EOS Lab — Cosmic Playground

Equation of State Lab

Compare stellar pressure channels with explicit cgs units and assumption framing.

LTE closure likely reasonable

Radiation pressure uses $P_{\rm rad}=\eta_{\rm rad}aT^4/3$ with default $\eta_{\rm rad}=1$ in optically thick LTE-like conditions.

Temperature $T$ (K) Density $\rho$ (g cm$^{-3}$)

Composition Controls

Hydrogen mass fraction $X$ Helium mass fraction $Y$

Computed metal fraction $Z = 1 - X - Y$:

Preset states (ASTR 201)

What to notice 3 prompts

At fixed density, radiation pressure climbs much faster with $T$ than gas pressure.
At high density, electron degeneracy can dominate even when thermal pressure is large.
Composition shifts $\mu$ and $\mu_e$, changing both gas and degeneracy channels.

Model notes cgs + assumptions

Quick summary: Three pressure channels compete inside every star.

Gas pressure ($P_{\rm gas}\propto\rho T$): hotter or denser gas pushes harder.
Radiation pressure ($P_{\rm rad}\propto T^4$): photon momentum; grows steeply with temperature, independent of density.
Degeneracy pressure ($P_{\rm deg}\propto\rho^{5/3}$ at low $T$): Pauli exclusion forces electrons into high-energy states even at zero temperature.

Technical details

Gas pressure: $P_{\rm gas}=\rho k_B T/(\mu m_u)$.
Radiation pressure: $P_{\rm rad}=\eta_{\rm rad} aT^4/3$ (default $\eta_{\rm rad}=1$).
Finite-$T$ electron EOS uses Fermi-Dirac pressure $P_{e,\rm FD}(n_e,T)$ with nonrelativistic and relativistic branches.
Displayed degeneracy channel is $P_{\rm deg,e}=\max\!\left(P_{e,\rm FD}-n_e k_B T,0\right)$.
Finite-$T$ proxy uses $1+\frac{5\pi^2}{12}\left(T/T_F\right)^2$ only when $x_F\ll1$ and $T/T_F\ll1$.
$\mu$ and $\mu_e$ for fully ionized mixtures: $\mu^{-1}\approx2X+\frac{3}{4}Y+\frac{1}{2}Z$, $\mu_e^{-1}\approx X+\frac{Y}{2}+\frac{Z}{2}$.
Future hook: the kernel accepts additional pressure terms and alternate degeneracy providers for pair-rich and neutron extensions.

Composition interpretation why $\mu$ matters

Smaller $\mu$ means more particles per gram, so gas pressure increases at fixed $(\rho,T)$.

Advanced diagnostics regime checks

Fermi momentum regime + extension pathway diagnostics.

$x_F=p_F/(m_e c)$

ratio

Fermi relativity regime

Sommerfeld factor $1+\frac{5\pi^2}{12}\left(T/T_F\right)^2$

factor

Neutron extension pressure

dyne cm$^{-2}$

Pressure Curves at Fixed Temperature

Move sliders in real time to see how $P_{\rm gas}$, $P_{\rm rad}$, and $P_{\rm deg,e}$ vary with density at the current temperature and composition.

Regime Map ($\log_{10}\rho$ vs $\log_{10}T$)

Each color shows which pressure channel dominates at a given $(\rho,T)$. The crosshair tracks your sliders—move them to explore different stellar regimes. The boundaries shift when you change composition.

P_gas dominant
P_rad dominant
P_deg,e dominant
Mixed dominance

Show solar model profile

$P_{\rm gas}$

dyne cm$^{-2}$

Thermal particle collisions; scales as $\rho T$.

$P_{\rm rad}$

dyne cm$^{-2}$

Photon momentum flux; scales as $T^4$ under LTE closure.

$P_{\rm deg,e}$

dyne cm$^{-2}$

Quantum correction beyond classical electrons; driven by Pauli exclusion.

Total pressure $P_{\rm tot}$: dyne cm$^{-2}$

Dominant channel:

$\mu$ $m_u$

$\mu_e$ $m_u$

$\beta=P_{\rm gas}/P_{\rm tot}$

$P_{\rm rad}/P_{\rm gas}$ ratio

$P_{\rm deg,e}/P_{\rm tot}$

$\chi_{\rm deg}=T/T_F$

$\Gamma_{\rm eff}$

Deep Dive: Gas Pressure cause + limits

Physical cause: momentum transfer from thermal particle collisions.

Limits: assumes ideal fully ionized plasma; no Coulomb or partial-ionization corrections.

Try this: Compare $P_{\rm gas}$ for pure hydrogen ($X=1$) vs. pure helium ($Y=1$) at the same $(T,\rho)$. Which gives higher pressure, and why?

Deep Dive: Radiation Pressure cause + limits

Physical cause: photon momentum flux from a near-thermal radiation field.

Limits: LTE closure can fail in optically thin or strongly non-equilibrium regimes.

Try this: Fix $\rho$ at the solar core value and increase $T$ until radiation pressure overtakes gas pressure. What temperature marks the crossover?

Deep Dive: Degeneracy Pressure cause + limits

Physical cause: Pauli exclusion fills momentum states even at low temperature.

Limits: finite-$T$ Fermi-Dirac electrons are included, but Coulomb/ion corrections and neutron-star matter channels are intentionally out of scope.

Try this: Select the white dwarf preset and watch $T/T_F$ in the readout strip. Now raise $T$ slowly—at what temperature does degeneracy give way to thermal pressure?