Gamma-ray bursts and tidal disruption candidates

A burst duration does not time every part of the source

A gamma-ray burst is detected through a rapid high-energy signal. Interpretations involve relativistic outflows and energy dissipation, while later afterglow radiation can arise as an outflow interacts with its surroundings. A radioactive kilonova or a supernova counterpart is another component with different physics.

The observed duration of prompt gamma radiation therefore need not equal the lifetime of the central engine, the time until an afterglow fades or the duration of radioactive ejecta emission. A time statistic is meaningful only after identifying which component and measured quantity it describes.

Traditional short and long burst categories summarize distributions of prompt properties. Berger discusses both overlap and instrument dependence. A commonly used boundary near two seconds is a phenomenological convention, not a deterministic rule that assigns a progenitor to every burst.

Rastinejad and collaborators’ study of GRB211211A is a dated example: follow-up evidence for a likely kilonova supported a compact-merger interpretation despite a long gamma-ray duration. Its role here is to test the universal duration-to-engine rule; no current event census or new observation is claimed.

T90 and T50 are intervals between specified fractions

The word fluence needs a specified quantity. Energy fluence is received energy per area, such as J m⁻², integrated over time in a declared band. Photon fluence is the number of incident photons per area integrated over time. Energy weighting can change the cumulative history when a spectrum evolves; neither quantity is identical to raw detector counts.

For a nonnegative known history of the declared band quantity with a finite positive total, define t_p as the first time its cumulative value reaches p percent of that total. Then T90 = t_95 − t_5 and T50 = t_75 − t_25. The starting times and the intervals are different quantities.

For a concrete measurement convention, the Fermi GBM catalogue method described by Paciesas and collaborators integrates a fitted, response-deconvolved photon-flux history over time for its duration quantiles. Its separately reported energy fluence adds energy weighting. Background subtraction, spectral fitting and the detector response connect the recorded counts to those inferred incident-photon quantities.

Real estimates depend on background, sensitivity, energy band, response and the selected analysis interval. The next exercise is explicitly an energy-fluence-based illustration of the quantile arithmetic, with its complete history stipulated exactly. It is not a calculation of a GBM catalogue T90 from energy fluence or a model of its fitted measurement uncertainties.

A complete toy history makes the quantiles reproducible

Choose an arbitrary positive fluence unit Φ0 in J m⁻². Define the observer-band fluence rate to be 5Φ0/s from 0 to 1 s, 20Φ0/s from 1 to 3 s, 10Φ0/s from 3 to 7 s and 5Φ0/s from 7 to 10 s, and zero elsewhere. Use half-open bins; isolated endpoint values do not change the integrals.

The bin fluences are 5, 40, 40 and 15 in units Φ0, for a total of 100Φ0. The 5% level is reached at 1 s. The 25% level is reached at 1 + (25 − 5)/20 = 2 s; the 75% level at 3 + (75 − 45)/10 = 6 s; and the 95% level at 7 + (95 − 85)/5 = 9 s.

Thus T90 = 9 − 1 = 8 s and T50 = 6 − 2 = 4 s. The signal has support over 10 s, which is different from either central-fluence interval. These invented values establish no physical progenitor.

For a separate corresponding-interval correction, stipulate cosmological redshift z = 1. Dividing by 1 + z gives rest-frame intervals of 4 s and 2 s. The observer band corresponds to rest-frame photon energies twice as large: this is not a remeasurement in one common rest-frame band, an engine lifetime or a unique distance inference.

Returning debris is one stage of the event

A tidal disruption event involves differential gravity acting across a star near a massive black hole. Comparing that differential pull with stellar self-gravity gives a Newtonian scale proportional to R_star(M_BH/M_star)^(1/3). The coefficient and outcome depend on stellar structure, orbit and rotation; strong-field encounters require relativistic treatment.

A close encounter can disrupt, partially strip or capture a star under different conditions. A simple tidal scale alone does not guarantee a luminous flare visible to an observer. The relevant debris and radiation have to survive capture and produce an escaping signal.

Some disrupted material can be unbound while other material follows bound orbits and returns. Returning streams can collide, dissipate orbital energy and circularize before material accretes. Outflows and delayed transport can further separate debris return from radiation reaching an observing band.

Gezari’s review explains both the observational candidates and the difficulty of connecting their light curves to simple fallback models. A galaxy-nuclear position or a smooth decline is useful context, but neither is a unique tidal-disruption signature.

The exponent depends on the debris distribution

For an original idealization, treat bound debris as test material in a Newtonian point-mass potential dominated by M_BH. Let ε be the positive magnitude of its negative specific orbital energy, in J kg⁻¹. Neglect interactions and assume each element leaves near pericentre and returns after one complete Kepler orbit.

The semimajor axis is a = GM_BH/(2ε), so the return time is t = 2π√[a³/(GM_BH)] = 2πGM_BH/(2ε)^(3/2). Time here is elapsed from the assumed disruption, not from the first detected light or an observed peak.

Write this relation as t = A ε^(−3/2), with A = 2πGM_BH/2^(3/2). Inverting gives ε = (A/t)^(2/3) and |dε/dt| = (2/3)A^(2/3)t^(−5/3). A constant mass per unit specific energy, dM/dε, then gives a debris-return rate proportional to t^(−5/3).

More generally, if the stipulated dM/dε is proportional to ε^k in the relevant weak-binding range, the return rate is (dM/dε)|dε/dt| proportional to t^[−(5 + 2k)/3]. This derivative connects an assumed mass distribution to a return rate; it supplies neither an accretion efficiency nor an optical radiation model.

The same initial rate need not mean the same returned mass

Construct two mathematical tails only for t ≥ t0 > 0, where t0 is a declared return-time scale and R0 a common positive mass-rate unit. Normalize both to return rate R0 at t0. The k = 0 model is Ṁ/R0 = (t/t0)^(−5/3); the k = 1 alternative is Ṁ/R0 = (t/t0)^(−7/3).

The table gives the full chosen sample set and the accompanying diagram uses common positive logarithmic axes. Zero is outside both axes. At 8t0 the ratios are 1/32 and 1/128, respectively. Decimal values report exact model arithmetic rounded for display.

Integrating these tails from t0 to infinity gives total model returned masses of 1.5R0t0 and 0.75R0t0. This follows from ∫₁^∞ x^(−p) dx = 1/(p − 1) for p > 1. Equal rates at one time do not make the integrated masses equal.

The k = 1 case is an invented distribution alternative, not a universal law for partial disruptions. Neither tail is defined here through t = 0, where an extrapolated power law would diverge. The energy distribution, assumed disruption time and ballistic conditions are essential parts of the construction.

A matching decline is a hypothesis test, not a unique label

Fallback, circularization, accretion, heating and wavelength-dependent radiation can have different time histories. The light curve may include reprocessing, obscuration or another component. A fitted optical exponent of −5/3 alone therefore does not uniquely identify a tidal disruption event.

Compare spectral evolution, the host environment, past and later variability and observations in other bands. Active-nucleus changes and supernovae can be alternatives to a nuclear transient interpretation. A poorly known onset can also alter an apparent fitted slope.

For both bursts and disruption candidates, retain the observed statistic alongside the proposed physical model. A clear duration or decay measurement remains valuable even when the engine is uncertain. The next lesson organizes the additional evidence needed to decide among classifications.

Worked examples

Try the reasoning

1. Is T90 necessarily the entire time interval over which the source emits in the chosen band?

   Show the answer and reasoning

   No.

   It measures the interval between defined cumulative fractions; the supplied history has 10 s support but T90 = 8 s.

2. With the stipulated z = 1, what corresponding rest-frame interval belongs to the toy observer T90 of 8 s?

   Show the answer and reasoning

   4 s.

   Divide by 1 + z = 2 while remembering that the corresponding rest photon-energy band also changes.

3. In this derivation, what is constant for the flat debris distribution?

   Show the answer and reasoning

   The mass per unit specific binding energy, dM/dε.

   Combining that constant with the Kepler energy-to-return-time derivative gives the −5/3 exponent.

4. Does a fitted optical −5/3 decline by itself uniquely prove a tidal disruption event?

   Show the answer and reasoning

   No.

   Debris return and band radiation require additional physical connections, and other transient explanations must be tested.

What to carry forward

• A prompt duration requires a band, cumulative quantity and measurement convention before physical interpretation.
• The Kepler fallback exponent is conditional on a debris-energy distribution and time origin.
• Return rates, accretion and observed light are different stages; spectra and temporal context test the proposed event class.