(chap:13)= # Explosions, transients, and multi-messenger quantum astronomy :::{admonition} Chapter opening :class: chapter-opening Transient sources compress astrophysical processes onto an observable timeline. Novae and supernovae couple expansion velocity, distance, and angular scale. GRB afterglows trace relativistic shocks, synchrotron radiation, and jet opening angles. Kilonovae combine gravitational-wave localization, $\gamma$-ray delay, color evolution, and $r$-process ejecta. Tidal disruption events connect black-hole mass, fallback rate, and a reprocessing photosphere to the data. An average light curve is not enough. An event table with time, frequency, polarization, spatial channel, and trigger provenance can constrain trigger epoch, background, expansion geometry, color evolution, and multi-messenger delay at the same time {cite:p}`2014Natur.514..339C,2014Sci...345..554A,2008Sci...321..223S,1998ApJ...497L..17S,2004RvMP...76.1143P,2017PhRvL.119p1101A,2017ApJ...848L..12A,2017Natur.551...75S,2021ARA&A..59...21G`. ::: ## Event tables and trigger likelihoods A transient observation begins with a trigger, but the trigger is already a statistical inference. For detector channel $k$, the event rate can be written as ```{math} :label: eq:ch13-rate-response \lambda_k(t)=b_k(t)+A_{{\rm eff},k}(t) \int R_k(\nu,p,t)\,F_\nu(t,p)\,d\nu , ``` where $\lambda_k$ is in $\mathrm{s^{-1}}$, $b_k$ is the background rate from sky, host galaxy, dark counts, and false triggers, $A_{{\rm eff},k}$ is effective area, $R_k$ is the instrument response including bandwidth, quantum efficiency, polarization selection, and temporal response, and $F_\nu(t,p)$ is the source photon flux at frequency $\nu$ and polarization state $p$. A single optical transient channel can range from below $1\,\mathrm{s^{-1}}$ to $10^6\,\mathrm{s^{-1}}$. High-energy detectors often record events in coarse energy channels; radio and optical fast detectors put more weight on sub-microsecond to nanosecond time stamps. For a fast-rise, slow-decay trigger window, a useful first model is ```{math} :label: eq:ch13-exponential-rate \lambda(t)=b+A\exp\!\left[-\frac{t-t_0}{\tau}\right]H(t-t_0), ``` where $b$ is background rate, $A$ is the initial excess source rate after trigger, $t_0$ is the physical start time, $\tau$ is the decay time, and $H$ is a step function. The time coordinate $t$ can be seconds, hours, or days, as long as $A$ and $b$ use the inverse of the same unit. For GRB prompt emission, $\tau$ can range from $10^{-2}$ to $10^2\,\mathrm{s}$. For optical afterglows, the effective decay time is usually hours to days. For novae and TDEs, it can be tens to hundreds of days. For the event table $\{t_j\}$, the log-likelihood of an inhomogeneous Poisson process is ```{math} :label: eq:ch13-poisson-likelihood \ln L=\sum_{j=1}^{N}\ln \lambda(t_j)-\int_{t_{\rm min}}^{t_{\rm max}}\lambda(t)\,dt . ``` The first term evaluates the model rate at the actual photon arrival times. The second subtracts the total number of counts predicted by the model over the observing window. This $\ln L$ can be computed from unbinned events. If the data are first placed into wide time bins, information about $t_0$ and the fast rise is averaged away. The same form works for events tagged by energy, frequency, and polarization after replacing $\lambda(t)$ with $\lambda(t,\nu,p)$ and extending the integral over the corresponding coordinates. ```{figure} ../_static/figures/generated/chapter_13/ch13_transient_event_likelihood.png :name: fig:chapter-13-event-likelihood :width: 92.0% An inhomogeneous Poisson event table constrains both trigger time and decay time. In the left panel, vertical marks are individual photon arrival times and the blue curve is the event-rate model. The right panel shows the relative likelihood surface in t₀ and τ for the same event table. A small number of early photons strongly constrains t₀; late background decides whether τ is overestimated. ``` A trigger usually compares a transient model $M_{\rm tr}$ with a background model $M_{\rm bg}$. Written as an odds ratio, ```{math} :label: eq:ch13-trigger-odds {\cal O}_{\rm tr,bg} = \frac{P(M_{\rm tr})}{P(M_{\rm bg})} \frac{L(D|M_{\rm tr})}{L(D|M_{\rm bg})}, ``` where $D$ is the event table or image-difference data and $P(M)$ is the prior event rate. If the expected background count in a window $\Delta t$ is $\mu_b=b\Delta t$, the single-window Poisson false-alarm probability is ```{math} :label: eq:ch13-false-alarm P_{\rm FA}(N\ge n)= \sum_{m=n}^{\infty}\frac{\mu_b^m e^{-\mu_b}}{m!}. ``` Real surveys must also account for trials: the number of time windows, sky pixels, energy channels, and templates all increase the false-trigger rate. FRB searches, $\gamma$-ray triggers, and optical image-difference surveys face the same statistical structure, although $\Delta t$ may range from milliseconds to days and the background may be instrumental noise, sky variation, or contaminating variables {cite:p}`2013Sci...341...53T,2017ApJ...835...29Y,2020Natur.587...59B,2020Natur.587...54C,2020Natur.581..391M`. ## Expanding sources: novae, supernovae, and angular scales Novae and supernovae share the physical picture of a velocity-stratified ejecta. The early photosphere is set by optically thick outer layers. Later, the line-forming region recedes inward and lower-opacity shells become visible. If the expansion is approximately homologous, so that radius and velocity obey $r=v(t-t_0)$, the angular radius is ```{math} :label: eq:ch13-angular-expansion \theta(t)=\frac{v_{\rm exp}(t-t_0)}{D} = 5.8\,{\rm mas} \left(\frac{v_{\rm exp}}{10^3\,{\rm km\,s^{-1}}}\right) \left(\frac{t-t_0}{10\,{\rm d}}\right) \left(\frac{D}{1\,{\rm kpc}}\right)^{-1}. ``` $\theta$ can be estimated from imaging, interferometry, or model visibility. $v_{\rm exp}$ is often estimated from Doppler line width or the minimum of an absorption trough. $D$ is distance. A nova at $D\sim1$--5 $\mathrm{kpc}$ with $v_{\rm exp}\sim10^3\,\mathrm{km\,s^{-1}}$ can reach milliarcsecond scales in days to tens of days. A Type Ia supernova at $D\sim10$--100 $\mathrm{Mpc}$ with $v_{\rm exp}\sim10^4\,\mathrm{km\,s^{-1}}$ has only a few microarcseconds of angular radius after ten days. A GW170817-like kilonova can reach $0.1$--$0.3c$, but at about $40\,\mathrm{Mpc}$ its early angular scale is still below the microarcsecond level. ```{figure} ../_static/figures/generated/chapter_13/ch13_expanding_fireball_resolution.png :name: fig:chapter-13-expansion :width: 92.0% The angular radius of an expanding source is set jointly by velocity, time, and distance. Galactic novae have lower speeds but are nearby, so their angular scales reach the milliarcsecond range quickly. Low-redshift Type Ia supernovae and kilonovae expand faster but remain at microarcsecond scales because they are distant. The horizontal dashed lines show the λ/B scale at $\lambda=0.5\,\mu{\rm m}$ for several optical baselines. ``` Spectral lines project velocity onto wavelength: ```{math} :label: eq:ch13-doppler-line v_{\rm los}\simeq c\,\frac{\lambda_{\rm obs}-\lambda_0}{\lambda_0}, ``` where $\lambda_0$ is the laboratory wavelength and $\lambda_{\rm obs}$ is the observed wavelength. The blue edge of P-Cygni absorption often approximates the fastest outer material; the absorption minimum is closer to the velocity layer of largest optical depth; emission-line half width mixes geometry, opacity, and density profile. If each spectral channel also has an angular scale or correlation function, the data become a set of $\theta(v_{\rm los})$, able to separate shells, bipolar cones, equatorial rings, and polar winds. In intensity interferometry, a circularly symmetric uniform disk gives $|V|^2$ through the second-order correlation. The corresponding first-order visibility amplitude is ```{math} :label: eq:ch13-uniform-disk-visibility |V(B)|= \left| \frac{2J_1(\pi B\Theta/\lambda)}{\pi B\Theta/\lambda} \right|, ``` where $\Theta=2\theta$ is angular diameter, $B$ is projected baseline, and $\lambda$ is observing wavelength. This formula assumes circular symmetry, monochromatic light, and no strong limb darkening. Real novae and supernovae more often require thin-shell, ring, ellipsoid, or multi-component models. Combining angular expansion with spectroscopic velocity gives an expansion parallax, ```{math} :label: eq:ch13-expansion-parallax D=\frac{v_{\rm exp}(t-t_0)}{\theta(t)}. ``` The distance $D$ is geometric only if the $v_{\rm exp}$ on the right side refers to the same material layer as $\theta$. A distance estimate becomes biased if the spectral line measures a high-velocity outer absorption component while the angular scale comes from the continuum photosphere. Classical novae are thermonuclear explosions on white-dwarf surfaces. Typical ejecta masses are $10^{-5}$--$10^{-4}M_\odot$, and velocities exceed $10^3\,\mathrm{km\,s^{-1}}$. After Fermi found novae as GeV $\gamma$-ray sources, the physical picture moved toward two-flow geometry: slow, dense equatorial ejecta shaped by the binary orbit, struck by a fast polar wind that accelerates particles in internal shocks. In V959 Mon, the $\gamma$-ray emission lasted about $12\,\mathrm{d}$. High-resolution radio imaging placed the shock at the interface between polar flow and equatorial material, with thermal ejecta mass of about $4\times10^{-5}M_\odot$ {cite:p}`2014Natur.514..339C,2014Sci...345..554A`. In such a source, $v_{\rm exp}$ contains both equatorial and polar velocity components; spectral lines, radio images, and photon event tables should be fitted together. Type Ia supernovae usually enter distance work through luminosity correction. The Phillips relation corrects peak absolute magnitude using the $B$-band decline rate $\Delta m_{15}(B)$. Cosmological applications then write the corrected distance modulus as ```{math} :label: eq:ch13-snia-distance-modulus \mu = m_B - M_B + \alpha x_1-\beta C+\Delta_{\rm host}, ``` where $m_B$ is observed peak magnitude, $M_B$ is standardized absolute magnitude, $x_1$ or a similar parameter describes light-curve width, $C$ is color, and $\alpha$, $\beta$, and $\Delta_{\rm host}$ are fitted from the sample. Nearby Type Ia supernovae with $m_B\sim10$--16 can be followed at high signal-to-noise with small telescopes; cosmological samples are often at $m_B\sim22$--25. If angular expansion could also be measured, it would provide a geometric cross-check on luminosity distance. The target is difficult because a Type Ia angular diameter after a few weeks is only of order microarcseconds. In long-baseline optical intensity interferometry, it is a demanding but well-motivated transient application {cite:p}`1993ApJ...413L.105P,1998AJ....116.1009R,1999ApJ...517..565P`. Polarization tests whether Type Ia ejecta are really close to spherical. Stokes parameters are often written as ```{math} :label: eq:ch13-stokes-polarization q=\frac{Q}{I},\qquad u=\frac{U}{I},\qquad P=\sqrt{q^2+u^2}, ``` where $q$, $u$, and $P$ are dimensionless and usually reported in percent. SN 1999by had $P\simeq0.3$--0.8% near maximum light, with a polarization change of about $0.4\%$ around Si II $6150\,\text{\AA}$. Models point to an overall asphericity of about $20\%$, with the Si layer and continuum roughly sharing the same symmetry axis {cite:p}`2001ApJ...556..302H`. The quantities $\Theta(t)$, $v_{\rm exp}$, and $P(t,\lambda)$ should be interpreted in a single geometric model: angular scale comes from photospheric shape, lines from velocity layers, and polarization from scattering geometry. The earliest window in a core-collapse supernova is shock breakout. Before the shock reaches the surface, radiation diffuses ahead of it and forms a radiative precursor. Let $d$ be the depth from the shock to the surface, $\rho$ the local density, and $\kappa$ the opacity. The optical depth is $\tau\simeq\kappa\rho d$, and the breakout condition is approximately ```{math} :label: eq:ch13-shock-breakout-condition t_{\rm diff}\simeq \frac{\tau d}{c}\simeq \frac{d}{v_s}, \qquad \tau\simeq \frac{c}{v_s}, ``` where $v_s$ is the shock velocity. For red supergiants, $v_s\sim1$--$2\times10^7\,\mathrm{m\,s^{-1}}$, and the early energy is mostly in the extreme-UV or soft X-ray. Optical discovery often comes days later. GALEX ultraviolet data for SNLS-04D2dc showed a radiative precursor before the shock reached the surface, constraining the envelope structure and progenitor radius {cite:p}`2008Sci...321..223S,2011ApJ...727..104C`. These hour-scale signals require trigger systems to compress alert latency, slew, acquisition, and first exposure into a short response chain. ## Kilonovae and multi-messenger delays GW170817 gave a clean timeline for a multi-messenger transient. The gravitational wave measured the coalescence time and distance of a binary neutron-star inspiral. Fermi/GBM and INTEGRAL saw GRB 170817A about $1.74\pm0.05\,\mathrm{s}$ later. The optical counterpart was localized within the next several hours, and the UV/optical/NIR colors then evolved rapidly from blue to red {cite:p}`2017PhRvL.119p1101A,2017ApJ...848L..13A,2017ApJ...848L..12A,2017ApJ...848L..17C,2017Natur.551...75S,2017ApJ...848L..27T,2017ApJ...848L..24V,2017ApJ...848L..18N,2017Sci...358.1565E,2017ApJ...851L..21V`. A multi-messenger delay is ```{math} :label: eq:ch13-mm-delay \Delta t_{a-b}=t_a-t_b . ``` $t_a$ and $t_b$ are arrival times on the same time-scale system. The interpretation of $\Delta t$ separates into three parts: ```{math} :label: eq:ch13-delay-budget \Delta t_{\rm obs} = \Delta t_{\rm engine} + \Delta t_{\rm prop} + \Delta t_{\rm clock}. ``` $\Delta t_{\rm engine}$ is an intrinsic emission delay, such as the time from merger to jet breakout. $\Delta t_{\rm prop}$ is propagation delay, which can constrain propagation-speed differences or effective masses. $\Delta t_{\rm clock}$ is timing and processing error in the instruments. If the source distance $D$ is known, the order-of-magnitude speed constraint is ```{math} :label: eq:ch13-speed-difference \frac{|\Delta v|}{c} \lesssim \frac{|\Delta t_{\rm prop}|}{D/c} = 2.4\times10^{-16} \left(\frac{\Delta t_{\rm prop}}{1\,{\rm s}}\right) \left(\frac{D}{40\,{\rm Mpc}}\right)^{-1}. ``` The full $1.74\,\mathrm{s}$ delay cannot be assigned directly to propagation, because the GRB emission may itself have started after merger. The value of the formula is that it puts any allowed new propagation effect on the right scale. ```{figure} ../_static/figures/generated/chapter_13/ch13_multimessenger_delay.png :name: fig:chapter-13-mm-delay :width: 92.0% Two readings of a multi-messenger delay. The left panel compares arrival times after merger: gravitational wave and γ-ray signals are on second scales, optical counterparts are often limited by localization and day-night cycles, and X-ray/radio emission can appear on day-to-month scales. The right panel converts propagation delay into the scale of |Δv|/c over 40 Mpc. The less certain the intrinsic emission delay, the more conservative the propagation constraint. ``` Kilonova luminosity comes from thermalized $r$-process radioactive decay in neutron-rich ejecta. A commonly used heating scale is ```{math} :label: eq:ch13-rprocess-heating \dot q(t)\simeq 2\times10^{10} \left(\frac{t}{1\,{\rm d}}\right)^{-1.3} {\rm erg\,s^{-1}\,g^{-1}}, ``` where $\dot q$ is the heating rate per unit mass. The luminosity receives $\epsilon_{\rm th}M_{\rm ej}\dot q$, with $\epsilon_{\rm th}<1$ the thermalization efficiency and $M_{\rm ej}$ the ejecta mass. Diffusion time sets the light-curve peak: ```{math} :label: eq:ch13-kilonova-diffusion t_{\rm pk}\simeq \left(\frac{\kappa M_{\rm ej}}{4\pi v_{\rm ej}c}\right)^{1/2} = 0.6\,{\rm d} \left(\frac{\kappa}{0.5\,{\rm cm^2\,g^{-1}}}\right)^{1/2} \left(\frac{M_{\rm ej}}{0.01M_\odot}\right)^{1/2} \left(\frac{v_{\rm ej}}{0.3c}\right)^{-1/2}. ``` $\kappa$ is opacity. Lanthanide-poor ejecta have $\kappa\sim0.3$--1 $\mathrm{cm^2\,g^{-1}}$, peak around a day, and are relatively blue. Lanthanide-rich ejecta have $\kappa\sim5$--30 $\mathrm{cm^2\,g^{-1}}$, peak after several to ten days, and move into the near-infrared. The early spectrum can often be approximated by a blackbody radius and temperature, ```{math} :label: eq:ch13-kilonova-blackbody L_{\rm bol}=4\pi R_{\rm BB}^2\sigma_{\rm SB}T_{\rm BB}^4 . ``` At about $0.6\,\mathrm{d}$, GW170817 was described by $T_{\rm BB}\simeq8300\,\mathrm{K}$, $R_{\rm BB}\simeq4.5\times10^{14}\,\mathrm{cm}$, and $L_{\rm bol}\simeq5\times10^{41}\,\mathrm{erg\,s^{-1}}$, corresponding to $v\simeq0.3c$. The SED then departed quickly from a single blackbody: UV/blue flux declined and NIR emission became relatively stronger. Two-component models typically require a blue component with $M_{\rm ej,blue}\sim0.01M_\odot$ and $v_{\rm blue}\sim0.27$--$0.3c$, plus a red component with $M_{\rm ej,red}\sim0.04M_\odot$ and $v_{\rm red}\sim0.1c$. Three-component models add a separate purple component with intermediate opacity {cite:p}`2017Natur.551...75S,2017ApJ...848L..17C,2017ApJ...848L..27T,2017Sci...358.1565E`. ```{figure} ../_static/figures/generated/chapter_13/ch13_kilonova_color_evolution.png :name: fig:chapter-13-kilonova :width: 92.0% Kilonova evolution from blue to red. The left panel uses three blackbody SEDs to show the peak wavelength shifting into the infrared as the temperature falls. The right panel uses the diffusion-time formula to show that higher opacity, larger mass, and lower velocity delay the peak. GW170817 had a fast, low-opacity blue component and a slower, high-opacity red component, so a single ejecta component cannot explain both the early blue light and the later NIR emission. ``` Gravitational waves also provide standard-siren distances. At low redshift, ```{math} :label: eq:ch13-standard-siren H_0\simeq \frac{cz_{\rm host}}{D_L}, ``` where $z_{\rm host}$ is the host-galaxy redshift and $D_L$ is the luminosity distance inferred from the gravitational-wave amplitude. In GW170817, the main degeneracy was inclination versus distance: a face-on binary is both brighter and harder to orient. The electromagnetic counterpart supplied the host, redshift, jet viewing angle, and ejecta geometry, reducing part of that degeneracy. A multi-messenger analysis puts these projections of the same physical event into one joint likelihood. ## GRB afterglows, jets, and rapid response GRB prompt emission gives the high-energy trigger. The afterglow gives a trackable external-shock problem. After a relativistic shell runs into the external medium, the observed time $t$ is related to radius $R$ and Lorentz factor $\Gamma$ approximately by ```{math} :label: eq:ch13-grb-arrival-time t\simeq \frac{(1+z)R}{4\Gamma^2c}. ``` At the same physical radius, relativistic beaming and light-travel-time effects compress the signal into a much shorter observed time. Typical early afterglows have $\Gamma\sim10$--300 and $R\sim10^{16}$--$10^{18}\,\mathrm{cm}$. Optical emission can enter a telescope field tens of seconds to minutes after trigger {cite:p}`1997ApJ...476..232M,1998ApJ...497L..17S,1998Natur.395..670G,1999PhR...314..575P,2004RvMP...76.1143P,2006ARA&A..44..507W`. If the electron distribution is ```{math} :label: eq:ch13-electron-powerlaw N(\gamma_e)\,d\gamma_e \propto \gamma_e^{-p}\,d\gamma_e, \qquad \gamma_e>\gamma_m, ``` the synchrotron spectrum has breaks near $\nu_m$ and $\nu_c$. A common observational form is ```{math} :label: eq:ch13-afterglow-powerlaw F_\nu(t)\propto t^{-\alpha}\nu^{-\beta}. ``` For a uniform external medium, adiabatic evolution, slow cooling, and $\nu_m<\nu<\nu_c$, ```{math} :label: eq:ch13-closure-relation \beta=\frac{p-1}{2}, \qquad \alpha=\frac{3(p-1)}{4}. ``` For $p=2.2$, this gives $\beta\simeq0.6$ and $\alpha\simeq0.9$. Above $\nu_c$, the temporal slope often changes to $\alpha=(3p-2)/4$. These closure relations assume a simple external medium, weak energy injection, slowly evolving microphysical parameters, and an observing band that does not cross a spectral break. Observationally, an optical afterglow is often $19$--20 mag one day after the burst. Polarization can reach a few percent and sometimes about $10\%$, tied to synchrotron emission and jet geometry {cite:p}`2004RvMP...76.1143P,2006ARA&A..44..507W`. ```{figure} ../_static/figures/generated/chapter_13/ch13_afterglow_synchrotron_breaks.png :name: fig:chapter-13-afterglow :width: 92.0% Synchrotron reading of a GRB afterglow. The left panel shows how ν_m and ν_c split the spectrum into regions with different slopes. The right panel shows the light curve steepening after a jet break. If optical and X-ray data are not in the same spectral segment, their α and β need not match; the first step is to locate the observing frequency relative to ν_m and ν_c. ``` When $\Gamma$ falls to about $1/\theta_j$, the observer begins to see the edge of the jet and the light curve develops an achromatic jet break. A common estimate is ```{math} :label: eq:ch13-jet-opening-angle \theta_j\simeq 0.10 \left(\frac{t_j}{1\,{\rm d}}\right)^{3/8} \left(\frac{1+z}{2}\right)^{-3/8} \left(\frac{E_{\rm iso,52}}{n_0}\right)^{-1/8}, ``` where $\theta_j$ is in radians, $t_j$ is the break time, $E_{\rm iso,52}=E_{\rm iso}/10^{52}\,\mathrm{erg}$, and $n_0$ is the external number density in $\mathrm{cm^{-3}}$. The coefficient depends on model details, but the weak exponents control error propagation: even an order-of-magnitude uncertainty in $E_{\rm iso}$ or $n$ changes $\theta_j$ by only tens of percent. The main risk is misclassification. Chromatic breaks, energy injection, density jumps, and supernova bumps can all be mistaken for a jet break {cite:p}`2003Natur.423..847H,2004ApJ...611.1005G`. GRB 980425/SN 1998bw and GRB 030329/SN 2003dh connected long GRBs with broad-lined Type Ic supernovae. The link between short GRBs and binary neutron-star mergers became direct multi-messenger evidence after GW170817. GRB follow-up systems have to handle extreme time hierarchy: a second-scale high-energy trigger, a minute-scale optical flash, hour-to-day afterglow, week-to-month radio calorimetry, and possible supernova or kilonova emission superposed on top. The event table should preserve not only flux, but also the time base and band connections for each stage. ## TDEs: black-hole mass, fallback rate, and the reprocessing photosphere A tidal disruption event occurs when a star passes close to a supermassive black hole. The tidal radius is ```{math} :label: eq:ch13-tidal-radius r_t\simeq R_\ast\left(\frac{M_\bullet}{M_\ast}\right)^{1/3}, ``` where $M_\bullet$ is black-hole mass, and $M_\ast$ and $R_\ast$ are the stellar mass and radius. The Schwarzschild radius is ```{math} :label: eq:ch13-schwarzschild-radius-tde r_s=\frac{2GM_\bullet}{c^2}. ``` If $r_t\lesssim r_s$, the star may be swallowed before producing a bright flare. Main-sequence TDEs are therefore most common around $M_\bullet\sim10^5$--$10^7M_\odot$, with stronger selection effects near $10^8M_\odot$ {cite:p}`1988Natur.333..523R,2021ARA&A..59...21G`. The earliest bound debris sets the fallback time, ```{math} :label: eq:ch13-tde-fallback-time t_{\rm min}\simeq 41\,{\rm d} \left(\frac{M_\bullet}{10^6M_\odot}\right)^{1/2} \left(\frac{M_\ast}{M_\odot}\right)^{-1} \left(\frac{R_\ast}{R_\odot}\right)^{3/2}. ``` In the idealized limit, the late-time fallback rate is ```{math} :label: eq:ch13-tde-fallback-rate \dot M_{\rm fb}\simeq \frac{M_\ast}{3t_{\rm min}} \left(\frac{t}{t_{\rm min}}\right)^{-5/3}. ``` $\dot M_{\rm fb}$ is often reported in $M_\odot\,\mathrm{yr^{-1}}$. The Eddington accretion rate is ```{math} :label: eq:ch13-eddington-rate \dot M_{\rm Edd} = \frac{L_{\rm Edd}}{\eta c^2} \simeq 0.022 \left(\frac{M_\bullet}{10^6M_\odot}\right) \left(\frac{\eta}{0.1}\right)^{-1} M_\odot\,{\rm yr^{-1}} , ``` where $\eta$ is radiative efficiency. For a $10^6M_\odot$ black hole, the peak fallback of a solar-type star can be strongly super-Eddington. The optical luminosity need not be super-Eddington, however, because circularization, wind, obscuration, and reprocessing redistribute inner-disk energy into UV/optical bands. ```{figure} ../_static/figures/generated/chapter_13/ch13_tde_fallback_reprocessing.png :name: fig:chapter-13-tde :width: 92.0% Fallback and reprocessing in a TDE. The left panel shows a t^−5/3 fallback curve entering the decline on a timescale of tens of days and remaining above the Eddington accretion rate of a 10⁶M_⊙ black hole for an extended period. The right panel shows a typical optical/UV blackbody photosphere: radius contracts from about 10¹⁵ cm to 10¹⁴ cm, while temperature rises from about 10⁴ K to several 10⁴ K. ``` Optical TDE blackbody radii are usually $10^{14}$--$10^{15}\,\mathrm{cm}$, much larger than a few $r_s$ for a $10^6M_\odot$ black hole. Temperatures are commonly several $10^4\,\mathrm{K}$, and the evolution is slower than that of ordinary supernovae. AT2018zr was the first ZTF TDE with good rise-to-peak coverage; the first ZTF detection was about $50\,\mathrm{d}$ before peak. Its optical/UV blackbody temperature was about $1.4\times10^4\,\mathrm{K}$ early and rose above $5\times10^4\,\mathrm{K}$ later, while the blackbody radius shrank from $10^{15.1}\,\mathrm{cm}$ to below $10^{14}\,\mathrm{cm}$. The X-ray luminosity was orders of magnitude below the simultaneous optical/UV blackbody luminosity, showing that obscuration or reprocessing cannot be ignored {cite:p}`2019ApJ...872..198V,2021ARA&A..59...21G`. Jetted TDEs such as Swift J1644+57 push the event table toward high energies and rapid variability: the X-ray light curve can flicker strongly, while radio emission traces the outflow interacting with the external medium {cite:p}`2011Sci...333..203B`. The multi-messenger association of IceCube-170922A with a blazar belongs to another source class; TDEs also have candidate associations with high-energy neutrinos {cite:p}`2021NatAs...5..510S`. Even for long-timescale transients, the event table should retain the full time information. Deciding whether a high-energy event belongs to an optical flare requires sky position, time delay, energy, directional uncertainty, and source evolution to be mutually consistent. ## Error budget for target-of-opportunity observations Transient observing is often won or lost by response time. A practical condition is ```{math} :label: eq:ch13-response-budget t_{\rm alert}+t_{\rm slew}+t_{\rm acq}+t_{\rm exp} < f_{\rm samp}\,t_{\rm evol}, ``` where $t_{\rm alert}$ is trigger-distribution latency, $t_{\rm slew}$ is telescope slew time, $t_{\rm acq}$ is target confirmation and guiding, $t_{\rm exp}$ is the first useful exposure block, and $t_{\rm evol}$ is the source-evolution timescale. To resolve a rising phase, $f_{\rm samp}\sim0.1$ is a sensible scale. A GRB optical flash can have $t_{\rm evol}<10^3\,\mathrm{s}$; the blue phase of a gravitational-wave counterpart is about one day; early radio or spectroscopic structure in a nova can change over days to weeks; TDE rise times are often tens of days. For ordinary imaging, the signal-to-noise ratio is ```{math} :label: eq:ch13-imaging-snr {\rm SNR}= \frac{N_s} {\left(N_s+N_b+n_{\rm pix}\sigma_{\rm read}^2\right)^{1/2}}, ``` where $N_s$ is source counts, $N_b$ is background counts, $n_{\rm pix}$ is the number of pixels in the aperture, and $\sigma_{\rm read}$ is the read noise per pixel. Intensity interferometry or photon-pair statistics also needs the number of pairs: ```{math} :label: eq:ch13-pair-statistics N_{\rm pair}\simeq r_1 r_2\,\Delta t\,T, \qquad \delta C\simeq N_{\rm pair}^{-1/2}, ``` where $r_1$ and $r_2$ are the event rates in two telescopes, $\Delta t$ is the correlation window, and $T$ is integration time. If $r_1=r_2=10^6\,\mathrm{s^{-1}}$, $\Delta t=100\,\mathrm{ps}$, and $T=1\,\mathrm{h}$, then $N_{\rm pair}\sim3.6\times10^5$, giving a pure Poisson correlation error of about $1.7\times10^{-3}$. That is only the statistical floor. Real errors also include time synchronization, dead time, spectral bandwidth, polarization leakage, and changing background. Host association has its own scale. If the transient localization has error radius $r$ and the background galaxy surface density to depth $m$ is $\Sigma(