Stellar Death - Low Mass

Part I

In another five billion years the hydrogen in the core of the Sun will become depleted. All stars will eventually run out of useable hydrogen reserves and this marks a dramatic change in the life of the star. More massive stars simply run out of hydrogen more quickly. Using the Sun as our prototype, we go to the point where the last hydrogen fusion shell is being consumed. The star performs the same steps as previously, with one important difference.

• Hydrogen becomes depleted in the last fusion shell.
• The reaction rate falls.
• Gravity is now stronger than radiation pressure (this zone cannot support its own weight).
• The core shrinks
• Compressing the ideal gasses of the core cause the temperature to increase.
• The increase in temperature provides energy to -
• Begin fusion reaction in the next shell
• Provide energy to the envelope
• The excess energy in the envelope lifts it against gravity

Notice that the star can no longer tap hydrogen reserves further out in the core. All of the energy produced as the core shrinks goes into lifting the envelope. The star grows dramatically to become a Red Giant. The Sun will expand to at least 100 times its present size. As the atmosphere expands (being an ideal gas) it cools and reddens. The evolution on the H-R Diagram is equally dramatic. Over the Main Sequence lifetime the position of the star on the diagram changed very little. Now the star moves quickly to the right and up on the diagram as it expands and cools.

Back at the core we still have a collapse occurring. Here we are compressing an ideal gas and thus heating it up. If the core can reach a temperature of 100,000,000 K then helium fusion (Triple Alpha Process) can happen. In helium fusion three helium nuclei (alphas) can combine into one carbon nucleus. Massive stars have no problem heating the core to the required temperature. But low mass stars like our Sun have a difficult time achieving temperatures that high. The matter becomes degenerate in the compression.

Degenerate gas works by different rules than ideal gasses. The electrons cause a change of state. Electrons are not friendly. As the core collapse continues, the electrons are forced to get closer and closer. Electrons repel one another in such a way that every available energy state eventually gets filled. When this happens there is an enormous outward pressure caused by the mutual repulsion of all of the electrons. We have a degenerate electron gas.

The core temperature in low mass stars (including the Sun) reaches the required 100 million Kelvin at just about the same time as the core becomes degenerate. We now have helium fusion starting in a degenerate gas. Unlike the forest fire analogy I used for fusion in an ideal gas, here there are two big differences:

• Fusion takes place all over the degenerate core simultaneously
• The energy supplied by the fusion process does not expand the core gasses as readily as in the ideal gas.

Ideal gasses act as a safety valve on the fusion process. If the reaction should overheat, the gas expands and cools, and the reaction rate slows. If the reaction rate becomes too slow, the gas will shrink and heat up, increasing the reaction rate. Fusion in the degenerate gas lacks this safety valve.

Helium fusion in a low mass star begins rather explosively then. It is referred to as the helium flash. Now that the star has found a nuclear energy source once again, the envelope settles down to close to its original size. As the fusion continues the core gasses do expand slightly and this slight expansion eventually breaks the degeneracy and the core reverts to being ideal again. The star is now in a quasi-stable period of helium fusion. The star may pulsate causing the light output to vary. Two effects nearly cancel out. The temperature of the star increases as the more energetic helium fusion takes hold. Higher temperature should lead to a brightening. But the envelope is shrinking back to its original size with a commiserate dimming. The consequence is that the evolution on the H-R Diagram is almost horizontal towards the left at this stage. This portion of the evolutionary track is called the horizontal branch.

Part II

We will take a short digression to search for real objects that match the predictions of the model for Helium fusion. Two classes of stars are located in this region of the H-R Diagram, both of which are pulsating variable stars. The first are the RR Lyrae variables. Variable stars get there names from the prototype, in the case, a variable in the constellation of Lyra called RR. They show a very characteristic light variation with periods less than a day, and brightness changes of 0.5 to 1 magnitude. The RR Lyrae stars are is the He fusion portion of their lifetimes. Clearly, He fusion is not always a stable process. But these stars are also useful as distance indicators. Because they are located in the horizontal branch, the absolute magnitude of all RR Lyrae stars is essentially the same (about -0.5). If we simply identify stars as belonging to this group of variable stars and measure the average apparent magnitude, we can compute the distance. Since they are giant or subgiant stars, RR Lyrae variables can be seen are great distance. Astronomers have used the RR Lyrae stars to gauge distances within the Milky Way.

Another type of variable star in He fusion is the Cepheid variables (the prototype is d Cephei). The periods of variation are several days and the brightness changes are about the same as the RR Lyrae variables. The Cepheids do not have the same absolute magnitudes, but, as Henrietta Leavett discovered, there is a relationship between the average brightness and the period for this group. The longer the period of the Cepheid, the brighter is the star. The procedure for extracting distance from the Cepheid variables is as follows:

• Discover variable stars that have the Cepheid properties
• Measure the period of the variation and the average apparent magnitude
• Using the Period-Luminosity relationship, determine the absolute magnitude
• With the apparent and absolute magnitude, we can compute the distance

An additional benefit of using Cepheid variables to measure distance is that they are supergiant stars that can be seen at tremendous distances. Cepheid variables have allowed astronomers to determine the distances to nearby galaxies.

Returning to the story of evolution, we go to the end of the He fusion stage. The core of the star is composed of almost pure carbon in the center, with small overlying fusion zones of both helium and hydrogen. The star repeats the scenario given earlier for the end of hydrogen fusion, namely:

• Helium becomes depleted in the last fusion shell.
• The reaction rate falls.
• Gravity is now stronger than radiation pressure (this zone cannot support its own weight).
• The core shrinks
• Compressing the ideal gasses of the core cause the temperature to increase.
• The increase in temperature provides energy to -
• Begin fusion reaction in the next shell
• Provide energy to the envelope
• The excess energy in the envelope lifts it against gravity

We predict a second giant phase for the star as helium fusion ends. The star will once again appear as a Red Giant, this time perhaps even brighter as the collapsing core heats up to very high temperatures and drives the envelope very far from itself. The core is trying to reach a temperature of 600 million K, which is the threshold temperature for Carbon fusion. Low mass stars like the Sun cannot reach these temperatures. There is simply too little gravity to provide the necessary compression.

Observationally, the Mira variables, located in the asymptotic branch, are likely in the second giant phase. These variables have very long periods (hundreds of days) and vary by many magnitudes.

The core of the star compresses until the material becomes degenerate again. At about the same time the star will eject a planetary nebula, in which most of the original envelope of the star is propelled away into space. The cause of the planetary nebula is not well established. It may be that the remaining He fusion zone becomes explosive as the core is collapsing below it. Another idea is that the recombination energy from hydrogen in the outer atmosphere provides the energy to lift the envelope away from the core. Many hundreds of planetary nebulae can be observed and each one is a low mass star like the Sun near the end of its lifetime.

The remaining star now is much smaller than before. We might predict that the star would appear fainter. But the remnant is also hotter - we are viewing the original core of the star with a very thin envelope remaining. Hotter means brighter. The two effects almost offset one another so that the star remains about the same brightness as it quickly evolves to the left of the H-R Diagram.

Part III

With the expulsion of the planetary nebula, the star also has considerably less mass than before. Recall that it is the mass of the envelope bearing down onto the core that maintains the very high temperature and pressure required for fusion reactions. Two mass loss mechanisms become significant in post-Main Sequence evolution:

• Enhanced stellar winds in the giant phase
• Planetary nebula

Should the mass of the star fall below the Chandresakhar Limit (1.4 Solar Masses) the star at this stage cannot continue to process nuclear fuel. This is also the maximum amount of mass that an electron degenerate core can support. Stars that begin their lives on the Main Sequence with less than 8-10 Solar Masses will all end their lives in the same way. A breakdown of the models is:

• Mass < 2.5 Solar Mass
• He fusion begins in a degenerate core (He flash)
• Nuclear fusion ends with He
• Mass between 2.5 and 8 Solar Mass
• He fusion begin when the core is an ideal gas (no He flash)
• Nuclear fusion ends at He
• Mass between 8 and 10 Solar Mass
• Carbon fusion may begin before mass loss causes the mass to fall below 1.4 Solar Mass.

All of these stars can be considered low mass and each will end its life with mass below the Chandresakhar limit. The electron degenerate core is supporting the weight (gravity) of the star. Recall that electron degeneracy is the mutual repulsion of the core electrons due to the electromagnetic force. Because there must be a balance between the gravity of the star and electron degeneracy pressure, we can make definite predictions about what the object looks like. The object should have a large carbon core with a thin atmosphere of hydrogen and helium. If we take a mass of one solar mass, the radius is about the radius of the Earth. Its small size will make it faint and very difficult to detect. The object is called a White Dwarf. The first white dwarf discovered was the companion of Sirius and was found to have all of these properties.

Review and associate the position of the low mass star on the H-R Diagram with the stage of life that the star is in.