© The Authors 2025

1. Introduction

Impacts of interstellar dust (ISD) grains on the Ulysses spacecraft allowed for the first successful in situ measurement of ISD (Grün et al. 1993). Long-term measurements of ISD with Ulysses and comparisons with numerical simulations (Landgraf 1998; Sterken et al. 2015) have since revealed previously unknown details of ISD. For example, Krüger et al. (2015) have analysed the ISD mass distribution in the Solar System and inferred the gas-to-dust mass ratio in the local interstellar medium (LISM), while Strub et al. (2015) have investigated the ISD time-variability due to the solar magnetic cycle and observed a change in the ISD flow direction in 2005–2006.

Ulysses’ orbit was highly inclined with respect to the Solar System’s ecliptic plane (Wenzel et al. 1992), in which interplanetary dust particles (IDPs) are predominant (e.g. Grün et al. 1997). The spacecraft’s orbital plane was oriented almost perpendicular to the inflow direction of ISD. Because zodiacal dust predominantly moves on prograde orbits, it can be easily distinguished from ISD during the majority of Ulysses’ orbit. Only close to the spacecraft’s perihelion did the direction of motion of ISD coincide with the direction of motion of particles on prograde orbits (Strub et al. 2015).

One difficulty when investigating ISD is to distinguish it from IDPs. The impact ionisation detector on board Ulysses (Grün et al. 1992) in combination with thorough calibrations (Göller & Grün 1989) allows for a reconstruction of the impact speed as well as the mass of the impacting particle (Grün et al. 1995a). Alongside the directionality of the impacting particle, which is approximately known from the detector boresight during the particle impact and the opening angle of the detector, the ability to reconstruct these aspects has allowed multiple sets of criteria to be established by which ISD can be discriminated from other dust species such as IDPs (Landgraf et al. 2003; Krüger et al. 2015; Strub et al. 2015). These different sets of criteria were devised to minimise biases for different types of analyses. As such, depending on the set of criteria, different biases are also induced on the resulting data subsets, and they may influence the ISD properties inferred from the Ulysses data (e.g. its particle flux, flow direction, or mass distribution) in different ways. Furthermore, different methodologies have been devised to determine the masses of the impacting dust particles (Grün et al. 1995a; Landgraf 1998; Krüger et al. 2015). These either depend entirely on properties of the impact signals measured by the detector or assume properties of the particles’ trajectories, such as the particle speed.

In this work, we investigate the different ISD selection criteria and mass determination methods. Our aim is to research the following: (1) how the choice of the ISD selection criteria influences the derived ISD particle flux or directionality, and (2) how the choice of the mass determination method influences the inferred ISD mass distribution.

The inferred gas-to-dust mass ratio in the LISM is dominated by the most massive ISD particles that were measured by Ulysses (Krüger et al. 2015). Furthermore, ISD particles of high mass are affected less by the Lorentz force within the heliosphere than lower-mass particles; they are not deflected from their trajectories as strongly (Sterken et al. 2012a). Thus, the most massive ISD particles are of particular relevance when investigating the direction of origin of the ISD. For example, if Ulysses had measured an additional interstellar inflow of dust particles that does not agree with the nominal ISD inflow direction¹, it would indicate another direction of origin of ISD particles, for example from within the G cloud (e.g. Redfield & Linsky 2008; Linsky et al. 2019, 2022a,b; Swaczyna et al. 2022).

For these reasons, we investigated the most massive ISD particles identified by Ulysses. Our aim is to research the following: (1) whether these most massive ISD particles are unambiguously interstellar or could have a different origin, (2) how this affects the inferred gas-to-dust mass ratio of the LISM, and (3) how the choice of the mass determination method influences this gas-to-dust mass ratio.

Before ISD particles can be measured by Ulysses or other in situ detectors within the Solar System, they must traverse both the heliosheath and the inner heliosphere, both of which modulate the size distribution of ISD. The modulation in the inner heliosphere can be numerically modelled (Landgraf 1998; Sterken et al. 2015; Strub et al. 2019), whereas the filtering by the heliosheath is less accurately known (Linde & Gombosi 2000; Slavin et al. 2012).

If a size distribution of ISD before it traverses the heliosphere is assumed, the size distribution measured by Ulysses in combination with numerical simulations of the inner heliosphere modulation can be used to infer the filtering in the heliosheath. This is illustrated and has been performed in this publication, for which we assumed an original ISD size distribution inferred from astronomical measurements (‘MRN distribution’; Mathis et al. 1977). The results of such an analysis can be used to generate predictions for the ISD environment that other spacecraft have encountered or will encounter, for example for DESTINY⁺ (Arai & DESTINY+ Team 2024; Hunziker et al., in prep.).

We introduce ISD and the Ulysses mission in Sect. 2, and we present the different ISD selection criteria and mass determination methods in Sects. 3 and 4, respectively. After investigating the research questions in Sects. 5, 6, and 7, we draw conclusions in Sect. 8.

2. Theoretical background

This section introduces the most relevant aspects of the physical forces that affect ISD in the heliosphere (Sect. 2.1) and of the Ulysses mission (Sect. 2.2). A detailed review of the former is given by, for example, Sterken et al. (2012a, Sterken et al. (2019). More information on the latter is given by, for example, Wenzel et al. (1992).

2.1. Trajectories of interstellar dust in the heliosphere

The Solar System moves with a relative speed of 26 km/s (Wood et al. 2015) through the surrounding Local Interstellar Cloud (LIC) in the direction of the G cloud (Redfield & Linsky 2008). Currently, it may reside in a mixed-cloud interstellar medium (ISM) in-between the LIC and the G cloud (Swaczyna et al. 2022).

It was concluded by Grün et al. (1993, 1994) that the inflow direction of ISD (from the retrograde direction at Ulysses’ aphelion) is compatible with the interstellar upstream direction and the helium inflow direction, which was also determined by Ulysses (Witte et al. 1993). This has been verified by Landgraf (1998). Krüger et al. (2015) confirmed an average impact speed of the dust grains on Ulysses of (24 ± 12) km/s. Throughout this work, we assumed that ISD homogeneously enters the heliosphere with the helium inflow, which comes from the direction of (l_He, b_He) = (255.41°, 5.03°) and with an ISD speed of ca. v_ISD, ∞ = 26 km/s (Swaczyna et al. 2018, 2023).

As these charged ISD particles travel through the magnetic field in the plasma interaction region of the Sun with the LISM, the so-called heliosphere (e.g. Richardson et al. 2023), they are filtered and modulated in different regions of the heliosphere by the Lorentz force. On their trajectories to the inner Solar System ISD particles must first cross the heliopause, which is the boundary between solar and interstellar plasma, and afterwards the termination shock, at which the solar plasma is abruptly decelerated and the interplanetary magnetic field diverges from a Parker spiral (Parker 1958). The region between the heliopause and the termination shock is called the (inner) heliosheath.

Inside the termination shock, meaning within the inner heliosphere, the forces that strongly affect ISD are well known (Sterken et al. 2012a): solar gravitation, F_G; solar radiation pressure, F_SRP; and the Lorentz force, F_L. Because the first two forces are both radial and both follow an inverse square law with respect to the heliocentric distance, their ratio, β = |F_SRP|/|F_G|, depends only on particle properties such as the mass, geometric cross section, and optical absorption and scattering efficiency. The Lorentz acceleration depends on the particle’s charge-to-mass ratio, Q/m. Together, the parameter set of (β, Q/m) is used to define different dust species in dynamical ISD simulations (Landgraf et al. 2000; Sterken et al. 2012a, 2013, 2015).

The relation between β and the particle mass, m, is described by a β-curve that depends on particle properties such as composition and morphology via its optical properties. Fig. 1 shows the calculated β-ratio against the mass for different particle compositions and porosities (see Gustafson 1994; Kimura & Mann 1999). Unless otherwise specified, we used the adapted astronomical silicates with a particle density of ρ_d = 2500 kg/m³ throughout this work (see Sterken et al. 2012a).

When β is greater than one, solar radiation pressure dominates over solar gravitation, and the ISD particles are deflected away from the Sun. Depending on how large the ISD particles’ β-ratio is, they cannot reach a paraboloid region around and behind the Sun, as seen from the ISD inflow direction. These exclusion zones are referred to as β-cones; for a given β_max only particles with smaller β < β_max can penetrate into these cones (see, e.g. Landgraf et al. 1999; Altobelli 2004). If a spacecraft’s orbit lies (partially or entirely) within a β-cone corresponding to β_max, the mass distribution of the ISD particles that impact the spacecraft will feature a gap that corresponds to the particles with β > β_max, the so-called β-gap.

2.2. The Ulysses mission

The Ulysses spacecraft was launched on 6 October 1990. After a swing-by manoeuvre at Jupiter in February 1992, it traversed the Solar System at an inclination of 79°, almost perpendicular to the ecliptic plane (Wenzel et al. 1992). The orbit of Ulysses is depicted in three dimensions in Fig. 2 and has been projected onto the heliocentric ecliptic coordinate xz- and yz-planes in Fig. 3.

Ulysses measured cosmic dust with a multi-coincidence impact ionisation detector (Grün et al. 1992). A hypervelocity impact on the detector’s target generated electric charges of both electrons and ions (Raizer 1960), and the amplitudes and rise times of the corresponding signals were measured. With aid of thorough calibration measurements (Göller & Grün 1989), the impact speed and mass of the impacting particle could be reconstructed (Grün et al. 1995a).

The dust detector was continuously active from 28 October 1990 until 30 November 2007, barring a few short time intervals that are listed in Table B.1 (see Krüger et al. 2010, Table 1); this is illustrated in Fig. 4. The surface area of the detector was 0.1 m². However, the effective sensitive surface area for measuring ISD depends on both the boresight direction and the field of view of the detector as well as on the relative speed between the spacecraft and the dust particle. At high relative speeds, the effective sensitive surface area can be larger than the detector’s physical surface area², ranging from 0 m² ≤ A_eff < 0.15 m² (cf. Appendix B.1).

The Ulysses spacecraft was spin-stabilised and its spin axis perpetually pointed towards Earth to ascertain stable communications via the high-gain antenna. The boresight of the dust detector rotated about this spin axis with approximately five revolutions per minute (Wenzel et al. 1992). The boresight of the detector can be described by the rotation angle, ϕ_rot, which gives the phase of the spacecraft’s spin. The two-dimensional pointing of the detector in heliocentric ecliptic coordinates thus must be reconstructed from the rotation angle and the vector of the spin axis at the time of impact. This is outlined in Appendix A.

3. Distinguishing ISD from IDPs and Jovian dust streams

Detailed information of every dust impact measured by the Ulysses dust instrument is publicly available in Grün et al. (2010)³. The dataset contains 6719 events, many of which are IDPs or particles from Jovian dust streams.

3.1. Excluding Jovian dust and IDPs from the dataset

One of the challenges of processing this dust dataset is to identify which events correspond to impacts of ISD, and which stem from IDPs or Jovian dust stream particles. During some sections of Ulysses’ orbit, for example when both ISD and IDPs come from roughly the same direction during Ulysses’ perihelion, it is not possible to unambiguously identify ISD. Therefore, certain time spans have been excluded entirely from the ISD dataset. There are three such general exclusion criteria (Krüger et al. 2015):

1. All events preceding Ulysses’ Jovian flyby on 8 February 1992 have been excluded because Ulysses was in the ecliptic plane.

2. All events recorded during Ulysses crossings of the ecliptic plane at the spacecraft’s perihelion (Ulysses’ ecliptic latitudes |b_U| < 60° and longitudes 247° < l_U < 360° & 0° < l_U < 67°) have been excluded.

3. All events corresponding to Jovian dust streams have been excluded. The time spans of these encounters have been taken from Baguhl et al. (1993, Table 2) and Krüger et al. (2006a, Table 1), and all data from 25 May 2004 until (exclusive) 12 September 2004 have been excluded (see Janisch 2021, Table A.2).

Applying these criteria to the datasets reduced the number of events to 1367.

Figure 4 displays the time intervals of the dust dataset that are missing or were excluded. Of the times where the dust detector was switched off, all events prior to the calendar year 2000 are due to spacecraft anomalies where all science instruments were turned off. After 2002 the detector was switched off because of software updates or to conserve power. The exclusion intervals in the ecliptic plane correspond to the time before the Jovian flyby and the three perihelion crossings.

3.2. The different selection criteria to identify ISD

Different criteria may be used to differentiate between ISD and IDPs/Jovian dust streams in the remaining data (Janisch 2021). These criteria bias the resulting data subsets in different ways. For example, the criterion that identifies ISD by its flow direction induces a direct bias of the flow direction; the resulting subset should not be used to investigate the directionality of ISD. Instead, another criterion is used that identifies ISD by the signal amplitude. However, because low-mass ISD is strongly affected by the Lorentz force, many of these particles are deflected from the original ISD flow direction. Thus, the flow direction of this ISD subset is indirectly biased as well.

Different ISD subsets have been compiled by using different ISD selection criteria. One of the main aspects of this study is the comparison of these ISD subsets and the effect that the choice of criteria has on the analyses (Sect. 5). We studied four subsets:

• 1.

  The ‘Krüger’ subset identified ISD primarily by its flow direction and was intended to be used for mass analyses. Krüger et al. (2015) propose two criteria:

  • Ia.

    In all years except 2005 and 2006, the impacting particle must have come from the direction of the He inflow; that is, the detector’s rotation angle, ϕ_rot, must have lain within 90° of the rotation angle associated with the He inflow, ϕ_He (see, e.g., Strub et al. 2015).

  • Ib.

    In the years 2005 and 2006, an additional range of 40° on the positive side of the rotation angle was allowed to account for the observed change in direction: ϕ_rot ∈ [ϕ_He − 90° ,ϕ_He + 130° ].

  • II.

    Electromagnetically accelerated fragments of IDPs at the solar poles (|b_U|≥60°) were excluded. These grains are associated with low ion amplitudes, Q_I < 10⁻¹³ C, which correspond in the dataset of Grün et al. (2010) to an ion amplitude code smaller than 8. Thus, all events at latitudes |b_U|≥60° with ion amplitude codes smaller than 8 were excluded.

    The Krüger subset contains 952 events. Krüger et al. (2015) found 987 events. The origin of this discrepancy mostly stems from a slightly different assumed ISD flow direction for the selection criterion (1.I)⁴. Additionally, the exclusion criterion (iii) of the Jovian dust streams was stricter in our investigation (Janisch 2021). Criterion (1.I) selected events by their direction and thus directly biased the directionality of the resulting subset. Because low-mass ISD is more strongly deflected by the Lorentz force, the resulting ISD mass distribution was also indirectly biased (see also criterion 1.II).

• 2.

  The ‘Strub’ subset primarily identified ISD by the signal amplitudes as a proxy for the particle masses and was intended for directionality analyses. Strub et al. (2015) propose three criteria:

  • I.

    Events of the lowest quality class (CLN = 0) were excluded as noise. Altobelli et al. (2004) found that particles with CLN ∈ {2, 3} more often come from the ISD inflow direction and thus more likely correspond to ISD particles.

  • II.

    Events with low ion signal amplitudes (Q_I < 10⁻¹³ C; ion amplitude code < 8) were excluded. These particles are scattered across the entire range of rotation, coming from both prograde and retrograde orbits (Landgraf 1998).

  • III.

    Events with impact speeds lower than half the dust inflow speed, v_msr < v_ISD, ∞/2, were excluded. This criterion excluded all particles with impact speeds slower than the bulk speed of interstellar dust, v_ISD, ∞, taking into account the error factor of the speed measurements, which is a factor of ∼2. However, Landgraf (1998) dismissed the use of this criterion because it neglects the relative velocity of the spacecraft to the ecliptic reference frame (i.e. the spacecraft’s orbital velocity). Furthermore, the criterion assumed that only a negligible amount of ISD particles is slower than the ISD bulk speed in the ecliptic reference frame, and it neglected that particles with β > 1 are decelerated as they draw closer to the Sun as well as any acceleration or deceleration by the Lorentz force.

    The Strub subset contains 533 events. Strub et al. (2015) found 580 events. The discrepancy mostly stems from the stricter handling of Jovian dust streams by exclusion criterion (iii), and from different assumptions of the ISD inflow speed, v_ISD, ∞, for criterion (2.III). Criterion (II.2) directly biased the mass distribution of the resulting ISD subset. However, because the Lorentz force affects low-mass particles most strongly, the criterion also indirectly biased the subset’s directionality.

• 3.

  The ‘union’ subset is the union of the Krüger subset (1) and the Strub subset (2) without the speed criterion (2.III), meaning that particles with low impact speeds were not categorically excluded. The union subset contains 1064 events.

• 4.

  The ‘intersect’ subset is the intersection of the Krüger subset (1) and the Strub subset (2) without the speed criterion (2.III). The intersect subset contains 517 events.

The influence of these different criteria on the inferred ISD flux, flow direction, and mass distribution has been evaluated in Sect. 5.

4. Determination of the ISD particle mass and speed

The Ulysses dust detector recorded for each impact the amplitudes and rise times of the ion and electron signals. The total emitted charge, Q, of a dust impact and therefore the measured ion and electron signal amplitude depend on the impacting particle’s mass, m, and its impact speed, v, following a power law:

$$\begin{array}{c}\hfill Q\propto m^{k_1}{v}^{k_2}.\end{array}$$(1)

Here, k₁ = 1 (Auer 2001, Ch. II.D.1), k₂ ≈ 3.5 (Balogh et al. 2001, Ch. 9.2), and the proportionality constant has to be determined with laboratory calibrations (e.g. Göller & Grün 1989). With this relation, the mass can be determined from the impact speed.

Three competing methods to determine the impact speeds, and thus the particle masses, are presented here (Landgraf 1998). The canonical method derived the impact speeds from the measured signal rise times, resulting in the impact speeds and particle masses listed in the dataset. This method is presented in Sect. 4.1. Alternatively, the ISD particle speeds could be assumed in order to reduce the effects of their measurement uncertainties. A method that assumed a homogeneous ISD inflow, corresponding to β ≡ 1, is presented in Sect. 4.2. Another alternative that iteratively refined the assumed ISD inflow speed and the resulting particle mass is presented in Sect. 4.3.

The influence of the choice of the mass determination method on the ISD flux and on the ISD flow direction is investigated in Sect. 5.3. The inferred gas-to-dust mass ratio in the LISM (Sect. 6.4) is also affected by the choice of the mass determination method.

4.1. Impact speeds and particle masses directly inferred from the measurements

The impact speed of a dust particle could be determined from the rise times of the signals measured by the detector (Grün et al. 1995a, 2010). With aid of detector calibration curves, the respective dust particle’s mass could be determined from this measurement-derived impact speed and the measured impact charge (Göller & Grün 1989).

The resulting particle masses and impact speeds are tabulated in the dataset of Grün et al. (2010). Because they were derived directly from measurements without any additional assumptions, they are referred to in the following as the measurement-derived particle masses and impact speeds, m_msr and v_msr, respectively.

The dust detector on board Ulysses was only sensitive to a certain interval of signal amplitudes, which introduced a detection bias: low-mass particles could only be detected if they impacted with sufficient speed, and low-speed particles could only be detected if they had sufficient mass. Similarly, fast and high-mass particles saturated the detector. However, this saturation only occurred for two particles, neither of which were selected by the ISD criteria.

Krüger et al. (1999) note that particle impacts on detector components other than the target, for example the inner wall or the entry grid, may artificially inflate signal rise times, leading to underestimated impact speeds and overestimated particle masses. This affected events where the measurement-derived impact speed was below v_msr < 3 km/s, which featured high masses more often than low masses due to the amplitude detection threshold and Eq. (1). The true impact speeds of these particles are unknown. Thus, we used neither the measurement-derived impact speeds nor the resulting particle masses for these particles unless we explicitly specify otherwise.

This method determined the particle masses from the impact speeds with Eq. (1). Because the measurement-derived impact speeds have an error factor of ∼2, this resulted in an error factor for the particle masses of ∼2^3.5 ≈ 10.

4.2. ISD particle masses inferred from assumed impact speeds corresponding to β≡ 1

To reduce the uncertainty of the particle masses that was introduced in the method of Sect. 4.1 by the derived impact speeds, Landgraf (1998) proposed an alternative that instead assumed that all particles move with v_ISD, ∞ = 26 km/s, corresponding to ISD particles with β ≡ 1. Strub et al. (2015, Table 1) provide a conversion from the measured ion signal amplitude, Q_i, to the ISD particle mass, m_ISD, based on the calibration curves by Grün et al. (1995a) and assuming an ISD impact speed of v_Strub = 23.2 km/s. Using this approximation and rescaling to the appropriate impact speeds, v_rel, Eq. (1) yields

$$\begin{array}{cc}& Q\approx {10}^{3.65}\frac{\mathrm{C}}{\mathrm{kg}}·m·{\left(\frac{v_{\mathrm{rel}}}{v_{\mathrm{Strub}}}\right)}^{3.5}\hfill \end{array}$$(2)

$$\begin{array}{cc}& \Rightarrow m_{\mathrm{ISD}}[\mathrm{kg}]\approx 13.465·{\left(v_{\mathrm{rel}}[\mathrm{km}/\mathrm{s}]\right)}^{-3.5}·Q_{\mathrm{i}}[\mathrm{C}].\hfill \end{array}$$(3)

Further assuming β ≡ 1 and an ISD inflow speed of v_ISD, ∞ = 26 km/s, the relative speed of the ISD particles of the union ISD subset ranges between 26 km/s < v_rel ≤ 41 km/s (see also Fig. 7).

This method reproduced the measurement-derived masses within an error interval of a factor of 10. It assumed that all ISD particles have a homogeneous ISD inflow speed, corresponding to β ≡ 1. Therefore, this method is referred to as the ‘β ≡ 1’ method. Any acceleration or deflection of the ISD particles, for example by solar gravitation, solar radiation pressure, or the Lorentz force, was neglected. Furthermore, this method assumed that all particles are ISD; unlike the measurement-derived method presented in Sect. 4.1 it would yield erroneous results for particles of other origins.

4.3. Iterative calculation of the ISD particle speed and mass

The method in Sect. 4.2 could be refined by replacing the heliocentric ISD inflow speed, v_ISD, ∞, with the local ISD speed, assuming a β-ratio that corresponds to the particle’s mass (see Fig. 1):

$$\begin{array}{c}\hfill v_{\mathrm{hel}}=\sqrt{\frac{2(1-\beta )G{M}_{\odot }}{r}+v_{\mathrm{ISD},\infty }^2},\end{array}$$(4)

where v_ISD, ∞ = 26 km/s is the homogeneous ISD inflow speed; G is the gravitational constant, M_⊙ is the solar mass, and β is the mass-dependent β-ratio. In this work we assume the β-curve of adapted astronomical silicates (cf. Fig. 1).

Thus, the ISD particle mass determined from Eq. (3) could be used to calculate the ISD particle’s heliocentric speed with Eq. (4), and the resulting relative speed could again be used to calculate a more accurate ISD particle mass. By alternately applying these two equations, the resulting values of the impact speed and particle mass could be iteratively refined until convergence (Landgraf et al. 2000; Krüger et al. 2015).

We note that this method only took into account the change in speed by solar gravitation and solar radiation pressure, but not any change in direction, which would affect the relative speed between the ISD particle and the dust detector. Furthermore, any effect of the Lorentz force was ignored. Nevertheless, large deviations of a given particle’s iteratively calculated values from its measurement-derived values may indicate that the particle does not agree with the assumption of the ISD inflow, meaning that it is not ISD (see, e.g., Fig. 9).

5. Investigating the influence of the ISD selection criteria and mass determination methods on the inferred ISD characteristics

This section demonstrates the effects of the choice of ISD selection criteria on the derived ISD particle flux (Sect. 5.1) and on the ISD flow direction (Sect. 5.2). We compare the different mass determination methods in Sect. 5.3 and investigate the influence of the choice of both the ISD selection criteria and the mass determination methods in Sect. 5.4.

5.1. Influence of the ISD selection criteria on the ISD particle flux

Fig. 5 shows the ISD particle fluxes, Eq. (B.1), of the union, Krüger, Strub, and intersect subsets measured by the Ulysses dust detector, binned with Δt = 4 months starting on 1 January 1992. The ISD inflow was assumed to come from a direction of (255.41° ,5.03° ) with a speed of 26 km/s (cf. Appendix B.1). The areas shaded in grey denote time intervals that were excluded because of perihelion crossings, Jovian stream encounters, or when the dust detector was switched off. Some of the time bins are almost completely covered by these excluded time intervals, for example in mid-2001 and 2007, leading to few measurements and large error bars. No data was available during the early 1995 and the early 2003 time bins due to these gaps.

Impacts against the detector’s inner wall can potentially also be measured (cf. Sect. 4.1), which increases the detector’s sensitive surface area (see Fig. B.1; cf. Altobelli et al. 2004). Here, wall impacts were taken into account, rescaling the measured particle flux to lower values (cf. Appendix B.1).

Due to the lower number of events contained in these subsets, the Strub and intersect subsets generally feature much lower fluxes than the other subsets, and the union subset features slightly higher fluxes than the Krüger subset. The magnitude of these differences directly corresponds to the number of events: the union subset contains 1064 events at an average ISD particle flux of 7.4 × 10⁻⁵ m⁻² s⁻¹, which is twice as high as the Strub subset with 533 events and an average flux of 3.6 × 10⁻⁵ m⁻² s⁻¹. Similarly, the Krüger subset features 952 events with an average flux of 6.6 × 10⁻⁵ m⁻² s⁻¹; about ∼10% less than the union subset. At 517 events and an average flux of 3.5 × 10⁻⁵ m⁻² s⁻¹ the intersect subset is similar to the Strub subset.

As is evident, the fluxes of all four ISD subsets follow similar general trends. In 1992 the fluxes are roughly twice as high for all four subsets as the respective all-time average flux before decreasing by roughly a third from 1992 until 1995. From 1995 to 2000 all four subsets have low fluxes at roughly half their respective all-time average. In 2000, the fluxes increase to a plateau roughly coincident with the respective all-time average, in 2005 all four subsets feature a notable peak at twice the all-time average, and this is followed by a stark decrease in 2006. The origin of these features is assumed to come from the solar cycle (Strub et al. 2015), as has been corroborated by simulations (Sterken et al. 2015).

The intersect subset typically features the lowest fluxes because it contains the fewest particles. However, at some times, for example in 1994 and from 2005 to 2006, the flux of the intersect subset is higher than the flux of the Strub subset. This is caused by the speed criterion (2.III), which excludes events with low impact speeds from the Strub subset but not from the intersect subset.

There is one notable outlier in mid-1998, where the Krüger subset shows a much higher flux compared to the surrounding time bins (+80%), and the Strub subset does not (−5%). During this time, when Ulysses was at its aphelion and crossed the ecliptic, an excess of low-mass particles was included by the directionality criterion (1.I). This is likely caused by IDPs that were not filtered out by that criterion (Janisch 2021).

In conclusion, Fig. 5 shows that the different ISD selection criteria result in similar global trends of the particle flux but can nevertheless strongly influence local features. This is mostly caused by the lack of small particles in the Strub and intersect subsets, and by the criteria falsely including IDPs or falsely excluding ISD at different times.

5.2. Influence of the ISD selection criteria on the ISD directionality

One of the most salient points of previous analyses of dust impact data from Ulysses is the directionality of the dust impacts (Landgraf 1998; Altobelli 2004; Krüger et al. 2015). The direction of origin of a given dust particle is only known to lie within the opening angle of the dust detector around the detector’s boresight, which is given by the spacecraft rotation angle, ϕ_rot. The ISD directionality was determined by the average deviation, ϕ_dev, of the rotation angle at the time the respective particle impacted from the rotation angle of the ISD inflow, ϕ_ISD (Appendix B.2).

Fig. 6 shows the ISD flow directionality (see Strub et al. 2015, Fig. 6) of the impacts of all four ISD subsets. The deviation was averaged within four month intervals beginning on 1 January 1992. Because the Strub and intersect subsets contain fewer particles, the uncertainties of the mean deviation from the ISD rotation angle are higher compared to the other two subsets⁵.

As noted in Sect. 3.2, the Krüger subset mainly selected ISD through its directionality, and it is thus directly biased for this analysis. The mainly mass-based Strub subset is indirectly biased as well. The low-mass particles are expected to have larger deviations due to the Lorentz force, and excluding these particles artificially restricts the ISD directionality.

Until 1999, the average ISD directionality is in reasonably good agreement for the four ISD subsets. The strong deviation of the Strub subset in late 1999 (ϕ_dev = +63°) from the values at the adjacent times and from the other ISD subsets (ϕ_dev < 7°) is caused by a single outlier measured at a highly different ϕ_rot that has a lower quality class; this also occurred in early 1997⁶. Although low-quality outliers are also included in the Krüger subset, its larger number of recorded events is more robust against such outliers⁷.

The different ISD subsets are not as closely grouped together post-2001 as they were pre-2001. This coincides with the solar maximum that separates the defocusing from the focusing phase of the solar magnetic cycle in 2000–2001. We speculate that the interplanetary magnetic field is least structured close to solar maxima and may chaotically deflect small dust particles. However, the larger deviations, for example in late 2003, are caused by large particles rather than small particles. For this reason, the Strub subset, which lacks the smallest particles and is thus influenced more strongly by large particles, deviates the most by ϕ_dev = −84°. The Krüger subset, which was filtered by direction and thus does not contain these particles, deviates significantly less by ϕ_dev = −25° (Janisch 2021).

The most notable feature of this analysis, as remarked by Strub et al. (2015), is the statistically significant change in direction of the dust inflow in 2005–2006, which can be observed in all four ISD subsets. Unfortunately, the following solar minimum in 2008–2009 coincided with the end of measurements, once again highlighting the necessity of long-term measurements of cosmic dust that cover at least a full solar cycle.

5.3. Comparison of the mass determination methods

In this section, we compare the three methods for the determination of the particle mass and impact speed. As a reminder, the measurement-derived speeds and masses were purely derived from the measured signal rise times and amplitudes (Sect. 4.1), the ‘β ≡ 1’ method assumed an identical heliocentric particle speed for all ISD (Sect. 4.2), and the iteratively calculated values assumed heliocentric speeds as per the β-ratios corresponding to the particles’ masses (Sect. 4.3).

In Fig. 7 the impact speeds and particle masses for all events of the union subset are plotted for each method. The measurement-derived values cover a much larger range of speeds and masses. The discrete speed bands of the measurement-derived values stem from the digitisation of the signal rise times into discrete bins. The speed bands with the lowest measurement-derived values, corresponding to v_msr < 3 km/s, contain 34 events with severely underestimated impact speeds and, thus, severely overestimated particle masses (Sect. 4.1). We treat these impact speeds and the corresponding masses as unknown in our analyses unless we explicitly specify otherwise. The lack of high-mass impacts with the ‘β ≡ 1’ method and the iteratively calculated values is directly caused by the lack of low-speed impacts as per Eq. (3): a higher impact speed for a given signal amplitude corresponds to a lower inferred particle mass, and vice versa.

All events have the same heliocentric particle speed determined by the ‘β ≡ 1’ method, but their impact speeds vary between ca. 26 km/s and 41 km/s due to the variation of the spacecraft’s heliocentric speed. The distributions of the ‘β ≡ 1’ method and the iterative calculations are more similar because both methods assumed similar heliocentric speeds instead of reconstructing the impact speeds from rise time measurements. The iterative calculations converged to iterative changes below 1% after eight iterations and to below 0.1% after eleven iterations.

The iteratively calculated heliocentric speeds range between 8 km/s < v_hel < 38 km/s for the assumed β-curve of adapted astronomical silicates⁸. These heliocentric speeds were converted to impact speeds in the spacecraft frame for comparison with the measurement-derived impact speeds, which is depicted in Fig. 7. As that figure indicates, the lowest speeds in both the spacecraft frame and the heliocentric frame correspond to those particles with the highest β, which are those with m ≈ 5 × 10⁻¹⁷ kg.

5.4. Influence of the mass determination methods on the ISD mass distribution

The inferred ISD mass distribution is plotted in Fig. 8 for all three determinations of the mass and all four ISD subsets⁹. The three methods for the determination of the mass agree in their major features: for the Krüger subset the majority of the particles have masses in the range 10⁻¹⁸ kg ≤ m ≤ 10⁻¹⁵ kg, and there is a notable dip in the distribution around m ≈ 10⁻¹⁷ kg that is likely an artefact in the instrument electronics (Grün et al. 1995a). The Strub and the intersect subsets lack the low-mass wing of the distribution, m < 10⁻¹⁷ kg, due to criterion (2.II), and show slightly fewer events in the other bins as well.

However, the three mass distributions of the Krüger subset differ in their minor features. For the measurement-derived masses, both flanks of the distribution are much gentler compared to the other two methods. This is a consequence of the assumed heliocentric speeds of the ‘β ≡ 1’ and the iterative method, which constrained the distribution of the impact speeds. The global maximum of the distribution is located at m_msr ≈ 10^−15.5 kg for the measurement-derived distribution and the iteratively calculated distribution, and at m = 10^−17.5 kg for the ‘β ≡ 1’ method.

Neither the ‘β ≡ 1’ method nor the iterative calculations feature any masses above m > 10^−12.5 kg, whereas the measurement-derived distribution does. This is, again, a consequence of the assumed heliocentric speed of the ‘β ≡ 1’ and the iterative method. This upper limit of the ISD mass distribution significantly affects the derived gas-to-dust mass ratio (cf. Sect. 6.4).

The mass distributions of the Krüger subset displayed in Fig. 8 are overall in good agreement with those of Krüger et al. (2015, Fig. 5) but do show some differences: the dip in the distribution at m_β ≡ 1 ≈ 10⁻¹⁷ kg was not resolved by the wider mass bins of Krüger et al. (2015) and may be an instrumental effect. Furthermore, their iteratively calculated mass distribution, which was determined with an assumed dust particle density of ρ_d = 3300 kg/m³ instead of 2500 kg/m³ as assumed in our analysis, features different heights of its local maxima.

The highest-mass particles of the union subset are separately examined in Sect. 6. Because all mass determinations are heavily dependent on the impact speed of the respective event, highly accurate impact speed measurements would drastically improve the accuracy and precision of any determined mass distribution.

6. Investigating the most massive ISD particles detected by Ulysses

The most massive ISD particles measured by Ulysses are of particular interest: they dominate the differential mass distribution and have the largest effect on the derived gas-to-dust mass ratio of the LISM. Furthermore, because their gyroradii are large, the effect of the heliospheric magnetic field on these particles’ trajectories is negligible compared to the effect of solar gravitation and solar radiation pressure. Thus, the most massive particles are best suited to investigate the particles’ direction of origin (see, e.g., Altobelli et al. 2005).

We compiled two subsets of massive ISD particles based on the union subset: Subset A contains the ten most massive particles, determined by either the measurement-derived or the iteratively calculated masses. Because the measurement-derived masses were most likely severely overestimated for events with impacts speeds below v_msr < 3 km/s, Subset B contains all 45 particles with measurement-derived masses above m_msr > 10⁻¹⁴ kg and impact speeds above v_msr > 3 km/s.

The properties of the particles of Subset A are presented in Sect. 6.1. These particles’ directions of origin is determined in Sect. 6.2. We examine Subset B in Sect. 6.3. Sect. 6.4 reports on the effect of including or excluding particles from Subsets A and B that are unlikely to be interstellar on the inferred gas-to-dust mass ratio in the LISM.

6.1. Properties of the most massive particles of Subset A

We used both the measurement-derived masses, m_msr, and the iteratively calculated masses, m_iter, to select Subset A. Fig. 9 shows both masses for all particles of the union subset. As expected, there are many particles where m_msr ≈ m_iter. However, there is also an extended population with m_msr ≫ m_iter, in particular at high masses.

Most striking in Fig. 9 is a cluster of six particles with m_msr > 10⁻¹⁰ kg, designated as A₁ and numbered 1–6. These particles have, by far, the highest measurement-derived masses; however, their iteratively calculated masses are considerably lower at 10⁻¹⁴ kg < m_iter ≲ 10⁻¹³ kg. Another cluster of two particles, designated as A₂ and numbered 7–8, features measurement-derived masses that are an order of magnitude lower, 10⁻¹¹ kg < m_msr ≤ 10⁻¹⁰ kg and even less massive as per the iteratively calculated masses, m_iter ≈ 4 × 10⁻¹⁵ kg.

These eight particles all feature measurement-derived impact speeds below v_msr ≤ 3 km/s (see Fig. 9), which were most likely severely underestimated (Sect. 4.1). Higher impact speeds would result in lower inferred masses. Thus, the eight particles of A_1, 2 are most likely not as massive as assumed¹⁰.

For this reason, Subset A also includes the two particles with m_msr > 10⁻¹² kg and v_msr > 3 km/s, designated as A₃ and numbered 9–10. These two particles have the highest and the eighth highest iteratively calculated mass, respectively. Their measurement-derived impact speeds are considerably higher, v_msr ≫ 3 km/s.

Properties of the ten particles of Subset A are tabulated in Table 1. There appears to be no pattern to the particles’ heliocentric distances or orbital positions. The iteratively calculated impact speeds are distributed over a wide range of values due to the variation of the spacecraft velocity and of the heliocentric distances at which these particles were detected.

The particles of A_1, 2 were detected from a wide range of ecliptic longitudes of the detector’s boresight, but appear to be confined to boresight latitudes of −31° ≤b_det ≤ −2°. In contrast, the two particles of A₃ were measured while the detector was pointing at fairly high positive latitudes. This disparity may indicate a systematic difference between the particles of A_1, 2 and those of A₃, for example a different origin of these particles.

6.2. Compatibility of the dust particles of Subset A with an interstellar origin

Different criteria can be used to investigate whether a particular dust particle can be of interstellar origin or not (cf. Sect. 3.2). Sect. 6.2.2 investigates whether the particles’ heliocentric speeds were sufficiently high to correspond to the assumed ISD inflow. This speed criterion only takes into account the (scalar) heliocentric speed of the respective particle, but cannot distinguish between inbound hyperbolic trajectories (i.e. ISD) or outbound hyperbolic trajectories (i.e. β-meteoroids). Therefore, Sect. 6.2.3 investigates whether the directionality of the particles of Subset A agrees with the assumed ISD inflow direction or a Sun-ward origin.

6.2.1. Heliocentric speed of the dust particles

The dust particle properties tabulated in Table 1 are generally insufficient to unambiguously distinguish ISD from IDPs. For example, dust particles from Oort cloud comets (OCCs) often have large eccentricities and come from all longitudes, including the direction of the ISD inflow (Poppe 2016). However, the orbits of OCCs, and thus the orbits of the majority of the dust particles originating from OCCs, are elliptic¹¹, whereas the orbits of ISD particles are hyperbolic. Thus, the local heliocentric speed of an OCC dust particle must be lower than the local escape speed, which is

$$\begin{array}{c}\hfill v_{\mathrm{esc}}=\sqrt{\frac{2(1-\beta )G{M}_{\odot }}{r_{\mathrm{U}}}},\text{for}\beta \le 1,\end{array}$$(5)

whereas the speed of an ISD particle is generally higher as per Eq. (4). We note that the Lorentz force is not taken into account here. While it does not play a significant role for these most massive particles due to their large gyroradii, it cannot be ignored for lower-mass particles.

The heliocentric speed, v_hel, of a dust particle encountered by Ulysses is not known; only the particle’s impact speed, v_msr, could be determined from the detector signal (cf. Sect. 4.1). This value has a multiplicative error factor of 1.9 (Grün et al. 2010); the true value of the impact speed lies within the interval [v_msr/1.9;  v_msr ⋅ 1.9].

Therefore, the particle’s heliocentric speed must be reconstructed from the particle’s and the spacecraft’s orbital velocities. However, the relative direction from which an impacting particle came is only known to have lain within the detector’s field of view, limiting how precisely the particle’s heliocentric speed can be reconstructed. Thus, the detector’s opening angle of 70° introduces an additional maximum error factor of the heliocentric speed of up to 1.36 at Ulysses’ aphelion and up to 3.8 at its perihelion (cf. Appendix C). For the configuration of Ulysses we assume that these error factors are roughly multiplicative¹². Therefore, assuming an opening angle of 70°, the heliocentric speeds could only be reconstructed with a total error factor between 1.9 ⋅ 1.36 ≈ 2.6 at the aphelion and 1.9 ⋅ 3.8 ≈ 7.2 at the perihelion.

To unambiguously distinguish ISD from dust on elliptic orbits, the heliocentric speed must be known with an error factor smaller than 1.73 at the heliocentric distance of Jupiter (cf. Appendix C). At 1 au the required precision is even finer with an error factor smaller than 1.18. To distinguish ISD from dust on circular orbits, these precision limits increase to 2.44 for Jupiter and 1.66 for Earth. The configuration of the dust detector on board Ulysses does not meet these requirements; in many cases ISD cannot be distinguished from OCC dust.

6.2.2. Compatibility of the particles’ heliocentric speed with ISD

Table 2 lists the most likely reconstructed heliocentric speed, v_rec, hel, for the particles of Subset A, using Eq. (C.1). The same table also gives confidence intervals, taking into account either only a finite opening angle of 70°, or both that finite opening angle and the error factor of the impact speed determination. For convenience, the heliocentric speed of Ulysses, as well as the heliocentric speeds of a particle on a circular orbit, on a parabolic orbit (i.e. the escape speed) according to Eq. (5), and for an ISD particle on a hyperbolic orbit according to Eq. (4) are tabulated as well, taking into account the β-ratio corresponding to either the measurement-derived or the iteratively calculated particle mass. We based the reconstruction of the most likely heliocentric speed on the measurement-derived impact speed, v_msr, only if v_msr > 3 km/s. Otherwise, the iteratively calculated relative speed, v_iter, was used. However, because the iterative calculation assumes that the particles are ISD, the resulting speeds cannot be used to confirm an ISD origin but can only reject this hypothesis should the resulting speeds be inconsistent with the local hyperbolic ISD speed.

Measurement-derived impact speeds, v_msr > 3 km/s, were only available for the particles of Subset A₃. For particle (9) the most likely reconstructed heliocentric speed, v_rec, hel ≈ 19.0 km/s, is slightly higher than the local escape speed, v_esc ≈ 17.9 km/s, but considerably lower than the local hyperbolic ISD speed, v_hel, ISD = 31.5 km/s. For particle (10), the local escape speed is higher than the reconstructed heliocentric speed, indicating that this particle is unlikely to be ISD.

Although the iterative calculation of the relative speed, v_iter, assumes that the particles of Subsets A_1, 2 are ISD, the resulting most likely heliocentric speeds are lower than the local escape speed for particle (1) and considerably lower than the local hyperbolic ISD speed for particles (1–3), (5), and (8), contradicting the assumption of an ISD origin. Only for particles (4) and (6–7) the most likely reconstructed heliocentric speeds are faster than the hyperbolic ISD speed.

When taking into account the uncertainties of the reconstructed heliocentric speed introduced by the opening angle of the detector and the error factor of the measurement-derived impact speed, the resulting ranges of reconstructed heliocentric speeds typically include both the local escape speed and the local hyperbolic ISD speed. The exceptions are particles (9–10), which may have been faster than the respective local escape speed but do not agree with the speed of a hyperbolic ISD particle. It is thus unlikely that particles (9–10) are ISD, although they may have come from interstellar objects other than the LISM.

Because the iteratively calculated impact speeds are based on the assumption of ISD, this analysis can only reject the hypothesis that a particle is ISD if the resulting speeds are inconsistent with this underlying assumption, but it cannot confirm whether particles (1–8) are ISD due to circular reasoning. Only for particles (4) and (6–7) the most likely reconstructed heliocentric speeds are consistent with the basic assumption of ISD. For the other fives particles of A_1, 2 the analysis is inconclusive due to the large uncertainty introduced by the wide opening angle of the detector.

We note that the Strub subset does not contain these ten massive particles because criterion (2.III) excluded all particles with impact speeds below v_msr < 13 km/s. As mentioned in Sect. 6.2.3, seven of these ten particles agree with the ISD inflow direction as per criterion (I.1), taking into account an opening angle of 90° around the detector’s boresight.

This analysis demonstrates that more precise measurements of the impact speed and particle directionality would enable the identification of ISD by the particles’ reconstructed heliocentric speeds. We estimate that the total error factor of the reconstructed heliocentric speed must lie below 1.73 to distinguish ISD from highly eccentric IDPs on spacecraft with Jupiter-like orbits (cf. Appendix C). For spacecraft on Earth-like orbits, the total error factor shrinks to 1.18¹³.

6.2.3. Compatibility of the particles’ direction with ISD

Fig. 10 shows the orbital position of Ulysses as well as the detector boresight at the time of impact for each particle of Subset A. Coloured areas indicate the outer edge of the cone of view corresponding to an opening angle of the detector of ±70°, meaning that wall impacts were not considered (cf. Appendix B.1). If wall impacts are considered, the opening angle is ±90°, meaning that the full hemisphere centred around the boresight would be sensitive to impacts. However, close to the outer edge of the cone of view, the effective sensitive surface area is almost zero. Therefore, it is far more likely that a particle came from closer to the boresight direction than from a direction close to the outer edge of the cone of the view.

The detector’s field of view takes the shape of a rotationally symmetric cone with an opening angle of 70° around the boresight (see Table 1) in the spacecraft’s frame of rest. However, because Ulysses and its dust detector were in motion with respect to the heliocentric reference frame, both the detector’s boresight and its field of view will appear geometrically distorted in the heliocentric reference frame. This was accounted for by a Galilean transformation from the spacecraft’s frame of rest to the heliocentric reference frame¹⁴,

$$\begin{array}{c}\hfill \widehat{\mathit{b}}\mapsto \frac{{\mathit{v}}_{\mathrm{U}}+v_i\widehat{\mathit{b}}}{|{\mathit{v}}_{\mathrm{U}}+v_i\widehat{\mathit{b}}|},\end{array}$$(6)

where $\widehat{\mathit{b}}$ is the detector boresight or a generatrix¹⁵ of its cone of view; v_i is the relative speed of the particle in the spacecraft’s frame of rest, which is assumed to be either the measurement-derived impact speed, v_msr, or the iteratively calculated impact speed, v_iter; their product, $-v_i\widehat{\mathit{b}}$, is the assumed dust particle velocity in the spacecraft’s frame of rest, which is subtracted from the heliocentric spacecraft velocity, v_U. This Galilean transformation results in the boresights and cones of view shown in Fig. 10.

The angles of deviation of the ISD inflow direction from the detector boresight are tabulated for all ten particles of Subset A in Table 2. Seven of these ten particles are part of the Krüger subset; they were measured when the detector’s boresight was within ±90° of the assumed ISD inflow direction, that is, under consideration of wall impacts. Nevertheless, due to the Galilean transformation, some of these particles lie close to the outermost edge or even outside of the detector’s cone of view when taking the spacecraft velocity into account. When assuming that the particles impacted with the measurement-derived impact speed, all ten particles of Subset A were measured when the ISD inflow direction lay outside the 70°-cone of view. However, the measurement-derived impact speeds are most likely spurious for all particles of Subsets A_1, 2. When using the iterative calculation of the impact speed, which assumes that the respective particle is ISD, the ISD inflow direction lay within the 70°-cone of view for only four of the ten particles: (3), (5), (8), and (10)¹⁶. For three of these four particles the ISD inflow direction lay close to the outer edge of the cone of view, $\mathrm{\Delta }{\varphi }_{\mathrm{ISD}}^{\mathrm{iter}}>60°$. This is in agreement with the reconstructed heliocentric speeds, which are not consistent with the local hyperbolic ISD speeds for $\mathrm{\Delta }{\varphi }_{\mathrm{ISD}}^{\mathrm{iter}}\approx 0°$ but are consistent for $\mathrm{\Delta }{\varphi }_{\mathrm{ISD}}^{\mathrm{iter}}\gg 0°$ (Sect. 6.2.2). Nevertheless, the detector’s sensitive surface area decreases with increasing impact angle (Fig. B.1). For any given impact, it is less likely that the particle came from close to the outer edge of the detector’s cone of view.

This indicates that three to four of the ten massive particles of Subset A may have been ISD but we cannot confirm it. The other six to seven massive particles may have come from cometary streams (Krüger et al. 2024a) or may have been more sporadic IDPs. More precise information on the particles’ direction of origin would be invaluable to more accurately distinguish ISD from IDPs.

This analysis can also be applied to other directions of origin: four of the ten particles of Subset A – including all ISD candidates – agree with a Sun-ward direction of origin even without considering wall impacts, indicating that they could be β-meteoroids (e.g. Wehry et al. 2004). However, these particles would be unusually massive for β-meteoroids.

6.3. Compatibility of the not-as-massive particles of Subset B with an interstellar origin

Infrared observations of the ISM and considerations of its elemental abundances suggest that ISD particles with masses of m > 10⁻¹² kg are unexpected (Draine 2009), which is consistent with the indication that the measurement-derived particle masses for Subsets A_1, 2 are most likely severely overestimated. Another caveat of the mass determination is that the instrument calibrations that derive the impact speeds from the signal rise times and the masses from the signal amplitudes (Sect. 4.1) all assume compact dust particles. Recent laboratory experiments have shown that porous (‘fluffy’) particles would yield different calibration curves depending on the densities of the dust particles and the impact target: a miscalibration may result in incorrect impact speeds and masses (Hunziker et al. 2022). Combining these considerations, it is expedient to not only investigate the particles of Subset A, but to also examine particles of lower but still reasonably high masses that also have reasonably high impact speeds, compiled as Subset B.

This subset of events was selected with an arbitrary lower mass threshold of m_msr > 10⁻¹⁴ kg, corresponding to particle radii of a ≳ 1 μm¹⁷. All particles with a measurement-derived impact speed below v_msr < 3 km/s (cf. Fig. 9) were excluded because the corresponding measurement-derived masses are most likely severely overestimated. Subset A₃ was removed as well because we have already scrutinised it in detail in the previous section. The resulting subset consists of 45 particles with masses within 10⁻¹⁴ kg ≤ m_msr ≤ 10⁻¹² kg and 10⁻¹⁶ kg ≤ m_iter ≤ 10⁻¹⁴ kg, and one particle with v_msr = 3.4 km/s and m_iter = 3.38 × 10⁻¹⁷ that was removed due to its comparably low iteratively calculated mass. The resulting Subset B is marked in Fig. 9. The particles’ properties and their reconstructed heliocentric speeds are tabulated in Tables E.1 and E.2, respectively.

Not a single particle of Subset B has a most likely reconstructed heliocentric speed, v_rec, hel, that is at least as fast as the local hyperbolic ISD speed, v_hel, ISD. However, one particle, numbered #31 in Table E.2, falls short by only 1%; the ISD inflow direction also deviates by less than $\mathrm{\Delta }{\varphi }_{\mathrm{ISD}}^{\mathrm{msr}}<70°$ from the detector boresight at the time particle #31 impacted. If the total error factors of the heliocentric speed reconstruction are taken into account, 20 of 45 particles both have reconstructed heliocentric speeds that come within 10% of the respective local hyperbolic ISD speed and agree with the ISD inflow direction. Relaxing the directionality criterion to ±90° increases this number further to 30 ISD candidates¹⁸.

Of the 45 particles of Subset B only eight both agree with the ISD inflow direction within 70° and are faster than the local escape speed, Eq. (5), without accounting for the total error factors of the reconstruction of the heliocentric speed. Taking both the total error factors and wall impacts into account increases the number of potentially interstellar particles to 41¹⁹. The remaining four particles agree with the speed criterion but not with the directionality criterion.

To summarise, for only one of the 45 particles within the mass range 10⁻¹⁴ kg ≤ m_msr ≤ 10⁻¹² kg and with impact speeds v_msr > 3 km/s an interstellar origin can be ascertained, and for only four of those particles it can be ruled out. As a rough estimate, one may assume that between a quarter and two thirds of the particles of Subset B may be interstellar.

6.4. Local interstellar gas-to-dust mass ratio

The gas-to-dust mass ratio in the LISM, R_g/d, is calculated by dividing the mass density, ρ_gas, of gas in the VLISM by the mass density, ρ_ISD, of the ISD measured by Ulysses:

$$\begin{array}{c}\hfill R_{\mathrm{g}/\mathrm{d}}=\frac{{\rho }_{\mathrm{gas}}}{{\rho }_{\mathrm{ISD}}}.\end{array}$$(7)

Details on how to calculate the gas mass density and the dust mass density are given in Appendix B.3. Here we assume a gas mass density of ρ_gas ≈ 5.14 × 10⁻²² kg/m³ (Krüger et al. 2015). The resulting gas-to-dust mass ratio will be slightly inaccurate because it does not take the filtering of ISD by the heliosheath into account (cf. Sect. 7.6): the mass distribution of ISD in the Solar System, where it can be measured in situ by, for example, Ulysses, is not expected to be identical to the pristine mass distribution of ISD in the LISM. However, massive particles, which contribute most strongly to the dust mass density, are the least affected by the filtering in the heliosheath (cf. Sect. 7).

Taking into account the measurement-derived masses of all particles of the Krüger subset, including those with v_msr < 3 km/s that most likely feature severely overestimates masses, results in an unrealistically low gas-to-dust mass ratio of $R_{\mathrm{g}/\mathrm{d}}\approx 0.{059}_{-0.023}^{+0.04}$ (see Table 3). If the particles with v_msr < 3 km/s are excluded, the gas-to-dust mass ratio increases by more than two orders of magnitude to $R_{\mathrm{g}/\mathrm{d}}\approx {22}_{-7}^{+11}$. For comparison, Slavin & Frisch (2008) estimate a hydrogen gas-to-dust mass ratio of R_g/d ∈ [149, 217] over the ∼130 pc-long sightline to ε CMa.

The comparably low value we determined stems from the influence of the most massive particles. As Sects. 6.2 and 6.3 show, many (if not most) of these particles are likely not of interstellar origin and should thus be excluded from the ISD mass distribution. Excluding all particles with measurement-derived masses above m_msr ≥ 10⁻¹² kg in accordance to the investigation into Subset A (see Sect. 6.2), and further removing an arbitrary half of all particles with measurement-derived masses above m_msr ≥ 10⁻¹⁴ kg in accordance to the investigation into Subset B (see Sect. 6.3) yields higher gas-to-dust mass ratios that are tabulated in Table 3. The inclusion of the possible ISD candidates of Subset A, particles (3, 5, 8, 10), severely influences the gas-to-dust mass ratio. For example, particle (10) by itself, which is the only ISD candidate of Subset A for which the measurement-derived mass, m_msr = 3.69 × 10⁻¹² kg, is not spurious, decreases the gas-to-dust mass ratio from $R_{\mathrm{g}/\mathrm{d}}={33}_{-13}^{+21}$ to ${19}_{-7}^{-12}$ if it is included.

If, as was done by Krüger et al. (2015), the iteratively calculated masses are used instead of the measurement-derived ones, the gas-to-dust mass ratio is considerably higher. Assuming that the majority of the particles of Subset A and about half the particles of Subset B are not interstellar, it is reasonable to remove an arbitrary half of all particles with iteratively calculated masses above m_iter ≥ 10⁻¹⁵ kg. The resulting dust mass densities and gas-to-dust mass ratios are tabulated in Table 3. Using the iteratively calculated masses, the gas-to-dust mass ratio that corresponds to the most likely selection of ISD – including only the ISD candidates of Subset A and excluding half of Subset B – is $R_{\mathrm{g}/\mathrm{d}}\approx {460}_{-190}^{+320}$.

This gas-to-dust mass ratio is significantly affected by the inclusion or exclusion of even a single massive ISD candidate. Excluding particle (5), which has an iteratively calculated mass of m_iter = 1.07 × 10⁻¹³, increases the gas-to-dust mass ratio from ${460}_{-190}^{+320}$ to ${670}_{-270}^{+440}$. This, as well as the large spread of the determined gas-to-dust mass ratios over more than one order of magnitude, shows how essential it is to be able to determine the origin and particle mass of the impacting dust particles with high accuracy. A more definitive selection of ISD particles would yield a more definitive gas-to-dust mass ratio. If the impact speeds were known with higher precision, the error factor of the determined masses and the uncertainty of the resulting gas-to-dust mass ratio could be reduced.

These gas-to-dust mass ratios are determined from the gas density in the VLISM and the dust density in the inner heliosphere. A correction for the modulation and filtering processes of ISD in the heliosheath (Sect. 7) would be required to more accurately determine the gas-to-dust mass ratio in the VLISM. However, these filtering effects are expected to act most severely on low-mass particles, which do not influence the gas-to-dust mass ratio as strongly as high-mass particles.

7. Investigating the filtering of interstellar dust in the heliosheath

Numerical simulations of ISD impacts on Ulysses were performed by Sterken et al. (2015). Their model did not include the heliosheath and could reproduce the measurements made by Ulysses either before or after ca. 2003 but not over the entire time span from 1992 to 2008. Sterken et al. (2015) ascribed this to a time-dependent filtering of ISD in the heliosheath, which is assumed to vary with the solar magnetic cycle through the Lorentz force.

By comparing numerical simulations similar to those by Sterken et al. (2015) to the measurements made by Ulysses, the time-dependent filtering function of ISD in the heliosheath can be constrained. The resulting heliosheath filtering function can then be applied to other spacecraft, allowing for more precise predictions of the encountered particle fluxes and measured size distributions of these other missions (see, e.g. Hunziker et al., in prep., for an application to DESTINY⁺).

7.1. Modulation of the LISM interstellar dust size distribution

As ISD moves from its unfiltered origin in the LISM, it experiences three general phenomena that can modulate its size distribution:

1. As ISD enters the heliosphere, it is filtered by the heliosheath through the Lorentz force, which varies with the solar magnetic cycle (see Table 4)²⁰. The exact details of this filtering are unknown but assumed to predominantly depend on particle properties such as the dust size, composition, and porosity. According to simulations, small particles (a ≲ 0.03 μm) cannot enter the heliosphere, with the exact cutoff size depending on the phase of the solar cycle (Slavin et al. 2012).

   Table 4.

   Minima and maxima of the solar magnetic cycle during the Ulysses mission.

2. On their trajectory from the heliopause to the detector ISD particles are affected by solar gravitation, solar radiation pressure, and the Lorentz force, the latter of which is highly dependent on the phase of the solar cycle, the particle size, composition, and porosity (‘fluffiness’).

3. What is measured by the detector depends, in addition to the ISD flux, on the effective sensitive surface area of the detector with respect to the ISD flow and on the relative speed between the dust particle and the spacecraft (cf. Appendix B.1); this is time-variable and size-dependent as well.

Assuming an, at this point, unspecified differential size distribution of ISD in the ISM, (dn/da)_ISM, the differential size distribution that is measured by Ulysses at a given time of observation can be described by

$$\begin{array}{c}\hfill {\left(\frac{\mathrm{d}n}{\mathrm{d}\mathrm{a}}\right)}_{\mathrm{obs}}=f_{\det }(a)·f_F(a)·f_{\mathrm{hs}}(a)·{\left(\frac{\mathrm{d}n}{\mathrm{d}\mathrm{a}}\right)}_{\mathrm{ISM}},\end{array}$$(8)

where f_hs is the filtering function of the heliosheath, f_F describes the modulation of the size distribution by the forces in the inner heliosphere, and f_det is the detector transfer function.

While the exact nature of the heliosheath filtering, f_hs, is unknown, the modulation by forces in the inner heliosphere, f_F, has been well investigated with numerical simulations (Sect. 7.3), and the detector transfer function, f_det, can be semi-analytically calculated (Sect. 7.4). Therefore, the heliosheath filtering function can be retrieved by (see Hunziker et al., in prep.)

$$\begin{array}{c}\hfill f_{\mathrm{hs}}(a)=\frac{1}{f_F(a)}·\frac{1}{f_{\det }(a)}·\frac{{\displaystyle {\left(\frac{\mathrm{d}n}{\mathrm{d}a}\right)}_{\mathrm{obs}}}}{{\displaystyle {\left(\frac{\mathrm{d}n}{\mathrm{d}a}\right)}_{\mathrm{ISM}}}}=:\frac{{\displaystyle {\left(\frac{\mathrm{d}n}{\mathrm{d}a}\right)}_{\mathrm{TS}}}}{{\displaystyle {\left(\frac{\mathrm{d}n}{\mathrm{d}a}\right)}_{\mathrm{ISM}}}},\end{array}$$(9)

where (dn/da)_obs is derived from spacecraft measurements, and (dn/da)_ISM must either be assumed or derived from remote astronomical observations. We term $f_F^{-1}(a)·f_{\det }^{-1}(a)·{(\mathrm{d}n/\mathrm{d}a)}_{\mathrm{obs}}=:{(\mathrm{d}n/\mathrm{d}a)}_{\mathrm{TS}}$ the inferred size distribution at the termination shock (cf. Sect. 7.5).

7.2. Measured size distribution of interstellar dust

The measurement-derived differential size distribution of the union subset in finite size bins, (Δn/Δa)_obs, as detailed in Appendix D.1, is plotted in Fig. 11a. For comparison, the extended MRN distribution (Mathis et al. 1977),

$$\begin{array}{c}\hfill {\displaystyle {\left(\frac{\mathrm{d}n}{\mathrm{d}a}\right)}_{\mathrm{MRN}}}=C_i{n}_{\mathrm{H}}{a}^{-3.5},\end{array}$$(10)

has been graphed as well, where C_i is a normalisation constant, C_C = 10^−25.13 cm^5/2/H for carbonaceous grains and C_SiO = 10^−25.11 cm^5/2/H for silicates (Draine & Lee 1984). The MRN distribution, which is valid only up to particle sizes of 0.25 μm for silicates and 1 μm for carbonaceous grains, has been extended over the entire particle size range of the available simulations, summing the distributions for silicates and carbonaceous grains, C_i = C_C + C_SiO. The number density of hydrogen nuclei, n_H = 0.1 cm⁻³, is uncertain; the resulting MRN distribution could be shifted to higher or lower values. We note that the MRN distribution has been derived from remote astronomical observations over long lines of sight through the Galaxy, assuming that the dust types and size distribution are similar in the LIC.

The measured differential size distribution has been plotted in Fig. 11a for four four-year subsets of the data to illustrate the influence of the solar cycle, and additionally for all sixteen years of measurements. In the determination of the length of the four-year time bins, the measurement gaps such as perihelion crossings, dust stream crossings, or instrument switch-offs have been taken into account; v_d = 26 km/s has been assumed throughout.

As is expected, the measured differential size distribution is reduced compared to the assumed MRN distribution and deviates more strongly for smaller particles than for larger ones. In the time period of 1996–2000, immediately following the solar minimum of the defocusing phase in August 1996 (cf. Table 4), the measured differential size distribution is strongly reduced compared to the MRN distribution, especially at small particle sizes, because fewer ISD particles were detected (Landgraf 2000; Sterken et al. 2012a)²¹. Conversely, in the time period of 2004–2008, preceding the solar minimum of the focusing phase in November 2008, the measured differential size distribution is considerably higher, especially for intermediate particles (a ≈ 0.1 μm).

The vertical error bars in Fig. 11a stem from the standard deviation of the number of measured particle impacts per size and time bin, Eq. (D.5). These errors are largest for the time period following the solar minimum of the defocusing phase, 1996–2000, during which the fewest particles were recorded. In order to avoid low statistics, it is inadvisable to divide the data into too many time intervals; therefore, no finer time resolution was chosen. Further uncertainties are discussed in Sect. 7.7.

7.3. Size-dependent, time-averaged modulation of interstellar dust by forces in the inner heliosphere

The modulation factor through the forces in the Solar System, f_F, has been investigated by numerical simulations of dust particle trajectories with the IMEX code (Sterken et al. 2012a; Strub et al. 2019). This code returns the time-dependent factor by which the number density, n, of ISD at a given particle size, a, is enhanced or reduced compared to a reference density, n_ISM, due to solar gravitation, solar radiation pressure, and the Lorentz force in the inner heliosphere:

$$\begin{array}{c}\hfill f_F(a;t)=\frac{n(a;t)}{n_{\mathrm{ISM}}}·\end{array}$$(11)

More details on the simulations are provided in Appendix D.2.

Fig. 11b shows the size-dependent, time-averaged modulation factor, ${\overline{f}}_F$, for the same four time periods as the measured differential size distribution; the time-dependent modulation is given in Appendix D.3. As Fig. 11b shows, the simulated modulation factor through the forces in the Solar System at the location of Ulysses approaches f_F ≈ 1 for large particles, and drops for smaller particles (f_F < 1). However, in the time period preceding the solar minimum of the defocusing phase (1996), particles of intermediate size (0.15 μm < a < 0.5 μm) were enhanced at Ulysses’ orbit.

7.4. Size-dependent, time-averaged detector transfer function

The effective sensitive surface area of Ulysses’ dust detector depends on the angle of incidence, α, between the detector’s boresight and the direction from which the dust particles came, and on the relative speed, |v_rel|, between the dust and the spacecraft. In analogy to Appendix B.1, the time-dependent and size-dependent detector transfer function is given by

$$\begin{array}{c}\hfill f_{\det }(a;t)=\frac{A_{\mathrm{eff}}(a;t,\alpha (t))}{A_{max}}=\frac{|{\mathit{v}}_{\mathrm{rel}}|(a;t)}{|{\mathit{v}}_{\mathrm{ISD}}|}\frac{A_{\mathrm{sens}}(\alpha (a;t))}{A_{max}},\end{array}$$(12)

where A_sens(α) is the angle-of-incidence-dependent sensitive surface area of the detector (see Fig. B.1), A_max = 0.1 m² is the detector’s total surface area, and v_rel(a;t) has been taken from the numerical simulations. This detector transfer function, f_det, indicates how strongly the measured particle flux appears enhanced or reduced compared to the particle flux at the spacecraft’s orbital position.

The time-averaged detector transfer function, ${\overline{f}}_{\det }$, has been averaged over the same time periods as ${\overline{f}}_F$; it is plotted in Fig. 11c. The time-dependent detector transfer function is shown in Appendix D.4.

The accuracy of this transfer function is limited by the averaging of the simulated particles within each simulation grid cell. However, because the transfer function remains within ${\overline{f}}_{\det }=0.35\pm 0.08$, the influence of its variation is not as strong as that of f_F, which changes by multiple orders of magnitude. Further uncertainties are discussed in Sect. 7.7.

7.5. Inferred size distribution of interstellar dust at the termination shock

The differential size distribution at the outer border of the inner heliosphere, the termination shock, was inferred by applying the inverse of ${\overline{f}}_F$ and ${\overline{f}}_{\det }$ to the measured differential size distribution, Eq. (9). This is plotted in Fig. 11d together with the MRN distribution, Eq. (10).

As Fig. 11d shows, the shapes of the inferred differential size distributions fit reasonably well to the MRN distribution for intermediate and large particles but deviate more strongly from a MRN-like power law for small particles. In the time period of 1992–1996 that precedes the solar minimum of the defocusing phase of 1996 (cf. Table 4), the inferred number density for small particles is lowest, whereas in the next time period, 1996–2000, immediately following this solar minimum, it is highest. This is a notable difference from the measured differential size distribution, where the opposite holds, and stems from the modulation factor due to inner heliosphere forces in relation to the measurements, ${\overline{f}}_F$, which causes the weakest depletion during 1992–1996 and the strongest depletion during 1996–2000 (see also Sterken et al. 2015).

The inferred differential size distribution agrees with the findings of Slavin et al. (2012), who found a strong depletion of small particles (a ≲ 0.1 μm) at the outer boundary of the inner heliosphere during the defocusing phase, and an enhancement locally at the heliosheath during the focusing phase.

The vertical error bars graphed in Fig. 11d correspond to the Gauss-propagated standard deviations of the measured size distribution, Eq. (D.7). Further uncertainties of this analysis are discussed in Sect. 7.7.

7.6. Filtering modulation of ISD in the heliosheath

The resulting time-averaged heliosheath filtering function, ${\overline{f}}_{\mathrm{hs}}$, Eq. (9), is calculated by dividing the inferred differential size distribution (Fig. 11d) by the MRN distribution, Eq. (10). Fig. 12a shows the heliosheath filtering function versus the particle size for the previously used four-year time periods. The vertical error bars correspond to the vertical error bars of the inferred differential size distribution, Eq. (D.9), which take into account only the (assumed) Poissonian standard deviation of the number of measured particle impacts per size and time bin. We discuss additional uncertainties in Sect. 7.7.

As expected, ${\overline{f}}_{\mathrm{hs}}\approx 1$ for large particles (a > 0.4 μm), and ${\overline{f}}_{\mathrm{hs}}<1$ for intermediate and smaller particles, except in the time period that immediately follows the solar minimum of the defocusing phase, 1996–2000 (cf. Table 4), where ${\overline{f}}_{\mathrm{hs}}>1$. This may be caused by the filtering modulation through inner heliosphere forces in the simulations, f_F, which reduces the number of small particles during this time period more strongly by at least an order of magnitude compared to other time periods (cf. Fig. 11b), in comparison with less filtering in the Ulysses dataset. This may indicate a focusing effect for small particles in the heliosheath during 1996–2000.

ISD particles measured by Ulysses in 1996–2000, which experienced a strong defocusing phase inside the Solar System, would have crossed the termination shock in ca. 1981–1985²². This time period coincides with the focusing phase surrounding the solar minimum of 1986, comparable to the time period of 2004–2008 preceding the solar minimum in 2008/2009, one ∼22 yr-cycle later. A similar focusing effect of ISD in the heliosheath has been simulated by Slavin et al. (2012) using a static focusing heliospheric magnetic field in the entire heliosphere including the heliosheath. This was illustrated by Sterken et al. (2015, Fig. 19) and later simulated by Godenko & Izmodenov (2024).

We note that the total travel time of ISD from the LISM to the inner Solar System is approximately 20–22 yr²³, which is comparable to the period of the solar magnetic cycle (Sterken et al. 2015). The width of the heliosheath varies over the solar magnetic cycle (Scherer & Fahr 2003). Thus, the time ISD spends in the heliosheath may vary with the phase of the solar magnetic cycle as well (see, e.g., Sterken et al. 2015, Fig. 19).

The time-averaged heliosheath filtering function that spans the full time range, 1992–2008, strongly resembles ${\overline{f}}_{\mathrm{hs}}$ for the time period preceding the solar minimum of the defocusing phase, 1992–1996. Nevertheless, that ${\overline{f}}_{\mathrm{hs}}$ deviates from its average so strongly for the time periods of 1996–2000 and 2000–2004 shows how important it is to treat this analysis in relation to the phase of the solar cycle. For a more accurate assessment, it would be essential to apply this methodology to a larger dataset that spans a full solar magnetic cycle or, ideally, even multiple cycles²⁴.

7.7. Uncertainties of the heliosheath filtering modulation

Due to its complexity and the large number of assumptions, the analysis illustrated in the previous sections has multiple sources of uncertainty.

In Figs. 11 and 12 the vertical error bars only correspond to the Poissonian standard deviation of the number of particle impacts per size and time bin (see Appendix D.5). There are, however, more uncertainties and assumptions that are not taken into account, such as the error factor of the measurement-derived particle mass, which is typically an order of magnitude (Grün et al. 1995a).

As Figs. D.2 and D.3 show, the time-dependent modulation by inner heliosphere forces, f_F, and the time-dependent detector transfer function, f_det, are not constant during the selected time intervals. Time-averaging these functions introduces uncertainties that are not taken into account by the error bars of Figs. 11 and 12. Smaller time intervals would reduce these uncertainties and result in a finer time resolution of the heliosheath filtering function²⁵. However, dividing the total number of particles into more bins would require a larger dataset. This also holds true for the bin width of the particle size.

Further uncertainties lie in the nature of the numerical model, for example via the assumption for the interplanetary magnetic field (see, e.g. Sterken et al. 2012a). Furthermore, in each model cell, the particle velocity is averaged over all particles, introducing another uncertainty. Yet another uncertainty stems from the assumed β-curve; if a different β-curve were assumed, different parameter sets of the numerical model, (β, Q/m), would have to be used. Succinctly put, the heliosheath filtering modulation derived by this method is only as good as the numerical model. Finally, uncertainties of the MRN distribution, including a possible deviation from the MRN distribution by the ISD size distribution in the LISM, are not taken into account in our analysis.

The choice of the ISD selection criteria (cf. Sect. 3.2) and the mass determination method (cf. Sect. 4) have an influence as well. This is investigated in Sects. 7.7.1 and 7.7.2, respectively.

7.7.1. Uncertainties introduced by the choice of the ISD selection criteria

The above analysis was performed with the union subset. For the Krüger subset, which lacks some of the particles in the intermediate-to-large size range but is otherwise similar to the union dataset (see Fig. 8), the inferred differential size distribution changes only negligibly.

For the Strub subset, low masses have been excluded; thus, no data is available for the lowest size bin, and the lower end of the differential size distribution is depleted more strongly. This is shown in Fig. 12b. The Strub subset contains only half as many events as the union subset. The resulting differential size distributions, and therefore the heliosheath filtering function also shows values that are lower by roughly a factor of two. However, the exclusion of low-mass particles in the Strub subset directly biases the ISD size distribution much more strongly than the indirectly biased Krüger subset does; the Strub subset was not devised to investigate the ISD mass or size distribution (Strub et al. 2015). Therefore, it is not advised to use the Strub subset for this type of analysis; here, we have used it merely for comparison with the other datasets.

7.7.2. Uncertainties introduced by the choice of the mass determination method

So far, the masses as derived from the measured signal rise times and amplitudes were used to calculate the particle sizes (cf. Sect. 4.1). If, instead, the approximate determination of the mass (‘β ≡ 1’, cf. Sect. 4.2) is used, the inferred differential size distribution shows only minor differences (not depicted).

If the iterative method (cf. Sect. 4.3) is used to determine the particle masses, the resulting inferred differential size distribution is also remarkably similar to the distributions generated with the masses of the other mass determination methods. This is shown by Fig. 12c. Nevertheless, some differences are apparent: large particles (a ≈ 0.54 μm) are depleted more strongly by half an order of magnitude for the iterative method compared to the measurement-derived mass determination, notably deviating from the extended MRN distribution even when accounting for statistical uncertainties. For small particles (a < 0.15 μm) the heliosheath filtering factor deviates by less than half an order of magnitude from the results for the measurement-derived mass determination, and at intermediate sizes the results are only negligibly dependent on the mass determination method. This holds true for all observed time periods.

8. Summary and conclusions

We have performed three distinct analyses of the dataset of ISD impacts on the Ulysses spacecraft. We investigated the influence of the choice of the different ISD selection criteria and of the competing mass determination methods on the ISD particle fluxes, directionality, and mass distribution (Sect. 5). We scrutinised the most massive particles detected by Ulysses to determine whether they are ISD, which affects the gas-to-dust mass ratio in the LISM inferred from in situ measurements (Sect. 6). We backcalculated the filtering effect of ISD in the heliosheath from the ISD size distribution measured by Ulysses with the aid of numerical simulations (Sect. 7). The resulting filtering modulation can be used to predict the fluxes on other spacecraft in the Solar System, including the effect of the heliosheath filtering without the need for a full heliosphere simulation (cf. Hunziker et al., in prep.). These investigations are summarised in Sect. 8.1, 8.2, and 8.3, respectively.

8.1. Conclusions of the investigation into the ISD selection criteria and mass determination methods

The influence of the different ISD selection criteria on the ISD particle flux and the ISD flow direction was investigated in Sects. 5.1 and 5.2. We make the following conclusions.

The different ISD selection criteria bias the resulting ISD subsets in different ways by design (Krüger et al. 2015; Strub et al. 2015). For example, the criterion that identifies ISD by its direction, (1.I), was devised by Krüger et al. (2015) to investigate the particles’ mass distribution, and it directly biases the directionality of the resulting Krüger subset. The criterion that excludes IDPs based on the measured signal amplitudes, (2.II), was devised by Strub et al. (2015) to investigate the particles’ directionality, and it directly biases the mass distribution of the resulting Strub subset. Nevertheless, criterion (1.I) also indirectly biases the mass distribution, and criterion (2.II) indirectly biases the directionality of the respective subsets.

The different criteria for ISD identification do not result in different long-term patterns in the particle flux or flow direction, except for a general factor of approximately two of the particle flux between the Krüger and the Strub subsets that directly stems from the number of particles contained in the respective subset. However, the different criteria do influence local features. This is, in most cases, caused by the lack of small particles in the Strub subset. For example, the Strub subset contains ∼44% fewer impacts than the Krüger subset, and the resulting particle flux is systematically lower by ∼45%. The shift in the ISD directionality in 2005–2006 is observed for all ISD subsets.

The most massive ISD particles measured by Ulysses dominate the differential mass distribution and have the largest influence on the gas-to-dust-mass ratio. Therefore, we compared the three different mass determination methods in Sect. 5.3 and the resulting mass distributions in Sect. 5.4. We came to the following conclusions.

The overall shape of the resulting mass distribution is similar for all three mass determination methods except that neither the mass distribution of the ‘β ≡ 1’ method nor of the iterative calculations feature any masses above m > 10^−12.5 kg, whereas the measurement-derived distribution does, thus confirming earlier results by Landgraf (1998) and Krüger et al. (2015). The distribution of the measurement-derived impact speeds covers a much wider range than for the other methods. The iterative method assumes an interstellar origin and thus cannot be used to determine whether a given particle is interstellar or not. The iterative method is generally flawed, as it does not take into account the directionality of the respective particle. This does not only pertain to the effects of the Lorentz force but also the deflection by solar gravitation and solar radiation pressure. We furthermore note that the inferred gas-to-dust mass ratio in the LISM (Sect. 6.4) is higher by orders of magnitude when using the iteratively calculated masses than when using the measurement-derived masses (see also Sect. 8.2).

8.2. Conclusions of the investigation into the most massive ISD particles detected by Ulysses

We investigated the most likely origin of the ten most massive ISD particles detected by Ulysses in Sect. 6.2 and conclude the following. Three to four of these ten particles may be compatible with the particle speed and direction of the ISD inflow. However, due to large measurement uncertainties, these particles cannot be unambiguously identified as ISD. Another two to three of these ten particles are ISD candidates only if wall impacts are taken into account. Four of these ten particles are inconsistent with an ISD origin.

We furthermore investigated 45 particles from the adjacent mass range of 10⁻¹⁴ kg ≤ m_msr ≤ 10⁻¹² kg, corresponding to particle radii of 1 μm ≲ a ≲ 5 μm, in Sect. 6.3. We conclude that four of these 45 particles are unlikely to be ISD, and only one of these particles is most likely ISD. The results are ambiguous for the remaining 40 particles.

Combined, these 55 most massive particles show the largest disagreement between their measurement-derived and their iteratively calculated masses, whereas for the lower-mass particles the two mass determination methods are more consistent (cf. Fig. 9). This indicates that the vast majority of the (lower-mass) particles identified by the ISD criteria do agree with an ISD origin.

We inferred the gas-to-dust mass ratio in the LISM in Sect. 6.4 and conclude the following. Following the approach by Krüger et al. (2015), we find a gas-to-dust mass ratio in the LISM of $R_{\mathrm{g}/\mathrm{d}}={310}_{-130}^{+210}$. This uses the iteratively calculated masses of all particles in the Krüger subset. Discounting the massive particles that were previously found unlikely to be ISD can increase the gas-to-dust mass ratio to values as high as $R_{\mathrm{g}/\mathrm{d}}={1000}_{-300}^{+500}$ for the iteratively calculated masses and $R_{\mathrm{g}/\mathrm{d}}={61}_{-26}^{+50}$ for the measurement-derived masses. The most likely gas-to-dust mass ratio is $R_{\mathrm{g}/\mathrm{d}}\approx {460}_{-190}^{+320}$. The exclusion of only a single massive ISD candidate can increase this value by almost 50%. The large spread of determined values of the gas-to-dust mass ratio highlights the necessity for more accurate determinations of the particles’ direction of motion and mass for micrometer-sized dust.

Within the scope of this analysis, we estimated the necessary accuracy by which the heliocentric particle speed must be known to distinguish ISD from IDPs (Sect. 6.2.1; Appendix C). We conclude that the error factor of the measurement of a dust particle’s impact speed for Ulysses is 1.9 (Grün et al. 2010). The detector opening angle of 70° introduces an uncertainty in a particle’s direction of motion, yielding a speed error factor of 1.36 at aphelion and 3.8 at perihelion. This results in a total error factor of the reconstructed heliocentric speed between 2.6 at aphelion and 7.2 at perihelion.

On a Jupiter-like orbit, the heliocentric particle speed must be known within a multiplicative error factor of 2.44 to distinguish ISD from IDPs on circular orbits and within an error factor of 1.73 to distinguish ISD from highly eccentric IDPs in the limit of parabolic orbits. On an Earth-like orbit, these error factors decrease to 1.66 to distinguish ISD from circular IDP orbits and to 1.18 for parabolic IDP orbits. This total error factor consists of the error factor introduced by the uncertainty of the impact speed and the one introduced by the uncertainty of the particle direction, which are roughly multiplicative. For example, if the impact speed is known with an error factor of 1.10, corresponding to roughly ${}_{-9\%}^{+10\%}$, the direction of origin must be known within an error factor of about 1.07, corresponding to an accuracy of 4°, to obtain a total error factor below 1.18. This shows that Ulysses cannot consistently and unambiguously distinguish ISD from IDPs for single particle impacts.

8.3. Conclusions of the investigation into the filtering of ISD by the heliosheath

We introduced a method to determine the ISD filtering effects of the heliosheath in Sect. 7 that is based on numerical simulations of ISD filtering within the Solar System, on in situ data observed by Ulysses inside the Solar System, and on an assumed ISD size distribution in the LISM. Its results do not only constrain future models of the heliosheath but can also be used to generate more accurate predictions of the ISD measurements for future missions. We conclude that assuming an MRN-like size distribution in the LISM and backcalculating the differential size distribution at the termination shock resulted in a MRN-like distribution for large particles. This size distribution is strongly modulated for small particles, and this modulation depends on the phase of the solar magnetic cycle. In addition to the defocusing effect of ISD in the heliosheath, focusing effects may also be possible (cf. Slavin et al. 2012; Sterken et al. 2015; Godenko & Izmodenov 2024).

This method serves as a proof of concept of backcalculating the filtering and focusing effects of ISD in the heliosheath by combining in situ spacecraft measurements with simulations. However, its application is hampered by the coarse time resolution, which is limited by the number of detected particles.

We conclude generally that almost every analysis would strongly benefit from more precise measurements of the particle origin and impact speed. This can be provided by, for example, a trajectory sensor (Auer 1975; Srama et al. 2005; Grün et al. 2019) or a segmented grid (Simolka et al. 2024). Similarly, a dust detector with a large surface area would yield better statistics and thus more precise results. Furthermore, measurements from spacecraft at multiple positions in the Solar System at the same time would give a more holistic view of ISD within the heliosphere. This type of setup is promised by the availability of dust detectors on multiple spacecraft (e.g. DESTINY⁺, IMAP, Lunar Gateway, Europa Clipper, JUICE²⁶, Comet Interceptor²⁷, SunCHASER/L₄) that will be active during the coming focusing phase of the solar magnetic cycle (see, e.g. Sterken et al. 2012b; Soja et al. 2013; McComas et al. 2018; Wozniakiewicz et al. 2021; Posner et al. 2021; Cho et al. 2023; Krüger et al. 2024b).

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Acknowledgments

V. J. Sterken, S. Hunziker, and L. R. Baalmann received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement N° 851544 – ASTRODUST. We would like to thank the anonymous referee for thought-provoking comments, and we are thankful for the statistical advice of Krzysztof Cybulski, Lukas Graz, James W. Kirchner, and Adrian Jäggi. This work makes use of the Julia programming language, v1.10.2 (Bezanson et al. 2017), and its packages BasicInterpolators.jl, Distributions.jl, Optim.jl (Mogensen & Riseth 2018), Plots.jl (Breloff 2023), and SPICE.jl (Acton 1996). Ulysses dust impact data was taken from the publicly available dataset of Grün et al. (2010): https://sbn.psi.edu/pds/resource/udds.html. Ulysses orbital data was taken from the publicly available Ulysses SPICE kernel: https://naif.jpl.nasa.gov/pub/naif/ULYSSES/. The planetary SPICE kernel de432s.bsp and the leapsecond SPICE kernel naif0012.tls were taken from the publicly available generic kernels: https://naif.jpl.nasa.gov/pub/naif/generic_kernels/. Author contributions: L.R.B. performed the data analysis and wrote and revised the manuscript. T.J. curated the data and performed the preliminary data analysis of Sect. 5 (Janisch 2021). S.H. and V.J.S. constructed the methodology of Sect. 7 (Hunziker et al., in prep.) and performed the simulations. P.S. and H.K. curated the original data and revised the manuscript. K.H. and M.S. performed preliminary data analysis (Hofstetter & Sieber 2021). V.J.S. initiated and supervised the project, conceptualised the methodology, revised the manuscript, and acquired funding.

References

Appendix A: The spacecraft rotation angle and the dust detector boresight

The boresight direction of Ulysses’ dust detector can be expressed in terms of the rotation angle, ϕ_rot, which gives the phase of the spacecraft’s rotation around its spin axis. This appendix presents how to calculate the detector boresight from the rotation angle and vice-versa.

The communication dish of Ulysses continuously pointed towards Earth, and the spacecraft-Earth axis is referred to as the spin axis. Let $\widehat{\mathit{s}}$ be the unit vector pointing from Earth towards Ulysses. The dust detector is fixed at an angle of γ_det = 95° with respect to the spin axis; it is tilted slightly towards Earth.

Because the detector’s angle with respect to $\widehat{\mathit{s}}$ is fixed, its pointing can be reconstructed from a single value, the rotation angle, ϕ_rot. As the name implies, this angle gives the phase of rotation around the spin axis.

The rotation angle, ϕ_rot, is defined as the angle between the pointing of the detector (‘boresight’), $\widehat{\mathit{b}}$, and the ecliptic north, $\widehat{\mathit{N}}$, projected onto the plane perpendicular to the spin vector, $\widehat{\mathit{s}}$, which is referred to as the spin plane. The spin plane can therefore be spanned by the vectors

$$\begin{array}{cc}\hfill {\widehat{\mathit{p}}}_y& =\widehat{\mathit{N}}\mathbf{\times }\widehat{\mathit{s}},\hfill \end{array}$$(A.1)

$$\begin{array}{cc}\hfill {\widehat{\mathit{p}}}_x& =\widehat{\mathit{s}}\mathbf{\times }{\widehat{\mathit{p}}}_y,\hfill \end{array}$$(A.2)

and thus the projection, $\widehat{\mathit{p}}$, of the boresight onto the spin plane is given by

$$\begin{array}{c}\hfill \widehat{\mathit{p}}={\widehat{\mathit{p}}}_{x}cos{\varphi }_{\mathrm{rot}}+{\widehat{\mathit{p}}}_{y}sin{\varphi }_{\mathrm{rot}}.\end{array}$$(A.3)

The boresight can be reconstructed from the rotation angle with

$$\begin{array}{c}\hfill \widehat{\mathit{b}}=\widehat{\mathit{s}}cos{\gamma }_{\det }+\widehat{\mathit{p}}sin{\gamma }_{\det }.\end{array}$$(A.4)

Because $\widehat{\mathit{s}}$ and $\widehat{\mathit{N}}$ are unit vectors, so are ${\widehat{\mathit{p}}}_{x,y}$, $\widehat{\mathit{p}}$, and $\widehat{\mathit{b}}$.

Vice versa, the rotation angle, ϕ_rot, corresponding to the detector pointing into a given direction, (l_det, b_det), can be calculated with:

$$\begin{array}{cc}\hfill \widehat{\mathit{r}}& ={(-1,l_{\det },b_{\det })}_{\mathrm{ecl}}\hfill \end{array}$$(A.5)

$$\begin{array}{cc}\hfill {\mathit{r}}_x& ={\widehat{\mathit{p}}}_x·\widehat{\mathit{r}}\hfill \end{array}$$(A.6)

$$\begin{array}{cc}\hfill {\mathit{r}}_y& ={\widehat{\mathit{p}}}_y·\widehat{\mathit{r}}\hfill \end{array}$$(A.7)

$$\begin{array}{cc}\hfill {\varphi }_{\mathrm{rot}}& =arctan({\mathit{r}}_y,{\mathit{r}}_x)\hfill \end{array}$$(A.8)

Here, $\widehat{\mathit{r}}$ is the unit vector in heliocentric ecliptic spherical coordinates, coming from the direction of (l, b); r_x,y are its projections onto the spin plane, spanned by ${\widehat{\mathit{p}}}_{x,y}$; and the used arctan function is the two-argument arcus tangens, often abbreviated as atan2. The notation (r, l, b)_ecl indicates that the vector is given in ecliptic coordinates; that is, it has the length r and points into the direction (l, b). The rotation angle corresponding to the ISD inflow direction of ϕ_ISD taken from Swaczyna et al. (2018), (255.41° ,5.03° ), which we introduce in Sect. 2.1, is graphed against time in Fig. A.1.

Appendix B: Derivation of ISD properties

This appendix gives details on how to derive the ISD particle flux (Appx. B.1), the ISD flow direction (Appx. B.2), and the gas-to-dust mass ratio (Appx. B.3) from the measurements made by Ulysses (Grün et al. 2010).

B.1. Derivation of the ISD particle flux

The ISD particle flux is derived from the number of particles measured by the detector, N, per unit time, Δt, and per unit effective sensitive surface area averaged over Δt, ${\overline{A}}_{\mathrm{eff}}$:

$$\begin{array}{c}\hfill F=\frac{N}{\mathrm{\Delta }t{\overline{A}}_{\mathrm{eff}}}.\end{array}$$(B.1)

For the Ulysses dust detector, the sensitive area, A_sens, depends on the impact angle, α, which is the angle between the trajectory of the incoming dust particle and the boresight of the detector, also referred to as the angle of incidence,

$$\begin{array}{c}\hfill \alpha =arccos\left(\frac{-\widehat{\mathit{b}}·{\mathit{v}}_{\mathrm{ISD},\infty }}{v_{\mathrm{ISD},\infty }}\right).\end{array}$$(B.2)

This is shown in Fig. B.1. Only for α = 0° the entire surface area of the detector is sensitive, $A_{\mathrm{sens}}(\alpha =0°)=A_{max}=0.1{\mathrm{m}}^2$, and decreases for larger α until the opening angle of the detector, α_max = 70°, is reached, A_sens(α_max) = 0 m².

As Altobelli et al. (2004) noted, not only impacts on the detector target but also against the detector’s inner wall may lead to a successful measurement of the dust impact. If wall impacts are taken into account, the sensitive surface area is significantly higher and non-zero up to an opening angle of α_max = 90° (see Fig. B.1).

However, the detector is not at rest with respect to the impacting particles. Therefore, its effective sensitive surface area,

$$\begin{array}{c}\hfill A_{\mathrm{eff}}=\frac{|{\mathit{v}}_{\mathrm{rel}}|}{|{\mathit{v}}_{\mathrm{ISD}}|}{A}_{\mathrm{sens}},\end{array}$$(B.3)

also depends on the magnitude of the relative velocity between the spacecraft and the dust particle,

$$\begin{array}{c}\hfill {\mathit{v}}_{\mathrm{rel}}={\mathit{v}}_{\mathrm{U}}-{\mathit{v}}_{\mathrm{ISD}},\end{array}$$(B.4)

where v_U is the heliocentric velocity of Ulysses.

Thus, the effective sensitive surface area depends on the impact angle and on the relative velocity between the spacecraft and the dust particle, which in turn depends on the detector boresight and on Ulysses’ orbital position and velocity. In total, A_eff is determined depending on the rotation angle and on the date and time of impact.

This is illustrated by Fig. B.2 (see also Fig. A.1). The effective sensitive surface area is given with respect to a preferential direction, which here is the ISD inflow direction, and has the inflow speed of ISD as a parameter. Thus, this determination assumes that the dust inflow comes precisely from the ISD inflow direction with exactly v_ISD, ∞, meaning not only that the ISD inflow should be perfectly collimated, but also that it should not be influenced by any forces such as solar gravitation, solar radiation pressure, or the Lorentz force.

To calculate the average ISD flux during a finite time interval, the effective sensitive surface area must be averaged over that time interval and over the full rotation angle. The time-averaging must take into account gaps during which no ISD could be detected, some of which are caused by the exclusion criteria (cf. Sect. 3.1): the period before the Jovian flyby, the ecliptic crossings at Ulysses’ perihelion, and the Jovian dust streams.

Multiple instrument switch-offs also occurred (see Table B.1), during which no measurements could be taken. The first of these periods corresponds to the time interval between the launch of the spacecraft and the initialisation of the dust detector, and the last of these periods similarly corresponds to the time interval between the shutdown of the dust detector and the end of the mission. The in-between time periods were either caused by anomalies where all scientific instruments were shut down, or in later years by power-sharing between the various instruments. All these gaps have the effective function of instrumental dead times, meaning that they reduce Δt in Eq. (B.1). This is especially relevant for the measurements taken before 1994 and since 2002, when many causes for gaps occurred.

The averaging of ${\overline{A}}_{\mathrm{eff}}$ in Eq. (B.1) complicates the calculation of the flux: analytically, ${\overline{A}}_{\mathrm{eff}}$ would be calculated with the integral over the time interval; numerically, it must be approximated by summation over finite steps of the time and rotation angle, Δt and Δϕ_rot, respectively. In practice, it has proven expedient to use Δt = 5 d and Δϕ_rot = 6°. Compared to calculations with Δt = 1 d and Δϕ_rot = 1° this introduces a maximum error of 0.2% while decreasing computational costs by one and a half magnitudes. Because the uncertainty of the flux (Appx. B.1.1) is generally much higher, this uncertainty is not taken into account. This determination of the ISD particle flux has been applied to the differently selected ISD subsets in Sect. 5.1.

B.1.1. Confidence interval of the ISD particle flux

The lower and upper limits, F_l, u, of the confidence interval of the ISD particle flux have been calculated in analogy to Eq. (B.1) by

$$\begin{array}{c}\hfill F_{\mathrm{l},\mathrm{u}}=\frac{{\lambda }_{\mathrm{l},\mathrm{u}}}{\mathrm{\Delta }t{\overline{A}}_{\mathrm{eff}}},\end{array}$$(B.5)

where λ_l, u are the lower and upper value of the expected number of ISD particle impacts during the respective time span, respectively. These values were determined under the assumption that ISD particle impacts are Poisson-distributed.

The Poisson distribution, Φ(λ, k), and its cumulative distribution function, CDF(Φ), are given by

$$\begin{array}{cccc}\hfill \mathrm{\Phi }(\lambda ,k)& =\frac{{\lambda }^k{e}^{-\lambda }}{k!},\hfill & \hfill \mathrm{CDF}(\mathrm{\Phi }(\lambda ,k))& ={\displaystyle \sum _{i=0}^k}\mathrm{\Phi }(\lambda ,i),\hfill \end{array}$$(B.6)

where k is the number of events (i.e. ISD particle impacts), and λ is a parameter that is identical to the variance of the distribution.

The cumulative distribution function is related to a given confidence level of $L_{\mathrm{C}}=0.8413\widehat{=}1\sigma$ by (Gehrels 1986):

$$\begin{array}{cccc}\hfill \mathrm{CDF}(\mathrm{\Phi }({\lambda }_{\mathrm{l}},N-1))& =L_{\mathrm{C}},\hfill & \hfill \mathrm{CDF}(\mathrm{\Phi }({\lambda }_{\mathrm{u}},N))& =1-L_{\mathrm{C}}.\hfill \end{array}$$(B.7)

In practice, λ_l, u are the λ that minimise the terms

$$\begin{array}{ccc}& |\mathrm{CDF}(\mathrm{\Phi }(\lambda ,N-1))-L_{\mathrm{C}}|,\hfill & \hfill |1-\mathrm{CDF}(\mathrm{\Phi }(\lambda ,N))-L_{\mathrm{C}}|,\end{array}$$(B.8)

respectively. We note that the equation for the lower limit of Eq. (B.7) does not hold for N = 0; λ_l(N = 0) = 0 is used instead.

B.2. Derivation of the ISD flow direction

The precise direction from which an impacting particle came is not known. However, the particle must have come from the spherical sector spanned by the detector’s opening angle centred around the detector boresight at the time of impact, which is given by the rotation angle. The detector’s opening angle is 70° if wall impacts are not considered; when taking wall impacts into account, the detector’s field of view amounts to a full hemisphere (cf. Appx. B.1; Altobelli et al. 2004). By averaging over multiple ISD impacts, a more precise estimate of the rotation angle from which the ISD particles came can be made (Grün et al. 2001, Ch. II.D).

ISD particles are assumed to come from the ISD inflow direction. A good measure to evaluate this assumption is the average deviation of the detector’s rotation angle from the rotation angle corresponding to the ISD inflow direction, ϕ_ISD (see Fig. A.1),

$$\begin{array}{c}\hfill {\varphi }_{\mathrm{dev}}=\overline{{\varphi }_{\mathrm{rot}}-{\varphi }_{\mathrm{ISD}}},\end{array}$$(B.9)

where the circular mean has been used for averaging.

The uncertainty of this measure is given by

$$\begin{array}{c}\hfill \mathrm{\Delta }{\varphi }_{\mathrm{dev}}={\varphi }_{\mathrm{rot}}(1\sigma )/\sqrt{N},\end{array}$$(B.10)

where ϕ_rot(1σ) is the angle at which the effective sensitive area reaches the value of 1σ ≈ 68.27% of its maximum, and N is the number of impacts over which was averaged. The 1σ-angle is ϕ_rot(1σ) = 46.7° if wall impacts are taken into account, and 31.1° if they are not. This uncertainty is similar but not identical to the tabulated values provided by Grün et al. (2001, Table 3), as noted by Janisch (2021, Table 3.1). This has been applied to the differently selected ISD subsets in Sect. 5.2.

B.3. Calculation of the gas-to-dust mass ratio

In order to calculate the gas-to-dust mass ratio in the LISM, R_g/d = ρ_gas/ρ_ISD, the gas mass density in the VLISM, ρ_gas, and the dust mass density measured by Ulysses, ρ_ISD, must be known. Krüger et al. (2015) calculated the gas mass density from the hydrogen number density, n_H, and the helium number density, n_He,

$$\begin{array}{c}\hfill {\rho }_{\mathrm{gas}}=n_{\mathrm{H}}{m}_{\mathrm{H}}+n_{\mathrm{He}}{m}_{\mathrm{He}},\end{array}$$(B.11)

where m_H, He are the atomic masses of hydrogen and helium, respectively. They adopted a hydrogen number density of n_H = 0.247 cm⁻³ (Slavin & Frisch 2008) and a helium number density of n_He = 0.015 cm⁻³ (Möbius et al. 2004), resulting in a gas mass density of ρ_gas ≈ 5.14 × 10⁻²² kg/m³. Their dust mass density of ρ_ISD = (2.1 ± 0.6)×10⁻²⁴ kg/m³ resulted in a gas-to-dust mass ratio of $R_{\mathrm{g}/\mathrm{d}}\approx {245}_{-54}^{+98}$ (Krüger et al. 2015).²⁸

Depending on which mass determination method is used (see Sect. 4), different values of R_g/d can be calculated. Following the approach of Landgraf (1998) and Krüger et al. (2015), the corresponding ISD subset is divided into sorted and ascending bins of the particle mass that count N = 30 particles each. The differential mass density distribution,

$$\begin{array}{cc}\hfill \mathcal{D}(m):=& \frac{\mathrm{d}m}{V{\mathrm{dlog}}_{10}(m[\mathrm{kg}])}\hfill \end{array}$$(B.12)

$$\begin{array}{cc}\hfill \approx & \frac{{\displaystyle \sum _{i\in \mathrm{bin}}}{m}_i}{V_{\det }\mathrm{\Delta }{({log}_{10}(m[\mathrm{kg}]))}_{\mathrm{bin}}}=:{\mathcal{D}}_{\mathrm{bin}},\hfill \end{array}$$(B.13)

is approximated by dividing the sum of all masses, m_i, in the respective bin by the swept volume of the detector, V_det, and the logarithmic width of the respective mass bin. The swept volume of the detector is calculated by integrating over the effective surface area of the detector during a given time interval, Eq. (B.3), multiplied with the distance the spacecraft has travelled in that time interval. From 29 October 1990 to 1 December 2007, taking into account the measurement gaps, this results in a swept volume of V_det ≈ 3.15 × 10¹¹ m³. For the measurement-derived and iteratively calculated masses of the Krüger subset the resulting differential mass density distribution is shown in Fig. B.3.

The uncertainty of the differential mass distribution, graphed as the vertical error bars of Fig. B.3, correspond to the uncertainty of the mass determination. Because the individual masses are known with an error factor of ∼10, that is, because their uncertainty is Δlog₁₀(m [kg]) ≈ 1, the uncertainty of the sum of N masses, $M={\displaystyle \sum _{i=1}^N}{m}_i$, is $\mathrm{\Delta }{log}_{10}(M[\mathrm{kg}])=1/\sqrt{N}$.

Integrating over the differential mass density distribution, which corresponds to summing the value of each mass bin multiplied with the logarithmic width of that bin, gives the mass density of ISD:

$$\begin{array}{cc}\hfill {\rho }_{\mathrm{ISD}}& ={\displaystyle \int \mathcal{D}}(m){\mathrm{dlog}}_{10}(m[\mathrm{kg}])\hfill \end{array}$$(B.14)

$$\begin{array}{cc}& \approx {\displaystyle \sum _{\mathrm{bin}}}{\mathcal{D}}_{\mathrm{bin}}\mathrm{\Delta }{({log}_{10}(m[\mathrm{kg}]))}_{\mathrm{bin}}\hfill \end{array}$$(B.15)

This has been applied to the measurement-derived and the iteratively calculated masses in Sect. 6.4.

Appendix C: Reconstruction of the heliocentric ISD particle speed

In general, the heliocentric speed, v_hel, of a dust particle encountered by Ulysses is not known; only the particle’s impact speed, v_msr, can be determined from the detector signal (cf. Sect. 4.1). If the heliocentric speed of the spacecraft, v_U, as well as the angle between the two velocity vectors, χ, are known, the heliocentric speed of the dust particle can be reconstructed:

$$\begin{array}{c}\hfill v_{\mathrm{rec},\mathrm{hel}}(\chi )=\sqrt{{(v_{\mathrm{msr}}-v_{\mathrm{U}}cos\chi )}^2+{(v_{\mathrm{U}}sin\chi )}^2},\end{array}$$(C.1)

where χ is the angle between the detector boresight and the spacecraft’s direction of motion (cf. Appx. A). An angle of χ = 0° corresponds to a particle impacting the spacecraft head-on with respect to the spacecraft’s direction of motion, meaning that the dust particle moves anti-parallel to the spacecraft. For χ = 180° the dust particle would impact the spacecraft’s aft. The precision of the determined impact speed, v_msr, is given by a multiplicative error factor, err_v, and the precision of χ is given by the opening angle of the detector, α_max.

C.1. Relevance of the detector’s opening angle

The role that these uncertainties play in the reconstruction of the particle’s heliocentric speed depends strongly on how the spacecraft speed, v_U, compares to the dust particle’s heliocentric speed, v_hel, and on the angle between the detector boresight and the spacecraft’s direction of motion, χ.

ISD particles, which follow hyperbolic heliocentric orbits, are usually faster than the spacecraft, which follows an elliptical orbit,²⁹v_hel > v_U. If the spacecraft speed is negligible compared to the speed of the dust particle, v_U ≪ v_hel, the detector’s opening angle is of little relevance, and the reconstructed heliocentric speed is almost identical to the determined impact speed, v_rec, hel ≈ v_msr, irrespective of χ. The reconstruction of the heliocentric speed will be as accurate as the impact speed determination is precise, and the opening angle of the detector only limits the precision by which the direction of origin is known.

However, if the speeds of the spacecraft and of the dust particle are similar, the reconstructed heliocentric speed of Eq. (C.1) strongly depends on the angle χ between the spacecraft’s and the dust particle’s velocity vectors. The finite opening angle of the detector introduces another uncertainty to the reconstructed heliocentric speed, Δv_rec, hel. How large this uncertainty is depends not only on the detector’s opening angle, α_max, but also on the heliocentric speed of the spacecraft, v_U, and on χ itself.

The uncertainty of the determined impact speed, v_msr, is given by a multiplicative error factor, err_v (cf. Sect. 3). The uncertainty of the reconstructed heliocentric speed that is introduced by the finite opening angle of the detector can be represented by an error factor as well, which we name err_α. Using Eq. (C.1) with a given impact speed, v_msr, and a given spacecraft speed, v_U, leads to the definition of this error factor as

$$\begin{array}{c}\hfill {\mathrm{err}}_{\alpha }:=\frac{v_{\mathrm{rec},\mathrm{hel}}(\chi +\alpha )}{v_{\mathrm{rec},\mathrm{hel}}(\chi )},\{\begin{array}{c}\chi \in [0^{°},{360}^{°}]\hfill \\ \alpha \in [0^{°},{\alpha }_{max}]\hfill \end{array},\end{array}$$(C.2)

which represents the factor by which the reconstructed heliocentric speed changes if the angle χ has been overvalued by α ∈ [0° ,α_max]. To reiterate, χ ∈ [0° ,360° ] is the angle between the spacecraft’s direction of motion and the direction from which the particle comes, and α ∈ [0° ,α_max] is the angle by which the detector’s boresight deviates from the particle’s incoming direction.

In most cases, for example on Ulysses, neither χ nor α are constrained more than is stated in Eq. (C.2), and the error factor varies for a given spacecraft speed, particle speed, and detector opening angle. Therefore, it is convenient to give the worst-case error factor, ${\mathrm{err}}_{\alpha }^{max}$; in other words, to maximise err_α over all χ and α. The angles that maximise err_α correspond to the spacecraft’s and the particle’s least favourable directions of motion for the purpose of accurately reconstructing the dust particle’s heliocentric speed from the determined impact speed.³⁰

C.2. Application to the orbits of Earth, Jupiter, and Ulysses

Examples of how the worst-case error factor of the reconstructed heliocentric speed, ${\mathrm{err}}_{\alpha }^{max}$, changes with the opening angle of the detector are plotted in Fig. C.1 for circular orbits of Earth and Jupiter for the cases where the spacecraft moves antiparallel to the ISD direction (particles impacting the bow of the spacecraft at high impact speed) and parallel to the ISD direction (particles impacting the aft of the spacecraft at low impact speed). The cases of Ulysses at its aphelion and perihelion, perpendicular to the ISD particles’ motion, are plotted as well.

In order to distinguish ISD from IDPs, the error factor must lie below an upper limit: assuming that ISD moves with v_hel as per Eq. (4) and that an extremely eccentric OCC particle moves with (almost) v_esc as per Eq. (5), the error factor must lie below v_hel/v_esc|_{r = 1 au} ≲ 1.18 at Earth and v_hel/v_esc|_{r = 5.2 au} ≲ 1.73 at Jupiter; the reconstructed heliocentric speed must be known within these accuracies. To distinguish an ISD particle from an IDP on a circular orbit, these accuracy limits would be 1.66 at Earth and 2.44 at Jupiter.

As Fig. C.1 shows, the worst-case error factor, ${\mathrm{err}}_{\alpha }^{max}$, increases almost linearly with the opening angle until approximately ${\alpha }_{max}\approx 90°$. The error factor for Ulysses with an opening angle of ${\alpha }_{max}=70°$ is 1.36 at its aphelion and 3.83 at its perihelion. For an opening angle of ${\alpha }_{max}=90°$, the worst-case error factors are 1.46 at the aphelion and 4.62 at the perihelion, assuming that the spacecraft’s motion is perpendicular to the ISD particle’s trajectory.

For spacecraft on orbits in or close to the ecliptic plane, the worst-case error factor changes during the spacecraft orbit: when the spacecraft moves roughly antiparallel to the ISD inflow, the relative speed between the detector and the dust particle is highest; that is, ISD particles impact at high speeds. Thus, as per Eq. (C.1), a wider opening angle is sufficient to distinguish ISD from IDPs. Conversely, during an orbital section where the spacecraft moves roughly parallel to the ISD inflow, a smaller opening angle is required to unambiguously distinguish ISD from IDPs.

For spacecraft on a roughly circular orbit similar to Earth, ISD particles can be distinguished from extremely eccentric cometary dust particles along the entire orbit of the spacecraft if the direction of origin of a given particle is known with a precision better than ${\alpha }_{max}\lesssim 8°$. If this precision is ${\alpha }_{max}\gtrsim 20°$, ISD particles and extremely eccentric cometary particles cannot be distinguished even when the spacecraft moves antiparallel to the ISD inflow direction. For spacecraft on Jupiter-like orbits, these critical angles increase to ${\alpha }_{max}\lesssim 25°$ and ${\alpha }_{max}\gtrsim 125°$, respectively. This assumes that the impact speed can be determined with perfect accuracy, or, in other words, that the error factor of the impact speed determination is err_v = 1.0, which is not the case.

C.3. Relevance of the uncertainty of the measurement-derived impact speed

The two error factors, ${\mathrm{err}}_{\alpha }^{max}$ and err_v, which are introduced by the finite opening angle and by the error factor of the impact speed determination, respectively, are at most multiplicative but not necessarily so. The faster the spacecraft speed is compared to the particle’s impact speed, the less influential the determined impact speed is in Eq. (C.1): for v_U ≫ m_msr the impact speed barely affects the reconstructed heliocentric speed, and the error factor of the determined impact speed has no significant bearing on the uncertainty of the reconstructed heliocentric speed. Thus, the total error factor, err_tot, is at least identical to the error factor introduced by the opening angle, ${\mathrm{err}}_{\alpha }^{max}\ge 1$, and at most the product of ${\mathrm{err}}_{\alpha }^{max}$ and the error factor introduced by the impact speed determination, err_v;

$$\begin{array}{c}\hfill {\mathrm{err}}_{\mathrm{tot}}\le {\mathrm{err}}_{\alpha }^{max}·{\mathrm{err}}_v.\end{array}$$(C.3)

Using the equality in Eq. (C.3) serves as a conservative estimate of the total error factors and is used in the following.

When using the product of both error factors, the precision of the reconstructed heliocentric speed is 1.36 ⋅ 1.9 ≈ 2.58 at Ulysses’ aphelion for an opening angle of 70°, which is insufficient to distinguish ISD particles from particles on parabolic or even circular orbits. However, depending on the detector boresight, the total error factor may be considerably smaller than this conservative estimate. Larger-number statistics are also an aid in narrowing down the direction of origin of many particles with similar trajectories (e.g. Gehrels 1986).

Nevertheless, the closer to the Sun, the more the total error factor increases, and the more the required precision tightens. At Ulysses’ perihelion, for example, the worst-case error factor is ${\mathrm{err}}_{\alpha }^{max}=3.8$, and thus the heliocentric particle speed can only be reconstructed within an accuracy of ${\mathrm{err}}_{\mathrm{tot}}^{max}\approx 3.8·1.9\approx 7.2$ for err_v = 1.9, which does not allow for any differentiation between ISD from dust on parabolic or even circular orbits.

This illustrates once again that a more precise determination of both the impact speed and the direction of origin would be an invaluable improvement, especially for distinguishing between different origins of dust (cf. Sect. 6.2.1).

Applying the above considerations to the DESTINY⁺ Dust Analyser (DDA), assuming an angular accuracy of 10° and a speed error factor of 1.1 on a roughly Earth-like orbit (Simolka et al. 2024), results in a total worst-case error factor of 1.08 when moving against the ISD inflow direction and 1.22 when moving with the ISD inflow direction. This should allow DDA to comfortably distinguish ISD from particles on circular orbits throughout the entire spacecraft orbit. Particles on highly eccentric (i.e. near-parabolic) orbits should be distinguishable from ISD during the half-orbit when DESTINY⁺ moves partially against the ISD inflow direction. During the other half-orbit where the spacecraft moves partially parallel to the ISD inflow, a clear identification of ISD by its reconstructed heliocentric speed may be more ambiguous.

We note that this analysis assumes that gravitation is dominant over solar radiation pressure and the Lorentz force (i.e. β ≈ 0 and Q/m ≈ 0). This is a reasonable assumption for silicate grains of all porosities but not for carbonaceous grains (see Fig. 1; see also Kimura 2017). The inclusion of solar radiation pressure will decrease the local escape speed and thus allow for larger error factors before ISD particles can no longer be identified. However, at least in the context of Ulysses’ most massive particles, this yielded no significant changes even for highly porous, carbonaceous grains (Sect. 6.2.2).

Appendix D: Details of the methodology to investigate the heliosheath filtering function

This appendix details how the measured size distribution was determined from Ulysses’ measurements (Appx. D.1) and how the numerical simulations were performed (Appx. D.2), and presents the time-dependent modulation of the ISD flux by forces in the inner heliosphere (Appx. D.3) and the time-dependent detector transfer function (Appx. D.4).

D.1. Details of the determination of the measured size distribution

A differential size distribution, dn/da, describes the number density of particles, n, in an infinitesimal size bin, a ∈ [a; a + da]. The number of particles within a finite size interval, (Δa)_i = [a_i, a_i + 1] that, during the time interval (Δt)_i = [t_i, t_i + 1], move through an area, A, perpendicular to their direction of motion with a speed of v_d is given by

$$\begin{array}{c}\hfill N_{{(\mathrm{\Delta }a)}_i,{(\mathrm{\Delta }t)}_i}={\displaystyle \underset{a_i}{\overset{a_{i+1}}{\int }}}{\displaystyle \underset{t_i}{\overset{t_{i+1}}{\int }}}\frac{\mathrm{d}n}{\mathrm{d}a}A{v}_{\mathrm{d}}\frac{\mathrm{d}t}{\mathrm{d}a},\end{array}$$(D.1)

or, assuming that measurements can only be taken in finite bins of size and time, by

$$\begin{array}{c}\hfill N_{{(\mathrm{\Delta }a)}_i,{(\mathrm{\Delta }t)}_i}\approx \frac{\mathrm{\Delta }n}{\mathrm{\Delta }a}A{v}_{\mathrm{d}}{(\mathrm{\Delta }t)}_i{(\mathrm{\Delta }a)}_i,\end{array}$$(D.2)

where Δn/Δa indicates the average of dn/da over the respective size bin and time bin. Correspondingly, the differential size distribution measured by a spacecraft, (Δn/Δa)_obs, in size bin (Δa)_i is

$$\begin{array}{c}\hfill {\left(\frac{\mathrm{\Delta }n}{\mathrm{\Delta }a}\right)}_{\mathrm{obs}}\approx \frac{N_{{(\mathrm{\Delta }a)}_i,{(\mathrm{\Delta }t)}_i}}{A_{max}{v}_{\mathrm{d}}{(\mathrm{\Delta }t)}_i{(\mathrm{\Delta }a)}_i},\end{array}$$(D.3)

where N_{(Δa)i, (Δt)i} is the number of particles measured within size bin (Δa)_i and time bin (Δt)_i, and A_max is the surface area of the detector.

In order to apply the filtering and modulation functions to this differential size distribution, the size bins of Eq. (D.3) must correspond to the sizes available for the numerical simulations. The simulated parameter sets, (β, Q/m), listed in Table D.1, are equidistant in log(a), forming the centres of equidistant logarithmic size bins of the width δlog(a [μm]) ≈ 0.1278.

D.2. Details of the numerical simulations used to infer the ISD filtering in the heliosheath

The modulation of the differential size distribution due to the forces in the inner heliosphere, f_F (cf. Sect. 7.3), is investigated with the numerical simulation code IMEX (Sterken et al. 2015; Strub et al. 2019), which has been verified by comparison with measurements taken by various spacecraft (Krüger et al. 2019).

In this instance, the code launches ISD particles from the heliocentric direction of (l, b) = (259° ,8° ) with a uniform speed of 26 km/s, with their initial positions randomised on a plane perpendicular to the velocity vector with a distance of 50 au to the Sun (Strub et al. 2019). The particles’ position and velocity are numerically advanced in time under the effect of solar gravitation, solar radiation pressure, and the Lorentz force.

For the Lorentz force, only the interplanetary magnetic field is taken into account. The magnetic field is simulated by a modified Parker spiral (Parker 1958). Its polarity depends on the phase of the solar magnetic cycle and the position of the dust particle with respect to the heliospheric current sheet, averaged over one solar rotation (Sterken et al. 2012a; Strub et al. 2019).

The simulation domain is separated into a heliocentric, equidistant, cubic grid with a spatial resolution of 0.25 au per grid cell edge, spanning a total volume of (20 au)³. The temporal resolution is 12.2 d.

For each item in a set of dust species, parametrised by their β-ratio and charge-to-mass ratio, (β, Q/m), 2 × 10⁹ trajectories were calculated. The time-resolved number densities, average particle velocities, and the velocity dispersion for each grid cell are saved.

Simulations were computed only for certain values of (β, Q/m), so simulations with the values closest to the assumed β-curve have to be selected to approximate the full size range of a particle distribution. Fig. D.1 shows the β-curves of Fig. 1, given in relation to the particle radius, a, and both the available and the selected tuples of (β, Q/m). For this, the charge-to-mass-ratio, Q/m, was converted to the particle radius, a, with the relation

$$\begin{array}{c}\hfill \frac{Q}{m}=\frac{3{\epsilon }_{0}U}{{\rho }_{\mathrm{d}}{a}^2},\end{array}$$(D.4)

where ε₀ is the electric constant and U = +5 V is the assumed surface potential of the dust particles. We note that no dynamic charging of the dust particles is included at this time, and Q/m is therefore constant. This is valid within ca. 50 au due to a decrease in the plasma density and the UV flux with the squared distance to the Sun. Eq. (D.4) follows directly from the equations for the surface charge of a spherical particle, Q = 4πε₀aU, and for the mass of a compact, spherical particle, m = 4π/3 ρ_da³.

The selected sets of (β, Q/m) are listed in Table D.1. For convenience, conversions to the particle mass and radius have been provided in that table as well. We note that two sets of (β, Q/m) were simulated for a ≈ 9.29 × 10⁻² μm because the assumed β-curve for that size lies in-between two available parameter sets. For this particle size, the results will be averaged over both sets.

The modulation of the differential size distribution due to the forces in the inner heliosphere, f_F, is the density enhancement factor, n/n_ISM, which is directly given as an output of the simulation, depending on both heliocentric coordinates, r, and time, t. Along the orbit of a spacecraft, for example Ulysses, the appropriate coordinates change with time, r(t), and thus f_F(a) depends only on time, Eq. (11).

D.3. Time-dependent modulation of the ISD flux by forces in the inner heliosphere

Fig. D.2 shows the change of the modulation due to inner heliosphere forces, f_F, over time for four selected particle sizes: a ≈ 2.30 μm, which are the largest simulated particles; a ≈ 4.07 × 10⁻¹ μm, which are the particles with β = 1; a ≈ 1.67 × 10⁻¹ μm, which are the particles with the highest β-ratio, β = 1.5; and a ≈ 5.15 × 10⁻² μm, which are the smallest simulated particles.

For the largest particles (β = 0.2, Q/m = 0.01 C/kg), the simulations result in virtually no modulation; the density enhancement factor is n/n_ISM ≈ 1, independent of the phase of the solar cycle.

The particles with β = 1, which do not experience solar gravitation or solar radiation pressure but are affected by the Lorentz force, show a dip to, at its minimum, 40% of the ISM density from 1997 to 2002, following the solar minimum of the defocusing phase of mid-1996 (Sterken et al. 2015; cf. Table 4), and an increase by a factor of two to three during the perihelion crossing of 2007.

The smaller particles are more strongly modulated. As expected, for β > 1 no particles reach into the β-cones (f_F = 0), as evidenced by the gaps surrounding the perihelion crossings. During the times of defocusing, beginning one to two years after the solar minimum of the defocusing phase in mid-1996, particles are strongly depleted (f_F < 0.1) but steadily increase until the solar minimum of the focusing phase in late 2008, which coincides with the final perihelion crossing of 2007 before the dust detector was switched off at the end of November 2007. Surrounding the perihelion crossing of 2001, slightly higher densities compared to the adjacent times can be observed because particles gather at the outer edges of the β-cones (Sterken et al. 2013). Preceding the first perihelion crossing, the particles are strongly enhanced (f_F ≈ 10). This can be attributed to the trailing flank of the solar minimum of the focusing phase of 1986.

The smallest particles (β = 0.50, Q/m = 20 C/kg) show a similar variation. As the solar cycle changes from its maximum in 1989 to the minimum of the defocusing phase in 1996, these particles are more and more depleted; and as the solar cycle changes to its minimum of the focusing phase in late 2008, this depletion becomes weaker. Only during the perihelion crossing in 2007, adjacent to the solar minimum of the focusing phase in late 2008, the particle density is enhanced up to a factor of f_F = 3.

The variability in time of the modulation by the inner heliosphere forces, f_F, illustrates that care must be taken when averaging over time, as will be necessary for most applications. A more detailed analysis of the modulation of particles of different sizes due to the forces in the inner heliosphere applied to Ulysses is given by Landgraf et al. (2000), Sterken et al. (2015).

D.4. Time-dependent detector transfer function

As stated in Sect. 7.4, the time-dependent detector transfer function depends on the angle of incidence between the detector boresight and the particle motion, and on the relative speed between the dust particle and the spacecraft, Eq. (12). The numerical simulations were used to determine the velocity vector of the dust particle. However, only the average velocity of all dust particles within the respective model cell is available. This average can strongly differ from the individual particle velocities. This happens, for example, when averaging two opposing velocity vectors in the gravitational focusing region downstream of the Sun.

The resulting transfer function has been plotted in Fig. D.3 for the same four parameter sets as in Fig. D.2. In analogy to Appx. B.1, A_eff was averaged over the rotation angle and over time. Because the ISD velocity vector has been taken from the numerical simulations, the transfer function cannot be derived if there is no simulation data available, for instance within the β-cone.

The detector transfer function typically lies between 0.2 ≤ f_det ≤ 0.5 but does reach significantly higher values for the smallest particles (β = 0.5, Q/m = 20 C/kg) surrounding the first perihelion crossing in early 1995. For the largest particles (β = 0.2, Q/m = 0.01 C/kg) and even more so for the particles with β = 1, the average velocity vector is similar to the ISD inflow vector. For smaller particles the Lorentz force deflects the particles from their initial orbits more strongly.

D.4. Determination of statistical uncertainties for Sect. 7

The determination of the heliosheath filtering function f_h has multiple sources of uncertainty. Some of these uncertainties are graphed as error bars in Figs. 11 and 12. These error bars are calculated as follows:

Fig. 11a shows the measured differential size distribution, (Δn/Δa)_obs, which is calculated with Eq. (D.3). There, N_{(Δa)i, (Δt)i} = :N is the number of particles within size bin (Δa)_i measured during time bin (Δt)_i, which we assume to be Poisson-distributed for the sake of simplicity. Hence, its uncertainty is the Poissonian standard deviation, which is the square root of the respective number of particles, $\sqrt{N}$. To calculate the uncertainty of (Δn/Δa)_obs, we Gauss-propagate this standard deviation of N through Eq. (D.3), which is identical to replacing N with $\sqrt{N}$ and results in

$$\begin{array}{c}\hfill \mathrm{\Delta }\left[{\left(\frac{\mathrm{\Delta }n}{\mathrm{\Delta }a}\right)}_{\mathrm{obs}}\right]\approx \frac{\sqrt{N_{{(\mathrm{\Delta }a)}_i,{(\mathrm{\Delta }t)}_i}}}{A_{max}·v_{\mathrm{d}}·{(\mathrm{\Delta }t)}_i·{(\mathrm{\Delta }a)}_i}.\end{array}$$(D.5)

These values have been plotted as vertical error bars in Fig. 11a.

Fig. 11d shows the inferred differential size distribution at the termination shock, (Δn/Δa)_TS, which was calculated following Eq. (9):

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{\mathrm{\Delta }n}{\mathrm{\Delta }a}}\right)}_{\mathrm{TS}}=\frac{1}{{\overline{f}}_F(a)}·\frac{1}{{\overline{f}}_{\det }(a)}·{\left({\displaystyle \frac{\mathrm{\Delta }n}{\mathrm{\Delta }a}}\right)}_{\mathrm{obs}}\end{array}$$(D.6)

Its uncertainty has been Gauss-propagated from Eq. (D.6),

$$\begin{array}{c}\hfill \mathrm{\Delta }\left[{\left(\frac{\mathrm{\Delta }n}{\mathrm{\Delta }a}\right)}_{\mathrm{TS}}\right]=\frac{1}{{\overline{f}}_F(a)·{\overline{f}}_{\det }(a)}·\mathrm{\Delta }\left[{\left({\displaystyle \frac{\mathrm{\Delta }n}{\mathrm{\Delta }a}}\right)}_{\mathrm{obs}}\right],\end{array}$$(D.7)

and plotted as vertical error bars in Fig. 11d.

Fig. 12 shows the size-dependent and time-averaged heliosheath filtering function,

$$\begin{array}{c}\hfill {\overline{f}}_{\mathrm{hs}}(a)=\frac{{(\mathrm{\Delta }n/\mathrm{\Delta }a)}_{\mathrm{TS}}}{{(\mathrm{d}n/\mathrm{d}a)}_{\mathrm{ISM}}},\end{array}$$(D.8)

where (dn/da)_ISM has been calculated with Eq. (10) for the geometric mean of the respective size bin. Its uncertainty follows directly from Eq. (D.8) by replacing (Δn/Δa)_TS with its uncertainty:

$$\begin{array}{c}\hfill \mathrm{\Delta }{\overline{f}}_{\mathrm{hs}}(a)=\frac{\mathrm{\Delta }\left[{(\mathrm{\Delta }n/\mathrm{\Delta }a)}_{\mathrm{TS}}\right]}{{(\mathrm{d}n/\mathrm{d}a)}_{\mathrm{ISM}}}\end{array}$$(D.9)

These values have been plotted as vertical error bars in Fig. 12. Additional uncertainties that have not been taken into account when calculating the error bars are discussed in Sect. 7.7.

Appendix E: Properties of Subset B

Table E.1 gives the properties of the particles of Subset B (Sect. 6.3) in analogy to Table 1. Table E.2 lists the reconstructed heliocentric speeds for the particles of Subset B in analogy to Table 2.

All Tables

All Figures