https://orcid.org/0000-0001-9329-0315
ABSTRACT
It is expected since the early 1970s that tenuous dust rings are formed by grains ejected from the Martian moons Phobos and Deimos by impacts of hypervelocity interplanetary projectiles. In this paper, we perform direct numerical integrations of a large number of dust particles originating from Phobos and Deimos. In the numerical simulations, the most relevant forces acting on the dust are included: Martian gravity with spherical harmonics up to fifth degree and fifth order, gravitational perturbations from the Sun, Phobos, and Deimos, solar radiation pressure, as well as the Poynting–Robertson drag. In order to obtain the ring configuration, simulation results of various grain sizes ranging from submicrometres to 100 μm are averaged over a specified initial mass distribution of ejecta. We find that for the Phobos ring grains smaller than about 2 μm are dominant; while the Deimos ring is dominated by dust in the size range of about 5–20 μm. The asymmetries, number densities, and geometric optical depths of the rings are quantified from simulations. The results are compared with the upper limits of the optical depth inferred from Hubble observations. We compare to previous work and discuss the uncertainties of the models.
1 INTRODUCTION
It is well known that Saturn, Jupiter, Uranus, and Neptune have ring systems. For Mars, no rings have been detected yet, but it was suggested by Soter (1971) that Mars should possess a ring system, which consists of dust grains originating from Phobos and Deimos. A number of studies contributed to the dynamical modelling of the suspected Martian dust rings, which are reviewed in several papers (Hamilton 1996; Krivov & Hamilton 1997; Krivov, Feofilov & Dikarev 2006; Zakharov et al. 2014). Attempts to detect the rings are described by Showalter, Hamilton & Nicholson (2006), Zakharov et al. (2014), and Showalter (2017). The Japan Aerospace Exploration Agency (JAXA) will launch the Martian Moons Exploration mission in 2024 with the Circum-Martian Dust Monitor (Kobayashi et al. 2018), an instrument that will conduct in situ dust measurements.
Recent studies of particle dynamics in the Martian dust rings were presented by Makuch, Krivov & Spahn (2005) and Krivov et al. (2006). In addition to solar radiation pressure and the Martian oblateness J2, Makuch et al. (2005) included in their model the Poynting–Robertson (P-R) drag which is important for grains with long lifetime. They integrated the orbit-averaged equations of motion for dust grains to obtain the spatial structure of the Deimos torus. Furthermore, the Martian eccentricity (eMars) combined with the three aforementioned perturbation forces was considered by Krivov et al. (2006), who also integrated the orbit-averaged equations of motion. From their numerical simulations the configurations of the Phobos and Deimos dust rings were obtained and the ring optical depths were estimated. The effect of eMars on the dust dynamics was also discussed by Showalter et al. (2006).
In this paper, we study the full dynamics of dust particles ejected from Phobos and Deimos in terms of direct numerical integrations of the equations of motion. Particle lifetimes and spatial configurations of number density are obtained for 12 different grain sizes ranging from |$0.5$| to |$100\, \mu \mathrm{m}$|. The steady-state size distributions, dominant grain sizes, cumulative grain densities, and geometric optical depths for the rings are derived by averaging over the initial mass distribution of ejecta. The major asymmetries of the rings are evaluated in a sun-fixed reference frame, allowing for quantitative predictions for future in situ measurements by forthcoming missions.
The paper is organized as follows. In Section 2, the mass production rates of dust from both source moons are estimated. The equations of motion and the numerical integrations are described in Section 3. The resulting lifetimes of various grain sizes are presented in Section 4, and the effect of the planetary odd zonal harmonic coefficient J3 is analysed in Section 5. In Section 6, we present the resulting spatial configuration of the Phobos and Deimos rings. A discussion of the model results and uncertainties, along with our conclusions, is given in Section 7.
2 DUST PRODUCTION RATES
3 NUMERICAL SIMULATIONS
A well-tested numerical code (see Liu et al. 2016; Liu & Schmidt 2018a, b) is used to integrate the evolution of dust grains. Grains of sizes ranging from submicrometres to 100 μm are simulated: |$0.5,\,1,\,2,\,5,\,10,\,15,\,20,\,25,\,30,\,40,\,60$|, and |$100 \, \mathrm{\mu m}$|, including the effects of Martian gravity, gravitational perturbations from the Sun, Phobos, and Deimos, solar radiation pressure, as well as P-R drag.
The shapes of Jupiter and Saturn are both nearly hemispherically and axially symmetric, which is not the case for Mars. Thus, in our model the Martian gravity field up to fifth degree and fifth order is considered. We use silicate as the material for dust and adopt a bulk density of 2.37 |$\mathrm{g \, cm^{-3}}$| (Makuch et al. 2005; Krivov et al. 2006).
The values of the gravity spherical harmonics are taken from the Mars gravity model MRO120D (Konopliv, Park & Folkner 2016). The formulas for solar radiation pressure and the P-R drag are taken from Burns, Lamy & Soter (1979). The size-dependent values of the solar radiation pressure efficiency Qpr are computed from Mie theory (Mishchenko et al. 1999; Mishchenko, Travis & Lacis 2002) for spherical grains (see fig. 3 in Liu & Schmidt 2018a), using the optical constants for silicate grains from Mukai (1989). Due to the small size of Phobos (RP ≈ 11 km) and Deimos (RD ≈ 6 km), the dust ejection velocity is higher than the escape velocity but much lower than the orbital velocity of the moons (Horanyi et al. 1990; Horányi et al. 1991). Therefore, it is a very good approximation to start grains directly from the orbits of Phobos and Deimos. It is known that the dynamical behaviour of dust particles strongly depends on the solar longitude (Martian season) at the launch time (Hamilton 1996; Krivov & Hamilton 1997; Makuch et al. 2005; Krivov et al. 2006). Thus, in our simulations 100 particles per grain size are launched with uniformly distributed initial mean anomalies of the Martian orbit around the Sun (Krivov & Hamilton 1997; Showalter et al. 2006).
The motions of dust grains are simulated until they hit Phobos, Deimos, Mars, they escape from the Martian system, or for a maximum of 100 000 yr. In order to save computation cost, we also stop the simulation when the fraction of grains remaining in orbit for certain sizes are less than 5 per cent (the integrations for the 100 particles per grain size run in parallel on a computer cluster provided by the Finnish CSC – IT Center for Science). When we check for collisions of the dust particles with a given target (Phobos, Deimos, and Mars), at each time-step of the integration we calculate the distance between the particle and the centre of the target. If this distance is smaller than the target’s radius, an impact occurs. When the grain is close to the target, in order to avoid overlooking the impact due to discrete time-steps of the numerical integrators, a cubic Hermite interpolation is adopted to calculate the minimum distance between the particle and the target’s centre approximately (see Chambers 1999; Liu et al. 2016). In our simulations, we store the slowly changing orbital elements including semimajor axis, eccentricity, inclination, argument of pericentre, longitude of ascending node, and true anomaly. Generally, we store 10 sets of orbital elements for one orbital period. The orbital segment between two consecutively stored sets of osculating elements is approximately considered as Keplerian. For denser output, each of these segments is further divided into intervals that are equidistant in time. Since dust particles are produced and removed continually (by hitting sinks or escape from the system), each discrete point corresponds to one particle in the steady-state ring configuration (see details in sections 3.4 and 4 in Liu et al. 2016 as well as section 4 in Liu & Schmidt 2018a).
4 PARTICLE LIFETIMES
The lifetimes for dust from Phobos and Deimos derived from our simulations are shown in Fig. 1. Overall, we confirm the picture drawn from previous work (see table 1 in Krivov et al. 2006, and references given there): grains larger than about |$5-$||$10\, \mu \mathrm{m}$| have lifetimes on the order of 10 000 yr if they come from Deimos and a few tens of years if they are ejected from Phobos. For grain sizes smaller than 5 |$\mathrm{\mu m}$| the lifetimes drop to a value of months.
Lifetimes for particles of various sizes originating from the Martian moons Phobos and Deimos.
Due to the low grain ejection velocity, which is negligible compared to the orbital velocity of the moons, the particles start their evolution with e0 ≈ 0. Since the Hamiltonian |$\mathcal {H}$| is conserved, the orbital evolution of dust particles follows in the phase plane of the Hamiltonian a curve with the value at the starting point |$\mathcal {H} (e=0, \phi _\odot)$|. Typical phase portraits, obtained as a contour plot from equation (3), are shown in Figs 2 and 3.
Phase portrait for 10.7 |$\mathrm{\mu m}$| particles from Phobos. The blue dashed circle denotes eimpact = 0.638 for Phobos grains.
Phase portrait for 6.0 |$\mathrm{\mu m}$| particles from Deimos. The blue dashed circle denotes eimpact = 0.855 for Deimos grains.
Particles larger than |$r_\mathrm{g}^\mathrm{impact}$| are safe from rapid collision with Mars. Their lifetime is limited mainly by collisions with their source moon and by the slow reduction of their semimajor axis, which, ultimately, increases again their chance to hit Mars.
5 THE EFFECT OF J3
For Jupiter and Saturn, because of their nearly hemispherically and axially symmetric shapes, the values of J3 are almost zero [J3 ≈ −4.2 × 10−8 for Jupiter (Iess et al. 2018), and J3 ≈ 5.9 × 10−8 for Saturn (Iess et al. 2019)], and thus J3 has negligible effect on the dynamics of particles in the Jovian and Saturnian rings. In contrast, Mars has a much larger value of J3 (≈3.15 × 10−5), exceeding the value of J3 for the Earth (≈−2.5 × 10−6, Pavlis et al. 2012), so that we might expect a noticeable effect on the dynamics of circum-Martian dust. From equation (5), J3 could be important for the evolution of the inclination for eccentric orbits. In Fig. 4 we show the evolution of a 20 μm particle from Phobos with and without the action of J3. The maximal effect of J3 on the inclination amounts to 10 per cent, roughly. The variations in inclination due to J3, in turn, alter the effects of solar radiation and Martian J2 on the evolution of semimajor axis and eccentricity. For a specific grain, the mildly altered dynamics can have a drastic effect on the lifetime (Fig. 4), while the overall, averaged effect on the lifetimes seems to remain small (see comparison to previous work in Section 4).
Evolution of inclination for a 20 μm particle from Phobos. The black line denotes the inclination evolution with all perturbation forces (see Section 3). The red line corresponds to the case with all perturbation forces except J3. Without J3 the particle re-impacts on the source moon at an earlier time.
Because of the factor sin i in equation (6), which is small due to the generally low orbital inclination, the direct effect of J3 on eccentricity remains negligible compared to that of solar radiation pressure.
The J3 effect decreases rapidly with increasing semimajor axis (equations 5 and 6). Thus, it is more important for grains from Phobos than for those from Deimos. The J5 gravitational coefficient has a similar, albeit much weaker effect on inclination. For particles from Phobos, the strength of the perturbation induced by J5 is roughly |$J_5/J_3 \times \left(R_\mathrm{M}/a_\mathrm{Phobos}\right)^2 \approx 2{{\ \rm per\ cent}}$| of the perturbation induced by J3.
6 PARTICLE SIZE DISTRIBUTION AND CONFIGURATION OF THE RINGS
(a) Steady-state differential size distribution (logarithmic scale) for the Phobos and Deimos rings obtained from numerical simulations (Section 3) combined with initial mass distribution of ejecta equation (7). A power law with slope q = 2.7 is also shown for reference. (b) Same as (a), but in linear scale for |$r_\mathrm{g}\lt 25\, \mu \mathrm{m}$|.
Since the lifetimes of larger grains from Phobos are not substantially longer than those of the small grains, we find that the Phobos ring is dominated by grains ≤2 |$\mathrm{\mu m}$|. A small, secondary peak around |$10\, \mu \mathrm{m}$| is visible best in logarithmic scale (Fig. 5). For the small grains that dominate the Phobos ring, solar radiation pressure is the most important perturbation force, which pushes nearly instantly the solar angle to 90° (Hamilton 1996; Hamilton & Krivov 1996). The subsequent rotation of the solar angle is induced by J2 and by the orbital motion of Mars (see discussion around equation 5 of Hamilton 1996). Because of the small semimajor axis of Phobos, and thus of the grains lifted from its surface, the effect of J2 dominates and the rotation is in anti-clockwise direction, assuming in principle values in the range 90° < ϕ⊙ < 270° (see Fig. 2). But the rotation of the solar angle takes place on time-scales that are longer than the lifetime of the grains. As a result, all grains develop their eccentricities with a sun-angle that remains close to 90° and the Phobos ring appears shifted perpendicular to the solar direction (Fig. 6).
![(a) Grain number density in the Phobos ring projected on to the Martian equatorial plane, vertically averaged over $[-0.3, 0.3] \, R_\mathrm{M}$ in a frame that keeps a fixed orientation with respect to the Sun. The plot is obtained by averaging simulation results of particles of 12 grain sizes ranging from $0.5$ to $100 \, \mathrm{\mu m}$ (Section 3) over an initial mass distribution of ejecta with the differential slope α = 0.9 (equation 7). See Fig. 5 for the steady-state differential size distribution for these ring particles. The positive xrot-axis points to the direction of the Sun. The blue line denotes the Martian radius, and the red dashed lines denote the orbits of Phobos (inner) and Deimos (outer). (b) Same as (a), but for the Deimos ring, vertically averaged over $[-2.0, 2.0] \, R_\mathrm{M}$.](https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/mnras/500/3/10.1093_mnras_staa3084/1/m_staa3084fig6.jpeg?Expires=1749258939&Signature=wqnwjyaNN6ZfosIc8JTajYhsj-~xBsf5m4YXzt~X3l2D8MLIzPf2mtmoi-9kn4F0~GYcrdK10hBVBhdQPitbqgkqP1j4PhWJcHOlWI-XGu0t3ukWgbX-WRe5hGl~9-JWSiDEfltzVep3K3qAvXbmwpKwfKB4MYO79tsQTdIXuEk2oMe5ShVxerFzbgbOmlJl~T~MzgP2haiVtjW6Fl3Qt6Uq~~lJxRaHrORObJp~9QocNRYaOBKJ7RMZD18xWl2WS1ndd-Jb9PuM7XmRPx3w6ZY0zx6n2i6OAXdSXWcyp8mq44IGWopcaRhLEXOCz2Fl0GwZFFZWYDt6qYhIm9mBNQ__&Key-Pair-Id=APKAIE5G5CRDK6RD3PGA)
(a) Grain number density in the Phobos ring projected on to the Martian equatorial plane, vertically averaged over |$[-0.3, 0.3] \, R_\mathrm{M}$| in a frame that keeps a fixed orientation with respect to the Sun. The plot is obtained by averaging simulation results of particles of 12 grain sizes ranging from |$0.5$| to |$100 \, \mathrm{\mu m}$| (Section 3) over an initial mass distribution of ejecta with the differential slope α = 0.9 (equation 7). See Fig. 5 for the steady-state differential size distribution for these ring particles. The positive xrot-axis points to the direction of the Sun. The blue line denotes the Martian radius, and the red dashed lines denote the orbits of Phobos (inner) and Deimos (outer). (b) Same as (a), but for the Deimos ring, vertically averaged over |$[-2.0, 2.0] \, R_\mathrm{M}$|.
The Deimos ring, in contrast, is dominated by larger grains (Fig. 5) in the size range of about 5–20 |$\mathrm{\mu m}$|, owing to the very large lifetimes of these particles (Fig. 1). This lifetime is longer than the period of the cycle in the evolution of the solar angle and the eccentricity (Fig. 3). For Deimos grains, the rotation of the solar angle induced by the orbital motion of Mars dominates over the effect of J2, so that the solar angle rotates clockwise (Hamilton 1996). The maximum eccentricity is attained at ϕ⊙ = 0°. Averaging over many grains, lifted from Deimos uniformly over one Martian year, the Deimos ring appears shifted away from the Sun (Fig. 6).
We find a peak number density (Fig. 6) for grains ≥0.5 μm for the Phobos ring of about |$4.7\times 10^{-5} \, \mathrm{m}^{-3}$| if vertically averaged over |$[-0.3, 0.3] \, R_\mathrm{M}$| (corresponding roughly to the thickness of the ring, Fig. 7), and for the Deimos ring of about |$9.2\times 10^{-6} \, \mathrm{m}^{-3}$| if vertically averaged over |$[-2.0, 2.0] \, R_\mathrm{M}$|. This allows us to estimate the number of dust impacts detected by an in situ instrument on a spacecraft that traverses the rings. For a vertical crossing of the rings with a 1 m2 detector (as the sensitive area of the detector onboard JAXA’s Martian Moons Exploration mission) one expects to record about 96 particles (≥0.5 μm) from the Phobos ring and about 125 particles from the Deimos ring.
(a) Geometric optical depth of the Phobos ring, when viewed from the opposite direction of the Martian vernal equinox point. The plot is obtained by averaging simulation results of particles of 12 grain sizes ranging from |$0.5$| to |$100 \, \mathrm{\mu m}$| (Section 3) over an initial mass distribution of ejecta with the differential slope α = 0.9 (equation 7). The z-axis is along the Martian spin axis. The x-axis (not shown in this plot) points to the Martian vernal equinox point, and the y-axis is perpendicular to the x-axis in the Martian equatorial plane. The blue line denotes the Martian radius, and the red dashed lines denote the orbital distances of Phobos (inner) and Deimos (outer). (b) Same as (a), but for the Deimos ring.
The non-detection of the Martian rings in observations with the Hubble Space Telescope in 2001 (Showalter et al. 2006) placed upper limits on their edge-on brightness. For a geometric albedo of 0.07, this translates into upper limits for the edge-on optical depth of ∼2 × 10−6 for the Phobos ring and ∼10−6 for the Deimos ring (Krivov et al. 2006). The edge-on geometric optical depths from our simulations are shown in Fig. 7. Here, the viewing direction is from the Martian vernal equinox point in the plane of the sky, that is, the ascending node of the Martian orbital plane in the Martian equatorial plane. For this configuration, we use a slope of α = 0.9 for the initial mass distribution (7) and we obtain an average edge-on geometric optical depth for the Phobos ring of about 3.5 × 10−8, and about 3.1 × 10−7 for the Deimos ring. If, alternatively, we use a value of α = 0.8 (Krivov et al. 2003), which is consistent with measurements of the dust cloud around the Galilean satellites by the Galileo Dust Detection System (Krüger, Krivov & Grün 2000), we get slightly larger values of about 3.7 × 10−8 for the Phobos ring and about 3.7 × 10−7 for the Deimos ring. In either case (α = 0.9 or 0.8), the average edge-on optical depth is lower than the upper limits inferred from the Hubble observations.
7 SUMMARY AND DISCUSSION
In this work, we present a comprehensive numerical model for the evolution of dust particles ejected from the Martian moons Phobos and Deimos, with the goal to construct a steady-state model for the configuration of the putative Martian dust rings. The new model ingredients are: (i) we perform direct numerical integrations of the equations of motion for a large number of particles, instead of integrating the orbit averaged evolution equations for the orbital elements. (ii) We use an array of 12 grain sizes, from submicrometres to 100 μm, to better resolve than previous studies the size-dependent lifetimes. We present results as averages over an initial ejecta mass distribution. (iii) We include the Martian gravity field up to fifth degree and fifth order and account for the gravitational perturbations of Phobos and Deimos. (iv) We check for impacts of grains on the source moons and with Mars directly at (and between) the time-steps of the integration, which allows for a more accurate evaluation of grain lifetimes than the probabilistic approach used in previous studies.
Our results are:
We evaluate the lifetimes of grains with radii between |$0.5$| and |$100\, \mu \mathrm{m}$|, confirming results obtained in the literature. For grains from both source moons a jump in lifetime occurs between 5 and 10 μm. Smaller grains have lifetimes of months up to one (Earth) year. Larger grains from Phobos have lifetimes of tens of years while grains from Deimos remain in orbit for 10 000 yr or more.
The gravity perturbation induced by the Martian north–south asymmetry has in our simulations a small but noticeable effect on the orbital evolution of the grains from Phobos, in particular on the evolution of the inclination. For Deimos, the effect is much smaller.
Taking into account the initial mass distribution of ejecta, we derive the steady-state size distribution of dust particles in both ring components. The Phobos ring is dominated by grains smaller than a few micronmetres. The Deimos ring is dominated by grains around 10 μm in size. This is consistent with (but refines) previous results.
Averaging over the initial mass distribution of ejecta and a large number of grains produced on the surfaces of the moons uniformly over one Martian year, we obtain a model for the steady-state configuration of the rings. For the Deimos ring, we confirm results from previous studies, in that the ring extends in the anti-sun direction, owing to an interplay of solar radiation pressure and the effect of Martian J2 and the orbital motion of Mars. The ring has a thickness of about 4 Martian radii. For the Phobos ring, we obtain the new result that the steady-state ring should have a solar angle of 90°. This configuration arises from the dominance of small grains in this ring component and the correspondingly small lifetimes. The grains do not have time to perform a full cycle of the evolution of the eccentricity, and the solar angle stays close to its initial value of 90°.
For a vertical traversal with a spacecraft, we estimate that a dust detector of 1 m2 area should record about 100 grains larger than |$0.5\, \mu \mathrm{m}$| from either ring.
We derive the edge-on geometric optical depth from our model, giving for the Phobos ring an estimate of τ ∼ 3.5 × 10−8 and about τ ∼ 3.1 × 10−7 for the Deimos ring, which is below the upper limits for the edge-on photometric optical depth inferred from observations of ∼2 × 10−6 for the Phobos ring and ∼10−6 for the Deimos ring.
Our model results are subject to fairly large uncertainties, as is the case for the models presented previously in the literature. The most uncertain parameter is the mass production rate of dust particles. On the one hand, the interplanetary projectile flux is still poorly constrained. On the other hand, not much is known about the surface properties of the source moons, which induces uncertainties in the ejecta yield, the bulk density of ejected grains, and in their dynamical response on solar radiation pressure and P-R drag. The values of the interplanetary flux and the ejecta yield affect the results linearly. The shape of the initial mass distribution, the bulk density of the particles, and their material have a size dependent effect, and their variation will affect the steady-state size distribution, the number densities, and the optical depth of the rings in a non-linear manner. The precise error limits of the model results induced by the uncertainties in the parameters is difficult to assess, but it might easily amount to an uncertainty of an order of magnitude, or even more. For the initial distribution of ejecta masses, we use a differential power law with a slope of α = 0.9, as it was inferred by in situ measurements in the lunar dust cloud. We also checked models with a slope of α = 0.8 (a value used in previous modelling of dust rings) and found that our main conclusions on the grain size distribution in the rings, the number densities, as well as their spatial configuration and the optical depths are robust. An additional complication might arise from the intermittency of the interplanetary flux (Horányi et al. 2015). For the small lifetimes of the grains that dominate the Phobos ring, this might result in a significant variability of the ring over months. Finally, we note that Krivov et al. (2006) pointed out the potential importance of grain–grain collisions as a sink, which we have not included in our modelling. Collisions might play a role especially for the Deimos ring, owing to the long particle lifetimes, leading to a depletion of the number density and optical depth.
Although the Martian rings escaped detection so far, there is little or no doubt that the dust tori of Phobos and Deimos exist. The mechanism of quasi-continuous dust production in impacts of interplanetary meteoroids has been confirmed by measurements (Krüger et al. 1999; Horányi et al. 2015), as well as the formation of dust rings by this mechanism (Burns et al. 1999, 2004; Hedman et al. 2009). The best chance to detect the rings might be in situ measurements with a dust detector onboard a spacecraft, or, high-phase angle imaging from an orbiter when the spacecraft is in the shadow of Mars.
ACKNOWLEDGEMENTS
This work was supported by the European Space Agency under the project Jovian Meteoroid Environment Model at the University of Oulu and by the Academy of Finland. We acknowledge CSC – IT Center for Science for the allocation of computational resources.
DATA AVAILABILITY
The data underlying this article will be shared on reasonable request to the corresponding author.
