Hydrostatic Equilibrium Explorer

Hydrostatic Reasoning Lab

Pressure support comes from a gradient, not just large pressure.

Move one shell first, then change compactness and structure to connect local force balance to global stellar support.

Step 1 of 7 Predict

Start with a prediction

If the same mass is squeezed into a smaller radius, what happens to the required central pressure?

Local shell thought experiment Toy model

Local shell force balance

Match the cutaway shell to the inset patch: equal pressure cancels, but a pressure difference can balance local weight.

$r/R = 0.55$ Uniform-density toy Balanced
Pressure forces Gravity Net response

Notice Student mode

What to notice now

    What changed

    Why

    Infer

    Reference hydrostatic profile Model context

    How that shell fits into the full star

    The shell guide marks the same radius shown in the cutaway, so one local observation can be placed in the whole star.

    Selected quantity
    $0$
    Current plot
    $P(r)$

    Thermal bridge Scale estimate

    Turn support into a temperature scale

    Separate the exact local law from the global scale estimate, then connect the support requirement to gas temperature.

    Exact local law: $dP/dr=-\rho g$ tells you what balance must hold at one shell.
    Global support scale: $P_c\sim GM^2/R^4$ estimates how demanding the full-star support problem is.
    Ideal-gas bridge: $P\sim \rho k_B T/(\mu m_p)$ turns support into a required thermal scale.

    Synthesis Explain the chain

    Write the full causal chain

    Use one local shell observation, one global scaling statement, and one thermal bridge statement.

    Mass M_sun
    $1.00$
    Radius R_sun
    $1.00$
    Mean density g cm^-3
    $1.41$
    Local gravity cm s^-2
    $0$
    Exact central pressure dyne cm^-2
    $0$
    Core-temperature scale K
    $0$
    Checkpoint deck synthesis below fold

    Checkpoint deck

    1 / 5

    Answer before checking. Use one local observation and one global scaling idea as evidence.

    What to notice 3 prompts
    • Uniform pressure does not create a net outward support force for a local gas patch. A pressure gradient does.
    • Switching from a uniform-density toy star to a centrally concentrated toy star raises the exact central pressure even when the total mass and radius stay fixed.
    • Halving the radius at fixed mass leaves the gravity problem much harder: the required pressure scale jumps by $16\times$.
    Model notes toy models + units
    • This demo uses toy models with CGS units internally: mass in g, radius in cm, density in g cm$^{-3}$, and pressure in dyne cm$^{-2}$.
    • Uniform-density toy model: $\rho(r)=\text{constant}$, so $M(r)\propto r^3$ and $g(r)\propto r$.
    • Centrally concentrated toy model: $\rho(r)=\rho_c\left[1-(r/R)^2\right]$ with normalization chosen so the total mass still equals the selected $M$.
    • The local shell inset uses a thin patch with equal face area, so it teaches the local meaning of $dP/dr=-\rho g$ directly.
    • Under-supported and over-supported modes perturb only the local shell patch. The global radial profiles stay hydrostatic so you can compare equilibrium to imbalance honestly.
    Support is not fusion common confusion

    Thermal pressure can support a star against gravity, but hydrostatic equilibrium by itself does not tell you where the heat came from. Main-sequence stars stay hot because nuclear fusion replenishes the energy that the star radiates away.

    That is why hydrostatic equilibrium does not mean the star is inert. It means the inward and outward forces nearly balance at each radius while the star evolves more slowly through changes in composition and energy flow.

    Derivation sketches compactness ladder

    Begin with the local force-balance law:

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

    Then estimate the order of magnitude by taking $dP/dr\sim P_c/R$ and $\rho\sim M/R^3$:

    $$\frac{P_c}{R}\sim \frac{G M}{R^2}\frac{M}{R^3} \quad \Rightarrow \quad P_c\sim\frac{G M^2}{R^4}$$

    If gas pressure supplies that support, use $P\sim \rho k_B T/(\mu m_p)$ with $\rho\sim M/R^3$:

    $$T_c\sim\frac{\mu G M m_p}{k_B R}$$