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I am confused with nucleosynthesis inside supernovae. I have read that the heavier elements are made through fusion of lighter element's namely hydrogen and helium.
Does the star "store" all the elements they fuse inside themselves or are they radiated out?
Example, A star fused hydrogen into helium and 2 helium atoms into carbon. Would it keep fusing until it hits the limit and store it inside only to be released as an explosion.
You are correct to say that all the heavier elements in the universe were formed in stars.
Stars like the sun fuse hydrogen into helium. When they get older they can fuse the helium into carbon (it actually takes 3 helium to make one carbon). Larger stars can fuse carbon into oxygen, and neon and elements in the first half of the periodic table.
When the star runs out of fuel, the outer layers are gently ejected in what is called a planetary nebula (though it has directly to do with actual planets). The ejected gas is enriched with the heavier elements that the star has fused. The heavier elements in the planetary nebula mix with the hydrogen and helium gas in outer space and can later go on to be found in new stars. Most of the carbon and oxygen and nitrogen on earth was formed by this process.
Very large stars will fuse all the way up to iron, and then collapse in a supernova. This releases a very large amount of energy, some of which is used to form elements heavier than iron. All the heavier elements (copper gold, uranium for example) are formed in supernovae.
The supernovae explosion distributes these elements back into space, mixed in with the debris from the star, as time goes on, and stars are born and die, the gas in space gets enriched with more and more heavier elements
So the heavier elements are formed in the cores of stars, and are stored there until the star dies, when some of the star's atoms are released into space. The heavier elements are not "radiated out" of the star, until the star dies. Some stars die in a supernova explosian, but most have a more peaceful death in a planetary nebula.
It's worth noting that elements with atomic mass greater than about 140 were likely created in the collision of neutron stars as opposed to supernovae. There is an interesting article on Physics.org and another on the Washington Post. There is also another here.
I too thought that all elements heavier than iron were created in supernova explosions, and when I first heard the theory that elements heavier than gold were likely created in neutron star collisions, I rejected that idea. But after reading these and other articles, I've come to accept it. Apparently, the energy generated in a supernova, while considerable, is not sufficient enough to synthesis the elements heavier than gold.
Edit:I found this interesting article on the Smithsonian Magazine's website. The theory that NS-NS collisions create elements heavier than gold came from data observed from such a collision, which only happen, on average, every 100,000 years. Apparently, this theory better answers questions that the supernova creation of heavy elements cannot. The observed NS-NS collision was observed on June 3, 2013
Does the star "store" all the elements they fuse inside themselves or are they radiated out?
It depends on the star. While the heaviest elements can only be produced (and released) by massive stars that undergo supernova, many other elements are created within relatively smaller stars that become asymptotic giant branch stars. These can undergo cycles of convection that bring the heavy elements in the core to the surface, and then release them in the stellar wind.
Restricting ourselves to stars that will end their lives as supernovae ($>8$ solar masses) then the answer is no; not all the products of nuclear fusion are stored inside the star until the final explosion.
There is a general conceptual picture that as a massive star evolves, that heavier elements are able to fuse more readily closer to the centre where the temperature is higher. This would result in an "onion-like" structure where you might have hydrogen fusing to helium in the outer part of the core of the star (which is round about the size of the Earth), with a layer inside this where helium is fusing into carbon and oxygen, a layer inside this where neon is being produced etc. At the very centre, just before the core-collapse and supernova explosion, there would be a phase of very rapid silicon burning which produces "iron-peak" elements including iron, nickel, cobalt etc. Outside the core there would be a much larger envelope consisting mainly of hydrogen and helium.
This view is very simplistic because it ignores two important facets of the evolution of high-mass stars - mixing and mass loss. Massive stars probably lose the majority of their mass during their lives. The more massive the star, the greater percentage of its mass is lost. The mass is lost through a powerful wind that is driven by radiation pressure. There are many examples that are observed to undergo this mass loss including objects like Eta Carina, Wolf-Rayet stars and Red Supergiants. Many of these objects have lost their entire outer envelopes and have essentially exposed the products of nuclear burning, or those products are mixed to the surface by convection or other more exotic mixing processes associated with their rotation. These products are then driven into space by their powerful winds.
This mechanism will be an important contributor to the carbon, nitrogen and oxygen abundances in the interstellar medium. The winds may also be enhanced in much heavier elements (even heavier than iron, such as Barium and Strontium) that have been produced via the s-process, where pre-existing iron-peak nuclei in the nuclear burning regions are able to capture neutrons. This process can also occur in stars with masses substantially lower than required to produce a supernova.
In addition to nucleosynthesis as the star ages, the action of core collapse into a supernova produces heavier elements too - supernova nucleosynthesis.
For example, the intense energy of the collapse causes fusion of some of the material surrounding the core. This material will escape, not be trapped in the core itself.
