Sublimation temperature of circumstellar dust particles and its importance for dust ring formation
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2011
Received: 10 November 2010
Accepted: 10 March 2011
Published: 2 February 2012
Dust particles in orbit around a star drift toward the central star by the Poynting-Robertson effect and pile up by sublimation. We analytically derive the pile-up magnitude, adopting a simple model for optical cross sections. As a result, we find that the sublimation temperature of drifting dust particles plays the most important role in the pile-up rather than their optical property does. Dust particles with high sublimation temperature form a significant dust ring, which could be found in the vicinity of the sun through in-situ spacecraft measurements. While the existence of such a ring in a debris disk could not be identified in the spectral energy distribution (SED), the size of a dust-free zone shapes the SED. Since we analytically obtain the location and temperature of sublimation, these analytical formulae are useful to find such sublimation evidences.
Refractory dust grains in orbit around a star spiral into the star by the Poynting-Robertson drag (hereafter P-R drag) and sublime in the immediate vicinity of the star. Because the particles lose their mass during sublimation, the ratio β of radiation pressure to gravity of the star acting on each particle ordinarily increases. As a result, their radial-drift rates decrease and the particles pile up at the outer edge of their sublimation zone (e.g., Mukai and Yamamoto, 1979; Burns et al., 1979). This is a mechanism to form a dust ring proposed by Belton (1966) as an accumulation of interplanetary dust grains at their sublimation zone. Ring formation of drifting dust particles is not limited to refractory grains around the sun but it also takes place for icy grains from the Edgeworth-Kuiper belt and for dust in debris disks (Kobayashi et al., 2008, 2010). Therefore, dust ring formation due to sublimation of dust particles is a common process for radially drifting particles by the P-R drag.
The orbital eccentricity and semimajor axis of a dust particle evolve by sublimation due to an increase in its β ratio as well as by the P-R drag. We have derived the secular evolution rates of the orbital elements (Kobayashi et al., 2009). The derived rates allow us to find an analytical solution of the enhancement factors for the number density and optical depth of dust particles due to a pile-up caused by sublimation. Our analytical solution is found to reproduce numerical simulations of the pile-up well but its applicability is restricted for low eccentricities of subliming dust particles. The analytical solution shows that the enhancement factors depend on dust shapes and materials as expected from previous numerical studies (cf. Kimura et al., 1997). Although the solution includes physical quantities for the shapes and materials, it does not explicitly show which quantity essentially determines the enhancement factors.
The goal of this paper is to derive simplified formulae that explicitly indicate the dependence of dust ring formation on materials and structures of dust particles. In this paper, we adopt a simple model for the optical cross sections of fractal dust particles and analytically obtain not only the enhancement factors but also the location of the pile-up and sublimation temperature. In addition, we extend the model of Kobayashi et al. (2009) by taking into account orbital eccentricities of subliming dust particles.
In Section 2, we derive the sublimation temperature as a function of the latent heat. In Section 3, we introduce the characteristic radius of fractal dust and derive the sublimation distance for that dust. In Section 4, we simplify the formulae of enhancement factors derived by Kobayashi et al. (2009) and obtain the new formulae that show explicitly the dependence on materials and structures of the particles. We provide a recipe to use our analytical formulae in Section 5, apply our simplified formulae to both the solar system and extrasolar debris disks, and discuss observational possibilities of dust sublimation in Section 6. We summarize our findings in Section 7.
2. Sublimation Temperature
A particle generated in a dust source initially spirals toward a star by the P-R effect. As it approaches the star due to the P-R inward drift, its temperature rises high and it finally starts active sublimation. The drift turns outward by sublimation when β increases with mass loss. The radial motion of the particles becomes much slower than the P-R drift alone, resulting in a pile-up of the particles. Note that other mass-loss mechanisms such as sputtering by stellar winds and UV radiation are negligible during active sublimation. 1 1
Here, we set m = 1.1 × 10−12g, α = 1/2, η = 1/3, Tsub = 1300 K and a = 15R⊙ with the solar radius R⊙ in the argument of the logarithmic function under the assumption of a spherical olivine dust particle around the sun; the other choice of these values does not change the result significantly because of the slowly-varying properties of the logarithmic function.
Material parameters: the material density, μ is the mean molecular weight, H is the latent heat of sublimation, and P0 is the saturated vapor pressure Pv in the limit of high temperature.
material density [g cm 3]
H [erg g−]
P0 [dyn cm−2]
3.21 × 1010
6.72 × 1014
9.60 × 1010
3.12 × 1011
7.12 × 1010
1.07 × 1014
7.27 × 1011
4.31 × 1016
2.97 × 1010a
5.00 × 104a
2.83 × 1010b
3.08 × 1013b
2.83 × 1010
2.67 × 1013b
3. Fractal Dust Approximation
3.1 Sublimation distance
4. Enhancement Factor
Dust particles with mass m0 in the range from m0min to m0max are mainly controlled by the P-R drag in their source and therefore spiral into the sublimation zone. As mentioned above, the smallest drifting dust with m0min corresponds to α = 1/2. If the drifting timescale of dust particles due to the P-R drag tPR is much shorter than the timescale of their mutual, destructive collisions tcol, the particles can get out of the dust source region by the P-R drag. The ratio of tPR to tcol increases with mass or size. Large dust particles with are collisionally ground down prior to their inward drifts. Therefore, the largest dust m0 max considered here roughly satisfies the condition tPR ~ tcol at the source region.
In the steady state, the number density of drifting dust particles is inversely proportional to the drift velocity (e.g., Kobayashi et al., 2009). Since the drift velocity of dust particles due to the P-R drag is proportional to α, their mass distribution is affected by the mass dependence η = −dlnα/dlnm. If the differential mass distribution of the dust source is proportional to m−b, that of drifting dust is modulated to m−b+η (e.g., Moro-Martin and Malhotra, 2003). Provided that successive collisions mainly produce dust particles in the dust source, we have b = (11 + 3p)/(6 + 3p) for the steady state of collisional evolution, where (Kobayashi and Tanaka, 2010). Here, is the specific impact energy threshold for destructive collisions and v is the collisional velocity. From the hydro-dynamical simulations and laboratory experiments, to m0 for small dust particles (Holsapple, 1993; Benz and Asphaug, 1999). Since p = −0.2 to 0 for a constant v with mass, b is estimated to be 1.8–1.9. This means that the smallest particles contribute most to the number density before dust particles start to actively sublime, while the largest particles dominate the optical depth prior to active sublimation.
When the temperatures of dust particles reach Tsub, they start to sublime actively. Their do not vanish perfectly, but they have very small relative to the initial P-R drift velocity. The magnitude of a pile-up due to sublimation is determined by the ratio of these drift rates (Kobayashi et al., 2009). Because the drift rate at the sublimation zone is independent of the initial mass and the initial P-R drift rate decreases with dust mass, the initially small dust piles up effectively. As a result, both the number density and the optical depth at the sublimation zone are determined by the initially smallest dust.
Because x is proportional to Tsub/ρ (see Eq. (9)), the enhancement factors increase with Tsub/ρ. Thus, materials with high sublimation temperature tend to pile up sufficiently. In addition, fluffy dust particles with D ≃ 2 cannot effectively pile up even though ρ is low (Kimura et al., 1997). This is explained by the low α = D − 2 in the function g(x). Particles with D = 3 produce the highest enhancement factors. In spite of D = 3, compact particles are not the best for the pile-up due to a high density. Dust particles composed by ballistic particle-cluster aggregation have D ≃ 3 but low effective densities relative to compact ones. Therefore, such porous particles with D ≃ 3 may produce high enhancement factors due to large x resulting from their low densities. In addition, high x around a luminous star brings the enhancement factors to increase with stellar luminosity, which is shown for dirty ice, obsidian, and carbon in (Kobayashi et al. 2008, 2009).
We briefly show a recipe to obtain the sublimation temperature Tsub, its distance asub, and the enhancement factors fN, f. At first, the sublimation temperature Tsub is available from Eq. (5) adopting the material properties μ, H, and P0 listed in Table 1. Then, we calculate the dimensionless parameter x through Eq. (9), applying the stellar luminosity and mass of interest and the bulk density listed in Table 1 for compact spherical dust. Note that we should adopt a lower density for porous particles, taking into account their porosity. Inserting x in Eq. (13), we derive the sublimation distance asub. We further need orbital eccentricities e1 of dust particles at the beginning of active sublimation to calculate the enhancement factors, f N and f τ . The dust particles resulting from collisions have eccentricities e0 ~ α at the distance a0 of the dust production region. Since particles with the highest α contribute most to a sublimation ring, we estimate e0 ~ 0.5. Because eccentricities are dumped by the P-R drag, we can calculate e1 from the relation (Wyatt and Whipple, 1950). Inserting x and e1 to Eqs. (14) and (15), we obtain f N and f τ .
The asteroid belt and the Edgeworth-Kuiper belt (EKB) are possible dust sources in the solar system. A dust counter on board spacecraft can measure the number density of dust particles. The sublimation of icy dust occurs at asub = 20 AU given by Eq. (13). Icy particles reaching the sublimation zone from the EKB still have high eccentricities e1≳ 0.1 (Kobayashi et al., 2008). Therefore, a substantial sublimation ring is unexpected because of f N = 1 for h(e1) = 0. Since the number density of dust particles decreases inside the sublimation zone, only a bump in the radial profile of the number density appears around asub (see Kobayashi et al., 2010). In contrast, dust particles composed of rocky, refractory materials actively sublime at several solar radii from the sun. Therefore, orbital eccentricities of dust particles coming from the asteroid belt drop to ~0.01 around the sublimation distance due to the P-R drag. Since we have h (0.01) ≃ 0.3 from Eq. (17) for b = 11/6 and D = 3, f N ≃ 1.3–3.0 is obtained from Eq. (14) for x ≳ 1. Therefore, future in-situ measurements of dust could find such a sublimation ring of refractory dust particles originating from the asteroid belt, but not a ring of icy dust particles from the EKB.
A dust ring was observed around 4R⊙ from the sun in the period 1966–1983, although it was not detected in the 1990s (Kimura and Mann, 1998 for a review). The optical depth is measured by dust emission observations. The enhancement factor fτ of the observed ring is estimated to be 2-3 (MacQueen, 1968; Mizutani et al., 1984). The mass distribution of drifting dust particles for b = 11/6 is consistent with the measurement of dust particles with masses ranging from m0min ~ 10−12g to m0max ~ 10−6g by spacecraft around the earth orbit (Grün et al., 1985). Using that, we estimate f τ ≲ 1.1 from Eq. (15) for e1 = 0.01. Thus, the enhancement by sublimation cannot account for the observed dust ring. However, this low f τ is mainly caused by the mass range of drifting dust particles. If m0 max decreased during the transport of dust particles from the earth’s orbit to the sublimation zone, higher f τ could be expected. For example, the largest particles in the mass distribution become smaller by the sputtering from the solar wind. The sputtering may decrease e1 as well as m0 max. Small particles with high eccentricities from the dust source are ground down by sputtering and blown out by the radiation pressure before reaching the sublimation zone, while large particles with low eccentricities gradually become small by sputtering without the increase of their eccentricities and drift into the sublimation zone. If the ratio of tPR to the timescale of decreasing size due to sputtering ranges in 0.1– 0.7, sublimation could form such a bright ring because of small m0max and e1. Indeed, the ratio derived by Mukai and Schwehm (1981) is consistent with the condition for the formation of a sublimation ring. That may be a clue to explain the observed ring.
Debris disks found around main sequence stars would be formed through collisional fragmentation in narrow planetesimal belts, which may resemble the asteroid and Edgeworth-Kuiper belts in the solar system. In young debris disks, fragments produced by successive collisions are removed from the disk by radiation pressure. We call such a disk a collision-dominated disk. Once the amount of bodies has significantly been decreased through this process, the P-R drag becomes the main removal process of fragments. Such a disk is referred to as a drag-dominated disk. We have investigated the dust ring formation in drag-dominated disks. To observe a sublimation ring requires a high enhancement factor f τ for the optical depth. As shown in Eq. (15), a small ratio of m0max to m0min yields high f τ . The condition of m0max ~ m0min is expected to form a bright ring. Since the drift time due to the P-R drag is comparable to the collisional time for bodies with m0max, the condition of m0max ~ m0min is achieved in transition from a collision-dominated disk to a drag-dominated one.
A significant sublimation ring consisting of icy particles is not expected in a debris disk due to high eccentricities if a planetesimal belt as a dust source is located within a few hundreds AU, similar to the solar system. On the contrary, dust particles composed of refractory materials have e1 ~ 0.01 or smaller if a planetesimal belt is around the distance of the asteroid belt or further outside. Furthermore, refractory dust particles have high sublimation temperatures and hence produce a higher enhancement factor. Recently, inner debris disks of refractory grains have been observed through interferometry around Vega, τ Cet, ζ Aql, and α Leo, and Formalhaut (Absil et al., 2006, 2008; Di Folco et al., 2007; Akeson et al., 2009). Such inner debris disks may have notable sublimation rings.
We provide formulae of enhancement factors for the number density and the optical depth due to a pile-up of dust particles caused by sublimation and its location and sublimation temperature, applying a simple model for optical cross sections of fractal dust particles.
High sublimation temperatures result in substantial enhancement factors, though the pile-up is insensitive to the optical properties of dust particles.
If we adopt the mass distribution measured around the earth, the enhancement factor for the optical depth near the sun is smaller than 1.1. Therefore, the enhancement cannot explain the solar dust ring detected in the epoch of 1966–1983, unless the largest particles were destroyed by sputtering.
The number-density enhancement factor is expected to be 1.4–3 in the vicinity of the sun. Therefore, a sublimation ring could be found by in-situ measurements by spacecraft around several solar radii from the sun.
Sublimation removes dust particles with temperatures higher than their sublimation temperature. In the spectral energy distribution, the flux density from a disk reduces with decreasing wavelength, if the wavelength is shorter than the peak one of the blackbody spectrum with the sublimation temperature. That could be seen as a sublimation evidence. However, it is difficult to find signs from a sublimation ring in the spectral energy distribution.
1We consider dust particles that can drift into their active sublimation zone. This is valid in the solar system, since the size decreasing timescale due to sputtering is longer than the drift time due to the P-R effect (Mukai and Schwehm, 1981). However, we note that icy particles may not come to their sublimation zone around highly luminous stars because of strong UV sputtering (Grigorieva et al., 2007).
2We apply the complex refractive indices of olivine from Huffman (1976) and Mukai and Koike (1990), of pyroxene from Huffman and Stapp (1971), Hiroi and Takeda (1990), Roush et al. (1991), and Henning and Mutschke (1997), of obsidian from Lamy (1978) and Pollak et al. (1973), of carbon from Hanner (1987), of iron from Johnson and Christy (1974) and Ordal et al. (1988), of ice and dirty ice from Warren (1984) and Li and Greenberg (1997).
3Note that τ0 depends on the collision and drift timescales, t c , tPR, in a planetesimal disk. Drag-dominated disks should satisfy the condition with the Keplarian velocity v k and the ratio γ of the P-R drag force due to the stellar wind to that due to the stellar radiation. The value of τ0 applied for fitting is much larger than that for the drag-dominated disk (τ0 ≲ 3 × 10−4), if we only consider the P-R drag by the stellar radiation. However, the mass loss rate of Vega is estimated to be less than 3.4 × 10−M⊙ yr−1 from radio-continuum observations (Hollis et al., 1985). For the upper limit of the mass loss rate, drag-dominated disks can have the optical depth τ0 ≲ 7 × 10−2 because of the P-R drag due to the stellar wind (γ ~ 300). Thus the disk around Vega may be a drag-dominated disk.
4The scattering of light from the disk around Vega is negligible around the wavelength ~1 μ m (Absil et al., 2006).
We appreciate the advice and encouragement of A. Krivov, M. Ilgner, and M. Reidemeister. The careful reading of the manuscript by the anonymous reviewers helps its improvement. This research is supported by grants from CPS, JSPS, and MEXT Japan.
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