Daniel D. DurdaDaniel D. Durda(Southwest Research Institute)(Southwest Research Institute)

Daniel D. DurdaDaniel D. Durda(Southwest Research Institute)(Southwest Research Institute)

Discontinuities in Size-Strength Scaling Laws: Another Source of Wavy Size Distributions

Discontinuities in Size-Strength Scaling Laws: Another Source of Wavy Size Distributions

Small-size cutoff in the particle populationSmall-size cutoff in the particle population: At small sizes, a particle cutoff may : At small sizes, a particle cutoff may result from the rapid removal of bodies by non-gravitational forces such as result from the rapid removal of bodies by non-gravitational forces such as Poynting-Robertson drag or the Yarkovsky effect. Poynting-Robertson drag or the Yarkovsky effect.

Fig. 8 from Durda and Dermott (1997)Fig. 8 from Durda and Dermott (1997)

Previously Identified Sources of Waves

Fig. 4a from Durda Fig. 4a from Durda et al.et al. (1998) (1998)

Distinct change in slope in a size-strength scaling lawDistinct change in slope in a size-strength scaling law: Results from the transition : Results from the transition from strength-scaling for small bodies to gravity-scaling for larger objects.from strength-scaling for small bodies to gravity-scaling for larger objects.

Previously Identified Sources of Waves

Discontinuities in the size-strength scaling law, as might occur at small sizes when Discontinuities in the size-strength scaling law, as might occur at small sizes when particles are no longer homogeneous over the scale of fragmentation, can also particles are no longer homogeneous over the scale of fragmentation, can also induce significant waves in evolved size distributions.induce significant waves in evolved size distributions.

Discontinuous Scaling Laws

At size scales small enough that material inhomogeneities At size scales small enough that material inhomogeneities due to the presence of chondrules or Ni-Fe metal blebs or due to the presence of chondrules or Ni-Fe metal blebs or the effects of mineral grain boundaries become important, the effects of mineral grain boundaries become important, impact strengths may be dictated by the failure of internal impact strengths may be dictated by the failure of internal flaws with a narrow range of critical tensile strengths. flaws with a narrow range of critical tensile strengths. Q*Q*DD

may still depend on size, but may change discontinuously may still depend on size, but may change discontinuously at specific sizes and/or have a different size dependence in at specific sizes and/or have a different size dependence in different size ranges.different size ranges.

Much attention has been focused on determining how Much attention has been focused on determining how Q*Q*DD

varies with size for objects larger than the ~10-cm size varies with size for objects larger than the ~10-cm size targets typically studied in laboratory impact experiments.targets typically studied in laboratory impact experiments.

Discontinuous Scaling Laws

The four hypothetical scaling laws were used within a numerical collision model to The four hypothetical scaling laws were used within a numerical collision model to determine catastrophic disruption lifetimes and resulting evolved size distributions. determine catastrophic disruption lifetimes and resulting evolved size distributions. Initial populations with a Initial populations with a DD = 200 km largest body and a power-law slope index of = 200 km largest body and a power-law slope index of pp = = 2.7 (the number of bodies larger than 2.7 (the number of bodies larger than DD is proportional to is proportional to DD-p-p; for scaling laws with ; for scaling laws with size-independent size-independent Q*Q*DD, equilibrium size distributions have , equilibrium size distributions have pp = 2.5) were followed = 2.5) were followed

through 4.5 Gyr of collisional evolution.through 4.5 Gyr of collisional evolution.

Collisional Model Results

An abrupt change in the slope of the log An abrupt change in the slope of the log Q*Q*DD vs. log vs. log DD relation (in this case from a strength- relation (in this case from a strength-

scaling-like scaling-like DD-0.4-0.4 size dependence to an energy-scaling, size-independent strength at size dependence to an energy-scaling, size-independent strength at smaller sizes) induces minor perturbations to the evolved size distribution. The smaller sizes) induces minor perturbations to the evolved size distribution. The equilibrium power-law slope index changes value at the size where the scaling law changes equilibrium power-law slope index changes value at the size where the scaling law changes slope, and relatively minor wave-like perturbations propagate up the size distribution from slope, and relatively minor wave-like perturbations propagate up the size distribution from that point.that point.

Scaling Law H2Scaling Law H2

Collisional Model Results

An abrupt change to An abrupt change to strongerstronger particles at smaller sizes leads to an immediate particles at smaller sizes leads to an immediate troughtrough in the in the size distribution for particles just larger than that at which the strength properties change, size distribution for particles just larger than that at which the strength properties change, since the stronger particles survive longer and represent an overabundant source of since the stronger particles survive longer and represent an overabundant source of projectiles to break up larger bodies.projectiles to break up larger bodies.

Scaling Law H3Scaling Law H3

Collisional Model Results

An abrupt change to An abrupt change to weakerweaker particles at smaller sizes leads to an immediate particles at smaller sizes leads to an immediate crestcrest in the in the size distribution, since the weaker particles do not survive as long and represent an size distribution, since the weaker particles do not survive as long and represent an underabundant source of projectiles.underabundant source of projectiles.

Scaling Law H4Scaling Law H4

Collisional Model Results

Now that we are beginning to understand the extent to which the details of small-particle Now that we are beginning to understand the extent to which the details of small-particle removal mechanisms and the effects of changes in size-strength scaling relations can removal mechanisms and the effects of changes in size-strength scaling relations can affect the evolved size distribution at even the largest sizes, the time has come for a better affect the evolved size distribution at even the largest sizes, the time has come for a better understanding of the impact properties of particles smaller than ~10 centimeters.understanding of the impact properties of particles smaller than ~10 centimeters.

Conclusion

Example: George Flynn and I have proposed to conduct a series of impact experiments at the Ames Vertical Gun Range to determine (and demonstrate experimentally) the nature and extent of slope changes and/or discontinuities in Q*D. We will simulate the effects of small-scale structural inhomogeneities that might be expected to occur within small natural targets by constructing centimeter- to decimeter-scale spherical targets of plaster with imbedded BB-like spheres composed of a different material.