REALISTIC DETECTABILITY OF CLOSE INTERSTELLAR COMETS

The currently accepted model of the origin of long-period comets was not always widely accepted. After some initial work by Öpik (1932), the beginnings of the modern model of an isotropic cloud of comets tenuously bound to the solar system emplaced by the planet formation process was originally posed as a "hypothesis" by Oort (1950). At that time, only ∼20 long-period comets had well-determined orbits after correcting for planetary perturbations. One major alternative hypothesis was that all (long-period) comets were interstellar, with perhaps an unseen stellar companion to the Sun helping to capture them (e.g., Valtonen & Innanen 1982). The distribution of long-period comets and other small bodies in the solar system now give overwhelming evidence for cometary origins in the Oort cloud, though the origin and history of this cloud is still under discussion (e.g., Levison et al. 2010; Kaib et al. 2011).

One of the earliest references to ICs in the context of the Oort cloud origin for long-period comets, is the estimate of Whipple (1975) on the frequency of ICs from their non-detection and from the frequency of gamma-ray bursts. As required for an unobserved population, Whipple (1975) made various assumptions to estimate the mass density of ICs in the galaxy to be less than $3\times {10}^{-4}\;{M}_{\odot }$ pc⁻³. Similarly, Sekanina (1976) estimated an upper limit to the mass density of comets of $\lesssim 6\times {10}^{-4}\;{M}_{\odot }$ pc⁻³ based on the non-detection of ICs up to that point.

Based on updated estimates of the number of Oort cloud comets, McGlynn & Chapman (1989) use a simple model to estimate that the number of comets is ∼10¹³ pc⁻³. If all of these are assumed to have ∼1 km in radius, this suggests a mass density of approximately ${10}^{-5}\;{M}_{\odot }$ pc⁻³. These estimates effectively took the number of Oort cloud comets expected from the formation of the solar system (∼10¹⁴ at the time) and multiplied this by the local density of stars. This is well explained by Stern (1990), who explicitly consider how the frequency of ICs can be used to infer properties of planet formation in the galaxy.

Upon finding that we have not seen the predicted number of ICs, McGlynn & Chapman (1989) attempt to draw strong conclusions, despite the simplicity of their model for the frequency and detectability of ICs. As a response, Sen & Rama (1993) suggested that the stellar density used by McGlynn & Chapman (1989) was an order of magnitude too high, bringing the expected frequency of detectable ICs down enough that they were no longer missing from the observations.

Using a much more detailed model and a survey simulator, Francis (2005) placed a limit of $3\times {10}^{12}$ ICs per cubic parsec based on a non-detection of ICs in the LINEAR survey. If these are all assumed to be ∼1 km in radius, this suggests a mass density of $3\times {10}^{-6}\;{M}_{\odot }$ pc⁻³, two orders of magnitude lower than the earliest estimates. Meinke et al. (2004) mention a study of the Spaceguard survey which found a 97% upper limit on 1 km ICs of 10¹⁴ ICs per cubic parsec, but included a power law distribution (corresponding to ${q}_{1}={q}_{2}=3.5$ in our nomenclature below).

The most recent study (Engelhardt et al. 2014) used Pan-STARRS 1 data, a careful analysis of detectability, a power-law distribution (with ${q}_{1}={q}_{2}=3.5$), and included the possibility of cometary activity. This most sophisticated method has similarities with our methodology described below. The 90% confidence upper limit on the number density of $\gtrsim 1$ km ICs per cubic parsec was $4.7\times {10}^{14}$ for inactive comets ("asteroids" in our nomenclature) and $1.6\times {10}^{13}$ for active comets. This is not as strong a limit as the Francis (2005) result, but is the result of a more detailed assessment. It is also able to roughly rule out the McGlynn & Chapman (1989) estimate.

M09 was the first theoretical estimate to use a more modern understanding of planet formation, accurate distributions for the frequency of stars of different masses in the Galaxy, and incorporating an IC size distribution to determine a much more realistic estimate for the frequency of ICs. Although their model makes many assumptions about the unknown properties of planet formation and IC properties, these are mostly included in the form of tunable parameters. M09 estimate the mass density of ICs to be $\sim 2\times {10}^{-7}\;{M}_{\odot }$ pc⁻³, which, when considering a size distribution for ICs, yields a number density of ICs larger than 1 km of 10^5–10 pc⁻³. This is between 3 and 8 orders of magnitude below the McGlynn & Chapman (1989) estimate and the Engelhardt et al. (2014) upper observational limit.

Following Alcock et al. (1986), Jura (2011) has also recently shown that the space density of ICs must be less than predicted by the earlier optimistic assessments by studying the chemical composition of unpolluted white dwarfs (which would have retained signatures of accreted ICs). They place an upper bound on the space density of ICs in agreement with M09 and far less than McGlynn & Chapman (1989). Zubovas et al. (2012) predict a similar frequency of ICs near the galactic center as a possible source of Sgr A* flares.

Besides illustrating the decline in IC frequency estimated over time, these studies show that simplifying assumptions and unknown properties can yield IC detectability estimates that span many orders of magnitude. Even intercomparing the results of these studies requires assumptions about the size distribution or typical size of detectable comet nuclei and the mass–radius–brightness relation of comet nuclei, neither of which are known very well. As in M09, we have tried to use tunable parameters to understand the importance of various assumptions. By using the most up-to-date theory from M09 with a model that incorporates realistic estimates of detectability, we have produced the most sophisticated estimate of the frequency of detectable of ICs to date.

Królikowska & Dybczyński (2013) and Dybczyński & Królikowska (2015) find that comet C/2007 W1 (Boattini) is a strong candidate IC. While some comets appear slightly hyperbolic due to interactions with the giant planets and/or non-gravitational forces, an analysis of these effects show that, in this case, they are much too small to explain C/2007 W1's orbit with an initial semimajor axis of −23,000 au ($1/a=-42.75\pm 2.34\times {10}^{-6}$ au), perihelion of 0.83 au, and inclination of 10° (Dybczyński & Królikowska 2015). Over 1000 observations of C/2007 W1 were obtained over 13 months, sufficient to produce a high quality orbit, as reflected on the Minor Planet Center, which has similar orbital estimates. Another candidate IC, C/1853 E1 (Secchi), relies on century-old astrometry and can be probably attributed to systematic errors (Branham 2012).

C/2007 W1 was observed spectroscopically and found to be an unusual comet chemically, as might be expected for an IC (Villanueva et al. 2011), although it was not completely out of the range of known comets. It was also the source of a somewhat unusual meteor shower (Wiegert et al. 2011).

Converting the excess energy in C/2007 W1 from Dybczyński & Królikowska (2015) to a velocity at infinity gives ${v}_{\infty }\simeq 0.27\pm 0.01$ km s⁻¹, much smaller than the expected ∼20 km s⁻¹ for an IC. Along the same lines, C/2007 W1 has a post-perihelion orbit that is bound to the solar system (semimajor axis of 1800 au, perihelion distance of 0.85 au) which seems unusual for an IC, though this is not an entirely separate argument, since the capture is highly enhanced due to the low ${v}_{\infty }$. In our simulations, we do not find a particularly enhanced preference for detecting low ${v}_{\infty }$ ICs, so this low relative velocity is seemingly an argument against the interstellar origin of C/2007 W1.

Altogether, the orbital and chemical evidence for the interstellar nature of C/2007 W1 is highly suggestive, but not conclusive. We, therefore, proceed without including this comet in our analysis.

During the lifetime of LSST (10 years), even ICs with unusually high (100 km s⁻¹) relative velocities to the Sun will only traverse approximately 200 au or ∼0.001 pc. Therefore, the dominant medium of ICs observable in the next several decades is the region of space extremely close to the Sun, in a galactic sense, and dominated by the so-called Circumheliospheric Interstellar Medium (CHISM or CISM), sometimes called the Very Local Interstellar Medium. This region is much smaller than the Local Interstellar Cloud (∼10 pc) or Local Bubble (∼100 pc), and not even much larger than the heliosphere (∼100 au). Indeed, any IC passing by Earth within the next decades is currently residing within the inner Oort cloud, far closer than the aphelion of Sedna and other Kuiper Belt objects (Brown et al. 2004). Here we briefly review the known properties of the IC source region. In situ samples of the CHISM as it flows into the inner solar system are taken by observing interstellar neutral atoms, interstellar pickup ions, interstellar dust grains (≲1 μm, entrained in the local gas flow), and interstellar micrometeorites (≳10 μm and decoupled from the gas). Observations of the CHISM over the last four decades have not shown any significant evolution, suggesting that the properties are homogeneous on the scales relevant to near-future IC detections (Frisch et al. 2011). The mass ratio of gas to dust in the CHISM is thought to be ∼100 (Frisch et al. 2011) and the dust mass density is about ∼100 times bigger than the mass in ICs estimated by M09 (Landgraf et al. 2000).

Dust smaller than ∼10 μm is observed in situ by dust detectors on NASA spacecraft (e.g., Ulysses, Cassini, Helios, and Stardust) in deep space. The observations show that these grains are fully entrained in the local interstellar flow, as expected since they are small enough to couple to the gas, even at the low CHISM gas densities (Frisch et al. 2011). The NASA Stardust mission returned samples from its Interstellar Dust Collector, which was designed to capture interstellar grains directly from this local flow (Westphal et al. 2014).

On the other hand, the trajectories of interstellar dust larger than ∼10 μm is dominated by solar gravity in the region of the Sun and can travel unperturbed for hundreds of parsecs from its source region where they are detected as Earth-impacting micrometeorites by ground-based optical and radar observations (Baggaley 2000; Murray et al. 2004; Musci et al. 2012). Very little is known about the frequency or properties of larger particles (Socrates & Draine 2009). Observations of these interstellar micrometeorites have detected a broad "background" source, but also a significant discrete source that may be associated with debris-disk and planet hosting star β Pictoris (Baggaley 2000; Lagrange et al. 2010). Murray et al. (2004) suggest that large grains may indeed be from discrete sources. However, there is a chance that these interstellar meteoroids are contaminated by solar system particles that have reached hyperbolic velocities due to planetary encounters (Wiegert 2014). Generally, these have lower velocities than the expected incoming stream of interstellar micrometeorites, but it does present a source of confusion. See Wiegert (2014), Hajduková et al. (2014) and references therein for additional discussion on interstellar micrometeorites and meteorites.

Direct observation of ICs with LSST or other optical surveys probes the local IC population well above 1 m sizes. M09 (and our own simulations) found that the most detectable objects will be the smallest ICs which, though intrinsically fainter, typically pass much closer than the larger ICs which more than compensates for their smaller area. Large (≳100 km) ICs are so rare, that they are very unlikely to be seen.

We are not considering objects in the ≳10,000 km size regime, as these unbound planet-sized objects (variously called free-floating planets, rogue planets, interstellar planets, or nomads) can have intrinsic luminosity and have been studied elsewhere (e.g., Kirkpatrick et al. 2011; Strigari et al. 2012).

3.1.1. Initial Position and Velocity

3.1.2. Orbital Motion and Gravitational Focusing

3.1.3. Broken Power Law Size Distribution and Mass Density

The number and size of the ICs that are initially placed in the simulation cube is determined by using a broken power law size distribution and an overall mass density (m_total). The size distribution defines the number of ICs we expect to exist for a given radius. Throughout this paper the "size" of an IC refers to the radius of the comet nucleus. The mass density is the amount of mass we expect exists in a given volume. The size distribution and mass density are combined by assuming that all ICs have the same density (nominally 0.5 g cm⁻³). As a result we can calculate the number of ICs that would exist in a given volume along with their respective radii.

Our nominal simulation adopts the mass density of M09, with the total mass of ICs of m_total = 2.2 × 10⁻⁷ M_⊙ pc⁻³ = 4.5 × 10²⁶ g pc⁻³ = 5.1 × 10¹⁰ g au⁻³. If the mass in a cubic au were concentrated into a single object with the density of water, it would only be 23 m in radius. This is ridiculously sparse, which explains why ICs have evaded detection thus far, even though their ∼4 au yr⁻¹ motion relative to the Sun is constantly replenishing the possibility of detection.

We note that, due to the uncertain nature of planet formation and the creation and ejection of ICs, the estimate of M09 could be substantially in error. Several effects could increase the mass density of ICs each by a factor of a few: an updated stellar density (Garbari et al. 2012), Oort cloud stripping from galactic tides (Veras & Evans 2013), ejection of Oort clouds due to the death of stars (Veras et al. 2011; Veras & Wyatt 2012), and many other possible effects. Since M09, the Kepler Space Telescope has also discovered entirely new classes of planetary systems, calling into question aspects of planet formation theory used to justify the estimates of M09.

Our simulations use the size distribution prescriptions suggested by M09. In particular, the size distribution is a broken power law, with a variable break radius $({r}_{b})$ and different size distribution slopes on each side of the break. Following M09, we define differential size distribution slopes q₁ and q₂, such that $n(r)\propto {r}^{-{q}_{1}}$ if $r\lt {r}_{b}$ and $n(r)\propto {r}^{-{q}_{2}}$ if $r\gt {r}_{b}$. We place practical limits on the minimum and maximum radii in our simulations (see below). The number densities as a function of different q₁ and q₂ values for ${r}_{b}=3$ km are shown in Figure 1. By detecting serendipitous KBO occultations, Schlichting et al. (2012) have estimated that ${q}_{1}\approx 2.8\pm 0.1$, while Fraser & Kavelaars (2009) and Fuentes et al. (2009) suggest that ${q}_{2}\approx 4.5$ with a break radius of ${r}_{b}\approx 75$ km in the Kuiper Belt (see also limits from the lack of detection of KBOs by WMAP, Ichikawa & Fukugita 2011). See M09 for more discussion on the possible size distribution relevant for ICs; here we simulate several different possibilities.

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Our "nominal" model uses typically assumed values for the variables that describe the properties of ICs except for the size distribution, where we use a very optimistic case. It is important to remember throughout that the observational and theoretical estimates of parameters in our "nominal" model are sometimes controversial and often with significant uncertainty. This reemphasizes the importance of leaving many variables as free parameters.

Finally, we note here that there was a small error in the number density equation as derived by M09. Their number density equation (Equation (5) in M09) is only valid for radii less than the break radius because of the limits of integration used during its derivation (A. Moro-Martín 2016, personal communication). A piecewise equation is needed to correctly define the number density for radii less than and greater than the break radius. We have made this correction and reproduced a plot of the number density of the ICs in Figure 1, showing the correction to their Figure 1 for their nominal mass density. Since the number density and observability of ICs is completely dominated by the small objects, their errors above the break radius have no consequence for the results reported in M09.

3.2.1. Asteroid Case

Based on the above calculations, we now have the position of the IC, Sun, and the Earth at any time. Determining the brightness is based initially on the standard apparent magnitude equation for asteroids (Bowell et al. 1989, pp. 524–556):

$$\begin{eqnarray}&&V=H+2.5[{\mathrm{log}}_{10}({{\rm{\Delta }}}_{\mathrm{sun}}^{2})+{\mathrm{log}}_{10}({{\rm{\Delta }}}_{\mathrm{earth}}^{2})]-2.5{\mathrm{log}}_{10}(\gamma )\end{eqnarray} \tag{ 1 }$$

where H is the (asteroid) absolute magnitude, related to the intrinsic brightness of the asteroid independent of the observing geometry and γ is the correction for photometric phase described below. Specifically, H is the brightness an asteroid would have if it were 1 au from the Sun and from the observer and observed with the Sun-asteroid-observer angle of zero (an observing geometry that is only possible if the observer is at the position of the Sun). The absolute magnitude is intrinsic to the body and determined by radius and albedo in the asteroid case. We use the standard definition

$$\begin{eqnarray}&&H=\displaystyle \frac{{\mathrm{log}}_{10}(2r)+0.5{\mathrm{log}}_{10}(p)-3.1236}{-0.2}\end{eqnarray} \tag{ 2 }$$

where r is the radius in km and p is the albedo. Following M09, our nominal assumed albedo is 0.06, but this is actually only applied to asteroids (the connection between comet brightness and nucleus radius is handled in a different way).

The photometric phase angle is the Sun-IC-Earth angle with the IC at the vertex. To encapsulate the effects of diminished reflection and non-uniform surface scattering and non-zero phase angles, we follow standard methods of using a phase function γ to adjust the brightness of the ICs based on its phase angle. The phase function we use is the standard function from Muinonen et al. (2010):

$$\begin{eqnarray}&&\gamma =(1-G){{\rm{\Phi }}}_{1}(\theta )+G{{\rm{\Phi }}}_{2}(\theta ),\end{eqnarray} \tag{ 3 }$$

where G is a slope parameter, ${{\rm{\Phi }}}_{1}$ and ${{\rm{\Phi }}}_{2}$ are basis functions for the phase curve (Equation (6) from Muinonen et al. 2010), and θ is the phase angle. The value of G controls how steep the phase curve is; values close to 0 indicate a steep curve and values close to 1 indicate a shallow curve. We use a steep curve with G = 0.15, the standard value used for objects with unmeasured phase curves. For the most part, the photometric phase angle correction is not significant unless observations are taken at large angles (i.e., far from opposition), when it can drop the brightness of an object by several magnitudes (mostly because of the smaller "day" side). This can be relevant for isotropically distributed ICs which may be seen in non-standard geometries.

Throughout, we refer to objects that follow this photometric prescription as "asteroids" although in practice they may be dormant or inactive comet nuclei.

3.2.2. Comet Brightening

As comets approach the Sun they can become active and have a significant increase in intrinsic brightness. There are two aspects of comet brightness to consider: how the radius of the comet nucleus relates to its brightness and how the intrinsic brightness grows in time as the comet approaches the Sun and becomes more active. In both cases, we use the standard empirical methods to determine the brightness of ICs.

To relate the radius of comet nuclei to their brightness, we use the comet absolute magnitude, which we call H_c in order to distinguish it from the asteroid absolute magnitude, which is defined differently. The relation to determine H_c for comets is

$$\begin{eqnarray}&&{H}_{c}=\displaystyle \frac{{\mathrm{log}}_{10}(2r)-{b}_{2}}{{b}_{1}},\end{eqnarray} \tag{ 4 }$$

where r is again the radius of the comet nucleus in km and b₁ and b₂ are empirical parameters determined from observational data. These terms absorb the albedo term seen in Equation (2); the asteroid case is equivalent to b₁ = −0.20 and ${b}_{2}=3.1236-0.5{\mathrm{log}}_{10}(p)=3.73$ using p = 0.06. Several different values of b₁ and b₂ have been estimated as noted in Table 1, therefore, we leave these as free parameters of our model. The nominal model uses ${b}_{1}=-0.13$ and ${b}_{2}=1.20$, the most recent estimates from Sosa & Fernández (2011).

Table 1.  List of the Empirical Comet Parameters that Relate Comet Nucleus Radius with Intrinsic Brightness using Equation (4)

Source	b1	b2
Kresak (1978)	−0.20	2.10
Bailey & Stagg (1988)	−0.17	1.90
Weissman (1996)	−0.13	1.86
Sosa & Fernández (2011)	−0.13	1.20
Asteroid with albedo 0.06	−0.20	3.73

Note. The influence on comet brightness from these parameters is given in Figure 2.

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After determining the absolute magnitude of the comets based on Equation (4), an additional term is needed to model how the intrinsic brightness of comets grows due to increased radiation from the Sun near perihelion. Following the standard in the comet community, we let the brightness vary as $1/{{\rm{\Delta }}}_{\mathrm{sun}}^{n}$ where n is an adjustable parameter called the photometric index (e.g., Sosa & Fernández 2011). When n = 2, this reduces to the case without comet brightening. Comets are typically modeled with two different values of n, one for the pre-perihelion approach and another for the post-perihelion orbit. We use a pre-perihelion n of 5.0 and a post-perihelion n of 3.5, standard for long period comets thought to originate from the Oort cloud, which are the best analog for ICs in this regard (Francis 2005). Large values of n correspond to steeper brightening functions. The photometric index is based on empirical observations of comets that are generally based on detections within $\lesssim 5\;{\rm{au}}$ and should be determined in conjunction with ${H}_{c},{b}_{1}$, and b₂ (e.g., Sosa & Fernández 2011). In our analysis, we allow H_c to be defined by the assumed nuclear radius (r) using Equation (4) independently of any correlation that may exist between $n,{b}_{1}$, and b₂. This affects the direct comparison to comet studies in the solar system, but is still a reasonable approximation in the face of other systematic errors and uncertainties in comet parameters. We also note here that the common comet absolute magnitude H₁₀ is equivalent to H_c under the assumption of a photometric index of n = 4, e.g., a brightness that depends on the Sun-comet distance to the fourth power.

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At distances far beyond Jupiter, the photometric index strongly penalizes the comet magnitude in an unrealistic way. Although it does not affect our results, we avoid this by requiring that the magnitude of an IC be always less than (e.g., brighter than) the magnitude of an interstellar asteroid with 6% albedo. Using this piecewise definition we ensure that a comet is never fainter than an asteroid of equivalent properties. A typical example of the brightness of a comet and asteroid in our model is shown in Figure 3.

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The full magnitude equation for comet brightening then becomes:

$$\begin{eqnarray}V & = & {H}_{c}+2.5\left[\displaystyle \frac{n}{2}\ {\mathrm{log}}_{10}({{\rm{\Delta }}}_{\mathrm{sun}}^{2})+{\mathrm{log}}_{10}({{\rm{\Delta }}}_{\mathrm{earth}}^{2})\right]\\ & & -2.5{\mathrm{log}}_{10}(\gamma )\end{eqnarray} \tag{ 5 }$$

which includes the most relevant effects. We do not consider intrinsic brightness variations due to rotational modulation or outbursts and we assume that any extended brightness due to a coma is smaller than the photometric aperture.

Using the above methods, we can take a specific population of ICs and determine their brightness at any time. The vast majority of ICs in the initial simulation are completely undetectable due to the large region of space modeled (10⁹ au³). To produce results that are robust from small number statistics, we use a Monte Carlo approach of simulating billions of ICs in order to ensure that a significant number (usually ≳100) of detectable ICs are generated. This necessitates the use of renormalization between simulated ICs and the actual proposed population of ICs, e.g., the orbital path of each simulated IC can represent a variable number of ICs that would actually be present, given the parameters of the simulation. We also calculate IC observations over a few thousand years and then average the non-transient portions of these observations in order to determine the rate of detectable ICs, even when this rate is very small.

The first step in narrowing down the large number of simulated ICs into those that are potentially detectable is an optimization routine that determines the minimum magnitude of a particular IC. We begin this routine with an initial guess that maximum brightness is at perihelion. In reality, there can be multiple maxima in brightness due to the orbital motion of the Earth. However, we determined that ICs that had a minimum magnitude of fainter than 28 in this first step will never be detectable by LSST.

The second step takes these potentially detectable ICs and computes their observational parameters on a much finer time grid that covers the possible range of observable times, based on the minimum magnitude found earlier. Every few hours, the brightness, solar elongation, airmass, and other parameters are determined. We consider an IC "detectable" or "observable" if it meets the following criteria during at least one timestep:

• 1.  
  IC magnitude less than the limiting magnitude
• 2.  
  Solar elevation less than −18° (end or beginning of astronomical twilight)
• 3.  
  IC airmass less than 2 as observed from Cerro Pachon (future site of LSST).

Note that the latter two effects automatically require the solar elongation to be greater than 48°. This is important since comet brightening is extremely and unrealistically enhanced when the Sun-IC distance is very small. We do not consider the effects of the Moon, the specific observing cadence, downtime, imaging fill factor, etc. In practice, we find that detectable ICs are often detectable over several days or weeks, so it is not likely that LSST will miss a substantial fraction of these objects due to these effects, as long as the above detectability criteria are met. We discuss the possible effects of trailing below, but otherwise assume that the survey is 100% efficient up to the limiting magnitude for simplicity. Note that the meaning of limiting magnitude in surveys is a 50% recovery rate, while our usage of the term implies a 100% recovery rate.

As discussed above, our nominal model uses standard (though uncertain and sometimes controversial) values for the input parameters, except for the size distribution, where we use the most optimistic case. Therefore, "nominal" should not be misconstrued as "best guess" and the nominal case is designed for comparison to M09. The specific values chosen are given in Table 2. It uses the mass density of M09 and the size distribution parameters (${q}_{1},{q}_{2},{r}_{b}$) that give the maximum number of small objects. We use the Sosa & Fernández (2011) radius–brightness relation and otherwise assume typical photometric parameters.

Table 2.  List of Input Parameters for the Simulation and Their Nominal Values

Parameter	Description	Nominal Value
mtotal	mass density of ICs	$4.5\times {10}^{26}$ g pc−3
q1	slope of the differential IC size distribution when $r\lt {r}_{b}$	3.5
q2	slope of the differential IC size distribution when $r\gt {r}_{b}$	5.0
rb	the break radius of the IC number density	3 km
rmin	minimum radius of detectable ICs	0.1 km
ρ	bulk density of IC nuclei	0.5 g cm−3
npre	pre-perihelion photometric index	5.0
npost	post-perihelion photometric index	3.5
b1	comet absolute magnitude parameter	−0.13
b2	comet absolute magnitude parameter	1.2
v0	velocity dispersion of ICs	30 km s−1
G	phase function steepness	0.15
p	albedo of asteroid	0.06
m	limiting magnitude	24.5
${{\rm{\Delta }}}_{\mathrm{earth}}$	minimum geocentric distance allowed	0 (au)

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Most comets seen in the inner solar system have a minimum nucleus radius of order 1 km. A common explanation for the lack of small comet nuclei is that these objects readily disintegrate and, effectively, evaporate, before they reach the inner solar system in a coherent form (Levison et al. 2002). ICs of this size may also be destroyed by the same mechanism in previous close stellar passages. Therefore, we consider it unrealistic to consider active ICs with nuclei sizes below 0.1 km, and choose our nominal model to have a minimum IC nucleus radius of ${r}_{\mathrm{min}}=0.1$ km. Even this is very optimistic, as comet nuclei smaller than ∼1 km may be far rarer than the size distribution would suggest. Based on this model, we expect LSST to be capable of observing on the of order 1 detectable IC over its 10 year lifetime (Table 3, Line 1). There is significant uncertainty and significant optimism in this result, but it is plausible for LSST to detect an IC. These conclusions are discussed further in Section 5.

Table 3.  The Number of ICs the LSST Could Observe in Its Lifetime for Various Different Cases as a Function of Limiting Magnitude

#	Label	Difference from Nominal Values			NLSST
			20.5	21.75	23	24.5	25.5	26.75
1	Nominal	none	0.15	0.23	0.35	0.57	0.85	1.2
2	Nominal without gravity	no gravity	⋯	⋯	⋯	0.51	⋯	⋯
3	Nominal ignoring the phase function	no phase function	⋯	⋯	⋯	0.95	⋯	⋯
4	Nominal using the comet density from M09	ρ = 1.5 g cm−3	⋯	⋯	⋯	0.20	⋯	⋯
5	Nominal excluding comets within 5 au	${{\rm{\Delta }}}_{\mathrm{earth}}\geqslant 5$	⋯	⋯	⋯	0.041	⋯	⋯
6	Moro-Martín	ρ = 1.5 g cm−3, ${n}_{\mathrm{pre}}=2,{n}_{\mathrm{post}}=2$,	⋯	⋯	⋯	0.0023	⋯	⋯
		${b}_{1}=-0.20,{b}_{2}=3.1236,{{\rm{\Delta }}}_{\mathrm{earth}}\geqslant 5$,
		$p=0.06$, no gravity, no phase function
7	McGlynn and Chapman mass density	${m}_{\mathrm{total}}=4.5\times {10}^{30}$ g pc−3	⋯	⋯	⋯	5900	⋯	⋯
8	Minimum size of 0.2 km	${r}_{\mathrm{min}}=0.2$	⋯	⋯	⋯	0.39	⋯	⋯
9	Minimum size of 0.5 km	${r}_{\mathrm{min}}=0.5$	⋯	⋯	⋯	0.22	⋯	⋯
10	Minimum size of 1 km	${r}_{\mathrm{min}}=1$	⋯	⋯	⋯	0.14	⋯	⋯
11	Realistic case	${q}_{1}=2.92,{r}_{\mathrm{min}}=1$ km	⋯	⋯	⋯	0.0012	⋯	⋯
12	Asteroid	${r}_{\mathrm{min}}=0.01$ km, ${n}_{\mathrm{pre}}=2,{n}_{\mathrm{post}}=2$,	⋯	⋯	⋯	0.94	2.9	6.8
		${b}_{1}=-0.20,{b}_{2}=3.1236,p=0.06$
13	Slow v0	${v}_{0}=5$ km s−1	0.15	0.23	0.35	0.64	0.86	1.2
14	Kresak comet parameters	${b}_{1}=-0.20,{b}_{2}=2.10$	0.21	0.33	0.55	1.0	1.42	2.1
15	Bailey & Stagg comet parameters	${b}_{1}=-0.17,{b}_{2}=1.90$	0.11	0.19	0.33	0.58	0.89	1.39
16	Weissman comet parameters	${b}_{1}=-0.13,{b}_{2}=1.86$	0.0045	0.023	0.079	0.20	0.37	0.76
17	Small number density	${q}_{1}=2.0,{q}_{2}=3.0$	0.00006	0.00010	0.00017	0.00029	0.00041	0.00060
18	Medium number density	${q}_{1}=2.5,{q}_{2}=3.5$	0.0012	0.0019	0.0031	0.0051	0.0075	0.011
19	Shallow phase function	G = 1	⋯	⋯	⋯	0.82	⋯	⋯

Note. N_LSST is the number of ICs the LSST could observe in 10 years at the listed limiting magnitude. Each row changes a few different parameters for different situations showing how N_LSST changes. Due to the Monte Carlo nature of our analysis, each value has approximately a ∼10% statistical uncertainty. The parameters and their nominal values are defined in Table 2. Note that the uncertainty in these parameters imply that the number of ICs detectable by LSST ranges by orders of magnitude.

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Although we cannot exactly reproduce the conditions used in the analytical estimates of M09, we can approximate this model by using equivalent parameters: turning off the mass of the Sun (which removes gravitational focusing), turning off comet brightening, turning off the phase function, and only considering ICs that are at least 5 au away from Earth. We still enforce the airmass and solar elevation constraints (a reduction of a factor of ∼3). This yields a detection frequency of ${N}_{\mathrm{LSST}}\simeq 0.002$ (Table 3, Line 6), similar to the estimate given in M09 (10⁻²–10⁻⁴).

Considering each of these factors individually (Lines 2–5), we find that the largest effect in the increase of N_LSST seen in our analysis is the inclusion of ICs that come closer to the Earth than 5 au. Not only does this makes ICs brighter because they are closer, but it makes objects that are smaller detectable. Since the size distribution is steep, decreasing the minimum size of detectable ICs creates almost a 10-fold increase in the number of expected detections. This can also be seen in the modest increase gained by decreasing the nuclear bulk density; since the mass density per unit volume is fixed, this has the effect of increasing the number of objects at a certain size (Line 4).

A large variety of tests show that the primary determinant of N_LSST is the number of objects at the smallest radii, unsurprising for a magnitude-limited survey of objects with steep size distributions. It also means that our results are strongly sensitive to parameters which affect the number density at the smallest radii. For a fixed number density, changing the size distribution slopes q₁ and q₂ strongly affects the number of objects at the smallest sizes (Figure 1) and N_LSST correlates with these changes. Similarly, increasing the total mass density of ICs to, for example, the overly optimistic estimate of McGlynn & Chapman (1989), increases the detection frequency significantly to ${N}_{\mathrm{LSST}}\approx 6000$.

We illustrate this point by changing the minimum detectable radius in our "nominal model" from 0.1 km to 0.2, 0.5, and 1 km in Lines 8–10, which decreases N_LSST from 0.57 to 0.39, 0.22, and 0.14, respectively. The use of different minimum radii mimics that of brighter limiting magnitudes.

A much more realistic case than the "nominal" values above draws from our understanding of solar system comets. This uses a differential nuclear radius distribution of ${q}_{1}=2.92$ from comets (Snodgrass et al. 2011) coupled with the fact that comets with sizes smaller than ∼1 km are unusually rare (e.g., ${r}_{\mathrm{min}}=1$ km). Although they are not as important, we also use ${q}_{2}=4.5$ and ${n}_{\mathrm{pre}}={n}_{\mathrm{post}}=4$ (to mitigate issues with the H_c–r–n distribution mentioned above). With these more realistic parameters, N_LSST becomes 10⁻³, much lower than the nominal case because of the less favorable (but more realistic) size distribution and cut-off. Using ${r}_{\mathrm{min}}=0.1$ would still only give ${N}_{\mathrm{LSST}}\simeq 0.005$.

Given the large uncertainties in the most important parameters, we have determined which parameters can be considered as minor effects. Based on our simulations, gravitational focusing, the specific photometric phase function, the chosen bulk nuclei density, the intrinsic IC velocity dispersion, the direction and velocity of the Sun's interstellar motion, and the different radius–brightness relations affect N_LSST at the factor of ≲2 level only (see Table 3). Figure 4 shows the effect of the different size–brightness relations for comets and asteroids (see Table 1), before observability criteria are enforced.

Depending on the parameters of planet formation, both rocky and icy bodies can be ejected into interstellar space (e.g., Weissman & Levison 1997; Shannon et al. 2015). If we consider the case of interstellar asteroids (or dead/inactive comets) that are not subject to disintegration, then we can probe to much smaller sizes where the objects are much more frequent (${r}_{\mathrm{min}}=0.01$). In this case, we move from a specific comet brightening law to using a particular albedo, nominally 0.06. However, we lose the advantage given by comet brightening. Considering the interstellar asteroid case down to a size of 10 m shows that these two effects roughly cancel with ${N}_{\mathrm{LSST}}\approx 0.9$. (Table 1, Line 12). Figure 4 shows how the comet radius–brightness relation and the asteroid case compare, though it is important to remember that these assume an overly optimistic value for the differential size distribution power law index. Note also that we are not considering a bimodal model of ICs, but rather assuming that the entire mass density is either in the comet or asteroid cases.

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As discussed below, interstellar asteroids of these sizes are moving very rapidly on the sky. Such objects will create a trail of significant length, even in LSST's short 15 s exposures. To account for this trailing, we adjusted the brightness of these objects down according to how long their trail would be compared to the expected FWHM of LSST detections (2''), effectively approximating a surface brightness of the trail as if it were a point source. This amounted to a small reduction of N_LSST down to 0.75, suggesting that trailing is just beginning to become important. An optimal detection algorithm would search for statistically significant trails, even if they fell below the point source detection limit (e.g., Vereš et al. 2012). In the case where ${r}_{\mathrm{min}}\approx 0.001$ km and ${N}_{\mathrm{LSST}}\approx 10$, such an algorithm would be essential, though there are unavoidable trailing losses that presumably mitigate the ability of detecting ∼1 m interstellar asteroids.

Changing the limiting magnitude modifies the typical distance at which the smallest ICs can be detected; the larger limiting magnitude of a deeper survey increases the volume where small ICs can be detected. In practice, the effect on N_LSST is not as strong as might be expected, due to the combination of actual IC motion with respect to the Earth and the Sun, comet brightening effects, and our requirement that ICs be greater than ${r}_{\mathrm{min}}=0.1$ km. Interestingly, shallower surveys have non-negligible sensitivity, suggesting that existing surveys can place interesting upper limits on the frequency of ICs, as seen in the work of Francis (2005), Meinke et al. (2004), and Engelhardt et al. (2014). However, keep in mind that N_LSST assumes a 10 year survey duration that is 100% efficient at finding any detectable IC brighter than the limiting magnitude, so existing ground-based surveys should be much less sensitive.

Another important consideration is that our frequency of detections assumes that the IC must be detected in a single image or exposure. It is possible that "shift-and-stack" techniques (e.g., Parker & Kavelaars 2010; Heinze et al. 2015) can use the same survey to effectively reach ∼2 mag deeper, which would certainly improve the detection rate as seen in Table 1.

The change in N_LSST as a function of magnitude suggests that it is better to spend survey time to go "wide" than it is to go deeper, all else being equal. This is partly due to the fact that the IC population is continuously replenishing due to solar motion through the Galaxy. However, the rarity of ICs means that even a survey wide enough to cover the whole sky every three days will still need to go deep in order to have a clear chance of detecting ICs. It also suggests that LSST's use of "Deep Drilling Fields" which are more heavily observed to reach a deeper co-added magnitude, are not likely to help unless combined with a shift-and-stack technique.

The asteroid case is much more sensitive as a function of magnitude. Very deep searches that are sensitive below ${r}_{\mathrm{min}}=0.01$ km could significantly benefit from going deeper, but would have to deal with larger amounts of trailing. A new technique which combines high-speed cameras with the shift-and-stack technique has successfully discovered Near Earth Objects (which we show below would have similar rates of motion as ICs would have) as small as ∼8 m (Shao et al. 2014; Zhai et al. 2014). However, as ICs are incredibly more sparse than NEOs, this method would have to be scaled up by orders of magnitude before ICs would be detected.

Through efficient programming and data control techniques, we were able to complete a full simulation of billions of ICs in, typically, several minutes, allowing for the computation of many possible models (Table 3). So that future studies can utilize and extrapolate our results, we have run several models with large variations in one or more parameters in order to see how N_LSST changes. We have found that the following multi-linear model gives an accurate estimate. However, we again caution that the true values of most relevant parameters are not well known and that the prediction of the number of ICs detectable by LSST or other surveys ranges over orders of magnitude.

Let $\mathrm{log}N=\mu ({\boldsymbol{\theta }})\times \mathrm{log}(r)+\beta ({\boldsymbol{\theta }})$ where N is the (differential, not cumulative) number of detectable ICs per year at V = 24.5 where r is the nuclear radius of the ICs in km. To go from N to N_LSST requires multiplying by 10 (due to the 10 year baseline) and removing objects that are not observable (e.g., airmass less than 2, solar elevation less than −18°), which is a factor of 10^0.5–2 depending on the specific populations.

At the small radius end of the distribution, the relationship between $\mathrm{log}N$ and $\mathrm{log}r$ is approximately linear (e.g., Figure 4). In that regime, we can describe the relationship between them using a slope and y-intercept, μ, and β, which are each themselves linear functions of the parameters ${\boldsymbol{\theta }}$. We describe the values of μ and β as a linear combination of these parameters, ${\boldsymbol{\theta }}=({q}_{1},{q}_{2},{b}_{1},{b}_{2},{v}_{0},\rho ,{n}_{\mathrm{pre}},{n}_{\mathrm{post}},$ and ${\mathrm{log}}_{10}({m}_{\mathrm{total}}))$ (in the units given in Table 2), plus a constant. The values of β are somewhat representative of the importance of this parameter, e.g., q₁ is much more important than v₀. Note that r_min is not included because the result gives N as a function of size, which can then be evaluated at the desired small size cutoff.

The coefficients of each of the parameters in ${\boldsymbol{\theta }}$ in this linear model are given in Table 4. They are obtained by performing a least squares fit for different runs of the simulation. A wide range of values for each parameter is used including varying multiple parameters at once, so as to make the model applicable to values beyond what is listed in Table 3. Although clearly not all of the figures listed are significant, using these values yields a correlation between this multi-linear approximation and the full model simulation with a high Pearson correlation coefficient (${R}^{2}\gt 0.95$).

Table 4.  Coefficients Needed for Extrapolation to Other Values of the Simulation Parameters

	μ	β
Constant	2.758	−30.399
q1	−0.2364	3.0801
q2	−0.4544	−1.167
b1	−0.28	−2.17
b2	−0.365	−0.802
v0	0.0029	0.0043
ρ	0.38	0.054
npre	0.0978	0.275
npost	−0.48	−1.166
${\mathrm{log}}_{10}({m}_{\mathrm{total}})$	0.0022	0.987

Note. The frequency of IC detections per year above a brightness of V = 24.5 (N) can be given by a power law for small values of the comet nucleus radius (r), as seen in Figure 4. We can then define $\mathrm{log}N=\mu ({\boldsymbol{\theta }})\times \mathrm{log}(r)+\beta ({\boldsymbol{\theta }})$ where μ, and β are each themselves linear functions of the parameters ${q}_{1},{q}_{2},{b}_{1},{b}_{2},{v}_{0},\rho ,{n}_{\mathrm{pre}},{n}_{\mathrm{post}},$ and ${\mathrm{log}}_{10}({m}_{\mathrm{total}}))$ (in the units given in Table 2), plus a constant. The number of significant figures displayed helps to ensure an accurate estimate. The values of β, which are related to the normalization of the power law, are somewhat representative of the importance of this parameter, e.g., the differential size distribution slope for small values (q₁) is much more important than the velocity dispersion (v₀). Note that r_min is not included because the result gives N as a function of size, which can then be evaluated at the desired small size cutoff.

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To illustrate, we can very unrealistically set each of these values equal to 10, which would give $\mu =2.758-2.364-4.544-2.8...\;+\;0.022=-10.571$ and $\beta =-30.399+30.801-11.67...\;+\;9.87=-39.445$ which implies that $N={10}^{-39.445}{r}^{-10.571}$, which can then be related to N_LSST as discussed above.

Using these results, future studies should be able to generate their own expectations for the number of detectable ICs by large scale surveys covering a wide range of observational and theoretical parameter space.

We have carefully created a model of the most detectable ICs under realistic observing conditions. Here, we present the orbital parameters of a representative sample of the most detectable ICs and interstellar asteroids. Table 5 shows these parameters for 20 ICs and 20 interstellar asteroids drawn from the nominal case.

Table 5.  Orbital Properties of Typical Detectable Interstellar Comets (First 20 Entries) and Interstellar Asteroids (Last 20 Entries)

Radius (km)	a (au)	e	q (au)	i (°)	Ω (°)	ω (°)	${v}_{\infty }$ (km s−1)
0.11	−0.72625	2.4336	1.0412	60.728	109.24	331.18	34.86
0.18	−6.1602	1.1571	0.96798	165.16	17.084	279.55	11.97
0.21	−0.69227	5.5729	3.1657	121.13	307.16	167.01	35.70
0.11	−4.8327	1.2481	1.1988	35.800	132.42	63.724	13.51
0.12	−0.99266	2.2125	1.2036	83.129	355.76	118.45	29.81
0.11	−0.43832	4.1940	1.4000	98.972	265.91	142.86	44.87
0.12	−1.7647	1.7781	1.3747	55.182	8.6212	331.63	22.36
0.20	−0.51516	4.8583	1.9876	49.244	293.03	125.17	41.39
0.12	−0.51014	5.1957	2.1404	24.874	267.36	140.49	41.59
0.11	−0.41419	3.8027	1.1608	51.671	293.28	115.89	46.16
0.10	−1.4211	1.1802	0.25631	35.660	227.27	120.63	24.91
0.14	−0.57002	3.4843	1.4161	50.452	326.48	103.23	39.34
0.11	−0.42728	4.7306	1.5940	136.89	310.14	111.82	45.44
0.39	−0.38712	9.1568	3.1577	124.25	309.47	125.98	47.74
0.43	−1.1024	3.5955	2.8612	76.358	221.06	56.088	28.29
0.12	−0.99311	1.5962	0.59213	106.96	217.56	106.32	29.81
0.11	−1.9871	1.1967	0.39085	20.689	204.23	15.098	21.07
0.15	−0.46875	3.5816	1.2101	67.981	80.129	278.54	43.39
0.11	−1.5640	1.0915	0.14310	105.47	78.959	217.24	23.75
0.11	−0.39364	6.9736	2.3515	22.201	207.98	163.29	47.34
0.032	−0.68023	1.9241	0.62862	124.65	156.97	258.53	36.02
0.069	−1.6503	1.6707	1.1068	75.355	7.4526	75.896	23.12
0.015	−3.5974	1.0529	0.19034	68.521	54.221	124.26	15.66
0.017	−0.95701	1.0238	0.022733	6.1824	359.55	229.32	30.36
0.041	−0.53558	2.2423	0.66536	167.17	258.52	77.446	40.59
0.053	−0.70879	2.1711	0.83010	108.78	315.23	149.74	35.28
0.11	−4.1081	1.4083	1.6775	19.809	170.96	268.51	14.66
0.034	−0.42086	2.6194	0.68155	114.89	304.18	117.75	45.79
0.012	−0.90621	1.5180	0.46943	86.207	249.46	94.273	31.20
0.033	−1.0996	1.4696	0.51641	109.64	209.49	81.180	28.33
0.018	−0.45199	2.3399	0.60561	94.492	296.68	110.37	44.18
0.050	−1.0242	1.0939	0.096220	166.45	292.78	97.257	29.35
0.016	−1.5610	1.4506	0.70345	169.23	16.931	136.11	23.77
0.011	−0.74861	1.3012	0.22552	121.60	118.92	250.08	34.33
0.028	−2.7367	1.0649	0.17765	90.813	27.294	133.56	17.96
0.016	−5.6296	1.1988	1.1192	120.03	342.69	2.0826	12.52
0.017	−6.2956	1.0560	0.35259	137.30	37.675	107.71	11.84
0.11	−0.50025	3.9496	1.4755	167.67	243.17	63.798	41.91
0.018	−4.3754	1.0949	0.41516	83.139	291.61	273.74	14.20
0.29	−1.0272	2.6104	1.6542	39.742	280.81	157.18	29.31

Note. Standard hyperbolic orbital elements in the ecliptic heliocentric reference frame are used: a is semimajor axis, e is eccentricity, i is inclination relative to the ecliptic, Ω is the longitude of the ascending node, ω is the argument of periapse, and ${v}_{\infty }$ is the excess velocity at infinity.

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It also serves as an ideal test population for future IC detection algorithms, similar in spirit to the Pan-STARRS Synthetic Solar System, which also contained an estimated interstellar object population (Grav et al. 2011).

In Table 5, we show the radius for the IC from our simulation; this is a randomly selected list of typical detectable IC properties from a Monte Carlo run that simulated a large population of ICs. As discussed above, the most detectable ICs typically have sizes just larger than the minimum detectable radius r_min (here 0.1 km), since these are most abundant.

Detected ICs have perihelia near the Earth's orbit, but detected interstellar asteroids tend to have perihelia less than 1 au. The distribution of orbital angles is effectively isotropic. ICs with lower incoming velocities are slightly favored, as expected from the enhancement due to gravitational focusing. However, they always have significant excess velocities, so will be clearly distinguishable from near-parabolic comets with ${v}_{\infty }\lessapprox 0$. This reinforces our caution of interpreting C/2007 W1 (Boattini) as a true IC (Section 2.2).

As input to future observational efforts, we have taken the set of the most detectable ICs shown in Table 5 and studied their astrometry for the purpose of orbit inversion. We estimate the amount of observational data required to determine that an object is an IC and to secure its orbit so that follow-up observations can be obtained.

To set the stage, Figure 5 shows the rate of motion of ICs the sky as a function of apparent brightness. ICs have rapid apparent motion, with typical rates of hundreds of arcseconds per hour, comparable to near-Earth objects (NEOs).

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First, we generate LSST-like synthetic astrometry for these ICs. Then the resulting astrometry is fed into a statistical orbit computation algorithm which provides an orbital solution. Finally we compute ephemerides based on the orbital solution. The software tools (less the shell scripts) are available in the OpenOrb package (Granvik et al. 2009).⁹ This code is different from the one developed to identify detectable ICs. Due to the difference in Earth's orbital position and a few other small details, not all of the ICs in Table 5 are detected (at our magnitude cutoff of V = 24.5) in these astrometric simulations, but this does not affect the results. This completely separate code also finds most of the ICs in Table 5 as detectable, partially validating our simulations.

For determining the astrometric orbits, we use the two-body approximation, no non-gravitational effects, JPL's DE405 planetary ephemerides, and a geocentric observer (as opposed to the topocentric observer used above). These approximations do not affect the astrometric results.

Synthetic astrometry is generated by propagating the ICs through their perihelion passage and recording their (R.A., decl.) coordinates twice with an interval of about 15 minutes every three days if the apparent V-magnitude $V\lt 24.5$, the solar elongation ${\epsilon }_{\odot }\gt 45^\circ$, and the lunar elongation ${\epsilon }_{\mathrm{Moon}}\gt 45^\circ$. Random Gaussian noise with σ = 01 is added to the coordinates to mimic astrometric uncertainty, though this may be an overestimate for a fully calibrated LSST.

We compute an orbital solution using the statistical ranging method (Virtanen et al. 2001; Granvik & Muinonen 2005) for each night that an object is detected twice. The process thus resembles the operation of a real survey where the orbital uncertainty of a given object gradually diminishes as more astrometry is obtained. The ranging method provides the full nonlinear orbital-element probability-density function (PDF) based on the synthetic astrometry—in practice a cloud of weighted orbital solutions that reproduce the synthetic astrometry within the limits set by the astrometric uncertainty. Using the PDF we assess whether elliptical solutions can be ruled out based on the synthetic astrometry.

The most obvious characteristic that separates an IC from its solar-system counterparts is its hyperbolic orbit with respect to the Sun. The typically high inclinations of ICs (Table 5) could also be utilized but this would possibly lead to a confusion as high inclinations and even retrograde orbits are known to exist for both near-Earth comets and near-Earth asteroids originating in the solar system (Greenstreet et al. 2012). To unambiguously determine whether an object is an IC we use the criterion that the minimum eccentricity within the orbital-element credibility region has to be greater than unity. In what follows, the credibility region encompasses 99.73% of an orbital solution's total probability mass thus corresponding to the 3σ limit of a one-dimensional Gaussian distribution. In astrometric analyses of solar system bodies, we have found that single night arcs can always be fit with a hyperbolic orbit model, therefore, we require at least 2 pairs of detections spaced by 3 nights (in our mock LSST-like cadence) before considering a hyperbolic detection secure. It is interesting to note that while all solar system objects appear to be moving retrograde at opposition due to the Earth's faster orbital velocity, ICs are an exception to this rule. Even at opposition, they can have a prograde motion, as a result of their excess velocity.

Each IC is typically detected tens or even hundreds of times during its perihelion passage. They are typically first "discovered" in the inner solar system at a heliocentric distance of ∼6 au. The minimum and median distance to a detectable IC are ∼1.5 and ∼3 au, respectively. If there were no lower limit on the size of an IC, i.e., if we allowed ${r}_{\mathrm{min}}\lesssim 0.1$ km, then we would detect a larger population of smaller bodies closer to the Earth (see the astrometric results for the asteroid case below).

Most ICs require 2–3 independent nights of observations during a timespan of 4–7 days to ensure that a hyperbolic orbit is the only viable solution. This rapid identification is due to the high rates of motion seen for typical ICs (∼200 arcsec hr⁻¹ or ∼1° day⁻¹), allowing for precise astrometric constraints over a short observational arc. The rapid motion is partially due to the higher orbital velocity experienced by unbound objects, but is also strongly due to the fact that a magnitude-limited survey will identify nearby objects that will have rapid apparent motion due to parallax from the Earth's motion. When the data are insufficient for a strong classification, ICs are usually confused with Aethra asteroids (Mars crossers).

In order to facilitate follow-up observations we compute the maximum sky-plane uncertainty 3 days and 14 days after the last detection. We assume that, e.g., photometric, spectrometric and polarimetric follow-up observations are feasible when the ephemeris uncertainty is below 30'' 3 days after the last observation. When this limit has been reached it essentially guarantees that the uncertainty does not grow with time as the ongoing survey provides additional astrometry every 3 days. Similarly, we assume that Target-of-opportunity-type observations can be planned when the ephemeris uncertainty is less than one degree within 2 weeks after the last detection. Most objects require 3 nights of observations spanning 7 days for the above criteria on the ephemeris uncertainty to be fulfilled. That is, the timeframe for establishing that an object is an IC is the same as the time frame for establishing its orbit sufficient for follow-up.

The ephemeris uncertainty drops quickly with increasing astrometry: at the time of fulfilling the above criteria the typical 3 day ephemeris uncertainty is only a few arcseconds, a few tens of arcseconds for the 14 day ephemeris uncertainty, and less than 0.2° 30 days into the future.

4.9.1. Asteroid Model

Throughout the analysis thus far, we have been considering an IC population that is completely isotropic. However, it is possible that the population of ICs is more heterogeneous. If we were passing through the Oort cloud of another star, for example, the IC density would go up significantly (Stern 1987). Recent stellar passages may be effective at temporarily stripping the Oort clouds of other stars, eventually resulting in an anisotropic IC source (e.g., Zheng & Valtonen 1999).

ICs from individual systems are ejected primarily at inclinations less than ∼30° with respect to the invariable plane (e.g., Duncan et al. 1989). However, these planes are oriented isotropically, so that the IC "luminosity" of a planet forming system is similar (proportional to the inverse square of distance) to conventional photon luminosity. Throughout, we have assumed that we are searching for ICs from, effectively, the "diffuse IC background." Here we consider the possibility of a single dominant discrete source of ICs.

As pointed out in Section 2.3, interstellar micrometeorites appear to have a discrete source possibly associated with edge-on debris-disk and planet hosting star β Pictoris (Baggaley 2000; Murray et al. 2004). A similar source for ICs is plausible and may significantly enhance their frequency over the estimates of M09.

As one of the nearest forming stars (19.4 pc), β Pic has been extensively studied. Since the discovery of micrometeories, observers have detected a collisionally active multi-component debris disk (Golimowski et al. 2006; de Vries et al. 2012) and a directly imaged planet (Lagrange et al. 2010; Chauvin et al. 2012) that was earlier predicted by theorists (e.g., Freistetter et al. 2007). There is every reason to expect that β Pic is actively ejecting ICs, some of which are currently passing through our solar system; the larger brethren of the already detected interstellar micrometeorites. (Its systemic radial velocity is ∼20 km s⁻¹ and therefore cannot be a source of C/2007 W1, which was verified by direct backwards integration (P. A. Dybczyski 2016, personal communication).)

To investigate this possibility, we considered a model where the ICs had no intrinsic velocity dispersion and thus come streaming in due only to the Sun's relative motion. This simulates what a single population of ICs from a discrete source may look like. Gravitational focusing can cause enhancements of interstellar particles at the antapex of the Sun's motion, and we did detect a weak spatial clustering of IC detections in this case. (Our nominal model showed no spatial clustering, as expected when the velocity dispersion of the ICs is larger or comparable to the solar velocity.)

The lower relative velocity also increases the importance of gravitational focusing and results in a slightly larger number of detectable objects to N_LSST of 0.61, all else being equal. A "high IC luminosity" discrete source may also enhance the mass density of ICs, which is not included here, but could easily be a significant effect.

It is tantalizing to note that if an IC is detected and its orbit recovered, backwards integration over several million years could reveal a very specific location for its original source, potentially identifying it as a planetesimal from a specific system like β Pic (Dybczyński & Królikowska 2015). Note that β Pic is here used as an example; the actual IC population may come from other discrete sources, including, potentially, multiple sources.