Pisces Galaxy Cluster Models

Pisces provides a suite of models for simulating and analyzing galaxy clusters. These models are idealized and generally rely on spherical symmetry and hydrostatic equilibrium assumptions; however, some more exotic models are in development. Below is a table of the available galaxy cluster models in Pisces:

Model	Geometry	EOS	Assumptions
SphericalGalaxyClusterModel	Spherical	IG	HSE
MagnetizedSphericalGalaxyClusterModel	Spherical	IG + MHD	HSE (w/ NTPS)

Note

Abbreviations used in the table:

• EOS = Equation of State

• IG = Ideal Gas

• HSE = Hydrostatic Equilibrium

• MHD = Magnetohydrodynamics

• NTPS = Non-Thermal Pressure Support

Galaxy Clusters Overview

Galaxy clusters are the largest gravitationally bound structures in the universe, containing hundreds to thousands of galaxies, vast amounts of hot gas, and an even larger reservoir of dark matter. For a classic review of cluster astrophysics, see [1].

A typical cluster’s mass budget is dominated by dark matter, which accounts for roughly 80–85% of the total mass [2][3]. The second-largest component is the intracluster medium (ICM), a hot, diffuse plasma that comprises about 10–15% of the cluster mass and emits strongly in the X-ray band via thermal bremsstrahlung and line emission [1][4][5]. The galaxies themselves make up only a few percent of the mass, but they are important tracers of the cluster potential and history of structure formation [6].

The ICM typically has temperatures of (10^7 - 10^8) K, making it observable primarily in X-rays. These observations reveal detailed information about the density, temperature, and dynamical state of the cluster [1][4].

Galaxy clusters are crucial for both cosmology and astrophysics: their abundance and growth rate provide sensitive tests of cosmological parameters such as the matter density and dark energy equation of state, while their internal structure offers insight into the physics of galaxy formation, feedback, and plasma processes [3][5].

Modeling

Galaxy cluster models in Pisces are based on the assumption of hydrostatic equilibrium (HSE), in which the pressure gradient of the intracluster medium balances the gravitational force of the cluster potential. Different model variants can include purely thermal support or incorporate additional non-thermal sources such as magnetic fields.

Fully Thermal Hydrostatic Models

In the simplest case, the intracluster medium (ICM) is modeled as an ideal gas in hydrostatic equilibrium within a spherically symmetric gravitational potential. The condition of hydrostatic equilibrium reads

\[\frac{dP}{dr} = -\rho_g(r) \frac{G M(<r)}{r^2},\]

where \(P(r)\) is the gas pressure, \(\rho_g(r)\) is the gas density, \(M(<r)\) is the total mass enclosed within radius \(r\), and \(G\) is Newton’s constant. It forms the basis of the SphericalGalaxyClusterModel.

Details:

Assuming an ideal gas equation of state (EOS),

\[P(r) = \frac{k_B}{\mu m_p} \rho_g(r) T(r),\]

with \(k_B\) Boltzmann’s constant, \(m_p\) the proton mass, and \(\mu\) the mean molecular weight, the density may be expressed as

\[\rho_g(r) = \frac{\mu m_p}{k_B T(r)} P(r).\]

Substituting this into the hydrostatic equilibrium equation gives

\[\frac{dP}{dr} = - \frac{\mu m_p}{k_B T(r)} P(r) \frac{G M(<r)}{r^2}.\]

Rearranging for the enclosed mass yields

\[M(<r) = - \frac{k_B T(r) r^2}{\mu m_p G P(r)} \frac{dP}{dr}.\]

It is often more convenient to express this in terms of logarithmic derivatives. Writing

\[\frac{dP}{dr} = \frac{d \ln P}{d \ln r} \, \frac{P(r)}{r},\]

we obtain the standard hydrostatic mass equation:

\[M(<r) = - \frac{k_B T(r) r}{\mu m_p G} \left( \frac{d \ln P}{d \ln r} \right).\]

Using the ideal gas EOS, one can also separate this into terms involving the gas density and temperature:

\[\frac{d \ln P}{d \ln r} = \frac{d \ln \rho_g}{d \ln r} + \frac{d \ln T}{d \ln r}.\]

Thus,

\[M(<r) = - \frac{k_B T(r) r}{\mu m_p G} \left( \frac{d \ln \rho_g}{d \ln r} + \frac{d \ln T}{d \ln r} \right).\]

This relation connects the cluster’s total mass profile directly to the observable radial gradients of the gas density and temperature profiles.

Non-Thermal Pressure Support

Observations and simulations suggest that in addition to thermal pressure, other sources of support such as turbulence, cosmic rays, and magnetic fields can contribute significantly to the equilibrium of clusters [7][8][9].

Magnetic Fields

In the magnetized hydrostatic case, the total pressure can be written as

\[P_{\text{tot}}(r) = P_{\text{th}}(r) + P_B(r),\]

where \(P_{\text{th}}\) is the thermal gas pressure and \(P_B = B^2/(8\pi)\) is the magnetic pressure associated with the magnetic field strength \(B(r)\).

The modified hydrostatic equilibrium equation becomes

\[\frac{d}{dr} \left[ P_{\text{th}}(r) + P_B(r) \right] = -\rho_g(r) \frac{G M(<r)}{r^2}.\]

This framework is implemented in the MagnetizedSphericalGalaxyClusterModel, which extends the purely thermal model to include magnetohydrostatic equilibrium (MHSE). Such models are important for studying the role of magnetic fields in regulating gas dynamics, stability, and heat transport in the ICM.

References

[1] (1,2,3)

Craig L. Sarazin. X-ray emission from clusters of galaxies. Cambridge University Press, 1988.

[2]

A. Vikhlinin, A. Kravtsov, W. Forman, C. Jones, M. Markevitch, S. S. Murray, and L. Van Speybroeck. Chandra Sample of Nearby Relaxed Galaxy Clusters: Mass, Gas Fraction, and Mass-Temperature Relation. \apj , 640(2):691–709, April 2006. arXiv:astro-ph/0507092, doi:10.1086/500288.

[3] (1,2)

Andrey V. Kravtsov and Stefano Borgani. Formation of Galaxy Clusters. \araa , 50:353–409, September 2012. arXiv:1205.5556, doi:10.1146/annurev-astro-081811-125502.

[4] (1,2)

G. Mark Voit. Tracing cosmic evolution with clusters of galaxies. Reviews of Modern Physics, 77(1):207–258, April 2005. arXiv:astro-ph/0410173, doi:10.1103/RevModPhys.77.207.

[5] (1,2)

G. W. Pratt, M. Arnaud, A. Biviano, D. Eckert, S. Ettori, D. Nagai, N. Okabe, and T. H. Reiprich. The Galaxy Cluster Mass Scale and Its Impact on Cosmological Constraints from the Cluster Population. \ssr , 215(2):25, February 2019. arXiv:1902.10837, doi:10.1007/s11214-019-0591-0.

[6]

A. Dressler. Galaxy morphology in rich clusters: implications for the formation and evolution of galaxies. \apj , 236:351–365, March 1980. doi:10.1086/157753.

[7]

Erwin T. Lau, Andrey V. Kravtsov, and Daisuke Nagai. Residual Gas Motions in the Intracluster Medium and Bias in Hydrostatic Measurements of Mass Profiles of Clusters. \apj , 705(2):1129–1138, November 2009. arXiv:0903.4895, doi:10.1088/0004-637X/705/2/1129.

[8]

V. Biffi, S. Borgani, G. Murante, E. Rasia, S. Planelles, G. L. Granato, C. Ragone-Figueroa, A. M. Beck, M. Gaspari, and K. Dolag. On the Nature of Hydrostatic Equilibrium in Galaxy Clusters. \apj , 827(2):112, August 2016. arXiv:1606.02293, doi:10.3847/0004-637X/827/2/112.

[9]

D. Eckert, V. Ghirardini, S. Ettori, E. Rasia, V. Biffi, E. Pointecouteau, M. Rossetti, S. Molendi, F. Vazza, F. Gastaldello, M. Gaspari, S. De Grandi, S. Ghizzardi, H. Bourdin, C. Tchernin, and M. Roncarelli. Non-thermal pressure support in X-COP galaxy clusters. \aap , 621:A40, January 2019. arXiv:1805.00034, doi:10.1051/0004-6361/201833324.