SIXTEEN YEARS OF ULYSSES INTERSTELLAR DUST MEASUREMENTS IN THE SOLAR SYSTEM. III. SIMULATIONS AND DATA UNVEIL NEW INSIGHTS INTO LOCAL INTERSTELLAR DUST

Astronomical observations of the wavelength dependence of the extinction of starlight over kiloparsec distances, polarization and emission in the infrared, and complementary modeling led to several models of the size distribution and composition of ISD (Trumpler 1930; Mathis et al. 1977; Draine & Lee 1984; Weingartner & Draine 2001). These models generally show a power-law distribution with an upper limit on the biggest particles constrained by observational limits and by the abundances of the material in the gas phase (with the solar composition as a reference). From the depletion of elements in the gas phase measured in the absorption spectra of starlight, the ISD must contain mainly C, Mg, Si, Fe, O, and perhaps H (Draine 2004). Since we do not know exactly the local ISD size distribution and composition in the LIC, we take an (extrapolated) ISD distribution modeled by Mathis et al. (1977) as a reference power-law distribution in our studies (Sterken et al. 2013a). This is the so-called MRN distribution from Mathis, Rumpl, and Nordsieck. It is the most commonly used mass distribution, consisting of two particle populations with a power law of −3.5 and an upper limit of 0.25 and 1 μm for silicate and graphite, respectively. One must keep in mind that these models are based on observations over long kiloparsec distances whereas the LIC is very local, only a few parsecs. Thus, the size distribution of ISD in the LIC may look different from these models, and one of our goals is to come closer to a realistic original ISD size distribution in the LIC (Grün & Landgraf 2000).

Bertaux & Blamont (1976) predicted that ISD can enter the solar system and calculated the effect of the solar gravity and solar radiation pressure forces on the ISD flux. Levy & Jokipii (1976) predicted that these particles have a net electric charge, and therefore their trajectories would be influenced by the interplanetary magnetic field (IMF) via the Lorentz force. Based on this, they predicted that some of the particles with a high charge-to-mass ratio would be excluded from the solar system. Gustafson & Misconi (1979) and Morfill & Grün (1979) calculated ISD trajectories and concluded that there are phases of focusing and phases of defocusing of dust with respect to the solar equatorial plane. These phases depend on the 22 yr Hale cycle of the Sun.

In 1993, a major milestone in the local ISD research occurred when ISD in the solar system was unambiguously detected by the Ulysses dust experiment (Grün et al. 1993). The Ulysses dust detector is an impact ionization detector that measured the velocity and mass of the impacting particles through the velocity of the ions and electrons and the total impact ionization charge after an impact of a dust particle on the detector (Grün et al. 1992). The distinction between interstellar and interplanetary dust was possible owing to the exceptional orbit geometry of the Ulysses mission—out of the ecliptic plane and almost perpendicular to the incoming stream of ISD. Moreover, during the longest parts of the Ulysses orbit (only except for perihelion), the ISD came from the opposite direction of interplanetary dust, allowing an easier distinction between the two. Today, the Ulysses ISD data set from 1992 until 2008 is the most unique ISD data set, with over 900 detected ISD particles spanning almost a full Hale cycle, and is described in Krüger et al. (2015) for the mass distribution and Strub et al. (2015) for the ISD flux and flow direction. The flow direction is derived from the pointing direction of the instrument at impact.

These Ulysses data from 1993 until 2002 were thoroughly analyzed by Landgraf et al. (2000), and extensive modeling was done by the same author to better understand the data set (Landgraf 2000). Also Galileo—which carried a twin of the Ulysses dust detector—provided ISD data, along with Helios and Cassini (Altobelli 2004). An overview of papers related to in situ ISD observations and their data reduction is presented in Table 1 of Sterken et al. (2012).

From the fit of the modeled flux and the flux from Ulysses and Galileo data from 1992 to 1998, Landgraf et al. (2003) concluded that the bulk of ISD particles have sizes of about 0.3 μm, assuming compact astronomical silicates with density 2.5 × 10³ kg m⁻³ for the ISD material. Astronomical silicates are hypothetical materials with optical properties corresponding to those as determined from ISD by Draine & Lee (1984). The flux from 1992 until 1998 could be reproduced in simulations for particles with Q/m between 0.5 and 1.4 C kg⁻¹.

Sterken et al. (2012) followed up on this work and did Monte Carlo simulations of the ISD trajectories for a whole range of material properties with a similar model (averaged IMF; see Section 2.2) in order to better understand the general flow and filtering of ISD in the solar system. The IMF was updated to have the heliospheric current sheet (HCS) rotation equal to the observed HCS rotation, but the standard model (sinusoidal HCS rotation) was still used to analyze the flow of ISD. In these studies, especially the Lorentz force leads to unexpected concentrations of dust at certain places in the solar system, depending on time. The updated ISD model was later used as a basis for inclusion in the ESA Interplanetary Meteoroid Environment for eXploration (IMEX) model, which is an engineering model for calculating the time-variable ISD flow and cometary dust streams to prepare future missions (Grün et al. 2013; Sterken et al. 2013b; Strub et al. 2013).

The influence of the heliopause (HP) on the ISD flux was calculated by Linde & Gombosi (2000) for the 1996 solar minimum, and they found a strong filtering for ISD smaller than 0.1 μm. Also Slavin et al. (2010) predicted a strong filtering at the HP for an IMF configuration with outward field structure ("positive field") in the northern hemisphere. Slavin et al. (2012) simulated the flow of ISD throughout the heliosphere boundaries and the inner heliosphere for both configurations of the IMF, but the effect of the changing IMF throughout the travel time of the ISD particles in the solar system was not taken into account. They concluded that particles as small as 0.03 μm can cross the HP. Although the results do not reflect the final distribution of ISD in the inner HP because of the lacking time dependence of the IMF, the authors did demonstrate that at the HP also focusing can occur, and that smaller particles can penetrate the HP at favorable configurations of the outer IMF.

Krüger et al. (2010) further analyzed the data of Ulysses and discovered an unexpected shift of 30° in the latitudinal direction of the dust away from the ecliptic plane with respect to its initial flow direction in 2005. This initial flow direction is assumed to be co-aligned with the neutral helium gas flow direction, supported by the argument that the overall dust direction from observations was found to be compatible with the helium flow direction (Baguhl et al. 1995; Landgraf 1998; Frisch et al. 1999) and that the main component of the dust velocity is determined by the high relative velocity of the solar system and the LIC. Finally, Strub et al. (2015) and Krüger et al. (2015) analyzed the complete ISD data set of Ulysses from 1992 February until 2007 November. They found that the small particles even have a larger shift in flow direction up to 50° in 2005.

Four other means of obtaining knowledge on the local ISD in the LIC are observations in the infrared, meteoroid observations, impact signatures picked up by spacecraft antennae, and sample return missions. Grogan et al. (1996a, 1996b) modeled the time-variable thermal emission of ISD with the aim to search for its signature in the Cosmic Background Explorer (COBE) and Infrared Astronomical Satellite (IRAS) data. Rowan-Robinson & May (2013) modeled the IR emission from the zodiacal cloud, including a steady component of the ISD from observations made by IRAS and COBE. They did not include the variations in ISD density in their models and conclude that there is "a need for a ground-based kinematic study of zodiacal dust" to test for this. The Herschel and Planck space telescopes have observed thermal emission and its evolution of dust in nearby molecular clouds, including cold and dense clouds, and the gas. ISD is often used as a tracer of their magnetic fields. Herschel data are also currently used in a survey of dust in different distant galaxies (Boselli et al. 2010). Although this dust is very far away, the link between in situ dust measurements and observations in the LIC, molecular clouds, or even other galaxies is important to get a full picture of the role of the dust in the universe.

Interstellar meteoroids were reported by Taylor et al. (1996), Hajdukova (1994), and Baggaley (2000), but much discussion remained about their true origin, and the results depended very strongly on measurement uncertainty (Hajduková et al. 2014). A maximum of about 20 ISD meteoroids were found in the data of the Japanese multistation video meteor network. These are macroscopic particles, typically larger than a millimeter.

Also radio and plasma wave instruments on STEREO detected ISD impacts (Belheouane et al. 2012): when the particles impact the spacecraft or the antennae, a cloud of plasma is generated from the dust and spacecraft material, which is then detected by the antennae. Calibrations of this mechanism were performed in the lab with different spacecraft materials (Collette et al. 2014). ISD detections using spacecraft antennae were also confirmed with the Wind/WAVES instrument (Malaspina et al. 2014).

In 2006, the Stardust mission returned a collector tray with contemporary ISD particles to Earth and started the "Interstellar Preliminary Examination (ISPE)": an effort to find and extract the first ISD particles, caught more or less intact in the aerogel, and analyze them for their chemical composition, crystallinity, size, and density. Two particles were found at the end of the track in the aerogel, and one track showed particle residues on the track walls but no end particle (Westphal et al. 2014). In the aluminum foils surrounding the aerogel, four craters were found with residues of ISD particles. The particles were diverse in chemical composition and crystallinity, were about 1 or 2 μm in size, and had a surprisingly low density of <0.4 g cm⁻³ and 0.7 g cm⁻³, respectively, for two of the particles (Westphal et al. 2014).

ISD in the LIC is fairly well characterized using the above-mentioned methods. However, still many questions remain open: (1) Why are astronomical observations and models of the dust mass distributions different from mass distributions derived from in situ measurements? (2) How does this comply to the constraints on the total mass of the ISD derived from abundances in the gas phase with respect to the Sun? (3) What causes the shift in dust flow direction in 2005 as reported by Krüger et al. (2010, 2015) and Strub et al. (2015)? (4) What are the characteristics of the ISD from the local interstellar cloud? (5) Can we fit the mass distributions and fluxes from astronomical observations and models, and what would we then learn about the dust particle properties?

In this paper we show and explain the simulated ISD flow for the Ulysses mission using the model of Sterken et al. (2012) and discuss the questions above in the context of the simulations and the Ulysses in situ ISD data. This is part of three related papers that aim at better understanding the complete Ulysses ISD data set, for fluxes, velocities, directions, and mass distributions throughout the whole mission (Krüger et al. 2015; Strub et al. 2015). The simulation program and the ISD flow characteristics are described in detail in Sterken et al. (2012). The filtering or enhancement of the ISD flux and changes in its size distribution throughout time are described for several places in the solar system in Sterken et al. (2013a). Those two papers give the background knowledge that is applied in this paper to understand the predicted fluxes, flow directions, and mass distributions at Ulysses.

We review the dust dynamics and assumptions of the model from Sterken et al. (2012) in Section 2 and then describe briefly the simulation results at the orbit of Ulysses in Section 3. A detailed description of the simulation results is given in Appendix A. The Ulysses data are briefly introduced in Section 4, after which the simulation results are discussed in the context of the observations in Section 5. This discussion is guided by five main research questions related to (1) big ISD particles, (2) small ISD particles, (3) the shift in dust flow direction in 2005, (4) particle properties, and (5) the ISD size distribution. The big- and small-particle discussions cover the first and second science questions mentioned above. In Section 6 we elaborate on possible improvements from proposed simulation extensions and calibration experiments. The paper is summarized in Section 7, and a conclusion and outlook are given in Section 8.

Throughout the paper we distinguish "big" particles from "small" particles at m ≈ 2 × 10⁻¹⁶ kg following the division in the Ulysses data from Strub et al. (2015). "Very big particles" or "the biggest particles" are generally micron sized and above. Because the density of the ISD particles is not yet constrained, an overview of particle masses with corresponding sizes for different densities is given in Table 1.

The trajectories of charged ISD particles in the solar system are shaped by solar gravity F_grav, solar radiation pressure force F_SRP, and the Lorentz force F_L due to interaction with the IMF. The first two forces cause a steady axisymmetric ISD flow. They both scale with the squared distance to the Sun, and thus their ratio $\beta ={F}_{\mathrm{SRP}}/{F}_{\mathrm{grav}}$ can be seen as a constant ISD property that depends on particle morphology, density, material composition, and size. ISD particles with β = 1 move on straight lines "undisturbed" through the solar system. Interstellar particles for which the gravity is larger than the solar radiation pressure force (β < 1) move in hyperbolic orbits around the Sun and back into interstellar space, leading to a focusing cone of higher dust densities downstream from the Sun. ISD with β > 1 cannot travel closer to the Sun than a paraboloid-shaped zone owing to the high solar radiation pressure forces. Such particles cannot penetrate this so-called β-cone. Such a cone exists for each β-value, and thus studying the detection of ISD in different regions in the solar system allows us to learn about β and to indirectly constrain the particle properties. Figure 1 shows Ulysses's second polar orbit about the Sun and the approximate positions of these β-cones.

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The relationship between β and the particle size is called the "β-curve" and is determined by theoretical computations of scattering of electromagnetic radiation (Mie theory) or by experimental measurements (Gustafson 1994). Some examples of β-curves from Kimura & Mann (1999) and Draine & Lee (1984) are shown in Figure 2.

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The third force included in the ISD trajectory calculations is the Lorentz force acting on charged particles that move through a magnetic field. The IMF strength, direction, and the particle charge-to-mass ratio Q/m determine the strength of the Lorentz force. In the solar system, the charge-to-mass ratio ranges typically between 0 and 10 C kg⁻¹. Particles with higher charge-to-mass ratio are considered not to be able to penetrate the HP. The Lorentz force thus depends on the 22 yr Hale cycle, resulting in a focusing and defocusing of the ISD with respect to the solar equatorial plane. The flow is not axisymmetric anymore and nonstationary. Numerical models are needed to predict density and velocity.

The general flow characteristics of ISD in the solar system can be summarized as follows.

Very big (compact) particles (m > 5 × 10⁻¹⁵ kg) have low β < 0.5 and a negligible charge-to-mass ratio. They are concentrated near and downstream from the Sun (with respect to the flow direction) by gravitational focusing. Their flow is stationary and axisymmetric.

Small particles (m < 10⁻¹⁸ kg) also have low β-values (β < 0.5, for silicates) but have higher charge-to-mass ratios. The flow of these dust particles is time dependent and focused or defocused at the equatorial plane of the Sun or at higher latitudes depending on the phase of the 22 yr Hale cycle. In this paper, we refer to these two phases as "the focusing phase of the Hale cycle" when the particles are focused near the equatorial plane of the Sun and "the defocusing phase of the Hale cycle" when the particles are concentrated at higher latitudes. Very small particles cannot enter the solar system owing to the Lorentz effects at the HP and heliosheath (see Section 2.2.6).

Intermediate-sized particles can have β-values higher than unity, leading to a β-cone around the Sun in the flow direction where no such particles can enter. Depending on the charge-to-mass ratio of the particle and on the phase in the Hale cycle, this cone is modified in shape. By comparing Ulysses in situ measurements with modeling, the maximum β-value of ISD was determined to be between 1.4 and 1.8 (Landgraf et al. 1999).

Sterken et al. (2012) give a general overview of how the flow is modified and show several plots of the shape modifications of the β-cone and the focusing and defocusing in the solar equatorial plane and at higher latitudes above the equatorial plane.

Understanding the interaction between the ISD and the heliosphere is a complex problem where several assumptions have to be made. Some of the parameters can be held open and be constrained in an iterative process, while others are taken to be constant. Numerical simulations of the flow of ISD were performed under the following assumptions.

2.2.1. Forces and Simulation Grid

2.2.2. Particle Charge

2.2.3. Interplanetary Magnetic Field

The IMF in the simulations is averaged over one solar rotation like in the model of Landgraf (2000). This is a simplified but valid approach for particles not too close to the Sun or for particles that are not too slow, like ISD. Furthermore, a steady sinusoidal change in the tilt angle of the HCS with time is assumed. The model was improved to include the angles of the HCS as given by the Wilcox Solar Observatory (WSO; Hoeksema 2015), and also the nonaveraged magnetic field was implemented, but in this paper we use the simplest model of the IMF for reasons of computation time. The dates of solar maxima and minima for the simplified model and WSO data are summarized in Figure 3. The model where the HCS turns at a steady rate is depicted as a sinusoidal wave (three Hale cycles), and the focusing and defocusing phases of the Hale cycle, as well as the solar maxima and minima, are indicated. The HCS angles (from 0° to 90°) from WSO solar magnetic field observations are shown in the bottom plot of Figure 3. The solid and dotted lines depict two different computation methods from the raw data, where the solid line is the preferred method (see Hoeksema 2015 for details). Solar maxima, minima, and the dust phases (defocusing, focusing) are indicated for comparison with the idealized model used in our simulations.

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To illustrate the effect of the IMF model, we show in Figures 4 and 5 the simulations of the Ulysses data from 1992 to 2008 for the main population of ISD according to Landgraf (2000): β = 1.1 and Q/m = 0.6 C kg⁻¹. We do this for the three different models (average Sun, rotating Sun, WSO Sun). The simulations agree well with the Landgraf simulations. Differences between the three models exist, but the influence of a missing HP will most likely be much more significant.

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2.2.4. Dust Properties

2.2.5. Undisturbed Flow

2.2.6. Boundary of the Heliosphere

Classical ISD models derived from observations over kiloparsec distances (Mathis et al. 1977; Draine & Lee 1984; Weingartner & Draine 2001) typically have a cutoff in maximum particle size. This is 0.25 μm for silicates and 1 μm for carbon in Mathis et al. (1977). The maximum particle size is determined by observational constraints,⁵  by constraints in cosmic abundances (material observed in the gas phase cannot be in the dust phase, with the solar abundances taken as a reference), and by timescales of coagulation and shattering due to shock waves. However, single ISD particles with masses above 2 × 10⁻¹³ kg (>5 μm in diameter for compact particles and greater for porous particles) were measured with the Ulysses dust detector in the solar system (Krüger et al. 2015). Also the Helios data showed a large proportion of big interstellar particles (Altobelli et al. 2006). Apart from local variations in the dust size distribution as a proposed cause of these observations (Grün & Landgraf 1997), there is more recent astronomical evidence of micron-sized ISD particles (Gall et al. 2014; Wang et al. 2014, but not yet in the LIC), and also the Stardust mission returned two local ISD candidates with sizes of about a micron. The existence of low-density big particles helps to explain the observations and constraints (Dwek 1997; Li 2005), but it is not sufficient to explain the most massive ISD particles in the Ulysses ISD data set. Understanding the big particles is very important for determining the gas-to-dust mass ratio.

The simulations in this paper suggest that if these big ISD particles indeed exist, there should be an increase in the flux of big ISD particles around the perihelia by a factor of up to 1.5 owing to gravitational focusing (see discussion in Section A.1). These big particles (m_d > 5 × 10⁻¹⁵ kg) are barely affected by solar radiation pressure force as they have a very low β-value and very small charge-to-mass ratios, and so also the Lorentz force is negligible. The large proportion of big particles in the Helios data was attributed to the Helios orbit being inside of the β-cone for β = 1.1, where gravitational focusing of the big particles plays a role and smaller particles with β > 1 are depleted (Altobelli et al. 2006). Also at perihelia, the flux of these particles in the Ulysses frame increases by a factor of up to 2 or 2.5, caused, in addition to the previously mentioned factor (1–1.5), by Ulysses's increased velocity. Strub et al. (2015) show in their Figure 10 the so-called mass index, which indicates the extent to which the ISD flow is dominated by larger or by smaller particles. The division between large and small is around 2 × 10⁻¹⁶ kg. The mass index should thus increase around the perihelia because there is a larger concentration of big particles (low β) and less midsize or smaller ones (β > 1). However, the effect is not so clear, probably because still a large portion of big particles have β > 1. There are local maxima in the mass index closer to the perihelia in Figure 10 of Strub et al. (2015) near 1995 and 2007, even if large parts of the perihelia data were disregarded in the data analysis because of difficulties in distinguishing between interstellar and interplanetary dust (Krüger et al. 2015; Strub et al. 2015).

The big particles in the excluded data at the perihelia that are consistent with ISD directions and of high speed could possibly be ISD (Krüger et al. 2015; Strub et al. 2015). However, also very slow big particles were found around the perihelia (Strub et al. 2015), which may be caused by an unknown instrumental effect, or it could be cometary stream particles. The ESA-IMEX dust stream model can be used to further explore this topic, which is beyond the scope of this paper. We propose in Section 6.1 another possible explanation for the presence of very big ISD particles in the Ulysses ISD data set, which is based on the calibration of the dust instrument for compact versus fluffy or porous particles.

The mass distribution in Krüger et al. (2015, Figure 4) shows a large discrepancy between the mass densities of the smaller particles in the MRN model and the mass densities measured by Ulysses. This was understood in previous studies as a filtering by the solar radiation pressure force and the Lorentz force in the inner solar system and at the boundary regions of the heliosphere. The filtering in the inner solar system has been well characterized (Landgraf et al. 2000; Sterken et al. 2013a), but for the heliosphere boundary, the time-dependent effect of this filtering still needs to be implemented in studies like the ones by Slavin et al. (2012) and Linde & Gombosi (2000). Czechowski & Mann (2003) did the first such study and concluded that the alternating polarities of the IMF close to the HCS are favorable for particles to pass through the heliosheath.

The Hale cycle dependence of the flux of smaller particles is seen in the simulations and in the measurements and corresponds well to the analysis of Landgraf et al. (2003) if the bulk of the particles have indeed a 0.3 μm radius with β = 1.1 and Q/m = 0.6 C kg⁻¹. Landgraf et al. (2003) concluded on this "bulk" population of 0.3 μm particles with smaller components of 0.4 and 0.2 μm particles by comparing ISD flux simulations with the Ulysses data.⁶ Only the flux was used for fitting the simulations to the data, and it only covered the period from 1992 until 2002.

Now not only are there three times as much data (1992 until 2008, Krüger et al. 2015), but we also consider the direction of the ISD flow (Strub et al. 2015). The general trend in the Hale cycle is still present in the full Ulysses data set, but the details cannot be fitted yet for the whole data set without additional modeling, as we explain in Sections 5.3 and 5.4. Note that in the data, particles smaller than 10⁻¹⁷ kg are measured before 1995 and after 2003 (Strub et al. 2015). In the simulations, these correspond to Q/m between 5 and 12 C kg⁻¹ and even beyond.

In order to understand the shift in dust flow direction measured by Ulysses, we studied the simulations for 2005 in greater detail (see Figure 18). Details of these simulations are described in Section A.6. The ISD flow latitude and flux are plotted for three different days in 2005 for all possible combinations of β and Q/m in order to leave the particle material and properties unconstrained (Figure 18). From these plots, we found that it is possible to have big latitudinal changes in dust flow direction owing to the Lorentz force in 2005, and that these occur first for smaller particles (beginning of 2005) of Q/m = 5–6 C kg⁻¹ and subsequently for larger particles of Q/m = 2–3 C kg⁻¹, as was also seen in the data (Strub et al. 2015). We conclude that Lorentz forces on the ISD particles inside the heliosphere are thus a plausible explanation for the shift in rotation angle reported by Krüger et al. (2007) and Strub et al. (2015). However, only a full fit of the flow direction and the flux at all times would allow a confirmation.

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5.3.1. Shift in Dust Flow Direction of the Big Particles

5.3.2. Shift in Dust Flow Direction of the Small Particles

The comparisons of Ulysses simulations with the data from Section 5.3 give us constraints on the density of the ISD dust particles. As shown in Figures 14 and 15, the shift in rotation angle for big particles in the data can best be explained using simulations for Q/m ≈ 0.5 C kg⁻¹ until 2003 and Q/m ≈ 4 C kg⁻¹ thereafter. The big particles in the data have masses of m ≥ 2 × 10⁻¹⁶ kg (Strub et al. 2015), which corresponds to a charge-to-mass ratio Q/m ≤ 0.7 C kg⁻¹ for compact silicates with particle densities of ρ = 3.3 g cm⁻³. This charge-to-mass ratio corresponds quite well to the fit of the bulk of the particles with Q/m ≈ 0.5 C kg⁻¹ before 2003. However, the big particles also showed a shift in rotation angle in 2005 for which Q/m ≈ 4 C kg⁻¹ fitted best. Particles of such high mass⁸ can only have such high Q/m if their density is very low, i.e., ~0.015 g cm⁻³.

This sounds unreasonably low, but ISD particles are also unlikely to be spherical particles, and they rather will have irregular surfaces (irregular or like aggregates). Such particles have higher charges in comparison to spherical particles, and their charge-to-mass ratio increases by up to twice the original value (Auer et al. 2007; Ma et al. 2013). Thus, fluffy particles of mass m ≥ 2 × 10⁻¹⁶ kg with a density of ρ ≈ 0.015 g cm⁻³ now have Q/m ≤ 8 C kg⁻¹, and particles with a density of ρ ≈ 0.120 g cm⁻³ would have Q/m ≤ 4 C kg⁻¹ like in the simulations that fit best to the data after 2003. Table 2 summarizes these and the above-mentioned constraints on the particle density (see also Section 6.3).

The outcome of new calibrations of the dust instrument with fluffy or porous dust will further constrain the fluffiness or porosity itself, as is described in Section 6.3.

Note that also the material of the particle plays a role for the particle charge (Kimura & Mann 1998), and we only considered silicate particles with a nominal surface potential U = +5 V in this discussion. A slightly higher grain potential as calculated by Slavin et al. (2012; ~30% higher) would simply correspond to a slightly higher Q/m for the same density, but it still cannot explain the strong shift in dust direction.

In short, the fluffiness or porosity of the particles is currently unknown, and only the outcome of the impact ionization experiments with low-density projectiles (Section 6.1) and a fit of all simulated ISD fluxes and directions to the data at all times can fully constrain the density of the observed ISD particles, which will require adequate modeling of the heliosphere boundary regions to confirm the hypothesis in Section 6.2. However, from the comparison of the simulations and data of the big dust particles, we find strong indications toward low-density fluffy particles that at some times enter the inner heliosphere and at other times are blocked out at its outer boundaries.

Particle porosity not only increases Q/m for an ISD particle but also tends to widen and flatten the β-curve for the same material (Kimura et al. 1999). Figure 17 shows the effect of lower porosity or higher fluffiness on the simulated size distribution of silicate and carbon particles using the β-curves of Kimura & Mann (1999) for carbon and the "adapted astrosilicates" for the silica-based particles (see Figure 2). Figure 17 also shows the Ulysses data from Krüger et al. (2015). A higher porosity seems to make the simulated mass distribution fit better to the data than for compact particles, but this is mainly because the original size distribution contains less mass (in a first-order approximation).

Also the composition of the particles plays a role. As explained in Section 2.1, only particles with a β-value smaller than the β-value of the β-cone at the orbit of Ulysses can be detected. Since Ulysses is always within the β-cone of β ≈ 1.9, all silicate particles would be able to reach Ulysses, but not all carbon particles. Compact carbon particles with a β-curve as shown in Figure 2 can only be detected by Ulysses if they are bigger than m ≈ 2 × 10⁻¹⁶ kg.

Comparing the ISD size distribution from the simulations with the data thus further constrains the particle properties like composition or different populations, but also here a full fit of the data is required, and hence extended simulations and instrument calibrations are needed. ISD populations of different materials and properties are briefly commented on in Section 6.4.

Impact ionization detectors like the ones on-board Ulysses or Cassini have always been calibrated using compact materials like iron, carbon (Göller & Grün 1989), or even platinum-coated (Hillier et al. 2009) or polypyrrole-coated minerals (Hillier et al. 2014). It was estimated theoretically that a 50% porosity particle with radius 0.3 μm, impacting at 26 km s⁻¹, would create a bigger charge in the plasma cloud that increases the derived particle mass by one order of magnitude as compared to a compact particle of the same mass (K. Hornung 2012, private communication; Sterken et al. 2012). Experiments are currently ongoing to verify this effect (Sterken et al. 2013b), which may imply a recalculation of the Ulysses ISD mass distribution. Especially the mass of the biggest particles that determine the gas-to-dust ratio would be reduced if this prediction can be verified by experiments and if the particles are porous or fluffy. The latter assumption is reasonable, since Stardust returned to Earth three contemporary ISD particles that showed surprisingly low densities (Westphal et al. 2014), and fluffy or porous ISD would help to match the ISD observations to the constraints from cosmic abundances. Moreover, the simulations in this paper also suggest low particle densities (see Section 5.4).

The data and simulations discussed in Section 5.3 tell us that if the Lorentz force is responsible for the shift in dust flow direction, a mechanism of "heliosheath filtering" is required that allows particles with high Q/m to pass through—or not—at certain times that are not necessarily in phase with the focusing or filtering in the inner heliosphere. These particles with higher Q/m can be small particles, porous particles, or fluffy aggregates. We illustrate this in Figure 19 and explain the mechanism in detail in Appendix B. From the qualitative analysis in Appendix B, we conclude that—if proven by detailed simulations—there are long periods where the smaller or fluffier particles cannot reach the inner solar system, and shorter periods, like between roughly 2005 and 2010, when they can. Therefore, the bulk size of the particles is found to vary according to the conditions of the Hale cycle: whereas Landgraf (2000) found a bulk particle size of 0.3 μm (assuming compact silicates) from the first half of the data, Strub et al. (2015, Figure 10) show that smaller particles make up the bulk of the ISD flow after 2003.

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An extra modulation in flux because of the heliosheath region shall thus indeed be taken into account in future studies, and a detailed simulation of the Lorentz forces in the heliosheath and across TS and HP is therefore needed.

If the impact charge of a porous or fluffy dust particle after impact on a dust instrument is indeed higher than after impact of a compact particle, this would further constrain the fluffiness or porosity of the particles themselves (see Section 5.4): the estimated low density would increase, as is the case for a nonspherical particle shape. For instance, if twice as much impact charge is released upon impact, then the estimated density of the particles is ρ ≈ 0.500 g cm⁻³ for Q/m ≤ 4 C kg⁻¹, which corresponds well to the densities of the particles derived from the two semi-intact Stardust-returned ISD samples, which had densities ρ < 0.4 g cm⁻³ and ρ = 0.7 g cm⁻³, respectively (Westphal et al. 2014).

Table 2 summarizes the constrained density values for different assumptions of (1) the particle fluffiness, thus the particle potential U related to the shape of the particle (aggregate, elongated, or spherical), and (2) the particle porosity or density, related to the ratio of total impact charge of fluffy particles versus compact particles upon impact on the instrument.

ISD may consist of a mix of materials (silicates, carbons), either combined in one particle or in two or more different populations. Also, their fluffiness or porosity can differ or be size dependent. A sum of two distinct populations of silicates and carbons is a possibility and could help to explain why the slope of the mass distribution has a "knee" around m = 6 × 10⁻¹⁶ kg: in addition to Lorentz force, the solar radiation pressure force keeps away (compact) carbon particles smaller than m ≈ 2 × 10⁻¹⁶ kg from Ulysses's orbit.

In Figure 6 we show the simulated density and flux in the Ulysses frame for the biggest particles (m_d > 5 × 10⁻¹⁵ kg, β < 0.1). These are normalized to the undisturbed flow, so the actual size distribution is not taken into account here. The size distributions are discussed in Section A.5. The simulated ISD flux for these particles increases at perihelion by a factor of 2–2.5, and the density increases by a factor of 1–1.5. As discussed in Section 2.1, particles with low β-values are concentrated downstream and near the Sun because of gravitational focusing. This effect is only moderate because Ulysses at perihelion is still 1.3 AU away from the Sun and is not downstream but rather "sidestream." The increase in simulated ISD flux is thus mainly caused by the increased velocity of Ulysses at perihelion and only very little by gravitational focusing. For those particles with low β and very low charge-to-mass ratios, the flux at Ulysses depends only on the Ulysses position and on its relative velocity with respect to the ISD. The simulated fluxes are equal at every orbit, which means that Ulysses should have been able to detect slightly more big and mainly fast particles (V_imp up to 50 km s⁻¹) at the perihelia than during other phases of its orbit.

In Figure 7 we show the normalized flux in the Ulysses frame for different particle parameters that roughly correspond to different particle radii of 0.7, 0.4, 0.3, and 0.2 μm. Note that these sizes are only rough indications for particle size, as we discuss in this paper the particle properties independently from the β-curve. The normalized flux is the enhancement or reduction factor in the flux with respect to the undisturbed flow. Here we take into account the relative velocity of Ulysses with respect to the ISD. The parameters β and Q/m are chosen for particle sizes along the β-curve as discussed in the assumptions (Section 2.2).

The purple line (β = 0,5, Q/m = 0.12 C kg⁻¹, a ≈ 0.7 μm) in Figure 7 has similar peaks in the ISD flux at the perihelia to those described for the biggest particles with negligible surface charge (see Figure 6), but even with the low value of $Q/m$ = 0.12 C kg⁻¹, the effect of the Lorentz force is already apparent: the flux at the second perihelion (2001) is reduced a bit in comparison to the two other perihelia in 1995 and 2007 because of the defocusing phase of the Hale cycle.

Reducing the particle size to a ≈ 0.4 μm along the assumed β-curve yields a particle with β = 1 and Q/m ≈ 0.5 C kg⁻¹. The light blue curve in Figure 7 is thus only determined by the relative velocity of Ulysses with respect to a stream of ISD with constant velocity of 26 km s⁻¹ and by the Lorentz force. All noncyclic changes are due to this varying Lorentz force. The trend is that during the defocusing phase of the Hale cycle (1991–2002 in this model), the flux is reduced with respect to the "zero-charge-to-mass" case. During the focusing phase (2002–2013), it is enhanced—with a time lag of a few years (Sterken et al. 2012). This effect is stronger for larger charge-to-mass ratios and thus for smaller particles. Around perihelion, the flux for these particles still shows a peak due to the increased relative velocity of Ulysses, but the differences in the flux between the focusing and defocusing phases are up to a factor of 2–2.5 at the perihelia owing to the Lorentz force only. Including the different relative velocities of Ulysses with respect to the ISD flow instead of the ecliptic frame yields a ratio of more than 10 in impact flux between, e.g., 1998 (aphelion + defocusing phase) and 2007 (perihelion + focusing phase). The lowest flux is seen in the simulations in 1998 for all particle sizes.

Particles with radius a ≈ 0.2 μm (β = 1.5 and Q/m = 1.5 C kg⁻¹) show even more variability owing to the Lorentz force, but they also show a "gap" in the flux at the perihelia. This is because Ulysses moves through the "β-cones" that were introduced in Section 2.1 (see also Figure 1). Particles with β = 1.3 do not show complete depletions because through these regions Ulysses only moved during a fraction of its orbit. Furthermore, the simulations were made using large binning boxes of 1.5 AU, which smoothen the results, especially at perihelion, where 1.5 AU boxes are very large. The very small particles (Q/m > 1.5 C kg⁻¹) are strongly reduced in number between 1997 and 2004, but they may as well not pass through the heliospheric boundary regions (Levy & Jokipii 1976; Linde & Gombosi 2000; Czechowski & Mann 2003; Slavin et al. 2012; see also discussion in Section 6.2).

The relative fluxes for all β-values and charge-to-mass ratios give an impression of the variability of the flux for all parameters—independently from the assumed β-curve. Except for β = 1.5 and β = 2, the fluxes are influenced primarily by the Lorentz force (and the Ulysses velocity if in the Ulysses frame) and less so by β. The higher the charges are, the less easy it is to understand the flux variations. Some of these can be due to "mirroring" effects upstream from the Sun (Sterken et al. 2012), especially for particles with Q/m = 3 C kg⁻¹, which makes the fluxes less intuitive to understand. The small particles then move in complicated patterns, sometimes upstream, and cause high concentrations of dust on locations in the solar system that one would not expect. Note the increase in relative particle flux around 2005 for Q/m = 1–3 C kg⁻¹ and the very high fluxes of particles with the same Q/m before 1995. These fluxes in particular are discussed in Section 5.3 in the context of the "shift of dust direction" and "HP" discussions. Also note that the smaller the particles, the more abundant they are in the undisturbed ISM because of the power-law size distributions from astronomical observations and modeling. The variability of the smallest particles has thus significantly more influence on the total dust flux at Ulysses.

Most of the simulated ISD impact velocities on Ulysses are between 20 and 50 km s⁻¹, which is important to know for the selection criteria of the data (Krüger et al. 2015; Strub et al. 2015) and important for improving the mass calibration of the instrument assuming the simulated ISD velocities instead of a constant velocity of 26 km s⁻¹. Krüger et al. (2015) report an average ISD flow velocity of 24 ± 12 km s⁻¹. It must be noted that this is the averaged velocity of the ISD trajectories modulated by the forces acting on the particles in the heliosphere, and not the undisturbed ISD velocity in the LIC with respect to the heliosphere. Depending on the properties of the bulk of the particles, the undisturbed ISD velocity will be slightly lower or slightly higher than the averaged impact velocity reported in Krüger et al. (2015).

Figure 8 shows the ISD velocities in the ecliptic frame from the simulations for four sets of β-Q/m parameters representative for declining particle sizes. The impact velocities (ISD velocity with respect to Ulysses) depend also on Ulysses's velocity, but the ISD velocities in the ecliptic frame depend only on its position in its orbit. We chose to show the velocities in the ecliptic frame because they give a clearer view on the influence of particle size (and thus parameters β and Q/m) on the ISD particle velocities. The velocities of particles with low β are higher, especially close to perihelion, whereas impact velocities of particles with β > 1 are lower, especially at the position of Ulysses close to (but outside of) the corresponding β-cone. The influence of the Lorentz force on the ISD velocity (in the ecliptic frame) is visible in Figure 8 in the light blue curve for β = 1, Q/m = 0.5 C kg⁻¹. This effect gets stronger for smaller particles (light green and red colors, Q/m = 1 and 1.5 C kg⁻¹).

Similar to the flux discussed in Section A.1, the changes in latitude for the very big particles (β = 0.1, Q/m ≈ 0 C kg⁻¹, m > 5 × 10⁻¹⁵ kg) depend only on the position and the velocity of Ulysses with respect to the ISD, and thus the pattern of directional variability repeats for every orbit. Figure 9 shows the latitude and longitude of the ISD flow in the ecliptic frame in order to show more clearly the differences between different particle parameters later throughout this paper and to be able to better understand these changes in latitude and longitude when considering the position of Ulysses in the flow of ISD (see Figure 1). The biggest changes of the "big-particle"–ISD flow direction are about 30° in latitude, peaking when Ulysses crosses the poles of the Sun, and 25° in (ecliptic) longitude, peaking at the perihelia themselves (see Figure 9). The ISD flow latitudinal direction increases before perihelion, flips around perihelion, and increases again until the next perihelion, whereas the longitude has a minimum value at aphelion, increases until perihelion, and then decreases again. At aphelion, far away from the solar radiation pressure and gravity from the Sun, the flow direction is close to the undisturbed flow direction toward −8° latitude and 79° longitude (thus from 7° and 259° latitude and longitude).

Figure 10 shows these changes for different particle properties: the smaller the particle, the larger the variability and the further the directional changes in flow direction extend toward the aphelia. The changes in latitude and longitude for gravity and solar radiation pressure only (Q/m ≈ 0 C kg⁻¹) are in opposite directions for β < 1 and β > 1. To understand the influence of the Lorentz force, we discuss the case for m ≈ 5 × 10⁻¹⁶ kg (β = 1, Q/m = 0.5 C kg⁻¹) in Figure 10.

We first discuss the latitudes:

Before the perihelion of 1995, when Ulysses is southward from the Sun, it finds itself in the defocusing phase of the Hale cycle. At that position, the latitudes of the ISD are shifted southward caused by the defocusing. At perihelion, this flips northward, as Ulysses is passing toward the northern hemisphere of the Sun, where the particles are deflected northward. This effect gets stronger with time, because the maximum of the modeled defocusing phase is in mid-1996. The maximum effect of this can be expected about 1–2 yr after the solar minimum of mid-1996, as can be seen in the curve. The latitudes decrease again toward the aphelion of 1998, where they should flip again southward owing to the defocusing phase of the Hale cycle, but not as much as before because Ulysses is much farther away from the Sun, and the effect of the defocusing decreases toward the solar maximum of mid-2001. Around that time, in 2001, Ulysses passes its next perihelion, where the latitudes flip again from southward to northward, just like at the first perihelion, but not as severe because of being closer to solar maximum. In that period, between solar maximum and the aphelion of 2004, there is not so much of an effect as the net Lorentz force is not so strong and Ulysses is again far away from the Sun. However, in 2007, Ulysses passes through perihelion while there is a solar minimum of the focusing phase. This means that before crossing the perihelion, the particles are deflected northward with respect to the undisturbed case (as can be seen between 2005 and 2007), and after the perihelion, they are directed southward as a result of the focusing. The effect is very strong, as Ulysses is at the perihelion during the solar minimum of the focusing phase. Had Ulysses been at this perihelion 1 yr later, this effect would have been even larger.

For the longitudes a similar reasoning can be applied: we first consider again the case for β = 1 and Q/m = 0.5 C kg⁻¹ in Figure 10 and recall from the general characteristics of the flow of dust (Sterken et al. 2012) that in the focusing phase of the Hale cycle, there is a slight defocusing for the longitudes and in the defocusing phase there is a slight focusing in the longitudes—contrary to the latitudes of the ISD flow directions! Before crossing the south pole of the Sun in mid-1994 (defocusing phase of the Hale cycle close before solar minimum), the ISD has a longitude smaller than its nominal longitude of 79° as it is directed slightly toward the Sun in longitudinal direction. In the orbit section between the south pole and the north pole of the Sun, this is the opposite and the longitudes are larger than 79°, again because the dust moves slightly into the direction of the Sun in longitudinal direction. As Ulysses approaches aphelion in 1998, the dust direction approaches its nominal longitude of 79°. The second perihelion (mid-2001) shows the same pattern in longitude as the first, but the effects are less severe, as the solar maximum between the defocusing and focusing phases is in 2002. At aphelion there is again little effect. Finally, at the last perihelion in mid-2007, there is a strong effect from the combination of perihelion with the solar minimum of the focusing phase of the Hale cycle. This leads to a defocusing in the longitudinal directions, and hence, before crossing the south pole of the Sun, the ISD longitudes are larger than 79°. This direction flips at the south pole of the Sun toward values smaller than 79°, and then it flips again at the north pole of the Sun toward larger than 79° (always away from the Sun in longitudinal direction). The same trends as described here are visible in any curve for β = 1 and Q/m > 0 C kg⁻¹.

For a combination of gravity, solar radiation pressure force, and Lorentz force, e.g., particles with β = 1.5 and Q/m = 1.5 C kg⁻¹ (about 0.2 μm for compact silicates), there are large changes in the ISD flow direction: at perihelion up to 60° in latitude and 40° in longitude from the nominal ISD direction. From the collection of simulation results made for this work, for different β and Q/m, and looking at the resulting flow directions, it turned out that the Lorentz force plays a major role, especially for particles with Q/m > 1 C kg⁻¹. Note that the flow direction latitudes near the perihelia for β > 1 are much bigger than for β < 1. The biggest shifts in dust directions occur around the perihelia. Also a significant shift in latitude of the flow is present in 1995–1997 and in 2005–2007 owing to the Lorentz force (see discussion in Section 5.3) for particles with higher charges (aggregates or smaller particles). Especially the period around the last perihelion is very interesting for further study.

In order to compare the simulations to the data of the Ulysses dust instrument, we need to transform ISD latitude and longitude in the ecliptic frame into directions in the spacecraft reference frame, and finally into the "rotation angle." This is the difference in angle between when a dust impact occurs and when the dust instrument points closest to the ecliptic north pole. The latter corresponds to 0° rotation angle, and a dust detector pointing parallel to the ecliptic plane corresponds to a rotation angle of 90°. A more detailed description of the spacecraft geometry and rotation angle is found in Grün et al. (1993), Landgraf (1998), and Strub et al. (2015).

It is largely the difference in latitude from the nominal latitude of the flow direction of the ISD (−8°) that determines the corresponding "shift in rotation angle" of Ulysses from the nominal rotation angle at impact of ISD. Figure 11 shows the varying rotation angle shifts for the same sizes of particles as shown in Figure 7. The advantage of a representation of the simulated ISD directions in "rotation angles" is that they are directly comparable to the data as they also take into account the velocity of Ulysses with respect to the ISD flow. The disadvantage is the loss in insight of how the ISD flows. Nevertheless, from Figure 11, the differences in rotation angle from the nominal rotation angle for particles with higher Q/m around 1996 and 2006 are well recognizable and correspond with the previously noticed shifts in latitude in the same periods. This is further discussed in Section 5.3.

The Lorentz force tends to widen the flow of ISD, especially in latitude (Sterken et al. 2012), and so small particles should experience a bigger spread of flow directions at certain times. Theoretically, there is a smaller width a few years after solar maximum and more variety (wider stream) a few years after solar minimum. The width is also smaller at the aphelia than at the perihelia. This is not only due to Lorentz force but also due to smaller effects from the solar radiation pressure force and gravity farther away from the Sun. Similar trends are also seen in the data (Strub et al. 2015) but are not very significant.

In this section we show the variations in ISD mass and size distribution at Ulysses orbit throughout the Hale cycle as derived from simulations. For this, we make two important assumptions: first, an initial undisturbed static flow with homogenous mass distribution is assumed, and second, a certain β-curve that relates particle mass with the sensitivity of the particles to solar radiation pressure force ($\beta ={F}_{{\rm{SRP}}}/{F}_{{\rm{grav}}}$) is chosen. The assumed undisturbed mass distribution is the MRN power-law distribution (Mathis et al. 1977), extrapolated to smaller and larger particles than in the MRN model. The reference β-curve used in this work is the curve for astronomical silicates, multiplied by a factor of 1.2 to have a maximum β of 1.6 that corresponds to the maximum β derived from Ulysses data of Landgraf et al. (1999), which is between 1.4 and 1.8, as already discussed in Section 2.2.4.

The left plot of Figure 12 shows the resulting mass distribution as a function of time. This plot illustrates the importance of the biggest particles for the total mass of the ISD flow, being equal to the integral under the mass distribution curve. The middle plot in Figure 12 shows the simulated number density distribution based on the assumption of an initial extrapolated MRN distribution. From this plot we learn that even if smaller particles are depleted with respect to the normal undisturbed flow, they still can outnumber the larger particles in total amount, unless they would get extra filtering at the HP. Figure 12 (right panel) shows how much the number of ISD particles is enhanced or reduced at Ulysses orbit with respect to the undisturbed case from 1993 until 2008 (f = 1, where f is the enhancement or reduction factor with respect to the undisturbed flow). The fluxes are given at one single day as indicated in the plot. The biggest masses show less variability than the smaller ones owing to the Lorentz force, and there is a noticeable dependence on the Hale cycle.

In order to better understand these fluctuations in the mass and size distributions, we plot them for the aphelia and perihelia of Ulysses in Figure 13, where the perihelia are in blue and the aphelia are in red. The mass distributions are shown for the average over the month in which the aphelia and perihelia took place. The plots show that there are indeed more "big" particles at the perihelia than at the aphelia, as was already mentioned in Section A.1, but also that the amount slightly depends on the phase in the Hale cycle (the particles were assumed to have Q/m = 0.125 C kg⁻¹).

In 1992, Ulysses is at aphelion, far away from the β-cones, and 1992 is only 1 yr after the solar maximum between the focusing and the defocusing cycle (see Figure 3). Since we are close to solar maximum (hence, the IMF averages out over one solar rotation and has little effect) and since the effect of the Lorentz force is most prominent only a few years after the actual phase of the Hale cycle¹¹ (Sterken et al. 2014), the enhancements or reductions on the fluxes are only moderate. In 1995, just before the solar minimum of the defocusing phase, the influence of the defocusing phase of the Hale cycle is already well visible as the particles with masses around 10⁻¹⁶ kg or less are reduced in flux. Moreover, the β-cone causes a gap in the flux between masses of about 10⁻¹⁷ and 10⁻¹⁶ kg (β > ≈1.2). However, still some of the particles that are even smaller (low β, around 10⁻¹⁷ kg) get through. In 1998 the effect of the defocusing phase of the Hale cycle is even bigger as the solar minimum of the defocusing cycle was about 1.5–2 yr before. There is a strong reduction in the flux for small particles, and particles smaller than about 8 × 10⁻¹⁷ kg do not even reach Ulysses anymore at aphelion. The orange curve for 1998 in Figure 13 is thus not related to the β-cones in any way. In mid-2001, at the next perihelion, there is still a strong reduction of small particles, this time also partially caused by the β-cone. In mid-2004, at the next aphelion, the strong effect of the defocusing phase has reduced in strength because from 2002, the next focusing phase of the modeled Hale cycle has started. Also, there is no β-gap as Ulysses is at perihelion. Finally, in 2007, there is an increase again in number of particles, although many midsized particles are filtered through the solar radiation pressure forces at perihelion. These plots show that also the mass distributions follow the logics of the flow of ISD, although some details remain difficult to understand, e.g., why the smallest particles (about 3 × 10⁻¹⁸ kg) are sometimes more abundant than the midsized particles (about 10⁻¹⁷ kg) at the aphelia.

In order to understand the reported shift in rotation angle of the Ulysses spacecraft for ISD impacts around 2005, we plotted the ISD flow latitudes in the ecliptic frame and the fluxes in the Ulysses frame for all particle parameters β and Q/m, essentially leaving the choice of the β-curve open (Figure 18). We explore the shifts in the latitude of the ISD flow because mainly the latitude determines the rotation angle at the time of a dust detection at the aphelion in 2005, since Ulysses's antenna points toward Earth and therefore lies in the ecliptic plane. We plot this for three dates in 2005: day 91 (Figure 18, top), day 273 (middle), and day 319 (bottom). The white region represents the undisturbed values for the flow direction (−8° in ecliptic latitude) and the normalized flux (in the Ulysses frame). The fluxes are only important to cross-check if the particles that are shifted in flow direction are not completely filtered out at that moment.

In 2005, day 91, the biggest dust flow latitudes seen are 36° and more for particles with Q/m around 5–6 C kg⁻¹ (Figure 18, top), meaning that the flow has shifted from its nominal direction (−8°) by more than 44° for these particles. The normalized fluxes with respect to Ulysses for these particles seem small (0.2–0.4 times the undisturbed flux) but nevertheless have a big influence on the total ISD flux because—assuming that these small particles can traverse the heliosphere boundary regions—the original size distribution is most likely a power law (see also Section A.5). Particles with a higher charge-to-mass ratio are (in the simulations) filtered out of the solar system.

On day 273 in 2005 (Figure 18, middle), the biggest latitudinal dust flow directions in the simulations are up to 28° latitude for particles with Q/m around 2–3 C kg⁻¹, corresponding to a shift of about 36° in latitude with respect to the undisturbed flow. The flux at Ulysses in the Ulysses frame is about equal to the undisturbed ISD flux (in the Ulysses frame). The simulated dust direction thus seems to have occurred for smaller particles first (Figure 18, top, Q/m = 5–6) and then for larger particles (Figure 18, middle, Q/m = 2–3 C kg⁻¹), as was also reported by Strub et al. (2015) from the data analysis. Finally, the bottom panels in Figure 18 show the latitudes and relative flux for the end of year 2005, day 319. The biggest latitudes are up to 32°–36°, corresponding to a shift of up to 40°–44° for the same particles with Q/m around 2–3 C kg⁻¹. The relative flux varies between 0.5 and 1.2 times the undisturbed flux, depending on the β-value of the particles.

These plots show that in principle it is possible to have high latitudinal changes in dust flow direction due to the Lorentz force in 2005, as was seen by Ulysses. The Lorentz forces in the solar system are thus a plausible explanation for the shift in rotation angle reported by Krüger et al. (2007) and Strub et al. (2015).