Transient stars sometimes appear on the sky. In this course, we will discuss ``transient observables,'' the mechanisms by which transients release the radiation we observe. It is a follow up of the mini-course on the transient physics required to understand known types of transients – novae, kilonovae, supernovae, hypernovae, X-ray and $γ$-ray bursts – and their aftermath.

For cases where mass is ejected due to strong energy deposition (dealt with in previous mini-course), the evolution usually includes the following stages:

1. Shock break-out;
2. Expansion (and thus cooling) with radiative losses (what we see!), possibly compensated with heating, e.g., by radioactive decay, recombination, a possible central power source, and/or internal shocks;
3. Interaction with surrounding matter (and thus reheating), e.g., a companion, left-over accretion disks, previously ejected shells, or simply the interstellar medium.

Exceptions to the above include transients where matter is confined. E.g.,

• Accretion disk instabilities (mostly confined gravitationally);
• X-ray bursts (idem);
• Magnetar flare afterglows (pair plasma confined magnetically).

At maximum brightness, a supernova has a typical luminosity of \(\sim\!10^{11}\,L_\odot\) and temperatures of \(\sim\!10^{4}{\rm\,K}\). Thus the effective radius of the emitting region is \(\sim\!(10^{5}/4)\,R_{\odot}\simeq2\times10^{15}{\rm\,cm}\simeq100{\rm\,AU}\).

The typical velocities are \(2\) to \(10{\rm\,Mm/s}\), implying large kinetic energy \(\sim1{\rm\,B}\,(M/M_\odot)(v/10{\rm\,Mm/s})^{2}\) (recall, \(1{\rm\,B}\equiv10^{51}{\rm\,erg}\)). It also implies an expansion time of \(\sim\!2\times10^{6}{\rm\,s}\), similar to the observed durations.

For this duration, the total radiated energy is \(\sim\!0.1{\rm\,B}\), i.e., large, but quite a bit smaller than the kinetic energy.

Since the radius is substantially larger than even the largest possible progenitors (say a red supergiant with a radius of \(\sim\!10^{3}\,R_{\odot}\)), the gas must have cooled quite a bit. Indeed, for a small progenitor such as a white dwarf, essentially no thermal energy would be left: thus there must be a longer-lived power source such as radio-active decay.

More generally, one can conclude that to obtain a transient as bright as a supernova, one needs to either

1. Start from a large star;
2. Have an internal energy source (lasting for of order weeks); or
3. Have external matter to interact with (at \(\sim\!100{\rm\,AU}\)).

Given the timescales, without heating, one would expect a transient to be at its brightest when the diffusion and expansion times match. With heating, as it becomes transparent, the luminosity will start to track the input heating rate.

With just diffusion and heating, one has \[ \frac{\frac{d}{dt}(TR)^{4}}{(TR)^{4}} = \frac{\dot\phi}{\phi}=\frac{1}{\tau_{\rm heat}} -\frac{1}{\tau_{\rm diff}} \qquad \Leftrightarrow \qquad \frac{\dot\phi}{\phi} = \left[\frac{\epsilon/\epsilon_{0}}{\tau_{\rm heat,0}\phi} -\frac{1}{\tau_{\rm diff,0}}\right]\frac{R}{R_0}, \] where we tried to write in terms of ratios on the right-hand side, with \(\epsilon/\epsilon_0\) capturing the time dependence of the heating process (and where we again implicitly assumed constant opacity).

Ignoring heating, an analytic solution is possible. Using that \(\tau_{\rm diff}=\tau_{\rm diff,0}(R_0/R)=\tau_{\rm diff}/(1+v_{\rm sc}t/R_0)\), and defining an expansion timescale \(\tau_{\rm exp}=R/v_{\rm sc}\), one finds \[ \phi = \exp\left(-\frac{t}{\tau_{\rm diff,0}} -\frac{t^2}{2\tau_{\rm exp}\tau_{\rm diff,0}}\right). \] Generally, \(\tau_{\rm diff,0}\gg\tau_{\rm exp}\), and thus for \(t>\tau_{\rm exp}\), the lightcurve is essentially a Gaussian, with a timescale that is the geometric mean of the expansion and diffusion times scales, \(\tau_{\rm lc}=\sqrt{\tau_{\rm exp}\tau_{\rm diff,0}} \propto \sqrt{\kappa M/v_{\rm sc}}\). Slower, more massive ejections lead to longer transients.

Including heating, the integration needs to be done numerically. However, generally, one expects maximum to occur when \(\dot\phi=0\), i.e., when \(1/\tau_{\rm heat}=1/\tau_{\rm diff}\) (of course, if heating is too small, this maximum after explosion never happens). From their definitions, the timescales match when \(L=\epsilon M\). Thus, maximum luminosity gives a measure of the total amount of radioactive decay – ``Arnett's rule.'' (This will be an underestimate if the opacity is decreasing with time – or if this is happening effectively due to recombination.)

At some temperature \(T_i\), material will recombine and become essentially transparent. If this happens inside the cloud, then this will effectively be at optical depth zero, and the photosphere would be at \(T_{\rm eff}^{4}\simeq2T_i^{4}\). As more matter recombines, the photosphere will move in, with recombination and advection (``freed'' radiation) giving additional sources of luminosity. At this time, one will have, \[ L_{\rm diff}+L_{\rm adv}+L_{\rm rec} = L_{\rm min} = 4\pi R_i^{2} \sigma 2T_i^{4}, \] where \(R_{i}=x_{i} R\) is the radius of the recombination front, and where we used the subscript ``min'' as a reminder that the luminosity cannot be lower than this value for this radius.

The luminosity due to recombination is \[ L_{\rm rec} = -4\pi R_i^2 \dot R_i\rho Q = -3x_i^{2}\dot x_{i}\frac{4\pi}{3}R^{3} \rho Q = -3x_i^{2}\dot x_{i} MQ, \] where \(Q\) is the energy release per unit mass due to recombination.

For the advection and diffusion terms, the results depend on whether the front moves slow or fast compared to the time to adjust the overall temperature structure. Generally, though, \(L_{\rm diff} = {\cal E}/\tau_{\rm diff}\) and, \[ L_{\rm adv} = -\dot x_i \frac{\partial{\cal E}}{\partial x_i} \] but the total thermal energy \({\cal E}\) and diffusion timescale \(\tau_{\rm diff}\) may now depend on \(x_i\). In consequence, not only the differential equation for \(\phi\) has to be solved, but also one for the recombination front position \(x_i\). The latter can be derived from the constraint that the additional luminosity \(L_{\rm rec}+L_{\rm adv}\) has to match the excess luminosity \(L_{\rm min}-L_{\rm diff}\), or \[ -\dot x_{i}\left[3x_{i}^{2} MQ + \frac{\partial\cal E}{\partial x_{i}}\right] = 4\pi R^2 x_i^2 2\sigma T_i^4 - \frac{\cal E}{\tau_{\rm diff}} \] Below, we will also use the timescale on which the initial energy would be radiated at an effective temperature of \(2^{1/4}T_i\), \[ \tau_{\rm i,0}\equiv\frac{{\cal E}_0}{L_{\rm min,0}} = \frac{\frac{4}{\pi}R_0^{3}aT_0^{4}}{4\pi R_0^{2}2\frac{ac}{4}T_i^4} = \frac{4R_0}{\pi^2 c}\frac{T_0^4}{2T_i^4}. \]

A full self-similar solution is possible for this phase, but the expected scaling can be seen from a one-zone model and energy conservation: \[ \frac{1}{2} M_{\rm tot} v^{2}_{\rm s} = E_{0}, \] where \(M_{\rm tot}\) is the total mass and \(v_{\rm s}=dR_{\rm s}/dt\) the shock velocity. Assuming the mass is dominated by swept up material with constant density \(\rho\), \[ M_{\rm tot} = \frac{4\pi}{3} R^{3}_{\rm s} \rho, \] one finds, \[ \frac{1}{2}\frac{4\pi}{3}\rho R^{3}_{\rm s} \left(\frac{dR_{\rm s}}{dt}\right)^2 = E_0, \] which can be integrated to give, \[ R_{\rm s} = \left(\frac{75}{8\pi}\frac{E_{0}}{\rho}\right)^{1/2} t^{2/5}. \] A more detailed calculation gives \(R_{\rm s}\simeq(2.026E_{0}/\rho)^{1/5}t^{2/5}\). For the shock velocity, one finds, \[ v_{\rm s} = \frac{2R}{5t}. \]

Assuming the energy from the shock is immediately radiated away, all that happens is that matter is slowed down as momentum is shared, i.e., \(M_{\rm tot}v_{\rm s}\) is constant. Inserting \(M_{\rm tot}\) as above, one finds, \[ R_{\rm s} \propto t^{1/4}, \] where the constant of proportionality depends on when the transition from the Sedov-Taylor solution happens.