Outflows in steep density gradients: diversity of behavior and implications for tidal disruption events and luminous fast blue optical transients

Abstract: Powerful explosions may undergo sustained energy injection as a central engine launches a wind into the surrounding gas, generating a forward and a reverse shock separated by a contact discontinuity. During the adiabatic phase, the dynamics depend strongly on the wind-to-ambient density ratio $f \equiv ρ<em>{\rm w} / ρ</em>{\rm a}$. For $f &lt;&lt; 1$, the reverse shock lies well inside the contact discontinuity, and the mechanical energy deposited by the wind is retained in a radially extended, approximately isobaric shocked-wind region whose pressure drives the swept-up ambient shell. For $f \gg 1$, the reverse shock remains close to the contact, and the expansion is governed by the ram-pressure interaction between the freely expanding wind and the swept-up ambient gas. We use analytic scalings and one-dimensional shock-capturing hydrodynamic simulations to determine how outflows in these two limits evolve in ambient density profiles $ρ<em>{\rm a} \propto r<sup>{-n}$, where $2 \leq n \leq 3$, and whether their shock structures accelerate or coast at constant velocity. For $n &gt; 2$, initially underdense outflows produce accelerating forward shocks whose radii evolve as $R</em>{\rm s} \propto t<sup>{3/(5-n)}$. Because $ρ<em>{\rm w} \propto r<sup>{-2}$, f increases with radius, causing the reverse-shocked wind region to contract relative to the contact position as the forward shock transitions toward constant-velocity expansion. This occurs when $f \sim$ a few at $t</em>{\rm dec} \propto f_0<sup>{1/(2-n)}$, where $f_0$ is the initial wind-to-ambient density ratio. By contrast, outflows initialized with $f_0 \gg 1$ do not develop an extended accelerating phase and remain approximately coasting throughout their adiabatic evolution. We discuss applications to tidal disruption event outflows and luminous fast blue optical transients, whose environments are often inferred to have steep density profiles with $n &gt; 2$.

• The paper shows that wind-driven bubbles in density profiles rho_a r^{-n} with 2 n 3 accelerate when initially underdense, then coast once the density contrast reaches order unity, while overdense winds coast throughout.
• The paper combines analytic scalings with one-dimensional simulations to verify that the transition time follows t_dec t_dyn f_0^{1/(2-n)}, making the duration of acceleration extremely sensitive to the initial density contrast when n is only slightly above 2.
• The paper finds that near-coasting tidal disruption event outflows are naturally explained by initially overdense ejecta-driven winds, whereas strongly accelerating luminous fast blue optical transients require evolving luminosity or structured ambient media beyond the constant-luminosity model.

Overview

This paper examines how wind-driven bubbles evolve when a time-steady, constant-velocity central-engine wind expands into a cold ambient medium with a steep power-law density profile, $\rho_{\rm a} \propto r^{-n}$ for $2 \leq n \leq 3$. The key control parameter is the instantaneous ratio of freely expanding wind density to ambient density at the contact discontinuity, $f(R_{\rm c}) = f_0 (R_{\rm c}/R_{\rm w,0})^{n-2}$. Because $\rho_{\rm w} \propto r^{-2}$, this contrast is constant at $n=2$ and grows with radius for $n>2$, which inverts the usual intuition from wind-bubble theory: steep gradients do not guarantee accelerating shocks. The authors combine analytic scalings with one-dimensional shock-capturing moving-mesh simulations to delineate when bubbles accelerate (pressure-driven) versus coast (ejecta-driven), and apply the results to tidal disruption events (TDEs) and luminous fast blue optical transients (LFBOTs).

Analytic framework

The two classical regimes are distinguished by $f(R_{\rm c})$. In the pressure-driven limit ($f \ll 1$), the reverse shock recedes well inside the contact discontinuity, leaving an extended, approximately isobaric shocked-wind reservoir that drives the swept-up shell via $p\,dV$ work; the generalized Weaver–Koo–McKee scaling gives

$$R_{\rm s}(t) = \left( \frac{L_{\rm w}}{\rho_{\rm a,0} R_{\rm w,0}^n} \right)^{1/(5-n)} t^{3/(5-n)},$$

so the expansion index $2 \leq n \leq 3$0 exceeds unity for $2 \leq n \leq 3$1. In the ejecta-driven limit ($2 \leq n \leq 3$2), the reverse shock hugs the contact discontinuity, the reverse-shocked shell has relative thickness $2 \leq n \leq 3$3, and the forward shock coasts at a velocity marginally above the wind speed per the self-similar solutions of Coughlin (2024). For $2 \leq n \leq 3$4, an initially ejecta-driven bubble becomes progressively more robustly coasting as it expands, since $2 \leq n \leq 3$5 grows; such a system cannot naturally relax into the pressure-driven regime.

The paper's central quantitative prediction concerns initially underdense winds: because $2 \leq n \leq 3$6 increases with radius, a pressure-driven bubble accelerates only until $2 \leq n \leq 3$7 reaches order unity, after which it transitions to coasting. Setting $2 \leq n \leq 3$8 within the pressure-driven scaling yields a deceleration time

$2 \leq n \leq 3$9

where $f(R_{\rm c}) = f_0 (R_{\rm c}/R_{\rm w,0})^{n-2}$0. The exponent diverges as $f(R_{\rm c}) = f_0 (R_{\rm c}/R_{\rm w,0})^{n-2}$1, so even modest changes in $f(R_{\rm c}) = f_0 (R_{\rm c}/R_{\rm w,0})^{n-2}$2 produce very large changes in the duration of the accelerating phase for profiles only slightly steeper than $f(R_{\rm c}) = f_0 (R_{\rm c}/R_{\rm w,0})^{n-2}$3 — a strong sensitivity with direct observational consequences.

Numerical results

Simulations use a relativistic shock-capturing moving-mesh code in which the computational domain tracks only the two shocked regions, with boundaries following the reverse shock, contact discontinuity, and forward shock. Runs adopt $f(R_{\rm c}) = f_0 (R_{\rm c}/R_{\rm w,0})^{n-2}$4, $f(R_{\rm c}) = f_0 (R_{\rm c}/R_{\rm w,0})^{n-2}$5, 800 zones per shocked subdomain, and evolve to $f(R_{\rm c}) = f_0 (R_{\rm c}/R_{\rm w,0})^{n-2}$6 for $f(R_{\rm c}) = f_0 (R_{\rm c}/R_{\rm w,0})^{n-2}$7 with $f(R_{\rm c}) = f_0 (R_{\rm c}/R_{\rm w,0})^{n-2}$8 (initially pressure-driven) or $f(R_{\rm c}) = f_0 (R_{\rm c}/R_{\rm w,0})^{n-2}$9 (initially ejecta-driven).

Three results stand out:

• Initially underdense winds accelerate, then coast. After a startup transient, the $\rho_{\rm w} \propto r^{-2}$0 runs enter an accelerating phase consistent with $\rho_{\rm w} \propto r^{-2}$1, then relax toward $\rho_{\rm w} \propto r^{-2}$2 as $\rho_{\rm w} \propto r^{-2}$3 grows. The transition occurs when $\rho_{\rm w} \propto r^{-2}$4 a few.
• The measured $\rho_{\rm w} \propto r^{-2}$5 follows the predicted scaling. Across all $\rho_{\rm w} \propto r^{-2}$6, the simulated deceleration times track $\rho_{\rm w} \propto r^{-2}$7, up to order-unity normalization offsets expected from the crude derivation. This confirms that the accelerating-phase duration is set by the time for the growing density contrast to reach unity.
• Initially overdense winds never accelerate. For $\rho_{\rm w} \propto r^{-2}$8, the reverse-shocked shell remains thin throughout, with $\rho_{\rm w} \propto r^{-2}$9 and $n=2$0 approaching the constant value of the interaction solutions. Fluid profiles match the ejecta-driven similarity solutions well at late times, though convergence slows as $n=2$1, where those solutions formally diverge.

An implication of the first result is that radio-emitting shocks in steep environments can show a light-curve slope change as the system crosses between regimes; the implication of the second is that observed near-coasting expansion does not by itself constrain $n=2$2 when $n=2$3, since parameters become degenerate.

Applications to TDEs

Radio modeling of several TDEs implies both steep circumnuclear profiles and near-coasting outflows: ASASSN-14li has $n=2$4 with $n=2$5 (Alexander et al., 2015), while AT2019dsg expanded as $n=2$6 at $n=2$7 through a profile steepening from $n=2$8 to $n=2$9 (Cendes et al., 2021). Within the pressure-driven picture these profiles would demand accelerating shocks and Rayleigh–Taylor-unstable shells, contrary to observation. The authors argue instead that TDE outflows have large initial overdensities ($n>2$0), placing them firmly in the ejecta-driven regime, where coasting is increasingly robust for $n>2$1. The near-coasting radio expansion of TDEs is therefore naturally explained without invoking fine-tuned ambient structure.

Applications to LFBOTs

LFBOTs show more diverse behavior. Mildly accelerating events such as AT2024wpp, with inferred $n>2$2–$n>2$3, are broadly compatible with energy-conserving evolution in a single power-law medium with plausible $n>2$4; a dimensionalized simulation with $n>2$5, $n>2$6, $n>2$7 reproduces the early shock radii reasonably well. Stronger accelerators are harder to accommodate: AT2023vth's early index $n>2$8 would require an effective $n>2$9 under constant luminosity, beyond the validity of the pressure-driven solutions. Relaxations include rising wind luminosity ($f(R_{\rm c})$0 with $f(R_{\rm c})$1 permits $f(R_{\rm c})$2 for $f(R_{\rm c})$3) or a broken power-law ambient profile, which could also explain the apparently sharp acceleration-to-coasting transitions — in the idealized single-power-law models here, the transition spans many orders of magnitude in time.

A further tension is physical realizability: the pressure-driven branch requires $f(R_{\rm c})$4, but fiducial LFBOT-like wind parameters ($f(R_{\rm c})$5, $f(R_{\rm c})$6cm, $f(R_{\rm c})$7, $f(R_{\rm c})$8) give $f(R_{\rm c})$9, implying coasting from the outset rather than initial acceleration. Events like CSS161010, with roughly constant mildly relativistic velocity (Coppejans et al., 2020), fit the ejecta-driven interpretation directly. For $f \ll 1$0, the absence of an observed transition could lower-bound $f \ll 1$1 and constrain $f \ll 1$2, but for $f \ll 1$3 no such constraints follow.

Caveats in radio-inferred shock dynamics

The paper devotes substantial attention to systematic uncertainties in the Chevalier (1998) synchrotron self-absorption formalism used to infer shock radii. Fixed microphysical parameters ($f \ll 1$4, $f \ll 1$5, filling factor, distance) shift only the radius normalization and not $f \ll 1$6 when applied consistently across epochs; the inferred dynamics are most sensitive to epoch-dependent errors in peak flux and frequency, non-simultaneous sampling, and choices about whether spectral slopes are fixed or fitted per epoch. Multi-zone structure or asphericity could cause successive fitted radii to trace different physical surfaces — plausibly relevant to the final epoch of AT2024wpp, where the inferred radius decreases, which is difficult to realize physically for a forward shock. The comparisons to observations should therefore be read as qualitative tests of accelerating versus coasting behavior rather than precise constraints on $f \ll 1$7, $f \ll 1$8, $f \ll 1$9, or $p\,dV$0; joint hydrodynamic-plus-radiative forward modeling is proposed as the more self-consistent approach.

Limitations and open questions

The calculations assume a constant-luminosity, spherical wind in a single power-law medium, adiabatic evolution, and one-dimensional geometry. Superbubble breakout, despite qualitative overlap with the accelerating solutions, involves radiative losses, mixing, and latitude-dependent density structure that require multidimensional treatment and is explicitly left outside the scope of the work. The ejecta-driven similarity solutions do not exist at $p\,dV$1, limiting analytic comparison there. Open questions raised by the paper include whether broken power-law or time-dependent-luminosity models can reproduce the abruptness of observed LFBOT transitions, what mechanism produces an apparently receding emitting region in events like AT2024wpp, and how degeneracies among $p\,dV$2, $p\,dV$3, and $p\,dV$4 can be broken observationally when $p\,dV$5.

Conclusion

This work establishes that in steep density gradients ($p\,dV$6), the evolutionary outcome of a wind-driven bubble is dictated primarily by its initial wind-to-ambient density contrast rather than by the slope alone. Underdense winds accelerate as $p\,dV$7 until $p\,dV$8 reaches order unity at $p\,dV$9, then coast; overdense winds coast throughout, with the approximation improving over time. Applied to transients, the framework explains near-coasting TDE radio outflows as overdense ejecta-driven flows, while showing that strongly accelerating LFBOTs cannot be described by constant-luminosity winds in single power-law media without additional ingredients such as rising energy injection or structured circumstellar environments.