The inside of a star must be hot, so that pressure can support it against gravitational collapse.
However, a star is constantly losing heat to outer space.
The competition between gravity and pressure drives the various stages of stellar evolution.
When a gas is compressed, its temperature rises.
This process is easy to understand in the case of a piston. The piston moves in to compress the gas, and when an atom bounces off the piston it moves faster than before it hit the piston.
As the piston moves in, the average speed of the atoms goes up, meaning that the temperature goes up.
This is the physical mechanism behind equation #5, Tc ~ M/Ravg.
A star has several possible sources of energy.
Thermal energy (or heat energy): energy stored in the motions of atoms and the energy of photons.
Nuclear energy: energy available from fusing light elements into heavier elements, e.g. hydrogen into helium.
Gravitational potential energy: energy that the star can tap by contracting.
A star is constantly losing its thermal energy supply to outer space, because it is luminous. It can replenish this thermal energy either by nuclear fusion or by contraction. A star heats up when it contracts, turning gravitational potential energy into thermal energy.
With the equation Tc ~ M/Ravg, we can build a simplified picture of star formation.
As long as the nuclear energy supply lasts, the star can shine without contracting further.
Note: The behavior in step (3) is completely different from that of a glowing piece of charcoal, which gets cooler as it radiates energy. As a protostar radiates, it gets hotter because of gravitational contraction.
This sketch of star formation captures some important features of the process, but it ignores rotation and magnetic fields, both of which play critical roles during the formation of real stars.
As a protostar contracts, its central temperature goes up, and the rate of nuclear fusion goes up. The rate at which photons can escape from the center of the star goes down.
A star stops contracting when the rate at which energy is produced by fusion just balances the rate at which energy is lost by escaping photons. The star has precisely the radius that allows nuclear energy to produce its luminosity.
This radius is stable:
Star expands slightly: Tc drops -> fusion rate drops, central pressure drops, star contracts
Star contracts slightly: Tc rises -> fusion rate rises, central pressure rises, star reexpands.
We can derive a rough approximation of the mass-radius relation by knowing that the rate of nuclear reactions is very sensitive to temperature. Raising the central temperature slightly raises the energy production enormously.
=> Stars with very different luminosities have quite similar central temperatures.
Since Tc ~ M/R, the condition Tc is approximately constant implies R ~ M.
For very hot stars, pressure is dominated by photons instead of by moving atoms, and the equation Tc ~ M/R does not apply. For these stars, a similar physical argument gives R ~ M1/2.
These approximate arguments reproduce the observed mass-radius relation of main sequence stars fairly well.
In 7.5, we saw how to compute the luminosity of a star given its mass and radius.
We now see how the mass of a star determines its radius, making it possible to compute the luminosity from the mass alone.
A more massive main sequence star is more luminous primarily because it is bigger. The larger core contains more photons and more thermal energy.
Calculations using these principles reproduce the observed mass-luminosity relation.
The mass of a main sequence star determines its radius and its luminosity.
Since L=(4`pi'`sigma')R2T4, the mass also determines the surface temperature T, hence the color and spectral type.
The mass of a main sequence star determines its position on the HR diagram.
Stars lie on a narrow sequence because luminosity and temperature climb steadily with mass.
A main sequence star is a star that is sustaining its luminosity by fusing hydrogen into helium in its core.
Hydrogen fusion can sustain a star for a long time, which is why most stars are found on main sequence.
As core turns from hydrogen into helium, the structure of the star changes slightly, and it evolves slightly away from ``zero age'' main sequence. However, it does not evolve up or down the main sequence.
Eventually, the supply of hydrogen in the core runs out. At this point, the star begins to evolve much more rapidly.
Sources of energy available to a star are thermal energy, nuclear energy, and gravitational potential energy.
As a protostar contracts it heats up, using gravitational energy to replenish thermal energy. When the core gets hot enough, hydrogen fusion begins.
A protostar stops contracting (reaches the main sequence) when the energy produced by hydrogen fusion matches the star's luminosity. The match of fusion rate to luminosity determines a star's radius.
The mass of a main sequence star determines its radius, its luminosity, and its temperature, thus its position in the HR diagram.
A star can live happily on the main sequence until the supply of hydrogen fuel in its core runs out.