Polytrope
A polytrope is a solution to the Lane–Emden equation, a key equation in astrophysics describing the structure of self-gravitating, spherically symmetric fluids, such as stars. More generally, a polytrope refers to an equation of state of the form:
P = Kργ
where:
- P is the pressure,
- ρ is the density,
- K is a constant, and
- γ is the polytropic exponent.
The polytropic exponent is often expressed in terms of the polytropic index, n, as γ = 1 + 1/n. Therefore, the equation of state can also be written as:
P = Kρ1 + 1/n
Polytropes provide simplified models for stellar structure and are useful for exploring the relationships between pressure, density, and radius within a star. Different values of the polytropic index n correspond to different types of physical systems. For example:
- n = 0 (γ = ∞) corresponds to an incompressible fluid.
- n = 1 (γ = 2) can approximate a white dwarf star.
- n = 3 (γ = 4/3) is relevant for stars supported by radiation pressure, as well as highly relativistic degenerate matter.
- n = 5 (γ = 6/5) represents the limiting case for stable polytropic spheres; polytropes with n > 5 are unstable.
While polytropes are simplifying approximations, they offer valuable insights into the behavior of more complex astrophysical objects and are used as building blocks for more sophisticated models. The polytropic equation of state also appears in other contexts beyond stellar structure, wherever the relationship between pressure and density can be approximated by a power law.