University of California, San Diego
Center for Astrophysics & Space Sciences

The evolutionary history of a star may be considered a story of the inexorable battle of the star against the force of gravity which, once the star begins its contraction out of the interstellar medium, attempts to pull it ever smaller into a more compact, more tightly bound sphere. Stars of solar-type masses will come to a compromise with gravity as they end their lives as compact, dense white dwarf stars with diameter about the size of the earth and density of order 1 ton/cm³. The most massive stars will lose this battle in spectacular fashion.

The Russell-Vogt Theorem

It follows that the single most important determinant of the life-history for a star is its mass; this principle is called the Russell-Vogt Theorem. Important mass regimes for stellar evolution:

• M_* < 0.01 M - Planet. Jupiter, for example, has a mass of about 0.001 x M. Jupiter's temperature is slightly warmer than would be expected from the amount of solar energy it receives; this is interpreted as due to gravitational potential energy stored as heat from Jupiter's contraction out of the proto-solar nebula. But the energy balance for Jupiter and other planets is largely determined by the energy received from the sun and central temperatures never come close to the 1 million K required for even the simplest nuclear reactions.

• 0.01 M < M_* < 0.085 M - Brown Dwarf; these objects will never become hot enough in their cores to ignite the P-P Chain. Release gravitational potential energy will cause them to heat up to core temperatures as hot as 3 million K, hot enough for the first stages of nuclear reactions, perhaps, but never hot enough to establish stable hydrogen burning. With atmospheric temperatures T_surface < 2000K, brown dwarfs will be very faint, radiating the vast majority of their luminous energy in the infrared, and very hard to detect. A new near-infrared survey of the sky called 2MASS has detected a large number of cool stars, now classified as L-stars which are likely to be brown dwarfs. Here's a press release of another possible brown dwarf detected at Palomar and the Hubble Space Telescope.

• 0.085 M < M_* < 0.4 M - these stars will be very long lived, but will never reach temperatures hot enough for the Triple-alpha process to occur. They will not have a helium flash in the red giant stage nor a helium-burning main-sequence phase.

• 0.4 M < M_* < 1.2 M - these stars like the sun will burn hydrogen to helium via the P-P Chain and will burn helium to crabon via the Triple-alpha process following a path through the H-R Diagram essentially like that outlined in the previous tutorial.

• M_* > 1.2 M - these stars will reach high enough core temperatures to burn hydrogen via the CNO cycle.

• M_* > 8 M - stars more massive than about 8 solar masses (this number is very uncertain compared with those above) will have a larger number of nuclear burning cycles and their cores will be more massive than the limiting mass of 1.4M, the largest mass that can be supported by electron degeneracy, and thus the largest possible mass for a white dwarf. As we shall see these stars end their lives with a cataclysmic explosion called a supernova.

As shown in the figure above the place where a star reaches the Main Sequence is directly related to the star's mass.

Massive Stars

We may crudely distinguish between stars more massive than the sun and solar type stars in their evolutionary characteristics:

1. Massive stars live their lives more rapidly than do solar-type stars -- they "live fast and die young." One can determine relatively straightforwardly from the balance between gravity, pressure and temperature that the luminosity of a star will be approximately proportional to the Mass^3.5. This is the Mass-Luminosity Relation which applies to all phases of stellar evolution:

   L M^3.5

   Since the available fuel is effectively the mass of the star, the lifetime will be approximately proportional to 1/Mass^2.5. A star of 10 solar masses can thus be expected to go through its life cycle about 300 times faster than the sun, with a main sequence lifetime of about 30 million years. (The most massive stars have lifetimes shorter than about a million years, while stars with masses less than about 3/4M have lifetimes longer than the age of the Universe!)

2. As described above massive stars require higher central temperatures to balance the greater pull of gravity. This means that massive stars produce helium from hydrogen via the CNO cycle rather than the P-P Chain.

3. Higher central temperatures and pressure dictate that the stellar core will not become electron degenerate at the onset of helium burning, so there will be no helium flash.

4. Because a massive star will reach higher core temperatures, massive stars will experience more advanced nuclear burning stages producing a wider range of nucleosynthesis products, up to iron.

5. As already mentioned, stars whose core is greater than 1.4 solar masses exceed the "Chandrasekhar limit" to the mass for a white dwarf. They will end their lives with a dramatic explosion, becoming either neutron stars or white dwarfs. Because stars lose considerable mass due to stellar winds in the later stages of evolution and in the planetary nebula phase, it is currently believed that stars with M < 8M end their lives as white dwarfs.

Confirmation of Stellar Theory -- Hertzsprung-Russell Diagrams

The "filmstrip" to the left shows the development of the H-R Diagram for a cluster of stars formed at a single epoch.

At an age of 1 million years the most massive stars have contracted to the Main sequence, lived out their hydrogen-burning lifetimes and are evolving off the Main Sequence. Lower mass stars like the sun are still in the Pre-Main Sequence phase. The youngest clusters observed in the Milky Way are estimated to have ages of a few million years. At 10 million years stars of 1 solar mass are still above the Main Sequence, just beginning nuclear reactions. They will be observed as T-Tauri stars. Stars with M ~ 20M are just moving off the Main Sequence. Such clusters will still be associated with regions of gas & dust from which they formed. At 100 million years most stars are on or nearing the Main Sequence, but stars with M > 5M are now moving off the Main Sequence. The Pleiades cluster is estimated to have an age of about 100 million years. With an age of a billion years, cluster stars with masses between 2--3 M are moving off the Main Sequence. The Main Sequence location at which stars are just beginning to exhaust the hydrogen fuel in their cores and move toward the Red Giant region is called the Main Sequence Turnoff The oldest clusters in the Milky Way, the globular clusters, are estimated to have ages of the order of 10-15 billion years and show H-R Diagrams like that at the left. Because the globular cluster stars have very low abundances of the elements heavier than helium (C,N,O ...) some corrections need to be made to compare their H-R diagrams to younger clusters with higher abundances. from Seeds Horizons © Wadsworth 1994

The theoretical H-R diagrams above can be compared with the schematic H-R Diagrams of a selection of clusters shown below.

Schematic H-R Diagrams for star clusters in the Milky Way.
The "Main Sequence Turnoff" is used to estimate the cluster age.

Globular Cluster H-R Diagrams

The H-R Diagram for a Globular Cluster, M3, in the galactic halo.

From considerations of the way in which the Milky Way formed, we believe that the globular clusters formed some 10-15 billion years ago, consistent with their ages determined from their H-R Diagrams.

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Conducted by Gene Smith, CASS/UCSD.
Comments? You may send email to hsmith@ucsd.edu

Prof. H. E. (Gene) Smith
CASS   0424   UCSD
9500 Gilman Drive
La Jolla, CA    92093-0424

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Last updated: 16 April 1999