Destruction of Molecular Hydrogen Ice and Implications for 1I/2017 U1 ('Oumuamua)

Heating by starlight and cosmic microwave background (CMB) radiation raises the surface temperature of H₂ ice. We assume that the local interstellar radiation field (ISRF) has the same spectrum as the ISRF in the solar neighborhood (Mathis et al. 1983) with a total radiation energy density of ${u}_{\mathrm{MMP}}\approx 8.64\times {10}^{-13}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$. We calibrate the strength of the local radiation field by the dimensionless parameter, U, so that the local energy density is ${u}_{\mathrm{rad}}={{Uu}}_{\mathrm{MMP}}$. The CMB radiation is blackbody with a temperature ${T}_{\mathrm{CMB}}=2.725(1+z)\,{\rm{K}}$ at a redshift z, and the radiation energy density is u_CMB = $\int 4\pi {B}_{\nu }({T}_{\mathrm{CMB}})$ dν/c = 4 $\sigma {T}_{\mathrm{CMB}}^{4}/c\approx 4.17\,\times \,{10}^{-13}{(1+z)}^{4}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$. At present, heating by the CMB is less important than by starlight.

The characteristic timescale for the evaporation of an H₂ molecule from a surface of temperature ${T}_{\mathrm{ice}}$ is

$$\begin{eqnarray}&&{\tau }_{\mathrm{sub}}={\nu }_{0}^{-1}\exp \left(\displaystyle \frac{{E}_{b}}{{{kT}}_{\mathrm{ice}}}\right),\end{eqnarray} \tag{ 1 }$$

where ${\nu }_{0}$ is the characteristic oscillation frequency of the H₂ lattice (Watson & Salpeter 1972). We adopt ${\nu }_{0}={10}^{12}\,{{\rm{s}}}^{-1}$ for H₂ ice (Hegyi & Olive 1986; Sandford & Allamandola 1993).

Assuming that the H₂ ice has a layered structure, the sublimation rate for an H₂ object of radius R is given by

$$\begin{eqnarray}&&\displaystyle \frac{{dR}}{{dt}}=-\displaystyle \frac{{\nu }_{0}}{{n}_{\mathrm{ice}}^{1/3}}\exp \left(-\displaystyle \frac{{E}_{b}}{{{kT}}_{\mathrm{ice}}}\right),\end{eqnarray} \tag{ 2 }$$

where ${n}_{\mathrm{ice}}\approx 3\times {10}^{22}\,{\mathrm{cm}}^{-3}$ is the molecular number density of H₂ ice with a mass density of ${\rho }_{\mathrm{ice}}=0.1\,{\rm{g}}\,{\mathrm{cm}}^{-3}$.

The sublimation time is then

$$\begin{eqnarray}&&{t}_{\mathrm{sub}}({T}_{\mathrm{ice}})=-\displaystyle \frac{R}{{dR}/{dt}}=\displaystyle \frac{{{Rn}}_{\mathrm{ice}}^{1/3}}{{\nu }_{0}}\exp \left(\displaystyle \frac{{E}_{b}}{{{kT}}_{\mathrm{ice}}}\right),\end{eqnarray} \tag{ 3 }$$

where dR/dt was substituted from Equation (2).

Plugging the numerical parameters into the above equation, we obtain

$$\begin{eqnarray}\begin{array}{rcl}{t}_{\mathrm{sub}}({T}_{\mathrm{ice}}) & \simeq & 2.95\times {10}^{7}\left(\displaystyle \frac{R}{1\,\mathrm{km}}\right)\\ & & \times \,\exp \left(100\,{\rm{K}}\left[\displaystyle \frac{1}{{T}_{\mathrm{ice}}}-\displaystyle \frac{1}{3\,{\rm{K}}}\right]\right)\mathrm{yr}\end{array}\end{eqnarray} \tag{ 4 }$$

for H₂ ice, and the ice temperature is normalized to ${T}_{\mathrm{ice}}=3\,{\rm{K}}$ expecting that starlight raises the surface temperature above ${T}_{\mathrm{CMB}}$. At the minimum temperature of the present-day CMB radiation, ${T}_{\mathrm{obj}}=2.725$ K, the sublimation time is ${t}_{\mathrm{sub}}\approx 0.85\,\mathrm{Gyr}$ for $R=1\,\mathrm{km}$.

The heating rate due to absorption of isotropic interstellar radiation and CMB photons is given by

$$\begin{eqnarray}&&\displaystyle \frac{{{dE}}_{\mathrm{abs}}}{{dt}}=\pi {R}^{2}c({{Uu}}_{\mathrm{MMP}}+{u}_{\mathrm{CMB}}){\epsilon }_{\star },\end{eqnarray} \tag{ 5 }$$

where ${\epsilon }_{\star }$ is the surface emissivity averaged over the background radiation spectrum.

The cooling rate by thermal emission is given by

$$\begin{eqnarray}&&\displaystyle \frac{{{dE}}_{\mathrm{emiss}}}{{dt}}=4\pi {R}^{2}{\epsilon }_{T}\sigma {T}^{4},\end{eqnarray} \tag{ 6 }$$

where ${\epsilon }_{T}=\int d\nu \epsilon (\nu ){B}_{\nu }(T)/\int d\nu {B}_{\nu }(T)$ is the bolometric emissivity, integrated over all radiation frequencies, ν.

The energy balance between radiative heating and cooling yields the surface equilibrium temperature

$$\begin{eqnarray}\begin{array}{rcl}{T}_{\mathrm{ice}} & = & {\left(\displaystyle \frac{c({{Uu}}_{\mathrm{MMP}}+{u}_{\mathrm{CMB}})}{4\sigma }\right)}^{1/4}{\left(\displaystyle \frac{{\epsilon }_{\star }}{{\epsilon }_{T}}\right)}^{1/4}\\ & \simeq & 3.59{\left(U+{[1+z]}^{3}\right)}^{1/4}{\left(\displaystyle \frac{{\epsilon }_{\star }}{{\epsilon }_{T}}\right)}^{1/4}\,{\rm{K}}.\end{array}\end{eqnarray} \tag{ 7 }$$

At this temperature, the sublimation time is short, less than $\sim 1\times {10}^{5}$ yr, according to Equation (4).

However, to access the actual temperature of the ice, we need to take account of evaporative cooling (Watson & Salpeter 1972; Hoang et al. 2015). The cooling rate by evaporation of H₂ is given by

$$\begin{eqnarray}&&\displaystyle \frac{{{dE}}_{\mathrm{evap}}}{{dt}}=\displaystyle \frac{{E}_{b}{{dN}}_{\mathrm{mol}}}{{dt}}=\displaystyle \frac{{E}_{b}{N}_{s}}{{\tau }_{\mathrm{sub}}({T}_{\mathrm{ice}})},\end{eqnarray} \tag{ 8 }$$

where ${{dN}}_{\mathrm{mol}}/{dt}$ is the evaporation rate, namely, the number of molecules evaporating per unit time, and ${N}_{s}=4\pi {R}^{2}/{r}_{s}^{2}$ is the number of surface sites with ${r}_{s}=10\,\mathring{\rm A}$ being the average size of the H₂ surface site (Sandford & Allamandola 1993).

The ratio of evaporative to radiative cooling rates is given by

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{{dE}}_{\mathrm{evap}}/{dt}}{{{dE}}_{\mathrm{emiss}}/{dt}}=\displaystyle \frac{{E}_{b}{\nu }_{0}\exp (-{E}_{b}/{{kT}}_{\mathrm{ice}})}{{\epsilon }_{T}\sigma {T}_{\mathrm{ice}}^{4}{r}_{s}^{2}}\\ \ \ \simeq \,\left(\displaystyle \frac{1.1}{{\epsilon }_{T}}\right){\left(\displaystyle \frac{3{\rm{K}}}{{T}_{\mathrm{ice}}}\right)}^{4}\exp \left(100\,{\rm{K}}\left[\displaystyle \frac{1}{{T}_{\mathrm{ice}}}-\displaystyle \frac{1}{3\,{\rm{K}}}\right]\right)\\ \ \ \times \,\left(\displaystyle \frac{{E}_{b}}{0.01\,\mathrm{eV}\,}\right){\left(\displaystyle \frac{10\mathring{\rm A} }{{r}_{s}}\right)}^{2},\end{array}\end{eqnarray} \tag{ 9 }$$

implying that the evaporative cooling dominates over radiative cooling for ${T}_{\mathrm{ice}}\gtrsim 3\,{\rm{K}}$.

For an H₂ object moving at a speed,${v}_{\mathrm{obj}}$, through the ISM, the heating rate by gas collisions is given by

$$\begin{eqnarray}&&\displaystyle \frac{{{dE}}_{\mathrm{coll}}}{{dt}}=\displaystyle \frac{1}{2}\pi {R}^{2}{n}_{{\rm{H}}}\mu {m}_{{\rm{H}}}{v}_{\mathrm{obj}}^{3},\end{eqnarray} \tag{ 10 }$$

where μ is the mean molecular weight of the ISM and m_H is the mass of a hydrogen atom. For the cosmic He abundance, μ = 1.4.

The ratio of collisional heating to radiative heating by starlight is given by

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{{dE}}_{\mathrm{coll}}}{{{dE}}_{\mathrm{abs}}} & = & \displaystyle \frac{({n}_{{\rm{H}}}\mu {m}_{{\rm{H}}}{v}_{\mathrm{obj}}^{3}/2)}{{{cUu}}_{\mathrm{MMP}}{\epsilon }_{\star }}\\ & \simeq & 1.2\left(\displaystyle \frac{{n}_{{\rm{H}}}}{{10}^{3}\,{\mathrm{cm}}^{-3}}\right){\left(\displaystyle \frac{{v}_{\mathrm{obj}}}{30\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{3}\left(\displaystyle \frac{U}{{\epsilon }_{\star }}\right),\end{array}\end{eqnarray} \tag{ 11 }$$

implying dominance of collisional heating if ${n}_{{\rm{H}}}\gtrsim {{cUu}}_{\mathrm{MMP}}/{(\mu {m}_{{\rm{H}}}{v}_{\mathrm{obj}}^{3}/2)\simeq 825U(30\mathrm{km}{{\rm{s}}}^{-1}/{v}_{\mathrm{obj}})}^{3}\,{\mathrm{cm}}^{-3}$, assuming ${\epsilon }_{\star }=1$. Thus, in GMCs, collisional heating is important and can destroy H₂ objects rapidly (see Section 5). For the diffuse ISM, collisional heating is negligible.

The final energy balance equation reads

$$\begin{eqnarray}&&\displaystyle \frac{{{dE}}_{\mathrm{abs}}}{{dt}}+\displaystyle \frac{{{dE}}_{\mathrm{coll}}}{{dt}}=\displaystyle \frac{{{dE}}_{\mathrm{emiss}}}{{dt}}+\displaystyle \frac{{{dE}}_{\mathrm{evap}}}{{dt}}.\end{eqnarray} \tag{ 12 }$$

We numerically solve the above equation for the equilibrium temperature, and obtain ${T}_{\mathrm{ice}}=2.996\approx 3\,{\rm{K}}$, assuming U = 1, the present CMB, and ${n}_{{\rm{H}}}=10\,{\mathrm{cm}}^{-3}$. At this temperature, Equation (4) implies the sublimation time of ∼30 Myr for R = 1 km. Passing near a region with enhanced radiation fields, e.g., near a star, would reduce the sublimation time significantly.

Next we estimate the lifetime of an icy H₂ object to ultraviolet (UV) photodesorption. Let ${Y}_{\mathrm{pd}}$ be the photodesorption yield, defined as the number of molecules ejected over the total number of incident UV photons. The rate of mass loss due to UV photodesorption is

$$\begin{eqnarray}&&\displaystyle \frac{{dm}}{{dt}}=\displaystyle \frac{4\pi {R}^{2}{\rho }_{\mathrm{ice}}{dR}}{{dt}}=-\bar{m}{Y}_{\mathrm{pd}}{F}_{\mathrm{UV}}\pi {a}^{2},\end{eqnarray} \tag{ 13 }$$

where $\bar{m}$ is the mean mass of ejected molecules, and ${F}_{\mathrm{UV}}$ is the flux of UV photons. This yields

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{dR}}{{dt}} & = & -\displaystyle \frac{\bar{m}{Y}_{\mathrm{pd}}{F}_{\mathrm{UV}}}{4{\rho }_{\mathrm{ice}}}\\ & \simeq & -262\left(\displaystyle \frac{{F}_{\mathrm{UV}}}{{10}^{7}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}}\right)\left(\displaystyle \frac{{Y}_{\mathrm{pd}}}{{10}^{3}}\right)\left(\displaystyle \frac{0.1\,{\rm{g}}\,{\mathrm{cm}}^{-3}}{{\rho }_{\mathrm{ice}}}\right)\,\mathring{\rm A} \,\,{\mathrm{yr}}^{-1},\end{array}\end{eqnarray} \tag{ 14 }$$

where $\bar{m}=2{m}_{{\rm{H}}}$, ${Y}_{\mathrm{pd}}=h\nu /{E}_{b}={10}^{3}$ for $h\nu =10\,\mathrm{eV}\,$, and ${F}_{\mathrm{UV}}={10}^{7}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$ for the ISRF.

We define $G={F}_{\mathrm{UV}}/{F}_{\mathrm{UV},\mathrm{MMP}}$ to calibrate the strength of background UV radiation, where ${F}_{\mathrm{UV},\mathrm{MMP}}={10}^{7}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$ is the UV flux of the standard ISRF. The photodesorption time for an object of radius R is

$$\begin{eqnarray}\begin{array}{rcl}{t}_{\mathrm{pd}} & = & -\displaystyle \frac{R}{{dR}/{dt}}\\ & \simeq & \displaystyle \frac{7.5\times {10}^{10}}{G}\left(\displaystyle \frac{R}{1\,\mathrm{km}}\right)\left(\displaystyle \frac{{10}^{3}}{{Y}_{\mathrm{pd}}}\right)\left(\displaystyle \frac{{10}^{7}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}}{{F}_{\mathrm{UV},\mathrm{MMP}}}\right)\mathrm{yr}.\end{array}\end{eqnarray} \tag{ 15 }$$

An enhancement of the local UV radiation near an OB association can increase the photodesorption rate by a factor of G.

At a characteristic speed of ${v}_{\mathrm{obj}}\sim 30\,\mathrm{km}\,{{\rm{s}}}^{-1}$, each ISM proton delivers an energy of ${E}_{p}={m}_{{\rm{H}}}{v}_{\mathrm{obj}}^{2}/2\approx 4.66\,\mathrm{eV}$ to the impact location. Thus, protons can eject H₂ out of the ice surface with a sputtering yield of ${Y}_{\mathrm{sp}}\sim {E}_{p}/{E}_{b}\sim 460$.

The destruction time of H₂ ice by sputtering is given by

$$\begin{eqnarray}\begin{array}{rcl}{t}_{\mathrm{sp}} & = & -\displaystyle \frac{R}{{dR}/{dt}}=\displaystyle \frac{4{\rho }_{\mathrm{ice}}R}{{n}_{{\rm{H}}}{m}_{{\rm{H}}}{v}_{\mathrm{obj}}{Y}_{\mathrm{sp}}}\\ & \simeq & 2.6\times {10}^{10}\left(\displaystyle \frac{0.1\,{\rm{g}}\,{\mathrm{cm}}^{-3}}{{\rho }_{\mathrm{ice}}}\right)\left(\displaystyle \frac{R}{1\,\mathrm{km}}\right)\left(\displaystyle \frac{10\,{\mathrm{cm}}^{-3}}{{n}_{{\rm{H}}}}\right)\\ & & \times \left(\displaystyle \frac{30\,\mathrm{km}\,{{\rm{s}}}^{-1}}{{v}_{\mathrm{obj}}}\right)\left(\displaystyle \frac{{10}^{3}}{{Y}_{\mathrm{sp}}}\right)\,\mathrm{yr},\end{array}\end{eqnarray} \tag{ 19 }$$

which is short only in GMCs and unimportant for the diffuse ISM. Moreover, most of proton's energy may go into forming a deep track instead of surface heating, reducing the sputtering effect.

Collisions of H₂ ice with the ambient gas at high speeds can heat the frontal area to a temperature ${T}_{\mathrm{evap}}$, resulting in transient evaporation. The volume of ice evaporated by a single collision, $\delta V$, can be given by

$$\begin{eqnarray}&&{n}_{\mathrm{ice}}\delta {{VE}}_{b}=\displaystyle \frac{1}{2}\mu {m}_{{\rm{H}}}{v}_{\mathrm{obj}}^{2},\end{eqnarray} \tag{ 20 }$$

where the impact kinetic energy is assumed to to be fully converted into heating.

The evaporation rate by gas collisions is given by

$$\begin{eqnarray}&&\displaystyle \frac{1}{V}\displaystyle \frac{{dV}}{{dt}}=-\displaystyle \frac{\pi {R}^{2}{n}_{{\rm{H}}}{v}_{\mathrm{obj}}\delta V}{V}=-\displaystyle \frac{3{n}_{{\rm{H}}}\mu {m}_{{\rm{H}}}{v}_{\mathrm{obj}}^{3}}{8{{Rn}}_{\mathrm{ice}}{E}_{b}}.\end{eqnarray} \tag{ 21 }$$

The evaporation time by gas collisions is then

$$\begin{eqnarray}\begin{array}{rcl}{t}_{\mathrm{evap},\mathrm{gas}} & = & -\displaystyle \frac{V}{{dV}/{dt}}=\displaystyle \frac{8{{Rn}}_{\mathrm{ice}}{E}_{b}}{3{n}_{{\rm{H}}}\mu {m}_{{\rm{H}}}{v}^{3}}\\ & \simeq & 6.5\times {10}^{9}\left(\displaystyle \frac{R}{1\,\mathrm{km}}\right){\left(\displaystyle \frac{30\mathrm{km}{{\rm{s}}}^{-1}}{{v}_{\mathrm{obj}}}\right)}^{3}\\ & \times & {\left(\displaystyle \frac{{n}_{{\rm{H}}}}{10{\mathrm{cm}}^{-3}}\right)}^{-1}\left(\displaystyle \frac{{E}_{b}}{0.01\,\mathrm{eV}\,}\right)\,\mathrm{yr},\end{array}\end{eqnarray} \tag{ 22 }$$

which is somewhat shorter than the sputtering time given in Equation (19).

Similarly, dust grains of mass m_gr deposit a kinetic energy of ${E}_{\mathrm{gr}}={m}_{\mathrm{gr}}{v}_{\mathrm{obj}}^{2}/2$ upon impact, resulting in transient evaporation. The evaporation rate by dust collisions is given by

$$\begin{eqnarray}&&\displaystyle \frac{1}{V}\displaystyle \frac{{dV}}{{dt}}=-\displaystyle \frac{\pi {R}^{2}{n}_{\mathrm{gr}}{v}_{\mathrm{obj}}\delta V}{V}=-\displaystyle \frac{3{n}_{\mathrm{gr}}{m}_{\mathrm{gr}}{v}_{\mathrm{obj}}^{3}}{8{{Rn}}_{\mathrm{ice}}{E}_{b}},\end{eqnarray} \tag{ 23 }$$

yielding a dust evaporation time,

$$\begin{eqnarray}&&{t}_{\mathrm{evap},{\rm{d}}}=-\displaystyle \frac{V}{{dV}/{dt}}=\displaystyle \frac{8{{Rn}}_{\mathrm{ice}}{E}_{b}}{3{n}_{\mathrm{gr}}{m}_{\mathrm{gr}}{v}_{\mathrm{obj}}^{3}}.\end{eqnarray} \tag{ 24 }$$

Assuming that all grains have the same size, a, and using the dust-to-gas mass ratio ${M}_{d/g}={n}_{\mathrm{gr}}4\pi {a}^{3}{\rho }_{\mathrm{gr}}/(3\mu {m}_{{\rm{H}}}{n}_{{\rm{H}}})$, one obtains the grain number density

$$\begin{eqnarray}\begin{array}{rcl}{n}_{\mathrm{gr}} & = & \displaystyle \frac{{M}_{d/g}(3\mu {m}_{{\rm{H}}}{n}_{{\rm{H}}})}{4\pi {a}^{3}{\rho }_{\mathrm{gr}}}\\ & \approx & 1.85\times {10}^{-11}\left(\displaystyle \frac{{M}_{d/g}}{100}\right)\left(\displaystyle \frac{{n}_{{\rm{H}}}}{10\,{\mathrm{cm}}^{-3}}\right)\\ & \times & {\left(\displaystyle \frac{0.1\mu {\rm{m}}}{a}\right)}^{3}\,{\mathrm{cm}}^{-3},\end{array}\end{eqnarray} \tag{ 25 }$$

where ${\rho }_{\mathrm{gr}}=3\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ is assumed.

Substituting ${n}_{\mathrm{gr}}$ into Equation (24) yields

$$\begin{eqnarray}\begin{array}{rcl}{t}_{\mathrm{evap},{\rm{d}}} & \simeq & 6.5\times {10}^{11}\left(\displaystyle \frac{R}{1\,\mathrm{km}}\right){\left(\displaystyle \frac{30\mathrm{km}{{\rm{s}}}^{-1}}{{v}_{\mathrm{obj}}}\right)}^{3}\\ & & \times \left(\displaystyle \frac{{n}_{{\rm{H}}}}{10\,{\mathrm{cm}}^{-3}}\right)\left(\displaystyle \frac{{E}_{b}}{0.01\,\mathrm{eV}\,}\right)\,\mathrm{yr}.\end{array}\end{eqnarray} \tag{ 26 }$$

The destruction by dust is less efficient than by gas due to a lower dust mass.

For a cold GMC of temperature ${T}_{\mathrm{gas}}\sim 3\,{\rm{K}}$, the thermal velocity of the gas is ${v}_{{\rm{T}}}={(3{{kT}}_{\mathrm{gas}}/{m}_{{\rm{H}}})}^{1/2}\sim 0.27{({T}_{\mathrm{gas}}/3{\rm{K}})}^{1/2}\,\mathrm{km}\,{{\rm{s}}}^{-1}$. Objects moving rapidly through the gas with ${v}_{\mathrm{obj}}\sim 30\,\mathrm{km}\,{{\rm{s}}}^{-1}\gg {v}_{T}$, will produce a bow shock if their radius is larger than the mean free path of gas molecules (Landau & Lifshitz 1959). The mean free path for molecular collisions is ${\lambda }_{\mathrm{mfp}}\sim 1/({n}_{{\rm{H}}}{\sigma }_{{\rm{H}}2})={10}^{4}({10}^{6}\,{\mathrm{cm}}^{-3}/{n}_{{\rm{H}}})({10}^{-15}\,{\mathrm{cm}}^{-2}/{\sigma }_{{\rm{H}}2})\,\mathrm{km}$ with ${\sigma }_{{\rm{H}}2}$ being the H₂ cross-section. Thus, for objects larger than $R={10}^{4}$ km, bow shocks are formed if the gas density ${n}_{{\rm{H}}}\gtrsim {10}^{6}\,{\mathrm{cm}}^{-3}$. The post-shock gas has a high temperature and can be efficient in thermal sputtering. However, bow shocks are not expected to form for objects of $R\lt {10}^{4}\,\mathrm{km}$ and ${n}_{{\rm{H}}}\lt {10}^{6}\,{\mathrm{cm}}^{-3}$.

Assuming that H₂ objects of various sizes are produced in a nearby GMC, we estimate the minimum size of objects that could reach the Earth. The closest GMC, W51, is located at a distance of ${D}_{\mathrm{GMC}}=5.2\,\mathrm{kpc}$. Thus, at a speed of $30\,\mathrm{km}\,{{\rm{s}}}^{-1}$, it takes ${t}_{\mathrm{trav}}\approx 1.6\times {10}^{8}$ yr for objects to reach the solar system.

Figure 1 compares the various destruction times with ${t}_{\mathrm{trav}}$ for different object radii at a typical speed. The sublimation time is obtained using the equilibrium temperature ${T}_{\mathrm{ice}}=2.9939\approx 3\,{\rm{K}}$ (see Section 2). We find that only very large objects of radius $R\gt 5\,\mathrm{km}$ could survive thermal sublimation and reach the solar system. The minimum size of the objects that can survive is obtained by setting ${t}_{\mathrm{sub}}={t}_{\mathrm{trav},\mathrm{ISM}}$, yielding

$$\begin{eqnarray}\begin{array}{rcl}{R}_{\min ,\mathrm{ISM}} & = & 5.4\left(\displaystyle \frac{{D}_{\mathrm{GMC}}}{5.2\,\mathrm{kpc}}\right)\left(\displaystyle \frac{30\,\mathrm{km}\,{{\rm{s}}}^{-1}}{{v}_{\mathrm{obj}}}\right)\\ & & \times \,\exp \left(100\,{\rm{K}}\left[\displaystyle \frac{1}{{T}_{\mathrm{ice}}}-\displaystyle \frac{1}{3\,{\rm{K}}}\right]\right)\,\mathrm{km}.\end{array}\end{eqnarray} \tag{ 27 }$$

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The total gas column density toward the densest GMC amounts to extinction of ${A}_{V}\sim 200$ (see e.g., Mathis et al. 1983), which corresponds to a hydrogen column density of ${N}_{{\rm{H}}}\sim 3.74\times {10}^{23}\,{\mathrm{cm}}^{-2}$ based on the scaling ${N}_{{\rm{H}}}/{A}_{V}\approx (5.8\times {10}^{21}/{R}_{V})\,{\mathrm{cm}}^{-2}/\mathrm{mag}$ with ${R}_{V}=3.1$ (Draine 2011). Assuming a mean GMC density ${n}_{{\rm{H}}}={10}^{4}\,{\mathrm{cm}}^{-3}$ and a GMC radius ${R}_{\mathrm{GMC}}\approx 12\,\mathrm{pc}$, the travel time is ${t}_{\mathrm{trav},\mathrm{GMC}}={R}_{\mathrm{GMC}}/{v}_{\mathrm{obj}}\approx 3.91\times {10}^{5}({R}_{\mathrm{GMC}}/12\,\mathrm{pc})(30\,\mathrm{km}\,{{\rm{s}}}^{-1}/{v}_{\mathrm{obj}})\,\mathrm{yr}$.

Equation (22) implies a destruction time by gas collisions of ${t}_{\mathrm{evap},\mathrm{gas}}\sim 9\times {10}^{6}(R/1\,\mathrm{km})({n}_{{\rm{H}}}/{10}^{4}\,{\mathrm{cm}}^{-3})\,\mathrm{yr}$, which is longer than the travel time.

As shown in Section 2, collisional heating is important in GMCs because of their high density, ${n}_{{\rm{H}}}\gtrsim {10}^{3}\,{\mathrm{cm}}^{-3}$. Assuming that a fraction η of the impinging proton's energy is converted into surface heating to a temperature below ${T}_{\mathrm{evap}}$, collisional heating raises the temperature of the surface to

$$\begin{eqnarray}\begin{array}{rcl}{T}_{\mathrm{ice}} & = & {\left(\displaystyle \frac{\eta {n}_{{\rm{H}}}1.4{m}_{{\rm{H}}}{v}_{\mathrm{obj}}^{3}/2}{4\sigma {\epsilon }_{T}}\right)}^{1/4}\\ & \simeq & 6.1{\left(\displaystyle \frac{{n}_{{\rm{H}}}}{{10}^{4}{\mathrm{cm}}^{-3}}\right)}^{1/4}{\left(\displaystyle \frac{\eta }{{\epsilon }_{T}}\right)}^{1/4}{\left(\displaystyle \frac{{v}_{\mathrm{obj}}}{30\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{3/4}\,{\rm{K}},\end{array}\end{eqnarray} \tag{ 28 }$$

which implies effective cooling by evaporation (see Equation (9)).

To find the actual equilibrium temperature, we solve Equation (12) and obtain ${T}_{\mathrm{ice}}\approx 3.26\,{\rm{K}}$ for ${n}_{{\rm{H}}}={10}^{4}\,{\mathrm{cm}}^{-3}$, assuming $\eta ={\epsilon }_{T}=1$. Substituting this typical temperature into Equation (3) yields

$$\begin{eqnarray}\begin{array}{rcl}{t}_{\mathrm{sub}}({T}_{\mathrm{ice}}) & \simeq & 2.1\times {10}^{6}\left(\displaystyle \frac{R}{1\,\mathrm{km}}\right)\\ & & \times \exp \,\left(100\,{\rm{K}}\left[\displaystyle \frac{1}{{T}_{\mathrm{ice}}}-\displaystyle \frac{1}{3.26\,{\rm{K}}}\right]\right)\,\mathrm{yr}.\end{array}\end{eqnarray} \tag{ 29 }$$

The minimum size of the objects that can survive is obtained by setting ${t}_{\mathrm{sub}}={t}_{\mathrm{trav},\mathrm{GMC}}$, yielding

$$\begin{eqnarray}\begin{array}{rcl}{R}_{\min ,\mathrm{GMC}} & \approx & 186\left(\displaystyle \frac{{R}_{\mathrm{GMC}}}{12\,\mathrm{pc}}\right)\left(\displaystyle \frac{30\,\mathrm{km}\,{{\rm{s}}}^{-1}}{{v}_{\mathrm{obj}}}\right)\\ & & \times \,\exp \left(100\,{\rm{K}}\left[\displaystyle \frac{1}{{T}_{\mathrm{ice}}}-\displaystyle \frac{1}{3.26\,{\rm{K}}}\right]\right)\,{\rm{m}}.\end{array}\end{eqnarray} \tag{ 30 }$$

We conclude that H₂ objects cannot survive their journey from their GMC birthplace to the ISM if their radius is below ${R}_{\min ,\mathrm{GMC}}$. The actual value ${R}_{\min ,\mathrm{GMC}}$ would be larger because the surface temperature is higher when objects are moving through the core of GMCs with higher gas density.

When 'Oumuamua entered the solar system, solar radiation heated the frontal surface to an equilibrium. We numerically solve Equation (12) and obtain the equilibrium temperatures ${T}_{\mathrm{ice}}=7.94$ and $7.15\,{\rm{K}}$ at heliodistances D = 0.25 and $0.5\,\mathrm{au}$. Substituting these temperatures into Equation (4), one obtains ${t}_{\mathrm{sub}}=3.18$ and 12.72 days, assuming $R=300\,{\rm{m}}$. These sublimation times are shorter than the travel time of ${t}_{\mathrm{trav}}=0.5\,\mathrm{au}/{v}_{\mathrm{obj}}=14.42$ days. The sputtering by the solar wind is less important than thermal sublimation.

Seligman & Laughlin (2020) suggested that H₂ ice objects can form by means of accretion and coagulation of dust grains in the densest region of a GMC where the gas density is ${n}_{{\rm{H}}}\sim {10}^{5}\,{\mathrm{cm}}^{-3}$ and temperature is ${T}_{\mathrm{gas}}\sim 3\,{\rm{K}}$. Below, we show that H₂-rich grains cannot form in the GMC due to destruction by collisional heating, preventing the formation of H₂ objects.

At low temperatures, the accretion of H₂ molecules from the gas phase onto a grain core is a main process enabling the formation of an H₂ mantle. The characteristic timescale for forming an icy grain of radius a is given by

$$\begin{eqnarray}\begin{array}{rcl}{t}_{\mathrm{acc}} & = & \displaystyle \frac{{m}_{\mathrm{gr}}}{1.05{s}_{{\rm{H}}}{n}_{{\rm{H}}}{m}_{{\rm{H}}}{v}_{\mathrm{th}}\pi {a}^{2}}=\displaystyle \frac{4{\rho }_{\mathrm{ice}}a}{3.15{s}_{{\rm{H}}}{n}_{{\rm{H}}}{m}_{{\rm{H}}}{v}_{\mathrm{th}}}\\ & \simeq & {10}^{2}\left(\displaystyle \frac{a}{1\,\mu {\rm{m}}}\right){\left(\displaystyle \frac{{n}_{{\rm{H}}}}{{10}^{5}{\mathrm{cm}}^{-3}}\right)}^{-1}{\left(\displaystyle \frac{{T}_{\mathrm{gas}}}{{T}_{\mathrm{CMB}}}\right)}^{1/2}\\ & & \times \,\left(\displaystyle \frac{{\rho }_{\mathrm{ice}}}{0.1\,{\rm{g}}\,{\mathrm{cm}}^{-3}}\right)\,\mathrm{yr},\end{array}\end{eqnarray} \tag{ 31 }$$

where the thermal velocity ${v}_{\mathrm{th}}={(8{{kT}}_{\mathrm{gas}}/\pi {m}_{{\rm{H}}})}^{1/2}$, the factor 1.05 accounts for $n(\mathrm{He})/{n}_{{\rm{H}}}=0.1$, and the sticking coefficient ${s}_{{\rm{H}}}=1$ is assumed.

The timescale to form H₂ ice from collisions between two icy grains of equal sizes a and relative velocity v_gg is given by

$$\begin{eqnarray}\begin{array}{rcl}{t}_{\mathrm{coag}} & = & \displaystyle \frac{1}{{n}_{\mathrm{gr}}{v}_{{gg}}\pi {a}^{2}}=\displaystyle \frac{4a{\rho }_{\mathrm{gr}}}{3{M}_{d/g}\mu {m}_{{\rm{H}}}{n}_{{\rm{H}}}{v}_{{gg}}{s}_{\mathrm{gr}}}\\ & \simeq & \displaystyle \frac{2.5\times {10}^{5}}{{s}_{\mathrm{gr}}}\left(\displaystyle \frac{a}{1\,\mu {\rm{m}}}\right)\left(\displaystyle \frac{{10}^{5}\,{\mathrm{cm}}^{-3}}{{n}_{{\rm{H}}}}\right)\\ & & \times \,\left(\displaystyle \frac{0.1\,\mathrm{km}\,{{\rm{s}}}^{-1}}{{v}_{{gg}}}\right)\left(\displaystyle \frac{0.01}{{M}_{d/g}}\right)\,\mathrm{yr},\end{array}\end{eqnarray} \tag{ 32 }$$

amounting to $\sim {10}^{4}$ yr for a density of ${n}_{{\rm{H}}}\sim {10}^{6}\,{\mathrm{cm}}^{-3}$, a sticking coefficient, ${s}_{\mathrm{gr}}=1$, and the grain number density, ${n}_{\mathrm{gr}}$, is given by Equation (25). In conclusion, the timescale to form micron-sized grains by coagulation is much longer than the formation time by gas accretion, in agreement with the estimate by Seligman & Laughlin (2020).

However, Seligman & Laughlin (2020) did not consider the destructive effect of icy H₂ grains by collisional heating by gas. Greenberg & de Jong (1969) noted that, at a density of ${n}_{{\rm{H}}}\gt {10}^{5}\,{\mathrm{cm}}^{-3}$, collisional heating might prevent the formation of H₂ ice. We calculate the grain temperature heated by gas with a minimum temperature ${T}_{\mathrm{gas}}={T}_{\mathrm{CMB}}=2.725\,{\rm{K}}$ as follows:

$$\begin{eqnarray}\begin{array}{rcl}{T}_{\mathrm{gr}} & = & {\left(\displaystyle \frac{1.05{n}_{{\rm{H}}}{v}_{\mathrm{th}}\times \pi {a}^{2}\times 2{{kT}}_{\mathrm{gas}}}{4\pi {a}^{2}\sigma \langle {Q}_{\mathrm{abs}}{\rangle }_{T}}\right)}^{1/4}\\ & \simeq & 3.02{\left(\displaystyle \frac{{n}_{{\rm{H}}}}{{10}^{5}{\mathrm{cm}}^{-3}}\right)}^{1/4}{\left(\displaystyle \frac{{T}_{\mathrm{gas}}}{{T}_{\mathrm{CMB}}}\right)}^{3/8}{\left(\displaystyle \frac{{10}^{-4}}{\langle {Q}_{\mathrm{abs}}{\rangle }_{T}}\right)}^{1/4}\,{\rm{K}},\end{array}\end{eqnarray} \tag{ 33 }$$

where $2{{kT}}_{\mathrm{gas}}$ is the mean kinetic energy of thermal particles colliding with the grain, and $\langle {Q}_{\mathrm{abs}}{\rangle }_{T}\approx 1.1\times {10}^{-4}{(a/1\mu {\rm{m}})(T/3{\rm{K}})}^{2}$ for silicate grains (Draine 2011). Gas collisions eventually lead to thermal equilibrium at ${T}_{\mathrm{gr}}={T}_{\mathrm{gas}}$.

Substituting this typical temperature and the grain size $a=1\,\mu {\rm{m}}$ into Equation (3) yields

$$\begin{eqnarray}&&{t}_{\mathrm{sub}}({T}_{\mathrm{gr}})\simeq 0.85\left(\displaystyle \frac{a}{1\,\mu {\rm{m}}}\right)\exp \left(100\,{\rm{K}}\left[\displaystyle \frac{1}{{T}_{\mathrm{gr}}}-\displaystyle \frac{1}{{T}_{\mathrm{CMB}}}\right]\right)\,\mathrm{yr},\end{eqnarray} \tag{ 34 }$$

which is much shorter than the accretion time ${t}_{\mathrm{acc}}$ given in Equation (31). The gas temperature in realistic GMCs is larger than ${T}_{\mathrm{CMB}}$ due to CR heating, resulting in a much shorter sublimation time. We therefore conclude that micron-sized H₂ grains cannot form in dense GMCs due to collisional heating.

Figure 2 shows the accretion time and sublimation time as functions of gas density for different emissivities $\langle {Q}_{\mathrm{abs}}{\rangle }_{T}$. For the typical value of $\langle {Q}_{\mathrm{abs}}\rangle \sim {10}^{-4}$, sublimation is faster than accretion for ${n}_{{\rm{H}}}\gt 2\times {10}^{4}\,{\mathrm{cm}}^{-3}$. On the other hand, in lower density regions, accretion is faster than sublimation, but heating by CRs and interstellar radiation could still be important for heating the gas above ${T}_{\mathrm{CMB}}$ and increase sublimation of H₂ ice.

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Assuming that H₂ objects could somehow form in the densest regions of GMCs, we found that sublimation by collisional heating inside the GMC would destroy the objects before their escape into the ISM. We also studied various destruction mechanisms of H₂ ice in the ISM. In particular, we found that H₂ objects are heated by the average interstellar radiation, so that they cannot survive beyond a sublimation time of ${t}_{\mathrm{sub}}\sim 10\,\mathrm{Myr}$ for R = 300 m (see Figure 1). Only H₂ objects larger than $5\,\mathrm{km}$ could survive.

Primordial snowballs were suggested as baryonic dark matter (White 1996). Previous studies considered collisions between snowballs as a destructive mechanism (Hegyi & Olive 1986; Carr & Sakellariadou 1999). Hegyi & Olive (1986) studied destruction of H₂ ice by the CMB and found that at redshift (1+z) = 3.5, sublimation would rapidly destroy H₂ ice. Later, White (1996) argued that the treatment of sublimation by Hegyi & Olive (1986) was inadequate because evaporative cooling was not taken into account. In this work, we have shown that the evaporative cooling is only important for ${T}_{\mathrm{ice}}\gtrsim 3\,{\rm{K}}$. Even when evaporative cooling is taken into account, thermal sublimation by starlight still plays an important role in the destruction of H₂ objects. The present CMB temperature ${T}_{\mathrm{CMB}}$ is not high enough to rapidly sublimate H₂ ice. However, at redshifts $z\gt 1$, the CMB temperatures of ${T}_{\mathrm{CMB}}\gt 5$ K, can rapidly destroy H₂ objects of $R\sim 1\,\mathrm{km}$ within ${t}_{\mathrm{sub}}\sim 48\,\mathrm{yr}$, based on Equation (4).

More importantly, we found that the formation of H₂ objects cannot occur in dense GMCs because collisional heating raises the temperature of dust grains, resulting in rapid sublimation of H₂ ice mantles. Thus, we find that large objects rich in H₂ ice are unlikely to form in dense clouds, in agreement with the conclusions of Greenberg & de Jong (1969).

Lastly, if H₂ objects form via a phase transition, as proposed by Füglistaler & Pfenniger (2018), they must be larger than ∼5 km to survive the journey from the GMC to the solar system.