Accretion

Matter from the companion star that falls into the Roche lobe of a compact object will not fall directly onto the object, as it will still possess a large amount of angular momen- tum from the initial circular motion of the binary system itself. Instead it will go into orbit around the object, forming an accretion disc through which angular momentum is transported outwards, allowing the matter in the centre of the disc to accrete onto the compact object (Pringle, 1981). This angular momentum is transported outwards through

Figure 1.13: An example of how the black-body-emitting segments of an accretion disc

(black) combine to produce a multicolour accretion disc spectrum (green).5

turbulence within the disc from magnetorotational instability, causing friction between the different orbits of matter and allowing them to move inwards towards the centre. This frictional heating causes the disc to emit radiation that, for stellar-mass compact objects, peaks in the X-ray regime. In the case of high opacity and relatively low accretion rate, the density of the accretion disc drops rapidly with the scale height, making the disc optically thick and geometrically thin (the ‘thin disc model’; Shakura & Sunyaev 1973).

The closer to the inner regions of the accretion disc, the faster the disc material is moving, and therefore the frictional heating is greater. The heated material emits thermal radiation, increasing in energy with the increase in temperature towards the centre of the disc. If each infinitesimal annulus that makes up the thin accretion disc is considered to radiate as a black body, then the total radiation from the accretion disc is the sum of all of these black bodies – a spectrum known as a multicolour disc (MCD; Mitsuda et al. 1984). The MCD spectrum has a power-law shape up to its peak, then takes on the Wien portion of the innermost part of the accretion disc at the highest energies (see Fig. 1.13). The energy of the peak of the spectrum then depends upon the innermost radius of the accretion disc. Since the luminosity of a black body depends upon its temperature, this also applies to the accretion disc, and the luminosity of the disc can be related to its

In the case of a NS, the accretion disc would extend to the surface of the star were it not for the magnetic field, which causes inflowing material to be dragged along the magnetic field lines instead, accreting along the poles of the NS (e.g. Davidson & Ostriker 1973). This causes the brightest X-ray radiation to be anisotropic, so that pulsations can be observed in the X-ray regime. For a BH, the accretion disc can extend down to the innermost stable circular orbit (ISCO; Misner et al. 1973) at:

RISCO=3Rs=

6GM

c2 (1.3.5)

For a spinning BH, the ISCO may be closer, as low asR_gfor a maximally spinning BH.

Inside the ISCO, particles will rapidly spiral into the BH. If it is assumed that an accretion disc extends to the ISCO, this allows the BH mass or spin to be estimated from the peak temperature of the MCD, although this only holds true for states in which the accretion disc is not truncated (see Section 1.4) and black-body-like emission is not expected from a different source such as outflowing winds (see Section 1.5.3).

1.3.3.1 The Eddington luminosity

The process of accretion is essentially the conversion of gravitational potential energy to kinetic energy to heat. The power generated by the accretion of free-falling matter (at a free-fall velocity ofv_ff=p

2GM/R) onto an object of massMat an accretion rate of ˙M

and at radiusRis given by:

L=1 2Mv˙ 2 ff= GMM˙ R = 1 2Mc˙ 2Rs R ≡ηMc˙ 2, (1.3.6)

whereη=Rs/2Ris an efficiency factor – the fraction of the rest mass energy of the matter being accreted that can be radiated away – depending on how compact the object is.

For a BH, this is the most efficient process in the Universe, withη∼0.1–0.4, depending

upon its spin.

Since the luminosity scales with the mass accretion rate, one could naively assume that the luminosity has no upper limit, as long as the mass accretion rate is high enough.

In reality, a limit is placed upon the mass accretion rate by the radiation pressure exerted

by the high luminosity. For a proton-electron pair at a distancerfrom an object of mass

M, the gravitational force is given by:

F_grav≈ GMmp

r2 , (1.3.7)

wherem_pis the mass of the proton, which dominates the total mass of the proton elec- tron pair. The force due to radiation pressure depends upon the Thomson cross-section, for which the term is dominated by the electron because of its far lower mass compared to the proton, and is thus given by:

F_rad≈ σTL

4πr2c, (1.3.8)

whereσTis the Thomson cross-section of the electron.

The proton-electron pair is bound by electrostatic forces, so both forces apply to the pair as a whole and can therefore be equated. Making the luminosity the subject of the equation gives the Eddington luminosity (e.g. Frank et al. 2002) – the theoretical max- imum luminosity that can result from an accreting source before it blows away the in- falling material and loses its fuel source:

LEdd=

4πGMmpc

σT (1.3.9)

This theoretical upper limit carries a number of assumptions, such as that the in-falling

material is made up purely of ionised hydrogen6, and that the system is spherically sym-

metric. In reality it is evident from the accretion disc alone that the system is anisotropic, and if accreting matter and outward radiation occur in different parts of the system and in different directions, this can allow the Eddington luminosity to be exceeded.

At luminosities that are a significant fraction of the Eddington luminosity, the accre- tion flow becomes geometrically thick due to internal radiation pressure and the thin disc model no longer holds, with a ‘slim disc’ model of accretion applying instead (Abramow- icz et al., 1988). In such a model, the time it takes for radiation to propagate through

6_{While the electron and proton we consider are not directly bonded as in atomic hydrogen, it is assumed}

Figure 1.14: An example of how multiple orders of Compton up-scattering (blue) of a MCD spectrum (red) combine to form power-law emission with a cut-off at high energies (green), from Done (2010).

the disc is greater than the time it takes for the matter to accrete, essentially trapping a proportion of the energy within the disc so that it is accreted rather than emitted. This ra- diative inefficiency allows for faster accretion of matter since the luminosity is lower than it would be for a thin disc. This process becomes important in models of super-Eddington accretion (see Section 1.5.3).