3. Understanding the Mode-5 Failure Response
Unlike failure response Modes 3 and 4, response Mode 5 (and also Mode 2) is not a direct
function of time from launch. For Modes 3 and 4, the mean point of impact (MPI) for each
debris class is fixed, once the failure time is established. At each instant there is only one
possible location for the :MPI for each debris class. On the other hand, the Mod~S impact-
density function for each debris class consists of a primary part and a secondary
superimposed part. The primary impact-density function accounts for impact variability
due to the erratic flight of the vehicle. It is used to determine the probability that the mean
piece in a debris class resulting from vehicle breakup falls in a given area (say on a building
or open field). The secondary density function accounts for debris dispersion due to
vehicle breakup and to aerodynamic effects during free fall. It is used to determine the
probability that fragments from the class actually hit a building or field. In other words, the
primary impact-density function is used to compute the probability that the secondary
function is centered in some specified area; the secondary function, which describes the
distribution of class pieces about the mean point, is then used to compute the probability
that one or more class pieces impacts on the specified population center or area.
The primary part of the Mod~S impact density function, which was presented as Eq. (9.5)
in Ref. [1], is reproduced here as Eq. (1):


                                                                                                       (1)



where R is the range from the launch point in miles, ~ is the angle in radians between the
uprange direction and a line fro:r,n the pad through the impact point, R is the impact-range
rate in miles per second. A and C are dimensionless shaping constants, and shaping-
constant D is in miles. For a Mod~S response, there is by definition an earliest time of
occurrence TP (pitch-over time) and a latest time of occurrence T5 (burnout, orbital injection,
or some other specified termination time). The specific time in this span at which a Mode-5
response manifests itself is of no consequence, although the duration of the span must be
considered in assigning a probability of occurrence for a Mod~S response.
Given that a Mod~S response has occurred, the probability that the center of the secondary
function lies in some region or on some building (population center) is determined by
integrating the primary impact-density function for the class over the region or building.
The primary function depends on range (R) and direction (q>) from the launch point to the
population center, but not directly on time from launch. The primary function does,



"' As an aid to understanding, the supplement of (j), designated as 0, is used in plots and tables in this
   report.


9/10/96                                                                                               RTI