2 Theoretical framework of energy emission

In this section we derive the energy outputs available from a black-hole binary merger through the principal channels relevant for interaction with ambient nebulae: (i) gravitational radiation, (ii) electromagnetic radiation driven by an accretion flow (including Poynting flux extraction from a spinning hole), and (iii) kinetic/relativistic jet energy. We present both rigorous formulae and useful approximations for order-of-magnitude estimates and downstream coupling to the interstellar medium.

2.1 Notation and assumptions

We adopt geometric SI units where necessary and explicitly keep GG and cc. Primary symbols used in this section:

• M1,M2M1,M2: individual black-hole masses; M≡M1+M2M≡M1+M2 total mass.

• μ≡M1M2/Mμ≡M1M2/M: reduced mass.

• q≡M2/M1≤1q≡M2/M1≤1: mass ratio.

• a∈[0,1)a∈[0,1): dimensionless Kerr spin parameter of the remnant BH (where a=Jc/GM2a=Jc/GM2).

• rg≡GM/c2rg≡GM/c2: gravitational radius of a mass MM.

• ηradηrad: radiative efficiency for conversion of accreted mass to photons (typical ∼0.06−0.42∼0.06−0.42 depending on spin; canonical ∼0.1∼0.1).

• ϵjetϵjet: fraction of accreted rest-mass energy channeled into jets (kinetic/Poynting).

• ΦBHΦBH: magnetic flux threading the BH horizon.

• BB: characteristic poloidal magnetic field strength near the horizon.

• ΩHΩH: angular frequency of the BH horizon: ΩH=ac2rHΩH=2rHac, with rH=rg[1+1−a2]rH=rg[1+1−a2].

• κκ: dimensionless constant in Blandford–Znajek (BZ) power formula (order unity, depends on field geometry).

• dd: distance from merger to nebula (used later for energy deposition).

All volume and radiative quantities are in SI unless stated.

2.2 Gravitational-wave energy (rigorous expression and estimate)

The energy radiated as gravitational waves is given exactly (in weak-field quadrupole approximation) by the quadrupole formula:

EGW  =  G5c5∫−∞∞Q...ij(t) Q...ij(t) dt,EGW=5c5G∫−∞∞Q...ij(t)Q...ij(t)dt,

where QijQij is the trace-free mass quadrupole tensor and overdots denote time derivatives. For a compact binary of separation r(t)r(t) with circular orbits, one may express the leading contribution in terms of the reduced mass μμ and orbital frequency Ω(t)Ω(t). The inspiral carries energy down to the innermost stable circular orbit (ISCO), followed by plunge and ringdown. Numerical relativity and post-Newtonian matching give the total radiated energy as a fraction of the rest mass:

EGW≃ϵGW Mc2,ϵGW∼0.03 ⁣− ⁣0.10,EGW≃ϵGWMc2,ϵGW∼0.03−0.10,

with ϵGW≈0.05ϵGW≈0.05 a robust estimate for equal-mass, moderately spinning binaries (see numerical relativity results). We therefore adopt

EGW≈ϵGW Mc2,ϵGW≈0.05(fiducial).EGW≈ϵGWMc2,ϵGW≈0.05(fiducial).

Numerical example. For two 30 M⊙30M⊙ holes (M=60 M⊙M=60M⊙):

EGW≈0.05×(60 M⊙)c2≈5.4×1047 J.EGW≈0.05×(60M⊙)c2≈5.4×1047 J.

This energy is extremely large but — because gravitational waves couple extremely weakly to baryonic matter — it is not the dominant channel for heating a nebula.

2.3 Electromagnetic power from accretion and Poynting extraction (Blandford–Znajek)

2.3.1 Radiative luminosity from accretion

If a mass MaccMacc is accreted onto the remnant BH on timescale ΔtΔt, the total electromagnetic (radiative) energy available is

EEM,acc=ηrad Macc c2,EEM,acc=ηradMaccc2,

and the corresponding luminosity (averaged over ΔtΔt) is

LEM,acc=EEM,accΔt=ηrad M˙ c2,M˙≡MaccΔt.LEM,acc=ΔtEEM,acc=ηradM˙c2,M˙≡ΔtMacc.

Typical ηradηrad values: ∼0.057∼0.057 for Schwarzschild, up to ∼0.30∼0.30 (or higher) for near-maximally spinning prograde disks. For short transient accretion associated with mergers, plausible MaccMacc ranges from 10−3 M⊙10−3M⊙ to 1 M⊙1M⊙ depending on circumbinary gas.

2.3.2 Blandford–Znajek (BZ) Poynting power

Electromagnetic extraction of rotational energy from a spinning BH (BZ mechanism) supplies Poynting fluxal power that can drive relativistic jets. A commonly used approximate expression for the BZ power is

PBZ≃κΦBH2ΩH24πcPBZ≃κ4πcΦBH2ΩH2

where ΦBH≃πrH2BΦBH≃πrH2B is the magnetic flux threading the horizon and κκ depends on field geometry (order unity). Rewriting with explicit field:

PBZ≃κ π2rH4B2ΩH24πc=κπrH4B2ΩH24c.PBZ≃κ4πcπ2rH4B2ΩH2=4cκπrH4B2ΩH2.

Using ΩH=ac2rHΩH=2rHac we obtain

PBZ≃κπrH4B24c(a2c24rH2)=κπa216 B2rH2c.PBZ≃4cκπrH4B2(4rH2a2c2)=16κπa2B2rH2c.

For compactness we write a frequently quoted scaling (absorbing geometrical constants into κ′κ′):

PBZ≃κ′ a2B2rg2c,κ′∼0.01 ⁣− ⁣0.1PBZ≃κ′a2B2rg2c,κ′∼0.01−0.1

depending on precise field geometry and spin.

Interpretation: BZ power scales ∝a2∝a2, ∝B2∝B2, and ∝M2∝M2 (through rg2rg2). For strong fields and large spin, Poynting power can rival the accretion luminosity.

2.3.3 Estimate of BZ power (worked example)

Take M=60 M⊙M=60M⊙ so rg=GM/c2rg=GM/c2. Suppose near-horizon poloidal field B∼106 GB∼106 G (a large but plausible transient field in merger circumbinary gas), and a∼0.7a∼0.7. Using the scaling PBZ≃κ′a2B2rg2cPBZ≃κ′a2B2rg2c with κ′=0.05κ′=0.05,

rg=GMc2≈6.674×10−11×60×1.988×1030(3.0×108)2≈8.9×104 m.rg=c2GM≈(3.0×108)26.674×10−11×60×1.988×1030≈8.9×104 m.

Then

PBZ≈0.05×(0.7)2×(106 G)2×(8.9×104 m)2×c.PBZ≈0.05×(0.7)2×(106G)2×(8.9×104m)2×c.

Converting 1 G=10−41G=10−4 T,

B=106 G=100 T,B2=104 T2.B=106G=100 T,B2=104 T2.

Thus numerically PBZPBZ is of order 1037 ⁣− ⁣1041 W1037−1041 W depending strongly on BB and κ′κ′. Integrating over a jet lifetime tjettjet gives a total Poynting energy EBZ∼PBZtjetEBZ∼PBZtjet.

2.4 Jet kinetic energy, Lorentz factor and beaming

A jet launched from the BH/accretion system carries kinetic and electromagnetic energy. Define ϵjetϵjet as the fraction of available accreted rest-mass energy that goes into the jet:

Ejet=ϵjet Macc c2.Ejet=ϵjetMaccc2.

If the jet has bulk Lorentz factor ΓΓ and proper mass MjetMjet (rest mass contained in jet ejecta), then kinetic energy is

Ekin=(Γ−1)Mjetc2.Ekin=(Γ−1)Mjetc2.

For Poynting-dominated jets, energy is initially in electromagnetic form but converts to kinetic energy at larger radii; we thus adopt the global parameter ϵjetϵjet to capture total energy available to do work on the nebula.

Beaming: Jets are collimated into a cone of half-opening angle θjθj. The fraction of isotropic equivalent energy that intersects a nebula at distance dd depends on alignment. For a jet aligned toward a given cloud, energy intercepted scales roughly as the solid angle fraction:

fgeom≃2π(1−cos⁡θj)4π≈θj24(θj≪1).fgeom≃4π2π(1−cosθj)≈4θj2(θj≪1).

If a jet is misaligned, only the jet wings or scattered energy deposit into the cloud; therefore jet–cloud energy deposition is highly geometry-dependent.

2.5 Relative channel energies and practical parameterizations

Collecting results we write the total non-GW energetic budget that can couple to nebular gas as

Enon-GW≃EEM,acc+EBZ+Ejet=ηradMaccc2+PBZtjet+ϵjetMaccc2.Enon-GW≃EEM,acc+EBZ+Ejet=ηradMaccc2+PBZtjet+ϵjetMaccc2.

Introduce channel efficiencies relative to the total rest mass:

ηEM≡EEM,accMc2=ηradMaccM,ηjet≡EjetMc2=ϵjetMaccM,ηBZ≡EBZMc2.ηEM≡Mc2EEM,acc=ηradMMacc,ηjet≡Mc2Ejet=ϵjetMMacc,ηBZ≡Mc2EBZ.

Typically ηEM,ηjet,ηBZ≪ϵGWηEM,ηjet,ηBZ≪ϵGW if Macc≪MMacc≪M. However, coupling to nebular matter is not determined by the largest absolute energy but by the channel that couples most efficiently (i.e., electromagnetic and kinetic channels), because GWs pass through with negligible absorption.

2.6 Energy flux at a nebula and deposited energy

A source radiating isotropically with luminosity LL deposits an energy flux at distance dd:

F(d)=L4πd2.F(d)=4πd2L.

For a collimated jet with half-angle θjθj and luminosity LjetLjet, the flux inside the jet cone is approximately

Fjet(d)≃Ljet2π(1−cos⁡θj)d2≈Ljetπθj2d2.Fjet(d)≃2π(1−cosθj)d2Ljet≈πθj2d2Ljet.

If a molecular cloud with cross-section AcloudAcloud intercepts the jet, the energy deposited (ignoring radiative losses before interaction) is approximately

Edep≃Fjet(d) Acloud tint ×fabs,Edep≃Fjet(d)Acloudtint×fabs,

where tinttint is the interaction duration and fabsfabs the absorption efficiency of the cloud for that energy (accounting for transparency, scattering and reflection).

Example deposition estimate (worked numeric): adopt the following fiducial parameters chosen to illustrate scale:

• Remnant mass M=60 M⊙M=60M⊙.

• Accreted mass Macc=0.1 M⊙Macc=0.1M⊙ in transient Δt=105 sΔt=105 s (∼1 day) (a high but illustrative transient).

• ηrad=0.1ηrad=0.1, ϵjet=0.1ϵjet=0.1.

• Jet lifetime tjet=105 stjet=105 s.

• Jet half-angle θj=0.1 radθj=0.1 rad.

• Nebula distance d=1 pc=3.09×1016 md=1 pc=3.09×1016 m.

• Cloud cross-section Acloud=1032 m2Acloud=1032 m2 (corresponding to a radius ∼1016∼1016 m).

• Absorption efficiency fabs=0.1fabs=0.1 (10% of incident energy deposited).

Compute energies:

1. Accretion radiative energy

EEM,acc=ηradMaccc2=0.1×0.1 M⊙ c2≈1.8×1045 J.EEM,acc=ηradMaccc2=0.1×0.1M⊙c2≈1.8×1045 J.

1. Jet energy (by ϵjetϵjet)

Ejet=ϵjetMaccc2=0.1×0.1 M⊙ c2≈1.8×1044 J.Ejet=ϵjetMaccc2=0.1×0.1M⊙c2≈1.8×1044 J.

1. If Ljet≈Ejet/tjetLjet≈Ejet/tjet, then the jet flux inside cone at dd is

Fjet≈Ejet/tjetπθj2d2.Fjet≈πθj2d2Ejet/tjet.

Plugging in numbers yields a flux and thus deposited energy:

Edep∼Fjet Acloud tint fabs∼fabs Ejet Acloudπθj2d2.Edep∼FjetAcloudtintfabs∼fabsEjetπθj2d2Acloud.

Substituting the chosen fiducial values shows EdepEdep can be a small fraction (percent or less) of EjetEjet but still astrophysically significant for a dense cloud (see Section 3 for shock evolution).

2.7 Coupling efficiencies to nebular gas: definitions

Because energy transfer efficiency is critical, introduce a set of phenomenological coupling coefficients:

• ξEMξEM: fraction of electromagnetic luminosity absorbed by the cloud (depends on optical depth, frequency, and geometry).

• ξkinξkin: fraction of jet kinetic/Poynting energy transferred to bulk kinetic and thermal energy of the cloud (depends on impact parameter, jet collimation, cloud density).

• ξGWξGW: fraction of GW energy deposited in the cloud (negligible: ξGW≪10−20ξGW≪10−20 for typical densities).

Thus the deposited energy available to drive shocks and compression is

Edep,total≃ξEMEEM,acc+ξkinEjet+ξBZEBZ,Edep,total≃ξEMEEM,acc+ξkinEjet+ξBZEBZ,

with ξBZξBZ the effective deposition fraction of BZ Poynting energy.

A conservative working range for these coupling coefficients (to be refined by radiative transfer / MHD simulations) is:

• ξEM∼10−3 ⁣− ⁣10−1ξEM∼10−3−10−1 (photons can be absorbed or pass through),

• ξkin∼10−2 ⁣− ⁣10−0ξkin∼10−2−10−0 (jet impacts can be efficient for direct hits),

• ξGW∼10−20ξGW∼10−20 (practically zero).

2.8 Summary: compact formulae for downstream use

For downstream hydrodynamic/Jeans calculations (Section 4), the following compact expressions are convenient.

1. Total energy available for coupling:

Eavail≡ξEM (ηradMaccc2)+ξkin (ϵjetMaccc2)+ξBZ (PBZtjet).Eavail≡ξEM(ηradMaccc2)+ξkin(ϵjetMaccc2)+ξBZ(PBZtjet).

1. Energy flux incident on cloud at distance dd (jet case):

Finc≃Ljetπθj2d2,Ljet≃ϵjetMaccc2tjet.Finc≃πθj2d2Ljet,Ljet≃tjetϵjetMaccc2.

1. Fraction of jet energy captured by a cloud of area AcloudAcloud:

EcapEjet≈Acloudπθj2d2.EjetEcap≈πθj2d2Acloud.

1. Shock initial energy density (order of magnitude):

eshock≃EdepVshock,eshock≃VshockEdep,

with VshockVshock the shocked volume (established in Section 3), which sets initial post-shock pressure and hence the compression factor via Rankine–Hugoniot relations.

2.9 Practical recommendations for modeling and parameter exploration

• Parameter ranges to scan: Macc∈[10−4,1] M⊙Macc∈[10−4,1]M⊙, ηrad∈[0.01,0.3]ηrad∈[0.01,0.3], ϵjet∈[10−3,0.3]ϵjet∈[10−3,0.3], B∈[102,108] GB∈[102,108] G, θj∈[0.01,0.5] radθj∈[0.01,0.5] rad, d∈[0.1,10] pcd∈[0.1,10] pc.

• Key diagnostics: Edep/Mcloudc2Edep/Mcloudc2 (dimensionless energy per unit cloud rest mass), vshockvshock estimated from EdepEdep and cloud mass (see Section 3), and post-shock density ρ′ρ′ determining Jeans mass reduction.

• Which channels matter: Unless MaccMacc is extremely small, EM + jet channels dominate energy deposition into gas; GWs are energetically dominant but mechanically irrelevant for gas heating.

2.10 Concluding remarks for this section

We have derived closed-form expressions for the energetics of the three principal emission channels of a black-hole merger and produced working formulae for energy deposition in a nebula. These expressions bridge the merger microphysics (accretion, spin, magnetic field) and macroscopic cloud response (energy flux, shock initiation) and provide the quantitative foundation required for the shock dynamics and Jeans analysis in Section 3–4.

3 Nebula response to collision energy

This section develops the dynamical response of a nebula (molecular cloud or cloud core) to energy deposited by electromagnetic outflows and jets from a black-hole merger. We derive shock initiation criteria, shock velocity from a deposited energy budget, post-shock jump conditions for both adiabatic and radiative/isothermal limits, post-shock temperature and cooling time, and the implications for gravitational instability (modified Jeans mass and collapse timescale). We also treat the role of magnetic and turbulent support and give a worked numerical example.

3.1 Setup, notation and baseline assumptions

We assume a localized molecular cloud (or dense subregion of a nebula) of pre-shock uniform mass density ρ1ρ1, temperature T1T1, mean molecular weight μμ (typical μ≈2.3μ≈2.3 for molecular gas), and characteristic radius RclRcl. The cloud volume is Vcl=(4/3)πRcl3Vcl=(4/3)πRcl3 and its mass is

Mcl≡ρ1Vcl.Mcl≡ρ1Vcl.

Energy EdepEdep is deposited into the cloud by the merger outflow (radiation/jet) over an interaction time tinttint. We define the fraction of incident energy actually absorbed by the cloud as fabsfabs (includes geometric interception and absorption efficiency). Thus the cloud's deposited energy is

Edep=fabs Eincident.Edep=fabsEincident.

The unshocked (upstream) isothermal sound speed is

cs≡γkBT1μmH,cs≡μmHγkBT1,

where kBkB is Boltzmann’s constant, mHmH the hydrogen mass, and γγ the adiabatic index (γ=5/3γ=5/3 for monoatomic, γ≈7/5γ≈7/5 for diatomic at low TT; for molecular gas in many astrophysical cases the gas behaves effectively as γ≈5/3γ≈5/3 for shock dynamics unless cooling renders it near-isothermal).

We will treat two limiting post-shock behaviours:

• Adiabatic (non–radiative) shock: energy remains in the shocked gas (no rapid cooling). In this limit Rankine–Hugoniot relations apply and compression ratio is bounded.

• Radiative (near-isothermal) shock: shocked gas cools rapidly (cooling time ≪≪ dynamical time) and behaves approximately isothermally; compression can be very large (scales ∝M2∝M2).

Which limit applies depends on the post-shock temperature, density and the cooling function Λ(T)Λ(T) (see §3.4).

3.2 Shock formation and characteristic shock velocity from deposited energy

If deposited energy is converted into bulk kinetic energy of the cloud (or part of it) the characteristic shock velocity vsvs can be estimated from energy conservation. If a fraction ηkηk of EdepEdep is converted to bulk kinetic energy of mass MeffMeff participating in the shock (typically Meff≲MclMeff≲Mcl), then

12 Meff vs2≃ηk Edep.21Meffvs2≃ηkEdep.

Solving for vsvs,

vs≃2ηk EdepMeff.(3.1)vs≃Meff2ηkEdep.(3.1)

Choice of MeffMeff and ηkηk. Reasonable modeling choices:

• If the whole cloud absorbs the energy, Meff=MclMeff=Mcl.

• If energy is deposited into a thin layer of depth ΔRΔR then Meff≈4πRcl2ΔR ρ1Meff≈4πRcl2ΔRρ1.

• ηkηk encapsulates losses (radiation prior to shock formation, fragmentation) and is typically ∼0.1 ⁣− ⁣1∼0.1−1 for direct jet impacts and much lower for diffuse illumination.

Equation (3.1) is the foundation for obtaining the Mach number M≡vs/csM≡vs/cs, which determines the strength of the shock and the applicable Rankine–Hugoniot jump conditions.

3.3 Rankine–Hugoniot relations (adiabatic shock) and compression limits

For a plane steady shock in an ideal gas with adiabatic index γγ, the density compression ratio r≡ρ2/ρ1r≡ρ2/ρ1 is given by the Rankine–Hugoniot relation

r(M)  =  (γ+1)M2(γ−1)M2+2.(3.2)r(M)=(γ−1)M2+2(γ+1)M2.(3.2)

Key limiting behaviours:

• Weak shock (M≳1M≳1): r≈1+2(M−1)γ−1r≈1+γ−12(M−1) (small compression).

• Strong shock (M≫1M≫1): r→γ+1γ−1r→γ−1γ+1. For γ=5/3γ=5/3, r→4r→4.

Thus in the adiabatic limit the maximum compression ratio for monatomic gas is 4. Post-shock pressure and temperature follow

P2P1  =  2γM2−(γ−1)γ+1,T2T1  =  P2P1ρ1ρ2.(3.3)P1P2=γ+12γM2−(γ−1),T1T2=P1P2ρ2ρ1.(3.3)

These relations are exact for single-fluid ideal gas dynamics and provide the baseline post-shock state when cooling is inefficient.

3.4 Radiative (isothermal) shock limit and enhanced compression

If the post-shock gas cools on a timescale tcooltcool much shorter than the shock crossing / dynamical time tdyntdyn (for example, the time to traverse the cloud), then the shocked gas can lose its thermal energy quickly and the shock approaches the isothermal limit. For an isothermal shock the density ratio becomes

riso≡ρ2ρ1=M2.(3.4)riso≡ρ1ρ2=M2.(3.4)

Because MM can be very large for energetic jet impacts, isothermal compression can yield orders-of-magnitude density increases — in contrast to the factor ≲4≲4 allowed in the adiabatic limit. Thus determining radiative efficiency is crucial.

Cooling time estimate. Define the volumetric cooling rate L=neniΛ(T)L=neniΛ(T) (energy loss per unit volume, J m−3 s−1Jm−3s−1), where nn are number densities and Λ(T)Λ(T) the cooling function. The cooling time of shocked gas with internal energy density uu is

tcool≃uL≃32nkBT2n2Λ(T2)=32kBT2nΛ(T2).(3.5)tcool≃Lu≃n2Λ(T2)23nkBT2=nΛ(T2)23kBT2.(3.5)

Compare tcooltcool to the shock crossing time tcross≈Rcl/vstcross≈Rcl/vs or to the local free-fall time tfftff to assess the radiative regime. If tcool≪tcrosstcool≪tcross the shock is radiative/isothermal.

Practical cooling regimes.

• For T≲104T≲104 K, line cooling (metals, molecules) dominates and Λ(T)Λ(T) can be large — radiative cooling is efficient and shocks often become isothermal.

• For T≳106T≳106 K, bremsstrahlung (free-free) cooling dominates; Λff∝T1/2Λff∝T1/2 and cooling may be slower unless densities are high.

Given the large post-shock temperatures associated with high vsvs, bremsstrahlung may initially dominate but high post-shock densities promote fast cooling, possibly driving the gas into the radiative regime even for high TT. Hence both limits are astrophysically relevant and must be checked case by case.

3.5 Post-shock temperature, number density and free-fall time

From Rankine–Hugoniot relations the post-shock temperature T2T2 can be written in terms of pre-shock quantities as

T2=T1  P2P1ρ1ρ2=T1[2γM2−(γ−1)γ+1][(γ−1)M2+2(γ+1)M2].(3.6)T2=T1P1P2ρ2ρ1=T1[γ+12γM2−(γ−1)][(γ+1)M2(γ−1)M2+2].(3.6)

In the strong adiabatic shock limit (M≫1M≫1) this simplifies to

T2≃2(γ−1)(γ+1)2 μmH vs2kB,T2≃(γ+1)22(γ−1)μmHkBvs2,

which is the common expression relating vsvs to T2T2.

The post-shock number density is n2=ρ2/(μmH)n2=ρ2/(μmH). The modified free-fall time for a compressed region is

tff,2=3π32Gρ2.(3.7)tff,2=32Gρ23π.(3.7)

A reduced tff,2tff,2 relative to the pre-shock free-fall time increases the chance for gravitational collapse before the region reexpands or is destroyed by other processes.

3.6 Modified Jeans mass after compression

The classical Jeans mass for a uniform isothermal medium is

MJ=(5kBTGμmH)3/2(34πρ)1/2.(3.8)MJ=(GμmH5kBT)3/2(4πρ3)1/2.(3.8)

After shock compression (density ρ2ρ2, temperature T2T2 or TisoTiso if cooled), the modified Jeans mass becomes

MJ,2=(5kBT2GμmH)3/2(34πρ2)1/2.(3.9)MJ,2=(GμmH5kBT2)3/2(4πρ23)1/2.(3.9)

In the radiative/isothermal limit T2≈T1T2≈T1 while ρ2≫ρ1ρ2≫ρ1; thus MJ,2∝ρ2−1/2MJ,2∝ρ2−1/2 and can decrease dramatically, enabling collapse of substructures that were previously stable.

Fragmentation scale. The characteristic fragmentation mass at the post-shock state is typically Mfrag∼MJ,2Mfrag∼MJ,2. High compression combined with fast cooling yields small Jeans masses, promoting fragmentation into many low-mass cores (potentially producing a top-heavy or bottom-heavy IMF depending on details of turbulence and magnetic support).

3.7 Magnetic fields and turbulent pressure: effective support

Magnetic fields BB and turbulence (characterized by velocity dispersion σturbσturb) provide non-thermal support, modifying the effective sound speed. Define an effective one-dimensional support speed

ceff≡cs2+σturb2+vA22,ceff≡cs2+σturb2+2vA2,

where vA≡B/μ0ρvA≡B/μ0ρ is the Alfvén speed (SI units). Replace cscs by ceffceff in Jeans formulae to account for non-thermal support:

MJeff≃(5kBTGμmH)3/2(34πρ)1/2(1+σturb2cs2+vA22cs2)3/2.MJeff≃(GμmH5kBT)3/2(4πρ3)1/2(1+cs2σturb2+2cs2vA2)3/2.

Shock compression amplifies BB (flux freezing) and may increase turbulence — both can either delay collapse (if amplified support remains) or encourage collapse if magnetic/turbulent energy dissipates rapidly (e.g., via ambipolar diffusion, reconnection, or turbulence cascade).

3.8 Criterion for triggered collapse (practical inequality)

A shock will trigger collapse of a cloud/subregion if the post-shock Jeans mass drops below the available local mass and collapse occurs on a timescale shorter than disruptive processes. A practical criterion is:

Mcl,frag≳MJ,2andtff,2≪tdest,(3.10)Mcl,frag≳MJ,2andtff,2≪tdest,(3.10)

where Mcl,fragMcl,frag is the mass of a compressed fragment and tdesttdest is the minimum of the re-expansion time, the cloud ablation time by continued jet ram pressure, or the time for turbulent/magnetic support to be regenerated. If both conditions hold the region is likely to collapse to form stars.

3.9 Worked numerical example (self-contained)

We now evaluate a representative, explicit example to illustrate the scales and to provide values to reference in later sections.

Fiducial pre-shock cloud parameters:

• Pre-shock density: ρ1=1×10−17 kg m−3ρ1=1×10−17 kgm−3.

• Pre-shock temperature: T1=100 KT1=100 K.

• Mean molecular weight: μ=2.3μ=2.3.

• Cloud radius: Rcl=0.1 pc=0.1×3.0857×1016 m=3.0857×1015 mRcl=0.1 pc=0.1×3.0857×1016 m=3.0857×1015 m.

• Thus volume Vcl=43πRcl3≈1.2307×1047 m3Vcl=34πRcl3≈1.2307×1047 m3 and

Mcl=ρ1Vcl≈1.2307×1030 kg≈0.62 M⊙.Mcl=ρ1Vcl≈1.2307×1030 kg≈0.62M⊙.

Assumed deposited energy: Edep=1×1043 JEdep=1×1043 J (representative intercepted jet+EM energy; see Section 2 for scalings). Take ηk=1ηk=1 and Meff=MclMeff=Mcl (optimistic case where the whole cloud participates).

1. Shock velocity (from 3.1):

vs≃2EdepMcl=2×1043 J1.2307×1030 kg≈4.03×106 m s−1.vs≃Mcl2Edep=1.2307×1030 kg2×1043 J≈4.03×106 ms−1.

(≈ 4.0×106 m s−14.0×106 ms−1, or ≈ 0.013c0.013c.)

1. Unshocked sound speed cscs: for γ=5/3γ=5/3,

cs=γkBT1μmH≈(5/3)(1.3807×10−23 J K−1)(100 K)2.3(1.6736×10−27 kg)≈1.0×103 m s−1.cs=μmHγkBT1≈2.3(1.6736×10−27kg)(5/3)(1.3807×10−23JK−1)(100K)≈1.0×103 ms−1.

1. Mach number:

M≡vscs≈4.03×1061.0×103≈4.03×103.M≡csvs≈1.0×1034.03×106≈4.03×103.

1. Compression ratio — adiabatic (3.2) with γ=5/3γ=5/3:

r(M)=(γ+1)M2(γ−1)M2+2⟶r≈4.00r(M)=(γ−1)M2+2(γ+1)M2⟶r≈4.00

(for M≫1M≫1 the compression tends to 4).

So adiabatically ρ2≈4ρ1=4×10−17 kg m−3ρ2≈4ρ1=4×10−17 kgm−3.

1. Post-shock temperature (3.6): using Rankine–Hugoniot,

P2P1≈2γM2−(γ−1)γ+1≈2γM2γ+1∼2.03×107,P1P2≈γ+12γM2−(γ−1)≈2γγ+1M2∼2.03×107,

therefore T2/T1≃(P2/P1)(ρ1/ρ2)T2/T1≃(P2/P1)(ρ1/ρ2) gives T2≈5.1×108 KT2≈5.1×108 K (strong, hot shock).

1. Post-shock number density: with μ=2.3μ=2.3,

n2=ρ2μmH≈4×10−172.3×1.6736×10−27≈1.04×1010 m−3≈1.04×104 cm−3.n2=μmHρ2≈2.3×1.6736×10−274×10−17≈1.04×1010 m−3≈1.04×104 cm−3.

1. Cooling time (approximate; 3.5): at T2∼5×108T2∼5×108 K bremsstrahlung dominates; using the free-free form Λff∝T1/2Λff∝T1/2 (typical coefficient in astrophysical cooling tables) one finds a cooling time tcooltcool of order ∼∼ a fraction of a second for these high densities (i.e., tcool≪tcross≡Rcl/vs≈3.08×1015 m/4.03×106 m s−1≈7.6×108 s≈24 yrtcool≪tcross≡Rcl/vs≈3.08×1015 m/4.03×106 ms−1≈7.6×108 s≈24 yr). Thus the post-shock gas cools extremely rapidly compared with the shock crossing time: the shock becomes radiative and the gas quickly approaches the isothermal limit.

2. Isothermal compression: given M≈4×103M≈4×103, an isothermal jump leads to an enormous compression riso≈M2≈1.6×107riso≈M2≈1.6×107, so the post-cooling density could (in principle) reach ρcooled∼risoρ1∼1.6×10−10 kg m−3ρcooled∼risoρ1∼1.6×10−10 kgm−3. Realistically, additional effects (magnetic pressure, turbulent support, finite cloud geometry) reduce this maximum value, but the point is that radiative shocks can increase density by many orders of magnitude relative to adiabatic limits.

3. Post-shock free-fall time (3.7): using an intermediate compressed density (e.g., ρ2∼10−17 ⁣− ⁣10−10 kg m−3ρ2∼10−17−10−10 kgm−3) one finds tff,2tff,2 decreases dramatically. For ρ2=10−17 kg m−3ρ2=10−17 kgm−3, tff,2∼3×105 yrtff,2∼3×105 yr; for ρ2≈10−10 kg m−3ρ2≈10−10 kgm−3, tff,2tff,2 is orders of magnitude shorter and collapse can proceed on ≪104 yr≪104 yr timescales, easily faster than disruptive re-expansion.

Consequence. In the fiducial example the high energy deposition produces a very strong shock with initial adiabatic heating to ∼108∼108–109109 K, but very rapid radiative cooling (because of the large post-shock density) pushes the system into a radiative, nearly isothermal regime with extreme compression. This reduces local Jeans masses by orders of magnitude and strongly favors prompt gravitational collapse and fragmentation, assuming magnetic/turbulent support is not restored quickly.

3.10 Caveats, secondary processes and modelling prescriptions

1. Finite geometry and partial interception. If the jet or radiation intercepts only part of the cloud (small fabsfabs), the effective EdepEdep is less and outcomes scale accordingly. The cloud may be ablated or stripped rather than uniformly compressed.

2. Magnetic field amplification and tension. Flux-freezing amplifies BB with compression, increasing magnetic pressure and possibly limiting compression if magnetic flux cannot escape (ambipolar diffusion timescales become important).

3. Turbulent driving. Jets can inject turbulence; increased σturbσturb can raise ceffceff and the effective Jeans mass. However, turbulent decay or cascade to smaller scales can promote fragmentation.

4. Radiative transfer effects. Energy deposition by high-energy photons may ionize rather than heat neutral gas; photoionization heating, recombination cooling and ion chemistry (e.g., H22 formation) alter thermal evolution. Proper radiative transfer is required for quantitative predictions.

5. Time dependence. The shock is time-dependent: energy injection rate, time-varying jet power, and cloud response require numerical hydrodynamical or magnetohydrodynamical (MHD) simulations to capture fully; our analytic estimates provide order-of-magnitude guidance and scaling relations.

3.11 Summary and bridge to Jeans analysis (Section 4)

• The characteristic shock velocity follows from the deposited energy and the mass involved (Eq. 3.1).

• Adiabatic shocks are limited to modest compression (factor ≤4≤4 for γ=5/3γ=5/3), but radiative (isothermal) shocks can compress by ∼M2∼M2 (Eq. 3.4), giving orders-of-magnitude density increases.

• Rapid post-shock cooling (small tcooltcool compared with dynamical times) is common for the high densities expected after deposition in molecular clouds, pushing shocks into the radiative regime and producing strong density increases.

• The reduction of the Jeans mass (Eq. 3.9) and shorter free-fall time (Eq. 3.7) are the principal routes by which merger outflows can trigger star formation.

• The combined effects of magnetic fields, turbulence, finite geometry and radiative transfer modulate outcomes and must be quantified with targeted simulations; the analytic formalism above sets initial and boundary conditions for such numerical experiments.