---
title: Accreting Compact Objects
draft: false
module: Binaries and Gravitational Waves
moduleNumber: 9
lessonNumber: 2
order: 902
summary: >
  Gas falling onto a compact object converts gravitational binding energy into
  radiation with an efficiency set by the depth of the potential well, up to tens of
  percent of the rest mass for a neutron star or black hole. Angular momentum forces
  the flow into a disk, and viscous dissipation gives a temperature profile that
  falls as radius to the minus three-quarters, producing a multicolor blackbody
  spectrum. Radiation pressure caps the steady luminosity at the Eddington limit.
  Unstable nuclear burning of the accreted fuel powers classical novae on white
  dwarfs and Type I X-ray bursts on neutron stars.
topics: [Binaries and Gravitational Waves]
sources:
  - book: Carroll & Ostlie
    ref: "Ch. 18 §18.4–18.6 — Accretion Disks, Novae, and X-ray Binaries"
  - book: Maoz
    ref: "Ch. 4 — Stellar Death and Remnants; Ch. 6 — Accretion"
---

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
$G M\,\d m / R$. For a steady accretion rate $\dot M$ the liberated power is the
**accretion luminosity**

$$
L_{\mathrm{acc}} = \frac{G M \dot M}{R} = \eta\,\dot M c^2,
\qquad
\eta \equiv \frac{G M}{R c^2}.
$$

The dimensionless efficiency $\eta$ 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.

> **Definition (Accretion efficiency).** The fraction of rest-mass energy released
> when matter settles onto a compact object, $\eta = G M / R c^2$, depends only on
> the compactness $M/R$:
> - **White dwarf** ($M \sim 0.6\,M_\odot$, $R \sim 10^4\ \mathrm{km}$):
>   $\eta \sim 3 \times 10^{-4}$.
> - **Neutron star** ($M \sim 1.4\,M_\odot$, $R \sim 10\ \mathrm{km}$):
>   $\eta \sim 0.1$–$0.2$.
> - **Black hole**: matter releases energy down to the innermost stable circular
>   orbit rather than a surface, giving $\eta \approx 0.057$ for a non-rotating hole
>   and up to $\eta \approx 0.42$ for a maximally rotating one.

Nuclear hydrogen burning converts $0.007$ of rest mass to energy. A neutron star's
$\eta \sim 0.1$ 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](/astrophysics-cosmology/stellar-death-and-compact-remnants/black-holes-schwarzschild-and-kerr).

$$
% caption: Accretion efficiency rises with compactness: matter falling onto a white
% dwarf releases a small fraction of its rest energy, a neutron star roughly a tenth,
% and a black hole down to the innermost stable orbit a comparable amount.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0.6,0.6) -- (8.4,0.6) node[right, black!70] {compactness M over R};
\draw[->, black] (0.6,0.6) -- (0.6,4.2) node[above, black!70] {energy yield};
% rising curve eta = GM/Rc^2 (linear in compactness)
\draw[acc, very thick] (0.9,0.75) -- (7.9,3.9);
\fill[black] (1.4,0.97) circle (1.8pt) node[below=3pt, black] {white dwarf};
\fill[black] (4.6,2.42) circle (1.8pt) node[right=3pt, black] {neutron star};
\fill[black] (7.2,3.58) circle (1.8pt) node[left=4pt, above=1pt, black] {black hole};
\end{tikzpicture}
$$

## 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

$$
D(r) = \frac{3 G M \dot M}{8\pi r^3}
       \left[\,1 - \left(\frac{R_\ast}{r}\right)^{1/2}\right],
$$

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

$$
T(r) = \left(\frac{3 G M \dot M}{8\pi \sigma r^3}\right)^{1/4} \propto r^{-3/4}.
$$

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.

$$
% caption: The thin-disk temperature falls as radius to the minus three-quarters
% away from the inner edge, and turns over near the inner radius where the boundary
% factor forces the dissipation to zero.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0.6,0.6) -- (8.4,0.6) node[right, black!70] {radius};
\draw[->, black] (0.6,0.6) -- (0.6,4.2) node[above, black!70] {temperature};
% rise from inner edge to a peak then r^{-3/4} decline
\draw[acc, very thick] (1.0,0.7) .. controls (1.4,2.4) and (1.7,3.4) .. (2.1,3.5)
  .. controls (3.2,3.5) and (4.6,2.0) .. (6.0,1.35)
  .. controls (7.0,1.05) and (7.6,0.95) .. (8.0,0.9);
\node[acc, anchor=south west] at (3.6,2.2) {T proportional to r to the minus 3/4};
\draw[black, densely dotted] (2.1,0.6) -- (2.1,3.5);
\node[black, anchor=north] at (1.4,0.55) {inner edge};
\end{tikzpicture}
$$

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

$$
F_\nu \propto \nu^{1/3},
$$

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 $\sigma_T$), 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
$r$:

$$
L_{\mathrm{Edd}} = \frac{4\pi G M m_p c}{\sigma_T}
                 \approx 1.26 \times 10^{31}
                 \left(\frac{M}{M_\odot}\right)\ \mathrm{W}.
$$

> **Definition (Eddington luminosity).** The maximum luminosity at which a source of
> mass $M$ can accrete spherically in steady state. Above $L_{\mathrm{Edd}}$ the
> outward radiation force on ionized gas exceeds gravity and drives the accreting
> matter away, choking the inflow. The limit scales linearly with mass and is set by
> fundamental constants and the Thomson cross section alone.

The corresponding maximum accretion rate follows from $L_{\mathrm{Edd}} = \eta \dot
M_{\mathrm{Edd}} c^2$. For a neutron star, $L_{\mathrm{Edd}} \sim 1.8 \times
10^{31}\ \mathrm{W}$, roughly $5 \times 10^4\,L_\odot$, 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](/astrophysics-cosmology/stellar-evolution/post-main-sequence-low-mass-evolution),
  until the layer expands, lifts degeneracy, and is ejected. The white dwarf
  survives, resumes accreting, and the cycle repeats over $10^4$ to $10^5$ years.

$$
% caption: A classical nova: hydrogen accreted onto a white dwarf builds a degenerate
% surface layer whose base ignites and burns unstably, expanding and ejecting the
% shell while the underlying white dwarf survives to accrete again.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% white dwarf core
\draw[black, very thick] (2.2,2.2) circle (0.85);
\fill[black] (2.2,2.2) circle (0.85);
\node[black, anchor=center] at (2.2,2.2) {white dwarf};
% accreted layer
\draw[acc, very thick] (2.2,2.2) circle (1.15);
\node[acc, anchor=south] at (2.2,3.45) {accreted layer};
% ignition at base (marker ring)
\draw[acc, densely dashed] (2.2,2.2) circle (0.98);
\node[acc, anchor=north] at (2.2,1.1) {base ignites};
% ejection arrows
\foreach \ang in {30,90,150,210,270,330}{
  \draw[black, ->] ($(2.2,2.2)+(\ang:1.25)$) -- ($(2.2,2.2)+(\ang:2.0)$);
}
\node[black, anchor=west] at (4.7,2.2) {shell ejected};
\end{tikzpicture}
$$

## 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.

$$
% caption: The two X-ray binary classes: a high-mass system fed by the wind of a
% massive companion, and a low-mass system fed by Roche-lobe overflow through an
% accretion disk onto the compact object.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% --- HMXB ---
\draw[very thick] (1.6,3.0) circle (0.8);
\node[black, anchor=center] at (1.6,3.0) {massive};
\fill[black!70] (4.0,3.0) circle (2pt);
\node[black, anchor=south] at (4.0,3.3) {compact};
% wind arrows toward compact
\foreach \dy in {-0.35,0,0.35}{
  \draw[black, ->] (2.5,{3.0+\dy}) -- (3.7,{3.0+0.4*\dy});
}
\node[black, anchor=north] at (2.8,2.5) {wind fed};
\node[black, anchor=west] at (5.0,3.0) {High-mass};
% --- LMXB ---
\draw[black, very thick] (1.4,0.9) circle (0.6);
\node[black, anchor=center] at (1.4,0.9) {donor};
% disk around compact
\draw[very thick] (3.9,0.9) ellipse (0.9 and 0.28);
\fill[black!70] (3.9,0.9) circle (1.8pt);
% stream from donor to disk
\draw[very thick, ->] (2.0,0.9) .. controls (2.7,0.95) and (2.9,1.05) .. (3.3,0.98);
\node[black, anchor=north] at (2.6,0.5) {Roche fed};
\node[black, anchor=west] at (5.0,0.9) {Low-mass};
\end{tikzpicture}
$$

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 $\eta \sim 0.1$ 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.

$$
% caption: A Type I X-ray burst light curve: unstable nuclear burning on the neutron
% star surface produces a fast rise of about a second and an exponential cooling
% tail over tens of seconds, superposed on the persistent accretion emission.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0.6,0.6) -- (8.4,0.6) node[right, black!70] {time in seconds};
\draw[->, black] (0.6,0.6) -- (0.6,4.0) node[above, black!70] {X-ray brightness};
% persistent level
\draw[black, densely dashed] (0.6,1.1) -- (8.2,1.1);
\node[black, anchor=south east] at (8.2,1.1) {persistent};
% fast rise then exponential decay
\draw[acc, very thick] (1.4,1.1) -- (1.9,3.6)
  .. controls (2.8,2.8) and (3.8,1.9) .. (5.0,1.45)
  .. controls (6.0,1.25) and (7.0,1.15) .. (8.0,1.12);
\node[acc, anchor=south west] at (2.0,3.5) {fast rise};
\node[black, anchor=south] at (4.6,1.8) {cooling tail};
\end{tikzpicture}
$$

## Summary

Accretion converts gravitational binding energy into radiation with efficiency
$\eta = GM/Rc^2$, ranging from $\sim 10^{-4}$ 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
$F_\nu \propto \nu^{1/3}$ intermediate band. Radiation pressure on ionized gas caps
steady spherical accretion at the Eddington luminosity
$L_{\mathrm{Edd}} = 4\pi G M m_p c/\sigma_T \approx 1.3\times10^{31}(M/M_\odot)$ 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.

[^co-accretion]: Carroll & Ostlie, Ch. 18 §18.4–18.5 — accretion disks, the disk temperature profile, and the release of gravitational energy in close binaries.
[^co-novae]: Carroll & Ostlie, Ch. 18 §18.6 — novae, cataclysmic variables, X-ray binaries, and Type I X-ray bursts.
[^maoz-eddington]: Maoz, Ch. 4 and Ch. 6 — accretion efficiency, the Eddington luminosity, and thermonuclear instabilities on accreting compact objects.