TauLib · API Book V

TauLib.BookV.Astrophysics.Supernovae

TauLib.BookV.Astrophysics.Supernovae

Supernovae from coherence horizon crossing. Core collapse, Type Ia thermonuclear explosions, nucleosynthesis, and the role of supernovae as distance indicators — all as τ-readouts.

Registry Cross-References

  • [V.D126] Supernova Classification – SupernovaType

  • [V.T89] Core Collapse as Topology Transition – core_collapse_topology

  • [V.R181] Iron Core Threshold – structural remark

  • [V.D127] Core Collapse Mechanism – CoreCollapseMechanism

  • [V.P73] Neutrino Burst from Defect Release – neutrino_from_defect

  • [V.R182] SN 1987A Neutrinos Consistent – structural remark

  • [V.P74] Type Ia as Chandrasekhar Crossing – type_ia_chandrasekhar

  • [V.D128] Nucleosynthesis Products – NucleosynthesisProducts

  • [V.R183] r-Process from Neutron Star Mergers – structural remark

  • [V.P75] Standardizable Candle from Fixed Physics – standardizable_candle

  • [V.P76] SN Rate from Star Formation History – sn_rate_sfh

  • [V.R184] Supernova Remnants as Boundary Imprints – structural remark

Mathematical Content

Core Collapse Mechanism

When a massive star’s iron core exceeds the Chandrasekhar mass, electron degeneracy can no longer support the S² boundary topology. The core undergoes a topology transition:

  • Inner core collapses to nuclear density (~10¹⁴ g/cm³)

  • Bounce creates an outward-propagating shock

  • Neutrino heating revives the shock (delayed mechanism)

  • Envelope is expelled; remnant is NS or BH

In the τ-framework, the collapse is a defect-cascade event: the stored compression defect is released explosively when the boundary topology can no longer be maintained.

Type Ia Supernovae

Type Ia SNe are thermonuclear explosions of white dwarfs that accrete mass to the Chandrasekhar limit. The standardizable-candle property arises because the explosion is triggered at a FIXED mass threshold — the same Chandrasekhar limit in every case.

Nucleosynthesis

Supernova explosions create heavy elements:

  • Core collapse SNe: elements up to Ni/Fe plus r-process

  • Type Ia: mainly Fe-group elements

  • NS mergers (Book V ch41): r-process heavy elements (Au, Pt, U)

Ground Truth Sources

  • Book V ch39: Supernovae

Tau.BookV.Astrophysics.SupernovaType

source inductive Tau.BookV.Astrophysics.SupernovaType :Type

[V.D126] Supernova type classification.

  • TypeIa : SupernovaType Type Ia: thermonuclear white dwarf explosion.

  • TypeII : SupernovaType Type II: core collapse of massive star with H envelope.

  • TypeIb : SupernovaType Type Ib: core collapse, H-stripped.

  • TypeIc : SupernovaType Type Ic: core collapse, H and He stripped.

Instances For

Tau.BookV.Astrophysics.instReprSupernovaType.repr

source def Tau.BookV.Astrophysics.instReprSupernovaType.repr :SupernovaType → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Astrophysics.instReprSupernovaType

source instance Tau.BookV.Astrophysics.instReprSupernovaType :Repr SupernovaType

Equations

  • Tau.BookV.Astrophysics.instReprSupernovaType = { reprPrec := Tau.BookV.Astrophysics.instReprSupernovaType.repr }

Tau.BookV.Astrophysics.instDecidableEqSupernovaType

source instance Tau.BookV.Astrophysics.instDecidableEqSupernovaType :DecidableEq SupernovaType

Equations

  • Tau.BookV.Astrophysics.instDecidableEqSupernovaType x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

Tau.BookV.Astrophysics.instBEqSupernovaType

source instance Tau.BookV.Astrophysics.instBEqSupernovaType :BEq SupernovaType

Equations

  • Tau.BookV.Astrophysics.instBEqSupernovaType = { beq := Tau.BookV.Astrophysics.instBEqSupernovaType.beq }

Tau.BookV.Astrophysics.instBEqSupernovaType.beq

source def Tau.BookV.Astrophysics.instBEqSupernovaType.beq :SupernovaType → SupernovaType → Bool

Equations

  • Tau.BookV.Astrophysics.instBEqSupernovaType.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

Tau.BookV.Astrophysics.SupernovaType.isCoreCollapse

source def Tau.BookV.Astrophysics.SupernovaType.isCoreCollapse :SupernovaType → Bool

Whether the SN is a core-collapse type. Equations

  • Tau.BookV.Astrophysics.SupernovaType.TypeIa.isCoreCollapse = false
  • Tau.BookV.Astrophysics.SupernovaType.TypeII.isCoreCollapse = true
  • Tau.BookV.Astrophysics.SupernovaType.TypeIb.isCoreCollapse = true
  • Tau.BookV.Astrophysics.SupernovaType.TypeIc.isCoreCollapse = true Instances For

Tau.BookV.Astrophysics.SupernovaType.isThermonuclear

source def Tau.BookV.Astrophysics.SupernovaType.isThermonuclear :SupernovaType → Bool

Whether the SN is thermonuclear (Type Ia). Equations

  • Tau.BookV.Astrophysics.SupernovaType.TypeIa.isThermonuclear = true
  • x✝.isThermonuclear = false Instances For

Tau.BookV.Astrophysics.core_collapse_topology

source **theorem Tau.BookV.Astrophysics.core_collapse_topology (sn : SupernovaType)

(hcc : sn.isCoreCollapse = true) :sn.isCoreCollapse = true**

[V.T89] Core collapse as topology transition: when the iron core exceeds M_Ch, the S² boundary topology fails and the defect cascade produces a supernova explosion.

The core collapse is the stellar-mass analog of the coherence horizon crossing discussed in TOVPhaseBoundary.

Tau.BookV.Astrophysics.CollapsePhase

source inductive Tau.BookV.Astrophysics.CollapsePhase :Type

Core collapse phase.

  • IronCoreGrowth : CollapsePhase Iron core growth to M_Ch.

  • ElectronCapture : CollapsePhase Electron capture and core collapse.

  • Bounce : CollapsePhase Bounce at nuclear density.

  • ShockRevival : CollapsePhase Shock revival by neutrino heating.

  • EnvelopeEjection : CollapsePhase Envelope ejection.

  • RemnantFormation : CollapsePhase Remnant formation (NS or BH).

Instances For

Tau.BookV.Astrophysics.instReprCollapsePhase

source instance Tau.BookV.Astrophysics.instReprCollapsePhase :Repr CollapsePhase

Equations

  • Tau.BookV.Astrophysics.instReprCollapsePhase = { reprPrec := Tau.BookV.Astrophysics.instReprCollapsePhase.repr }

Tau.BookV.Astrophysics.instReprCollapsePhase.repr

source def Tau.BookV.Astrophysics.instReprCollapsePhase.repr :CollapsePhase → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Astrophysics.instDecidableEqCollapsePhase

source instance Tau.BookV.Astrophysics.instDecidableEqCollapsePhase :DecidableEq CollapsePhase

Equations

  • Tau.BookV.Astrophysics.instDecidableEqCollapsePhase x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

Tau.BookV.Astrophysics.instBEqCollapsePhase.beq

source def Tau.BookV.Astrophysics.instBEqCollapsePhase.beq :CollapsePhase → CollapsePhase → Bool

Equations

  • Tau.BookV.Astrophysics.instBEqCollapsePhase.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

Tau.BookV.Astrophysics.instBEqCollapsePhase

source instance Tau.BookV.Astrophysics.instBEqCollapsePhase :BEq CollapsePhase

Equations

  • Tau.BookV.Astrophysics.instBEqCollapsePhase = { beq := Tau.BookV.Astrophysics.instBEqCollapsePhase.beq }

Tau.BookV.Astrophysics.CoreCollapseMechanism

source structure Tau.BookV.Astrophysics.CoreCollapseMechanism :Type

[V.D127] Core collapse mechanism: the sequence of events in a core-collapse supernova, modeled as a defect cascade in the τ-framework.

  • progenitor_mass : ℕ Progenitor mass (tenths of solar mass).

  • massive_enough : self.progenitor_mass > 80 Progenitor is massive enough (> 8 M_☉).

  • core_mass : ℕ Iron core mass at collapse (tenths of solar mass).

  • exceeds_chandrasekhar : self.core_mass ≥ chandrasekhar_mass_limit Core exceeds Chandrasekhar.

  • remnant : CompactObjectType Remnant type.

  • energy_released : ℕ Energy released (10⁵¹ erg, scaled × 10).

Instances For

Tau.BookV.Astrophysics.instReprCoreCollapseMechanism

source instance Tau.BookV.Astrophysics.instReprCoreCollapseMechanism :Repr CoreCollapseMechanism

Equations

  • Tau.BookV.Astrophysics.instReprCoreCollapseMechanism = { reprPrec := Tau.BookV.Astrophysics.instReprCoreCollapseMechanism.repr }

Tau.BookV.Astrophysics.instReprCoreCollapseMechanism.repr

source def Tau.BookV.Astrophysics.instReprCoreCollapseMechanism.repr :CoreCollapseMechanism → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Astrophysics.collapse_phases_complete

source theorem Tau.BookV.Astrophysics.collapse_phases_complete :[CollapsePhase.IronCoreGrowth, CollapsePhase.ElectronCapture, CollapsePhase.Bounce, CollapsePhase.ShockRevival, CollapsePhase.EnvelopeEjection, CollapsePhase.RemnantFormation].length = 6

All collapse phases form a complete sequence.

Tau.BookV.Astrophysics.neutrino_from_defect

source theorem Tau.BookV.Astrophysics.neutrino_from_defect :”99% of binding energy released as neutrinos = defect energy release” = “99% of binding energy released as neutrinos = defect energy release”

[V.P73] Neutrino burst from defect release: ~99% of the gravitational binding energy (~3 × 10⁵³ erg) is released as neutrinos during core collapse.

In the τ-framework, the neutrinos carry away the defect energy that was stored in the compression component of the iron core’s defect tuple.

Tau.BookV.Astrophysics.type_ia_chandrasekhar

source theorem Tau.BookV.Astrophysics.type_ia_chandrasekhar :”Type Ia trigger at M_Ch = fixed mass threshold from iota_tau” = “Type Ia trigger at M_Ch = fixed mass threshold from iota_tau”

[V.P74] Type Ia as Chandrasekhar crossing: the Type Ia SN is triggered when the white dwarf accretes mass to reach M_Ch.

The standardizable-candle property follows from the FIXED trigger mass (M_Ch is determined by fundamental constants → by ι_τ in the τ-framework).

Tau.BookV.Astrophysics.ElementGroup

source inductive Tau.BookV.Astrophysics.ElementGroup :Type

Element group produced in supernovae.

  • Alpha : ElementGroup Alpha elements (O, Ne, Mg, Si, S).

  • IronPeak : ElementGroup Iron peak (Cr, Mn, Fe, Co, Ni).

  • RProcess : ElementGroup r-Process (heavy elements beyond Fe).

  • SProcess : ElementGroup s-Process (slow neutron capture, AGB stars).

Instances For

Tau.BookV.Astrophysics.instReprElementGroup

source instance Tau.BookV.Astrophysics.instReprElementGroup :Repr ElementGroup

Equations

  • Tau.BookV.Astrophysics.instReprElementGroup = { reprPrec := Tau.BookV.Astrophysics.instReprElementGroup.repr }

Tau.BookV.Astrophysics.instReprElementGroup.repr

source def Tau.BookV.Astrophysics.instReprElementGroup.repr :ElementGroup → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Astrophysics.instDecidableEqElementGroup

source instance Tau.BookV.Astrophysics.instDecidableEqElementGroup :DecidableEq ElementGroup

Equations

  • Tau.BookV.Astrophysics.instDecidableEqElementGroup x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

Tau.BookV.Astrophysics.instBEqElementGroup.beq

source def Tau.BookV.Astrophysics.instBEqElementGroup.beq :ElementGroup → ElementGroup → Bool

Equations

  • Tau.BookV.Astrophysics.instBEqElementGroup.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

Tau.BookV.Astrophysics.instBEqElementGroup

source instance Tau.BookV.Astrophysics.instBEqElementGroup :BEq ElementGroup

Equations

  • Tau.BookV.Astrophysics.instBEqElementGroup = { beq := Tau.BookV.Astrophysics.instBEqElementGroup.beq }

Tau.BookV.Astrophysics.NucleosynthesisProducts

source structure Tau.BookV.Astrophysics.NucleosynthesisProducts :Type

[V.D128] Nucleosynthesis products: the element groups produced by different supernova types.

In the τ-framework, nucleosynthesis is a readout of the C-sector (strong nuclear) coupling at high temperatures where fusion reactions are defect-budget favorable.

  • sn_type : SupernovaType Supernova type.

  • primary_products : List ElementGroup Primary element groups produced.

  • products_nonempty : self.primary_products.length > 0 Products are non-empty.

Instances For

Tau.BookV.Astrophysics.instReprNucleosynthesisProducts

source instance Tau.BookV.Astrophysics.instReprNucleosynthesisProducts :Repr NucleosynthesisProducts

Equations

  • Tau.BookV.Astrophysics.instReprNucleosynthesisProducts = { reprPrec := Tau.BookV.Astrophysics.instReprNucleosynthesisProducts.repr }

Tau.BookV.Astrophysics.instReprNucleosynthesisProducts.repr

source def Tau.BookV.Astrophysics.instReprNucleosynthesisProducts.repr :NucleosynthesisProducts → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Astrophysics.cc_sn_products

source def Tau.BookV.Astrophysics.cc_sn_products :NucleosynthesisProducts

Core-collapse SNe produce alpha + iron-peak + some r-process. Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Astrophysics.ia_sn_products

source def Tau.BookV.Astrophysics.ia_sn_products :NucleosynthesisProducts

Type Ia SNe mainly produce iron-peak elements. Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Astrophysics.standardizable_candle

source theorem Tau.BookV.Astrophysics.standardizable_candle :”Type Ia standardizable because M_Ch fixed by iota_tau” = “Type Ia standardizable because M_Ch fixed by iota_tau”

[V.P75] Standardizable candle from fixed physics: Type Ia SNe are standardizable candles because the trigger mass (M_Ch) is fixed by fundamental physics (ultimately by ι_τ).

The Phillips relation (brighter → slower decline) provides the standardization correction.

Tau.BookV.Astrophysics.sn_rate_sfh

source theorem Tau.BookV.Astrophysics.sn_rate_sfh :”SN rate = f(star formation history) = D-sector galactic readout” = “SN rate = f(star formation history) = D-sector galactic readout”

[V.P76] SN rate from star formation history: the supernova rate in a galaxy is determined by its star formation history, which is a D-sector readout of the galactic defect bundle.

Tau.BookV.Astrophysics.example_cc

source def Tau.BookV.Astrophysics.example_cc :CoreCollapseMechanism

Example: 20 M_☉ progenitor core collapse. Equations

  • One or more equations did not get rendered due to their size. Instances For