NEW DEBRIS DISKS AROUND YOUNG, LOW-MASS STARS DISCOVERED WITH THE SPITZER SPACE TELESCOPE

We derive X-ray apparent flux densities from ROSAT PSPC detected count rates using the conversion of Schmitt et al. (1995):

$$\begin{eqnarray} \mbox{Apparent Flux} &=& \mbox{no. of counts/s} \times (8.31 + 5.3\times \mbox{HR1})\nonumber\\ && \times 10^{-12}\;{\rm erg\;cm}^{-2}\;{\rm s}^{-1} \end{eqnarray} \tag{ 2 }$$

where no. of counts/s is the ROSAT count rate, and HR1 is the ROSAT hardness ratio no. 1. We require that optical source positions fall within the 1σ ROSAT position uncertainties to associate the ROSAT detections with our sample stars. We infer the fractional X-ray luminosity L_X/L_* and the X-ray flux density F_X at the stellar photosphere using the derived stellar properties (radius, luminosity) from synthetic spectral fitting in Section 4.1.

We use five main methods to estimate ages for the stars in our sample—young moving group membership (<100 Myr), X-ray activity, Ca ii H&K activity, rotation periods, and fitting to theoretical isochrones. First, we assign ages based on cluster and association memberships to AU Mic, AT Mic, and HD 358623 (Beta Pic moving group; 12 Myr), to TWA association sources (8 Myr), and AB Dor moving group members AB Dor, GJ 3400A and LO Peg (75 ± 25 Myr). For sources with measured X-ray activity, rotation periods or Ca ii H&K activity, we estimate ages using activity–age correlation relations derived in Mamajek & Hillenbrand (2008). We use their Equations (A3), (12)–(14), and (3), respectively, with appropriate limits (e.g., B − V < 1.2, −5 < log(L_X/L_*) < −4.3). When two or three of these activity indices are measured, we adopt the mean and standard deviation as the final age estimate and uncertainty. When only a single indicator is available, we adopt an uncertainty of 20% for Ca ii H&K and X-ray activity. For rotation-based age estimates, we propagate the uncertainties in the coefficients of the fit in Mamajek & Hillenbrand (2008) to Equations (12)–(14), the relation first presented in Barnes (2007). We obtain age estimates for 46 of our 70 target stars using these first four age determination methods.

For eight of the remaining stars with T_*< 6000 K and X-Ray activity implying ages <300 Myr, we fit the derived stellar radii and temperature in Section 4.1 to the Seiss et al. (2000) isochrones. We adopt average ages and uncertainties from the range of isochrone grid points that are consistent to within 5% of the derived stellar radius and temperature. The 5% criterion is chosen based on the estimated radius uncertainties in the fits in Section 4.1, the S/N of Hipparcos measured parallax, and to recover reasonable age uncertainties (∼50%). If a source is an unresolved binary, we adjust the derived radius appropriately before fitting to isochrones to avoid systematically underestimating the age. Three of the eight stars have measured lithium abundances that support the estimated ages. For stars that are implied to be young from measured activity indicators, the activity-based ages are consistent with isochrone fitting results. For 14 stars with T_* > 6000 K, we use the average age estimates of Song (2000), Song et al. (2001), and J. Stauffer (2000, private communication) from fitting Stromgren photometry.

We are unable to estimate ages for two target stars using the above five methods. For the possible Pleiades moving group member RE 1816+541 lacking a trigonometric parallax, we assign an age of 115 Myr based upon the Pleiades age estimates from Messina et al. (2003) and Stauffer et al. (1998). Finally, we obtain an age estimate of 1.1 Gyr from Nordstrom (2004) for HD 17240 and adopt an age uncertainty of 50%.

In Figure 5, we plot the age distribution for our sample as a function of effective stellar temperature. The K and M stars in our sample are primarily selected from young moving groups whereas the A-G type stars are primarily selected from other indicators. As a result of these selection criteria, the stars in our sample cluster in age as a function of spectral type in Figure 5. The T_* < 5000 K low-mass K and M dwarfs are younger than the 5000–6000 K G dwarfs and the T_* > 6000 K stars, with median ages of 12.25 Myr, 340 Myr, and 200 Myr, respectively. This age bias limits comparisons across spectral types in our sample.

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We wish to consider stellar wind drag from stellar wind protons as a disk dissipation mechanism (Section A.1). In order to evaluate this mechanism, we estimate the stellar wind mass-loss rates for stars in our sample. If F_X < 8 × 10⁵ erg s⁻¹ cm⁻² at the stellar surface (stars approximately older than 700 Myr), we can use the relation of Wood et al. (2002, 2005) to estimate stellar mass-loss rates, $\dot{M}_{\rm sw}$:

$$\begin{equation} \dot{M}_{\rm sw} = \dot{M}_\odot \left(\frac{F_X }{ F_{X\odot }}\right)^{1.34} \left(\frac{R_*}{R_\odot }\right)^2 \end{equation} \tag{ 3 }$$

where $\dot{M}_\odot =2\times 10^{-14}\; M_\odot$ yr⁻¹. Seven stars with X-ray detections meet these criteria in our sample. We caution against estimating mass-loss rates from X-Ray activity for stars younger than ∼700 Myr or with F_X > 8 × 10⁵ erg s⁻¹ cm⁻² (Wood et al. 2005). HD 135599 has an abnormally low X-ray luminosity given its estimated age of 214 ± 17 Myr, and also possesses a strong 70 μm excess.

We estimate stellar masses for our sample using the estimated stellar ages, effective temperatures, and the Seiss et al. (2000) isochrones. We use the estimated stellar mass in the calculations of dust disk properties and timescales. For a given age and the corresponding Seiss et al. (2000) isochrone, we linearly interpolate between the Seiss et al. (2000) stellar masses as a function of stellar effective temperature.

We investigate the possibility that the small excesses we detect at 24 μm are not due to re-processed radiation from dusty debris disks. HD 10008, HD 73350, HD 112429, and HD 135599 also possess 70 μm excesses, supporting the assertion that our data and analysis are sensitive to ∼10% excesses at 24 μm. For the seven sources with 24 μm excesses and without 70 μm detections, the upper limits at 70 μm are consistent with a debris disk SED. Furthermore, excesses at 24 μm are more likely at younger ages. The stars with 24 μm excesses we identify are generally much younger than our overall sample age distribution.

However, an unknown low-mass spectroscopic companion—e.g., an M5 companion to a K0 star—could produce an apparent ∼10% flux excess at 24 μm. The occurrence of spectroscopic binaries in the field with the mass ratio required to reproduce the 24 μm excess is much less than the 10% (7/70) of stars for which we observe this excess (Hogeveen 1992). Nonetheless, we do not rule out low-mass companions for these sources. BO Mic and GJ 3400A possess published IRAC photometry, but the [8]–[24] colors are inconclusive as to which scenario is favored (Chen et al. 2005b). Positionally coincident background AGN and serendipitous asteroids could also account for the observed 70 and 24 μm excess, but are statistically unlikely for any given star (<1%). >0.1 mag stellar variability at the K_s-band could also produce an apparent 24 μm excess for noncontemporaneous observations.

We note that the distribution of K_s–[24] photospheric colors for low-mass stars is less constrained than it is for warmer stars. We compare the formal errors derived from the 24 μm S/N and K_s-band uncertainty, and compare it to the empirical K_s–[24] distribution after subtracting off the predicted photospheric color. We derive an empirical distribution uncertainty of 0.03 mag from a Gaussian fit, which is consistent with our formal propagated errors. Nine of the 11 24 μm excess sources lie >3σ in excess above the fit to the empirical distribution. The 10th, HD 112429, also possesses a 70 μm excess, and the 11th at 2.5σ, is 12.5 Myr 4240 K GJ 907.1. A 24 μm excess for GJ 907.1 is entirely plausible, but the excess for Gl 907.1 is the least robust detection of a 24 μm excess that we include in our results. For a sample size of 70 stars, we might expect ∼1 false-positive detection of excess from statistical fluctuations.

At 70 μm, we can detect debris disks that are generally >100% brighter than the expected photosphere (Figure 2). For the 15 stars in our sample with T_* > 6000 K, we find a disk fraction of 13.3^+13.3_−4.6%. This is smaller than the disk fraction derived from analogous studies of A-type stars (33% ± 5%; Su et al. 2006), but the results are consistent to within 1σ given the small number statistics. In our sample 70 μm excesses are ∼6 times less common around stars with effective temperatures of less than 5000 K (3.7^+7.6_−1.1%) than around stars with effective temperatures between 5000 K and 6000 K (21.4^+9.5_−5.7%). These two disk fractions differ at the ∼2σ level. We observe this discrepancy even though the cooler stars have a younger median age in our sample (12.25 Myr versus 340 Myr). At younger ages, debris disks might be expected to be brighter by factors of a few. This paucity of late-type star debris disks at 70 μm has been found by, and is consistent with, several other more detailed studies (Hillenbrand et al. 2008; Gautier et al. 2008; Trilling et al. 2008; Beichman et al. 2006a, and references therein). The interpretation of our result is complicated and limited by several factors and selection effects in our sample that we consider in more detail in Section A.3.

At 24 μm, we can detect much smaller (∼10%–15%) excesses relative to the photosphere (Figure 3). When we include the detections of excesses at 24 μm, the disk fraction dependence on spectral type disappears. We find an overall disk excess fraction of 22.2^+9.8_−5.9% for stars with effective temperatures of less than 5000 K, and 25.0^+9.6_−6.3% for stars with effective temperatures between 5000 K and 6000 K. This result suggests that debris disks are just as common around cool stars as they are around hotter stars, but the dust disks around hotter stars may generally be more massive and brighter relative to the stellar photosphere. This result is consistent with the frequency of ∼40 Myr M dwarfs with 24 μm excesses identified in Forbrich et al. (2008).

Our sample includes 24 known binary systems (Table 1), and constitutes a sample that has been fairly well studied for AO, wide separation, and spectroscopic companions. The binary completeness of our sample, however, is undoubtedly <100%. The incompleteness of the known binaries in our sample limits our interpretation, but we can make a tentative comparison. We do not identify a statistically significant difference between the disk fractions for binaries and single stars. We derive a single star disk fraction of 22.7^+3.0_−5.0% (10/44), and a binary disk fraction of 19.2^+9.8_−5.4% (5/26). Trilling et al. (2007) carried out an investigation of debris disks around A- and F-type binaries, and concluded that binaries have a higher disk fraction on average than single stars. We do not confirm this result when investigating lower mass stars, with the aforementioned caveats. Trilling et al. (2007) identified that tight (separations <3 AU) and wide separation (>50 AU) binaries have a higher disk fraction than intermediate separation binaries (3–50 AU). Our sample of binaries is evenly split between intermediate separation (12) and wide separation binaries (14), and we find 3 and 2 debris disks in each category, respectively.

Since the discovery of a resolved disk around AU Mic in Kalas et al. (2004), much work has been done to characterize and model AU Mic's debris disk (Fitzgerald et al. 2007; Strubbe & Chiang 2006; Quillen et al. 2007; Krist et al. 2005a, 2005b; Metchev et al. 2005). Our updated 70 μm flux density measurement should not significantly alter interpretations. We update the application of the stellar wind drag steady-state model to AU Mic in Plavchan et al. (2005). The discovery in Wood et al. (2005) that X-ray activity does not correlate with stellar wind mass-loss rates for F_X > 8 × 10⁵ erg s⁻¹ cm⁻² invalidates the application in Plavchan et al. (2005) of a steady-state stellar wind drag model to AU Mic; AU Mic possesses an X-ray surface flux density of ∼1.8 × 10⁷ erg s⁻¹ cm⁻². We estimate the grain–grain collision timescale within the "birth ring" at 43 AU (the location of the dust-producing ring of parent bodies; Strubbe & Chiang 2006) to be ∼10⁴ times shorter than the stellar wind drag timescale for $\dot{M}_{\rm sw}=10\;\dot{M}_{\odot }$, a dust disk mass of 10⁻² M_⊕, and a disk width of 4.3 AU (Equation (A23), Section A.2; Fitzgerald et al. 2007; Metchev et al. 2005; Liu et al. 2004; Augereau & Beust 2006). The evolution of the AU Mic debris disk is currently dominated by grain–grain collisions and the presence of any approximately Jovian mass planets (Section A.2).

For this K0 star (∼1/3 L_☉), the inferred mass-loss rate implies that the drag timescale from stellar wind is ∼10 times shorter than Poynting–Robertson (P–R) drag (Equations (A8)–(A10), Sections A.1 and A.2). The stellar wind drag timescale is ∼0.4 times the grain–grain collision timescale (Equations (A14) and (A23)) for the dust radius (12 AU) and dust mass (6.6 × 10⁻⁵ M_moon) estimated in Table 4 (Section 4.3). This implies that stellar wind drag is dynamically important in the evolution of the dust in this disk, and makes HD 135599 a canonical example to study the transition between disk evolution dominated by grain–grain collisions and drag-force-dominated disk evolution (Sections 6.2 and A.2.3). The Spitzer IRS spectra of HD 135599 are consistent with the "power-law" excess predicted for a drag-force-dominated debris disk (e.g., F_ν(dust)∝ν^α, Sections 6.2 and A.2 in this paper; Lawler et al. 2009). A more accurate dust mass measurement from submillimeter observations and high-contrast imaging to obtain radial surface brightness profiles is warranted to clarify this interpretation further.

A power-law extrapolation of the IRS excess for HD 135599 predicts a flux at 70 μm that is smaller than what is observed (Lawler et al. 2009). This implies that the entire disk for HD 135599 is not described by a single dominant dust removal mechanism. Due to the different power-law dependences on orbital distance for drag forces and grain–grain collisions, and because the collisional timescale depends on the local dust disk density, the evolution of dust at different orbital radii can be dictated by different dust removal mechanisms (Section A.2). For HD 135599, it is plausible that there is a ring of parent bodies producing dust through mutual collisions where the dust collision timescales are short relative to the stellar wind drag timescale, but not significantly shorter. This would produce the "excess excess" seen at 70 μm. Then, a fraction of the dust produced in the ring spirals inward under the action of stellar wind drag into smaller orbital radii where the collisional timescales become longer than the drag force timescales. This would then explain the observed IRS power-law excess.

We consider the large fractional infrared excesses (F₇₀/F_* > 10) observed for HD 7590, HD 73350, HD 92945, and HD 135599 (the "bright disk stars"). For HD 7590, HD 73350 and HD 135599, the stellar ages inferred from X-ray activity and R'_HK tend toward older ages, whereas ages inferred from rotation periods tend toward younger ages.⁷ This trend is not present for stars with no excess, and follows directly by comparing the ratio of the three activity indices. Within our sample, HD 7590, HD 73350, and HD 135599 are outliers when computing the ratio of the rotation period to both log(R'_HK) and log(L_X/L_*) (Figure 9). Given the small size of our sample, this result is inconclusive and may be spurious. A more rigorous treatment with a larger debris disk sample of solar-type stars with measured rotation periods and activity indices is warranted.

If HD 7590, HD 73350, and HD 135599 are younger than we estimate from X-ray and R'_HK activity, then the activity for these stars are abnormally low. This presents the possibility that stars with large excesses are more likely to be less active at a given age. This anticorrelation between activity and excess is consistent with the possibility that stellar wind drag is the dominant grain removal mechanism (Section A.1). At a fixed age, stellar wind drag predicts that less active stars will have brighter disks due to the longer wind drag timescales (Section A.2; Chen et al. 2005a). However, at these ages of <700 Myr, X-ray activity is not a reliable proxy for stellar wind mass-loss rate as discovered in Wood et al. (2005). Alternatively, this anti-correlation is also consistent with massive debris disks evolving from massive primordial disks, which in turn regulated the host star angular momentum (Currie et al. 2008a).

A second hypothesis is that the "bright disk stars" are undergoing an epoch of enhanced planetesimal collisions akin to the Late Heavy Bombardment within our own solar system when it was ∼700 Myr old (Kleine et al. 2002; Chambers 2004; Gomes et al. 2005; Strom et al. 2005). HD 10008, HD 59967, and HD 123998 also fall within the estimated age range of 200–500 Myr.

The average time for a grain to collide with another grain is estimated with the expression in Minato et al. (2006); Najita & Williams (2005) as

$$\begin{eqnarray} \tau _{\rm coll} &= 2\times 10^3\;\mbox{yr} \left(\frac{r}{50\,\mbox{AU}}\right)^{7/2}\left(\frac{a}{\mu \mbox{m}}\right)\left(\frac{\rho _g}{\mbox{g cm}^{-3}}\right)\left(\frac{\pi a^2}{S_z}\right)\nonumber\\ &\hspace{10pt} \times\left(\frac{M_*}{M_\odot }\right)^{-1/2}\left(\frac{M_d}{10^{-3}\,M_\oplus }\right)^{-1}, \end{eqnarray} \tag{ A1 }$$

where a is the grain radius, r is the dust orbital distance, ρ_g is the dust grain density, M_* is the stellar mass, M_d is the dust mass, M_☉ is the solar mass, M_⊕ is the Earth's mass, and (πa²/S_z) = 1/Q_coll, the ratio of the effective grain–grain collision cross section to the geometric cross section (Q_coll ∼ 1).

Equation (A1) relies on two important assumptions. First, the grains are assumed to be sufficiently inclined in their orbits to see the full vertical optical depth of the disk at the orbital radius of the grain (the grain traverses the full optical depth of the disk twice as it orbits the parent star). For optically thin debris disks, the average grain inclination should approximate the ratio of the scale height of the dust disk to the orbital distance, h/r.

For the second assumption, the dust is assumed to be evenly distributed over a disk surface area πD²_OUT. This second assumption is only valid when the entire dust mass M_d is interior to the orbital distance under consideration, i.e., r = D_OUT. In this sense, τ_coll in Equation (A1) represents the timescale for all grains at the disk outer radius D_OUT of size a and density ρ_g in an entire disk of mass M_d to undergo a collision.

With approximately a dozen bright, spatially resolved disks, the observed dust geometries are primarily narrow rings (Kalas et al. 2008; Fitzgerald et al. 2007; Krist et al. 2005a, 2005b; Telesco et al. 2005; Marsh et al. 2005; Stapelfeldt et al. 2004; Kalas et al. 2004; Holland et al. 2003; Wahhaj et al. 2003; Heap et al. 2000; Schneider et al. 1999; Jayawardhana et al. 1998). Here, we update Equation (A1) and do away with the second assumption to avoid misapplication of collisional timescales as a function of orbital radius, and to make more realistic estimates.

We define the radial surface density profile Σ_r(r) such that

$$\begin{equation} M_d = \int _{D_{\rm IN}}^{D_{\rm OUT}} \Sigma _{r}(r)\times r \;dr, \end{equation} \tag{ A2 }$$

where D_IN and D_OUT are the inner and outer disk radii, respectively, and we have assumed azimuthal symmetry. Note we have incorporated the constant from the azimuthal integration into Σ_r(r). Then the general form for Equation (A1) becomes

$$\begin{eqnarray} \tau _{\rm coll} &= 2\times 10^3\;\mbox{yr} \left(\frac{r}{50\,{\rm AU}}\right)^{3/2}\left(\frac{a}{\mu \mbox{m}}\right)\left(\frac{\rho _g}{\mbox{g cm}^{-3}}\right)\left(\frac{\pi a^2}{S_z}\right)\nonumber\\ &\hspace{10pt} \times\left(\frac{M_*}{M_\odot }\right)^{-1/2}\left(\frac{2}{\Sigma _{r}(r)} \times \frac{10^{-3}\,M_{\oplus }}{(50\,{\rm AU})^2}\right) \end{eqnarray} \tag{ A3 }$$

From Equation (A2), if we set D_IN = 0, and Σ_r(r) = Σ'₀, we get that Σ_r(r) = Σ'₀ = 2M_D/D²_OUT, and we recover Equation (A1).

We consider a generic radial surface density profile of

$$\begin{equation} \Sigma _{r}(r) = \Sigma _0 r^\alpha \end{equation} \tag{ A4 }$$

with inner and outer disk radii D_IN and D_OUT, which can be set appropriately depending on α. From Equation (A2), we find

$$\begin{equation} \Sigma _0= \left\lbrace \begin{array}{@{}lr} M_D(\alpha +2) / (D_{\rm OUT}^{\alpha +2} - D_{\rm IN}^{\alpha +2}),& \alpha \ne -2 \vspace*{4pt}\\ M_D/\ln (D_{\rm OUT}/D_{\rm IN}), &\alpha =-2 \end{array} \right. \end{equation} \tag{ A5 }$$

and Equation (A3) becomes

$$\begin{eqnarray} \tau _{\rm coll} &=& 2\times 10^3\;\mbox{yr} \;\mbox{$\gamma $} \left(\frac{r}{50\,{\rm AU}}\right)^{3/2-\alpha }\left(\frac{a}{\mu \mbox{m}}\right)\left(\frac{\rho _g}{\mbox{g cm}^{-3}}\right)\nonumber\\ &&\times\,\left(\frac{\pi a^2}{S_z}\right)\left(\frac{M_*}{M_\odot }\right)^{-1/2}\left(\frac{M_d}{10^{-3}\,M_\oplus }\right)^{-1} \end{eqnarray} \tag{ A6 }$$

where

$$\begin{equation} \mbox{$\gamma $} = \left\lbrace \begin{array}{lr} \left(\frac{2}{\alpha +2}\right)\left[\left(\frac{D_{\rm OUT}}{50\,AU}\right)^{\alpha +2}-\left(\frac{D_{\rm IN}}{50\,AU}\right)^{\alpha +2}\right], & \alpha \ne -2 \vspace*{4pt}\\ 2\ln (D_{\rm OUT}/D_{\rm IN}), & \alpha =-2 \end{array}. \right. \end{equation} \tag{ A7 }$$

The gamma factor in Equations (A6) and (A7), encapsulating the disk geometry, is important for estimating grain lifetimes. For example, consider a ring of dust at radius D_ring and width ΔD_ring = 0.1D_ring. The collision timescale estimate in Equation (A1) will differ from the actual collision timescale by a factor of γ ∼ 2ΔD_ring = 0.2 for α = −2 or 0, and overestimate the collision frequency for any of the dust that is scattered or dragged out of the ring. As a second example, we consider α = −1.5 with D_IN = 0 AU and D_OUT = 200 AU. The collision time as a function of orbital distance will scale as r³, and the collision timescales will be γ = 8 times longer.

A drag force removes angular momentum from an orbiting dust grain, causing it to spiral into the star. The time for a dust grain to spiral into a star under the action of P–R drag is estimated in Minato et al. (2006) as

$$\begin{eqnarray} \tau _{\rm PR} &=& 2\times 10^6\;\mbox{yr}\left(\frac{r}{50\,{\rm AU}}\right)^{2}\left(\frac{a}{\mu \mbox{m}}\right)\left(\frac{\rho _g}{\mbox{g cm}^{-3}}\right)\nonumber\\ &&\times\left(\frac{\pi a^2}{C_{ph}}\right)\left(\frac{L_*}{L_\odot }\right)^{-1} \end{eqnarray} \tag{ A8 }$$

where (πa²/C_ph) = 1/Q_rad, where Q_rad is the radiative coupling coefficient, the ratio of the effective to geometric cross sections. Similarly, the time for a dust grain to spiral into a star under the action of stellar wind drag is estimated in Minato et al. (2006) as

$$\begin{eqnarray} \tau _{\rm SW} &=& 6\times 10^6\;\mbox{yr}\left(\frac{r}{50\,{\rm AU}}\right)^{2}\left(\frac{a}{\mu \mbox{m}}\right)\left(\frac{\rho _g}{\mbox{g cm}^{-3}}\right)\nonumber\\ &&\times\left(\frac{\pi a^2}{C_{\rm sw}}\right)\left(\frac{\dot{M}_*}{\dot{M}_\odot }\right)^{-1} \end{eqnarray} \tag{ A9 }$$

where (πa²/C_sw) = 1/Q_SW, where Q_SW is the stellar wind coupling coefficient, the ratio of the effective to geometric cross sections. While Plavchan et al. (2005) assumed Q_SW = 1, this assumption is confirmed with modeling of dust grains in Minato et al. (2006).

From Equations (A8) and (A9) it follows

$$\begin{equation} \frac{\tau _{\rm SW}}{\tau _{\rm PR}} = 3\frac{Q_{\rm rad}}{Q_{\rm SW}}\frac{L_*}{L_\odot }\frac{\dot{M}_\odot }{\dot{M}_*} \end{equation} \tag{ A10 }$$

as in Plavchan et al. (2005) with Q_rad/Q_SW = 1 assumed. We note that Equation (3) in Plavchan et al. (2005) is more fundamental than Equation (A10) since the solar wind mass-loss rate is variable, but this is a good approximation. Using the X-ray mass-loss rate relation in Wood et al. (2005), we can re-express (A10) as

$$\begin{equation} \frac{\tau _{\rm SW}}{\tau _{\rm PR}} = 3\frac{Q_{\rm rad}}{Q_{\rm SW}}\frac{L_*}{L_\odot }\left(\frac{L_{X,\odot }}{L_{X,*}}\right)^{1.34}\left(\frac{R_{*}}{R_\odot }\right)^{0.68} \end{equation} \tag{ A11 }$$

For the Sun, L_X/L_☉ ∼ 5 × 10⁻⁷. When τ_SW/τ_PR < 1, P–R drag timescales are longer than stellar wind drag timescales, and stellar wind drag is relevant. Using A11 and the approximation that 1.34 = 4/3, stellar wind drag dominates P–R drag when

$$\begin{eqnarray} &&\frac{L_X}{L_*} > 3^{0.75}\left(\frac{L_{X,\odot }}{L_\odot }\right)\left(\frac{T_\odot }{T_*}\right) \nonumber \\ &&\hphantom{\frac{L_X}{L_*}}> 1.13\times 10^{-6}\left(\frac{T_\odot }{T_*}\right). \end{eqnarray} \tag{ A12 }$$

Equations (A11) and (A12) are subject to the condition that F_X < 8 × 10⁵ erg s⁻¹ cm⁻², which can be rewritten as

$$\begin{equation} \frac{L_X}{L_*}<1.26\times 10^{-5}\left(\frac{T_\odot }{T_*}\right)^4. \end{equation} \tag{ A13 }$$

For "saturated" stars with F_X > 8 × 10⁵ erg s⁻¹ cm⁻², stellar wind mass loss can still dominate the P–R drag, but the mass-loss rates cannot be estimated accurately from the X-ray activity.

Stellar wind drag and P–R drag are additive, and can be combined as done in Strubbe & Chiang (2006) and Fitzgerald et al. (2007):

$$\begin{eqnarray} \tau _{\rm drag} &=& 2\times 10^6\;\mbox{yr}\left(\frac{r}{50\,{\rm AU}}\right)^{2}\left(\frac{a}{\mu \mbox{m}}\right)\left(\frac{\rho _g}{\mbox{g cm}^{-3}}\right)\nonumber\\ &&\times\left(\frac{1}{P_{\rm CPR}}\right)\left(\frac{L_*}{L_\odot }\right)^{-1}, \end{eqnarray} \tag{ A14 }$$

where P_CPR is defined in Strubbe & Chiang (2006):

$$\begin{equation} P_{\rm CPR} \equiv Q_{\rm rad} + Q_{\rm SW}\frac{\dot{M}_* c^2}{L_*}, \end{equation} \tag{ A15 }$$

which we can re-express as

$$\begin{equation} P_{\rm CPR} =Q_{\rm rad}\left(1+\frac{\tau _{\rm PR}}{\tau _{\rm SW}}\right). \end{equation} \tag{ A16 }$$

Drag forces predict a disk with constant vertical optical depth (≡surface density). In a steady state, the mass must cross an annulus which is constant,

$$\begin{equation} M_{\rm {crossing}} = \mbox{constant} = 2\pi r\; \Sigma _r(r) \; v_r. \end{equation} \tag{ A17 }$$

The average radial inward drift velocity of a grain goes as v_r = r/τ_drag ∝ r⁻¹. Hence, Σ_r(r) must also be constant as a function of orbital radius. An infrared spectra consistent with a power-law follows directly from this.

While stellar wind drag and P–R drag forces can have similar magnitudes, the same is not true for the strength of the radial or "blowout" force unless $(\dot{M}_* / \dot{M}_\odot)(L_\odot /L_*)\sim 3000$ (Plavchan et al. 2005; Gustafson 1994). Such high mass-loss rates are not supported by recent theoretical arguments (Holzwarth & Jardine 2007). The additional factor of ∼1000 arises since the ratio of drag forces compared to the ratio of radial forces for P–R and stellar wind is enhanced by a factor of (c/v_sw), where v_sw is the stellar wind velocity. Strubbe & Chiang (2006) writes the grain blowout radius from the combined effects of radiation and stellar wind as

$$\begin{equation} a_b = \frac{3}{8\pi }\frac{L_*P_{\rm SWR}}{G M_* c \rho _g}, \end{equation} \tag{ A18 }$$

where

$$\begin{equation} P_{\rm SWR} \equiv Q_{\rm rad} + Q_{\rm SW}\frac{\dot{M}_* c \; v_{\rm sw}}{L_*}. \end{equation} \tag{ A19 }$$

We estimate the minimum mass of the parent bodies—asteroid and comet proto-planetary analogs—by considering how quickly the dust mass must be replenished to maintain a steady state as in Chen et al. (2005b). Grain–grain collision timescales can be shorter than both P–R and stellar wind drag timescales. The shortest timescale dictates which mechanism is responsible for removing dust, but that shortest timescale will never be longer than the P–R drag timescale for the orbital dust distances typically under consideration. Hence, we use the P–R drag timescale to estimate minimum total parent body mass, but note that the total parent body mass can be substantially larger if the grain–grain collision timescale or the stellar wind drag timescale are significantly shorter (Chen et al. 2005b; Plavchan et al. 2005; Minato et al. 2006). The minimum total parent body mass is given by

$$\begin{equation} M_{\rm PB} > 4 L_{\rm IR} t_*/c^2 \end{equation} \tag{ A20 }$$

as in Chen & Jura (2001), where t_* is the age of the star.

Finally, throughout our analysis, we assume $\dot{M}_*$ to be the average mass-loss rate. As noted in Augereau & Beust (2006), flaring activity can substantially enhance the stellar wind drag force relative to a quiescent state. For a star that flares ∼10% of the time with mass-loss rated enhanced by a factor of ∼100 during flared states with respect to quiescent activity, the average mass-loss rate can be ∼10 times greater than the quiescent mass-loss rate, and hence grain removal timescales can be ∼10 times shorter. The stellar activity cycle can be assumed to be much shorter than the ∼Myr drag-force removal timescale, and hence individual grain lifetimes will be insensitive, on average, to the flared state of a star. However, the methods in Wood et al. (2005) and Schmidt et al. (2007) to estimate mass-loss rates are sensitive to the flared state of a star, and a correction factor based on the activity level of the star should be applied to estimate the average mass-loss rate.

From Equations (A14) and (A6), we derive

$$\begin{eqnarray} \frac{\tau _{\rm drag}}{\tau _{\rm coll}} &=& \frac{1000}{\gamma }\left(\frac{r}{50\,{\rm AU}}\right)^{1/2+\alpha }\left(\frac{Q_{\rm coll}}{P_{\rm CPR}}\right)\left(\frac{M_d}{10^{-3}\,M_\oplus }\right)^{1}\nonumber\\ &&\times\left(\frac{L_*}{L_\odot }\right)^{-1}\left(\frac{M_*}{M_\odot }\right)^{1/2}. \end{eqnarray} \tag{ A21 }$$

We consider the dust mass estimate from the fractional infrared excess (Section 4.4; Chen et al. 2005b; Jura et al. 1995):

$$\begin{equation} M_d \ge \frac{16}{3}\pi \frac{L_{\rm IR}}{L_*}\rho _g D^2_{\rm {min}}\langle a\rangle. \end{equation} \tag{ A22 }$$

If we set r = D_min = D_ring, and if we assume that the dust is originating in a ring with relative width of 0.1, then Equation (A21) approximates the local dust mass in the ring. Furthermore, Equation (A14) represents the total time for a dust grain to spiral into the star. The average time for a dust grain to escape the ring by being dragged inward before colliding with another grain is further reduced by a factor of ∼ΔD_ring/(2D_ring) = 0.05. If we assume 〈a〉 = 1 μm, ρ_g = 2.5 g cm⁻³, and α = 0, we get

$$\begin{eqnarray} \frac{\tau _{\rm drag}}{\tau _{\rm coll}}(\mbox{ring}) &=& \frac{L_{\rm IR}/L_*}{9\times 10^{-6}} \left(\frac{D_{\rm ring}}{50\,{\rm AU}}\right)^{5/2}\left(\frac{Q_{\rm coll}}{P_{\rm CPR}}\right)\nonumber\\ &&\times\left(\frac{L_*}{L_\odot }\right)^{-1}\left(\frac{M_*}{M_\odot }\right)^{1/2} \end{eqnarray} \tag{ A23 }$$

For a solar-type star with a young Kuiper–Belt analog at 50 AU and $\dot{M}_{\rm sw}= \dot{M}_\odot$ such that Q_coll/P_CPR = 0.75, Equation (A22) predicts that debris disks are drag-force dominated (and collision-force dominated otherwise) when

$$\begin{equation} \frac{L_{\rm IR}}{L_*}<1.2\times 10^{-5}. \end{equation} \tag{ A24 }$$

Increased stellar winds increase the fractional infrared excess luminosity transition in Equation (A23) by a factor of ∼$\frac{3}{4}(1+\frac{1}{3}(\dot{M}_{\rm sw}/\dot{M}_\odot))$ for a solar-type star. If $\dot{M}_{\rm sw}=10\;\dot{M}_{\odot }$ as might be reasonable for young, low-mass stars, the transition fractional infrared excess luminosity is 4 × 10⁻⁵.

When discussing the role of different grain removal mechanisms in debris disks, the effect of planets on the dynamics cannot be neglected. Dust trapped in orbital resonances with planets are invoked to explain clumps and nonazimuthally symmetric disk structures seen in resolved disks such as Vega and Eps Eri (Wyatt 2003; Greaves et al. 2005; Krivov 2007; Krivov et al. 2007). Herein, we consider the dynamically cold model of Wyatt (2003), with planetesimal (and dust) eccentricities of <1%, to approximate when the presence of a planet is important for disk dynamical evolution relative to the processes already discussed.

Wyatt (2003) considers a migrating planet with migration rate $\dot{a}_p$ encountering planetesimals to estimate resonance capture probabilities. Planetesimals will be captured in resonances with a probability Pr given by the functional form from Wyatt (2003):

$$\begin{equation} \mbox{\rm PR} = \left[1+\left(\frac{\dot{a}_p}{\dot{a}_{0.5}}\right)^{\gamma }\right]^{-1}, \end{equation} \tag{ A25 }$$

where $\dot{a}_{0.5}$ is the 50% capture probability migration rate and γ dictates how fast the probability turns over as a function of $\dot{a}_p$. Wyatt (2003) determined that $\dot{a}_{0.5}$ takes the form

$$\begin{equation} \dot{a}_{0.5} = \frac{1}{X}\left(\frac{M_p}{M_\oplus }\frac{M_\odot }{M_*}\right)^{u}\sqrt{\frac{M_*}{M_\odot }\frac{\mbox{AU}}{D_{\rm pl}}}, \end{equation} \tag{ A26 }$$

depending on the orbital resonance ($\dot{a}_{0.5}$ in units of AU Myr⁻¹; D_pl is the planet orbital radius). For the 2:1 orbital resonance, Wyatt (2003) finds through computer simulations X = 5.8 and u = 1.4; for the 3:2 resonance, X = 0.37 and u = 1.4; for the 4:3 resonance X = 0.23 and u = 1.42; and higher order resonances have weaker capture probabilities (bigger X). If $\dot{a}_p < \dot{a}_{0.5}$, planetesimals will have a >50% of being caught in an orbital resonance and vice versa.

We relate the planet–planetesimal interaction analysis of Wyatt (2003) to planet–dust interaction as in Krivov et al. (2007). First, we replace the planetesimals with dust grains. Dust grains will orbit on slightly larger orbits than the Keplerian orbits of the planetesimals due to the effect of radiation pressure. As a result, resonant orbital semimajor axes will be slightly larger and grain-size dependent. The correction factor to a resonant semimajor axis is (1 − β)^1/3, where β is the ratio of the radiation pressure force to gravity and is dependent on grain properties.

Second, inward migrating dust under the action of a drag force is equivalent to an outward migrating planet. We can replace $\dot{a}_p$ with the dust inward drift velocity v_r when $|v_r|\gg |\dot{a}_p|$. In the example of Vega in Wyatt (2003), a planet migration rate of $\dot{a}_p=0.45$ AU Myr⁻¹ is derived to model the resolved Vega submillimeter excess. For comparison, from Section A.1.2, the dust migration rate in units of AU Myr⁻¹ from the combined effects of P–R and stellar wind drag goes as

$$\begin{eqnarray} \hspace{-1pc}v_r (\mbox{AU/Myr}) &=& \left(\frac{r}{{\rm AU}}\right)\left(\frac{\mbox{Myr}}{\tau _{\rm drag}}\right) = 25 \left[\left(\frac{r}{50\,{\rm AU}}\right)\left(\frac{a}{\mu \mbox{m}}\right)\right.\nonumber\\ \hspace{-1pc}&&\left.\times\left(\frac{\rho _g}{\mbox{g cm}^{-3}}\right)\left(\frac{1}{P_{\rm CPR}}\right)\left(\frac{L_*}{L_\odot }\right)^{-1}\right]^{-1}. \end{eqnarray} \tag{ A27 }$$

This is a factor of ∼50 larger than the inferred planet migration rate for Vega for the default values of grain size, density, orbital distance, etc. Hence, replacing the planet migration rate with the dust inward drift velocity is appropriate.

With $|v_r|\gg |\dot{a}_p|$ (and consequently τ_coll/τ_drag > 1), the disk would be drag-force dominated excepting for the presence of a planet. We expect the dust to spiral inward toward the sublimation radius unless it is trapped in an orbital resonance, or scattered from the system in the "resonance overlap region."

Dust will be trapped into an orbital resonance with >50% probability if

$$\begin{equation} v_r < \dot{a}_{0.5} = \frac{1}{X}\left(\frac{M_p}{M_\oplus }\frac{M_\odot }{M_*}\right)^{u}\sqrt{\frac{M_*}{D_{\rm pl}}}, \end{equation} \tag{ A28 }$$

which we can re-express as

$$\begin{eqnarray} &&\frac{M_p}{M_\oplus } > \left[177\;X\;P_{\rm CPR}\;\left(\frac{M_*}{M_\odot }\right)^{u-0.5}\left(\frac{L_*}{L_\odot }\right)\left(\frac{D_{p+q:q}}{50\,{\rm AU}}\right)^{-0.5} \right.\nonumber\\ &&\hphantom{\frac{M_p}{M_\oplus } >}\times\! \left.\left(\frac{a}{\mu \mbox{m}}\right)^{-1} \left(\frac{\rho_g}{\mbox{g cm}^{-3}}\right)^{-1} \right]^{1/u}. \end{eqnarray} \tag{ A29 }$$

If we set u = 1.4, we can rewrite this in terms of Jovian masses (M_J):

$$\begin{eqnarray} &&\frac{M_p}{M_J}> 0.13 \;X^{0.71}\;P_{\rm CPR}^{0.71}\;\left(\frac{M_*}{M_\odot }\right)^{0.64}\left(\frac{L_*}{L_\odot }\right)^{0.71}\left(\frac{D_{p+q:q}}{50\,{\rm AU}}\right)^{-0.64}\nonumber\\ &&\hphantom{\frac{M_p}{M_J}>}\times\left(\frac{a}{\mu \mbox{m}}\right)^{-0.71} \left(\frac{\rho _g}{\mbox{g cm}^{-3}}\right)^{-0.71}. \end{eqnarray} \tag{ A30 }$$

In Equations (A28) and (A29), D_{p + q:q} corresponds to the semimajor axis of the orbital resonance p+q:p, which can be related to the planet semimajor axis as in Krivov et al. (2007):

$$\begin{equation} D_{p+q:q} = D_{\rm pl}\left(1-\beta \right)^{1/3}\left(\frac{p+q}{p}\right)^{2/3}. \end{equation} \tag{ A31 }$$

From Equation (A29), we conclude that a proto-planet >0.05 M_J at 50 AU will trap a majority of 1 μm dust (2.5 g cm⁻³) around a solar-type star in the 3:2 orbital resonance ($\dot{M}_{\rm sw}/\dot{M}_{\odot }=1$ assumed). At 10 AU, the minimum planet mass is 0.14 M_J, and at 5 AU the minimum planet mass is 0.22 M_J. Since these planet masses are comparable to known extrasolar planet masses, we conclude that dust trapped in orbital resonances with Jovian planets should be a common phenomenon. In our own solar system, resonant trapping of a small fraction of inward migrating dust has been observed for the Earth (Dermott et al. 1994; Reach et al. 1995), and would be observable for the Jovian planets if we were not embedded in our own zodiacal cloud.

Krivov et al. (2007) goes into more detail about what happens to the dust once it is trapped into resonance. The resulting observability is determined by the survival time of grains in resonance relative to the collision and drag timescales. While in resonance, the dust eccentricities will be pumped up until the dust escapes resonance or collides with other dust. If the dust escapes resonance, it can be scattered inward and continue spiraling into the star, or be scattered out of the system, and this is again dependent on the planet mass. Additionally, the orbits of dust quickly become chaotic for planets of a few Jovian masses, providing an upper limit to the planet mass for observable resonant trapping in addition to the lower limit in Equation (A29) (Kuchner & Holman 2003). We note that the minimum planet mass required to capture dust is proportional to P^0.7_CPR. If the stellar wind mass-loss rate is substantially larger than solar, it affects the planet–dust interaction and disk dynamics overall. Stars with high mass-loss rates are less likely to have observable dust in resonance with planets of a similar mass compared to stars with small mass-loss rates.

Dust in nonresonant orbits can enter the "resonance overlap region" and be scattered out of the system on short timescales

$$\begin{equation} \left|\frac{r}{D_{\rm pl}}-1\right| < 1.3 \left(\frac{M_p}{M_*}\right)^{2/7}, \end{equation} \tag{ A32 }$$

where r and D_pl are the dust and planet semimajor axes, respectively, and M_p and M_* are the planet and stellar masses in the same units (see also Quillen 2006). Again, if the planet is massive enough, the planet can effectively scatter a majority of the inspiraling dust out of the system, creating a disk with an inner hole. This is the "disk with inner hole" model is widely used to explain infrared spectra with no short-wavelength "hot excess," but longer wavelength "cold excess." Such a disk would have a constant surface density truncated at R_IN = D_pl and R_OUT equal to the orbital radius of the colliding parent bodies producing the dust. Conversely, the presence of a warm excess where collisional timescales are long relative to drag timescales implies the lack of a planet of sufficient mass interior to where dust is being produced to halt the inward migration.

In summary, if τ_coll/τ_drag < 1, then the radial inward drift velocity of dust is effectively zero. Since dust grains undergo destructive collisions before they can drift into a planet, dust will be confined to where the parent bodies are located, such as in a ring or in resonance with a planet (if the planet is migrating). The presence of a planet can likely be inferred from the presence of clumpy disk structure. If τ_coll/τ_drag > 1, the presence of Jovian planets halts or substantially arrests the inward migration of dust under drag forces. If the planet is on the order of a Jovian mass, the dust will also likely be trapped in clumpy resonances that could dominate the fractional infrared excess. The presence of nonazimuthal disk structures for disks with L_IR/L_* < 10⁻⁴ can be used to infer the presence of Jovian planets without directly detecting the planets themselves. Conversely, a power-law infrared excess can be used to exclude the presence of Jovian planets at the orbital radii of the emitting dust.