Elsevier

New Astronomy

Volume 85, May 2021, 101493
New Astronomy

Accretion discs with non-zero central torque

https://doi.org/10.1016/j.newast.2020.101493Get rights and content

Highlights

  • Accretion discs are central to much of modern astronomy

  • The disc structure is sensitive to the inner accretion or decretion boundary

  • Here we provide the theory for a more general inner disc boundary condition

  • For many astrophysical systems such a boundary condition is more appropriate

  • We provide analytical and numerical solutions to the disc evolution

Abstract

We present analytical and numerical solutions for accretion discs subject to a non-zero central torque. We express this in terms of a single parameter, f, which is the ratio of outward viscous flux of angular momentum from the inner boundary to the inward advected flux of angular momentum there. The standard “accretion” disc, where the central boundary condition is zero-torque, is represented by f=0. A “decretion” disc, where the radial velocity at the inner boundary is zero, is represented by f → ∞. For f > 0 a torque is applied to the disc at the inner boundary, which feeds both angular momentum and energy into the disc. This can arise, for example, in the case of a circumbinary disc where resonances transfer energy and angular momentum from the binary to the disc orbits, or where the disc is around a rotating magnetic star which can allow the disc orbits to be accelerated outwards at the magnetospheric radius. We present steady-state solutions to the disc structure as a function of f, and for arbitrary kinematic viscosity ν. For time-dependent discs, we solve the equations using a Green’s function approach for the specific case of ν ∝ R and provide an example numerical solution to the equations for the case of ν ∝ R3/2. We find that for values of f ≲ 0.1 the disc solutions closely resemble “accretion” discs. For values of f ≳ 10 the solutions initially resemble “decretion” discs, but at sufficiently late times exhibit the properties of “accretion” discs. We discuss the application of this theory to different astrophysical systems, and in particular the values of the f parameter that are expected in different cases.

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 R=Rin.

For a steadily accreting disc of this kind, as shown by Shakura and Sunyaev (1973), the energy dissipated per unit area is given byD(R)=3GMM˙4πR3[1(RinR)1/2],where M is the mass of the central object and M˙ is the steady accretion rate through the disc. Thus the total luminosity released in such a disc isL=Rin2πRD(R)dR=GMM˙2Rin.This 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:Σt+1RR(RΣVR)=0,where VR is the radial velocity, and, second, conservation of angular momentumt(ΣR2Ω)+1RR(R3ΣΩVR)=1RR(νΣR3Ω).Here Ω(R) is the angular velocity of the disc material, ν the vertically averaged kinematic viscosity, and Ω=dΩ/dR.

By manipulating these we find thatVR=R(νΣR3Ω)ΣR(R2Ω).

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 M˙ 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 Ω2=GM/R3. To simplify the algebra we defineS=νΣR1/2.

The usual boundary conditions are either zero torque (S=0) or zero radial velocity (dS/dR=

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 (f=0) 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, ΩK=(GM/R3)1/2 (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 (f=0) 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

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