Reconciling the 16.35-day period of FRB 20180916B with jet precession

We take SS 433 as an example to introduce the accretion disk-driven jet precession model. SS 433 is an X-ray binary, containing an $8\ M_{\sun}$ spinning BH and a $24\ M_{\sun}$ nondegenerate star (Han & Li, 2020). In this model, it is assumed that the outer regions of the accretion disk are tilted to the orbital plane. The spin axis of the BH and the binary axis are misaligned. The inner regions of the accretion disk and BH are gravitationally coupled and can precess together due to the Lense-Thirring effect (Sarazin et al., 1980). If the precession period is shorter than the viscous timescale in the disk, the precession process is approximate to the conservation of angular momentum. Following the spirit of Sarazin et al. (1980), we assume a ring in the disk with the width $dr$ has an angular momentum per logarithmic interval of radius $J(r)=dJ/d(\ln{r})=2\pi r^{3}\Sigma v_{\phi}$, where $\Sigma$ is the surface density of the disk and $v_{\phi}$ is the rotational velocity of the disk. Meanwhile, the angular momentum of the BH is $J_{\ast}=a_{\ast}GM^{2}_{\ast}/c$, where $M_{\ast}$ is the BH mass, $c$ is the speed of light, $G$ is the gravitational constant, and $a_{\ast}$ is the dimensionless specific angular momentum of BH, $0\leqslant a_{\ast}<1$. Then, there exists a critical radius $r_{\rm{prec}}$ (the precession radius) such that $J(r_{\rm{prec}})=J_{\ast}$. The outer portions of the disk ($r>r_{\rm{prec}}$) with more angular momentum than the central BH can cause the precession of the BH and the central parts of the disk ($r\ll r_{\rm{prec}}$). If the disk is adequately massive ($10^{-5}-10^{-2}\ M_{\sun}$ (Sarazin et al., 1980)), the outer portions of the disk with sufficient angular momentum will maintain their orientation. Part of the accreted inner region materials are supposed to be ejected out along the BH spin axis, forming a precessing jet.

The precession frequency is $\Omega_{\rm{prec}}=2GJ(r)/c^{2}r^{3}$ (Sarazin et al., 1980). Since $J(r)/r^{3}$ decreases with increasing $r$ in the outer disk, the fastest precession rate is produced at the radius $r=r_{\rm{prec}}$ with $J(r_{\rm{prec}})=J_{\ast}$:

By using the standard disk model (SSD) of Shakura & Sunyaev (1973), the precession radius $r_{\rm{prec}}$ and the precession period $P_{\rm{prec}}$ can be derived as (also see Equations (2) and (5) in Sarazin et al., 1980)

where $\alpha$ is the dimensionless viscous parameter. Note that, $10^{-8}\ M_{\sun}\ \rm{yr^{-1}}$ is the Eddington accretion rate of a $1\ M_{\sun}$ compact object. Recalling that $M_{*}=8\ M_{\odot}$ and $\dot{M_{\ast}}\sim 10^{-4}\ M_{\sun}\ \rm{yr^{-1}}$, according to Equation (3), the required $\alpha$ to reconcile the observed precession period is $\sim 10^{-8}$ (see the solid line in Figure 1; also see Sarazin et al., 1980) as long as $a_{*}>0.1$; this value is seven orders of magnitude smaller than the value inferred from the observations (e.g., the AGN optical variability; see Sun et al., 2020). The reasons for an extremely low viscous parameter are as follows. The outer regions of the disk should have enough angular momentum to force the BH and the inner disk to precess. To obtain enough angular momentum, a high surface density $\Sigma$ of the disk is needed in the model. For fixed accretion rate, according to the mass conservation equation $\dot{M}_{\ast}=-2\pi r\Sigma v_{r}$, the radial velocity $v_{r}$ should be extremely small in the outer portions of the disk. The required slow radial velocity, which scales as $v_{r}\propto\alpha^{4/5}$ (Shakura & Sunyaev, 1973), indicates that $\alpha$ should also be unusually small.

Given the fact that the accretion rate in SS 433 is extremely super-Eddington, the application of SSD to this system might not be self-consistent. The slim disk model (Abramowicz et al., 1988) is likely a better choice. In the slim disk model, the flow with $v_{\phi}$ is sub-Keplerian $v_{\phi}\sim V_{\rm{K}}/\sqrt{5}$ (Wang & Zhou, 1999). The radial velocity of the disk material $v_{r}$ is roughly equal to $-\alpha V_{\rm{K}}/\sqrt{5}$. Thus, the precession radius and the precession period can be expressed as

From Equation (5), the viscous parameter $\alpha$ at precession radius $r_{\rm prec}$ for the 162.5-day precession period of SS 433 is $\sim 10^{-9}$ (see the dashed line in Figure 1). According to Equation (6) of Sarazin et al. (1980), it is quite straightforward to see that the precession period $P_{\rm{prec}}$ depends upon the ratio of the radial to angular velocities (for fixed accretion rate). In the slim disk model, the ratio of $v_{\rm{r}}$ to $v_{\rm{\phi}}$ is larger than that in SSD. In order to produce the same precession period, for fixed accretion rate $\dot{M}_{\ast}$, the required $\alpha$ for the slim disk model should be smaller than that for SSD.

Although the disk-driven jet precession is unlikely to account for the observed periods for SS 433, this model is able to interpret the long timescale precession of AGNs (e.g., Lu, 1990; Lu & Zhou, 2005) and the burst timescale of GRBs (e.g., Liu et al., 2010; Sun et al., 2012; Hou et al., 2014).

Katz (2017) proposed that the repeating FRBs can be produced by a wandering jet based on the chaotic accretion (i.e., the accreting gas has random angular momentum) in an intermediate-mass ($\thicksim 10^{2}-10^{6}\ M_{\sun}$) BH accretion system. A repeating FRB is observed when the jet repeatedly sweeps across our line of sight. In addition, for the disk-driven jet precession in an intermediate-mass BH system, an extremely low viscous parameter ($\alpha\approx 10^{-12}$ for $\dot{M}\approx 10^{-7}\ M_{\sun}\ \rm{yr}^{-1}$; $\alpha\approx 10^{-8}$ for $\dot{M}\approx 10^{-1}\ M_{\sun}\ \rm{yr}^{-1}$) is necessary to account for the 16.35-day period of FRB 180916 regardless of the BH spin, as implied by Equations (3) and (5). Therefore, whether the central BH is a stellar mass or an intermediate mass, the disk-driven jet precession model is unlikely to account for the periodic activity of FRB 180916.

Another mechanism involves the jet precession of a binary system due to the tidal force of the companion star. Katz (1973) first proposed this model to explain the precession period of Her X-1. Then, Katz (1980) and Leibowitz (1984) used this model to explain the precession period of SS 433. The model can explain the precession of binary system of T-Tauri stars and some X-ray binary systems, such as Cyg X-1 and LMC X-1 (e.g., Papaloizou & Terquem, 1995; Larwood, 1997, 1998; Caproni et al., 2006).

If the observed period is indeed caused by the tidal force-driven jet precess, we might immediately infer some appropriate characteristics of FRB 180916. Larwood (1998) and Caproni et al. (2006) showed that the ratio of the precession period $P_{\rm{prec}}$ and orbital period $P_{\rm{orb}}$ decreases with increasing mass ratio of two stars in the tidal force-driven jet precession model. For SS 433, the mass ratio of two stars is $q=M_{\rm{2}}/M_{\ast}=3$ (Han & Li, 2020) and the period ratio is $P_{\rm{prec}}/P_{\rm{orb}}>10$, where $M_{\rm{2}}$ is the mass of the secondary star. Considering the period ratio of SS 433 and the observed 16.35-day precession period of FRB 180916, we can infer that the orbital period of the binary system for FRB 180916 is less than one day since the relevant $q$ is unlikely to be larger than 3. That is, FRB 180916 may come from such a system with super-Eddington accretion rate, relativistic jets, and a short orbital period ($<$1 day). Under the condition of stable mass transfer of the binaries, the orbital periods of most double neutron star systems (DNSs) are more than one day (Tauris et al., 2017) and the accretion rates of double white dwarfs (DWDs) and BH-NS binaries are much less than the Eddington accretion rates (Marsh et al., 2004). Hence, we rule out the DNSs, DWDs, and BH-NS models. According to the above speculations, the system of FRB 180916 can be either an NS-WD binary (Gu et al., 2016, 2020) or a BH-WD binary (Dong et al., 2018). Note that the mass ratio of a BH/NS-WD binary $q\ll 3$.

The schematic diagram of our model is shown in Figure 2. In this model, we consider the binary system in a circular orbit. We denote the mass of the central compact star is $M_{1}$ and the mass of the companion star (WD) is $M_{2}$ ($M_{\rm{WD}}$). The central object is surrounded by a non-self-gravitating and axisymmetric disk. The outer regions of the disk are tilted with respect to the orbital plane whereas the inner regions of the disk are coplanar with orbit. The spin axis of the central object is misaligned with the binary axis. In such a system, the mass transfer occurs when the WD fills its Roche lobe. These accreted materials form a ring at the edge of the outer portions of the central star’s disk. Then, the plane of the ring will precess due to the tidal force of the WD (Katz, 1973). The tidal torque applied to the outer edge of the accretion disk will transfer inward by bending wave transport or by viscous stresses, or by a combination of both effects (Larwood, 1998). There is a gradient of the tilt angle in the disk with different radii, which leads to the precession of the whole disk. Owing to the Bardeen-Petterson effect (Bardeen & Petterson, 1975), the central object along with the central part of the disk precesses together. Part of the accreted materials from the WD is supposed to be ejected out along the BH spin axis, forming a precessing jet.

The precession frequency $\Omega_{\rm{prec}}$ of the disk induced by the tidal force is taken the form (Papaloizou & Terquem, 1995; Larwood, 1997)

where $\Sigma$ is axisymmetric surface density, $\Omega$ is Keplerian angular velocity and $a$ is the distance between two stars. The parameter $\theta$ indicates the orbital inclination with respect to the plane of the accretion disk. In the circular orbit, the jet forms a precession cone with half opening angle equal to the angle of orbital inclination $\theta$ (Caproni & Abraham, 2004). The minus sign indicates that the precession is retrograde. That is, the precession is opposite to the rotation direction of the accretion disk.

Following Larwood (1997), in order to avoid the detailed investigation on the energy balance, a simple polytropic relation between gas pressure $P$ and density $\rho$ is adopted, i.e., $P=K\rho^{1+1/n}$, where $n$ is the polytropic index. Such a polytropic relation was widely used in literatures on accretion systems, such as the Bondi accretion and the vertical structure of accretion disks (e.g., Chapter 2.2.1 and 7.2.1 of Kato et al., 2008). By integrating Equation (6) from the inner radius $R_{\rm{in}}$ of the disk to the outer radius $R_{\rm{out}}$ of the disk , we can obtain (Larwood, 1998)

where $q$ is mass ratio defined as $q\equiv M_{\rm{WD}}/M_{1}$ , and $\Omega_{\rm{out}}$ is the Keplerian angular velocity at the outer radius $R_{\rm{out}}$. For $n=3/2$, the precession period is written as (Larwood, 1998)

where $P_{\rm{orb}}$ is the orbital period. The size of the accretion disk is a fraction of the primary star’s Roche-lobe radius $R_{\rm{L1}}$, which is defined as $R_{\rm{out}}=\beta R_{\rm{L1}}$. The Roche-lobe radius of the primary star can be expressed as (Eggleton, 1983)

Thus, Equation (8) can be rewritten as

where $\mathcal{R}=R_{\rm{L1}}/a$. From Equation (10), the precession period $P_{\rm{prec}}$ can be derived if the values of $\beta$, the mass ratio $q$, the orbital inclination $\theta$, and the orbital period $P_{\rm{orb}}$ are given. Paczynski (1977) calculated the value of $\beta$ for mass ratio $0.03<q<30$. For mass ratio $0.03<q<2/3$, $\beta$ is a mean value of 0.86. For mass ratio $2/3<q<30$, $\beta$ is a function of mass ratio $q$

In our model, the binary system should keep the state of stable accretion and gravitational balance, so the mass ratio should $q<2/3$ (King et al., 2007a). Consequently, $\beta$ can take a constant value of 0.86. Here we take an average half opening angle of the jet about $\theta=20{\arcdeg}$ (Caproni et al., 2006; Larwood, 1998).

The WD fills its Roche lobe, i.e., the radius of the Roche lobe is equal to the radius of WD, $R_{\rm{L2}}=R_{\rm{WD}}$. $R_{\rm{WD}}$ can be expressed as (Verbunt & Rappaport, 1988)

where $M_{\rm{Ch}}$ is the Chandrasekhar mass limit $M_{\rm{Ch}}=1.44\ M_{\sun}$ and $M_{\rm{p}}=0.00057\ M_{\sun}$. The radius of the Roche lobe for WD $R_{\rm{L2}}$ can be expressed as (Eggleton, 1983)

The Kepler’s third law for this system is

Once $M_{1}$ and $M_{\rm{WD}}$ are given, we can obtain $P_{\rm{orb}}$ and $a$ by Equations (12)-(14). Then, we can infer the precession period $P_{\rm{pre}}$ by Equation(10).

As a second step, we estimate the mass transfer rate of the binary system. SS 433 has a pair of precessing jets with a super-Eddington accretion rate. We consider that our model should also have high accretion rates in order to produce the collimated jets. Dong et al. (2018) proposed a BH-WD binary system with an extremely super-Eddington accretion rate which has violent accretion to explain long GRBs without detection of supernova association. On the contrary, under the case of stable accretion, the mass of BH/NS-WD binary transfers through Roche-lobe overflow. The mass transfer rate from the WD to its companion is (Dong et al., 2018)

where $M=M_{1}+M_{\rm{WD}}$ is the total mass of the primary and secondary star. Then, a dimensionless parameter, specific accretion rate, is defined as $\dot{m}=\dot{M}/\dot{M}_{\rm{Edd}}$, where $\dot{M}_{\rm{Edd}}=10L_{\rm{Edd}}/c^{2}$ is the Eddington accretion rate for the primary star. Thus,

where the opacity $\kappa_{\rm{es}}=0.34\ \rm{cm}^{2}\ \rm{g}^{-1}$.

We now need to evaluate the appropriate accretion rate for FRB 180916 in BH/NS-WD binary system. The isotropic radio luminosities of FRBs are estimated to be in a range $L_{\rm{radio}}\ \sim 10^{38}-10^{42}\ \rm{erg}\ \rm{s}^{-1}$ (Cordes & Chatterjee, 2019). However, due to our poor knowledge of radiative efficiency $\eta$ and the bolometric luminosity, we cannot estimate the accretion rate. Based on the simulation of Sadowski & Narayan (2015), it can be showed that SS 433 has a pair of radiatively driven jets, which is baryon-loaded, kinetically dominated, and collimated. For a $10\ M_{\sun}$ BH accreting at $\dot{m}=100$ in SS 433, the jet power is roughly $0.01\dot{M}c^{2}\approx 2\times 10^{40}\ \rm{erg\ s^{-1}}$ (Sadowski & Narayan, 2015). This is sufficient to explain the kinetic luminosity of the jets observed in SS 433. Thus, if the power of the jet in our model is roughly the same as that in the SS 433, we reckon the range of specific accretion rate $\dot{m}=10-1000$ for the binary system of FRB 180916. For an NS-WD binary, as the mass of NS is relatively small, a large accretion rate, $\dot{m}\sim 1000$, is needed to generate the observed radio luminosity. For a BH-WD binary, an appropriate specific accretion rate, $\dot{m}\sim 10$, can meet the luminosity criteria because the BH mass is relatively larger than that of an NS.

Combining Equations (10) and (17), we can obtain the relationship between the precession period $P_{\rm{prec}}$ and the masses of two stars in the range of specific accretion rate $\dot{m}=10-1000$ (Figure 3). There is a critical relationship for the mass of two stars (the green line in Figure 3) which corresponds to the boundary between stable and unstable accretion of the binary systems (see Equation (9) in Dong et al., 2018). Once the masses of the primary stars and WDs beyond the critical mass, the unstable and extremely violent accretion will occur (Dong et al., 2018). Then, the mass transfer process might be accompanied by strong outflows. The red line in Figure 3 represents the precession period of FRB 180916. Under the range of $\dot{m}=10-1000$, the mass of the primary star can be from $M_{1}\sim 2.8\ M_{\sun}$ (a massive NS) to $M_{1}\sim 15\ M_{\sun}$ (a stellar-mass BH) to reproduce the 16.35-day period in FRB 180916.

In summary, the tidal force driven-jet precession model can explain the periodic activity of FRB 180916 in BH/NS-WD binary system with a high accretion rate.

If the tidal force-driven jet precession model is correct, we might be able to infer the masses and types of the central stars of other repetitive and periodic FRBs. Combining Equations (10) and (17), if $P_{\rm{prec}}\lesssim 1\ \rm{day}$, the central objects are massive NSs; if $P_{\rm{prec}}\gtrsim 20\ \rm{day}$, FRBs can be observed in stellar-mass BH-WD binaries.

We can estimate the duration (denoted as $T$) of a periodic FRB in a binary system with a high accretion rate. To estimate the duration $T$ of a periodic FRB, we fix $M_{1}=1.4\ M_{\sun}$ for the NS-WD binary. For the BH-WD binary, we fix $M_{1}=3\ M_{\sun}$ and $10\ M_{\sun}$. In all binaries, the WDs are assumed to have an initial mass of $0.6\ M_{\sun}$ (Kepler et al., 2007). In our model, the duration $T$ of a periodic FRB is equal to the evolution time of the WD which evolves from the initial mass ($M_{\rm{WD}}=0.6\ M_{\sun}$) to the final mass (e.g., $\dot{m}=10,M_{\rm{WD}}\sim 0.14\ M_{\sun}$; see Equation (17) when $M_{1}=1.4\ M_{\sun}$). The mass transfer rate from the WD to BH/NS, $\dot{M}=dM_{\rm{WD}}/dT$, can be calculated from Equation (15), where the unknown parameter $a$ depends on $M_{\rm{WD}}$ and $M_{1}$ (see Equations (12) and (13)). As $M_{1}$ is usually much larger than $M_{\rm{WD}}$, we can reasonably assume that $M_{1}$ remains constant during the mass transfer process. The time interval between $M_{\rm{WD}}=0.6\ M_{\sun}$ and $M_{\rm{WD}}=0.14\ M_{\sun}$ is estimated as follows.

The results are shown in Figure 4. It shows that the duration $T$ of the periodic FRBs increases (decreases) with increasing $M_{\rm{WD}}$ (the mass of the BH/NS $M_{1}$) for all binaries. As mentioned above, FRBs can be produced for NS-WD or BH-WD binaries with $\dot{m}>10$. As the specific accretion rate $\dot{m}$ drops below 10, the system might evolve into UCXBs. Hence, the duration time of FRB 180916 generation ranges from thousands of years to tens of thousands of years. This duration time is only a small portion of the evolution time of binaries.

The event rate of FRBs may be affected by a number of factors, such as redshift, luminosity, and the types of FRBs (repeating or one-off). For example, the volumetric rate of FRBs is roughly $2\times 10^{3}\ \rm{Gpc}^{-3}\ \rm{yr}^{-1}$ at redshift $z\sim 1$ (Petroff et al., 2019), whereas Cao et al. (2018) showed that the local event rate is about $(3-6)\times 10^{4}\ \rm{Gpc}^{-3}\ \rm{yr}^{-1}$. With the luminosity function of 46 known FRBs, the event rate of FRBs was found to be $\sim 3.5\times 10^{4}\ \rm{Gpc^{-3}\ yr^{-1}}$ above $10^{42}\ \rm{erg\ s^{-1}}$, $\sim 5.0\times 10^{3}\ \rm{Gpc^{-3}\ yr^{-1}}$ above $10^{43}\ \rm{erg\ s^{-1}}$, and $\sim 3.7\times 10^{2}\ \rm{Gpc^{-3}\ yr^{-1}}$ above $10^{44}\ \rm{erg\ s^{-1}}$ (Luo et al., 2020). In addition, Wang & Zhang (2019) estimated the volumetric rate of repeating FRBs about $500\ \rm{Gpc}^{-3}\ \rm{yr}^{-1}$ over $0<z<0.5$. Hence, it seems that most FRBs are non-repeating FRBs.

According to our model, the event rate of the periodic FRBs $R_{\rm{FRB}}$ equals to the volumetric rate of BH/NS-WD binary systems with high specific accretion rates $\dot{m}\geq 10$. For the BH-WD binary, according to Fryer et al. (1999), the formation rate of BH-WD binaries is estimated to be $\sim 0.15\ \rm{Myr}^{-1}\ \rm{galaxy}^{-1}$. Then, we use the relationship between the number density of galaxy and the age of the Universe (Conselice et al., 2016) to estimate the number of galaxies in $0<z<1$. The total number of galaxies within $0<z<1$ can be expressed as (Conselice et al., 2016)

where the volume of the whole sky is interated through $4\pi$ in steradians, $D_{A}$ is the angular size distance, $D_{H}=c/H_{0}(H_{0}\approx 70\ \rm{km}\ {s}^{-1}\ \rm{Mpc}^{-1})$, $E(z)=\left(\Omega_{M}\left(1+z\right)^{3}+\Omega_{\lambda}\right)^{1/2}$($\Omega_{M}\approx 0.3,\Omega_{\lambda}\approx 0.7$), and $t$ is the age of the universe in units of $\rm{Gyr}$ at redshift $z$. The number density of galaxies $\phi_{T}(t)$ can be fitted as (Conselice et al., 2016)

By multiplying Equation(19) and the formation rate of BH-WD binaries, we can get the volumetric rate of BH-WD binaries $\sim 340\ \rm{Gpc}^{-3}\ \rm{yr}^{-1}$. For the NS-WD binary, Zhao et al. (2021) calculate the number of the NS-WD binaries evolved from a unit mass stellar population, $f_{\rm{NSWD}}\sim 2.1\times 10^{-5}\ M_{\sun}^{-1}$. Then, we need the cosmic star formation rate (CSFR) to estimate the volumetric rate of NS-WD binaries. The CSFR is taken the form (Madau & Dickinson, 2014)

Combining $f_{\rm{NSWD}}$ and Equation (21), we can evaluate the volumetric rate of NS-WD binaries in redshift $0<z<1$, i.e., $\sim 4500\ \rm{Gpc}^{-3}\ \rm{yr}^{-1}$.

According to our model, not all BH/NS-WD binaries produce the periodic FRBs. Only in BH/NS-WD binaries with high accretion rates ($\dot{m}\geq 10$), can we observe periodic FRBs. Not all BH/NS-WD binaries have mass transfer processes. That is, only a fraction (denoted as $f_{\rm{acc}}$) of BH/NS-WD binaries can power periodic FRBs. Only 14 UCXBs have been confirmed in the Galaxy (van Haaften et al., 2012). The donor stars in these binary systems must be WDs or helium-burning stars that fill their Roche lobes since the orbital periods of these binaries are less than one hour. Therefore, it seems that $f_{\rm{acc}}$ is very small. The total number of periodic FRBs should also be proportional to the duration time $T$ and anti-correlate with the jet beaming factor $f_{\rm{b}}$. Thus, the event rate of the repetitive and periodic FRBs is evaluated to be $\sim 0.05\ f_{\rm{acc}}\ (f_{\rm{b}}/100)^{-1}\ (T/1\ \rm{kyr})\ \rm{Gpc}^{-3}\ \rm{yr}^{-1}$.