Abstract
Cataclysmic variables (CVs) are binary star systems in which a white dwarf accretes from a low-mass star. Such systems typically become observationally invisible when their orbital periods decrease to between 3 and 2 hours, also known as the period gap (Schreiber et al. 2024). The period gap is thought to be a result of a once efficient magnetic braking process becoming inefficient as the binary’s orbit shrinks, causing orbital decay and subsequently mass transfer to slow down in the gap. The lack of accretion novae on the surface of the white dwarf is a factor in the CV’s invisibility. In this work we use 1-D stellar evolution code MESA (Modules for Experiments in Stellar Astrophysics) to determine whether simulation results reflect the current paradigm for the CV period gap. Using MESA, we construct and evolve a grid of compact object and hydrogen main sequence stars (CO-HMS) whose mass ratio and initial orbital period are varied. We conclude that simulations do support the current period gap paradigm, as angular momentum lost via magnetic braking becomes inefficient, in agreement with current theories. Future work includes running similar grids through population synthesis codes (such as POSYDON) to evolve the CVs beyond the common envelope phase, and running similar grids with different magnetic braking prescriptions which more accurately reflect single and binary star physics (Gossage et al. 2023).
Introduction
A main sequence star and a white dwarf orbiting each other is known as a cataclysmic variable (CV) system. Few CV systems with orbital periods between about 2-3 hours are noted in observational rotation period distributions (Schreiber et al. 2024). The lack of systems within this orbital period range (known as the period gap) occurs in low-mass main sequence donor stars (or stars with a convective envelope, usually those below 1.3M⊙), and is likely due to inefficient magnetic braking. Magnetic braking is thought to be the primary mechanism through which CV systems initially lose orbital angular momentum, thereby shrinking the system’s orbit, facilitating mass transfer. Once the system’s orbit shrinks and reaches the period gap, the donor star is thought to transition to an inefficient magnetic braking process, which stops rapid orbital decay and hinders mass transfer (Howell et al. 2001). Mass transfer stops as the binary’s period shrinks to under 3 hours, and starts again as the binary’s period shrinks below 2 hours (Howell et al. 2001). Above the 3-hour mark, the CV orbital period shrinks via magnetic braking (Howell et al. 2001). Magnetic braking occurs when the material flowing from the donor star is trapped by the it’s surface magnetic field, causing the moment of inertia to increase, slowing the star down. Eventually, the material is lost from the CV entirely, and as the donor star loses angular momentum through magnetic braking, the orbit shrinks via tidal forces to conserve angular momentum. Throughout this process, the donor star continues to transfer mass to the white dwarf. Eventually the donor star becomes fully convective (instead of having a convective envelope and radiative core), meaning all the material within the star is moving as a means of heat transfer. It is thought that the star becoming fully convective means that magnetic braking is likely no longer as efficient, given that the star does not have a radiative core alongside a convective envelope (the accepted star structure for generating a magnetic field, which is thought to be a key ingredient in the generation of strong surface magnetic fields via an alpha-omega (Solar-like) dynamo). Since magnetic braking plays a diminished role in angular momentum loss, the orbital decay rate decreases. Subse- quently, the mass transfer rate slows down, and the donor star returns to thermal equilibrium, allowing it to shrink back within its Roche lobe radius, stopping mass transfer and making the CV less visible. The cessation of mass trans- fer means that any novae or luminosity via accretion heating on the surface of the white dwarf caused by accreting mass (that would make it more visible) are less likely (Knigge et al. 2011). Alongside the relatively small time spent evolving through the period gap, this decrease in visibility of CVs is thought to give rise to the period gap. Without magnetic braking, the binary does not shrink as rapidly, but angular momentum is still lost via gravitational wave radiation and winds. Once the binary shrinks past around 2 hours, the angular momentum lost through gravitational wave radiation increases, and the binary orbit undergoes in-spiral. (see Paxton et al. 2015 for more on how these angular momentum losses are calculated). Here, we attempt to investigate and explain the reasons behind the period gap in CV systems using data collected from Modules in Experimental Astrophysics (MESA) (Paxton et al. 2011; Paxton et al. 2015) and Population Synthesis with Detailed Binary-Evolution Simulations (POSYDON) (Fragos et al. 2023).
Methods
MESA is a 1-D stellar evolution code that evolves single and binary stars over time. MESAbinary is the module used to evolve binary systems, employing the MESAstar module to simulate an interacting, two-star system (Paxton et al. 2015). In the case of our binary grids where one member is a compact object (such as a white dwarf), the compact object is evolved as a point mass to simplify calculations. For this work, MESA version r11701 is used (Paxton et al. 2019). POSYDON is a binary population synthesis code that interpolates MESA data to calculate the evolution of a grid of binary systems (Fragos et al. 2023). Using computed MESA data, POSYDON interpolates the data, and evolves the binary further. For this work, POSYDON version 2 is used (Andrews et al., in prep). We use the population synthesis code POSYDON to analyze the theoretical orbital period distribution and investigate its mechanisms. To do this, we construct grids of white dwarfs (compact objects or CO) and hydrogen main sequence stars (HMS) with varying initial mass ratios and orbital periods at solar metallicity (Asplund et al. 2009). The grids are generated using binary stellar evolution models computed by MESA. Multiple grids of binary systems were constructed, however, our present analysis is focused on the CO-HMS grid, since CVs are likely to be a white dwarf accreting from a low-mass main sequence star. In order to characterize the CV period gap, we plot the orbital period of the CO-HMS binaries over time, which shows many of the binaries have orbital periods that steadily decrease over time and that enter and evolve though the period gap.
Results
Plotting the loss of angular momentum via magnetic braking, the mass transfer rate, and the donor mass of these select systems versus the orbital period (where the orbital period decreases from left to right on the x-axis), as shown in Figure 1, reveals that these systems appear to follow the paradigm described in 1. The angular momentum loss rate via magnetic braking becomes zero, and the donor mass also decreases significantly, while entering the gap at 3 hours. The donor mass reaches the fully convective limit (around 0.3 solar masses (Jao et al. 2018)) just before the period gap, showing the donor star is fully convective within the period gap. Both artifacts correspond to the lesser efficiency of magnetic braking as binaries enter the gap. The combination of the decreasing angular momentum loss via magnetic braking and simultaneously decreasing donor mass confirms that select binaries in the CO-HMS grid are CV systems, which experience the period gap around 2-3 hours.
Discussion
Simulation data supports the current accepted explanation for magnetic braking. The magnetic braking prescription used by MESA (and in this work) involves a calculation of the approximate torque caused by magnetic braking on the donor star, which is a power law relationship between donor mass, donor angular velocity, and donor radius (Rappaport et al. 1983). While this prescription is accurate for binaries, it inaccurately predicts the way single stars slow down over time (Gossage et al. 2023). Furthermore, the prescription remains unclear about the manner in which magnetic braking becomes inefficient. Future runs should investigate magnetic braking models that can be applied to single and binary star systems accurately, such as modeling the magnetic fields of the secondary star, and empirically derived descriptions of how magnetic braking efficiency should scale with stellar properties (Garraffo et al. 2018; Matt et al. 2015). By accurately modeling magnetic braking to reflect the observed properties of single stars, binary star magnetic braking calculations can be improved, and the magnetic braking prescription constrained, mainly the mechanism by which magnetic braking becomes inefficient and its implications for the stellar dynamo. Furthermore, modeling magnetic breaking will improve models of the galactic population of CVs. By modeling the population of CVs in the galaxy, we can better understand how many signals to expect once upcoming missions such as LISA begin detecting finer gravitational waves, since CVs are expected to be a major source of gravitational wave signals in the future (Scaringi et al. 2023).
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Acknowledgements
This material is based upon work supported by the National Science Foundation under grant No. AST 2149425, a Research Experiences for Undergraduates (REU) grant awarded to CIERA at Northwestern University. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
This research was supported in part through the computational resources and staff contributions provided for the Quest high performance computing facility at Northwestern University which is jointly supported by the Office of the Provost, the Office for Research, and Northwestern University Information Technology.
I would like to thank my research advisors Dr. Seth Gossage and Dr. Vicky Kalogera and my REU advisors Dr. Aaron Geller and Dr. Chase Kimball for their guidance, my REU cohort for all the fun, and my family for their support.
