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https://github.com/fgittins/relmodpy

Computing the oscillation modes of relativistic stars in Python.
https://github.com/fgittins/relmodpy

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Computing the oscillation modes of relativistic stars in Python.

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# RelModPy

Computing the oscillation modes of relativistic stars in Python.



For the uninitiated, a good textbook on oscillation modes in the context of Newtonian gravity is Ref. [1]. This implementation follows closely Ref. [2].



## Usage

`relmodpy` depends on `numpy` and `scipy`. It uses `numpy` for mathematical functions and linear-algebra operations. `scipy` is used for solving differential equations.



An example of its use is shown in `example.py`. Throughout its implementation, geometric units, where $G = c = 1$, are assumed.



### Equation of state

To calculate oscillation modes, the user must supply, first and foremost, an equation of state for the stellar matter. This will be described by an `EOS` object that contains the following thermodynamic relationships as methods:

* `epsilon(p)`: the energy density [km^-2] as a function of pressure [km^-2],

* `Gamma(p)`: the adiabatic index associated with the background [dimensionless] as a function of pressure [km^-2],

* `Gamma1(p)`: the adiabatic index associated with the perturbations [dimensionless] as a function of pressure [km^-2].



`relmodpy` provides some simple examples in `eos.py`, one of which is used in `example.py`. These are assumed to be barotropic, where $\Gamma_1 = \Gamma$, and are called `Polytrope` and `EnergyPolytrope`. These are not intended to be accurate descriptions of neutron-star matter.



### Stellar background

The perturbations are calculated on top of a spherically symmetric background. This must be initialised before the modes are computed. 



The background is contained in a `Star` object, which has two parameters: an `EOS` object and `pc` central pressure [km^-2]. It can be initialised as

```

from relmodpy import Star

from relmodpy.eos import EnergyPolytrope



eos = EnergyPolytrope(1, 100)

star = Star(eos, 5.52e-3)

```



### Searching for oscillations

Provided the equation of state and the computed background, one is in a position to obtain the oscillation modes. The `Mode` object is initialised as

```

from relmodpy import Mode



mode = Mode(star)

```

Note: `mode` accesses the equation of state from `star`, since the background and mode will use the same equation of state.



Searching for the quadrupole *f*-mode can be done as

```

l = 2

omegaguess = (0.171 + 6.2e-5j)/star.M

mode.solve(l,

           (omegaguess,

            1.001*omegaguess.real + 1j*omegaguess.imag,

            omegaguess.real + 1.001j*omegaguess.imag))

```

From `mode`, the user can access the eigenfrequency `omega`, as well as the eigenfunctions `(H1, K, W, X)`.



## Testing



### Disclaimer

I do not claim that this code is suited to finding the infinite spectrum of oscillation modes of a star, since this is a substantially challenging, computational task. Furthermore, it should be noted that for oscillations that are damped extremely weakly (*g*-modes), the imaginary parts of their eigenfrequencies can be below machine precision. In `relmodpy`, an artifical cut-off frequency is assumed, below which only the real parts are calculated. I have, however, made an effort to test this code on established results in the literature [3,4]. These can be found in `test.py`.



### Running the tests

For example, to run the tests on `mode.py`, write

```

python -m unittest tests.test_mode

```

in the root directory. Warning: this can take quite a while...



## References

[1] Unno *et al.* (1989), "Nonradial oscillations of stars," University of

    Tokyo Press, Tokyo.



[2] Detweiler and Lindblom (1985), "On the nonradial pulsations of general

    relativistic stellar models," *Astrophys. J.* **292**, 12.



[3] Andersson, Kokkotas and Schutz (1995), "A new numerical approach to the

    oscillation modes of relativistic stars," *Mon. Not. R. Astron. Soc.*

    **274** (4), 1039.



[4] Krüger (2015), "Seismology of adolescent general relativistic neutron

    stars," PhD thesis, University of Southampton.