Mmod Shielding Sizing

MMOD Shielding Sizing (space-systems/subsystems/mmod-shielding-sizing)

Use when the task is sizing the hypervelocity meteoroid and orbital debris (MMOD) impact protection of a spacecraft wall with the NASA/JSC ballistic-limit equation family: the single-wall ballistic limit through the Cour-Palais cratering equation, and the Whipple shield through the Christiansen new-non-optimum ballistic limit equations. It pairs with space-systems/mission-design/radiation-debris for the debris flux and collision-probability environment inputs, and with space-systems/subsystems/thermal-design and space-systems/subsystems/propellant-tank-sizing as sibling spacecraft bus subsystem sizing leaves.

Units convention (the handbook units, stated once): thicknesses and diameters in cm, densities in g/cm^3, velocities in km/s, yield stress in ksi, impact angle in degrees from the target normal, mass in g.

Domain quick reference

• Single-wall Cour-Palais cratering, eqs (4-1)/(4-2): for the density ratio r = rho_p / rho_t, P_inf = 5.24 d^(19/18) BHN^(-1/4) r^n (V cos(theta) / C_t)^(2/3), with n = 1/2 for r < 1.5 and n = 2/3 for r >= 1.5. C_t is the target sound speed, BHN the target Brinell hardness.
• Single-wall damage thresholds, eqs (4-3) to (4-5): required thickness t = k P_inf with k = 1.8 (perforation, the default damage mode), 2.2 (detached spall), 3.0 (incipient spall); eq (4-6) inverts the relation to the critical diameter dc at a given wall thickness.
• Whipple bumper design, eq (4-21): tb = cb d rho_p / rho_b, with cb = 0.25 for standoff ratio S/d < 30 and cb = 0.20 for S/d >= 30.
• Whipple rear-wall design, eq (4-22), valid only for a normal impact velocity Vn = V cos(theta) >= 7 km/s: tw = 0.16 d^(1/2) (rho_p rho_b)^(1/6) Mp^(1/3) Vn (70/sigma)^(1/2) / S^(1/2), with Mp the projectile mass (defaulting to the sphere mass (pi/6) rho_p d^3) and sigma the rear-wall yield stress in ksi.
• Whipple ballistic limit, three Vn regimes with the impact angle capped at 65 degrees (eq 4-26) before Vn is formed: the low-velocity equation (eq 4-24) for Vn <= 3 km/s, an exact linear interpolation (eq 4-25) between the low-velocity anchor at Vn = 3 and the hypervelocity anchor at Vn = 7 for 3 < Vn < 7, and the hypervelocity equation (eq 4-23) dc = 3.918 tw^(2/3) rho_p^(-1/3) rho_b^(-1/9) Vn^(-2/3) S^(1/3) (sigma/70)^(1/3) for Vn >= 7. The hypervelocity regime depends on Vn only, not on V and theta separately.
• Per-impact verdict: margin = d / dc; margin >= 1 is PENETRATION, margin < 1 is NO_PENETRATION.
• Mission rollup: penetration_probability(lambda) = 1 - exp(-lambda), the Poisson probability of at least one penetrating impact, where lambda (the expected number of penetrating impacts) is a GIVEN input assembled from the environment leaf's flux at and above dc, not computed here.
• All equations are the published closed forms of Christiansen et al., NASA TM-2009-214789 (JSC-64399), summary-only; the aluminum-on- aluminum single wall and the aluminum Whipple shield are the only configurations modeled.

Workflow

1. Set the projectile and target parameters (diameter d, densities rho_p and rho_t, impact velocity v, obliquity theta) for a single aluminum wall and compute the semi-infinite crater depth with cour_palais_penetration_depth.
2. Size the single-wall damage-threshold thickness with single_wall_required_thickness at the chosen k (K_PERFORATION by default), and invert it with single_wall_critical_diameter to recover the critical diameter a given wall thickness stops.
3. For a Whipple shield, size the bumper thickness with whipple_bumper_thickness from the standoff S and the S/d branch.
4. Size the rear-wall design thickness with whipple_rear_wall_thickness at the design projectile and its Vn >= 7 km/s design impact, using sphere_mass_g when no explicit projectile mass is given.
5. Evaluate the shield's ballistic limit over velocity with whipple_critical_diameter, which dispatches across the low, intermediate-blend, and hypervelocity regimes with the 65 degree obliquity cap.
6. Grade a single impact with penetration_verdict for a single wall or whipple_penetration_verdict for a Whipple shield, comparing the projectile diameter against the critical diameter margin.
7. Roll the per-impact verdict over the mission with penetration_probability from the expected number of penetrating impacts, a GIVEN input from the space environment assessment.
8. Confirm the deterministic checks with the contract test scripts/test_mmod-shielding-sizing.py.

Worked example

A thin-aluminum Whipple shield for a 1.0 cm spherical aluminum projectile (rho_p = 2.70 g/cm^3) at 7 km/s normal impact (theta = 0 deg): rear wall Al 6061-T6 class (sigma = 40.0 ksi), aluminum bumper (rho_b = 2.70 g/cm^3), standoff S = 11.43 cm (S/d = 11.43 < 30).

• Bumper: whipple_bumper_thickness(1.0, 2.7, 2.7, 11.43) = 0.25 cm (cb = 0.25 since S/d < 30).
• Projectile mass: sphere_mass_g(1.0, 2.7) = 1.413716694115407 g.
• Rear wall: whipple_rear_wall_thickness(1.0, 2.7, 2.7, 11.43, 40.0, 7.0, 0.0) = 0.684892875727985 cm.
• Ballistic limit at the design point: whipple_critical_diameter of the sized shield at Vn = 7 km/s = 1.000075994715671 cm, matching the closed form 3.918 * 0.16^(2/3) * (pi/6)^(2/9) = 1.000075994715671 (the eq 4-22 sizing is the exact inverse of the eq 4-23 performance equation at Vn = 7 km/s). The design case verdict is NO_PENETRATION with margin d / dc = 0.999924011059087.
• Same shield at 10 km/s: dc = 0.7884334285317389 cm; the hypervelocity scaling dc(10)/dc(7) = 0.7883735163105243, matching (7/10)^(2/3) = 0.7883735163105242. Verdict PENETRATION with margin 1.2683379012255367: a shield sized at 7 km/s does not stop the same projectile at 10 km/s.
• Single aluminum wall at 10 km/s: a 0.48 cm monolithic Al 6061-T6 wall (BHN = 95, rho_t = 2.70 g/cm^3, C_t = 5.0 km/s) against the 1.0 cm projectile: cour_palais_penetration_depth = 2.664324077003382 cm, single_wall_required_thickness = 4.795783338606088 cm (= 1.8 * P_inf), and single_wall_critical_diameter(0.48, ...) = 0.11297781552560618 cm, so the 0.48 cm wall stops only sub- millimeter-class projectiles at 10 km/s.
• Mission rollup: penetration_probability(0.5) = 0.3934693402873666.

Verification

• Confirm the design self-consistency: whipple_critical_diameter of the sized Case-1 shield at Vn = 7 km/s equals the closed form 3.918 * 0.16^(2/3) * (pi/6)^(2/9) = 1.000075994715671 within 1e-9 relative.
• Confirm the single-wall round trip: single_wall_critical_diameter of single_wall_required_thickness(d, ...) equals d within 1e-9 relative on both density branches (aluminum-on-aluminum and steel-on- aluminum).
• Confirm regime continuity at Vn = 3 km/s and Vn = 7 km/s, and the hypervelocity scaling dc(V2)/dc(V1) = (V1/V2)^(2/3) for Vn >= 7.
• Confirm the obliquity cap: dc at theta >= 65 degrees equals dc at theta = 65 degrees for the same velocity.
• Confirm every non-positive thickness, density, hardness, sound speed, yield stress, mass, and velocity raises ValueError, theta outside [0, 90) raises ValueError, a rear-wall design below Vn = 7 km/s raises ValueError, negative lambda raises ValueError, and any boolean argument raises ValueError.
• Run the contract test offline: python3 scripts/test_mmod-shielding-sizing.py (35 tests, deterministic).

Related leaves

• space-systems/mission-design/radiation-debris: the space environment leaf that produces the debris flux, fluence, and collision probability this leaf consumes as GIVEN inputs; it never estimates a penetration outcome.
• space-systems/subsystems/thermal-design: the thermal control sizing sibling of the spacecraft bus.
• space-systems/subsystems/propellant-tank-sizing: another spacecraft bus subsystem sizing leaf sharing the ECSS lifecycle context.

Pitfalls

• Reading the debris collision probability as the penetration probability: radiation-debris's collision probability is the chance of any impact; this leaf's penetration_probability is the chance of an impact exceeding the ballistic limit, a downstream, smaller quantity that needs the critical diameter first.
• Using the bumper thickness in the hypervelocity regime: eq (4-23) does not take tb as an input; the equation assumes the bumper is adequate to disrupt the projectile, so a resized bumper alone never changes dc above 7 km/s.
• Forgetting the obliquity cap: feeding theta above 65 degrees into the raw cos(theta) terms without capping first understates Vn and overstates the ballistic limit; cap theta at 65 degrees before forming Vn.
• Assuming a shield sized at one velocity stays adequate at another: the Case-1 shield sized for a 7 km/s design impact drops to margin 1.268 (PENETRATION) at 10 km/s, since dc falls as Vn^(-2/3) in the hypervelocity regime.
• Applying the rear-wall design equation below 7 km/s: eq (4-22) assumes hypervelocity impact; whipple_rear_wall_thickness raises ValueError below Vn = 7 km/s rather than returning an unvalidated thickness.
• Summing yearly penetration probabilities instead of compounding: penetration_probability uses 1 - exp(-lambda) over the whole mission lambda, not a sum of per-year probabilities.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_mmod-shielding-sizing.py

The test covers the worked-example contract (bumper and rear-wall design thicknesses, the design self-consistency identity, the single- wall penetration depth and required thickness, the critical diameter at both wall configurations, and the mission penetration probability), the single-wall round trip on both density branches, regime continuity at the Vn = 3 and Vn = 7 km/s boundaries, the hypervelocity scaling and obliquity-cap identities, the per-impact verdict semantics, mission probability monotonicity, determinism, and ValueError rejection of every non-physical or boolean input.

Compliance

• Standards referenced, not reproduced: ECSS-E-ST-10C (systems engineering general requirements) frames the spacecraft subsystem design lifecycle; NASA TM-2009-214789 (JSC-64399) and the Cour-Palais and Christiansen ballistic-limit equation family are cited summary-only per standards-map.yaml, never reproduced as design data.
• compliance: STANDARDS-REF, gated: false.