From fallback to infrared emission

An infrared echo is a delayed, wavelength-dependent record of radiation absorbed and re-emitted by dust. This model follows that process from a stellar fallback history to the actual WISE W1 and W2 measurements.

Read this page for the physical assumptions and equations, Numerical method for their discretisation, Data & parameters for the actual inputs, and References for the papers and original data products behind them.

1. Fallback and accretion

A tidal disruption can return stellar debris to a massive black hole over an extended interval. Instead of prescribing a single analytic decline, the calculation selects a tabulated hydrodynamical fallback history from STARS. Its stellar structure and encounter identifier remain fixed within each candidate model. Fallback is the mass-return rate, not automatically the instantaneous accretion rate.

The tables use a reference black-hole mass of 10⁶ M_⊙. For the adopted UGC 11487 mass, M_BH = 10^6.8 M_⊙, the implementation applies:

Stretching time and reducing the rate by reciprocal factors preserves the integrated returned mass. This is an adopted rescaling of a selected table, not a new hydrodynamical disruption simulation. The time origin is shifted to the maximum of the scaled, unsmoothed fallback rate. Interpolation is linear and the table contributes zero outside its supplied support.

Accretion is represented by causal exponential smoothing, a prescription used in TDE light-curve modelling by Mockler, Guillochon & Ramirez-Ruiz:

Here H is the Heaviside step function: the response cannot precede the fallback. The kernel has unit integral, so complete temporal support preserves mass; a finite calculation can omit a late tail. The limit τ_visc = 0 uses the unsmoothed history. The parameter describes effective smoothing, not separately resolved circularisation and inward transport. A further constant delay is fixed to zero because it would be degenerate with the event epoch.

The radiative efficiency η is constant within a model. The luminosity and total emitted energy follow from the selected history and η; there is no imposed target-energy normalisation and no Eddington luminosity cap. Mass loss and a time-varying efficiency are not calculated separately.

2. The illuminating spectrum

The source is isotropic, with a fixed blackbody shape at T_src = 30,000 K. Only its luminosity changes:

B_λ is the Planck spectral radiance. The denominator is evaluated on the actual transport wavelength grid, so the discretised source fractions sum to one. The source temperature is an assumption, not a temperature measured from the unobserved primary optical/UV flare. No changing source colour or anisotropic central illumination is introduced.

3. Geometry and grain populations

The reprocessor contains two disjoint radial populations, compact (C) and extended (E). Their grain number density, integrated over size, follows n_j(r) ∝ r^−p_j between R_in,j and R_out,j. Both occupy the same equatorial angular domain:

The polar angle θ and the viewing inclination i are measured from the symmetry axis. Within the occupied angular region the prescribed density has no polar or azimuthal dependence. The central cavity, gap between the populations and polar openings contain no dust. The geometrical sky fraction covered as seen from the source is cos θ_open; it is not generally equal to the fraction of source energy absorbed, which also depends on optical depth and wavelength.

Archived Case A has a compact population from 0.25 to 0.50 pc with p = 0, and an extended population from 1.0 to 2.0 pc with p = 2. A constant grain density in the compact population does not mean equal grain numbers in equal radial intervals: shell volume grows with radius. The calculation represents thermal quantities on an axisymmetric (r, μ) mesh while tracing radiation along three-dimensional paths.

Both populations use graphite grains with the power-law number distribution associated with Mathis, Rumpl & Nordsieck:

The fixed choices are q = 3.5 and a = 0.1–1.0 μm, represented by 41 logarithmically spaced size nodes. The interval and pure-graphite composition are model choices, not the original full interstellar grain mixture. Number weights do not already contain an a² area factor; that factor belongs in the absorption cross-section. The finite-volume implementation fixes the physical support at the stated size endpoints.

4. Optical properties and absolute normalisation

Wavelength- and size-dependent absorption efficiencies Q_abs(λ,a) come from the Draine graphite table Gra_81.gz, associated with Draine & Lee (1984) and Laor & Draine (1993). These spherical-grain optical properties use the orientation-averaged 1/3–2/3 approximation for graphite. The model uses absorption, not scattering extinction:

C_abs is an area. C̄_λ is its grain-number average, and C̄_src additionally averages over the illuminating spectrum. The two fitted optical depths define the radial grain columns:

The coefficient α_λ has units of inverse length. In the discrete mesh the normalisation is imposed using the same piecewise-constant density that the rays traverse. Multiplying density by cell volume and size weight gives the number of emitting grains. This fixes the absolute emission scale: no separate W1 or W2 amplitude is fitted and no spectrum is rescaled after solving the temperatures.

Three different uses of “tau” must stay separate: τ_C and τ_E are dimensionless absorption optical depths; τ_visc is a rest-frame smoothing time; an observed echo lag is an outcome of the source history, geometry and transfer. Neither optical depth is a time lag.

5. Absorption and grain temperatures

Along a path with optical-depth increment Δτ_λ, the surviving fraction is exp(−Δτ_λ) and the absorbed fraction is 1−exp(−Δτ_λ). Foreground absorption matters: the extended population is illuminated by source radiation already filtered by intervening compact dust, not by a second independent flare.

Each grain size in each thermal cell is placed in radiative equilibrium with the local radiation field:

J_λ is the local mean intensity, including direct and secondary radiation. The common factor 4π cancels between absorbed and emitted bolometric powers in the equilibrium equation; it remains in the single-grain spectral luminosity P_λ. Summing P_λ over the actual grain numbers produces the cell luminosity.

At each wavelength, absorbed cell power is partitioned among grain sizes in proportion to w_ℓC_abs(λ,a_ℓ). Dividing each share by its grain count gives the power per grain. The temperature then follows from the Planck equilibrium relation, separately for every cell, size and time. W1/W2 colour evolution is therefore a prediction of the evolving radiation field and grain properties, not a fitted temperature curve.

Thermal equilibration is assumed fast compared with the resolved variability. Grain heat storage and stochastic heating are omitted. The fixed host flux is added to photometry later; it does not heat the grains. A zero or very low pre-flare temperature in the stored transient solution must not be interpreted as the absolute quiescent dust temperature of the galaxy.

6. Secondary heating and self-absorption

Dust emission can escape, be absorbed elsewhere, or be re-absorbed within its own finite cell. Absorbed thermal radiation heats the grains again. V4.8 iterates this coupled emission and absorption field, including the travel time for each path. The same geometry, density and absorption cross-sections determine direct heating, secondary transport and attenuation toward the observer.

This is not an optically thin sum of independent shells. Self-obscuration and internal recycling affect the amount, spectrum, timing and viewing-angle dependence of the escaping emission. Scattering, a dynamically evolving torus and resolved individual clouds are not included. The transport kernels are numerical quadratures through an effective medium, not a list of physical clumps.

7. Keep the three time coordinates separate

Let r be distance from the central source, t_local a local rest-frame time, and n̂ the direction toward the observer. Direct illumination depends on the source-retarded coordinate u, while arrival at the observer also depends on position along the line of sight:

The thermal arrays are stored in u. For direct reprocessing, the arrival delay relative to direct light emitted by the source at the same source time is:

For radiation travelling from position r₁ to r₂, the additional delay in the two cells’ stored coordinates is:

Bold positions are vectors; the scalar radii r₁ and r₂ are distances from the centre. The magnitude of their vector difference gives the path length. Non-negativity follows from the triangle inequality. Secondary reprocessing adds actual intervening path lengths, not an adjustable extra lag.

The fitted epoch t₀ aligns source time zero with MJD. Because that origin is the unsmoothed fallback peak, t₀ is not an independently measured disruption date. A simultaneous local snapshot at time t_local samples u = t_local−r/c at each radius; an equal-arrival-time observer image is a different slice. The website’s 0–5000 d animation shows the former. Moving its slider changes only the displayed stored slice, not the solver’s time window.

8. Static opacity and the sublimation boundary

The calculation keeps grain numbers and cross-sections fixed. It checks all computed grain temperatures against the graphite threshold implemented from the density-based prescription of Baskin & Laor (2018):

Here ln is the natural logarithm and the numerical n_H is supplied in cm⁻³. At 10⁹ cm⁻³, the threshold is 1793.30 K. The prescription assumes the gas conditions described in the cited work; this density is an input to the threshold, not the fitted grain-density normalisation. The implemented gate has no grain-size dependence, even though different sizes acquire different temperatures.

If any evaluated grain reaches or exceeds the threshold, the run is outside this solver’s domain. It is stopped or recorded as unsupported_sublimation, not converted into a zero-emission solution. The solver does not destroy grains, move a sublimation front, reform dust or recalculate opacity after destruction. Archived Case A reaches 1143.58 K, below the adopted gate; that supports the static-opacity approximation for this particular prescription, not survival of every conceivable dust population.

9. From escaping light to the WISE measurement

The observer calculation sums thermal emission after wavelength-dependent absorption along the viewing direction and adds the transmitted central-source contribution. At redshift z = 0.01895 and luminosity distance D_L = 82.34 Mpc:

L_λ^iso is the directional isotropic-equivalent spectral luminosity, not necessarily the angle-integrated escaping luminosity. The redshift factor and wavelength convention must be kept consistent. The code evaluates the grain spectra at the rest wavelengths corresponding to the native WISE response nodes.

WISE catalog-equivalent flux density is a calibrated band integral, not simply the monochromatic model spectrum at 3.4 or 4.6 μm:

The adopted files are the official W1/W2 equal-energy responses and CALSPEC alpha_lyr_stis_011.fits. No additional photon-counting factor λ/(hc) is applied to these already equal-energy responses. Target and Vega integrals use consistent wavelength and flux-density units. The implementation uses zero points 309.54 Jy (W1) and 171.79 Jy (W2), the latter rounded from the 171.787 Jy calibration reference. See calibration provenance.

The predicted total flux is host plus the full transient response, including the transmitted source term. The host levels are fixed at approximately 8.072 mJy in W1 and 5.215 mJy in W2. They are not free offsets and do not supply a background heating field. The likelihood compares the total prediction directly with the observed fluxes.

10. Energy conservation without double-counting

One source photon can be absorbed and re-emitted more than once. Adding every intrinsic thermal emission therefore counts internally recycled energy repeatedly. The quantity to compare with directly absorbed source power is the energy that ultimately escapes as thermal radiation:

These forms assume negligible stored thermal energy and complete enough temporal coverage. A finite window also requires radiation pending absorption or further reprocessing. The energy ledger explicitly distinguishes those terms, the full driver from its retained window, and eventual escape from emission already computed. A small budget residual checks conservation but does not prove spatial resolution or a unique fit.

11. What to infer, and what not to infer

• The broad response comes from the driver, light-travel paths, distributed temperatures and secondary processing. It is not obtained by fitting independent band lags, widths or amplitudes.
• A characteristic lag is response-weighted. It need not equal an inner radius divided by c, and it is not a direct image of a unique shell.
• The 3D points sample the temperature field. Their count is not a fitted cloud count and their visual density does not directly encode grain-number density.
• Two photometric bands constrain the observed response, but cannot uniquely reconstruct a general three-dimensional dust distribution, source spectrum or grain mixture.
• Inference is conditional on the selected fallback history, geometry family, grain properties, source assumptions, fixed hosts, error model and numerical resolution. Passing the sublimation gate and conserving energy do not remove those conditions.

Follow the calculation

The original six-step reading path remains available below. Expand a step to connect its physical meaning with the detailed chapters and implementation.

For exact implementation, start at the archived Case A entry point, then the production solver and the angular transfer implementation. The source map distinguishes these calls from retained legacy workflows.