proposition 52.1 Hydrostatic equilibrium of a self-gravitating sphere

open in the book · parts/05-general-relativity-cosmology/11-stellar-structure-nucleosynthesis.tex:46 · p. 1387

Rests on

depends_on proposition 31.16 Hydrostatic balance ¶
- depends_on definition 31.1 Fluid ¶
  - depends_on theorem 30.18 Cauchy: the stress tensor exists ¶
    - depends_on definition 30.17 Traction ¶
      - depends_on definition 7.126 Line and surface integrals ¶
      - depends_on definition 30.2 Representative volume element; continuum field ¶
    - depends_on definition 13.5 Tensor under orthogonal transformations ¶
      - depends_on definition 13.4 Orthogonal coordinate transformation ¶
    - proves proof ch:13-continuum-elasticity@proof-7 ¶
- depends_on theorem 30.21 Cauchy's equation of motion ¶
  - depends_on theorem 7.133 Gauss ¶
    - depends_on definition 7.127 Simple regions ¶
      - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 6.9 Compact set ¶
    - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on definition 7.125 Multiple integral ¶
      - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
    - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - depends_on definition 7.41 Antiderivative ¶
      - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - proves proof ch:05-real-analysis@proof-25 ¶
    - proves proof ch:05-real-analysis@proof-79 ¶
  - depends_on theorem 30.18 Cauchy: the stress tensor exists ¶ ↺
  - proves proof ch:13-continuum-elasticity@proof-9 ¶
- proves proof ch:14-fluid-dynamics@proof-7 ¶
proves proof ch:11-stellar-structure-nucleosynthesis@proof-1 ¶

Supports

depends_on phenomenon 52.12 The mass–luminosity relation ¶
- depends_on phenomenon 52.17 The Hertzsprung–Russell diagram ¶
depends_on proposition 49.3 Lane–Emden form of hydrostatic equilibrium; restated from the stellar-structure chapter ¶
- depends_on theorem 49.7 The Chandrasekhar mass ¶
  - depends_on phenomenon 49.6 A maximum mass for a cold degenerate star ¶
depends_on theorem 52.2 Virial theorem for a self-gravitating sphere ¶
- depends_on phenomenon 52.4 The Sun has shone for billions of years ¶
  - depends_on phenomenon 52.23 The Sun's energy budget audited reaction by reaction ¶
- depends_on proposition 52.18 Schönberg–Chandrasekhar limit ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Hydrostatic balance	declared	parts/05-general-relativity-cosmology/11-stellar-structure-nucleosynthesis.tex:55
`depends_on`	←	The mass–luminosity relation	declared	parts/05-general-relativity-cosmology/11-stellar-structure-nucleosynthesis.tex:615
`depends_on`	←	Lane–Emden form of hydrostatic equilibrium; restated from the stellar-structure chapter	declared	parts/05-general-relativity-cosmology/08-compact-stars.tex:252
`depends_on`	←	Virial theorem for a self-gravitating sphere	declared	parts/05-general-relativity-cosmology/11-stellar-structure-nucleosynthesis.tex:100
`proves`	←	ch:11-stellar-structure-nucleosynthesis@proof-1	declared	parts/05-general-relativity-cosmology/11-stellar-structure-nucleosynthesis.tex:58