What If Planet Nine Has Satellites?

Currently, there are eight planets officially identified in our solar system. Most of the newly discovered large astronomical objects outside Neptune are dwarf planets or large asteroids called trans-Neptunian objects (TNOs). In view of the TNOs, the new discovery of 2012 VP113 and some potential members of the inner Oort Clouds has revealed a strange clustering in orbital elements (Trujillo & Sheppard 2014). The perihelion distance has arguments of perihelia ω clustered approximately around zero (Trujillo & Sheppard 2014; Batygin & Brown 2016a). Later analysis shows that the chance for this strange clustering being random is just 0.0007% (Batygin & Brown 2016a). Therefore, a dynamical mechanism involving a new planet located at more than 100 au has been suggested (Batygin et al. 2019). Many studies have constrained the mass and the orbital properties of the hypothesized Planet Nine (P9; Batygin & Brown 2016b; Gomes et al. 2016; Sheppard & Trujillo 2016; Becker et al. 2018; Sheppard et al. 2019). Current benchmark models suggest that P9 has mass M₉ ∼ 5–10 M_⊕, orbital semimajor axis a₉ ∼ 400–800 au, and eccentricity e₉ ∼ 0.2–0.5 (Batygin et al. 2019). However, the in situ formation of P9 is strongly disfavored, so that P9 might be a captured planet from the free-floating objects near the solar system (Kenyon & Bromley 2016; Batygin et al. 2019). A more detailed assessment of the probability of capture can be found in Li & Adams (2016).

Current benchmark models of P9 suggest that it has a temperature ∼40 K and a radius ∼3–4 R_⊕ (Batygin et al. 2019). The possible location of P9 in the celestial sphere is also constrained (Fienga et al. 2016; Batygin et al. 2019; Socas-Navarro 2022). Based on these properties, various observations, such as optical and microwave/infrared observations, have been deployed to observe the hypothesized P9 (Meisner et al. 2017, 2018; Naess et al. 2021). However, no electromagnetic wave signal has been detected for P9 (Linder & Mordasini 2016; Meisner et al. 2017, 2018). Careful examinations based on previous optical surveys also do not reveal the existence of P9 (Linder & Mordasini 2016). Therefore, these null results have made the P9 hypothesis more mysterious.

In view of these problems, some of the studies have suggested that P9 is a dark object (dark P9), such as a compact object made by dark matter (Wang et al. 2022) or a primordial black hole (PBH; Scholtz & Unwin 2020). In particular, the proposal of the PBH P9 has attracted many discussions because many studies beyond the standard models have already proposed the existence of PBHs with mass ∼M_⊕. There are various mechanisms that can generate PBHs in the early universe (Carr et al. 2021). However, the direct signals emitted by the PBH P9 (e.g., Hawking radiations) are too small to detect (Arbey & Auffinger 2006). Even if we assume that dark matter can distribute around the PBH P9, the resulting gamma-ray signals might be smaller than the current observation limits (Scholtz & Unwin 2020). Besides, a recent innovative proposal suggests that a small laser-launched spacecraft with a velocity of order 0.001c could reach the PBH P9 to detect its gravitational field, though we would need to wait roughly a decade for the measurement (Witten 2020).

Nevertheless, there are a lot of TNOs orbiting about the Sun inside the scattered disk region (∼100–1000 au), located between the inner Oort Clouds and the Kuiper Belt. These TNOs are also known as detached objects. Most of them are scattered from either the central solar system or the Kuiper Belt region. In fact, we have already observed at least 47 large TNOs with orbital semimajor axis larger than 100 au and size larger than 100 km. Therefore, it is possible that these large TNOs would be captured by P9 to become satellites of P9. Many dwarf planets, such as Pluto, and TNOs outside Neptune have satellite systems (Brown et al. 2006; Grundy et al. 2019). If these small objects can have satellites, it can be conceived that the more massive P9 might also have a number of satellites. In this article, we discuss some important observable features if P9 has captured satellites. For large satellites with small orbital semimajor axis, the tidal heating effect due to P9 would be important. It can be shown that these satellites would give an observable standard thermal radio spectrum. If P9 is a dark object, observing the satellites would be another kind of investigation to examine the P9 hypothesis in the near future. In the following, we assume that P9 is a dark object, and we follow the benchmark model of P9 with mass M₉ = 5 M_⊕, eccentricity e₉ = 0.2, orbital inclination i = 20°, and semimajor axis a₉ = 450 au (Batygin et al. 2019). We simply take the semimajor axis a₉ = 450 au as the average distance to the dark P9 from Earth.

There are many large TNOs moving in the scattered disk region (∼100–1000 au), such as 2018 AG37, 2018 VG18, and 2020 BE102. It is quite likely that some of the large TNOs (e.g., with size D ∼ 100 km) could be captured by the dark P9. In fact, many of the Kuiper Belt dwarf planets have at least one satellite. For example, the satellite of the dwarf planet Eris has radius R ∼ 700 km and semimajor axis a ∼ 4 × 10⁴ km (Brown & Butler 2018).

In general, when a TNO has a close encounter with a planet, energy will be lost in the capturing process owing to the inverse of the gravitational slingshot mechanism (Napier et al. 2021). The maximum capturing distance between the dark P9 and any TNOs can be characterized by the impact parameter b (Napier et al. 2021):

$$\begin{eqnarray}&&b\sim \displaystyle \frac{{M}_{9}}{{M}_{\odot }}{\left(\displaystyle \frac{{{GM}}_{\odot }}{{a}_{9}}\right)}^{3/2}{v}^{-3}{a}_{9},\end{eqnarray} \tag{ 1 }$$

where v is the incoming relative velocity between the dark P9 and any TNOs. Here b can be regarded as the closest distance between the dark P9 and the TNOs for the capturing process. Therefore, the relative velocity between the dark P9 and the TNOs is given by

$$\begin{eqnarray}&&v\sim \sqrt{\displaystyle \frac{{{GM}}_{\odot }}{{a}_{9}}}-\sqrt{\displaystyle \frac{{{GM}}_{\odot }}{{a}_{9}\pm b}}\cos {\rm{\Delta }}i,\end{eqnarray} \tag{ 2 }$$

where Δi is the orbital inclination difference between the dark P9 and the TNOs. As b ≪ a₉, the relative velocity is

$$\begin{eqnarray}&&v\sim \sqrt{\displaystyle \frac{{{GM}}_{\odot }}{{a}_{9}}}(1-\cos {\rm{\Delta }}i).\end{eqnarray} \tag{ 3 }$$

Putting Equation (3) into Equation (1), we get

$$\begin{eqnarray}&&b\sim {a}_{9}{\left(1-\cos {\rm{\Delta }}i\right)}^{-3}\left(\displaystyle \frac{{M}_{9}}{{M}_{\odot }}\right).\end{eqnarray} \tag{ 4 }$$

The benchmark orbital inclination of the dark P9 is i = 20° (Batygin et al. 2019). Based on the catalog compiled by the International Astronomical Union, ¹ the orbital inclinations of the TNOs (with semimajor axis a > 100 au) are quite close to i = 20°, except for three with i > 100°. The average difference between the orbital inclinations of P9 and the TNOs is about Δi = 18°. Including the possible uncertainty of the benchmark orbital inclination of the dark P9 δ i = 5° (Batygin et al. 2019), we take a conservative choice of Δi = 25°, which gives b ∼ 8.2 au.

On the other hand, we can also apply the radius of influence R_in discussed in Bate et al. (1971) to characterize the value of the impact parameter (i.e., b ≈ R_in). The radius of influence defines the region where the incoming TNO switches from a two-body problem with central mass M_⊙ to a two-body problem with central mass M₉ in the matched conics approximation (Napier et al. 2021). Based on this approximation, the impact parameter is given by (Bate et al. 1971)

$$\begin{eqnarray}&&b={R}_{\mathrm{in}}={a}_{9}{\left(\displaystyle \frac{{M}_{9}}{{M}_{\odot }}\right)}^{2/5}.\end{eqnarray} \tag{ 5 }$$

Using our benchmark parameters, the dark P9 can capture any TNOs moving within the distance of b ∼ 5.3 au. To get a more conservative estimation, in the following we adopt the value of b = 5.3 au as the impact parameter. In view of this, the dark P9 can create a "capturing volume" when it is orbiting about the Sun. All of the TNOs inside this capturing volume would likely be captured by the dark P9. The capturing volume is given by

$$\begin{eqnarray}&&\begin{array}{l}V=\left(2\pi {a}_{9}\sqrt{1-\displaystyle \frac{{e}_{9}^{2}}{2}}\right)(\pi {b}^{2})=2{\pi }^{2}{b}^{2}{a}_{9}\sqrt{1-\displaystyle \frac{{e}_{9}^{2}}{2}}\\ \approx 2.5\times {10}^{5}\,{\mathrm{au}}^{3}.\end{array}\end{eqnarray} \tag{ 6 }$$

Generally speaking, very large TNOs (with size ≥ 500 km) would be easier for us to identify. Based on the catalog compiled by the International Astronomical Union, there are four TNOs with size ≥500 km (assuming a standard asteroid albedo p = 0.1) and orbital semimajor axis a = 100–1000 au. The number of very large TNOs can provide a standard reference for estimating the number of TNOs with different sizes inside the scattered disk region.

Consider the region of the scattered disk for a = 100–1000 au. Based on the TNO catalog, all of the reported TNOs with a ≤ 1000 au are located within a scale disk thickness of 72.5 au above and below the P9 orbital plane. We therefore consider the volume of the scattered disk V_d ∼ (2 × 72.5)π(1000² − 100²) ≈ 4.5 × 10⁸ au³. Assuming that the distribution of asteroid size is the same as that in the Kuiper Belt, ${dN}/{dD}\propto {D}^{-q}$ (Fraser et al. 2014). This size distribution in the Kuiper Belt is well represented by a broken power law in D for large and small Kuiper Belt objects. For cold Kuiper Belt objects, the slope q for large objects (with size D ≥ 140 km) is q = 8.2 ± 1.5, while q = 2.9 ± 0.3 for D < 140 km (Fraser et al. 2014). Since there are four TNOs with size ≥500 km, taking q = 8.2, the average number density of TNOs with size D ≥ 140 km inside V_d is 8.5 × 10⁻⁵ au⁻³.

Since the capturing volume is 2.5 × 10⁵ au³, the average number of TNOs with size D ≥ 140 km captured is about 20. Note that this number is close to the typical number of satellites found in Jovian planets. In fact, the Jovian planets are somewhat close to each other, so that the gravitational perturbation effect is significant. This would reduce the capturing volume and the number of satellites. However, there is almost no massive perturber for P9. The closest massive object Sedna (semimajor axis a ∼ 500 au) has a relatively small mass ∼10⁻³ M_⊕ only, which cannot affect the capturing volume significantly. Therefore, we expect that there are a considerable number of captured TNOs to form a satellite system for P9, like the satellite systems in Jovian planets.

Consider a fiducial radius of the satellite R = D/2 = 100 km. For simplicity, let us assume that the satellite is spherical in shape. The tidal force on the satellite is large when the satellite is close to P9. The Roche limit is ∼2 × 10⁴ km if we assume the density of the satellite to be ρ = 1 g cm⁻³. For Uranus and Neptune, which have masses similar to the dark P9, the range of the orbital semimajor of the satellites is a_s ∼ 5 × 10⁴ km to 5 × 10⁷ km. In the following, we will mainly consider the range of the orbital semimajor axis a_s = 10⁵–10⁶ km. Note that captured objects generally have large semimajor axis and eccentricity initially (Goulinski & Ribak 2018; Napier et al. 2021). However, orbital evolution through tidal effects would further decrease the values of semimajor axis and eccentricity (see the discussion below).

The equilibrium temperature due to solar luminosity is approximately given by

$$\begin{eqnarray}&&T\approx 54.8\sqrt{\displaystyle \frac{26}{{a}_{9}}}\,{\rm{K}},\end{eqnarray} \tag{ 7 }$$

where we have neglected the albedo and the phase integral (Stansberry et al. 2008). For a₉ = 450 au, we get T = 13 K. However, if the satellite is very close to P9, the tidal heating effect would be very significant. The tidal heating model has been discussed for more than 50 yr (Goldreich & Soter 1966). In general, the tidal heating rate can be calculated by (Segatz et al. 1988; Lainey et al. 2009; Renaud & Henning 2018)

$$\begin{eqnarray}&&\dot{E}=\displaystyle \frac{21C}{2}\displaystyle \frac{{\left({Rn}\right)}^{5}{e}_{s}^{2}}{G},\end{eqnarray} \tag{ 8 }$$

where $n=\sqrt{{{GM}}_{9}/{a}_{s}^{3}}$ is the mean orbital motion and e_s is the eccentricity of the satellite orbit (Segatz et al. 1988). Here the constant C is related to the Love number k₂ and the quality factor Q, which reflects the physical properties (e.g., elastic rigidity) of the satellite (Segatz et al. 1988; Lainey et al. 2009; Hussmann et al. 2010). However, the value of C for the satellite is uncertain. Theoretical prediction shows that the value of C should be lower than 0.06 for the high-density satellite core (Kervazo et al. 2022). We adopt the value revealed from the observational data of Jupiter's moon Io, C ≈ 0.02 (Lainey et al. 2009). In equilibrium, the tidal heating rate would be equal to the radiation cooling rate. Therefore, we have

$$\begin{eqnarray}&&T={\left(\displaystyle \frac{\dot{E}}{4\pi {\sigma }_{s}{\epsilon }_{\nu }{R}^{2}}\right)}^{1/4},\end{eqnarray} \tag{ 9 }$$

where σ_s is the Stefan–Boltzmann constant and _ν is the gray emissivity. For simplicity, we assume _ν = 1 here.

In Figures 1 and 2, we plot the equilibrium temperature as a function of a_s  for different values of R and e_s , respectively. We can see that the temperature can be quite high for some values of a_s , R, and e_s . Generally speaking, smaller values of a_s and larger values of R and e_s can give a higher equilibrium temperature. For the fiducial values of a_s = 10⁵ km, R = 100 km, and e_s = 0.5, we get $\dot{E}=1.4\times {10}^{12}$ W. The equilibrium temperature of the satellite is about 119 K, which can emit a significant amount of radio radiation with frequency ν > 100 GHz. Besides, we can estimate the time required for the satellite to heat up from 10 to 100 K. Assuming a typical specific heat capacity for the satellite c_s = 1000 J kg⁻¹ K⁻¹, the time required is ∼10⁴ yr for the fiducial parameters used.

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In the following, we estimate the thermal radio flux emitted by the satellite with the fiducial parameters. The thermal radio flux density is given by

$$\begin{eqnarray}&&{S}_{\nu }=\int \displaystyle \frac{2h{\nu }^{3}}{{c}^{2}({e}^{h\nu /{kT}}-1)}d{\rm{\Omega }}\approx \displaystyle \frac{2\pi h{\nu }^{3}}{{c}^{2}({e}^{h\nu /{kT}}-1)}{\left(\displaystyle \frac{R}{{a}_{9}}\right)}^{2}.\end{eqnarray} \tag{ 10 }$$

Therefore, we can get the expected thermal radio flux density as a function of ν for the fiducial parameters (see Figure 3). The radio flux density is ∼2 μJy for ν = 300 GHz. The observable limit for the most sensitive submillimeter interferometer (e.g., Atacama Large Millimeter/submillimeter Array, ALMA) is around 1 μJy at ν = 100–300 GHz. Hence, it is feasible to observe this small flux using current observational technologies. For lower frequencies, the expected radio flux density is S_ν ≈ 10 nJy at ν = 20 GHz. This can be observable by the future SKA radio interferometer.

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Moreover, the thermal radio flux density S_ν is proportional to the frequency ν². This can be differentiable from the normal background radio flux, which is usually modeled by S_ν ∝ ν^−α with α > 0. In other words, by obtaining the radio spectrum emitted from the region of the dark P9, if we can detect a relatively strong thermal radio spectrum (S_ν ∝ ν²), this would be solid evidence to verify the P9 hypothesis because there is no other astrophysical mechanism that can increase the temperature of a distant object to more than 50 K. For the conventional P9 model (not a dark object), the expected radio flux emitted by P9 should be ∼mJy at 200 GHz (Naess et al. 2021), which is 1000 times larger than that of a satellite. In any case, if we can detect either a millijansky signal from P9 or a microjansky signal from the satellite, the P9 hypothesis can be verified. Besides, if there is any potential signal received from P9 or the satellites, we can track the source for a couple of years to see whether the signal would follow a nearly Keplerian orbit over time or not. This can further provide smoking-gun evidence to verify the P9 hypothesis.

Previous studies have constrained the possible range of location for P9 (Fienga et al. 2016; Batygin et al. 2019; Socas-Navarro 2022). A recent study has further constrained the exact location of P9 to R.A. = 482 ± 4° and decl. = 103 ± 18 (Socas-Navarro 2022). Such a small constrained region can make the observation much easier. The telescopes or interferometers used can focus on the target region for a very long exposure time to gain enough sensitivity to detect the potential thermal signals.

Note that the tidal heating rate gained by the satellite originates from the loss rate of the gravitational potential energy of the P9–satellite system. The eccentricity would gradually decrease so that the tidal heating rate would also decrease. The eccentricity fractional change rate is given by

$$\begin{eqnarray}&&\displaystyle \frac{| {\dot{e}}_{s}| }{{e}_{s}}=\left(\displaystyle \frac{{e}_{s}^{2}-1}{2{e}_{s}^{2}}\right)\displaystyle \frac{\dot{E}}{E}.\end{eqnarray} \tag{ 11 }$$

The timescale for the eccentricity shrinking is $\tau \sim | {e}_{s}/{\dot{e}}_{s}|$, which is about 0.6 Myr for the fiducial parameters. This timescale is short compared to the age of the solar system. In fact, there is a compromise between having the orbital parameters of the satellites such that the radio emission is detectable (e.g., with small a_s ) and sufficiently long lived to make the higher detection a probability (e.g., with large a_s ). Here the range of a_s we considered (a_s = 10⁵–10⁶ km) is almost optimal for examination. Nevertheless, the relatively short eccentricity shrinking timescale would not be a big problem if the satellite capture event were a recent event. In addition, as we have shown that the satellite capture is not a rare event, there would be more than one satellite with size >140 km at a_s ∼ 10⁵ km. Therefore, we expect that such a thermal radio signal of the satellite may still be observed.

In this article, we have demonstrated a theoretical framework to predict the possible observable signal from the P9–satellite system. If the dark P9 has a satellite system, the only current feasible observation is to detect the possible signals from the satellites. We have shown that for a satellite with a typical size ∼100 km with average orbital radius a_s ∼ 10⁵ km from the dark P9 the temperature can be as large as ∼100 K owing to the tidal heating effect. For such a high temperature, the satellite can emit a strong enough thermal radio flux (∼1 μJy at 100–300 GHz) that can be observed by ALMA. Moreover, the specific thermal radio spectrum S_ν ∝ ν² could be easily differentiable from the background radio flux so that it can provide smoking-gun evidence for the P9 hypothesis. The only possible reason for the existence of a ∼100 K object at ∼450 au from the Sun is that it is a satellite of a host planet. This is because a host dwarf planet or a minor planet does not have enough mass to heat up the satellite to ∼100 K.

As we have shown above, there are a lot of TNOs with size >140 km in the scattered disk region. Therefore, the chance for these large TNOs (with R ∼ 100 km) being captured by P9 is not low. Besides, based on the example of Uranus (≈14 M_⊕), at least 13 satellites are located within 10⁵ km, which suggests that our fiducial value of a_s = 10⁵ km is a reasonable choice of consideration. For the eccentricity, simulations show that most of the captured objects would be orbiting with a very high eccentricity ≈1 (Goulinski & Ribak 2018). Therefore, our fiducial value e_s = 0.5 is a conservative choice of estimation.

Since no optical and radio signals have been detected so far for P9, the suggestion of P9 being a PBH has become a hot topic recently. There are some suggestions to send detectors to visit the alleged PBH P9 (Witten 2020; Hibberd et al. 2022). It would be very exciting because this may be our only chance to visit a black hole within our approachable distance. Nevertheless, we would need to wait for at least 10 yr for the detectors to arrive at the PBH P9. Some other studies have proposed to detect P9 by gravitational lensing (Philippov & Chobanu 2016; Schneider 2017; Domènech & Pi 2022). However, the mass of P9 is very small, so it requires a very sensitive measurement for the short-lived lensing event, from which it may not be very easy to get any good confirmation. A recent study has proposed narrow possible locations of P9 (Socas-Navarro 2022). If P9 is a dark object and it has a satellite system, our proposal can directly observe the potential thermal signals emitted by the satellites now. Therefore, this would be a timely and effective method to confirm the P9 hypothesis and verify whether P9 is a dark object or not.

The work described in this paper was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (project No. EdUHK 18300922).