Input Parameters
Each ExpCGM model for a galactic atmosphere depends on a user-specified gravitational potential function $\varphi(\mathbf{r})$ and a user-specified shape function $\alpha(\mathbf{r})$ for the atmosphere’s pressure profile. Both functions can depend on a three-dimensional position vector $\mathbf{r}$. However, spherically symmetric models depending only on $r = | \mathbf{r} |$ are often sufficient.
This introductory page focuses on spherically symmetric models. The ExpCGM framework will eventually include triaxial models among the Extensions.
This page is still preliminary. If you have suggestions, please send them to galacticatmospheres@gmail.com.
Contents
Potential and Shape Functions
Most of the input parameters for ExpCGM models describe the two required parametric functions:
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Gravitational Potential, $\varphi( ~r~ | ~v_\varphi , r_{\rm s} , … ~)$: The gravitational potential model requires only one parameter, the maximum circular velocity $v_\varphi$ of the confining halo’s gravitational potential. It may also include a scale radius $r_{\rm s}$ determining how the potential’s circular velocity $v_{\rm c}$ depends on $r$. Descriptions of more detailed potential wells may require additional parameters.
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Shape Function, $\alpha( ~r~ | ~\alpha_{\rm in} , \alpha_{\rm out} , r_\alpha , … ~)$: The atmosphere’s shape function can be just a constant value of $\alpha$. It can also shift from one limiting value $\alpha_{\rm in}$ at small radii to another one $\alpha_{\rm out}$ at large radii in the vicinity of a crossover radius $r_\alpha$. The Pressure Profiles page discusses how $\alpha (r)$ is related to the processes that shape a galaxy’s atmosphere and provides parametric expressions for several physically motivated options.
Once $\alpha(r)$ has been specified, the pressure profile of an ExpCGM atmosphere model depends on just the normalization factor $P_0$ at a user-chosen fiducial radius $r_0$. The atmosphere’s pressure profile becomes \(P(r) = P_0 \cdot \exp \left[ - \int_1^{r/r_0} \frac {\alpha(x)} {x} dx \right]\) Dividing $v_{\rm c}^2 (r)$ by $\alpha(r)$ gives the temperature profile for a thermally supported atmosphere: \(kT(r) = \frac {\mu m_p v_{\rm c}^2(r)} {\alpha(r)}\) The atmosphere’s density profile then depends on just $\varphi(r)$, $\alpha(r)$, and $P_0$: \(\rho(r) = \frac {\alpha(r) P(r)} {v_{\rm c}^2(r)}\) However, $T(r)$ and $\rho(r)$ may also depend on two additional parametric functions representing the atmosphere’s thermalization fraction $f_{\rm th}$ and force modification factor $f_\varphi$. (See the Essentials page for the definitions of $f_{\rm th}$ and $f_\varphi$.)
Isothermal Atmosphere
Users seeking to minimize degrees of freedom can opt for an isothermal potential well with constant circular velocity ($v_{\rm c} = v_\varphi$) that confines an atmosphere with constant $\alpha$. Setting $f_{\rm th}$ and $f_\varphi$ to unity then gives an atmospheric temperature $kT = \mu m_p v_\varphi^2 / \alpha$ that is independent of radius, and the pressure and density are power laws: \(P(r) = P_0 \left( \frac {r} {r_0} \right)^{-\alpha} ~~~,~~~ \rho(r) = \frac {\alpha P_0} {v_\varphi^2} \left( \frac {r} {r_0} \right)^{-\alpha}\) This isothermal power-law atmosphere model has three degrees of freedom $(P_0,v_\varphi,\alpha)$, because $r_0$ is degenerate with $P_0$.
Double Power-Law Atmospheres
Cosmological structure formation produces gaseous atmospheres in which $\alpha(r)$ increases with radius. An ExpCGM model can describe that increase with the four-parameter fitting formula \(\alpha(r) = \alpha_{\rm in} + ( \alpha_{\rm out} - \alpha_{\rm in} ) \left[ \frac {(r / r_\alpha)^{\alpha_{\rm tr}}} {1 + (r / r_\alpha)^{\alpha_{\rm tr}}} \right]\) If all four parameters are left free, the resulting model for an atmosphere in an isothermal potential well has six degrees of freedom instead of three. Its pressure profile is \(P(r) \propto \left( \frac {r} {r_\alpha} \right)^{-\alpha_{\rm in}} \left[ 1 + \left( \frac {r} {r_\alpha} \right)^{\alpha_{\rm tr}} \right]^{-(\alpha_{\rm out} - \alpha_{\rm in}) / {\alpha_{\rm tr}}}\) and its gas temperature declines from $kT \approx \mu m_p v_\varphi^2 / \alpha_{\rm in}$ at small radii toward $kT \approx \mu m_p v_\varphi^2 / \alpha_{\rm out}$ at large radii as $\alpha (r)$ rises.
Restricting the double power-law atmosphere model by choosing $\alpha_{\rm in} = 0$ and $\alpha_{\rm tr} = 2$ results in \(P(r) \propto \left[ 1 + \left( \frac {r} {r_\alpha} \right)^2 \right]^{-\alpha_{\rm out} / 2}\) This relation is nearly equivalent to the classic “beta model” for galaxy-cluster atmospheres, but with thermal pressure replacing gas density. However, the model has a drawback: It results in a gas-temperature profile that diverges at small $r$ in an isothermal potential well with constant $v_\varphi$. This undesirable feature is less problematic in gravitational potentials with small values of $v_{\rm c}$ at small radii.
NFW-like Models
Cosmological structure formation also produces halo potential wells that are not quite isothermal. Typically, the circular velocity profile of a cosmological halo rises with radius at small $r$ and declines with radius at large $r$. The most common fitting formula accounting for that feature is the Navarro-Frenk-White (NFW) model, which has a circular velocity profile \(v_{\rm NFW}^2(x) = A_{\rm NFW} v_\varphi^2 \left[ \frac {\ln (1 + x)} {x} - \frac {1} {1 + x} \right]\) with $x = r / r_{\rm s}$ and $A_{\rm NFW} = 4.625$.
The Essentials page presents a simple example with four degrees of freedom $(P_0,v_\varphi,r_{\rm s}, \alpha)$ describing a power-law atmosphere in an NFW potential well. Expanding that example using all four parameters of a double power-law atmosphere yields a model with seven degrees of freedom $(P_0,v_\varphi,r_{\rm s},\alpha_{\rm in},\alpha_{\rm out},\alpha_{\rm tr},r_\alpha)$. Users can reduce those degrees of freedom to five by keeping $\alpha_{\rm tr}$ fixed and by choosing to make $r_\alpha$ a constant multiple of $r_{\rm s}$.
The double power-law atmosphere model is equivalent to what the astronomical literature sometimes calls a “generalized NFW model.”
NFW Halo + Central Galaxy
An NFW halo model coupled with a shape function model that has $\alpha_{\rm in} > 0$ results in an atmospheric temperature that formally approaches zero at small radii. Adding a central galaxy to the gravitational potential model helps to mitigate that potentially problematic issue.
One option is to use a Hernquist model to represent the galaxy’s contribution to the potential’s circular velocity profile: \(v_{\rm H}^2(r) = \frac {G M_* r} { ( r + r_{\rm H})^2 }\) In this expression, $M_*$ represents the galaxy’s total stellar mass and the Hernquist radius $r_{\rm H}$ is a scale radius determining how the galaxy’s mass profile converges toward $M_{\rm H}$.
The model parameters $M_*$ and $r_{\rm H}$ can be free, or they can be fixed at values consistent with the observed stellar mass and effective radius of the halo’s central galaxy. If both $M_*$ and $r_{\rm H}$ are allowed to be free, then adding a central galaxy to the gravitational potential model adds two more degrees of freedom to the input parameter space.
Most central galaxies have a maximum circular velocity $(G M_* / 4 r_{\rm H})^{1/2}$ similar to the maximum circular velocity $v_\varphi$ of the surrounding halo. It is therefore reasonable to reduce the dimensionality of the parameter space by applying the restriction $r_{\rm H} = G M_* / 4 v_\varphi^2$, so that $\max (v_{\rm H}) = v_\varphi$. However, applying that restriction is unwise for galaxy-cluster models, because the maximum circular velocity of a central cluster galaxy is significantly smaller than the maximum circular velocity of its halo.
NFW Halo + Central Galaxy + BH
A supermassive black hole may dominate the central galaxy’s gravitational potential at the smallest radii. Its contribution can be included in an ExpCGM model using the Newtonian formula \(v_{\rm BH}^2 (r) = \frac {G M_{\rm BH}} {r}\) where $M_{\rm BH}$ is the central black hole’s mass. Adding a black hole to the potential model so that \(v_{\rm c}^2 (r) = v_{\rm NFW}^2 (r) + v_{\rm H}^2 (r) + v_{\rm BH}^2 (r)\) then ensures that the model atmosphere’s temperature does not formally go to zero at the center.
Thermalization Fraction
An atmosphere’s thermalization fraction $f_{\rm th}$ can also be a free parameter of an ExpCGM model. Observational constraints can be placed on $f_{\rm th}$ if the overall data set contains information complementary to the atmospheric temperature profile.
An ExpCGM user interested in thermalization may choose to express $f_{\rm th}$ as a parametric model that depends on radius. Combining that model with $\alpha (r)$ gives \(\alpha_{\rm eff} (r) = \alpha (r) + \frac {d \ln f_{\rm th}} {d \ln r}\) The predicted temperature profile then becomes \(kT(r) = \frac {\mu m_p v_{\rm c}^2 (r)} {\alpha_{\rm eff} (r)} f_{\rm th} (r)\) Furthermore, assuming that isotropic turbulence provides the rest of the support needed for force balance leads to the prediction \(\sigma_{\rm 1D}^2 (r) = \frac {2} {3} \frac {v_{\rm c}^2 (r)} {\alpha_{\rm eff} (r)} \left[ 1 - f_{\rm th} (r) \right]\) in which $\sigma_{\rm 1D}$ is the one-dimensional velocity dispersion of turbulent support.
Fitting such an ExpCGM atmosphere model to a data set containing information about both $T(r)$ and $v_{\rm c} (r)$ then constrains $f_{\rm th}(r)$ and $\sigma_{\rm 1D}$. Likewise, having information about both $T(r)$ and $\sigma_{\rm 1D}$ constrains $v_{\rm c} (r)$ and $f_{\rm th}(r)$.
Auxilliary Input Parameters
The output of an ExpCGM model may contain predictions for observable features that depend on a halo’s redshift $z$, the collective abundance $Z$ of atmospheric elements other than hydrogren and helium (relative to the solar proportion), and the frequency band $[\nu_{\rm in},\nu{\rm out}]$ being observed. Those auxiliary parameters $(z,Z,\nu_{\rm in},\nu_{\rm out})$ can be specified as part of the input parameter set.
Summary of Input Parameters
Essential Parameters
| Parameter | Description |
|---|---|
| $r_0$ | Fiducial radius (chosen by user and remains fixed) |
| $P_0$ | Thermal pressure normalization at $r_0$ |
| $v_\varphi$ | Halo circular velocity (constant for isothermal potential) |
| $\alpha$ | Shape function (constant for power-law atmosphere model) |
Shape Function Parameters
| Parameter | Description |
|---|---|
| $\alpha_{\rm in}$ | Asymptotic value of $\alpha$ at small radii |
| $\alpha_{\rm out}$ | Asymptotic value of $\alpha$ at large radii |
| $r_\alpha$ | Transitional radius from $\alpha_{\rm in}$ to $\alpha_{\rm out}$ |
| $\alpha_{\rm tr}$ | Parameter determining sharpness of transition |
Potential Well Parameters
| Parameter | Description |
|---|---|
| $v_\varphi$ | Maximum circular velocity of halo potential |
| $r_{\rm s}$ | Scale radius of NFW halo potential well |
| $M_*$ | Stellar mass of central galaxy |
| $r_{\rm H}$ | Hernquist scale radius of stellar potential well |
| $M_{\rm BH}$ | Mass of central supermassive black hole |
Force Balance Parameters
| Parameter | Description |
|---|---|
| $f_{\rm th}$ | Thermalization fraction (may be constant or a parametric function of $r$) |
| $f_\varphi$ | Force modification factor (may be constant or a parametric function of $r$) |
Auxilliary Parameters
| Parameter | Description |
|---|---|
| $z$ | Halo redshift |
| $Z$ | Heavy-element abundance in units of $Z_\odot$ (default: 1) |
| $\nu_{\rm min}$ | Minimum frequency of spectral band (default: 0) |
| $\nu_{\rm max}$ | Maximum frequency of spectral band (default: $\infty$) |