temp_ratio

plasmapy.formulary.collisions.helio.collisional_analysis.temp_ratio(*, r_0: Quantity, r_n: Quantity, n_1: Quantity, n_2: Quantity, v_1: Quantity, T_1: Quantity, T_2: Quantity, ions: str | int | integer | Particle | CustomParticle | Quantity = ('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)[source]

Calculate the thermalization ratio for a plasma in transit, taken from Maruca et al. [2013] and Johnson et al. [2023]. 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 (Quantity, keyword-only) – Starting position of the plasma in units convertible to astronomical units.

  • r_n (Quantity) – Final position of the plasma in units convertible to astronomical units.

  • n_1 (Quantity) – The primary ion number density in units convertible to m-3.

  • n_2 (Quantity) – The secondary ion number density in units convertible to m-3.

  • v_1 (Quantity) – The primary ion speed in units convertible to km s-1.

  • T_1 (Quantity) – Temperature of the primary ion in units convertible to temperature K.

  • T_2 (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 Hellinger et al. [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 Hellinger et al. [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 Hellinger et al. [2011].

Returns:

theta – The dimensionless ion temperature ratio prediction for the distance provided.

Return type:

float

Raises:

  • TypeError – If applicable arguments are not instances of Quantity or cannot be converted into one.

  • 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 [Verscharen et al., 2019]. The relative temperature between constituent plasma ion species is given as:

\[\theta_{21} = \frac{T_{2}}{T_{1}} \, ,\]

where \(T_{1}\) and \(T_{2}\) are the scalar temperatures for the primary ion of interest and the secondary ion, respectively. The scalar temperature defined as:

\[T_{\rm i} = \frac{2T_{{\rm i}, \perp} + T_{{\rm i}, \parallel}}{3} \, ,\]

where \(T_{{\rm i}, \perp}\) and \(T_{{\rm i}, \parallel}\) are the temperature of the \({\rm i}\)-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 [Maruca et al., 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 Maruca et al. [2013] is given below:

\[\frac{d \theta_{21}}{dr} = A \left ( \frac{n_1}{v_1 T_1^{3/2}} \right ) \frac{\left( \mu_{1} \mu_{2} \right )^{1/2} Z_{1} Z_{2} \left( 1 - \theta_{21} \right ) \left(1 + \eta_{21}\theta_{21} \right )}{\left( \frac{\mu_{2}}{\mu_{1}} + \theta_{21} \right )^{3/2}} \lambda_{21}\]

and

\[\lambda_{21} = 9 + \ln \left| B \left ( \frac{T^{3}_{1}}{n_{1}} \right )^{1/2} \left( \frac{Z_{1}Z_{2}(\mu_{1} + \mu_{2}) }{\theta_{21} + \frac{\mu_{2}}{\mu_1}} \right ) \left( \frac{n_{2}Z_{2}^{2}}{n_{1}Z_{1}^{2}} + \theta_{21} \right)^{1/2}\right |\]

With \(\eta = \frac{n_{2}}{n_{1}}\), \(\theta = \frac{T_{2}}{T_{1}}\), \(A = 2.60 \times 10^{7} \, {\rm cm}^{3} \, {\rm km} \, {\rm K}^{3/2} \, {\rm s}^{-1} \, {\rm au}^{-1}\), and \(B = 1 \, {\rm cm}^{-3/2}{\rm K}^{-3/2}\).

The thermalization is from Coulomb collisions, which assumes “soft”, small-angle deflections mediated by the electrostatic force Baumjohann and Treumann [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 Richardson [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 Hellinger et al. [2011].

\[n(r) \propto r^{-1.8}\ , \hspace{1cm} v_{r}(r) \propto r^{-0.2}\ , \hspace{0.5cm} {\rm and} \hspace{0.5cm} T(r) \propto r^{-0.74}\]

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...]