calculations.

Other aspects of planet formation are also important. If the cores of gas
giant planets form before their parent stars reach the main sequence, we
expect gas giants at 20–30 AU around 2–3 M⊙ stars (e.g., Kennedy & Kenyon,
2008). Gas giants rapidly remove debris in the inner disk and impose structure
in the debris beyond 30 AU (e.g., Wilner et al., 2002; Moro-Martín & Malhotra,
2005). Because gas giants are common around evolved A stars (Johnson et al.,
2007), gas giants probably play a significant role in the evolution of debris
disks around A stars.

Catastrophic collisions may also produce diversity among A star debris disks
(e.g., Wyatt & Dent, 2002; Su et al., 2005). Although debris from complete
disruption of colliding planetesimals is unobservable in our simulations (see
also Kenyon & Bromley, 2005), dynamical events similar to those that produced
the Late Heavy Bombardment in the Solar System probably are visible (e.g.,
Gomes et al., 2005). Testing this idea requires numerical calculations that
link the dynamics of massive planets with the collisional evolution of smaller
objects (e.g., Charnoz & Morbidelli, 2003; Kenyon & Bromley, 2006).

We conclude that our debris disk models can explain the overall time evolution
of the IR excesses and IR colors of A-type main sequence stars at 24 $\mu$m
and at 70 $\mu$m. Our calculations for disks with $x_{m}$ = 1/3 to 3 around 2
M⊙ stars fit the overall level of the excesses and the trends with stellar
age. Explaining the full range of observed IR excesses and IR colors requires
a set of disks with $x_{m}$ = 0.01–3, as suggested from observations of disks
around the youngest stars. Matching other properties of these stars –
including the relative amount of emission from a warm inner disk, an
outflowing wind of small grains, and a large outer disk – requires
calculations that include a broader range of initial disk radii and gas giant
and terrestrial planet formation at $a_{i}<$ 30 AU (e.g., Kenyon & Bromley,
2005; Nagasawa et al., 2007; Kennedy & Kenyon, 2008; Ida & Lin, 2008; Kretke
et al., 2008).

###  5.3 Debris Disks Around Solar-type Stars

Although most of the debris disks discovered with IRAS and ISO have A-type
central stars, a few have F-type or G-type central stars with masses of 1–1.5
M⊙ (Backman & Paresce, 1993; Lagrange et al., 2000; Decin et al., 2003; Song
et al., 2005; Rhee et al., 2007a). More recent Spitzer observations reveal
debris disks around many solar-type stars (Bryden et al., 2006; Beichman et
al., 2006; Meyer et al., 2006; Trilling et al., 2008; Hillenbrand et al.,
2008). Although several Spitzer programs concentrate on older solar-type stars
as preparation for detailed planet searches, the range of ages is large enough
to provide an initial test of our predictions.

To compare our model predictions with observations, we consider data for
nearby solar-type stars from Beichman et al. (2006) and Hillenbrand et al.
(2008). Beichman et al. (2006) observed $\sim$ 80 solar-type stars at 24
$\mu$m and at 70 $\mu$m using MIPS on Spitzer. Hillenbrand et al. (2008)
analyze $\sim$ 30 stars with 70 $\mu$m excesses out of a sample of 328 stars
from the Spitzer Legacy Science Program, “Formation and Evolution of Planetary
Systems” (Meyer et al., 2006). After eliminating K-type and M-type stars from
the Beichman et al. (2006) study, the two programs contain $\sim$ 80 stars
with ages of $\sim$ 10 Myr to $\sim$ 10 Gyr. Most of these stars have masses
of 0.8–1.5 M⊙. Thus, we compare these data with our results for debris disk
evolution around 1 M⊙ stars.

The observed 70–160 $\mu$m excesses of solar-type stars show trends similar to
those observed in the evolution of A-type stars at 24–70 $\mu$m (Fig. 25–26).
Although the statistics for solar-type stars are poor for the youngest stars,
the data suggest a rise in the 70–160 $\mu$m excess at 10–100 Myr. The maximum
in the 70 $\mu$m excess is comparable in magnitude but a factor of $\sim$ 10
later in time than the maximum 70 $\mu$m excess for A-type stars. At 70 $\mu$m
and at 160 $\mu$m, the excess follows a roughly power-law decline with time
for older stars. Solar-type stars also have a large range in excess at all
ages, with $F_{70}/F_{70,0}\approx$ 1–300 at 100 Myr and
$F_{70}/F_{70,0}\approx$ 1–30 at 1–3 Gyr. For stars with similar ages, the
range in the 70 $\mu$m excess is larger for solar-type stars than for A-type
stars.

Our models match the observed trends for the IR excesses of solar-type stars.
At 70 $\mu$m, 65%–75% of the observations lie within model predictions; at 160
$\mu$m, more than half of the observations are within model predictions. For a
range of initial disk masses ($x_{m}\approx$ 0.01–3) and outer radii
($a_{out}\approx$ 70–150 AU), we predict a large range of excesses at all
ages, as observed. These model also explain the larger 70 $\mu$m excesses
observed for solar-type stars relative to A-type stars, the apparent maximum
in the 70–160 $\mu$m excess at 30–100 Myr, and the general power-law decline
in the excess flux for the oldest stars.

Despite this general success, however, the models underpredict the largest
observed excesses. At 70 $\mu$m, the brightest disks are a factor of 5–10
brighter than disks with $x_{m}$ = 3. At 160 $\mu$m, the brightest systems are
3–5 times brighter than our most luminous disks. Although the sample of 160
$\mu$m sources is small, our models underpredict the largest observed fluxes
at all ages.

Changing two assumptions in our models yields a better match to the observed
fluxes at 70–160 $\mu$m. For realistic grain properties, Burns, Lamy, & Soter
(1979) show that radiation pressure from the Sun cannot eject small grains
from the Solar System. Reducing the minimum stable grain size from $r_{2}$ = 1
$\mu$m to $r_{2}$ = 0.1 $\mu$m increases our predicted 70 $\mu$m (160 $\mu$m)
fluxes by a factor of 2–3 (1.5–2). Submm observations of several debris disks
imply $q\approx$ 0.6–1 for the slope of the radiative emissivity. If we adopt
$q=0.7$ instead of $q=1$, our predicted 70–160 $\mu$m fluxes increase by
factors of 2–4. Combining these two modifications increase our predicted
fluxes by a factor of $\sim$ 5–10 at both wavelengths.

Several observations could check whether these modifications of our standard
model are reasonable. By analogy with the Spitzer Vega data, detection of an
outflowing wind of small grains in a debris disk around a solar-type star
provides a clean measurement of $r_{2}$ and a better constraint on our
predicted IR excesses. Measurements of $q$ for larger samples of debris disks
allows a better assessment of our assumptions for the grain emissivity.

##  6 CONCLUSIONS

Our calculations provide a robust picture for the formation of planets and
debris disks from a disk of icy planetesimals and set the context for the
evolution of dusty debris in a dynamic system of planets. The results of this
study provide a framework for interpreting existing observations of debris
disks around 1–3 M⊙ stars and suggest new observational tests of this picture.

We describe a suite of numerical calculations of planets growing from
ensembles of icy planetesimals at 30–150 AU in disks around 1–3 M⊙ stars.
Using our hybrid multiannulus coagulation code, we solve for the evolution of
sizes and orbits of objects with radii of $\sim$ 1 m to $\gtrsim$ 1000 km over
the main sequence lifetime of the central star. These results allow us to
constrain the growth of planets as a function of disk mass, stellar mass, and
semimajor axis.

Debris disk formation is coincident with the formation of a planetary system.
All calculations of icy planet formation at 30–150 AU lead to a collisional
cascade which produces copious amounts of dust on timescales of 5–30 Myr. This
dust is observable throughout the lifetime of the central star. Because we
consider a broad range of input parameters, we derive the time evolution of
(i) dust produced in the collisional cascade and (ii) the IR and submm
emission from this dust as a function of disk mass, stellar mass, and time.

We divide the rest of this section into (i) theoretical considerations, (ii)
observable consequences, and (iii) observational tests. The theoretical
considerations build on the highlights of icy planet formation in §3.4.
Observable consequences of the calculations follow from the discussion in §4.
The observational tests of the models are described in §5.

###  6.1 Theoretical Considerations

  * •

Icy planet formation at 30–150 AU is self-limiting. Starting with a swarm of
$\lesssim$ 1 km planetesimals, runaway growth produces a set of 100–500 km
protoplanets. As the protoplanets grow, they stir up leftover planetesimals
along their orbits. When the leftovers reach high $e$, collisions produce
debris instead of mergers. Because protoplanets cannot accrete leftovers
rapidly, a cascade of destructive collisions grinds the leftovers to dust.
Poynting-Robertson drag and radiation pressure then remove the dust from the
disk.

  * •

The maximum sizes of icy planets at 30–150 AU are remarkably independent of
initial disk mass, stellar mass, and stellar age. For disks with $x_{m}$ = 1/3
to 3 around 1–3 M⊙ stars with ages $t=0.1-1t_{ms}$, the typical planet has
$r_{max}\sim$ 1750 km and $m_{max}\sim 0.005$ M⊕. These objects contain
$\lesssim$ 3%–4% of the initial disk mass. Although this result is also
independent of the fragmentation parameters, the finite main sequence
lifetimes of 1–3 M⊙ stars limits the formation of many large planets in the
outer disk. Thus, the inner disk produces many more Pluto-mass planets than
the outer disk (Tables 2 and 3).

  * •

For stars close to the main sequence turnoff, stellar lifetimes and the
collisional cascade limit the mass in solid objects at 30–150 AU. In the inner
disk, the collisional cascade removes most of the leftover planetesimals
before the central star evolves off the main sequence. Thus, the typical mass
in small objects is $\sim$ 10% of the initial mass at 30–40 AU. In the outer
disk, smaller collision rates produce a slower cascade. Thus, the central star
evolves off the main sequence with $\sim$ 50% of the initial mass remaining in
1–10 km planetesimals at 125–150 AU.

  * •

The collisional cascade produces copious amounts of dust. Dust begins to form
during the transition from runaway to oligarchic growth ($t$ = 5–10 Myr),
peaks when the first objects reach their maximum sizes ($t$ = 10–30 Myr), and
then slowly declines $t\gtrsim$ 30–50 Myr). The peak mass in 0.001–1 mm
(0.001–1 m) particles is $\sim$ 1–2 lunar masses ($\sim$ 1 M⊕). Disks with
initial masses $x_{m}$ = 1/3 to 3 reach these peak masses when the age of the
star is $\sim$ 10% to 20% of its main sequence lifetime. Because the timescale
to form dust is short ($\sim$ 10–20 Myr), stars are surrounded by large disks
of debris at 30–150 AU throughout their main sequence lifetimes.

  * •

Radiative processes remove large amounts of mass from debris disks. Radiation
pressure produces a radial wind of small particles containing $\sim$ 60% to
70% of the mass removed from the disk. Poynting-Robertson drag pulls the rest
of the lost mass into the inner disk. Because radiation pressure is more
important than Poynting-Robertson drag when collision rates are large, we
expect more wind (inner disk) emission earlier (later) in the evolution.

###  6.2 Observable Consequences

We derive clear observational consequences of the collisional cascade.

  * •

The dusty debris from the collisional cascade is directly observable. For
disks around 1–3 M⊙ stars, the maximum fractional dust luminosity of
$L_{d}/L_{\star}\sim 2\times 10^{-3}$ is comparable to the maximum dust
luminosities of known debris disks (Backman & Paresce, 1993; Rieke et al.,
2005; Su et al., 2006; Rhee et al., 2007a). The dust temperature at the inner
edge of a 30–150 AU disk scales with the temperature of the central star;
thus, the predicted 24 $\mu$m excess is very sensitive to the stellar mass. At
70 $\mu$m, the predicted excesses scale roughly linearly with disk mass and
stellar mass. The predicted 160–850 $\mu$m excesses depend on the disk mass
but are nearly independent of the stellar mass.

  * •

For systems with little or no emission from terrestrial dust ([5.8]–[8]
$\lesssim$ 0.1), mid-IR color-color diagrams clearly distinguish debris disks
around stars of different masses. In a [8]–[24] vs [24]–[70] diagram, 2–3 M⊙
(1–2 M⊙) stars have red (blue) [8]–[24] and blue (red) [24]–[70] (Fig 20). In
both cases, the color scales with the initial disk mass (Fig 21). Optical
colors and spectra generally provide good estimates for stellar mass; thus,
these diagrams provide good tests of our model predictions.

###  6.3 Observational Tests

We compare our predictions with observations of A-type stars and solar-type
stars.

  * •

For A-type stars, our calculations are the first to explain the observed rise
and fall of debris disk fluxes at 24 $\mu$m (Fig. 22; Currie et al., 2008a,
b). In our picture, the rise in debris disk emission corresponds to the
transition from runaway growth – when mergers of small planetesimals produce
larger protoplanets – to oligarchic growth – when the collisional cascade
begins to grind leftover planetesimals into dust. When oligarchs in the inner
disk are close to their maximum sizes of $\sim$ 1750 km, the collisional
cascade produces a maximum in debris disk emission. For a wide range of
initial conditions, this maximum occurs at 10–20 Myr. As the collisional
cascade moves out through the disk, smaller collision rates produce less dust
which emits at lower temperatures. Thus, the 24 $\mu$m excess falls with time.
The predicted rate of decline, $t^{-n}$ with $n\approx$ 0.6–0.8, is close to
the observed rate ($n\approx$ 0.5–1; Greaves & Wyatt, 2003; Rieke et al.,
2005; Rhee et al., 2007a).

  * •

At longer wavelengths, the maximum excess is larger and lasts longer than at
24 $\mu$m. Predicted mid-IR colors also increase slowly with time. Although
larger samples of A-type stars with 8 $\mu$m photometry would provide a better
test of our models, current data for the 70 $\mu$m excess and the evolution of
the [24]–[70] color agree with our predictions (Fig. 23–24). For 2–3 M⊙ stars
near the main-sequence turnoff, our calculations also yield a clear maximum in
the 850 $\mu$m flux. Large surveys, such as the proposed JCMT Legacy Survey
(Matthews et al., 2007) and submm observations with ALMA, Herschel and SOFIA,
can test this prediction.

  * •

For solar-type stars, our models match observations of most sources. The
predicted evolution of the 70–160 $\mu$m excesses follows the observed rise at
10–100 Myr, the peak at $\sim$ 30–100 Myr, and the decline at $\gtrsim$ 300
Myr. Although $\sim$ 70% (55%) of observed debris disks have fluxes that lie
within model predictions, our models underpredict fluxes for the brightest
sources by a factor of 5–10. Fluxes for models with $r_{2}\approx$ 0.1 $\mu$m
and $q\lesssim$ 0.7 provide better matches to these observations. To guide our
choices for $r_{2}$ and $q$, we require spatially resolved images and submm
fluxes for these objects.

  * •

For 1–3 M⊙ stars with ages $\sim$ 0.1–1 Gyr, current data suggest that solar-
type stars have a larger range of far-IR excesses than A-type stars. In our
models, faster debris disk evolution around A-type stars produces a smaller
dispersion in far-IR excesses and colors for stars with ages of 100 Myr to 1
Gyr. Larger samples of debris disks can test this prediction in more detail.

  * •

We also consider observations of Vega, the prototypical debris disk. If we
adopt models with a blowout radius $r_{2}$ = 10 $\mu$m, we can match
observations with a standard collisional cascade within a broad torus at
80–200 AU. If the torus contains $\sim$ 1–5 M⊕ in large objects with
$r\gtrsim$ 1 cm, the cascade can generate (i) the observed ensemble of grains
with $r\sim$ 200–300 $\mu$m within the torus and (ii) an outflowing wind of
small grains with $r\sim$ 1–50 $\mu$m.

This conclusion differs from Su et al. (2006), who postulate a recent
catastrophic collision between two large objects as the source of the dusty
Vega wind. Although the complete destruction of two large icy objects can
produce a massive outflowing wind, our results suggest that the dusty wind is
short-lived and cannot be rapidly replenished by the observed population of
larger objects. We show that a steady-state collisional cascade can explain
the Spitzer data (see also Kenyon & Bromley, 2005). If our interpretation is
correct, sensitive observations at 1–10 mm should detect our proposed
reservoir of larger objects.

Matching other observations of debris disks requires more realism in our
planet formation calculations. Adding binary companions and giant planets
provides ways to modify the evolution of the collisional cascade and to impose
structure on rings and tori (e.g. Wilner et al., 2002; Moro-Martín & Malhotra,
2005; Quillen, 2006). Extending the coagulation calculations to smaller sizes
allows studies of the formation of winds and other large structures. Although
these calculations have been prohibitively expensive in computing time, rapid
advances in computing technology will make these additions possible in the
next few years.

Based on the results described here and in Kenyon & Bromley (2004b), we
conclude that debris disks are the inevitable outcome of icy planet formation
in a disk of solid objects. The basic structures produced by this model –
broad tori and narrow rings of dust that propagate out through the disk
(Kenyon & Bromley, 2004b) – are consistent with observations (e.g.
Jayawardhana et al., 1998; Kalas, 2005; Su et al., 2006; Fitzgerald et al.,
2007). The model also explains the time evolution of mid-IR colors and fluxes
for debris disks around A-type and solar-type stars.

We acknowledge a generous allotment, $\sim$ 1000 cpu days, of computer time on
the 1024 cpu Dell Xeon cluster ‘cosmos’ at the Jet Propulsion Laboratory
through funding from the NASA Offices of Mission to Planet Earth, Aeronautics,
and Space Science. We thank M. Werner for his strong support of this project.
We also acknowledge use of $\sim$ 250 cpu days on the CfA cluster ‘hydra.’
Advice and comments from T. Currie, M. Geller, G. Kennedy, M. Meyer, G. Rieke,
K. Su, and an anonymous referee greatly improved our presentation. Portions of
this project were supported by the NASA  Astrophysics Theory Program, through
grant NAG5-13278, the NASA TPF Foundation Science Program, through grant
NNG06GH25G, and the Spitzer Guest Observer Program, through grant 20132.

##  Appendix A APPENDIX

###  A.1 Growth rates

In standard coagulation theory, protoplanets accrete material from a swarm of
planetesimals at a rate (e.g., Safronov, 1969; Lissauer, 1987; Wetherill &
Stewart, 1993)

$\dot{M}\propto\Sigma~{}\Omega~{}r^{2}~{}\left[1+(v_{esc}/v)^{2}\right]~{},$
(A1)

where $r$ is the radius of a planetesimal, $\Omega$ is the angular frequency
of material in the disk, $v$ is the random velocity of planetesimals, and
$v_{esc}$ is the escape velocity of the protoplanet. The $1+(v_{esc}/v)^{2}$
term is the gravitational focusing factor.

To derive the accretion time, we set $t=M/\dot{M}$ and substitute the orbital
period for the angular frequency,

$t\propto(\rho~{}r~{}P/\Sigma)~{}\left[1+(v_{esc}/v)^{2}\right]^{-1}~{},$ (A2)

where $\rho$ is the mass density of a planetesimal. Throughout runaway growth
and the early stages of oligarchic growth $v_{esc}/v\gg 1$. Because we are
interested in the time to produce planets with the same $r$ and $\rho$ in
disks with different $P$ and $\Sigma$, we eliminate $\rho$ and $r$. Thus, the
growth time is roughly

$t\propto(P/\Sigma)(v/v_{esc})^{2}~{}.$ (A3)

This equation sets the typical timescale for planet growth in a disk of
planetesimals. If $\Sigma\sim\Sigma_{0}x_{m}a^{-3/2}$ (Eq. (27)) and
$v/v_{esc}$ $\sim$ constant (Fig. 1; Wetherill & Stewart, 1993; Goldreich,
Lithwick, & Sari, 2004),

$t\propto a^{3}x_{m}^{-1}\Sigma_{0}^{-1}~{}.$ (A4)

This result is close to the $t\propto a^{3}x_{m}^{-1.15}\Sigma_{0}^{-1}$
derived for the formation of the first 1000 km object in our calculations
(e.g., Eq. (41)).

To evaluate possible sources for the extra factor of $x_{m}^{-0.15}$ in our
derived accretion times, we consider the random velocity $v$ of accreted
planetesimals. Shorter growth times require smaller random velocities. Thus,
we consider processes that damp planetesimal velocities. In our calculations,
collisions and gas drag can reduce $v$; dynamical friction and viscous
stirring increase $v$. At 30–150 AU, gas drag damps random velocities $\sim$
10–20 times more rapidly than collisions (Goldreich, Lithwick, & Sari, 2004).
Thus, we ignore collisional damping and concentrate on gas drag.

Rafikov (2004) investigated the dynamics of small planetesimals and growing
protoplanets in a gaseous nebula. For the early stages of oligarchic growth,
the random velocity of planetesimals is

$v/v_{esc}\propto\Sigma_{gas}^{-\gamma_{1}}~{}.$ (A5)

Substituting this expression into Eq. (A4) and adopting a constant gas-to-dust
ratio, $\Sigma_{gas}\sim\Sigma$, we derive

$t\propto a^{3}x_{m}^{-\gamma 2}\Sigma_{0}^{-1}~{},$ (A6)

with $\gamma_{2}=2\gamma_{1}+1$. For typical conditions in planetesimal disks,
Rafikov (2004) derived $\gamma_{1}\approx$ 1/6 to 1/5. Thus,
$\gamma_{2}\approx$ $1.3$ to $1.4$, close to the exponent of $1.15$ derived in
our calculations.

Our treatment of gas drag probably reduces the exponent of $x_{m}$ in Eq. (A6)
from the predicted $1.3$–$1.4$ to $1.15$. In our simulations, we assume the
gas density declines exponentially on a timescale $t_{gas}$ = 10 Myr. With
typical growth times of 20–40 Myr, the gas density is $\sim$ 1% to 10% of its
initial value when the first 1000 km objects form in the inner disk. Thus, gas
drag cannot reduce planetesimal random velocities as efficiently as predicted
in Eq. A6. Reducing drag lowers the exponent. With gas depletion timescales
$\sim$ 25% to 50% of the growth time, we expect an exponent of
$\gamma_{2}\approx$ $1.1$ to $1.2$, similar to the $\gamma_{2}=1.15$ in our
calculations.

###  A.2 Radiation from dust

In the Appendix of Kenyon & Bromley (2004a), we briefly described our simple
algorithm for the evolution of particles with sizes smaller than the smallest
object – $r\sim$ 1 m – followed in the multiannulus coagulation code. This
algorithm yields the optical depth in very small grains ejected from the
system and the optical depth in larger grains evolving under the influence of
collisions and Poynting-Robertson drag. The optical depth in both grain
populations allows us to derive the time evolution of the disk luminosity and
surface brightness in bolometric units. Here, we describe the derivation of
grain temperature for these populations that yields the predicted time
evolution of the broadband spectral energy distributions of debris disks.

As in Kenyon & Bromley (2004a), we divide objects with sizes smaller than
$\sim$ 1 m into very small grains, small grains, and large grains. In each
annulus $k$ of our calculation, radiation pressure ejects very small grains
with radii between $r_{1}$ and $r_{2}$. If $\rho_{g}$ is the mass density of
these grains and $\dot{M}_{k}$ is the production rate of very small grains in
each annulus, the very small grains have an integrated optical depth

$\tau_{s}=\frac{3(\sqrt{r_{2}/r_{1}}-1)}{8\pi\rho_{g}r_{2}(1-\sqrt{r_{1}/r_{2}})}\sum_{i=1}^{N}~{}\left[\sum_{k=1}^{i}\left(\frac{\dot{M}_{k}}{v_{Kk}h_{k}}\right)\left(\frac{1}{a_{b,k}}-\frac{1}{a_{b,k+1}}\right)\right]~{},$
(A7)

where $a_{b,k}$ is the inner boundary of an annulus centered at $a_{k}$,
$h_{k}$ is the vertical scale height in units of the semimajor axis, and
$v_{K,k}$ is the orbital velocity in annulus $k$.

For small ($r$ = $r_{2}$ to 1 mm) and large grains ($r$ = 1 mm to 1 m), we
derive the optical depth $\tau_{k}$ in each annulus. To derive the radial
surface brightness and total disk luminosity, we follow Kenyon & Hartmann
(1987) and derive the amount of stellar radiation absorbed by each annulus. We
assume a spherical, limb-darkened star with radius $R_{\star}$, luminosity
$L_{\star}$, and limb-darkening parameter $\epsilon_{0}$ = 0.6. For a point
$P$ at the outer boundary of annulus $k$ with height $h_{P}$ above the disk
midplane, rays from the star enter the annulus at a scale height $h_{in}$
above (below) the midplane. We compute the length $l$ of the path through the
disk and derive the optical depth along this path as $\tau_{p}$ = ($l/\Delta
a_{k}$)$\tau_{k}$, where $\Delta a_{k}$ is the width of the annulus. The
radiation absorbed along this path is $e^{-\tau_{p}}I_{0}$, where $I_{0}$ is
the flux incident on the boundary of the annulus. Numerical integrations over
the stellar surface and the vertical extent of an annulus yield the amount of
flux absorbed by each annulus, which we convert to relative surface
brightness. A final numerical integration over the radial extent of the disk
yields the ratio of the disk luminosity to the stellar luminosity,
$L_{d}/L_{\star}$.

To derive the spectral energy distribution of the disk, we make several
assumptions. Consistent with observations of scattered light from resolved
debris disks (Backman & Paresce, 1993; Lagrange et al., 2000), we adopt a
single albedo $\omega$ = 0.25 for all grains. For all $\lambda$, the
luminosity in scattered light is then $\omega L_{d}/L_{\star}$; the thermal
luminosity emitted by all grains is $(1-\omega)L_{d}/L_{\star}$. In each
annulus $k$, we assume grains emit at a temperature $T_{i,k}$, where the index
$i$ refers to discrete bins in grain size. To derive equilibrium temperatures
for these grains, we assume the grains have an absorption efficiency
$\epsilon_{a}\propto(\lambda/\lambda_{0})^{p}$ and radiative efficiency
$\epsilon_{r}\propto(\lambda/\lambda_{0})^{q}$. For most grains in our
calculations, the grain size is larger than the peak wavelength of radiation
emitted by the central star. Thus, the grains efficiently absorb stellar
photons and $p$ = 0. Large grains with $r\gg\lambda$ emit as blackbodies and
have $q$ = 0. Smaller grains radiate less efficiently and have $q\approx$ 1.

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Table 1: Grid of Debris Disk CalculationsaaNumber of independent calculations for each combination of $M_{\star},x_{m}$ | Stellar Mass in $M_{\odot}$
---|---
$x_{m}$ |  1.0bb32 annulus models at 30–70 AU and at 70–150 AU |  1.0cc64 annulus models at 30–150 AU |  1.5 |  2.0 |  2.5 |  3.0
0.33 | 41 | 19 | 15 | 20 | 18 | 18
0.50 | 49 | 18 | 18 | 17 | 17 | 17
1.00 | 49 | 15 | 22 | 17 | 15 | 15
2.00 | 41 | 15 | 30 | 17 | 16 | 15
3.00 | 45 | 18 | 12 | 22 | 15 | 21
$t_{ms}$ddMain sequence lifetime in Gyr Demarque et al. (2004) | 10.00 | 10.00 | 2.90 | 1.22 | 0.65 | 0.39

Table 2: Median number of Plutos at t = $t_{ms}$/3 for disks around 1 M⊙ stars $x_{m}$ | 30–37 AU | 37–45 AU | 45–55 AU | 55–67 AU | 67–82 AU | 82–100 AU | 100–123 AU | 123–146 AU
---|---|---|---|---|---|---|---|---
0.33 | 40 | 44 | 33 | 32 | 19 | 12 | 3 | 1
0.50 | 62 | 65 | 39 | 49 | 33 | 25 | 10 | 1
1.00 | 111 | 110 | 73 | 73 | 47 | 55 | 26 | 5
2.00 | 172 | 194 | 134 | 155 | 116 | 84 | 58 | 33
3.00 | 165 | 260 | 172 | 251 | 137 | 109 | 85 | 44

Table 3: Median number of Plutos at t = $t_{ms}$/3 for disks around 1.5–3 M⊙ stars $x_{m}$ | 30–37 AU | 37–45 AU | 45–55 AU | 55–67 AU | 67–82 AU | 82–100 AU | 100–123 AU | 123–146 AU
---|---|---|---|---|---|---|---|---
1.5 M⊙
0.33 | 46 | 46 | 45 | 28 | 16 | 3 | 1 | 1
0.50 | 69 | 85 | 68 | 48 | 53 | 25 | 2 | 1
1.00 | 102 | 136 | 128 | 101 | 98 | 67 | 28 | 2
2.00 | 158 | 243 | 240 | 261 | 211 | 123 | 169 | 47
3.00 | 201 | 239 | 301 | 295 | 381 | 165 | 198 | 97
2.0 M⊙
0.33 | 55 | 51 | 48 | 30 | 2 | 1 | 1 | 3
0.50 | 115 | 87 | 85 | 51 | 20 | 10 | 1 | 2
1.00 | 172 | 187 | 231 | 123 | 82 | 50 | 1 | 3
2.00 | 261 | 230 | 366 | 236 | 324 | 204 | 77 | 5
3.00 | 259 | 262 | 398 | 264 | 446 | 295 | 171 | 48
2.5 M⊙
0.33 | 105 | 92 | 63 | 55 | 19 | 1 | 5 | 2
0.50 | 152 | 139 | 123 | 100 | 53 | 10 | 5 | 5
1.00 | 164 | 174 | 234 | 198 | 125 | 90 | 7 | 6
2.00 | 223 | 230 | 278 | 243 | 158 | 108 | 10 | 8
3.00 | 353 | 434 | 490 | 495 | 477 | 615 | 313 | 157
3.0 M⊙
0.33 | 133 | 121 | 62 | 36 | 6 | 5 | 1 | 1
0.50 | 127 | 153 | 129 | 103 | 50 | 2 | 2 | 1
1.00 | 199 | 230 | 239 | 258 | 171 | 66 | 6 | 5
2.00 | 224 | 353 | 376 | 392 | 342 | 265 | 173 | 5
3.00 | 428 | 598 | 479 | 657 | 570 | 756 | 578 | 272

Table 4: Predicted Excesses for Disks Around 1 M⊙ StarsaaThe electronic version of this paper contains the complete version of this Table. log $t$ (yr) | log $L_{d}/L_{\star}$ | log $F_{24}/F_{24,0}$ | log $F_{70}/F_{70,0}$ | log $F_{160}/F_{160,0}$ | log $F_{850}/F_{850,0}$
---|---|---|---|---|---
$x_{m}$ = 0.33
5.05 | -4.57 | 0.000 | 0.053 | 0.145 | 0.090
5.15 | -4.57 | 0.000 | 0.053 | 0.145 | 0.090
5.25 | -4.63 | 0.000 | 0.047 | 0.131 | 0.082
5.35 | -4.65 | 0.000 | 0.045 | 0.127 | 0.080
5.45 | -4.67 | 0.000 | 0.043 | 0.123 | 0.077
$x_{m}$ = 1.0
5.05 | -4.18 | 0.000 | 0.096 | 0.255 | 0.172
5.15 | -4.18 | 0.000 | 0.096 | 0.255 | 0.172
5.25 | -4.29 | 0.000 | 0.090 | 0.243 | 0.164
5.35 | -4.31 | 0.000 | 0.088 | 0.238 | 0.161
5.45 | -4.33 | 0.000 | 0.086 | 0.233 | 0.158
$x_{m}$ = 3.0
5.05 | -4.02 | 0.000 | 0.162 | 0.464 | 0.357
5.15 | -4.06 | 0.000 | 0.150 | 0.410 | 0.307
5.25 | -4.10 | 0.000 | 0.138 | 0.355 | 0.257
5.35 | -4.12 | 0.000 | 0.133 | 0.354 | 0.248
5.45 | -4.14 | 0.000 | 0.127 | 0.332 | 0.238

Table 5: Predicted Excesses for Disks Around 1.5 M⊙ StarsaaThe electronic version of this paper contains the complete version of this Table. log $t$ (yr) | log $L_{d}/L_{\star}$ | log $F_{24}/F_{24,0}$ | log $F_{70}/F_{70,0}$ | log $F_{160}/F_{160,0}$ | log $F_{850}/F_{850,0}$
---|---|---|---|---|---
$x_{m}$ = 0.33
5.05 | -4.40 | 0.001 | 0.100 | 0.202 | 0.110
5.15 | -4.45 | 0.001 | 0.090 | 0.184 | 0.099
5.25 | -4.51 | 0.001 | 0.078 | 0.163 | 0.088
5.35 | -4.52 | 0.001 | 0.074 | 0.157 | 0.085
5.45 | -4.53 | 0.001 | 0.070 | 0.151 | 0.082
$x_{m}$ = 1.0
5.05 | -4.01 | 0.001 | 0.162 | 0.321 | 0.188
5.15 | -4.19 | 0.001 | 0.148 | 0.299 | 0.178
5.25 | -4.22 | 0.001 | 0.140 | 0.287 | 0.170
5.35 | -4.24 | 0.001 | 0.133 | 0.275 | 0.163
5.45 | -4.24 | 0.001 | 0.128 | 0.266 | 0.157
$x_{m}$ = 3.0
5.05 | -3.73 | 0.002 | 0.206 | 0.406 | 0.261
5.15 | -4.01 | 0.002 | 0.199 | 0.394 | 0.251
5.25 | -4.04 | 0.002 | 0.192 | 0.382 | 0.242
5.35 | -4.05 | 0.002 | 0.181 | 0.361 | 0.227
5.45 | -4.08 | 0.002 | 0.167 | 0.339 | 0.211
Table 6: Predicted Excesses for Disks Around 2.0 M⊙ StarsaaThe electronic version of this paper contains the complete version of this Table. log $t$ (yr) | log $L_{d}/L_{\star}$ | log $F_{24}/F_{24,0}$ | log $F_{70}/F_{70,0}$ | log $F_{160}/F_{160,0}$ | log $F_{850}/F_{850,0}$
---|---|---|---|---|---
$x_{m}$ = 0.33
5.05 | -4.28 | 0.002 | 0.128 | 0.215 | 0.109
5.15 | -4.34 | 0.002 | 0.118 | 0.201 | 0.102
5.25 | -4.41 | 0.002 | 0.108 | 0.191 | 0.096
5.35 | -4.43 | 0.002 | 0.103 | 0.183 | 0.092
5.45 | -4.46 | 0.002 | 0.098 | 0.175 | 0.087
$x_{m}$ = 1.0
5.05 | -3.89 | 0.003 | 0.195 | 0.334 | 0.185
5.15 | -4.00 | 0.003 | 0.186 | 0.320 | 0.177
5.25 | 4.11 | 0.003 | 0.177 | 0.309 | 0.170
5.35 | -4.16 | 0.003 | 0.175 | 0.307 | 0.168
5.45 | -4.18 | 0.003 | 0.161 | 0.286 | 0.156
$x_{m}$ = 3.0
5.05 | -3.71 | 0.005 | 0.248 | 0.421 | 0.250
5.15 | -3.98 | 0.005 | 0.239 | 0.409 | 0.240
5.25 | -4.02 | 0.004 | 0.227 | 0.389 | 0.226
5.35 | -4.02 | 0.004 | 0.207 | 0.358 | 0.205
5.45 | -4.07 | 0.004 | 0.192 | 0.333 | 0.188
Table 7: Predicted Excesses for Disks Around 2.5 M⊙ StarsaaThe electronic version of this paper contains the complete version of this Table. log $t$ (yr) | log $L_{d}/L_{\star}$ | log $F_{24}/F_{24,0}$ | log $F_{70}/F_{70,0}$ | log $F_{160}/F_{160,0}$ | log $F_{850}/F_{850,0}$
---|---|---|---|---|---
$x_{m}$ = 0.33
5.05 | -4.18 | 0.004 | 0.170 | 0.259 | 0.120
5.15 | -4.22 | 0.004 | 0.153 | 0.233 | 0.108
5.25 | -4.34 | 0.004 | 0.137 | 0.207 | 0.096
5.35 | -4.36 | 0.004 | 0.130 | 0.197 | 0.093
5.45 | -4.38 | 0.004 | 0.123 | 0.188 | 0.086
$x_{m}$ = 1.0
5.05 | -3.97 | 0.008 | 0.237 | 0.352 | 0.181
5.15 | -4.08 | 0.007 | 0.219 | 0.329 | 0.168
5.25 | -4.09 | 0.000 | 0.000 | 0.000 | 0.000
5.35 | -4.14 | 0.006 | 0.201 | 0.306 | 0.154
5.45 | -4.14 | 0.005 | 0.187 | 0.287 | 0.144
$x_{m}$ = 3.0
5.05 | -3.83 | 0.010 | 0.285 | 0.424 | 0.230
5.15 | -3.95 | 0.009 | 0.270 | 0.403 | 0.217
5.25 | -3.98 | 0.009 | 0.258 | 0.387 | 0.206
5.35 | -4.03 | 0.009 | 0.248 | 0.372 | 0.197
5.45 | -3.48 | 0.087 | 0.749 | 0.887 | 0.558

Table 8: Predicted Excesses for Disks Around 3.0 M⊙ StarsaaThe electronic version of this paper contains the complete version of this Table. log $t$ (yr) | log $L_{d}/L_{\star}$ | log $F_{24}/F_{24,0}$ | log $F_{70}/F_{70,0}$ | log $F_{160}/F_{160,0}$ | log $F_{850}/F_{850,0}$
---|---|---|---|---|---
$x_{m}$ = 0.33
5.05 | -4.08 | 0.011 | 0.173 | 0.229 | 0.099
5.15 | -4.11 | 0.010 | 0.164 | 0.213 | 0.091
5.25 | -4.14 | 0.009 | 0.152 | 0.198 | 0.083
5.35 | -4.16 | 0.008 | 0.140 | 0.183 | 0.076
5.45 | -4.19 | 0.007 | 0.134 | 0.174 | 0.072
$x_{m}$ = 1.0
5.05 | -3.96 | 0.015 | 0.254 | 0.326 | 0.150
5.15 | -4.01 | 0.014 | 0.239 | 0.310 | 0.142
5.25 | -4.04 | 0.013 | 0.227 | 0.297 | 0.135
5.35 | -4.07 | 0.012 | 0.216 | 0.282 | 0.127
5.45 | -4.10 | 0.011 | 0.204 | 0.269 | 0.120
$x_{m}$ = 3.0
5.05 | -3.83 | 0.020 | 0.318 | 0.410 | 0.199
5.15 | -3.88 | 0.019 | 0.301 | 0.390 | 0.189
5.25 | -3.92 | 0.030 | 0.321 | 0.389 | 0.187
5.35 | -3.38 | 0.113 | 0.661 | 0.740 | 0.416
5.45 | -3.01 | 0.226 | 0.934 | 0.987 | 0.592

Table 9: Debris disk loci in color-color space $M_{\star}$ (M⊙) |  x0,y0 |  xu,yu |  xl,yl
---|---|---|---
1.0 | 0.00,0.0 | 4.0,0.1 | 5.00,0.00
1.5 | 1.50,0.0 | 4.0,1.5 | 4.50,0.25
2.0 | 1.25,0.0 | 2.5,2.5 | 3.50,0.50
2.5 | 1.00,0.0 | 2.0,4.0 | 3.00,1.00
3.0 | 1.00,0.0 | 1.5,5.0 | 2.25,1.75

Table 10: Vega Debris Disk ModelaaFluxes ($F$) are in units of Jy; dust mass in 1–50 $\mu$m particles ($M_{d,1-50}$) is in units of $10^{-3}$ M⊕; dust production rate ($\dot{M}$) is in units of $10^{21}$ g yr-1.  Source | $F_{24,\star}$ | $F_{70,\star}$ | $F_{160,\star}$ | $F_{850,\star}$ | $F_{24,disk}$ | $F_{70,disk}$ | $F_{160,disk}$ | $F_{850,disk}$ | $M_{d,1-50}$ | $\dot{M}$
---|---|---|---|---|---|---|---|---|---|---
Vega | 7.2 | 0.8 | 0.16 | 0.006 | 1.5 | 7.0 | 4.0 | 0.091 | 3.0 | 30bbDust production rate from Su et al. (2005). Our analysis suggests a smaller dust production rate, $\dot{M}\gtrsim 0.3\times 10^{21}$ g yr-1.
Model 1ccDebris disk model with $M_{\star}$ = 2 M⊙, $x_{m}$ = 1/3, $t$ = 200 Myr | 7.2 | 0.8 | 0.16 | 0.006 | 4.2 | 10.0 | 3.0 | 0.05 | 0.9 | 0.2
Model 2ddAs in note (c) for $M_{\star}$ = 2 M⊙, $x_{m}$ = 1, $t$ = 200 Myr | 7.2 | 0.8 | 0.16 | 0.006 | 7.2 | 23.0 | 8.0 | 0.11 | 3.8 | 0.6
Model 3eeAs in note (c) for $M_{\star}$ = 2.5 M⊙, $x_{m}$ = 1/2, $t$ = 200 Myr | 7.2 | 0.8 | 0.16 | 0.006 | 10.9 | 15.0 | 4.0 | 0.05 | 2.4 | 0.3
Model 4ffAs in note (c) for $M_{\star}$ = 2.5 M⊙, $x_{m}$ = 1, $t$ = 200 Myr | 7.2 | 0.8 | 0.16 | 0.006 | 15.5 | 23.0 | 5.0 | 0.07 | 4.8 | 0.8

Figure 1:  Evolution of a multiannulus coagulation model with
$\Sigma=0.18(a_{i}/{\rm 30~{}AU})^{-3/2}$ g cm-2 at 30–37 AU around a 1 M⊙
star. Left: median cumulative mass distribution at times indicated in the
legend. Right: median eccentricity distributions at $t$ = 0 (light solid
line), $t$ = 10 Myr (filled circles), $t$ = 100 Myr (open boxes), $t$ = 1 Gyr
(filled triangles), and $t$ = 10 Gyr (open diamonds). As large objects grow in
the disk, they stir up the leftover planetesimals to $e\sim$ 0.1. Disruptive
collisions then deplete the population of 0.1–10 km planetesimals, which
limits the growth of the largest objects.

Figure 2: Evolution of the radius of the largest object in each annulus for a
MMSN disk around a 1 $M_{\odot}$ star. The number to the right of each set of
points indicates log $t$ in yr from the start of the calculation. Large
objects with $r\sim$ 1000 km form at the inner edge of the disk in 10–100 Myr,
in the middle of the disk in 0.3–1 Gyr, and at the edge of the disk in 10 Gyr.
Figure 3: Median radii of the largest objects at 1 Gyr for disks around a 1
$M_{\odot}$ star. The numbers to the right of each set of points indicate
$x_{m}$, the disk mass in units of the MMSN. Planets form earlier in the inner
portions of the most massive disks. Icy planet formation saturates at maximum
radii $r\sim$ 1500 km.

Figure 4: Median cumulative mass distributions at 1 Gyr for annuli at 30–37 AU
(left panel) and at 123–146 AU (right panel) around a 1 M⊙ star. The legend
indicates $x_{m}$, the initial disk mass in units of the scaled MMSN. In the
inner disk, many large planets form and the collisional cascade removes nearly
all of the material in objects with $r\lesssim$ 1–10 km. In the outer disk,
few large planets form; collisions are inefficient at removing material in
small objects.

Figure 5: Radius of the largest object in each annulus at 10 Myr and at 100
Myr for a MMSN disk around a 1 $M_{\odot}$ star. The black points indicate
results for calculations with 64 annuli; the magenta and cyan points show
results for calculations with 32 annuli. The good agreement between the 32
annulus and 64 annulus calculations shows that planet formation is not
sensitive to the size of the grid.  Figure 6: Median production rate of 0.01–1
$\mu$m objects as a function of time for 30–150 AU disks around a 1 M⊙ central
star. The legend indicates $x_{m}$, the initial disk mass in units of the
scaled MMSN. As large objects grow during the early stages of the evolution,
the dust production rate declines. Once large objects start to stir up the
leftover planetesimals, debris production rises rapidly. After dust production
peaks, the collision rate and dust production decline slowly with time. For
all stars, more massive disks eject more material into a wind of small
particles.  Figure 7: Median mass in 0.001–1 mm objects as a function of time
for 30–150 AU disks around a 1 M⊙ central star. The legend indicates $x_{m}$,
the initial disk mass in units of the scaled MMSN. During runaway growth, the
median dust mass is small and roughly constant in time. As planet formation
propagates through the disk, the dust mass grows with time. Once planets form
in the outer disk, collisions and Poynting-Robertson drag removes small grains
from the disk.  Figure 8: Time evolution of the radius of the largest object
at 40 AU and at 100 AU for identical disks around 1 M⊙ (dot-dashed curves), 2
M⊙ (solid curves), and 3 M⊙ (triple dot-dashed curves) stars. Planets grow
faster around more massive stars and in the inner disks of all stars.

Figure 9: Median radius of the largest object at 100 Myr in each annulus for a
scaled MMSN disk ($x_{m}$ = 1) around 1–3 $M_{\odot}$ stars. The number to the
right of each set of points indicates the stellar mass in M⊙. At all disk
radii, large objects form faster around more massive stars. At 30–60 AU,
planet formation saturates at radii $r\sim$ 1000–2000 km on relatively short
timescales, $t\sim$ 100 Myr (see also Eq. (41)). At larger disk radii, planets
form more slowly and do not reach the maximum radius unless the formation time
is shorter than the main sequence lifetime.

Figure 10: Median cumulative mass distributions at 100 Myr for planet
formation calculations at 30–37 AU around 1–3 M⊙ stars. Left: Results for
models with $\Sigma=0.18~{}(a_{i}/{\rm 30~{}AU})^{-3/2}$ g cm-2. The light
solid line indicates the initial mass distribution. The dashed (1 M⊙, $x_{m}$
= 1), dot-dashed (2 M⊙, $x_{m}$ = 1/2), and heavy solid (3 M⊙, $x_{m}$ = 1/3)
lines show median results for the same initial conditions. Right: Results for
models with a scaled surface density ($x_{m}$ = 1), $\Sigma=0.18~{}(a_{i}/{\rm
30~{}AU})^{-3/2}$ $(M_{\star}/M_{\odot})$ g cm-2, and different stellar masses
(1 M⊙: dashed line, 2 M⊙: dot-dashed line, and 3 M⊙: heavy solid line).
Although more massive planets form around more massive stars, the collisional
cascade leads to a small dispersion in total disk mass at late times.

Figure 11: Median cumulative mass distributions at $t=t_{ms}$ for annuli at
30–37 AU (left panel) and at 123–146 AU (right panel) for identical disks
($\Sigma=0.18~{}(a_{i}/{\rm 30~{}AU})^{-3/2}$ g cm-2) around 1–3 M⊙ stars. The
legend indicates the stellar mass in M⊙. In the inner disk, many large planets
form and the collisional cascade removes a large fraction of the material in
objects with $r\lesssim$ 1–10 km. In the outer disk, few large planets form;
collisions are inefficient at removing material in small objects.

Figure 12: Median production rate of 0.01–1 $\mu$m objects at 30–150 AU as a
function of time for scaled MMSN ($x_{m}$ = 1) around 1–3 M⊙ central stars.
The legend indicates the stellar mass in M⊙. For scaled MMSN, disks around
more massive stars eject much more material in very small grains at early
times ($t\lesssim$ 10–100 Myr). In an ensemble of stars with a variety of disk
masses, there is wide range of dust production rates.  Figure 13: Median mass
in 0.001–1 mm objects as a function of time for scaled MMSN ($x_{m}$ = 1) at
30–150 AU around 1–3 M⊙ central stars. The legend indicates the stellar mass
in M⊙. For $t\lesssim$ 1–3 Myr, icy planet formation produces little dust. At
10–100 Myr, the mass in small grains is $\sim$ 1 lunar mass for most disks. At
late times, the mass in small grains slowly declines to currently undetectable
levels.

Figure 14: Time evolution of the median $L_{d}/L_{\star}$ (dust luminosity
relative to the luminosity of the central star) for disks surrounding a 1
$M_{\odot}$ star. The legend indicates the disk mass in units of the MMSN.
More massive disks reach larger peak dust luminosities earlier than less
massive disks. The typical peak dust luminosity is comparable to the dust
luminosity of the most luminous debris disks associated with solar-type stars.

Figure 15: As in Fig. 14 for the median 70 $\mu$m excess (left panel) and the
median 850 $\mu$m excess (right panel). At both wavelengths, dust emission
begins to increase at 5–10 Myr. Peak dust emission occurs at 30–100 Myr (70
$\mu$m) and 100–300 Myr (850 $\mu$m). When the central star evolves off the
main sequence, the typical excess at 70 $\mu$m (850 $\mu$m) is $\sim$ 2–3
(3–10) times the flux from the central star.

Figure 16: Variation of dust excess with disk size. Left panel: time evolution
of the median 70 $\mu$m excess for MMSN disks with outer radii of 70 AU
(dashed line) and 150 AU (solid line). At late times, smaller disks produce
smaller IR excesses. it Right panel: as in the left panel for the median 850
$\mu$m excess.

Figure 17: Time evolution of the median $L_{d}/L_{\star}$ for MMSN disks
surrounding 2 $M_{\odot}$ stars (left panel) and 3 $M_{\odot}$ stars (right
panel). The legend indicates the initial disk mass in units of the scaled
MMSN. The typical maximum dust luminosity, $L_{d}/L_{\star}\sim 10^{-3}$ is
comparable to the dust luminosity of the brightest debris disks around A-type
stars.  Figure 18: Time evolution of median IR excesses for MMSN disks around
1–3 $M_{\odot}$ stars. The legend in the lower left panel indicates the
stellar mass in solar masses for each curve in all panels. Lower left panel:
24 $\mu$m excess. Upper left panel: 70 $\mu$m excess. Upper right panel: 160
$\mu$m excess. Lower right panel: 850 $\mu$m excess. At 24 $\mu$m, the peak
excess increases dramatically with the temperature of the central star. Thus,
hotter stars produce much larger 24 $\mu$m excesses. At longer wavelengths,
the magnitude of the excess is correlated with the mass of the central star.
Roughly independent of stellar mass, the magnitude of the excess at 24–850
$\mu$m peaks at 10–30 Myr as observed in debris disks around A-type stars
(Currie et al., 2008a). At late times, the 160–850 $\mu$m excesses for all
stars are $\sim$ 3–5 times the flux from the stellar photosphere.  Figure 19:
Time evolution of the median [24]–[70] color as a function of time for 1.5 M⊙
(dashed line), 2.0 M⊙ (solid line), and 3.0 (triple dot-dashed line) M⊙ stars.
Debris disks around lower mass stars have redder [24]–[70] colors than disks
around more massive stars. For massive stars ($\gtrsim$ 2 M⊙), the [24]–[70]
color increases slowly throughout the main sequence lifetime and then declines
just before central star evolves off the main sequence. For lower mass stars,
the [24]–[70] color reaches a broad maximum at 300 Myr to 1 Gyr and then
declines.  Figure 20: Evolution of debris disks with $x_{m}$ = 1 in color-
color space. For icy planet formation at 30–150 AU, debris disks around
massive stars are hotter than debris disks around less massive stars. Thus,
debris disks around stars of different masses occupy specific regions of the
[8]–[24] vs [24]–[70] color-color diagram.  Figure 21: As in Fig. 20 for disks
with different initial masses around 2 M⊙ stars. Boxes: $x_{m}$ = 1/3, filled
circles: $x_{m}$ = 1, diamonds: $x_{m}$ = 3. Although more massive disks have
redder [8]–[24] and [24]–[70], the shape of the color-color track is
independent of mass. Thus, the color-color diagram isolates stars of different
masses.  Figure 22: Observations of the 24 $\mu$m excess for nearby A-type
stars with known ages (Rieke et al., 2005; Su et al., 2006). The lines show
the predicted evolution of the excess for debris disk models around 2 M⊙ stars
(dot-dashed line: $x_{m}$ = 1/3; solid line: $x_{m}$ = 1, triple dot-dashed
line: $x_{m}$ = 3). Observations for all but four stars (including one 8 Myr
old star with log $F_{24}/F_{24,0}\approx$ 2) fall within loci defined by our
debris disk calculations. Model predictions are also consistent with
observational evidence for a peak in the 24 $\mu$m excess at 10–20 Myr (see
also Currie et al., 2008a).  Figure 23: As in Fig. 22 for the 70 $\mu$m
excess. Observations for all but 2–3 stars fall within the loci defined by the
model tracks. Consistent with model predictions, the data suggest a peak in
the 70 $\mu$m excess at 10–20 Myr.  Figure 24: Observations of the [24] - [70]
color for nearby A-type stars with known ages (Su et al., 2006). The lines
show the predicted evolution of the excess for debris disk models around 1.5
M⊙ stars (triple dot-dashed line: $x_{m}$ = 2) and for 2 M⊙ stars (dot-dashed
line: $x_{m}$ = 1/3; solid line: $x_{m}$ = 1). Observations for all but 2–3
stars fall within the model predictions.  Figure 25: Observations of the 70
$\mu$m excess for nearby solar-type stars with known ages (Beichman et al.,
2006; Hillenbrand et al., 2008). The lines show the predicted evolution of the
excess for debris disk models around 1 M⊙ stars (dot-dashed line: $x_{m}$ =
1/3; solid line: $x_{m}$ = 1, triple dot-dashed line: $x_{m}$ = 3). Most stars
fall within the loci defined by the calculations, but many stars are 3–10
times brighter than model predictions.  Figure 26: As in Fig. 25 for the 160
$\mu$m excess.