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AST-25

Stellar hydrostatic snapshot

dP/dr = − G M ρ / r². Pressure gradient that holds a star up.

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StarsHydrostatic

Governing equation

dPdr=−GMρr2\dfrac{dP}{dr}=-\dfrac{GM\rho}{r^2}

r∈[0.05,10]r \in \left[0.05, 10\right]

where

M
Enclosed mass (M_\odot)
\rho
Density (kg/m³)
r
Radius (R_\odot)
dP/dr
Pressure gradient (Pa/m)

The figure, part by part

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AST-25 · star
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①

Governing law — red

dPdr=−GMρr2\dfrac{dP}{dr}=-\dfrac{GM\rho}{r^2}

r∈[0.05,10]r \in \left[0.05, 10\right]

r=0.05r = 0.05dP/dr=−1.54×108dP/dr = -1.54\times 10^{8}

②

Domain of the variable — blue

rmin⁡=0.05,rmax⁡=10r_{\min} = 0.05,\quad r_{\max} = 10

r∈[0.05,10]r \in \left[0.05, 10\right]

r=0.05r = 0.05r=0.05r = 0.05

③

Response curve — green

dPdr=−GMρr2\dfrac{dP}{dr}=-\dfrac{GM\rho}{r^2}

r∈[0.05,10]r \in \left[0.05, 10\right]

r=0.05r = 0.05dP/dr=−1.54×108dP/dr = -1.54\times 10^{8}

④

Rate of change — amber

ddP/drdr\frac{\mathrm{d}dP/dr}{\mathrm{d}r}

r∈[0.05,10]r \in \left[0.05, 10\right]

r=0.113r = 0.113dP/dr′=1.13×109dP/dr^{\prime} = 1.13\times 10^{9}

Lecture brief

Historical brief

Hubble expansion, Jeans collapse, Eddington luminosity, Bondi accretion and Stefan–Boltzmann stars are the first astrophysical budgets. The sheets scale a star, a cloud and a horizon. This sheet (AST-25 — Stellar hydrostatic snapshot) is the form associated with Hydrostatic. Working symbols: MM, ρ\rho, rr →\rightarrow dP/drdP/dr. The first stellar-structure equation. Combined with mass continuity and energy, it builds a star.

Purpose

Purpose: compute dP/drdP/dr from MM, ρ\rho, rr in Astrophysics via dPdr=−GMρr2\dfrac{dP}{dr}=-\dfrac{GM\rho}{r^2} dP/dr = − G M ρ / r². Pressure gradient that holds a star up. Use it when a real astrophysics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given M=1.000 ModotM = 1.000\,\mathrm{M_odot}, ρ=1400.000 kg/m3\rho = 1400.000\,\mathrm{kg/m^{3}}, r=0.500 Rodotr = 0.500\,\mathrm{R_odot}, the governing relation dPdr=−GMρr2\dfrac{dP}{dr}=-\dfrac{GM\rho}{r^2} yields dP/dr=−1.536e+6 Pa/mdP/dr = -1.536e+6\,\mathrm{Pa/m}. A star, a radial pressure arrow, gravity inward. Move a slider: the numbers are this situation, not a canned story.

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Inputs

Outputs

  • Pressure gradient dP/dr-1535567.059 Pa/m
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AST-25 · star
00:0 / 00:08

Narration of this film

A star, a radial pressure arrow, gravity inward.

The first stellar-structure equation. Combined with mass continuity and energy, it builds a star.

Reading speed

Watch on YouTube