## 1 Introduction  
A key component in a probabilistic risk assessment for spacecraft being designed to operate in the micrometeoroid and orbital debris (MMOD) environment is a ballistic limit equation, or BLE. This is an equation used to determine if a spacecraft component will suffer a critical failure following an on-orbit high-speed impact. One type of BLE is derived from a damage predictor equation that is itself obtained from a curve-fit of a damage measurements, such as crater depth or hole diameter, in terms of impact parameters, material properties, and target configuration. The other kind of BLE is basically a \"hand-drawn\" discriminant line that, for example, separates (projectile diameter, impact velocity) combinations that cause failure from those that do not. It is these types of BLE that were used to design the MMOD shielding on the International Space Station (see, e.g., [1] for more information on how such BLEs were developed). A frequently used BLE for dual-wall configurations (also known as \"Whipple Shields\") is the New Non-Optimum, or NNO, BLE [2]. These wall designs are frequently used on spacecraft to protect them against the threats posed by MMOD particles impacts. The NNO BLE consists of three parts in terms of increasing impact velocity--a low velocity portion (through approx. 3 km/s), a high velocity portion (above approx. 7 km/s), and a linear interpolation between the BLE values at the end of and the start of the low and high velocity portions, respectively. Quite frequently, a BLE is needed for a new structural system or element that has not yet been tested, but resembles to a fair degree a dual-wall system for which the NNO BLE would be applicable. It such cases it is common practice to equivalence the materials, thicknesses, etc., of the system or element of interest (but for which a BLE does not yet exist) to the materials, thicknesses, etc., of a dual-wall system. In this manner, the NNO BLE can be used to estimate the failure/non-failure response characteristics for the new system of interest without having to expend significant resources to generate a BLE for that new system. One such structural wall system for which a BLE does not yet exist, but is also seeing an increase in application, is a dual-wall system that is completely filled with a lightweight foam material (i.e., there are no discernable air gaps between either wall of the dual-wall system and the foam filling the space between them). The outer and inner walls in such a system could be aluminum, or made out of a composite material. Since such a system bears a close resemblance to a more standard dual-wall system (where the space between the outer and inner walls is empty), it could be considered appropriate to re-cast the foam-stuffed dual-wall system as an \"empty\" all-aluminum Whipple Shield with wall thicknesses that take into account any non-aluminum wall materials as well as the presence of the foam stuffing in the original system. In such a dual-wall configuration, the foam between the outer and inner walls can be either metallic (e.g., lightweight aluminum foams), or non-metallic (e.g., lightweight polymer foams like polyurethane). The focus of the study described herein was on lightweight non-metallic polymer-based foams. Even then, the space between the outer and inner walls could be either fully filled or partially filled. We again focus our attention on dual-wall systems that are fully filled with foam. In this manner, the foam in the configuration we studied not only could affect the protective capability of the dual-wall system, but it also would provide some structural support to keep the outer wall at a constant distance away from the inner wall. This is especially important in applications where the outer and inner walls might be made of extremely flexible materials, such as composite fabrics or very thin metallic plates. As will be seen shortly, we found that the NNO BLE in its original form frequently over- or under-predicted the response of such a system.