Lecture 29 - Stellar Evolution (3/29/96)

Seeds: Chapter 10

1. Low Mass Stars
   • Stars less than 1/12 solar mass are called brown dwarfs. The central temperature of these stars never gets high enough to fuse hydrogen. These are "failed stars" and are very faint. They can be thought of as large "Jupiters".
   • There may be very many brown dwarfs in our galaxy, though we wouldn't see them easily. Recently, the first brown dwarfs were discovered in binary systems, where their masses could be deduced.
   • Stars between 1/12 and 0.4 solar masses are called red dwarfs. These are main sequence hydrogen burning stars, but cooler, smaller, and fainter than the Sun (hence "red" and "dwarf").
   • Red dwarfs are much less luminous than the Sun. The approximate mass-luminosity relation (L/Lsun) = (M/Msun)^3.5 leads to the lifetime (t/tsun) = (M/L)/(Msun/Lsun) = (Msun/M)^2.5, so a star with mass 0.1 Msun would have a lifetime of 300 times the Sun's lifetime of 10 billion years, or 300 billion years! Red dwarf stars burn their fuel so slowly that they live a very long time (the ultimate conservationists).
   • Red dwarf stars, because of their low luminosities, are fully convective - that is they transport heat from their cores to the surface by gas currents. This also mixes the stars fully so that they can use all of the hydrogen in the star to burn, and the helium made in the core will be mixed throughout the star.
   • When a red dwarf has used up all its hydrogen, it will simply collapse under gravity until it becomes a helium white dwarf (see below). Then it will fade away fainter and cooler until it is a cold dead ball of gas.
2. Medium Mass Stars
   • Stars from about 0.4 to 3 solar masses are normal hydrogen fusing main sequence stars like our Sun for most of their lives.
   • Stars with masses less than about 1.1 solar masses have no convection in their cores. Heat transport is purely radiative out into the envelope. Most importantly, there is therefore no mixing of unburnt envelope hydrogen into the core. Once the core uses up its fuel, it has no recourse.
   • Only about the inner 10% to 13% of the mass of the star is available for fusion in the hot core. This is why the lifetime of the Sun on the main sequence is 10 billion years, not 80 billion years. Gas outside the core is not available for fuel.
   • Stars from 1.1 to 3 solar masses have some convection in their core region. However, this can only mix in around a few percent of the mass near the core, so this does not make much difference in the lifetimes.
   • In all these cases, when the core hydrogen has been fused to helium, we are left with an inert helium core.
   • The helium core cannot maintain pressure against the weight of the star above it, and to generate heat and thus pressure it contracts releasing gravitational energy. This is slow contraction by about 40 meters per year to maintain 1 solar luminosity of energy.
   • The increase in temperature from the gravitational contraction of the core raises the temperature of the hydrogen containing layers just outside the core, causing fusion to occur. At this point hydrogen fusion is going on in a shell around the contracting core. This stage is called hydrogen shell burning.
   • The energy from the core contraction and the shell burning flows out into the envelope. The structure of the envelope was set up to deal with the main sequence luminosity which was approximately constant. Now it cannot handle this extra luminosity so it expands outward becoming larger in radius, less dense, and thus lower temperature ( T^4 proportional to L/R^2 with R increasing).
   • At first only a little of the extra luminosity gets out, with most of the energy going into expanding the envelope. Eventually the stare grows in size by around 25 times its initial radius, and it becomes low enough density so that the radiation can flow out at the rate it is being generated. The luminosity now has grown by a factor of 10 or so.
   • The star is now cooler ("redder") and larger ("a giant"), and is called a red giant.
   • Red giants are located to the right and up from the main sequence. They form a locus of stars called the giant branch.
   • On the giant branch, the helium core is contracting and getting hotter and hotter. These stars are massive enough that the core eventually gets hot enough (100 million K) to fuse helium into carbon (and nitrogen and oxygen) in the triple-alpha reaction.
   • How this fusion begins depends upon the state of the core at this time. Stars with masses below about 2 solar masses have cores that are degenerate (see below) when helium fusion begins. This means that the helium fusion begins all at once in a big flash called the helium flash. In about 1 second the core emits the equivalent luminosity of 10 billion suns! You dont notice anything from the outside because this energy is absorbed by the core.
   • Stars with more the 2 solar masses do not have degenerate cores at the time of helium fusion, so they start fusion gradually.
   • In any event, helium fusion in the core stops the contraction, and expands the core slightly, and the envelope contracts a little increasing the temperature. The star moves to the left a little and slightly down on the HR diagram after reaching the top of the giant track at the helium flash. This is called the horizontal branch, since it extends to the left horizontally from the giant branch.
   • The helium is burning in the core and hydrogen is burning in the shell around the core for stars on the horizontal branch.
   • After the helium is done burning in the core, the core, which is now made of carbon, nitrogen and oxygen, contracts again heating up and starting helium burning in a shell (with a hyrogen shell further out).
   • Thus, with helium and hydrogen burning in concentric shells, the star moves up the giant branch again, becoming even larger. This track parallels but is slightly to the left (hotter) than the giant branch, and is called the asymptotic giant branch. This branch extends a factor of 10 to 100 times the luminosity of the giant branch, and stars on this branch are called red supergiants.
   • For these medium mass stars with less than 3 Msun, the core never gets hot enough for the carbon to fuse. The helium core contracts until it becomes degenerate.
3. Degenerate Matter
   • We have talked about degenerate matter as if it is some special state that stuff turns into at high density. What is it?
   • Upon collapse, the helium and C-N-O cores collapse to about 0.01 of their initial radius, thus increasing the density by a factor of 10^6.
   • The electrons and nuclei of the ionized gas are squeezed together tighter and tighter. Do you remember what happened when we tried to confine an electron into a small orbit around the nucleus ?
   • We found that we had to deal with the wave nature of the electron. Interference effects allowed us to have only integer numbers of waves in the orbit -> energy levels.
   • The requirement that we be able to distinguish electron led us to allowing only one electron of a given spin (either spin "up" or spin "down") in each wavelength (orbital).
   • The same thing happens here. If you confine an electron into a radius r, the smallest you can make this is 2pi r = L where L is the wavelength: L = h/ mv. (See Lecture 16)
   • Thus, we are limited to confining the electron into a space L = 2pi r = h/ mv
   • The rms velocity of the particles in the gas is proportional to the temperature with < v^2 > = 3kT/m thus for a given velocity, the electron can be confined to a space no smaller than r v = h/2pi m
   • This relation is usually written as DX DP > h/2pi where DX represents the localization in space, and DP the localization in momentum (velocity). Note that this also means that if you know the position of a particle to some DX, you cannot know its velocity to an accuracy better than DV = DP/m (and visa versa). This is known as the Heisenberg Uncertainty Principle and is one of the foundations of quantum mechanics. This shows that there is an inherent unpredictability at the wavelength scale of any system.
   • Now the distinguishability enters: you can have only two electrons confined to a given wavelength-sized space, one of each allowed spin. You can think of the gas as a huge parking lot, with lots of available spaces (wavelengths = energy levels) for cars (electrons). Its sort of a strange parking lot, since you can put two cars in a given space, as long as one is parked forward and the other backward (spin s = +1/2 and spin s = -1/2)! As you compress the gas, it is like filling up the parking lot until it is completely full (all available energy levels below some level have 2 electrons). To add more electrons they have to go on the outskirts of the lot (higher energy levels) where there is free space (emptly "orbitals"). A full lot (energy levels all full) is said to be degenerate.
   • If you try to press the electrons closer together than this, they resist very strongly, exerting a stiff pressure against further compression. This is degeneracy pressure.
   • When the core contracts to the point of degeneracy, the pressure is no longer supplied by the temperature but by the electron degeneracy. Furthermore, if you add heat into this degenerate gas the pressure does not increase since the heat goes into the nuclei motions, not the electrons (which are jammed into the parking lot and moving one means moving all in the way or blasting it all the way to the free spaces at the edge).
   • To repeat: you can heat up a degenerate gas without increasing the pressure.
   • If the helium core is degenerate or nearly degenerate when its temperature reaches the threshold for helium fusion to occur, then something catastrophic occurs. When the fusion starts, it releases energy and heats the core further, but this does not increase the pressure which would expand the core lowering the temperature, which is the usual negative feedback control. If the core is degenerate, the increase in temperature causes the core to grow hotter, which increases the fusion rate, which releases more energy, which increases the temperature, and so on. This is a case of positive feedback which leads to uncontrolled fusion - the helium flash.
   • In a few seconds the helium core ignites uncontrolled by thermal pressure, releasing huge amounts of energy. For this short time the core produces the luminosity of a billion suns. However, this energy goes into removing the core degeneracy (unjamming the parking lot by throwing cars from the inside way out to the edge where there is space) and the rest gets absorbed in the envelope.
   • The helium flash does not disrupt the star, and there is no outward sign that it occured. The star continues now with a non-degenerate helium burning core (and hydrogen burning shell).
4. Planetary Nebulae
   • While the star has a contracting C-N-O core with helium and hydrogen burning in shells, it is ascending the giant branch again. These two shells are pumping out lots of energy into the envelope distending it even further than before. This begins to blow off the outermost layers compeletely, beginning the process of mass loss of a large fraction of the stars mass.
   • Also, the helium shell burning is somewhat unstable, with sudden flares of burning causing shells of material to blow off the star.
   • This process accelerates as the mass is lost, until the entire envelope is blown off and we are left with the bare core!
   • All this envelope is blown away into shells around the naked core, forming a planetary nebula.
   • During the planetary nebula phase, the remaining star becomes hotter and hotter as lower high temperature layers are exposed. The luminosity remains the same, since that is generated in the core. The star moves rapidly to the left in the H-R diagram, until the naked core is exposed at 10^7 K.
5. White Dwarfs
   • After all burning has stopped in the core, and the envelope has been stripped away, we are left with a degenerate C-N-O core with a temperature of millions of degrees. This star is very hot, "white-hot", and is called a white dwarf.
   • The white dwarfs produced by low mass stars are made of helium since only hydrogen fusion occured -> He white dwarf
   • The white dwarfs produced by medium mass stars are made of carbon, nitrogen and oxygen from helium fusion -> CNO white dwarfs
   • A white dwarf has the mass of 0.1 to 1.4 Msun and radii about the size of the Earth (10^4 km). The densities are about 10^6 times that of the Earth.
   • The companion star to Sirius, called Sirius B, is a white dwarf with a mass of 0.98 Msun, but a radius of about 2.7 Rearth. Q: Calculate the mean density and surface gravity of Sirius B.
   • White dwarfs slowly cool down, radiating their vast heat through their small surface areas. Eventually they fade away to become black dwarfs.
   • When you add mass to a white dwarf, it shrinks! This is because adding mass increases the gravity, and the spacing between electrons decreases (just like increasing the charge of the nucleus for the ground state orbit). When the white dwarf mass reaches 1.4 Msun, the electron degeneracy pressure can no longer withstand gravity, and the core will collapse again.
   • The limit of 1.4 Msun is known as the Chandrasekhar Limit. You cannot have a white dwarf with a mass above this limit.
   • We believe that stars with initial of about 8 solar masses or less can get rid of enough of their mass to form a white dwarf with mass 1.4 Msun or less. This limit is uncertain, and at least some stars with masses in range of 8 to 10 Msun can form a white dwarf also (these would be made of oxygen-magnesium-neon as the products of carbon fusion).

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