Orbital and Precession Periods in Repeating FRB 20121102A

Li {\it et al.\/} (2024) reported a 4.605 day period in the repeating FRB 20121102A in addition to the previously reported 157 day modulation of its activity. This note suggests that the shorter period is the orbital period of a mass-transferring star orbiting a black hole, possibly of intermediate mass, and that the 157 day period is the precession period of an accretion disc around the black hole. The mass-losing star must be evolved.

1. Introduction Li et al. (2024) recently reported the discovery of a 4.605 day period in the bursts of the repeating FRB 20121102A, previously known (Rajwade et al. 2020;Cruces et al. 2021) to have its activity modulated with a period of 157 days.It has been proposed (Katz 2022b) that the 16.3 day periodic activity modulation of the repeating FRB 20180916B may be attributed to precession of an accretion disc around a black hole, with fast radio bursts (FRB) emitted from the accretion funnel near the direction of the angular momentum axis.Such a model implies the presence of a binary companion as the source of the accreted matter and of the torque that drives the precession, hence the existence of an orbital period.This note identifies the newly discovered 4.605 day period of FRB 20121102A as the orbital period of its source.

Comparison to Other Precessing Binaries
Her X-1 was the first binary discovered to have a superorbital period.This was soon attributed (Katz 1973) to Newtonian precession of the accretion disc's axis under the influence of its companion star, in analogy to the precession of the Moon's orbit under the influence of the Sun's gravity.A precessing disc in SS 433 (Katz 1980;Margon 1984) accelerates a subrelativistic jet along its axis that emits hydrogen lines whose Doppler shifts permit quantitative measurement of the disc's precession (Katz et al. 1982).Larwood (1998) summarized the parameters of X-ray binaries with well-established superorbital periods attributed to disc precession.The ratios of superorbital to orbital periods range from 13 to 22.The two periods reported for FRB 20121102A have a ratio of 34, somewhat outside the range of the other systems, but qualitatively similar, supporting the identification of its periods as orbital and precession.
Theoretical calculations (Katz 1973(Katz , 2024) ) of the period ratio if the disc is treated as a ring at the Roche Circularization Radius (RCR) indicate much larger values, especially if the binary mass ratio is large.However, such calculations ignore the fact that viscosity broadens the ring to a disc after it forms.The period ratio depends on the unknown distribution of mass in the disc (Larwood 1998), and is much smaller than for a ring at the RCR because the outward expansion of the disc to its last stable orbit (Bahcall et al. 1974) increases the torque on it.The period ratio of 34 observed in FRB 20121102A is likely consistent with attribution of its 157 day period to disc precession and of its 4.605 day period to its orbit.

Periodicity vs. Periodic Modulation of Activity
The bursts of FRB 20121102A are not periodic, despite the presence of periodic signals in the Phase-folding probability Binomial Analysis (PBA).Only the envelope of burst activity is periodic, as also observed for the repeating FRB 20180916B (CHIME/FRB 2020; Pleunis et al. 2021; Mckinven et al. 2023) with a period of 16.3 d (there is yet no evidence for another period in this system).This explains why the PBA can reveal the periodicities of FRB 20121102A while periodograms, χ 2 , H tests and quadratic mutual information analysis, that are sensitive to periodicities but not to more subtle temporal structure, do not (Li et al. 2024).

Discussion
No repeating FRB has shown periodic bursts and upper limits are constraining (Katz 2022a;Niu et al. 2022;Du et al. 2023).As a result, it has been proposed that their sources are not magnetic neutron stars, whose emission would be expected to be periodic at their rotational periods, but rather the throats of accretion discs around black holes for which no periodicity is expected (Katz 2017).The source of mass for accretion might be interstellar gas, in which case again no periodicity would be expected, but it might alternatively be a binary companion, in which case both orbital and precessional periods exist and might be observable.The viewing angle of the throat of a precessing disc varies at the precession period, naturally modulating the burst rate (Katz 2022b).It is harder to see how the orbital period might be manifested observationally, but it is known that in SS 433 there are intercombination frequencies in the orientation of the jet, and hence of the funnel from which it emerges (Katz et al. 1982).
The detection of jitter (about the modulation period) in the times of bursts from FRB 20180916B with fractional amplitude similar to that of jitter in SS 433 (Katz 2022b) strengthened the case for accretion discs as the sources of repeating FRB.The recent discovery (Li et al. 2024) of a second period in FRB 20121102 demonstrates the existence of the second period predicted by precessing disc models, with a period ratio comparable to those of known precessing discs in binary X-ray sources (Larwood 1998).
In the hypothetical semi-detached mass-transferring binary system with orbital period 4.605 days the density ρ of the mass-losing star may be estimated from Kepler's law and approximations (Eggleton 1983) to the size of Roche lobes.If the ratio of the mass of the accreting black hole to that of the mass-losing star q ≫ 1, the result is nearly independent of q and independent of the masses if q is specified.This is about two orders of magnitude less than the density of any main sequence star, implying an evolved star.The origin of such a system is a matter for speculation, but it might be either the remains of a massive binary one of whose members has collapsed to a black hole or an intermediate mass black hole that captured, by either tidal dissipation or three-body -6interaction, a star.
This model suggests that PBA analysis of the times of bursts from FRB 20180916B may reveal another shorter period in addition to its known 16.3 d period.By analogy, such an orbital period might be O(0.5 d).The same suggestion applies to any other FRB whose activity is periodically modulated.