EOS Lab: Pressure Support in Stars

Name: ________________________________ Section: __________ Date: __________

Station: __________ Group members: ________________________________________________

Goal: Use the demo to make a claim about stellar equations of state supported by (1) at least one number/readout and (2) at least one sanity check.

Station card: Equation-of-State Lab (10–12 minutes) Demo setup: Set composition to solar ($X = 0.70$, $Y = 0.28$, $Z = 0.02$). Start at the solar core defaults ($T = 1.5 \times 10^7$ K, $\rho = 150$ g cm$^{-3}$). Tip: Click Station Mode to add EOS anchor rows and export your data table.

Your station artifact (fill in):

1. Control(s): temperature $T$ (K), density $\rho$ (g cm$^{-3}$), composition $X / Y / Z$

2. Observable(s): $P_\text{gas}$ (dyn cm$^{-2}$), $P_\text{rad}$ (dyn cm$^{-2}$), $P_\text{deg}$ (dyn cm$^{-2}$), dominant channel, $T / T_F$, $\chi_\text{deg}$

3. Governing relationships:

   $$P_\text{gas} = \frac{\rho k_B T}{\mu m_H}, \quad P_\text{rad} = \frac{1}{3} a T^4, \quad P_\text{deg} = K \left(\frac{\rho}{\mu_e m_H}\right)^{5/3}$$

4. Sanity check: The solar core should be gas-pressure dominated ($P_\text{gas} \gg P_\text{rad}, P_\text{deg}$). A white dwarf interior ($T \sim 10^7$ K, $\rho \sim 10^6$ g cm$^{-3}$) should be degeneracy-dominated.

5. Connection sentence: “The dominant pressure channel matters for stellar structure because…”

Data Collection Tasks (Station Mode)

1. Click Station Mode. Add the three EOS anchors using the preset buttons: Solar core, Red giant core, White dwarf. Three rows appear in the table.

2. Analyze the solar core row. Which pressure channel dominates? Compute the ratio $P_\text{rad} / P_\text{gas}$. Is the gas ideal or degenerate? (Check $T / T_F$: if $T / T_F \gg 1$, the gas is classical.)

3. Analyze the white dwarf row. Which pressure channel dominates now? Compute $P_\text{deg} / P_\text{gas}$. What is $T / T_F$? Describe in one sentence why this gas is qualitatively different from the solar core.

4. Scaling experiment — temperature. Starting from the solar core defaults, raise $T$ by one decade (to $\sim 1.5 \times 10^8$ K) while holding $\rho$ fixed. Click Add Row. Which pressure grew fastest? Explain why, using the exponents in the governing equations.

5. Scaling experiment — density. Reset to solar core defaults. Now raise $\rho$ by two decades (to $\sim 1.5 \times 10^4$ g cm$^{-3}$) while holding $T$ fixed. Click Add Row. Which pressure channel changed the most?

6. Click Export to copy the table, then paste it into your lab document.

Data table

Environment	$T$ (K)	$\rho$ (g cm$^{-3}$)	$P_\text{gas}$	$P_\text{rad}$	$P_\text{deg}$	Dominant	$T/T_F$
Solar core	$1.5 \times 10^7$	150
Red giant core
White dwarf		$10^6$
Expt: raise $T$	$1.5 \times 10^8$	150
Expt: raise $\rho$	$1.5 \times 10^7$	$1.5 \times 10^4$

Word bank

• $P_\text{gas}$ (ideal gas pressure): pressure from thermal motion of ions and electrons; $\propto \rho T$.
• $P_\text{rad}$ (radiation pressure): pressure from photons; $\propto T^4$ — grows much faster than $P_\text{gas}$ with temperature.
• $P_\text{deg}$ (electron degeneracy pressure): quantum-mechanical pressure from the Pauli exclusion principle; depends on density, not temperature ($\propto \rho^{5/3}$ in the non-relativistic limit).
• $\mu$ (mean molecular weight): average mass per particle in units of $m_H$; depends on composition and ionization.
• $T_F$ (Fermi temperature): the temperature scale where quantum degeneracy sets in; $T / T_F \gg 1$ means classical, $T / T_F \lesssim 1$ means degenerate.
• $\chi_\text{deg}$ (degeneracy parameter): a dimensionless measure of how degenerate the electron gas is; related to $T / T_F$.
• LTE (local thermodynamic equilibrium): the assumption that matter and radiation share a single local temperature; valid in stellar interiors.

Sanity checks:

• Solar core: gas-dominated, classical ($T / T_F \gg 1$), $P_\text{rad} / P_\text{gas} \sim 0.01$.
• White dwarf: degeneracy-dominated ($P_\text{deg} \gg P_\text{gas}$), $T / T_F \lesssim 1$.
• Raising $T$ by one decade at fixed $\rho$: $P_\text{rad}$ grows by $10^4$ while $P_\text{gas}$ grows by $10^1$.
• Raising $\rho$ by two decades at fixed $T$: $P_\text{deg}$ grows by $10^{10/3} \approx 2{,}150\times$ while $P_\text{gas}$ grows by $10^2 = 100\times$.