Equations of state available in Phantom

The following is a list of equations of state currently implemented in phantom. For full details have a look in the source code

Equations of state implemented in phantom

ieos	Description
1	Isothermal eos \(P = c_s^2 \rho\) where \(c_s^2 \equiv K\) is a constant stored in the dump file header
2	Adiabatic equation of state (code default) \(P = (\gamma - 1) \rho u\) if the code is compiled with ISOTHERMAL=yes, ieos=2 gives a polytropic eos: \(P = K \rho^\gamma\) where K is a global constant specified in the dump header
3	Locally isothermal disc as in Lodato & Pringle (2007) where \(P = c_s^2 (r) \rho\) sound speed (temperature) is prescribed as a function of radius using: \(c_s = c_{s,0} r^{-q}\) where \(r = \sqrt{x^2 + y^2 + z^2}\)
4	Isothermal equation of state for GR, enforcing cs = constant Warning this is experimental: use with caution
6	Locally isothermal disc centred on sink particle As in ieos=3 but in this version radius is taken with respect to a designated sink particle (by default the first sink particle in the simulation)
7	Vertically stratified equation of state sound speed is prescribed as a function of (cylindrical) radius R and height z above the x-y plane Warning should not be used for misaligned discs
8	Barotropic equation of state \(P = K \rho^\gamma\) where the value of gamma (and K) are prescribed functions of density
9	Piecewise Polytropic equation of state \(P = K \rho^\gamma\) where the value of gamma (and K) are a prescribed function of density. Similar to ieos=8 but with different defaults and slightly different functional form
10	MESA equation of state a tabulated equation of state including gas, radiation pressure and ionisation/dissociation. MESA is a stellar evolution code, so this equation of state is designed for matter inside stars
11	Equation of state with pressure and temperature equal to zero \(P = 0\) useful for simulating test particle dynamics using SPH particles
12	Ideal gas plus radiation pressure \(P = (\gamma - 1) \rho u\) but solved by first solving the quartic equation: \(u = \frac32 \frac{k_b T}{\mu m_H} + \frac{a T^4}{\rho}\) for temperature (given u), then solving for pressure using \(P = \frac{k_b T}{\mu m_H} + \frac13 a T^4\) hence in this equation of state gamma (and temperature) are an output
13	Locally isothermal eos for generic hierarchical system Assuming all sink particles are stars. Generalisation of Farris et al. (2014; for binaries) to N stars. For two sink particles this is identical to ieos=14
14	Locally isothermal eos from Farris et al. (2014) for binary system uses the locations of the first two sink particles
15	Helmholtz equation of state for fully ionized gas Warning not widely tested in phantom, better to use ieos=10
16	Shen (2012) equation of state for neutron stars this equation of state requires evolving temperature as the energy variable Warning not tested: use with caution
20	Gas + radiation + various forms of recombination from HORMONE, Hirai+2020, as used in Lau+2022b
21	HII region two temperature “equation of state” flips the temperature depending on whether a particle is ionised or not, use with ISOTHERMAL=yes
22	Same as ieos=21 but sets the thermal energy for use when u is stored (ISOTHERMAL=no)
23	Tillotson (1962) equation of state for solids (basalt, granite, ice, etc.) Implementation from Benz et al. (1986), Asphaug & Melosh (1993) and Kegerreis et al. (2019) In the compressed (\(\rho > \rho_0\)) or cold (\(u < u_{\rm iv}\)) state gives \(P_c = \left[a + \frac{b}{(u/(u_0 \eta^2) + 1}\right]\rho u + A \mu + B\mu^2\) where \(\eta = rho/rho_0\), \(\mu = \eta - 1\), u is the specific internal energy and a,b,A,B and \(u_0\) are input parameters chosen for a particular material In the hot, expanded state (\(\rho < \rho_0\) and \(u > u_{\rm iv}\)) gives \(P_e = a\rho u + \left[\frac{b\rho u}{u/(u_0 \eta^2) + 1} + A\mu \exp{-\beta \nu} \right] \exp(-\alpha \nu^2)\) where \(\nu = \rho/\rho_0 - 1\). In the intermediate state pressure is interpolated using \(P = \frac{(u - u_{\rm iv}) P_e + (u_{\rm cv} - u) P_c}{u_{\rm cv} - u_{\rm iv}}.\) When using this equation of state bodies should be set up with uniform density equal, or close to the reference density \(\rho_0\), e.g. 2.7 g/cm^3 for basalt
24	Tabulated EoS of Stamatellos et al. 2007 (includes opacities) Tabulated equation of state with opacities from Lombardi et al. 2015. For use with icooling = 9, the radiative cooling approximation (Young et al. 2024).