jnu_mixed.cu

Documentation for the file jnu_mixed.cu. This file contains functions related to the thermal synchrotron emissivity calculation.

Functions

Functions

__host__ __device__ double jnu_synch(const double nu, const double Ne, const double Thetae, const double B, const double theta, const double K2)

Calculates the thermal synchrotron emissivity \(j_\nu(\nu, \theta)\) following Leung et al.(2011).

This function implements the analytical fit for the emissivity of a thermal distribution of electrons: \( j_\nu(\nu, \theta) = \frac{\sqrt{2}\pi e^2 n_e \nu_s}{3 c K_2(1/\Theta_e)} (X^{1/2} + 2^{11/12} X^{1/6})^2 \exp(-X^{1/3}) \)

Parameters:

• nu – Photon frequency in the plasma frame.

• Ne – Electron number density ( \(n_{\rm e}\)).

• Thetae – Dimensionless electron temperature ( \(\Theta_{\rm e}\)).

• B – Magnetic field strength.

• theta – Angle between the magnetic field and the wave vector ( \(\theta\)).

• K2 – Precomputed value of the modified Bessel function \( K_2(1/\Theta_e) \) for efficiency.

Returns:

The local emissivity \( j_\nu \) in CGS units.

__host__ __device__ double int_jnu_thermal_synch(double Ne, double Thetae, double Bmag, double nu, double K2)

Calculates the angle-integrated thermal synchrotron emissivity \( J_\nu \) in CGS units.

This function evaluates the total energy emitted per unit time, per unit volume, and per unit frequency, integrated over all solid angles \( \int j_\nu d\Omega \).

• Implements: \( J_\nu = \frac{\sqrt{2} e^3 n_e B \Theta_e^2}{27 m_e c^2 K_2(1/\Theta_e)} F(\Theta_e, B, \nu) \)

Parameters:

• Ne – Electron number density \( n_e \).

• Thetae – Dimensionless electron temperature \( \Theta_{\rm e} \).

• Bmag – Magnetic field strength \( B \).

• nu – Photon frequency \( \nu \) in the plasma frame.

• K2 – Precomputed value of the modified Bessel function \( K_2(1/\Theta_e) \) for efficiency.

Returns:

The integrated emissivity in \( \text{erg} \cdot \text{s}^{-1} \cdot \text{cm}^{-3} \cdot \text{Hz}^{-1} \).

double jnu_integrand(double th, void *params)

Provides the integrand for calculating the angle-integrated thermal synchrotron emissivity.

This function is used by the GSL integrator to evaluate the integral over solid angles \( \int j_\nu \sin\theta d\theta \).

The integrand is given by:

\[ \sin^2\theta \left( (K/\sin\theta)^{1/2} + 2^{11/12}(K/\sin\theta)^{1/6} \right)^2 \exp\left[-(K/\sin\theta)^{1/3}\right] \]

Parameters:

• th – The angle \( \theta \) between the magnetic field and the photon wave vector.

• params – A pointer to the dimensionless frequency parameter \( K \).

Returns:

The value of the integrand at the given angle.

__host__ void init_emiss_tables(void)

Initializes lookup tables for the angle-integrated emissivity function \( F \) and the modified Bessel function \( K_2 \).

Precomputes \( F(K) \) by integrating the angular dependence of synchrotron emission over the solid angle using GSL integration and stores the second-order modified Bessel function \( K_2(1/\Theta_e) \) in log-space for rapid retrieval during the simulation.

The integration goes as:

\(F[k] = \ln \left( 4\pi \int_{0}^{\pi/2} \sin^2\theta \left[ \sqrt{\frac{K}{\sin\theta}} + 2^{11/12} \left( \frac{K}{\sin\theta} \right)^{1/6} \right]^2 \exp\left[ -\left( \frac{K}{\sin\theta} \right)^{1/3} \right] d\theta \right)\)

Returns:

void

__host__ __device__ double K2_eval(const double Thetae)

Evaluation of the modified Bessel function of the second kind, order 2, \( K_2(1/\Theta_e) \).

This function retrieves the value of \( K_2(1/\Theta_e) \) from precomputed lookup tables for efficient computation during simulations.

Note

for large \( \Theta_e \), we can use the asymptotic expansion \( K_2(1/\Theta_e) \approx 2 \Theta_e^2 \).

Note

TMAX and THETA_MIN define the valid range for \( \Theta_e \) in the lookup tables.

Parameters:

Thetae – Dimensionless electron temperature \( \Theta_e \).

Returns:

The value of \( K_2(1/\Theta_e) \).

__host__ __device__ double F_eval(const double Thetae, const double Bmag, const double nu)

Evaluates the frequency-dependent component \( F(K) \) of the angle-integrated emissivity. This function calculates the dimensionless frequency parameter \( K \) and retrieves the corresponding value of the precomputed integral \( \int j_\nu d\Omega \).

• If \( K > K_{\rm MAX} \): Returns 0 due to the exponential decay of the synchrotron spectrum.

• If \( K < K_{\rm MIN} \): Uses a power-law approximation based on the asymptotic behavior of the emissivity at low frequencies.

• Otherwise: Performs a linear interpolation using the precomputed F table.

Parameters:

• Thetae – Dimensionless electron temperature \( \Theta_e \).

• Bmag – Magnetic field strength \( B \).

• nu – Photon frequency \( \nu \) in the plasma frame.

Returns:

The value of the integrated emissivity function \( F \).

__host__ __device__ double linear_interp_F(const double K)

Performs linear interpolation on the precomputed emissivity function \( F(K) \).

Parameters:

K – Dimensionless frequency parameter \( K \).

Returns:

The interpolated value of \( F(K) \).

__host__ __device__ double linear_interp_K2(const double Thetae)

Performs linear interpolation on the precomputed modified Bessel function \( K_2(1/\Theta_e) \).

Parameters:

Thetae – Dimensionless electron temperature \( \Theta_e \).

Returns:

The interpolated value of \( K_2(1/\Theta_e) \).

__host__ __device__ double jnu_bremss(const double nu, const double Ne, const double Thetae)

Calculates the thermal bremsstrahlung emissivity \( j_\nu \) for a Maxwellian distribution of electrons.

Note

This method is from Method from Rybicki & Lightman, ultimately from Novikov & Thorne.

Parameters:

• nu – Photon frequency \( \nu \) in the plasma frame.

• Ne – Electron density.

• Thetae – Dimensionless electron temperature \( \Theta_e \).

Returns:

The value of the bremsstrahlung emissivity \( j_\nu \).

__device__ double jnu_total(const double nu, const double Ne, const double Thetae, const double B, const double theta, const double K2)

Calculates the total emissivity \( j_\nu \) by summing contributions from synchrotron and bremsstrahlung processes.

Parameters:

• nu – Photon frequency \( \nu \) in the plasma frame.

• Ne – Electron density.

• Thetae – Dimensionless electron temperature \( \Theta_e \).

• B – Magnetic field strength \( B \).

• theta – Angle between the magnetic field and the line of sight.

• K2 – Precomputed value of the modified Bessel function \( K_2(1/\Theta_e) \) for efficiency.

Returns:

The value of the total emissivity \( j_\nu \).

__host__ __device__ double int_jnu_bremss(const double Ne, const double Thetae, const double nu)

Calculates the angle-integrated thermal bremsstrahlung emissivity \( J_\nu \) for a Maxwellian distribution of electrons.

Parameters:

• Ne – Electron density.

• Thetae – Dimensionless electron temperature \( \Theta_e \).

• nu – Photon frequency \( \nu \) in the plasma frame.

Returns:

The value of the angle-integrated bremsstrahlung emissivity \( J_\nu \).

__host__ __device__ double int_jnu_total(const double Ne, const double Thetae, const double Bmag, const double nu, const double K2)

Calculates the total angle-integrated emissivity \( J_\nu \) by summing contributions from synchrotron and bremsstrahlung processes.

Parameters:

• Ne – Electron density.

• Thetae – Dimensionless electron temperature \( \Theta_e \).

• B – Magnetic field strength \( B \).

• nu – Photon frequency \( \nu \) in the plasma frame.

• K2 – Precomputed value of the modified Bessel function \( K_2(1/\Theta_e) \) for efficiency.

Returns:

The value of the total angle-integrated emissivity \( J_\nu \).

__device__ double jnu_ratio_brems(const double nu, const double Ne, const double Thetae, const double B, const double theta, const double K2)

Calculates the ratio of bremsstrahlung to total emissivity. It serves to realocate it in the correct bins when saving the final spectrum.

Parameters:

• nu – Photon frequency \( \nu \) in the plasma frame.

• Ne – Electron density.

• Thetae – Dimensionless electron temperature \( \Theta_e \).

• B – Magnetic field strength \( B \).

• theta – Angle between the magnetic field and the line of sight.

• K2 – Precomputed value of the modified Bessel function \( K_2(1/\Theta_e) \) for efficiency.

Returns:

The ratio of bremsstrahlung to total emissivity.