Stellar Nucleosynthesis

Stars live on the edge. Throughout their life, they constantly walk a line between blowing themselves apart and gravitational collapse. Gravity is constantly trying to pull inwards on the mass of the star. Without a counter force, the star would collapse. As gravity causes collapse, the gas in the core is heated to tremendous temperatures. Eventually, the temperature is so high, and atoms are moving so fast that the natural repulsion of protons, the Coulomb force, is overcome and the strong nuclear force dominates and allows fusion to occur. The energy released by these reactions pushes outwards and balances the force of gravity which keeps the star stable.
 As stars age, the fusion reactions that keep the star stable get increasingly complex. Starting with the most simple form of fusion, hydrogen fusion, new elements are created and build up in the core. As the reactions progress, their by-products begin to build up in the core and eventually choke the reaction like soot in a combustion reaction. As the reaction is choked off, the core cools because the reactions are no longer as efficient as they once were. Gravity gains a foothold and the fragile balance is broken. The core begins to collapse.
 As the core starts to collapse, it squeezes the matter in the core causing the temperature to rise. Eventually, the temperature rises so high that the by-products from the previous reaction is able to fuse. This in turn causes the cycle to begin again whereby heavier elements are forged, they eventually build-up, choke the reaction, collapse ensues, heating results, and new fusion occurs. The cycle continues until the core produces iron which is incapable of fusion. 
 At this point, fusion in the central core halts and there is no longer a repulsive force in the central core to prevent collapse. The core violently collapses with such pressure that the electrons and protons in the core merge to form a ball of neutrons. A critical density is reached, neutron degeneracy, and the core can be compressed no further. The collapse halts suddenly, creating a shockwave. The combination of the shockwave and a massive flood of neutrinos created from the formation of the neutrons in the core rip the outer layers of the star apart and a supernova explosion results. What remains of the original star is in the form of the neutron rich core, also known as a neutron star. 
 No nuclear reactions are taking place, but the star retains tremendous temperatures. Neutron stars can be up to 2-3 times the mass of the sun, but all of this matter is squeezed into a ball about 10 km across. Neutron stars often retain the magnetic field and angular momentum of the parent star. When squeezed to such a small size, the rotation is amplified like an ice skater spinning as he or she pulls in their arms. Rotation rates of neutron stars are often hundreds of times per second. Densities of neutron stars approach 3 x (10^14)g/(cm^3). The magnetic field strength of a neutron star can reach as high as 1 x (10^15) gauss in extreme cases (the earth's magnetic field measures 1 gauss for comparison).

Fusion Processes

The fusion processes in a high mass star can be quite exotic. All stars begin by fusing hydrogen into helium. The most common form of hydrogen fusion in low mass stars is the proton-proton chain. Larger stars make use of a more efficient process known as the CNO cycle. We start by visualizing the proton-proton chain.

Proton Proton Chain (Hydrogen Burning)

This reaction dominates in lower mass stars like the sun. It involves 2 protons in a chain of reactions that result in helium.

(^1)H+(^1)H->(^2)H+(SuperPlus[β])+ν
(^2)H+(^1)H->(^3)He+γ
(^3)He+(^3)He->(^4)He+(^1)H+(^1)H

phase1a=Table[Graphics3D[{Red,Specularity[White,50],Sphere[{-x,0,1},1],Sphere[{x,0,1},1],White,Text[Style["p","Label",18,Bold],{-x,0,1}],Text[Style["p","Label",18,Bold],{x,0,1}],Text[Style["(^1)H+(^1)H->(^2)H+(SuperPlus[β])+ν","Label",18,Bold],Scaled[{.5,.5,-.5}]]},Boxed->False,PlotRange->{{-5,5},{-5,5},{-6,6}},Background->Black],{x,4,1,-.1}];

phase1b=Table[Graphics3D[{Red,Specularity[White,50],Sphere[{-1,0,1},1],Blue,Sphere[{1,0,1},1],Green,Sphere[{x,0,x+1},1],Yellow,Sphere[{x,0,-x},1],White,Text[Style["p","Label",18,Bold],{-1,0,1}],Text[Style["n","Label",18,Bold],{1,0,1}],Text[Style["(SuperPlus[β])","Label",18,Bold],{x,0,x+1.75}],Text[Style["ν","Label",18,Bold],{x,0,-x-1.75}],Text[Style["(^1)H+(^1)H->(^2)H+(SuperPlus[β])+ν","Label",18,Bold],Scaled[{.5,.5,-.5}]]},Boxed->False,PlotRange->{{-5,5},{-5,5},{-6,6}},Background->Black],{x,1,8,.1}];

phase2a=Table[Graphics3D[{Red,Specularity[White,50],Sphere[{-1,0,1},1],Sphere[{0,0,z+1},1],Blue,Sphere[{1,0,1},1],White,Text[Style["p","Label",18,Bold],{-1,0,1}],Text[Style["n","Label",18,Bold],{1,0,1}],Text[Style["p","Label",18,Bold],{0,0,z+1}],Text[Style["(^2)H+(^1)H->(^3)He+γ","Label",18,Bold],Scaled[{.5,.5,-.5}]]},Boxed->False,PlotRange->{{-5,5},{-5,5},{-6,6}},Background->Black],{z,4,1,-.1}];

phase2b=Table[Show[Graphics3D[{Red,Specularity[White,50],Sphere[{-1,0,1},1],Sphere[{0,0,2},1],Blue,Sphere[{1,0,1},1],White,Text[Style["p","Label",18,Bold],{-1,0,1}],Text[Style["n","Label",18,Bold],{1,0,1}],Text[Style["p","Label",18,Bold],{0,0,2}],Text[Style["γ","Label",18,Bold],{t+2,0,1.75}],Text[Style["(^2)H+(^1)H->(^3)He+γ","Label",18,Bold],Scaled[{.5,.5,-.5}]]},Boxed->False,PlotRange->{{-5,5},{-5,5},{-6,6}},Background->Black],ParametricPlot3D[{x,0,1+.5 Sin[10 x-t]},{x,0+t,π/2+t},PlotStyle->{White,Thickness[.007]},Mesh->None,MaxRecursion->0]],{t,0,5,.1}];

phase3a=Table[Graphics3D[{Red,Specularity[White,50],Sphere[{1,0,z-1},1],Sphere[{0,0,z-2},1],Sphere[{-1,0,1},1],Sphere[{0,0,2},1],Blue,Sphere[{-1,0,z-1},1],Sphere[{1,0,1},1],White,Text[Style["p","Label",18,Bold],{1,0,z-1}],Text[Style["n","Label",18,Bold],{-1,0,z-1}],Text[Style["p","Label",18,Bold],{0,0,z-2}],Text[Style["p","Label",18,Bold],{-1,0,1}],Text[Style["n","Label",18,Bold],{1,0,1}],Text[Style["p","Label",18,Bold],{0,0,2}],Text[Style["(^3)He+(^3)He->(^4)He+(^1)H+(^1)H","Label",18,Bold],Scaled[{.5,.5,-.5}]]},Boxed->False,PlotRange->{{-5,5},{-5,5},{-6,6}},Background->Black],{z,-4,-1,.1}];

phase3b=Table[Graphics3D[{Red,Specularity[White,50],Sphere[{1,0,-1},1],Sphere[{x,0,-2-x},1],Sphere[{-1,0,1},1],Sphere[{x,0,x+2},1],Blue,Sphere[{-1,0,-1},1],Sphere[{1,0,1},1],White,Text[Style["p","Label",18,Bold],{1,0,-1}],Text[Style["n","Label",18,Bold],{-1,0,-1}],Text[Style["p","Label",18,Bold],{x,0,-2-x}],Text[Style["p","Label",18,Bold],{-1,0,1}],Text[Style["n","Label",18,Bold],{1,0,1}],Text[Style["p","Label",18,Bold],{x,0,2+x}],Text[Style["(^3)He+(^3)He->(^4)He+(^1)H+(^1)H","Label",18,Bold],Scaled[{.5,.5,-.5}]]},Boxed->False,PlotRange->{{-5,5},{-5,5},{-6,6}},Background->Black],{x,0,5,.1}];

fusion=Join[phase1a,phase1b,phase2a,phase2b,phase3a,phase3b];

Length[fusion]

SetDirectory[NotebookDirectory[]];

Export["proton-proton.gif",fusion,"GIF",AnimationRepetitions->∞,ImageSize->400];

Additional steps can also occur involving Beryllium, Boron, and Lithium. In all cases, the result is a build-up of Helium in the core.

CNO Cycle (Hydrogen Burning)

This reaction dominates in high-mass stars with heavier element impurities. Protons react with carbon, nitrogen, and oxygen in a cycle that results in a build-up of helium.

(^1)H+(^1)H->(^4)He
(^12)C+(^1)H->(^13)H+γ
(^13)N->(^13)C+(SuperPlus[β])
(^13)C+(^1)H->(^14)N+γ
(^14)N+(^1)H->(^15)O+γ
(^15)O->(^15)N+(SuperPlus[β])
(^15)N+(^1)H->(^12)C+(^4)He

Triple Alpha process (Helium Burning)

This process starts when helium burning is ignited in the core at a temperature of approximately 100 x (10^6) Kelvin. It involves 2 alpha particles fusing to form an unstable Beryllium atom which fuses with a third alpha particle before it has a chance to decay. The result is a build-up of carbon in the core. Helium burning in low mass stars can be explosive since the density of the core is so high that the electrons are in a degenerate phase. This means that as the core heats up, it doesn't expand and the temperature rises unchecked in an event known as a Helium flash. For larger stars, the triple alpha process begins before electron degeneracy takes hold and so there is a gradual progression to the fusing of higher mass elements.

(^4)He+(^4)He->(^8)Be
(^8)Be+(^4)He->(^12)C+γ

It is also possible for the carbon to capture an additional alpha particle and results in oxygen. The ratio of the carbon to oxygen created through this set of reactions can quite literally determine the future of the star.

(^12)C+(^4)He->(^16)O

Carbon/Oxygen Burning

Once the helium in the core is replaced by carbon and the core collapses due to the reduced energy output, the gravitational energy given up heats the core until carbon can fuse with carbon. The is reaction results in several possible outcomes, but in general results in a build-up of Magnesium, Neon, and Oxygen in the core at a temperature of approximately 600 x (10^6) Kelvin. There are also large numbers of neutrinos created in these reactions. In stars between 3 and 9 solar masses, the core becomes degenerate before Carbon fusion can occur. With the core degenerate, a normal pressure-temperature thermostat is turned off. The core doesn't expand as the temperature rises. When carbon fusion ignites, its does so explosively and can blow such stars apart.

(^12)C+(^12)C->(^24)Mg
(^12)C+(^12)C->(^20)Ne+(^4)He
(^12)C+(^12)C->(^16)O+2 (^4)He

There are a number of other reactions also possible.

(^12)C+(^12)C->(^23)Na+(^1)H
(^12)C+(^12)C->(^20)Ne+(^4)He
(^16)O+(^16)O->(^31)P+(^1)H
(^16)O+(^16)O->(^28)Si+(^4)He
(^16)O+(^16)O->(^31)S+n
(^16)O+(^12)C->(^27)Al+(^1)H
(^16)O+(^12)C->(^24)Mg+(^4)He
(^16)O+(^12)C->(^27)Si+n

Neon Burning

If a star is more than 9 solar masses, fusion continues in the core. The energy in the core prevents the core from becoming degenerate and neon fusion can take place. Neon burning involves the photodisintegration of Neon into Oxygen and also the fusing of Neon with an alpha particle to create Magnesium. This results in a build-up of Oxygen and Magnesium in the core at a temperature of approximately 1.2 x (10^9) Kelvin.

(^20)Ne+γ->(^16)O+(^4)He
(^20)Ne+(^4)He->(^24)Mg+γ

Oxygen Burning

Oxygen burning involves the fusion of oxygen atoms into a variety of products. The dominant result is a build-up of Silicon in the core at a temperature of approximately 2 x (10^9) Kelvin.

(^16)O+(^16)O->(^32)S+γ
(^16)O+(^16)O->(^31)S+n
(^16)O+(^16)O->(^31)P+(^1)H
(^16)O+(^16)O->(^28)Si+(^4)He
(^16)O+(^16)O->(^24)Mg+((2 )^4)He

Silicon Burning

Silicon burning involves the fusion of two Silicon atoms and two decays and results in a build-up of iron in the core at a temperature of approximately 3 x (10^9) Kelvin. In actuality, the Coulomb barrier is probably too high to allow the fusion of two Silicon atoms directly. It is more likely that a complex set of reactions involving the photodisintegration of Silicon and the subsequent capture of protons and neutrons gradually builds up to the elements near iron. For brevity, we assume the Silicon fusion happens directly through the following reaction.

(^28)Si+(^28)Si->(^56)Ni+γ
(^56)Ni->(^56)Co+(SuperPlus[β])+ν
(^56)Co->(^56)Fe+(SuperPlus[β])+ν

Since traditional fusion of iron is not possible except by endothermic reactions involving the capture of neutrons, silicon fusion is the final phase of fusion that can take place in a star. Although some fusion continues to take place in shells surrounding the core, fusion reactions in the central core cease. The structure of the star resembles an onion with concentric shells of fusion of various elements from earlier phases. Since the energy output of the fusion was the only force countering the inward pull of gravity, gravity dominates and the core collapses. During the collapse process, the electrons and protons in the central core are forced together to form neutrons at a temperature of approximately 5 x (10^9) Kelvin.

p+(SuperMinus[e])->n+ν

When the density of the neutrons is so high that neutron degeneracy is achieved, the core rebounds suddenly. The neutrino surge is so large that the shockwave is enhanced tremendously by the neutrino pressure and the star is ripped apart in a supernova explosion. Typically, the remains of the core are in the form of a large ball of neutrons known as a neutron star. In extreme cases, if the mass of the star is high enough, the collapse can result in a stellar mass black hole. As the shockwave progresses outwards, it heats the surrounding shells tremendously causing exotic forms of nucleosynthesis, primarily through neutron capture at a very high rate. This creates the elements higher than iron on the periodic table.

The following code generates a visualization of the structure of the star just before core collapse.

Stellar Core Structure

shell[{r1_,r2_},label_,color_,opts___]:=Show[ParametricPlot3D[{r2 Cos[theta]Sin[phi],r2 Sin[theta]Sin[phi],r2 Cos[phi]},{theta,0,3Pi/2},{phi,0,Pi},opts,PlotPoints->30,PlotStyle->color,Mesh->None,MaxRecursion->0],ParametricPlot3D[{rad Cos[3Pi/2]Sin[phi],rad Sin[3Pi/2]Sin[phi],rad Cos[phi]},{rad,r1,r2},{phi,0,Pi},PlotPoints->30,PlotStyle->color,Mesh->None,MaxRecursion->0],ParametricPlot3D[{rad Cos[0]Sin[phi],rad Sin[0]Sin[phi],rad Cos[phi]},{rad,r1,r2},{phi,0,Pi},PlotPoints->30,PlotStyle->color,Mesh->None,MaxRecursion->0],Graphics3D[{Text[Style[label,Bold,Black,12,"Label"],{(r1+r2)/2,0,0},{0,-1},{0,1}]}],PlotRange->All,SphericalRegion->True,Axes->False]

gr[zoom_]:=Show[shell[{.6,2},"Hydrogen",{Orange,Specularity[White,10]}],shell[{.5,.6},"Helium",{RGBColor[.49,0,0],Specularity[White,10]}],shell[{.4,.5},"Carbon",{RGBColor[.25,.43,.82],Specularity[White,10]}],shell[{.3,.4},"Neon",{RGBColor[.48,.11,.56],Specularity[White,10]}],shell[{.2,.3},"Oxygen",{RGBColor[.6,.73,.36],Specularity[White,10]}],shell[{.1,.2},"Silicon",{White,Specularity[White,10]}],Graphics3D[{Specularity[White,50],Black,Sphere[{0,0,0},.1],Black,Text[Style["Iron",Bold,White,12,"Label"],{0,0,-.02}]}],Boxed->False,Axes->False,Background->Black,ImageSize->600,Lighting->{{"Ambient",GrayLevel[.7]},{"Directional",GrayLevel[1],ImageScaled[{0,1,1}]}},ViewAngle->zoom]

gr[π/10]

Here is a zoomed in view of the core.

gr[π/25]

Here, we generate an animation that zooms into the core region for more detail.

anim=Table[gr[z],{z,π/10,π/25,-0.005}];

Finally, we can export the animation.

SetDirectory[NotebookDirectory[]];

Export["core.gif",anim,"GIF",ImageSize->400];

References

For classic work on nucleosynthesis see:
Burbidge, E.M., Burbidge, G.R., Fowler, W.A.& Hoyle, F., Rev.Mod.Phys. 29, 547 (1957)

For a short overview of stellar nucleosynthesis, see:
 Krane, Kenneth S., Introductory Nuclear Physics, John Wiley & Sons, p.769-780, 1988

For an overview of stars, including the fusion processes, see ScienceWorld:
 http://scienceworld.wolfram.com/astronomy/Star.html