SIGNATURES OF PLANETS AND PROTOPLANETS IN THE GALACTIC CENTER: A CLUE TO UNDERSTANDING THE G2 CLOUD?

The tidal shear of the SMBH can unbind a planet from its star if the initial semimajor axis of the planet orbit is

$$\begin{eqnarray}{a}_{p} & \geqslant & 19\;\mathrm{AU}\left({\displaystyle \frac{d}{0.01\;\mathrm{pc}}}\right){\left({\displaystyle \frac{{m}_{*}}{10\;{M}_{\odot }}}\right)}^{1/3}{\left({\displaystyle \frac{4\times {10}^{6}\;{M}_{\odot }}{{M}_{\mathrm{BH}}}}\right)}^{1/3},\end{eqnarray} \tag{ 1 }$$

where d is the periapse of the star orbit around the SMBH, ${m}_{*}$ is the star mass, and ${M}_{\mathrm{BH}}$ is the SMBH mass. Figure 1 shows ${a}_{p}$ as a function of d, for ${m}_{*}=1$ and 10 M${}_{\odot }$.

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One of the two members of the split binary (generally the most massive one) receives a kick that makes it more bound to the SMBH, while the other member (generally the less massive one) becomes less bound. The less bound object might be ejected, while the remaining one is captured by the SMBH (e.g., Ginsburg et al. 2012). On the other hand, the typical variation $\delta {}_{v}$ of the velocity of the planet is (Pfahl 2005)

$$\begin{eqnarray}\delta {}_{v} & \sim & \sqrt{2}\;{\left({\displaystyle \frac{G\;{m}_{*}}{{a}_{p}}}\right)}^{1/2}\;{\left({\displaystyle \frac{{M}_{\mathrm{BH}}}{{m}_{*}}}\right)}^{1/6}\\ & \sim & 170\;\mathrm{km}\;{{\rm s}}^{-1}\;{\left({\displaystyle \frac{{m}_{*}}{1\;{M}_{\odot }}}\right)}^{1/2}\;{\left({\displaystyle \frac{10\;\mathrm{AU}}{{a}_{p}}}\right)}^{1/2}\\ & & \times {\left({\displaystyle \frac{{M}_{\mathrm{BH}}/{m}_{*}}{4\times {10}^{6}}}\right)}^{1/6},\end{eqnarray} \tag{ 2 }$$

since $\delta {}_{v}$ is lower than the Keplerian velocity around the SMBH at d = 0.01 pc ($\sim 1300$ km s⁻¹), it is plausible that both the planet and the star remain bound to the SMBH. Dynamical simulations are necessary to quantify how many planets will be ejected and how many will be captured by the SMBH. If the planet is captured, its semimajor axis ${a}_{\mathrm{cap}}$ and eccentricity ${e}_{\mathrm{cap}}$ would then be (Hills 1991; Perets et al. 2009)

$$\begin{eqnarray}{a}_{\mathrm{cap}} & \simeq & 1.30\;\mathrm{pc}\;\left({\displaystyle \frac{{a}_{p}}{19\;\mathrm{AU}}}\right){\left({\displaystyle \frac{{M}_{\mathrm{BH}}}{4\times {10}^{6}\;{M}_{\odot }}}\right)}^{2/3}\;{\left({\displaystyle \frac{1\;{M}_{\odot }}{{m}_{*}}}\right)}^{2/3},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{e}_{\mathrm{cap}}=1-{\displaystyle \frac{d}{{a}_{\mathrm{cap}}}}\sim 0.99.\end{eqnarray} \tag{ 4 }$$

The tidal split of planets from their stars can be substantially enhanced by planet–planet scatterings, which were shown to be able to scatter Jupiter-mass planets on orbits with semimajor axis $\gt 50$ AU (Chatterjee et al. 2008, 2011; Marois et al. 2008). The planet could even be ejected from its initial system by planet–planet scattering: Veras et al. (2009) estimate that $\sim 40\%$ of planets in a multiple-planet system are ejected by planet–planet scatterings in $2\times {10}^{8}$ Myr. The ejection of Jupiter-like planets from their planet systems is supported by the observation of "freely floating" giant planets (Sumi et al. 2011).

It is even possible that starless planets form directly from gravitational instabilities in accretion disks around SMBHs (Shlosman & Begelman 1989; Nayakshin 2006). Similarly, starless planets could have formed in the same star formation episode that gave birth to the clockwise disk: according to one of the most popular scenarios, a molecular cloud disrupted by the SMBH might have settled into a parsec-scale dense gaseous disk. Gravitational instabilities in the gaseous disk might have led to the formation of the young massive stars that lie in the clockwise disk (e.g., Paczynski 1978; Kolykhalov & Syunyaev 1980; Shlosman & Begelman 1987; Collin & Zahn 1999; Gammie 2001; Goodman 2003; Tan & Blackman 2005; Nayakshin et al. 2007; Bonnell & Rice 2008; Mapelli et al. 2008, 2012, 2013; Hobbs & Nayakshin 2009; Alig et al. 2011; Lucas et al. 2013), and also to the formation of starless giant planets (Shlosman & Begelman 1989). In such case, the initial orbit of the planets would be inside the clockwise disk, and then gravitational interactions with stars or other planets might plunge the planets on a more radial orbit. For example, Murray-Clay & Loeb (2012) predict that $\sim 1/100$ of the entire population of low-mass objects in the clockwise disk could have been delivered onto highly eccentric orbits by two-body interactions with massive stars.

What happens to a planet that is captured by the SMBH on a very eccentric orbit? Tidal disruption is unlikely, as the planet should have a radius ${r}_{p}$ larger than

$$\begin{eqnarray}{r}_{t} & = & 1.3\times {10}^{13}\;\mathrm{cm}\;{\displaystyle \frac{d}{0.01\;\mathrm{pc}}}\\ & & \times {\left({\displaystyle \frac{4\times {10}^{6}\;{M}_{\odot }}{{M}_{\mathrm{BH}}}}\right)}^{1/3}\;{\left({\displaystyle \frac{{m}_{p}}{{10}^{-3}\;{M}_{\odot }}}\right)}^{1/3},\end{eqnarray} \tag{ 5 }$$

i.e., $\sim 2000\;(d/0.01\;\mathrm{pc})$ times the radius of Jupiter. Figure 1 shows the behavior of ${r}_{t}$ as a function of d, for a planet mass ${m}_{p}={10}^{-3}$ M${}_{\odot }$ and for a brown dwarf of mass ${m}_{p}=2\times {10}^{-2}$ M${}_{\odot }$.

The planet will suffer continuous atmosphere evaporation by the UV field of the young stars in the central parsec. According to Murray-Clay et al. (2009) (using the simplified Equation (19)), the mass loss rate by atmosphere evaporation is

$$\begin{eqnarray}&&{\stackrel{\dot{}}{M}}_{\mathrm{MC}}\simeq {\displaystyle \frac{\epsilon \pi \;{r}_{p}^{2}\;{L}_{\mathrm{UV}}/(4\;\pi \;{D}^{2})}{G\;{m}_{p}/{r}_{p}}},\end{eqnarray} \tag{ 6 }$$

where ${r}_{p}$ is the planet radius, ${L}_{\mathrm{UV}}\sim {10}^{40}$ erg s⁻¹ is the ionizing luminosity of massive stars in the GC, is the fraction of the UV luminosity that goes into heat ($\simeq \tilde{\phi }/{\phi }_{0}-1$, where $\tilde{\phi }$ is the average energy of ionizing photons, and ${\phi }_{0}\simeq 13.6\;\mathrm{eV}$ is the ionization energy for atomic H), and D is the distance of the massive stars from the rogue planet. We assume that $D\simeq 0.1\;\mathrm{pc}$, since this is the outer rim of the clockwise disk in the GC (Yelda et al. 2014).

For large values of ${r}_{p}$, Equation (6) implies that the number of ionizations is larger than the number of ionizing photons reaching the planet surface,

$$\begin{eqnarray}&&{Q}_{p}={\displaystyle \frac{\pi \;{r}_{p}^{2}\;{L}_{\mathrm{UV}}/(4\;\pi \;{D}^{2})}{{\phi }_{0}/(1-\epsilon )}}.\end{eqnarray} \tag{ 7 }$$

Therefore, we employ an evaporation rate

$$\begin{eqnarray}\stackrel{\dot{}}{M} & = & \mathrm{min}({\stackrel{\dot{}}{M}}_{\mathrm{MC}},{m}_{\mathrm{prot}}\;{Q}_{p})\\ & = & {\displaystyle \frac{{r}_{p}^{2}\;{L}_{\mathrm{UV}}}{4\;{D}^{2}}}{\displaystyle \frac{{m}_{\mathrm{prot}}}{{\phi }_{0}}}\mathrm{min}\left[\epsilon {\displaystyle \frac{{\phi }_{0}/{m}_{\mathrm{prot}}}{(G\;{m}_{p})/{r}_{p}}},(1-\epsilon )\right]\\ & & \simeq 2.0\times {10}^{11}\;{\rm g}\;{{\rm s}}^{-1}{\left({\displaystyle \frac{{r}_{p}}{{10}^{10}\;\mathrm{cm}}}\right)}^{2}\left({\displaystyle \frac{{L}_{\mathrm{UV}}}{{10}^{40}\;\mathrm{erg}\;{{\rm s}}^{-1}}}\right){\left({\displaystyle \frac{D}{0.1\mathrm{pc}}}\right)}^{-2}\\ & & \times \mathrm{min}\left[0.98\;\epsilon {\left({\displaystyle \frac{{m}_{p}}{{10}^{-3}\;{M}_{\odot }}}\right)}^{-1}\left({\displaystyle \frac{{r}_{p}}{{10}^{10}\;\mathrm{cm}}}\right),(1-\epsilon )\right],\end{eqnarray} \tag{ 8 }$$

where ${m}_{\mathrm{prot}}\simeq 1.67\times {10}^{-24}\;{\rm g}$ is the proton mass. The bottom panel of Figure 2 (dotted line) shows the behavior of $\stackrel{\dot{}}{M}$ as a function of the planet radius.

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The average kinetic energy of the evaporating ions is of the order of $\sim \tilde{\phi }-{\phi }_{0}=\epsilon \;[{\phi }_{0}/(1-\epsilon )]$, implying a velocity of the order of ${v}_{g}\sim 30\;\mathrm{km}\;{{\rm s}}^{-1}$ (for $\epsilon \sim 0.3$). This velocity is generally of the same order of magnitude as the escape velocity ${v}_{\mathrm{esc}}=\sqrt{(2\;G\;{m}_{p})/{r}_{p}}$.

Thus, the density of the ionized gas that evaporates from the star is

$$\begin{eqnarray}{n}_{+} & = & {\displaystyle \frac{\stackrel{\dot{}}{M}}{4\;\pi \;{m}_{\mathrm{prot}}\;{r}_{p}^{2}\;{v}_{g}}}\\ & = & {10}^{7}\;{\mathrm{cm}}^{-3}\left({\displaystyle \frac{\stackrel{\dot{}}{M}}{6\times {10}^{10}\;{\rm g}\;{{\rm s}}^{-1}}}\right)\\ & & \times \left({\displaystyle \frac{30\;\mathrm{km}\ {{\rm s}}^{-1}}{{v}_{g}}}\right){\left({\displaystyle \frac{{10}^{10}\;\mathrm{cm}}{{r}_{p}}}\right)}^{2}.\end{eqnarray} \tag{ 9 }$$

The central panel of Figure 2 (dotted line) shows ${n}_{+}$ (obtained assuming ${v}_{g}={v}_{\mathrm{esc}}=\sqrt{(2\;G\;{m}_{p})/{r}_{p}}$), as a function of the planet radius.

If the velocity of the evaporating matter is close to the escape velocity from the planet, the density profile becomes $n(r)\sim {n}_{+}{(r/{r}_{p})}^{-3/2}$, and the recombination rate can be estimated as

$$\begin{eqnarray}R & = & {{\displaystyle \int }}_{{r}_{p}}^{{r}_{\mathrm{max}}}4\;\pi \;{r}^{2}{\alpha }_{{\rm B}}\;{n}_{+}^{2}\;{\left({\displaystyle \frac{r}{{r}_{p}}}\right)}^{-3}{dr}\\ & = & 4\pi \;{\alpha }_{{\rm B}}\;{n}_{+}^{2}\;{r}_{p}^{3}\;\mathrm{ln}{\displaystyle \frac{{r}_{\mathrm{max}}}{{r}_{p}}}\\ & & \simeq 4\;\pi \;{\alpha }_{{\rm B}}\;{n}_{+}^{2}\;{r}_{p}^{3}\;\mathrm{ln}{\left({\displaystyle \frac{{n}_{+}\;{m}_{\mathrm{prot}}}{{\rho }_{h}}}\right)}^{2/3}\\ & & \sim 3\times {10}^{33}\;{{\rm s}}^{-1}{\left({\displaystyle \frac{{n}_{+}}{{10}^{7}\;{\mathrm{cm}}^{-3}}}\right)}^{2}\\ & & \times {\left({\displaystyle \frac{{r}_{p}}{{10}^{10}\;\mathrm{cm}}}\right)}^{3}{\displaystyle \frac{\mathrm{ln}\left[{\displaystyle \frac{({n}_{+}\;{m}_{\mathrm{prot}})}{{\rho }_{h}}}\right]}{9.7}},\end{eqnarray} \tag{ 10 }$$

where ${\alpha }_{{\rm B}}\simeq 2.6\times {10}^{-13}\;{\mathrm{cm}}^{3}\;{{\rm s}}^{-1}$ is the case B recombination coefficient for Hydrogen (at a temperature $\sim {10}^{4}\;{\rm K}$), and we have chosen ${r}_{\mathrm{max}}$ as the radius where the density of the evaporated gas drops to the density of the hot medium: ${m}_{\mathrm{prot}}\;n({r}_{\mathrm{max}})\equiv {\rho }_{h}$ (the normalization of the logarithmic term is appropriate for ${n}_{+}={10}^{7}\;{\mathrm{cm}}^{-3}$, ${\rho }_{h}={10}^{-21}\;{\rm g}\;{\mathrm{cm}}^{-3}$). The corresponding emission measure (EM) is EM$\;=\int {n}_{e}^{2}\;{dV}\sim {10}^{45}$ cm⁻³, for ${r}_{p}={10}^{10}$ cm and ${n}_{+}={10}^{7}$ cm⁻³.

Equation (10) assumes that the evaporated gas is nearly completely ionized. This is a good approximation, because the recombination time scale

$$\begin{eqnarray}{t}_{\mathrm{rec}}(r) & = & {\displaystyle \frac{1}{{\alpha }_{{\rm B}}{n}_{+}(r)}}\\ & & \simeq 3.8\times {10}^{5}\;{\rm s}{\left({\displaystyle \frac{{n}_{+}}{{10}^{7}\;{\mathrm{cm}}^{-3}}}\right)}^{-1}{\left({\displaystyle \frac{r}{{r}_{p}}}\right)}^{3/2}\end{eqnarray} \tag{ 11 }$$

is longer than the ionization time scale

$$\begin{eqnarray}{t}_{\mathrm{ion}} & \sim & {\displaystyle \frac{1}{\sigma (\tilde{\phi }){L}_{\mathrm{UV}}/(4\pi {D}^{2}\tilde{\phi })}}\\ & \simeq & 1800\;{\rm s}\left({\displaystyle \frac{{10}^{40}\;\mathrm{erg}\;{{\rm s}}^{-1}}{{L}_{\mathrm{UV}}}}\right){\left({\displaystyle \frac{D}{0.1\;\mathrm{pc}}}\right)}^{2},\end{eqnarray} \tag{ 12 }$$

where we assumed that the the ionization cross section of neutral Hydrogen is $\sigma (\tilde{\phi })\simeq 6.3\times {10}^{-18}\;{\mathrm{cm}}^{2}{(\tilde{\phi }/{\phi }_{0})}^{-3}\simeq 2.1\;\times {10}^{-18}\;{\mathrm{cm}}^{2}$, which is appropriate for $\epsilon =0.3$, i.e., $\tilde{\phi }=\phi {}_{0}/0.7\;\simeq 19.4\;\mathrm{eV}$.

Furthermore, in this case, we can ignore the possibility of shocks with the hot medium, because the relatively low values of ${v}_{g}$ often lead to a Mach number $\mathcal{M}={v}_{g}/{c}_{s}\lesssim 1$, where ${c}_{s}$ is the sound speed (slow-wind case). In the Appendix, we discuss the case in which $\mathcal{M}\gg 1$ (fast-wind case). Photoevaporation in the fast-wind approximation leads to a recombination rate about one order of magnitude lower with respect to the slow-wind approximation, but shocks occurring in the fast-wind case enhance the recombination rate by a factor depending on $\mathcal{M}$ (see the Appendix).

In the following, we refer to CASE 1 (see Table 1) as the model where $\stackrel{\dot{}}{M}$, ${n}_{+}$ and R have been calculated from Equations (8)–(10), respectively (i.e., R has been calculated in the slow-wind approximation, with ${v}_{g}=\sqrt{2\;G\;{m}_{p}/{r}_{p}}$ and ${L}_{\mathrm{UV}}={10}^{40}$ erg s⁻¹). Figure 2 (dotted line) shows $\stackrel{\dot{}}{M}$, ${n}_{+}$ and R in CASE 1, as a function of the planet radius.

Table 1.  Summary of Model Properties

Name	$\stackrel{\dot{}}{M}$ (g s−1)	${n}_{+}$ (cm−3)	R (s−1)	Tidal Stripping
CASE 1	from Equation (8)	from Equation (9)	from Equation (10)	no
CASE 2	from Equation (15)	from Equation (14)	from Equation (10)	no
CASE 3	⋯	⋯	from Equation (20)	yes

Note. Columns 2–4 specify how $\stackrel{\dot{}}{M}$, ${n}_{+}$ and R were calculated in each model. CASE 1 corresponds to a photoevaporating planet (Equation (8), Section 2) in the slow-wind approximation (Equation (10)), with ${m}_{p}={10}^{-3}$ M${}_{\odot }$, ${v}_{g}=\sqrt{2\;G\;{m}_{p}/{r}_{p}}$ and ${L}_{\mathrm{UV}}={10}^{40}$ erg s⁻¹. CASE 2 corresponds to a photoevaporating cloud (Equation (15), Section 3) in the slow-wind approximation (Equation (10)) with ${m}_{p}={10}^{-3}$ M${}_{\odot }$, ${v}_{g}={c}_{s}=10$ km s⁻¹ and ${L}_{\mathrm{UV}}={10}^{40}$ erg s⁻¹. CASE 3 corresponds to a photoevaporating cloud undergoing tidal stripping (${r}_{p}\geqslant {r}_{t}$), with ${v}_{g}={c}_{s}=10$ km s⁻¹ and ${L}_{\mathrm{UV}}={10}^{40}$ erg s⁻¹. If the cloud is being stripped, we do not need to derive $\stackrel{\dot{}}{M}$ and ${n}_{+}$ in order to calculate R, since $R={Q}_{\mathrm{tid}}$ (see Equation (20) and the discussion in the text).

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For a radius of the PE ${r}_{p}\lt {r}_{t}\sim 1.3\times {10}^{13}\;\mathrm{cm}\;(d/0.01\;\mathrm{pc})$ (see Equation (5)), the PE will avoid tidal disruption during the capture by the SMBH. From Equation (8), we infer that the mass loss by evaporation for a PE is $\stackrel{\dot{}}{M}\sim 3.5\times {10}^{16}$ g s⁻¹ for ${r}_{p}=5\times {10}^{12}$ cm (and $\epsilon =0.3$), corresponding to ${n}_{+}\sim 2.9\times {10}^{8}$ cm⁻³ for ${v}_{g}=\sqrt{2\;G\;{m}_{p}/{r}_{p}}$. Thus, the recombination rate for an evaporating PE would be $R\sim 3.1\times {10}^{44}$ s⁻¹ (using Equation (10)), corresponding to an EM $\sim {10}^{57}$ cm⁻³. However, this result neglects the optical depth encountered by the ionizing photons before reaching the PE surface,

$$\begin{eqnarray}\tau & = & {{\displaystyle \int }}_{{r}_{p}}^{{r}_{\mathrm{max}}}\sigma (\tilde{\phi }){n}_{+}{\left({\displaystyle \frac{r}{{r}_{p}}}\right)}^{-3/2}{\displaystyle \frac{{t}_{\mathrm{ion}}}{{t}_{\mathrm{rec}}(r)}}\;{dr}\\ & \simeq & 5\times {10}^{-4}{\left({\displaystyle \frac{{n}_{+}}{{10}^{7}\;{\mathrm{cm}}^{-3}}}\right)}^{2}\left({\displaystyle \frac{{r}_{p}}{{10}^{10}\;\mathrm{cm}}}\right)\\ & & \times \left({\displaystyle \frac{{10}^{40}\;\mathrm{erg}\;{{\rm s}}^{-1}}{{L}_{\mathrm{UV}}}}\right){\left({\displaystyle \frac{D}{0.1\;\mathrm{pc}}}\right)}^{2},\end{eqnarray} \tag{ 13 }$$

where we approximated the neutral fraction in the evaporating gas at radius r as ${t}_{\mathrm{ion}}/{t}_{\mathrm{rec}}(r)$ (which is correct as long as ${t}_{\mathrm{ion}}\ll {t}_{\mathrm{rec}}(r)$).

If we use the CASE 1 approximations at ${r}_{p}=5\times {10}^{12}\;\mathrm{cm}$, we get $\tau \sim 200$, implying that the estimate of R is too high, since the ionizing photons would be unable to photoevaporate the surface of the PE. In general, our approximations break down when $\tau \gtrsim 1$, i.e., when ${r}_{p}\gtrsim 1\;\mathrm{AU}\;{({n}_{+}/{10}^{7}\;{\mathrm{cm}}^{-3})}^{-2}({L}_{\mathrm{UV}}/{10}^{40}\;\mathrm{erg}\;{{\rm s}}^{-1}){(D/0.1\;\mathrm{pc})}^{-2}$; in particular, CASE 1 holds only if ${r}_{p}\lesssim 2\times {10}^{11}\;\mathrm{cm}$.

Therefore, it is more appropriate to derive the mass loss rate $\stackrel{\dot{}}{M}$ with the assumption that the PE is a pure gas cloud (such as a protoplanetary disk). Following Murray-Clay & Loeb (2012), the mass loss rate of a gas cloud with radius ${r}_{p}$, which is undergoing photoevaporation, can be expressed as $\stackrel{\dot{}}{M}\sim 4\;\pi \;{r}_{p}^{2}\;{m}_{\mathrm{prot}}\;{n}_{+}\;{c}_{s}$, where the number density of the photoevaporated ions (${n}_{+}$) can be calculated assuming balance between recombinations and photoionizations at the base of the wind,

$$\begin{eqnarray}&&{n}_{+}\approx ({L}_{\mathrm{UV}}/{(4\;\pi \;\tilde{\phi }\;{D}^{2}))}^{1/2}\;{({\alpha }_{{\rm B}}\;{r}_{p})}^{-1/2},\end{eqnarray} \tag{ 14 }$$

where $\tilde{\phi }$ is the average energy of ionizing photons. Thus, the mass loss rate is

$$\begin{eqnarray}&&\stackrel{\dot{}}{M}\sim {\left({\displaystyle \frac{4\;\pi }{{\alpha }_{{\rm B}}}}\right)}^{1/2}\;{m}_{\mathrm{prot}}\;{c}_{s}\;{r}_{p}^{3/2}\;{\left({\displaystyle \frac{{L}_{\mathrm{UV}}}{\tilde{\phi }\;{D}^{2}}}\right)}^{1/2}\\ &&\quad \sim 6.7\times {10}^{14}\;{\rm g}\;\;{{\rm s}}^{-1}\;\left({\displaystyle \frac{{c}_{s}}{10\;\mathrm{km}\;{{\rm s}}^{-1}}}\right){\left({\displaystyle \frac{{r}_{p}}{{10}^{12}\;\mathrm{cm}}}\right)}^{3/2}\\ &&\quad \times {\left({\displaystyle \frac{{L}_{\mathrm{UV}}}{{10}^{40}\;\mathrm{erg}\;\;{{\rm s}}^{-1}}}\right)}^{1/2}\;{\left({\displaystyle \frac{19.4\;\mathrm{ev}}{\tilde{\phi }}}\right)}^{1/2}\;\left({\displaystyle \frac{0.1\;\mathrm{pc}}{D}}\right),\end{eqnarray} \tag{ 15 }$$

and the optical depth due to the evaporating gas becomes $\tau \sim 0.5$, independent of all the considered parameters, apart from $\tilde{\phi }$. This result shows that Equation (14) can be applied for a PE, since the absorption in the photoevaporative wind does not stop the photoevaporation itself.

Furthermore, if we substitute Equation (14) into (10), the recombination rate becomes

$$\begin{eqnarray}R & \simeq & {\displaystyle \frac{4\pi {r}_{p}^{2}{L}_{\mathrm{UV}}}{4\pi {D}^{2}\tilde{\phi }}}\mathrm{ln}\left({\displaystyle \frac{{r}_{\mathrm{max}}}{{r}_{p}}}\right)\\ & = & \left[4\mathrm{ln}\left({\displaystyle \frac{{r}_{\mathrm{max}}}{{r}_{p}}}\right)\right]{Q}_{p},\end{eqnarray} \tag{ 16 }$$

where ${Q}_{p}$ is the number of ionizing photons reaching the PE surface in the optically thin approximation (Equation (7)). We have $R\gt {Q}_{p}$ because ${r}_{\mathrm{max}}\gg {r}_{p}$, so that the number of available ionizing photons is much larger than ${Q}_{p}$.

The central and bottom panel of Figure 2 (solid line) show the behavior of ${n}_{+}$ and $\stackrel{\dot{}}{M}$, as derived from Equations (14) and (15), respectively. Finally, the top panel of Figure 2 (solid line) shows the recombination rate R obtained combining Equation (14) with (10) (or, equivalently, using Equation (16)). In the following, we define this model as CASE 2 (see Table 1).

The tidal radius ${r}_{t}$ of a PE can become similar (or smaller) than its radius ${r}_{p}$ for $d\lesssim 0.01$ pc (Figure 1); in such case, mass loss by tidal stripping might become non-negligible at some point in the orbit. When ${r}_{t}\leqslant {r}_{p}$, we can estimate the mass-loss rate by tidal stripping as (Murray-Clay & Loeb 2012)

$$\begin{eqnarray}{\stackrel{\dot{}}{M}}_{\mathrm{tid}} & \sim & 4\pi {r}_{p}^{2}\;\rho ({r}_{p}){({\displaystyle \frac{{m}_{p}}{3{M}_{\mathrm{BH}}}})}^{1/3}{v}_{\perp }\\ & \simeq & {r}_{p}^{-1}\;{\displaystyle \frac{{m}_{p}^{4/3}}{{(3{M}_{\mathrm{BH}})}^{1/3}}}{v}_{\perp }\\ & \simeq & 8.7\times {10}^{22}\;{\rm g}\;{{\rm s}}^{-1}{({\displaystyle \frac{{r}_{p}}{{10}^{12}\;\mathrm{cm}}})}^{-1}{({\displaystyle \frac{{m}_{p}}{{10}^{-3}{M}_{\odot }}})}^{4/3}\\ & & \times \left({\displaystyle \frac{{v}_{\perp }}{{10}^{3}\;\mathrm{km}\;{{\rm s}}^{-1}}}\right){({\displaystyle \frac{{M}_{\mathrm{BH}}}{4\times {10}^{6}{M}_{\odot }}})}^{-1/3},\end{eqnarray} \tag{ 17 }$$

where we assume that the surface density of the PE is $\rho ({r}_{p})\simeq {m}_{p}/(4\pi {r}_{p}^{3})$ (appropriate in the case of an isothermal density profile), and ${v}_{\perp }$ is the radial component of the orbital velocity ${v}_{p}$ (Murray-Clay & Loeb 2012).

Which is the fate of the stripped material? How could we observe it? The tidally stripped material might undergo shocks with the high-temperature medium. The stagnation radius ${r}_{s}$ where ram pressure is balanced between the bow shock of the stellar wind and the hot medium is (Burkert et al. 2012; Scoville & Burkert 2013)

$$\begin{eqnarray*}{r}_{s} & = & {\left({\displaystyle \frac{\stackrel{\dot{}}{M}\;{v}_{g}}{4\;\pi \;{\rho }_{h}\;{v}_{p}^{2}}}\right)}^{1/2}\\ & = & 2\times {10}^{16}\;\;\mathrm{cm}{\left({\displaystyle \frac{{\stackrel{\dot{}}{M}}_{\mathrm{tid}}}{8.7\times {10}^{22}\;{\rm g}\;{{\rm s}}^{-1}}}\right)}^{1/2}{\left({\displaystyle \frac{{v}_{g}}{10\;{\mathrm{kms}}^{-1}}}\right)}^{1/2}\end{eqnarray*}$$

$$\begin{eqnarray} & & \times \;{\left({\displaystyle \frac{{10}^{-21}\;{\rm g}\;{\mathrm{cm}}^{-3}}{{\rho }_{h}}}\right)}^{1/2}\left({\displaystyle \frac{1300\;\;\mathrm{km}\;{{\rm s}}^{-1}}{{v}_{p}}}\right),\end{eqnarray} \tag{ 18 }$$

where ${v}_{p}$ ($\sim 1300\;{\mathrm{kms}}^{-1}$) is the Keplerian velocity of a planet orbiting the SMBH at 0.01 pc, and $\rho {}_{h}={10}^{-21}$ g cm⁻³ is the density of the hot medium from X-ray measurements (Yuan et al. 2003).

Thus, the maximum luminosity that can be emitted by the tidally stripped material in such shocks is

$$\begin{eqnarray}{L}_{\mathrm{max}} & = & \pi \;{r}_{s}^{2}\;\left({\displaystyle \frac{{\rho }_{h}}{{m}_{\mathrm{prot}}}}\;{v}_{p}\right){\displaystyle \frac{1}{2}}\;{m}_{\mathrm{prot}}\;{v}_{p}^{2}\\ & = & {\displaystyle \frac{1}{8}}\;{\stackrel{\dot{}}{M}}_{\mathrm{tid}}\;{v}_{g}\;{v}_{p}\\ & \simeq & 1.4\times {10}^{36}\;\mathrm{erg}\;{{\rm s}}^{-1}\;\left({\displaystyle \frac{{\stackrel{\dot{}}{M}}_{\mathrm{tid}}}{8.7\times {10}^{22}\;{\rm g}\;\;{{\rm s}}^{-1}}}\right)\\ & & \times \left({\displaystyle \frac{{v}_{g}}{10\;\mathrm{km}\;{{\rm s}}^{-1}}}\right)\left({\displaystyle \frac{{v}_{p}}{1300\;\mathrm{km}\;{{\rm s}}^{-1}}}\right),\end{eqnarray} \tag{ 19 }$$

where we used Equations (17) and (18) for ${\stackrel{\dot{}}{M}}_{\mathrm{tid}}$ and ${r}_{s}$, respectively. This luminosity is very high but can be achieved only if: (i) the stripped material reaches the stagnation radius ${r}_{s}$ (but we estimate that the stripped gas needs $t\gtrsim 300$ years to reach the stagnation radius, ${r}_{s}\approx {10}^{16}\;\mathrm{cm}$, if it travels at ${v}_{g}\sim 10\;\mathrm{km}\;{{\rm s}}^{-1}$); (ii) all of the kinetic energy is converted into (observable) luminosity. Thus, we expect that the luminosity due to these shocks is much lower than ${L}_{\mathrm{max}}$, and we will neglect it in the rest of this paper.

On the other hand, the stripped material will be exposed to photoevaporation by the massive young stars (Murray-Clay & Loeb 2012). The presence of the stripped material modifies the geometry of the PE and might strongly enhance photoevaporation (as discussed in Murray-Clay & Loeb 2012). To correctly evaluate the new geometry is beyond the aims of the current paper, but we can account for this correction in the following way. The surface of the stripped material will take part in photoevaporation, and will produce an approximately spherical wind.

The recombination rate in the wind will be of the same order of magnitude as the rate of ionizing photons that can be absorbed by the tidally stripped material, that is

$$\begin{eqnarray}&&{Q}_{\mathrm{tid}}={\displaystyle \frac{4\pi \;{r}_{\mathrm{str}}^{2}\;{L}_{\mathrm{UV}}}{4\pi {D}^{2}{\phi }_{0}}},\end{eqnarray} \tag{ 20 }$$

where ${r}_{\mathrm{str}}$ is the maximum radius reached by the tidally stripped material. We evaluate ${r}_{\mathrm{str}}$ as

$$\begin{eqnarray}&&{r}_{\mathrm{str}}\sim {r}_{p}+{t}_{t}\;{v}_{g},\end{eqnarray} \tag{ 21 }$$

i.e., as the radius of the PE, plus the distance travelled by the stripped material (moving at velocity ${v}_{g}$) during the time ${t}_{t}$ between the instant when the PE radius becomes equal to the tidal radius (${r}_{t}={r}_{p}$), and the time of the periapse passage.

Figure 3 shows the recombination rate⁴ R (assumed to be equal to ${Q}_{\mathrm{tid}}$) and the radius ${r}_{\mathrm{str}}$, as derived from Equations (20) and (21), respectively. In Figure 3, we assume ${v}_{g}=10$ km s⁻¹, periapse distance = 200 AU and eccentricity e = 0.976, i.e., the same periapse and eccentricity as the G2 object. If ${r}_{p}\gt {r}_{t}$ along the entire orbit, we assume ${t}_{t}=0.5\;{T}_{\mathrm{orb}}$ (where ${T}_{\mathrm{orb}}$ is the orbital period). In principle, this kind of calculation can be applied also to planets, but for the orbit of the G2 cloud ${r}_{t}$ is always larger than ${r}_{p}$ if ${r}_{p}\lesssim 1.5\times {10}^{12}$ cm (for ${m}_{p}={10}^{-3}$ M${}_{\odot }$). In the following, we refer to the model presented in Figure 3 as CASE 3 (see Table 1).

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Do we have any chances of detecting rogue planets or PEs in the GC with current or forthcoming facilities? The K (2.1 $\mu {\rm m}$) and L' (3.8 $\mu {\rm m}$) magnitudes of a rogue planet at the distance of the GC ($\sim 8$ kpc) are ${m}_{K}\sim 32$ and ${m}_{L\prime }\sim 26$ for a mass ${m}_{p}\sim {10}^{-3}$ M${}_{\odot }$ (Allard et al. 2001, 2007), respectively. Therefore, such rogue planet would be invisible for current and forthcoming facilities (30 m class telescopes are expected to observe stars down to ${m}_{K}\sim 24$). On the other hand, some processes might take place that enhance the chances of observing a planet, such as tidal disruption, atmosphere evaporation and bow shocks. In the previous section, we analyzed these processes for both planets and PEs.

Figure 4 shows the luminosity of the Br $\gamma$ line (2.166 $\mu {\rm m}$) derived from ${L}_{\mathrm{Br}\gamma }=2.35\times {10}^{-27}\;\mathrm{erg}\;{{\rm s}}^{-1}\;R/\alpha {}_{{\rm B}}$, where the recombination rate R was calculated in the previous section. The most optimistic prediction for a Jupiter-like planet (${m}_{p}={10}^{-3}$ M${}_{\odot }$ and ${r}_{p}={10}^{10}$ cm) is a Brγ luminosity ${L}_{\mathrm{Br}\gamma }\approx 5\times {10}^{18}$ erg s⁻¹, if the planet atmosphere is undergoing photoevaporation. At the distance of the GC, this corresponds to a flux $\lesssim {10}^{-27}\;\mathrm{erg}\;{{\rm s}}^{-1}\;{\mathrm{cm}}^{-2}$, which is far below the sensitivity of existing and forthcoming instruments (a 24 hr integration with VLT SINFONI can theoretically detect a Brγ flux $\sim 5\times {10}^{-18}\;\mathrm{erg}\;{{\rm s}}^{-1}\;{\mathrm{cm}}^{-2}$ with a signal-to-noise ratio (S/N) of ∼10).

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Thus, the only chance of detecting a rogue planet in the GC is that it is disrupted and accretes onto the SMBH. If a portion of the planet mass is accreted by the SMBH, this might lead to a flare with bolometric luminosity $\leqslant 2\times {10}^{41}$ erg s⁻¹ (from Equation (45) of Zubovas et al. 2012). This event is very unlikely, because the tidal disruption occurs only if the periapse distance from the SMBH is $\lesssim 1.6\;\mathrm{AU}\;({r}_{p}/{10}^{10}\;\mathrm{cm})$. We expect a planet disruption rate ${R}_{\mathrm{dis}}\sim {10}^{-5}{f}_{p}$ yr⁻¹ (where ${f}_{p}$ is the number of planets per star in the GC, and 10⁻⁵ yr⁻¹ is the tidal disruption rate of stars estimated by Alexander 2005). This process is very rare, but might be relevant for explaining flares in other galactic nuclei.

Theoretical models (e.g., Wuchterl 1999; Boss 2005; Helled et al. 2006; Helled & Bodenheimer 2010) suggest that a 0.001 M${}_{\odot }$ PE does not collapse immediately (because it is optically thick) and needs to cool for $\lesssim 1$ Myr at a bolometric luminosity $\sim {10}^{-6}$ L${}_{\odot }\sim {10}^{27}\;\mathrm{erg}\;{{\rm s}}^{-1}$. At the distance of the GC, this luminosity corresponds to ${m}_{L\prime }\gtrsim 25$, which is far below the limits of current and forthcoming observational facilities.

The photoevaporation of the "atmosphere" of a PE (with a radius of ${r}_{p}=5\times {10}^{12}$ cm) could lead to a Brγ luminosity of $\sim 5\times {10}^{27}$ erg s⁻¹, considering CASE 2 estimates for the mass loss by photoevaporation, assuming ${L}_{\mathrm{UV}}={10}^{40}$ erg s⁻¹. If the PE is undergoing tidal stripping, this luminosity might be boosted by more than two orders of magnitude. A ${L}_{\mathrm{Br}\gamma }\approx {10}^{30}$ erg s⁻¹ can be observed with current 8 m telescopes, and is not far from the actual luminosity of the G2 cloud ($\sim 6\times {10}^{30}$ erg s⁻¹).

Tidal disruption of a PE occurs if the distance of the PE from the SMBH is $d\sim 750\;\mathrm{AU}\;({r}_{p}/5\times {10}^{12}\;\mathrm{cm})$, suggesting that the cross section for PE disruption is a factor of $\approx {10}^{5}$ larger than the cross section for planet disruption (neglecting gravitational focusing). This leads to a disruption rate of $\sim {f}_{\mathrm{PE}}$ yr⁻¹, where ${f}_{\mathrm{PE}}$ is the number of PEs per each star.

How many PEs can form in the GC and for how long do they survive? This question contains a number of major uncertainties and we can just suggest a few hints. PEs are quite elusive objects and are expected to be relatively short-lived before they contract to Jupiter-like size ($\sim {10}^{3-6}$ years, Wuchterl 1999; Helled et al. 2008; Forgan & Rice 2013), but in the GC they could be efficiently separated from their parent star, and the strong UV flux might substantially slow down their cooling (and collapse). Furthermore, we could reverse our question, and use the non-detection of ${L}_{\mathrm{Br}\gamma }\approx {10}^{31}$ erg s⁻¹ objects to constrain the frequency of PEs in the GC.

The G2 cloud is the only observed object (so far) that shares similar properties with a photoevaporating and partially stripped PE. G2 probably originated from the clockwise disk in the GC, and has a very high eccentricity, $e\sim 0.976\pm 0.007$ (Pfuhl et al. 2015). Both these constraints on the orbit are fairly consistent with the hypothesis of tidal capture of a PE (initially formed in the clockwise disk) by the SMBH, or of a dynamical ejection of the PE from the clockwise disk. It is now clear that G2 survived its periapse passage at a distance of $\sim 200$ AU from the SMBH (Witzel et al. 2014), but some of its material was tidally stripped. A PE with radius $\sim {10}^{12}$ cm is tidally stripped if the periapse is $\sim 160$ AU: such PE would not be completely tidally disrupted at a distance of $\sim 200$ AU from the SMBH, but it would suffer some tidal stripping.

Figure 4 shows that the Br $\gamma$ luminosity of a PE (as derived from photoevaporation and partial tidal stripping) matches the one observed for G2, if the PE has ${m}_{p}\approx {10}^{-3}$ M${}_{\odot }$ and ${r}_{p}\approx 2\times {10}^{13}$ cm. If the main trigger of the Br $\gamma$ emission is photoevaporation by the intense UV background in the GC, we expect the Br $\gamma$ luminosity to remain roughly constant during the PE orbit. This is fairly consistent with the fact that the Br $\gamma$ luminosity of G2 remained nearly constant over 10 years (Pfuhl et al. 2015).

G2 was detected also in the L' band (${m}_{L\prime }\sim 14$, Pfuhl et al. 2015). Gillessen et al. (2012) suggest that the continuum L' emission comes from small ($\sim 20$ nm), transiently heated dust grains with a total warm dust mass of $\sim 2\times {10}^{23}$ g. The grains might be warmed up by an inner source (e.g., a low-mass star, Ballone et al. 2013; Scoville & Burkert 2013; Witzel et al. 2014), or by some external mechanism (UV heating, shocks, etc.). In the PE scenario, the minimum PE radius necessary to reach ${L}_{L\prime }\sim 2.1\times {10}^{33}\;\mathrm{erg}\;{{\rm s}}^{-1}$ (Witzel et al. 2014) is

$$\begin{eqnarray}{r}_{\mathrm{min}} & \sim & 5.5\times {10}^{12}\;\mathrm{cm}{\left({\displaystyle \frac{{L}_{L}\prime }{2.1\times {10}^{33}\;\mathrm{erg}\;{{\rm s}}^{-1}}}\right)}^{1/2}{\left({\displaystyle \frac{{T}_{\mathrm{dust}}}{560\;{\rm K}}}\right)}^{-2},\end{eqnarray} \tag{ 22 }$$

where ${T}_{\mathrm{dust}}\sim 560$ K is the estimated dust temperature (from $L\prime -M\prime \sim 0.3$, Gillessen et al. 2012). We note that ${r}_{\mathrm{min}}$ strongly depends on ${T}_{\mathrm{dust}}$ and that the estimate of ${T}_{\mathrm{dust}}$ is very uncertain.⁵

While the scenario of a wind-enshrouded low-mass star would naturally explain the continuum L' emission (Witzel et al. 2014), we cannot reject the hypothesis that a PE, embedded in the hot dense medium of the GC, might host sufficient warm dust to power the observed L' emission. Thus, a rogue PE might be a viable scenario to explain G2 and other G2-like objects (e.g., Pfuhl et al. 2015 suggest that the object G1 is related to G2).

The analytic model discussed in this paper relies on the possibility that planets form in the GC. The GC might be a hostile environment for planet formation, given the high UV background, the high temperature and the strong tidal field by the SMBH. Discussing the chances that planets form in the GC is beyond the aims of this paper. Here, we just mention that planetesimals and asteroids were recently investigated as possible sources of SMBH flares (Cadez et al. 2008; Kostić et al. 2009; Zubovas et al. 2012; Hamers & Portegies Zwart 2015), and there are even some observational hints for the existence of protoplanetary disks in the innermost parsec (Yusef-Zadeh et al. 2015). Further observational evidence is necessary, to confirm that planets and protoplanets form in the GC.

Planets cannot be directly observed with current and forthcoming facilities. Only photoevaporating PEs are sufficiently bright to be detected. PEs are theoretically predicted objects but have not yet been observed. They can form only if gravitational instabilities in a protoplanetary disk lead to fragmentation, and to the formation of self-gravitating clumps. Recent work has shown that gravitational instabilities can lead to the formation of PEs only in the outermost regions of a protoplanetary disk ($\gt 100$ AU, Boley 2009), where self-gravity is stronger. This enhances the probability that PEs become unbound with respect to their initial system, but might imply that their formation is endangered by the strong UV field in the GC. Furthermore, even if hydrodynamical simulations show that clumps can form, they cannot predict self-consistently whether these clumps survive further evolution.

Recent estimates suggest that PEs remain in the pre-collapse phase (the one considered in this paper) for $\sim {10}^{5}$ years and for $\sim {10}^{4}$ years if their mass is $\sim {10}^{-3}$ and 10⁻² M${}_{\odot }$, respectively (Helled et al. 2006; Helled & Bodenheimer 2010). From an observational perspective, while the detection of massive giant planets ($\gtrsim {10}^{-3}$ M${}_{\odot }$) at distance $\gg 10$ AU from the central star might favor the gravitational instability scenario (Marois et al. 2008; see Figure 3 of Pepe et al. 2014), there is no strong evidence supporting the existence of PEs. Since PEs are such elusive objects, it is quite difficult to quantify the uncertainties in our model.

Besides the debate on the very existence of PEs, Figure 5 shows the impact of the main sources of uncertainties on the predicted Br-$\gamma$ luminosity of a photoevaporating PE. In particular, we focus on (1) the adopted approximation for the wind (fast or slow-wind approximation), (2) the flux of the UV background (from 10³⁸ to 10⁴⁰ erg s⁻¹ in the innermost 0.1 pc), (3) the mass of the PE.

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Figure 5 shows that the Br-$\gamma$ luminosity of a photoevaporating PE is uncertain by several orders of magnitude. We stress that the recombination rate (and thus the Br-$\gamma$ luminosity) does not depend on the mass of the PE in CASE 2. The mass of the PE is important only if we assume that the PE can be tidally stripped by the SMBH (CASE 3), because ${r}_{t}$ depends on the PE mass. In the fast-wind approximation (i.e., the gas is ejected with ${v}_{g}\gg {v}_{\mathrm{esc}}$, as discussed in detail in the Appendix), the Br-$\gamma$ luminosity due to photoevaporation (from Equation (23)) is lower by about one order of magnitude with respect to the slow-wind approximation, but the gas may undergo shocks with the hot medium, and this enhances the Br-$\gamma$ luminosity to a value (from Equation (24)) comparable with (or even higher than) the slow-wind approximation.