[Paper Review] The opacity limit

Michael Y Grudić, Philip F. Hopkins|arXiv (Cornell University)|Aug 30, 2023

Astrophysics and Star Formation Studies4 citations

TL;DR

This paper re-evaluates the opacity limit in star formation, deriving self-consistent expressions for the thermal evolution of dust-cooled collapsing gas clumps under varying radiation fields and dust opacity laws. It shows that the minimum Jeans mass $M_{ m J}^{ m min}$ scales as $A_{ m d}^{-1/11}$ to $A_{ m d}^{-1/15}$ in weak radiation or low-dust environments, but as $A_{ m d}^{1/3}$ in strong radiation and high-dust conditions—implicating a less bottom-heavy initial mass function in dense, dusty regions like galactic centers or high-redshift galaxies.

ABSTRACT

The opacity limit is an important concept in star formation: isothermal collapse cannot proceed without limit, because eventually cooling radiation is trapped and the temperature rises quasi-adiabatically, setting a minimum Jeans mass $M_{ m J}^{ m min}$. Various works have considered this scenario and derived expressions for $M_{ m J}^{ m min}$, generally $\sim 10^{-3}-10^{-2}M_\odot$ in normal star-forming conditions, but with conflicting results about the scaling with ambient conditions and material properties. We derive expressions for the thermal evolution of dust-cooled collapsing gas clumps in various limiting cases, given a general ambient radiation field ($u_{ m rad}$, $T_{ m rad}$) and a general power-law dust opacity law $σ_{ m d} = A_{ m d} T^β$. By accounting for temperature evolution self-consistently we rule out a previously-proposed regime in which the adiabatic transition occurs while the core is still optically-thin. If the radiation field is weak or dust opacity is small, $M_{ m J}^{ m min}$ is insensitive to dust properties/abundance ($\sim A_{ m d}^{-\frac{1}{11}}-A_{ m d}^{-\frac{1}{15}}$), but if the radiation field is strong and dust is abundant it scales $\propto A_{ m d}^{1/3}$. This could make the IMF less bottom-heavy in dust-rich and/or radiation-dense environments, e.g. galactic centers, starburst galaxies, massive high-$z$ galaxies, and proto-star clusters that are already luminous.

Motivation & Objective

• To resolve conflicting results in the literature regarding the scaling of the minimum Jeans mass ($M_{ m J}^{ m min}$) with dust properties and ambient conditions in star formation.
• To correct flawed assumptions in prior works—particularly the idea that adiabatic transition occurs when optical depth $\tau \gtrsim 1$—by self-consistently modeling temperature evolution during collapse.
• To derive general expressions for the opacity-limited Jeans mass and transition density ($n_{\rm ad}$) in various asymptotic regimes, accounting for both compression heating and radiation field effects.
• To assess the implications of environmental variation in $M_{\rm J}^\rm{min}$ for the initial mass function (IMF), especially in extreme environments like galactic centers or high-redshift star clusters.
• To provide a unified, physically consistent framework for the opacity limit using calibrated approximations validated against radiation hydrodynamics simulations.

Proposed method

• Derives the thermal evolution of dust-cooled, self-gravitating gas clumps under a general power-law dust opacity law $\sigma_{\rm d} = A_{\rm d} T^{\beta}$ and ambient radiation field ($u_{\rm rad}, T_{\rm rad}$).
• Identifies two critical regimes: a compression-heated limit (analogous to Low & Lynden-Bell 1976) and a radiation-heated limit (generalizing Masunaga & Inutsuka 1999), with the larger determining $M_{\rm J}^\rm{min}$.
• Uses asymptotic approximations for temperature evolution in the high-density, optically thick limit, avoiding the assumption that $\tau \gtrsim 1$ alone triggers adiabatic behavior.
• Applies dimensional analysis and scaling arguments to derive analytic expressions for $M_{\rm J}^\rm{min}$ and $n_{\rm ad}$, validated against 1D radiation hydrodynamics simulations.
• Evaluates the scaling of $M_{\rm J}^\rm{min}$ with dust abundance $A_{\rm d}$, radiation field strength, and metallicity, distinguishing regimes of weak vs. strong radiation fields.
• Integrates results into a broader context of IMF formation, comparing with turbulent fragmentation and protostellar feedback as alternative regulators of fragment mass.

Figure 1: Map of the parameter space of density versus temperature (left) or Jeans mass $M_{\rm J}$ (right) for collapsing cores cooled by dust emission. Solid curves plot the various simulation results for the central temperature in the \al@masunaga1998 radiation hydrodynamics simulations, using th

Experimental results

Research questions

• RQ1 What is the correct physical condition for the onset of quasi-adiabatic evolution in collapsing, dust-cooled gas clumps, and why is the $\tau \gtrsim 1$ criterion insufficient?
• RQ2 How does the minimum Jeans mass $M_{\rm J}^\rm{min}$ scale with dust abundance $A_{\rm d}$ and ambient radiation field strength in different astrophysical environments?
• RQ3 In what regimes does the opacity limit dominate over other fragmentation mechanisms like turbulent fragmentation or protostellar feedback in setting the IMF?
• RQ4 Why do previous works report conflicting scalings of $M_{\rm J}^\rm{min}$ with dust opacity, and what assumptions lead to unphysical results like $M_{\rm J}^\rm{min} \propto A_{\rm d}^{-1}$?
• RQ5 How do environmental factors such as metallicity and radiation field intensity affect the opacity-limited Jeans mass, and what are the implications for the IMF in extreme star-forming regions?

Key findings

• The adiabatic transition does not occur at $\tau \gtrsim 1$; instead, it is governed by the balance between compressional heating and radiative energy diffusion, invalidating a previously proposed regime where adiabatic evolution begins in optically thin cores.
• In weak radiation fields or low-dust environments, $M_{\rm J}^\rm{min} \propto A_{\rm d}^{-1/11}$ to $A_{\rm d}^{-1/15}$, indicating weak sensitivity to dust abundance.
• In strong radiation fields with abundant dust, $M_{\rm J}^\rm{min} \propto A_{\rm d}^{1/3}$, implying that $M_{\rm J}^\rm{min}$ increases with dust abundance, potentially making the IMF less bottom-heavy in such environments.
• The minimum Jeans mass is determined by the larger of two limits: a compression-heated limit (Low & Lynden-Bell 1976) and a radiation-heated limit (generalizing Masunaga & Inutsuka 1999), with the latter dominating in high-radiation, high-dust conditions.
• The derived scaling of $M_{\rm J}^\rm{min}$ with $A_{\rm d}$ and radiation field strength suggests that the opacity limit may play a more significant role in regulating the IMF in dense, dusty regions such as galactic centers or high-redshift star clusters.
• The results reconcile conflicting literature findings by showing that the $A_{\rm d}^{-1}$ scaling from Masunaga & Inutsuka (1999) is unphysical and arises from incorrect assumptions about the adiabatic transition condition.

Figure 2: Variation of the minimum Jeans mass $M_{\rm J}^{\rm min}$ as a function of the Solar-normalized dust opacity parameter $Z_{\rm d}\hat{\sigma}_{\rm d}$ and the radiative equilibrium temperature $T_{\rm bb}$ (Eq. 12 ). We assume a $\beta=1.5$ dust opacity model and estimate $M_{\rm J}^{\rm m