Stellar Structure and Evolution

How do mass, composition and fuel change a star? Gravity compresses a star until its pressure gradient supports the overlying gas. Mass and composition also change fuel consumption and heat transport, so stars with more fuel can burn through it sooner.

Drag a shell through saved stellar interiors. Compare main-sequence fuel, independent helium burning and a cooling white dwarf.

Radius inside the grid $0.25$

Shell temperature $1.2\times10^{7}\,\mathrm{K}$

Shell density $0.764\,\mathrm{g\,cm^{-3}}$

Shell pressure $1.29\times10^{15}\,\mathrm{dyn\,cm^{-2}}$

Look beneath the surface

Saved stellar interiorShell depth

r/R_{\mathrm{grid}}

Cutaway colours

Drag the gold shell on the exposed face. Choose Core view to reach the compact centre.

Synthetic measurement and prior

Age is inferred only for the selected known main-sequence mass and composition. Temperature and luminosity errors are fixed in log space.

\sigma_{\log T}=0.01,\qquad\sigma_{\log L}=0.05

Random seedAge prior

Model, scale and sources

The cutaway has real depth and uses the source’s volume-equivalent radial coordinate. Colours describe the selected diagnostic; they are not a photograph. Each probe selects an actual native shell.

The coloured cut faces sample the native grid, with fixed physical colour scales across cases. The probe and curves use the full saved grid.

The stellar histories have fixed physical prescriptions. The helium-burning and white-dwarf clocks restart from imported models.

The independent eleven-Sun age test is overconfident despite a good HR fit. The late white-dwarf global energy comparison differs by about six percent; local source agreement does not remove it.

Original model and CC BY data Read the stellar evolution course

Open the original polytrope model

Model assumptions

The atlas reads licensed saved one-dimensional MESA models. Four rotating main-sequence mass/composition cases share fixed mixing and wind prescriptions. He-burning and white-dwarf examples have separate reset clocks. The cutaway uses volume-equivalent radius and diagnostic colours. Age inference uses synthetic HR measurements conditional on the selected known native main-sequence model. Saved interiors are discrete; continuous mass/composition interpolation, progenitor-to-remnant clocks, complete energy conservation and precise WD cooling ages are not inferred. The original static polytrope mode remains available.

Ten Suns · Solar mixture

Saved MESA interiors

Source MS clock $17.4\,\mathrm{Myr}$ · $10\,M_{\odot}$ · $6.55\,R_{\odot}$ ·MESA model 236

This rotating main-sequence track begins with source relaxation.

Temperature, logarithmic colour scale

3.5\;\longrightarrow\;8.5

The saved state on its own HR track

\log_{10}(T_{\mathrm{eff}}/\mathrm{K}),\quad\log_{10}(L/L_{\odot})

Blue: native track. Gold: the selected saved interior. Pink: synthetic HR measurement. Temperature increases toward the left.

Fuel and ashes inside the star

m/M,\quad X_i

Teal: hydrogen. Gold: helium. The mass coordinate spreads out the compact core.

Temperature with depth

r/R_{\mathrm{grid}},\quad\log_{10}(T/\mathrm{K})

Orange: native temperature. The gold point follows the radial probe.

Age from a synthetic HR measurement

t/\mathrm{Myr},\quad p(t\mid\mathrm{HR},M,Z)/p_{\max}

Purple: relative age probability density for this known native model. The observed point is synthetic.

$t_{16,50,84}=(14.4,16.2,17.7)\,\mathrm{Myr}$

Zsolt Keszthelyi · Zenodo 7069766 · CC BY 4.0 · Twelve significant digits in the view; original bytes in the scientific tutorial.

Read the discoveries behind these models →

Focus question: How do mass, composition and fuel change a star?
One-sentence intuition: A stellar clock belongs to a specified physical model. More initial fuel can be consumed sooner when the required luminosity is larger.

Core mathematical model

Support in the spherical limit

\frac{dP}{dm}=-\frac{Gm}{4\pi r^4}

Here $m$ is enclosed mass and $r$ is radius. The four rotating main-sequence sources also record a rotation correction. Their radial coordinate is volume-equivalent.

The surface on an HR diagram

L=4\pi R^2\sigma_{\mathrm{SB}}T_{\mathrm{eff}}^4

Luminosity, effective temperature and photospheric radius describe one selected native state. The gold HR point always matches that interior.

Usable fuel and energy demand

t_{\mathrm{fuel}}\sim\frac{E_{\mathrm{usable}}}{L}

This estimate holds the luminosity fixed. A calculated lifetime additionally follows mixing, changing composition, wind loss and luminosity.

Common difficulties

A large fuel tank

Typical misconceptionMore hydrogen must imply a longer life.

Better mental modelThe time depends on usable core fuel and the rate at which energy leaves the star. A more luminous star can consume a larger supply sooner.

Joining independent clocks

Typical misconceptionThe helium and white-dwarf cases complete the massive star’s timeline.

Better mental modelThey come from separate calculations with imported structures and reset clocks. The atlas preserves those origins.

A good fit and a correct model

Typical misconceptionA small HR residual proves that the inferred age is reliable.

Better mental modelAn independently withheld eleven-Sun model gives overconfident ages even with small residuals. Age inference here is conditional on a known selected native model.

Run the experiment

01
Read a saved hydrogen-burning core
Begin with Ten Suns, Solar mixture. Drag the gold shell through the exposed cut face.
What to observe: Temperature and pressure rise inward. The hydrogen and helium curves reveal a mixed core and a different envelope.
02
Compare an actual second model
Choose Twelve Suns, Solar mixture, then Hot metal-poor star. Read each model number and its own gold HR point.
What to observe: The four main-sequence structures are discrete outputs with fixed physical prescriptions. Their composition comparison also changes helium and element ratios.
03
Open an independent later-stage case
Choose the helium core and cooling remnant demos. Use Core view for the compact helium-burning region.
What to observe: Helium, carbon and oxygen trace different fuel histories. The imported helium and white-dwarf models have separate reset clocks; the white dwarf is not the next frame of the displayed massive star.
04
Separate inference from a saved interior
Return to a main-sequence case. Open Synthetic measurement and prior, draw one point, then change the proper age prior.
What to observe: The synthetic age density changes. The selected interior and gold native HR marker keep their original model number and source clock.

Focus question: What holds a star up against its own gravity?
One-sentence intuition: A pressure difference between the two faces of each layer balances its weight.

Core mathematical model

Balance at each radius

\frac{dP}{dr}=-\rho(r)\frac{Gm(r)}{r^2}

Pressure P decreases outward. Density is rho, the mass inside radius r is m, and G is the gravitational constant. The pressure gradient supplies an outward force per volume equal to the inward gravitational force.

A polytropic pressure law

P=K\rho^{1+1/n}

The index n sets the shape of the pressure–density relation. The constant K follows from the selected mass and radius. The uniform reference is the incompressible limit at index zero.

Common difficulties

Pressure and force

Typical misconceptionHigh pressure pushes the gas outward by itself.

Better mental modelPressure acts from every direction. The outward force comes from the difference between the stronger pressure below a layer and the weaker pressure above it.

Run the experiment

01
Look below the surface
Choose Broad interior and drag the bright shell toward the center.
What to observe: The pressure and enclosed mass change smoothly. The two arrows stay equal because every layer is in equilibrium.
02
Concentrate the mass
Choose Dense core. Compare its density profile with Broad interior at the same mass and radius.
What to observe: Much more mass sits near the center. The central density rises relative to the mean density.
03
Compress the star
Reduce Radius in Suns while leaving the mass fixed.
What to observe: The central pressure rises steeply. A smaller sphere needs stronger internal pressure to support the same mass.

Stellar Structure and Evolution

Ten Suns · Solar mixture

The saved state on its own HR track

Fuel and ashes inside the star

Temperature with depth

Age from a synthetic HR measurement

How the star changes with depth

Density and pressure

How much mass lies inside a shell

Pressure, heat and the fuel inside a star

Core mathematical model

Support in the spherical limit

The surface on an HR diagram

Usable fuel and energy demand

Common difficulties

A large fuel tank

Joining independent clocks

A good fit and a correct model

Run the experiment

Read a saved hydrogen-burning core

Compare an actual second model

Open an independent later-stage case

Separate inference from a saved interior

Why the pressure rises toward the center

Core mathematical model

Balance at each radius

A polytropic pressure law

Common difficulties

Pressure and force

Run the experiment

Look below the surface

Concentrate the mass

Compress the star

Related experiments and articles

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