Atmospheric loss and supply by an impact-induced vapor cloud: Its dependence on atmospheric pressure on a planet
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB 2010
Received: 29 July 2009
Accepted: 8 June 2010
Published: 31 August 2010
Hypervelocity impact would vaporize the impactor and part of planetary surface and create a rock vapor cloud. Results from previous studies suggest that the energetic impact would have a role to blow off and cause a large-scale loss of the planetary atmosphere through expansion of the vapor cloud. Impact also has been considered as a material source. Numerous, repeated impact events during the heavy bombardment period could greatly affect the amount of volatiles and the atmospheric pressure on the planetary surface in either way. To discuss the evolution of the atmospheric pressure by impacts, we carried out hydro-calculations with a two-dimensional hydrodynamic code and investigated the dependence of the loss and supply of the atmosphere on the atmospheric pressure. We integrated both effects by impacts over impactor size distribution and assessed the evolution of the atmospheric pressure on early Mars. Using this approach, we found that the numerous impacts likely increase the atmosphere monotonically or control the atmospheric pressure to some value, rather than causing the monotonic decrease as the previous study suggested.
Key wordsImpact erosion late heavy bombardment Mars atmosphere hydrodynamic calculation
The presence of numerous craters on the Moon and Mars suggest that the terrestrial planets experienced intense series of impact events after the main stage of their formation. The dating analysis of lunar rocks suggests that a ‘heavy bombardment period’ occurred about 3.8 billion years ago, when the terrestrial planets had already grown to nearly their present size. At this time, impact velocity would be high enough to vaporize fully or partially the impactor and part of the planetary surface. Such energetic impact events could affect the volatile budget on the planet.
Cameron (1983) suggested the possibility that impacts would cause the loss of planetary atmospheres: hypervelocity impact events create cloud of rock vapor that could blow off a fraction of the planetary atmosphere via energetic expansion of the vapor cloud. The process is referred to ‘impact erosion’. Melosh and Vickery (1989) analytically estimated the condition for the large-scale loss of the atmosphere and integrated effects of the atmospheric loss by impacts over impactor size distribution. Their results suggested that the atmospheric pressure would significantly decrease through the heavy bombardment period on Mars. The atmospheric mass eroded from the planet has been extensively studied and revised by several groups with more sophisticated approaches: an analytical model in Vickery and Melosh (1990) and hydrodynamic calculations of vapor expansion in Newman et al. (1999). Recently, Shuvalov and Artemieva (2002) performed full simulations for impacts of asteroids and comets on the present-day Earth. In these three papers (Vickery and Melosh, 1990; Newman et al., 1999; Shuvalov and Artemieva, 2002), however, the calculations are carried out only under the current atmospheric pressure on the Earth, namely, 1 [bar] air. It is not enough to discuss the atmospheric pressure change through the bombardment because the atmospheric pressure may greatly change (up to two orders of magnitude) by impacts through the heavy bombardment period on Mars, as discussed by Melosh and Vickery (1989). The change in the atmospheric pressure could, in turn, affect the loss efficiency of the atmosphere by impacts.
Impacts also contribute to the supply of atmospheric volatiles. A fraction of the volatiles in the asteroids and comets does not escape and is retained on the planet. Also, there may be volatiles buried in the planet that are liberated but do not escape. The competition between the atmospheric loss by vapor expansion and the volatile supply by retention of the vapor cloud would affect the volatile budget. The mass lost from the planet and its atmospheric pressure dependence, however, has not been fully investigated by hydrodynamic calculation. Melosh and Vickery (1989) did not account for the supply of volatiles, while Chyba (1990) and Zahnle (1993) assumed the rock vapor to be totally retained on the planet in their discussion of the volatile budget on Earth and Mars, respectively. To discuss the influence in volatile budget by impacts, it is necessary to consider the effects of both the atmospheric blow-off and vapor retention, and their atmospheric pressure dependence.
In this paper, we develop a two-dimensional (2-D) hydro-dynamic code and carry out calculations of vapor expansion over a wide range of atmospheric pressure on a planet to investigate the pressure dependence of the behavior of the atmosphere and vapor cloud. Our hydrocode is equivalent to that by Newman et al. (1999) and may be rather primitive compared with the full simulations in Shuvalov and Artemieva (2002). However, it enables us to carry out calculations over the wider parameter range because of the less computational load. Also, our simple model setting helps us to understand the physics which controls the induced flow and the mass of the eroded atmosphere and retained vapor cloud. We integrated the impact-induced loss and supply of the atmosphere over a plausible impactor mass distribution for Mars, taking the derived atmospheric pressure dependence into consideration. Our results indicate that the integrated effect of numerous impact events is to monotonically increase or to control to some value the atmospheric pressure on Mars during the heavy bombardment period, instead of the monotonic decrease suggested by Melosh and Vickery (1989).
2. Numerical Method
2.1 Model setting
2.2 Formulation and computational model
We used a staggered grid system with the finite difference scheme. The initial vapor cloud is resolved by 10 × 10 regular zones. For the other region, the spatial grid interval is set to increase by a geometric series in which the ratio of successive terms is 1.05. The entire computational region is resolved into 180 × 180 rectangular zones and may become as large as about 1 × 108 times the initial vapor size. In the case that the ratio of successive terms is 1.02, the mass of the atmospheric and vapor loss increases by about 4%. We applied solid boundary conditions to the planetary surface and z-axis and non-reflective boundary conditions to the edges of the atmosphere. The time interval is determined at each time step from the Courant-Friedrichs-Lewy (CFL) condition. We set the CFL number at 0.2.
3. Impact Erosion and Supply
3.1 Definition of the “blown-off mass” for the atmosphere and vapor cloud
In Fig. 1, our result is plotted with numerical result reported in Newman et al. (1999). From Fig. 1 we can see that their computation time is not enough for the escaping mass to level off. It is possible that these researchers calculated the flow at the very early stage, during which the shock wave plays a dominant role in atmospheric acceleration. In such an early stage, a larger portion of the vapor cloud, which has most of the momentum, remains near the planetary surface. Also, our calculation shows that the escaping vapor cloud accounts for a large fraction of the total escaping mass. Though they did not distinguish the escaping vapor from the escaping atmosphere, it is speculated that the total escaping mass in Newman et al. (1999) mainly represents the escaping mass not of the atmosphere, but of the vapor cloud.
3.2 Atmospheric-pressure dependence of the vapor-cloud loss
The vapor mass retained on the planet is derived from subtracting the mass of its loss from its total mass. The results indicate that the vapor mass retained on the planet is almost proportional to the vapor mass but independent of the ambient pressure at the limit of minimum retention, and that the supply of the volatiles would be more effective for the larger vapor mass.
3.3 Atmospheric-pressure dependence of the mass of atmospheric loss
3.4 The shape of atmospheric escape region
The shape of the escape region reflects the dynamic flow of the atmosphere. The atmosphere is strongly compressed by the passage of the preceding shock wave so that a high-pressure shell of the shocked atmosphere is formed ahead of the vapor expansion front. Since the atmospheric mass in the shell is large for a massive atmosphere, the vapor cloud is required to transfer more momentum to expand. Thus, the vapor cloud loses its momentum at the very early stage while the flow is almost radial outward from the impact point. This process results in the spherical shape of the high-pressure shell. The shape of the escape region is simply determined with the critical zenith angle θ from the vertical, which is smaller than that the vapor cloud can push away the atmosphere in the azimuthal bin, resulting in the cone-shaped escape region (Fig. 4(a)).
In contrast, the bowl- or truncated-cone-shaped escape region results from the deviation of the vapor flow from the radial direction. As the atmospheric pressure decreases, the vapor cloud can expand further against the ambient atmosphere. The velocity of the shock wave is zenith angle-dependent in the stratified atmosphere: that is, the higher the velocity, the smaller the zenith angle. The velocity differences between the zenith angles changes the shape of the high-pressure shell from the closed spherical shape to an open bowl or truncated-cone profile, and the atmosphere and vapor cloud within the shell flows along the shell wall. As a result, the deflection of the radial-vapor flow induces the arc-like flow in the atmosphere with the large zenith angle (near surface), which results in the truncated-cone shaped escape region (Fig. 4(b)).
The shape of the escape region allows us to speculate on the pressure dependence of the escaping atmospheric mass. Although the atmospheric density is proportional to the atmospheric pressure, the blown-off mass of atmosphere varies as pressure to the ∼0.3 power for small atmospheric pressures, as shown in Fig. 3 and the previous section. Under such an atmospheric pressure, the escape region is truncated-cone-shaped, and its bottom radius determines the mass of the atmospheric loss. The vapor cloud can expand further against the lower atmosphere and, as a result, the bottom radius of the escape region spreads out. This compensates for the increase in atmospheric density with atmospheric pressure and weakens the sensitivity of the blown-off mass to the pressure.
3.5 The optimum loss pressure and an empirical formula of the eroded atmospheric mass
3.6 Comparison with the hemispheric blow-off model
We also applied the hemispheric blow-off model in Vickery and Melosh (1990) to calculate and compare their pressure dependence (Figs. 2 and 3). The pressure dependence is in remarkably agreement with our hydrodynamic calculations for low atmospheric pressure. However, at some large pressures in the hemispheric blow-off model, both the loss of the atmosphere and vapor cloud rapidly decrease and shut off, while our results show more gradual decreases. The difference results from the difference in the criteria for escape from the planet. In the model of Vickery and Melosh (1990), it is required that the radial velocity averaged in the azimuthal sector exceeds the planetary escape velocity to blow off the gases. In actual conditions, however, radial velocity distribution exists, and the front of the expanding vapor cloud is faster than that of its most inner portion. The criterion using the averaged velocity in the sector therefore underestimates atmospheric and vapor loss by the impacts.
The mass of the atmospheric loss estimated by Vickery and Melosh (1990) is less than that of our hydrodynamic calculations by some factor, while the pressure dependence is in good agreement between both models. For the parameter set used at the calculations in Figs. 2 and 3, the factor is almost 0.5. Although the value of the factor varies with the parameter sets, it is found that the value stays below unity over the ranges of the parameters considered.
3.7 The implication to the atmospheric pressure change on early Mars
The atmospheric pressure dependence derived from our calculations can be summarized as follows. The eroded mass increases with the power-law dependence against the atmospheric pressure, while the retained fraction of the vapor cloud is nearly constant, when the impact satisfies condition (16), which means the impact is energetic enough against the ambient atmospheric mass. If the impact is less energetic, the eroded mass of both the atmosphere and vapor cloud rapidly decreases with the atmospheric pressure and finally becomes zero.
The mass is given by the empirical formula (18) forthe atmospheric loss and by the interpolation from the calculation results for the vapor cloud retention, respectively. In this paper, we treated the planetary surface as a planeparallel one. The planetary surface and atmosphere, however, have a curvature, and this would be more important for the larger impactor. The assumption of the plane-parallel atmosphere causes an over estimation of the mass of the atmospheric loss because in the plane-parallel model the infinite atmospheric mass is assumed on the surface. To avoid this overestimation, we set the upper limit for the atmospheric mass eroded from the planet by the atmospheric mass above the tangent plane to the impact point.
Although most of the fraction of the retained vapor would condense with expansion and be removed from the atmosphere, some volatile elements would remain in the gas phase, leading to the increase of the atmospheric mass. We treated its supply efficiency as a free parameter, w, which denotes the mass fraction of the gas supplied to the atmosphere after the cooling of the vapor cloud. Although the parameter w is not well constrained at present, it may be inferred from the mass fraction of carbon dioxide supplied in the impactor. Carbon content is 0.03–0.05 for CI chondrites (Kerridge, 1985) and 0.001–0.002 for ordinary chondrites (Jarosewich, 1990). These values correspond to 0.11–0.18 and 0.04–0.07 of carbon dioxide in mass fraction, respectively, if simply assumed that the all the carbon becomes carbon dioxide.
The positive value of Δ a in Fig. 5 denotes an increase in the atmospheric mass and the negative one denotes a decrease. At the atmospheric pressure where the value of Δ a equals 0, the loss and supply are in balance so that the atmospheric pressure keeps its value. The results in Fig. 5 suggest two possible ways in which the atmospheric pressure evolves by impacts. One is the monotonic increase, and the other is the regulation to a certain pressure.
The monotonic increase occurs at relatively large w and/or very large atmospheric pressure. At a large w value the curve has no intersection with the x-axis, such as the case with w = 0.03 in Fig. 5(b). In this case, the volatiles supplied by impacts overwhelm the atmospheric loss, and the impacts increase the atmospheric mass on the planet regardless of its atmospheric pressure. At very large atmospheric pressure, the curve has an intersection at which the value of Δ a changes from the negative to positive with increasing atmospheric pressure, although such a large atmospheric pressure is out of range in Fig. 5. Then, the atmospheric mass would monotonically increase if the atmospheric pressure is larger than the value at the intersection.
The regulation of pressure occurs in those cases when curves in Fig. 5 intersect with the x axes at the point where the sign of Δ a changes from a positive to a negative value with increasing atmospheric pressure. (For example, the curves with w = 0.01−0.1 in Fig. 5(a) and w = 0.001 and 0.01 in Fig. 5(b)). We refer to the atmospheric pressure at such an intersecting point as pctl. Let us now consider the behavior of the atmospheric pressure by the impacts around pctl. If the atmospheric pressure is less than pctl, the value Δ a is positive so that the atmospheric pressure increases and approaches pctl by the impacts. One can have a similar result for the opposite case. This indicates that the impacts would play a role in controlling the atmospheric pressure to approach to pctl, not unilaterally to take it away.
This behavior is a natural result of the difference in the pressure dependence between the atmospheric loss and the retained vapor cloud. The eroded mass of the atmosphere shows the power-law dependence (Eq. (18)) to the atmospheric pressure and has the maximum value at the optimum atmospheric pressure. On the other hand, the mass of the retained vapor cloud is almost independent of the atmospheric pressure as long as the impact is energetic enough to satisfy condition (16). As a result, the atmospheric pressure decrease by impacts, if any, has the minimum value.
It should be noted that whether the final atmospheric pressure after the heavy bombardment reaches pctl or not depends on the total impactor mass through the heavy bombardment. Also, the value of pctl and its actual existence also depend on the value of w and the vapor energy (the impact velocity). We treated the maximum impactor mass mimax as an independent variable. Assuming that the Poisson distribution can be applied to the number of the largest impacts and that the expected value of the number of the largest impact is equal to 1, the value of mimax could vary from 0.651 × 1018 [kg] to 2.16 × 1018 [kg] by using 1σ, and then the value of pctl could change by a factor of about 2. This change is mainly caused by the change in the retained vapor mass because the mass of the retained vapor cloud is approximately proportional to mimax, while the change of the mass of the atmospheric loss is relatively less, especially under the smaller atmospheric pressure.
Zahnle et al. (1992) also considered the competition between impact erosion and supply of atmospheres over Titan, Ganymede and Callisto. They found two regimes of atmospheric evolution; the erosive regime and the accumulative regime. The erosive regime seems to be corresponding to the curves with the intersections in our Fig. 5. These authors also suggested that the equilibrium between impact erosion and supply would occur under certain conditions in this erosive regime. Their equilibrium pressure would be qualitatively equivalent to pctl in our paper. Their equilibrium pressure, however, decreases to zero with time, probably because the maximum impactor mass is supposed to decrease to zero with time in their model. The atmospheric evolution in their paper seems to have a more erosive history than that suggested by our results. Even in their accumulative regime, an initially thick atmosphere decreases its mass by the impacts. Such an erosive evolution is not only because they used quite different parameters from ours (such as the target planets, the composition and velocity of the impactor, the maximum mass of the impactor and etc.), but also because they assumed the tangent-plane mass atmospheric erosion model by Melosh and Vickery (1989), which assumes the mass of the atmospheric loss to be proportional to the atmospheric pressure. Hence, in some cases, the thicker atmosphere tends to lose the more mass and have more erosive history.
In order to discuss the time evolution and the final pressure of the atmosphere, the random impacts should be considered with Monte Carlo calculations, as Griffith and Zahnle (1995) performed, because it is plausible that the order in which each impact falls onto the planet may be one of the factors influencing the final atmospheric pressure on the planet. For example, if the largest impactor falls at the relatively late stage, the impactor would supply the enormous amount of volatiles so that the subsequent impacts cannot reduce it, and the final atmospheric pressure would be greater than pctl. As such, it would be necessary to discuss the distribution of the final atmospheric pressure.
Our results suggest that it is possible that the terrestrial planets could have the thin atmosphere under which the impact erosion is balanced against the impact supply. Especially if the initial atmosphere is thin, the planet would acquire some volatiles by the impacts and the atmospheric pressure would reach pctl. For example, Fig. 5 shows that even under the erosive conditions such that the supply efficiency w is 0.01 and the impact velocity is about 19 [km/s], Mars could get 0.03 [bar] of atmosphere with the sufficient amount of the veneer. If this happens, the composition of the atmosphere should reflect the aftermath, being similar to the volatile composition of the impactors. If Mars had lost its atmosphere by impacts, its atmospheric composition would likely modified by the volatile component in the impactors, unless they were very dry. Xe isotopic abundances in the Martian atmosphere, however, are not coincident with those of any asteroids as Zahnle indicated in his paper in 1993. It is still unknown what this means. It may suggest that Mars did not experience the intense bombardment that researchers thought or that the veneer composition was quite different from the existing asteroids-comets or very dry, volatile poor planetesimals.
4.1 The effect of the wake and the initial vapor velocity
Impactor penetration accelerates the atmosphere in the entry path and creates a hot wake in which the atmospheric density is lower by a few orders of magnitude than the ambient atmosphere. The recent full-simulation by Shuvalov and Artemieva (2002) suggests that the occurrence of the wake has an important role in preventing the ambient atmosphere from escaping because the vapor cloud could go preferentially through the wake. One of their conclusions is that the difference in the shape of the induced wake contributes to more atmospheric loss by an oblique impact than by a normal impact. Unfortunately, they did not examine the mass of the vapor retained on the planet and the atmospheric pressure dependence. We therefore performed additional experiments on the effects by the wake and the initial velocity distribution within the vapor cloud. We compared the mass of the atmospheric loss for the specific cases under which they performed the numerical simulations.
The most remarkable feature is that the direction of the initial vapor velocity could greatly change the mass of the atmospheric loss. The isotropic initial velocity produces the same results as with condition (a), in which all of the energy of the vapor cloud is given as a thermal one, while the upward initial velocity resulted in less atmospheric loss at low atmospheric pressures and more loss at its onset under the large pressure. Also, Fig. 7 shows the same power-law dependence on the atmospheric pressure regardless of the initial conditions for the non-dimensional atmospheric pressure σ−1 of less than 10−6.
These results mean that one of the key parameters for the impact-erosion problem is the azimuthal distribution of the initial vapor energy instead of its form (thermal or kinetic). The direction of the initial velocity would be different between the normal and oblique impacts. The suggestion in Shuvalov and Artemieva (2002) that the oblique impacts cause more atmospheric loss than a normal impact may be due to the difference in the direction of the initial velocity, instead of the shape of the wake. Further investigation is necessary to gain a better understanding of the effects of oblique impacts.
The comparison results show that our model with the above impact-vapor relations (Eqs. (19), (20) and (27)) predicts a greater loss of atmospheric mass than that in Shuvalov and Artemieva (2002) by a factor of 150–300 for the asteroids and 1.2–30 for the comets. The differences are more remarkable for the lower velocity impacts or the asteroidal impacts rather than the cometary impacts—both are impacts that create less vapor cloud. From the results of our preliminary experiments, which suggest that the existence of the wake is not an influential factor, the quantitative discrepancy would be primarily attributed to the impact-vapor relation that we used. The impact-vapor relation assumed in this paper (Eqs. (19), (20) and (27)) may overestimate the mass and/or the energy of the vapor cloud. Alternatively, it may be that the vapor volume is underestimated in Shuvalov and Artemieva (2002) because Pierazzo et al. (1997) reported that the vapor volume is underestimated by using the equation of state they used, ANEOS.
It should be noted that, even if our impact-vapor relation leads to an overestimation of the mass and energy of the vapor cloud, the scenario that the atmosphere decreases monotonically by impacts is less plausible because we tend to overestimate the effect of impact erosion. The atmospheric-pressure dependence that we derived would hold and the discussion about the pressure control mechanism involving pctl is valid qualitatively—unless the ambient atmosphere affects the vaporization process of the impactor and target surface by an impact.
4.2 Condensation of a vapor cloud
5. Concluding Remarks
Hypervelocity impact would have great effects in the volatile budget on the planetary surface: its loss is due to the explosive vapor expansion, and its supply is due to the vapor retention. Numerous impact events during the heavy bombardment period would have the great consequences for the atmospheric evolution, especially for the atmospheric pressure change on the planets. In this paper, we performed the hydrodynamic calculations of the expansion stage of the impact-induced vapor cloud and examined the atmospheric pressure dependence of the mass that the vapor cloud would blow off and be retained, respectively. The effect of the atmospheric pressure depends on whether the impact is sufficiently energetic against the ambient atmospheric mass. If energetic enough, the mass of the blown-off atmosphere shows the power-law dependence, and the retained vapor mass is nearly independent of the atmospheric pressure. We also discussed the atmospheric pressure change by the impacts on early Mars, taking the derived atmospheric pressure dependences into consideration. We then found that the numerous impact events could change the atmospheric pressure in two possible ways: the monotonic increase or the regulation to a certain atmospheric pressure value. Which way the atmospheric pressure would evolve, however, is strongly dependent on the vapor energy and mass and the supply efficiency of volatiles in the vapor cloud. It is likely that impacts in the early stage of the terrestrial planets contribute to the supply of volatile material rather than its loss, though the quantitative discussions for the evolution of the atmospheric pressure requires more detailed full calculations, including proper EOS and the composition and content of volatile materials in impactors.
The constructive comments by K. Zahnle and an anonymous reviewer greatly helped us to improve this paper. We also appreciate the support of H. St. C. O’Neill to continue this study. This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for JSPS Fellows.
- Cameron, A. G. W., Origin of the atmospheres of the terrestrial planets, Icarus, 56, 195–201, 1983.View ArticleGoogle Scholar
- Chyba, C. F., Impact delivery and erosion of planetary oceans in the early inner solar-system, Nature, 343, 129–133, 1990.View ArticleGoogle Scholar
- Davis, S. S., An analytical model for a transient vapor plume on the Moon, Icarus, 202, 383–392, 2009.View ArticleGoogle Scholar
- Griffith, C. A. and K. Zahnle, Influx of cometary volatiles to planetary moons: The atmospheres of 1000 possible Titans, J. Geophys. Res, 100, 16907–16922, 1995.View ArticleGoogle Scholar
- Jarosewich, E., Chemical-analyses of meteorites—a compilation of stony and iron meteorite analyses, Meteoritics, 25, 323–337, 1990.View ArticleGoogle Scholar
- Kerridge, J. F., Carbon, Hydrogen and nitrogen in carbonaceous chondrites—Abundances and isotopic compositions in bulk samples, Geochim. Cosmochim. Acta, 49, 1707–1714, 1985.View ArticleGoogle Scholar
- Melosh, H. J. and A. M. Vickery, Impact erosion of the primordial atmosphere of Mars, Nature, 338, 487–489, 1989.View ArticleGoogle Scholar
- Newman, W. I., E. M. D. Symbalisty, T. L. Ahrens, and E. M. Jones, Impact erosion of planetary atmospheres: Some surprising results, Icarus, 138, 224–240, 1999.View ArticleGoogle Scholar
- Ohkubo, T., M. Kuwata, B. Luk’yanchuk, and T. Yabe, Numerical analysis of nanocluster formation within ns-laser ablation plume, Appl. Phys. a-Materials Sci. Process., 77, 271–275, 2003.Google Scholar
- Pierazzo, E., A. M. Vickery, and H. J. Melosh, A reevaluation of impact melt production, Icarus, 127, 408–423, 1997.View ArticleGoogle Scholar
- Shuvalov, V. V. and N. A. Artemieva, Atmospheric erosion and radiation impulse induced by impacts, Geol. Soc. Am. Spec. Pap., 356, 695–703, 2002.Google Scholar
- Takewaki, H. and T. Yabe, The cubic-interpolated pseudo particle (CIP) method: application to nonlinear and multi-dimensional hyperbolic equations, J. Comput. Phys., 70, 355–372, 1987.View ArticleGoogle Scholar
- Vickery, A. M. and H. J. Melosh, Atmospheric erosion and impactor retention in large impacts, with application to mass extinctions, Geol. Soc. Am. Spec. Pap., 247, 300–389, 1990.Google Scholar
- Yabe, T. and T. Aoki, A universal solver for hyperbolic equations by cubic-polynomial interpolation I. One-dimensional solver, Comput. Phys. Commun, 66, 219–232, 1991.View ArticleGoogle Scholar
- Yabe, T., T. Ishikawa, P. Y. Wang, T. Aoki, Y. Kadota, and F. Ikeda, A universal solver for hyperbolic equations by cubic-polynomial interpolation II. Two- and three-dimensional solvers, Comput Phys Commun., 66, 233–242, 1991.View ArticleGoogle Scholar
- Yabe, T., F. Xiao, D. Zhang, S. Sasaki, Y. Abe, N. Kobayashi, and T. Terasawa, Effect of EOS on break-up of Shoemaker-Levy 9 entering Jovian atmosphere, J. Geomag. Geoelectr., 46, 657–662, 1994.View ArticleGoogle Scholar
- Yabe, T., F. Xiao, and H. Mochizuki, Simulation Technique for Dynamic Evaporation Processes, Nuclear Eng. Design, 155, 45–53, 1995.View ArticleGoogle Scholar
- Zahnle, K. J., Xenological Constraints on the Impact Erosion of the Early Martian Atmosphere, J. Geophys. Res, 98, 10899–10913, 1993.View ArticleGoogle Scholar
- Zahnle, K., J. B. Pollack, and D. Grinspoon, Impact-Generated Atmospheres over Titan, Ganymede, and Callisto, Icarus, 95, 1–23, 1992.View ArticleGoogle Scholar