ON THE EXISTENCE OF A DISTANT SOLAR COMPANION AND ITS POSSIBLE EFFECTS ON THE OORT CLOUD AND THE OBSERVED COMET POPULATION

There is a huge space between the outer fringe of the known solar system, where the trans-Neptunian bodies are located, and the interstellar limit set by the dynamical stability of bound solar system bodies against external forces (passing stars, Galactic tides). We know that this is not an empty region, but it is populated by a huge number of comets, distributed around the Sun in a spherical swarm known as the Oort Cloud. Are there also massive bodies out there? This issue generated wide attention a few decades ago when the presence of a solar companion star—called Nemesis—on a 26 Myr period orbit was invoked to explain presumed periodicities in biological mass extinctions and the impact cratering record (Davis et al. 1984; Whitmire & Jackson 1984). Yet, the Nemesis theory was called into serious question because the statistical evidence for periodicities was found to be very weak and Nemesis's orbit would be dynamically unstable over timescales much shorter than the solar system age (e.g., Tremaine 1986).

Even though a stellar companion of the Sun at Oort Cloud distances can by now be ruled out given its negative detection in multiple sky surveys, it is still possible to find smaller objects there, such as brown dwarfs or Jupiter-sized bodies. At a distance of 10³ AU, Jupiter and Neptune would have apparent visual magnitudes of ∼20 and ∼23, respectively, i.e., they would appear on images taken by the largest telescopes as very dim dots. At ∼10⁴ AU, they would be hopelessly beyond detection by any optical telescope available today. Another possible detection mechanism is astrometric microlensing (Gaudi & Bloom 2005) in which the apparent motion of the massive body along the parallactic ellipse deflects the angular position of a background star. Such a detection mechanism might be feasible only with an all-sky astrometric mission, such as Gaia, which is expected to be launched in 2012 to conduct a census of about 10⁹ stars of our Galaxy.

Because low-temperature bodies are more efficient emitters in the infrared, the infrared surveys are better adapted to look for such putative solar companions. There have been a few such ground-based infrared surveys, in particular the Two-Micron All-Sky Survey (2MASS) that led to the discovery of hundreds of brown dwarfs, but given the high atmospheric extinction in the IR, it is much more appropriate to study the infrared sky from space. There have already been several such space-based surveys, starting with the Infrared Astronomical Satellite (IRAS) in 1983, followed by the Cosmic Background Explorer (COBE), Spitzer, the Infrared Space Observatory (ISO), Akari, and very recently, Herschell and the Wide-field Infrared Survey Explorer (WISE). The latter mission will provide an all-sky survey from 3 to 25 μm with high sensitivity. It is expected to uncover all nearby brown dwarfs, and Jupiter-sized and Saturn-sized bodies at distances smaller than ∼2 × 10⁴ AU and ∼10⁴ AU, respectively. Going to smaller sizes, a Neptune-sized object could be uncovered only if its distance is ≲600 AU. Therefore, once the WISE IR map is complete, we will be able to set more stringent constraints on the mass–distance parameter space allowed for putative solar companions.

For the time being, another approach to the detection of and constraints on putative massive solar companions is through their perturbations on nearby comets. Such perturbations may add some extra comets to the steady injection rate of Oort Cloud comets (OCCs) in the inner planetary region caused by the action of external perturbers (passing stars, Galactic tidal forces). Interestingly, the aphelion directions of such extra comets should appear concentrated in a narrow strip along the companion's trajectory around the Sun, stretching along a great circle in the sky, which will make them more easily noticeable against the randomly scattered background of the aphelion directions of field comets. There have already been some claims made about the existence of massive solar companions based on alleged concentrations of comet aphelia on certain sky areas, tending to circle the sky (e.g., Matese et al. 1999; Murray 1999). The clustering of aphelion points along a great circle by a bound perturber could be distinguished from clusterings produced by unbound perturbers (passing stars), the latter being confined to a short arc around the point of the closest approach of the star to the Sun (Lüst 1984; Fernández 1992). Even though a closer scrutiny of the comet sample, considering observational biases's and more recent data, disproves Murray's (1999) planet X hypothesis, Matese et al. (1999) alleged comet clustering seems to deserve further critical analysis (Horner & Evans 2002). We will next discuss the arguments that give theoretical support to the presence of massive bodies in the Oort Cloud region (distances ≳1000 AU) and then estimate the efficiency of the injection mechanism of OCCs into the inner planetary region by distant solar companions of different masses.

Several dwarf planets with diameters >1000 km have already been discovered in the trans-Neptunian region, and there is little doubt that this is the tip of the iceberg; many more bodies of similar or larger sizes are likely waiting to be discovered. Sedna is one of these dwarf planets that has the peculiarity of being completely decoupled from the planetary region given its high perihelion distance of q = 76 AU. Brown et al. (2004) estimate that ∼500 bodies on Sedna-like orbits should exist, with a total mass of ∼5 Earth masses. A recent wide-field survey of distant Kuiper Belt objects carried out by Schwamb et al. (2010)—that led to the recovery of Sedna but not to new detections of Sedna-like objects—allowed them to place the estimated size of the Sedna-like population at about a few hundred objects, with a large uncertainty (from a few tens to about a couple thousand objects). Schwamb et al. (2010) estimate the total mass enclosed in this population at about 1–10 M_⊕. A simple extrapolation, from an assumed power-law distribution of sizes, will place the largest members of the Sedna-like population in the terrestrial-planet scale. Planetary perturbations alone do not seem to be capable of placing a body coming from the planetary zone on a Sedna-like orbit (Gomes et al. 2005), so an external perturber such as, for instance, a passing star is required (Morbidelli & Levison 2004). Another interesting possibility of our direct concern here was analyzed by Gomes et al. (2006). From numerical simulations they found that a rogue planet of Neptune's size with a semiminor axis b_P ⩽ 2000 AU, or one of Jupiter's size with b_P ⩽ 5000 AU, would be capable of raising the perihelion of an undetached scattered disk object to values comparable to Sedna's.

What cosmogonic reasons can we invoke to support the presence of massive objects at Oort Cloud distances? We have at least two possible mechanisms.

• 1.  
  The latest accretion stages of the Jovian planets were followed by the massive scattering of residual planetesimals, and even some protoplanets that ended up crossing the orbit of one of the Jovian planets during the phase of planet migration (Fernández & Ip 1984; Tsiganis et al. 2005). Ip (1989) argued that the macro-accretion of Uranus and Neptune could have led to the scattering of terrestrial-scale objects. A similar result was later found by Goldreich et al. (2004) who showed that the formation of Uranus and Neptune was followed by the ejection of a substantial amount of mass (up to ∼100 M_⊕), comprising several Earth-mass bodies or larger. We may further conjecture that once such bodies reached semimajor axes ∼10³–10⁴ AU, their perihelia could have been raised by external perturbers such as, for instance, nearby stars, thus acquiring dynamically stable orbits. A recent numerical model by Ford & Chiang (2007) shows that a primordial more packed ensemble of as many as five Neptune-sized bodies, formed between 15 and 25 AU, may reproduce, through mutual gravitational interactions and interactions with Jupiter and Saturn, the current Uranus–Neptune configuration and, as a byproduct of the process, the ejection of the extra Neptune-sized bodies.
• 2.  
  The previous mechanism may provide an endogenous source of distant massive bodies gravitationally bound to the Sun. An alternative or complementary source can be found if the Sun formed in a star cluster, as most stars appear to form, in which case the strong perturbations from nearby cluster stars could have shaped the distribution of OCCs (Gaidos 1995; Fernández 1997) and exchanged comets with other stars of its birth cluster (assuming that they possessed their own comet clouds; Levison et al. 2010). In this scenario, the Sun could have also very well captured Jupiter-sized intracluster bodies or snatched any of them from other cluster stars during encounters. Formation of wide binary systems was also found to be a frequent outcome during the stage of cluster dissolution, which may explain the existence of wide binary stars in the field, with separations in the range 10³ AU–0.1 pc (Kouwenhoven et al. 2010). This mechanism can very easily be extended to consider the pairing of bodies of different masses, as for instance star–brown dwarf or star–gas giant planet.

Let us consider a comet moving on a near-parabolic orbit at a distance r from the Sun. The perihelion distance q is related to the transverse component of the velocity v_T through

$$\begin{equation} q \simeq \frac{v_T^2r^2}{2\mu }, \end{equation} \tag{ 1 }$$

where μ = GM_☉, with G the gravitational constant and M_☉ the Sun's mass.

Let us now assume that the comet is perturbed by a body of mass M_P that passes at a closest distance D with a relative velocity $\vec{u} = \vec{v}_P - \vec{v}_c$, where $\vec{v}_P$ and $\vec{v}_c$ are the orbital velocities of the perturber and the comet, respectively. We have

$$\begin{equation} u^2 = (\vec{v}_P - \vec{v}_c)^2 = v_P^2 + v_c^2 -2v_pv_c\cos {\alpha }, \end{equation} \tag{ 2 }$$

where α is the angle between $\vec{v}_P$ and $\vec{v}_c$. If we assume that the orientations of both the comet and the perturber are random, then cos α can adopt any value between −1 and +1, so we can adopt in Equation (2) an average 〈cos α〉 = 0. By considering that the encounter occurs at that distance r, we have

$$\begin{equation} u^2 \simeq \mu \left(\frac{2}{r} - \frac{1}{a_P}\right) + \mu \left(\frac{2}{r} - \frac{1}{a}\right), \end{equation} \tag{ 3 }$$

where a_P and a are the semimajor axes of the perturber and the comet, respectively. If we assume r ∼ a_P, and that a can vary from values a ∼ a_P to a ≫ a_P, we find as a reasonable average

$$\begin{equation} u \simeq \left(\frac{2.5\mu }{r}\right)^{1/2}. \end{equation} \tag{ 4 }$$

The perturber will impart an impulse on the comet which can be approximately expressed by

$$\begin{equation} \vec{\Delta v}_c = \frac{2G\!M_P}{uD} \frac{\vec{D}}{D}, \end{equation} \tag{ 5 }$$

which is valid for Δv_c ≪ u (e.g., Harris & Ward 1982). The previous condition is generally fulfilled in our case. For instance, a 1 M_J perturber at ∼5000 AU will meet a random comet at a relative velocity u ∼ 0.7 km s⁻¹. For a typical target radius ∼30 AU (cf. Equation (10)), it will impart an impulse on the comet Δv_c ∼ 0.09 km s⁻¹, i.e., nearly one order of magnitude smaller than the relative velocity.

The impulse $\vec{\Delta v}_c$ can be split into two components: one on the plane perpendicular to $\vec{r}$ and the other one along $\vec{r}$ (Figure 1). The latter will not have any effect on the comet's angular momentum (and thus on q), whereas the first one will add to the transverse component of the comet's velocity, leading to a new component

$$\begin{equation} {v^{\prime }}_T^2 = v_T^2 + 2v_T\Delta v_c\sin {\theta }\cos {\beta } + \Delta v_c^2\sin ^2{\theta }, \end{equation} \tag{ 6 }$$

where θ is the angle between the vector $\vec{r}$ and the impulse $\vec{\Delta v}_c$ and β is the angle between the projection of $\vec{\Delta v}_c$ onto the plane perpendicular to $\vec{r}$ and the transverse component v_T.

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We can now define a new parameter A such that Δv_c = Av_T, namely A expresses the modulus of the perturbation Δv_c in units of v_T. We will consider for A values in the range 0.1–100. By replacing v_T and v'_T in Equation (6) by the corresponding perihelion distances q and q' by means of Equation (1), we get

$$\begin{equation} q^{\prime } = q(1 + 2A\sin {\theta }\cos {\beta } + A^2\sin ^2{\theta }). \end{equation} \tag{ 7 }$$

Let us now consider OCCs whose perihelia are initially beyond Neptune, namely with q > 30 AU. We want to know which among them will suffer a drastic reduction of their perihelion distances by a massive perturber in order to become potentially observable, say in Jupiter-crossing orbits, i.e., q' < a_Jup, where a_Jup is Jupiter's semimajor axis. For a certain A, there will be orientations of $\vec{\Delta v}_c$ that will decrease v_T to values v'_T such that q' = (v'_Tr)²/2μ < a_Jup. Such orientations of $\vec{\Delta v}_c$ will fall within a "cone"¹ whose central axis points to the direction opposite to $\vec{v}_T$.

Figure 2 shows the fraction f_J of comets with an initial perihelion distance q = 50 AU that get q' < a_Jup as a function of the parameter A. The values of f_J were computed numerically considering samples of 10⁴ impulses $\vec{\Delta v_c}$ of a given modulus A between 0.1 and 100, angles β taken at random in the interval (0, π), and random values of cos θ in the interval (−1, + 1). We repeated the computations for 29 different values of A in order to plot the curve of Figure 2. We see in the figure that f_J rises very sharply from zero at A_I ≃ 0.75, so we can take this value as that associated with the "injection" target radius D_I (cf. Equation (10)). Actually, A_I varies slightly with q, from A_I ≃ 0.65 for q ≃ 30 AU to A_I ≃ 0.85 for q ≃ 150 AU. For simplicity, we assume in the following an average value A_I = 0.75 for any initial q.

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Summing up the above discussion, we can say that for values A ≲ A_I no solutions are possible (f_J = 0), as Δv_c is too small to decrease v_T below a threshold v_Tt = (2μa_Jup)^1/2/r, namely that corresponding to an orbit with q = a_Jup; while for A ≳ A_I we can obtain solutions q < a_J. Actually, for A ≳ 5 f_J → 0, as very large impulses imparted by the perturber on the interacting comets are very unlikely to drive them to very small-q orbits. Note that A ∝ 1/D, so increasing values of A correspond to ever smaller values of the distance of closest approach of the comet to the perturber. Therefore, f_J is a function of D and also of the initial q, i.e., f_J = f_J(q, D).

The D-averaged fraction F_J(q), for a sample of comets of a given initial q encountering the perturber at random distances D ⩽ D_I (or A ⩾ A_I), is given by

$$\begin{equation} F_J(q) = \frac{1}{\pi D_I^2} \int _0^{D_I} 2 \pi D f_J(q,D) dD. \end{equation} \tag{ 8 }$$

Equation (8) was integrated numerically for 14 different initial values of q, between 30 and 500 AU. The computed values of F_J(q) are shown in Figure 3. As seen in the figure, the fraction F_J decreases very quickly with increasing initial q of OCCs, as the fraction of the phase space of the angles (β, θ) available for the impulse $\vec{\Delta v}_c$ to reduce q below a_Jup shrinks to negligible values.

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We want to stress that the distribution functions f_J and F_J, shown in Figures 2 and 3, are independent of the semimajor axes a of OCCs, as long as we assume that OCCs move on near-parabolic orbits (a ≫ q), and that their semimajor axes are around or above a_P (see explanation preceding Equation (4)); in other words, the orbit of the OCC is not entirely interior to the perturber's orbit.

Let us now analyze the distribution function of perihelion distances of OCCs, f_q(q)dq. If we assume that the Oort Cloud is a population that has been thermalized by external perturbers, then the fraction of comets with eccentricities in the range (e, e + de) is f_e(e)de = 2ede (see, e.g., Fernández 2005, pp. 126–130), and bearing in mind that q = a(1 − e), we can derive the q-distribution leading to

$$\begin{equation} f_q(q)dq = 2\left(1-\frac{q}{a}\right) \frac{dq}{a} \sim \frac{2dq}{a}, \end{equation} \tag{ 9 }$$

provided that we are considering comets in highly eccentric orbits, so q ≪ a.

By introducing the injection target radius D_I in Equation (5), and bearing in mind that A = A_I for D = D_I, we can establish

$$$ {\Delta v}_c = \frac{2G\!M_P}{uD_I} = A_Iv_T = A_I\frac{\sqrt{2\mu q}}{r} $$$

and solving for D_I we get

$$\begin{equation} D_I = \frac{\sqrt{2\mu }}{A_I} \mbox{ } \frac{M_P}{M_{\odot }} \mbox{ } \frac{r}{q^{1/2}u}. \end{equation} \tag{ 10 }$$

The perturber will encounter a number of comets d²(n_e) with perihelion distances in the range (q, q + dq), within the cross section of radius D_I, during a time dt given by

$$\begin{equation} d^2(n_e) = \sigma u \nu (r) dt f_q(q)dq, \end{equation} \tag{ 11 }$$

where σ ≃ πD²_I (we neglect gravitational focusing) and ν(r) is the density profile of OCCs.

Numerical simulations of Oort Cloud formation and evolution carried out by different authors (e.g., Duncan et al. 1987; Dybczyński et al. 2008; Kaib & Quinn 2009) show that the number density profile of OCCs follows a law of the form

$$\begin{equation} \nu (r) = Cr^{-s}, \end{equation} \tag{ 12 }$$

where C is a normalization factor that depends on the total Oort Cloud population N_OC, and for the exponent s the previous authors found values in the range ∼3.4–4. We have

$$\begin{equation} N_{\rm {OC}} = \int _{r_b}^{r_{\rm {OOC}}} \nu (r) 4\pi r^2 dr, \end{equation} \tag{ 13 }$$

where r_b is the inner boundary of the inner core of the Oort Cloud and r_OOC is the radius of the outer Oort Cloud. Under the assumption that r_b ≪ r_OOC, that s > 3, and that the same value of s holds for all the range (r_b, r_OCC), we obtain

$$\begin{equation} C \simeq \frac{(s-3)N_{\rm {OC}}}{4\pi r_b^{(3-s)}}. \end{equation} \tag{ 14 }$$

By substituting Equations (9), (10), and (12) into Equation (11), we obtain

$$\begin{equation} d^2(n_e) = \pi \left[\frac{\sqrt{2\mu }}{A_I} \frac{M_P}{M_{\odot }} \frac{r}{q^{1/2}u}\right]^2 u \mbox{ }C r^{-s} dt \mbox{ } \frac{2dq}{a}. \end{equation} \tag{ 15 }$$

We can assume that the comets that encounter the perturber at a distance r have on average semimajor axes a = r/1.5 (this comes from the computation of the time-average heliocentric distance $\bar{r} = a(1+e^2/2)$, where for highly eccentric orbits e ∼ 1). We can then substitute a for r in Equation (15). It is more convenient to take the true anomaly f as the independent variable instead of the time t. From Kepler's second law, we have

$$\begin{equation} dt = \frac{r^2df}{h_P}, \end{equation} \tag{ 16 }$$

where h_P is the perturber's specific angular momentum which is given by $h_P = \ssty\sqrt{\mu a_P(1-e_P^2)}$, e_P being the eccentricity of the perturber's orbit. Furthermore, from Kepler's first law, we have

$$\begin{equation} r = \frac{a_P\big(1-e_P^2\big)}{1+e_P\cos {f}}. \end{equation} \tag{ 17 }$$

Actually, we are interested in the fraction of comets encountering the perturber that are deflected to orbits with q' < a_Jup, which is given by Equation (8), namely

$$\begin{equation} d^2(n_{eJ}) = d^2(n_e) \times F_J(q). \end{equation} \tag{ 18 }$$

This expression will give the number of OCCs with perihelion distances (q, q + dq) injected into Jupiter-crossing orbits by the perturber when it moves within true anomalies (f, f + df). By substituting Equations (14), (16), and (17) into Equation (15), then substituting Equation (15) into Equation (18), and finally replacing u by the expression found in Equation (4), we get

$$\begin{eqnarray} d^2(n_{eJ}) &=& \frac{0.95(s-3)}{A_I^2} \mbox{ } \frac{N_{\rm {OC}}}{r_b^{(3-s)}} \left(\frac{M_P}{M_{\odot }}\right)^2 \mbox{ } \big[a_P\big(1-e_P^2\big)\big]^{(3-s)} \mbox{ } \nonumber \\ && \times (1 + e_P\cos {f})^{(s-7/2)}df \mbox{ } F_J(q)\frac{dq}{q}. \end{eqnarray} \tag{ 19 }$$

By integrating along the perturber's orbit and for all cloud comets with perihelion distances q₁ < q < q₂, we finally get the number n_eJ of OCCs encountering the perturber that are injected into Jupiter-crossing orbits per orbital revolution of the perturber

$$\begin{eqnarray} n_{eJ} \!\!\!\! &=& \!\!\!\! 1.7(s-3) \mbox{ } \frac{N_{\rm {OC}}}{r_b^{(3-s)}} \left(\frac{M_P}{M_{\odot }}\right)^2 \mbox{ } \big[a_P\big(1-e_P^2\big)\big]^{(3-s)} \mbox{ } \nonumber \\ && \!\! \!\! \times \ 2\int _0^{\pi }(1 + e_P\cos {f})^{(s-7/2)}df \mbox{ } \int _{q_1}^{q_2} F_J(q)\frac{dq}{q}, \end{eqnarray} \tag{ 20 }$$

where we have introduced the value A_I = 0.75. For the range of perihelion distances of OCCs we adopt the values q₁ = 30 AU and q₂ = 250 AU. The upper limit is justified bearing in mind that the condition q ≪ a has to be fulfilled, and that, in any case, the contribution of OCCs with q ≳ 250 AU to n_eJ becomes negligible.

We can instead compute the number of OCCs injected into Jupiter-crossing orbits per year by dividing Equation (20) by the orbital period of the perturber, i.e.,

$$\begin{equation} \dot{n}_{eJ} = \frac{n_{eJ}}{a_P^{3/2}}. \end{equation} \tag{ 21 }$$

Figure 4 (left panel) shows the computed influx rate of comets injected by the perturber into Jupiter-crossing orbits (i.e., q < 5.2 AU). For the total Oort Cloud population we have adopted: N_OC = 10¹², including its inner core, which is consistent with previous estimates based on the observed flux of new comets and numerical models of the Oort Cloud formation (e.g., Weissman 1996; Dones et al. 2004; Kaib & Quinn 2009), and an exponent for the density profile s = 3.5, valid for a range of heliocentric distances stretching from 10³ AU up to the limit of dynamical stability of OCCs. This model implies that about two-thirds of the Oort Cloud population would be enclosed in its inner core between 10³ and 10⁴ AU. As a comparison (horizontal dashed line), we also show the steady flux of OCCs injected into Jupiter-crossing orbits by the combined action of Galactic tides and nearby stars (which will be referred to as field OCCs). This flux is estimated to be of about 5 yr⁻¹ for an absolute total magnitude H = 9 (which roughly corresponds to a nucleus radius R_N = 1 km; Fernández 2005, 2010).

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As shown in Figure 4, the injection rate triggered by a 5 M_J perturber at distances to the Sun ≲8000 AU can be substantially greater than the steady flux of field OCCs. The same perturber, but at distances between ∼8000 AU and ∼20,000 AU can still inject a significant fraction (≳10%) of the total population reaching Jupiter's orbit. A 1 M_J perturber will contribute much less to the point that the perturber-driven fraction of comets will be significant (≳10%) only if the perturber were located at distances ≲(2–3) × 10³ AU to the Sun. The situation is even more problematic for a Neptune-mass perturber (M_P = 10⁻⁴M_☉) for which the input rate of extra comets driven by it is always below ∼0.01 of the steady flux of field OCCs for any distance in the range 1000–20,000 AU.

We are concerned not only with the input rate of extra comets injected by a putative perturber, and how it compares with the steady flux of field OCCs, but also with the distribution of their aphelion points on the sky. Field OCCs have essentially a random Poisson distribution on the whole sky, leaving aside a certain concentration at mid-Galactic latitudes due to the dependence of the tidal force of the Galactic disk on the Galactic latitude (Delsemme 1987), and some observational biases, in particular the one favoring discovery of comets reaching perihelion in the Northern hemisphere as compared to those that reach perihelion in the Southern hemisphere (Horner & Evans 2002). By contrast, comets injected by the perturber will concentrate in a narrow strip along the perturber's track of angular width ∼2D_I/a_P (cf. Equation (10)). As we see in Figure 4 (right panel), the sky density of aphelion points of comets injected by the perturber is quite prominent. Even a Neptune-mass perturber at less than a few 10³ AU can give a sky density about one order of magnitude greater than that of field OCCs, so a small number of a few tens of extra comets injected by the perturber will appear as a prominent feature.

The injected extra comets will show not only a clustering of aphelion points along a narrow strip on the sky, but also some concentration of semimajor axes around ∼a_P (or an energy ∼x_P = 1/a_P). We note that the energies of the observed field OCCs are concentrated in a narrow spike in the range 10 ≲ x ≲ 50, where the number of observed comets raises to ∼2.5 per unit energy (our unit is 10⁻⁶ AU⁻¹), while it decreases to ∼0.1 per unit energy in the range 80 ≲ x ≲ 1000 (Figure 5). Therefore, a few tens of extra comets from the inner core of the Oort Cloud, say at 5000 AU (x ≃ 200), will show a concentration of energies which will stand up against the sparse energy distribution there. Yet the same amount of extra comets will make little difference if their semimajor axes heap up around, say 2.5 × 10⁴ AU, i.e., energies falling within the Oort spike, where they will mix and dilute with the much more numerous field OCCs. The advantage of the strategy of looking first for clusterings of aphelion points is that the prominence of this effect is independent of the typical orbital energies of the companion-driven comets. Should a clustering of aphelion points be detected, then we can look next for a correlated concentration of orbital energies to check the reality of such a feature.

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We must now ask the following question: could a massive companion moving in the Oort Cloud cause a serious depletion of comets throughout the solar system lifetime? There are three possible loss mechanisms associated with the perturbing action of the companion: (1) injection of comets in the planetary region where they are quickly removed from the Oort Cloud by planetary perturbations; (2) ejection to interstellar space; or (3) collision with the companion. Let us analyze these three effects. With regard to injection into the planetary region by the companion, the estimate of the injection rate can be done by means of Equation (21), considering an enlarged target area of radius D_plan, instead of D_I, such that a comet with impact parameter D < D_plan can be injected into an orbit with q < 15 AU. We shall assume that comets getting perihelion distances <15 AU will be removed from the Oort Cloud by the strong perturbations of Jupiter and Saturn. Although a sharp boundary at q = 15 AU may seem somewhat arbitrary, the typical energy change for q ≲ 15 AU is larger than the typical bound energy of OCCs, while the opposite occurs for q ≳ 15 AU (Fernández 1981). By recomputing the injection rate of Equation (21) with D_plan instead of D_I, we obtain influx rates about a factor of four larger than those shown in Figure 4.

Hyperbolic ejection of the comet after an encounter with the companion is another possible loss mechanism. Let us consider again a comet of semimajor axis a, and orbital energy x = 1/a at a heliocentric distance r, the orbital velocity is

$$\begin{equation} v_c^2 = \mu \left(\frac{2}{r} - x\right), \end{equation} \tag{ 22 }$$

so the change in orbital energy due to an impulse Δv_c imparted by the companion is

$$\begin{equation} \Delta x = -\frac{2v_c\Delta v_c}{\mu }. \end{equation} \tag{ 23 }$$

To be ejected, the comet has to experience an energy change of the order Δx ≃ 1/a, so substituting this in the previous equation, and Δv_c into Equation (5), we obtain

$$\begin{equation} D_{ej} \simeq 4a_P\left(\frac{v_c}{u}\right)\left(\frac{M_P}{M_{\odot }}\right), \end{equation} \tag{ 24 }$$

where we have considered that most of the comets that interact with the companion have semimajor axes a ∼ a_P.

With regard to the target radius for collision, including gravitational focusing, it will be given by

$$\begin{equation} D_{{\rm col}} = R_P \left(1+\frac{v_{{\rm esc}}^2}{u^2} \right)^{1/2}, \end{equation} \tag{ 25 }$$

where R_P is the radius of the companion and v_esc = (2GM_P/R_P)^1/2 is the escape velocity from its surface.

The cross section for injection into the planetary region (q < 15 AU) turns out to be about one order of magnitude larger than the cross section for ejection, which in turn is a factor of a few tens larger than the cross section for collision. Therefore, the first process dominates the loss rate of comets due to the perturbing action of the companion. Let Φ_loss be the comet loss rate (measured as a fraction of the Oort Cloud population). The decrease of the Oort Cloud population, dN, during dt will then be given by

$$\begin{equation} dN = -\Phi _{{\rm loss}}Ndt, \end{equation} \tag{ 26 }$$

where N is the population at t. By integrating this expression, we get the decrease of the population from an initial one N₀ at t = 0, i.e.,

$$\begin{equation} N = N_0 \exp \! {(-\Phi _{{\rm loss}}t).} \end{equation} \tag{ 27 }$$

The half-life of the comet population would be

$$\begin{equation} t_{{\rm h}\hbox{-}{\rm l}} = - \frac{\ln {(1/2)}}{\Phi _{{\rm loss}}}. \end{equation} \tag{ 28 }$$

For instance, for a massive companion of mass M_P = 5 × 10⁻³ M_☉ and semimajor axis a = 2000 AU, the relative loss rate is Φ_loss = 4 × 10⁻¹⁰ yr⁻¹, which leads to a half-life of the Oort Cloud t_h-l = 1.73 × 10⁹ yr, i.e., such a putative companion would have severely depleted the Oort Cloud through the solar system age (∼4.6 × 10⁹ yr). The same companion but at a distance a_P = 20,000 AU would deplete the outer Oort Cloud with a half-life 5.8 × 10¹⁰ yr, i.e., one order of magnitude longer than the solar system age. Therefore, only very massive companions (several Jupiter masses) at distances ≲10⁴ AU would have caused a serious erosion of the Oort Cloud.

The previous results show that only very massive perturbers (≳a few M_J) can produce a statistically significant number of extra comets (from ∼0.1 to ∼10 times the number of field comets). Should such a massive perturber exist, at least 20 of the ∼200 observed new and young long-period comets with good quality orbits would have been injected by it, with their aphelion points concentrated along the perturber's trajectory. On the other hand, if the semimajor axis of such a massive perturber was ≲10⁴ AU, it would have severely eroded the inner Oort Cloud. We will come back to this point below. A 1 M_J perturber would produce a detectable clustering within the current sample of observed comets only if its distance to the Sun were less than a few 10³ AU. Perturbers of even smaller masses (say Neptune's to Saturn's) would not produce enough extra comets to be discernible within the current comet sample for any distance above ∼10³ AU to the Sun. These results are of course a function, among other things, of the total number of OCCs, including those in the inner core.

We also stress that our estimated Oort Cloud population is for nucleus radii R_N > 1 km, namely the sizes of most discovered comets. If new sky surveys (e.g., Pan-STARRS) allow us to reach a high discovery completeness down to, say R_N = 0.1 km, and as far as, say Saturn's distance, the available comet sample may increase by about one to two orders of magnitude with respect to the current one of about a couple hundreds of new and young long-period comets with good quality orbits. Should a Neptune-mass perturber be present at less than about a couple thousand AU, within a decade or so we may thus have an available comet sample large enough to pick up a few dozens of extra perturber-driven comets that might reveal its presence.

The consequence of detecting concentrations of comets along narrow belts will be twofold: (1) to uncover the presence of a solar companion at a distance that may be inferred from the distribution of semimajor axes (or orbital energies) of the perturbed comets (from here we can see that it is necessary not only to have a good record of comet passages, but also a good orbit determination) and (2) to derive the space density of comets at the companion's distance. In the near future, WISE will set more stringent conditions on the distances and masses of putative companions.

Matese et al. (1999) argued that a 3 M_J companion at a distance of ∼2.5 × 10⁴ AU is responsible for an anomalous clustering of aphelion points of new comets along a great circle nearly perpendicular to the Galactic plane, which crosses the Galactic equator at longitudes 135°(315°) ± 15°, yet, an extrapolation of our results of Figure 4 to such a distance suggests that a 3 M_J companion would produce less than 10% of extra comets, which means ≲10 in a sample of 82 (the sample used by Matese et al.), too few to allow a meaningful statistical study. The situation will ameliorate if we consider a somewhat more massive companion (say 5 M_J) closer to the Sun, which would lead to a number of extra comets more in line with that estimated by Matese et al. We can then conclude that Matese et al.'s solar companion hypothesis is only marginally consistent with our calculations. With a sample of observed comets two orders of magnitude greater, which is within reach in the foreseeable future, we will be able to confirm or reject the reality of this clustering, or others that might appear as the comet sample grows. We will also be able to discriminate between clusterings caused by bound perturbers and those caused by unbound ones (passing stars), the latter case being more narrowly concentrated around the point of closest approach of the star to the Sun. Comets are then very suitable natural probes for learning about the past or present presence of massive bodies in the outskirts of the Sun's sphere of influence.

The author thanks Douglas Hamilton for a fruitful discussion on the erosion of the Oort Cloud by a massive bound perturber.