  (Abridged) A numerical model of a circumstellar debris disk is developed and
applied to observations of the circumstellar dust orbiting beta Pictoris. The
model accounts for the rates at which dust is produced by collisions among
unseen planetesimals, and the rate at which dust grains are destroyed due to
collisions. The model also accounts for the effects of radiation pressure,
which is the dominant perturbation on the disk's smaller but abundant dust
grains. Solving the resulting system of rate equations then provides the dust
abundances versus grain size and over time. Those solutions also provide the
dust grains' collisional lifetime versus grain size, and the debris disk's
optical depth and surface brightness versus distance from the star. Comparison
to observations then yields estimates of the unseen planetesimal disk's radius,
and the rate at which the disk sheds mass due to planetesimal grinding.
  The model is then applied to optical observations of the edge-on dust disk
orbiting beta Pictoris, and good agreement is achieved when the unseen
planetesimal disk is broad, with 75<r<150 AU. If it is assumed that the dust
grains are bright like Saturn's icy rings, then the cross section of dust in
the disk is A_d~2x10^20 km^2 and its mass is M_d~11 lunar masses. In this case
the planetesimal disk's dust production rate is quite heavy, dM_d/dt~9
earth-masses per Myr, implying that there is or was a substantial amount of
planetesimal mass there, at least 110 earth-masses. But if the dust grains are
darker than assumed, then the planetesimal disk's mass-loss rate and its total
mass are heavier. In fact, the apparent dearth of any major planets in this
region, plus the planetesimal disk's heavy mass-loss rate, suggests that the
75<r<150 AU zone at beta Pic might be a region of planetesimal destruction,
rather than a site of ongoing planet formation.
