Detecting Interstellar Objects Through Stellar Occultations

Draft version February 12, 2020 Typeset using LATEX twocolumn style in AASTeX62

Amir Siraj1 and Abraham Loeb1

1Department of Astronomy, Harvard University, 60 Garden Street, Cambridge, MA 02138, USA

ABSTRACT Stellar occultations have been used to search for Kuiper Belt and Oort Cloud objects. We propose a search for interstellar objects based on the characteristic durations (∼ 0.1s) of their stellar occultation signals and high inclination relative to the ecliptic plane. An all-sky monitoring program of all ∼ 7×106 stars with R . 12.5 using 1-m telescopes with 0.1 s cadences is predicted to discover ∼ 1 interstellar object per year.

Keywords: Minor planets, asteroids: general – meteorites, meteors, meteoroids

1. INTRODUCTION The data from such a search would calibrate popu- ‘Oumuamua was the first interstellar object (ISO) re- lation parameters for ISOs crucial for constraining for- ported in the Solar System (Meech et al. 2017; Micheli mation theories of exoplanetary systems (Duncan et al. et al. 2018). Follow-up studies of ‘Oumuamua were con- 1987; Charnoz & Morbidelli 2003; Veras et al. 2011, ducted to better understand its origin and composition 2014; Pfalzner et al. 2015; Do et al. 2018; Raymond et al. (Bannister et al. 2017; Gaidos et al. 2017; Jewitt et al. 2018; Hands et al. 2019; Pfalzner & Bannister 2019; Siraj 2017; Mamajek 2017; Ye et al. 2017; Bolin et al. 2017; & Loeb 2019d). Below we derive the physical character- Fitzsimmons et al. 2018; Trilling et al. 2018; Bialy & istics and rate of the expected ISO occultation events. Loeb 2018; Hoang et al. 2018; Siraj & Loeb 2019a,b; 2. THEORY Seligman et al. 2019; Sekanina 2019). ‘Oumuamua’s size The measured intensity at wavelength λ from a diffrac- was estimated to be 200 m, based on Spitzer Space . tion pattern created by a spherical object of radius R Telescope constraints on its infrared emission given its L at a distance D is, expected surface temperature based on its orbit (Trilling L et al. 2018). In addition to ‘Oumuamua, CNEOS 2014-01-08 (Siraj  2 2 U0 (ρ, t/tF ) + U1 (ρ, t/tF ), t/tF ≤ ρ, & Loeb 2019a) was tentatively the first interstellar me-   2 2 1 + U1 (ρ, t/tF ) + U2 (ρ, t/tF ) teor discovered larger than dust, and 2I/Borisov (Guzik Iρ(t) = et al. 2019) was the first confirmed interstellar comet. −2U (ρ, t/t ) sin π (ρ2 + (t/t )2)  1 F 2 F Transient diffraction patterns caused by the occul-  +2U (ρ, t/t ) cos π (ρ2 + (t/t )2), t/t ≥ ρ, tation of a distant star due to an intervening small 2 F 2 F F (1) body have been proposed and used to search for Kuiper where ρ = (R /F ) is the radius of the object in Belt objects (KBOs) and Oort Cloud objects (OCOs) L units of the Fresnel scale F = pλD /2, and t is in the Solar System (Bailey 1976; Dyson 1992; Roques L F the Fresnel scale crossing time for the object (Nihei arXiv:2001.02681v2 [astro-ph.EP] 11 Feb 2020 & Moncuquet 2000; Nihei et al. 2007; Schlichting et al. et al. 2007; Roques et al. 1987). For a transverse 2009, 2012; Arimatsu et al. 2019). Here, we propose an speed, v , at a distance D , the Fresnel time is t = analagous search for interstellar objects (ISOs), flagged ⊥ L F 0.12s (D /20 AU)1/2(v /10 km s−1)−1(λ/µm)1/2 and by their unusual inclinations and unique kinematics L ⊥ the Fresnel scale is F = 1.2km (D /20 AU)1/2(λ/µm)1/2. leading a distribution of characteristic distribution of L The Lommel functions are defined as, durations, the peak of which lies between that of Kuiper ∞ belt objects and that of Oort cloud objects. X µn+2k U (µ, ν) = (−1)k J (πµν) , (2) n ν n+2k k=0 [email protected], [email protected] where Jn is a Bessel function of order n. With regard to the impact parameter b, we assume b = 0 for sim- 2 Siraj & Loeb

= 0.22 = 0.32

1.2 1.10

1.1 1.05 ) )

t t 1.0

( 1.00 ( I I

0.95 0.9

0.90 0.8

0.85 = 0.66 = 3 1.2 1.4

1.2 1.0 1.0

0.8 0.8 ) ) t t ( ( I I 0.6 0.6 0.4

0.2 0.4 0.0 4 2 0 2 4 4 2 0 2 4

Figure 1. Occultation lightcurves for the values of ρmin (ratio of ISO radius to the Fresnel scale) in Table1 (dotted green lines), and sampled lightcurves at temporal resolution, tF (solid red lines). For a transverse speed v⊥ at a distance DL, the 1/2 −1 −1 time unit, tF = 0.1s (DL/20 AU) (v⊥/10 km s ) . plicity. Figure1 shows lightcurves for different values RS and distance DL, the three conditions for an ISO of ρ, as well as lightcurves sampled as at a spatial reso- occultation event are expressed as follows: lution of F . While the calculated lightcurves represent r monochromatic light, the finite filters used in practice λD R > ρ L , (3) slightly reduces the statistical power. L min 2 For a solar-type star with an R-magnitude of 12, the 5 −2 −1 R R flux of R-band photons at Earth is ∼ 2.8×10 m s . L > S , (4) For an intensity dip caused by an occultation event, a DL DS 2 signal-to-noise ratio of & 10 for a telescope with 1 m R < 10 km . (5) collecting area with a temporal resolution of tf would L require the intensity dip to be & 6%, corresponding to Figure2 illustrates a parameter space in which all three ρ & 0.22. The appropriate values of ρmin as a function conditions are satisfied. of magnitude are listed in Table1. We approximate the cumulative number density of In addition to ρ ≥ ρmin for a strong signal, we also ISOs as a power law with exponent -3.3, calibrated by require the occulting object to subtend a larger angular −3 the number density (nO ∼ 0.2 AU ) of ‘Oumuamua size on the sky than the star to avoid dilution of the sig- size (RO ∼ 100 m) objects (Siraj & Loeb 2019c; Do nal by the finite source size. We adopt the conservative et al. 2018; Landgraf et al. 2000). condition RL ≤ 10 km for ISOs. Given a star of radius We adopt the three–dimensional velocity dispersion of stars in the thin disk of the Milky Way as a proxy for the kinematics of ISOs, each corresponding to the Detecting Interstellar Objects Through Stellar Occultations 3

4 ) m

/ 3 L R ( 0 1

g 2 o l

RL = ( F), = 0.22 1 (RL/DL) = (RS/DS) RL = 10 km

0 1 2 3 log10(DL/AU)

Figure 2. Parameter space in which conditions (3), (4), and (5), are satisﬁed for a R = 12 solar-type star. RS ∼ R and DS ∼ 300 pc for an R = 12 solar-type star.

Table 1. Background photon ﬂux, maximum dip in inten- where the limits of the integrals, L1 and L2, are deﬁned sity, and minimum value of ρ for occultation events with as follows: solar-type background stars, assuming a 1 m2 collecting area and a temporal resolution of tf .  ρ2 λD2 ρ2 λD2 R Magnitude R-Band Photon Flux (m−2s−1) ∆I (%) ρ min S min S RmaxDS min min  2 , 2 ≤ , 5 2R 2R RS 12 2.8 × 10 6 0.22 L1 = S S (7) 2R2 ρ2 λD2 14 4.4 × 104 15 0.32 max min S RmaxDS  ρ2 λ , 2R2 ≥ R , 16 6.9 × 103 38 0.66 min S S 18 1.1 × 103 95 3

 2 2 RmaxDS ρminλDS RmaxDS , 2 ≤ ,  RS 2R RS standard deviation of a Gaussian distribution about the L2 = S (8) ρ2 λD2 −1 min S RmaxDS local standard of rest (LSR): σ = 35 km s , σ = 0, 2 ≥ . x y 2RS RS 25 km s−1, σ = 25 km s−1 (Bland-Hawthorn & Ger- z The resulting ISO occultation rate per star as a func- hard 2016). The resulting distribution of observed tran- tion of magnitude is shown in Figure4, and corresponds sit speeds (once the motion of the Earth is subtracted) to ∼ 0.1 Myr−1 at R = 12. has a mean value ofv ¯ ∼ 40 km s−1. obs To determine how easily discernable ISO, KBO, and The rate of occultations per star is given by, OCO occultation signatures are from each other, and to understand the distribution of timescales on which ISO occultations occur, we numerically simulated the distribution of expected Fresnel distance crossing times for each population, assuming a R = 12 solar-type star. 2 We draw distances from the distribution P (DL) ∝ DL with bounds of 1 AU to (R D /R ) for ISOs, bounds ˙ max S S NO,? ≈ of 30 to 50 AU for KBOs, and bounds of 2 × 103 to  !−3.3  5 Z L1 p −3.3 10 AU for OCOs. We draw velocities from the afore- ρmin λDL/2 Rmax 2π nO − mentioned kinematics for ISOs; we compute the or-  R R  1 AU O O bital speed corresponding the distance for KBOs and v¯obsDLRS for OCOs. We do not consider gravitational focusing dDL + DS and acceleration by the Sun for ISOs as these eﬀects " −3.3 −3.3# would only be signiﬁcant at D 1 AU. We sample Z L2 R D R L . 2π n S L − max random points in the Earth’s orbit to obtain the motion O D R R L1 S O O of the observer relative to the object. We draw sizes for v¯obsDLRS each population from a power law distribution with in- dDL , DS dex ∼ −3, with the appropriate bounds for each popula- (6) tion. If conditions (4) and (5) are satisifed, we compute 4 Siraj & Loeb

0 ) 1

r 1 y M

0 2 1 / ,

O 3 N ( 0 1

g 4 o l

5 12 14 16 18 R

Figure 3. ISO occultation rate per star as a function of R magnitude for solar-type stars.

0 ) 1 r

y 1 M 0 1 / 2 , O N ( 0

1 3 g o l

2 1 0 1 2 log10(R/R )

Figure 4. ISO occultation rate per star as a function of stellar radius, for a R = 12 star. the Fresnel distance crossing time, tF = (F/v⊥). Oth- resolutions of 0.1 s should yield a discovery rate of ∼ 1 erwise, we re-draw distance and size. The resulting dis- interstellar object per year. This would be a significant tributions of tF are shown in Figure5. ISO occultation improvement on the current discovery rate of an ISO ev- events have have an even distribution with distance be- ery few years. Our method supplements direct detection tween 1 AU and ∼ 103 AU. We find that the timescales through reflected sunlight but nearby events could ben- for ISO and OCO occultation events are distinct. The efit from both methods of detection. By measuring each ISO timescale distribution peaks at tF = 0.1 s, and a occultation in two or three colors, the radius and dis- survey can avoid KBOs by pointing away from the plane tance of the occulting object can be constrained (Dyson of the ecliptic. 1992). The data from such a survey would provide invalu- 3. CONCLUSIONS able new information on the size distribution, compo- 6 There are ∼ 7×10 stars in a with magnitude R . 12.5 sition, and possible origin of ISOs. Such information (McMillan 2018). An all-sky network of 1-m telescopes would be particularly valuable given the puzzle of the continuously monitoring all R = 12 stars with a time first two confirmed interstellar objects, ‘Oumuamua and Detecting Interstellar Objects Through Stellar Occultations 5

1.0 ) F t (

P 0.5

0.0 3 2 1 0 1 2 log10(tF/s)

Figure 5. Expected probability distribution of Fresnel crossing times during stellar occultations observed from Earth for ISOs (blue), KBOs (green), and OCOs (orange), for a R = 12 solar-type star. Borisov, having such diﬀerent physical characteristics high implied abundance of ISOs relative to previous pre- (Meech et al. 2017; Guzik et al. 2019), and given the dictions (Moro-Martin 2009).

ACKNOWLEDGEMENTS This work was supported in part by a grant from the Breakthrough Prize Foundation.

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