Stellar Modeling

Stellar Modeling#

Notes based on HKT Chapter 7.1-7.2.

To err is human, but to really foul things up requires a computer.

Friday, Feb. 21, 2025

astrophysics of stars and planets - spring 2025 - university of arizona, steward observatory

Today’s Agenda#

  1. Announcements (2m)

  2. Reading Overview/Key Points (5m)

  3. ICA (25m)

  4. Debrief + Share Results (5m)

The Equations of Stellar Structure#

We begin with the need to construct a model that describes the pressure, density, temperature and luminosity and their derivatives:

\[\begin{split} \Large{ P = P(\rho,T,\bf{X}) \\ E = E(\rho,T,\bf{X}) \\ \kappa = \kappa(\rho,T,\bf{X}) \\ \epsilon = \epsilon(\rho,T,\bf{X}) } \end{split}\]

where \(\bf{X}\) is shorthand for composition (as in a specification of nuclear species).

../../_images/stellar_structure.jpg

We also have our various \(\nabla\) relations leading to the creation of a fourth-order differential equation in space or mass requiring 4 boundary conditions.

How to solve these equations is not trival. But we can make some simplifications that lead to the use of Polytropes. Polytropes are pseudo-stellar models for which power law equations of pressure versus density are assumed a priori but where no reference to heat transfer or thermal balance is made.

Polytropic Equations of State and Polytropes#

A polytropic stellar model is defined as

Definition 105

\[ \large P(r) = K \rho^{1+1/2}(r) \]

where \(n\) is the polytropic index and K is a constant of proportionality.

From Eqn 7.16-7.26 lead to the Lane-Emden equation,

a dimensionless form of Poisson’s equation where \(\xi\) is a dimensionless radius and \(\theta\) relates to the density.

Models corresponding to solutions of this equation for a chosen \(n\) are called “polytropes of index n” and the solutions themselves are “Lane–Emden solutions” and are denoted by \(\theta_{n}(\xi)\).

Solutions to the LE equation are often restricted to \(0\lesssim n \lesssim 5\).

Some relevant examples for Polytropes#

  1. The pressure of the completely degenerate but nonrelativistic electron gas goes as \(\rho^{5/3}\). Hence, by the definition of the polytropic equation of state (7.16), \(n\) for this case is 1.5 (or “a three-halves polytrope”).

  2. The density exponent for the fully relativistic case is 4/3 and thus \(n = 3\) (or “an n equal three polytrope”).

  3. Recall that \(P\propto\rho^{5/3}\) in an ideal gas convection zone. If no ionization is taking place (almost a contradiction for a real convection zone) then \(\Gamma_{2} = 5/3\) and \(n = 3/2\) again.

These represent common scenarios for polytropes and rely on the conditions to which we are comparing.

Solving the Stellar Structure Equations#

Thorough discussion in MESA 1 Section 6 for more details.

In-Class Assignment 13#

In-Class Assignment 13 is here.

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