A SELF-CONSISTENT MODEL OF THE CIRCUMSTELLAR DEBRIS CREATED BY A GIANT HYPERVELOCITY IMPACT IN THE HD 172555 SYSTEM

To estimate the stability of fine dust, we calculate the ratio of the radiation pressure force to gravitational force as a function of particle size

$$\begin{equation} \beta (a) = \frac{{3\,L_{\star} \langle Q_{\rm pr} (a)\rangle}}{{16\pi GM_ \star ca\rho }}, \end{equation} \tag{ 1 }$$

where L_⋆ and M_⋆ are the luminosity and mass of HD 172555, c is the velocity of light, ρ is the density of the grains, a is the radius of the grains, and 〈Q_pr(a)〉 is the radiation pressure coefficient for particles of radius a averaged over the stellar spectrum (Artymowicz 1988). β Pictoris, a fellow 12 Myr old A5V member of the β Pictoris moving group (BPMG), has the same effective temperature T_eff = 8000 K as HD 172555. Thus, for particles of the same composition and density, the β values for HD 172555 are given by the following scaling $\beta (a) = \beta (a)_{\beta _{\rm pic} } (M_{\beta _{\rm pic} } /M_ \star)(L_ \star /L_{\beta _{\rm pic} })$, where $L_{\beta _{\rm pic} } = 6.5\,L_{\odot}$,$M_{\beta _{\rm pic} } = 1.7\,M_{\odot}$, and $\beta (a)_{\beta _{\rm pic} }$ are the luminosity, mass, and β values for β Pictoris (Artymowicz 1988), and L_⋆ = 9.5 L_☉ and M_⋆ = 2 M_☉ are the luminosity and mass of HD 172555 (Li2009). The calculated values for β(a) in HD 172555 are shown in Figure 3.

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We can use the value of Beta for a dust particle to determine its long-term dynamical stability versus the forces of stellar radiation pressure and gravity. For example, dust with β < 1/2, produced from parent bodies on circular orbits having β = 0, will have stable orbits. On the other hand, when β > 1/2, the dust, produced from parent bodies on circular orbits having β = 0, will move outward with a terminal radial velocity

$$\begin{equation} v_r \approx \left[ {\left({\frac{{2GM_ \star }}{R}} \right)\left({\beta - \frac{1}{2}} \right)} \right]^{\frac{1}{2}} \end{equation} \tag{ 2 }$$

(Su et al. 2005). For HD 172555, R ∼ 6 AU is the radius at which the dust is created, G is the gravitational constant, and M_⋆ = 2 M_☉ (Li2009).

Using the optical properties of obsidian, a majority amorphous silica species with ∼75% SiO₂ and ∼14% Al₂O₃ by mole (Artymowicz 1988 and references therein), a composition that is comparable to the ∼69% SiO₂, 10% MgO, and 9% Al₂O₃ mixture of Bediasite and obsidian found for the "fine dust" in HD 172555 (Li2009), Figure 3 shows that dust between 0.02 μm and 1.5 μm in radius will be removed from HD 172555 by radiation pressure. To estimate the rate at which dust is removed from the torus we consider the crossing time,

$$\begin{equation} \tau _c \approx \frac{r}{{v_r }}, \end{equation} \tag{ 3 }$$

where r is the minor radius of the torus. A best-fit model of the 18 μm resolved emission gives a disk with a width dr ≈ 9.5 AU (Smith et al. 2012; for detailed description the debris geometry, see Figure 4). Assuming that the disk can be approximated by a torus, the minor radius of the torus is about half of the disk's width or r ≈ 4.75 AU. Smith et al. (2012) estimate that the lower limit for disk width is dr ≈ 1 AU or r ≈ 0.5 AU. Additionally, the surface area of the dust requires r > 0.01/sin(i) AU, where i is the inclination of the disk or torus (Li2009). If r = 0.01/sin(i) AU, the torus would be optically thick and emit as a blackbody. The Spitzer IRS spectrum is not consistent with blackbody emission, therefore, the torus must be optically thin with r significantly greater than r = 0.01/sin(i) AU. Thus, we assume that r is on the order of 1 AU and leave it as a variable in all of our calculations. The imaging observations also show that mid-infrared emission comes dominantly from a limited spatial region described by a disk or torus and emission from dust outside of this torus or disk produces a negligible portion of the observed infrared excess (Pantin & Di Folco 2011; Smith et al. 2012). Thus, when assessing the photometric stability of the system we can safely neglect any emission from dust that is blown out of the torus.

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We find that dust between 0.02 μm and 1.5 μm in radius will be removed from the torus in τ_c < 0.4 (r/AU) yr (Figure 5). The lifetime of submicron dust, τ_c < 0.4 (r/AU) yr, seems to be inconsistent with observational evidence that the infrared excess of HD 172555, which is dominated by emission from submicron dust, has appeared photometrically stable over the ∼27 years from its IRAS discovery in 1983 to its re-observation by WISE in 2010 (Figure 1), let alone its stability on the Myr timescales of the system's age and the estimated circumstellar disk lifetime calculated by Jackson & Wyatt (2012). However, we show next that it is possible to construct a PSD that is consistent with the calculation of dust that will be blown out by radiation pressure and the Spitzer IRS spectra of HD 172555 (Figure 6, bottom).

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New spectral modeling work, using the same dust composition found in Li2009 and the same methods of Li2009, shows that additional PSDs are capable of giving a better than 95% confidence level fit to the Spitzer IRS spectrum of HD 172555 (Figure 6, top). The mid-infrared spectral modeling is degenerate in that any collection of dust particles that satisfies the following integral equation will produce emission that is essentially identical to the observed Spitzer spectrum. We have

$$\begin{equation} F_\lambda = \frac{1}{{\Delta ^2 }} \sum _i \int_0^\infty B_\lambda \left({T_i \left({a,R} \right)} \right)Q_{{\rm abs},i} \left({a,\lambda } \right)\pi a^2 \frac{{dn_i \left(R \right)}}{{da}}da, \end{equation} \tag{ 4 }$$

where the subscript denotes each individual dust component of the ith composition (e.g., olivine, silica, etc.), T_i(a, R) is the particle temperature for a particle of radius a and composition i at a distance R from the primary, Δ is the distance from Spitzer to the dust, B_λ is the blackbody radiance at wavelength λ, Q_{abs, i}(a, λ) is the emission efficiency of the particle of composition i at wavelength λ and particle size a, dn/da is the differential PSD, the integral is over all sizes, and the sum is over all compositions. In practice, for a spectrum with strong infrared emission features, the compositional mix of materials is relatively uniquely defined. The overall run of temperature with particle size is as well, which helps specify the location of the dust with respect to the central star. The size distribution of the dust can have some play or flexibility in it; a population of 1–10 μm silicaceous dust, for example, will produce both strong features and some mid-infrared continuum; a population of extremely small (<1 μm) particles, which produces only strong emission features plus a population of extremely large (>100 μm) particles, which produces only continuum, may produce similar model infrared spectra, and depending on the quality of the observed spectrum, it may be difficult to distinguish between the two.

The non-unique nature of the spectral modeling allows us to test additional PSDs that can potentially yield a better than 95% confidence level fit to the Spitzer IRS spectrum (as did original PSD published by Li2009; Figure 6). For a single power-law PSD, the best-fit PSD is dn/da∝a^{−3.95  ±  0.10} (this is the same PSD published by Li2009; Figure 6, top). Figure 6 (middle) shows a broken power law, with dn/da∝a^−3.55 below 1 μm and dn/da∝a^−4.02 above 1 μm. Motivated by calculation of β for obsidian grains, we show that a PSD with dn/da∝a^−3.95 with a reduction in the number of grains between 0.1 μm and 1 μm is also viable (Figure 6, bottom).

Although the three PSD's in Figure 6 look very different, they have many similarities. All of the PSDs are very steep with approximately dn/da∝a^{−3.95 ± 0.10} compared to the canonical steady-state distribution given approximately by dn/da∝a^−3.5 for particles with dispersal thresholds that are independent of particles size (Backman & Paresce 1993). Although the PSD's have particles up to 100 μm in them, these steep PSDs have most of their mass contained in the small particles and all of these PSDs correspond to 10¹⁹–10²⁰ kg of submicron dust. After removal of the emission created by the PSDs there is still a large 200 K blackbody signature, as described in Li2009 (Figure 2). Following Li2009 and assuming that this emission is created by the smallest particles capable of emitting as blackbodies ∼100 μm, all three PSDs require the addition of 10²¹–10²² kg of large "rubble." Following Li2009 we call the warm 340 K dust described by the PSDs "fine dust" and the larger particles emitting as 200K blackbodies, which are not included in the PSDs, "rubble." The temperature difference between the "rubble" and "fine dust" is expected, because the large optically thick "rubble" particles co-located with the "fine dust" are much better at radiating heat and needs only to reach a color temperature of ∼200 K to establish energy balance with the incident starlight.

Lab work by Takasawa et al. (2011) shows that a much steeper-than-collisional size distribution is consistent with PSDs created by hypervelocity impacts. Takasawa et al. (2011) measured the size distribution for seven hypervelocity shots with impact velocities ranging from 9 to 61 km s⁻¹. The average slope of the PSDs for the seven shots gives dn/da∝a^{−4.4  ±  0.8} (Takasawa et al. 2011). Although they use a range of impact velocities, there is no obvious correlation between the slope of the PSD and the impact velocity.

We believe the steeper than collisional PSDs found for hypervelocity impacts in the laboratory indicates that the steeper than collisional PSDs derived for the "fine dust" in HD 172555 by Li2009 implies that most of the "fine dust" around HD 172555 is sourced by hypervelocity impacts. However, we do not believe that this "fine dust" was created by the initial planetary scale hypervelocity impact, but by subsequent hypervelocity impacts between sub-millimeter-sized particles similar to those used by Takasawa et al. (2011). Other support for a constant hypervelocity grinding scenario is produced by dynamical modeling of the circumstellar debris disk created by the Moon forming impact, which shows that typical collision velocities in the circumstellar disk are ⩾5 km s⁻¹ (Jackson & Wyatt 2012). Although this velocity depends on the details of the orbital elements of the debris, the study of Jackson & Wyatt (2012) shows that hypervelocity grinding is not unprecedented.

In addition to being sourced by hypervelocity impacts, these steep PSDs also require that the particles have not had time to become a steady-state collisional distribution. The time it takes for particles to relax to the steady-state collisional distribution is their collisional lifetime. Following Wyatt et al. (1999) we find the collisional lifetime of the smallest particles that dominate the surface area of the torus is given by

$$\begin{equation} \tau _{\rm col} = \frac{{t_{\rm orb} }}{{4\pi f_A }}, \end{equation} \tag{ 5 }$$

where t_orb ∼ 10 yr is the orbital timescale at 6 AU, and f_A = A_dust/2πR2r is the fractional cross-sectional area of the torus that the particles occupy. The cross-sectional area of the dust is A_dust = (1 − 5)  ×  10²¹ m² (Li2009). Combining this we find

$$\begin{equation} \tau _{\rm col} = ({300 - 1400})\left({\frac{r}{{\rm AU}}} \right)\;{\rm yr}. \end{equation} \tag{ 6 }$$

To observe a steep PSD with dn/da∝a^{−3.95  ±  0.10} the fine dust must have only been created within a collisional timescale. As previously stated, the canonical steady-state distribution with dn/da∝a^−3.5 is for particles with dispersal thresholds that are independent of particle size (Backman & Paresce 1993). The dispersal threshold, Q_D, is the specific incident energy required to cause a catastrophic collision, where a catastrophic collision is one where the mass of the largest object remaining after a collision is half the mass of the largest object before the impact. It is possible that we are observing a steady-state PSD if the dispersal threshold Q_D∝a^−0.92 (Wyatt et al. 2011). This dependence is significantly steeper than typical dispersal threshold size dependences with approximately Q_D∝a^−0.4 (Benz & Asphaug 1999).

Based on the β values from Figure 3, we expect that the PSD agreement with laboratory studies should only apply to dust larger than ∼1 μm, as the submicron dust is still removed on a timescale of τ_c < 0.4 (r/AU) yr by radiation pressure. Thus the dn/da∝a^−3.95 PSD (Figure 6, top) corresponds to a system in which the submicron dust is somehow bound to the system and long lived. If the debris torus is optically thick enough to reduce the effective luminosity of HD 172555 from L_⋆ = 9.5L_☉to L_eff ≈ L_☉ within the torus, then β values for obsidian (Figure 3) would be ∼10 times smaller. This would make β < 0.5 for dust of all sizes meaning no radiation pressure blowout would occur. Without any reason to believe the debris torus would be optically thick at the peak wavelength in the HD 172555 spectral energy distribution, we conclude that this situation is unlikely.

The PSD with dn/da∝a^−3.95 and a reduction in the number of grains between 0.1 μm and 1 μm (Figure 6, bottom) reconciles the apparent contradiction of mid-infrared emission that is photometrically stable over 27 years (Figure 1) and dust between 0.02 μm and 1.5 μm in radius being removed from the torus in τ_c < 0.4 (r/AU) yr by radiation pressure (Figure 5). As previously discussed, this PSD provides a better than 95% confidence level fit with the Spitzer IRS spectrum and should remain stable over a lifetime equal to the collisional lifetime of the torus, which is determined by the lifetime of the largest object in the debris disk and could be longer than 25 Myr (Jackson & Wyatt 2012). For obsidian, Figure 3 illustrates that β > 0.5 for particles between 0.02 μm and 1.5 μm and thus they should be quickly ejected from the system by radiation pressure. We argue that the difference between calculated size of the blowout grains (Figure 3) of 0.02–1.5 μm and the blowout grain size of 0.1–1 μm obtained from the spectral modeling (Figure 6, bottom) are trivial considering the uncertainties associated with each calculation.

The calculation of β for obsidian assumes that the particles are ∼75% SiO₂ and ∼14% Al₂O₃ by mole (Artymowicz 1988 and references therein) while the spectral modeling of Li2009 shows that the "fine dust" is dominated by a combination of Bediasite and obsidian with an average composition that is ∼69% SiO₂, 10% MgO, and 9% Al₂O₃. Although we have neglected species with mole percent less than 5%, this comparison shows that the obsidian used in Figure 3 is only an approximate match to the "fine dust" in HD 172555. As shown in Equation (1), β∝1/ρ, where ρ is the bulk density of the grains, so if ρ is larger than the 2370 kg m⁻³ used by Artymowicz (1988), β could be somewhat smaller than the values shown in Figure 3. Another source of uncertainty comes from the degeneracy associated with the PSDs and spectral modeling described at the beginning of this section. Although the submicron obsidian grains with β < 0.5 will spiral into the primary in 10⁴–10⁶ yr due to Poynting Robertson drag, as long as the fine dust is replenished by collisions, the emission will remain stable over a lifetime equal to the collisional lifetime of the torus, which could be longer than 25 Myr (Jackson & Wyatt 2012).

Assuming the SiO vapor is in thermal equilibrium with the SiO₂ dust, we can estimate the equilibrium vapor pressure of SiO above solid SiO_x at a temperature of ∼340 K, the approximate temperature of the fine silica dust found by Li2009. We make the estimate using an empirical formula derived by Ferguson & Nuth (2008) for the equilibrium vapor pressure of SiO above solid SiO_x with a stoichiometric x = 1,

$$\begin{equation} {\log _{10} ({{P}_{{\rm SiO}} /{\rm Pa}} ) = 13.29 \pm 0.39 - (17740 \pm 550)}/{({\rm T}/{\rm K})}. \end{equation} \tag{ 7 }$$

At 340 K the formula gives P_SiO ∼ 10⁻³⁹Pa. If we use T = 200 K, the equilibrium vapor pressure is even lower, P_SiO ∼ 10⁻⁷⁵Pa. To convert pressure to number of molecules, we assume that SiO vapor is the only vapor present in the system and treat it as an ideal gas with pressure,

$$\begin{equation} P_{\rm SiO} \approx \frac{{N_{\rm SiO} k_b T}}{{V_{\rm torus} }}, \end{equation} \tag{ 8 }$$

where N_SiO the number of SiO vapor molecules k_b is Boltzmann's constant. Assuming the SiO vapor is in equilibrium with the fine dust, it will have a temperature T = 340 K. Lastly, we assume that the SiO vapor is co-spatial with the dust and in a torus with a volume

$$\begin{equation} V_{\rm torus} = 2\pi R \times \pi r^2 \approx 4 \times 10^{35} \left({\frac{r}{{\rm AU}}} \right)^2\;{\rm m}^3 , \end{equation} \tag{ 9 }$$

where R ≈ 6 AU is the major radius of the torus and r, the minor radius of the torus.

This leads to

$$\begin{equation} N_{\rm SiO} \approx 10^{17} \left({\frac{r}{{\rm AU}}} \right)^2 \;{\rm molecules}, \end{equation} \tag{ 10 }$$

which is 30 orders of magnitude less than the 10⁴⁷ molecules of SiO vapor predicted by Li2009. Thus, if there are 10⁴⁷ molecules of SiO vapor in the system, they are not in equilibrium with the dust. A model of condensation of glassy silicate material in an impact produced vapor plume predicts that a significant portion of the vaporized material does not condense during its initial expansion into space (Johnson & Melosh 2012a, 2012b). Thus, the presence of abundant SiO vapor, which is out of equilibrium and still condensing onto amorphous silica dust, is an expected outcome for material condensed from an impact produced vapor plume. This process requires impact velocities higher than ∼15 km s⁻¹ for a non-porus silica impactor and target with lower impact velocities required for less refractory and/or more porous materials (Melosh 2007; Johnson & Melosh 2012a).

As discussed in Section 3.2, dynamical modeling of the circumstellar debris disk created by the Moon-forming impact shows that typical collision velocities in the disk are ⩾5 km s⁻¹ (Jackson & Wyatt 2012). These high collisional velocities are a product of the inclined and eccentric orbits of the dust particles that are scattered when making close encounters with Earth. Assuming a similar-sized planet is scattering particles in the HD 172555 debris torus, we expect that particles should collide with velocities of v_col ⩾ 5 km s⁻¹. Although vapor created in an initial hypervelocity impact starts with an initial eccentricity and inclination, we expect that any initial eccentricity or inclination should be quickly damped down. Thus dust particles will initially move through the vapor with relative velocities of v_col. The initially large relative velocity of dust particles with respect to the vapor will decay in a time τ_damp, such that the dust particle hits a mass of vapor equivalent to its mass, or

$$\begin{equation} M_{\rm dust} = \frac{4}{3}\pi a^3 \rho = mv_{\rm col} n_{\rm SiO} \pi a^2 \tau _{\rm damp} , \end{equation} \tag{ 11 }$$

where ρ = 3000 kg m⁻³ is the assumed density of the dust, v_col = 5000 m s⁻¹ is the assumed collision velocity, a is the radius of the dust, and all other variables have been previously defined. Solving Equation (11) using the estimate of 10⁴⁷ molecules of SiO made by Li2009, we find

$$\begin{equation} \tau _{\rm damp} = \frac{{4a\rho }}{{3mv_{\rm col} n_{\rm SiO} }} \approx 1\left({\frac{a}{{{\rm \mu m}}}} \right)\left({\frac{r}{{{\rm AU}}}} \right)^2 {\rm yr}. \end{equation} \tag{ 12 }$$

For submicron dust, any inclination or eccentricity causing a relative velocity with respect to the vapor should decay in less than a year. We expect that most of the vapor will condense onto the submicron dust, which dominates the surface area of the debris (Section 3.2). Because this dust has little or no relative velocity with respect to the vapor, we expect that condensation will occur primarily by diffusion. Assuming a sticking coefficient of unity, we can estimate the time it takes the vapor to condense as

$$\begin{equation} \tau _{\rm cond} = \frac{{N_{\rm SiO} }}{{SA_{\rm dust} \Phi }}, \end{equation} \tag{ 13 }$$

where N_SiO is the number of SiO molecules and SA_dust is the surface area of the dust particles. Then Φ is the molecular flux given by Φ = (n_SiOv_th/4), where n_SiO is the number density of SiO given by n_SiO = (N_SiO/V_torus), and the thermal velocity v_th ≈ 360 m s⁻¹ at T = 340 K. Li2009 estimated the surface area of the fine dust to be SA_dust = 4 × 10²¹ − 2 × 10²² m². Combining all of this we find

$$\begin{equation} \tau _{\rm cond} = \frac{{4V_{\rm torus} }}{{SA_{\rm dust} v_{\rm th} }} \approx ({10^3 - 10^4 })\left({\frac{r}{{\rm AU}}} \right)^2\; {\rm yr}. \end{equation} \tag{ 14 }$$

This result is independent of the total amount of SiO vapor in the system. This estimate also assumes that the SiO vapor is at the same temperature, ∼340 K, as the warmest fine dust. This assumption is likely a lower limit on the SiO vapor temperature, making our lifetime estimate a maximum condensation lifetime. Thus, if we are observing HD 172555 within 10³–10⁴ yr of the initial impact that created the dust torus, we cannot rule out the existence of SiO vapor using arguments based on the condensation timescale.

The long condensation lifetime of SiO vapor in HD 172555 allows us to treat material in the vapor phase and dust as a completely separate chemical system, simplifying the chemistry substantially. Although SiO vapor has a long condensation lifetime, we must also estimate the rate of the removal of SiO via the photo-dissociation reaction SiO + hν → Si + O. The flux of photons able to dissociate the SiO molecule at 6 AU from the HD 172555 primary is given by

$$\begin{equation} \Phi _{(\lambda < \lambda _0 = 1500{\AA})} = \int_0^{\lambda _o } F_\lambda \frac{\lambda }{{hc}}d\lambda \approx 1.5 \times 10^{16} m^{ - 2} s^{ - 1} , \end{equation} \tag{ 15 }$$

where F_λ is the spectral flux density of HD 172555 at a distance of 6 AU, and λ₀ = 1500 Å is the maximum wavelength photon that can dissociate the SiO molecule (dissociation energy = 8.26 eV; Tarafdar & Dalgarno 1990). Following Ch2006, we assume that the HD 172555 primary has solar metallicity, log(g) = 4.5, and an effective temperature T_eff = 8000 K. We then estimate the stellar photospheric flux, which yields F_λ, using the appropriate stellar photospheric emission model (Kurucz 1992). Combining this, we find the value of $\Phi _{(\lambda < \lambda _0 = 1500{\AA})} \approx 1.5 \times 10^{16}\, m^{ - 2}\, s^{ - 1}$ quoted in Equation (15).

Following Tarafdar & Dalgarno (1990), we use $\sigma _{(\lambda < \lambda _0 = 1500\,{\bf {\AA}})} \approx 2 \times 10^{ - 21}$ m² to estimate the SiO photo-dissociation cross-section at λ ⩽ 1500 Å. The lifetime of a free SiO molecule at 6 AU from the primary is then given by

$$\begin{equation} {\rm \tau }_{{\rm dis}} = \frac{1}{{{\rm \Phi }_{\left({{\rm \lambda } < {\rm \lambda }_0 = 1500\,{\bf {\AA}}} \right)} \sigma _{({\rm \lambda } < {\rm \lambda }_0 = 1500\,{\bf {\AA}})} }} = 0.01\;{\rm yr}. \end{equation} \tag{ 16 }$$

Thus, if SiO is not optically thick at λ ⩽ 1500 Å, it will be destroyed in less than 0.01 yr by photo-dissociation. To estimate whether we expect 1.4 × 10⁴⁷SiO molecules to be optically thick at λ ⩽ 1500 Å, we must compare the cross-sectional area of the SiO molecules to the cross-sectional area of the torus containing those molecules. The total cross-sectional area of 1.4 × 10⁴⁷ SiO molecules is

$$\begin{equation} A_{\rm SiO(\lambda < 1500{\AA})} = \sigma _o N_{\rm SiO} \approx 3 \times 10^{26}\, {\rm m}^2. \end{equation} \tag{ 17 }$$

While the total cross-sectional area of the circumstellar torus is

$$\begin{equation} A_{\rm torus} = 2\pi R \times 2r = 1.7 \times 10^{24} \left({\frac{r}{{{\rm AU}}}} \right){\rm m}^2 , \end{equation} \tag{ 18 }$$

where R = 6 AU. With A_{SiO(λ < 1500 Å)} ≫ A_torus we expect that 1.4 × 10⁴⁷ SiO molecules should be optically thick at λ ⩽ 1500 Å, and every photon of λ ⩽ 1500 Å should dissociate one SiO molecule. This increases the photo dissociation lifetime estimate to

$$\begin{equation} \tau _{\rm dis} = \frac{{N_{\rm SiO} }}{{\Phi _{\rm dis} A_{\rm torus} }} = 0.2\left({\frac{r}{{{\rm AU}}}} \right)^{ - 1}\; {\rm yr}. \end{equation} \tag{ 19 }$$

This lifetime implies that to be observing 10⁴⁷ molecules of SiO vapor, we must either be observing HD 172555 within 0.2 yr of the original impact, which would be extremely fortuitous, or SiO is being replenished at an unrealistically high rate. This short lifetime is also inconsistent with observations of system's mid-infrared photometric flux, which has remained constant within 4% over the last 27 years (Figure 1).

So far we have only considered the photo-dissociation of SiO neglecting much of the possible chemistry of the system. In Section 4.3, we showed that the SiO vapor will be optically thick at wavelengths λ ⩽ 1500 Å. In this section, we consider the possibility that SiO is self-shielded against photo-dissociation. Note that "self-shielding" does not mean that there is no photolysis occurring, only that it is confined to the surface of an optically thick torus and that the re-formation back reaction creates SiO at the same rate it is destroyed.

The equilibrium chemistry of a self-shielded SiO torus is dominated by four reactions. The first reaction is the photo-dissociation reaction

$$\begin{equation} SiO{\rm } + {\rm }h{\rm \nu } \to {\rm Si} + {\rm O,} \end{equation} \tag{ 20 }$$

which has a rate coefficient

$$\begin{equation} \Gamma _{\rm dis} = \frac{1}{{\tau _{\rm dis} }} = \frac{{\Phi _{({\lambda < 1500\,{\AA}})} A_{\rm torus} f_{\rm SiO} }}{{N_{\rm SiO} }} \end{equation} \tag{ 21 }$$

where f_SiO is the fraction of the surface area of the torus occupied by SiO, as seen by photons of wavelength λ < 1500 Å. This term must be added because atomic Si vapor may also shield the SiO at λ < 1500Å and vice versa. SiO is dissociated by photons with λ < 1500 Å while the first ionization of Si occurs at λ < 1520 Å. The two species also have very similar photo-cross-sections, with $\sigma _{\rm SiO(\lambda < 1500\,{\rm {\AA}})} = 2 \times 10^{ - 21}\, m^2$ and $\sigma _{\rm Si(\lambda < 1520\,{\rm {\AA}})} = (1 - 4) \times 10^{ - 21}\, m^2$ for wavelengths λ = 912 − 1521 Å (van Dishoeck 1988). To make order of magnitude estimates for species abundance, we approximate

$$\begin{equation} f_{\rm SiO} \approx \frac{{N_{\rm SiO} }}{{N_{\rm Si} + N_{\rm SiO} }}, \end{equation} \tag{ 22 }$$

which implies f_Si = (1 − f_SiO), where f_Si is the fraction of the surface area of the torus occupied by Si, as seen by photons of wavelength λ < 1500 Å. This assumes that f_O = 0, where f_O is the fraction of the surface area of the torus occupied by O, as seen by photons of wavelength λ < 1500 Å. This assumption is valid because O, which has a first ionization energy of ∼13.62 eV requiring photons of wavelength λ < 910 Å, has a maximum value of f_O given by f_O = (Φ_{(λ < 910 Å)}/Φ_{(λ < 1500 Å)}) ≈ 10⁻⁸. For similar reasons, we have also neglected the ionization of SiO, which occurs at λ < 1084 Å (Tarafdar & Dalgarno 1990). For our estimates, we also approximate

$$\begin{eqnarray} \Phi _{({\lambda < 1500\,{\bf {\AA}}})} &=& 1.5 \times 10^{16}\; {\rm photons\,s}^{ - 1}\; {\rm m}^{ - 2} \approx \Phi _{({\lambda < 1520\,{\bf {\AA}}})}\nonumber\\ &=& 2.1 \times 10^{16}\; {\rm photons\,s}^{ - 1}\; {\rm m}^{-2}. \end{eqnarray} \tag{ 23 }$$

The second reaction we consider is the radiative association reaction

$$\begin{equation} {\rm Si} + {\rm O} \to {\rm SiO} + {\rm h}{\rm \nu ,} \end{equation} \tag{ 24 }$$

which occurs with a rate coefficient k_a = 5.52 × 10⁻²⁴T^0.31 m³ s⁻¹ (Andreazza et al. 1995). At a temperature of ∼340 K the association rate coefficient is k_a ∼ 3.4 × 10⁻²³ m³ s⁻¹.

The third reaction we consider if the photoionization of Si

$$\begin{equation} {\rm Si} + {\rm h\nu } \to {\rm e} + {\rm Si}^ + , \end{equation} \tag{ 25 }$$

which has a rate coefficient

$$\begin{equation} \Gamma _{\rm Ion} = \frac{{\Phi _{({\lambda < 1520\,{\AA}})}\, A_{\rm torus} ({1 - f_{\rm SiO} })}}{{N_{\rm Si} }}. \end{equation} \tag{ 26 }$$

As we will show below, Si is largely neutral and self-shielded from photoionization. Thus, we do not consider the higher ionization levels of Si.

The fourth and final reaction we consider is the radiative recombination reaction

$$\begin{equation} {\rm e}^ - + {\rm Si}^ + \to {\rm Si} + {\rm h\nu }, \end{equation} \tag{ 27 }$$

where the recombination rate α = 7 × 10⁻¹⁸ m³ s⁻¹ (Nahar 1995). The equilibrium between the photo-dissociation of SiO and its radiative association reaction gives the following

$$\begin{equation} k_a [O][{Si}] = \Gamma _{\rm dis} [{SiO}] = \frac{{\Phi _{({\lambda < 1520\,{\AA}})}\, A_{\rm torus}\, f_{\rm SiO} }}{{N_{\rm SiO} }}[{SiO}], \end{equation} \tag{ 28 }$$

where [X] = (N_X/V_torus) is the concentration of species X and N_X is the number of species X in the torus. The equilibrium between the ionization of Si and the radiative recombination gives the following

$$\begin{eqnarray} \Gamma _{\rm Ion} [{Si}] &=& \frac{{\Phi _{(\lambda < 1520\,{\AA})} \times A_{\rm torus} \times (1 - f_{\rm SiO})}}{{N_{\rm Si} }}[{Si}]\nonumber\\ &=& \alpha [{Si^ + }][{e^ -}] \approx \alpha [{Si^ +}]^2 , \end{eqnarray} \tag{ 29 }$$

which assumes that [e⁻] = [Si⁺] since Si⁺ is the only ion we have considered and we have neglected other possible sources of electrons like the stellar wind and easily ionized metal atoms. Solving these equations, we find

$$\begin{equation} N_{\rm Si} = \frac{{({1 - f_{\rm SiO} })N_{\rm SiO} }}{{f_{\rm SiO} }}, \end{equation} \tag{ 30 }$$

$$\begin{equation} N_O = \frac{{f_{\rm SiO}^2 \Phi _{({\lambda < 1520\,{\AA}})}\; A_{\rm torus}\; V_{\rm torus} }}{{k_a (1 - f_{\rm SiO})}}N_{\rm SiO} , \end{equation} \tag{ 31 }$$

and

$$\begin{equation} N_{\rm Si^ + } = \sqrt {\Phi _{({\lambda < 1520\,{\AA}})}\; V_{\rm torus}\; A_{\rm torus} \frac{{1 - f_{\rm SiO} }}{\alpha }}. \end{equation} \tag{ 32 }$$

Figure 7 shows that if SiO vapor is produced at the same rate it is destroyed by photo-dissociation in the system, then for 10⁴⁷ SiO molecules to be present, there must also be abundant atomic Si and O vapor in the system. The figure also shows that the total number of ions, atoms, and molecules in the vapor phase must be $N_{\rm tot} = N_{\rm Si} + N_O + N_{\rm SiO} + N_{\rm Si^ + } > 5 \times 10^{48}$, corresponding to more than 3 × 10²³ kg or 0.05 M_⊕ of vapor in the system. Although it is possible to have 0.05 M_⊕of vapor in the system, this would make the HD 172555 debris disk truly unique.

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So far we have also neglected the constraints that the initial composition of vaporized material give us. In Section 4.1, we briefly mentioned the model of Johnson & Melosh (2012a), which predicts that impact produced silicate vapor does not condense efficiently during its initial expansion into space. Assuming the vaporized material is initially composed dominantly of a combination of silica (SiO₂), olivine ([Mg, Fe]₂SiO₄), and pyroxene ([Mg, Fe]SiO₃), the uncondensed vapor will have N_Si ⩽ N_O ⩽ 4N_Si, where N_Si is the total number of Si atoms in the system and N_O is the total number of O atoms in the system. If there are no major mechanisms to remove Si or O from the system, the constraints of N_Si and N_O coupled with the results illustrated in Figure 7, imply that

$$\begin{equation} N_O \sim N_{\rm Si} \sim 10^{48}\; {\rm molecules}. \end{equation} \tag{ 33 }$$

So far we have neglected the possible effects of the radiation force on the vapor molecules and atoms. The ratio of the radiation force to the gravitational force β has been calculated for atomic Si, O, and Si⁺ in the β Pictoris system (Fernández et al. 2006). Scaling the β values as described in Section 3.1, we find

$$\begin{eqnarray} \beta _{\rm Si} &=& 7.5 \pm 0.7,\quad \beta _{\rm Si^ + } = 11.2 \pm 11.2,\quad {\rm and}\nonumber\\ \beta _O &=& ({4.1 \pm 0.2}) \times 10^{ - 4}. \end{eqnarray} \tag{ 34 }$$

For β > (1/2), the particles will quickly be removed from the system by the radiation force. Thus, atomic O will not be removed from the system by the radiation force, but Si atoms will. Using the radial velocity equation (Equation (2)), we estimate the time it will take Si to be removed from the system to $\tau _{F_{\rm rad} }\,{\approx}$ 0.1 yr. The large uncertainty in $\beta _{\rm Si^ + }$ makes it impossible to estimate what effect the radiation force will have on Si⁺.

These estimates on the effect of the radiation force are only valid if the torus is optically thin at the wavelengths corresponding to the atomic transitions. Following Hilborn (1982) and Fernández et al. (2006), the wavelength-integrated cross-section of a line transition is

$$\begin{equation} \sigma _0 = \frac{1}{{8\pi c}}\left({\frac{{g_k }}{{g_i }}} \right)A_{\rm ki} \lambda _{\rm ki}^3 , \end{equation} \tag{ 35 }$$

where g_k and g_i are the degeneracies of levels k and i, A_ki is the Einstein coefficient for spontaneous emission from level k to i, and λ_ki is the wavelength of the transition. Then the radiation force on a single atom is given by

$$\begin{equation} F_{\rm rad} = \frac{1}{c}\mathop \sum \limits_{i < k} F_\lambda \lambda _{\rm ki} \sigma _0 , \end{equation} \tag{ 36 }$$

where F_λ is the stellar flux per unit wavelength at the line center. Fernández et al. (2006) argue that the low temperatures and densities in the β Pictoris disk imply that radiative de-excitations are much faster than collisional or radiative excitations, implying that i = 0 or all absorptions are from the ground state. Using this assumption, and atomic data from the NIST Atomic Spectra Database, version 5.0 (Kramida et al. 2012) we find the λ_ki = 2514 Å transition of Si will dominate the radiation force on Si. It is a good example of a strong transition with A_ki = 7.39 × 10⁷ s⁻¹, g_k = 3, and g_i = 1, with a typical σ₀ = 5 × 10⁻²² m². The Si atoms in the torus will be optically thick at line transition wavelengths as long as N_Siσ_o ≫ A_torus ≈ 10²⁴ m², or N_Si ≫ 2 × 10⁴⁵. Thus, if N_Si ⩽ 2 × 10⁴⁵, Si will be quickly removed from the torus in $\tau _{F_{\rm rad} }\,{\approx}$ 0.1 yr unless efficient braking by collisions with species with β < (1/2) occurs.

To estimate the conditions necessary for efficient braking to take place, we estimate the mean free path of vapor molecules in the torus as $\lambda = ({\sqrt 2 n_{\rm tot} \sigma })^{ - 1}$, where σ ≈ 10⁻¹⁹ m² is a typical molecular collisional crossection and is assumed to be approximately equal for all the vapor species and n_tot = (N_tot/V_torus) is the total number density of vapor. The mean free path will be much less than the size of the torus as long as N_tot(r/AU) ≫ 10⁴³. Thus, if N_tot(r/AU) ≫ 10⁴³, an Si atom will undergo many collisions with particles with β < (1/2) before it can leave the torus. These collisions will act as a drag force on the Si, keeping Si atoms bound to the torus and reducing its effective Beta (Fernández et al. 2006). Thus, for the N_O ∼ N_Si ∼ 10⁴⁸ molecules required to keep 10⁴⁷ molecules of SiO vapor stable against photo-dissociation, Si will not be blown out of the torus by radiation pressure.

One of the most exciting possible sources of the observed mid-IR emission we have considered is the direct detection of emission from a magma ocean-covered world. The kilometers-deep ocean could easily be created in the giant impacts required to form rocky worlds of Earth-size or larger. However, we can rule out this possibility, as it would be short lived and is unable to produce the total luminosity observed by Spitzer. In this section, we also consider the possibility of a self-luminous circumplanetary magma disk. Another way of saying self-luminous is that the planet or disk is not in thermal equilibrium with the central star. These systems are attractive because the fine dust and vapor would naturally be stable as long as the planet or disk is still hot. We can estimate the lifetime of such a system by comparing the total amount of energy an impact brings into the system to the rate at which the system is radiating energy.

The total luminosity implied by the IR excess seen in HD 172555 is 2 × 10²⁴ W (Li2009). We can then estimate the lifetime of the system as shown below

$$\begin{equation} \tau \sim \frac{{{\rm energy}\;{\rm from}\;{\rm impact}}}{{{\rm Observed}\;{\rm radiation}\;{\rm power}}}\sim \frac{{\frac{1}{2}\,{M}_{\rm imp}\, {V}_{\rm imp}^2 }}{{2 \times 10^{24}\, {\rm W}}}. \end{equation} \tag{ A1 }$$

Even if we assume an extremely high impact velocity of ∼50 km s⁻¹ and an impactor mass, M_imp = 6 × 10²³ kg, or 0.1 M_⊕, we find τ ∼ 24 yr. We know that IR photometry of HD 172555 has been stable since its IRAS discovery in 1983 to its re-observation by WISE in 2010 implying a minimum lifetime of ∼27 yr (Figure 1). This minimum lifetime is a strong argument against all but the most extreme self-luminous sources.

In addition to the estimated lifetime of such a system, we can also estimate the size and temperature of such a system required to radiate energy at the rate we observe. We assume that the body radiates as a black body and find

$$\begin{equation} 2 \times 10^{24} {\rm W} = 4\pi R^2 \sigma T^4 , \end{equation} \tag{ A2 }$$

where R is the radius of the body, σ is the Stefan–Boltzmann constant, and T is the temperature of the body. The black body temperature of the disk of a magma ocean planet is probably less than T ∼ 3000 K (Miller-Ricci et al. 2009). This means such a system would have to have a physical radius, R > 2 × 10⁵ km. This size is several orders of magnitude larger than any known planet therefore ruling out the possibility of magma ocean planet. This large physical size is still consistent with a circumplanetary disk, but the temperature we used T ∼ 3000 K is much higher than the temperature implied by spectroscopic observations. If we instead assume the black body temperature of the disk is ∼200 K, or thermal equilibrium at 6 AU as shown by Li2009, we find the disk must have a radius R > 2 × 10⁷ km. As we will show in the next section, this size disk is consistent with the size of the Hill sphere of a Jovian planet located at ∼6 AU from the primary.

We have ruled out a magma ocean planet as a possible source for the IR excess because such a body could not possibly radiate enough energy to explain the flux seen by Spitzer, even at temperatures much higher than the dust temperature implied by the Spitzer spectra. In addition to ruling out a magma ocean planet as a possible source, the short lifetime of self-luminous source radiating thermal energy at the levels seen in HD 172555 argues against them as a possible source for the IR radiation seen by Spitzer. As such, the only systems that are consistent with observations are those in thermal equilibrium with the central star.

Another system that could possibly explain the Spitzer observations of HD 172555 is a much larger circumplanetary disk whose energy budget is driven by the central star's radiation. In this system the disk is gravitationally bound to a central planet, which means the fine dust in the system can be longer lived than the radiation pressure blowout lifetime. We can estimate the size of such a disk by assuming the disk perfectly re-radiates 100% of the impinging radiation as the IR that Spitzer sees. This means that the total luminosity implied by Spitzer data is equal to the cross-sectional area of the disk multiplied by the flux from the central star at 6 AU, or

$$\begin{equation} \frac{{L_{\rm spitzer} }}{{L_ \star }} = 5 \times 10^{ - 4} = \frac{{\pi r_{\rm disk}^2 }}{{4\pi ({6\,{\rm AU}})^2 }} \end{equation} \tag{ A3 }$$

and

$$\begin{equation} r_{\rm disk} \ge 4 \times 10^7\,{\rm km}. \end{equation} \tag{ A4 }$$

We can then calculate the size of a planet needed to contain a circumplanetary disk of this size in its Hill sphere. If we ignore eccentricity and assume that the planet is 6 AU from a central star of mass M_⋆ = 2 M_☉.

$$\begin{equation} {M}_{{\rm planet}} \ge 3\frac{{{R}_{{\rm Hill}}^3 }}{{{R}_{{\rm orbit}}^3 }}{\rm M}_ \star \sim 10^{27}\; {\rm kg}{.} \end{equation} \tag{ A5 }$$

This means a planet with a mass comparable to Jupiter is required to stabilize the large cold circumplanetary disk. One possibility that could lead to such a system is a large impact onto one of the moons orbiting a Jovian planet. Recent observations using the APP coronograph of VLT/NACO to search for planetary companions by Quanz et al. (2011) rule out a Jovian planet larger than about four Jupiter masses at a distance of 11 AU from HD 172555. The 3σ upper limit of this non-detection is not enough to rule out a 1 M_Jup planet at ∼6 AU, however.

Another possible source for dust in a circumplanetary disk is a tidally heated moon. Currently the Io-Jupiter system is thought to be in a quasi-steady state where ∼1000 kg s⁻¹ leaves Io and ∼1000 kg s⁻¹ leaves the Jovian Hill sphere (Thomas et al. 2004). Using the estimated age of HD 172555 as a BPMG member at ∼12 Myr, and the estimated mass of dust in the system of 10²¹ kg (Li2009), we find the minimum rate at which an Io-like moon would have to lose mass to produce the dust seen by Spitzer is at least 10⁷ kg s⁻¹ (∼10⁴ times greater than any known comet or Moon in today's solar system) for 10⁷ yr in order to produce the debris seen. 10²¹ kg of material is also a huge mass for a moon to lose (Ganymede, the largest moon in the solar system, is only 1.5 × 10²³ kg). Moreover, Io's mass loss is primarily ionized SO₂, not silicate material (Dessler 1980; Strobel & Wolven 2001).

Another possible source for the observed dust is motivated by work done by Canup (2010). This work argues that giant planet satellite systems form and stabilize via a series of large body accretions and subsequent migrations. The stable systems seen today are the product of a series of moon formation, migration, tidal disruption, and planetary accretion events. It is possible that at only ∼12 Myr, a giant planet in HD 172555 at ∼6 AU has recently (within the last ∼1 Myr) completely lost a moon due to tidal disruption and breakup. We argue against this possible source using the large spatial extent the dust must have, r_disk ⩾ 4 × 10⁷ km. For a Jovian planet, the Roche limit is on the order of R_roche ∼ 10⁵ km. The Roche limit scales as the radius of the planet, so we would need a planet more massive than the Sun to have a tidal disruption create a disk of the required radius. It is remotely possible that through the process of angular momentum transport, a small fraction of this material is transported to larger radii but analysis that is more detailed is required to determine if this idea is plausible.

Although some of the circumplanetary disk models may be able to explain the spectroscopic observations of HD 172555, tentative observations of the system including both interferometric visibility and photometric imaging are inconsistent with a localized dust source (Smith et al. 2012; Pantin & Di Folco 2011). This rules out all of the circumplanetary disk models, unless we consider a leaky Hill sphere scenario. In this scenario, material is only weakly bound to the Jovian Hill sphere and is constantly being lost to the circumstellar torus. This scenario does not change any of the analysis presented in the main text, it only allows for a somewhat different and arguably less probable creation scenario for the circumstellar debris disk or torus of HD 172555.