51 lines
2.9 KiB
Markdown
51 lines
2.9 KiB
Markdown
# Explicit rotating n=3 physical-family control
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Branch `research/n3-compatible-family-20260919`; parent `6d423cb`.
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This changes experimental physical inputs, not solver equations or production
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defaults. It must not be presented as a preconditioner speedup comparison.
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The driver normally chooses K for a nonrotating mass-one Lane–Emden star.
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The sharp gravitational-energy inequality and the steady virial identity
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require, for n=3 and a nonnegative exact-EOS density,
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beta = T/|W| <= [1 - (M_LE(K)/M)^(2/3)]/2.
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At exactly M=M_LE(K), nonzero rotation is impossible. An independent DOP853
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Lane–Emden integration with five tolerance/step/start controls gives the
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actual driver's K=0.36393905995216302 a Lane–Emden mass
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0.9999999918534539. Thus the numerical K is slightly on the permissible
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side, but beta cannot exceed approximately 2.71552e-9. The control-to-control
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mass spread is 1.95e-14; this is a numerical sensitivity observation, not a
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rigorous interval enclosure. The intended moderate rotation cannot be reached
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while retaining that essentially critical mass/K combination.
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The optional experiment flag `--n3-K-factor f` multiplies K by a positive
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factor, with index required to be exactly three when f differs from one.
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The default is one. The analytic seed central density is multiplied by
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f^(3/2), keeping its radius equal to the unchanged reference radius. Its
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mass becomes approximately f^(3/2), while the target mass remains one.
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This is a deliberately off-constraint initial guess, not a density rescaling
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during Newton. The public EOS and independent density formulation are unchanged.
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The seed's Bernoulli scalar continues to use the target mass; its discrepancy
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is part of the initial nonlinear residual and is not hidden by this control.
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The bounded first case uses f=.99, J=.035, no central border, the existing
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gravity-Schur preconditioner and gravity-curvature trial path, 10 updates,
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eta=.03 and absolute nonlinear target 1e-12. The necessary beta bound is then
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0.00500000269. This does not guarantee existence, convergence, or identify
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a fraction of critical rotation. J already breaks the nonrotating n=3
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homology, so this case does not add an extra central-density phase condition.
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The energy inequality's assumptions are not automatically satisfied by the
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discrete independent rho/P(h) fields, approximate gravity, or a central
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artificial forcing. This compatibility argument selects a sensible continuum
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target; it is not a new density limiter or a substitute for numerical tests.
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Primary source: Cheng, Cheng and Lin, equations (1.14)–(1.15) and Theorem 3.1,
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[Expanding solutions near unstable Lane–Emden stars](https://arxiv.org/html/2305.10218#S3.SS1).
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The displayed beta bound is our consequence of their inequality and
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2T+W+3 integral(P)=0. Their G=1 convention matches this diagnostic.
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Independent integration evidence is retained in
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`../../diagnostics/n3_mass_compatibility_2026-09-19/`.
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