Probing Turbulent Scattering Effects on Suprathermal Electrons in the Solar Wind: Modeling, Observations, and Implications

We describe the evolution of the gyrophase-averaged electron phase space distribution function f(v, μ, r ) by the "focused transport equation" introduced by Skilling (1971) to describe cosmic ray diffusion, extended by Isenberg (1997) to the context of solar wind pick-up ions, and extensively discussed by Zank (2014) or le Roux & Webb (2012). We use the formalism of the latter reference:

$$\begin{eqnarray}&&\begin{array}{l}\frac{\partial f}{\partial t}+\left({\boldsymbol{V}}+\mu v{\boldsymbol{b}}\right)\cdot {\boldsymbol{\nabla }}f+{\left\langle \frac{{dv}}{{dt}}\right\rangle }_{\phi }\frac{\partial f}{\partial v}\\ +{\left\langle \frac{d\mu }{{dt}}\right\rangle }_{\phi }\frac{\partial f}{\partial \mu }=\nu { \mathcal L }(f).\end{array}\end{eqnarray} \tag{ 1 }$$

In this equation, b the local unit vector along the magnetic field, V the solar wind velocity field, v is the modulus of an electron velocity vector and $\mu ={\boldsymbol{v}}\cdot {\boldsymbol{b}}/v=\cos \theta$ its pitch angle cosine. The distribution function is defined such that the number of particles in an infinitesimal phase space volume is ${dN}=f(v,\mu ,{\boldsymbol{r}})d{\boldsymbol{r}}d{\boldsymbol{v}}$, with dv = 2π v² d μ dv. The phase-averaged pitch angle evolution is given by

$$\begin{eqnarray}&&\begin{array}{l}{\left\langle \displaystyle \frac{d\mu }{{dt}}\right\rangle }_{\phi }=\displaystyle \frac{1-{\mu }^{2}}{2}\\ \times \left(v{\boldsymbol{\nabla }}\cdot {\boldsymbol{b}}+\mu {\boldsymbol{\nabla }}\cdot {\boldsymbol{V}}-3\mu {\boldsymbol{b}}{\boldsymbol{b}}:{\boldsymbol{\nabla }}{\boldsymbol{V}}-\displaystyle \frac{2{\boldsymbol{b}}}{v}\cdot \left(\displaystyle \frac{\partial {\boldsymbol{V}}}{\partial t}+{\boldsymbol{V}}\cdot {\boldsymbol{\nabla }}{\boldsymbol{V}}\right)-\displaystyle \frac{2{{eE}}_{\parallel }}{{mv}}\right)\end{array}\end{eqnarray} \tag{ 2 }$$

where E_∥ it the electric field component parallel to the magnetic field line, and e > 0 and m the electron charge and mass. The phase-averaged velocity modulus evolution is given by

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{1}{v}{\left\langle \displaystyle \frac{{dv}}{{dt}}\right\rangle }_{\phi }=-\displaystyle \frac{1-{\mu }^{2}}{2}{\boldsymbol{\nabla }}\cdot {\boldsymbol{V}}+\displaystyle \frac{1-3{\mu }^{2}}{2}{\boldsymbol{b}}{\boldsymbol{b}}:{\boldsymbol{\nabla }}{\boldsymbol{V}}\\ -\displaystyle \frac{\mu {\boldsymbol{b}}}{v}\cdot \left(\displaystyle \frac{\partial {\boldsymbol{V}}}{\partial t}+{\boldsymbol{V}}\cdot {\boldsymbol{\nabla }}{\boldsymbol{V}}\right)-\displaystyle \frac{\mu {{eE}}_{\parallel }}{{mv}}.\end{array}\end{eqnarray} \tag{ 3 }$$

Finally, the right-hand term of Equation (1) is an operator describing the isotropic diffusion of the electron velocity in pitch angle over a timescale ν⁻¹, with ${ \mathcal L }$ the Lorentz diffusion operator ${ \mathcal L }(f)={\partial }_{\mu }(1-{\mu }^{2})/2{\partial }_{\mu }f$ (see, e.g., Helander & Sigmar 2005).

Solving Equation (1) is obviously a complicated task, but we will argue that only a few of these terms play a significant role in the evolution of the suprathermal pitch angle distribution function. For this we look at the ordering of the different terms appearing in Equations (2) and (3). The first term of Equation (2) describes the focusing of electrons along a magnetic field line and is of the order of ν_focus ∼ v/L, where L is a typical gradient length of the system (here, strictly speaking, of the magnetic field, but we will consider the gradient scales of all relevant quantities to have the same order of magnitude). The various terms involving V describe effects related to the action of inertial forces on the electrons (the velocity vectors of which are defined in the solar wind frame, which is non-Galilean). Their order of magnitude is ν_inert ∼ V/L or ∼(V/v)V/L. Finally, the last terms of Equations (2)–(3) describe the effect of the parallel electric field on the electrons. The interplanetary parallel electric field is E_∥ ∼ kT_e /eL, where T_e is the electron temperature, and L is the typical length of the electron pressure gradient. Therefore the order of magnitude of these terms is ν_E ∼ kT_e /mvL. In this paper, we focus on the evolution of the suprathermal electrons, which by definition fulfill the condition v ≫ v_the ≫ V. For this population, ${\nu }_{\mathrm{inert}}/{\nu }_{\mathrm{focus}}={ \mathcal O }(V/v)$ or ${ \mathcal O }({V}^{2}/{v}^{2})$, which are both small compared to one, and ${\nu }_{E}/{\nu }_{\mathrm{focus}}={ \mathcal O }({{kT}}_{e}/{{mv}}^{2})$, which is also small compared to one: the inertial forces and the electric field act on the suprathermal part of the VDF on much longer timescales than the magnetic focusing does. This analysis shows that the evolution of the distribution of the modulus of the suprathermal electrons velocity vector v (which involves only terms ∼ν_inert and ∼ν_E ) occurs on a quite longer timescale than that of the distribution of the pitch angle distribution. Keeping only leading order terms in Equation (1), we see that the evolution of the latter is governed by

$$\begin{eqnarray}&&\displaystyle \frac{\partial f}{\partial t}+\mu v\displaystyle \frac{\partial f}{\partial s}+\displaystyle \frac{(1-{\mu }^{2})v}{2{L}_{B}(s)}\displaystyle \frac{\partial f}{\partial \mu }=\displaystyle \frac{\partial }{\partial \mu }\displaystyle \frac{(1-{\mu }^{2})\nu }{2}\displaystyle \frac{\partial f}{\partial \mu }\end{eqnarray} \tag{ 4 }$$

where s is the curvilinear coordinate along the field line and ${L}_{B}(s)=-{\left(d\mathrm{ln}B/{ds}\right)}^{-1}$ is the characteristic length of the magnetic field gradient along the field line. This equation, which will be used in the rest of the paper, describes the evolution of an electron with a single effective degree of freedom (μ, s), bound to a field line, and subject to two competing physical effects: magnetic focusing acting on a typical length L_B , and diffusion in pitch angle, acting on a typical length λ = v/ν, that we shall call after the isotropization mean free path. Both of these processes act at constant velocity vector modulus, and from now on, v will only have the role of a constant parameter. We note that, by neglecting everywhere the solar wind speed V compared to the electron speed v, we place ourselves in a "static field line approximation," in which the time-dependent nature of the field line due to its advection by the solar wind is completely neglected. This is convenient because it provides a one-to-one, time-independent correspondence between the curvilinear coordinate s and the heliocentric distance r (see Appendix B, Equation (B1)), but some effects due to the proper motion of the field line, especially at distances far from the Sun, or involving the very long term history of particles, will be absent from the present treatment.

The Knudsen number Kn(s, v) = λ(s, v)/L_B (s) is the dimensionless parameter that locally measures the relative importance of the focusing and the diffusion effects. For small values of Kn, the particles will be diffused and isotropized before they can sense much change in the magnetic field (and therefore experience any perceptible focusing): the electrons distribution is in this case expected to behave locally (i.e., the distribution at coordinate s will be determined by the value of Kn at s). At the opposite, large values of Kn will imply that the electron trajectories are essentially deterministic, and that they can sense large changes in L_B before undergoing any deflection from the isotropization process. We expect, in this case, the electrons to behave in an essentially nonlocal way, the pitch angle distribution at a given coordinate s not being determined by the local values of λ and L_B , but by the whole profile of these quantities "from zero to infinity."

As already noted by Roelof (1969), an exact steady state (i.e., ∂_t f = 0) solution to Equation (4) can easily be found if both λ and L_B are independent of the position s—and the magnetic field modulus is therefore $\propto \exp (-s/{L}_{B})$. The phase space distribution is in this case

$$\begin{eqnarray}&&f(\mu ,s)={{Ae}}^{-s/{L}_{B}}{e}^{\mathrm{Kn}\mu },\end{eqnarray} \tag{ 5 }$$

where A is an integration constant. In the following of this paper, and in particular for comparison with observations, we shall be interested in the evolution of the normalized pitch angle distributions $\bar{f}(\theta )$, defined such that the probability dp of observing an electron with a pitch angle between θ and θ + d θ is ${dp}=\bar{f}(\theta )\sin \theta d\theta$. According to Equation (5), this distribution is independent of s and given by

$$\begin{eqnarray}&&\bar{f}(\theta )=\displaystyle \frac{\mathrm{Kn}{e}^{\mathrm{Kncos}\theta }}{2\mathrm{sinhKn}},\end{eqnarray} \tag{ 6 }$$

with θ between 0 and π.

Of course, this solution is not precisely what we are looking for, since we know that the solar wind magnetic field profile is not exponential. However, it is interesting in that it provides a clear interpretation of the strahl/halo as a steady-state structure resulting from a competition between magnetic focusing and diffusion. The outcome of this competition is, in this particular case, solely determined by the local value of the Knudsen number: if Kn ≪ 1, diffusion dominates, Equation (6) simplifies into $\bar{f}(\theta )\sim 1/2$ and the velocity distribution is essentially isotropic. At the opposite, when Kn ∼ 1 or larger, the diffusion is not fast enough to cancel the focusing effect, which manifests as an important excess of electrons with field-aligned velocities, that one may call a strahl. Importantly, the distribution (6) does not depend on any boundary condition at the Sun level: the strahl emerges here as a statistical feature, stemming from the fact that an electron, undergoing a random walk in pitch angle space, spends in average more time field aligned than not, because of the focusing effect.

Apart from its interest in providing us with some intuitive picture of the origin of the strahl/halo structure, solution (6) is important in that it gives an analytical expression for the steady-state distribution valid in an arbitrary magnetic field profile, provided that the Knudsen number is small enough. Still following Roelof (1969), we can estimate (6) to hold whenever dKn/d(s/λ) ≪ 1, or to reformulate it as an upper limit on the Knudsen number,

$$\begin{eqnarray}&&\mathrm{Kn}\ll | {{dL}}_{B}/{ds}{| }^{-1/2}.\end{eqnarray} \tag{ 7 }$$

This inequality can be seen as a condition for a local behavior to hold, and Equation (6) as an expression for the pitch angle distribution valid when this locality condition is fulfilled. In the case of interest for the interplanetary medium, this condition is essentially equivalent to Kn ≪ 1, as discussed in Appendix B.

In this section, as in the previous, we consider the situation in which λ is independent of s, but we now look at a situation more directly relevant to solar wind electrons, with a magnetic field following a Parker spiral. B is then expressed in a polar coordinate system centered on the Sun as

$$\begin{eqnarray}&&{\boldsymbol{B}}=\mathrm{const}.\left(\displaystyle \frac{{{\boldsymbol{u}}}_{{\bf{r}}}}{{r}^{2}}+\displaystyle \frac{{{\boldsymbol{u}}}_{{\boldsymbol{\phi }}}}{{r}_{* }r}\right)\end{eqnarray} \tag{ 8 }$$

where const. is a constant depending on the boundary conditions, which determines the amplitude of the field. Since we are interested only in its gradient scale, we leave it unspecified. The characteristic length of the spiral is ${r}_{* }=V\sin {\rm{\Theta }}/\omega$, where V is the solar wind's velocity, assumed radial and constant in modulus, Θ the solar colatitude, and ω the Sun's rotation angular frequency.

There is no straightforward way to obtain a steady-state analytical solution to the transport Equation (4) in this case. Therefore, we integrated the equation numerically, using a pseudoparticle Monte Carlo method described in Appendix A. Figures 1 and 2 show the result of this integration for parameters r_* = 1 au and λ = 0.6 au. The boundary condition was set by imposing an isotropic normalized distribution $\bar{f}(\theta )=1$ for θ ∈ [0, π/2] (and 0 elsewhere), at a distance s₀ = 0.01 au from the Sun. The distributions are obtained by binning the pitch angles and positions of N = 10⁶ pseudoparticles on a N_θ × N_s = 30 × 100 grid, with distances ranging from 0 to 3 au. Figure 2 shows normalized pitch angle distributions at four different distances, together with the local solutions (6), that one expects to be valid when Kn is small enough. The distributions are plotted as a function of the "outward pitch angle," that is, the electron pitch angle defined with respect to a field line oriented outward from the Sun (since the transport model involves averaging on the electron gyromotion, the orientation of the field line does not play any explicit role). So, an electron having a 0° (resp. 180°) outward pitch angle has a velocity vector parallel to the field line, pointing in the direction outward from (resp. toward) the Sun.

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As a first general comment on these figures, we can see that the strahl-halo picture commonly observed in the solar wind is well reproduced, with an excess of electrons having field-aligned velocity vectors directed outward from the Sun. The strahl pitch angle width, as well as the fraction of isotropized electrons (the "halo electrons"), both increase with increasing distance from the Sun. The steady-state solution to the transport Equation (4) with constant λ therefore reproduces, at least qualitatively (we shall see in Section 3 that the agreement is also remarkably quantitative), the main features of the strahl/halo radial evolution reported in previous observational papers. Importantly, we note that the results presented in Figures 1 and 2 are essentially independent of the boundary condition chosen at s₀ = 0.01 au. Simulations using various boundary conditions (for instance, using $\bar{f}(\theta ,{s}_{0})=\delta (\theta -{\theta }_{0})$ with various values of θ₀) were performed and gave as a result steady-state phase space distributions indistinguishable from the one presented in this section. The reason for this is discussed below.

To understand how the pitch angle distributions are shaped, we note that, since the mean free path λ is here independent of s while L_B (s) is an increasing function of s, the Knudsen number decreases with distance to the Sun, as can be seen on the bottom panel of Figure 1 (see Appendix B for the explicit calculations). Far enough away from the Sun, Kn(s) ≃ λ/2s (or as a function of r, Kn(r) ≃ λ r_*/r²), and one necessarily reaches a distance where Kn is small enough for the locality condition (7) to be fulfilled. We can see on the bottom panel of Figure 2 that, indeed, the local solution is a good approximation to the numerical solution at a distance r ≃ 2 au, at which Kn ≃ 0.16. The three other panels show pitch angle distributions at closer distances from the Sun, where the values of Kn are larger. These distributions do not fit to the local solution, as expected: they are clearly the result of nonlocal phenomena. This nonlocality is manifested by the presence of an isotropized component even at close distances from the Sun, where it is clear that the diffusion process did not have the time to scatter the electrons directly coming from the Sun by a large angle. This can be seen from a simple order of magnitude estimate: the variation of pitch angle due to the local action of the diffusion process on an electron traveling a distance s from the Sun is ${\rm{\Delta }}\theta \sim \sqrt{s/\lambda }$, which is very small if s ≪ λ (and this order of magnitude does not even take into account the effect of magnetic focusing, which will further reduce the spread in θ).

Figure 3 presents the trajectory of a pseudoparticle and sheds some light on the origin of the strahl and halo components, as well as on the reason why the phase space distribution does not depend on the boundary condition in the corona. First, notice that close to the Sun, the focusing length is very small, and the Knudsen number is consequently very high (∼200): in this region, the scattering of electrons can be, to a good approximation, neglected. Therefore their dynamics are essentially determined by the focusing effect that occurs on a typical length L_B ≃ s/2 ≃ 0.005 au (the latter numerical estimate at position s₀). After a few L_B , the electrons are well aligned with the magnetic field direction, and a very collimated beam is already formed at a few solar radii from the Sun, regardless of how the initial condition is chosen at s₀. The red line in Figure 3 shows the deterministic trajectory μ(s) that can be obtained from Equation (4) (or equivalently Equations (A2)–(A3)) in the limit ν → 0,

$$\begin{eqnarray}&&\mu (s)=\sqrt{1-\left(1-\mu {\left({s}_{0}\right)}^{2}\right)\displaystyle \frac{B({s}_{0})}{B(s)}}.\end{eqnarray} \tag{ 9 }$$

This trajectory is closely followed by the electrons leaving the Sun, and illustrates both how the strahl is formed by magnetic focusing of electrons streaming out of the Sun, and why the distributions at distances larger than a few solar radii do not depend on the boundary condition at s₀.

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So, the first part of the trajectory after the electrons are released from the corona basically consists in focusing, and then free streaming of electrons along the field line in a weakly scattering, high Kn environment: electrons in this part of their history constitute the strahl. However, Figure 3 also illustrates that the free-streaming motion changes into a more chaotic one when the electron enters the region of small Knudsen numbers. There, the velocity vector of the particle is isotropized, and the electron follows a random walk with a small drift outward from the Sun: the distribution's behavior is here essentially local. The origin of the halo at intermediate distances, where the Knudsen number is high, is well illustrated by the portions of trajectory around ν t ≃ 5 and ν t ≃ 80: here the electron, after having been deflected at a position of rather low Kn, has acquired a negative value of μ. As a consequence, it travels in the direction of the Sun and now encounters a magnetic field intensity which increases along its way: its parallel velocity decreases (in absolute value) until the electron gets mirrored back to a positive value of μ. The two successive mirroring motions at ν t ≃ 80 can quite clearly be seen on the top panel as a loop in phase space with a mirror position around s ≃ 0.15 au. This provides a picture of the origin of electrons traveling toward the Sun at distances where Kn is large: these electrons did not come straight out of the Sun but first traveled to outer heliospheric regions where Kn is small. Here they got scattered back into the large Kn region where they are observed. We shall not, in this paper, go much further than this qualitative description. However, we want to stress this important point: whereas the strahl is somehow a local component (in the sense that it is determined only by the fields situated between the Sun and the position where it is observed), the halo observed in the high Kn region is an intrinsically nonlocal feature, in that its density is determined by the values taken by the Knudsen number in outer regions, far from the location where it is observed.

In the previous sections, we focused on the situation in which the mean free path λ ≡ λ_turb = v/ν_turb is independent of the distance from the Sun. But it can be expected, especially close to the Sun where the plasma density is high, that Coulomb collisions play a role in shaping the distribution function. For instance, their effect on small pitch angle particles streaming out of the Sun has been studied by Horaites et al. (2018, 2019), showing that they clearly play a role in determining the strahl angular width.

The aim of this section is to evaluate the effect of competition between Coulomb collisions and a turbulent scattering mechanism characterized by a mean free path which, as in previous sections, is assumed not to vary with distance. This study can be made from Equation (4), using the effective scattering frequency ν(r) = ν_turb + ν_col(r), where ν_col is the Coulomb collision frequency. For the results presented in this section, we used for ν_col the scattering frequency of a test electron against static targets,

$$\begin{eqnarray}&&{\nu }_{\mathrm{col}}(r)=\displaystyle \frac{3}{2}\displaystyle \frac{4\pi n(r){q}_{e}^{4}{\rm{\Lambda }}(r)}{{m}_{e}^{2}{v}^{3}}\end{eqnarray} \tag{ 10 }$$

where ${q}_{e}=e/\sqrt{4\pi {\epsilon }_{0}}$ and ${\rm{\Lambda }}=\mathrm{ln}({\lambda }_{D}/{\lambda }_{L})$ is the logarithm of the ratio of the Debye to the Landau radius—${\lambda }_{D}^{2}={\epsilon }_{0}{{kT}}_{e}/{ne}$ and ${\lambda }_{L}=2{q}_{e}^{2}/{m}_{e}{v}^{2}$ (Rax 2005). Here n(r) is the background plasma density profile, which was taken according to the density model of Sittler & Guhathakurta, M (1999). T_e (r) is the electron temperature, the variation of which has little importance given its only appearance in the logarithmic factor Λ. It was taken as a power law with index −0.7 (see for instance Issautier et al. 1998), and a 10 eV value at r = 1 au. The factor 3/2 accounts for both ion–electron and electron–electron collisions. Neglecting the thermal motion of the background particles provides a satisfactory approximation in our case, for only electrons with energies quite larger than the thermal energy are considered. More complete expressions (e.g., Equation (10) from Scudder & Olbert 1979) would here only provide marginal corrections.

Figure 4 shows the evolution of the Knudsen number with radial distance for an electron of energy 150 eV, and a turbulent scattering mean free path λ_turb = 1 au. It is dominated, up to around 0.1 au, by Coulomb collisions (where the plasma density is high, and where Kn_turb ≃ 2λ_turb/s ≫ 1), while turbulent scattering dominates from distances of around 0.3 au (since the scattering mechanism is assumed independent of the position, Kn_turb ≃ λ_turb/2s ≪ 1, see Appendix B). The Knudsen number reaches its maximum Kn ∼ 10 around 0.15 au. The figure displays three different layers at the heliospheric scale: a most internal one, labeled "region I" up to a few 10⁻² au, where electrons are strongly scattered by Coulomb collisions; a most external one, labeled "region III," beyond around 1 au, where electrons are strongly scattered by the turbulent process; and in between, a "ballistic" layer, where the electron dynamics is mainly determined by deterministic processes (magnetic focusing in our case), labeled "region II." Of course, the boundaries between these regions are vague and depend on the criteria chosen to describe strong diffusion. Defining the boundaries as the locations where $\mathrm{Kn}={\mathrm{Kn}}^{\mathrm{lim}}$, the position r_I of the most internal boundary is a solution of ${r}_{{\rm{I}}}{\nu }_{\mathrm{col}}({r}_{{\rm{I}}})=2v/{\mathrm{Kn}}^{\mathrm{lim}}$—it corresponds to the usual exobase of exospheric models, which depends on the electron energy, as discussed by Brandt & Cassinelli (1966). The location of the external boundary can be found by inverting Equation (B5). Assuming that this location s_III ≫ r_*, one would have ${s}_{\mathrm{III}}\simeq {\lambda }_{\mathrm{turb}}/2{\mathrm{Kn}}^{\mathrm{lim}}$, or, in terms of radial distances from the Sun, ${r}_{\mathrm{III}}\simeq {\left({r}_{* }{\lambda }_{\mathrm{turb}}/{\mathrm{Kn}}^{\mathrm{lim}}\right)}^{1/2}$, which may also depend, through λ_turb, on the particle's energy. We finally note that, as discussed in the previous sections, the local solution Equation (6) should be a good approximation of the pitch angle distributions in the regions I and III, provided we chose ${\mathrm{Kn}}^{\mathrm{lim}}$ small enough.

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Figure 5 presents the results of the numerical integration of the transport equation when Coulomb collisions are taken into account. In order to provide an easily readable illustration of the effect of Coulomb collisions, we restricted the representation to the first statistical moment of the normalized distributions

$$\begin{eqnarray}&&\left\langle \mu (s)\right\rangle ={\int }_{-1}^{1}f(\mu ,s)\mu d\mu .\end{eqnarray} \tag{ 11 }$$

The first moment provides a convenient measurement of the anisotropy of the velocity distributions, since, as seen in previous sections, these distributions are all shaped according to a similar strahl and halo structure—the first order anisotropy parameter being, strictly speaking, defined as ${A}_{1}=3\left\langle \mu \right\rangle$, see for instance Brüdern et al. (2022). Then, a value $\left\langle \mu \right\rangle \to 1$ indicates a very peaked distribution at small pitch angles, while $\left\langle \mu \right\rangle \to 0$ indicates a nearly isotropic velocity distribution. Figure 5 clearly shows that Coulomb collisions play a role in isotropizing the distribution functions close to the Sun, especially at relatively low energies. In the first distance bin, corresponding to s = 0.045 au, values of $\left\langle \mu \right\rangle$ are systematically reduced compared to the collisionless case illustrated by the black curve, with the level of isotropy increasing with decreasing energy, as expected from the energy dependency of the collision frequency. We note that the effect of Coulomb collisions is, even in the closest bin, nearly negligible for 600 eV electrons, which appears as a threshold above which collisional effects will not be observable. For lower energy channels, the curves $\left\langle \mu (s)\right\rangle$ all show a maximum at a location close to the maximum of the Knudsen number (which goes further out when energy decreases): this is the location at which the velocity distributions are the most peaked—a point that has recently been identified with Parker Solar Probe observations (Romeo et al. 2022). Once this point has been crossed, the distributions all converge toward the collisionless solution, showing that at large distances, the effect of Coulomb collisions near the Sun has been forgotten and that the distributions are shaped only by the turbulent scattering. This latter point is the most important for the following of this article, where we shall be interested in the experimental determination of the turbulent scattering mean free path λ_turb: beyond a certain distance d_col(E) from the Sun, the electron distributions depend only on λ_turb, and the results of the numerical integration of Equation (4) in the collisionless regime can be used to fit the data. d_col(E) has been determined empirically from the curves presented in Figure 5 as the distance at which the relative difference between collisional and collisionless curves passes below the 3% level. The obtained points for d_col(E) are plotted in Figure 6, together with an exponential curve that we shall use to interpolate the value of d_col at any needed energy. This empirical curve is ${d}_{\mathrm{col}}(E)\simeq 0.5\exp \left(-5.4\times {10}^{-4}(E-80)\right)$, with E expressed in eV and d_col in au.

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Figure 7 presents an example of electron velocity distribution measured by PSP's electrostatic analyzers (Whittlesey et al. 2020). The top panel presents the differential energy flux integrated over pitch angles, showing its energy dependence. The first bump, around 6 eV, is due to the flux of photoelectrons and secondary electrons emitted by the spacecraft or produced inside the instrument; the second, around 70 eV shows the thermal electron flux. At energies above ∼1 keV, the measured flux artificially flattens as the physical particle flux decreases below the noise level of the instrument. In this study, we shall therefore limit ourselves to energy channels comprised between 100 eV (above the thermal level) and 1 keV, as marked by the dashed red lines in Figure 7. The middle panel shows the differential energy flux in the $({v}_{\parallel }=v\cos \theta ,{v}_{\perp }=v\sin \theta )$ plane. The plot was reconstructed from energy and pitch angle measurements and contains the full available information on the VDF (assuming gyrotropy around the magnetic field). The color code goes from dark to light (blue to yellow) on a logarithmic scale, and one can clearly see the presence of an anisotropy extended along the magnetic field direction in our range of energies. That said, the important variation of flux from one energy channel to another hides the details of the anisotropy structure, and the study of the latter is better done by looking at the fluxes normalized per energy bin. This is done in the bottom panel, which shows the contour levels of

$$\begin{eqnarray}&&\bar{f}(E,\theta )=\displaystyle \frac{F(E,\theta )}{{\int }_{0}^{\pi }F(E,\theta )\sin \theta d\theta }\end{eqnarray} \tag{ 12 }$$

where F(E, θ) is the differential energy flux presented in the middle panel. In a given energy channel, $\bar{f}(E,\theta )$ is just the normalized pitch angle distribution that we have been modeling and discussing in Section 2. We can see that this distribution is rather isotropic below the thermal energy and that a strong anisotropy develops from around 100 eV (inner red dashed line). This anisotropy consists of the excess of field-aligned electrons—the very visible strahl—and to a corresponding depletion in the other directions. Energy-wise, the depletion starts at the same energy at which the strahl appears (being its counterpart), that is, essentially at the energy at which the Coulomb collisions are not efficient enough to counterbalance the effect of magnetic focusing and regulate the distribution's isotropy: the energy at which the electron behavior transitions from thermal to nonthermal (Landi et al. 2012). At energies above 1 keV the distribution is dominated by the noise and is thus isotropic. At slightly smaller energies, an increase of the isotropy might be observed, together with an increase of the strahl angular width. This effect may be related to a decrease of the turbulent scattering mean free path at high energies, which will be discussed in the following sections—but it must be taken with care on this particular figure, since the interpolation of the (E, θ) distribution function on the (v_∥, v_⊥) grid can produce artificial visual effects.

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Our goal is to determine the turbulent mean free path λ_turb from the data. In order to do so, we selected quiet intervals of study, in which the observed distribution functions will be fitted to the results provided by the transport model presented in Section 2, varying λ_turb as a free parameter. The intervals were selected manually, at different distances from the Sun, as periods were the magnetic field measured by the FIELD fluxgate magnetometer (Bale et al. 2016) is relatively constant and where the pitch angle distribution does not undergo strong fluctuations. Figure 8 presents an example of such an interval. Note that all the pitch angles shown in this section are "outward pitch angles," as defined in Section 2.3. All selected intervals are presented in Table 1 in Appendix C.

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For each of the selected intervals, the pitch angle distribution was averaged in time, normalized, and then compared to the results of the numerical integration of the transport equation. For this purpose, the numerical integration of Equation (4) was performed for 19 values of the turbulent mean free paths ranging in λ_turb = 0.01–3.5 au and a Parker spiral scale of r_* = 1 au. The integrations were performed using N = 10⁵ pseudoparticles initialized as an isotropic distribution at s₀ = 0.01 au, and the phase space distribution f(θ, s) was computed on a N_θ × N_s = 30 × 50 grid, corresponding to a resolution of Δθ = 6° and Δs = 0.06 au. This corresponds to a total of 19 × 50 = 950 numerically obtained pitch angle distributions. For each of those, the squared residuals defined as

$$\begin{eqnarray}&&{R}^{2}({\lambda }_{\mathrm{turb}},s)=\displaystyle \frac{1}{{N}_{\theta }}\displaystyle \sum _{i\,=\,1}^{{N}_{\theta }}{\left({\bar{f}}_{{\lambda }_{\mathrm{turb}},s}({\theta }_{i})-{\bar{f}}_{\mathrm{obs}}({\theta }_{i})\right)}^{2}\end{eqnarray} \tag{ 13 }$$

were computed. Here ${\bar{f}}_{{\lambda }_{\mathrm{turb}},s}({\theta }_{i})$ is the theoretical normalized distribution function obtained from Equation (4), and ${\bar{f}}_{\mathrm{obs}}({\theta }_{i})$ is the observed distribution. θ_i is the center value of the ith pitch angle bin. Since the bins of the simulations and the observations were not matching, the observed and simulated distributions were linearly interpolated on the same pitch angle grid for the purpose of calculating R². The best least-square fit corresponds, by definition, to the value of the parameters couple (λ_turb, s) that minimizes R².

The bottom panel of Figure 9 presents R²(λ_turb, s) in a color plot. One can note the existence of an "R² valley" in the (λ_turb, s) plane. This degeneracy stems from the fact that increasing λ_turb while keeping s constant or increasing s with λ_turb constant has a similar effect of increasing the width of the distribution function—more technically, as λ_turb is here a constant, Equation (4) can be expressed only as a function of the normalized distance σ = s/λ_turb and of the function Kn(σ). Because of this degeneracy, it will frequently occur that the absolute minimum of R²(λ_turb, s) occurs for a value of the position s not matching with the actual position of the spacecraft when the measurement was performed—inducing a strong error on the estimated value of λ_turb. Figure 9 presents a rather extreme example of such a behavior, using measurements from SolO/SWA/EAS. Here, the minimum of R² occurs at s ≃ 1.6 au and λ_turb ≃ 3.25 au, whereas SolO was at 0.9 au (shown by the vertical red line) when performing the measurement. In order to overcome this problem, the best value of λ_turb retained was the one minimizing R² at the position of the spacecraft during the measurement. The top panel of Figure 9 shows the distribution measured by SolO, together with the absolute best fit (black curve) and the best fit at SolO's position (red curve). The value retained here is λ_turb = 1.5 au. The similarity between the black and red curves illustrates the degeneracy just discussed.

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Figure 10 concludes this section on the fitting of the data by presenting data measured by PSP/SWEAP/SPAN-E in the same energy bin at three different distances from the Sun. The typical behavior is illustrated with an increase of the halo level, and a broadening of the angular distribution with increasing radial distance. All of these plots present an excellent agreement between the data and the solutions of Equation (4), on the whole 0°–180° pitch angle range, obtained by varying the only free parameter λ_turb. This leaves little doubt that the electron distribution functions observed in the solar wind are indeed determined by the processes described in Section 2: a competition between magnetic mirroring and a turbulent scattering mechanism acting even far away from the Sun, with a rather constant mean free path.

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Figures 11, 12, and 13 show the values of λ_turb determined using the fitting procedure described in Section 3.2. In Figures 11 and 12, λ_turb is plotted as a function of the energy in eight different intervals (four for each spacecraft), corresponding to eight different distances from the Sun. In Figure 13, it is plotted as a function of distance in four different energy channels, for all selected intervals. Since the energy channels of SolO/SWA and PSP/SWEAP do not match exactly, the closest channels were selected to produce the figure.

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We can observe that the turbulent scattering mean free path is of the order of the astronomical unit and does not depend strongly on the energy—the energy dependence observed close to the Sun at low energies being due to the non-negligible effect of the Coulomb collisions in the yellow-shaded region (as defined by Figure 6). A decrease in the mean free path seems to be occurring in the higher energy bins of several of the selected intervals. This effect will have to be confirmed and further studied in future works.

We also note some variability of λ_turb from one interval to another, even for intervals within close distance. This is particularly striking in the distance range from 0.5 to 0.8 au, where the distribution of λ_turb spreads from 0.1 to 3.5 au. This variability is likely to be explained by a dependence of the scattering mechanism on parameters specific to the solar wind flux tube in which the electrons are propagating, which can strongly vary from one interval to another. A parametric study of how λ_turb correlates to these parameters is beyond the scope of this paper but should be undertaken in the future. Parameters are known to influence the strahl pitch angle width, such as the plasma β (Berčič et al. 2019), the level of magnetic fluctuations (Pagel et al. 2007), the magnetic field amplitude, and solar wind speed (Owen et al. 2022), among others, should be considered.

This variability is illustrated in a particular manner by the events noted with a (*) in Table 1, as for instance the one recorded at 0.53 au by SolO, and appearing on the lower-mid panel of Figure 12. The turbulent mean free path derived goes in this case up to the maximum value retained for the numerical integrations λ_turb = 3.5 au, and saturates there for a few energy channels. Such an interval exhibits very weak diffusion and could be labeled as "scatter free"—as is done for solar energetic particle events, for which weakly diffusive behaviors are also commonly observed (Lin 1970, 1974). In this event, the mean free path clearly increases with energy from the lowest energies to somewhere around 400 eV: for such an event, the strahl angular width decreases with increasing energy up to around 400 eV, before increasing again. This is consistent with previous studies of the strahl pitch angle width, in particular the recent one by Berčič et al. (2019), where a decrease of the pitch angle width with energy is sometimes observed but where, more generally, the strahl angular width seems mostly independent of the energy (just as λ_turb is, for most intervals, independent of the energy in ours).