main.py

"""Modelling the Variable X-ray Spectrum of an Accreting Black Hole Binary System.

A Monte-Carlo Radiative Transfer (MCRT) simulation of the spectrum of X-Ray photons 
emitted from a Narrow-Line Seyfert 1 (NLS1) Active Galactic Nuclei (AGN).

Photons are subjected to various scattering and absorbtion probabilities 
as they traverse through a circumnuclear gas distribution.
"""

import concurrent.futures
from itertools import accumulate, product
import math
import random
from time import time
from typing import Union

import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import numpy as np
import pandas as pd

from helpers import import_cross_sections_dataframe


class Constants:
    """Class containing Physics constants to be used in the simulation.

    Class attributes:
        1) SPEED_OF_LIGHT
        2) GRAVITATIONAL_CONSTANT
        3) SOLAR_MASS
        4) THOMPSON_CROSS_SECTION
        5) ELECTRON_REST_MASS
        6) INTERACTION_CROSS_SECTIONS
    """

    # Constants.
    SPEED_OF_LIGHT: float = 2.9979 * 10 ** 8  # (m s^-1)
    GRAVITATIONAL_CONSTANT: float = 6.6741 * 10 ** -11  # (N m^2 kg^-2)
    SOLAR_MASS: float = 1.988 * 10 ** 30  # (kg)
    THOMPSON_CROSS_SECTION: float = 6.6525 * 10 ** -29  # (m^2)
    ELECTRON_REST_MASS: float = 511.0  # (keV)

    # Absorbtion cross-section DataFrame.
    INTERACTION_CROSS_SECTIONS: pd.core.frame.DataFrame = (
        import_cross_sections_dataframe()
    )

    @classmethod
    def plot_absorbtion_cross_section(self) -> None:
        """Class method to produce graph of various
        interaction cross-sections against photon energy.

        Interaction cross-sections:
            1) Total Absorbtion Cross-Section
            2) FeK Absorbtion Cross-Section
            3) Non-FeK Absorbtion Cross-Section
            4) Klein-Nishina Scattering Cross-Section
        """

        # Filter to range of observable photon energies and
        # plot interaction cross-sections against photon energies on a loglog scale.
        self.INTERACTION_CROSS_SECTIONS.loc[
            (self.INTERACTION_CROSS_SECTIONS.index >= Photon.MIN_ENERGY)
            & (self.INTERACTION_CROSS_SECTIONS.index <= Photon.MAX_ENERGY)
        ].plot(
            y=[
                "Total Absorbtion Cross-Section",
                "FeK Absorbtion Cross-Section",
                "Non-FeK Absorbtion Cross-Section",
                "Klein-Nishina Scattering Cross-Section",
            ],
            loglog=True,
        )

        # Add Legend.
        plt.legend(loc="best")

        # Save figure.
        plt.savefig(
            "./graphs/interaction_cross_sections.png",
            dpi=300,
            bbox_inches="tight",
        )


class Photon:
    """Class representing Photon in the MCRT simulation.

    Class attributes:
        1) MIN_ENERGY
        2) MAX_ENERGY
        3) GAMMA
        4) FEK_BETA_TO_ALPHA_RATIO
        5) FLUORESCENCE_YIELD
        6) FEK_ALPHA_ENERGY
        7) FEK_BETA_ENERGY
        8) MAX_MOVEMENT_PER_ITERATION
        9) STEP_SIZE
        10) MAX_ITERATIONS
    """

    # Range of possible photon energies.
    MIN_ENERGY: float = 0.4  # (keV) Taken from Miller & Turner (2011), approximately the minimum observable energy with telescopes.
    MAX_ENERGY: float = 400.0  # (keV) Highest energy that can contribute observable energy bands via Compton down-scattering.

    # Exponent in photon flux power law.
    GAMMA: float = 2.38  # Taken from Miller et al. (2007)
    FEK_BETA_TO_ALPHA_RATIO: float = 0.120  # Taken from Han et al. (2007)
    FLUORESCENCE_YIELD: float = 0.358  # Taken from Han et al. (2007)
    FEK_ALPHA_ENERGY: float = 6.407  # Yaqoob et al. (2007)
    FEK_BETA_ENERGY: float = 7.034  # Yaqoob et al. (2007)

    # Photon movement parameters.
    MAX_MOVEMENT_PER_ITERATION: int = 50  # The maximum length over which scattering probabilities are evaluated within an iteration.
    STEP_SIZE: int = 1  # Steps over which scattering probabilities are evaluated.
    MAX_ITERATIONS: int = 15  # Number of iterations to perform on a photon in the box before it is discarded.

    _tiny_float: float = np.finfo(
        float
    ).eps  # Smallest float to avoid using log(0) in probability calcs.

    def __init__(self, box_length: int) -> None:
        """Initialisation method for instance of class Photon.

        Args:
            box_length (int): Non-dimensional length of box containing circumnuclear gas.
            Should be set to Gas.box_length.
        """

        # Initial photon position in center of the box.
        self._x: int = box_length // 2 - 1
        self._y: int = box_length // 2 - 1
        self._z: int = 0

        # Random initial direction of travel in spherical coords.
        self._theta: float = 2 * np.pi * random.uniform(0.0, 1.0)  # Polar angle
        self._phi: float = math.acos(random.uniform(0.0, 1.0))  # Azimuthal angle

        # Random initial photon energy and weighting factor
        # using artificially flattened power-law distribution.
        self._energy: float = self.MIN_ENERGY + (
            self.MAX_ENERGY - self.MIN_ENERGY
        ) * random.uniform(0.0, 1.0)

        self._weight: float = (
            (self.GAMMA - 1.0)
            * (self.MAX_ENERGY - self.MIN_ENERGY)
            * (self._energy ** (-self.GAMMA))
            / (
                self.MIN_ENERGY ** (1.0 - self.GAMMA)
                - self.MAX_ENERGY ** (1.0 - self.GAMMA)
            )
        )

        # Initial distance travelled by photon is zero.
        self._distance: float = 0.0

        # Initially set event flag as None.
        self._event: str = "None"

        # Store box length in __dict__ for easy access.
        self._box_length: int = box_length

    def _display_photon_attributes(self) -> None:
        """Method to print attributes of photon instance to the console.

        Not used in main() but useful for debugging/development.
        """

        # Iterate through __dict__ and print to console.
        print("Photon Params: ")
        for param, value in vars(self).items():
            param = param.lstrip("_")
            print(f"{param}: {value}")

    def output_photon_attributes(self) -> str:
        """Method to output attributes of photon instance as a single line of a .csv file.

        This will be used to store outputs of MCRT simulation in a single .csv file per process.

        Returns:
            str: String in the format of a single line in a .csv file.
        """

        return (
            ", ".join(
                [
                    str(value)
                    for param, value in vars(self).items()
                    if param != "_box_length"
                ]
            )
            + "\n"
        )

    def _bin_photon_energy(self) -> float:
        """Method to bin photon energies into buckets consistent
        with the available interaction cross-section data.

        Returns:
            float: Binned photon energy.
        """

        if self._energy < 10:
            return round(self._energy, 3)
        elif self._energy < 20:
            return round(self._energy, 1)
        else:
            return round(self._energy, 0)

    def _calculate_possible_coords(self) -> np.ndarray:
        """Create arrays of possible x, y, and z coordinates
        for a single iteration of the simulation.

        Returns:
            np.ndarray: Numpy arrays of possible x, y and z coords.
        """

        # Generate array of possible radial movements.
        steps = np.arange(0, self.MAX_MOVEMENT_PER_ITERATION, self.STEP_SIZE)

        # Project radial movements onto the x-axis and clip values to the edge of the box.
        possible_x = (
            self._x + math.sin(self._phi) * math.cos(self._theta) * steps
        ).astype(int, copy=False)
        possible_x.clip(0, self._box_length - 1, out=possible_x)

        # Project radial movements onto the y-axis and clip values to the edge of the box.
        possible_y = (
            self._y + math.sin(self._phi) * math.sin(self._theta) * steps
        ).astype(int, copy=False)
        possible_y.clip(0, self._box_length - 1, out=possible_y)

        # Project radial movements onto the z-axis and clip values to the edge of the box.
        possible_z = (self._z + math.cos(self._phi) * steps).astype(int, copy=False)
        possible_z.clip(0, self._box_length // 2 - 1, out=possible_z)

        return possible_x, possible_y, possible_z

    def _generate_random_interaction_log_probability(self) -> float:
        """Method to generate the negative log of a random number
        in the range (0, 1] from a uniform distribution.

        Returns:
            float: negative log of random number in range (0, 1]
        """

        # Generate random number from a uniform distribution.
        # An infinitessimal number is used as the lower bound to prevent
        # a potential log(0) error.
        random_interaction_probability = random.uniform(self._tiny_float, 1.0)

        return -math.log(random_interaction_probability)

    def _calculate_event_index(self, log_probability_along_path: np.ndarray) -> int:
        """Method to evaluate the index along the radial vector at
        which an interaction first occurs for a given interaction type.

        Args:
            log_probability_along_path (np.ndarray): A NumPy array of cumulative log probabilies
            along the radial vector for a given interaction type.

        Returns:
            int: Index along the radial vector at which an event first occurs.
            Will return endpoint index if no event occurs.
        """

        # Call _generate_random_interaction_log_probability
        random_interaction_log_probability = (
            self._generate_random_interaction_log_probability()
        )

        # If any events occur then return first index of event, else return endpoint index.
        if (log_probability_along_path >= random_interaction_log_probability).any():
            return (
                log_probability_along_path >= random_interaction_log_probability
            ).argmax()
        else:
            return log_probability_along_path.shape[0] - 1

    def _determine_event(
        self, KN_event_index: int, FeK_event_index: int, non_FeK_event_index: int
    ) -> int:
        """Method to compare event indices of various interaction types
        to determine which event, if any, occurs.

        Args:
            KN_event_index (int): Klein-Nishina scattering event index.
            FeK_event_index (int): Fe-K absorbtion event index.
            non_FeK_event_index (int): non-Fe-K absorbtion event index.

        Returns:
            int: Smallest event index.
        """

        # Find smallest event index and corresponding event,
        # defaulting to endpoint of radial vector and no event occuring.
        minimum_event_index = self.MAX_MOVEMENT_PER_ITERATION - 1
        event = "None"
        if FeK_event_index < minimum_event_index:
            minimum_event_index = FeK_event_index
            event = "FeK_Absorb"
        if KN_event_index < minimum_event_index:
            minimum_event_index = KN_event_index
            event = "KN_Scatter"
        if non_FeK_event_index < minimum_event_index:
            minimum_event_index = non_FeK_event_index
            event = "Non_FeK_Absorb"

        # Update photon event flag.
        self._event = event

        return minimum_event_index

    def _update_energy(self) -> None:
        """Method to update photon energy and weighting factor,
        depending on which event has occured.
        """

        # If photon is at the edge of the box then it
        # exits the box with no further attribute updates.
        if (
            (self._x in (0, self._box_length - 1))
            | (self._y in (0, self._box_length - 1))
            | (self._z == self._box_length // 2 - 1)
        ):
            self._event = "Box_Exit"
            return None

        # If photon hits the AGN accretion disk then it is
        # absorbed and removed from the simulation.
        if self._z == 0:
            self._event = "Disc_Absorb"
            self._weight = 0.0
            return None

        # If photon encounters non-FeK absorbtion then it is
        # absorbed and removed from the simulation.
        if self._event == "Non_FeK_Absorb":
            self._weight = 0.0
            return None

        # Klein-Nishina scattering.
        if self._event == "KN_Scatter":

            # Generate new random direction of travel.
            new_theta = 2 * math.pi * random.uniform(0.0, 1.0)
            new_phi = math.acos(random.uniform(-1.0, 1.0))

            # Determine cosine of scattering angle
            cosine_scattering_angle = math.cos(new_theta) * math.cos(
                self._theta
            ) + math.sin(new_theta) * math.sin(self._theta) * math.cos(
                new_phi - self._phi
            )

            # Calculate ratio of photon energy to electron rest mass energy.
            energy_to_rest_mass_ratio = self._energy / Constants.ELECTRON_REST_MASS

            # Calculate Klein-Nishina angular factor.
            KN_angular_factor = 1.0 / (
                1.0 + energy_to_rest_mass_ratio * (1.0 - cosine_scattering_angle)
            )

            # Update energy of photon post-scattering.
            self._energy = self._energy * KN_angular_factor

            # Calculate divisor for Klein-Nishina weight adjustment.
            weight_divisor = (
                3.0
                / (8.0 * energy_to_rest_mass_ratio)
                * (
                    (
                        1
                        - 2
                        * (1 + energy_to_rest_mass_ratio)
                        / energy_to_rest_mass_ratio ** 2
                    )
                    * math.log(1 + 2 * energy_to_rest_mass_ratio)
                    + 0.5
                    + 4.0 / energy_to_rest_mass_ratio
                    - 1 / (2 * (1 + 2 * energy_to_rest_mass_ratio) ** 2)
                )
            )

            # Update photon energy weighting to account for
            # angular dependence of Klein-Nishina scattering.
            self._weight = (
                self._weight
                * 0.75
                * KN_angular_factor ** 2
                * (
                    KN_angular_factor
                    + 1.0 / KN_angular_factor
                    - 1
                    + cosine_scattering_angle ** 2
                )
                / weight_divisor
            )

            # Update photon direction of travel post-scattering.
            self._theta = new_theta
            self._phi = new_phi

        # FeK absorbtion.
        if self._event == "FeK_Absorb":

            # Generate new random direction of travel.
            self._theta = 2 * np.pi * random.uniform(0.0, 1.0)
            self._phi = math.acos(random.uniform(-1.0, 1.0))

            # Randomly assign event to FeK Alpha or Beta absorbtion and update energy.
            if random.uniform(0.0, 1.0) < self.FEK_BETA_TO_ALPHA_RATIO:
                self._energy = self.FEK_BETA_ENERGY
            else:
                self._energy = self.FEK_ALPHA_ENERGY

            # Update photon energy weight to account for potential Auger electron emission.
            self._weight = self.FLUORESCENCE_YIELD * self._weight

        # If photon drops out of observable energy range then remove from simulation.
        if (self._energy < self.MIN_ENERGY) | (self._energy > self.MAX_ENERGY):
            self._event = "Unobservable_Absorb"
            self._weight = 0.0

        return None

    def iterate(self, gas_array: np.ndarray) -> None:
        """Method to perform a single iteration of the MCRT simulation for a photon.

        Args:
            gas_array (np.ndarray): 3-D NumPy array giving the density of
            circumnuclear gas at each point in the Cartesian box.
            Will be passed in via simulate method.
        """

        # Calculate possible x, y, & z coordinates.
        possible_x, possible_y, possible_z = self._calculate_possible_coords()

        # Evaluate cumulative density at each point along the generated radial vector.
        cumulative_density_along_path = np.cumsum(
            gas_array[possible_x, possible_y, possible_z]
        )

        # Calculate binned photon energy to look up corresponding interaction cross-sections.
        binned_energy = self._bin_photon_energy()

        # Calculate cumulative log probabilities along generated radial vector.
        KN_log_probability_along_path = (
            cumulative_density_along_path
            * Constants.INTERACTION_CROSS_SECTIONS.at[
                binned_energy, "Klein-Nishina Scattering Cross-Section"
            ]
        )

        FeK_log_probability_along_path = (
            cumulative_density_along_path
            * Constants.INTERACTION_CROSS_SECTIONS.at[
                binned_energy, "FeK Absorbtion Cross-Section"
            ]
        )

        non_FeK_log_probability_along_path = (
            cumulative_density_along_path
            * Constants.INTERACTION_CROSS_SECTIONS.at[
                binned_energy, "Non-FeK Absorbtion Cross-Section"
            ]
        )

        # Determine index of radial vector at which each event first occurs.
        KN_event_index = self._calculate_event_index(KN_log_probability_along_path)
        FeK_event_index = self._calculate_event_index(FeK_log_probability_along_path)
        non_FeK_event_index = self._calculate_event_index(
            non_FeK_log_probability_along_path
        )

        # Determine event that occurs first and return corresponding index.
        minimum_event_index = self._determine_event(
            KN_event_index, FeK_event_index, non_FeK_event_index
        )

        # Determine new coordinates.
        new_x = possible_x[minimum_event_index]
        new_y = possible_y[minimum_event_index]
        new_z = possible_z[minimum_event_index]

        # Update distance travelled.
        self._distance += math.sqrt(
            (new_x - self._x) ** 2 + (new_y - self._y) ** 2 + (new_z - self._z) ** 2
        )

        # Update coordinates.
        self._x = new_x
        self._y = new_y
        self._z = new_z

        # Update energy and weight of photon.
        self._update_energy()

    def _continue_simulation(self) -> bool:
        """Method to determine whether or not to continue with simulation for photon.

        Returns:
            bool: Boolean indicating whether or not to continue with simulation.
        """

        # Dictionary of continue conditions.
        continue_dict = {
            "Box_Exit": False,
            "Disc_Absorb": False,
            "Non_FeK_Absorb": False,
            "Unobservable_Absorb": False,
            "None": True,
            "KN_Scatter": True,
            "FeK_Absorb": True,
        }

        return continue_dict[self._event]

    def simulate(self, gas_array: np.ndarray):
        """Main method to perform MCRT simulation on a single photon.

        This method will repeatedly loop iterations of MCRT simulation
        until the simulation is exited or the max number of iterations is reached.

        Args:
            gas_array (np.ndarray): 3-D NumPy array giving the density of
            circumnuclear gas at each point in the Cartesian box.
            Should be set to Gas().generate_conical_gas_array().gas_array.
        """

        # Initialise iteration counter.
        iteration_counter = 0

        # Loop iterate method until the simulation encounters a break event.
        while self._continue_simulation():
            self.iterate(gas_array)

            # Increment iteration counter and break loop
            # if max iteration number is exceeded.
            iteration_counter += 1
            if iteration_counter == self.MAX_ITERATIONS:
                break

        return self


class Gas:
    """Class for circumnuclear gas distribution surrounding an AGN.

    N.B the dimensions of the circumnuclear gas are much greater than the radius of the black hole.
    Therefore consider the black hole as a point like source at the center of the box.

    Class attributes:
        1) BLACK_HOLE_MASS
        2) GRAVITATIONAL_RADIUS
        3) COMPTON_THICK_COLUMN_DENSITY
    """

    # Mass of black hole at the center of AGN.
    BLACK_HOLE_MASS: float = (
        Constants.SOLAR_MASS * 2 * 10 ** 6
    )  # Consistent with De Marco et al. (2013)

    # Gravitational radius of black hole.
    GRAVITATIONAL_RADIUS: float = (
        Constants.GRAVITATIONAL_CONSTANT
        * BLACK_HOLE_MASS
        / Constants.SPEED_OF_LIGHT ** 2
    )

    # Compton thick column density of circumnuclear gas.
    COMPTON_THICK_COLUMN_DENSITY: float = (
        10.0 ** 28 / 10.0
    )  # Taken from Tatum et al. (2013).
    # N.B. Divide by 10 added to reduce number of scatterings per photon.

    def __init__(
        self,
        min_radius: Union[int, float] = 100.0,
        max_radius: Union[int, float] = 400.0,
        opening_phi: float = np.pi / 4,
        box_length: int = 100,
    ) -> None:
        """Initialisation method of Gas class.

        Args:
            min_radius (Union[int, float], optional): Minimum radius of circumnuclear
            gas distribution in terms of gravitational radii. Defaults to 100.0.
            max_radius (Union[int, float], optional): Minimum radius of circumnuclear
            gas distribution in terms of gravitational radii. Defaults to 400.0.
            opening_phi (float, optional): Opening angle of conical gas distribution. Defaults to np.pi/4.
            box_length (int, optional): Number of cells in Cartesian box. Defaults to 100.
        """

        # Initialise attributes of gas distribution.
        self._min_radius = min_radius * self.GRAVITATIONAL_RADIUS
        self._max_radius = max_radius * self.GRAVITATIONAL_RADIUS
        self._opening_phi = opening_phi

        # Initialise Cartesian box.
        self._box_length = box_length
        self._unit_cell_density = self.COMPTON_THICK_COLUMN_DENSITY

        # Initialise empty NumPy array.
        self._gas_array = np.zeros(
            shape=(box_length, box_length, box_length // 2),
            dtype=float,
        )
        self._gas_array_type = "empty"

    @property
    def box_length(self) -> int:
        """Getter for box_length attribute.

        Returns:
            int: Box length in non-dimensional units.
        """
        return self._box_length

    @property
    def gas_array(self) -> np.ndarray:
        """Getter for gas_array attribute.

        Returns:
            np.ndarray: 3-D NumPy array giving the density of
            circumnuclear gas at each point in the Cartesian box.
        """
        return self._gas_array

    def generate_conical_gas_array(self):
        """Method to generate the standard disk wind geometry distribution
        of circumnuclear gas used in Sim et al. (2008).
        """

        # Create empty gas array.
        gas_array = np.zeros(
            shape=(self._box_length, self._box_length, self._box_length // 2),
            dtype=float,
        )

        # Calculate minimum radius in terms of Cartesian box coords.
        scaled_min_radius = (self._min_radius / self._max_radius) * (
            self._box_length // 2
        )

        # Find mid-point of box
        midpoint_position_adjustment = self._box_length // 2 - 1

        # Calculate tan of conical opening angle.
        tan_opening_phi = math.tan(self._opening_phi)

        # Iterate through all cells in the Cartesian box.
        for x, y, z in product(
            range(self._box_length),
            range(self._box_length),
            range(self._box_length // 2),
        ):

            # If the opening angle of the current point is greater than
            # the conical opening angle then set the density at the
            # current point equal to the unit cell density.
            # N.B. can compare tan of the angles as tan is strictly
            # increasing in the range [0, 90].
            tan_phi = np.sqrt(
                (x - midpoint_position_adjustment) ** 2
                + (y - midpoint_position_adjustment) ** 2
            ) / (z + scaled_min_radius / tan_opening_phi)

            if tan_phi > tan_opening_phi:
                gas_array[x, y, z] = self._unit_cell_density

        # Set density of the outside of the box to zero to ensure photons
        # exit the gas distribution into the vacuum of space.
        gas_array[0, :, :] = 0.0
        gas_array[:, 0, :] = 0.0
        gas_array[:, :, 0] = 0.0
        gas_array[self._box_length - 1, :, :] = 0.0
        gas_array[:, self._box_length - 1, :] = 0.0
        gas_array[:, :, self._box_length // 2 - 1] = 0.0

        # Update gas array attribute.
        self._gas_array = gas_array
        self._gas_array_type = "conical"

        return self

    def plot_gas_array(self) -> None:
        """Method to create and save a 3-D voxel plot of the gas distribution."""

        # Create 3-D axis and set viewing angle.
        fig = plt.figure()
        ax = fig.add_subplot(111, projection="3d")
        ax.view_init(elev=50.0, azim=-60.0)

        # Create voxel plot.
        ax.voxels(self._gas_array)

        # Save the plot.
        plt.savefig(
            f"./graphs/{self._gas_array_type}_gas_distribution.png",
            dpi=300,
            bbox_inches="tight",
        )


def main(process_number: int) -> None:
    """Main function of MCRT simulation to be run once per process.

    Creates and simulates 5 million photons per process.
    The results of all the simulations are then saved to
    a .csv file corresponding to the input process_number.

    Args:
        process_number (int): Unique integer for the process calling main.
    """

    # Initialise instance of Gas class and generate conical gas array.
    gas = Gas()
    gas.generate_conical_gas_array()

    # Simulate 5 million photons.
    t1 = time()
    results = [Photon(gas.box_length).simulate(gas.gas_array) for i in range(5000000)]
    t2 = time()
    print(f"{t2-t1}s")

    # Iterate through results of simulations and save to a .csv file.
    with open(f"./results/results{process_number}.csv", "w") as results_file:

        results_file.write("x, y, z, theta, phi, energy, weight, distance, event\n")
        for result in results:
            results_file.write(result.output_photon_attributes())


if __name__ == "__main__":

    # Create 4 processes and run main in each process.
    with concurrent.futures.ProcessPoolExecutor(max_workers=4) as executor:
        result = executor.map(main, [1, 2, 3, 4])