Rick's Cosmology Tutorial: Chapter 15 Abstract

In this Chapter lower and upper bounds are derived for the mass, central temperature and central density of stars. This is done using simple approximations. The objective is to derive closed-form expressions which can be used to examine the scope for stars to exist in alternative universes in which the universal constants take different values.

The hydrostatic equilibrium of a star requires that the inward force of gravity be balanced by pressure. There are three sources of pressure: gas pressure, degeneracy pressure and radiation pressure. An expression is derived which relates the electron degeneracy pressure to the electron number density. It is proportional to the electron number density to the power 5/3 for non-relativistic electrons, but to power 4/3 for extreme relativistic electrons.

It will be seen that the lower bound mass is determined by the onset of electron quantum degeneracy. Such stars are called white dwarfs.

The upper bound mass is determined by stability. Larger stars become increasingly unstable due to radiation pressure. It will be seen that radiation pressure is unimportant for stars of solar mass or less. However, the importance of radiation pressure increases as mass-squared, and hence eventually dominates for stars of larger mass.

In this universe, i.e. for the actual values of the universal constants, we derive the lower bound stellar mass to be ~7% of solar mass, and the upper bound ~40 times solar mass. This mass range does indeed encompass almost all observed stars. The bounds are also derived in terms of arbitrary values of the universal constants. As a multiple of the nucleon mass, the bounds on the stellar mass depend upon the electromagnetic and gravitational fine structure constants and the ratio of the proton to electron masses.

Bounds on the central temperature and density of stars are derived for an arbitrary stellar mass and arbitrary values of the universal constants. Substitution of the upper bound for the stellar mass provides upper limits for the central temperature and density. The lower temperature and number density limits depend upon the proton mass and the electromagnetic fine structure constant. The upper temperature and number density limits depend upon the electron mass only (other than c and h). This leads to a requirement that the proton mass be less than 6 million times the electron mass in any universe with stars. This is easily achieved in this universe, of course, by 3 orders of magnitude.

In this universe the lower and upper bound central temperatures for a stable star self-sustained by nuclear fusion reactions are ~7 million K and ~30 billion K respectively. The corresponding central density range is from little more than that of terrestrial atmospheric air up to 10 million times the density of terrestrial steel.

Picture of drifts on Mars, taken by the Mars Rover exploration vehicle, Opportunity (NASA)