PHOTOMETRIC MEASUREMENTS OF H₂O ICE CRYSTALLINITY ON TRANS-NEPTUNIAN OBJECTS^*

From the measured $H-$ NB1657 index, we obtain the fraction of the crystalline phase in the H₂O ice spectrum, hereinafter called the "crystallinity factor" (f_crys). In this paper, the reflectance spectrum of phase-mixed H₂O ice, ${S}_{{{\rm{H}}}_{2}{\rm{O}}}(\lambda )$ (λ is wavelength in μm), is represented by a linear sum of the crystalline and amorphous spectra (${S}_{{\rm{crys}}}(\lambda )$ and S_amor(λ), respectively) as defined in Newman et al. (2008), i.e.,

$$\begin{eqnarray}&&{S}_{{{\rm{H}}}_{2}{\rm{O}}}(\lambda )={f}_{{\rm{crys}}}\,{S}_{{\rm{crys}}}(\lambda )+(1-{f}_{{\rm{crys}}})\,{S}_{{\rm{amor}}}(\lambda ).\end{eqnarray} \tag{ 1 }$$

Modeling reflectance spectra of the target objects is required to convert the 1.65 μm absorption strength derived from Δ($H-$ NB1657) into the crystallinity factor.

We assume that near-infrared spectra of those bodies are represented by a simple model consisting of the H₂O ice spectrum and a linear continuum presented by Brown et al. (2012). The model spectrum, S(λ), is given as

$$\begin{eqnarray}S(\lambda ) & = & {f}_{{{\rm{H}}}_{2}{\rm{O}}}\,{S}_{{{\rm{H}}}_{2}{\rm{O}}}(\lambda )+(1-{f}_{{{\rm{H}}}_{2}{\rm{O}}})\\ & & \times [{m}_{{\rm{cont}}}(\lambda -1.74\,\mu {\rm{m}})+0.49],\end{eqnarray} \tag{ 2 }$$

where ${f}_{{{\rm{H}}}_{2}{\rm{O}}}$ is the spectral fraction of H₂O ice and m_cont is a continuum slope. The H₂O ice spectra were generated from the geometric albedo A_p described by the radiative transfer model of Hapke (1993) as

$$\begin{eqnarray}&&{A}_{p}\simeq {r}_{0}\left(\displaystyle \frac{1}{2}+\displaystyle \frac{1}{6}{r}_{0}\right)+\displaystyle \frac{w}{8}[(1+{B}_{0})p(0)-1],\end{eqnarray} \tag{ 3 }$$

where w is the single-scattering albedo given from the optical constants and grain size, r₀ is the diffusive reflectance given by $\tfrac{1-\sqrt{1-w}}{1+\sqrt{1-w}}$, B₀ is the total amplitude of the opposition surge, and p(0) is the phase function at zero phase angle. We used B₀ = 0.67, a typical value among icy satellites (Verbiscer & Helfenstein 1998, p. 157) as in Merlin et al. (2009), and isotopic scattering, i.e., p(0) = 1.0. The optical constants of amorphous and crystalline H₂O ices were derived from the laboratory data provided by Mastrapa et al. (2008), Mastrapa (2012).

The grain size is one of the most sensitive parameters for determining the reflectance spectrum but still unknown. We tentatively assumed uniform grain particles with a diameter of d = 50 μm as in Brown et al. (2012). The uncertainty due to the dependence on grain size is discussed in Section 3.3.

The absorption spectrum of H₂O ice, especially the 1.56 μm and 1.65 μm bands in crystalline H₂O ice, varies with temperature (e.g., Grundy & Schmitt 1998). The surface temperature of airless solid bodies depends on the solar flux, rotation, and surface properties. Stansberry et al. (2008, p. 161) presented temperatures of 49 TNOs/Centaurs derived from 24 μm and 70 μm flux data collected by the Spitzer Space Telescope. Figure 2 shows the color temperatures as a function of target distance from the Sun at the time of the observations. The plot is well approximated by the equation

$$\begin{eqnarray}&&T=(389\pm 14){r}_{{\rm{h}}}^{-1/2}+(-7.1\pm 3.3),\end{eqnarray} \tag{ 4 }$$

where T and ${r}_{{\rm{h}}}$ are the temperature in kelvin and the heliocentric distance in au, respectively. Here, we consider the thermal temperatures derived from this equation assuming isothermal blackbodies to be their ice temperatures. The appropriateness of this is assessed below. Mastrapa's data set contains the optical constants of crystalline H₂O ice at temperatures from 20–150 K every 10 K and those of amorphous H₂O ice at temperatures higher/lower than 70 K. Based on the estimated temperatures, we drew the spectral reflectance of crystalline/amorphous ices from the optimum optical constant data for each target object as seen in Table 4. The model spectra of the target objects were generated from Equation (2) with the synthetic spectra of H₂O ice as mixtures of crystalline and amorphous phases given by Equation (1).

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Table 4.  Estimated Crystallinity Factors with Grain Size of 50 μm Diameter

Object	rha	Tb	H2O Ice Spectrac	Crystallinity Factord
	(au)	(K)
(90482) Orcus	48.0	49	crys_50K, amorph_low	${0.53}_{-0.09}^{+0.08}$
(315530) 2008 AP129	37.9	56	crys_60K, amorph_low	${1.00}_{-0.47}^{+0.00}$
(42355) Typhon	19.1	82	crys_80K, amorph_high	${0.79}_{-0.27}^{+0.21}$
(136108) Haumea	50.8	48	crys_50K, amorph_low	${0.77}_{-0.05}^{+0.06}$
(38628) Huya	28.6	66	crys_70K, amorph_low	${0.93}_{-0.93}^{+0.07}$
(50000) Quaoar	43.1	52	crys_50K, amorph_low	${0.82}_{-0.15}^{+0.15}$

Notes.

^aHeliocentric distance at the time of observation. ^bThermal temperature from Equation (4). ^cThe optical constant data set derived from laboratory spectra provided by Mastrapa et al. (2008). ^dThe errors show the 1σ uncertainty.

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The spectral fraction of H₂O ice ${f}_{{{\rm{H}}}_{2}{\rm{O}}}$ and the continuum slope m_cont were assumed to be those suggested by Brown et al. (2012) as listed in Table 1. Note that these parameters have been estimated from simple modeling with the optical constants of fully crystallized ice. We evaluated the variability of ${f}_{{{\rm{H}}}_{2}{\rm{O}}}$ determined through the model depending on crystallinity. The synthetic spectra from Equation (2) were compared between pure crystalline ice and amorphous-dominated/mixed ice by least-squares optimization under the same conditions as Brown et al. (2012), i.e., temperatures of 50 K, grain size of 50 μm, and wavelength ranges of 1.45–1.80 μm and 1.95–2.30 μm. Table 5 shows the best-fit ${f}_{{{\rm{H}}}_{2}{\rm{O}}}$ values for the spectra created with f_crys of 0.00, 0.25, and 0.50. Since there are only small differences from the given values for any f_crys, one can see that ${f}_{{{\rm{H}}}_{2}{\rm{O}}}$ has little dependence on crystallinity. This indicates the validity of our crystallinity measurement based on the published ${f}_{{{\rm{H}}}_{2}{\rm{O}}}$ values.

Table 5.  Dependence of Estimated H₂O Ice Fraction ${f}_{{{\rm{H}}}_{2}{\rm{O}}}$ on Crystallinity f_crys

	fcrys	${f}_{{{\rm{H}}}_{2}{\rm{O}}}$
Given	1.00	0.70	0.50	0.30	0.10
	0.50	0.69	0.50	0.30	0.10
Best fit	0.25	0.69	0.49	0.29	0.10
	0.00	0.68	0.48	0.29	0.10

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To compare with the photometric data, the model spectra were converted into Δ($H-$ NB1657) using

$$\begin{eqnarray}&&{\rm{\Delta }}(H-{\rm{NB1657}})=\\ &&2.5\times \mathrm{log}\left[\displaystyle \frac{\int {R}_{{\rm{NB}}}(\lambda )S(\lambda )\lambda {F}_{\lambda ,\odot }(\lambda )d\lambda /\int {R}_{H}(\lambda )S(\lambda )\lambda {F}_{\lambda ,\odot }(\lambda )d\lambda }{\int {R}_{{\rm{NB}}}(\lambda )\lambda {F}_{\lambda ,\odot }(\lambda )d\lambda /\int {R}_{H}(\lambda )\lambda {F}_{\lambda ,\odot }(\lambda )d\lambda }\right],\end{eqnarray} \tag{ 5 }$$

where R_H(λ) and R_NB(λ) are the response functions for the H and NB1657 bands, respectively. ${F}_{\lambda ,\odot }(\lambda )$ is the wavelength flux density of the Sun. The response functions include atmospheric transmission at the summit of Maunakea generated by ATRAN modeling software (Lord 1992) assuming an airmass of 1.0 and water vapor column of 1.0 mm. We used the solar reference spectrum distributed by STScI Calibration Database System.⁷

By comparing the obtained Δ($H-$ NB1657) index with the model spectra, we determined the crystallinity factors of H₂O ice for each target object. Figure 3 shows Δ($H-$ NB1657) derived from our observation and modeling with respect to the crystallinity factor. The point where the model curve intersects with the measured value represents a plausible crystallinity factor. The resulting crystallinity factors are listed in Table 4. The accuracy of determination depends on photometric precision and the slope of the model curve given by the abundance of H₂O ice. For bright objects including Orcus, Haumea, and Quaoar, the crystallinity factor has been fixed with an accuracy of ∼0.1. Huya is also bright, but the error is very large due to low ${f}_{{{\rm{H}}}_{2}{\rm{O}}}$ (0.08 ± 0.02) causing little variation in Δ($H-$ NB1657) with crystallinity factor. This implies that ${f}_{{{\rm{H}}}_{2}{\rm{O}}}\sim 0.1$ is the limit of application of this technique for TNOs in Subaru/MOIRCS observation.

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Under hypothetical ice conditions with a temperature based on thermal flux and a grain size of 50 μm, the model matching indicates abundant crystalline H₂O ice on the surfaces of Haumea and Quaoar, as well as a moderate amount of the crystalline phase on Orcus. This result agrees with previous spectroscopic works showing a significant feature of the 1.65 μm absorption on those objects (e.g., Jewitt & Luu 2004; de Bergh et al. 2005; Trujillo et al. 2007). Typhon also shows high crystallinity, consistent with the spectral model presented by Guilbert et al. (2009) (5% crystalline and 0% amorphous).

We evaluated the validity of this technique through comparison with published near-infrared spectra of Haumea and Quaoar obtained with Keck/NIRC⁸ (Barkume et al. 2008). Figure 4 shows those spectra as well as the spectral models generated from Equation (2) with the crystallinities measured by this work. The models reproduce the observed spectra well, including the 1.65 μm feature in both objects. This supports the idea that our photometric method is useful in constraining the ratio of crystalline to amorphous phases for an icy small body whose ${f}_{{{\rm{H}}}_{2}{\rm{O}}}$ is known. The effect of the assumed ice properties, i.e., temperature and grain size, on the determination of the crystallinity factor is examined in the following section.

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The depth and center wavelength of the 1.65 μm band vary sensitively with temperature and grain size (Fink & Larson 1975; Clark 1981; Grundy & Schmitt 1998; Taffin et al. 2012); the two parameters were given as the thermal temperature estimated from Equation (4) and d = 50 μm, respectively, in the above modeling. Grundy et al. (1999) presented disk-averaged H₂O ice temperatures of Jovian, Saturnian, and Uranian satellites derived from their near-infrared spectra. They pointed out that the ice temperature is generally lower than the brightness temperature derived from thermal emission, which is sensitive to warm regions on the surface with a typical difference of ∼10 K. The target objects could actually have a lower surface temperature.

We conducted additional modeling with the same processes as presented in Section 3.1 but using the optical constant data of crystalline H₂O ice at temperatures 10 K lower than those in Table 4. The results, shown in Table 6, indicate no significant change in the obtained crystallinity factor compared with the original model spectrum in any objects. Such a difference in the given temperature causes a systematic error of no more than ∼10% for our crystallinity measurements.

Table 6.  Estimated Crystallinity Factors with Lower Temperatures

Object	Ta	H2O Ice Spectrab	Crystallinity Factorc
	(K)
(90482) Orcus	42	crys_40K, amorph_low	${0.51}_{-0.08}^{+0.09}$
(315530) 2008 AP129	48	crys_50K, amorph_low	${1.00}_{-0.47}^{+0.00}$
(42355) Typhon	70	crys_70K, amorph_high	${0.72}_{-0.24}^{+0.24}$
(136108) Haumea	41	crys_40K, amorph_low	${0.75}_{-0.05}^{+0.06}$
(38628) Huya	56	crys_60K, amorph_low	${0.89}_{-0.89}^{+0.11}$
(50000) Quaoar	44	crys_40K, amorph_low	${0.80}_{-0.15}^{+0.15}$

Notes.

^aReduced thermal temperature (see text). ^bThe optical constant data set derived from laboratory spectra provided by Mastrapa et al. (2008). ^cThe errors show the 1σ uncertainty.

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On the other hand, grain size can induce considerable uncertainty in the modeling. Although the dominant size of the ice grains on the surface layer of TNOs is still unknown, Barucci et al. (2011) reported models of the surface composition of 12 TNOs/Centaurs based on their near-infrared spectroscopy data, showing that most of their surfaces contain H₂O ice (in crystalline and/or amorphous states) with particle diameters of ∼10–200 μm. Many of the previous studies of TNO/Centaur spectra showed the best-matched grain sizes to be in this range. Note that H₂O ice with larger grains enhances the absorption coefficients, but the enhancement saturates at the diameter of ∼1000 μm. In contrast, if the grains are smaller than ∼10 μm, the absorptions are too weak to measure the 1.65 μm feature via the Δ($H-$NB1657) index. Thus, our technique is applicable to objects whose surfaces are dominated by H₂O ice with grain sizes of ∼10–1000 μm.

We examined the variation in crystallinity factor for our observed objects among the spectral models with grain sizes from 10 to 200 μm in diameter (see Figure 5). A uniform size between the crystalline and amorphous particles was assumed in each pattern. The results are shown in Figure 6. One can see that the crystallinity factor decreases monotonically with increasing grain size. This is because of the optical characteristic of H₂O ice whereby larger grains increase the amount of 1.65 μm absorption rather than that of the entire absorption covered by the H band, as pointed out in Grundy & Schmitt (1998). Although the crystallinity factor varies greatly between the smallest and largest grain sizes, it remains larger than ∼0.5 to within the error in Haumea, Quaoar, Typhon, and 2008 AP₁₂₉. Orcus has a slightly lower crystallinity in the case of the largest grain sizes, but definitely contains a certain level of crystalline ice. This result indicates that all of the five objects whose crystallinity has been precisely determined are likely to be covered by a surface containing crystalline-dominant H₂O ice or one comparable to it.

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The model spectrum given by Equation (2) approximates the surface reflectance spectrum of the target objects as a combination of H₂O ice and other materials for which the total spectrum shows a continuum with a linear slope and no absorption feature. For the H-band data, this assumption seems to be reasonable because most non-H₂O ices that potentially exist on the TNOs' surface, e.g., CH₃OH, CO, CO₂, N₂, and NH₃, have no or only slight absorptions over the wavelength range (Cruikshank et al. 1984); Gerakines et al. 2005. However, CH₄ ice could be influential in determining H₂O ice crystallinity because it exhibits several absorption features from 1.5 μm to 1.8 μm (Pearl et al. 1991). In particular, the strong absorption band at 1.67 μm possibly makes a significant negative contribution to the Δ($H-$ NB1657) index.

Various near-infrared spectroscopic studies suggest the presence of CH₄ ice on several TNOs with an H₂O ice-rich surface including Quaoar (Schaller & Brown 2007; Dalle Ore et al. 2009) and Orcus (Barucci et al. 2008; Delsanti et al. 2010; Carry et al. 2011). According to previous studies of spectra from the two objects, the average fraction of CH₄ ice is ∼0.05, corresponding to CH₄/H₂O ∼ 0.15. To evaluate the effect of CH₄ ice mixing on measurements of H₂O ice crystallinity, we calculated the Δ($H-$NB1657) index with several patterns of the particle number ratio (CH₄/H₂O = 0.05–0.20) and size of CH₄ ice grains (d = 10, 50, and 100 μm). The optical constants of CH₄ ice used for modeling the spectrum are as follows: (i) the real part of the refractive index n is given by $n=1.38-0.03\lambda$ (0.67 μm $\leqslant$ λ $\leqslant$ 2.0 μm; de Bergh et al. 2008, p. 483), (ii) the imaginary part of the refractive index k is converted from the absorption coefficients of phase I CH₄ ice (stable between 20 K and 90 K) presented by Grundy et al. (2002). The synthetic model spectra were generated from Equation (2) by replacing the reflectance spectrum of H₂O ice, ${S}_{{{\rm{H}}}_{2}{\rm{O}}}$, with that of mixed ices of H₂O and CH₄ derived from the intimate mixture model developed by Hapke (1993). The grain size of H₂O ice was fixed to d = 50 μm.

The results are shown in Figure 7. If CH₄ ice grains are of equal size to or smaller than H₂O ice grains, the H₂O ice crystallinity factors are larger than 0.50 for Quaoar and 0.25 for Orcus. The situation is similar if CH₄/H₂O ≲ 0.05. In contrast, the absorption due to the larger-grain CH₄ ice on the surface with CH₄/H₂O ≳ 0.1 dominates at ∼1.65 μm and could cause great uncertainty in the H₂O ice crystallinity.

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We recognize that Dalle Ore et al. (2009) and Carry et al. (2011) provide the most reliable spectral modeling for Quaoar and Orcus, respectively, among the published works. The former suggests that Quaoar's surface contains fine CH₄ ice grains of d ∼ 10 μm with a slightly high fraction of CH₄/H₂O ∼ 0.3. The latter suggests that Orcus's surface contains coarse CH₄ ice grains of d ∼ 100 μm with a tiny fraction of CH₄/H₂O < 0.03. If those parameters are simply applied to our modeling, they indicate that the potential presence of CH₄ ice is unlikely to induce significant overestimation of H₂O ice crystallinity in Quaoar and Orcus.

Prior to discussions of the thermal evolution, we should confirm the variations in surface temperature of the target objects over an orbit. Amorphous ice would be crystallized on surfaces that have been much warmer than 70 K over the timescale of the age of the solar system or less (Kouchi et al. 1994; Jenniskens & Blake 1996). Even if there is no heat source except solar insolation, the transition from amorphous to crystalline occurs on the surface of bodies sufficiently close to the Sun. All the target objects other than Typhon have perihelion distances larger than 28.5 au, which are converted into surface temperatures of less than 66 K by Equation (4). Assuming that there has been no significant migration since their formation, their surfaces have been cold enough for amorphous ice to be continuously stable.

However, Typhon has a high orbital eccentricity (0.54), allowing it to approach up to 17.5 au from the Sun. Around its perihelion, the sunward surface would be warmed up to ∼86 K, while the ordinary temperature is ∼50 K during most of its orbit. In the model of Jenniskens & Blake (1996), the time to onset of crystallization at that temperature is ∼10⁶ yr. The insolation heating can induce surface crystallization if Typhon has retained such an eccentric orbit over the age of the solar system and the intermittent contribution has accumulated monotonically.

The decay of radioactive isotopes contained in dust mixed with ice is the most important heat source for the internal thermal evolution of small icy bodies in the outer solar system. It allows the crystallization to proceed outward from the deep interior of the body.

In Prialnik & Podolak (1995), thermal evolutionary calculations of icy bodies larger than 40 km in diameter (D) with heat sources of ²⁶Al and four long-lived radioactive isotopes indicate that H₂O ice on the outer layer seems to be left in the amorphous form. Choi et al. (2002) also reported that ²⁶Al decay could not raise the surface temperature of TNOs with D = 20–1000 km higher than ∼50 K and not induce H₂O ice crystallization at the surface layers. Those models cannot explain the surface crystallization of objects with D > 100 km.

However, assuming that the objects were formed closer to the Sun than their present orbits, radiogenic heating can potentially crystallize the surface ice because of fast accretional growth, and thus sufficient radiogenic heating by ²⁶Al. The simulations of Merk & Prialnik (2006) showed that icy bodies above D ∼ 30 km originally located at around 20 au could be covered by a crystalline ice surface. McKinnon et al. (2008, p. 213) showed that if icy bodies formed at a heliocentric distance less than ∼32 au, then the crystalline/amorphous ice boundary reaches the surface.

More recent work by Guilbert-Lepoutre et al. (2011) presented three-dimensional simulations to compute the thermal evolution of icy small bodies with heat sources of several short- and long-lived radioactive isotopes. Their results suggest:

• 1.  
  When the growth advanced quickly (∼1 Myr) and short-lived isotope decay effectively contributes to the thermal evolution, objects with D > 100 km retain only a thin (< 2 km) layer of amorphous H₂O ice on their surface, which may possibly be removed by impact excavation. In addition, provided such objects are formed close enough to the Sun (10–15 au), crystallization would be triggered at the surface by insolation.
• 2.  
  When the thermal evolution is dominated by long-lived isotope decay, only large objects (D > 600 km) with a bulk density of at least 1.5 g cm⁻³ should contain crystalline H₂O ice close to the surface.

Our findings regarding the presence of crystalline H₂O ice on Typhon and 2008 AP₁₂₉ agree with the first scenario. In contrast, Typhon and possibly 2008 AP₁₂₉ (see Table 1) are inconsistent with the second scenario. Based on the model of Guilbert-Lepoutre et al. (2011), at least those small objects should have grown in a sufficiently short timescale for the surfaces to be heated up to the crystallization temperature by the decay of short-lived isotopes unless they were formed close to the Sun (10–15 au).

Such a rapid formation as to induce sufficient heating for surface ice crystallization due to the decay of short-lived isotopes can also occur at a smaller heliocentric distance than the current location. Actually, several theoretical studies support this model. Weidenschilling (2004, p. 97) presented coagulation simulations of planetesimal growth in the outer solar system and showed the formation of objects with D > 100 km only inside ∼30 au. More recently, Kenyon & Bromley (2012) performed N-body coagulation calculations for the formation and evolution of TNOs, indicating that objects at 15–18 au grow up to 100 km in diameter within ∼1 Myr at earliest, while it takes more than 10 Myr at 33–40 au. Outward transport of planetesimals from the inner regions via planetary migration has been believed to take place based on numerical simulations for the dynamical evolution of giant planets and TNOs (e.g., Gomes 2003; Levison & Morbidelli 2003; Hahn & Malhotra 2005; Levison et al. 2008). Rapid growth around 20 au from the Sun is a plausible mechanism for crystallization of surface ice from the early radioactive heating on TNOs as small as Typhon.

The thermal evolution model may also be able to explain the low fraction of crystalline ice on Orcus compared with Haumea and Quaoar. One of the important parameters that the heating rate depends on is the bulk density of the body. A higher density increases the abundance of dust materials containing radioactive isotopes and thus enhances the heating efficiency. The difference in bulk density possibly causes the variation in crystallinity among objects of comparable size. Guilbert-Lepoutre et al. (2011) suggest that a density of at least 1.5 g cm⁻³ is required for the presence of crystalline ice on the surface of objects with D ≳ 600 km. Haumea is well known to have a high density (∼ 2.6 g cm⁻³: Rabinowitz et al. 2006; Lacerda & Jewitt 2007). Quaoar's density has also been estimated to be as high as ${2.18}_{-0.36}^{+0.43}$ g cm⁻³ from thermal flux (Fornasier et al. 2013) or 1.99  ±  0.46 g cm⁻³ from occultation (Braga-Ribas et al. 2013). In contrast, Orcus exhibits an obviously lower density of ∼1.5 g cm⁻³ (Brown et al. 2010), just about the same value as the criterion for surface ice crystallization shown by Guilbert-Lepoutre et al. (2011). This fact suggests the possibility that temperature of the upper layer of Orcus failed to rise sufficiently to crystallize most of the surface H₂O ice due to a deficiency in the amount of radioactive isotopes. The potential correlation between crystallinity and bulk density could support the hypothesis that radiogenic heating is the dominant effect on surface crystallization for TNOs.

Some surface ice may not have been formed in situ but may have been supplied from the interior by geological activity of volcanic eruptions of ice, so-called cryovolcanism. Actually, atmospheric plumes have been seen on several icy satellites of giant planets, such as Triton (Smith et al. 1989; Kargel 1994), Enceladus (Porco et al. 2006; Spencer et al. 2009, p. 683), and Europa (Roth et al. 2014), as possible evidence of cryovolcanic events. The occurrence of a substantial flow of liquid H₂O to the surface requires some sort of continual heating process to melt the subsurface ice. For the satellites, tidal energy dissipation is likely to be a major heat source (e.g., Ruiz & Tejero 2000). Even though inclusion of NH₃ in H₂O ice allows the melting point to drop from 273 K to 176 K at minimum (Kargel et al. 1991) and decreases the thermal conductivity, it is still uncertain whether the cryovolcanic activity can be driven on TNOs primarily by long-lived radioactive isotopes without tidal heating.

Cook et al. (2007) suggested that a body with D > 1200 km composed of rock with a mass fraction ≳ 0.7 (corresponding to a bulk density ≳1.8 g cm⁻³) and NH₃-rich H₂O ice (∼0.15 by weight) could develop cryovolcanism and be steadily resurfaced by crystalline H₂O ice at a sufficient rate to be detected at near-infrared wavelengths. Haumea, with a mean diameter of 1200–1300 km (Lellouch et al. 2010; Fornasier et al. 2013) and a density of 2.6 g cm⁻³ (Lockwood et al. 2014), satisfies those conditions. Quaoar is slightly smaller than the required size (∼1000–1100 km) and has a high density (1.99 ± 0.46 g cm⁻³: Braga-Ribas et al. 2013; ${2.18}_{-0.36}^{+0.43}$ g cm⁻³: Fornasier et al. 2013). Orcus is likely to be comparable in size to Quaoar (∼850–950 km: Brown et al. 2010; Lim et al. 2010; Fornasier et al. 2013), while its density is relatively low (${1.53}_{-0.13}^{+0.15}$ g cm⁻³: Fornasier et al. 2013). As pointed out by Desch et al. (2009), ice containing CH₃OH along with NH₃ freezes at a lower temperature (153 K; Kargel 1994) and therefore is likely to reduce the minimum diameter of a TNO that can retain liquid H₂O to 800–1000 km. Cryovolcanism may be active on Quaoar and Orcus if they consist of CH₃OH–NH₃–H₂O ice.

Unfortunately, our results cannot provide explicit constraints on the source of crystalline H₂O ice at the surfaces of Haumea, Quaoar, and Orcus. On the other hand, Typhon and 2008 AP₁₂₉ are likely to be too small to exhibit cryovolcanic activity even if antifreeze compounds such as NH₃ and CH₃OH are sufficiently mixed in the subsurface ice.

Instead of the internal heat sources as mentioned above, H₂O ice crystallization on surfaces of airless bodies in the trans-neptunian region could be induced by impacts of meteorites such as interplanetary dust particles (IDPs). Cook et al. (2007) assessed possible mechanisms with micrometeorites to explain the presence of crystalline H₂O ice on Charon's surface, namely, impact gardening and annealing. They approximated the mass flux of IDPs from Pioneer 10 observations as 2.4 × 10⁻¹⁷ kg s⁻¹ m⁻² at 18 au and concluded that those mechanisms make little contribution to surface renewal on either Charon or other TNOs. In contrast, Porter et al. (2010) expected the IDP flux at Uranus to be 1.2 × 10⁻¹⁶ kg s⁻¹ m⁻² and claimed that impact annealing could be effective for surface crystallization on TNOs.

As seen in Table 2 of Porter et al. (2010), the impact velocity of IDPs and required dust flux for annealing are similar among objects in the main trans-neptunian belt. Assuming a homogeneous IDP distribution over the region, the surface crystallinity would be almost uniform among TNOs if micrometeorite impacts are the primary factor. However, considering our result that indicates a low crystallinity of Orcus compared with Haumea and Quaoar, micrometeorite annealing seems not to be a major mechanism of surface crystallization, at least for icy objects as large as Orcus.

Finally, we focus on the unique situation of 2008 AP₁₂₉, which has an orbit consistent with the Haumea collisional family (Brown et al. 2007). The velocity relative to the estimated center of mass of the collision, ∼140 m s⁻¹, also agrees with the velocity dispersion of the known family members (50–300 m s⁻¹), suggesting a dynamical association with the Haumea family (Volk & Malhotra 2012). The spectra of family members uniquely show significantly high fractions of H₂O ice as well as the crystalline feature (Barkume et al. 2008; Barucci et al. 2011; Brown et al. 2012). However, 2008 AP₁₂₉ contains only a minor fraction of H₂O ice (${f}_{{{\rm{H}}}_{2}{\rm{O}}}$ = 0.14 ± 0.09: Brown et al. 2012), which prevents characterization of the membership. If 2008 AP₁₂₉ originates from a collisional fragment as an ice-poor family member, the presence of crystalline H₂O ice on the surface is natural because the body is likely to derive from the icy mantle of Haumea. The lack of ice could be explained by inhomogeneous compositions of the partial differentiation (Volk & Malhotra 2012) although the actuality is still unknown.