Cluster Models#

The model.ClusterModel class is one of the core structures available in cluster_generator. These objects hold all of the data related to a single cluster being modeled. The user can use a model.ClusterModel object to generate particle distributions, enforce hydrostatic equilibrium, virialize halos, and generate initial conditions. Furthermore, the user can generate model.ClusterModel objects using a variety of pre-built protocols using radial_profiles.RadialProfile objects to provide necessary fields of data.

Cluster Models: Mathematical Overview#

Reader Suggestion

To learn more about the various gravity theories discussed here, see the gravity page.

The Intracluster Medium (ICM)#

Assuming the intracluster medium of galaxy clusters can be modeled as an ideal fluid, the momentum density \(\rho{\bf v}\) of the gas obeys the Euler momentum equation (here written in conservative form and ignoring magnetic fields, viscosity, etc.):

\[\frac{\partial({\rho_g{\bf v}})}{\partial{t}} + \nabla \cdot (\rho_g{\bf v}{\bf v}) = -\nabla{P} + \rho_g{\bf g}\]

where \(\rho_g\) is the gas density, \({\bf v}\) is the gas velocity, \(P\) is the gas pressure, and \({\bf g}\) is the gravitational acceleration. The assumption of hydrostatic equilibrium implies that \({\bf v} = 0\) and that all time derivatives are zero, giving:

\[\nabla{P} = \rho_g{\bf g}\]

We therefore define the “hydrostatic vector”, \(\Gamma\) of the system to be

\[\Gamma = -\frac{\nabla P}{\rho_g} = \nabla \Phi.\]

In the context of galaxy clusters, we may typically assume an ideal gas behavior for the ICM, and therefore,

\[\Gamma = -\frac{\nabla P}{\rho_g} = \frac{-k_b T}{m_p \eta} \left[\frac{d\ln(\rho_g)}{dr} + \frac{d\ln(T)}{dr} \right].\]

As such, we recognize that from \(\rho_g\) and \(T\), we can generally obtain the acceleration and therefore the dynamical mass of the system. Because the dynamical mass need not be equivalent to the gasseous mass, it follows that finding the dynamical mass will also yield the halo mass of the system.

It is worth noting throughout the remainder of this document that for \(r\to \infty\), if \(\rho \sim r^\alpha\) and \(T\sim r^\beta\), then

\[\Gamma \sim r^{\beta-1}.\]

This point becomes important as we begin discussing the interaction between the gas dynamics above and the potential of the system.

Gravity#

In the preceding section, we described the basic mathematics of hydrostatic equilibrium; however, to find the dynamical mass and other related quantities, an expression for \(\Gamma\) must be obtained independently. In each gravitational theory implemented in CG, the dynamical field may be determined in terms of \(\Gamma\) and can then be used to construct the correct mass profiles. In the window below, we’ve included 3 archetypal examples:

In Newtonian gravity, the dynamical mass equation derived from the Poisson equation is

\[M_{\mathrm{dyn}}(<r) = \frac{r^2}{G}\Gamma = \frac{-r^2 k_b T}{G m_p \eta} \left[\frac{d\ln(\rho_g)}{dr} + \frac{d\ln(T)}{dr} \right].\]

In the AQUAL flavor of MOND, the dynamical mass is different from the Newtonian case by a factor of \(\mu(a)\).

\[M_{\mathrm{dyn}} = \frac{r^2}{G}\mu\left(\frac{| \Gamma |}{a_0}\right)\Gamma.\]

In the QUMOND flavor of MOND, the equations cannot be solved independently and therefore numerical methods are required to solve the implicit equation

\[\nu\left(\frac{|\nabla \Psi |}{a_0}\right)\nabla \Psi = \Gamma,\;\;\nabla \Psi = \frac{GM_{\mathrm{dyn}}}{r^2}\]

Each of these theories has a different asymptotic behavior in terms of the temperature. Therefore, the required temperature behavior at large radii is constrained by the following table. This should be considered when choosing a model.

Gravity Type

\(\Gamma\) and \(\Phi\) relationship

Mass Stability Condition

Newtonian

\(\Gamma = \nabla \Phi = GM_{\mathrm{dyn}}(<r)/r^2\)

\(T\sim r^{-1}\)

AQUAL

\(\mu\left(\frac{|\Gamma|}{a_0}\right)\Gamma = GM_{\mathrm{dyn}}(<r)/r^2\)

\(T\sim r^{0}\)

QUMOND

\(\Gamma = \nu\left(\frac{|\nabla \Psi|}{a_0}\right) \nabla \Psi\) where \(\nabla \Psi = GM_{\mathrm{dyn}}(<r)/r^2\)

\(T\sim r^{0}\)

Generating a model.ClusterModel Using Radial Profiles#

Based on the mathematics above, there are a variety of ways to produce model.ClusterModel objects. Most of the common approaches that see use in practice are built into the cluster_generator package; however, the from_arrays() class method can be used to generate a model.ClusterModel object manually. The available generation approaches are listed as follows:

Method

Function

Description

From \(\rho_g\) and \(\rho_{\mathrm{dyn}}\)

from_dens_and_tden()

Generates the galaxy cluster from the gas and dynamical density profiles. Computes temperature / grav. field.

From \(\rho_g\) and \(T_g\)

from_dens_and_temp()

Generates the galaxy cluster from the gas density and temperature profiles. Computes total mass, dm, stellar etc.

From \(\rho_g\) and \(\rho_{\mathrm{dyn}}\)

from_dens_and_entr()

Generates the galaxy cluster from the gas density and entropy profiles. Computes total mass, dm, stellar etc.

ClusterModel from Gas Density and Gas Temperature Profiles#

The user may generate a model.ClusterModel from the model.ClusterModel.from_dens_and_temp() method, which requires radial_profiles.RadialProfile objects for the temperature and the gas density. Using the condition of hydrostatic equilibrium, the potential, pressure, dynamical mass, and other necessary fields are automatically computed based on the chosen gravity theory.

The pressure \(P(r)\) is provided via the ideal gas law:

\[P(r) = \frac{\rho_g(r) k_b T(r)}{m_p \eta},\]

where \(\eta\) is the mean-molecular mass (generally 0.6 for galaxy clusters).

Once the pressure is determined, Euler’s Equations can be used for an incompressible fluid, yielding

\[\Gamma = \frac{-\nabla P(r)}{\rho_g} = \nabla \Phi\]

Once the potential has been determined, the corresponding gravity theory is applied to determine the dynamical mass of the system and by extension determine the necessary halo component.

\[M_{dm} = M_{\mathrm{dyn}} - M_{\mathrm{bary}}.\]

Provided

Computed

\(T_g\) \(\rho_{g}\) \(\rho_{\star}\)

\(\rho_{dm}, M_{dm}, M_{dyn}\) \(S, \Phi, \nabla \Phi, P\)

ClusterModel from Gas Density and Total Density Profiles#

In the case where the dynamical density is already known, the halo mass function is trivially obtained from

\[\rho_{\mathrm{halo}} = \rho_{\mathrm{dyn}} - \rho_{\mathrm{bary}},\]

where the baryonic component may contain a stellar distribution if the user supplies one. The difficult in this case is to efficiently determine the temperature profile which yields HSE. Recalling that

\[\Gamma = -\frac{\nabla P}{\rho_g} = \frac{-k_b T}{m_p \eta} \left[\frac{d\ln(\rho_g)}{dr} + \frac{d\ln(T)}{dr} \right].\]

The differential equation may be inverted, yielding

\[T(r) = \frac{-m_p \eta}{k_b \rho_g} \int_{r_0}^r \rho_g(r') \nabla \Phi(r') dr' + \frac{\rho_g(r_0)}{\rho_g(r)}T(r_0).\]

The most efficient approach here is to take \(r_0 = \infty\), in which case,

\[T(r) = \frac{m_p \eta}{k_b \rho_g} \int_{r}^\infty \rho_g(r') \nabla \Phi(r') dr'.\]

From this, the temperature profile is obtained.

ClusterModel from Gas Density and Gas Entropy Profiles#

Users may also generate a model.ClusterModel object from an entropy profile instead of the corresponding temperature profile. There is a 1-to-1 correspondence between entropy and temperature given that the gas density is fixed, therefore, the procedure follows exactly from the previous section; however, the entropy is converted first to a temperature profile using the widely accepted formula

\[S(r) = k_b T_g(r)n_e(r)^{-2/3}.\]

ClusterModel Without Gas#

Not all of the model.ClusterModel objects are generated with a gas density profile. If one wants to produce a gas-less model, the model.ClusterModel.no_gas() method is available. Users must provide a dynamical density profile, and are optionally allowed to provide a stellar component. See the API reference for syntactical information.

Checking the Hydrostatic Equilibrium#

In the vast majority of cases, the aim of the model.ClusterModel object is to produce a model which is in hydrostatic equilibrium. To check how successful the algorithm was in generating a hydrostatic galaxy cluster, one can use the model.ClusterModel.check_hse() method, which will provide the maximal relative deviation of the system from hydrostatic equilibrium.

This is done by recalling that the hydrostatic equilibrium condition requires that

\[-\frac{\nabla P}{\rho_g} = \nabla \Phi.\]

This, we define the hydrostatic variable \(\xi\) such that

\[\xi = \frac{\nabla \Phi \rho_g + \nabla \Phi}{\nabla \Phi \rho_g}.\]

clearly, \(\xi \approx 0\) indicates a successfully equilibrated cluster.

Setting a Magnetic Field Strength Profile#

Magnetic fields play a major role in the dynamics of the ionized plasma modeled in hydro simulations of galaxy clusters. In cluster_generator, there are currently two approaches for generating a magnetic field for a galaxy cluster.

Setting a Magnetic Field from \(\beta\)#

Users can specify the magnetic field profile using the set_magnetic_field_from_beta(), which allows the user to specify the magnetic pressure factor \(\beta\), which is defined such that

\[\beta = \frac{p_{\mathrm{thermal}}}{p_{\mathrm{magnetic}}}.\]

Setting a Magnetic Field from plasma density#

Another common approach for initializing magnetic fields is to let the magnetic field be proportional to a power of the gas density. Most commonly, this value is \(\eta = \frac{2}{3}\); however, the user may specify whichever exponent they choose. To initialize a cluster with a magnetic field using this approach, use the set_magnetic_field_from_density().

Adding Other Fields#

Users may add additional fields to the ClusterModel instance using the set_field() method, which takes a name and value and loads the corresponding field into the model.

Attention

User’s should be aware that value’s being attributed as fields must be unyt_array’s, not generic list types or np.ndarray. Furthermore, the length of the value being specified must match ClusterModel.num_elements.

Reading and Writing model.ClusterModel Objects to and from Disk#

User Advice

Generating model.ClusterModel instances can be slow. As many cases as possible, it is worthwhile to write models to disk in one of the formats discussed below so that they can be read more easily.

Many different protocols for writing and reading model.ClusterModel instances to / from disk have been implemented. The following table summarizes the options available; however, we advice consulting the API reference for each of the methods to get more information about specifics regarding each of the available formats.

Format

Write Method

Read Method

Notes

ASCII

write_model_to_ascii()

Not Yet Implemented

None

HDF5

write_model_to_h5()

from_h5_file()

None

BINARY

write_model_to_binary()

Not Yet Implemented

None

Non-Physical Profiles#

Sometimes, for a variety of reasons, generated model.ClusterModel instances might have profiles which are non-physical. Sometimes, this is caused by a poor choice of radial profile for the given gravity theory, or is caused because some other physical constraint is not respected. CG has a model (correction) entirely focused on fixing these problems. For a comprehensive description of these issues and the algorithms used to solve them, see Correcting Non-Physical Behavior .