Deepanshu Gurjar

Supervisor: Dr. Luke Chamandy

In this project, we aim to investigate the impact of a neutron star jet on the Common Envelope Evolution (CEE) process. The Common Envelope Evolution occurs in binary star systems where one star expands and engulfs its companion within its envelope. This interaction leads to the formation of a close binary system, which can undergo various evolutionary paths.

Common Envelope Evolution (CEE)

During the common envelope phase, the orbital energy of the binary system is transferred to the envelope, resulting in the inspiral which can either lead to ejection of the envelope or merger of the core and companion. The CEE process plays a significant role in the formation of compact binaries, such as binary neutron star systems.

Methods

In this section, we describe the methodology and setup employed in our study of Common Envelope Evolution using Adaptive Mesh Refinement (AMR) techniques with the ASTROBEAR code. The simulation setup involved a detailed consideration of the stellar parameters, computational domain, and resolution levels.

Stellar Parameters

Jet material The primary star in our simulation has a mass of 1.96 and a radius of 48.1R_⊙. Within the primary star, the core contains a mass of 0.37M_⊙. The secondary star has a mass of 0.98M_⊙ and is situated at a distance of 49.0R_⊙ from the primary star.

Computational Domain

The computational domain, or simulation box, was set to dimensions of 1150R_⊙ × 1150R_⊙. This adequately encapsulates the physical region of interest for the common envelope evolution process, ensuring that the interaction between the two stars and the ensuing envelope ejection can be accurately simulated.

Adaptive Mesh Refinement (AMR)

We employed Adaptive Mesh Refinement (AMR) techniques to ensure that our simulations achieved both high accuracy and computational efficiency. The highest resolution utilized in our simulations was 0.140R_⊙, which allowed us to capture intricate details of the common envelope evolution. Additionally, we used a base resolution of 2.25R_⊙ in regions where a coarser grid was sufficient, optimizing our computational resources.

Ambient Medium Properties

We specify the density and pressure of the ambient medium such that we can resolve the pressure height scale with the availaible computatinal resources. The density of the ambient medium was set to 6.7 × 10⁻⁹g cm⁻³ . The pressure was chosen to be 1.0 × 10⁵dyn cm⁻² .

JET MODEL

Jet Geometry

• The jet axis is oriented perpendicular to the orbital plane of the binary system.
• The jet has half-opening angle denoted by \(\theta_h\) which is set to \(\theta_h = 15\)° This angle defines the extent of the jet’s angular spread.
• The distribution of density and radial velocity within the jet varies with polar angle. It is strongly peaked at polar angles less than θ_h/6 , emphasizing the concentration of jet material in a particular direction.
• The majority of the jet mass is concentrated at a distance approximately equal to half the distance between the boundary of jet’s spherical sector and the secondary star. This distance, denoted as r_j , was defined as 2.25R_⊙.
• Following the framework established by Fedderath, our jet model features a high-momentum spine with a narrow opening angle. This spine represents a focused, collimated portion of the jet with high momentum.
• In addition to the high-momentum spine, our model also includes a lower-momentum component known as the ‘wide angle wind.’ This component is characterized by a broader angular spread, contributing to a more diffuse region of the jet.

Initial Conditions

• To initiate the jet material, we provided it with an additional velocity component equal to the instantaneous orbital velocity of the secondary star.
• The temperature of the jet material was set to T_j = 10⁴ Kelvin (10⁴ K). It is worth noting that the results of our simulations are independent of the jet’s temperature due to the supersonic nature of the jet.

Super-Eddington Accretion

As the envelope material starts accreting on to the secodnary, we can calculate this rate of accretion. Using the virial theorem, which states that half the liberated potential energy is added to the accretor in the form of thermal energy. Assuming the opacity due to electron, we find the following expression for the mass accretion rate on to the secondary

where \(R_*\) is the radius of the companion and \(\tilde{\lambda}\) is the efficieny parameter that depends on the density profile

Envelope Unbinding

Following definition for gas "unbound" has been used

where

• \(E_{kin, gas}\) = Kinetic energy density of bulk motion
• \(E_{int, gas}\) = Internal energy density
• \(E_{pot, gas-gas}\) = Potential energy density due to self-gravity of the gas
• \(E_{pot, gas-1}\) = Potential energy of gas with particle 1, the RGB core
• \(E_{pot, gas-2}\) = Potential energy of gas with particle 2, the companion

Gas is considred to be unbound if \(E_{unb} \) > 0 .

Simulation Runs

The following table shows the details of the simulation runs conducted in this project:

Run	Significance	\(\dot{M}_j (10^{-5} M_\odot)\) yr\(^{-1})\)	\(v_j\) (km/s)	\(\dot{M}_\mathrm{Edd}\) \(M_\odot)\ yr\(^{-1}\)	\(\dot{M}_{acc} (M_\odot) \) yr(\{-1})	MinDensity (g/cc)	\(\dot{M}_j / \dot{M}_\mathrm{Edd}\)	Restart_time(frame)	MaxLevel	Status
J8	WD run	2	8640	2.1 × 10⁻⁵	0.2	1e-10	0.95	0 d	4	Complete
001	Higher \(\dot{M}_j\)	20	8640	-	0.2 - 1.0	1e-10	-	19.91 d (86)	4	Complete
003	NS run (High \( \dot{M}_j \) and \(v_j\))	20	30000	3.4 × 10⁻¹⁰	0.1 to 1.0	1e-10	\(5.9 \times 10^{6}\)	19.91 d(86)	4	Complete
004	No-Accretion NS run	20	30000	3.4 × 10⁻¹⁰	0	1e-10	\(5.9 \times 10^{6}\)	19.91 d(86)	4	Complete
005	Lower MinDensity	20	30000	3.4 × 10⁻¹⁰	-	1e-12	\(5.9 \times 10^{6}\)	33.80 d(146)	4	Complete
006	High Resolution	20	30000	3.4 × 10⁻¹⁰	-	1e-12	\(5.9 \times 10^{6}\)	19.91 d(86)	5	frame 93
007	Low Resolution	20	30000	3.4 × 10⁻¹⁰	-	1e-12	\(5.9 \times 10^{6}\)	19.91 d(86)	3	Complete
008	Started at an earlier time	20	30000	3.4 × 10⁻¹⁰	-	1e-14	\(5.9 \times 10^{6}\)	14 d(60)	4	Complete
009	Higher \(\dot{M}_j\)	200	30000	3.4 × 10⁻¹⁰	-	1e-12	\(5.9 \times 10^{7}\)	20 d(60)	4	frame 102
011	No Accretion, Started at an earlier time	20	30000	3.4 × 10⁻¹⁰	-	1e-12	\(5.9 \times 10^{6}\)	14 d(60)	4	Complete