Tidal Disruption Events

A tidal disruption event is what a dormant black hole looks like when a single unlucky star wanders too close. In the nucleus of an ordinary galaxy a supermassive black hole sits dark and starved, betraying itself only through the orbits of nearby stars. Two-body relaxation occasionally nudges one of them onto a nearly radial "loss-cone" orbit that plunges almost straight at the hole; if its pericenter falls inside the tidal radius — where the differential pull across the star exceeds its own self-gravity — the star is pulled apart in a single passage, converted from a self-bound sphere into an elongated ribbon of gas. Two competitions govern everything that follows. The first is tides versus self-gravity, which fixes the tidal radius and, through a race against the event horizon, sets a maximum black-hole mass above which the star is swallowed whole with no flare. The second is the debris energy spread versus Keplerian mechanics: the tidal field freezes a small spread of binding energies into the stream, roughly half bound and half unbound, and Kepler's third law stretches the return of the bound half over months to years — a decline that follows the celebrated $t^{-5/3}$ power law, the fingerprint of a disruption. Turning returning gas into light demands that the streams shock, circularize, and accrete, often far above the Eddington limit. The result is a luminous, months-long ultraviolet and soft-X-ray transient in a galaxy that was quiet the year before — a flashbulb briefly illuminating an otherwise invisible black hole, letting us weigh it, count its kind, and watch super-Eddington accretion switch on in real time.

Key takeaways

A tidal disruption event (TDE) occurs when a star passes inside the tidal radius of a supermassive black hole, where the hole's tidal field overwhelms the star's self-gravity and shreds it along its orbit.
The critical distance is $r_{\mathrm{t}}\simeq R_\star (M_{\mathrm{BH}}/M_\star)^{1/3}$ ; because it grows only as the cube root of mass while the horizon grows linearly, disruption is visible only for $M_{\mathrm{BH}}\lesssim 10^8\,M_\odot$ — the Hills mass.
About half the debris is bound and returns on highly eccentric orbits with a mass-return rate that famously scales as $\dot M\propto t^{-5/3}$ , producing a characteristic slowly-declining flare.
Stream self-intersection and shocks circularize the returning gas into an accretion flow that radiates near or above the Eddington limit, peaking in the ultraviolet and soft X-rays.
TDEs light up otherwise-dormant black holes, weigh them where reverberation mapping cannot reach, probe super-Eddington accretion, and occasionally launch relativistic jets, as in Swift J1644+57.
Key idea: a TDE is fixed by two competitions — tides versus self-gravity (giving the tidal radius $r_{\mathrm{t}}$ ) and the debris energy spread versus Keplerian orbital mechanics (giving the $t^{-5/3}$ fallback). Only when $r_{\mathrm{t}}$ exceeds the horizon does a visible flare, rather than a silent plunge, occur.

The core insight

A TDE is set by two competitions: tides versus self-gravity, which gives the tidal radius $r_{\mathrm{t}}$ , and the spread of debris energy versus Keplerian mechanics, which gives the $t^{-5/3}$ fallback. Only when $r_{\mathrm{t}}$ exceeds the horizon does a visible flare, rather than a direct plunge, occur.

Where this lives today

Tidal disruption has moved from a theorist's prediction to an industrial observational discipline. Wide-field optical surveys — ASAS-SN, ZTF, and soon Vera Rubin's LSST — now find tens of TDEs per year as slow, blue nuclear transients, a count set to reach thousands per year at full LSST cadence. Each event is a rare chance to weigh a dormant black hole: the fallback timescale, peak luminosity, and light-curve shape all encode $M_{\mathrm{BH}}$ , reaching the low- and intermediate-mass regime that reverberation mapping cannot touch and testing the low-mass extrapolation of the $M$ – $\sigma$ relation. The early super-Eddington phase makes TDEs clean laboratories for outflow-dominated accretion, and the puzzle that many events shine far brighter in the optical and UV than a bare $r_{\mathrm{t}}$ -scale disk allows has spawned a subfield of reprocessing and stream-collision models. Jetted events like Swift J1644+57 connect disruptions to jet launching and black-hole spin, while quasi-periodic X-ray eruptions and repeating partial TDEs — stars surviving multiple grazing passages — open a route to millihertz gravitational-wave sources detectable by LISA.

The physical picture

Picture the star as a swarm of test particles sharing one orbit. Near pericenter the black hole tugs the near side harder than the far side, and as long as the star's own gravity holds this differential in check the swarm moves as one body. Once the tidal difference exceeds the star's surface gravity the swarm stops cohering — every parcel follows its own Keplerian orbit and the star unspools into a stream. The distance where this tips over is the tidal radius, and whether it lies outside the event horizon decides whether we see anything at all.

Figure

A star on a radial loss-cone orbit crosses the tidal radius r_t and is torn into a debris stream: about half the mass is flung out unbound while half stays bound and returns to pericenter. For a 10⁶ M_⊙ black hole shredding the Sun, r_t ≈ 100 R_⊙ ≈ 23 r_s, safely outside the horizon

The energy spread frozen in at disruption is minute next to the orbital energy but enormous next to the star's own binding energy. Mapped through Kepler's third law $t\propto|E|^{-3/2}$ , it stretches the debris return over months to years and produces a smooth power-law decline rather than a single flash: the most tightly bound gas returns first, at $t_{\min}$ , and ever less-bound gas trickling back behind it settles the mass-return rate onto the universal $t^{-5/3}$ slope. The second figure turns that into the fallback curve, with real numbers for a Sun-like star and a million-solar-mass hole.

Figure

The fallback rate for a solar-type star onto a 10⁶ M_⊙ black hole: after the most-bound debris returns at t_min ≈ 41 days the rate peaks near 3 M_⊙ per year, then declines with slope −5/3. It sits far above the Eddington rate ≈ 0.02 M_⊙ per year for the first ≈ 2 years, so the early flare is strongly super-Eddington

Whether we see anything at all is a race between two radii: the tidal radius climbs only as $M_{\mathrm{BH}}^{1/3}$ , but the Schwarzschild radius climbs as $M_{\mathrm{BH}}$ . Below about $10^8\,M_\odot$ the tidal radius wins and the star is shredded outside the horizon; above it a Sun-like star is swallowed whole and dark, unless spin shifts the prograde horizon inward and rescues visibility. The flare that follows is fuel-limited by fallback but radiation-capped: gas returning faster than Eddington is expelled or advected inward, so the light curve often hugs Eddington before tracking the decline.

The core equations

The single length that decides a star's fate is the tidal radius: the distance at which the black hole's differential pull across the stellar diameter matches the star's own surface gravity. Inside it the star is destroyed, outside it survives — and crucially it grows only as the cube root of the black-hole mass.

r_{\mathrm{t}} \simeq R_\star\left(\frac{M_{\mathrm{BH}}}{M_\star}\right)^{1/3} \qquad (1)

where: $r_{\mathrm{t}}$ — tidal disruption radius; $R_\star$ — stellar radius; $M_\star$ — stellar mass; $M_{\mathrm{BH}}$ — black-hole mass. Reading: a hole $1000\times$ heavier disrupts at only $10\times$ the distance — a slow dependence that will collide with the horizon. As the star is destroyed near $r_{\mathrm{t}}$ , its near and far sides sit at slightly different depths in the black-hole potential and freeze in with slightly different specific binding energies. This spread $\Delta E$ is tiny next to the orbital energy but huge next to the star's own binding energy, and it is what smears the debris return over time.

\Delta E \simeq \frac{G M_{\mathrm{BH}} R_\star}{r_{\mathrm{t}}^2} = \frac{G M_\star}{R_\star}\left(\frac{M_{\mathrm{BH}}}{M_\star}\right)^{1/3} \qquad (2)

where: $\Delta E$ — spread in specific orbital energy across the star at disruption. Reading: because $\Delta E \gg G M_\star/R_\star$ , roughly half the debris is left with $E<0$ (bound) and half with $E>0$ (unbound). Kepler's third law turns that flat energy spread into a distribution of return times. The most-bound debris comes back first, at $t_{\min}$ ; thereafter the mass-return rate to pericenter declines as a pure power law, with a prefactor fixed by demanding the time-integral equal the bound half $M_\star/2$ .

\dot M(t) = \frac{M_\star}{3\,t_{\min}}\left(\frac{t}{t_{\min}}\right)^{-5/3}, \qquad t_{\min} = \frac{2\pi G M_{\mathrm{BH}}}{(2\,\Delta E)^{3/2}} \qquad (3)

where: $\dot M(t)$ — mass return rate to pericenter; $t_{\min}$ — return time (orbital period) of the most tightly bound debris. Reading: the shape $t^{-5/3}$ is universal; only $t_{\min}$ and the normalization depend on the star and hole. Evaluating $t_{\min}$ for a solar-type star gives the flare's onset and rise time. Its scaling as $M_{\mathrm{BH}}^{1/2}$ means heavier holes flare later, over weeks to months.

t_{\min} \sim 41\,\mathrm{d}\left(\frac{M_{\mathrm{BH}}}{10^6 M_\odot}\right)^{1/2}\left(\frac{M_\star}{M_\odot}\right)^{-1}\left(\frac{R_\star}{R_\odot}\right)^{3/2} \qquad (4)

where: the scaling combines $t_{\min}\propto M_{\mathrm{BH}}^{1/2}$ (from Kepler) with the stellar structure factors. Reading: the slow, weeks-to-months rise is a defining observable of the class. Requiring the tidal radius to sit outside the horizon caps the black-hole mass that can produce a visible flare. This is the Hills mass, and it is why TDEs preferentially select lower-mass, harder-to-weigh black holes.

M_{\mathrm{BH}}^{\max} \simeq \left(\frac{c^2 R_\star}{2 G}\right)^{3/2} M_\star^{-1/2} \approx 1.1\times10^{8}\,M_\odot\left(\frac{R_\star}{R_\odot}\right)^{3/2}\left(\frac{M_\star}{M_\odot}\right)^{-1/2} \qquad (5)

where: $M_{\mathrm{BH}}^{\max}$ — maximum non-spinning black-hole mass giving an observable disruption of the given star. Reading: giants (large $R_\star$ ) can be disrupted by heavier holes; spin raises the cutoff further. The luminosity the flare can radiate is capped by the Eddington limit, where radiation pressure on ionized gas balances gravity. Because peak fallback far exceeds the corresponding accretion rate, the emitted power saturates near this value.

L_{\mathrm{Edd}} = \frac{4\pi G M_{\mathrm{BH}} m_p c}{\sigma_{\mathrm{T}}} \approx 1.3\times10^{44}\,\mathrm{erg\,s^{-1}}\left(\frac{M_{\mathrm{BH}}}{10^6 M_\odot}\right) \qquad (6)

where: $L_{\mathrm{Edd}}$ — Eddington luminosity; $m_p$ — proton mass; $\sigma_{\mathrm{T}}$ — Thomson cross section. Reading: the surplus fuel beyond $L_{\mathrm{Edd}}$ is advected inward or expelled in radiation-driven winds. The accretion luminosity is the fallback rate times a rest-mass efficiency, capped near Eddington. The temperature of the $r_{\mathrm{t}}$ -scale emitting region places the flare firmly in the ultraviolet and soft X-rays.

L_{\mathrm{peak}} \sim \eta\,\dot M_{\mathrm{peak}}\,c^2 \;\;\text{(capped near }L_{\mathrm{Edd}}\text{)}, \qquad T_{\mathrm{eff}} \sim \left(\frac{L}{4\pi r_{\mathrm{t}}^2 \sigma_{\mathrm{SB}}}\right)^{1/4} \sim \text{a few}\times10^{4\text{–}5}\,\mathrm{K} \qquad (7)

where: $\eta\sim0.1$ — radiative efficiency; $\dot M_{\mathrm{peak}}$ — peak fallback rate; $\sigma_{\mathrm{SB}}$ — Stefan–Boltzmann constant; $T_{\mathrm{eff}}$ — effective temperature. Reading: reprocessing by an optically thick envelope can shift some of this power into the optical.

Derivation AThe tidal radius and the frozen-in energy spread

The goal is to pin down the single length that decides a star's fate — the tidal radius — by balancing the black hole's differential (tidal) pull across the stellar diameter against the star's own surface gravity, and then to read off the spread in orbital binding energy that the same tidal field freezes into the debris. These two results, $r_{\mathrm{t}}$ and $\Delta E$ , are the seeds of everything downstream: the mass cutoff, the fallback timescale, and the shape of the light curve.

Step 1. Expand the black hole's gravitational acceleration $g(r)=GM_{\mathrm{BH}}/r^2$ across the stellar diameter. The difference between the near and far sides, separated by $\sim 2R_\star$ , is the tidal acceleration $a_{\mathrm{tidal}}\sim |dg/dr|\,R_\star \sim 2GM_{\mathrm{BH}}R_\star/r^3$ . Why: tides are a differential, not the total pull — it is the gradient of gravity, falling as $r^{-3}$ , that stretches an extended body.

Step 2. Write the star's own restraining acceleration, its surface self-gravity, as $a_{\mathrm{self}}\sim GM_\star/R_\star^2$ . Why: this is the glue holding the star together; disruption is precisely the moment the tidal stretch overpowers it.

Step 3. Set the two equal to locate the tipping point, $GM_{\mathrm{BH}}R_\star/r^3 \sim GM_\star/R_\star^2$ , and solve for $r$ to get $r_{\mathrm{t}}\sim R_\star (M_{\mathrm{BH}}/M_\star)^{1/3}$ . Why: inside this radius tides dominate and the star cannot survive; the cube-root dependence on mass is the robust, structure-independent result.

Step 4. For the energy spread, note the specific binding energy varies across the star as $\Delta E \sim |d(GM_{\mathrm{BH}}/r)/dr|\,R_\star = GM_{\mathrm{BH}}R_\star/r^2$ , evaluated at $r=r_{\mathrm{t}}$ . Substituting equation (1) gives $\Delta E \simeq (GM_\star/R_\star)(M_{\mathrm{BH}}/M_\star)^{1/3}$ . Why: this dwarfs the star's binding energy $GM_\star/R_\star$ by the factor $(M_{\mathrm{BH}}/M_\star)^{1/3}\sim 10^2$ , so the star is torn cleanly into a nearly symmetric bound/unbound split.

Go deeper · Derivation B · 1 minThe minus five-thirds fallback law

Derivation BThe minus five-thirds fallback law

The goal is to explain why the debris does not all come back at once but instead returns with a rate that falls as a fixed power of time, and to derive that exponent exactly. The only ingredients are a nearly flat spread of specific orbital energies frozen in at disruption and Kepler's third law relating each bound orbit's binding energy to its return time; together they force the celebrated minus-five-thirds decline — a clean change of variables from energy to time.

Step 1. Each bound debris element of specific energy $E<0$ moves on a Keplerian ellipse of semimajor axis $a$ with $E=-GM_{\mathrm{BH}}/2a$ , and returns to pericenter once per orbital period. Why: after disruption every parcel is a free test particle; its whole future is fixed by its conserved energy.

Step 2. Kepler's third law gives the period $P=2\pi\sqrt{a^3/GM_{\mathrm{BH}}}$ ; eliminating $a$ with Step 1 yields $t\approx P = 2\pi GM_{\mathrm{BH}}(2|E|)^{-3/2}$ . Why: this is the bridge between energy and time — the more tightly bound a parcel, the sooner it returns, as $t\propto|E|^{-3/2}$ .

Step 3. Invert to get $|E|\propto t^{-2/3}$ , and differentiate to find $|dE/dt|\propto |E|^{5/2}\propto t^{-5/3}\cdot t^{5/3}\cdots$ — more directly, $dt/dE\propto|E|^{-5/2}$ so $|dE/dt|\propto |E|^{5/2}$ . Why: the rate at which parcels of a given energy peel off in time is what carries the power law.

Step 4. Assemble the mass-return rate by the chain rule, $\dot M = (dM/dE)\,|dE/dt|$ . With $dM/dE\approx$ constant (a flat energy distribution) this is $\dot M \propto |E|^{5/2}\propto (t^{-2/3})^{5/2}=t^{-5/3}$ . Why: a flat spread in energy plus the Keplerian mapping is all it takes; the exponent is a pure consequence of $t\propto|E|^{-3/2}$ , independent of the messy hydrodynamics.

Step 5. Fix the normalization by requiring $\int_{t_{\min}}^{\infty}\dot M\,dt = M_\star/2$ , the bound half. This yields $\dot M(t)=(M_\star/3t_{\min})(t/t_{\min})^{-5/3}$ , with the most-bound debris returning at $t_{\min}$ . Why: the prefactor is not free — conservation of the bound mass ties the peak rate directly to $t_{\min}$ , and hence to $M_{\mathrm{BH}}$ .

Go deeper · Derivation C · 1 minThe Hills mass and the visibility cutoff

Derivation CThe Hills mass and the visibility cutoff

The goal is to find the heaviest black hole that can still put on a show — the Hills mass — by demanding that the tidal radius lie outside the event horizon, so the star is shredded before it is swallowed. Because the tidal radius grows only as the cube root of the black-hole mass while the Schwarzschild radius grows linearly, their ratio must fall below one at some mass, and setting it to unity fixes the cutoff. This is why TDEs are found around $10^6$ – $10^7\,M_\odot$ holes and essentially never around the most massive ones.

Step 1. Form the ratio of the two relevant radii, $r_{\mathrm{t}}/r_{\mathrm{s}} = R_\star(M_{\mathrm{BH}}/M_\star)^{1/3} \big/ (2GM_{\mathrm{BH}}/c^2)$ , using $r_{\mathrm{s}}=2GM_{\mathrm{BH}}/c^2$ . Why: a visible disruption requires the star to be torn apart outside the horizon, i.e. $r_{\mathrm{t}}\gtrsim r_{\mathrm{s}}$ .

Step 2. Collect the mass dependence: $r_{\mathrm{t}}/r_{\mathrm{s}} = (R_\star c^2/2GM_\star^{1/3})\,M_{\mathrm{BH}}^{-2/3}$ . Why: the ratio decreases as $M_{\mathrm{BH}}^{-2/3}$ , so there is a maximum mass beyond which the horizon overtakes the tidal radius.

Step 3. Set the ratio to unity and solve, $M_{\mathrm{BH}}^{\max} = (c^2 R_\star/2G)^{3/2} M_\star^{-1/2}$ . Why: at this mass the disruption happens exactly at the horizon; heavier holes swallow the intact star.

Step 4. Insert solar values, $R_\star=R_\odot$ , $M_\star=M_\odot$ : $M_{\mathrm{BH}}^{\max}\approx 1.1\times10^{8}\,M_\odot$ . Why: this Newtonian estimate matches full relativistic calculations to tens of percent; spin shrinks the prograde horizon and pushes the cutoff higher, which is how the heaviest TDE hosts constrain spin.

Limiting cases

TDE phenomenology depends sharply on black-hole mass, stellar structure, and how deeply the star plunges — the last set by the penetration factor $\beta \equiv r_{\mathrm{t}}/r_{\mathrm{p}}$ , the ratio of tidal radius to orbital pericenter.

Case	Result	Meaning
Massive-hole limit	$M_{\mathrm{BH}}\gtrsim 10^{8}\,M_\odot$	Tidal radius falls inside the horizon for solar-type stars; the star is swallowed whole with no flare. Spin can extend visibility to a few $\times10^8\,M_\odot$ .
Deep penetration	$\beta \gg 1$	Pericenter well inside $r_{\mathrm{t}}$ : the star is fully destroyed and strongly compressed at pericenter. A partial disruption leaving a surviving core occurs near $\beta\simeq1$ .
Late-time fallback	$\dot M\propto t^{-5/3}$	Long after peak, $t\gg t_{\min}$ , the return rate approaches the canonical power law set by the flat debris energy distribution.
Compact-star disruption	$r_{\mathrm{t}} < r_{\mathrm{s}}$	White dwarfs need lower-mass ( $\lesssim 10^5\,M_\odot$ ) intermediate-mass black holes to be disrupted outside the horizon; deep passages can compress and even thermonuclearly ignite them.

Common trapThe t⁻⁵ᐟ³ curve is the fallback rate, not the light curve

It is tempting to read a measured light curve directly off the $t^{-5/3}$ law and "reject" a disruption when the two disagree. That inverts the logic. The $t^{-5/3}$ law describes the mass fallback rate to pericenter under idealized assumptions — a flat energy spread, instantaneous circularization, no radiative cap. Between fallback and emitted light sit several filters: the streams must self-intersect and circularize (a viscous time that can delay or smear the peak), the accretion may be capped near Eddington for months, and a reprocessing envelope can reshape the spectrum. A light curve that flattens, plateaus, or declines as $t^{-9/4}$ is therefore fully consistent with a disruption — a probe of the intervening physics, not a falsification. The clean $-5/3$ slope is best regarded as the late-time asymptote of the fuel supply, not a promise about the flux.

Worked examples

1 · A solar-type star meets a million-solar-mass hole (fully worked)

A Sun-like star ( $M_\star=M_\odot$ , $R_\star=R_\odot$ ) is scattered onto a nearly radial orbit around a $10^6\,M_\odot$ black hole and disrupted at $\beta\simeq1$ . Find the tidal radius in horizon units, the fallback timescale, the peak fallback rate, and compare it to the Eddington accretion rate.

Step 1. Tidal radius: $r_{\mathrm{t}}=R_\odot(10^6)^{1/3}=100\,R_\odot=6.96\times10^{10}$ m. The horizon is $r_{\mathrm{s}}=2GM_{\mathrm{BH}}/c^2=2.96\times10^{9}$ m, so $r_{\mathrm{t}}/r_{\mathrm{s}}\approx\mathbf{24}$ — the star is destroyed comfortably outside the horizon, and a flare is guaranteed.

Step 2. Fallback timescale from equation (4): with all factors unity, $t_{\min}\approx\mathbf{41\ days}$ . This is the delay before the flare rises and roughly sets the time to peak — an intrinsically slow, months-long transient, quite unlike a supernova's days.

Step 3. Peak fallback rate: $\dot M_{\mathrm{peak}}\approx M_\star/(3t_{\min}) = M_\odot/(3\times41\ \mathrm{d})\approx\mathbf{3\ M_\odot\,yr^{-1}}$ . The bound half of the star, $0.5\,M_\odot$ , is delivered over a few $t_{\min}$ .

Step 4. Eddington comparison: $L_{\mathrm{Edd}}=1.3\times10^{44}\,\mathrm{erg\,s^{-1}}$ (equation 6), so at $\eta=0.1$ the Eddington accretion rate is $\dot M_{\mathrm{Edd}}=L_{\mathrm{Edd}}/\eta c^2\approx\mathbf{0.02\ M_\odot\,yr^{-1}}$ . The peak fallback exceeds this by a factor $\sim\mathbf{130}$ — the early flare is strongly super-Eddington. Fallback drops back below Eddington only after $t\sim t_{\min}(\dot M_{\mathrm{peak}}/\dot M_{\mathrm{Edd}})^{3/5}\approx\mathbf{2\ years}$ . So: r_t ≈ 24 r_s, t_min ≈ 41 d, peak ≈ 3 M_⊙/yr, super-Eddington by ~130× for ~2 yr.

2 · Why a billion-solar-mass black hole shows no flare

Estimate the Hills mass for the Sun and explain what happens to a solar-type star that falls toward a $10^9\,M_\odot$ black hole.

Step 1. From equation (5), $M_{\mathrm{BH}}^{\max}=(c^2R_\odot/2G)^{3/2}M_\odot^{-1/2}\approx\mathbf{1.1\times10^{8}\ M_\odot}$ — above this the tidal radius sinks inside the horizon for a Sun-like star.

Step 2. Check the ratio at $10^9\,M_\odot$ : $r_{\mathrm{t}}/r_{\mathrm{s}}=(R_\odot c^2/2GM_\odot^{1/3})M_{\mathrm{BH}}^{-2/3}$ scales as $(M_{\mathrm{BH}}/M^{\max})^{-2/3}=(10^9/1.1\times10^8)^{-2/3}\approx\mathbf{0.23}$ . The tidal radius is only a quarter of the horizon, so the star crosses the horizon while still self-bound.

Step 3. Physical outcome: the star is swallowed whole with no electromagnetic flare — a silent plunge. Two escapes exist: a spinning (Kerr) hole has a smaller prograde horizon, raising the cutoff to several $\times10^8\,M_\odot$ , and a red giant's far larger radius raises its Hills mass by $(R_\star/R_\odot)^{3/2}\sim10^3$ . So: solar-type stars flare only for M_BH ≲ 10⁸ M_⊙; heavier holes require spin or a giant star to be seen.

Physics of the portrait

TDEs knot together three threads of twentieth-century astrophysics — stellar dynamics near a central mass, accretion onto black holes, and the demographics of galactic nuclei. One theorist did more than anyone to weave them into a predictive picture of an observable flare.

Origins · 1 minMartin Rees and how this was discovered

Martin Rees

born 1942 · the fallback-and-flare theory of tidal disruption events

Martin Rees, Astronomer Royal and one of the architects of modern high-energy astrophysics, gave tidal disruption its decisive theoretical form. The idea had been floated earlier — Jack Hills argued in 1975 that such disruptions should occur in galactic nuclei, and Lacy, Townes and Hollenbach modelled the debris in 1982 — but it was Rees, in a 1988 Nature paper, who assembled the modern synthesis: the star is disrupted at the tidal radius, roughly half the debris is bound and returns, and the fallback of that bound material powers a luminous flare fading over months to years. He and Phinney (1989) worked out the frozen-in energy spread that yields the $t^{-5/3}$ decline now used as the signature of the class. Having already laid much of the groundwork for accretion-powered active nuclei and the loss-cone problem of feeding central black holes, Rees saw TDEs as its natural culmination — a prediction that even dormant black holes would occasionally flare and betray themselves. It sat largely untested for two decades until X-ray and then optical surveys began finding exactly the nuclear transients he had described.

History

1975

— Hills predicts stars tidally disrupted by massive black holes in galactic nuclei

1982

— Lacy, Townes and Hollenbach model the debris and flare from a disrupted star

1988

— Rees derives the debris fallback and its −5/3 power-law flare

1989

— Phinney fixes the frozen-in energy spread underlying the −5/3 law

2011

— Swift J1644+57 reveals a jetted tidal disruption event in real time

Going further

Texts. Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed. — black holes, accretion, and the Eddington luminosity · Binney & Tremaine, Galactic Dynamics, 2nd ed. — loss-cone dynamics, relaxation, and the tidal radius · Mo, van den Bosch & White, Galaxy Formation and Evolution — nuclear black holes and their demographics · Dodelson, Modern Cosmology — background for the cosmic growth of the black holes TDEs probe.

Provenance. The tidal radius, debris energy spread, and Hills mass follow from Newtonian tidal balance against self-gravity and the Schwarzschild radius (Carroll & Ostlie; Binney & Tremaine). The $t^{-5/3}$ fallback derivation uses only Kepler's third law and a flat $dM/dE$ (Rees, Nature 333, 523, 1988; Phinney 1989). Numerical anchors were recomputed from fundamental constants: for a solar star and a $10^6\,M_\odot$ hole, $r_{\mathrm{t}}=100\,R_\odot\approx24\,r_{\mathrm{s}}$ , $t_{\min}\approx41$ d, $\dot M_{\mathrm{peak}}\approx3\,M_\odot\,\mathrm{yr^{-1}}$ , $L_{\mathrm{Edd}}\approx1.3\times10^{44}\,\mathrm{erg\,s^{-1}}$ , and the Hills mass $M_{\mathrm{BH}}^{\max}\approx1.1\times10^{8}\,M_\odot$ .

Prerequisites. Supermassive Black Holes · Eddington Luminosity and Accretion Limits · Thin Accretion Disks and the Shakura–Sunyaev Model · The M–sigma Relation. Leads to. Advection-Dominated Accretion Flows · Relativistic Jets and Launching Mechanisms · Black Hole Spin and the Ergosphere · Active Galactic Nuclei: the Unified Model.

Ask the tutor: Ask me to walk through the $t^{-5/3}$ derivation line by line, or to work out the tidal radius, fallback time, and Hills mass for any star and black-hole mass you choose.

Is the t⁻⁵ᐟ³ law the shape of the observed light curve?

Not directly. The $t^{-5/3}$ law is the rate at which mass falls back to pericenter, derived from the debris energy spread and Kepler's third law. The luminosity we observe depends on what happens after fallback: the streams must self-intersect and circularize into an accretion flow, which takes a viscous time and can delay or smooth the peak; the accretion is often capped near the Eddington limit for months; and an optically thick envelope reprocesses the emission. Many real TDEs do show a decline close to $-5/3$ at late times, but a plateau, a steeper slope, or a bumpy curve is perfectly consistent with disruption. Treat $-5/3$ as the asymptotic behaviour of the fuel supply, not a template for the flux.

Where does the energy that powers the flare actually come from?

Ultimately from gravitational binding energy released as the bound debris accretes. Falling toward the innermost stable circular orbit, gas converts a fraction $\eta\sim0.1$ of its rest mass into radiation, so accreting even $0.1\,M_\odot$ releases $\sim10^{52}$ erg — ample to power a $10^{44}\,\mathrm{erg\,s^{-1}}$ flare for a year. A prompt contribution also comes from shocks where the incoming and outgoing streams collide near pericenter; in some models these stream collisions, rather than the inner disk, dominate the early optical light. Which mechanism leads is still debated.

Why are TDEs seen only around lower-mass black holes?

Because of the Hills mass. The tidal radius grows only as $M_{\mathrm{BH}}^{1/3}$ while the event horizon grows as $M_{\mathrm{BH}}$ , so above roughly $10^8\,M_\odot$ a Sun-like star crosses the horizon before tides can shred it, and it is swallowed whole with no flare. This makes TDEs an unusually clean selector of $10^5$ – $10^7\,M_\odot$ black holes — exactly the mass range that is hardest to weigh by other means. Two effects push the ceiling higher: rapid black-hole spin shrinks the prograde horizon, and evolved giant stars have far larger radii and hence much larger Hills masses. The observed upper cutoff in TDE host masses is therefore itself a diagnostic of black-hole spin.

How do we know a nuclear transient is a TDE and not an AGN flare or supernova?

Several signatures combine. TDEs sit at the exact photometric centre of a previously quiescent galaxy, ruling out most supernovae, which explode off-nucleus in star-forming regions. Their spectra are nearly featureless blue continua with broad, transient hydrogen and helium (sometimes nitrogen-rich) lines, and their colours stay hot as they fade — unlike supernovae, which cool. Their light curves rise over weeks and decline smoothly over months to years. And unlike a persistent active nucleus, a TDE is a single smooth flare from a galaxy with no prior accretion activity, lacking the stochastic, recurrent variability of an AGN. No single feature is decisive; the combination is.

Can a black hole disrupt something other than a main-sequence star?

Yes. White dwarfs can be disrupted, but only by intermediate-mass black holes ( $\lesssim10^5\,M_\odot$ ), because a white dwarf's small radius places its tidal radius inside the horizon of any heavier hole — so white-dwarf TDEs are a sought-after signpost of that elusive population, and deep passages can compress the star enough to ignite thermonuclear burning. Giant stars, at the opposite extreme, have enormous radii and huge Hills masses, so they can be disrupted even by the most massive nuclear black holes, though their low-density envelopes make for faint, slow events. The disrupted star's composition is imprinted on the flare spectrum, which is why nitrogen-rich lines can betray a chemically evolved donor.

Why do many TDEs shine so brightly in the optical when the theory points to X-rays?

This is one of the field's open puzzles. Equation (7) puts the emitting region near $r_{\mathrm{t}}$ at a few $\times10^{4\text{–}5}$ K, peaking in the ultraviolet and soft X-rays, and some TDEs are indeed luminous soft-X-ray sources. Yet many are optically bright and X-ray-faint, with blackbody radii far larger than $r_{\mathrm{t}}$ . Two ideas dominate: in the reprocessing picture an optically thick envelope or wind driven by super-Eddington accretion absorbs the inner X-rays and re-emits them cooler in the optical and UV; in the stream-collision picture the early optical light comes not from the inner disk but from shocks where the debris streams intersect at large radius. Both are probably at work, and disentangling them is a major goal of current research.