X-ray burstsLast time we talked about one of the major differences between NS and BH: NS havestrong magnetic fields. That means that “hot spots” can be produced near the magneticpoles, leading to pulsations (this general statement is true for both rotation-powered andaccretion-powered pulsars). Now we’ll talk about one manifestation of another crucialdifference: NS have surfaces, unlike black holes.So far we’ve talked about two sources of energy associated with neutron stars: rotationand accretion. Suppose we have a neutron star that is accreting hydrogen and helium froma companion. Ask class: what is another potential source of energy, given that the surfacetemperature is typically 107K? Since the accreted matter is hydrogen and helium, at suchhigh temperatures nuclear fusion may progress. Deeper down, where the temperature anddensity are even higher, fusion will definitely happen. Therefore, nuclear burning is anothersource of energy.Ask class: if nuclear burning happens continuously during accretion, would it bevisible as another source of energy? No. The reason is that nuclear burning is much lessefficient than accretion onto a neutron star. Conversion of hydrogen into helium releases afraction ≈ 7 × 10−3of the mass-energy, whereas accretion onto the stellar surface releases≈ 0.2 of the mass-energy, and the uncertainty in the accretion efficiency is a lot more than7 × 10−3! Therefore, if burning happens continuously, it will never be noticed.Moving now to the observational story, starting in 1974 there were a number ofdetections of bursts of X-rays from different NS LMXBs. The properties varied, buttypically the burst would last about 10 seconds, with a r ise time that was often less thanone second. The time between bursts was anywhere between hours and weeks, dependingon the source. In addition, if one defined the parameter α as the ratio of the time-integratedenergy in the persistent emission to the time-integrated energy in bursts, the ratio wasα ∼ 20 − 300. Ask class: what could cause this? The total energy release in the bursts isconsistent with nuclear bu rning. It was t herefore suggested that here was an example ofunstable nuclear burning; somehow, the fuel is stored up until some critical moment, thenit all burns at once. If the bur ning happens rapidly, then both the rise time and the typicalduration can be explained by the radiative transfer time necessary to get, respectively, thefirst photons out and the last photons out. Some bursts are longer: up to hundreds ofseconds. This is also consistent with the nuclear flash interpretation, if one assumes that itis hydrogen that is being burned (there are weak decays involved, which slows down theprocess).For relatively dim persistent sources, the luminosity can go up by a factor of ∼ 100during a burst. The luminosity is, however, limited by the Eddington luminosity. If theluminosity is much higher than that, it creates a wind of matter that streams out andobscures the burst. This is the explanation (first proposed informally by Fred Lamb) forwhy it is that in high energy photon bins many bursts seem to be “double-peaked”. That is,the flux rises, but when it gets high enough to eject matter, the effective radiative surfacearea increases dramatically (because the surface of optical depth unity is now at a muchlarger radius). As a result, the emission is much cooling, so although the total luminosityis still high, in the higher energies it decreases. When the luminosity goes well belowEddington, the atmosphere settles down and the high-energy emission increases.The preceding discussion indicates th at a wind of matter can be ejected during a burst.Ask class: considering the bulk of th e matter involved in the burning, when there isa flash do you expect most of the matter to be ejected to infinity or to be bound fairlyclosely to the surface? The majority of the matter stays close to the surface, because theburst does not release enough energy to eject more than a small fraction to infinity (thisis one consequence of the gravitational binding energy being much larger than the nuclearbinding energy). Ask class: so what would one expect from the same situation on a whitedwarf, where there is a nuclear burst? In that case, the radius is 1000x larger than a NS,so the ratio is reversed: nuclear ener gy dominates over gravitational energy. Therefore,in such a burst one expects the entire envelope to be ejected. This is what is thought tohappen in classical novae. When the burning starts, the matter is lifted away from thesurface. This decreases both the density and the temperature, so the rate of burning dropsdramatically. This also means that in novae, unlike in X-ray bursts, burning is thought tobe very incomplete. In X-ray bursts, the matter all burns to completion.Instability and different burning regimesLet’s now look more closely at the instability itself. Ask class: in general, what doesit mean that a system is unstable? It means that a slight perturbation will cause thesystem to change its state dramatically. Ask class:, so what does it mean for the nuclearburning to be unstable? It means that if there is a slight perturbation, the r ate of nuclearburning will change quickly. In particular, if for some reason the temperature goes up, thenfor instability the nuclear burning rate must rise by enough that the temperature goes upfurther, leading to a runaway process.This means, first of all, that the rate of nuclear burning must be sensitive to temperature.This is not always the case. Generically, nuclear burning requires that the wavefunctionsof nuclei overlap enough for a reaction to happen. In normal thermonuclear burning (suchas in stars), this happens because there is a small fraction of nuclei with enough energyto penetrate the Coulomb barrier. L et’s treat this classically first. Specifically, imaginethat there is some barrier energy Ebthat must be overcome for the reaction to occur. Ifthe temperature is T , then a fraction ∼ exp(−Eb/kT ) of the nuclei will have the requiredenergy. If kT ≪ Eb, this reaction is therefore extremely sensitive to the temperature, sinceexp(−Eb/kT ) goes up dramatically when T increases. In reality, the process is not classical,but quantum, and the actual reaction rate is dominated by quantum tunneling. In a veryrough way, the effect is to lower the barrier energy (that is, a nucleus can react with amuch lower energy than one would have thought