What Sets the Radial Locations of Warm Debris Disks?

If the warm dust is produced in-situ from an exo-asteroid belt, it is expected to occur near the primordial snow line. Several mechanisms predict an enhancement of solid material and planetesimal formation at (or near) the snow line in a protoplanetary disk. Water vapor diffusing outward through the disk will condense at the snow line and increase the density of solid material there (Stevenson & Lunine 1988). Icy, roughly meter-size bodies from the outer disk will migrate inward due to gas drag and will sublimate at the snow line, creating an enhancement of vapor and solids (Cuzzi & Zahnle 2004). The increase in the dust-to-gas ratio at the snow line can create a region of lower turbulence in the disk. This leads to lower collisional velocities, promoting planetesimal growth, and creates a gas pressure maximum that traps inward migrating solids (Kretke & Lin 2007; Brauer et al. 2008). The resulting population of planetesimals may become the parent bodies in the debris disk, or this may trigger the formation of a giant planet that will stir the planetesimals around it and prevent them from coalescing into a planet. The continued stirring will cause planetesimal collisions that yield warm dust for an extended period of time. This is similar to how the gravitational influence of Jupiter stirs the asteroid belt in the solar system (e.g., Petit et al. 2001). In any case, the location of the resulting warm dust is linked to the primordial snow line.

The temperature in the midplane of a protoplanetary disk is set primarily by viscous heating. Min et al. (2011) give the following relation for the location of the primordial snow line:

$$\begin{eqnarray}&&{r}_{\mathrm{SL}}\propto {{M}_{\star }}^{1/3}{\dot{M}}^{4/9}{{\kappa }_{R}}^{2/9}{f}^{-2/9}{\alpha }^{-2/9}{{T}_{\mathrm{ice}}}^{-10/9},\end{eqnarray} \tag{ 1 }$$

where $\dot{M}$ is the mass accretion rate, κ_R is the Rosseland mean opacity, f is the gas-to-dust ratio, α is the turbulent mixing strength, and T_ice is the ice sublimation temperature. Of these parameters, only $\dot{M}$ is believed to vary significantly with stellar mass and thus is relevant for estimating the form of the r_SL–M_⋆ relation. The mass accretion rate has been found to vary with stellar mass as $\dot{M}\propto {M}_{\star }^{2}$ over a large range of stellar masses, including the masses of the stars in our sample (Calvet et al. 2004; Muzerolle et al. 2005; Natta et al. 2006). This implies that ${r}_{\mathrm{SL}}\propto {{M}_{\star }}^{1.2}$. As we will emphasize later, this relation is shallower than that for the current snow line. Other investigations into the location of the primordial snow line also predict the r_SL–M_⋆ relation to be significantly shallower than the current snow line relation (Kennedy & Kenyon 2008; Martin & Livio 2013a).

If, instead, the warm dust is transported inward from an outer reservoir during the present debris disk phase, it is expected to reside at the current snow line. There are two plausible mechanisms for the inward transport. The first mechanism is analogous to that described by Nesvorný et al. (2010), who found that most of the warm dust in the inner region of the solar system originates from the disruption of Jupiter family comets (JFCs). JFCs are dynamically controlled by Jupiter and have orbits with significantly lower eccentricities and smaller aphelia than Halley-type comets or long-period comets. Simulations show that JFCs likely originate in the Kuiper Belt and are dynamically passed inward by the giant planets (Levison & Duncan 1997), with some JFCs arriving on asteroid-like orbits inside that of Jupiter (Rickman et al. 2017). When JFC orbits cross the current snow line, they begin to sublimate and eventually disintegrate, releasing dust. Simulations of generic planetary systems also show that a chain of several planets is required to efficiently transport planetesimals inward from an outer reservoir (Bonsor et al. 2012, 2014). This is consistent with the idea that the region between the snow line and the cold belt is maintained by one or more planets (Su et al. 2013).

For the second mechanism, dust generated by collisions in the outer parent body belt flows inward due to Poynting–Robertson (P–R) drag and stellar wind drag. While Wyatt (2005) argued that most disks we can detect are collision-dominated rather than drag-dominated (that is, grains are destroyed by mutual collisions faster than they can move inward), Kennedy & Piette (2015) noted that some inward transport is inevitable unless planets are present interior to the cold belt to remove the inflowing dust. Since these grains originate in the outer part of the system, they may contain a mixture of icy and refractory material. When the grains reach the snow line, the ices sublimate, reducing the grain size and consequently increasing the ratio of the radiation force to the gravitational force on the grain (β). This halts the grain's inward motion and eventually causes it to be expelled outward. The net result is a pile-up of grains at the location of the current snow line (Kobayashi et al. 2008). However, numerical simulations (van Lieshout et al. 2014) indicate that the inward flow via this mechanism may be inadequate to maintain the amount of warm dust required for detectable infrared excesses with Spitzer, even with the snow line pile-up.

The location of the current snow line is determined by the incident stellar flux, so it scales as ${r}_{\mathrm{SL}}\propto {L}_{\star }^{1/2}$, where L_⋆ is the stellar luminosity. Combining this with ${L}_{\star }\propto {M}_{\star }^{4}$, the typical relation between stellar luminosity and mass yields ${r}_{\mathrm{SL}}\propto {M}_{\star }^{2}$. Importantly, the current snow line relation is steeper than the primordial snow line relation (index of 2.0 versus 1.2).

In this paper, we analyze the SEDs of a sample of debris disks with warm components and infer the stellocentric locations of the warm dust. The dust location derived solely from an SED is subject to many uncertainties and cannot be determined absolutely for any given system. Therefore we focus our attention on the relative behavior of dust location with stellar mass, while holding all other parameters (e.g., grain materials) constant—except for the minimum grain size, which is known to vary systematically with stellar properties. We examine the r_dust–M_⋆ trend and compare the slope with those predicted by the primordial and current snow lines, providing insight into which snow line sets the dust location. From this we can deduce the the origin of the warm dust components.

For our sample, we used the systems with a warm component found by Ballering et al. (2013), where "warm" was defined as warmer than 130 K. All of these systems have data available from the Multiband Imaging Photometer for Spitzer (MIPS; Rieke et al. 2004) at 24 and 70 μm and from the Spitzer Infrared Spectrograph (IRS; Houck et al. 2004).

Throughout the analysis, we separated the systems with only a warm component from those that also possess a detected cold component. The systems without a detected cold component should provide less ambiguous results, since these warm components could not arise from particles moving inward from cold belts via P–R drag, although they could still arise from comets originating in cold components that are below the current detection limit (Wyatt et al. 2007). In addition, modeling systems with a single dust component is less complicated and the results have less uncertainty.

Ballering et al. (2014) discovered silicate emission features in the IRS spectra of 22 of these systems. These features revealed the presence of exozodiacal dust, which is believed to reside at a different location than the typical warm component. Ballering et al. (2014) found that, besides the exozodiacal dust, an additional colder component was also required to fit the full IRS spectra of these sources. Whether this remaining excess consists of one or multiple components is difficult to determine. Thus, to ensure a pure sample of warm components, we excluded these 22 targets. If, however, the dust components giving rise to these features are a natural extension of standard warm components, then excluding these targets may introduce a bias to our sample.

We also excluded HIP 32435 (HD 53842) because the IRS data may have been contaminated by background sources (Donaldson et al. 2012). We removed additional targets in the course of our fitting procedure, as described in Section 2.6. The remaining 83 targets used for our analysis are listed in Table 1.

Table 1.  Target Properties

HIP	Spectral	V	J	H	K	T⋆	L⋆	M⋆	aBOS	d	IRS	xLL1	xSL1	xSL2
Identifier	Type	(mag)	(mag)	(mag)	(mag)	(K)	(L⊙)	M⊙	(μm)	(pc)	AOR
Single-component Systems
HIP 9902	F8V	7.5	6.53	6.3	6.2	6294	1.62	1.12	1	44.17	13621504	1.00	1.07	1.05
HIP 14684	G0V	8.49	7.16	6.79	6.7	5528	0.52	0.88	0.5	37.41	5340672	1.00	1.05	...
HIP 17549	A0	6.88	6.88	6.93	6.92	9820	34.15	2.26	7.5	140.84	14148096	0.94	1.19	1.08
HIP 18217	A5m	5.79	5.45	5.41	5.37	7667	9.73	1.69	3.1	50.51	14160640	1.00	1.02	1.02
HIP 18481	A2Vn	6.08	5.97	5.97	5.92	8801	15.12	1.88	4.2	70.22	14139648	0.99	1.00	0.99
HIP 23871	A5V	5.28	5.11	5.08	5.05	8308	20.32	2.01	5.2	58.11	14139904	1.00	1.10	1.06
HIP 28103a	F2V	3.72	3.06	2.98	2.99	7207	7.88	1.60	2.7	14.88	15998720	1.00	1.04	1.04
HIP 41967	G5V	8.26	7.05	6.77	6.7	5875	0.89	0.99	0.7	45.07	6596864	0.94	1.00	...
HIP 49593	A7V	4.49	4.27	4.05	4	8000	9.98	1.70	3.2	28.24	14141184	1.00	1.06	1.00
HIP 54879	A2V	3.32	3.12	3.19	3.08	8080	92.43	2.92	15.5	50.61	16177152	1.00	1.00	1.00
HIP 57524	G4Vp	9.07	7.89	7.57	7.51	5887	1.75	1.14	1	91.66	21799936	1.00	1.06	1.04
HIP 57950	F2IV/V	8.25	7.47	7.31	7.28	6730	3.89	1.37	1.7	98.14	21800448	0.95	1.12	1.10
HIP 60348	F5V	8.78	7.95	7.76	7.67	6515	2.15	1.19	1.2	93.72	21801728	0.97	1.12	1.11
HIP 62134	F2V	8.63	7.88	7.73	7.71	6658	3.79	1.36	1.7	115.61	21802240	1.00	1.12	1.11
HIP 63836	F7	9	8.09	7.89	7.87	6501	2.41	1.23	1.3	107.41	21803008	1.00	1.13	1.11
HIP 64053	B8V	5.7	5.82	5.75	5.73	9800	51.38	2.52	10.1	100.1	22803712	1.00	1.03	1.02
HIP 64877	F5V	8.47	7.62	7.41	7.41	6448	5.21	1.47	2.1	125	22806784	1.00	1.07	1.07
HIP 67068	F3V	8.46	7.69	7.52	7.47	6787	2.82	1.27	1.4	91.58	26361856	1.00	0.99	0.96
HIP 67230	F5V	8.03	7.14	6.93	6.89	6476	8.73	1.65	2.9	131.75	22802688	1.00	1.08	1.08
HIP 70455	B8V	6.96	7.03	7.07	7.08	10460	48.73	2.49	9.7	165.02	14148608	1.00	1.02	1.00
HIP 74824	A3Va	4.07	3.93	3.81	3.88	7694	17.73	1.95	4.8	30.55	14141952	1.00	1.00	1.00
HIP 77315	A0V	6.92	6.69	6.72	6.67	8880	34.38	2.27	7.6	147.28	22804992	1.00	1.05	1.05
HIP 79710	F0V	8.4	7.77	7.65	7.61	7000	5.45	1.48	2.1	127.39	26314752	1.00	0.98	0.98
HIP 79881	A0V	4.79	4.86	4.94	4.74	9210	17.93	1.95	4.7	41.29	21809408	1.00	1.09	1.15
HIP 89770	F5	6.68	5.85	5.7	5.62	6599	4.87	1.44	2	53.22	15016960	1.00	1.00	0.99
HIP 95560	A0V	5.59	5.58	5.63	5.61	8673	25.02	2.11	6.1	72.89	14143488	0.98	1.04	1.04
HIP 106783	A2V	6.18	6.13	6.13	6.11	9158	22.37	2.05	5.6	87.64	21812992	0.99	1.10	1.11
HIP 112542	B9V	5.68	5.75	5.79	5.73	10016	50.69	2.51	10	97.37	14144000	1.00	1.05	1.05
HIP 115738a	A0p...	4.93	5.32	4.98	4.9	9836	37.72	2.34	8.1	47.06	14144256	0.99	1.11	1.05
Two-component Systems
HIP 345	A0V	6.39	6.28	6.25	6.26	8936	37.34	2.31	8.1	124.84	12720128	1.00	1.01	1.01
HIP 682	G2V	7.59	6.42	6.15	6.12	5962	1.23	1.05	0.8	39.08	5268736	1.00	1.07	...
HIP 1473	A2V	4.52	4.34	4.42	4.46	9005	22.94	2.06	5.7	41.32	14160384	1.00	1.03	1.03
HIP 1481	F9V	7.46	6.46	6.25	6.15	6273	1.50	1.10	0.9	41.55	21788160	0.97	1.07	1.06
HIP 2472a	A0V	4.77	4.67	4.77	4.7	9459	29.59	2.22	6.7	52.97	21788672	1.00	1.09	1.11
HIP 2710	F2	6.91	6.04	5.85	5.75	6400	2.28	1.21	1.2	40.92	25673472	1.02	...	...
HIP 7805	F2IV/V	7.61	6.84	6.69	6.63	6796	3.27	1.32	1.5	67.25	21789952	1.00	1.13	1.11
HIP 7978	F9V	5.52	4.79	4.4	4.34	6000	1.58	1.11	1	17.43	16029952	0.99	1.05	1.06
HIP 8241	A1V	5.04	4.99	5.03	4.96	9478	32.98	2.25	7.4	62.03	14139392	1.00	1.06	1.04
HIP 11360	F2	6.8	6.03	5.86	5.82	6500	3.18	1.31	1.5	45.23	10885632	1.00	1.18	1.08
HIP 11477	A2V	5.13	5.12	5.03	4.94	8900	16.00	1.90	4.4	46.6	21790976	1.00	1.11	1.13
HIP 11847	F2V	7.49	6.7	6.61	6.55	6922	3.38	1.33	1.5	63.49	10886144	0.99	1.07	1.08
HIP 13141a	A2V	5.26	5.14	5.16	4.97	8390	17.48	1.95	4.7	50.45	21791744	1.00	1.11	1.11
HIP 18859a	F5V	5.38	4.71	4.34	4.18	6315	2.59	1.24	1.3	18.83	5308416	1.00	1.01	...
HIP 20901	A7V	5.01	4.79	4.66	4.53	7592	17.44	1.94	4.7	48.85	14146816	1.00	1.03	1.03
HIP 22192	A3IV	6.18	5.8	5.73	5.71	7901	8.41	1.64	2.8	56.18	14147584	0.99	1.12	1.06
HIP 22226	F3V	7.86	7.1	6.95	6.89	6828	3.75	1.36	1.7	80.26	10887424	1.02	1.16	1.12
HIP 23451	A0	8.14	7.69	7.62	7.59	7957	6.19	1.53	2.3	112.36	15903744	1.00	1.05	1.12
HIP 26453	F3V	7.25	6.47	6.29	6.28	6758	3.28	1.32	1.5	56.79	5306880	1.00	1.08	...
HIP 26796	A0V	6.34	6.41	6.45	6.43	10130	69.94	2.72	12.6	153.61	12713728	0.99	1.10	1.07
HIP 34276	A0V	6.52	6.49	6.48	6.48	9400	23.76	2.08	5.8	102.35	21795584	1.00	1.07	1.07
HIP 36948	G8Vk+?	8.22	6.91	6.58	6.46	5598	0.61	0.91	0.5	35.35	5267200	1.00	1.02	...
HIP 41152	A3V	5.54	5.25	5.29	5.25	8077	12.05	1.78	3.6	50.43	14140416	1.00	1.10	1.06
HIP 41373	A0V	6.05	5.94	5.91	5.89	8839	15.39	1.88	4.3	69.44	14140672	1.00	1.12	1.08
HIP 45167	A0V	6.14	6.14	6.16	6.12	9152	29.56	2.19	6.8	99.3	12710400	1.00	1.04	1.03
HIP 47135	G2V	8.59	7.47	7.24	7.16	6050	1.46	1.09	0.9	67.98	25677056	1.00	...	...
HIP 48423	G5	7.73	6.47	6.14	6.09	5199	0.80	0.97	0.6	32.8	5399808	1.00	1.04	...
HIP 55485	A7Vn	6.44	6.08	6.02	5.99	8000	13.83	1.84	4	80.84	14141440	1.00	1.08	1.03
HIP 58720	B9V	5.88	6.01	6.08	6.08	10000	55.31	2.57	10.6	105.71	22799360	0.99	1.08	1.07
HIP 60074a	G2V	7.07	5.87	5.61	5.54	5809	1.11	1.04	0.8	27.46	5312256	1.00	0.97	...
HIP 61684	A9V	8.09	7.41	7.27	7.2	7000	5.84	1.51	2.2	111.86	22801152	1.00	1.06	1.04
HIP 62657	F5/F6V	8.87	8	7.83	7.72	6417	2.65	1.25	1.3	108.58	13621248	1.01	1.06	1.05
HIP 73145	A2IV	7.86	7.6	7.56	7.52	8281	8.75	1.65	2.9	122.7	22803968	0.98	1.14	1.15
HIP 74499	F4V	8.74	7.88	7.73	7.65	6545	2.07	1.18	1.1	89.93	21806336	0.98	1.08	1.06
HIP 75077	A1V	7.16	6.98	7.04	6.96	8599	18.99	1.98	5	131.58	14151168	1.00	0.97	0.97
HIP 75210	B8/B9V	6.64	6.75	6.83	6.76	10642	45.95	2.45	9.3	136.24	14147840	1.00	1.05	1.04
HIP 76736	A1V	6.42	6.3	6.34	6.27	8769	13.56	1.83	3.9	78.49	14142208	1.00	1.10	1.12
HIP 77432	F5V	8.97	8.11	7.94	7.87	6594	1.98	1.17	1.1	96.34	21808128	1.00	1.12	1.10
HIP 77464	A5IV	5.53	5.33	5.27	5.26	8248	14.07	1.84	4	54.02	14142720	1.00	1.03	1.03
HIP 78043	F3V	8.95	8.15	7.97	7.94	6639	4.31	1.40	1.8	144.3	26313728	1.00	0.99	0.99
HIP 78045	A3V	5.75	5.65	5.66	5.57	8777	17.84	1.95	4.7	66.01	14142464	1.00	1.05	1.05
HIP 79516	F5V	8.9	8.02	7.85	7.79	6495	4.04	1.38	1.8	133.69	15554560	1.00	1.05	1.04
HIP 79742	F5	9.16	8.28	8.06	8.06	6516	3.80	1.36	1.7	146.2	15555328	1.00	1.00	0.98
HIP 83187	A5IV-V	5.65	5.32	5.26	5.19	7800	11.42	1.76	3.5	51.81	14160896	1.00	1.03	1.06
HIP 85537	A8V	5.42	4.81	4.88	4.8	7201	18.72	1.97	5	59.63	27224064	1.00	0.98	1.00
HIP 85922	A5V	5.62	5.25	5.25	5.14	7800	10.23	1.71	3.2	48.1	14142976	1.00	1.06	1.03
HIP 87108	A0V	3.75	3.59	3.66	3.62	8517	25.39	2.11	6.1	31.52	4931328	1.00	1.01	1.02
HIP 94114	A2Va	4.1	4.09	3.92	4.05	8400	26.47	2.13	6.3	38.43	14145536	1.00	1.05	1.01
HIP 95270	F5/F6V	7.04	6.2	5.98	5.91	6502	3.34	1.32	1.5	51.81	3564032	1.00	1.05	1.07
HIP 101612a	F0V	4.76	4.28	4.02	4.04	7233	8.10	1.61	2.8	27.79	21812224	1.00	1.10	1.11
HIP 101800	A2V	5.43	5.41	5.37	5.3	9131	19.67	1.99	5.1	57.94	14161408	0.98	1.08	1.08
HIP 106741	F4IV	7.17	6.38	6.25	6.18	6786	2.92	1.28	1.4	51.81	15022592	1.00	1.10	...
HIP 114189	A5V	5.95	5.38	5.28	5.24	7033	4.82	1.44	2	39.4	28889856	1.00	1.00	1.00
HIP 117452a	A0V	4.58	4.8	4.64	4.53	9673	33.98	2.29	7.4	42.14	14161664	1.00	1.02	1.02

Note.

^aThese targets not listed in McDonald et al. (2012); we inferred their stellar properties from their $V-K$ color.

A machine-readable version of the table is available.

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The stellar temperature (T_⋆), luminosity, and distance from Earth (d) of most of our targets were taken from McDonald et al. (2012), who derived T_⋆ and L_⋆ by fitting the visible and near-IR photometry of these systems with stellar SED models. We then obtained the stellar mass (M_⋆) from L_⋆ using the (broken) power-law relation by Eker et al. (2015). McDonald et al. (2012) assumed a 10% uncertainty on the photometry they used to derive L_⋆, so we also assumed a 10% uncertainty on L_⋆. Combining this with the intrinsic 25%–38% scatter in the L_⋆ values Eker et al. (2015) used to derive their relations yields a ∼6%–10% uncertainty on our M_⋆ values. For the targets not listed in McDonald et al. (2012; denoted with an "a" after the target name in Table 1), we inferred their stellar properties from their $V-K$ color using the tabulated values maintained online³ by E. Mamajek as an expanded and updated version of Table 5 in Pecaut and Mamajek (2013).

We required a model spectrum of the stellar photosphere for each of our targets, both for modeling the photospheric contribution to the observed SED and for calculating the temperature of dust grains when generating model spectra of the dust emission. We used an ATLAS9 (Castelli & Kurucz 2004) photosphere model with log g = 4.0, solar metallicity, and T_⋆ closest to that for each target (at most a difference of 125 K). These photosphere model spectra were modeled only out to 160 μm, so for completeness we extended them to 10,000 μm by extrapolating with a Rayleigh–Jeans power-law. We normalized the integrated spectra to L_⋆ for each target. (Although during the fitting process we allowed the amplitude of the model photosphere to vary by a small amount to improve agreement with the photometry; see Section 2.6.)

While many infrared excesses have been identified from photometric measurements alone (Rieke et al. 2005; Su et al. 2006; Wyatt 2008; Matthews et al. 2014b; Sierchio et al. 2014), accurately measuring the temperature/location of the emitting dust requires the wide spectral coverage offered by IRS.

We obtained low-resolution IRS spectra for our targets from the LR7 release of The Combined Atlas of Sources with Spitzer IRS Spectra⁴ (CASSIS; Lebouteiller et al. 2011). Both long–low orders (LL1: 19.5–38.0 μm; LL2: 14.0–21.3 μm) were available for all targets, and one or both of the short-low orders (SL1: 7.4–14.5 μm; SL2: 5.2–7.7 μm) were also available for most of the targets. The IRS Astronomical Observation Requests numbers (AORs) for our targets are given in Table 1.

We removed outlying points more than 3σ away from a third-degree polynomial fit to the measurements in each spectral order. To remove offsets between orders, we multiplied the LL1, SL1, and SL2 flux density values by correction factors (determined by eye), to align them with the LL2 order and to each other. The choice to pin the other orders to LL2 was arbitrary but had no effect on the results because, as described in Section 2.6, the amplitude of the whole IRS spectrum was varied as part of the fitting process. These correction factors (designated x_LL1, x_SL1, and x_SL2) are listed in Table 1. During this process we opted to remove HIP 79631 from our sample because the offsets between the orders were much greater than for any other target, suggesting the data may be unreliable.

In addition to the IRS data, we included in our SEDs photometry from MIPS at 24 and 70 μm plus additional photometry at wavelengths ≥70 μm from the literature. These data are listed in Table 2. Upper limits are at the 3σ level.

Careful restriction of the model characteristics was necessary to avoid degeneracies that would undermine our ability to determine the trend of warm dust location with stellar mass. We assumed that the dust lies in a circular ring. To test the sensitivity of our conclusions to the particular ring geometry we used two different models: one with a constant ring width (independent of radius) and a second with a ring width that was a constant fraction of the ring radius. We discuss the first set of models here and the second set in Section 4.1, where we explore the robustness of our results. For these models, we assumed that the ring has r_out = r_in + 2 au (r is the stellocentric distance). We modeled the surface number density of grains as Σ(r) ∝ r^−p with p = 1, but varying this parameter within reasonable bounds had virtually no effect on our results. We assumed a power-law grain size distribution (n(a) ∝ a^−q, where a is the grain radius) with size index q = 3.65 (Gáspár et al. 2012), minimum grain size a_min = a_BOS (a_BOS is the blowout size), and maximum grain size a_max = 1000 μm. Grains larger than a_max contribute negligibly to the overall emission. We assumed a grain composition of 60% astronomical silicates and 40% organic refractory material by volume (Ballering et al. 2016). To test how robust our conclusions are to grain composition, we report on similar models with grains composed entirely of astronomical silicates in Section 4.1. The only free parameters in our dust belt models were r_in and the total mass of dust.

Fixing the minimum grain size to the predicted blowout size was an important step in the modeling because a_BOS varies systematically with stellar type and a_min can have a substantial effect on the derived dust location (a smaller value of a_min leads to a larger inferred r_in). That is, we do not simply link the dust temperature and location using the blackbody temperature. The choice to use a_min = a_BOS is justified theoretically and also empirically. Detailed fits to debris disk spectra exhibiting silicate emission features were able to constrain both the dust location and minimum grain size; these fits showed the minimum spherical grain size to be consistent with the blowout size (Ballering et al. 2014). Furthermore, Booth et al. (2013) compared the sizes of cold debris disks, as measured from their resolved Herschel images and as derived from fitting blackbody functions to their SEDs, and found that the differences could be largely explained by models assuming a_min = a_BOS, although in a similar analysis Pawellek et al. (2014) found a_min values somewhat larger than a_BOS.

The blowout size is the largest grain size for which β > 0.5, where β is the ratio of the radiation force to the gravitational force on a grain. For spherical grains, β is given by

$$\begin{eqnarray}&&\beta =\displaystyle \frac{3{L}_{\star }}{16\pi {{GM}}_{\star }{ac}\rho }\displaystyle \frac{{\int }_{0}^{\infty }{Q}_{\mathrm{pr}}(\lambda ,a){F}_{\lambda \star }(\lambda )\,d\lambda }{{\int }_{0}^{\infty }{F}_{\lambda \star }(\lambda )\,d\lambda },\end{eqnarray} \tag{ 2 }$$

where F_λ⋆(λ) is the stellar flux density, Q_pr(λ, a) is the radiation pressure efficiency on the grain, G is the gravitational constant, c is the speed of light, and ρ = 2.34 g cm⁻³ is the grain density (2.7 g cm⁻³ for the astronomical silicates and 1.8 g cm⁻³ for the organic refractory material). We computed Q_pr(λ, a) and the absorption efficiency, Q_abs(λ, a) (needed to model the dust emission as discussed later) from the optical constants using the Mie theory code miex (Wolf & Voshchinnikov 2004). The optical constants of this grain composition mixture are given in Table 3 of Ballering et al. (2016). a_BOS for each target is given in Table 1.

To compute the model dust emission, we needed the temperature of the grains (T_dust) as a function of their radial location and size. We calculated this by computing

$$\begin{eqnarray}&&r({T}_{\mathrm{dust}},a)=\displaystyle \frac{1}{4\pi }\sqrt{\displaystyle \frac{\int {Q}_{\mathrm{abs}}(\lambda ,a){L}_{\lambda \star }(\lambda )\,d\lambda }{\int {Q}_{\mathrm{abs}}(\lambda ,a){B}_{\lambda }(\lambda ,{T}_{\mathrm{dust}})\,d\lambda }},\end{eqnarray} \tag{ 3 }$$

and then inverting it to solve for T_dust(r, a). L_λ⋆(λ) is the stellar spectral luminosity, and B_λ(λ, T_dust) is the Planck function. Equation (3) is derived from balancing the heating and cooling power on the grain. Finally, we calculated the emission spectrum from each grain,

$$\begin{eqnarray}&&{F}_{\nu }(\lambda ,r,a)={\left(\displaystyle \frac{a}{d}\right)}^{2}{Q}_{\mathrm{abs}}(\lambda ,a)\pi {B}_{\nu }(\lambda ,{T}_{\mathrm{dust}}),\end{eqnarray} \tag{ 4 }$$

and combined these spectra into a single spectrum weighted by the spatial distribution and grain size distribution of the model.

We fit the observed SED of each target, including photometric points at V, J, H, and K, the IRS spectrum, and the additional photometry listed in Table 2. The stellar photosphere accounts for virtually all of the observed flux density in the visible and near-IR, but it also contributes significantly at longer wavelengths. We modeled the photospheric contribution to the SED as described in Section 2.1, and we allowed small adjustments in the photospheric level to optimize the fits. To fit the excess emission from the debris disk, we used three different models: (1) a single modified blackbody (described later), which we used to double-check for systems that could be fit best by a single cold component; (2) a single warm dust belt, as described in Section 2.5, with the location of the dust (r_warm = r_in) and the mass of dust in the belt (M_warm) as free parameters; and (3) a warm dust belt plus a modified blackbody, with the blackbody accounting for a possible cold component. We did not fit the cold components with our dust belt model because their locations were not of interest for our analysis.

For the modified blackbody, we followed the formulation used by Kennedy & Wyatt (2014):

$$\begin{eqnarray}&&{F}_{\nu }(\lambda )={c}_{\mathrm{BB}}{B}_{\nu }(\lambda ,{T}_{\mathrm{cold}})X{(\lambda )}^{-1},\end{eqnarray} \tag{ 5 }$$

where c_BB is a constant (amplitude), T_cold is the temperature of the cold component, and

$$\begin{eqnarray}X(\lambda )=\left\{\begin{array}{ll}1 & \lambda \lt {\lambda }_{0}\\ {(\lambda /{\lambda }_{0})}^{\tilde{\beta }} & \lambda \gt {\lambda }_{0}\end{array}\right..\end{eqnarray} \tag{ 6 }$$

The modification to the blackbody, X(λ), models the steeper than Rayleigh–Jeans fall off at long wavelengths due to grains not emitting efficiently at wavelengths longer than their size. The free parameters for the modified blackbody were c_BB, T_cold, λ₀, and $\tilde{\beta }$. In the fitting we required 50 μm < λ₀ < 500 μm, $0\lt \tilde{\beta }\lt 2$, and T_cold < 130 K when part of a two-component fit.

In the fitting process, we also allowed for a small amplitude adjustment to the IRS data (c_IRS), which we allowed to take values between 0.8 and 1.2. This effectively corrected any systematic calibration error. This procedure yielded good results with c_IRS constrained at two points: first, the short end of the IRS data needed to match the photosphere model, which in turn had to match the visible/near-IR photometry; second, the IRS data needed to match the MIPS photometry point at 24 μm in order for the best fit model to pass through both the MIPS and IRS data at this wavelength.

In practice, we performed a grid search over r_in (in steps of 0.1 au) and c_IRS (in steps of 0.01). At each point in the grid we then found best values for the rest of the free parameters with a Levenberg–Marquardt algorithm (the Matlab function lsqcurvefit). The best fit was the model that minimized the standard χ² metric. When calculating χ², we enhanced the weights of the photometry points at ≥70 μm by a factor of 25 to balance their influence on the fit against the large number of points in the IRS spectra. Without this extra weighting, the behavior of the model in the far-IR/sub-mm often reflected an extrapolation from the longest wavelength IRS points, rather than fitting to the data in this wavelength regime. Upper limit photometry measurements were not included in the χ² calculation, but we inspected the best-fitting models to ensure they were consistent with these measurements. Parameters λ₀ and $\tilde{\beta }$ were often not well-constrained by the fitting, except for the targets with accurately measured far-IR/sub-mm photometry.

We inspected the results of our single blackbody fits to ensure that we only included genuine warm components in our sample. Targets that were fit well by a single blackbody with temperature <130 K were discarded from our sample. These included HIPs 544, 2072, 9141, 16852, 17764, 24947, 46843, 51194, 59072, 59960, 61960, 65728, 66065, 90936, and 107649. Some of these targets had single warm components with temperatures just above the 130 K cutoff according to Ballering et al. (2013), but with the additional far-IR photometry and the new fitting procedure used here, they now fell below this cutoff. For others, Ballering et al. (2013) had found two components with the warm component being relatively weak, but here we found that the warm component was no longer necessary to fit the IRS data.

We also excluded HIP 6276 because it has an extremely weak warm excess (and no evidence for a cold component) from which we could not place any meaningful constraints on the dust location. Finally, we excluded HIP 53954 and HIP 65109, because any model fit to the IRS data significantly over-predicted the far-IR photometry. A similar steep decline of the disk flux in the far-IR has been noted in a few other disks (Ertel et al. 2012), requiring a very unusual distribution of grain sizes to model.

We examined the fits of the remaining targets to determine which were best fit with a single warm belt and which required a cold component as well. In many cases, the requirement for a cold component came from the far-IR photometry, with the IRS data fitting well with a single warm belt. We included a cold component when the warm-only model under-predicted the far-IR data by more than 2σ (with the offset from multiple far-IR points combined in quadrature) and the addition of a cold component improved the fit. For most systems, the designations agreed with those of Ballering et al. (2013). Five systems that previously had been fit with a single warm component now were fit with two components (HIPs 1473, 1481, 77432, 78045, and 85922), and two systems that previously had been fit with two components were now fit best with a single warm belt (HIP 63836 and HIP 112542). The best fit cold component of HIP 117452 had T_cold = 130 K (the upper bound allowed by the fit), suggesting this system has an unusually warm cold component. However, the model fit the data well, so a significantly higher value of T_cold is likely not required. An unusually warm cold component does not, however, impact the need for a separate warm belt, as we found that a single component could not fit all the data.

From our χ² metric, we found that the median statistical uncertainties on r_in were 5% and 13% for the single-component and two-component systems, respectively. We calculated the luminosities of the best fit model components by integrating under the model spectra. We then derived the fractional luminosity of each component: f_warm = L_warm/L_⋆ and f_cold = L_cold/L_⋆. The results of the fitting are given in Tables 3 and 4. The best fit model SED for each target is shown in Figures 1 and 2. Our final sample had 29 systems with single warm components and 54 systems with two components.

Table 4.  Two-component Fit Results

HIP	rwarm	Mwarm	fwarm	Tcold	fcold	λ0	$\tilde{\beta }$	cIRS
Identifier	(au)	(×10−5 M⊕)	(×10−5)	(K)	(×10−5)
HIP 345	8.00	3.07	3.33	56.17	6.09	100.00	1.00	0.96
HIP 682	5.40	0.59	6.18	51.03	35.47	160.00	0.55	0.92
HIP 1473	6.00	0.44	0.99	101.53	0.56	237.43	0.11	0.91
HIP 1481	1.10	0.04	4.91	109.63	4.14	100.00	1.00	0.93
HIP 2472	16.20	3.09	1.02	60.06	0.84	100.00	1.00	0.91
HIP 2710	4.30	0.16	1.84	51.24	11.48	100.00	1.00	0.93
HIP 7805	5.40	0.95	6.42	57.67	29.95	100.00	1.00	0.92
HIP 7978	36.00	16.09	4.23	46.37	24.06	70.00	0.56	0.98
HIP 8241	6.80	1.37	2.10	57.79	7.85	154.25	1.20	0.94
HIP 11360	7.10	1.28	5.34	53.79	45.39	61.15	0.61	0.99
HIP 11477	7.30	0.76	1.44	80.76	3.15	100.00	1.00	0.89
HIP 11847	22.60	160.33	77.27	67.11	120.06	71.42	0.80	0.91
HIP 13141	8.50	0.66	0.90	54.75	4.68	136.05	0.93	0.91
HIP 18859	1.70	0.06	2.76	71.68	7.92	196.02	0.76	0.96
HIP 20901	11.30	2.53	2.08	62.27	3.71	100.00	1.00	0.91
HIP 22192	6.30	0.40	1.32	64.70	3.63	71.42	1.12	0.94
HIP 22226	26.50	37.44	12.21	58.26	74.10	83.52	0.82	0.88
HIP 23451	3.20	5.29	62.92	80.41	482.70	87.34	0.47	0.96
HIP 26453	5.80	1.40	8.31	77.43	24.96	100.00	1.00	0.91
HIP 26796	6.70	3.00	3.39	85.61	3.55	100.00	1.00	0.92
HIP 34276	12.00	1.78	1.13	61.12	25.38	71.42	0.16	1.01
HIP 36948	36.10	67.92	28.64	49.64	203.69	50.00	0.41	0.93
HIP 41152	9.30	1.89	2.63	57.42	6.61	71.42	0.41	0.91
HIP 41373	3.70	0.50	3.05	73.92	8.80	100.77	0.97	0.89
HIP 45167	4.00	1.16	4.69	70.44	4.96	100.00	1.00	0.94
HIP 47135	17.50	3.81	4.32	45.97	16.51	100.00	1.00	0.93
HIP 48423	3.20	0.18	5.68	73.49	7.84	100.00	1.00	0.96
HIP 55485	3.70	0.65	4.23	75.57	2.82	100.00	1.00	0.94
HIP 58720	4.50	2.87	7.19	112.95	7.08	100.00	1.00	0.91
HIP 60074	8.70	1.05	4.74	48.28	93.20	349.97	0.71	1.02
HIP 61684	5.90	4.16	18.42	86.22	34.30	100.00	1.00	0.91
HIP 62657	20.70	125.70	79.60	56.17	135.68	166.66	0.69	0.86
HIP 73145	6.10	22.66	77.36	71.83	240.75	194.79	0.73	1.00
HIP 74499	5.80	2.27	16.80	66.50	83.11	100.00	1.00	0.92
HIP 75077	6.70	1.00	2.01	50.94	4.88	100.00	1.00	1.01
HIP 75210	3.20	1.60	7.62	120.77	3.52	100.00	1.00	0.91
HIP 76736	7.30	4.41	8.96	59.72	41.58	100.00	1.00	0.98
HIP 77432	0.50	0.03	8.68	101.45	13.05	100.00	1.00	0.91
HIP 77464	6.70	0.71	1.67	60.46	7.54	100.00	1.00	0.97
HIP 78043	6.00	2.78	13.70	60.41	62.21	114.20	0.70	1.02
HIP 78045	2.90	0.26	2.25	91.42	2.03	100.00	1.00	0.96
HIP 79516	24.90	295.37	104.69	57.92	205.73	50.53	0.48	0.91
HIP 79742	14.30	37.63	39.87	69.64	182.99	62.45	0.42	0.96
HIP 83187	17.30	5.02	2.25	53.10	5.01	100.00	1.00	0.95
HIP 85537	22.80	10.67	2.25	44.59	5.73	93.26	0.98	0.95
HIP 85922	1.50	0.14	4.11	99.92	1.77	100.00	1.00	0.93
HIP 87108	12.90	5.04	2.72	64.53	7.75	127.18	1.01	0.98
HIP 94114	5.70	0.91	2.12	77.84	0.41	100.00	1.00	0.96
HIP 95270	33.80	324.22	72.14	60.43	215.48	55.16	0.50	1.00
HIP 101612	17.10	2.57	1.38	43.90	7.87	71.42	0.96	0.89
HIP 101800	12.90	3.36	2.03	61.59	1.58	100.00	1.00	0.92
HIP 106741	8.00	0.59	2.09	52.27	33.65	152.58	0.62	0.96
HIP 114189	3.90	0.46	4.36	44.51	29.16	200.82	1.18	1.09
HIP 117452	3.30	0.32	1.68	130.00	0.88	100.00	1.00	0.94

A machine-readable version of the table is available.

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View extended figure (3 panels)

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View extended figure (5 panels)

We first considered the systems with a single warm component. Figure 3 plots r_warm versus M_⋆ and shows clear evidence for a positive trend. To quantify this trend, we fit the function $\mathrm{log}({r}_{\mathrm{warm}}/\mathrm{au})=\,a+b\mathrm{log}({M}_{\star }/{M}_{\odot })$ to these points. The best fit values of b and a were computed as

$$\begin{eqnarray}&&b=\displaystyle \frac{\overline{\mathrm{log}({M}_{\star })\cdot \mathrm{log}({r}_{\mathrm{warm}})}-\overline{\mathrm{log}({M}_{\star })}\cdot \overline{\mathrm{log}({r}_{\mathrm{warm}})}}{\overline{\mathrm{log}{({M}_{\star })}^{2}}-{\overline{\mathrm{log}({M}_{\star })}}^{2}}\end{eqnarray} \tag{ 7 }$$

and $a=\overline{\mathrm{log}({r}_{\mathrm{warm}})}-b\cdot \overline{\mathrm{log}({M}_{\star })}$. We found b = 1.08, a = 0.566 (i.e., r_warm/au = 3.68 (M_⋆/M_⊙)^1.08). This best fit trend line is plotted with the data points in Figure 3.

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Considering the substantial amount of scatter evident in the data around the trend line, we used a bootstrap procedure to quantify the significance of our derived value of b. We used 10,000 trials for the bootstrap procedure. For each trial we randomly selected 29 points (with replacement) from our sample and recomputed the best fit b using Equation (7). Figure 4 shows the distribution of b values found by the bootstrap procedure. The distribution was fit by a normal distribution with mean = 1.08 and σ = 0.21 using the Matlab function fitdist.

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We found that b for the targets with a single warm component was consistent (within 0.6σ) with the value predicted by the primordial snow line (1.2) but was not consistent ($\gt 4.3\sigma$) with the relation predicted by the current snow line (2.0). Thus we conclude that these warm components are more likely to arise from dust produced by exo-asteroid belts than from disintegrating comets or dust dragged inward from an outer belt that is below our current detection limit.

Next we examined the r_warm–M_⋆ trend for the two-component systems, shown in Figure 5. There was more scatter than for the single-component systems, and no simple relation was evident that represented all the points. For comparison, we added to this plot the best fit trend found for the single-component systems (green dashed line from Figure 3). We found that many of the warm components in the two-component systems were consistent with this same trend, but there were also several systems both above and below the trend.

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We subtracted the values of r_warm predicted by the trend line for the single-component systems from the measured r_warm values in both samples. Fitting a normal distribution to residuals of the single-component systems (using fitdist) gave a mean = 0.0 and σ = 0.16 dex in $\mathrm{log}({r}_{\mathrm{warm}}/\mathrm{au})$. This provided a measure of the inherent scatter around this trend for systems that likely follow the primordial snow line relation. The 1σ region around the trend due to this scatter is depicted in Figures 3 and 5 with thin dotted green lines.

Figure 6 shows the distribution of the residual warm dust locations from the trend (green for the single-component systems, magenta for the two-component systems). The distribution for the two-component systems peaks around the trend (no residual). We note that 32/54 = 59% of these systems fall within the 2σ tolerance of the trend for the systems with only warm components. The histogram shows for the two-component systems a decreasing tail of warm dust locations below the trend (negative residual), whereas the warm dust locations above the trend show signs of a separate population of systems. It may be that some warm components in two-component systems have locations set by the primordial snow line while others end up off of this trend for various reasons. Alternatively, these warm components may have locations scattered more or less randomly, with some inevitably falling along the trend set by the single-component systems. In Section 4.2 we discuss various possibilities for the nature of the warm components in two-component systems.

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An essential feature of our analysis is that we compare the trend of warm dust location with stellar mass to the predicted trend for the two possible snow lines. By comparing trends rather than absolute values, we expect that systematic errors due to our choice of model disk geometry or grain composition (optical constants) will not influence our results. To test what role our model assumptions may have had on our results, we repeated our analysis twice with different assumptions. First, we re-fit the SEDs with dust belts with fractional widths of 0.4 r_in (in contrast to the fixed belt width of 2 au we used in the main analysis). The results agreed almost exactly with those of our main analysis: the single-component systems showed a clear trend in belt location with stellar mass while the two-component systems showed considerable scatter. A bootstrap analysis of the single-component power-law index yielded b = 0.95 ± 0.19. Second, instead of using a mixture of astronomical silicates and refractory organic material for the dust composition, we used 100% astronomical silicates (a common first-order assumption in debris disk modeling). Again, we found very similar results to our main analysis with b = 1.26 ± 0.21. The r_warm vs, M_⋆ trends for all three sets of model assumptions are plotted together in Figure 7. Thus neither of these modifications to our model assumptions affected our main conclusion that the locations of single-component systems are consistent with the primordial snow line and inconsistent with the current snow line.

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We have shown that warm dust in the single-component systems resides at locations consistent with being set by the primordial snow line and not by the current snow line, and thus is likely to originate from exo-asteroid belts. The warm dust locations in two-component systems are much more scattered, although many reside at similar locations to those in single-component systems. Here we turn our attention to the nature of the two-component systems, especially those with warm components that clearly do not trace the primordial snow line location.

4.2.1. Planets?

4.2.2. Inward Transport?

As discussed earlier, the inward transport of material from the cold outer belt is expected to result in warm dust at a preferential location—the current snow line. However, these warm components span a large range of locations for a given stellar mass, so it is unlikely that inward transport could explain all of these systems.

The specific scenario of inward transport by drag forces predicts that the warm component of dragged in material should be much fainter than the cold reservoir from which it originates (Kennedy & Piette 2015). We looked for this in our sample by plotting the fractional luminosities of the warm⁵ and cold components—and their ratio—against the residual warm dust location from the trend (Figure 8). We found that the f_warm/f_cold ratio (bottom panel) is roughly the same across our entire sample (or perhaps somewhat larger in systems below the trend). Thus this provides no additional support for the drag scenario. In fact, Geiler & Krivov (2017) found that the f_warm/f_cold ratios for two-component debris disks are consistent with the steady-state collisional evolution of inner and outer parent body belts originating from protoplanetary disks with reasonable radial density profiles.

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Later-type stars may be able to drag in a substantial amount of dust to generate a bright warm component if their luminosity is low enough such that no grains are blown out of the system by radiation pressure (there is no a_BOS). Drag is also enhanced in later-type stars by stronger stellar winds. With no significant radiation force on the grains, there would also be no pile-up of grains when the icy constituents sublimate, so the warm components would not trace any snow line location. The systems in our sample, however, have sufficiently high luminosities that they should be able to blow grains out, so we deem this situation unlikely.

4.2.3. Contamination by Exozodiacal Dust?

4.2.4. Contamination by Outer Dust?

4.2.5. Conclusion