Mechanical and thermal equilibrium

Notes following Chapter 2 of Pols Lectures here, Sec. 1-2 of
Kippehahn 2012, and HKT, Chapter 1, Available Online.

Day 2 - September, 2, 2025

Agenda:

• Updates/Reminders (2m)
• Lecture (25m)
• In-Class Activity 2 - Groups of 2 - Due end of day (30m)
• Report out on ICA (10m)

Hydrostatic Equilibrium

Consider the following assumptions of an object:

• spherically symmetric
• nonrotating
• nonmagnetic
• single star on which there are no net forces acting and, hence, no accelerations

Can we describe this "star" in equations?

Hydrostatic Equilibrium

First, some conventions:

• radius: radial distance measured from the stellar center (cm)
• density: is the mass density at (g cm)
• temperature: is the temperature at (K)
• pressure: is the pressure at (dyne cm = erg cm)
• mass: is the mass contained within a sphere of radius (g)
• luminosity: , the rate of energy through a sphere at (erg s)
• local gravity: local acceleration due to gravity (cm s)

Conservation of Mass and Coordinate Systems

Applying mass conservation to the mass d of a spherical shell of thickness d at radius gives us

integrating this gives us the total mass of the star .

It is often more convenient to describe stellar structure in a Lagrangian as opposed to Eulerian coordinate system - .

Conservation of Mass and Coordinate Systems

Applying this transformation,

allows us to rewrite our equation for mass conservation:

This brings us to our first stellar structure equation!

Hydrostatic Equilibrium

Next, we to describe the equation of motion for a gas element inside the star:

For a static star, no net acceleration, and this equation reduces to

Hydrostatic Equilibrium

This is known as the equation of hydrostatic (or mechanical) equilibrium (HSE), we can also write in our Lagrangian formulation

Since , it follows that , and the pressure must decrease from the center outwards, everywhere.

This brings us to our second stellar structure equation!

The Virial Theorem

connects two important energy reservoirs of a star and allows
predictions and interpretations phases in the evolution of stars.

Starting from HSE and multiplying by the volume and integrating:

reducing to

The Virial Theorem

We can identify the right hand side (RHS) of the equation as the familiar gravitational potential energy of the star, the work performed by the gravitational force to bring all mass elements from infinity to their current radius:

The LHS requires integration by parts (im sorry):

The Virial Theorem

assuming the pressure at the surface is negligible , we can reduce to

The Virial Theorem

We typically see it written in its final form as

or more commonly, using ,

This is the general form of the virial theorem.

The Virial Theorem

The LHS is (for an ideal gas!) related to the internal energy,

leading to the final relation

The Virial Theorem

General form:

For an ideal monotomic gas:

• The virial theorem tells us that a more tightly bound star must have a higher internal energy, i.e. it must be hotter.

Timescales of stellar evolution

We will explore and use different relevant estimated timescales for stellar evolution.

Dynamical timescale
timescale on which a star reacts to a perturbation of hydrostatic equilibrium

Q: What is the dynamical timescale of the sun?

Timescales of stellar evolution

Thermal or Kelvin-Helmholtz timescale

• describes how fast changes in the thermal structure of a star can occur, time for star without a nuclear burning source to contract and radiate away its internal energy content ().

Timescales of stellar evolution

Nuclear timescale - Timescale that a star can remain in thermal equilibrium due to its nuclear burning and fuel supply

where, is a fraction of the rest mass in the nuclear reaction, and is the fraction of the mass of the star which may serve as fuel.

In-Class Assignment 2

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.

After Class: End of day today, September, 2, 2025

• Submit as a PDF, preferrably using nbconvert to D2L, the progress you have made.

ICAs are not always designged to be completed but rather worked on in class, submit what you have when you leave the class even if you did not make much progress.