Radiative-to-Gravitational Force Ratio

• Radiative-to-gravitational force ratio is a dimensionless measure that quantifies the strength of outward radiation pressure relative to inward gravitational pull in various astrophysical contexts.
• Simulations decompose UV and IR contributions using dust opacities and fluxes to reveal that radiative driving rarely unbinds gas due to photon leakage and dust destruction.
• In binary systems, post-Newtonian expansions capture the minimal impact of radiation-reaction forces until near-merger phases where gravitational wave effects become significant.

The radiative-to-gravitational force ratio quantifies the comparative strength of radiative and gravitational forces acting on astrophysical systems. It serves as a diagnostic for regimes where radiation pressure is dynamically significant relative to gravity, with applications ranging from radiatively-driven outflows in galaxies to the impact of gravitational radiation-reaction in binary systems. The ratio formalizes conditions for equilibrium, such as local Eddington parameters in dusty interstellar media, and gauges the efficiency of gravitational wave back-reaction in compact binaries.

1. Definition and Physical Context

The radiative-to-gravitational force ratio is defined as the dimensionless quotient of the magnitude of forces (or accelerations) due to radiative processes and those arising from gravity at a given location. In the context of astrophysical gas dynamics—especially in environments influenced by intense ultraviolet (UV), optical, or infrared (IR) radiation fields—the ratio is commonly denoted by the Eddington parameter $\Gamma(r)$: $$\Gamma(r) \equiv \frac{a_{\rm rad}(r)}{a_{\rm grav}(r)}$$ where $a_{\rm rad}(r)$ is the outward radiative acceleration (typically due to absorption and scattering by dust or electrons) and $a_{\rm grav}(r)$ is the inward acceleration from gravity. For gravitating binaries emitting gravitational waves, a corresponding ratio can be formed as

$R \equiv \frac{|F_{\rm rad}|}{F_{\rm grav}}$

where $F_{\rm rad}$ is the radiation-reaction force from gravitational wave emission, and $F_{\rm grav}$ is the Newtonian gravitational attraction.

This ratio distinguishes regimes where radiative processes are dynamically dominant ($\Gamma > 1$ or $R \gtrsim 1$) from those in which gravity prevails.

2. Formulation in Astrophysical Gas Dynamics

In simulations of active galactic nuclei (AGN) and starburst-driven outflows, the radiative acceleration imparted to dusty gas can be decomposed into UV/optical and IR bands: $$a_{\rm rad,UV}(r) = \frac{\kappa_{\rm UV}(r)\,F_{\rm UV}(r)}{c}, \quad a_{\rm rad,IR}(r) = \frac{\kappa_{\rm IR}(r)\,F_{\rm IR}(r)}{c}$$ yielding a total radiative acceleration

$$a_{\rm rad}(r) = a_{\rm rad,UV}(r) + a_{\rm rad,IR}(r)$$

where $\kappa_{\rm UV},\,\kappa_{\rm IR}$ are dust opacities and $F_{\rm UV},\,F_{\rm IR}$ are energy fluxes in the respective bands. The gravitational acceleration under spherical symmetry is

$a_{\rm grav}(r) = \frac{G\,M_{\rm enc}(r)}{r^2}$

with $M_{\rm enc}(r)$ as the mass enclosed within radius $r$.

The local Eddington parameter governing the force ratio is thus

$$\Gamma(r) = \frac{\kappa_{\rm UV}F_{\rm UV}+\kappa_{\rm IR}F_{\rm IR}}{c}\frac{r^2}{G\,M_{\rm enc}(r)}$$

This formalism captures both optically thin and thick limits—where photon mean free paths are long or short relative to system size—as well as the transition regime via closure relations that interpolate the transfer equation solution (Novak et al., 2012).

3. Limiting Regimes and Physical Dependence

The force ratio's behavior is controlled by the optical depth in each radiation band:

• Optically Thin Limit ($\tau \ll 1$):

Radiative accelerations scale as

$$a_{\rm rad,UV} \simeq \frac{\kappa_{\rm UV} L_{\rm UV}}{4\pi r^2 c}\,\tau_{\rm UV},\quad a_{\rm rad,IR} \simeq \frac{\kappa_{\rm IR} L_{\rm IR}}{4\pi r^2 c}\,\tau_{\rm IR}$$

The force ratio is linearly proportional to optical depth.

• Optically Thick Limit:
  • For UV, photon momentum $L_{\rm UV}/c$ is fully deposited at a thin absorptive shell, and subsequent energy is re-emitted as IR.
  • For IR, multiple scatterings can multiply the momentum input by $\tau_{\rm IR}$, boosting $a_{\rm rad,IR}$ up to $\sim\tau_{\rm IR}L_{\rm IR}/c$.
• Intermediate Regime:

Full two-moment radiative transfer equations with closure parameter $t(r)$ interpolate between point-source and isotropic emission limits, allowing computation of $\Gamma(r)$ across all optical depth regimes.

In realistic systems, several physical effects limit the efficacy of radiative driving:

• Photon Leakage: Optically thin windows allow IR photons to escape, preventing the full $\tau_{\rm IR}L_{\rm IR}/c$ momentum input.
• Dust Destruction: Rapid dust destruction (sputtering) in hot ISM lowers $\kappa_{\rm UV},\,\kappa_{\rm IR}$, reducing $a_{\rm rad}$ (Novak et al., 2012).

4. Simulation-Based Quantification and Observational Implications

Numerical simulations calibrated to Milky Way dust-to-gas ratios ($\sim1/150$) yield opacities $\kappa_{\rm UV} \simeq 1.37\times 10^3$ cm$^2$/g, $\kappa_{\rm IR} \simeq 2.3$ cm$^2$/g. In early-type galaxies, resulting profiles of $\Gamma(r)$ peak at $0.1-0.2$ and rarely exceed unity. Only brief phases resembling ultra-luminous infrared galaxies (ULIRGs), characterized by high central gas masses ($\sim3\times 10^{10} M_\odot$) and intense star formation (SFR $\sim$ 200 $M_\odot$/yr), temporarily achieve $\Gamma \sim 0.5-1$. Even in such episodes, the time-averaged value of $\Gamma$ remains below unity, and momentum input is of order $L/c$ rather than $\tau_{\rm IR}L/c$.

The momentum transfer from radiation is therefore generally insufficient to unbind gas except during rare, extreme bursts. Black hole growth is suppressed by a factor $\sim5-7$ compared to dust-free cases during these high-$\Gamma$ intervals (Novak et al., 2012).

5. Radiative-to-Gravitational Ratio in Gravitational Wave Back-Reaction

For non-relativistic binary systems emitting gravitational waves, the leading-order (LO) radiation-reaction force is given by the Burke–Thorne potential: $$U_{BT}(t, x^k) = - \frac{G}{5c^5}I^{(3)}_{ij}(t)x^ix^j$$ with radiation-reaction acceleration

$$a^i_{RR}(t) = \frac{2G}{5c^5m_A}x^j \frac{d^3}{dt^3}Q_{ij}(t)$$

and the Newtonian force

$F_{\rm grav}(r) = \frac{G m_1 m_2}{r^2}$

The dimensionless ratio for a Keplerian circular orbit is

$$R_{LO} = \frac{2}{5}\frac{m_1}{M}\left( \frac{G M}{c^2 r} \right)^{3/2}$$

For equal-mass binaries, $R_{LO} = \frac{1}{5} x^{3/2}$, where $x \equiv G M / (c^2 r)$. The radiation-reaction force is thus suppressed by a steep post-Newtonian factor $x^{3/2}$ relative to gravity, remaining dynamically negligible until the near-merger regime (Birnholtz et al., 2015).

Corrections at next-to-leading order (+1PN) introduce $O(v^2/c^2)$ enhancements. For equal masses,

$$R_{1PN} = R_{LO}\left[1 + \frac{9}{8}\frac{v^2}{c^2} + \cdots \right]$$

with $v$ as the orbital velocity. The dominance of gravity persists except at relativistic velocities.

6. Key Limiting Factors and Regimes of Dominance

Several mechanisms constrain the radiative-to-gravitational force ratio in realistic astrophysical systems:

• IR Optical Depths: In present-day ellipticals, $\tau_{\rm IR}$ is typically $\lesssim 1$, curbing enhancements to the radiative force.
• Dust Survival: Sputtering timescales in the hot ISM (e.g., $t_{\rm sput} \sim 10^5$ yr for ISM densities and grain sizes) are shorter than dynamical timescales, suppressing $\kappa_{\rm UV},\,\kappa_{\rm IR}$ and thus $\Gamma$.
• Potential Depth: Deep galactic potentials require $\Gamma \gtrsim 1$ for outflows to escape, a condition rarely met outside ULIRG phases.

A plausible implication is that, despite high dust opacities in the UV—often $10^3$ times electron scattering—the net momentum input in simulations remains at $L/c$ due to leakage and destruction processes. Thus, radiative driving alone rarely enables gas escape except during exceptional short-lived episodes (Novak et al., 2012).

7. Broader Significance and Theoretical Implications

The radiative-to-gravitational force ratio functions as a threshold criterion for outflow launching, AGN feedback efficacy, starburst regulation, and the suppression of black hole accretion. In binaries, it sets the scale of non-conservative dynamical evolution due to gravitational wave emission. Across these domains, careful accounting for wave transport, absorptive processes, and matter composition is essential. Theoretical modeling must therefore incorporate both radiative transfer effects (opacity, anisotropy, spectral redistribution) and gravitational structure to accurately capture the dynamical consequences of their competition (Novak et al., 2012, Birnholtz et al., 2015).