Accretion discs with non-zero central torque
Introduction
In a standard accretion disc, all of the material arriving at the inner radius, Rin, is accreted by the central object, along with all its angular momentum. This is equivalent to setting a zero viscous torque boundary condition at .
For a steadily accreting disc of this kind, as shown by Shakura and Sunyaev (1973), the energy dissipated per unit area is given bywhere M is the mass of the central object and is the steady accretion rate through the disc. Thus the total luminosity released in such a disc isThis is one half of the available gravitational energy. The other half is advected through the inner boundary in the form of kinetic energy of circular motion.
The lack of a central torque also implies that all the angular momentum arriving at the inner boundary is also advected inwards. This is a good approximation, for example, for a thin disc around a star which is not rotating close to break up (e.g. Pringle, 1981). However, this may not always be the case. Examples of where the inner torque on the disc might not be zero to a good approximation include:
- 1.
an accretion disc which is truncated by a stellar magnetosphere,
- 2.
a “decretion” disc around a Be star, and
- 3.
a circumbinary disc.
The torque at the inner boundary has also been discussed in the context of the appropriate boundary condition to apply at the innermost stable circular orbit (ISCO) of a disc around a black hole.
Here we explore disc solutions in the presence of a non-zero torque at the inner boundary of the disc. In Section 2 we provide the underlying equations. In Section 3 we provide steady state solutions to the disc structure. In Section 4 we provide analytical and numerical solutions to the time-dependent disc evolution. In Section 5 we discuss different astrophysical systems for which these models may be appropriate, and we conclude in Section 6.
Section snippets
Disc equations
The disc equations describing the time dependence of surface density, Σ(R, t) where R is radius and t time, are (see, for example, Pringle, 1981), first, conservation of mass:where VR is the radial velocity, and, second, conservation of angular momentumHere Ω(R) is the angular velocity of the disc material, ν the vertically averaged kinematic viscosity, and .
By manipulating these we find that
Henceforth we
Steady disc
To illustrate the implications of an effective inner torque, we first consider a steady disc between radii Rin and Rout with matter being added at a constant rate at a radius Radd, where Rin < Radd < Rout. We are also interested in the infinite disc, which is the limiting case Rout → ∞. For simplicity we consider the case of a Keplerian disc, for which . To simplify the algebra we define
The usual boundary conditions are either zero torque () or zero radial velocity (
Time-dependent discs
We next consider the time-dependent behaviour of discs with non-zero inner torque. We consider the case in which the viscosity is a given function of radius. This keeps things simple as the evolution equation is then linear in Σ, but the basic physics is unaltered.
Discussion
We have noted that the standard zero inner torque boundary condition () is an adequate approximation for discs around non-magnetic stars for which the stellar angular velocity, Ω⋆, is not close to the break-up speed, (Pringle, 1977, Pringle, 1981, Tylenda). If Ω⋆ becomes very close to ΩK (assuming that the star remains stable, which it probably does not) then accretion can still continue, but as Ω⋆ → ΩK, the star provides a net torque and f → 1 (Pringle, 1989, Popham, Narayan,
Conclusions
We have presented analytical and numerical calculations of accretion discs with a non-zero torque inner boundary condition. We have defined the parameter f as the ratio of the outward viscous flux of angular momentum to the inward advected flux at the inner disc boundary. Our results approach the standard cases of accretion discs () and decretion discs (f → ∞) in the appropriate limits. For f > 0 both energy and angular momentum are fed into the disc through the inner disc boundary, and we
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
We thank the referee, Stephen Lubow, for helpful input. We thank Steven Balbus for useful correspondence. We thank Eric Coughlin for useful comments on the manuscript. CJN is supported by the Science and Technology Facilities Council (grant number ST/M005917/1). CJN acknowledges funding from the European Unions Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 823823 (Dustbusters RISE project). This research used the ALICE High Performance
References (57)
- et al.
Nature
(2019) - et al.
ApJ
(2003) - et al.
ApJ
(2001) - et al.
ApJ
(1991) - et al.
ApJ
(1994) - et al.
ApJ
(1996) - et al.
ApJ
(2017) - et al.
MNRAS
(2018) - et al.
MNRAS
(1974) - et al.
Nature
(1980)
MNRAS
MNRAS
ApJ
Computational differential equations
ApJ
ApJ
ApJ
ApJ
Table of integrals, series and products
ApJ
ApJ
MNRAS
ApJ
ApJ
ApJ
ApJ
ApJ
Cited by (26)
Disc precession in Be/X-ray binaries drives superorbital variations of outbursts and colour
2024, Monthly Notices of the Royal Astronomical Society: LettersDecomposing the Spectrum of Ultraluminous X-Ray Pulsar NGC 300 ULX-1
2023, Astrophysical Journal