Transient stars sometimes appear on the sky. In this course, we will discuss ``transient physics,'' the physics we require to understand known types of transients – novae, kilonovae, supernovae, hypernovae, X-ray and $γ$-ray bursts – and their aftermath. It sets the stage for the companion mini-course where we will compare predicted behaviour with observations. We will also discuss ``predicted transients'' – both those we know must exist and those that seem not impossible (and therefore may well exist – ``everything not forbidden in compulsory'').

Two possible definitions of an ``interesting'' transient:

1. large (>100?) increase in brightness, lasts much less than human lifetime, and recurs infrequently;
2. A significant fraction (>1%?) of the total energy available is used in a short time.

One may also want to distinguish between disruption (one-off) and eruption (can repeat).

Mass conservation and hydrostatic equilibrium \[ \frac{dM_r}{dr} = 4\pi{}r^{2}\rho \qquad \Rightarrow\qquad \rho\propto\frac{M}{R^3}, \] \[ \frac{dP}{dr} = \frac{GM_{r}\rho}{r^2}\qquad \Rightarrow\qquad P\propto\frac{M^2}{R^4}. \] Dynamical timescale \[ t_{\rm dyn} = \frac{1}{\sqrt{G\rho}}. \]

For a star to be stable to a density perturbation (upon, e.g., compression), pressure has to increase faster than gravity, i.e., faster than \(P\propto{}R^{-4}\).

For a polytrope, \(P\propto\rho^{\gamma}\propto R^{-3\gamma}\), with \(\gamma\equiv{}C_P/C_V\) the adiabatic index (where \(C_P\) and \(C_V\) are the specific heats at constant volume and pressure, resp.). In terms of the degrees of freedom \(f\) of the gas particles, one has \(\gamma=1+2/f\). Hence, for a monatomic ideal gas, one has \(f=3\) for the three spatial motions and \(\gamma=5/3\), for a diatomic one at low enough temperature for vibration not to be important, but the two rotations to be excited, \(f=5\) and \(\gamma=7/5\), and at higher temperature \(f=6\) and \(\gamma=4/3\). One also has \(\gamma=4/3\) for radiation-pressure dominated gas. For degenerate gasses, \(\gamma=5/3\) for the non-relativistic case, \(\gamma=4/3\) for the extremely relativistic case.

Thus, for realistic cases, stars are stable, though they are closer to instability if they are composed of molecules, are radiation dominated, or relativistic. The way to get \(\gamma\) to drop, is ionisation (in its most general form), i.e., let part of the work done compressing matter go into ionisation rather than increasing the kinetic energy of the constituent particles. Examples: molecular dissociation, ionisation, pair creation, and nuclear dissociation. Capture of energetic electrons on protons (to form neutrons and neutrinos, with the latter escaping) has a similar effect.

For a star to be stable to a temperature perturbation, upon, e.g., an increase in fusion rate, the star has to expand and cool, i.e., have negative gravothermal specific heat (see KW, Ch. 25). For an ideal gas, the virial theorem shows this is the case; for a degenerate gas, though, \(E_{\rm kin}\) (like pressure) does not depend on temperature, and thus fusion is unstable.

Note that if cooling increases with increasing temperature (like for neutrino cooling, but unlike radiative losses), the situation reverses: this is unstable for an ideal gas, stable for a degenerate one.

Another way to understand the stability is by considering the increase in fusion in steps: first, one adds some energy and thus (for a non-degenerate gas) increases the temperature and pressure in some small central region with radius \(r\). Then, the star will expand until the pressure in the central region and that required to support the rest of the star match. For homologous expansion with a fractional change in radius \(\delta{}R\), one gets a final fractional change in pressure \(\delta{}P/P=-4\delta{}R/R=-4\delta{}r/r\). Furthermore, since the mass of the inner region of \(m=\frac{4}{3}\pi{}r^3\) is constant, the density will have changed as \(\delta{}\rho/\rho=-3\delta{}r/r\). Thus, in the inner region, \[ \frac{\delta{}T}{T}=\frac{\delta{}P}{P}-\frac{\delta\rho}{\rho} = -\frac{\delta{}r}{r}. \] Hence, the temperature has decreased relative to the initial state, and thus fusion in cores is stable.

Now consider nuclear fusion in a thin shell around a core with radius \(r\) with thickness \(D\ll{}r\) (think why shells naturally will tend to become thin). Again assume a small increase in energy and thus in pressure and temperature. This will cause expansion of the part of the star above the shell (but not of the core), i.e., some change in radius \(\delta{}R\) until hydrostatic equilibrium has been regained. In the new equilibrium, the pressure that the shell needs to provide will then have been reduced by \(\delta{}P/P=-4\delta{}R/R=-4\delta{}r/r=-4\delta{}D/r\). The difference with the above, however, is the behaviour of the density: the shell has mass \(m\simeq4\pi{}r^{2}D\) and can expand in only one direction, and hence after expansion \(\delta\rho=-\delta{}D/D\). Thus, for a thin shell, the density decreases much faster than the pressure (as, e.g., even a doubling of the thickness of the shell hardly changes the size of the star and thus hardly affects the weight of the layers above). For the temperature, one finds that in the new equilibrium, \[ \frac{\delta{}T}{T}=\frac{\delta{}P}{P}-\frac{\delta\rho}{\rho} = -4\frac{\delta{}D}{r}+\frac{\delta{}D}{D}=\left(1-4\frac{D}{r}\right)\frac{\delta{}D}{D}. \] Hence, in the new equilibrium, the temperature will be above the initial one for \(D < 4/4\). (See KW, §33.2 for more detail.) This instability leads to so-called thermal pulses on the asymptotic giant branch.

A molecular cloud is roughly isothermal, i.e., \(P\propto\rho\) and thus \(\gamma=1\). Hence, the initial collapse is unstable until the cloud becomes opague and further contraction is adiabatic (with \(\gamma=7/5\) for H₂). As H₂ is dissociated, and later H ionised, γ drops below \(4/3\) and further collapse occurs.

For (cores of) stars below the Chandrasekhar mass (for their \(\mu_e\)), contraction brings them closer to degeneracy – \(P_{\rm ideal}=P_{\rm deg}\) occurs at a line with slope \(T\propto\rho^{2/3}\) since for non-relativistic degenerate gas one has \(P_{\rm deg}\propto\rho^{5/3}\). Once a star becomes degenerate, further loss of energy does not change the pressure or density, but just cools the object.

Above the Chandrasekhar mass, contraction continues to lead to increased heating (see track C in Fig. 1). The separation corresponds to a maximum temperature reachable by a degenerate object of \(1.3\times10^{9}{\rm\,K}\) (for \(\mu_e=2\); see Arnett, §6.1, and Fig. 2).

Below the Chandrasekhar mass, the maximum temperature along a track that a star can reach scales with \(M^{4/3}\), and thus for each fusion stage there is a minimum mass – the ignition mass – that determines whether a given (core of a) star can reach it. This is \(\sim\!0.08\,M_\odot\) for hydrogen, \(\sim\!0.3\,M_\odot\) for helium, \(\sim\!0.8\,M_\odot\) for carbon, and \(\sim\!1.36\,M_\odot\) for neon (Nomoto et al. 1984ApJ...277..791N). All heavier fuels are only burnt by stars (or cores of stars) above the Chandrasekhar mass.

In practice, for the helium cores of single stars to contract and ignite directly, requires a main-sequence mass above \(\sim\!2\,M_\odot\) (see Fig. 3), while stars with masses above \(\sim7.5\,M_\odot\) ignite C directly, and stars above \(\sim8\,M_\odot\) ignite all further stages (latter two numbers uncertain).

For reference, the burning stages are (from Arnett, his table 6.2):

• Hydrogen burning (\(2\times10^7{\rm\,K}\), \((5\ldots8)\times10^{18}{\rm\,erg\,g^{-1}}\)): mostly helium (via p-p or CNO; all CNO converted to N).
• Helium burning (\(1.5\times10^8{\rm\,K}\), \(7\times10^{17}{\rm\,erg\,g^{-1}}\)): mostly carbon and oxygen (triple alpha, plus \({}^{12}{\rm C}(\alpha,\gamma){}^{16}{\rm O}\), with final abundance ratio strongly dependent on rate, and on extent to which fresh helium is brought in at late stages (which would lead to more oxygen).
• Carbon burning (\(8\times10^8{\rm\,K}\), \(5\times10^{17}{\rm\,erg\,g^{-1}}\)): mainly O, Ne, Mg, Si.
• Neon burning (\(1.5\times10^9{\rm\,K}\), \(1.1\times10^{17}{\rm\,erg\,g^{-1}}\)): mainly O, Mg. Starts with photo-desintegration: \({}^{20}{\rm Ne}(\gamma,\alpha){}^{16}{\rm O}\), followed by \({}^{20}{\rm Ne}(\alpha,\gamma){}^{24}{\rm Mg}\).
• Oxygen burning (\(2\times10^9{\rm\,K}\), \(5\times10^{17}{\rm\,erg\,g^{-1}}\)): mainly Si, S.
• Silicon burning (\(3.5\times10^9{\rm\,K}\), \((0\ldots3)\times10^{17}{\rm\,erg\,g^{-1}}\)): iron-peak.

In a shell, the density and pressure drop very quickly, and hence their properties depend mostly on the properties of the core. Analogously to main-sequence stars, one can use homology arguments to show how \(\rho\), \(P\), \(T\), and \(L\) scale with \(M_{\rm core}\) and \(R_{\rm core}\). For instance, for the pressure, \[ P \approx \frac{GM_{\rm core}}{R_{\rm core}}\int_{\rm shell}\rho{}dr \approx \frac{GM_{\rm core}}{R_{\rm core}}\rho{}H, \] where homology requires that the scaleheight \(H\propto{}R_{\rm core}\). Here, calculating \(\rho\) requires information on the fusion process and the opacities (for details, see KW, §32.2), but this drops out for the temperature, for which, assuming an ideal gas, one finds \[ T \propto \frac{P}{\rho} \propto \frac{1}{\rho}\frac{GM_{\rm core}}{R_{\rm core}^2}\rho H \propto \frac{M_{\rm core}}{R_{\rm core}}. \] Since fusion is generally a steep function of temperature, this means the luminosity will depend sensitively on the core properties. As the envelopes of red giants are mostly convective, and the photospheres have roughly fixed effective temperature, the radius is also a strong function of core mass.

The degenerate core will be heated by its surrounding shell, the temperature of which will increase as the core grows in mass and shrinks in size. Furthermore, the contraction will release energy as well. Since degenerate matter conducts efficiently, the core will be nearly isothermal, with a slightly lower temperature in the centre due to neutrino losses (which are stronger at higher density).

When \(M_{\rm core}\simeq0.45\,M_\odot\), the temperature becomes hot enough for ignition, and runaway fusion starts off-centre. This region is only moderately degenerate, which limits the maximum temperature and luminosity reached and thus avoids a dynamic event (timescales for entropy increase always remain larger than the dynamical timescale). In the end, the star settles down as a core-helium burning giant, with most of the luminosity still due to the hydrogen shell. See Fig. 4 for an overview. We see such sources as red-clump stars in metal-rich populations, and horizontal-branch stars in metal-poor ones.

When two or more shells are present, they do not necessarily evolve at the same rate, leading to changing separation (in mass coordinates), and to thin shells that are geometrically unstable (see above). This underlies the thermal pulses on the asymptotic giant branch (see Fig. 5).

Absent mass loss, for a sufficiently massive star, eventually the core density and temperature would increase sufficiently to ignite carbon. This happens as one approaches the Chandrasekhar mass, and in the core, which has a lower ignition temperature due to its very high density (note that the competition between neutrino cooling and compressional heating also makes the temperature distribution less uniform than was the case for a helium core). As one is at very high degeneracy, the run-away destroys the whole core (and surrounding envelope). In reality, this does not happen as the luminosity becomes so high that mass is lost in the AGB super wind.

The local heating time is given by: \[ t_{\rm h,local} = \frac{c_{P}T}{\epsilon}. \] The convection zone, however, will carry away heat to the surroundings, and the temperature will thus increases on a slower timescale, of \[ t_{\rm h, cvz} = \frac{\int_{\rm cvz} c_{P}T dm}{\int \epsilon dm} \approx \frac{M_{\rm cvz}}{M_{\rm fusion}} t_{\rm h, local}, \] where \(dm=4\pi{}r^{2}\rho{}dr\) and \(\int\epsilon{}dm\) is the luminosity \(L\). Given the strong temperature-dependence of fusion, the ratio of masses is typically large, at least ten.

A fusing region will truly run away either when the convection can no longer carry away the heat sufficiently efficiently, or when a blob passing through the burning region increases its heat faster than it is cooled by rising to a region of lower pressure. In non-degenerate matter, these two conditions give about the same answer, that this happens when the convective velocities approach the velocity of sound, but for degenerate matter where the velocities are lower, the second condition is satisfied first. This happens when along the path, the integral \[ \int_{\rm path} \left[\left.\frac{dT}{dr}\right|_{\rm ad} + \frac{\epsilon}{c_{P}v_{\rm conv}}\right] dr, \] diverges. Approximating crudely, this happens when \[ \frac{\Delta{}T}{\Delta{}R} < \frac{\epsilon}{c_{P}v_{\rm conv}} \qquad \Rightarrow \qquad \frac{c_{P}\Delta{}T}{\epsilon} < \frac{\Delta{}R}{v_{\rm conv}}. \] where \(\Delta{}R\) is the size of the burning region and \(\Delta{}T\) the associated temperature difference. The first term is similar to the local heating time, and the second is the crossing time through the burning region. Hence, a sufficient condition is that \[ t_{\rm h, local} < t_{\rm cross}. \] In practice, run-away will happen earlier, since there will be a distribution of bubbles, some of which will already be a little hotter or move more slowly. For details, see Woosley et al. 2004ApJ...607..921W.

Once small region runs away, it initially keeps on rising. It just expands as it is heated until \(t_{\rm heat} > t_{\rm dyn}\), when a flame is born. Its overpressure will not be that high: at \(\rho=10^{9}{\rm\,g/cm^3}\), the Fermi energy is \(\epsilon_F=cp_F=ch(3\rho/(4\pi{}g\mu_e))^{1/3}\simeq4{\rm\,MeV}\), while the burning releases about 1 MeV/nucleon, or about 2 MeV/electron. Hence, at least initially a deflagration front is born, in which the now very hot bubble will ignite neighbouring material by thermal conduction. The speed at which the burning front proceeds is sub-sonic.

To trigger a detonation does not appear to be easy. One way is to have a temperature gradient in which the phase speed at which neighbouring regions run away, \(v_{\rm phase}=(d\tau/dr)^{-1}\), is comparable to the sound speed. For a nice description, see Seitenzahl et al. 2009ApJ...696..515S.

If reactions are sufficiently fast, as in a detonation, there is no time for weak reactions and thus proton and neutron number are conserved separately. Hence, the initial \(\rm^{12}_{6}C\) and \(\rm^{16}_{8}O\) end up in the most stable element with equal numbers of protons and neutrons, \(\rm^{56}_{28}Ni\). Only subsequently does this decay, via \[ \rm^{56}_{28}Ni \rightarrow \rm^{56}_{27}Co + e^{+} + \nu_e \qquad (\tau_{1/2}=6.077~d) \] \[ \rm^{56}_{27}Co \rightarrow \rm^{56}_{26}Fe + e^{+} + \nu_e \qquad (\tau_{1/2}=77.27~d) \] These decays power the lightcurves of most supernovae. Note that there are also gamma rays emitted, as the decays leave the daughter nuclei in excited states.

In contrast, in a slow deflagration, there is time for electon captures (remember that the Fermi energy is very high) and hence in the densest regions one forms stable elements with one proton turned into a neutron, in particular \(\rm^{58}_{28}Ni\) and \(\rm^{54}_{26}Fe\).

For a detonation, all one has is the excess number of neutrons initially present. This is turn depends on the amount of C, N, and O, that the star initially had, as one can see from considering the H and He burning cycles:

H burning: all H to He, (essentially) all C, N, O to \(\rm^{14}_{7}N\) (since the proton capture on \(\rm^{14}_{7}N\) is the slowest step).

He burning: all He to C, O; but also \[ \rm^{14}_{7}N(\alpha,\gamma)^{18}_{9}F(e^{-},\nu)^{18}_{8}O(\alpha,\gamma)^{22}_{10}Ne \] Here, the electron capture (or inverse β decay) of fluorine produces an extra neutron. During a detonation, the \(\rm^{22}_{10}Ne\) would be turned into \(\rm^{58}_{28}Ni\) and \(\rm^{54}_{26}Fe\). Hence, a prediction of detonating little-processed white dwarfs is that the amount of stable nickel should be proportional to the initial metallicity.

In detail, the neutronisation is also affected by the simmering. See, e.g., Chamulak et al. 2008ApJ...677..160C.