SCIENCE · INVESTIGATE AND EXPLAIN

Integrate stellar support rather than equating force with pressure

Hydrostatic equilibrium balances a pressure gradient against inward gravity: dP/dr=−GM(r)ρ/r². Pressure itself is not a net force; differences across a layer provide the force per volume. A stable average structure can still contain microscopic thermal motion, radiation transport and convection.

Learning outcome: Check center, midpoint and surface pressure, then compare mass and radius scalings. Retain the separate structural, equation-of-state and energy-supply assumptions.

Starting recap

Keep geometry, luminosity, support, energy, evolutionary stage and population assumptions distinct.

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Try

At M=2 and R=1, density is 3/(2π), central pressure 3/(2π), and normalized central temperature 1. At half radius the pressure is three quarters of its central value. Surface pressure reaches the stipulated zero boundary.

Open the experiment

Predict

Why must the enclosed mass, rather than the whole star’s mass, enter an interior gravitational acceleration?

Your prediction and reason

Experiment

Check center, midpoint and surface pressure, then compare mass and radius scalings. Retain the separate structural, equation-of-state and energy-supply assumptions.

Integrate a hydrostatic pressure gradient

Worked teaching model · synthetic values · Original uniform density, Newtonian G=1, surface pressure zero: P(r)=3M²/(8piR⁴)[1−(r/R)²]. Ideal-gas pressure/density ratio uses normalized gas constant 1; uniform density is not a self-consistent actual stellar structure model.

Uniform sphere mass: 2 normalized mass Sphere radius: 1 normalized length

Reset keeps your prediction, captures and practice history.

Hydrostatic support is a pressure gradient balancing gravity, not a lack of internal motion.

Original uniform density, Newtonian G=1, surface pressure zero: P(r)=3M²/(8piR⁴)[1−(r/R)²]. Ideal-gas pressure/density ratio uses normalized gas constant 1; uniform density is not a self-consistent actual stellar structure model.

Original conditional model ledger
Quantity	Value	Unit
Uniform mean density	0.4774648	mass/length³
Central pressure	0.4774648	pressure
Central ideal pressure/density temperature	1	normalized temperature
Surface pressure	0	pressure

Read plotted point values

Radius / supplied length	Pressure
0	0.4774648
0.1000000	0.4726902
0.2000000	0.4583662
0.3000000	0.4344930
0.4000000	0.4010705
0.5000000	0.3580986
0.6000000	0.3055775
0.7000000	0.2435071
0.8000000	0.1718873
0.9000000	0.09071832
1	0

Original model records, not measured stars, a self-consistent stellar evolution code or independent observing competence.

Hydrostatic support is a pressure gradient balancing gravity, not a lack of internal motion.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

1. Define the stellar inference

2. Connect mechanism and assumptions

The original uniform-density sphere has M(r)=M(r/R)³, density 3M/(4πR³), surface pressure zero and G=1. Integrating gives P(r)=3M²/(8πR⁴)[1−(r/R)²]. Central pressure therefore scales as M²/R⁴; using total M instead of enclosed M(r) at every interior radius would produce the wrong profile.

A separate ideal-gas relation with normalized gas constant one gives central temperature Pc/ρ=M/(2R). The uniform-density approximation and this ideal relation are teaching constraints, not a self-consistent real stellar equation of state or an energy-transport solution. Radiation and degeneracy pressure require different terms where important.

Fusion supplies energy that can sustain a main-sequence thermal structure, but the immediate mechanical support is the pressure gradient. Energy flows outward through radiative transfer and, where unstable, convection. Hydrostatic balance alone neither identifies the energy source nor gives the star’s remaining lifetime.

3. Work the supplied case

4. Predict and investigate

Why must the enclosed mass, rather than the whole star’s mass, enter an interior gravitational acceleration?

Check center, midpoint and surface pressure, then compare mass and radius scalings. Retain the separate structural, equation-of-state and energy-supply assumptions.

Apply the evidence to a changed case

At M=2 and R=2, compare central pressure and temperature with the R=1 case.

Retain the evidence frame, calculations and assumptions before opening public feedback.

Compare a worked response

Doubling R at fixed mass divides central pressure by 16 and density by 8, so Pc/ρ and central temperature halve. These are conditional uniform-sphere scalings, not a complete expansion track.

Source and boundaries

Primary source. Checked 31 October 2026. All numerical records and diagrams here are original supplied models, not measured stellar populations or a full stellar-evolution solver. No direct observing or physical experiment is requested.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

At M=2 and R=2, compare central pressure and temperature with the R=1 case.

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

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Previous: Distinguish bound cloud energy from actual star formation Next: Separate a star’s fuel inventory from its mass defect

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