The Secret Math Lives of Stars

Hydrostatic Equilibrium

One of the basic principles underlying how a star works is hydrostatic equilibrium, the balance of thermal pressure and gravity.

In the core of a star, hydrogen atoms combine to form helium through nuclear fusion reactions, releasing a tremendous amount of energy. This released energy leads to extremely high temperatures, causing the gas particles in the core to move rapidly, resulting in great outward thermal pressure.

This outward thermal pressure is balanced by the star's inward gravitational force, a state known as hydrostatic equilibrium. This balance is self-regulating: if the fusion rate increases, the pressure increases and the core expands, decreasing density and temperature, which in turn reduces the fusion rate and pressure. Similarly, if the fusion rate decreases, the core contracts, increasing the density and temperature, thereby increasing the fusion rate and pressure. Thus, a star is kept stable in size and prevented from expanding or collapsing.

This equilibrium can be modeled mathematically as:

The Life of a Star

A star can remain in this state of hydrostatic equilibrium for billions of years. During this stage, it is called a main sequence (MS) star. But what happens when the star starts running out of hydrogen in its core? The outward gas pressure from fusion decreases, causing gravity to compress the core, which then heats up enough to start fusing helium into carbon.

As helium fusion starts in the core, hydrogen fusion begins in a shell surrounding the core. The increased energy production from the outer shell causes the star's outer layers to expand and cool, making it appear red and significantly increasing its overall size. At this stage, the star is called a red giant.

Stars more massive than the Sun continue this process, fusing heavier elements up to iron, which then triggers a massive supernova explosion. This explosion distributes heavy elements throughout the universe.

A star that is comparable to our sun, at the end of its helium-burning red giant phase, sheds its outer layers to form a planetary nebula and leaves behind a white dwarf. The white dwarf eventually cools into a black dwarf, in theory, after trillions of years (since the universe is only 13.8 billion years old now, there are no black dwarfs in existence yet).

The Hertzsprung-Russell Diagram

The Hertzsprung-Russell Diagram (or H-R diagram for short) is a scatter plot of stars based on their luminosity and temperature or color. It organizes stars into distinct groups like main sequence, red giants, white dwarfs, etc. It is used by astronomers to study stars and their lifecycles (called stellar evolution).

A simple Hertzsprung-Russell diagram showing the positions of main sequence, giant, supergiant, hypergiant, and white dwarf stars, as well as some example star positions (credit: starparty)

MESA Stellar Evolution Software

As you can probably imagine, there is a tremendous amount of math involved in stellar evolution. MESA (Modules for Experiments in Stellar Astrophysics) is a popular and advanced stellar evolution software tool that incorporates all of this math! It is used by researchers to simulate the evolution of stars; it provides tools to model stellar properties and evolutionary processes, enabling a wide range of research in astrophysics, such as studying binary stars, pulsating variable stars, and supernovae. You can check out a simple web interface to this amazing tool here: MESA-Web.