed the period minimum are predicted to exist but are absent in observed samples remains puzzling (Inight et al. 2023a). Perhaps the latter two issues are related as the late appearance of white dwarf magnetic fields may generate extended detached phases in period bouncers (Schreiber et al. 2023).

Acknowledgments

We thank B.T. Gänsicke and K. Inight for lively discussions and for providing the SDSS samples of cataclysmic variables. This research was supported by the National Science Foundation under grant No. NSF PHY-1748958 and the Deutsche Forschunsgemeinschaft (DFG) under grants EXC-2094–390783311 and Schw536/37-1. M.R.S. and D.B. are supported by FONDECYT (grant numbers 1221059 and 3220167). M.R.S. further acknowledges support from ANID, – Millennium Science Initiative Program – NCN19_171.

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All Tables

Table 1.

Characteristics of the probability density derived using the kernel density estimation method adopting Gaussian kernels with the critical bandwidth calculated according to Silverman (1981).

In the text

All Figures

thumbnail Fig. 1.

Period distribution of both polars (red line) and non-polar cataclysmic variables (black line and grey histogram) from the SDSS I–IV sample of cataclysmic variables recently published by Inight et al. (2023b). A dearth of systems between 147 and 191 min (light blue shaded region) in the sample of non-polars can clearly be detected. We claim that the reduced number of non-polar cataclysmic variables in this period range represents the (in)famous period gap of cataclysmic variables. The dashed vertical lines indicate the boundaries of the period gap as identified by Knigge et al. (2011), while the upper boundary perfectly fits with what we found here, the lower edge of the period gap in the SDSS I–IV sample seems to be located at longer periods. The period distribution of polars does not show evidence for a reduction in the number of systems in the gap.
In the text
thumbnail Fig. 2.

Probability density functions based on kernel density estimations with the Gaussian kernel and adopting the critical bandwidth of Silverman (1981) of the period distribution of different sub-samples of cataclysmic variables. From the top to bottom panels, we excluded the polars, the polars plus the nova likes, and finally the non-polars. The vertical dashed lines indicate the location of the modes assuming bimodal distributions (top and middle panels) and a unimodal distribution (bottom panels).
In the text
thumbnail Fig. 3.

Period distributions of non-polar cataclysmic variables from SDSS I–IV (Inight et al. 2023b), the plate survey that is part of SDSS V (Inight et al. 2023a), the incomplete but volume-limited (300 pc) Gold sample (Inight et al. 2021), as well as the largely complete but small 150 pc sample (Pala et al. 2020). While both SDSS samples show a clear period gap in the range between 147 and 191 min, the volume-limited samples contain too few cataclysmic variables with periods exceeding ∼120 min to derive meaningful constraints concerning the period gap. This result illustrates that the SDSS samples are still significantly biased against cataclysmic variables with periods shorter than two hours.
In the text

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