DiffusiveShockAccelerationModels.jl

This package provides a number of efficiency models for Diffuse Shock Acceleration (DSA). It provides a number of functions to calculate what fraction of the energy dissipated at a shock is used to accelerate Cosmic Rays (CRs). If you use this implementation in a publication please cite Böss et al. (2023).

Install

As usual with Julia just run

] add DiffusiveShockAccelerationModels

Mach number dendent efficiency models

Different authors found a number of models that describe the acceleration efficiency of CRs at shocks dependent on the sonic Mach number. Here we implemented the following DSA models, as summarized in Böss et al. (2023):

DiffusiveShockAccelerationModels.Kang07 — Type

Kang07(X_cr::T = 0.3, η_max::T = 0.57) where T

Efficiency model by Kang, Ryu, Cen, Ostriker 2007, http://arxiv.org/abs/0704.1521v1

Values

X_cr: Pcr / Pth defined in model for re-acceleration. Basis for interpolation between acceleration and re-acceleration efficiency.
η_max: Maximum efficiency defined in the model

DiffusiveShockAccelerationModels.KR13 — Type

KR13(X_cr::T=0.05, η_max::T=0.0348) where T

Efficiency model by Kang&Ryu 2013: ApJ, 764, 95.

Values

X_cr: Pcr / Pth defined in model for re-acceleration. Basis for interpolation between acceleration and re-acceleration efficiency.
η_max: Maximum efficiency defined in the model

DiffusiveShockAccelerationModels.CS14 — Type

    CS14(X_cr::T=0.05, η_max::T=0.5*0.2055) where T

Efficiency model by Caprioli&Spitkovsky 2014 efficiency.

Values

X_cr: Pcr / Pth defined in model for re-acceleration. Basis for interpolation between acceleration and re-acceleration efficiency.
η_max: Maximum efficiency defined in the model

DiffusiveShockAccelerationModels.Ryu19 — Type

    Ryu19(X_cr::T=0.05, η_max::T=0.0348) where T

Efficiency model by Ryu et al. 2019: https://arxiv.org/pdf/1905.04476v2.pdf.

Values

X_cr: Pcr / Pth defined in model for re-acceleration. Basis for interpolation between acceleration and re-acceleration efficiency.
η_max: Maximum efficiency defined in the model

Implemented DSA models

On top of that you can find the DSA model by Kang (2024):

DiffusiveShockAccelerationModels.Kang24_p — Type

Kang24_p(X_cr::T=1.0, η_max::T=0.03) where T

Efficiency model for protons by Kang (2024).

Values

X_cr: Pcr / Pth defined in model for re-acceleration. Basis for interpolation between acceleration and re-acceleration efficiency.
η_max: Maximum efficiency defined in the model

DiffusiveShockAccelerationModels.Kang24_e — Type

Kang24_e(X_cr::T=1.0, η_max::T=0.0004871685) where T

Efficiency model for electrons by Kang (2024).

Values

X_cr: Pcr / Pth defined in model for re-acceleration. Basis for interpolation between acceleration and re-acceleration efficiency.
η_max: Maximum efficiency defined in the model

Kang 2024 DSA model

Version v0.2 included the model by Gupta et al. (2025)

DiffusiveShockAccelerationModels.Gupta24_e — Type

Gupta24_e(X_cr::T=1.0, η_max::T=0.0004871685) where T

Efficiency model for electrons by Gupta, Caprioli & Sptikovsky (2024)

Values

X_cr: Pcr / Pth defined in model for re-acceleration. Basis for interpolation between acceleration and re-acceleration efficiency.
η_max: Maximum efficiency defined in the model

You can also use the constant efficiency used by Pfrommer et al. (2017):

DiffusiveShockAccelerationModels.P16 — Type

    P16(X_cr::T=0.05, η_max::T=0.5) where T

Constant efficiency as in Pfrommer+ 2016, doi: 10.1093/mnras/stw2941

These efficiency models can be used with::

DiffusiveShockAccelerationModels.η_Ms — Function

η_Ms(η_model::AbstractShockAccelerationEfficiency, M::T, X_cr::T) where {T<:Real}

Calculates the sonic Mach Number dependent η_model as a linear interpolation between acceleration and reacceleration.

You can also disentangle this for either pure acceleration:

DiffusiveShockAccelerationModels.η_Ms_acc — Function

η_Ms_acc(η_model::Kang07, M::Real)

Initial acceleration efficiency for model from Kang, Ryu, Cen, Ostriker 2007, http://arxiv.org/abs/0704.1521v1

η_Ms_acc(η_model::KR13, M::T) where T

Efficiency model from Kang&Ryu 2013, doi:10.1088/0004-637X/764/1/95

η_Ms_acc(η_model::CS14, M::T) where T

Efficiency from Caprioli&Spitkovsky 2015, doi: 10.1088/0004-637x/783/2/91 Same simplified approach as Vazza+16 -> is roughly half the efficiency of Kang&Ryu 2013.

η_Ms_acc(η_model::Ryu19, M::T) where T

Efficiency model from Ryu et al. 2019, https://arxiv.org/abs/1905.04476 Values for 2.25 < M <= 5.0 extrapolated to entire range

η_Ms_acc(η_model::P16, M::T) where T

Constant efficiency as in Pfrommer+ 2016, doi: 10.1093/mnras/stw2941

η_Ms_acc(η_model::Kang24_p, M::Real)

Initial acceleration efficiency model for protons by Kang (2024).

η_Ms_acc(η_model::Kang24_e, M::Real)

Initial acceleration efficiency model for electrons by Kang (2024).

η_Ms_acc(η_model::Gupta24_e, M::Real)

Initial acceleration efficiency model for electrons by Gupta, Caprioli & Sptikovsky (2024)

Or re-acceleration:

DiffusiveShockAccelerationModels.η_Ms_reacc — Function

η_Ms_reacc(η_model::Kang07, M::Real)

Reacceleration efficiency for model from Kang, Ryu, Cen, Ostriker 2007, http://arxiv.org/abs/0704.1521v1

η_Ms_reacc(η_model::KR13, M::T) where T

Efficiency model from Kang&Ryu 2013, doi:10.1088/0004-637X/764/1/95

η_Ms_reacc(η_model::CS14, M::T) where T

Efficiency from Caprioli&Spitkovsky 2015, doi: 10.1088/0004-637x/783/2/91 Same simplified approach as Vazza+16 -> is roughly half the efficiency of Kang&Ryu 2013.

η_Ms_reacc(η_model::Ryu19, M::T) where T

Efficiency model from Ryu et al. 2019, https://arxiv.org/abs/1905.04476 Values for 2.25 < M <= 5.0 extrapolated to entire range

η_Ms_reacc(η_model::P16, M::T) where T

Constant efficiency as in Pfrommer+ 2016, doi: 10.1093/mnras/stw2941

η_Ms_reacc(η_model::Kang24_p, M::Real)

Reacceleration efficiency for model by Kang (2024), identical to acceleration efficiency.

η_Ms_reacc(η_model::Kang24_e, M::Real)

Reacceleration efficiency for model by Kang (2024), identical to acceleration efficiency.

η_Ms_reacc(η_model::Gupta24_e, M::Real)

Reacceleration efficiency for model by Kang (2024), identical to acceleration efficiency.

Defining your own acceleration efficiency

You can also define your own acceleration efficiency as a subtype of

DiffusiveShockAccelerationModels.AbstractShockAccelerationEfficiency — Type

AbstractShockAccelerationEfficiency

Abstract type for shock acceleration efficiencies

We provide a helper function for the fit used in Kang+2007 in case it's useful for your acceleration model:

DiffusiveShockAccelerationModels.kr_fitting_function — Function

kr_fitting_function(x::Real, a0::Real, a1::Real, a2::Real, a3::Real, a4::Real)

Helper function to use the fitting function from KR07.

Magnetic field angle dependent efficiency models

Another parameter in the acceleration efficiency is the shock obliquity.

DiffusiveShockAccelerationModels.η_B — Function

η_B(θ_B::Real, θ_crit::Real)

Calculate B angle dependent efficiency component following Pais et. al. 2018, eq. 7, https://arxiv.org/pdf/1805.00128.pdf

η_B(η_model::Gupta2024_t, θ_B::Real, θ_crit::Real)

Calculate B angle dependent efficiency component following Pais et. al. 2018, eq. 7, https://arxiv.org/pdf/1805.00128.pdf

Here we provide the results from Pais et al. (2019) who fit a functional form to the data by Caprioli&Spitkovsky (2014).

DiffusiveShockAccelerationModels.Pais2018_t — Type

Pais2018_t(X_cr::T=0.05) where T

Efficiency model by Pais et al. 2018: MNRAS, 478, 4, 5278–5295, https://arxiv.org/pdf/1805.00128.pdf

Values

X_cr: Pcr / Pth defined in model for re-acceleration. Basis for interpolation between acceleration and re-acceleration efficiency.

Or for electrons we supply the model by Gupta et al. (2024).

DiffusiveShockAccelerationModels.Gupta2024_t — Type

Gupta2024_t(X_cr::T=0.05) where T

Efficiency model by Gupta2024_t, Caprioli & Spitkovsky 2024: MNRAS, 478, 4, 5278–5295, https://arxiv.org/pdf/1805.00128.pdf

Values

X_cr: Pcr / Pth defined in model for re-acceleration. Basis for interpolation between acceleration and re-acceleration efficiency.

Ions

Ions are found to be accelerated primarily at quasi-parallel shocks. We provide two helper functions for this.

DiffusiveShockAccelerationModels.ηB_acc_p — Function

ηB_acc_p(θ_B::Real)

Magnetic field geometry dependent efficiency for protons at initial acceleration.

ηB_acc_p(η_model::Gupta2024_t, θ_B::Real)

Magnetic field geometry dependent efficiency for protons at initial acceleration.

DiffusiveShockAccelerationModels.ηB_reacc_p — Function

ηB_reacc_p(θ_B::Real)

Magnetic field geometry dependent efficiency for protons at reacceleration.

ηB_reacc_p(η_model::Gupta2024_t, θ_B::Real)

Magnetic field geometry dependent efficiency for protons at reacceleration.

Electrons

Electron acceleration can be modeled via

DiffusiveShockAccelerationModels.ηB_acc_e — Function

ηB_acc_e(θ_B::Real)

Magnetic field geometry dependent efficiency for electrons at initial acceleration.

ηB_acc_e(η_model::Gupta2024_t, θ_B::Real)

Magnetic field geometry dependent efficiency for electrons at initial acceleration.

DiffusiveShockAccelerationModels.ηB_reacc_e — Function

ηB_reacc_e(θ_B::Real)

Magnetic field geometry dependent efficiency for electrons at reacceleration.

ηB_reacc_e(η_model::Gupta2024_t, θ_B::Real)

Magnetic field geometry dependent efficiency for electrons at reacceleration.

Usage

To use for example the mach number dependent model by Kang & Ryu (2013), combined with the shock obliquity model by Pais et. al. (2019)

using DiffusiveShockAccelerationModels

ηM_model = KR13()       # Mach number dependent model
ηM_model = Pais2018_t() # Obliquity dependent model
Mach = 5.0              # we assume a Mach 5 shock
θ_B  = 0.1π             # angle between shock normal and magnetic field vector
X_cr = 0.0         # X_cr = P_cr / P_th -> in this case no pre-existing CRs

# magnetic field angle dependent acc. efficiency
ηB   = ηB_acc_p(ηB_model, θ_B)  

# Mach number dependent acc. efficiency
ηM   = η_Ms(ηM_model, Mach, X_cr)

# total efficiency
η_tot = ηB * ηM