STIS CORONAGRAPHIC IMAGING OF FOMALHAUT: MAIN BELT STRUCTURE AND THE ORBIT OF FOMALHAUT b

The stellar companions Fomalhaut B or C could disturb the Fomalhaut A system either by a close flyby interaction (Larwood & Kalas 2001; Kenyon & Bromley 2002; Ardila et al. 2005; Reche et al. 2009; Malmberg et al. 2011) or a secular perturbation (Augereau & Papaloizou 2004; Wyatt 2005). For example, a flyby interaction studied numerically for the β Pic debris disk gives a geometry that qualitatively resembles that of Fomalhaut A, Fomalhaut B, and the main belt (Larwood & Kalas 2001). An initially symmetric circumprimary disk of material perturbed by a close stellar encounter results in eccentric belts of material (technically, tightly wound spiral arms) where the apastra of the belts point toward the direction of the perturber's periastron. This means that the apastron of the perturber (or its post-flyby trajectory in a hyperbolic orbit) and the apastra of the eccentric belts are pointed in opposite directions. This simple geometrical picture is consistent with the present epoch location of Fomalhaut B south of Fomalhaut A, and the main belt and Fomalhaut b apastra to the north of Fomalhaut A (Figure 27). However, the critical problem is that if Fomalhaut B is bound to Fomalhaut A, and given a system age ≈400 Myr, there are repeated periastron passages that would wipe out the belt structure created by the first periastron passage.

Instead of a flyby, Fomalhaut B and C may influence the Fomalhaut A system via a secular perturbation. The Kozai resonance has been invoked as one mechanism to explain highly eccentric exoplanets (Wu & Murray 2003; Takeda & Rasio 2005). For Fomalhaut B or C to be responsible for a Kozai resonance, it must have a mutual inclination of >392 relative to the orbital planes of either Fomalhaut b or the main belt. If Fomalhaut b and the main belt are not coplanar, then it is possible that the Kozai mechanism operates only on one component of the Fomalhaut system. The approximate period for a Kozai oscillation (Ford et al. 2000) is

$$\begin{equation} P_K \simeq P_{b} \frac{m_A + m_b}{{m_B}} \left (\frac{a_B}{{a_b}}\right)^3 \left(1 - e_B^2\right)^{3/2}. \end{equation} \tag{ 2 }$$

Here the subscripts A, b, and B refer to the respective components of the Fomalhaut system. Using the approximate values of P_b = 10³ yr, m_A = 2 M_☉, m_B = 1 M_☉, m_b = 0, a_B = 6 × 10⁴ AU, a_b = 10² AU and e = 0.5, the Kozai period is of order 10¹¹ yr (and longer for the more distant Fomalhaut C). The Kozai resonance is therefore relatively ineffective for separations as large as observed between Fomalhaut A and B.

Instead of Fomalhaut B or C, the main belt mass may be responsible for a secular perturbation on Fomalhaut b. This scenario has been studied by Terquem & Ajmia (2010), but the initial conditions presume that a planet begins with a mutual inclination ≳ 30° relative to the main belt. To reach this starting point, the scenario needs to invoke an additional dynamical interaction with some other body, e.g., a Fomalhaut c, and therefore the planet-belt Kozai effect does not give a dynamical history consistent with no other massive bodies.

In addition to the problems of explaining Fomalhaut b's dynamically hot orbit, the properties of the main belt are left without adequate explanation. The eccentric orbit of Fomalhaut b tends to exclude the possibility that it dynamically sculpts the inner edge of the belt since only a small fraction of the belt could be disturbed during each orbital period of Fomalhaut b. Moreover, the secular perturbation theory invoked to explain the main belt stellocentric offset is second order with respect to eccentricity (Wyatt et al. 1999) and breaks down at high eccentricity. If secular theory is applicable, then Fomalhaut b's high eccentricity would predict that the main belt's eccentricity should be larger than observed. On the other hand, apsidal alignment between Fomalhaut b and the main belt continues to be indicated by the new orbit determination (Table 5). Future work needs to determine if the orbital parameter space presented here could be consistent with secular theory and the observed stellocentric offset. In any case, the main belt's sharp inner edge and the azimuthal gap are consistent with the existence of another planet orbiting near the main belt.

To summarize, the observed Fomalhaut inventory (Section 9.1) does not appear to be sufficient for explaining Fomalhaut b's high eccentricity and the main belt morphology. Other perturbing objects must be present in the dynamical history of the Fomalhaut system.

Permitting the existence of additional perturbers in the past and/or present epochs allows a variety of plausible dynamical histories that are consistent with the current observables. Three classes of dynamical paradigms could focus on endogenic perturbations, exogenic perturbations, or a blend of both. For example, the dynamical paradigm of Oort cloud comets is a blend that involves the increasing of minor body semi-major axes and aphelia by close-encounters with gas giant planets, followed by the raising of perihelia by passing stars and molecular clouds (Oort 1950; Duncan et al. 1987).

In the endogenic class of paradigms, Fomalhaut b's eccentric orbit was produced by an interaction with at least one other planet in the system. The general idea for planet–planet dynamical interactions is that two or more planets initially form in relative isolation from each other, but subsequent migration mechanisms lead to unstable orbital configurations. Two planets may enter within a few times their mutual Hill sphere (Gladman 1993; Chambers et al. 1996; Rasio & Ford 1996; Levison et al. 1998; Marzari & Weidenschilling 2002; Adams & Laughlin 2003; Veras & Armitage 2004; Chatterjee et al. 2008; Jurić & Tremaine 2008; Veras et al. 2009), or their orbits may cross into an unstable resonance due to planetesimal-driven migration (Tsiganis et al. 2005; Thommes et al. 2008). An instability that modifies the orbital elements of two planets in the system may then lead to unstable orbits for other planets in the system, producing a "global" instability.

Because planet–planet scattering evolution is chaotic, the initially closest planet may end up the farthest and vice versa. Overall, the surviving (i.e., not ejected) planet that ends up with the largest semi-major axis will also have higher eccentricity if its mass is less than or equal to the planet it interacted with. Unfortunately, the upper mass limit of Fomalhaut b (⩽1 M_J) is not particularly helpful for constraining the expected mass of another surviving planet. Moreover, it is also possible that a hypothetical Fomalhaut c was ejected, leaving behind Fomalhaut b as the interior planet. For example, numerical tests by Ford & Rasio (2008) involving two planets indicate that the largest eccentricities are obtained for near equal mass planets, where the surviving (bound) planet has e = 0.624 ± 0.135, but the second planet is lost. Jurić & Tremaine (2008) find that 20% of simulations that begin with multiple planets end with only one bound planet. However, the majority of systems have at least two surviving planets after chaotic evolution, agreeing with simulations of three-planet systems conducted by Chatterjee et al. (2008). Therefore, the detection of Fomalhaut b as a large-a, high-e exoplanet makes the existence of another comparably massive exoplanet in the system more likely than not. This would also mean that the orbit of Fomalhaut b may undergo further dynamical interactions that will evolve the orbits of both Fomalhaut b and Fomalhaut c.

The observational avenue for constraining the problem clearly rests on detecting Fomalhaut c and determining its orbital properties. Direct imaging surveys to date have not detected a second companion at infrared wavelengths (Kalas et al. 2008; Marengo et al. 2009; Kenworthy et al. 2009, 2013; Janson et al. 2012). Since planet–planet scattering or other instabilities may involve a Fomalhaut c with a Jupiter mass or below, the mass limits explored by these surveys, >1 M_J, are not adequate to rule out the existence of a Fomalhaut c.

In the exogenic class of perturbations, an interloping star could have been responsible for perturbing the Fomalhaut system. Deltorn & Kalas (2001) searched for Fomalhaut "nemesis" encounters among 21,497 stars where space motions could be derived from radial velocity and Hipparcos information. The strongest perturbation was from HD 16895 (HIP 12777; SpT = F7V) $474^{+20}_{-19}$ Myr ago. The age of HD 16895 is ∼9 Gyr (Ng & Bertelli 1998), and therefore it did not form as part of the Fomalhaut system. The closest approach distance was $1.15^{+0.41}_{-0.34}$ pc, at which time $105^{+21}_{19}$ kAU represents the approximate radius of a sphere centered on Fomalhaut A where a particle experiences equivalent gravitational forces from Fomalhaut A and HD 16895. Given the present-day separation of 158 kAU (0.77 pc) between Fomalhaut A and C, there is a possibility that Fomalhaut C was gravitationally perturbed by HD 16895. Therefore there is some empirical evidence for a possible exogenic disturbance to the system that could propagate inward, resulting in a global dynamical instability on a secular timescale (e.g., Zakamska & Tremaine 2004). The availability of expanded position, proper motion, and radial velocity catalogs may be used to identify other potential perturbers in future work, and the effect of the galactic tides should also be incorporated in new calculations (Kaib et al. 2013; Veras & Evans 2013).

A blend of endogenic and exogenic perturbations requires a comprehensive analytical and numerical analysis. To gain a rough picture concerning the dynamical lifetime and outcomes of the current orbital configuration, we used the numerical simulator AstroGrav to evolve the orbits of several test cases for 440 Myr. The simulations include Fomalhaut A and B, two planets orbiting Fomalhaut A, but have no test particles representing the belt to minimize the simulation times. One of the test planets represents Fomalhaut b with a = 177 AU and e = 0.8, and the mutual inclination with the second planet is either i = 0° or i = 20°. The second planet is either at a = 30 AU or a = 120 AU. We tested a combination of various masses for the two planets representing Jupiter, Saturn, and Neptune. Fomalhaut B has mass 1.45 × 10³⁰ g, a = 57400 AU, e = 0.0 and i = 0°.

The general outcome is that if Fomalhaut b is coplanar with Fomalhaut c, it is ejected from the system on <10⁷ yr timescales, though there are exceptions where Fomalhaut b survives for the age of the system. In a small fraction of cases, Fomalhaut b approaches Fomalhaut B as its semi-major axis evolves to large values, but capture is unlikely. The exogenic influence of Fomalhaut B appears minor given the fixed assumption of a = 57,400 AU, e = 0.0. Fomalhaut b remains bound to Fomalhaut A for >10⁷ yr timescales in the test cases where the mutual inclination is 20°. We also found cases where Fomalhaut c has high eccentricity (e ∼ 0.8) after Fomalhaut b is ejected. This confirms the previously stated notion that the observed Fomalhaut b could have been a planet on a low-eccentricity orbit, as originally envisioned to account for the main belt properties, but recently acquired high eccentricity via a planet–planet scattering event.

The overall picture is that given the uncertainties concerning the orbital parameters of Fomalhaut b, Fomalhaut B, and the existence of other Fomalhaut A planets, there are configurations where Fomalhaut b obtained its high eccentricity >10⁷ yr ago at early epochs, particularly in the non-coplanar cases. However, there are circumstances where the configuration is younger than 10⁷ yr. A test of which scenario should be favored could look into how likely the belt is to survive in either case, though in such a scenario the assumed mass of Fomalhaut b is increasingly relevant. In the next section we consider the belt survival timescales and other physics that would be implied by a co-planar case, given a variety of masses for Fomalhaut b up to one Jupiter mass.

Here we assume that Fomalhaut b is a low-mass planetesimal that is optically bright because of reflected light from a fresh dust cloud surrounding it. For example, it could be a planetesimal that was recently disrupted by forces associated with its recent periaston passage. Fomalhaut b is unlikely to be only a dust cloud (i.e., only ⩽1 mm sized grains) because the size of the object required to account for the grain scattering surface area is at least 10 km in size (Kalas et al. 2008). Therefore it resides within the gravity regime of planetesimal collision physics, where "catastrophic" collisions are defined as retaining 50% of the precursor mass in a largest remnant.

An alternative to a collision is tidal, thermal, and/or spin breakup of a weak planetesimal (e.g., Jewitt 2012). Fomalhaut b's precursor could be an analog to Shoemaker–Levy 9 (SL9), tidally disrupted by passing within the Roche radius of the hypothetical Fomalhaut c. Alternately, the analogy may be to a Sun-grazing comet that breaks up near periastron due to thermal and tidal stresses, or elsewhere due to spin (Marsden 2005). One empirical test of this idea is to search for debris along Fomalhaut b's orbital path (Figure 20). Unfortunately, the current data are dominated by speckle noise in most of the region closer to the star than Fomalhaut b's current location. One might classify the Fomalhaut b phenomenon as cometary, but the inferred dust mass and stellocentric distance places it in the "giant" comet category with activity involving supervolatiles, as inferred for the activity of comet Halley and other icy objects at large heliocentric distances (Sekanina et al. 1992; Jewitt 2009).

Does a Fomalhaut b dust cloud survive the belt crossing as it collides with main belt material? A key consequence of Fomalhaut b's e ∼ 0.8 orbit is that the relative velocity of Fomalhaut b with respect to material in the belt is greater than previously assumed. Therefore the collision lifetime, t_cc, of cloud particles will be shorter by some scaling factor compared to the lifetime of particles that collide with each other in the main belt. The collision lifetime of dust grains within the main belt has been analytically determined by Wyatt & Dent (2002): 10 μm grains have 10⁵ yr ⩽ t_cc ⩽ 10⁶ yr. Therefore our task is to estimate the appropriate scaling factor pertinent to Fomalhaut b's relative velocity.

The relative velocity of particles orbiting within the belt is (Wyatt & Dent 2002)

$$\begin{equation} v_{{\rm rel}} = f(e,I) v_k = (1.25 e^2 + I^2) ^{1/2} v_k, \end{equation}$$

where v_k is the Keplerian orbital velocity. The collisional belt model adopted by Wyatt & Dent (2002) assumes that the belt lies between 125 and 175 AU radius, and the average inclinations and eccentricities of belt particles are I = 5° and e = 0.065. At 150 AU, v_k = 3.4 km s⁻¹, yielding v_rel = 0.4 km s⁻¹.

To calculate v_rel between Fomalhaut b and the belt as it enters the belt two decades from now, we assume that the incidence angle is 45° in the prograde sense. The model belt is 50 AU wide and the oblique path through the belt has length 71 AU. The entry point is 150 AU from Fomalhaut A (recall the stellocentric offset), where the orbital velocity of Fomalhaut b is ∼3.7 km s⁻¹, and hence the belt crossing requires ∼100 yr. The velocity components of Fomalhaut b are 2.6 km s⁻¹ both parallel and orthogonal to the disk velocity vector. Thus, while cutting diagonally through the main belt, Fomalhaut b is rear-ended by belt material moving faster in the parallel direction at v_k = 3.4 km s⁻¹, or v_rel = 0.8 km s⁻¹. Fomalhaut b will also undergo head-on collisions with an orthogonal component v_rel = 2.6 km s⁻¹. In the reference frame of Fomalhaut b, $v_{{\rm rel}} = \sqrt{(}0.8^2 + 2.6^2)$ = 2.7 km s⁻¹ from the lower right (cf. inset of Figure 20).

Given that v_rel for Fomalhaut b is six to seven times greater than for belt particles, the catastrophic collision timescale could then be taken as proportionally shorter for dust grains surrounding the planetesimal compared to dust grains colliding with each other in the belt. The dependence is in fact stronger because smaller and smaller particles become catastrophic impactors as the relative velocity increases. Under the assumption that smaller impactors are present, the collision timescale (s):

$$\begin{equation} t_{{\rm cc}} \propto v_{{\rm rel}}^{-(1+2\alpha) / 3} \propto v_{{\rm rel}}^{-8/3}, \end{equation}$$

where α is the exponent for the particle size distribution dependence, taken here as α = 7/2 (Wyatt et al. 2007, 2010, 2011). Therefore the collision timescale for Fomalhaut b dust-cloud particles passing through the belt is (2.7/0.4)^8/3 = 163 times shorter than the collision timescale of particles within the belt. This value has significant uncertainties that depend on how the grain size distribution of the planetesimal cloud differs from that of the belt.

Counterbalancing this is the fact that Fomalhaut b spends only ∼200 yr in the belt per orbital period, which is ∼15% of the orbital period of a belt particle at 150 AU. Therefore the collision timescale of Fomalhaut b is one order of magnitude shorter than a belt particles instead of two orders of magnitude.

This estimate for the scaling factor suggests that the catastrophic collision timescale for a 10 μm grain bound to Fomalhaut b is 10⁴ yr ⩽ t_cc ⩽ 10⁵ yr. A Fomalhaut b dust cloud would survive belt passages for 10–100 orbital periods.

The survival of a Fomalhaut b dust cloud after many crossings through the main belt appears counterintuitive. We therefore conduct an order of magnitude check based on the observables in the optical data, rather than the above extrapolation from the analytical analysis given by Wyatt & Dent (2002). We begin by assuming that the lifetime of Fomalhaut b as a dust cloud is roughly equal to the timescale for intercepting its own mass in main belt dust grains. For this simple scenario we assume that in fact all of the dust cloud interacts with main belt material, and that both cloud and belt have a uniform number density of objects for any given grain size.

Regardless of whether or not the scattering grains are in a small cloud, large cloud, ring, or any other geometry, the optical photometry constrains the geometric scattering cross sections of grains that comprise the structure. We use the relationship derived by Kalas et al. (2008),

$$\begin{equation} m_p = -2.5 {\rm log}(\sigma _pQ_s) + 70.2 \ {\rm mag}, \end{equation}$$

where σ_p is the projected geometric surface area of scattering grains in m², and Q_s is a scattering efficiency factor, such as the geometric albedo. Observations give m_p ∼ 25 mag in the optical. Therefore if the cloud material has a relatively low albedo such that Q_s = 0.1, then the projected geometric surface area of grains within the Fomalhaut b cloud is σ_p = 1.2 × 10¹⁹ m². Turning now to the main belt, Kalas et al. (2005) give the model dependent total grain scattering cross section σ_mb = 5.5 × 10²² m², given an assumed albedo = 0.1. The planetesimal cloud therefore has a total grain cross section that is 2.2 × 10⁻⁴ of the entire optically detected main belt.

As an aside, these results are testable in the submillimeter, where the flux of the entire main belt is 81 mJy (Holland et al. 1998), yielding a predicted flux from Fomalhaut b of ∼18 μJy (if the circumplanetary grains have similar properties as grains in the main belt). The current ALMA data have an rms noise of ∼60 μJy beam⁻¹ (Boley et al. 2012), which to first order excludes a circumplanetary dust cloud significantly more massive than in our prediction above. Vetting various dust-cloud models will require additional future observations with ALMA.

For a single belt crossing, the cloud will encounter only a fraction of the main belt volume. If Fomalhaut b requires 100 yr to cross the belt, then the volume of the main belt that is encountered by the grains within the cloud is 1.1 × 10³² m³. Adopting the assumptions from Wyatt & Dent (2002) that the belt has an inner and outer radius of 125 and 175 AU, respectively, and adopting a fixed vertical width of ∼10 AU, the volume of the model belt is 1.6 × 10³⁹ m³. Thus one encounter volume is 6.9 × 10⁻⁸ of the total main belt volume. Multiplying σ_mb by this factor, the geometric surface area of main belt grains encountered by cloud grains in a single belt crossing is 3.8 × 10¹⁵ m². For the dust-cloud grains to intercept an equal surface area of belt grains requires 3 × 10³ belt crossings. Given two belt crossings per 1700 yr orbital period, the lifetime of the cloud is ∼10⁶ yr.

To summarize, both the empirically based, order-of-magnitude approach and the analytic approach of Wyatt & Dent (2002) confirm that a dust cloud is not destroyed by a single belt crossing, and in fact survives for a minimum of 10 orbital periods.

Is there a shorter timescale by which the cohesiveness of the dust cloud could be lost? If the cloud is not gravitationally bound to itself, then two possibilities are orbital shearing between the portions of the cloud closest and farthest from the star, and the velocity dispersion of grains. For shearing, if the semi-major axis of one side of the cloud differs from the opposite side by 2 AU (260 mas; 5 STIS pixels) then the orbital velocities differ by 10⁻² km s⁻¹. In 10 yr the separation along the direction of orbital motion increases by ∼0.02 AU (3 mas), which is not detectable. However, in one orbital period (∼2000 yr) the cloud shears by 4.6 AU (600 mas). A spherically symmetric cloud of grains with no self-gravity therefore spreads into a triaxial spheroid and eventually into a trail within a few orbital periods.

Figure 28 demonstrates the shearing of a spherical cloud that begins with radius 0.05 AU at a position −90° from periapse. We use the AstroGrav numerical model to study the evolution of a dust cloud composed of 1000 massless particles, with a stellocentric motion that follows the nominal Keplerian orbit of Fomalhaut b (a = 177 AU, e = 0.8, i = 0°). Shearing at periapse extends the structure such that by 200 yr when it is in Fomalhaut b's position near belt crossing, the diameter is 0.5 AU. This corresponds to 65 mas and would be unresolved by the observations. However, after the second periapse passage the structure is 9 AU in length, resembling a trail of material along the path of the orbit. Thus, even though Fomalhaut b as a dust cloud could survive many belt crossings, it shears into a trail of particles in one orbit if it is not gravitationally bound to a more massive object. We therefore consider it unlikely that Fomalhaut b is only a dust cloud, because it requires a fortuitous timing in discovering it a few centuries after it was created.

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How massive does a central object have to be so that an 0.05 AU radius cloud is not disrupted by shearing at periapse? Using the numerical simulation, we found a mass of 1.0 × 10²³ kg is sufficient (1.4 lunar mass). The cloud can be smaller, since Kalas et al. (2008) suggested that a circumplanetary disk with ∼30 R_J (2.1 × 10⁹ m or 0.014 AU) is consistent with the detected optical flux. For this smaller radius, the cloud is stable against shearing if the central object has mass 5 × 10²¹ kg (5× Ceres mass). Therefore a dwarf planet between Ceres and Pluto in mass may retain a system of satellite dust and moons that is stable against shearing.

The dust-cloud model in Kalas et al. (2008) specifies that the dust cloud's scattering surface area corresponds to the disruption of a minimum 10 km radius object. Since dwarf planets are much larger, ∼500 km radius objects, the mass in the dust cloud could have been launched from a single cratering impact.

Finally, given a scenario that Fomalhaut b is a dwarf planet, is it possible that its mass is still in the process of increasing significantly due to the accretion of belt material during belt passages? The mass accretion rate (kg s⁻¹) enhanced by gravitational focusing is (Kennedy & Wyatt 2011)

$$\begin{equation} dM/dt = (M_{{\rm mb}} / V_{{\rm mb}}) \pi R_b^2 \left(1+v_{{\rm esc}}^2 \left/ v_{{\rm rel}}^2\right) v_{{\rm rel}}/right., \end{equation}$$

where the main belt total mass and volume are estimated as M_mb = 75 M_⊕ and V_mb = 10³⁹ m³, and the dwarf planet radius and escape velocity are R_b = 10⁶ m and v_esc = 1000 m s⁻¹, respectively. Therefore for a relative velocity v_rel = 2700 m s⁻¹, we find dM/dt = 1.4 × 10¹¹ kg yr⁻¹. Since each orbital period consists of only 200 yr spent within the belt, the mass accretion rate is equivalently expressed as 2.8 × 10¹³ kg orbit⁻¹. If we assume that 90% of this accreted material adds to the mass of the central object and 10% adds to the mass of the planetesimal cloud, then we have the following results: in ∼3000 orbits (5 × 10⁶ yr) Fomalhaut b has accreted ∼10¹⁶ kg of additional mass into the planetesimal cloud, which is equivalent to the 10 km sized object that was originally envisioned to explain the grain scattering cross section. However, the central mass has increased by only a factor of 10⁻⁵. The 5 × 10⁶ yr timescale is of order the lifetime we might expect for the coplanar case where Fomalhaut b crosses through the planetary region, leading to an eventual strong scattering event with another planet. We therefore conclude that even though the coplanar, belt-crossing orbit is most likely short-lived, it is long enough for a dwarf planet to capture a surrounding cloud, but not long enough for the dwarf planet to increase its mass significantly.

The circumplanetary dust disk hypothesis presented by Kalas et al. (2008) received a measure of plausibility with the discovery of Saturn's Phoebe ring at >200 R_p (Verbiscer et al. 2009). The basic physical mechanism is that the surface of a small (radius ∼100 km), distant (a = 215 R_p) planetary moon is bombarded by interplanetary meteoroids, launching ejecta that spirals toward the planet due to Poynting–Robertson drag. Verbiscer et al. (2009) estimate that the normal optical depth of the Phoebe ring is ∼2 × 10⁻⁸, which could be attributed to material ejected from a single, 1 km diameter crater.

One consequence of the highly eccentric orbit for Fomalhaut b is that it is less likely to capture additional outer satellites compared to nested planets such as Saturn because its velocity relative to nested orbiting objects is high, and stellar gravitational shearing forces at periastron are significant. However, the Phoebe ring scenario only requires the existence of a single distant satellite, and certainly the prospect that Fomalhaut b previously had a lower eccentricity orbit is not ruled out. Moreover, a planet–planet scattering event that could account for Fomalhaut b's high eccentricity does not necessarily lead to the loss of the moon orbiting the scattered planet (Debes & Sigurdsson 2007; Nesvorný et al. 2007).

Instead of a single Phoebe-like satellite, Kennedy & Wyatt (2011) study the capture of many irregular satellites around Fomalhaut b that collisionally produce a circumplanetary dust cloud. This collision concept is different because the Phoebe ring is produced when a satellite is "stranded" far from the next innermost satellite, and a large fraction of impactors that strike the moon originate from outside the system. If there are numerous irregular satellites like Phoebe, then this "swarm" is self-eroding via many mutual collisions. The morphology of the dust swarm resembles an hourglass instead of a ring or torus due to the instability of high inclination moons (Hamilton & Burns 1991; Nesvorný et al. 2003). Krivov et al. (2002) present evidence that Jupiter is surrounded by a cloud of dust particles between 50 and 300 R_J, except that given the relatively few irregular satellites at this late epoch, the erosion mechanism is mainly the external meteoroid flux, as with Phoebe, instead of self-erosion.

The e ∼ 0.8, a = 177 AU orbit for Fomalhaut b changes the Kennedy & Wyatt (2011) scenario in that the Hill radius is effectively three times smaller at periastron, and so too is the region of satellite stability. The Hill radius depends on the planet and stellar masses, and the planet semi-major axis, a_pl:

$$\begin{equation} R_H = a_{{\rm pl}} (m_{{\rm pl}} / 3\ M_\star)^{1/3}. \end{equation}$$

However, in the case of an eccentric orbit, the instantaneous star–planet separation, ρ, should be adopted instead of a_pl. At an assumed periastron of q = ρ = 32 AU, and with m_pl = 1 M_J and M_⋆ = 2 × 10³ M_J, the Hill radius is 1.76 AU. For comparison, the Hill radius for our Jupiter and Neptune are 0.33 AU and 0.77 AU, respectively, and a perihelion scaling reduces these values by only a few percent. As Fomalhaut b intersects the belt at ∼150 AU in the coplanar case, the instantaneous Hill radius is 8.24 AU.

Fomalhaut b therefore represents an interesting case study for the dynamical evolution of moons and the observational consequences when the host planet has a highly eccentric orbit. To gain some rough insights, we used the N-body package AstroGrav to study the evolution of 500 moons, randomly assigned 0.01 AU ⩽ a_moon ⩽ 10 AU, 0.0 ⩽ e_moon ⩽ 0.1, with a spherical distribution of orbits around a Jupiter-mass planet that has Fomalhaut b's orbital properties. After 2 × 10⁵ yr, approximately 50 moons remain bound to the planet with 0.02 ⩽ a_moon ⩽ 0.91 AU, with median values a = 0.37 AU. The maximum value of 0.91 AU is 52% of the Hill radius calculated at periastron. This result is consistent with previous observational and theoretical studies concerning the dynamical evolution of distant satellites orbiting asteroids and planets (Hamilton & Burns 1992; Jewitt & Haghighipour 2007; Shen & Tremaine 2008)

The majority of moons lost near periastron orbit the star at a reduced semi-major axis and eccentricity, forming an eccentric disk interior to the main belt, and apsidally aligned with Fomalhaut b. It is possible that such lost moons exist if Fomalhaut b's orbit before the dynamical instability had a larger Hill sphere because the orbit was initially farther from the star. An instability that subsequently reduces the star–planet separation would then result in a smaller Hill sphere and lost moons orbiting the star instead of the planet. The implication is that the interbelt dust disk may consist of material with different origins: (1) bodies that formed there, (2) moons lost by planet instabilities, and (3) material perturbed inward from the outer disk (cf. Section 9.3.3). This scenario also invokes the possibility that if the perturbed planet is not coplanar with the main belt, the eccentric disk of lost moons would also orbit and collisionally evolve in the planet's orbital plane and not the main belt plane. The system would therefore appear to have two inclined debris disks. The β Pic system shows evidence for a secondary, inclined disk that is significantly less massive and prominent than the primary disk (Heap et al. 2000; Golimowski et al. 2006). The concept of lost moons after a planetary dynamical instability could serve as an alternate model to the current paradigm that inserts an inclined planet into a pre-existing disk, creating a vertical disk warp that propagates outward (Mouillet et al. 1997).

Since Fomalhaut b and the lost moons continue to have a similar periapse, a bottleneck of orbits is evident near periapse. Even though Fomalhaut b's Hill radius is at a minimum here, the volume number density of lost moons is greatest near periapse. We find that moons lost at periastron may be recaptured near periastron (ejection from the system or collision with the planet are also possible). In fact, the recapture epoch begins after periastron when the Hill sphere of the planet is expanding but the bottleneck is still providing a relatively high volume number density of objects. Recaptured moons tend to be captured into eccentric (e ≳ 0.3) orbits around the planet, which means that they are loosely bound and lost again as the planet approaches the next periastron passage.

Even if the recapture of moons does not occur, the outer moons still bound to Fomalhaut b are dynamically heated by periastron passage. The eccentricities found in the surviving, bound moons are 0.0 ⩽ e_moon ⩽ 0.9, with median e_moon = 0.1. Thus there are at least three mechanisms that could increase collisional dust production surrounding Fomalhaut b at periastron and soon after periastron: objects have energetic collisions with Fomalhaut b at periastron, the collisional grinding of moons bound to Fomalhaut b is enhanced at periastron, and soon after periastron additional moons may be recaptured on highly eccentric orbits that would collide with the bound moon system. Fomalhaut b is currently observed ∼120 yr after periastron, and the simulation supports the concept that moon capture may have recently activated the collisional erosion of a planetary moon system.

Two more epochs of enhanced collisionally activity may occur at the ascending and descending nodes relative to the belt plane. When Fomalhaut b crosses the orbital plane of the belt or the interbelt dust disk, the external meteoroid flux is enhanced again.

Observationally, the optical depth of the Fomalhaut b ring/cloud system will increase during enhanced erosion, the grain size distribution will shift temporarily to small sizes as fresh dust below the radiation pressure blowout radius is released, and as a result we might observe a brighter and bluer scattered light signature from Fomalhaut b. On the other hand, a dust cloud could also become optically thick, which would make light scattered toward the observer sensitive to viewing geometry. In other words, shadowing due to too rapid dust production could decreases the brightness.

Though this discussion focuses on the erosion of satellites from both circumplanetary and circumstellar impactors, generating fresh circumplanetary dust that is observable in reflected light, other processes may be at work that have distinct observable signatures. When the bound moons have their eccentricities pumped by periastron passage, this could increase the planetary tidal heating of the hypothetical moons (Peale & Cassen 1978; Peale et al. 1979; Cassen et al. 1979). Tidal heating and melting has an infrared signature (e.g., Peters & Turner 2013 and references therein). A clone of Jupiter's regular moon Io would also generate an optically detectable sodium cloud. Jupiter's sodium cloud has been detected in Sun scattered light to at least 400 R_J (0.2 AU; Mendillo et al. 1990). Thus, in addition to dust scattered light and Hα emission as possible explanations for Fomalhaut b's anomalously high optical flux (Kalas et al. 2008), a sodium cloud could also contribute at 0.59 μm. This lies in the F606W bandpass of the 2004 and 2006 HST/ACS observations.

Finding definitive evidence for such a cloud would have many significant implications, such as showing that Fomalhaut b has a magnetic field similar to Jupiter's. Clearly a spectrum of Fomalhaut b is required, but the issue can also be examined via imaging. For example, Jupiter's sodium cloud is variable in size and brightness. These variations are correlated to the volcanic activity of Io (Mendillo et al. 2004). Therefore, the characteristic timescales of variability from imaging Fomalhaut b may be more geophysical than astrophysical—Fomalhaut b may episodically appear brighter and more extended on timescales measured in months.

We have argued that even though variability and extended morphology exist in the HST optical data (Figures 2 and 4), they can also be attributed to instrumental noise (Section 2.2; Figure 6). However, Galicher et al. (2013) claim that the extended morphology of Fomalhaut b in the 2006 ACS/HRC data is not instrumental in the F814W image. This is puzzling because the F606W image taken at the same epoch and with slightly greater sensitivity does not appear to be extended. This is difficult to reconcile with a model where optical light arises from grain scattering—the F606W image should also show extended morphology. However, the more complicated model involving the evolution of atomic and molecular species from moon volcanism to cirumplanetary magnetospheres could yield a solution. For example, Io is also the source of a circumplanetary potassium cloud that emits at 0.77 μm (Trafton 1975), and this lies within the F814W bandpass. Jupiter's potassium cloud is significantly weaker and less extended than the sodium cloud, but a different geochemistry for the hypothetical Fomalhaut b moon could conceivably produce a potassium cloud that adds a halo of extended light in the F814W images.

A key question is whether or not a planet mass passing through the belt disrupts the morphology of the belt. The maximum radius of gravitational influence could be taken as approximately three times the Hill radius (Section 9.3.2) at 150 AU. For a Jupiter mass this corresponds to 25 AU (65) diameter. This size is 3.2 times smaller for a 10 M_⊕ planet, but the corresponding 20 angular scale is still resolvable with current instrumentation. In principle, the local dynamical stirring should enhance dust production, shifting the grain size distribution to favor smaller grains with larger surface area, and produce a transient brightening of the belt in scattered light (e.g., Kenyon & Bromley 2001; Dominik & Decin 2003).

To explore the cumulative effects of Fomalhaut b's dynamical perturbations on belt parent bodies over many belt crossings and under a variety of assumptions, we use the three-dimensional, N-body simulator AstroGrav. Our model of the Fomalhaut system begins with a nested Jupiter-mass planet (Fomalhaut c) with a = 120 AU, e = 0.10, i = 0 and ϖ = 178°. We add two populations of main belt objects. First, an effectively massless population of 8000 objects are randomly assigned 140 AU ⩽ a ⩽ 160 AU, 0.09 ⩽ e ⩽ 0.11, and −15 ⩽ i ⩽ 15. The orbits are apsidally aligned with the nested planet and randomly distributed in orbital phase. The second population is 200 objects between Ceres and Pluto in mass and radius (total mass = 0.057 M_⊕), distributed randomly throughout the belt as in the first population. However, the masses are not negligible and will gravitationally perturb other objects. Collisions are treated as mergers.

The first part of the simulation does not contain Fomalhaut b (assumed to be on a circular orbit well within the orbit of Fomalhaut c and dynamically negligible). The goal is to reach a quasi-steady state with respect to the dynamical sculpting of the belt's inner edge by Fomalhaut c. The entire system is coplanar and integrated for 1.5 × 10⁵ yr (∼1500 orbits). The model qualitatively reproduces the numerical experiment of Chiang et al. (2009) where Fomalhaut c maintains a sharp inner edge. A key difference is that Chiang et al. (2009) proceeded to model the dynamics of dust particles in addition to the parent bodies, where radiation pressure instantaneously increases the eccentricities of the observed dust population. Since we are studying only the parent bodies, the timescales given below may be considered upper limits to the eccentricity evolution of observable particles. Another difference is that particles approaching within a few times the Hill radius are not removed from our simulation, and therefore we observe the capture (and loss) of satellites.

The second part of the simulation assumes that after 1.5 × 10⁵ yr, Fomalhaut b is strongly perturbed from the inner part of the system (by interaction with a third planet) and appears as a rogue planet with a = 177 AU, e = 0.8, i = 0°, and apsidally aligned with Fomalhaut c. We assume three cases where the rogue planet has a Jupiter, Saturn, and Neptune mass. We use the "roque" terminology to designate bound planets with large a and e such that they cross the orbits of other planets and belts in the system.

The impulse imparted on belt material by a planet crossing through the belt does not create a visually noticeable disruption of the overall belt morphology. Inspection of the belt particle orbital elements shows no statistically significant difference before and after planet crossing. For example, after the nested Jupiter-mass Fomalhaut c has been sculpting the belt for the first 1.5 × 10⁵ yr, the mean eccentricity distribution of belt particles has evolved to e = 0.1077 ± 0.0747. Fomalhaut b as a rogue Jupiter mass would have the most significant dynamical effect, yet after a single crossing the eccentricity distribution is e = 0.1072 ± 0.0746. Measured another way, before the belt crossing, 8.07% of belt particles have e > 0.20. After the single crossing of a Jupiter, 8.17% have e > 0.20.

The cumulative effect of many belt crossings is to gradually spread the belt radially. Over 10²–10⁴ belt crossings, the only belt morphology that is noticeably "disrupted" is the sharpness of the belt boundaries. The belt becomes a disk.

Kalas et al. (2006) noted that debris disks appear either as extended disks or narrow belts. The numerical models studied for the Fomalhaut system suggest that when a planet transitions from a nested to a rogue orbit, belts become disks. The rogue can subsequently evolve away from a belt-crossing orbit, and the disk will then resemble a belt again by interaction with the nested planets, though with reduced mass due to the scattering of objects during the rogue phase. The Fomalhaut system could be in transition from belt to disk, depending on the mass and orbit of Fomalhaut b.

The rogue planet in our model competes with the nested planet in shaping the inner edge (Figure 29). The qualitative result is that Fomalhaut b as a coplanar Saturn erodes the belt edge significantly after only 10⁵ yr (Figure 30). A coplanar Neptune mass Fomalhaut b, on the other hand, does not erode the belt inner edge on timescales approaching 10⁶ yr. One reason we stop the simulations before ∼10⁶ yr is that the coplanar geometry leads to a significant planet–planet scattering that alters the orbit of Fomalhaut b and the belt edge erosion timescales have to be reconsidered. For example, some encounters reduce Fomalhaut b's apoastron so that it resides in the belt. With Fomalhaut b spending a greater fraction of time in the belt, a Neptune mass becomes a significant disruptor of pre-existing morphology and the belt becomes a disk in <10⁶ yr.

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These timescales are likely lower limits because we have been assuming that Fomalhaut b is coplanar with both the main belt and with Fomalhaut c, maximizing the probability of strong interactions. When Fomalhaut b's orbital plane is inclined by 20° relative to Fomalhaut c and the main belt, the belt still spreads radially in the Saturn case, but the timescale is 500 kyr instead of the 75 kyr observed in the coplanar case (Figure 29). Inclining a Neptune-mass Fomalhaut b by 20° decreases the probability of a strong interaction with a nested Jupiter. However, many weak interactions over ∼10⁶ yr timescales will evolve the Neptune orbit. For example, by 6 Myr Fomalhaut b's inclination has increased to ∼39° and eccentricity has decreased to ∼0.5. Kozai-like oscillations between inclination and eccentricity means that the belt tends to be protected from significant interactions with Fomalhaut b: high I, low e excursions mean that Fomalhaut b enters the system only near periastron in the inner regions of the system that miss the belt, whereas low I, high e means that Fomalhaut b passes closer to the main belt, but with a shorter timescale due to the high e.

A more thorough exploration of the orbit and mass parameter space is required and could establish a lower limit to the age of the current orbital configuration. Certainly any model must also test the origin of Fomalhaut b's eccentric orbit, with the possibility that multiple massive planets are located in the system. Future observations can also search for evidence that the belt was disturbed by the previous belt crossing east of the star.

In summary, a planet mass for Fomalhaut b is not excluded by arguments concerning belt disruption because (1) belts are not "wiped out" by a single belt crossing of a Jupiter-mass planet, and we do not know how many belt crossings have already occurred; (2) Saturn masses and below can cross through the belt hundreds of times before the belt edges are eroded; (3) belt crossings erode the sharpness of belt edges, but nested planets may compensate by maintaining the edges sharp; and (4) a mutual inclination of Fomalhaut b relative to the main belt increases belt edge erosion timescales significantly.

One indirect way to infer the mass of Fomalhaut b is if impacts become evident during a belt crossing. If Fomalhaut b has the mass of a gas giant, then we could expect phenomena similar to the SL9 (D/1993F2) impacts with Jupiter (e.g., Graham et al. 1995; Zahnle & Mac Low 1995; Anic et al. 2007). The greater energies involved would manifest as significant optical and infrared variability, with a different characteristic spectral energy distribution and time dependence than impacts on a dwarf planet. Careful analysis could even yield information concerning its atmosphere and composition (e.g., Bjoraker et al. 1996).

For the SL9 events, Carlson et al. (1997) report ∼0.5 × 10²⁵ erg from the G-impact fireball in a 60 s interval. Temperatures are 8000 K at the beginning, cooling to ∼1000 K after 80 s, giving roughly 8 × 10²² erg s⁻¹ = 8 × 10¹⁵ W. If this event were located at the heliocentric distance, D, for Fomalhaut, then

$$\begin{equation} f_{G} = L / (4 \pi D^2) = 1.1\times 10^{-20}\ \mathrm{ W\ m}^{-2}. \end{equation}$$

Relative to the luminosity of Fomalhaut A:

$$\begin{eqnarray} \Delta \mathrm{mag} &=& m_G - m_\star = -2.5 \mathrm{log} (N\times f_G / f_\star) \nonumber \\ &=& {-}2.5 {\rm log}(N) + 29.8 \ {\rm mag}. \end{eqnarray} \tag{ 3 }$$

Here we assume that the received flux from Fomalhaut A is f_⋆ = 8.9 × 10⁻⁹ W m⁻², and N is some tuning factor, such as N fireballs. For N > 100, Δmag < 24.8 mag and Fomalhaut b would appear significantly brighter. However, the peak of emission quickly (i.e., in seconds) shifts from an optical flash to a relatively long-lived emission at near-infrared wavelengths.

Are N > 100 fireballs plausible for Fomalhaut b's encounter with the belt? In Section 9.3.1 we calculated the accretion rate with gravitational focusing onto a dwarf planet. If we instead assume a Jupiter mass, with R_b = 6.99 × 10⁷ m and v_esc = 159 km s⁻¹, then dM/dt = 6.4 × 10¹⁰ kg s⁻¹, or about one comet Halley per hour.

This rough calculation suggests that optical flashes may be observable as Fomalhaut b crosses through the belt. The energies involved will help constrain the mass of Fomalhaut b, but its atmosphere will also be heated and excavated. The infrared luminosity would therefore rise and molecular features in a spectrum may become observable and display variability as conditions change on the planet.

Extending the impact theme even further, is it possible that Fomalhaut b collided with a hypothetical second planet, Fomalhaut c, and the main belt is now the remnant debris of Fomalhaut c? Giant impacts that can produce transient circumstellar dust rings have recently been invoked to explain the properties of several debris disks (e.g., Lisse et al. 2009; Melis et al. 2012). Such a hypothesis is exceedingly unlikely at the large stellocentric distances represented in the Fomalhaut system. On the other hand, the hypothesis could account for several observed properties. The stellocentric offset of the belt is explained as the elliptical orbit of the precursor object, Fomalhaut c. The belt is narrow because it is recently created and has not had time to collisionally evolve and spread radially. Fomalhaut b is belt crossing because the belt would not exist otherwise. While most of the mass in Fomalhaut c is dispersed along its orbit, most of Fomalhaut b's mass is retained in a circumplanetary disk in the process of reaccreting onto the planet or forming moons, but temporarily making it bright in reflected light. The fraction of Fomalhaut b's mass that has been lost comprises a more tenuous stream of co-orbital material that manifests as the interbelt 141° arc of 450 μm emission. In the next section we also study whether or not the main belt gap could be explained by this model.

The critical problem given by Boley et al. (2012) is that the impact speed for collisional erosion has to be significantly greater than the mutual escape speed of the two bodies, but the orbital velocities are small at great distances (150 AU) from the star. Erosive impacts, require v_i/v_esc ⩾ 1.5 (Asphaug 2009; Marcus et al. 2010; Stewart & Leinhardt 2012; Leinhardt & Stewart 2012), where the impact velocity is $v_i = \sqrt{\vphantom{A^A}\smash{{{ v_{{\rm esc}}^2+v_{{\rm rel}}^2}}}}$.

For a Moon mass, v_esc = 2.4 km s⁻¹ and the 45° entry of Fomalhaut b into the belt in the prograde sense gives v_rel = 2.7 km s⁻¹. Therefore, v_i = 3.6 km s⁻¹, which is a factor of 1.3 greater than v_rel. If the collision is in the retrograde sense, then v_rel = 6.6 km s⁻¹, and v_i/v_esc = 2.6.

Therefore the collision of two Moon mass objects would be an erosive event in the retrograde sense. For lower mass objects (e.g., Pluto) the prograde collision becomes erosive (v_i/v_esc ≈ 3) but the objects do not represent enough mass to account for the main belt mass. For higher mass objects, v_esc becomes too large to be consistent with erosive impacts. Therefore the likely mass ranges of the colliding objects are in the Moon regime and a retrograde collision may be necessary.

The timescale for the debris to spread into a circumstellar belt is given by Wyatt & Dent (2002): $\Delta t = 2 \pi / (2 \sqrt{3} v_\infty a)$. For a = 150 AU and the range 10 m s⁻¹ <v_∞ < 100 m s⁻¹, we find 10⁴ yr < Δt < 10³ yr. Thus the collision may have occurred relatively recently. Since the collision lifetime for 10 μm sized grains is 10⁵ yr (Wyatt & Dent 2002), the giant impact scenario allows stripped material from Fomalhaut c to evolve into a belt on shorter timescales than the collision lifetime of grains.

The current snapshot that Fomalhaut b is about to cross through the belt near the 331° gap invokes the idea that the gap is caused by material scattered away from the belt when the planet crosses through. In the previous sections we argued that this is not true in the case of Fomalhaut, though other astrophysical disks that are thin, gaseous, self-gravitating, and/or shadowed by optically thick material closer to the star may display such morphology (e.g., the azimuthal gap in the circumbinary belt surrounding the pre-main sequence system GG Tau; Roddier et al. 1996; Krist et al. 2005).

One possibility for explaining Fomalhaut's main belt gap is that a planet orbits within the belt, and the gap represents tadpole or horseshoe orbits of the co-orbital planetesimals and dust. The analogy is to the dynamics of the giant planet Trojan populations observed in our solar system (Chiang & Lithwick 2005; Nesvorný & Vokrouhlický 2009; Lykawka et al. 2011). The Earth is also known to trap in-spiralling dust grains near a 1:1 resonance, producing a ∼1 AU radius dust ring orbiting the Sun, except near the planet, where grain dynamical lifetimes are short (Jackson & Zook 1992; Dermott et al. 1994).

A second possibility is that the gap is related to the giant impact scenario (Section 9.3.5). For ∼10³ orbital periods all particles created at the collision origin point return back to the collision origin point. A snapshot of debris particle location reveals a gradual pinching of the belt toward and away from the collision origin point (Jackson & Wyatt 2012). This tapered morphology resembles the main belt gap. Fomalhaut b would return to the collision origin point if a grazing, hit-and-run collision ejects mantle material, but the planet core stays close to the pre-encounter orbit. Therefore, one of Fomalhaut b's two apparent belt crossing per orbital period should occur near this point, which should coincide with the radially tapered section of the belt. The recent, giant impact thereby ties together the apparent proximity of Fomalhaut b to the main belt gap to the north of it.

Contradicting this scenario is that even though the orbital paths for debris tapers toward and away from the collision origin point, the quantity of dust is not changing. This scenario therefore does not account for diminishing the scattered light in the belt gap region, though more complex effects mentioned below may come into play. Another significant problem is that the belt pinch requires a low velocity dispersion of debris (100–150 m s⁻¹), which counters the relatively high velocities needed to disrupt the required precursor mass (A. Jackson 2012, private communication).

One general solution is to suppose that the giant impact occurred recently (<10⁴ yr) and the debris is still spreading along the orbit of the precursor object—more time is required to create an azimuthally uniform ring. We cannot rule this out, and certainly future observations should search for other structure in the belt consistent with a debris field that is dynamically young. A second solution is that the gap arises from a combination of a relatively young ring (∼10⁴ yr) and a period commensurability between the planet and the debris (see Figure 6 in Jackson & Wyatt 2012).

We propose a third possibility concerning a geometrical effect that results if apsidal alignment is accompanied by nodal alignment, presumably because of dynamical interactions with a low-eccentricity planet. If a large ensemble of belt particles have both non-zero orbital inclinations and nodal alignment, the belt pinches vertically toward the midplane at the ascending and descending nodes. Figure 31 demonstrates this hypothetical configuration. We note that the 331° is roughly 180° away from our estimate for the ascending node. In other words, the belt gap is near the descending node where one of the two pinch areas occurs. We find that due to projection effects, the pinch area is obscured at the ascending node due to foreground and background material contained in the line of sight to the ascending node. The orbital configuration of particles is similar to the giant impact scenario, except that there are two pinch points.

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As with the giant impact debris field, the problem with this scenario is that material is confined, but not necessarily removed from the pinch area. Therefore the surface brightness should not diminish significantly. On the other hand, the material at a vertical pinch point has a very flat spatial distribution, so that belt particles are more likely to be self-shadowing. Self-shadowing is invoked to explain why Uranus' ring is fainter at periapse than at apoapse—the ring optical thickness increases at periapse and self-shadows (Karkoschka 1997).

Future work is needed to quantitatively study the cumulative effect of these factors on the scattered light appearance of debris belts near pinchpoints. The theory of apsidally and nodally locked planetary rings depends on the interplay between collisions, self-gravity, and the quadropole field of the planet (Chiang & Culter 2003). For a debris belt, collisions are important, but self-gravity is assumed to be insignificant, and planet–belt dynamics include significant secular effects. Observationally, the origin of Fomalhaut's 331° main belt gap could be explored by ALMA observations. Self-shadowing would be irrelevant at mm wavelengths such that a belt gap in a mm map would indicate an absence of grains, thereby supporting the horseshoe/tadpole orbit hypothesis.