Polytropic stellar models

Notes following Ch. 4 of Pols Lectures here & HKT, Ch. 7, Available Online.

Day 5 - September, 11, 2025

Agenda:

• Updates/Reminders - HW1 Due Thur. Sept. 18 before class (2m)
• Lecture (20m)
• In-Class Activity 5 - Not for credit (20m)
• Report out on ICA in ica Slack Channel (10m)

Recap

Mass continuity:

Hydrostatic equilibrium:

Recap

Pressure of a mixture of gas and radiation

If the electrons are not degenerate (they can be described classically as in Pols 3.19),

In practice, stellar evolution code often rely on tabulated EOS, which account for many non-ideal effects.

Today, we will talk about additional terms to the total pressure and when quantum mechanic effects might play a role.

Polytropic EOS

• a special case of barotropic EOS, which are all the EOSs for which the density depends only on the pressure and not on the temperature () or composition (): .

These have applications beyond stellar physics (e.g., for atmospheric physics). A polytropic EOS is usually written (following the general form of Eq. 4.1 Pols/ HKT Eqn. 7.16:

where K is a constant, and by definition . Note in Pols () and is the polytropic index.

Polytropic EOS

Goal: to derive the analytic theory of polytropes and construct polytropic models, and study to which kind of stars they correspond.

Q: How can we take this equation for the pressure of the gas to determine other properties of the star?

• A: By combining with mass continuity and assumptions of HSE!

1. Multiply HSE by () and take derivative ()

Polytropic EOS

2. Simplifying

3. Recalling that

We find the resulting relation is Poisson's Equation.

Polytropic EOS

We want to make a series of substitutions to make a dimensionaless version of this equation called the Lane-Emden Equation.

1. where

2. Plug into polytrope equation

where we have set the central pressure

Polytropic EOS

3. Use these definitions and plug into Poisson's Equation

We can simplify by using the fact that

Polytropic EOS

To further simplify we need to define a few more things:

• dimensionless radial coordinate , defined as

with the is equal to the prefactor (units of length squared)

Polytropic EOS

We can now substitute this to transform from to the dimensionaless variable to arrive at the Lane Emden Equation:

• A solution for for a value of can be found by solving this Eqn. and are called polytropes of index . Analytic solutions are sometimes called E-solutions.

Q: Do we have everything we need to solve this Equation?

Polytropic EOS

Boundary Conditions:

as well as

leading to the total dimensional radius or

Solutions to the Lane-Emden Equation

Analystic E-solutions exist for and . While numerical solutions are required for general .

gives the solution for a constant-density model

with

To compute we need to define K,n, and or .

Summarizing Polytropic EoS

• Recall , using our Polytrope equation for this case, a three-halves prolytrope.

• Recall , using our Polytrope equation for this case, an n equals three prolytrope.

Applicable to certain stellar environments including White Dwarfs.

In-Class Assignment 5

In class: Work on ICA here with partner, I will ask one or two people to share and describe plots at the end of class.

Not for Credit

If you've missed a previous ICA, but make progress on this one today, you can upload this one in place of the missed one and recieve half credit.