Synthetic Impactor Populations

• Synthetic impactor populations are computationally generated datasets that replicate the statistical properties, temporal evolution, and size–frequency distributions of impactors on lunar and terrestrial surfaces.
• The methodology employs empirical SFDs, piecewise and power-law parameterizations, and pi-scaling laws to simulate both primary (LHB and NEO) and secondary impact events with controlled uncertainties.
• This framework enhances our understanding of planetary surface dating and dynamical evolution, providing actionable insights for reconstructing cratering histories and flux rates in the inner Solar System.

Synthetic impactor populations are algorithmically constructed datasets that mimic the statistical properties, size–frequency distributions (SFDs), and temporal evolution of objects that have impacted inner Solar System bodies, principally the Moon and terrestrial planets. These synthetic populations are central to quantitative analyses of planetary cratering histories, planetary surface dating, and dynamical evolution studies by yielding reproducible, controlled "ground-truth" scenarios. The construction of such populations is underpinned by the empirical identification of two principal primary impactor groups—Population 1 (linked to the Late Heavy Bombardment, LHB) and Population 2 (post-LHB, dominated by near-Earth objects, NEOs)—and a secondary crater subpopulation ("Population S") (Strom et al., 2014).

1. Empirical Basis and Distinct Impactor Populations

The cratering record on lunar and martian terrains demarcates two temporally and morphologically distinct primary impactor populations. Population 1, dominating the early Solar System (<~3.9 Ga), matches the current main asteroid belt in size distribution and underlies the complex, multi-sloped R-plot signatures of heavily cratered surfaces. This population arose from main belt asteroids ejected in a size-independent fashion, plausibly by gravitational resonance sweeping induced by planetary migration, manifesting as the LHB (Strom et al., 2014).

Population 2 emerges post-LHB (since ~3.8–3.7 Ga), bearing an SFD identical to that of NEOs. Its cratering imprint is characterized by a single power-law slope, consistent with a near-steady-state delivery of NEO-derived bodies, where size-dependent non-gravitational mechanisms dominate source population depletion (Strom et al., 2014).

A third, non-primary group—Population S—comprises secondary craters. These originate as high-velocity ejecta from primary impacts and can dominate the record at small diameters.

2. Size–Frequency Distributions and Parameterizations

The quantitative synthesis of impactor populations employs piecewise or single power-law constructions for the differential SFDs:

$\frac{dN}{dd} = k \cdot d^{-q},$

with cumulative

$N(>d) = \int_{d}^{d_{max}} \frac{dN}{dd'} dd'.$

For Population 1 (LHB), the SFD is parameterized by three contiguous diameter regimes:

Diameter Interval (d)	Differential Slope (q)	Functional Form
$[0.5 \ \mathrm{km}, 3 \ \mathrm{km})$	$2.2$	$k_1 \cdot (d/3 \ \mathrm{km})^{-2.2}$
$[3 \ \mathrm{km}, 6 \ \mathrm{km})$	$3.0$	$k_1 \cdot (d/3 \ \mathrm{km})^{-3.0}$
$[6 \ \mathrm{km}, 20 \ \mathrm{km}]$	$4.0$	$$k_1 \cdot (6/3)^{-3.0} \cdot (d/6 \ \mathrm{km})^{-4.0}$$

Normalization $k_1$ is fixed at a reference diameter, $d_{ref} = 1 \ \mathrm{km}$, via $N_1(>1 \ \mathrm{km}) = F_{1,ref}$, typically with $$F_{1,ref} = 10^{-14} \ \mathrm{km}^{-2} \ \mathrm{yr}^{-1}$$; $k_1 = (q-1)F_{1,ref}d_{ref}^{q-1}$, applied piecewise (Strom et al., 2014).

For Population 2 (NEO-like, post-LHB):

• Single power-law over $d \in [0.05, 10] \ \mathrm{km}$,
• $q_2 = 2.8$ (projectile), normalization $k_2 = 1.8 F_{2,ref}$ with $$F_{2,ref} = 3 \times 10^{-15} \ \mathrm{km}^{-2} \ \mathrm{yr}^{-1}$$,
• $dN_2/dd = k_2 (d/1 \ \mathrm{km})^{-2.8}$.

Uncertainties in normalizations are typically within a factor of two; exponents are uncertain by $\pm 0.1$ for Population 1 and $\pm 0.05$ for Population 2.

3. Crater-to-Impactor Scaling: Pi-Group Formalism

Transformation of observed crater SFDs to projectile SFDs relies on the $\pi$-scaling law, which encapsulates the dependence of final crater diameter $D_{cr}$ on impactor and target material properties, impact velocity, and gravity:

$$D_{cr} = K \cdot \frac{(\rho_{imp} / \rho_{tgt})^{1/3} \ d^{\mu} \ v^{\nu}}{g^{\beta}},$$

where:

• $K \approx 1.03$, $\mu \approx 0.55$, $\nu \approx 0.43$, $\beta \approx 0.22$ (competent rock; Melosh & Beyer)
• $$\rho_{imp} = \rho_{tgt} = 3000 \ \mathrm{kg \ m}^{-3}$$
• $v$ = median velocity (18.9 km/s Moon, 12.4 km/s Mars)
• $g$ = surface gravity (1.62 m/s² Moon, 3.71 m/s² Mars)

Given $D_{cr}$, inversion to $d$ is required; this process defines the break-diameters ($d_1$, $d_2$) and underpins observed shifts in SFD downturns across planetary bodies.

4. Temporal Evolution of Impact Flux

The synthetic model implements the impact flux history as two superposed, temporally distinct components:

• Population 1: A two-stage exponential decay captures the LHB, parameterized as

$$f_1(t) = \begin{cases} F_{1,peak} \exp[-(t-t_p)/\tau_1], & t_0 < t < t_1 \ F_1(t_1)\exp[-(t-t_1)/\tau_2], & t_1 < t < t_{end} \end{cases}$$

with $t_0 \approx 4.1$ Ga, $t_1 \approx 3.9$ Ga, $t_{end} \approx 3.7$ Ga, $\tau_1 \approx 50$ Myr, $\tau_2 \approx 300$ Myr, and $F_{1,peak}$ normalized to yield $\sim 10$ large lunar basins.

• Population 2: Post-LHB NEO flux is approximated as constant, $f_2(t) \approx F_{2,ref}$, with $\pm 20\%$ secular drift permitted over Gyr intervals.

Uncertainties in LHB decay timescales are factors of two.

5. Modeling Secondary Crater Populations

Each primary impact with crater diameter $D_{cr}$ generates a distribution of secondaries ("Population S") following a steep power-law:

$$\frac{dN_s}{dd_s} = A_s(D_{cr}) \left( \frac{d_s}{D_{cr}} \right)^{-q_s}, \qquad q_s \in [3.6, 4.0],$$

where $A_s \propto D_{cr}^2$ modulates the total mass yield. In practical calculations, secondaries are restricted in size, commonly omitting those with $d_{min}>0.5$ km (Moon) or $>10$ km (Mercury) to avoid overwhelming the small crater record with secondaries (Strom et al., 2014).

6. Synthetic Population Generation Algorithm

Generation of a synthetic impactor sample proceeds as follows:

1. Interval Discretization: The chronology $[t_{start}, t_{stop}]$ is divided into bins $\Delta t$ (e.g., 1 Myr).
2. Primary Event Sampling: For each bin:
   • Compute $F_1(t)$, $F_2(t)$.
   • Integrate $$N_{p,prim} = \Delta t \int_{d_{min}}^{d_{max}}\frac{dN_p}{dd} dd$$ for each population.
   • Draw $N_{p,prim}$ from Poisson$(\lambda = N_{p,prim})$.
   • For each primary:
     • Sample impactor diameter $d$, impact velocity $v$ (empirical distribution), and impact angle $\theta$ (PDF $\sin 2\theta$ over $[0^\circ, 90^\circ]$).
     • Compute $D_{cr}$ via the $\pi$-scaling law.
     • Optionally, generate secondary cluster: sample $N_s$ and assign spatial positions.
3. Population Accumulation: All events are cataloged to yield a synthetic crater or impactor-flux record.

Parameter tuning of $F_{1,peak}$, $\tau_1$, $\tau_2$, and adopting the power-law slopes and breakpoints produces R-plot statistics and fluxes consistent with observed cratering on Mercury, Moon, and Mars.

7. Sources of Uncertainty and Model Limitations

Uncertainties in this approach stem from both measurement and dynamical modeling constraints. The SFD power-law exponents are typically uncertain by ±0.1 for Population 1 and ±0.05 for Population 2, normalization in flux by a factor of two, and break-diameters by ±10%. LHB decay timescales are uncertain by about 50%. Normalization of secondary cratering is sensitive to both scaling models and primary impactor energy. These limitations affect absolute model calibration, but the prescription suffices to reproduce the empirical multi-sloped (LHB) and single-sloped (NEO) crater statistics and observed R-plot breakpoints for inner Solar System surfaces (Strom et al., 2014).