Rick's Cosmology Tutorial: Chapter 16 Abstract

Derivation of the Stellar Hydrogen Opacity

The opacity of the stellar medium to electromagnetic radiation is crucial to the structure and lifetime of stars. The rate of heat loss from the core must balance against the rate of nuclear heat production within the core if the star is to be stable. An opaque stellar medium implies a low rate of heat transport and hence a low rate of nuclear heat production. This requires lower core temperatures, pressures and densities. Conversely, a very low opacity implies high rates of heat transport and hence high rates of nuclear heat prouction. This requires higher core temperatures, pressures and densities. Thus, whilst an estimate of the lifetime of a star can be made by assuming its central temperature, together with the nuclear reaction rate at this temperatue, it is actually the opacity which determines the lifetime, since it is the opacity which constrains the central temperature.

In this Chapter we derive the opacity due to atomic hydrogen and free electrons from first principles. Approximate, closed form expressions for the opacity are derived. These are plotted against temperature for a range of densities. They are compared with solar opacities from the literature. Given the great sensitivity of the opacity to both temperature and density, the qualitative agreement between our opacity estimates and the literature is good.

In particular, the opacity peaks between temperatures of about 10,000 K and 100,000 K. The reason for this is clear from our derivation. For bound-free atomic electron transitions, the temperature must be high enough for sufficient numbers of photons to be able to ionise the atom. On the other hand, the temperature must be low enough for significant numbers of neutral atoms to occur un-ionised. The temperature range 10,000-100,000 K represents the compromise between these two competing effects.

Our derivation also subsumes a first-principles demonstration of the origin of the Kramers opacity. This applies in the region between the opacity peak and about 10^7 K for which the opacity is proportional to density and proportional to temperature raised to the power -7/2. This rapidly decreasing opacity terminates in a lower bound opacity due purely to free electron scattering by photons (Thompson scattering), which applies above about 10^7 K..

However, our simple estimate of the opacity, based on hydrogen bound-free transitions and free electrons alone, is a significant underestimate in general. The reason is that several contributions to the total opacity have not been addressed. These are listed herein. Possibly the chief amongst these is the opacity due to the star's metal content.

Read Chapter 16 (pdf): Opacity Of The Stellar Medium

Go to Next Chapter Abstract

Go back to Chapter Selection page

Go to Rick's Cosmology FAQ

Go to Rick's Critique of the Cosmic Coincidences

Go back to Rick's Home Page (Main Menu)

Contact me and other links

Picture of Mars from the Viking Orbiter