Instructor notes: Hydrostatic Equilibrium Explorer

Navigation

• Instructor hub: Instructor hub
• Student demo: Student demo
• This demo: Model · Activities · Assessment · Backlog

This guide is instructor-facing Student demo: /play/hydrostatic-equilibrium-explorer/
Target model code: packages/physics/src/hydrostaticEquilibriumModel.ts
Target demo shell: apps/demos/src/demos/hydrostatic-equilibrium-explorer/main.ts

Where to go next

• Model + math + assumptions: model.md
• In-class activities: activities.md
• Assessment bank: assessment.md
• Planning backlog: backlog.md

Why this demo exists

This demo gives students a concrete way to reason through the stellar structure chain: gravity demands a pressure gradient, the pressure gradient sets a pressure scale height, the ideal gas law turns that balance into a temperature scale, and the core temperature becomes an inference instead of a mystery number.

The pedagogical target is not memorizing the hydrostatic equation. It is learning to ask, “If the star is heavier, smaller, or chemically different, what must the interior do to stay in balance?”

Learning goals

• Explain hydrostatic equilibrium as a local balance, not a static object.
• Read pressure scale height as a length scale for how quickly support changes with radius.
• Use the ideal gas law to connect composition and temperature at fixed pressure.
• Infer how core temperature scales with $M/R$ in an order-of-magnitude way.
• Use readouts and sanity checks to justify one physical claim.

10-15 minute live-teach script

1. Start with the balanced baseline and ask: “What would have to change for gravity to win?” Let students name mass, radius, and composition before moving any control.

2. Increase mass at fixed radius. Point to the steeper pressure gradient and ask: “What happened to the pressure scale height?” Surface the idea that stronger gravity forces a shorter scale height.

3. Shrink the radius at fixed mass. Ask: “Does the same pressure now have to support more weight per unit area?” Connect that to the core-temperature estimate.

4. Change $\mu$. Ask: “If the same density has fewer particles per gram, what happens to the temperature required for the same pressure?” Use the ideal gas readout to close the loop.

5. Finish with a one-sentence explanation: gravity, scale height, and ideal-gas closure are the whole story the demo is trying to build.

Common misconceptions

• “Hydrostatic equilibrium means nothing is happening.”
  Counter: it means the net force is near zero, not that pressure or gravity disappear.

• “Pressure only matters at the center.”
  Counter: the whole point of the pressure gradient is that support is distributed through the star.

• “A larger core temperature is measured directly.”
  Counter: the demo uses balance plus an equation of state to infer a scale.

What to notice / readouts

• Gravity readout in cm s$^{-2}$ or equivalent local support units.
• Pressure scale height $H_P$ in cm or km.
• Pressure gradient or balance indicator as a sign of under-support or over-support.
• Mean molecular weight $\mu$ and its effect on the ideal-gas temperature estimate.
• Core temperature $T_c$ in K as an inferred scale.

Related pages

• Activities
• Assessment
• Model notes
• Backlog

Navigation

• Instructor hub: Instructor hub
• Back to guide: Guide
• Student demo: Student demo
• This demo: Model · Activities · Assessment · Backlog

0-2 min: Predict

Prompt: “If gravity gets stronger but the star’s size stays the same, what must happen to the pressure gradient and core temperature?”

Student action: Write one prediction sentence and one reason.

Expected observation: Students usually predict “more pressure” but miss that the gradient has to steepen, not just the center pressure.

Instructor move: Ask them to say whether the change is local or global before revealing the readouts.

2-6 min: Play the balance

Prompt: “What does balanced support look like in the readout?”

Student action: Start from the solar-like baseline and identify the balance indicator, $g$, and $H_P$.

Expected observation: Students notice that the supported state is not flat; it has a specific gradient scale.

Instructor move: Point to the local nature of the balance and connect it to $\frac{dP}{dr}$.

6-10 min: Scale height reasoning

Prompt: “What happens to $H_P$ when the star is more compact?”

Student action: Shrink the radius or raise the mass and compare the new $H_P$ to the baseline.

Expected observation: $H_P$ decreases as gravity strengthens.

Instructor move: Translate the visual change into the statement $H_P = P/(\rho g)$.

10-13 min: Ideal gas and temperature

Prompt: “If the same pressure must be carried by fewer particles per gram, what changes?”

Student action: Adjust $\mu$ and watch the inferred $T_c$.

Expected observation: Higher $\mu$ requires a higher temperature for the same pressure support.

Instructor move: Make the composition link explicit and separate it from the gravity link.

13-15 min: Explain

Prompt: “Use one sentence to connect gravity, scale height, and the ideal gas law.”

Student action: Share a claim with one number or one readout from the station card.

Expected observation: Students can usually say “stronger gravity needs a steeper pressure gradient,” but they may need help naming the scale-height step.

Instructor move: Close with the support chain in order: gravity -> $\frac{dP}{dr}$ -> $H_P$ -> ideal gas -> $T_c$.

Station version (8-10 min)

Use apps/site/src/content/stations/hydrostatic-equilibrium-explorer.md as the printable student artifact.

Navigation

• Instructor hub: Instructor hub
• Back to guide: Guide
• Student demo: Student demo
• This demo: Model · Activities · Assessment · Backlog

Clicker prompts

1. If $g$ increases at fixed $P$, then the pressure scale height $H_P$ should:
   A. increase
   B. decrease
   C. stay the same
   D. become zero
   Correct: B

2. In hydrostatic equilibrium, the correct interpretation is:
   A. pressure is absent
   B. gravity is absent
   C. the pressure gradient balances gravity locally
   D. the core temperature is directly measured
   Correct: C

3. At fixed pressure and density, a larger $\mu$ implies:
   A. lower $T$
   B. higher $T$
   C. no change in $T$
   D. higher $g$
   Correct: B

Short-answer checks

• Explain why a star can be in hydrostatic equilibrium and still have a large pressure gradient.
• Use $H_P = P/(\rho g)$ to explain what happens when gravity gets stronger.
• Write one sentence showing how the ideal gas law turns a support problem into a temperature estimate.

Exit ticket

1. State the hydrostatic-equilibrium equation in words or symbols.
2. Name one way the pressure scale height changes when the star becomes more compact.
3. In one sentence, explain why core temperature is an inference rather than a direct measurement in this demo.

Navigation

• Instructor hub: Instructor hub
• Back to guide: Guide
• Student demo: Student demo
• This demo: Model · Activities · Assessment · Backlog

Core support chain

The teaching goal is to move from a force-balance statement to a temperature scale:

$$\frac{dP}{dr} = -\frac{G M(r)\rho(r)}{r^2}$$

where $P$ is pressure, $r$ is radius, $G$ is the gravitational constant, $M(r)$ is enclosed mass, and $\rho(r)$ is local density.

The companion mass-continuity relation is:

$$\frac{dM}{dr} = 4\pi r^2 \rho(r)$$

Together these say that the required pressure gradient is set by the mass already inside the shell.

Pressure scale height

The pressure scale height is the local distance over which pressure changes by a factor of $e$:

$$H_P = -\left(\frac{d\ln P}{dr}\right)^{-1} = \frac{P}{\rho g}$$

with

$$g = \frac{G M(r)}{r^2}$$

This is the best bridge between “gravity got stronger” and “the star needs a steeper internal profile.”

Ideal gas closure

For the pedagogical closure used in this demo:

$$P = \frac{\rho k_B T}{\mu m_u}$$

so

$$T = \frac{\mu m_u}{k_B}\frac{P}{\rho} = \frac{\mu m_u}{k_B} g H_P$$

This is the key interpretive move: if the model shows a shorter scale height at similar density, the temperature scale has to respond.

Core-temperature estimate

A one-zone estimate for the core temperature should be presented as a scaling relation, not as an exact stellar-structure solution:

$$T_c \sim C\,\frac{\mu m_u}{k_B}\frac{G M}{R}$$

where $C$ is an order-unity structure factor that depends on the chosen density profile and boundary condition.

Useful instructor language:

• “The demo is showing the support temperature scale, not a nuclear-burning calculation.”
• “The exact value depends on structure, but the scaling with $\mu M/R$ is the physical takeaway.”
• “Hydrostatic equilibrium is local; it does not require flat pressure or zero pressure.”

What the demo should not imply

• It should not imply the star is literally static in every sense; hydrostatic equilibrium is a force balance, not a motion detector.
• It should not imply the core temperature is directly observed.
• It should not collapse composition into a vague modifier; $\mu$ must explicitly change the ideal-gas temperature inference.

Recommended sanity checks during class

• If $M$ rises at fixed $R$, then the required support should rise and $H_P$ should shrink.
• If $R$ shrinks at fixed $M$, then $g$ should rise roughly like $R^{-2}$ at the surface scale.
• If $\mu$ increases, the same pressure requires a higher temperature.
• A balanced state should still have a nonzero $\frac{dP}{dr}$.
• The core-temperature estimate should move in the same direction as $M/R$.

Simplified assumptions to state explicitly

• Spherical symmetry.
• 1D radial structure.
• Snapshot balance, not time-dependent collapse or expansion.
• Ideal-gas closure for the temperature inference.
• No nuclear burning, transport, rotation, or magnetic fields in the core inference loop.

Navigation

• Instructor hub: Instructor hub
• Back to guide: Guide
• Student demo: Student demo
• This demo: Model · Activities · Assessment · Backlog

P0 (blocking before launch-ready promotion)

• Run one classroom pilot and record timing, confusion points, and the exact sentence students used when they explained the support chain.
• Capture keyboard and screen-reader smoke notes so the live-region and readout language are classroom-safe.
• Verify that every visible readout names its units and that the core-temperature readout is clearly labeled as an inference.

P1 (important, non-blocking for candidate)

• Add one comparison preset for a more compact star so the gravity-to-temperature scaling is obvious in a single click.
• Add a short “what changed?” reflection card after the balance move to strengthen predict-play-explain cadence.
• Add export-friendly row labels for $g$, $H_P$, and $T_c$ if the demo runtime exposes a copy-results payload.

P2 (nice to have)

• Add an optional “composition microscope” that shows how $\mu$ changes the ideal-gas closure.
• Add a challenge prompt that compares two stars with the same mass but different radii.
• Add a one-page worksheet variant for stations that need a faster turnaround.