Source code for plasmapy.formulary.collisions.helio.collisional_analysis

"""
Module containing the Collisional Analysis formulation.
"""

__all__ = ["temp_ratio"]

import logging
import numbers

import astropy.units as u
import numpy as np

from plasmapy.particles import ParticleLike, ParticleList
from plasmapy.utils.decorators import validate_quantities


[docs] @validate_quantities( T_1={"can_be_negative": False, "equivalencies": u.temperature_energy()}, T_2={"can_be_negative": False, "equivalencies": u.temperature_energy()}, ) def temp_ratio( # noqa: C901 *, r_0: u.Quantity[u.au], r_n: u.Quantity[u.au], n_1: u.Quantity[u.cm**-3], n_2: u.Quantity[u.cm**-3], v_1: u.Quantity[u.km / u.s], T_1: u.Quantity[u.K], T_2: u.Quantity[u.K], ions: ParticleLike = ("p+", "He-4++"), n_step: int = 100, density_scale: float = -1.8, velocity_scale: float = -0.2, temperature_scale: float = -0.74, verbose: bool = False, ): r""" Calculate the thermalization ratio for a plasma in transit, taken from :cite:t:`maruca:2013` and :cite:t:`johnson:2023a`. This function allows the thermalization of a plasma to be modeled, predicting the temperature ratio for different ion species within a plasma at a different point in space. Parameters ---------- r_0 : `~astropy.units.Quantity`, |keyword-only| Starting position of the plasma in units convertible to astronomical units. r_n : `~astropy.units.Quantity` Final position of the plasma in units convertible to astronomical units. n_1 : `~astropy.units.Quantity` The primary ion number density in units convertible to m\ :sup:`-3`. n_2 : `~astropy.units.Quantity` The secondary ion number density in units convertible to m\ :sup:`-3`. v_1 : `~astropy.units.Quantity` The primary ion speed in units convertible to km s\ :sup:`-1`. T_1 : `~astropy.units.Quantity` Temperature of the primary ion in units convertible to temperature K. T_2 : `~astropy.units.Quantity` Temperature of the secondary ion in units convertible to temperature K. ions : |particle-list-like|, default: ``("p+, "He-4 2+")`` Particle list containing two (2) particles, primary ion of interest is entered first, followed by the secondary ion. n_step : positive integer The number of intervals used in solving a differential equation via the Euler method. density_scale : real number, default: -1.8 The value used as the scaling parameter for the primary ion density. The default value is taken from :cite:t:`hellinger:2011`. velocity_scale : `float`, default: -0.2 The value used as the scaling parameter for the primary ion velocity. The default value is taken from :cite:t:`hellinger:2011`. temperature_scale : `float`, default: -0.74 The value used as the scaling parameter for the primary ion temperature. The default value is taken from :cite:t:`hellinger:2011`. Returns ------- theta : `float` The dimensionless ion temperature ratio prediction for the distance provided. Raises ------ `TypeError` If applicable arguments are not instances of `~astropy.units.Quantity` or cannot be converted into one. ~astropy.units.UnitTypeError If applicable arguments do not have units convertible to the expected units. Notes ----- The processes by which Coulomb collisions bring ion temperatures into local thermal equilibrium (LTE) has received considerable attention :cite:p:`verscharen:2019`. The relative temperature between constituent plasma ion species is given as: .. math:: \theta_{21} = \frac{T_{2}}{T_{1}} \, , where :math:`T_{1}` and :math:`T_{2}` are the scalar temperatures for the primary ion of interest and the secondary ion, respectively. The scalar temperature defined as: .. math:: T_{\rm i} = \frac{2T_{{\rm i}, \perp} + T_{{\rm i}, \parallel}}{3} \, , where :math:`T_{{\rm i}, \perp}` and :math:`T_{{\rm i}, \parallel}` are the temperature of the :math:`{\rm i}`-particles along the axes perpendicular and parallel to the ambient magnetic field. In order to determine how extensively an individual parcel of plasma has been processed by Coulomb collisions :cite:p:`maruca:2013` introduced an approached called collisional analysis. This paper seeks to quantify how collisions affect the plasma's departures from LTE, the equation for collisional thermalization from :cite:t:`maruca:2013` is given below: .. math:: \frac{d \theta_{21}}{dr} = A \left ( \frac{n_1}{v_1 T_1^{3/2}} \right ) \frac{\left( \mu_{1} \mu_{2} \right )^{1/2} Z_{1} Z_{2} \left( 1 - \theta_{21} \right ) \left(1 + \eta_{21}\theta_{21} \right )}{\left( \frac{\mu_{2}}{\mu_{1}} + \theta_{21} \right )^{3/2}} \lambda_{21} and .. math:: \lambda_{21} = 9 + \ln \left| B \left ( \frac{T^{3}_{1}}{n_{1}} \right )^{1/2} \left( \frac{Z_{1}Z_{2}(\mu_{1} + \mu_{2}) }{\theta_{21} + \frac{\mu_{2}}{\mu_1}} \right ) \left( \frac{n_{2}Z_{2}^{2}}{n_{1}Z_{1}^{2}} + \theta_{21} \right)^{1/2}\right | With :math:`\eta = \frac{n_{2}}{n_{1}}`, :math:`\theta = \frac{T_{2}}{T_{1}}`, :math:`A = 2.60 \times 10^{7} \, {\rm cm}^{3} \, {\rm km} \, {\rm K}^{3/2} \, {\rm s}^{-1} \, {\rm au}^{-1}`, and :math:`B = 1 \, {\rm cm}^{-3/2}{\rm K}^{-3/2}`. The thermalization is from Coulomb collisions, which assumes "soft", small-angle deflections mediated by the electrostatic force :cite:t:`baumjohann:1997`. It is assumed that there is no relative drift between the ion species and that it is a mixed ion collision, the Coulomb logarithm for a mixed ion collision is given by :cite:t:`nrlformulary:2019`. The density, velocity and temperature of the primary ion can be radially scaled, as seen below. The values for the scaling can be altered, though the default values are taken from :cite:t:`hellinger:2011`. .. math:: n(r) \propto r^{-1.8}\ , \hspace{1cm} v_{r}(r) \propto r^{-0.2}\ , \hspace{0.5cm} {\rm and} \hspace{0.5cm} T(r) \propto r^{-0.74} Application is primarily for the solar wind. Examples -------- >>> import astropy.units as u >>> from plasmapy.formulary.collisions import helio >>> r_0 = [0.1, 0.1, 0.1] * u.au >>> r_n = [1.0, 1.0, 1.0] * u.au >>> n_1 = [300, 400, 500] * u.cm**-3 >>> n_2 = [12, 18, 8] * u.cm**-3 >>> v_1 = [450, 350, 400] * u.km / u.s >>> T_1 = [1.5 * 10**5, 2.1 * 10**5, 1.7 * 10**5] * u.K >>> T_2 = [2.5 * 10**6, 1.8 * 10**6, 2.8 * 10**6] * u.K >>> ions = ["p+", "He-4++"] >>> helio.temp_ratio( ... r_0=r_0, r_n=r_n, n_1=n_1, n_2=n_2, v_1=v_1, T_1=T_1, T_2=T_2, ions=ions ... ) [2.78928645832..., 1.04007368797..., 1.06914450183...] """ # Validate ions argument if not isinstance(ions, list | tuple | ParticleList): ions = [ions] ions = ParticleList(ions) # Validate number of ions if len(ions) != 2: raise ValueError( "Argument 'ions' can only take two (2) input values. " f"Instead received {len(ions)} input values." ) if not all(ions.is_category("ion")): raise ValueError( f"Particle(s) in 'ions' must be ions, received {ions=} " "instead. Please renter the 'ions' input parameter." ) # Validate n_step argument if not isinstance(n_step, numbers.Integral): raise TypeError( "Argument 'n_step' is of incorrect type, type of " f"{type(n_step)} received instead. While 'n_step' must be " "of type int." ) # Validate scaling arguments for arg in (density_scale, velocity_scale, temperature_scale): if not isinstance(arg, numbers.Real): raise TypeError( "Scaling argument is of incorrect type, type of " f"{type(arg)} received instead. Scaling argument " "should be of type float or int." ) # Define differential equation function def df_eq( r_0, r_n, n_1_0, n_2, v_1_0, T_1_0, T_2, ions, n_step, density, velocity, temperature: float, ): # Initialize the alpha-proton charge and mass ratios. z_1 = ions[0].charge_number mu_1 = ions[0].mass_number z_2 = ions[1].charge_number mu_2 = ions[1].mass_number # Initialise. d_r = (r_n - r_0) / n_step # Define constants A = 2.6 * 10**7 * (u.cm**3 * u.km * (u.K**1.5)) / (u.s * u.au) B = 1 / (u.cm * u.K) ** 1.5 # Define Coulomb log for mixed ion collisions, see docstring def lambda_ba( theta: float, T_1, n_1, n_2, z_1, z_2, mu_1, mu_2, ): a = np.sqrt(T_1**3 / n_1) b = (z_1 * z_2 * (mu_1 + mu_2)) / (theta + mu_2 / mu_1) c = np.sqrt((n_2 * z_2**2 / n_1 * z_1**2) + theta) return 9 + np.log(B * a * b * c) theta = T_2 / T_1_0 for i in range(n_step): r = r_0 + ((i + 1) * d_r) n_1 = n_1_0 * (r / r_n) ** density v_1 = v_1_0 * (r / r_n) ** velocity T_1 = T_1_0 * (r / r_n) ** temperature eta = n_2 / n_1 alpha = n_1 / (v_1 * (T_1**1.5)) beta = ( np.sqrt(mu_1 * mu_2) * (z_1 * z_2) ** 2 * (1 - theta) * (1 + eta * theta) ) / (np.sqrt((mu_2 / mu_1) + theta) ** 3) l_ba = lambda_ba(theta, T_1, n_1, n_2, z_1, z_2, mu_1, mu_2) d_theta = d_r * alpha * l_ba * A * beta theta = theta + d_theta return theta.value variables = [r_0, r_n, n_1, n_2, v_1, T_1, T_2] d_type = [bool(hasattr(var, "__len__")) for var in variables] var = all(d_type) if not var: return df_eq( r_0, r_n, n_1, n_2, v_1, T_1, T_2, ions, n_step, density_scale, velocity_scale, temperature_scale, ) try: all(len(variables[0]) == len(z) for z in variables[1:]) res = [] for i in range(len(variables[0])): res.append( df_eq( r_0[i], r_n[i], n_1[i], n_2[i], v_1[i], T_1[i], T_2[i], ions, n_step, density_scale, velocity_scale, temperature_scale, ) ) if verbose: logging.info(f"\r {(i / len(variables[0])) * 100:.2f} %") # noqa: G004 return res # noqa: TRY300 except Exception as e: # noqa: BLE001 raise ValueError( "Argument(s) are of unequal lengths, the following " "arguments should be of equal length: 'r_0', 'r_n', " "'n_1', 'n_2', 'v_1', 'T_1' and 'T_2'." ) from e