• Barbara Jesline

Do You Know: Fundamentals of Stellar Structure

What exactly do we mean by a ‘star’? A useful definition for the purpose of this course is as follows: a star is an object that radiates the energy from an internal source and second is bound by its own gravity. This definition excludes objects like planets and comets because they do not comply with the first criterion. In the strictest sense, it also excludes brown dwarfs, which are not hot enough for nuclear fusion. The second criterion excludes trivial objects that radiate, e.g. glowing coals.

An important implication of this definition is that stars must evolve (why?). A star is born out of an interstellar molecular gas cloud, lives for a certain amount of time on its internal energy supply, and eventually dies when this supply is exhausted. As we shall see, a second implication of the definition is that stars can have only a limited range of masses, between ∼0.1 and ∼100 times the mass of the Sun. The life and death of stars form the subject matter of this course. We will only briefly touch on the topic of star formation, a complex and much less understood process in which the problems to be solved are mostly very different than in the study of stellar evolution.

Stellar structure describes the internal structure of a star in detail and makes predictions about the luminosity, the color and the future evolution of the star. Different classes and ages of stars have different internal structures, reflecting their elemental makeup and energy transport mechanisms. There is a list of processes through which the Stellar Structure of any star determined. Few of them are mentioned below:

• Stars are held together by gravitation — attraction exerted on each part of the star by all other parts.

• Collapse is resisted by internal thermal pressure.

• Gravitation and internal thermal pressure must be in balance.

• Stars continually radiate into space; for thermal properties to be constant, a continuous energy source must exist.

• Theory must describe the origin of energy and its transport to the surface.

Commonly 4th process is used. For stars which are isolated, static and spherically symmetric, there are four equations to describe structure. All physical quantities depend only on the distance from the center of the star:

• Equation of Hydrostatic Equilibrium – at each radius, forces due to pressure difference balance gravity.

• Conservation of mass.

• Conservation of energy – at each radius, the change in the energy flux is the local rate of energy release.

• Equation of Energy Transport – the relation between the energy flux and the local temperature gradient.

Different layers of the stars transport heat up and outwards in different ways. In stars with masses of 0.3–1.5 solar masses (M☉), including the Sun, hydrogen-to-helium fusion occurs primarily via proton-proton chains, which do not establish a steep temperature gradient. Thus, radiation dominates in the inner portion of solar-mass stars. The outer portion of solar mass stars is cool enough that hydrogen is neutral and thus opaque to ultraviolet photons, so convection dominates. Therefore, solar mass stars have radiative cores with convective envelopes in the outer portion of the star.

In massive stars (greater than about 1.5 M☉), the core temperature is above about 1.8×107 K, so hydrogen-to-helium fusion occurs primarily via the CNO cycle. In the CNO cycle, the energy generation rate scales as the temperature to the 15th power, whereas the rate scales as the temperature to the 4th power in the proton-proton chains. Due to the strong temperature sensitivity of the CNO cycle, the temperature gradient in the inner portion of the star is steep enough to make the core convective. In the outer portion of the star, the temperature gradient is shallower but the temperature is high enough that the hydrogen is nearly fully ionized, so the star remains transparent to ultraviolet radiation. Thus, massive stars have a radiative envelope.

The most important fundamental property, the mass, cannot be measured directly for a single star. To measure stellar masses one needs binary stars showing radial velocity variations (spectroscopic binaries). Radial velocities alone can only yield masses up to a factor sin i, where i is the inclination angle of the binary orbit. To determine absolute mass values one needs information on i, either from a visual orbit (visual binaries) or from the presence of eclipses (eclipsing binaries). In particular, for so-called double-lined eclipsing binaries, in which the spectral lines of both stars vary, it is possible to accurately measure both the masses and radii (with 1–2 % accuracy in some cases) by fitting the radial-velocity curves and the eclipse lightcurve. Together with a photometric or, better, spectroscopic determination of Teff also the luminosity of such binaries can be measured with high accuracy, independent of the distance. For more details see the Master course on Binary Stars.

Chemical reactions can be ruled out as a possible source for stellar energy because:

• It was shown above that the Sun is made up of largely ionized material. Hydrogen, in particular, is almost completely ionized except in the atmosphere. There are therefore very few atoms or ions having the bound electrons needed for chemical reactions to proceed.

• The energy source needs to provide, in the solar case, the energy equivalent of at least 10−4 M over ∼ 109 years. Chemical reactions such as the combustion of fossil fuels release the energy equivalent of ∼ 5 × 10−10 M in the same period.

To understand the structure and evolution of stars, and their observational properties, using known laws of physics. This involves applying and combining ‘familiar’ physics from many different areas (e.g. thermodynamics, nuclear physics) under extreme circumstances (high temperature, high density), which is part of what makes studying stellar evolution so fascinating.