O2scl_eos Documentation

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All equations of state inherit from eos (except for the TOV solver tov_solve and cold_nstar). The hal_eos library contains all of the classes mentioned in this documentation.


User's Guide


Hadronic equations of state

The hadronic equations of state are all inherited from hadronic_eos: schematic_eos, skyrme_eos, rmf_eos, apr_eos, and gen_potential_eos.

hadronic_eos includes several methods that can be used to calculate the saturation properties of nuclear matter. These methods are sometimes overloaded in descendants when exact formulas are available.

There is also a set of classes to modify the quartic term of the symmetry energy: rmf4_eos, apr4_eos, skyrme4_eos, and mdi4_eos all based on sym4_eos_base which can be used in sym4_eos.


Equations of state of quark matter

The equations of state of quark matter are all inherited from quark_eos: bag_eos is a simple bag model, nambujl_eos is the Nambu--Jona-Lasinio model.


Solution of the Tolman-Oppenheimer-Volkov equations

The class tov_solve provide a solution to the Tolman-Oppenheimer-Volkov (TOV) equations given an equation of state. This is particularly useful for static neutron star structure: given any equation of state one can calculate the mass vs. radius curve and the properties of any star of a given mass. An adaptive integration is employed and calculates the gravitational mass, the baryonic mass (if the baryon density is supplied), and the gravitational potential. The remaining columns is the equation of state are also interpolated into the solution, e.g. if a chemical potential is given, then the radial dependence of the chemical potential for a 1.4 solar mass star can be automatically computed. The equation of state may be specified in arbitrary units so long as an appropriate conversion factor is supplied. An equation of state for low densities (baryon density < 0.08 $ \mathrm{fm}^{-3} $ ) is provided and can be automatically appended to the user-defined equation of state.

This is still experimental.


Naive Cold Neutron Stars

There is also a class to calculate zero-temperature neutron stars: cold_nstar. It uses tov_solve to compute the structure, given a hadronic equation of state (of type hadronic_eos). It also computes the adiabatic index, the speed of sound, and determines the possibility of the direct Urca process as a function of density or radius.

This is still experimental.


Non-relativistic Finite Temperature Approximations

This is taken from the Prakash87.

The entropy is

\[ s = -\sum_k \left[ n_k \ln n_k + \left(1-n_k\right) \ln \left(1-n_k\right)\right] \]

The low-temperature (degenerate) approximation to the entropy is

\[ s=\pi^2/3 N(0) T \]

where the density of states at the Fermi surface is

\[ N(0)=\sum_k \delta(\epsilon_k -\mu) = \frac{3 \rho }{k_F v_F} \]

where the Fermi velocity is

\[ v_F = \left.\frac{\partial \epsilon_k}{\partial k}\right|_{k_F} = \frac{k_F}{m^{*}} \]

The latter equation defines the effective mass. The level density parameter is given by

\[ a=\frac{\pi^2 N(0)}{6 \rho} \]

Defining the Fermi temperature:

\[ T_F=\frac{1}{2} k_F v_F = k_F^2/2/m^{*} \]

another expression for the entropy is

\[ s=\frac{\pi^2}{2} \rho (T/T_F) \]

Expressions for the remaining quantities are

\[ P=P(T=0)+\frac{\rho}{3} a T^2 \left(1+\frac{d \ln v_F}{d \ln k_F }\right) \]

\[ E/A=E/A(T=0)+a T^2 \]

\[ \mu=\mu(T=0)-\frac{1}{3} a T^2\left(2-\frac{d \ln v_F}{d \ln k_F }\right) \]

Typically, the leading correction to

\[ s=\frac{\pi^2}{2} \rho (T/T_F) \]

is of order $ (T/T_F)^2 $ unless soft collective modes give rise to a $ (T/T_F)^3 \ln (T/T_F) $ correction.

At high temperature (non-degenerate approximation), a stationary phase approximation gives

\begin{eqnarray*} \rho(T) &\sim& \frac{\gamma}{2 \pi^2} e^{\mu/T} \cdot k^2 e^{-\epsilon_k/T} \sqrt{2 \pi} \left[\frac{2}{k^2}+ \frac{1}{T}\frac{\partial v_k}{\partial k}\right]^{-1/2} \\ &=& e^{\mu/T} \left.f(T)\right|_{k=k_{\rho}} \end{eqnarray*}

where $ \gamma $ is the spin and isospin degeneracy and the velocity function is $ v_k=\partial \epsilon_k/\partial k $ . The function $ f(T) $ is evaluated at momentum $ k_{\rho} $ which is obtained by solving $ T-k v_k/2=0 $ . The chemical potential is obtained by inverting the above relation for $ \rho(T) $ :

\[ \mu \sim T \ln \rho - T \ln f(T) \]

From this value of $ \mu $ we can derive the entropy density using

\[ T s \sim \sum_k n_k \epsilon_k + \rho T - \mu \rho \]

Using the stationary phase method:

\begin{eqnarray*} \sum_k n_k \epsilon_k &\sim& \frac{\gamma}{2 \pi^2} e^{\mu/T} \cdot k^2 \epsilon_k e^{-\epsilon_k/T} \sqrt{2 \pi} \left[\frac{2}{k^2}-\left(\frac{1}{\epsilon_k}-\frac{1}{T}\right) \frac{\partial v_k}{\partial k}+ \left(\frac{v_k}{\epsilon_k}\right)^2\right]^{-1/2} \\ &=& e^{\mu/T} \left.g(T)\right|_{k=k_E} \end{eqnarray*}

where $ k_E $ is the solution of

\[ \frac{2}{k}+v_k \left(\frac{1}{\epsilon_k}-\frac{1}{T}\right)=0 \]

This provides a first approximation to the energy and together with the thermodynamic identity gives the pressure.

Other Todos

Todo:
Right now, the equation of state classes depend on the user to input the correct value of non_interacting for the particle inputs. This is not very graceful...

Document the "n15" models. What where they for?


Bibliography

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