# Numerical code and data for the stellar structure and dynamical instability analysis of generalised uncertainty white dwarfs

## Citation

Mathew, Arun; Nandy, Malay K. (2021), Numerical code and data for the stellar structure and dynamical instability analysis of generalised uncertainty white dwarfs, Dryad, Dataset, https://doi.org/10.5061/dryad.dncjsxkzt

## Abstract

The enclosed code and dataset correspond to the numerical solution of  Tolman Oppenheimer Volkoff (TOV) equation (3.3) and (3.4) and the dynamical instability scheme given by equations 4.13 to 4.16 in the research article Existence of Chandrasekhar's limit in generalized uncertainty white dwarfs'' by the same authors. The dataset is generated for a wide range of central Fermi momentum $xi_c$ supplemented by the equation of state (2.6) and (2.11)  parametrized by the Fermi momentum $\xi$. For a given value of central Fermi momentum, the solution gives the total mass and radius of the white dwarfs. These solutions facilitate the evaluation of the integrals 4.14--4.16 giving the eigenfrequency of the fundamental mode in equation (4.13).

## Methods

The research article entitled Existence of Chandrasekhar's limit in generalized uncertainty white dwarfs'' by the same authors requires a numerical solution of the Einstein equation for spherically symmetric white dwarf stars.

A single dataset in Figures (1) and (2) in the research article corresponds to solving Tolman Oppenheimer Volkoff (TOV) equations for a range central Fermi momenta with a particular choice of GUP parameter $\beta_0$. The first-order differential equations (see equations 3.3 and 3.4 in the article) are solved numerically with the aid of C programming using the fourth-order Runge-Kutta method with boundary conditions as described in the article.

The dataset for the eigenfrequency of the fundamental mode is obtained from the dynamical instability scheme as described in the article. Integrations in equations 4.14-4.16 carried out employing the Trapezoidal method. This yields the eigenfrequency corresponding to a range of central Fermi momentum (or central density) for a particular choice of the GUP parameter $\beta_0$.

## Usage Notes

There is a total of 17 datasets to produce all the Figures in the article. There are mainly two different data files: GUP White Dwarf Mass-Radius (GUPWD_M-R) data and GUP White Dwarf Profile (GUPWD_Profile) data.

The file GUPWD_M-R gives only the Mass-Radius relation with Radius (km) in the first column and Mass (solar mass) in the second.

On the other hand GUPWD_Profile provides the complete profile with following columns.

column 1: Dimensionless central Fermi Momentum $\xi_c$
column 2: Central Density $\rho_c$ ( Log10 [$\rho_c$ g cm$^{-3}$] )
column 3: Radius $R$ (km)
column 4: Mass $M$ (solar mass)
column 5: Square of fundamental frequency $\omega_0^2$ (sec$^{-2}$)

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Figure 1 (a) gives Mass-Radius (M-R) curves for $\beta_0=10^{42}$, $10^{41}$ and $10^{40}$.
The filenames of the corresponding dataset are

GUPWD_M-R[Beta0=E42].dat
GUPWD_M-R[Beta0=E41].dat
GUPWD_M-R[Beta0=E40].dat

=====================================================================================

Figure 1 (b) gives Mass-Radius (M-R) curves for the high value of $\beta_0=10^{44}$ which is
numerically obtained in the dataset with the filename

GUPWD_M-R[Beta0=E44].dat.

The Figure also plots analytically obtained M-R relation. For low $\xi_c$ values (inset), the mass-radius curve is given by the expression (3.9) and (3.12) in the article, the corresponding data is given in the file ''GUPWD_M-R_Asym_Low.dat''. Note that Mass-radius the curve is independent of the GUP parameter for low values of central Fermi momentum. For high $\xi_c$ values, the mass-radius curve is given by the expression (3.20), and it is a function of the value $\beta_0$. The corresponding data is given in the file ''GUPWD_M-R_Asym_High[Beta0=E44].dat'' for $\beta_0=10^{44}$.

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Figure 2 (a) plots Mass-Radius (M-R) curves for $\beta_0=6.3\times 10^{39}$, $6.0\times 10^{39}$, $5.38\times 10^{39}$, $5.0\times 10^{39}$ and $4.5\times 10^{39}$.

The filenames of the corresponding dataset are

GUPWD_M-R[Beta0=6.30E39].dat
GUPWD_M-R[Beta0=6.00E39].dat
GUPWD_Profile[Beta0=5.38E39].dat
GUPWD_Profile[Beta0=5.00E39].dat
GUPWD_M-R[Beta0=4.50E39].dat

=====================================================================================

Figure 2 (b) plots Mass-Radius (M-R) curves for $\beta_0=10^{39}$,  $\beta_0=10^{38}$ and  $\beta_0=0$.

The filenames of the corresponding dataset are

GUPWD_M-R[Beta0=E39].dat
GUPWD_Profile[Beta0=1.00E38].dat
GUPWD_Profile[Beta0=0.0].dat

=====================================================================================

Figure 3 plots the square of the eigenfrequency of the fundamental mode as the function of central density.

The filenames for the corresponding dataset is

GUPWD_Profile[Beta0=1.00E40].dat
GUPWD_Profile[Beta0=5.60E39].dat
GUPWD_Profile[Beta0=5.38E39].dat
GUPWD_Profile[Beta0=5.00E39].dat
GUPWD_Profile[Beta0=1.00E38].dat
GUPWD_Profile[Beta0=0.0].dat

## Funding

Indian Institute of Technology Guwahati, Guwahati 781039, India

Indian Institute of Technology Guwahati, Guwahati 781039, India