Tolman–Oppenheimer–Volkoff Equation Solver in Julia

Overview

This project provides a Julia implementation for solving the Tolman–Oppenheimer–Volkoff (TOV) equations. It uses equation of state (EoS) to model compact stars (e.g. neutron star, quark star) and calculates their mass, radius, and tidal deformability (cf. the PiecewisePolytrope_test.ipynb). Input EoS should be in the geometrized unit system, where distances are expressed in centimeters [cm]. If the energy_density and pressure are given in [g/cm^3] unit, the inputs of the MainModule should be energy_density/unit_g and pressure/unit_g [1/cm^3]. In addition, this project includes the Julia code to generate a piecewise polytrope EoS for a given parameters.

Files and Modules

1. main.jl

• Purpose: Acts as the main entry point for calculations.
• Key Functions:
  • make_eos_monotonic(e, P): Processes energy density (e) and pressure (P) arrays to ensure monotonicity by removing non-physical regions.
  • out_RMT(ε, pres; ...): Solves the TOV equations to compute radius, mass, and tidal deformability for given energy densities (ε) [1/cm^3] and pressures (pres) [1/cm^3].
• Dependencies:
  • solver_code.jl for solving the TOV equations.

2. solver_code.jl

• Purpose: Contains the numerical solvers and tools to compute stellar properties.
• Key Functions:
  • Debug(input_list; base_filename): Logs data for debugging.
  • tidal_deformability(y, M, R): Computes tidal deformability given dimensionless compactness and stellar properties.
  • solveTOV_RMT(center_idx, ε, pres, debug_flag): Solves the TOV equations for a given central density [1/cm^3] and pressure [1/cm^3] using an ODE solver.
  • TOV_def!(...): Defines the TOV differential equations.
• Dependencies: Relies on DifferentialEquations.jl for ODE solving.

3. piecewise_polytrope_eos_cgs.jl

• Purpose: Implements the piecewise polytropic EoS framework in the CGS unit system!
• Key Features:
  • Defines fixed crust parameters for the EoS.
  • Provides functions to calculate pressure [g/cm^3], energy density [g/cm^3], and polytropic constants for different density segments.
• Key Functions:
  • get_all_params(log_p1, Gamma; p_c=fixed_crust): Computes all parameters required to describe the equations of state.
  • make_polyEos(rho_lim_arr, a_arr, ...): Constructs energy density and pressure arrays for the entire density range.

4. constants.jl

• Purpose: Contains physical and conversion constants used throughout the code.

Usage

Prerequisites

• Julia version 1.6 or higher.
• Install required Julia packages:

  using Pkg
  Pkg.add(["DifferentialEquations", "Dates"])

Running the Code

1. Include the necessary modules in your Julia environment:

   include("main.jl")
   using .MainModule

2. Prepare energy density and pressure data (in this example, energy_density and pressure are given in [g/cm^3]):

   energy_density, pressure = MainModule.make_eos_monotonic(energy_density, pressure)

   

3. Call out_RMT to calculate stellar properties (This input EoS should be in geometrized unit system, where distances are expressed in centimeters [cm]). If energy_density and pressure are given in [g/cm^3] unit, the inputs should be energy_density/unit_g and pressure/unit_g [1/cm^3]:

   RMT, solution = MainModule.out_RMT(energy_density/unit_g, pressure/unit_g)

4. Plot results contained in RMT (cf. the PiecewisePolytrope_test.ipynb):

   

Key Concepts

1. Tolman–Oppenheimer–Volkoff (TOV) Equations

The TOV equations describe the structure of a spherically symmetric star in hydrostatic equilibrium, accounting for general relativity.

2. Piecewise Polytropic EoS

The piecewise polytropic model approximates the neutron star EoS by dividing it into segments, each described by a polytropic relation: [ P(\rho) = K \rho^\Gamma ]

3. Tidal Deformability

Tidal deformability quantifies how easily a compact star deforms under an external gravitational field, crucial for gravitational wave studies.

References

• T. Hinderer, B. D. Lackey, R. N. Lang and J. S. Read, Phys. Rev. D 81, 123016 (2010) doi:10.1103/PhysRevD.81.123016 [arXiv:0911.3535 [astro-ph.HE]].
• J. S. Read, B. D. Lackey, B. J. Owen and J. L. Friedman, Phys. Rev. D 79, 124032 (2009) doi:10.1103/PhysRevD.79.124032 [arXiv:0812.2163 [astro-ph]].
• Y. Fujimoto, K. Fukushima, K. Hotokezaka and K. Kyutoku, Phys. Rev. Lett. 130, no.9, 091404 (2023) doi:10.1103/PhysRevLett.130.091404 [arXiv:2205.03882 [astro-ph.HE]].
• G. Baym, T. Hatsuda, T. Kojo, P. D. Powell, Y. Song and T. Takatsuka, Rept. Prog. Phys. 81, no.5, 056902 (2018) doi:10.1088/1361-6633/aaae14 [arXiv:1707.04966 [astro-ph.HE]].
• B. P. Abbott et al. [LIGO Scientific and Virgo], Phys. Rev. Lett. 121, no.16, 161101 (2018) doi:10.1103/PhysRevLett.121.161101 [arXiv:1805.11581 [gr-qc]].

Contributing

Feel free to submit issues or pull requests to improve this project.

License

This project is licensed under the MIT License. See the LICENSE file for details.