Be Star & Decretion Disk

A star throwing off its own equator

Picture a hot blue star spinning so fast that its equator is hurtling around at a few hundred kilometres a second — fast enough that the centrifugal push there almost cancels its own gravity. That star can't quite hold onto its outer equatorial layers. Every so often it lets a parcel of gas go, and instead of falling back, the gas spreads outward into a thin, glowing ring in the equatorial plane. That ring is a decretion disk, and a B-type star surrounded by one is a Be star — the "e" tagged onto the spectral type because the disk shines in hydrogen emission lines.

The "classical" Be stars are non-supergiant stars of spectral type roughly O9 to A0 (overwhelmingly B) whose Balmer lines show emission, produced not by the star but by the disk. The defining property is rapid rotation. A normal B star is a tidy ball with absorption lines; a Be star spins near the brink of breakup and surrounds itself with a gaseous Keplerian disk that comes and goes. The same star can show strong emission one decade, lose its disk and look like an ordinary B star the next, then rebuild. About 17 percent of all B stars are Be stars at some point, with the fraction peaking near spectral type B1–B2.

Critical rotation and the breakup velocity

The whole phenomenon hinges on how close the star spins to critical (breakup) rotation — the rate at which centrifugal acceleration at the equator equals the effective surface gravity. Setting the two equal,

v_crit = √( G M / R_eq )

with the oblate equatorial radius R_eq = 1.5 R_pole at exact breakup

For a typical Be star of mass M ≈ 8 M☉ and polar radius R_pole ≈ 5 R☉, the critical equatorial velocity comes out around 450–550 km/s. Be stars are observed to spin at roughly 70 to 100 percent of that value. Measured projected rotation speeds (v sin i, the line-of-sight component) cluster between 200 and 400 km/s; since sin i ≤ 1, the true equatorial speeds are higher still. Compare that with the Sun's leisurely 2 km/s equatorial rotation and you see why these stars live on a knife edge.

Spinning that fast deforms the star. At breakup the equatorial radius is 1.5 times the polar radius; real Be stars are oblate ellipsoids flattened by 20–50 percent. Achernar (α Eridani), the brightest Be star in the sky at apparent magnitude 0.5 and about 140 light-years away, was resolved by the VLTI interferometer and found to be about 1.45 times wider at its equator than between its poles — the most flattened star ever directly imaged at the time.

Gravity darkening: why fast rotators get the gas out

Rapid rotation does more than flatten the star. By von Zeipel's theorem, the radiative flux emerging from a rotating star's surface is proportional to the local effective gravity:

T_eff(θ) ∝ g_eff(θ)^β     (von Zeipel, 1924)

β ≈ 0.25 for a fully radiative envelope
g_eff = g_grav − Ω² R sin θ   (gravity minus centrifugal term)

The bulging equator has a larger radius and a stronger centrifugal term, so its effective gravity is low — which means it is cooler and dimmer (gravity darkening), while the poles are hot and bright. For Achernar the pole is roughly 20,000 K and the equator nearer 10,000 K. A cool, low-gravity equator that is barely held down is exactly the launch pad you want for shedding material. The star doesn't need to reach the full breakup velocity on its own — rotation gets it most of the way, and a small extra push (a pulsation, a tidal nudge from a companion) lifts gas off the equator into orbit.

Decretion versus accretion: same physics, opposite direction

The disk that forms is governed by the same viscous angular-momentum bookkeeping as an accretion disk — but run in reverse. In an accretion disk, viscosity carries angular momentum outward so that mass can spiral in and fall onto the central object. In a Be decretion disk, the star injects gas at the inner edge already carrying nearly Keplerian angular momentum; viscosity transports that angular momentum outward and the mass spreads outward with it. The star is offloading spin, not gaining mass. The standard framework is the Viscous Decretion Disk (VDD) model of Lee, Saio & Osaki (1991), built directly on Shakura–Sunyaev α-disk theory.

Property	Accretion disk	Be decretion disk
Net mass flux	Inward, onto central body	Outward, away from star
Angular momentum role	Removed from gas so it can fall in	Supplied by star, carried out by gas
Energy source	Gravitational potential released as light	Stellar UV reprocessed; rotational KE of star
Inner boundary	ISCO / stellar surface — sink	Stellar equator — source
Viscosity α	0.01 – 0.3 (MRI)	0.1 – 1 (observed, MRI-driven)
Temperature	Up to 10⁷–10⁹ K (compact objects)	~0.6 × T_eff ≈ 7,000–15,000 K (isothermal)
Geometry	Thin, Keplerian	Thin, nearly Keplerian, flaring H ∝ r¹·⁵
Lifetime	Steady or secular	Builds and dissipates over months–years

The disk is geometrically thin near the star — vertical scale height H ∝ r^(1.5) for a roughly isothermal gas, so it flares outward — and rotates very nearly on Keplerian orbits, V_φ ∝ r^(−1/2). That Keplerian rotation is the clinching observational fact: it is read directly from the double-peaked profiles of the emission lines, whose peak-to-peak velocity separation maps the rotation speed at the disk's outer emitting edge.

How we know the disk is there

We never resolve most Be disks directly (though interferometers now do for the nearest, like ζ Tauri and δ Scorpii). Almost everything is read from the light:

• Emission lines. The disk gas absorbs ultraviolet photons from the star and re-emits them as hydrogen recombination lines. Hα at 656.3 nm is the strongest and the defining feature; Hβ, Hγ, and Paschen and Brackett lines in the infrared also appear. When the disk is seen at an intermediate angle the line is double-peaked (Doppler shift of the two sides of the rotating disk); seen pole-on it is single-peaked; seen edge-on it develops a deep central absorption (the shell phase).
• Infrared excess. Free–free (bremsstrahlung) and free–bound radiation from the warm ionised disk add a smooth excess that rises toward longer wavelengths, often a magnitude or more in the near-IR. The disk is brightest relative to the star in the IR because the star peaks in the UV.
• Linear polarisation. Starlight scattered off the flattened disk is partially polarised, typically at the 0.5–2 percent level, with the polarisation angle pinned to the disk's projected major axis. This is one of the cleanest geometric probes of disk orientation.
• Photometric outbursts. A fresh injection of mass brightens the star by a few hundredths to a few tenths of a magnitude over days to weeks as the inner disk densifies; the Kepler and TESS space photometers have caught many such micro-outbursts and tied them to beating non-radial pulsation modes.

Real numbers: sizes, masses, and timescales

A concrete sense of scale. A representative classical Be star and its disk:

Quantity	Typical value	Note
Stellar mass	3 – 20 M☉	Spectral type O9–B9, mostly early B
Stellar radius	4 – 10 R☉	Equatorial radius larger than polar by 20–35%
Effective temperature	10,000 – 30,000 K	Pole hotter than equator (gravity darkening)
Luminosity	10³ – 10⁵ L☉
Equatorial rotation	200 – 450 km/s	0.7 – 1.0 of critical velocity
Disk outer (Hα) radius	5 – 20 R★	Hα forms farther out than Brackett-γ
Disk mass	~10⁻⁹ – 10⁻⁸ M☉	Tiny — a fraction of a lunar mass of gas
Mass-loss rate	10⁻¹¹ – 10⁻⁹ M☉/yr	Into the disk, episodic
Disk density (inner)	~10⁻¹¹ g/cm³	Drops as ρ ∝ r⁻³·⁵ outward
Build/dissipation time	Months to years	Set by viscous timescale at α ≈ 0.1–1

The disk mass is startlingly small — often less than the mass of Earth's Moon spread over a region several stellar radii across. That is why the disk responds so quickly: there is little material to add or remove, so the structure builds and drains on a viscous timescale of order

t_visc ≈ R² / ν = R² / (α c_s H)

for α ≈ 0.5, R ~ 10 R★, this is months to a few years

Famous Be stars and where they show up

• γ Cassiopeiae — the prototype, the first emission-line star ever recorded (Secchi, 1866) and the namesake of the "γ Cas analogues." It is a B0.5 star with a hot companion and shows anomalously hard, variable X-ray emission whose origin (white-dwarf accretion versus star–disk magnetic interaction) is still debated. About 550 light-years away, apparent magnitude variable around 2.
• Achernar (α Eridani) — at magnitude 0.5 the brightest Be star and the ninth-brightest star in the night sky. The VLTI measured its 1.45:1 oblateness, the textbook image of a star deformed by rotation.
• Pleione (28 Tauri) — a member of the Pleiades cluster that has run through complete disk-loss and disk-rebuilding cycles in the last hundred years, oscillating between Be, Be-shell, and normal B appearances. The poster child for the transience of decretion disks.
• δ Scorpii — a binary that switched on a strong disk around its 2000 periastron passage and has been monitored ever since; its disk growth was triggered by the close approach of the companion.
• Be X-ray binaries. When a Be star shares an eccentric, wide orbit with a neutron star, the system becomes a BeXRB — the most populous class of high-mass X-ray binaries. Near periastron the neutron star skims the disk, captures gas, and flares. A0535+26 (companion to the Be star HDE 245770) and the Magellanic-Cloud systems found by RXTE and Swift are canonical examples; the eccentric binary PSR B1259−63 pairs a Be star with a radio pulsar and lights up in gamma rays at each periastron.

Type I and Type II outbursts in Be X-ray binaries

In a Be X-ray binary the neutron star spends most of its eccentric orbit far from the Be disk and the system is quiescent. Two kinds of outburst punctuate that quiet:

• Type I (normal) outbursts. Once per orbit, at or near periastron, the neutron star passes close to or through the outer decretion disk, captures gas, funnels it onto its magnetic poles, and radiates accretion luminosity of order 10³⁶–10³⁷ erg/s in X-rays. These outbursts are periodic at the orbital period (days to months) and modest in amplitude.
• Type II (giant) outbursts. When the Be disk grows large, becomes misaligned, or is warped, the neutron star can capture far more gas and reach near-Eddington luminosities of 10³⁸ erg/s or more, lasting weeks and not locked to orbital phase. These are how many BeXRBs are first discovered.

The connection runs both ways: the X-ray behaviour is a direct readout of the decretion disk's size and density. A disk-loss episode in the Be star shuts off the outbursts; a disk-growth episode can set off a giant flare. The neutron star also truncates the outer disk gravitationally, which is part of why BeXRB disks tend to be denser and more compact than those around isolated Be stars.

What triggers an outburst — pulsations and companions

Rapid rotation explains why a Be star can shed gas, but not when it does. Two mechanisms supply the trigger that lifts gas the last small step into orbit:

• Non-radial pulsations. Nearly all early Be stars pulsate in low-order gravity and pressure modes (they overlap the SPB and β Cephei instability strips). When two closely spaced modes beat together, the surface velocity amplitude swells periodically; space photometry (Kepler, TESS, the CoRoT study of HD 49330) shows mass-ejection outbursts coinciding with these beat maxima. The pulsations add the few tens of km/s the rotation alone falls short by.
• Binary interaction. Many Be stars are or were in binaries. A companion on an eccentric orbit can tidally trigger ejections at periastron (as in δ Scorpii). More fundamentally, the leading explanation for why Be stars spin so fast at all is past mass and angular-momentum transfer from a now-stripped companion — a scenario in which today's Be star is the spun-up "gainer," and the stripped donor has become a hot subdwarf, white dwarf, or neutron star. This neatly explains why so many Be stars turn up with compact companions and as BeXRBs.

Common misconceptions and edge cases

• Be stars are not accreting. The disk is fed from the star outward. Calling it an accretion disk reverses the physics; "decretion" was coined precisely to mark the outward mass flux.
• The disk is not the stellar wind, and it is not a shell. Hot-star radiation-driven winds are fast, low-density, and roughly spherical or polar; the Be disk is slow, dense, equatorial, and rotationally supported on Keplerian orbits. "Be-shell star" is not a different object — it is a classical Be star seen nearly edge-on, so the disk gas projects onto the stellar disk and adds sharp absorption cores.
• Classical Be stars are not Herbig Ae/Be stars. Herbig Ae/Be stars are young pre-main-sequence stars with leftover protoplanetary accretion disks falling inward. Classical Be stars are evolved or main-sequence objects with outflowing decretion disks. Same letters, opposite disks.
• Losing the emission doesn't mean the star changed. A Be star with no current disk is spectroscopically a normal B star, but it is still a Be star — the rotation that drives the cycle hasn't gone anywhere, and the disk will return.
• Rotation alone may not reach breakup. Many Be stars spin "only" at 70–80 percent of critical, not 100 percent. That is why a trigger (pulsation beating, tidal forcing) is needed; the star sits close enough that a small extra kick suffices.