Numerical Model of Formation of Ejecta Faculae on Ceres

Ceres is a dwarf planet between the orbits of Mars and Jupiter. It is a dark body with geometric albedo ~0.09 (Ciarnello et al., 2017). Ceres' radius is ~469 km, density is 2162 kg/m³, and surface gravity is 0.284 m/s² (Park et al. 2019, Hargitai and Kereszturi, 2015). It is a partially differentiated body. It has a mainly rocky part and the crust. The crust consists of rock, with less than 40% water ice, salts, and clathrates (Scully et al., 2020; Ermakov et al., 2017). Note that the possible liquid water in Ceres interior now or in the past has high importance for astrobiology (e.g., Domagal-Goldman et al., 2016).

There are many bright spots (known as faculae [singular facula]) on the surface of Ceres discovered in 2015. They are built from bright material. Albedo of faculae could be four times higher than the average albedo of Ceres (Hargitai and Kereszturi, 2015). According to Palomba et al. (2019), the faculae are almost randomly distributed on the dwarf planet (Figure 1). The faculae contain mainly NH₄-chloride, Al-phyllosilicates, and Na-carbonate.

Figure 1.

The locations of bright spots (faculae) on the dwarf planet Ceres. There are more than 300 faculae. The bright ejecta are blue in this map. PIA21914.jpg. Image credit: NASA / JPL-Caltech / UCLA / MPS / DLR / IDA / PSI / Caltech.

The dark area contains Mg-phyllosilicates, Mg-carbonate, and NH₄-phyllosilicates (Raponi et al., 2019). For the bright matter, the term “salt” is often used here, without specification its chemical composition.

On Ceres, there are four types of faculae: (a) floor faculae (they are on the floor inside of impact craters), (b) faculae on Ahuna Mons, (c) rim/wall faculae found on craters' rims or walls, and (d) ejecta faculae in the form of bright ejecta blankets. With the exception of case (b), faculae are (in some way) associated with impact craters (Stein et al., 2019). We consider here mainly the ejecta faculae type (d) – Figure 2. The thickness of faculae is probably in the range ~2–50 m (Scully et al., 2020).

Figure 2.

The small (~10 km) bright crater Oxo on Ceres in perspective view. The elevation has been exaggerated by a factor of 2. The resolution is ~35 m/px. Recorded by NASA's Dawn mission. Note the irregular distribution of bright matter on the surface. PIA20916.jpg. Image credit: NASA/JPL-Caltech/UCLA/MPS/DLR/IDA

The aim of the work is to present and discuss a hypothesis explaining the formation of faculae type (d) as a result of grain separation due to the interaction of regolith with gas streams expanding during the formation of impact craters.

There is a hypothesis that floor faculae (type a) were formed by evaporation of water from brine or by sublimation of ice from frozen brine (Nathues et al., 2022; Schröder et al., 2021). According to this hypothesis, the release of H₂O to the surface would be the result of crumbling of the regolith and its heating during the meteorite impact. Floor faculae could be a direct result of this process (Thomas et al., 2018; Ruesch et al., 2016; Ruesch et al., 2019; Castillo et al., 2019).

However, this hypothesis has some drawbacks. It requires the recent (in geological terms) existence of substantial water reservoirs close to the surface on Ceres (e.g., compare the age of Ceres surface) (Neesemann et al., 2019; Bowling et al., 2019) or transferring large impact energy deep into the planet. The energy should melt some volume of brine (if necessary) and give rise to the mechanism (e.g., compressed gases) that transport the brine to the surface several Myr after the impact (e.g., in Occator crater).

Therefore, Czechowski (2023a, b), based on his experiments, proposed another hypothesis. It states that evaporation and/or sublimation of water have occurred on Ceres mainly before the formation of faculae. These processes took place in the regolith. After evaporation (or sublimation), grains of salt had been left dispersed in regolith. The concentration of salt on the surface (in the form of the present floor faculae) was the result of grain segregation occurring later by gas jets. This hypothesis has some advantages: it does not require substantial water reservoirs to exist several Myr after the impact. It only requires jets that can be a result of a fracture extending from the surface to a gas (e.g., CO₂) reservoir.

Of course, this hypothesis can be considered also as a complement to the hypotheses presented by Nathues et al. (2022), Schröder et al. (2021), Thomas et al. (2018), Ruesch et al. (2016), and Ruesch et al. (2019). It is possible that some floor facula arose as a direct result of evaporation on the surface, while others are the result of the mechanism proposed by Czechowski (2023a, b). Moreover, it is possible that both mechanisms may contribute when forming a given facula.

The origin of ejecta faculae of type (d) is the main subject of the present paper. According to some hypotheses, faculae (d) were formed as a result of meteoroid impacts (e.g., Stein et al., 2019), which exposed previously covered deposits of bright matter and ejected them outside the crater. However, this hypothesis has several drawbacks: 1)

The initial bright deposits should have been covered before exposure. Ceres does not have typical mechanisms for covering sediments (e.g., covering by material brought by water or wind) known from Earth.

Points (1–4) indicate that it is worth to discuss further hypothesis. We propose our mechanism subsequently. We suggest that initially, grains of salt had been dispersed in regolith (Section 2). However, during meteorite impact, volatiles from regolith evaporate and form intensive jets. These jets lead to segregation of grains and formation of deposits of bright matter.

Of course, contribution from the mechanism given in points 1–4 and from our mechanism is possible. In this sense, we propose an additional possible mechanism that may be responsible for the formation of some faculae of type (d) (on bodies with regolith containing significant amounts of volatile substances).

Our investigations mentioned in Section 2 concerning formation of floor faculae (type a) indicate intensive segregation of granular matter by means of a gas stream. However, they do not take into account the process during formation of impact craters. During the meteoroid impact, the gas stream is much more intense than during formation of faculae of type (a). Moreover, the geometry of impact gas streams is different. Therefore, we developed the numerical model here.

The formation of impact craters is developing in several phases. Melosh (2011) indicated the following phases: (1) contact and compression of the impactor and the surface of celestial body, (2) excavation (i.e., excavation flow), (3) early modification of the transient crater, and (4) later modification by erosion and accumulation in the long term. Vaporization of some matter may also take place. Even silicate might be vaporized, but the mass of such vapor is small due to the properties of the silicates (Gustavo et al., 2017; Bu et al., 2019).

The above scheme is based mainly on the study of impact craters on bodies with a low content of volatile substances. There are substantial differences for impact processes with a large amount of volatile substances (e.g., Hörz, 1982; Schenk et al., 2020; Sturm et al., 2013).

During impact on regolith containing significant amount of volatiles, the mass of the vaporized volatiles may be large (see also, e.g., Vickery, 1986; Silber et al., 2018). This gas expands in the space above the surface of the planet as well as in the fractured regolith. These gas streams may be agents of grain separation according to size, density, or shape.

The above considerations are confirmed by field studies. Three major facies are found in Ries Crater ejecta, indicating intensive processes of segregation (Hörz, 1982). Formation of facies (1) (of moldavite tektites) required drag forces in a rapidly ascending cloud of vaporized materials. The gravitational field alone does not give separation in the case of free motion of grains.

In the present work, we consider the model of interaction of gas with granular matter during the formation of an impact crater. We consider here only radially moving gas and grains’ motion above the surface of the celestial body because this is the process that may be mainly responsible for forming ejecta blankets and faculae of type (d).

During this interaction, the grains could have been segregated according to weight, density, size, and shape. Because grains of bright matter are composed of a different substance than the rest of the regolith, they may be separated and may form faculae of type (d) (in the form of bright ejecta blankets).

According to our hypothesis, the formation of ejecta faculae type (d) is the result of grains’ separation during formation of the impact crater. Separation is the result of the interaction of gas streams on grains with different size, shape, density, etc. To describe this process, we use the numerical model described below.

Let us consider the process of interaction of grains and a 2D flow of gas. Axial symmetry of flow is assumed (y-axis is the axis of symmetry); therefore, the process could be considered only for (x > 0) and (y > 0) and (x² + y² > R_hsp²). We assume radial motion of the gas above the flat surface (in the upper half-space) (Figure 3).

Figure 3.

The sketch of the situation considered. Gas flows radially from point (0, 0). The y-axis is the axis of symmetry. At the hemisphere of radius R_hsp we assume the radial velocity of the gas to be v_0gas. The gas velocity decreases in proportion to (R_hsp/R)², where R is the distance from (0, 0). The gas interacts with the grain (dark dot). The acceleration of gravity is g = (0, −g).

We use the following formula for the drag force vector for a grain of characteristic size r moving with velocity v in a gas of density ρ_gas and velocity v_gas: (1) Fdrag=−12CDρgasAv−vgasv−vgas, {{\boldsymbol {F}}_{{\boldsymbol {drag}}}} = - {1 \over 2}{C_D}{\rho _{gas}}A\left| {{\boldsymbol {v}} - {{\boldsymbol {v}}_{gas}}} \right|\left( {{\boldsymbol {v}} - {{\boldsymbol {v}}_{gas}}} \right), where cross-section area of the grain is A = πr² and drag coefficient is C_D. Its value depends mainly on the shape; for a perfect sphere, C_D = 0.47 and for a cube, it is larger. |v − v_gas| denotes the absolute value of a vector v − v_gas. Eq. (1) is a 2D form of typical formula of drag force for large Reynolds number Re (e.g., for Re > 10²) (e.g., Czechowski, 2014). Note that (2) Re=ρgasvs/η, {Re} = {\rho _{gas}}vs/\eta , where η is dynamical viscosity [Pa s] and s is a size of grains [m] (e.g., Czechowski et al., 2023). For simplicity, we use the same formulas for mass m and cross section A of grains, as for spherical bodies. Of course, it is just an approximation. The equation of motion for a given grain (a test particle) is (3) m dv/dt=Fdrag+mg, m\;d{\boldsymbol {v}}/dt = {{\boldsymbol {F}}_{{\boldsymbol {drag}}}} + m{\boldsymbol {g}}, where the mass of grain m = (4/3)πr³ ρ_grain and mg is the gravity force. After substituting (1) into (3) and dividing by mass m, we obtain (4) dv/dt=−Dv−vgasv−vgas+g, d{\boldsymbol {v}}/dt = - D\left| {{\boldsymbol {v}} - {{\boldsymbol {v}}_{gas}}} \right|\left( {{\boldsymbol {v}} - {{\boldsymbol {v}}_{gas}}} \right) + {\boldsymbol {g}}, where the coefficient D is (5) D=3CDρgas/8rρgrain. D = 3{C_D}{\rho _{gas}}/\left( {8r{\rho _{grain}}} \right).

We assume that gas flows radially from a hemisphere of radius R_hsp located on the surface of the planet (Figure 3). This is a simple model of gas outflow from the region of impact.

On this hemisphere, we assume the velocity of gas v_0gas. Only the radial component of v_0gas is nonzero. Assuming the law of mass conservation, v_gas decreases proportionally to (R_hsp/R)², where R is the distance from the origin of coordinate system (0, 0).

Initial position and velocity of grain are given by R₀, α₀, v₀. The initial position is x₀ = R₀ cos α₀, y₀ = R₀ sin α₀. For initial velocity v₀, we assume that v_0x=v_0y.

In equation (4), most of the parameters are included in coefficient D. The number of independent parameters is therefore smaller. The system contains the following parameters: D, R₀, α₀, v₀, g, R_hsp, v_0gas. Note that v_0gas and v₀ give only two scalar parameters because only the radial component of v_gas is nonzero and we assume that v_0x = v_0y.

There are several independent parameters in our problem. Grain parameters are r [m], ρ_grain [kg/m³], drag coefficient C_D (Ciarnello et al., 2017), initial position of the grain (a test particle) R₀, α₀ and its velocity v₀. The gas parameters are ρ_gas, R_hsp, v_0gas. The planet parameter is gravity: g = (0, −g) [m/s²].

The number of parameters can be further reduced by introducing the dimensionless form of equation (4) and dimensionless form of initial/boundary conditions. For this purpose, let us introduce the following natural units (n.u.): of length L = R_hsp and time τ = (L/g)^1/2. The n.u. of velocity is ɛ = (Lg)^1/2. Using these units, equation (4) takes the form (6) dv′/dt′=−Cv′−v′gasv′−v′gas+1, d{\boldsymbol {v}'}/dt' = - C\left| {{\boldsymbol {v}'} - {{{\boldsymbol {v}'}}_{gas}}} \right|\left( {{\boldsymbol {v}'} - {{{\boldsymbol {v}'}}_{gas}}} \right) + {\boldsymbol {1}}, where dimensionless parameter C = DL and primes denote dimensionless quantities (time t′ and velocity v′). Dimensionless forms of parameters are (7) C=DL, R′0=R0/Rhsp, α0=α0′,v′0=v0/ε,g′=1, R′hsp=1, v′0gas=v0gas/ε. \matrix{{C = DL,\;\;\;{{R}^\prime_0} = \left( {{R_0}/{R_{hsp}}} \right),\;\;\;{\alpha _0} = {\alpha _0}^{\prime},{{{\boldsymbol {v}'}}_0} = {{{\boldsymbol v}}_0}/\varepsilon ,} \cr {g' = 1,\;\;\;{{R}'_{hsp}} = 1,\;\;\;{{{\boldsymbol {v}'}}_{{0_{gas}}}} = {{\boldsymbol {v}}_{{0_{gas}}}}/\varepsilon .} \cr }

Note that now g′ = 1 and R′_hsp = 1. We stay with only five parameters: C, R′₀, α₀, v′₀, and v′_0gas. Let us remember that for the initial grain velocity v′₀, we assume that v′_0x = v′_0y. Reducing the number of independent parameters makes it easier to consider a wide range of cases.

As one can see, the introduction of a dimensionless system of equations reduced the number of parameters. In our case, this is mainly the result of the appearance of parameter C, which includes several dimensional parameters. When transforming equations into dimensionless form, such parameters often appear (e.g., Qing-Ming, 2011). Some of them have their own names (e.g., Reynolds number, Rayleigh number, etc.). In hydrodynamics alone, one can find over 70 such dimensionless numbers, but there are actually many more in the literature. For example, a few additional Rayleigh numbers for thermal convection need to be defined (depending on the method of heating or the type of rheology) (Czechowski and Kossacki, 2012). Generally, the number and form of the resulting dimensionless numbers depend on the problem and the choice of n.u. by the scientist (see also, e.g., Czechowski and Kossacki, 2012; Gritsevich, 2009; Moilanen et al., 2021). For a given problem, one can generally select several sets of n.u. and obtain several forms of the system of dimensionless equations. This often depends on the needs of the individual researcher.