I always hear the narrator of documentaries say that a star exploded because it ran out of fuel. Usually things explode when they have too much fuel, not when they run out of fuel. Please explain...

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    $\begingroup$ A (large enough) star has a lot of matter. Gravity tries to pull all this matter together at the center of mass, so something needs to push back. For a star the fusion process in the core producing light is the one pushing back. At one point the star runs out of fuel and the "push out" vanishes so everything collapses into the center very rapidly. Then it explodes. $\endgroup$ – Thorbjørn Ravn Andersen May 3 '17 at 9:41
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    $\begingroup$ @ThorbjørnRavnAndersen A key point is that it isn't everything collapsing. If it were then the released gravitational potential energy would be insufficient even to reverse the collapse, let along cause an explosion. Only the core collapses. The envelope remains blissfully unaware of the collapse until it is blown into space. $\endgroup$ – Rob Jeffries May 3 '17 at 12:02
  • $\begingroup$ Are "answers as a comment" allowed on this SE? $\endgroup$ – dav1dsm1th May 3 '17 at 17:49
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    $\begingroup$ @dav1dsm1th No, it is not allowed on any SE. However, it is a fairly common practice; not everyone has the time to write up a full fledged answer, so they jot down whatever they can and hope that someone can come along to flesh it out into a full answer. $\endgroup$ – Setsu May 3 '17 at 18:20
  • $\begingroup$ @Setsu Good to hear. Hopefully these comments will get cleaned up at some point (including my noise). $\endgroup$ – dav1dsm1th May 3 '17 at 21:21

Short answer:

A tiny fraction of the gravitational potential energy released by the very rapid collapse of the inert iron core gets transferred to the outer layers and this is sufficient to power the observed explosion.

In more detail:

Consider the energetics of an idealised model star. It has a "core" of mass $M$ and initial radius $R_0$ and an outer envelope of mass $m$ and radius $r$.

Now suppose the core collapses to a much smaller radius $R \ll R_0$ on such a short timescale that it decouples from the envelope. The amount of gravitational potential energy released will​ be $\sim GM^2/R$.

A fraction of this released energy can be transferred to the envelope in the form of outward moving shocks and radiation. If the transferred energy exceeds the gravitational binding energy of the envelope $\sim Gm^2/r$ then the envelope can be blown into space.

In an exploding star (a type II core collapse supernovae) $R_0\sim 10^4$ km, $R\sim 10$ km and $r \sim 10^8$ km. The core mass is $M \sim 1.2M_{\odot}$ and the envelope mass is $m \sim 10M_{\odot}$. The dense core is mostly made of iron and supported by electron degeneracy pressure. The star is said to have "run out of fuel" because fusion reactions with iron nuclei do not release significant amounts of energy.

The collapse is triggered because nuclear burning continues around the core and so the core mass is gradually increased and as it does so it gradually shrinks (a peculiarity of structures supported by degeneracy pressure), the density increases and then an instability is introduced either by electron capture reactions or photodisintegration of iron nuclei. Either way, electrons (which are what is providing the support for the core) are mopped up by protons to form neutrons and the core collapses on a free fall timescale of $\sim 1$ s!

The collapse is halted by the strong nuclear force and neutron degeneracy pressure. The core bounces; a shock wave travels outwards; most of the gravitational energy is stored in neutrinos and a fraction of this is transferred to the shock before the neutrinos escape, driving away the outer envelope. An excellent descriptive account of this and the previous paragraph can be read in Woosley & Janka (2005).

Putting in some numbers. $$GM^2/R = 4\times 10^{46}\ {\rm J}$$ $$Gm^2/r = 3\times 10^{44}\ {\rm J}$$

So one only needs to transfer of order 1% of the collapsing core's released potential energy to the envelope in order to drive the supernova explosion. This is actually not yet understood in detail, though somehow supernovae find a way to do it.

A key point is that the rapid collapse takes place only in the core of the star. If the entire star collapsed as one, then most of the gravitational potential energy would escape as radiation and neutrinos and there would be insufficient energy even to reverse the collapse. In the core collapse model, most (90%+) of the released gravitational energy is lost as neutrinos, but what remains is still easily sufficient to unbind the uncollapsed envelope. The collapsed core remains bound and becomes either a neutron star or black hole.

A second way to cause a star (a white dwarf) to explode is a thermonuclear reaction. If the carbon and oxygen can be ignited in nuclear fusion reactions then enough energy is released to exceed the gravitational binding energy of the white dwarf. These are type Ia supernovae.

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    $\begingroup$ It's worth noting that models of core collapse supernovae have generally failed to consistently produce supernovas. In simulations the shock usually stalls, and even when this does not happen, simulations usually have difficulty matching the observed luminosities. The introduction to this paper presents a good introduction to some of the difficulties in the field: adsabs.harvard.edu/abs/2012ApJ...746..106P $\endgroup$ – J. O'Brien Antognini May 3 '17 at 18:56
  • $\begingroup$ My question would be broadly why does it explode rather than transition uneventfully as the point of stability wanders through whatever parameter space. Is the key point that when you have enough temperature/density to jam protons and electrons together, that all of a sudden removes what's holding everything up, so it falls, can increase density further, removes more...but then again why isn't that a process that can "slowly" ramp up and maintain some stability? Surely the star doesn't go from no electron captures to all the electron captures? $\endgroup$ – Nick T May 3 '17 at 19:47
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    $\begingroup$ @J.O'BrienAntognini Indeed, models can struggle to work out how to transfer the 1% of energy required - as I alluded to above. But real stars have figured it out and nobody disputes what the source of energy is. $\endgroup$ – Rob Jeffries May 3 '17 at 20:24
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    $\begingroup$ @NickT it is indeed a runaway instability. Electron capture occurs at a threshold density because the degenerate electrons have a distinct, density-dependent maximum energy (they don't have a Maxwellian distribution). This disappearance of electrons reduces the pressure, so the star collapses, increasing the density and hence the maximum energy of the degenerate electrons, allowing more and more of them to participate in neutronisation. The result is total collapse within a second of the onset. $\endgroup$ – Rob Jeffries May 3 '17 at 20:30
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    $\begingroup$ @RobJeffries This is true, although it should also be noted that it could very well be that a substantial fraction of massive stars which have failed supernovae! So while a few stars have certainly figured it out, it is not necessarily the case that they all have! There are some loose constraints which put the fraction of failed supernova at somewhere between 5 and 50%: adsabs.harvard.edu/abs/2016arXiv161002402A $\endgroup$ – J. O'Brien Antognini May 3 '17 at 22:22

To give an answer in more simple turns. (Yes very simplified, but it should introduce the basic concept).

A Star "burns" by nuclear fusion between lighter elements such as Hydrogen turning to Helium. The heat and energy of that burning constantly pushes on the matter inside the star holding it up. The fusing hydrogen generates enough energy to stop it from being able to collapse down into the center.

As the star starts running out of fuel that "fire" gets colder, and the pushing out gets weaker.

Eventually the push isn't enough to keep the star apart and it all rushes back together. That collapse releases a huge amount of energy which causes the explosion.

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    $\begingroup$ "As the star starts running out of fuel that "fire" gets colder, and the pushing out gets weaker." The temperature at the core of a star continues to increase throughout it's life right up until the supernova explosion. $\endgroup$ – Rob Jeffries May 3 '17 at 10:52
  • $\begingroup$ @RobJeffries I don't claim to be an expert but my understanding is that that is due to gravitational collapse releasing potential energy rather than heat from the ongoing fusion? The "fire" has grown colder but other factors are taking over. $\endgroup$ – Tim B May 3 '17 at 11:01
  • $\begingroup$ Conversion of gravitational potential energy into heat is minimal at best. The increase in temperature is actually due to the continued fusion of heavier and heavier elements in the core. For example, read this wiki page. $\endgroup$ – zephyr May 3 '17 at 13:06
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    $\begingroup$ Nice and short, but I would add the term "bounce" to that description as Rob Jeffries did. It is vivid and would finish off your description more nicely than "causes the explosion" $\endgroup$ – Mike Wise May 3 '17 at 16:25
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    $\begingroup$ @TimB Rob is right about the temperature thing. The easiest way to see that it must be so is to note that massive stars go through a series of burning stages each requiring higher temperatures than the last. As a star exhausts the fuel for stage $n$, it collapses and warms until it gets hot enough for stage $n+1$ burning to kick in. It's worth reading about the Virial theorem in this context, because what is happening there is potential energy is being converted to thermal energy. $\endgroup$ – dmckee --- ex-moderator kitten May 3 '17 at 21:06

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