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Numerical Model of Formation of Ejecta Faculae on Ceres

Leszek Czechowski

Leszek Czechowski

Centrum Badań Kosmicznych Polskiej Akademii NaukWarsaw, Poland
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Czechowski, Leszek
  
31 dic 2024
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Pubblicato online: 31 dic 2024
Pagine: 127 - 142
Ricevuto: 20 feb 2024
Accettato: 28 ott 2024
DOI: https://doi.org/10.2478/arsa-2024-0009
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© 2024 Leszek Czechowski, published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

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 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.

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 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

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 Rhsp we assume the radial velocity of the gas to be v0gas. The gas velocity decreases in proportion to (Rhsp/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).
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 Rhsp we assume the radial velocity of the gas to be v0gas. The gas velocity decreases in proportion to (Rhsp/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).

Figure 4.

Trajectories of test particles (in n.u.) for C in the range 0.016–524 (see the line colors in the legend), v′0gas = 50 (only the radial component), v′0 = (5,5), and R′0 = 1. Parameters are in n.u.The initial positions of the test particle are x′0 = cos α0 and y′0 = sin α0, where α0 = 20°,30°,40°,50°,60°,70°. Farthest from the point (0, 0), in a wide range from x′ = ~30 to ~50, the particles with the largest C land (blue lines). This is the result of the strong influence of gas motion on this group of particles. Inside this region, a concentrated group of particles with the smallest C = 0.016 lands. In this group of particles, the effect of drag of gas is less important and the trajectories are mainly determined by the force of gravity. Particles from the other groups (C = 0.128–65.536) land in the region of x′ = ~10–30, creating rather complicated relationship between the x' of landing and the particle parameters.
Trajectories of test particles (in n.u.) for C in the range 0.016–524 (see the line colors in the legend), v′0gas = 50 (only the radial component), v′0 = (5,5), and R′0 = 1. Parameters are in n.u.The initial positions of the test particle are x′0 = cos α0 and y′0 = sin α0, where α0 = 20°,30°,40°,50°,60°,70°. Farthest from the point (0, 0), in a wide range from x′ = ~30 to ~50, the particles with the largest C land (blue lines). This is the result of the strong influence of gas motion on this group of particles. Inside this region, a concentrated group of particles with the smallest C = 0.016 lands. In this group of particles, the effect of drag of gas is less important and the trajectories are mainly determined by the force of gravity. Particles from the other groups (C = 0.128–65.536) land in the region of x′ = ~10–30, creating rather complicated relationship between the x' of landing and the particle parameters.

Figure 5.

Trajectories of test particles for C in the range 0.016–524 (see the line colors in the legend), v′0gas = 20 (only the radial component), v′0 = (10,10), and R′0 = 3. The initial positions of the test particle are x′0 = 3 cos α0, y′0 = 3 sin α0, where α0 = 20°, 30°, 40°, 50°, 60°, and 70°. Here, we observe a far ejection of particles for small C = 0.016 (red lines, x′end = ~70). It is a result of high initial velocity v′0. The remaining groups fall near x′end = 7–30, creating a relatively complicated dependence on particle parameters, similar but not identical to Figure 4.
Trajectories of test particles for C in the range 0.016–524 (see the line colors in the legend), v′0gas = 20 (only the radial component), v′0 = (10,10), and R′0 = 3. The initial positions of the test particle are x′0 = 3 cos α0, y′0 = 3 sin α0, where α0 = 20°, 30°, 40°, 50°, 60°, and 70°. Here, we observe a far ejection of particles for small C = 0.016 (red lines, x′end = ~70). It is a result of high initial velocity v′0. The remaining groups fall near x′end = 7–30, creating a relatively complicated dependence on particle parameters, similar but not identical to Figure 4.

Figure 6.

Trajectories of test particles for C in the range 0.016–524 (see the legend), v′0gas = 50 (only the radial component), v′0 = (10,10), and R′0 = 1. The initial positions of the test particle are x′0 = cos α0, y′0 = sin α0, where α0 = 20°,30°,40°,50°,60°, and 70°. Here, we observe a far ejection of particles for small C = 0.016 (red lines, x′end = ~70). These trajectories are almost identical to red trajectories in Figure 5. The remaining groups fall near x′end = ~10–40, creating a relatively complicated dependence on particle parameters, similar but not identical to Figures 4 and 5. The difference is a result of initial positions of the test particles.
Trajectories of test particles for C in the range 0.016–524 (see the legend), v′0gas = 50 (only the radial component), v′0 = (10,10), and R′0 = 1. The initial positions of the test particle are x′0 = cos α0, y′0 = sin α0, where α0 = 20°,30°,40°,50°,60°, and 70°. Here, we observe a far ejection of particles for small C = 0.016 (red lines, x′end = ~70). These trajectories are almost identical to red trajectories in Figure 5. The remaining groups fall near x′end = ~10–40, creating a relatively complicated dependence on particle parameters, similar but not identical to Figures 4 and 5. The difference is a result of initial positions of the test particles.

Figure 7.

The parameters are similar to Figure 5, but the largest particles are excluded (with C = 0.016) and the smallest are added (C = 5242) – see the legend. The rest of the parameters are the same as in Figure 5: v′0gas = 20, v′0 = (10,10), and R′0 = 3. The initial positions of the test particle are x′0 = 3 cos α0, y′0 = 3 sin α0, where α0 = 20°,30°,40°,50°,60°, and 70°. The smallest particles (i.e. added particles with C = 5242) land at x′end > 30.
The parameters are similar to Figure 5, but the largest particles are excluded (with C = 0.016) and the smallest are added (C = 5242) – see the legend. The rest of the parameters are the same as in Figure 5: v′0gas = 20, v′0 = (10,10), and R′0 = 3. The initial positions of the test particle are x′0 = 3 cos α0, y′0 = 3 sin α0, where α0 = 20°,30°,40°,50°,60°, and 70°. The smallest particles (i.e. added particles with C = 5242) land at x′end > 30.

Figure 8.

Trajectories of test particles for C = 0.016–524 (see the legend), v′0gas = 50, v′0 = (0,0), and R′0 = 2. The initial positions of the test particle are x′0 = 2 cos α0, y′0 = 2 sin α0, where α0 = 20°,30°,40°,50°,60°, and 70°. Note zero initial velocity of the particles. Particles with C = 0.128 and 1.024 land close one to another. For other values of C, we observed that the smaller the C, the smaller the x′end.
Trajectories of test particles for C = 0.016–524 (see the legend), v′0gas = 50, v′0 = (0,0), and R′0 = 2. The initial positions of the test particle are x′0 = 2 cos α0, y′0 = 2 sin α0, where α0 = 20°,30°,40°,50°,60°, and 70°. Note zero initial velocity of the particles. Particles with C = 0.128 and 1.024 land close one to another. For other values of C, we observed that the smaller the C, the smaller the x′end.

Figure 9.

Another case with initial zero velocity of the test particles. However, the initial positions of these particles are closer to (0, 0), so the initial acceleration is also higher than in Figure 8. Trajectories for test particles with C = 0.016–524 (see the legend), v′0gas = 50, v′0 = (0,0), and R′0 = 1.The initial positions of the test particles are x′0 = cos α0, y′0 = sin α0, where α0 = 20°,30°,40°,50°,60°,70°. Particles with C = 0.016 – 8.192 land close one to another. Only for C = 65.536 and 524.288, particles land further.
Another case with initial zero velocity of the test particles. However, the initial positions of these particles are closer to (0, 0), so the initial acceleration is also higher than in Figure 8. Trajectories for test particles with C = 0.016–524 (see the legend), v′0gas = 50, v′0 = (0,0), and R′0 = 1.The initial positions of the test particles are x′0 = cos α0, y′0 = sin α0, where α0 = 20°,30°,40°,50°,60°,70°. Particles with C = 0.016 – 8.192 land close one to another. Only for C = 65.536 and 524.288, particles land further.

Figure 10.

Next case with initial zero velocity of the test particles. Particles with a range of C = 0.016–524 (see the legend), v′0gas = 20, v′0 = (0,0), and R′0 = 2. The initial positions of the test particles are x′0 = 2 cos α0, y′0 = 2 sin α0, where α0 = 20°,30°,40°,50°,60°,70°. The gas drag (lower than in Figure 9 because of lower gas velocity v′0gas = 20 and R′0 = 2) and gravity are the only accelerating forces. Note especially low acceleration of large particles (red lines). Their final velocity is low and landing points x′end = ~2. Note that this is the only case where the distance of landing is monotonically increasing with parameter C.
Next case with initial zero velocity of the test particles. Particles with a range of C = 0.016–524 (see the legend), v′0gas = 20, v′0 = (0,0), and R′0 = 2. The initial positions of the test particles are x′0 = 2 cos α0, y′0 = 2 sin α0, where α0 = 20°,30°,40°,50°,60°,70°. The gas drag (lower than in Figure 9 because of lower gas velocity v′0gas = 20 and R′0 = 2) and gravity are the only accelerating forces. Note especially low acceleration of large particles (red lines). Their final velocity is low and landing points x′end = ~2. Note that this is the only case where the distance of landing is monotonically increasing with parameter C.

The considered ranges of values of dimensionless parameters, corresponding to ranges presented in Table 1 expressed in n_u_

C R′hsp R′init v′0gas Radial component v′0 v′0x = v′0y α′0 = α0 g′
min 0.0881 1 1 1.84 0 0 1
max 15,000 1 5 58.4 58.4 90 1

Considered ranges of dimensional parameters of the system

Rhsp [m] v0gas [m/s] ρgas [kg/m3] CD [1] ρgrain [kg/m3] r [m] α0 [deg] v0grain [m/s] Rinit [m] g [m/s2]
min 1000 100 0.01 0.47 500 0.0001 0 0 1000 0.284
max 10,000 1000 0.1 2 2000 0.01 90 1000 5000 0.284

The ranges of values of natural units (n_u_) of length, time, and velocity expressed in SI units

L = Rhsp n.u. of length [m] τ = (L/g)1/2 n.u. of time [s] ɛ = (Lg)1/2 n.u. of velocity [m/s]
min 1000 54.42 17.11
max 10,000 184.74 54.12
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