Are There Terrestrial Planets Lurking in the Outer Solar System?

The first free-floating planets were discovered over two decades ago (Lucas & Roche 2000). Since then, microlensing searches have become the favored method for free-floating-planet detection (Sumi et al. 2011; Mróz et al. 2017, 2019). In recent years, the first low-mass free-floating planets have been detected (Mróz et al. 2020; Gould et al. 2022). Most recently, Sumi et al. (2023) measured the mass function of free-floating planets down to low-mass planets. Here, we use this mass distribution to ask the question: could there exist yet-undiscovered terrestrial planets in the outer solar system that were gravitationally captured by the solar system early in its history?

The possibility explored here is entirely distinct from Planet Nine and from other hypothetical planets (Lykawka & Mukai 2008; Batygin & Brown 2016; Whitmire 2016; Volk & Malhotra 2017; Silsbee & Tremaine 2018; Lykawka & Ito 2023). Planet Nine, a proposed ∼6 M_⊕ planet with a semimajor axis of ∼400 au, is a possibility motivated by the observed clustering of extreme trans-Neptunian objects in the outer solar system (Batygin & Brown 2016; Brown & Batygin 2016; Batygin et al. 2019; Brown & Batygin 2021). The potential for its existence is debated (Shankman et al. 2017; Bernardinelli et al. 2020a, 2020b; Clement & Kaib 2020; Zderic & Madigan 2020; Napier et al. 2021). Several searches for Planet Nine have been and continue to be conducted, including ones with the Wide-field Infrared Survey Explorer (Meisner et al. 2018), the Transiting Exoplanet Survey Satellite (Rice & Laughlin 2020), the Atacama Cosmology Telescope (Naess et al. 2021), the Zwicky Transient Facility (Brown & Batygin 2022), and the Dark Energy Survey (Belyakov et al. 2022). Here, we present theoretical arguments for yet-undiscovered sub-Earth-mass planets, motivated by novel statistics on free-floating planets. This contrasts with the Planet Nine hypothesis' basis in the posited clustering of solar system objects, and Planet Nine's mass several times that of the Earth.

In this Letter, we consider the implications of recently discovered free-floating planets for the possibility that the solar system hosts terrestrial planets at large distances from the Sun. In Section 2, we create a theoretical framework for exploring this possibility and explore the masses of such planets. In Section 3, we evaluate the detectability of such planets. Finally, in Section 4, we discuss the results in the context of the literature and highlight key areas for future work.

Parker et al. (2017) studied the capture of free-floating planets by the solar system in its birth cluster and considered three scenarios for the birth environment: subvirial (partially collapsing), slightly supervirial (gently expanding), or highly supervirial (unbound, rapidly expanding). Batygin et al. (2019) noted that planet capture is enhanced for supervirial clusters but also that solar system enrichment of short-lived radiogenic isotopes is suppressed for those clusters (and enhanced for subvirial ones, in which planet capture is suppressed). ¹ To be as conservative as possible, we analyze the subvirial case, in which planet capture is suppressed, but observed meteoric enrichment is more easily explained.

The numerical approach adopted in Parker et al. (2017) improves on the earlier calculations of planetary capture (Li & Adams 2016; Mustill et al. 2016) in part by including Gaussian kinematic substructure, as opposed to assuming a Maxwellian distribution. We adopt the recently quantified mass function of free-floating planets (Sumi et al. 2023) to estimate the mass of a captured free-floating planet with an expected abundance of unity in the solar system as a function of semimajor axis,

$$\begin{eqnarray}&&{M}_{p}=8{M}_{\oplus }{\left[{2.18}_{-1.40}^{+0.52}{f}_{{ffp}}f(\lt a)\right]}^{1/\alpha },\end{eqnarray} \tag{ 1 }$$

where the fraction of free-floating planets that are captured is f_ffp ∼ 2%, ² and where the cumulative fraction of captured planets out to a certain semimajor axis a, f( < a), is adopted from Figure 2 of Parker et al. (2017). We adopt the power-law slope of $\alpha ={0.96}_{-0.27}^{+0.47}$, which was applied by Sumi et al. (2023) down to Mars-mass planets. In Equation (1) and throughout this Letter, unless otherwise stated, we adopt ${dN}/d\mathrm{log}M$, meaning that the quoted abundance of planets corresponds to planets masses within a logarithmic bin centered on the quoted mass.

Survival in the field is essentially guaranteed given survival in the birth cluster since the low stellar density and high velocity dispersion vastly outweigh the longer timescale (Batygin et al. 2019).

To evaluate the expected mass of the most massive captured free-floating planet as a function of the maximum semimajor axis, given the arguments presented in Section 2, we implement a Monte Carlo simulation. In order to compute M_p for a given choice of a, as described in Equation (1), we draw the normalization factor interior to the brackets from a split Gaussian distribution centered at 2.18 with σ_u = 0.52 and σ_l = 1.40, and we draw α from a split Gaussian distribution centered at 0.96 with σ_u = 0.47 and σ_l = 0.27. As mentioned in Section 2, we adopt f_ffp = 2% and the f( < a) distribution for the subvirial case in Parker et al. (2017). We evaluate M_p 10⁶ times out to each value of a (which is sampled 10³ times in log-space) to obtain their probability distributions. The median and 50% confidence interval are displayed in Figure 1.

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We use a Monte Carlo simulation to explore the detectability of captured planets in the outer solar system. First, we draw instances of semimajor axis a and eccentricity e for the probability distributions in Parker et al. (2017) corresponding to the subvirial case. Next, we draw a value for the planet's mean anomaly M from a uniform distribution between 0 and 2π (implying that the observation occurs at an arbitrary time). We then calculate the planet's eccentric anomaly by solving Kepler's equation $M=E-e\sin E$, and subsequently compute the planet's heliocentric distance, $a(1-e\cos E)$. We run this routine 10⁸ times and record the heliocentric distance for each run. Figure 2 shows the cumulative probability distribution of heliocentric distances (lower panel), as well as this distribution multiplied by the capture fraction, f_ffp ∼ 2%, and by the best-fit power-law for ${dN}/d\mathrm{log}M$ from Sumi et al. (2023) evaluated at the masses of the four terrestrial planets, to give the expected number of planets within logarithmic mass bins centered on the quoted mass (upper panel). To directly evaluate detectability, we calculate the distribution of apparent magnitudes using the distribution of heliocentric distances, the geometric cross sections of the four terrestrial planets, and a nominal albedo of 0.2. Figure 3 shows the cumulative probability distribution of apparent magnitudes for each planet size (lower panel), as well as the expected number of planets (upper panel), calculated the same way as in Figure 2. Note that the upper panels of both Figures 2 and 3 adopt the best-fit values in Sumi et al. (2023), which correspond to the solid blue curve in Figure 1.

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We estimate that the number of captured free-floating planets in the outer solar system with mass strictly greater than that of Mars is ∼1.2 and that the number of such planets with a strict cutoff at the mass of Mercury is ∼2.4. ³ When we instead adopt logarithmic bins centered at the Mars mass and the Mercury mass, respectively, we find that the expected number of such planets is ∼2.7 for mass comparable to that of Mars and ∼5.2 for mass comparable to that of Mercury. These planets would have a median heliocentric distance of ∼1400 au, with ∼half of them existing in the range 600–3500 au.

LSST's 10 yr coadded detection limits in the g and r bands are 26.9. ⁴ We note that the magnitude limit for detecting a planet in the outer solar system would be lower than this value, in reality, to compensate for inevitable false positives encountered during the blind shift-and-stack search procedure. Simply for the purposes of illustrating the effect of such a penalty, we will adopt a somewhat arbitrary value of 1 mag.

At the 10 yr coadded detection limit of LSST, 26.9 mag, the expectation values are ∼1.4 Mercury-size planets and ∼0.9 Mars-size planets. If we include a fiducial blind shift-and-stack penalty of 1 mag (bringing the limit to 25.9 mag), the expectation values decrease, respectively, to ∼1.0 Mercury-size planets and ∼0.7 Mars-size planets. Note that these numbers would change if the albedo of these planets were taken to be a value other than the nominal one adopted here, 0.2. Any captured planets discovered by LSST would likely exist at a heliocentric distance in the range of 400–700 au.

LSST will only survey about half of the celestial sphere, so if the closest captured planet were located in the other half of the sky, it would not be detected. Future work should explore the potential for telescopes in the Northern Hemisphere to contribute to the search for the closest captured planet. Additionally, in contrast to Planet Nine, we do not expect the parameter space for a Mercury- or Mars-mass planet in the outer solar system to overlap with the detection capabilities of the Atacama Cosmology Telescope (Naess et al. 2021).

Perets & Kouwenhoven (2012) numerically estimated that 3–6 × (N_ffp /1)% of all stars capture a planetary companion at a distance of ∼few × 100–10⁶ au, where N_ffp is the number of free-floating planets per star in the birth clusters. Using the Sumi et al. (2023) result down to ${M}_{\min }=0.4{M}_{\oplus }$, this implies ∼1–2 planets with mass ≳0.4M_⊕ captured per star. We note that this estimate is in excellent agreement with our calculations, which suggest that there should be ∼1 planet with mass ≳0.4M_⊕ in the Oort cloud.

In recent years, proposed outer-solar-system planets have primarily been motivated by posited patterns in the orbital elements of small bodies in and beyond the Kuiper Belt (Lykawka & Mukai 2008; Batygin & Brown 2016; Whitmire 2016; Volk & Malhotra 2017; Lykawka & Ito 2023). In contrast, this Letter presented a theoretical argument motivated by new observational constraints on the abundance of free-floating planets as a function of mass (Sumi et al. 2023). While other papers have explored the implications of free-floating planets for the possibility of bound, undiscovered planets in the outer solar system, to date they have produced very low likelihoods for the existence of such planets (Li & Adams 2016; Mustill et al. 2016; Parker et al. 2017). We showed, based on a straightforward theoretical argument, that captured terrestrial planets are likely to exist in the outer solar system. This prediction is unique in terms of the combination of semimajor axes and masses of the proposed planets. Additionally, the prediction avoids the numerous pitfalls involved in identifying patterns in the orbital elements of Kuiper Belt objects (e.g., Shankman et al. 2017; Van Laerhoven et al. 2019; Bernardinelli et al. 2020a, 2020b; Napier et al. 2021).

Future work should include simulations studying in greater detail the capture and retention of free-floating planets as well as planets bound to other stars. In addition, simulations can shed light on the probability distribution for the orbital plane and position in the sky for captured planets. Future work should also explore other observational tests for the existence of captured planets. If the closest captured planet is currently in the Southern sky and has favorable orbital conditions for detection, it could potentially be discovered in LSST images with a sufficiently advanced blind shift-and-stack algorithm. The detectability of captured planets with surveys such as LSST should be explored in more detail. We note that in the future, discoveries of free-floating planets with the Nancy Grace Roman Telescope (Johnson et al. 2020) and detections of extreme trans-Neptunian objects with LSST (Ivezic et al. 2004) will help refine the estimates made in this Letter. If the statistics from Sumi et al. (2023) hold to significantly lower masses, it may even be possible to detect captured dwarf planets (with masses comparable to the mass of Pluto) early in the LSST observing program.

We thank the anonymous referee for insightful comments that greatly improved the quality of the manuscript.