Accretion Stream and Hot Spot: Where the Ballistic Jet Hits the Disk

What the hot spot is: a collision, not an object

The hot spot is not a body but a standing collision front. In a semi-detached binary, the cooler donor star fills its Roche lobe — the teardrop-shaped equipotential surface bounding its gravitational domain. Gas at the lobe's tip, the inner Lagrangian point L1, feels essentially zero net force and leaks toward the white dwarf primary.

Because that gas carries the donor's orbital angular momentum, it cannot fall straight in. It swings around the white dwarf, and viscous friction spreads it into a flat accretion disk. But fresh gas keeps arriving from L1 along a curved ballistic stream, and this stream overshoots and strikes the already-formed disk from outside. At that impact the stream's ordered bulk motion is randomized into heat.

• The impact is highly supersonic — the stream's sound speed is tiny next to its orbital velocity.
• A shock forms, gas is compressed and heated, then re-radiates as a localized bright spot fixed in the rotating binary frame.

The mechanism: ballistic flight and a supersonic shock

Once through L1, the gas is cold (sound speed ~10 km/s) but quickly moving fast, so pressure is negligible and the flow is essentially free-particle (ballistic). Its path is set by three forces in the rotating frame: the white dwarf's gravity, the donor's gravity, and the Coriolis force. The trajectory curves and the stream reaches a minimum distance from the primary before looping back — the classic teardrop-into-spiral geometry.

The disk's outer radius is set by where infalling angular momentum can be accommodated; the circularization radius R_circ (where a ring of the stream's specific angular momentum would orbit) scales roughly as:

R_circ / a ≈ (1 + q) · (0.5 − 0.227 log q)^4, with q = M_donor / M_WD.

The disk grows a bit larger than R_circ. Where the ballistic stream meets that rim, the relative velocity is a large fraction of the local Keplerian speed. The flow crosses the shock at Mach 5–10, converting bulk kinetic energy to thermal energy: kT ~ (3/16) μ m_H v_shock². The dense, optically-thick core cools to ~6,000–15,000 K, while a shocked halo can be far hotter and briefly X-ray/UV bright.

Characteristic numbers and a worked example

Take a representative dwarf nova: white dwarf M_WD = 0.8 M_sun, donor M_2 = 0.15 M_sun (q ≈ 0.19), orbital period P = 1.6 h, so the binary separation a ≈ 4.9 × 10^8 m (~0.7 R_sun).

• Circularization radius: plugging q ≈ 0.19 gives R_circ ≈ 0.23 a, and the disk rim sits near R_d ≈ 0.3 a ≈ 1.5 × 10^8 m.
• Impact speed: the ballistic stream reaches the rim at ~600 km/s in the WD frame; the relative speed onto the disk (accounting for the disk's own Keplerian rotation, ~500 km/s at that radius) is a few hundred km/s.
• Shock temperature: for v_shock ≈ 300 km/s, post-shock kT corresponds to ~10^6 K in the thin halo, though most reprocessed light emerges in the optical/UV.
• Luminosity: for a mass-transfer rate Ṁ ~ 10^16 g/s, the impact dissipates L_spot ~ (1/2) Ṁ v² ~ 10^32 erg/s — comparable to the quiescent disk itself.

That is why the hot spot dominates the optical light curve of many quiescent, high-inclination systems.

How it is observed: orbital humps and stepped eclipses

The hot spot is compact and fixed in the binary frame, so as the system rotates it presents different faces to us. Its emitting surface is foreshortened except when it points toward Earth, producing a broad brightening — the orbital hump — that peaks around orbital phases 0.6–1.0, just before the donor eclipses the disk.

In high-inclination systems, the donor star eclipses the primary components in sequence. Because the white dwarf, disk, and hot spot lie at different positions, the eclipse is structured: the white dwarf ingress/egress is sharp and central, while the hot spot, offset to one side of the disk, is covered and uncovered at a different phase. The result is a famous stepped or asymmetric egress — the light curve visibly jumps as the bright spot reappears.

• Timing the white-dwarf and hot-spot contact points yields precise mass ratios and radii.
• Eclipse mapping, developed by Keith Horne in 1985, inverts the smooth eclipse light curve using maximum-entropy methods to reconstruct a 2-D brightness image of the disk — the hot spot appears as a compact bright patch on the rim.

How it compares: streams, spots, and magnetic funnels

The stream-disk hot spot is one member of a family of impact and reprocessing features across interacting binaries:

• Non-magnetic CVs (dwarf novae, nova-likes): a full disk forms and the classic rim hot spot appears. Cases like Z Cha, HT Cas, and OY Car are textbook examples.
• Polars (AM Her stars): the white dwarf's ~10–200 MG field is so strong that no disk forms. The stream is captured mid-flight and funneled onto a magnetic pole, where it shocks against the surface — the hot spot becomes an X-ray/cyclotron-emitting accretion column, not a rim bright spot.
• Intermediate polars: a truncated disk plus magnetic curtains — a hybrid.
• Low-mass X-ray binaries (neutron star/black hole accretors): the same ballistic-stream/disk-rim collision occurs, but the far deeper potential makes the disk X-ray luminous and the rim structure drives dips rather than a simple optical hump.

Debate persists over whether some systems show no distinct shock hot spot at all — a few simulations argue the stream can merge smoothly into a disk 'overflow' that skims over the rim and re-impacts closer in, producing a secondary bright region.

Why it matters and what is still uncertain

The hot spot is a diagnostic goldmine. Because its brightness tracks Ṁ, it is a direct probe of the mass-transfer rate driving binary evolution. Its eclipse timing anchors the most precise white-dwarf and donor masses in all of stellar astrophysics — the work of Hessman, Wood, Horne, and later the ULTRACAM team (Dhillon, Marsh) on systems like OU Vir and GY Cnc has delivered mass ratios to a few percent.

• Stream overflow: how much gas skims over the disk rim versus shocking at it remains debated, with implications for SW Sextantis stars, whose peculiar emission may come from stream material overflowing onto the disk face.
• Superhumps and precession: in SU UMa systems the bright spot's phasing helps trace the eccentric, precessing disk during superoutburst.
• Flickering: much of a CV's rapid optical flickering originates at or near the impact region, and its physical origin is still not fully modeled.

Laboratory laser experiments have even reproduced the reverse radiative shock of stream-disk impact, tying this cosmic collision to high-energy-density plasma physics on Earth.

Emission components in a quiescent, high-inclination dwarf nova and their characteristic properties

Component	Typical temperature	Size scale	Light-curve signature
White dwarf	10,000–40,000 K	~10,000 km (Earth-sized)	Sharp, deep central eclipse
Hot spot / bright spot	6,000–15,000 K (optically thick core, shocked halo hotter)	Compact, ~few % of disk radius	Orbital hump (phase 0.6–1.0) + stepped eclipse egress
Accretion disk	Radial: ~50,000 K inner to ~3,000 K rim	0.2–0.5 × separation	Broad, shallow eclipse; double-peaked lines
Donor (secondary) star	3,000–5,000 K (M/K dwarf)	Roche-lobe filling	Ellipsoidal modulation, none eclipsed