TauLib · API Book V

TauLib.BookV.Astrophysics.AccretionJets

TauLib.BookV.Astrophysics.AccretionJets

Accretion disk physics and jet formation from bipolar topology. The Eddington limit, disk structure, and relativistic jets emerge from the τ-framework’s bipolar lemniscate boundary L = S¹ ∨ S¹.

Registry Cross-References

  • [V.P77] Accretion as Defect Infall – accretion_as_defect_infall

  • [V.D129] Accretion Disk Structure – AccretionDiskStructure

  • [V.R185] Shakura-Sunyaev from D-Sector Viscosity – structural remark

  • [V.D130] Eddington Limit Data – EddingtonLimitData

  • [V.P78] Eddington from Sector Balance – eddington_sector_balance

  • [V.T90] Bipolar Jet Theorem – bipolar_jet_theorem

  • [V.R186] Lemniscate Topology Forces Bipolarity – structural remark

  • [V.T91] Jet Power from Spin Readout – jet_power_from_spin

  • [V.R187] Blandford-Znajek as Readout Mechanism – structural remark

  • [V.D131] Jet Collimation Data – JetCollimationData

  • [V.R188] AGN Unification from Viewing Angle – structural remark

  • [V.P79] Quasar Luminosity from Accretion Rate – quasar_luminosity

  • [V.R189] Feedback from Jet-ISM Interaction – structural remark

  • [V.D132] AGN Classification – AGNType

  • [V.T92] Accretion Efficiency Bound – accretion_efficiency_bound

  • [V.R190] Efficiency Exceeds Nuclear – structural remark

  • [V.R191] Accretion-Jet Cycle as Defect Pump – structural remark

Mathematical Content

Accretion Disk Structure

Accretion disks form when matter spiraling toward a compact object has sufficient angular momentum. In the τ-framework, the disk is a steady-state defect flow where:

  • Gravitational defect (D-sector) drives inward flow

  • Angular momentum defect resists compression

  • Viscous defect transport mediates angular momentum outward

Bipolar Jet Formation

Jets are bipolar because the lemniscate boundary L = S¹ ∨ S¹ has TWO lobes. The disk plane coincides with the crossing point of L. Material can only escape along the two polar directions — the lobe axes. This is a TOPOLOGICAL prediction, not a dynamical one.

Eddington Limit

The Eddington luminosity L_Edd = 4πGMm_pc/σ_T is the balance between D-sector (gravity) and B-sector (EM radiation pressure). Above L_Edd, radiation pressure disrupts the accretion flow.

Accretion Efficiency

Gravitational accretion can convert up to ~40% of rest mass to radiation (for maximally spinning BH), far exceeding nuclear fusion efficiency (~0.7%). This is the most efficient energy extraction mechanism in nature.

Ground Truth Sources

  • Book V ch40: Accretion and Jets

Tau.BookV.Astrophysics.accretion_as_defect_infall

source theorem Tau.BookV.Astrophysics.accretion_as_defect_infall :”Accretion = defect-bundle infall along D-sector coupling gradient” = “Accretion = defect-bundle infall along D-sector coupling gradient”

[V.P77] Accretion as defect infall: matter accreting onto a compact object is defect-bundle material flowing down the D-sector coupling gradient.

The accretion rate is determined by the defect-transport rate through the angular-momentum barrier.

Tau.BookV.Astrophysics.DiskModel

source inductive Tau.BookV.Astrophysics.DiskModel :Type

Disk model type.

  • ThinDisk : DiskModel Thin disk (Shakura-Sunyaev): H/R « 1.

  • ThickDisk : DiskModel Thick disk (torus/ADAF): H/R ~ 1.

  • SlimDisk : DiskModel Slim disk: intermediate, radiation-trapped.

Instances For

Tau.BookV.Astrophysics.instReprDiskModel

source instance Tau.BookV.Astrophysics.instReprDiskModel :Repr DiskModel

Equations

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

Tau.BookV.Astrophysics.instReprDiskModel.repr

source def Tau.BookV.Astrophysics.instReprDiskModel.repr :DiskModel → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instDecidableEqDiskModel

source instance Tau.BookV.Astrophysics.instDecidableEqDiskModel :DecidableEq DiskModel

Equations

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

Tau.BookV.Astrophysics.instBEqDiskModel.beq

source def Tau.BookV.Astrophysics.instBEqDiskModel.beq :DiskModel → DiskModel → Bool

Equations

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

Tau.BookV.Astrophysics.instBEqDiskModel

source instance Tau.BookV.Astrophysics.instBEqDiskModel :BEq DiskModel

Equations

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

Tau.BookV.Astrophysics.AccretionDiskStructure

source structure Tau.BookV.Astrophysics.AccretionDiskStructure :Type

[V.D129] Accretion disk structure: parametrization of the steady-state accretion disk around a compact object.

All disk properties are readouts of the D-sector coupling combined with angular momentum conservation.

  • central_object : CompactObjectType Central object type.

  • model : DiskModel Disk model.

  • inner_radius : ℕ Inner disk radius (Schwarzschild radii, scaled × 10).

  • inner_pos : self.inner_radius > 0 Inner radius positive.

  • accretion_rate : ℕ Accretion rate (scaled, 10⁻⁸ M_☉/yr × 100).

  • efficiency_permil : ℕ Radiative efficiency (percent × 10).

Instances For

Tau.BookV.Astrophysics.instReprAccretionDiskStructure.repr

source def Tau.BookV.Astrophysics.instReprAccretionDiskStructure.repr :AccretionDiskStructure → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instReprAccretionDiskStructure

source instance Tau.BookV.Astrophysics.instReprAccretionDiskStructure :Repr AccretionDiskStructure

Equations

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

Tau.BookV.Astrophysics.EddingtonLimitData

source structure Tau.BookV.Astrophysics.EddingtonLimitData :Type

[V.D130] Eddington limit data: the maximum luminosity at which radiation pressure (B-sector) balances gravity (D-sector).

L_Edd = 4πGMm_pc / σ_T ≈ 1.26 × 10³⁸ (M/M_☉) erg/s.

  • mass_tenth_solar : ℕ Mass of the accreting object (tenths of solar mass).

  • mass_pos : self.mass_tenth_solar > 0 Mass positive.

  • l_edd_scaled : ℕ Eddington luminosity (10³⁸ erg/s, scaled × 100).

  • is_super_eddington : Bool Whether the source is super-Eddington.

Instances For

Tau.BookV.Astrophysics.instReprEddingtonLimitData.repr

source def Tau.BookV.Astrophysics.instReprEddingtonLimitData.repr :EddingtonLimitData → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instReprEddingtonLimitData

source instance Tau.BookV.Astrophysics.instReprEddingtonLimitData :Repr EddingtonLimitData

Equations

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

Tau.BookV.Astrophysics.eddington_sector_balance

source theorem Tau.BookV.Astrophysics.eddington_sector_balance :”Eddington limit = D-sector (gravity) balanced by B-sector (radiation)” = “Eddington limit = D-sector (gravity) balanced by B-sector (radiation)”

[V.P78] Eddington from sector balance: the Eddington limit is the balance point between D-sector (gravity) and B-sector (electromagnetic radiation pressure). Two sectors, one limit.

Super-Eddington accretion is possible when photon trapping reduces the effective radiation pressure (slim disk regime).

Tau.BookV.Astrophysics.bipolar_jet_theorem

source theorem Tau.BookV.Astrophysics.bipolar_jet_theorem :”Jets are always bipolar: 2 lobes of L = S^1 v S^1” = “Jets are always bipolar: 2 lobes of L = S^1 v S^1”

[V.T90] Bipolar jet theorem: relativistic jets from accreting compact objects are ALWAYS bipolar (two opposing jets) because the lemniscate boundary L = S¹ ∨ S¹ has exactly two lobes.

The disk plane contains the crossing point of L. The jet axes align with the lobe axes.

This is a topological prediction: jets cannot be unipolar or have more than two lobes in the τ-framework.

Tau.BookV.Astrophysics.jet_power_from_spin

source theorem Tau.BookV.Astrophysics.jet_power_from_spin :”P_jet proportional to a^2B^2M^2 = spin readout of T^2 rotation” = “P_jet proportional to a^2B^2M^2 = spin readout of T^2 rotation”

[V.T91] Jet power from spin readout: the mechanical power of a relativistic jet scales with the spin of the central BH:

P_jet ∝ a² · B² · M²

where a = dimensionless spin parameter, B = magnetic field strength, M = BH mass.

In the τ-framework, the spin is a rotation index of the torus vacuum T², and the jet power is its D-sector readout.

Tau.BookV.Astrophysics.JetCollimationData

source structure Tau.BookV.Astrophysics.JetCollimationData :Type

[V.D131] Jet collimation data: the opening angle and extent of a relativistic jet, determined by the lemniscate boundary geometry and the ambient pressure profile.

  • half_angle : ℕ Opening half-angle (degrees × 10).

  • extent_kpc : ℕ Jet extent (kpc, scaled × 10).

  • lorentz_factor : ℕ Lorentz factor (bulk).

  • lorentz_ge_one : self.lorentz_factor ≥ 1 Lorentz factor at least 1.

  • is_relativistic : Bool Whether the jet is relativistic (Γ > 2).

Instances For

Tau.BookV.Astrophysics.instReprJetCollimationData

source instance Tau.BookV.Astrophysics.instReprJetCollimationData :Repr JetCollimationData

Equations

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

Tau.BookV.Astrophysics.instReprJetCollimationData.repr

source def Tau.BookV.Astrophysics.instReprJetCollimationData.repr :JetCollimationData → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.AGNType

source inductive Tau.BookV.Astrophysics.AGNType :Type

[V.D132] AGN classification: active galactic nuclei classified by accretion rate and viewing angle.

In the τ-framework, the AGN “zoo” is a single phenomenon (accretion + jets around a supermassive BH) viewed from different angles and accretion states.

  • Seyfert1 : AGNType Seyfert 1: face-on view, broad lines visible.

  • Seyfert2 : AGNType Seyfert 2: edge-on, broad lines obscured.

  • Quasar : AGNType Quasar: high-luminosity AGN.

  • Blazar : AGNType Blazar: jet pointed at observer.

  • RadioGalaxy : AGNType Radio galaxy: powerful jets, large lobes.

  • LINER : AGNType LINER: low-ionization nuclear emission region.

Instances For

Tau.BookV.Astrophysics.instReprAGNType.repr

source def Tau.BookV.Astrophysics.instReprAGNType.repr :AGNType → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instReprAGNType

source instance Tau.BookV.Astrophysics.instReprAGNType :Repr AGNType

Equations

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

Tau.BookV.Astrophysics.instDecidableEqAGNType

source instance Tau.BookV.Astrophysics.instDecidableEqAGNType :DecidableEq AGNType

Equations

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

Tau.BookV.Astrophysics.instBEqAGNType.beq

source def Tau.BookV.Astrophysics.instBEqAGNType.beq :AGNType → AGNType → Bool

Equations

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

Tau.BookV.Astrophysics.instBEqAGNType

source instance Tau.BookV.Astrophysics.instBEqAGNType :BEq AGNType

Equations

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

Tau.BookV.Astrophysics.quasar_luminosity

source theorem Tau.BookV.Astrophysics.quasar_luminosity :”L_quasar = etaMdotc^2, eta ~ 0.1 for thin disk accretion” = “L_quasar = etaMdotc^2, eta ~ 0.1 for thin disk accretion”

[V.P79] Quasar luminosity from accretion rate: quasar luminosities (up to 10⁴⁷ erg/s) derive from accretion onto supermassive BH (10⁸-10⁹ M_☉) at near-Eddington rates.

L_quasar = η · M_dot · c² where η ~ 0.1 for a thin disk.

Tau.BookV.Astrophysics.nuclear_efficiency

source def Tau.BookV.Astrophysics.nuclear_efficiency :ℕ

Nuclear fusion efficiency (percent × 10). Equations

  • Tau.BookV.Astrophysics.nuclear_efficiency = 7 Instances For

Tau.BookV.Astrophysics.max_accretion_efficiency

source def Tau.BookV.Astrophysics.max_accretion_efficiency :ℕ

Maximum accretion efficiency (percent × 10). Equations

  • Tau.BookV.Astrophysics.max_accretion_efficiency = 400 Instances For

Tau.BookV.Astrophysics.accretion_efficiency_bound

source theorem Tau.BookV.Astrophysics.accretion_efficiency_bound :nuclear_efficiency < max_accretion_efficiency

[V.T92] Accretion efficiency bound: gravitational accretion efficiency (up to ~40% for max spin) greatly exceeds nuclear fusion efficiency (~0.7%).

This is the most efficient energy extraction mechanism and explains why quasars outshine their host galaxies despite accreting modest mass rates.

Tau.BookV.Astrophysics.stellar_bh_disk

source def Tau.BookV.Astrophysics.stellar_bh_disk :AccretionDiskStructure

Example: thin disk around stellar-mass BH. Equations

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

Tau.BookV.Astrophysics.m87_jet

source def Tau.BookV.Astrophysics.m87_jet :JetCollimationData

Example: M87 jet. Equations

  • Tau.BookV.Astrophysics.m87_jet = { half_angle := 10, extent_kpc := 500, lorentz_factor := 6, lorentz_ge_one := Tau.BookV.Astrophysics.m87_jet._proof_2, is_relativistic := true } Instances For

Tau.BookV.Astrophysics.ToroidalFluxIntegral

source structure Tau.BookV.Astrophysics.ToroidalFluxIntegral :Type

[V.D285] Toroidal Flux Integral: magnetic flux through a meridional cross-section of the torus, measuring the poloidal field component.

  • surface : String Description of the flux surface.

  • flux_x1000 : ℕ Flux value (arbitrary units × 1000).

  • flux_nonneg : self.flux_x1000 ≥ 0 Flux is non-negative.

Instances For

Tau.BookV.Astrophysics.instReprToroidalFluxIntegral

source instance Tau.BookV.Astrophysics.instReprToroidalFluxIntegral :Repr ToroidalFluxIntegral

Equations

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

Tau.BookV.Astrophysics.instReprToroidalFluxIntegral.repr

source def Tau.BookV.Astrophysics.instReprToroidalFluxIntegral.repr :ToroidalFluxIntegral → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.PoloidalFluxIntegral

source structure Tau.BookV.Astrophysics.PoloidalFluxIntegral :Type

[V.D286] Poloidal Flux Integral: magnetic flux through the torus hole, topologically protected in ideal MHD. Requires genus ≥ 1.

  • surface : String Description of the flux surface.

  • flux_x1000 : ℕ Flux value (arbitrary units × 1000).

  • topo_protected : ℕ Topologically protected (1 = yes).

Instances For

Tau.BookV.Astrophysics.instReprPoloidalFluxIntegral.repr

source def Tau.BookV.Astrophysics.instReprPoloidalFluxIntegral.repr :PoloidalFluxIntegral → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instReprPoloidalFluxIntegral

source instance Tau.BookV.Astrophysics.instReprPoloidalFluxIntegral :Repr PoloidalFluxIntegral

Equations

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

Tau.BookV.Astrophysics.flux_threading_theorem

source theorem Tau.BookV.Astrophysics.flux_threading_theorem :”Φ_pol(T²) topologically protected; Φ_hole(S²) = 0” = “Φ_pol(T²) topologically protected; Φ_hole(S²) = 0”

[V.T228] Flux Threading Theorem: both toroidal and poloidal fluxes are conserved on T². Poloidal flux is topologically protected. On S², there is no topological flux (H_1(S²) = 0).

Tau.BookV.Astrophysics.homology_rank_t2_vs_s2

source theorem Tau.BookV.Astrophysics.homology_rank_t2_vs_s2 :2 > 0

H_1(T²) ≅ Z² (rank 2 homology), H_1(S²) = 0 (rank 0).

Tau.BookV.Astrophysics.FluxRatio

source structure Tau.BookV.Astrophysics.FluxRatio :Type

[V.P153] Flux Ratio: Φ_pol/Φ_tor ~ ι_τ ≈ 0.341 from area ratio.

  • phi_pol_x1000 : ℕ Poloidal flux (units × 1000).

  • phi_tor_x1000 : ℕ Toroidal flux (units × 1000).

  • tor_pos : self.phi_tor_x1000 > 0 Toroidal flux is positive.

  • ratio_x1000 : ℕ Ratio × 1000 (should be ≈ 341).

Instances For

Tau.BookV.Astrophysics.instReprFluxRatio

source instance Tau.BookV.Astrophysics.instReprFluxRatio :Repr FluxRatio

Equations

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

Tau.BookV.Astrophysics.instReprFluxRatio.repr

source def Tau.BookV.Astrophysics.instReprFluxRatio.repr :FluxRatio → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.JetPoloidalField

source structure Tau.BookV.Astrophysics.JetPoloidalField :Type

[V.D289] Jet Poloidal Field: axial B-field component in the jet, originating from topologically protected flux through the torus hole.

  • source : String Source name.

  • b_z_base_x100 : ℕ Poloidal field at base (Gauss × 100).

  • b_phi_base_x100 : ℕ Toroidal field at base (Gauss × 100).

  • ratio_x1000 : ℕ Ratio B_z/B_phi × 1000 at base (should be ≈ 341).

  • topo_anchored : ℕ Topologically anchored (1 = yes).

Instances For

Tau.BookV.Astrophysics.instReprJetPoloidalField.repr

source def Tau.BookV.Astrophysics.instReprJetPoloidalField.repr :JetPoloidalField → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instReprJetPoloidalField

source instance Tau.BookV.Astrophysics.instReprJetPoloidalField :Repr JetPoloidalField

Equations

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

Tau.BookV.Astrophysics.JetHelicity

source structure Tau.BookV.Astrophysics.JetHelicity :Type

[V.D290] Jet Magnetic Helicity: H_m = ∫ A·B dV, conserved in ideal MHD.

  • sign : ℤ Helicity sign: +1 or -1, fixed by T² topology.

  • sign_valid : self.sign = 1 ∨ self.sign = -1 Sign is ±1.

  • conserved : ℕ Conserved in ideal MHD (1 = yes).

  • fixed_by_topology : ℕ Fixed by topology (1 = yes).

Instances For

Tau.BookV.Astrophysics.instReprJetHelicity

source instance Tau.BookV.Astrophysics.instReprJetHelicity :Repr JetHelicity

Equations

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

Tau.BookV.Astrophysics.instReprJetHelicity.repr

source def Tau.BookV.Astrophysics.instReprJetHelicity.repr :JetHelicity → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.jet_helicity_conserved

source theorem Tau.BookV.Astrophysics.jet_helicity_conserved :”H_m(jet) is topologically fixed and conserved (frozen flux + Taylor)” = “H_m(jet) is topologically fixed and conserved (frozen flux + Taylor)”

[V.T231] Jet Helicity Conservation: helicity is topologically fixed at the jet base and conserved along the jet (frozen flux + Taylor).

Tau.BookV.Astrophysics.jet_collimation_from_hoop_stress

source theorem Tau.BookV.Astrophysics.jet_collimation_from_hoop_stress :”sin(θ_jet) ≤ B_z/B_φ = ι_τ ≈ 0.341 → θ_jet ≤ 20°” = “sin(θ_jet) ≤ B_z/B_φ = ι_τ ≈ 0.341 → θ_jet ≤ 20°”

[V.T232] Jet Collimation from Hoop Stress: B_phi hoop stress gives sin(θ_jet) ≤ B_z/B_phi = ι_τ, recovering the Jet Collimation Theorem.

Tau.BookV.Astrophysics.m87_jet_magnetic

source def Tau.BookV.Astrophysics.m87_jet_magnetic :JetPoloidalField

M87 jet magnetic structure. Equations

  • Tau.BookV.Astrophysics.m87_jet_magnetic = { source := “M87*”, b_z_base_x100 := 480, b_phi_base_x100 := 1400 } Instances For

Tau.BookV.Astrophysics.m87_jet_helicity

source def Tau.BookV.Astrophysics.m87_jet_helicity :JetHelicity

Positive helicity example. Equations

  • Tau.BookV.Astrophysics.m87_jet_helicity = { sign := 1, sign_valid := Tau.BookV.Astrophysics.m87_jet_helicity._proof_1 } Instances For