Neutron Star F-Mode Solver project.

📚 Academic & Research Keywords

• Primary Field: Computational Astrophysics / General Relativity
• Specific Object: Compact Objects (Neutron Stars)
• Physics Concepts: Hydrostatic Equilibrium, Radial Oscillations, f-mode Eigenfrequencies, Geometrized Units ($G=c=1$)
• Data Source: CompOSE (CompStar Online Supernovae Equations of State)
• Methods: Shooting Method, Boundary Value Problems (BVP), Monte Carlo Uncertainty Quantification

🧠 Concept Categories

1. Physics & Astronomy

• Stellar Structure: Solving the Tolman-Oppenheimer-Volkoff (TOV) equations.
• Perturbation Theory: Linearized radial pulsation equations.
• Nuclear Physics: Interpreting tabulated Equations of State (EoS) for dense matter.

2. Numerical Methods

• ODE Integration: Using Runge-Kutta methods (via solve_ivp) for coupled differential equations.
• Root Finding: Using the "Shooting Method" to find eigenvalues (frequencies) that satisfy boundary conditions.
• Interpolation: Handling tabular data (interp1d).

3. Data Science & Python

• Stochastic Modeling: Using Monte Carlo simulations to generate population statistics.
• Data Visualization: Using Seaborn for violin plots and correlation heatmaps to analyze physical properties.

🌌 TOV Solver with Malliavin Calculus

Uncertainty Quantification for Neutron Star Models

This project provides a Python-based solver for the Tolman-Oppenheimer-Volkoff (TOV) equations, enhanced with Malliavin Calculus for stochastic uncertainty quantification.

Designed for Google Colab, this script allows physicists and researchers to upload Equation of State (EoS) files, generate Mass-Radius relations, and mathematically verify the stability of the numerical solutions against stochastic noise.

🚀 Key Features

• Stochastic TOV Solver: Solves hydrostatic equilibrium equations using the Euler-Maruyama method with injected noise.
• Malliavin Calculus: Computes the sensitivity of the Mass/Radius solution to the underlying Brownian motion to establish a theoretical variance bound.
• Smart Auto-Detection: Automatically detects Pressure and Density columns (including Log10 density) in uploaded data files.
• Consistency Check: Verifies if the empirical variance of the Monte Carlo simulations respects the Malliavin theoretical bound ($Var[F] \le E[|DF|^2]$).
• Visualization: Plots Mass-Radius curves with uncertainty clouds and consistency ratio bar charts.

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📦 Dependencies

This script is optimized for Google Colab.

• numpy: Numerical computations and array handling.
• matplotlib: Plotting Mass-Radius relations and statistical charts.
• pandas: Robust file parsing and data ingestion.
• google.colab: For file upload widgets.

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📖 How to Run

1. Open in Google Colab: Copy the script into a notebook cell.
2. Run the Cell: Execute the code.
3. Upload Files:
   • A prompt will appear: Upload your EoS files now...
   • Click "Choose Files" and select your .dat, .txt, or .thermo EoS files.
4. Automatic Analysis:
   • The script will parse the files.
   • It will run 200 stochastic simulations per file.
   • It will generate plots and a statistical summary table.

Note: If you do not upload any files, the script will automatically generate synthetic sample data to demonstrate functionality.

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📄 Input Data Format

The script uses a heuristic autodetect_eos_columns function to read standard EoS files. Ideally, your files should look like this:

• Format: ASCII text (columns separated by spaces or tabs).
• Columns: Must contain Pressure (P) and Density ($\rho$).
• Units: The solver assumes consistent nuclear units (e.g., $fm^{-4}$ or $MeV/fm^3$).
• Logarithmic Data: The script detects if density is provided as $\log_{10}(\rho)$ (usually by checking for negative values) and converts it automatically.

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📊 Interpreting the Results

The script outputs a summary table with a "Consistency Ratio". Here is how to read it:

The Malliavin Bound

Malliavin calculus provides a derivative operator $D$. The fundamental inequality checked is: $$ \text{Empirical Variance} \approx E[(F - E[F])^2] \le E[|D F|_{H}^2] $$

Output Columns

• Max Mass: The maximum mass supported by the EoS (in $M_{\odot}$).
• Empirical Variance: The actual variance observed across the 200 Monte Carlo paths.
• Malliavin Bound: The theoretical upper bound calculated via the sensitivity Jacobian.
• Ratio: $\frac{\text{Empirical Variance}}{\text{Malliavin Bound}}$

Status Codes

• 🟢 PASS (Ratio $\le$ 1.05): The model is mathematically consistent. The numerical noise is well-behaved.
• 🔴 FAIL (Ratio > 1.05): The empirical variance exceeds the theoretical bound. This suggests numerical instability, overly stiff EoS regions, or integration steps ($dr$) that are too coarse.

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⚠️ Troubleshooting

• "Failed to confidently detect columns": Your file might have too many header lines or non-standard formatting. Ensure standard space-delimited columns.
• Step Size: The default radial step is dr = 0.01. If your simulation fails, try reducing this value in the code.
• Noise Magnitude: The default noise is sigma = 1e-3. If the variance is too high, reduce sigma.

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📜 License

Open source. Free to use for academic and research purposes.

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🌟 Neutron Star F-Mode Solver & Monte Carlo Analysis

Physics-Informed Machine Learning & Asteroseismology

This project is a high-precision numerical toolkit designed to solve the Tolman-Oppenheimer-Volkoff (TOV) equations and the Linearized Radial Pulsation equations.

It takes tabulated Equation of State (EoS) data specifically from the CompOSE database format and calculates the fundamental oscillation frequency (f-mode), Mass, and Radius of neutron stars. It then performs a Monte Carlo simulation to explore the stability landscape of various nuclear models.

🚀 Key Features

• Geometrized Unit System: Internal calculations are performed using $G=c=1$ for numerical stability, with automatic conversion from Nuclear units (MeV/fm³) and SI units.
• Eigenfrequency Shooting: Solves the boundary value problem for radial pulsations to find the fundamental eigenfrequency $\omega_0$.
• Monte Carlo Sampling: Instead of calculating a single star, the script samples a distribution of central pressures ($P_c$) to generate full Mass-Radius-Frequency curves with statistical uncertainty.
• CompOSE Compatibility: Native support for .thermo files from the CompOSE nuclear physics database.
• Advanced Visualization: Generates 4 distinct plots including M-R relations colored by frequency, violin plots of stability, and correlation heatmaps.

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📦 Dependencies

This script is optimized for Google Colab and uses the scientific Python stack:

• numpy & scipy: For ODE integration (solve_ivp), root finding (root_scalar), and interpolation.
• pandas: Data management for Monte Carlo results.
• seaborn & matplotlib: Statistical data visualization.
• google.colab: For file upload widgets.

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⚙️ Input Data Format (CompOSE)

The code is pre-configured to read CompOSE .thermo files.

• File Extension: .thermo
• Header: Skips the first 2 lines.
• Units: Assumes input data is in MeV/fm³.
• Column Indices:
  • Pressure ($P$): Column Index 12
  • Energy Density ($\epsilon$): Column Index 13

⚠️ Important Configuration: If your data files use different column indices, locate this section in the __main__ block and adjust the integers:

# 0-indexed column numbers
P_SI_index = 12
eps_SI_index = 13

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📖 How to Run

1. Start the Environment: Open the script in a Google Colab notebook.
2. Execute: Run the cell.
3. Upload: A file upload widget will appear. Select one or more .thermo files from your local machine.
4. Wait for Computation:
   • The code converts units to Geometrized form.
   • It runs 50 Monte Carlo iterations per EoS file.
   • It solves the ODEs for hydrostatic equilibrium and pulsation.
5. View Results: The script will automatically output four analytical plots.

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📊 Outputs & Visualization

The tool generates the following visualizations to help interpret the nuclear physics:

1. M-R Relation (Bubble Plot):

   • X-Axis: Mass ($M_{\odot}$)
   • Y-Axis: Radius (Geometric Units)
   • Color: Different EoS models.
   • Size: Magnitude of the f-mode frequency ($\omega$).

2. Violin Plot:

   • Displays the distribution of fundamental frequencies ($\omega$) allowed by each Equation of State.

3. 2D Density Histogram:

   • A heatmap showing the most probable regions in the Mass vs. Frequency phase space.

4. Correlation Matrix:

   • Quantifies the linear relationships between Central Pressure, Mass, Radius, and Frequency.

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📐 Physics Background

1. TOV Equations (Structure): Determines the static structure of the star: $$ \frac{dP}{dr} = - \frac{(\epsilon + P)(m + 4\pi r^3 P)}{r(r-2m)} $$ $$ \frac{dm}{dr} = 4\pi r^2 \epsilon $$

2. Radial Pulsation (Stability): Determines the oscillation eigenfrequencies by solving for the displacement $\xi$ and pressure perturbation $\Delta P$: $$ \frac{d\xi}{dr} = -\left(\frac{3}{r} + \frac{P'}{\Gamma_1 P}\right)\xi - \frac{\Delta P}{\Gamma_1 P} $$ ...coupled with the equation for $\Delta P$. The solver "shoots" for $\omega^2$ such that the boundary conditions at the surface are satisfied.

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📜 License

Open Source. Free for academic and educational use in relativistic astrophysics.