Table E1.
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Table of key variables used in our analytical model.

VariableDescription
|$\dot{M}_{\mathrm{d}}$|⁠, |$\dot{M}_{\bullet }$|⁠, f•, LAGN, τ•Accretion rates in AGN discs and onto the SMBH, scaling, AGN luminosity, duration
|$\ \mathrm{M}_\odot$|⁠, R⊙, L⊙, G, c, zγSun’s mass, radius, luminosity, gravitational constant, speed of light, and redshift
Ω, Rd, R, rpc, QAGN disc’s angular frequency, size, radius, normalized by pc, gravitational stability
R•, M•, m8SMBH’s gravitational radius, mass in physical units and normalized by |$10^8 \ \mathrm{M}_\odot$|
ρcore, ρc, Σg, Tc, TeStar’s core density, disc’s mid-plane and surface density, mid-plane and effective temperature
cs, H, h, ν, ανMid-planet sound speed, thickness, aspect ratio, viscosity, turbulent efficiency
ϵ•, ϵHe, ϵαEnergy conversion efficiency due to accretion, hydrogen burning, triple-α & α-chain reaction
Q−, |$Q^+ _\nu$|⁠, |$Q^+ _\star$|⁠, s⋆, |${\dot{s}}_\star$|⁠, |${\dot{N}}_\star$|Disc’s cooling flux, viscous, stellar heating flux, stars’ surface density, formation flux, rate
κes, κdust, κγElectron scattering, dust, and general opacity
P⋆, ρ⋆, T⋆, F⋆, μ⋆, |${\dot{\mu }}_\star$|Stars’ internal pressure, density, temperature, radiative flux, molecular weight, changing rate
LE•, LE⋆, LE⊙, λ•, λ⋆, τSalSMBH’s, stars’, and Sun’s Eddington luminosity and factor, Salpeter time-scale
V⋆, L⋆, M⋆, m⋆, mgapStars’ escape speed, luminosity, mass and gap-forming mass normalized by |$\ \mathrm{M}_\odot$|
Leq, Meq, meq, |${\bar{L}}_{\rm eq}$|⁠, |${\bar{M}}_{\rm eq}$|Equilibrium luminosity, mass, normalized, averaged equilibrium luminosity, mass
τ⋆, ΔtMS, ΔtPostMS, N⋆Stars’ life span, duration of MS H and PostMS He depletion, total population
R⋆, R⋆, r⋆, r⋆Star’s physical, gravitational radius, and normalized by solar radius
RH, |${\dot{M}}_{\rm Bondi}$|⁠, τB, RB, rBHill’s radius, Bondi accretion rate, time-scale, radius, and normalized with solar radius
|$\dot{M}_{\rm wind, \Gamma }$|⁠, |${\dot{M}}_{\rm Bondi, \Gamma }$|Mass-loss and accretion rates modified by the Eddington limit
|$\dot{M}_{\rm wind}$|⁠, |${\dot{M}}_{\rm He}$|⁠, |${\dot{\mathcal {M}}}_{\rm He}$|⁠, f↺Mass loss rate, He release rate by evolving and immortal stars, retention efficiency
Aλ, τwind, ρwindWind loss efficiency, time-scale, and density at the launch base
Pgas, Prad, U⋆Gas and radiation pressure and flow velocity
Aμ, Ax, Ay, AzPower index of equilibrium-mass distribution with respect to μ, X⋆, Y⋆, and Z⋆
M0, M1, M2, M3Nominal model’s initial, onset-of-radiative-zone, PostMS-transition, pre-collapse masses
X⋆, Y⋆, Zα, ZFe, XaccStars’ H, Helium, α, and Fe mass fraction, H mass fraction in accreted gas
X0, Y0, Y1, Y2, ΔY, ΔZInitial H, He fraction, onset-of-radiative-zone, PostMS-transition, change in He, metal fraction
ΔMH, ΔMZ, ΔMHeTotal mass released to the disc in H, metals, He
|${\dot{\rm X}}_{\star }$|⁠, |${\dot{\rm Y}}_\star$|⁠, Xd, Yd, |${\mathcal {Z}}_\alpha$|⁠, |${\mathcal {Z}}_{\rm Fe}$|Net change rates of stars’ H and He fraction, disc’s H, Helium, α, and Fe mass fraction
VariableDescription
|$\dot{M}_{\mathrm{d}}$|⁠, |$\dot{M}_{\bullet }$|⁠, f•, LAGN, τ•Accretion rates in AGN discs and onto the SMBH, scaling, AGN luminosity, duration
|$\ \mathrm{M}_\odot$|⁠, R⊙, L⊙, G, c, zγSun’s mass, radius, luminosity, gravitational constant, speed of light, and redshift
Ω, Rd, R, rpc, QAGN disc’s angular frequency, size, radius, normalized by pc, gravitational stability
R•, M•, m8SMBH’s gravitational radius, mass in physical units and normalized by |$10^8 \ \mathrm{M}_\odot$|
ρcore, ρc, Σg, Tc, TeStar’s core density, disc’s mid-plane and surface density, mid-plane and effective temperature
cs, H, h, ν, ανMid-planet sound speed, thickness, aspect ratio, viscosity, turbulent efficiency
ϵ•, ϵHe, ϵαEnergy conversion efficiency due to accretion, hydrogen burning, triple-α & α-chain reaction
Q−, |$Q^+ _\nu$|⁠, |$Q^+ _\star$|⁠, s⋆, |${\dot{s}}_\star$|⁠, |${\dot{N}}_\star$|Disc’s cooling flux, viscous, stellar heating flux, stars’ surface density, formation flux, rate
κes, κdust, κγElectron scattering, dust, and general opacity
P⋆, ρ⋆, T⋆, F⋆, μ⋆, |${\dot{\mu }}_\star$|Stars’ internal pressure, density, temperature, radiative flux, molecular weight, changing rate
LE•, LE⋆, LE⊙, λ•, λ⋆, τSalSMBH’s, stars’, and Sun’s Eddington luminosity and factor, Salpeter time-scale
V⋆, L⋆, M⋆, m⋆, mgapStars’ escape speed, luminosity, mass and gap-forming mass normalized by |$\ \mathrm{M}_\odot$|
Leq, Meq, meq, |${\bar{L}}_{\rm eq}$|⁠, |${\bar{M}}_{\rm eq}$|Equilibrium luminosity, mass, normalized, averaged equilibrium luminosity, mass
τ⋆, ΔtMS, ΔtPostMS, N⋆Stars’ life span, duration of MS H and PostMS He depletion, total population
R⋆, R⋆, r⋆, r⋆Star’s physical, gravitational radius, and normalized by solar radius
RH, |${\dot{M}}_{\rm Bondi}$|⁠, τB, RB, rBHill’s radius, Bondi accretion rate, time-scale, radius, and normalized with solar radius
|$\dot{M}_{\rm wind, \Gamma }$|⁠, |${\dot{M}}_{\rm Bondi, \Gamma }$|Mass-loss and accretion rates modified by the Eddington limit
|$\dot{M}_{\rm wind}$|⁠, |${\dot{M}}_{\rm He}$|⁠, |${\dot{\mathcal {M}}}_{\rm He}$|⁠, f↺Mass loss rate, He release rate by evolving and immortal stars, retention efficiency
Aλ, τwind, ρwindWind loss efficiency, time-scale, and density at the launch base
Pgas, Prad, U⋆Gas and radiation pressure and flow velocity
Aμ, Ax, Ay, AzPower index of equilibrium-mass distribution with respect to μ, X⋆, Y⋆, and Z⋆
M0, M1, M2, M3Nominal model’s initial, onset-of-radiative-zone, PostMS-transition, pre-collapse masses
X⋆, Y⋆, Zα, ZFe, XaccStars’ H, Helium, α, and Fe mass fraction, H mass fraction in accreted gas
X0, Y0, Y1, Y2, ΔY, ΔZInitial H, He fraction, onset-of-radiative-zone, PostMS-transition, change in He, metal fraction
ΔMH, ΔMZ, ΔMHeTotal mass released to the disc in H, metals, He
|${\dot{\rm X}}_{\star }$|⁠, |${\dot{\rm Y}}_\star$|⁠, Xd, Yd, |${\mathcal {Z}}_\alpha$|⁠, |${\mathcal {Z}}_{\rm Fe}$|Net change rates of stars’ H and He fraction, disc’s H, Helium, α, and Fe mass fraction
Table E1.
Open in new tab

Table of key variables used in our analytical model.

VariableDescription
|$\dot{M}_{\mathrm{d}}$|⁠, |$\dot{M}_{\bullet }$|⁠, f•, LAGN, τ•Accretion rates in AGN discs and onto the SMBH, scaling, AGN luminosity, duration
|$\ \mathrm{M}_\odot$|⁠, R⊙, L⊙, G, c, zγSun’s mass, radius, luminosity, gravitational constant, speed of light, and redshift
Ω, Rd, R, rpc, QAGN disc’s angular frequency, size, radius, normalized by pc, gravitational stability
R•, M•, m8SMBH’s gravitational radius, mass in physical units and normalized by |$10^8 \ \mathrm{M}_\odot$|
ρcore, ρc, Σg, Tc, TeStar’s core density, disc’s mid-plane and surface density, mid-plane and effective temperature
cs, H, h, ν, ανMid-planet sound speed, thickness, aspect ratio, viscosity, turbulent efficiency
ϵ•, ϵHe, ϵαEnergy conversion efficiency due to accretion, hydrogen burning, triple-α & α-chain reaction
Q−, |$Q^+ _\nu$|⁠, |$Q^+ _\star$|⁠, s⋆, |${\dot{s}}_\star$|⁠, |${\dot{N}}_\star$|Disc’s cooling flux, viscous, stellar heating flux, stars’ surface density, formation flux, rate
κes, κdust, κγElectron scattering, dust, and general opacity
P⋆, ρ⋆, T⋆, F⋆, μ⋆, |${\dot{\mu }}_\star$|Stars’ internal pressure, density, temperature, radiative flux, molecular weight, changing rate
LE•, LE⋆, LE⊙, λ•, λ⋆, τSalSMBH’s, stars’, and Sun’s Eddington luminosity and factor, Salpeter time-scale
V⋆, L⋆, M⋆, m⋆, mgapStars’ escape speed, luminosity, mass and gap-forming mass normalized by |$\ \mathrm{M}_\odot$|
Leq, Meq, meq, |${\bar{L}}_{\rm eq}$|⁠, |${\bar{M}}_{\rm eq}$|Equilibrium luminosity, mass, normalized, averaged equilibrium luminosity, mass
τ⋆, ΔtMS, ΔtPostMS, N⋆Stars’ life span, duration of MS H and PostMS He depletion, total population
R⋆, R⋆, r⋆, r⋆Star’s physical, gravitational radius, and normalized by solar radius
RH, |${\dot{M}}_{\rm Bondi}$|⁠, τB, RB, rBHill’s radius, Bondi accretion rate, time-scale, radius, and normalized with solar radius
|$\dot{M}_{\rm wind, \Gamma }$|⁠, |${\dot{M}}_{\rm Bondi, \Gamma }$|Mass-loss and accretion rates modified by the Eddington limit
|$\dot{M}_{\rm wind}$|⁠, |${\dot{M}}_{\rm He}$|⁠, |${\dot{\mathcal {M}}}_{\rm He}$|⁠, f↺Mass loss rate, He release rate by evolving and immortal stars, retention efficiency
Aλ, τwind, ρwindWind loss efficiency, time-scale, and density at the launch base
Pgas, Prad, U⋆Gas and radiation pressure and flow velocity
Aμ, Ax, Ay, AzPower index of equilibrium-mass distribution with respect to μ, X⋆, Y⋆, and Z⋆
M0, M1, M2, M3Nominal model’s initial, onset-of-radiative-zone, PostMS-transition, pre-collapse masses
X⋆, Y⋆, Zα, ZFe, XaccStars’ H, Helium, α, and Fe mass fraction, H mass fraction in accreted gas
X0, Y0, Y1, Y2, ΔY, ΔZInitial H, He fraction, onset-of-radiative-zone, PostMS-transition, change in He, metal fraction
ΔMH, ΔMZ, ΔMHeTotal mass released to the disc in H, metals, He
|${\dot{\rm X}}_{\star }$|⁠, |${\dot{\rm Y}}_\star$|⁠, Xd, Yd, |${\mathcal {Z}}_\alpha$|⁠, |${\mathcal {Z}}_{\rm Fe}$|Net change rates of stars’ H and He fraction, disc’s H, Helium, α, and Fe mass fraction
VariableDescription
|$\dot{M}_{\mathrm{d}}$|⁠, |$\dot{M}_{\bullet }$|⁠, f•, LAGN, τ•Accretion rates in AGN discs and onto the SMBH, scaling, AGN luminosity, duration
|$\ \mathrm{M}_\odot$|⁠, R⊙, L⊙, G, c, zγSun’s mass, radius, luminosity, gravitational constant, speed of light, and redshift
Ω, Rd, R, rpc, QAGN disc’s angular frequency, size, radius, normalized by pc, gravitational stability
R•, M•, m8SMBH’s gravitational radius, mass in physical units and normalized by |$10^8 \ \mathrm{M}_\odot$|
ρcore, ρc, Σg, Tc, TeStar’s core density, disc’s mid-plane and surface density, mid-plane and effective temperature
cs, H, h, ν, ανMid-planet sound speed, thickness, aspect ratio, viscosity, turbulent efficiency
ϵ•, ϵHe, ϵαEnergy conversion efficiency due to accretion, hydrogen burning, triple-α & α-chain reaction
Q−, |$Q^+ _\nu$|⁠, |$Q^+ _\star$|⁠, s⋆, |${\dot{s}}_\star$|⁠, |${\dot{N}}_\star$|Disc’s cooling flux, viscous, stellar heating flux, stars’ surface density, formation flux, rate
κes, κdust, κγElectron scattering, dust, and general opacity
P⋆, ρ⋆, T⋆, F⋆, μ⋆, |${\dot{\mu }}_\star$|Stars’ internal pressure, density, temperature, radiative flux, molecular weight, changing rate
LE•, LE⋆, LE⊙, λ•, λ⋆, τSalSMBH’s, stars’, and Sun’s Eddington luminosity and factor, Salpeter time-scale
V⋆, L⋆, M⋆, m⋆, mgapStars’ escape speed, luminosity, mass and gap-forming mass normalized by |$\ \mathrm{M}_\odot$|
Leq, Meq, meq, |${\bar{L}}_{\rm eq}$|⁠, |${\bar{M}}_{\rm eq}$|Equilibrium luminosity, mass, normalized, averaged equilibrium luminosity, mass
τ⋆, ΔtMS, ΔtPostMS, N⋆Stars’ life span, duration of MS H and PostMS He depletion, total population
R⋆, R⋆, r⋆, r⋆Star’s physical, gravitational radius, and normalized by solar radius
RH, |${\dot{M}}_{\rm Bondi}$|⁠, τB, RB, rBHill’s radius, Bondi accretion rate, time-scale, radius, and normalized with solar radius
|$\dot{M}_{\rm wind, \Gamma }$|⁠, |${\dot{M}}_{\rm Bondi, \Gamma }$|Mass-loss and accretion rates modified by the Eddington limit
|$\dot{M}_{\rm wind}$|⁠, |${\dot{M}}_{\rm He}$|⁠, |${\dot{\mathcal {M}}}_{\rm He}$|⁠, f↺Mass loss rate, He release rate by evolving and immortal stars, retention efficiency
Aλ, τwind, ρwindWind loss efficiency, time-scale, and density at the launch base
Pgas, Prad, U⋆Gas and radiation pressure and flow velocity
Aμ, Ax, Ay, AzPower index of equilibrium-mass distribution with respect to μ, X⋆, Y⋆, and Z⋆
M0, M1, M2, M3Nominal model’s initial, onset-of-radiative-zone, PostMS-transition, pre-collapse masses
X⋆, Y⋆, Zα, ZFe, XaccStars’ H, Helium, α, and Fe mass fraction, H mass fraction in accreted gas
X0, Y0, Y1, Y2, ΔY, ΔZInitial H, He fraction, onset-of-radiative-zone, PostMS-transition, change in He, metal fraction
ΔMH, ΔMZ, ΔMHeTotal mass released to the disc in H, metals, He
|${\dot{\rm X}}_{\star }$|⁠, |${\dot{\rm Y}}_\star$|⁠, Xd, Yd, |${\mathcal {Z}}_\alpha$|⁠, |${\mathcal {Z}}_{\rm Fe}$|Net change rates of stars’ H and He fraction, disc’s H, Helium, α, and Fe mass fraction
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