Periodically Modulated FRB as Extreme Mass Ratio Binaries

The activity of at least one repeating Fast Radio Burst (FRB) source is periodically modulated. If this modulation is the result of precession of the rotation axis and throat of an accretion disc around a black hole, driven by a companion that is also the source of accreted mass, then it may be possible to constrain the mass of the black hole. The dynamics is analogous to that of superorbital periods in ordinary mass-transfer binaries in which the accreting object may be a stellar-mass black hole, a neutron star or a white dwarf, but in the FRB source it may be an intermediate mass black hole. In a semi-detached (mass-transferring) binary the orbital period is related to the mean density of the mass-losing star. Assuming a value for its density and identifying the observed modulation period as a disc precession period would determine the mass ratio and the mass of the black hole. This model and magnetar-SNR models make distinguishable predictions of the evolution of the FRB rotation measure that may soon be tested in FRB 121102.


INTRODUCTION
Fast radio burst sources may be divided into two classes, frequent repeaters, with intervals observed by the most sensitive telescopes of ∼ 1 min during periods of intense activity, and apparent non-repeaters.These classes differ in other properties (Katz 2022a) and may be disjoint.The apparent nonrepeaters likely repeat at long intervals because the rate of catastrophic events appears to be insufficient to explain them (Hashimoto et al. 2020).Katz (2017); Sridhar et al. (2021) suggested that the frequent repeaters are produced by accretion discs around intermediate mass (10 2 -10 6 M ) black holes.This hypothesis is supported by the explanation of the periodic modulation, and its jitter around strict periodicity, of the activity of FRB 2018916B as a result of disc precession (Katz 2022b).
In such a model the binary system is semi-detached, or nearly so; the nondegenerate companion is in (or very close to) contact with its Roche lobe and mass transfer is continuous, although its rate may vary.The bursts themselves must be the result of brief plasma storms superposed on the more slowly varying disc orientation and mass flow rate.
Precession of the disc angular momentum axis about the orbital axis is the cause of many or most of the "superorbital" periods ubiquitous in mass-transfer binaries.Precession is a weakly damped eigenmode of dynamical excitation about the disc's lowest energy state, in which it is circular and coplanar with the orbit.Excitation of this mode is generally attributed to "turbulence" (fluctuating torques) in the accretion disc or fluctuating deviations of the mass-transfer stream from the orbital plane, perhaps driven by turbulence E-mail katz@wuphys.wustl.eduor spin mis-alignment in the mass-losing star.The amplitude of excitation cannot be predicted, but in the most favorable systems may be measured, at least approximately.Precession may be manifested as periods in which the accreting object is eclipsed by the disc (Her X-1; Katz (1973); Levine & Jernigan (1982)) or directly measured in Doppler shifts of a disc's axial jet (SS 433;Katz (1980);Milgrom (1981); Katz, Anderson, Margon et al. (1982); Margon (1984); Fabrika (2004)).Four such systems are well established (Larwood 1998) with ratios of their precession to orbital periods in the range 12.5-22:1.These have primary to secondary mass ratios O(1), though only approximately determined.

PRECESSION RATE
The precession rate ωpre of a circular ring of radius R in orbit around an object of mass Mp, that itself has a secondary object of mass Ms in a circular orbit of radius a R and angular rate Ω orb , is (Katz 1973;Larwood 1998) ωpre where q = Ms/Mp and δ is the inclination between the orbital and ring angular momentum axes (cos δ is generally taken as unity).This is a classical result whose origin is Newton's theory of the precession of the plane (recession of the nodes) of the Moon's orbit under the gravitational influence of the Sun.Larwood (1998), making plausible assumptions for the distribution of mass in a disc extending inward from R to the compact object, calculated that the factor of 3/4 in Eq. 1 should be replaced by 3/7 because the effective value of R for a continuous accretion disc is less than its outer radius1 .Accretion discs may be accompanied by and coupled to excretion discs extending beyond the RCR, increasing the coefficient in Eq. 1, offsetting the reduction calculated by Larwood (1998).At some radius > R but < a the disc is truncated as its orbits become unstable (Bahcall et al. 1974), limiting the possible value of the coefficient.
R is usually taken as the Roche Circularization Radius (RCR) (Katz 1973) at which a stream of matter spilling over the L1 point from the mass-losing secondary circularizes its motion by dissipatively colliding with itself, conserving angular momentum.In the absence of angular momentum transport and further viscous dissipation, a circular ring (with δ = 0) is the lowest energy state.Viscous dissipation more slowly turns the ring into a disc.

EXTREME MASS RATIOS
Eq. 1 is only valid in the limit R/a → 0. This is a good approximation if q 1. Calculations of binary accretion discs (Warner & Peters 1972;Katz 1973;Frank, King & Raine 2002) have usually considered Ms Mp2 .This is appropriate for many binary stars undergoing mass transfer, but not for extreme recycled binary pulsars and cataclysimic variables in which the mass-losing star may have a mass of hundredths of M .
A widely used fit to the RCR (Eq.4.20 of Frank, King & Raine ( 2002)) breaks down for q < 0.1, for which it would imply a value greater than the separation between the stars.Yet we are concerned with accretion from ordinary stars onto intermediate mass black holes, so that q 1, perhaps by orders of magnitude.
It is straightforward to calculate the Roche Circularization Radius by integrating the orbits of freely falling particles (or gas streams with low sound speeds) from the inner Lagrange Point L1, making the usual Roche assumption of an aligned and synchronously rotating mass-losing (secondary) star with a cool (small scale height) atmosphere.The circularization criterion where L1 is the specific angular momentum about the primary (black hole), must be evaluated in an inertial frame, although the orbit calculation is conveniently done in the corotating frame.For q = O(1) this is a small correction, but for q 1 it is essential because the circularized matter orbits the primary at an angular rate that approaches Ω orb in the limit q → 0.
The location of the inner Lagrange point L1 and RCR are shown in Fig. 1 as a function of q.The location of L1, obtained by numerically finding the maximum of the effective potential (including the centrifugal force) in the corotating frame, agrees in the limit q → 0 with the asymptotic expression of Kopal (1977).The orbits of freely falling particles or cool gas streams from L1 to their RCR are shown in Fig. 2.
For q 1, R a and Eq. 1 is not a valid approximation for the precession rate.Then the precessional torque on an inclined ring (δ = 0) is nearly impulsive, concentrated on mass elements as they pass near L1 at distances a.For q 1 it must be evaluated by numerical integration.It is assumed that δ 1 − R/a 1.The results are shown in Fig. 3 as the ratio of the precession period to the binary orbital period in the limit δ → 0. In the regime q 1 the results are well fit by the expression Ppre P orb ≈ 6.86q −2/3 . (3)

APPLICATION TO INTERMEDIATE MASS BLACK HOLE BINARIES
The difficulty of applying Eq. 3 to observations is that while it can be argued that in at least one repeating FRB (Katz 2022b) Ppre is observed as an activity modulation period, neither the value of P orb nor the secondary (mass-losing star) mass Ms is known.Hence it is not possible to use Eq. 3 directly to determine q and the black hole mass.We suggest some possible ranges of these values to indicate how the method may be used if more data become available.In a fair approximation to Roche geometry for q 1 and a Periodically Modulated FRB 3 Figure 2. Orbits of particles or cool matter spilling over the secondary's Roche lobe at L1 for several values of q 1.The paths are truncated at the RCR, and these radii are illustrated by circles centered at the primary.The origin is taken at the primary (not the center of mass), and the stellar separation is taken as unity.
The paths begin at L1 on the abscissa, and are bent by the Coriolis force so it is only necessary to show one quadrant.
where Rs would be the radius of the secondary star were it not perturbed by the gravitational field of the primary.
Using Kepler's law for a circular orbit, Eq. 4 implies, nearly independent of q, where ρ is the mean density of the secondary star.
If the secondary is a white dwarf then the orbital period is O(10 s).This may be excluded because its lifetime to gravitational radiation would be shorter than the observed multiyear lower bounds on the lifetimes of repeating FRB sources.
The mean densities of nondegenerate stars range from O(100 g/cm 3 ) for low mass main sequence stars to O(10 −6 g/cm 3 ) for supergiants.This is consistent with a very wide range of P orb and an even wider range of Ppre (because q is not known a priori).Very long Ppre are unobservable.

SECONDARY DENSITY AND BLACK HOLE MASSES
If the 16.3 day periodic modulation of the activity of FRB 180916B (Pleunis et al. 2021) is interpreted (Katz 2022b) as a disc precession period, a coupled constraint on q and ρ can be set.Combining Eqs. 3 and 5, ρ ≈ 0.3q −4/3 g-cm −3 .
(6) For q 0.2 (a necessary condition for the applicability of Eq. 6), ρ 3 g-cm −3 ; the secondary star must be a low mass main-sequence star; see Eq. 7. The same interpretation of the 159 day periodicity of FRB 121102 (Rajwade et al.

2020) implies ρ
3.5 × 10 −5 q −4/3 g-cm −3 , consistent with q 0.2, a subgiant secondary star, or both.If the secondary is on the zero-age main sequence its mean density may be approximated where m ≡ ms/M and ρ ≈ 1.4 g-cm −3 .Combining these results, the black hole mass mBH This is plotted in Fig. 4. Lower MBH are possible if the secondary has evolved from the main sequence and its density has decreased below the value of Eq. 7.

DISCUSSION
It is not evident how to test directly the hypothesis that FRB emerge from the throat of an accretion disc.In some cases (Katz 2022b) indirect arguments may be used, but these may be challenged as relying on unproved models.The results of this paper do not solve that problem because they only imply relations between pairs of quantities (mBH and m, or mBH Curves are labeled by the precessional period.All masses are in units of M .Curves are only plotted for q ≤ 0.2 for which Eq. 3 is valid. and q, or mBH and P orb , etc.), neither of which is (at present) directly measurable, or between Ppre and a quantity ( ρ , P orb , q, m, etc.) that is not directly measurable.If many more periodically modulated FRB sources are discovered it may be possible to compare their statistics to the predictions of population synthesis and the model developed here.
A test of a leading alternative model, that FRB are produced by young magnetars in young supernova remnants, may be possible, at least for one source (FRB 121102) with periodically modulated activity (Rajwade et al. 2020).FRB 121102 has an extraordinarily large and rapidly decreasing rotation measure (Michilli et al. 2018;Hilmarsson et al. 2021;Plavin et al. 2022;Feng et al. 2023).In a naïve magnetar-SNR model, the rotation measure decreases ∝ r −4 ∝ t −4 , where r is the radius of the SNR and t is the time since the explosion, because the column density and a frozen-in magnetic field each decrease ∝ r −2 ∝ t −2 .The observed rapid decrease would imply the source was formed 10 years before the first measurements of its rotation measure in 2016.This power law decay would continue until only residual Galactic and host galaxy contributions, orders of magnitude less than the observed rotation measure, remain.
In the model proposed here, the rotation measure is produced by a chaotic plasma environment of the accretion disc (and perhaps the disc itself), and fluctuates irregularly, like the dispersion measure of FRB 190520B (Katz 2022c).In contrast to the power law decay predicted by the magnetar-SNR model, the accretion disc model predicts that the rotation measure may (but would not necessarily) reverse sign, or that its decay may be replaced by a period of growth.Continued observation of the rotation measure of FRB 121102 thus may be a definitive test of the magnetar-SNR model, even though it would not be a direct test of the accretion disc model.

Figure 1 .
Figure1.Roche Circularization Radius (RCR) and the distance of the inner Lagrange point L1 from the center of mass, in units of the binary separation a, as functions of the mass ratio q for q ≤ 1.

Figure 3 .
Figure3.The precession period in the Roche model for q ≤ 1, as a multiple of the binary orbital period.

Figure 4 .
Figure 4. Black hole mass as a function of the secondary mass if the secondary is on the main sequence with ρ given by Eq. 7.Curves are labeled by the precessional period.All masses are in units of M .Curves are only plotted for q ≤ 0.2 for which Eq. 3 is valid.