Public API

Basic model types and methods

density(model::DensityModel, r)

Local volume density at r.

density2d(model::DensityModel, R)

Local surface density at R. If not defined for a subtype of DensityModel, then calculate numerically as the Abel transform from the volume density profile.

mass(model::DensityModel, r)

Enclosed mass within r. If not defined for a subtype of DensityModel, then calculate numerically as the integral of the volume density.

update(model)

Create a new model from the specified parameters.

Collection of anisotropy models.

• IsotropicModel: isotropic model
• ConstantBetaModel: constant beta model
• RSBetaModel: flexible beta model from Read & Steger
• beta: orbital anisotropy profile

ConstantBetaModel(beta)

Model that assumes beta is constant with galactocentric radius.

IsotropicModel()

Parameter-less model representing an isotropic system (i.e. beta = 0).

RSBetaModel(beta0, betaInf, rbeta, nbeta)

Flexible velocity anisotropy model from Read & Steger 2017 (eq. 9)

• beta0: inner asymptotic anisotropy
• betaInf: outer asymptotic anisotropy
• rbeta: transition radius
• nbeta: transition sharpness, higher is a faster transition

beta0, betaInf, nbeta = 0, 1, 2 corresponds to the Osipkov-Merritt profile

beta0, betaInf, nbeta = 0, 0.5, 1 corresponds to the Mamon-Lokas profile

beta(model::AnisotropyModel, r)

Orbital anisotropy as a function of radius. The orbital anisotropy is defined as

Collection of tracer density models.

SersicModel : Representation of a Sersic surface density profile

SersicModel(Re, n, Mtot)

Representation of a Sersic surface density profile.

• Re: the effective radius in kpc.
• n: the Sersic index
• Mtot: the total mass associated to the profile.

For the purposes of using the SersicModel to track the tracer population's density distribution, Mtot is somewhat arbitrary. For using it as a component of the total mass profile, this should take units of solar masses.

Dark matter halo models and relations.

Significant credit to Benedikt Diemer for developing Colossus.

While much of this is inspired by Colossus; any implementation faults are my own.

The default halo mass definition is "200c", i.e. the virial radius of the halo is defined to be the radius that encloses a density that is 200 times the critical density of the universe.

The default cosmology (from Slomo.CosmologyTools) is currently set to the values from Planck 2018 (h = 0.6766, Neff = 3.046, OmegaM = 0.3111).

Common HaloModel methods

• scale_radius: scale radius of the halo
• virial_radius: virial radius of the halo
• virial_mass: virial mass of the halo
• concentration: halo concentration (defined as rvir / rs)

HaloModel subtypes

• NFWModel: Navarro, Frenk, and White (1997) profile
• CoreNFWModel: Core-NFW (Read et al. 2016) profile
• GNFWModel: generalized NFW model (i.e., αβγ where α = 1, β = 3)
• ABGModel: α-β-γ profile (double power law)
• EinastoModel: Einasto (1965) profile
• SolNFWModel: Soliton + NFW model (Marsh & Pop 2015)
• SolABGModel: Soliton + αβγ model (Wasserman et al. 2019)

Halo relations

• hmcr: halo mass-concentration relation from Dutton & Maccio (2014)
• shmr: stellar-halo mass relation from Rodriguez-Puebla et al. (2017)
• abg_from_logshm: αβγ scaling with log(Mstar / Mhalo) from DiCintio et al. (2014)

ABGModel(rs, rhos, alpha, beta, gamma)

αβγ double-power law density model.

Merritt et al. 2006

• rs: scale radius in kpc
• rhos: scale density in Msun / kpc3
• alpha: transition sharpness
• beta: negative outer log slope
• gamma: negative inner log slope

CoreNFWModel(rs, rhos, rc, nc)

coreNFW halo density model from Read et. al (2016)

Read, Agertz, and Collins 2016

• rs: scale radius in kpc
• rhos: scale density in Msun / kpc3
• rc: core radius in kpc
• nc: core index (0 -> no core, 1 -> full core)

EinastoModel(rs, rhos, alpha)

Einasto halo model

• Einasto 1965

• rs: scale radius in kpc

• rhos: scale density in Msun / kpc3

• alpha: Einasto shape index, lower is more cored

GNFWModel(rs, rhos, gamma)

Generalized NFW (gNFW) halo density model. This is a specific instance of a generalized Hernquist (i.e. αβγ) model with the transition parameter α = 1, the outer slope β = 3, and the inner slope, γ, left as a free parameter.

• Hernquist 1990

• Zhao 1996

• rs: scale radius in kpc

• rhos: scale density in Msun / kpc3

• gamma: negative of inner density log slope

NFWModel(rs, rhos)

NFW halo density model. Default constructor will make a halo with Mvir = 1e12 Msun.

• rs: scale radius in kpc
• rhos: scale density in Msun / kpc3

SolABGModel(rs, rhos, alpha, beta, gamma, rsol, rhosol, repsilon)

Soliton + αβγ model. For consistency, the αβγ density profile must match the soliton density profile at repsilon. If that is not the case, the constructor will update repsilon.

• rs: NFW scale radius in kpc
• rhos: NFW scale density in Msun / kpc3
• alpha: transition sharpness
• beta: negative outer log slope
• gamma: negative inner log slope
• rsol: soliton scale radius in kpc
• rhosol: soliton scale density in kpc
• repsilon: transition radius

SolNFWModel(rs, rhos, rsol, rhosol, repsilon)

Soliton NFW model. For consistency, the NFW density must match the soliton density at repsilon. If that is not the case, the constructor will update repsilon.

• Marsh & Pop 2015

• Robles, Bullock, and Boylan-Kolchin 2019

• rs: NFW scale radius in kpc

• rhos: NFW scale density in Msun / kpc3

• rsol: soliton scale radius in kpc

• rhosol: soliton scale density in kpc

• repsilon: transition radius

ABG_from_virial(Mvir, cvir, alpha, beta, gamma; <keyword arguments>)

Construct a ABGModel halo from the virial mass, concentration, and shape parameters.

Arguments

• mdef::AbstractString = default_mdef: halo mass definition (e.g., "200c", "vir")
• cosmo::AbstractCosmology = default_cosmo: cosmology under which to evaluate the overdensity
• z::Real = 0.0: redshift at which to evaluate the overdensity

ABG_from_virial(Mvir, cvir, Mstar; <keyword arguments>)

Construct a ABGModel halo from the virial mass, concentration, and stellar mass, using the scaling relations from DiCintio et al. 2014

Arguments

• mdef::AbstractString = default_mdef: halo mass definition (e.g., "200c", "vir")
• cosmo::AbstractCosmology = default_cosmo: cosmology under which to evaluate the overdensity
• z::Real = 0.0: redshift at which to evaluate the overdensity

CoreNFW_from_virial(Mvir, cvir, Re, t_sf; <keyword arguments>)

Construct a CoreNFWModel from the scaling relations in Read et al. (2016).

• Mvir: virial mass in Msun
• cvir: halo concentration
• Re: effective radius of stellar distribution, in kpc
• t_sf: time since start of star formation, in Gyr

Arguments

• mdef::AbstractString = default_mdef: halo mass definition (e.g., "200c", "vir")
• cosmo::AbstractCosmology = default_cosmo: cosmology under which to evaluate the overdensity
• z::Real = 0.0: redshift at which to evaluate the overdensity

Einasto_from_virial(Mvir, cvir, alpha; <keyword arguments>)

Construct an Einasto halo from the halo mass, concentration, and shape parameter.

Arguments

• mdef::AbstractString = default_mdef: halo mass definition (e.g., "200c", "vir")
• cosmo::AbstractCosmology = default_cosmo: cosmology under which to evaluate the overdensity
• z::Real = 0.0: redshift at which to evaluate the overdensity

GNFW_from_virial(Mvir, cvir, gamma; <keyword arguments>)

Construct a gNFW halo from the virial mass and concentration.

Arguments

• mdef::AbstractString = default_mdef: halo mass definition (e.g., "200c", "vir")
• cosmo::AbstractCosmology = default_cosmo: cosmology under which to evaluate the overdensity
• z::Real = 0.0: redshift at which to evaluate the overdensity

NFW_from_virial(Mvir, cvir; <keyword arguments>)

Construct an NFW halo from the virial mass and concentration.

Arguments

• mdef::AbstractString = default_mdef: halo mass definition (e.g., "200c", "vir")
• cosmo::AbstractCosmology = default_cosmo: cosmology under which to evaluate the overdensity
• z::Real = 0.0: redshift at which to evaluate the overdensity

SolABG_from_virial(Mvir, cvir, alpha, beta, gamma, m22; <keyword arguments>)

Construct a Soliton + αβγ halo from the halo mass, concentration, shape parameters, and axion mass.

Arguments

• rsol = nothing: for default, use the soliton size-halo mass scaling relationship
• mdef::AbstractString = default_mdef: halo mass definition (e.g., "200c", "vir")
• cosmo::AbstractCosmology = default_cosmo: cosmology under which to evaluate the overdensity
• z::Real = 0.0: redshift at which to evaluate the overdensity
• x_start::Real = 2.0: factor of radius scale to start the search for the root
• maxevals:Int = 100: maximum number of function evalutions for the root finding

SolNFW_from_virial(Mvir, cvir, m22; <keyword arguments>)

Construct a Solition-NFW halo from the halo mass, concentration, and axion mass.

Arguments

• rsol = nothing: for default, use the soliton size-halo mass scaling relationship
• mdef::AbstractString = default_mdef: halo mass definition (e.g., "200c", "vir")
• cosmo::AbstractCosmology = default_cosmo: cosmology under which to evaluate the overdensity
• z::Real = 0.0: redshift at which to evaluate the overdensity
• x_start::Real = 2.0: factor of radius scale to start the search for the root
• maxevals:Int = 100: maximum number of function evalutions for the root finding

abg_from_logshm(logshm)

αβγ scalings with log10(Mstar / Mhalo) from DiCintio+2014a.

concentration(halo::HaloModel; <keyword arguments>)

Compute the halo concentration.

Arguments

• mdef::AbstractString = default_mdef: halo mass definition (e.g., "200c", "vir")
• cosmo::AbstractCosmology = default_cosmo: cosmology under which to evaluate the overdensity
• z::Real = 0.0: redshift at which to evaluate the overdensity
• xstart::Real = 10.0: factor of the scale radius to start the search for the root
• rtol::Real = 1e-2: relative error tolerance for the root finding
• maxevals:Int = 100: maximum number of function evalutions for the root finding

hmcr(Mvir; <keyword arguments>)

Compute the expected halo concentration from halo mass using the relation of Dutton & Maccio 2014 (equations 7, 10-13).

Arguments

• mdef::AbstractString = default_mdef: halo mass definition (e.g., "200c", "vir")
• cosmo::AbstractCosmology = default_cosmo: cosmology under which to evaluate the overdensity
• z::Real = 0.0: redshift at which to evaluate the overdensity

hmcr_prior(Mvir, cvir; <keyword arguments>)

Compute the probability of drawing a halo with virial parameters (Mvir, cvir), using the halo mass-concentration relation of Dutton & Maccio 2014.

Arguments

• mdef::AbstractString = default_mdef: halo mass definition (e.g., "200c", "vir")
• cosmo::AbstractCosmology = default_cosmo: cosmology under which to evaluate the overdensity
• z::Real = 0.0: redshift at which to evaluate the overdensity
• sigma_logc::Real = 0.16: scatter in log_concentration at fixed Mvir (in dex)

scale_radius(halo::HaloModel)

Return the scale radius of the halo. The convention adopted here is that the scale radius occurs where the logarithmic slope of the halo density profile is equal to -2. For the standard NFW model this is equal to rs, but other halo models may compute some other value to stay consistent with this convention (e.g., the ABGModel).

shmr(Mvir; <keyword arguments>)

Compute the expected stellar mass from halo mass using the relation of Rodriguez-Puebla et al. 2017.

See equations 25-33 for the functional form and 49 - 55 for the parameters.

Arguments

• mdef::AbstractString = default_mdef: halo mass definition (e.g., "200c", "vir")
• cosmo::AbstractCosmology = default_cosmo: cosmology under which to evaluate the overdensity
• z::Real = 0.0: redshift at which to evaluate the overdensity

shmr_prior(Mvir, Mstar; <keyword arguments>)

Compute the probability of drawing a halo mass and stellar mass pair (Mvir, Mstar).

Arguments

• mdef::AbstractString = default_mdef: halo mass definition (e.g., "200c", "vir")
• cosmo::AbstractCosmology = default_cosmo: cosmology under which to evaluate the overdensity
• z::Real = 0.0: redshift at which to evaluate the overdensity
• sigma_logMstar::Real = 0.15 scatter in log_Mstar (in dex)

virial_mass(Rvir::Real; <keyword arguments>)

Compute the virial mass from the virial radius.

Arguments

• mdef::AbstractString = default_mdef: halo mass definition (e.g., "200c", "vir")
• cosmo::AbstractCosmology = default_cosmo: cosmology under which to evaluate the overdensity
• z::Real = 0.0: redshift at which to evaluate the overdensity

virial_mass(halo::HaloModel; <keyword arguments>)

Compute the virial mass via root finding.

Arguments

• mdef::AbstractString = default_mdef: halo mass definition (e.g., "200c", "vir")
• cosmo::AbstractCosmology = default_cosmo: cosmology under which to evaluate the overdensity
• z::Real = 0.0: redshift at which to evaluate the overdensity
• xstart::Real = 10.0: factor of the scale radius to start the search for the root
• rtol::Real = 1e-2: relative error tolerance for the root finding
• maxevals:Int = 100: maximum number of function evalutions for the root finding

virial_radius(halo::HaloModel, <keyword arguments>)

Compute the virial radius of the halo via root finding.

Arguments

• mdef::AbstractString = default_mdef: halo mass definition (e.g., "200c", "vir")
• cosmo::AbstractCosmology = default_cosmo: cosmology under which to evaluate the overdensity
• z::Real = 0.0: redshift at which to evaluate the overdensity
• xstart::Real = 10.0: factor of the scale radius to start the search for the root
• rtol::Real = 1e-2: relative error tolerance for the root finding
• maxevals:Int = 100: maximum number of function evalutions for the root finding

Jeans model with line-of-sight velocity dispersion predictions.

• JeansModel: collection of mass model, tracer density model, and anisotropy model
• sigma_los: calculate the line-of-sight velocity dispersion
• kappa_los: calculate the line-of-sight velocity kurtosis

JeansModel(mass_model, tracer_model, anisotropy_model)

Represents the key ingredients for the Jeans modeling. The first argument can be an array of DensityModel instances, in which case we will consider the mass model to be the sum of all its components. The second argument refers to the density profile of the kinematic tracer population (e.g., the stars or star clusters).

kappa_los(model::JeansModel, R; <keyword arguments>, parameters...)

Calculate the line-of-sight velocity kurtosis by numerically integrating the spherical Jeans equations.

Examples

model = JeansModel(NFWModel(), SersicModel(), ConstantBetaModel())
R = collect(1:0.5:10)
kappa = kappa_los(model, R)

Arguments

• n_interp::Int = 10: number of points on the interpolation grid
• fudge::Real = 1e-6: fudge factor for tweaking integration bounds
• interp::Bool = true: whether or not to interpolate from a grid of R
• return_sigma::Bool = false: whether or not to additionally return the velocity dispersion
• rmax::Real = 1e2: maximum radius (in kpc) for integrating the Jeans equation
• parameters: keywords used to update JeansModel parameter values

See also: sigma_los

sigma_los(model::JeansModel, R; <keyword arguments>, parameters...)

Calculate the line-of-sight velocity dispersion by numerically integrating the spherical Jeans equation.

Examples

model = JeansModel(NFWModel(), SersicModel(), ConstantBetaModel())
R = collect(1:0.5:10)
sigma = sigma_los(model, R)

Arguments

• n_interp::Int = 10: number of points on the interpolation grid
• fudge::Real = 1e-6: fudge factor for tweaking integration bounds
• interp::Bool = true: whether or not to interpolate from a grid of R
• rmax::Real = 1e2: maximum radius (in kpc) for integrating the Jeans equation
• parameters: keywords used to update JeansModel parameter values

See also: sigma_los_parallel, kappa_los

sigma_los_parallel(model::JeansModel, R, parameter_sets; <keyword arguments>)

Calculate the line-of-sight velocity dispersion by numerically integrating the spherical Jeans equation in parallel for different sets of parameters.

parameter_sets should be a list of dictionaries where each dictionary contains an update to the model parameters.

Examples

model = JeansModel(NFWModel(), SersicModel(), ConstantBetaModel())
R = collect(1:0.5:10)
betas = collect(0.1:0.1:0.9)
parameter_sets = [Dict(:beta => b) for b in betas]
sigmas = sigma_los(model, R, parameter_sets)

Arguments

• n_interp::Int = 10: number of points on the interpolation grid
• fudge::Real = 1e-6: fudge factor for tweaking integration bounds
• interp::Bool = true: whether or not to interpolate from a grid of R
• rmax::Real = 1e2: maximum radius (in kpc) for integrating the Jeans equation

See also: sigma_los