Gamma-Ray Burst Jets in Circumstellar Material: Dynamics, Breakout, and Diversity of Transients
Abstract
Abstract Recent observations indicate that stripped-envelope core-collapse supernovae are often surrounded by dense circumstellar material (CSM). Motivated by this, we develop an analytic model to systematically study the dynamics of long gamma-ray burst (LGRB) jet propagation in various CSM environments. We derive a general expression for the jet head velocity ( β h ) and breakout time ( t b ) valid across Newtonian, relativistic, and intermediate regimes, accounting for a previously unrecognized dependence on 1 − β h . Our results highlight a fundamental distinction between jet propagation in massive stars, where β h ≪ 1, and in extended CSM, where 1 − β h ≪ 1. We establish an analytic success/failure criterion for jets and express it in terms of jet and CSM parameters, revealing a strong dependence on CSM radius. To quantify the relativistic nature of the jet-cocoon system, we introduce the energy-weighted proper velocity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mrow> <mml:mo mathvariant="normal">Γ</mml:mo> <mml:mi>β</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy="true">¯</mml:mo> </mml:mrow> </mml:mover> </mml:math> . We identify three possible jet outcomes—(a) successful jets ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mrow> <mml:mo mathvariant="normal">Γ</mml:mo> <mml:mi>β</mml:mi> </mml:mrow> <mml:mo stretchy="true">¯</mml:mo> </mml:mover> <mml:mo>∼</mml:mo> <mml:mn>10</mml:mn> <mml:mo>-</mml:mo> <mml:mn>100</mml:mn> </mml:math> ), (b) barely failed jets ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mrow> <mml:mo mathvariant="normal">Γ</mml:mo> <mml:mi>β</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy="true">¯</mml:mo> </mml:mrow> </mml:mover> <mml:mo>∼</mml:mo> <mml:mn>1</mml:mn> </mml:math> ), and (c) completely failed jets ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mrow> <mml:mo mathvariant="normal">Γ</mml:mo> <mml:mi>β</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy="true">¯</mml:mo> </mml:mrow> </mml:mover> <mml:mo>∼</mml:mo> <mml:mn>0.1</mml:mn> </mml:math> )—and constrain their respective jet/CSM parameter spaces. We show that, in (b) and (c), large CSM radii can result in luminous fast blue optical transients via cocoon cooling emission. This theoretical framework provides a basis for future observational and theoretical studies to understand the link between LGRBs, intermediate GRBs, low-luminosity LGRBs, and their environments.