Accreting Compact Objects

The gas that streams through the inner Lagrange point in a close binary carries angular momentum and cannot fall straight onto its destination. It circularizes into an accretion disk, spirals inward as viscosity drains its angular momentum, and releases gravitational binding energy as heat and light. Accretion onto a compact companion is the most efficient sustained energy source in stellar astrophysics, outproducing nuclear fusion per unit mass by more than an order of magnitude when the accretor is a neutron star or black hole. This lesson derives the accretion luminosity and its efficiency, the disk temperature profile and spectrum, the Eddington limit that caps steady accretion, and the thermonuclear instabilities, novae and X-ray bursts, that ignite the accreted fuel.

Accretion luminosity and efficiency

A mass element $d m$ falling from rest at infinity to the surface of a compact object of mass $M$ and radius $R$ releases its gravitational binding energy $GM d m / R$ . For a steady accretion rate $\dot{M}$ the liberated power is the accretion luminosity

L_{acc} = \frac{GM M ˙}{R} = η \dot{M} c^{2}, η \equiv \frac{GM}{R c ^{2}} .

The dimensionless efficiency $η$ measures how deep the potential well is in units of $c^{2}$ . It is the ratio of the gravitational redshift at the surface to unity, and it grows as the object becomes more compact.

Nuclear hydrogen burning converts $0.007$ of rest mass to energy. A neutron star's $η \sim 0.1$ therefore beats fusion by more than a factor of ten, which is why accreting neutron stars and black holes are the brightest compact X-ray sources in the Galaxy. The efficiency for a black hole follows from the binding energy at the innermost stable circular orbit, developed in the black-hole lesson.

Accretion efficiency rises with compactness: matter falling onto a white dwarf releases a small fraction of its rest energy, a neutron star roughly a tenth, and a black hole down to the innermost stable orbit a comparable amount.

The accretion disk and its temperature profile

Angular momentum forces the accreted gas into a thin, nearly Keplerian disk. Each annulus orbits at the local circular speed, and neighboring annuli shear against one another. Viscous stress transports angular momentum outward, lets most of the mass spiral inward, and dissipates orbital energy locally as heat. In steady state the viscous power dissipated per unit disk area (both faces) at radius $r$ is

D (r) = \frac{3 GM M ˙}{8 π r ^{3}} [1 - (\frac{R _{*}}{r})^{1/2}],

where $R_{*}$ is the inner disk radius set by the accretor. Each annulus radiates as a blackbody, so $σ T^{4} = D (r)$ fixes the local temperature. Far from the inner edge the bracket approaches unity and

T (r) = (\frac{3 GM M ˙}{8 π σ r ^{3}})^{1/4} \propto r^{- 3/4} .

The disk is hottest in its inner regions and cools outward as a power law. The factor of three over the naive estimate reflects that an annulus radiates not only its own dissipation but energy transported from further in.

The thin-disk temperature falls as radius to the minus three-quarters away from the inner edge, and turns over near the inner radius where the boundary factor forces the dissipation to zero.

Because the disk spans a wide range of temperatures, its integrated spectrum is not a single blackbody but a superposition, a multicolor blackbody. Summing Planck spectra weighted by the annular areas gives, in the intermediate frequency band between the outer (cool) and inner (hot) cutoffs, a characteristic flat power law

F_{ν} \propto ν^{1/3},

bracketed by the Rayleigh-Jeans rise of the coolest annulus at low frequency and the Wien cutoff of the hottest at high frequency. The inner-edge temperature sets where the spectrum turns over: for a white dwarf the disk peaks in the ultraviolet, for a neutron star or black hole in soft-to-hard X-rays.

The Eddington limit

Radiation escaping an accreting object pushes back on the infalling gas. For fully ionized hydrogen the outward force is radiation pressure scattering off electrons (Thomson cross section $σ_{T}$ ), while gravity pulls on the protons to which the electrons are electrostatically bound. Balancing the two at radius $r$ , the luminosity at which radiation pressure halts spherical accretion is independent of $r$ :

L_{Edd} = \frac{4 π GM m _{p} c}{σ _{T}} \approx 1.26 \times 1 0^{31} (\frac{M}{M _{⊙}}) W .

The corresponding maximum accretion rate follows from $L_{Edd} = η \dot{M}_{Edd} c^{2}$ . For a neutron star, $L_{Edd} \sim 1.8 \times 1 0^{31} W$ , roughly $5 \times 1 0^{4} L_{⊙}$ , radiated almost entirely in X-rays. The Eddington limit is not absolute; it assumes spherical symmetry and steady state, and disk accretion can exceed it locally along the disk plane while radiation escapes along the poles. It nonetheless sets the natural luminosity scale for accreting compact objects and, applied to supermassive black holes, caps the growth rate of quasars.

Cataclysmic variables and novae

A cataclysmic variable is a close binary in which a white dwarf accretes hydrogen- rich gas from a low-mass main-sequence companion that fills its Roche lobe. The accreted hydrogen accumulates in a surface layer on the white dwarf, compressed and heated by the degenerate star beneath it. Two instabilities produce outbursts.

Dwarf novae arise from a thermal-viscous instability in the disk itself. The disk cycles between a cool, low-viscosity state that stores mass and a hot, high-viscosity state that dumps it onto the white dwarf, brightening by a few magnitudes every few weeks. No nuclear burning is involved.
Classical novae are thermonuclear. As the accreted hydrogen layer grows, its base is compressed to degenerate conditions. When the temperature reaches hydrogen ignition, burning begins in degenerate matter, where pressure does not respond to the rising temperature. The burning runs away, exactly as in the helium flash, until the layer expands, lifts degeneracy, and is ejected. The white dwarf survives, resumes accreting, and the cycle repeats over $1 0^{4}$ to $1 0^{5}$ years.

A classical nova: hydrogen accreted onto a white dwarf builds a degenerate surface layer whose base ignites and burns unstably, expanding and ejecting the shell while the underlying white dwarf survives to accrete again.

X-ray binaries and thermonuclear bursts

When the accretor is a neutron star or black hole, the disk radiates in X-rays and the system is an X-ray binary. Two populations divide by the mass of the donor.

High-mass X-ray binaries (HMXBs) pair a compact object with a massive O or B star. The compact object accretes from the strong stellar wind or from focused Roche-lobe overflow. These are young systems, concentrated in the Galactic plane and star-forming regions.
Low-mass X-ray binaries (LMXBs) pair a compact object with a low-mass star that transfers mass by Roche-lobe overflow through a disk. These are old systems, found in the bulge and in globular clusters, and they are the sites where neutron stars are spun up to millisecond periods.

The two X-ray binary classes: a high-mass system fed by the wind of a massive companion, and a low-mass system fed by Roche-lobe overflow through an accretion disk onto the compact object.

On the surface of an accreting neutron star the accreted hydrogen and helium accumulate, compress, and ignite. When helium burning ignites unstably in the accreted layer, it burns through in seconds, producing a Type I X-ray burst: a sharp rise in X-ray flux over about a second followed by a cooling decay over tens of seconds, recurring on timescales of hours to days as the fuel replenishes. The burst energetics match the nuclear energy release of the accreted layer, and the ratio of persistent accretion luminosity to time-averaged burst luminosity, roughly the ratio of gravitational $η \sim 0.1$ to nuclear $0.007$ , confirms that the persistent emission is accretion powered and the bursts are nuclear. Bursts that reach the Eddington limit, where radiation pressure lifts the neutron-star photosphere, standardize the peak luminosity and serve as distance indicators.

A Type I X-ray burst light curve: unstable nuclear burning on the neutron star surface produces a fast rise of about a second and an exponential cooling tail over tens of seconds, superposed on the persistent accretion emission.

Summary

Accretion converts gravitational binding energy into radiation with efficiency $η = GM / R c^{2}$ , ranging from $\sim 1 0^{- 4}$ for a white dwarf to $\sim 0.1$ – $0.4$ for a neutron star or black hole, so accreting compact objects outshine fusion per unit mass. Angular momentum forces the flow into a disk whose viscous dissipation gives $T (r) \propto r^{- 3/4}$ and a multicolor-blackbody spectrum with an $F_{ν} \propto ν^{1/3}$ intermediate band. Radiation pressure on ionized gas caps steady spherical accretion at the Eddington luminosity $L_{Edd} = 4 π GM m_{p} c / σ_{T} \approx 1.3 \times 1 0^{31} (M / M_{⊙})$ W. Accreted fuel ignites unstably: degenerate hydrogen burning drives classical novae on white dwarfs, and unstable helium burning drives Type I X-ray bursts on neutron stars, with HMXBs fed by winds and LMXBs by Roche-lobe overflow through a disk.