3.1. Ions and electrons parameters

The ions and electrons parameters are inferred from the sweep mode of the Langmuir probe, one of the main components of Cassini's RPWS consortium (Gurnett et al. Reference Gurnett, Kurth, Kirchner, Hospodarsky, Averkamp, Zarka, Lecacheux, Manning, Roux and Canu2004). The 5 cm spherical probe is mounted on an approximately $1.5 \, \textrm {m}$ boom and measures the current of the charged particles in two different modes, the voltage sweep mode and a high-resolution 20 Hz continuous mode (Gurnett et al. Reference Gurnett, Kurth, Kirchner, Hospodarsky, Averkamp, Zarka, Lecacheux, Manning, Roux and Canu2004). During the sweep mode, and for the considered ring-grazing orbits, the probe measures the plasma current across the bias voltage interval of ${\pm }$32 V every 48 s. From the current–voltage curve it is possible to derive the electron ($n_e$) and ion densities ($n_i$) as well as the electron temperatures ($T_e$) and the spacecraft potential ($U_{sc}$) (Fahleson Reference Fahleson1967; Wahlund et al. Reference Wahlund, André, Eriksson, Lundberg, Morooka, Shafiq, Averkamp, Gurnett, Hospodarsky and Kurth2009; Morooka et al. Reference Morooka, Wahlund, Eriksson, Farrell, Gurnett, Kurth, Persoon, Shafiq, André and Holmberg2011; Holmberg et al. Reference Holmberg, Wahlund, Morooka and Persoon2012; Shebanits et al. Reference Shebanits, Wahlund, Edberg, Crary, Wellbrock, Andrews, Vigren, Desai, Coates and Mandt2016; Holmberg et al. Reference Holmberg, Shebanits, Wahlund, Morooka, Vigren, André, Garnier, Persoon, Génot and Gilbert2017).

3.2. Dust parameters

None of the Cassini particles instruments was designed to directly measure sub-micrometre charged grains. The CDA could only detect particles of radii $a \gtrsim 1 \, \mathrm {\mu }$m (Srama et al. Reference Srama, Kempf, Moragas-Klostermeyer, Helfert, Ahrens, Altobelli, Auer, Beckmann, Bradley and Burton2006; Kempf et al. Reference Kempf, Beckmann, Moragas-Klostermeyer, Postberg, Srama, Economou, Schmidt, Spahn and Grün2008), and the RPWS electric antennae were only sensitive to charged grains larger than a few micrometres (Gurnett et al. Reference Gurnett, Kurth, Kirchner, Hospodarsky, Averkamp, Zarka, Lecacheux, Manning, Roux and Canu2004; Ye et al. Reference Ye, Gurnett, Kurth, Averkamp, Kempf, Hsu, Srama and Grün2014; Ye, Gurnett & Kurth Reference Ye, Gurnett and Kurth2016; Ye et al. Reference Ye, Kurth, Hospodarsky, Persoon, Gurnett, Morooka, Wahlund, Hsu, Seiß and Srama2018). Nevertheless, due to the absorption of the electrons by all the grains, including the nanoparticles, the existence of the nanometre-sized grains and their total charge densities ($z_dn_d$) can be inferred directly from the LP-derived ion and electron densities as $z_dn_d = n_i-n_e$ as a consequence from the charge neutrality (Wahlund et al. Reference Wahlund, André, Eriksson, Lundberg, Morooka, Shafiq, Averkamp, Gurnett, Hospodarsky and Kurth2009; Shafiq et al. Reference Shafiq, Wahlund, Morooka, Kurth and Farrell2011; Shebanits et al. Reference Shebanits, Wahlund, Mandt, Ågren, Edberg and Waite2013; Engelhardt et al. Reference Engelhardt, Wahlund, Andrews, Eriksson, Ye, Kurth, Gurnett, Morooka, Farrell and Dougherty2015; Shebanits et al. Reference Shebanits, Wahlund, Edberg, Crary, Wellbrock, Andrews, Vigren, Desai, Coates and Mandt2016; Morooka et al. Reference Morooka, Wahlund, Andrews, Persoon, Ye, Kurth, Gurnett and Farrell2018,  Reference Morooka, Wahlund, Hadid, Eriksson, Edberg, Vigren, Andrews, Persoon, Kurth and Gurnett2019). The general expression for the total dust density ($n_d$) of radius $a$ can be obtained from a power law differential size distribution of the dust (Yaroshenko et al. Reference Yaroshenko, Ratynskaia, Olson, Brenning, Wahlund, Morooka, Kurth, Gurnett and Morfill2009):

(3.1)\begin{equation} {\mathrm d}n_d = Ka^{-\mu} \,{\mathrm d}a, \end{equation}

where $a$ is the size of the grain in the interval [$a_\textrm {min},a_\textrm {max}$], $\mu$ is the power law index and $K$ is a normalisation constant. Then $n_d$ is given by

(3.2)\begin{equation} {n_{d} = \frac{q_e(n_e-n_i)}{4{\rm \pi} \epsilon_0 U_{sc}}\left(\frac{2-\mu}{1-\mu}\right)\left({\frac{a_{\rm max}^{1-\mu}-a_{\rm min}^{1-\mu}}{a_{\rm max}^{2-\mu}-a_{\rm min}^{2-\mu}}}\right)}, \end{equation}

where $q_e$ is the elementary charge, $\epsilon _0$ the vacuum permittivity and

(3.3)\begin{equation} K={\frac{q_e(n_e-n_i)(2-\mu)}{4{\rm \pi}\epsilon_0 U_{sc}(a_{\rm max}^{2-\mu}-a_{\rm min}^{2-\mu})}}. \end{equation}

Assuming a spherical shape for the dust particles, and considering that water ice grains are dominant in Saturn's rings (mass density of the water ice, $\rho =920 \, \textrm {kg}\,\textrm {m}^{-3}$), the mass density of the dust particles of radius ranging from $a_\textrm {min}$ to $a_\textrm {max}$, can be estimated as

(3.4)\begin{equation} {n_{d}m_d = \int_{a_{{\rm min}}}^{a_{\rm max}} \frac{4}{3}{\rm \pi}\rho K a^{3-\mu}\,{\mathrm d}a = \frac{\rho q_e(n_e-n_i)}{3\epsilon_0 U_{sc}}\left(\frac{2-\mu}{4-\mu}\right)\left({\frac{a_{\rm max}^{4-\mu}-a_{\rm min}^{4-\mu}}{a_{\rm max}^{2-\mu}-a_{\rm min}^{2-\mu}}}\right)}. \end{equation}

4.1. RPWS/LP data

Using the data collected by the RPWS/LP instrument (§ 3) during Cassini's F-ring-grazing orbits, we focus the analysis on the near-equatorial plane ($\vert \textrm {Z} \vert \lesssim 0.4\, \textrm {R}_{s}$, latitude $< 10^{\circ }$) and select the cases for which the inferred ion and electron density profiles were smooth (absence of large fluctuations) and were converging to similar values (as expected from quasi-neutrality for $\vert \textrm {Z} \vert \gtrsim 0.1\, \textrm {R}_{s}$). This constraint leads to five orbits: revs 253, 254, 266, 267 and 268.

In figure 1, we show the plasma data for rev 254 that has been discussed in detail in Morooka et al. (Reference Morooka, Wahlund, Andrews, Persoon, Ye, Kurth, Gurnett and Farrell2018). In figure 1(a), the green and blue curves represent the ion ($n_i$) and electron ($n_e$) LP-derived charge densities, respectively, plotted versus the vertical distance ($\textrm {Z}$) from the ring plane. Briefly summarising the observations from (Morooka et al. Reference Morooka, Wahlund, Andrews, Persoon, Ye, Kurth, Gurnett and Farrell2018), one can see that, away from the equator ($\vert \textrm {Z} \vert \gtrsim 0.1\, \textrm {R}_{s}$), the electron and ion number densities are very comparable and increase gradually for decreasing $\textrm {Z}$. A clear difference between $n_i$ and $n_e$, with $n_e < n_i$, can be noted close to Saturn's equatorial plane between around $-0.12 \, \textrm {R}_{s} < \textrm {Z} < 0.11\, \textrm {R}_{s}$ (grey shaded area), where $n_i$ increases sharply to approximately $200 \, \textrm {cm}^{-3}$ whereas $n_e$ remains lower at approximately $50 \, \textrm {cm}^{-3}$. The authors have shown that this deficiency in the electron number density was caused by the dominance of the negatively charged nanometre-sized grains in this region ($\vert \textrm {Z} \vert \lesssim 0.1\, \textrm {R}_{s}$) with a total charged dust density ($n_d$) of approximately $100\, \textrm {cm}^{-3}$. The dust number density inferred from the LP measurements is represented by the solid red curve. Using the same parameters as in Morooka et al. (Reference Morooka, Wahlund, Andrews, Persoon, Ye, Kurth, Gurnett and Farrell2018), $n_d$ is computed from (3.2) for a dust particle of radius $a$ varying from $a_\textrm {min}=1 \, \textrm {nm}$ to $a_\textrm {max}=100 \, \textrm {nm}$ and a power law distribution index $\mu =5$ (see Morooka et al. (Reference Morooka, Wahlund, Andrews, Persoon, Ye, Kurth, Gurnett and Farrell2018) for more details). The data in dashed lines are calculated assuming an exponential decay of the dust densities, $n_d=n_{d0}\exp (-\textrm {Z}/\textrm {H})^2$, where $n_{d0}$ represents the charged dust number density for $\textrm {Z}=0$ and $\textrm {H}=0.054 \, \textrm {R}_{s}$ is the estimated scale height of the nanometre-sized grains (Morooka et al. Reference Morooka, Wahlund, Andrews, Persoon, Ye, Kurth, Gurnett and Farrell2018). In the absence of dust ($\vert \textrm {Z} \vert \gtrsim 0.1\, \textrm {R}_{s}$), the plasma is quasi-neutral and the measured difference between $n_e$ and $n_i$ is due to the low signal-to-noise ratio of the LP in this region, where the plasma density ($\sim 10 \, \textrm {cm}^{-3}$) approaches the measurement limits of the LP (Morooka et al. Reference Morooka, Wahlund, Andrews, Persoon, Ye, Kurth, Gurnett and Farrell2018). Therefore, to avoid numerical errors from the noise in the derivation of $\boldsymbol {E_{\|}}$, we strictly enforce the quasi-neutrality, approximating the ion densities to be equal to the electron densities for $\vert \textrm {Z} \vert \gtrsim 0.1\, \textrm {R}_{s}$. These approximated ion densities ($n_{i,\,approx}$) are represented by the dashed yellow line. The modelled dominant molecular hydrogen and oxygen ion and neutral densities (Tseng et al. Reference Tseng, Johnson and Elrod2013) are given in figure 1(b) for reference. As expected, the neutral densities dominate close to the ring plane. In particular, $n_{\textrm {H}_{2}}$ reaches a maximum value of approximately $10^3 \, \textrm {cm}^{-3}$, about one order of magnitude larger than the oxygen ion ($n_{\textrm {O}_{2}^+}$) densities. The electron temperature ($T_e$, figure 1(c)) increases from approximately $0.5 \, \textrm {eV}$ to approximately $2 \, \textrm {eV}$ away from the equatorial plane, due to convection. The local increase at the equator is attributed to the presence of the negatively charged dust that induces electron heating in their potential well (Farrell et al. Reference Farrell, Kurth, Gurnett, Persoon and MacDowall2017; Morooka et al. Reference Morooka, Wahlund, Andrews, Persoon, Ye, Kurth, Gurnett and Farrell2018).

Figure 1. Plasma parameters for rev 254 versus the vertical distance from the ring plane ($\textrm {Z}$). (a) The LP-derived number density of the electrons ($n_e$) is shown in blue and the ions ($n_i$) in green. In order to enforce quasi-neutrality outside the dusty region, we set $n_i=n_e$ for $\vert \textrm {Z} \vert \gtrsim 0.1\, \textrm {R}_{s}$ shown in yellow dashed line as the approximated ion density ($n_{i,\,\textrm {approx}}$). The total dust densities ($n_d$, solid red line) is computed from (3.2) for a nanometre-sized particle of size $a$ varying from $a_\textrm {min}=1 \, \textrm {nm}$ to $a_\textrm {max}=100 \, \textrm {nm}$ and a power law index $\mu =5$. For $\vert \textrm {Z} \vert > 0.1\, \textrm {R}_{s}$ (dashed red line), $n_d$ is calculated assuming an exponential decay with a scale height value $\textrm {H}=0.054\,\textrm {R}_{s}$. (b) The modelled ions ($\textrm {O}_{2}^+$, $\textrm {H}_{2}^+$) and neutral ($\textrm {O}_{2}$, $\textrm {H}_{2}$) densities adapted from Tseng et al. (Reference Tseng, Johnson and Elrod2013). (c) The electron temperature inferred from the LP. The grey area highlights the region where the negatively charged dust dominate ($-0.12 \, \textrm {R}_{s} < \textrm {Z} < 0.11\, \textrm {R}_{s}$). The grey and black dashed lines represent respectively the magnetic and kronographic equators.

4.2. Ambipolar electrostatic field estimation

We start by calculating the parallel electrostatic field terms in the presence ($\boldsymbol {E}_{\parallel \textrm {tot}}$) and absence ($\boldsymbol {E}^{\prime }_{\parallel \textrm {tot}}$) of negatively charged dust for rev 254, as expressed in (3.5)–(3.7). In figure 2(a) and 2(b), we show the relative importance of the parallel electrostatic field terms versus $\textrm {Z}$. In the no dust case (figure 2a), close to the ring plane in the grey shaded area, the pressure gradient term ($\boldsymbol {E}^{\prime }_{\parallel p}$, in green) dominates over the gravitational and centrifugal terms ($\boldsymbol {E}^{\prime }_{\parallel g}$, in blue and $\boldsymbol {E}^{\prime }_{\parallel \textrm {cf}}$, in orange), whereas at higher latitudes, the centrifugal force $\boldsymbol {E}^{\prime }_{\parallel \textrm {cf}}$ takes over the other terms, as expected. On the other hand, in the dust case (figure 2b), the contribution of the gravitational ($\boldsymbol {E}_{\parallel g}$, in blue), and the centrifugal terms ($\boldsymbol {E}_{\parallel \textrm {cf}}$, in orange) are larger than the pressure gradient term ($\boldsymbol {E}_{\parallel p}$, in green) at any distance from the equator. It is worth noting that in this narrow equatorial region, the centrifugal force still dominates over the pressure gradient and gravitational forces, in agreement with the diffusive model of Persoon et al. (Reference Persoon, Gurnett, Santolik, Kurth, Faden, Groene, Lewis, Coates, Wilson and Tokar2009) for the heavy plasma species. More importantly, one can see that the pressure gradient force in the absence (figure 2a) and presence (figure 2b) of dust have similar magnitudes but opposite directions. This highlights the importance of taking into account the dust particles in the estimation of the parallel electrostatic field as discussed more in details in § 5.

Figure 2. Ambipolar electrostatic field terms of (a) $\boldsymbol{E}^{\prime}_{\parallel \textrm {tot}}$ without dust (3.7) and (b) $\boldsymbol {E}_{\parallel \textrm {tot}}$ with dust (3.5) plotted versus the distance from the ring plane $Z$, and (c) the corresponding total fields. The grey shaded area highlights the region where the negatively charged dust dominates ($-0.12 \, \textrm {R}_{s} < \textrm {Z} < 0.11\, \textrm {R}_{s}$). The grey and black dashed lines represent the magnetic and kronographic equators, respectively.

In figure 2(c), we show the estimation of the total parallel electrostatic field. In the absence of dust, $\boldsymbol {E}^{\prime }_{\parallel \textrm {tot}} \sim 10^{-7} \, \textrm {V}\, \textrm {m}^{-1}$ (consistent with previous modelling results of Maurice et al. Reference Maurice, Blanc, Prangé and Sittler1997) and is positive in the northern and negative in the southern hemisphere (black curve), or in vector terms, it is directed towards the kronographic equator, acting to remove the electrons from it. However, in the presence of dust, $\boldsymbol {E}_{\parallel \textrm {tot}}$ (red curve) is amplified by two orders of magnitude (approximately $10^{-5} \, \textrm {V}\, \textrm {m}^{-1}$, in the southern hemisphere) and exhibits an asymmetry around the magnetic equator, with larger values southward. This asymmetry with respect to the magnetic equator is consistent with the northward offset of Saturn's magnetic field around $0.047\, \textrm {R}_{s}$ (Dougherty et al. Reference Dougherty, Cao, Khurana, Hunt, Provan, Kellock, Burton, Burk, Bunce and Cowley2018) further enhanced by the ring-plane bound ions and dust grains. Furthermore, a clear change in the direction of $\boldsymbol {E}_{\parallel \textrm {tot}}$ can be noted inside the dusty region with respect to the magnetic equator: it is parallel (positive) for $-0.12 \, \textrm {R}_{s} < \textrm {Z} <0.047 \, \textrm {R}_{s}$ and anti-parallel (negative) for $0.047 \, \textrm {R}_{s} < \textrm {Z} <0.11 \, \textrm {R}_{s}$. Outside the dusty region $\boldsymbol {E}_{\parallel \textrm {tot}}$ converges to the same amplitude and sign of the no-dust case. In vector terms, in the dusty region ($\vert \textrm {Z} \vert \lesssim 0.11 \, \textrm {R}_{s}$), $\boldsymbol {E}_{\parallel \textrm {tot}}$ is directed away from the magnetic equator, acting to confine the electrons on it. That is reversed compared with the case of neglecting the presence of dust ($\boldsymbol {E}^{\prime }_{\parallel \textrm {tot}}$).

4.3. Ambipolar electrostatic potential estimation

Another way to illustrate the effect of $\boldsymbol {E}_{\parallel \textrm {tot}}$ on confining the electrons to the magnetic equator, is to plot the electrostatic potential with respect to the ring plane, $\varPhi =-\int _{z_0}^{z} \boldsymbol {E}_{\parallel \textrm {tot}} \,{\mathrm d} z$ with dust (figure 3a) and without dust (figure 3b). In the dust-free environment, $\varPhi$ would be characterised by a local peak close to the kronographic equator. The electrons are therefore in an unstable equilibrium and will be accelerated towards higher latitudes. However, in the presence of negatively charged dust, $\varPhi$ is characterised by a potential well centred on the magnetic equator with asymmetrical barrier, with much higher values southward ($\Delta \textrm {V} \approx 110 \, \textrm {V}$) than northward ($\Delta \textrm {V} \approx 10 \, \textrm {V}$). To put this into perspective, the electron temperature in the same region (figure 3b) is of the order of 1 eV, implying that the electrons cannot escape the potential well towards higher latitudes. This result puts constraints on the plasma influx from the F ring into the ionosphere, known as ‘ring rain’ (Northrop & Connerney Reference Northrop and Connerney1987; O'Donoghue et al. Reference O'Donoghue, Moore, Connerney, Melin, Stallard, Miller and Baines2019) for particles with energy $\lesssim 10 \, \textrm {eV}$ (or $\lesssim 110 \, \textrm {eV}$ south of the magnetic equator).

Figure 3. Electrostatic potential versus $\textrm {Z}$ (a) including and (b) excluding dust. (c) Electron temperature inferred from the LP (same as in figure 1). The grey area highlights the region where the negatively charged dust dominate ($-0.12 \, \textrm {R}_{s} < \textrm {Z} < 0.11\, \textrm {R}_{s}$). The horizontal grey and black dashed lines represent the magnetic and kronographic equators, respectively.

4.4. Statistical results

In figure 4, we show the total dust density (figure 4a), $\boldsymbol {E}_{\parallel \textrm {tot}}$ in the presence (coloured) and absence (black) of dust (figure 4b), and the corresponding electrostatic potential in the presence (coloured) and absence (black) of dust (figure 4c), for all the analysed orbits revs 253, 254, 266, 267 and 268. The dark grey area ($-0.12 \, \textrm {R}_{s} \lesssim \textrm {Z} \lesssim 0.11 \, \textrm {R}_{s}$) highlights the region within which $\boldsymbol {E}_{\parallel \textrm {tot}}$ is amplified for the lowest charged dust distribution (rev 254). The outermost light grey areas ($0.11 \, \textrm {R}_{s} \lesssim \textrm {Z} \lesssim 0.2 \, \textrm {R}_{s}$ and $-0.13 \, \textrm {R}_{s} \lesssim \textrm {Z} \lesssim -0.12 \, \textrm {R}_{s}$) indicate the extended regions where $\boldsymbol {E}_{\parallel \textrm {tot}}$ is amplified due to the variation of the dust density distribution. This highlights the direct effect of the charged dust densities on $\boldsymbol {E}_{\parallel }$: the larger the distribution of $n_d$, the more extended the effect of $\boldsymbol {E}_{\parallel }$ (e.g. rev 267). As one can see, the magnitude and direction of $\boldsymbol {E}_{\parallel \textrm {tot}}$ and the electrostatic potential are consistent for all the analysed orbits: revs 253, 254, 266, 267 and 268.

Figure 4. (a) Total dust density, (b) total $\boldsymbol {E}_{\parallel \textrm {tot}}$ in the presence (coloured) and absence (black) of dust and (c) the corresponding electrostatic potential in the presence (coloured) and absence (black) of dust, for all the analysed orbits revs 253, 254, 266, 267 and 268. The dark grey area ($-0.12 \, \textrm {R}_{s} \lesssim \textrm {Z} \lesssim 0.11 \, \textrm {R}_{s}$) highlights the region within which $\boldsymbol {E}_{\parallel \textrm {tot}}$ is amplified for the lowest charged dust distribution (rev 254). The outermost light grey areas ($0.11 \, \textrm {R}_{s} \lesssim \textrm {Z} \lesssim 0.2 \, \textrm {R}_{s}$ and $-0.13 \, \textrm {R}_{s} \lesssim \textrm {Z} \lesssim -0.12 \, \textrm {R}_{s}$) indicate the extended regions where $\boldsymbol {E}_{\parallel \textrm {tot}}$ is amplified due to the variation of the dust density distribution. The horizontal grey and black dashed lines represent respectively the magnetic and kronographic equators.

These results indicate the important role of the negatively charged dust grains (where present) in the motion and distribution of the plasma along the magnetic field lines. Because the ambipolar electric field is set up by the dominant light species, there is an important distinction between the dust-free and dusty plasmas. If there is no dust, $\boldsymbol {E^\prime }_{\parallel }$ is set up by the electrons and can be derived from the electron momentum equation (2.6). In the dusty case, the electrons are depleted by the dust grains and $\boldsymbol {E}_{\parallel }$ is instead set up by the positive ions and can be then derived directly from the ion momentum equation (2.5, discussed in more detail in § 5). As a consequence, one would expect the electrons to be pushed towards the magnetic equator, in the opposite direction of $\boldsymbol {E}_{\parallel }$, and the lighter ions away from the magnetic equator, in the direction of $\boldsymbol {E}_{\parallel }$. We note that no dust levitation effects were observed near the F ring. A likely reason is that the measured ambipolar electrostatic field is not strong enough to pull the charged dust off the ring plane. Nevertheless, on average the charged dust are observed between $-0.13 \, \textrm {R}_{s} \lesssim \textrm {Z} \lesssim 0.2\, \textrm {R}_{s}$, slightly shifted towards the northern hemisphere. However, to confirm this offset as the effect of the ambipolar electrostatic field, the current dataset is not sufficient.

This overall picture is in agreement with the LP measurements presented by Morooka et al. (Reference Morooka, Wahlund, Andrews, Persoon, Ye, Kurth, Gurnett and Farrell2018) (see figure 2 in their work) who showed that the ions are concentrated on the equatorial plane (below the magnetic equator), whereas the electrons are more dominant above the ring plane than below it. However, the reason for the electron density asymmetry with respect to the kronographic equator was not clear until now.

4.5. Indications of electron confinement towards the northern hemisphere

The confinement of the electrons and ions towards the magnetic and kronographic equators, respectively, is readily illustrated by the density profiles. The ion density profiles of included orbits are shown in figure 5 together with their respective inverted (mirrored) profiles with respect to the kronographic (figure 5a and 5c) and magnetic (figure 5b and 5d) equators. The electron density profiles are shown in the same fashion in figure 5(e)–5(h). It is clear that the peaks of the ion densities are centred on the kronographic equator and tend to be symmetric around it. Another way to show this, is to calculate the difference between the measured and mirrored profiles, and its averaged standard deviation (figure 5c and 5d), which are significantly smaller for the inversionaround the kronographic equator (figure 5c). The results from the electron measurements are not as convincing as the ion measurements, because no clear peak in the density can be detected. Nevertheless, the alignment of the profiles in figure 5(f) and the difference in the average standard deviation indicates that the electrons are confined above the ring plane at the magnetic equator.

Figure 5. (a), (b) Ion density profiles ($n_{i}$, solid lines) for all the analysed orbits compared with their mirror image ($n_{i,\, \textrm {inverted}}$, coloured dashed lines) with respect to the (a) kronographic equator and the (b) magnetic equator. (c), (d) Differences between the density profiles ($n_{i}$–$n_{i,\, \textrm {inverted}}$) for (c) the equatorial plane symmetry and (d) the magnetic equator symmetry. (e)–(h) Same as (a)–(d), respectively, but for the electrons. The averaged standard deviation is given by $\langle \sigma \rangle$. The grey area highlights the dusty region defined as in figure 4. The grey and black dashed lines represent the magnetic and kronographic equators, respectively.