Setting an Observational Upper Limit to the Number Density of Interstellar Objects

8/13/2019 Setting an Observational Upper Limit to the Number Density of Interstellar Objects

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Earth Oriented Space Science and Technology

Masters Thesis

Setting an Observational Upper Limit to theNumber Density of Interstellar Objects

in collaborationwith the

by

Toni [email protected]

- Supervisors -Dr. Robert Jedicke (IfA), Prof. Urs Hugentobler (TUM)

January 31 st , 2014


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Abstract

Over the course of this work three major accomplishments could be booked. With the aidof Pan-STARRS 1 data comprising 27 month of observations from three large asteroid-and comet surveys (3 , Medium Deep Field and Solar System Survey) the main goal couldbe completed to rst ever set an observational upper limit to the number density of In-terstellar Objects (ISOs) based on actual telescope pointings (elds), system efficiency and

the fact of non-detection. The 90% Poisson Condence of the limit versus slope parameter and limiting absolute magnitude H of the Size Frequency Distribution (SFD) has beencomputed and plotted for inert objects (no activity) as well as the case of cometary activitysimilar to Oort cloud comets.

For this task a sophisticated ISO model was developed. It includes gravitational focusingeffects and maintains correctness over a 10 year timeframe starting January 1st, 2005,covering the entire Pan-STARRS survey as well as the Catalina Sky Survey (CSS), to datethe two most successful experiments regarding asteroid- and comet discoveries. The modelwas validated in a detailed analysis of velocity, energy and orbital element distribution of the synthetic objects. It could be shown that ISO orbits are more likely to have high orbitalinclination and that the eccentricity distribution peaks at higher values for larger accessiblevolumes. Findings like these help to improve search patterns for future surveys dedicatedto ISO discovery.

Furthermore, the efficiency of the current conguration of the Pan-STARRS systemwas determined with which it recognises objects in exposures depending on their apparentmagnitude. Efficiencies were assessed for each of the 6 spectral lters on 3 levels of processingwithin Pan-STARRS Moving Object Processing system (MOPS). These tools were used toconduct a simulation replicating the Pan-STARRS survey and determine the volume thathas been efficiently searched for ISOs over the telescope lifetime. With the assumptionof inert objects it amounts to only 42 .9AU 3 while the inclusion of Oort cloud comet-likeactivity yields an observed volume of 1 , 411AU 3, using a slope parameter of 0.5 and alimiting apparent magnitude H of 19 ( 1km diameter) for the SFD. The numbers can bedirectly converted to an upper limit for the ISO number density of 5 .4

10 02AU 3 for

inert objects and 1 .6 10 03AU 3 respectively for activite ISOs.These results dismantle expectations of a much tighter limit to be set by Pan-STARRS,encouraged by previous estimations on the number density limit with less powerful telescopesurveys like for instance the Lincoln Near-Earth Asteroid Research (LINEAR) . The directconclusion is that the capability of state-of-the-art telescopes to discover ISOs was over-estimated in the past.


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Contents

1 Introduction 1

2 Interstellar Objects (ISOs) 32.1 Distribution & Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Optical Properties and Activity . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 ISO Number Density Estimates . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Pan-STARRS 1 (PS1) telescope 113.1 Gigapixel Camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Fill Factor F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Photometric System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4 PS1 Surveys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Moving Object Processing System (MOPS) 174.1 Image Processing Pipeline (IPP) . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Detections, Tracklets & Derived Objects . . . . . . . . . . . . . . . . . . . . 18

4.3 Digest Score . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.4 System Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.5 Pan-STARRS Survey Simulations . . . . . . . . . . . . . . . . . . . . . . . . 27

5 Interstellar Object Model 295.1 Object Generation and Propagation . . . . . . . . . . . . . . . . . . . . . . 295.2 Orbit Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.3 ISO Model Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6 Observational Number Density Limit for ISOs 416.1 Poisson Statistics of a Non-Detection . . . . . . . . . . . . . . . . . . . . . . 426.2 Determination of

V with a MOPS simulation . . . . . . . . . . . . . . . . . 44

6.3 Assignment of a Size Frequency Distribution (SFD) . . . . . . . . . . . . . . 466.4 Pre-computation of Digest Scores . . . . . . . . . . . . . . . . . . . . . . . . 476.5 90% Condence Limit (C.L.) . . . . . . . . . . . . . . . . . . . . . . . . . . 48

7 Results & Discussion 53

8 Outlook 55

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Acronyms

C.L. Condence Limit.CCD Charge Coupled Device.CSS Catalina Sky Survey.

IEOV Independent Effectively Observed Volume.IfA Institute for Astronomy Hawaii.IPP Image Processing Pipeline.ISO Interstellar Object.

LINEAR Lincoln Near-Earth Asteroid Research.LSST Large Synoptic Survey Telescope.

MBO Main Belt Object.MOPS Moving Object Processing System.MPC Minor Planet Center.

NEO Near Earth Object.NIR Near Infrared.NSB Night Sky Background.

OTA Orthogonal Transfer Array.OTCCD Orthogonal Transfer Charge-Coupled Device.

Pan-STARRS Panoramic Survey Telescope and Rapid Re-sponse System.

PS1 Pan-STARRS 1.

S/N Signal-to-Noise ratio.S3M Synthetic Solar System Model.SFD Size Frequency Distribution.

TTI Transit Time Interval.

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to date zero Interstellar Objects have been discovered...

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x Acronyms


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Chapter 1

Introduction

It was not until recently that science set focus on asteroids and comets in the night sky.Single objects were studied for centuries, discovered by accident or during a close approach toEarth. But their faint luminosity at larger distances require modern light sensor technologyand high computation capacities to achieve reasonable detection rates in order to studytheir distribution. With the evolution of CCD chips and tens of thousand discoveries everyyear the picture of our solar system has quite changed since Edmond Halley computed theorbit of his famous comet in the beginning of the 18 th century [ 15]. The Asteroid Belt wasdiscovered, Jupiters Trojans and the Kuiper Belt, just to mention a few of the many knownfamilies of celestial bodies that orbit the Sun.

Today several experiments like the Lincoln Near-Earth Asteroid Research (LINEAR ),the Catalina Sky Survey (CSS), the Panoramic Survey Telescope and Rapid Response Sys-tem (Pan-STARRS) and the Large Synoptic Survey Telescope (LSST), which is expectedto see rst light in the early 2020s are dedicated to nd and catalog representatives of theasteroid- and comet populations in the solar system. The prime motivation for the recentendeavours in discovering especially Near Earth Objects (NEOs) certainly is the hazardthat is constantly imposed to Earths inhabitants in the case of an impact as just recentlyimpressively demonstrated by a rather small meteor exploding over the city of Chelyabinsk,Russia, in February 2013 [35] or the Chicxulub meteorite that supposedly initiated the ex-tinction of the dinosaurs 66 million years ago [16][33]. But it is not only the thread theycause which makes asteroids and comets worthwhile to study. Their distribution and com-position comprises valuable information about the dynamical evolution of our solar system,from its birth to the state we currently observe. Rather new theories suggest that cometsmight have brought water to the surface of our planet or even life itself, hitchhiking on anasteroid billions of miles from another world.

In 1950 Jan Hendrik Oort [ 27] published a revolutionary work about a hypothetic spher-ical reservoir of objects orbiting the Sun in a distance of 50,000 to 150,000 AU, more than1,000 times further than the Kuiper Belt. He therewith rst explained the random orien-tations of orbital planes that were observed for long-periodic comets with nearly parabolictrajectories. While there is still no denite proof the existence of the so-called Oort Cloud isnowadays widely accepted in the scientic community. Simulations conducted by Charnozand Morbidelli [ 4][5] reconstructing the evolution of the solar system support Oorts work

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2 CHAPTER 1. INTRODUCTION

showing that during orbit migration of the gas giants Jupiter, Saturn, Uranus and Neptunethe majority of all planetesimals 1 were ejected from the inner solar system onto highly ec-centric orbits. McGlynn and Chapman [ 24] comprise results from several publications andconclude that the ejection process to form the Oort cloud was very inefficient and that forevery planetesimal reaching stable Oort cloud orbit 30-100 objects have been lifted ontohyperbolic trajectories, leaving the solar system ultimately for interstellar space.

This work is the rst ever attempt to set an upper limit to the number density of ISOsin the local neighbourhood of our solar system based on actual observations and the factof non-detection. The space density of ISOs yields valuable information about the ejectionprocesses occurring during the formation of solar systems. It can for example be an indicatorfor the fraction of stars that harbour giant planets capable of ejecting planetesimals.

In chapter 2 I will give an overview of ISOs, what we know about them and relatedwork. Chapters 3 and 4 introduce the Pan-STARRS 1 telescope and its image processingsystem and show how the total system efficiency was determined. Chapter 5 describes howa new ISO model was created suitable for the determination of an observational limit of the

number density of ISOs as depicted in chapter 6.

1 The term planetesimal originates from the planet formation process in a protoplanetary disk wherecosmic dust particles collide to form larger objects. All objects left over from this process that did notcollide with a planet or the Sun are referred to as planetesimal.


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Chapter 2

Interstellar Objects (ISOs)

Per Denition the term ISO comprises all celestial bodies that are gravitationally not boundto a star. In commonly used terminology however and in this thesis it is used only for asubset of these objects, namely interstellar asteroids and comets within the Milky Waygalaxy. They might orbit the galactic center but traverse solar systems they encounter inalmost all cases on hyperbolic trajectories with an eccentricity signicantly larger than 1(Figure 2.1).

Figure 2.1: Principal sketch of different orbit types, depending on eccentricity e . An orbit with e = 0 iscalled circular (not illustrated here). 0 < e < 1 results in an elliptical orbit, e = 1 in a parabolic trajectory

and every eccentricity greater than 1 in a hyperbolic one.

So far not a single ISO was detected directly as such. Despite the fact of non-detectionthe latest models of the distribution of ISOs predict number density values that exceed thepopulations of planets and stars in the galaxy by several magnitudes. Due to their faintluminosity state-of-the-art telescopes (see section 1) can detect ISOs only during a passagethrough the inner solar system. The simulation conducted for this work shows that out of 1 million ISOs with diameter larger than 1 km passing the solar system within 50 AU overa timespan of 10 years a 842 day survey ( 27 month) conducted by Pan-STARRS - the

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4 CHAPTER 2. INTERSTELLAR OBJECTS (ISOS)

most powerful comet hunting telescope to date - would only reveal less than 100 of themassuming no activity and less than 1 , 000 if cometary activity is included in the model (thisis with slope parameter = 0 .5 and limiting absolute magnitude of the SFD H = 19, seesection 6.5, gures 6.5 and 6.6).

Due to the technical limits of detecting passing hyperbolic objects another approach hasbeen conceived. In very rare occasions an ISO can be captured by a giant planet via three-body interaction. Simulations conducted by Torbett [40] show that in our solar systemonly Jupiter is capable of scattering a hyperbolic object into a bound orbit. With a numberdensity estimation of 1.110

3AU 3 they calculated that such an event would occur withan average rate of about 60 million years. Objects with unusual orbits frequently draw theattention of astronomers. Comet 96P/Machholz is the most famous candidate for a capturedISO. It is the only known short-periodic comet with both high orbital inclination and higheccentricity [ 32] and besides its unusual orbit it also has a nearly unique composition. Itwas found to be carbon- and cyanogen-depleted [31], which implies an origin different fromother known long-period comets.

Figure 2.2: Comet 96P/Machholz as seen by STEREO-A in April 2007.Image taken from Wikimedia.

Nevertheless, backward orbit propagation is very inaccurate over long timespans. Nar-row keyholes decide whether an object came from interstellar space, the Oort Cloud orsomewhere else. Uncertainties in the orbit determination iterated over many revolutionsbecome too large so that so far neither 96P/Machholz nor any other object could unam-biguously be identied as a captured ISO.


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2.1. DISTRIBUTION & VELOCITY 5

2.1 Distribution & Velocity

The spatial as well as the velocity distribution of ISOs throughout our galaxy are notknown. Still, all authors referenced in this thesis assume a homogenous spatial distribution

at least in the local neighbourhood of our solar system. The basis for this considerationis the assumption that our solar system is an average star system and that most of thesurrounding star systems also eject planetesimals into interstellar space. While the absolutenumber of ejected objects remains speculative due to various factors outlined in section 2.3,the assumption of a homogenous distribution has been established and was adopted for theISO model generated in this work (chapter 5).

When an ISO is ejected from a star system it is decelerated due to the gravitational pullof the host star. Once the object has left the gravitational sphere of inuence of the hoststar its relative velocity with respect to the host star is reduced to a rather small value if one compares it to the velocity of the host star with respect to the Sun. We therefore canconclude that the velocity distribution of ISOs is comparable to the velocity distribution

of surrounding stars relative to the Sun, which was computed by Dehnen and Binney [6].With Hipparcos data they measured velocities in the order of 10 40km/s .Additionally to spatial and velocity distribution there is a third one usually associatedwith asteroid and comet populations, the Size Frequency Distribution (SFD). It gives arelative measure for the probability of an ISO having a certain size and is of great importancesince the size of an object determines its brightness. Dohnanyi [ 8] created a collisional modelof interplanetary debris and found that after a certain time an equilibrium state is reachedwhere the SFD does not change anymore. It can be parameterised as an exponential functionof the form

(H ) = 010 (H H 0 ) (2.1)

where H is the absolute brightness of an object. It corresponds to its visual brightness if it were observed from the heliocenter in 1 AU distance from Sun. It is given in the logarithmicscale of magnitudes [mag ]. The smaller H the brighter the object ( < H < ). Givenshape and albedo (reectivity of the object) brightness is a direct measure of the objects size(see section 2.2, gure 2.4). 0 is the density of ISOs with absolute magnitude H 0. Togetherthese two parameters determine the scale of the SFD. is the so-called slope parameter, itdetermines the steepness of the function. Dohnanyi [ 8] computed the theoretical value of = 0 .5 as equilibrium state between grinding and fusion of planetesimals. Parker et al. [ 28]measured slightly different SFDs for various families of Main Belt Objects (MBOs). Due tosimplicity however for this work the simple model given in equation 2.1 was used togetherwith various different values for . Figure 2.3 gives some example SFD curves for different

slope parameters.

2.2 Optical Properties and Activity

Since no ISO has ever been observed their composition, their optical properties and theactivity to expect when they approach the Sun are unknown. However, following the argu-ments that lead to the assumption of their existence - namely the theory that the evolution


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Figure 2.3: Illustration of different slope parameters. The curves are labeled with values from 0 .3 to0.7. 0 was selected to be 1 for this plot at H 0 = 19 .1, which corresponds to a diameter of 1 km for an

albedo of 0.04.

of most other star systems is comparable to the one of our own - we can conclude that theirproperties are comparable to the ones of objects that were ejected from our own solar sys-tem. As outlined in the introduction not all objects that were ejected left the solar systemfor good. A small fraction is still orbiting the Sun in the Oort cloud and now and thenperturbations induced by passing large objects ( e.g. stars, brown dwarfs, giant planets) orgalactic tides send them back to the inner solar system where we can observe them andanalyse their composition. These objects were found to be composed mostly of hydrogen- oroxygen-based ices such as water, methane, ethane, carbon monoxide and hydrogen cyanide[13]. Sweeping up cosmic dust and rocks from the protoplanetary disc 1 surrounding theirhost star the comets become matt and dark so that their reectivity (albedo) is typicallyvery low. For all calculations throughout this thesis a value of 0.04 has been used. With axed albedo and the common simplication of all asteroids and comets being spherical theirbrightness is only dependent on diameter. Fowler and Chillemi [ 10] derived the following

1 A protoplanetary disk is a rotating circumstellar disk of dense gas surrounding a young newly formedstar, a T Tauri star, or Herbig Ae/Be star [Wikipedia]


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2.2. OPTICAL PROPERTIES AND ACTIVITY 7

relation for inert asteroids

d = 1347.4

p 10H 5 km (2.2)

where d is the diameter of the asteroid, p represents the reectivity (albedo) and H itsabsolute brightness. The equation is illustrated in gure 2.4 and also applicable for cometsif the heliocentric distance is large.

Figure 2.4: Brightness versus Size for astroids according to [10], with an albedo p = 0 .04. The plot wasgenerated with equation 2.2.

Once the comet approaches the Sun it is likely that it will show activity. Incident solarradiation causes some of the frozen material to melt and vaporise, creating a halo aroundthe comet. The so-called coma reects sunlight and can increase the visual brightnessof a comet by several magnitudes. In this case equation 2.2 looses integrity and has tobe adapted. There is no universal recipe to predict the activity of a comet accurately.While some show high activity already at heliocentric distances larger than 9 AU, likefor instance comet ISON, others become active only at closer distances or not at all, e.g.2005 VX3. In general the activity of a comet decreases with every close approach to theSun. Fresh comets, which get close to the heliocenter for the rst time usually show


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very high activity as observed for instance for Oort cloud comets with nearly hyperbolictrajectories. Short-periodic comets in contrary increase only little in brightness since theyalready lost big amounts of outgasing material over many revolutions. Just like Oort cloudcomets ISOs approach the Sun for the rst time and we can therefore assume high activity.The following estimation for ISO activity was suggested by Prof. Alan Fitzsimmons fromthe Queens University Belfast. He is an expert in the eld of comets and a collaboratorfor a paper that will be released about this work. He used equation 2.2 together with anestimation on the correlation between sublimation rate and absolute brightness to derive acorrected absolute magnitude for the case of high activity

H a = 1 .6H 19.9 4.1log10 f (2.3)where f is the fraction of the sun-facing area of the comet sublimating material. For

highly active ISOs f = 1. Equation 2.3 therefore reads

H a = 1 .6H

19.9 (2.4)

The equation does not include any dependency on heliocentric distance and thereforedelivers absurd results if we do not constrain it. From comets like ISON we have learnedthat highly active comets show activity from distances around 9 AU . Applying equation2.4 for heliocentric distances smaller than 10 AU and assuming no cometary activity at allfor everything further was regarded as suitable approximation. Therefore, throughout thecontext of this thesis we use the corrected absolute magnitude

H a = (1 .6H 19.9) H(10AU R) (2.5)for all calculations incorporating cometary activity, where H denotes the Heaviside-function and R the heliocentric distance of the object.

2.3 ISO Number Density Estimates

Different approaches have been pursued to estimate the number density of ISOs in inter-stellar space. The most common method is to estimate the number of objects that wereejected from our own solar system and associate it with the number density of star systemsin the galaxy. Using this simple technique McGlynn and Chapman [24] derived a numberdensity of 1.1 10

3AU 3. With the same approach Jewitt [ 19] calculated a similarnumber of 10 3AU 3 while Sen and Rama [34] claim to use a more advanced estimate forthe number of stars and predict an ISO density of 1.610

4AU 3. Francis [11] used a 3year sample of the LINEAR survey (1999-2002) and the non-detection of ISOs to set a 95%Poisson Condence Limit on the maximum number density of ISOs. Depending on the usedcomet population he derived two different values, 6 10

4AU 3 for the population givenby Hughes [18] and 9 10

4AU 3 for a population according to Everhart [ 9], both of whichare derived from Oort cloud comet observations. He estimates that an extension of thesample of observations until the end of 2004 could reduced the limit to 3 4.510

4AU 3.He also used a reduced Oort cloud population and therefore a reduced ISO ejection rateto compute an estimate of the ISO number density with a method conceived by Stern [36].


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Figure 2.5: glsISO number density estimates over time. The blue shaded areas show a 95% PoissonCondence Limit on the maximum number density of ISOs determined by Francis [11] with a 3 year

sample of the LINEAR survey (1999-2002). The darker blue patch corresponds to the limit computed withthe actual data sample. It is slightly variable depending on the used comet population. The bottom edge

represents the limit for a Hughes [ 18] population of 6 10 4 AU 3 and the top edge the limit for aEverhart [9] population of 9 10 4 AU 3 . He suggests that since there hasnt been a discovery with

LINEAR until the end of 2004 with a comparable sky coverage the limit could be adjusted to lower valuesbetween 3 4.5 10 4 AU 3 , represented in the gure with the lighter blue patch.


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Chapter 3

Pan-STARRS 1 (PS1) telescope

Pan-STARRS 1 (PS1) is the rst operational prototype telescope of the Panoramic SurveyTelescope and Rapid Response System (Pan-STARRS ), currently under development bythe Institute for Astronomy (IfA) of the University of Hawaii. It is located atop MaunaHaleakal a on the Island of Maui, at an elevation of 10,023ft (3,055m) above sea level [ 1].

Figure 3.1: Pan-STARRS 1 telescope atop Mauna Haleakal a during twilight.Image taken from the PS1 Science Consortium website: http://ps1sc.org.

Its primary mission is to detect potentially hazardous asteroids. For this delicate taskPS1 combines a wide eld of view (7 square degree) with a high resolution of 1.4 Giga pixels.It operates in the visible spectrum and Near Infrared (NIR) and was designed according toeconomic prospects [21]. In general the price for a telescope grows exponentially with itsprimary mirror diameter, offset by the cost for the camera and focusing unit. The minimumof the performance versus cost curve resulted for the mission objectives of Pan-STARRSin a design of 4 similar telescopes with an aperture of 1.8m, f/4.4 focusing on the sameportion of sky [17]. The result is equivalent to a 3.6m telescope, but much cheaper. At themoment it is not clear if the complete array will ever be funded but its predecessor Pan-

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12 CHAPTER 3. PAN-STARRS 1 (PS1) TELESCOPE

STARRS 2 is completely assembled and already saw rst in the end 2013. The coupled useof both telescopes by the beginning of 2014 will signicantly increase the limiting magnitudeof currently 21 .6mag . But already in the current conguration Pan-STARRS is the mostsuccessful asteroid- and comet hunting telescope in the world. After almost 3 years of operation it discovered more than 800 NEOs, roughly 40,000 MBOs and almost 50 comets.It also reported 7.2 million asteroid positions of 560,000 distinct asteroids and thereforeobserved 89% of the known asteroid population [ 41]. Traditionally PS1 telescope pointingsor bore sites are called elds.

3.1 Gigapixel Camera

The heart of PS1 is a sophisticated camera designed to cover a large area of the sky whileproviding high sensitivity and resolution. These quantities directly derive from mirrorquality, aperture diameter, number of pixels on the photo-electric chip and its thermal

noise, known as dark current. The camera is composed of 60 Orthogonal Transfer Arrays(OTAs) arranged in a square of 8 by 8 missing one in each corner (see gure 3.2) [26]. Thecorner pixels were omitted since they fall completely out of the illumination circle which isdened by the circular primary mirror (see gure 3.3).

Figure 3.2: Pan-STARRS 1 camera composed of 60 OTAs.Image taken from the Pan-STARRS website: http://pan-starrs.ifa.hawaii.edu.

The OTA technology was developed by the MIT Lincoln Laboratory to correct atmo-spheric distortions in wide-eld telescopes where conventional tip-tilt mirror adaptive opticsdevices overrun their limits [3]. Each OTA is composed of an 8x8 array of Orthogonal Trans-fer Charge-Coupled Devices (OTCCDs) that are capable of shifting the charge in a pixel toeither of the 4 neighbouring pixels at rates of up to about 30 Hz . This technique is moreor less an electronic version of the tip-tilt mirror correction and allows the compensationof atmospheric phase distortion and telescope motion. Together the 60 OTAs count 1.4billion pixels and compile the largest camera ever built, capturing visible light and NIR (seesection 3.3).


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3.2. FILL FACTOR F 13

3.2 Fill Factor F

Technological limitations force the design layout of the camera to have little gaps between

the single OTAs . Additionally, there are bad cells on the OTCCDs , affected by crosstalk 1or saturation due to bright stars illuminating parts of the chip. All these effects reduce theeffectively illuminated area on the chip. The ratio between the effectively usable and totalilluminated area is called ll factor F . Figure 3.3 shows an image mask simulating the llfactor, which is varying but in average roughly 75% [ 7]. It is measured and determined withthe aid of reference asteroids and accounted for with the system efficiency (section 4.4).

Figure 3.3: The gure shows the grid-like gaps between OTAs and a simulated typical distribution of badcells on the OTCCDs . Image taken from [7].

1 Crosstalk is any phenomenon by which a signal transmitted on one circuit or channel of a transmissionsystem creates an undesired effect in another circuit or channel. Crosstalk is usually caused by undesiredcapacitive, inductive, or conductive coupling from one circuit, part of a circuit, or channel, to another.[Wikipedia]


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3.3 Photometric System

The PS1 photometric system is similar to a sloan lter system [12]. It consists of 5 multi-spectral bands gP S 1, rP S 1, iP S 1, zP S 1, yP S 1 plus a wide-band wP S 1 comprising the gP S 1,

r P S 1 and iP S 1 spectrum as described by Tonry et al. [ 39]. Compared to the sloan system ithas the ultra-violet band replaced with an additional NIR band shifting the spectral rangeto 400 1000nm . The transmission for each of the lters is given in gure 3.4.

Figure 3.4: PS1 lter transmission curves as a function of eld angle, in 0 .15 steps from 0 to 1.65

(grey lines), with the area-weighted average in red. Image taken from [39].

Characteristic for the system are wide spectral bands optimised for the detection of faintobjects while still providing good contrast to the Night Sky Background (NSB). The totalspectral telescope throughput dened by aperture, photo sensor efficiency (see section 3.1)and lter transmission is given in gure 3.5 as cross section for a standard air mass of 1.2.

3.4 PS1 SurveysAmong several others PS1 is conducting three surveys suitable for asteroid- and cometdetection. The so-called 3 -survey covers the largest portion of sky with an solid angle of 3 steradians (4 represents the entire sphere) in 5 sloan-like lters (see section 3.3), gP S 1,r P S 1, iP S 1, yP S 1 and zP S 1. It takes up 56% of PS1 observation time and covers the entiresky North of declination 30

[30]. The huge coverage allows not more than maximal 4


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are mainly below 20 elevation from the ecliptic and in chunks around the opposition andso-called morning- and evening sweet spots that are characterised with low solar elongationsbetween 60 and 90 degrees. For the number density limit estimation of ISOs in this thesis(see chapter 6) all 181,388 elds produced from these 3 surveys until June 12th, 2013 areused. In the following I will call them together the PS1 survey.

Figure 3.6: Sky coverage for lter gP S 1 in the 3 survey, observing cycle 162.The coverage for lters r P S 1 and i P S 1 are similar.

Figure 3.7: Sky coverage for lter z P S 1 in the 3 survey, observing cycle 162.The coverage for lter yP S 1 is similar.


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Chapter 4

Moving Object Processing System(MOPS)

The Moving Object Processing System (MOPS) was developed for Pan-STARRS to extendbasic object recognition performed by the Image Processing Pipeline (IPP). It links multipleobservations of the same object together to estimate preliminary orbit parameters andcategorise the object. In the following a short overview of the system is given to introduceessential principles of determining the system efficiency as described in section 4.4. Adetailed report of the MOPS system was written and published by Denneau et al. [ 7].

4.1 Image Processing Pipeline (IPP)

The IPP essentially performs the entire image analysis of Pan-STARRS raw data to thepoint where it can be used by scientic analysis tools. The core requirements for the IPPare robustness and the capability to cope with the enormous data amounts produced bythe gigapixel camera (section 3.1). The image processing of an observation cycle has to benished before the next observation cycle starts. In general every night observations areconducted. The image analysis procedure was explained in detail by Magnier et al. [ 23].It can be summarized in two major steps. In Phase 1) the raw images are combined withmetadata to perform astrometric and photometric calibrations. It ends with calibratedimages and a table of objects they contain. Exposures of the same portion of sky arecombined in Phase 2) to a so called stack. The result is a high quality image with cosmeticdefects removed and a better Signal-to-Noise ratio (S/N). To detect asteroids and cometstwo images of the same region are subtracted. Static objects like stars and galaxies areremoved and only moving objects remain in the difference image with underlying noise. Toimprove this technique a Static Sky map is currently under development. A Static Skyimage is the average of tens of stacked images over a long period of time. The more imagesare added up the fainter is the limiting magnitude and the better is the S/N . Eventually themoving object detection will be accomplished for all images with the subtraction of a stackimage and the Static Sky image. So far this technique is only available for the MediumDeep survey (section 3.4). The MOPS processing chain described in the following chapter4.2 starts with the output of the IPP system.

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20 CHAPTER 4. MOVING OBJECT PROCESSING SYSTEM (MOPS)

extinction. For PS1 specically the ll factor described in section 3.2 and the quota of theIPP to recognise an object if it is in an image additionally reduce the detection efficiency.

Analytical solutions to calculate the detection efficiency are complex. Therefore refer-ence measurements with known objects are used to directly measure the detection efficiencyon a nightly basis for each lter and t an efficiency curve. Since the detection efficiencyis depending on the apparent magnitude V the reference objects are sorted into bins witha width of 0.25 magnitudes. For each night the detection efficiency is determined dividingthe number of known objects in a certain V -bin that were in a certain lter and admittedto the detection database through the number of known objects that were in PS1 eldsthat night in the same bin and lter. The detection efficiency depending on V follows theempirical function

d (V ) = 0

1 + eV L

w(4.1)

with the tting parameters 0, L and w to be determined. 0 is the maximum efficiencythat can be reached for bright objects at a certain night. It stays constant with increasing V until the limiting magnitude L is approached. A smooth symmetric drop-off to zero occursaround L that is dened as the V value at which the efficiency is exactly 50% of 0. Theparameter w is a measure for the width in V -domain over which the drop-off takes place.

In order to generate only trustworthy efficiency curves two requirements are imposed ona nightly sample of reference measurements before the set goes into the tting algorithm.At least 10 known objects have to be present in a V -bin to form an efficiency measurementand at least 10 of those efficiency measurements have to be available per night. Experi-ments with different tting functions 2 lead to the use of the Levenberg-Marquardt algorithmimplemented in Pythons curve t function. It performs comparable to more complex al-gorithms for this task while being much faster. It has shown that for all tested algorithms

- local and global optimisation - a weighting of the efficiency measurements is necessary inorder to retrieve satisfying results. The most common method, which was also applied here,is a weighting with the inverse of the uncertainty.

The efficiency measurements are created dividing two histograms representing the num-ber of objects that are known to be in an exposure N and a subset of N representing theknown objects that have been detected, denoted k. The measurements are separated intodifferent bins i depending on the apparent magnitude V of the object.

d (V i ) = k(V i )N (V i )

(4.2)

Two methods are common to estimate the uncertainty of such measurement series de-

pending on the underlying distribution: Poisson Errors and Binomial Errors. Both of thesemethods deliver absurd results in limiting cases when the efficiency approaches 0 or 1 as de-scribed by Paterno [ 29]. He developed a method based on Bayes Theorem that supposedlyavoids this problem. He developed a C++ software to calculate uncertainties according to

2 tested were Nelder-Mead, L-BFGS-B, Simulated Annealing, Powell, Conjugate Gradient, Newton-CG, COBYLA and Sequential Linear Squares Programming from the python custom library lmt:http://cars9.uchicago.edu/software/python/lmt/


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4.4. SYSTEM EFFICIENCY 21

his method, which is available upon request per email: [email protected] . For this projectthe implementation of his software was considered too time consuming and not in relationto the cause. Therefore an empirical x has been implemented together with the binomialerror estimation. The binomial error for the efficiency stated in equation 4.2 is given by

d (V i ) = d (V i ) (1 d (V i ))N (V i ) (4.3)and delivers reasonable uncertainty values for efficiencies in the range from 0.02 to0.98. The developed Python software computes uncertainties in this range according to

equation 4.3, otherwise uses

d (V i ) = 1N (V i ) (4.4)This method is empirical but delivers good ts for almost all nights as shown in an

example in gure 4.2. Additionally to an appropriate weighting reasonable initial valueshave to be admitted to the optimisation algorithm for all tting parameters. They wereselected empirical the to be

0,init Linit winit0.5 21 5

A software has been developed that downloads all reference measurements from theMOPS database and computes tting parameters for all nights PS1 operated. All ttingparameters are collected in a table and stored in a le for this project and further use.The tting algorithm works ne in most cases, but not all. To avoid corrupt ts require-ments have been imposed on the tting algorithm. It has to deliver 3 real-valued, positiveparameters with the corresponding -values (standard deviation) not exceeding the upperlimits

0 ,max L,max w,max0.1 3 1

All limits were set empirically. Corrupt ts were agged and not admitted to the nalefficiency table. Figure 4.3 shows the distribution of the ags that were set for each nightof the PS1 survey in the w-lter. The corresponding distribution of the parameters in thenal table is given in gure 4.4.

Tracklet Efficiency

The tracklet efficiency is dened as the number of objects in a certain number of elds thatcreated at least 1 tracklet in the MOPS tracklet database over the total number of objectsthat were in those elds.

t = 1N

i

t i (4.5)


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Figure 4.2: Example of an efficiency t curve for the w-lter. The original efficiency measurements aredisplayed in orange with corresponding error bars in grey. The boxes in the top right corner give the

computed tting parameters together with their standard deviation estimated by Pythons curve t function. The limiting magnitude L is represented by a solid dark grey vertical line accompanied by twolight grey lines indicating the width of the drop-off or w respectively.

where the index t i is a boolean which is 1 if a tracklet was created for the object with idi and 0 if no tracklet was created. N is the total number of objects that were in the elds.The relation between detection efficiency and tracklet efficiency is not trivial. Trackletscan be created out of 2, 3 or 4 detections all affected by the ll factor (see section 3.2)and potentially originating from different lters. Analytical solutions to determine thecapability of MOPS to link detections into tracklets are complex and afflicted by large errorbars. Hence, the most accurate method to determine the tracklet efficiency is to measure

it directly with reference objects as well.So far tracklet efficiency measurements are only implemented for the w-band and only

for a fraction of the nights with detection efficiency measurements available. Therefore thetting parameters for the detection- and tracklet efficiencies in the w-lter were averagedto create functions for average detection- and average tracklet efficiency. Correction factorswere derived for each tting parameter to convert it from detection efficiency to trackletefficiency.


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Figure 4.3: Distribution of ags set by the efficiency t algorithm for the w-lter. The red bars to theleft of GOOD FIT represent ags that originate from errors encountered by Python in the tting process

or from not fullled criteria required for a tting approach. pcov innity is set when values in thecovariance matrix go against innity. tting error is set when the tting algorithm returns a not furtherspecied error, not enough bins indicates that less than 10 bins were available and that therefore

according to specications no t was created. no data is set when no measurements were available for agiven night at all. The yellow bars on the right of GOOD FIT indicate if a tting parameter was

negative or its standard deviation is exceeding the maximum values specied above.

c 0 = 0,tf

0,df (4.6)

cL =L tf L

df

(4.7)

cw = wtf

wdf (4.8)

where the index tf denotes the relation to tracklets and df the relation to detections in acertain lter f . There is no physical or mathematical proof that the same correction factorsapply to all lters but due to missing measurements it was regarded as the most reasonableestimate to use the w-lter correction for all lters. To convert a tting parameter for a


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Figure 4.4: Distribution of efficiency tting parameters and corresponding standard deviation for thew-lter.


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given lter from average detection efficiency to average tracklet efficiency it is multipliedby the corresponding correction factor from the w-lter

0, tf = 0, df c 0 (4.9)L tf = Ldf cL (4.10)wtf = wdf cw (4.11)

where again the index tf denotes the relation to tracklets and df the relation to detec-tions in lter f . The tracklet efficiency is therefore computed as

t = 0,d c 0

1 + eV L d cL

w d cw

(4.12)

The resulting efficiency curves for each lter are illustrated in gure 4.5.

ISO Efficiency

In the current conguration of Pan-STARRS only a fraction of the created tracklets make itinto tracks (see section 4.2). All other tracklets receive a digest rating (see section 4.3) andare submitted to the MPC if they are considered real. The MPC publishes the submittedobject candidates on a public website 3 where other telescope projects as well as amateurastronomers around the world can access them and freely choose tracklets of their interestto follow-up. If one of these follow-up efforts is successful and the object can be recovereda precise orbit is generated. In case the object is not already known the MPC announcesa new discovery. In general the follow-up approaches are guided by the digest score, butthere is no guarantee that an object with a high score is followed-up. On the other hand,sometimes tracklets with a score of only 50 or even lower are pursued. For simplicity acut was set at a digest score of 90. We assume that all tracklets with a score of 90 orhigher are followed-up and produce a new object, knowing that some of them will not.As compensation all objects that have a score lower than 90 and might actually still berecovered are neglected.

Even though ISOs usually receive a high score there are some that are eliminated bythe digest score procedure. A detailed analysis on how many objects are rejected is still inprogress. The efficiency we are concerned about for the estimation of the number densitylimit of ISOs is the efficiency that an object created a tracklet, received a high digest score,was submitted to the MPC, was successfully followed-up and was announced as new object.I will refer to this combined efficiency as ISO efficiency. Since objects can create more than

one tracklet and frequently do so the digest score modied tracklet efficiencies have to becombined to give the actual ISO efficiency.The probability of an event happening in at least one case out of n is

P combined = 1 n

[ 1P n ] (4.13)3 http://www.minorplanetcenter.net/iau/NEO/ToConrm.html


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Figure 4.5: Detection- and Tracklet Efficiency Functions depending on apparent magnitude V . The solidlines represent the detection efficiency curves for all six PS1 lters and the tracklet efficiency curve for the

w-lter tted through actual measurements while the dashed lines represent the estimated trackletefficieny curves for all other lters derived with the correction factors described above.


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4.5. PAN-STARRS SURVEY SIMULATIONS 27

where P n is the probability that the event n has a positive outcome. Sequentially, theefficiency that a certain object in a PS1-eld is found is given by

ISO,i = 1 N t

n i =1[ 1( tf (V i ) H(Dn i 90) ) ] (4.14)

where the index i refers to a unique object id and n i to a unique tracklet id of a trackletthat has the object id i. tf is the average tracklet efficiency in a certain lter f and N t thetotal number of tracklets generated. Dn represents the digest score for the tracklet n. Hdenotes the Heaviside step function.

4.5 Pan-STARRS Survey Simulations

In order to test MOPS and assess the efficiency with which it nds representatives of certainobject populations a simulation mode is implemented. Objects can be inserted in real or

arbitrary elds before they are run through MOPS processing. Grav et al. [14] created aSynthetic Solar System Model (S3M) with optimised orbits so that they are likely found inthe PS1 survey. It includes: Near Earth Objects (NEOs ), Main Belt Objects (MBOs), Jo-vian Trojans, Centaurs, Trans-Neptunian Objects (TNOs), Jupiter-Family Comets (JFCs),Long Periodic Comets (LPCs) and also Interstellar Objects (ISOs). For the purpose of thisthesis however the ISOs featured in the S3M are not suitable since they are not related toa real population. A new ISO-model was therefore generated as input for the simulationwith requirements tailored to the task (see chapter 5). If MOPS is run in simulation modean additional table is created called S3M containing the positions and properties of thesynthetic objects. A comparison between the S3M-table and for instance the tracklets-table shows how many of the objects appeared in PS1 elds and were linked into tracklets.

The system is smart enough here to assign corresponding S3M IDs to all detections andtracklets so that for each object one can trace whether it was linked or not. The efficiencyof linking synthetic objects is always 100% presumed they appear in a PS1-eld, which iswhy the real efficiency has to be determined and assigned separately.


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Chapter 5

Interstellar Object Model

This section describes how a new ISO model was developed according to state-of-the-artknowledge of distribution, velocity (see section 2.1) and properties (see 2.2) of ISOs. Themodel has an object density several magnitudes higher than the real one expected for thesolar system. This is necessary to retrieve valid statistics from a simulation that coversonly 27 month. Normalised spatial distribution and SFD however stay the same. Theequations derived later will show that the use of a correct ISO distribution is essential forthe conceived method. The better the synthetic model resembles the real distribution of ISOs the more accurate the derived estimate of the number density limit.

5.1 Object Generation and Propagation

To generate a realistic ISO model including gravitational focusing homogeneously randomdistributed ISOs were initialised in a sphere with radius r init . Velocity vectors with randomGaussian-distributed norms were assigned to all objects to dene their orbits and adddynamics to the model. The velocity distribution was adapted from Grav et al. [14] whoused numbers derived by Kresak [ 22] and Dehnen and Binney [6]. They estimate therelative velocity of ISOs with respect to the Sun to be in the same order as the velocity of surrounding stars. The used Gaussian distribution around = 25km/s with = 5km/splaces 99.7% of the generated objects between vmin = 10km/s and vmax = 40km/s .

A 2-body orbit propagation (object-Sun) introduces gravitational focusing to the modelover the propagation time interval T p. It has to be long enough that all objects which wereinitialised in the Suns sphere of inuence and therefore have a distribution that does nottake gravitational focusing into account get enough time to leave the sphere of inuenceand do not distort the model. Additionally, objects outside the sphere need to have timeto replace them. Since no objects enter the initialisation sphere while there is a constantoutow the model density decreases over time. From the beginning of orbit propagation asub-sphere emerges within the initialisation sphere harbouring a valid model while the outerareas become adulterated. This sphere shrinks over time and will be denoted valid spherewith radius r valid . Considering both of these effects as well as the computation time whichgrows exponentially with r init a minimisation problem arises. To keep the computation timewithin feasible limits r init has to be as small as possible while still providing a valid model

29


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30 CHAPTER 5. INTERSTELLAR OBJECT MODEL

within rvalid . The time of initialisation T init is dictated by the time frame T v in which themodel has to be valid. It was set to January 1st, 2005 (53371 MJD) until January 1st, 2015(57387 MJD). This includes the Pan-STARRS survey and CSS, which will be included inthe number density limit estimation in the future.

r valid was set to 50AU , a distance where only objects with a diameter of several hundredkilometres can be detected by PS1. Leaving them out assuming an exponential SFD (seesection 6.3) introduces a negligible error to the number density limit estimation. Thepreparation time was estimated as

T p = 2r valid

vmin(5.1)

where is the margin factor. The calculated time span gives the slowest objects enoughtime to cross the entire valid sphere without accounting for the acceleration due to thegravitational pull of the Sun. A margin factor of 1.5 gives a T p of 71 years. r init was

estimated according to

r init = rvalid + vmax (T v + T p) (5.2)

so that the fastest objects would just reach the valid sphere if they were on a straightpath towards the heliocenter. With this choice of r init = 742AU it is guaranteed thatthroughout the entire survey the inow of ISOs into the valid sphere equals the outow.The margin factor introduced in equation 5.1 is automatically incorporated in r init .

After all objects were initialised at T init = 27399MJD (beginning of PS1-survey - T p)they have to be propagated to the beginning of the PS1-survey before they can be admittedto the MOPS simulation. To achieve a number of 1,000,000 objects in the simulationappearing in the valid sphere during the survey lifetime 1,674,293,112 objects had to begenerated. Propagating this enormous amount of objects takes a long time. Therefore onlyselected objects were propagated fullling the following selection criteria. Only objects withperihelion distance smaller than r valid and a perihelion passing time greater than T init wereselected. Additionally, for each object fullling both of these criteria the time instants werecomputed at which the object enters the valid sphere and leaves it. A good approximationcan be accomplished using the hyperbolic Kepler equation. For small perihelion distancesq 3.5AU however the algorithm runs into problems and delivers NaN values. As a simplesolution for the problem all objects delivering NaN values are admitted to the model so thatit will eventually include objects that do not enter the valid sphere at all during the surveylifetime. Their numbers are negligible though. This x is visible in the model statisticsillustrated in section 5.3. With 1,000,000 objects in the model a little bit more than aquarter of them is situated in the valid sphere at any given time during the survey givingan average number density of roughly 0 .65AU 3 (see gure 5.12).

In the model generation there is no step included verifying that the generated objectsare hyperbolic. An analysis of the eccentricity shows that the model includes objects withe < 1. These objects are eliminated in the implementation setting a requirement of e > 1when the tracklets are retrieved from the simulation output in the MySQL database.


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5.2. ORBIT ANALYSIS 31

5.2 Orbit Analysis

To characterise and validate the orbits of ISOs generated with the procedure described insection 5.1 sample objects have been generated and propagated over 50,000 days (roughly137 years) from the time of initialization. Figure 5.1 and 5.2 show the trajectories with 100day sampling. It is clearly visible that the trajectories are bent due to the gravitationalinuence of the Sun. The effect is dependent on perihelion distance and therefore small formost objects according to the perihelion distribution given in gure 5.5.

Figure 5.3 shows the orbital velocity of the sample objects depending on heliocentricdistance R. It is clearly visible how the velocity increases according to a power law while theobject approaches the Sun. The velocity gain is becoming signicant at distances smallerthan roughly 50 AU . Figure 5.4 shows the specic energy of the objects comprising thepotential energy and the kinetic energy normalised by the unit mass. The independence of heliocentric distance veries the conservation of energy and therefore the correctness of themodel in that sense. The orbital parameters are investigated in the next section in order

to characterise and further validate the ISO model.

5.3 ISO Model Statistics

In the following the distributions of orbital elements in the ISO model are analysed andvalidated against requirements for the project. Figure 5.5 shows the normalised periheliondistance distribution of all generated objects (black) and the selection eventually incorpo-rated in the model (red). The latter features a cut-off at the radius of the valid spherer valid induced by the selection criterion described in section 5.1 after which only objectsare selected with q < r valid . It also shows a peak in the rst bin which is related to thespecial treatment of NaN values occurring for complex solutions of the hyperbolic Keplerequation mentioned in section 5.1. Due to the x applied the rst bin includes all objectsthat enter the valid sphere at any given time, not only objects that are present in the validsphere during at least a fraction of the specied time frame as it is the case for the rest of the selected objects.

Figure 5.6 shows the normalised distribution of the Right Ascension of the AscendingNode (RAAN) with the same colour coding as gure 5.5 and all following plots. As ex-pected the direction from which the ISOs approach the Sun is absolutely random and thedistributions therefore at for all generated objects as well as the selected batch.

Figure 5.7 shows the normalised eccentricity distribution for all generated objects aswell as for the selection while gure 5.8 gives only the selected distribution on a smallerscale for better resolution. Due to the correlation of perihelion distance and eccentricitythe peak in the rst bin as seen in gure 5.5 is also present here. The eccentricity of anobject is directly related to its perihelion distance and its velocity. The further away anobject passes the Sun the less its orbit gets curved by gravitational acceleration and thehigher its eccentricity. An object passing the Sun outside its sphere of inuence will followa straight trajectory and per denition have an eccentricity of innity. Also, the faster anobject moves the less time it is exposed to gravitational acceleration, the less its orbit iscurved and the higher its eccentricity. At each given perihelion distance the eccentricity


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distribution is similar to the velocity distribution of the objects, in this case a Gaussian.From gure 5.5 we can therefore conclude that the eccentricity distribution is an overlapof Gaussians with a more or less linear increasing maximum up to the point where theperihelion distance corresponding to the eccentricity reaches the radius of the initialisationsphere/valid sphere. From this point the distribution drops off partly due to the Gaussiandrop-off from the last bins and partly due to the drop-off in the distribution of periheliondistance.

Figure 5.9 shows the normalised distribution of the perihelion passing time . For thetotal generated population is normal distributed around the time of initialisation whilefor the selected objects the distribution is centred around the survey time. This behaviouris just as intended and validates the selection process.

Figure 5.10 gives the normalised distribution of the objects inclination. Its cosine shapecan be explained the following way. Only objects that are located in the ecliptic plane andadditionally have an velocity vector parallel to the ecliptic have an inclination of 0 or 180 .These congurations are rather rare. On the other hand, every object that has a velocity

vector perpendicular to the ecliptic has an inclination of 90

regardless of its position. Theplot veries what was expected from a direction independent ISO distribution.

Figure 5.11 shows the number of objects selected for the model depending on heliocentricdistance R. Additional to gravitational focusing the relation of the volume of a sphericalshell to its radius is incorporated. The gure demonstrates that in the volume with gooddetection probability even for small objects from approximately 2 5AU (0 1AU is thevolume between Earth and Sun and therefore not observed) more than 100 objects arepresent over the survey lifetime.

Figure 5.12 gives the model density versus R and shows that the distribution of objectsin the model goes asymptotically against a homogenous distribution at large heliocentricdistances. Towards R = 0 the model density increases drastically due to gravitational

focusing. Within R = 1AU the model density is almost 5 times higher than at 50 AU . FromR roughly greater 10 AU however gravitational focusing does not have big impact on themodel density. It was therefore regarded as valid for the calculations in chapter 6 that themodel density at R = 50AU of 0.66AU 3 is a good approximation for the model density ininterstellar space S,IS .

Figure 5.13 is the result of an orbit propagation of all 1 million selected objects throughthe valid time frame (10 years) computing ephemeris in 120 day intervals. The model densityuctuates the stronger the smaller the heliocentric distance R. This effect can be explainedwith the number of objects per bin, which is increasing with R. In the bins representing0AU < R < 1AU zero to less than ve objects are used to calculate the density while thereare several hundreds to several thousands in bins representing greater heliocentric distances.In general however the dynamic of the model does not alter the model density at a given Rover time apart from uctuations that should be compensated over the integration time.


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5.3. ISO MODEL STATISTICS 33

Figure 5.1: Trajectories of 10 sample objects. The orange sphere represents the valid sphere with radiusr valid = 50AU . Objects outside of the sphere are the more opaque the closer they are to the heliocenter

while objects inside the sphere are coloured in red.

Figure 5.2: Trajectories of 10 sample objects [zoomed]. This plot is equivalent to gure 5.1.


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Figure 5.3: Orbital velocity of sample objects depending on the heliocentric distance.

Figure 5.4: Specic energy of sample objects comprising potential and kinetic energy.


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Figure 5.5: Normalised distribution of perihelion distance q for all generated objects (black) and selectedobjects (red).

Figure 5.6: Normalised distribution of Right Ascension of the Ascending Node (RAAN) for all generatedobjects (black) and selected objects (red).


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Figure 5.7: Normalised distribution of eccentricity e for all generated objects (black) and selected objects(red).

Figure 5.8: Normalised distribution of eccentricity e for the selected objects.


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Figure 5.9: Normalised distribution of perihelion passing time t p for all generated objects (black) andselected objects (red).

Figure 5.10: Normalised distribution of inclination i for all generated objects (black) and selected objects(red).


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Figure 5.11: Number of objects in the model N S versus heliocentric distance R on January 1st, 2005(53371 MJD), beginning of the simulation.

Figure 5.12: ISO model density S versus heliocentric distance R on January 1st, 2005 (53371 MJD),beginning of the simulation.


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Chapter 6

Observational Number DensityLimit for ISOs

While several attempts have been pursued to tie down the upper limit for the number densityof interstellar objects (section 2.3), this is the rst ever accomplished incorporating actualobservations with corresponding efficiency measurements (section 4.4). It is based on thePoisson statistics of a non-detection (section 6.1) of ISOs in the PS1 survey. Assuming thatthe number density of ISOs underlies statistical uctuations but the average is constant overa time interval much larger than the time of observation the steady-state number densityis given as

=N V

(6.1)

where N is the average number of ISOs located within V . To determine the steady-statenumber density of ISOs within a certain volume of space it has to be either very large ormonitored over a long time to detect signicantly more than zero ISOs, which is requiredto provide valid statistics. Assuming that the ISOs are distributed homogeneously in thegalaxy with exception of the direct vicinity of stars, where gravitational focusing acts, anarbitrary volume can be used as representative for the entire interstellar space. A volumeclose to a star can also be used as shown in this chapter if an appropriate correction is appliedto compensate gravitational focusing. Compared to the time it takes ISOs to travel the vastdistances in the galaxy the time frame of a typical sky survey is extremely short. Also theobserved volume for a ground-based survey, which is limited by the telescope performanceand Earths atmosphere is relatively small and xed at the heliocenter, dictated by Earthsorbit. PS1 observations conducted year-round in wide elds mainly close to opposition sothat a big portion of the night sky is covered (section 3.4). The observed volume thereforecan be seen from a simplied point of view as a large fraction of a heliocentric sphere witha radius determined by the limiting magnitude of PS1, leaving out the innermost sphericalregion between Earth and Sun where no observations are conducted. This volume is referredto as accessible volume V A .

The number density of ISOs is so low that not a single one could unambiguously beidentied as such in the history of space surveillance. Since the PS1 survey is only a tiny

41


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42 CHAPTER 6. OBSERVATIONAL NUMBER DENSITY LIMIT FOR ISOS

sample of a much larger volume the non-detection of ISOs does not out-rule their existence,but it puts a constraint on the maximum density we can expect. The upper limit for thenumber density of ISOs is directly proportional to the observed volume, which is inuencedby the efficiency of the entire Pan-STARRS system and the dynamics of ISOs. Since ISOsmove through space and shuffle two measurements of the same volume can be regardedas independent random samples of the entire interstellar space if the timespan between thetwo measurements is long enough to assure that all objects within the volume leave it whilenew objects move in to replace them. The necessary shuffling time is dependent exclusivelyon the velocity distribution of ISOs.

Exploiting these two aspects observing the same volume in space more than once addsadditional volume to the total volume observed. Due to efficiencies less than 1 objects inthe observed volume might not be spotted at the rst observation but at a subsequent one.Additionally, over time, parts of the observed volume or even the entire volume becomeindependent from the previous observation and all other observations made because of theshuffling effect that brings in objects from outside the accessible sphere. To distinguish

between the volume accessible by PS1 and the volume that was actually observed the latteris referred to as Independent Effectively Observed Volume (IEOV) and represented by V .It can be seen as a representative equivalent volume that was entirely observed with anefficiency of 1. The observational form of the theoretical equation 6.1 therefore reads

=N V

(6.2)

While N is unknown and has to be populated with a statistical estimator derived froma random sample (section 6.1) V can be determined numerically with the aid of a MOPSsimulation (section 6.2).6.1 Poisson Statistics of a Non-DetectionThe non-detection of ISOs in the PS1 survey is only a tiny random sample of a much largervolume and therefore has to be regarded as such. With the assumptions made about thedistribution and properties of ISOs in chapter 2 we can conclude that the number of ISOsdetected in the PS1 survey or any other similar survey follows a Poisson Distribution andcan therefore be represented by statistical estimator derived in this section.

The Poisson Distribution is a discrete approximation for the probability P (k) of a givennumber of events k happening in a xed interval of time or space if the average number of events occurring in a large dataset of repeated measurements conducted under identicalconditions is small.

P (k) = k

k!e (6.3)

The expected number of ISOs to be found if the volume V determined in section 6.2is observed we rearrange equation 6.3. It is irrelevant here if the PS1 survey is seen as arandom sample from a time series of a xed volume or a random sample volume taken froma much larger volume where the measurement series is xed at a certain time interval. This


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6.1. POISSON STATISTICS OF A NON-DETECTION 43

equality arises from the assumption that the distribution of ISOs in interstellar space acrossthe galaxy is homogenous and time invariant over a time scale much larger than the timeof observation. Solving the equation for with given probability P (k) for a certain k givesin this case simply

= ln (P (0)) (6.4)With only one measurement available, namely zero discoveries in the PS1 survey, the

probability of nding zero ISOs cannot be estimated very well. However, it can be impliedthat in case of a representative measurement with a probability referred to as condencel the expectation value does not exceed a certain value C.L. without predicting an out-come probability for a non-detection smaller than 1 l. In other words, with the concretecondence of l = 90%: If the expectation value of the number of ISOs in the observedvolume

V would exceed

C.L. = ln(1 l) = ln(1 0.9) = 2 .30 (6.5)the probability of detecting zero ISOs in a PS1-like survey would be less than 1 l =10%. This Poisson Probability Distribution for a condence of 90% is illustrated in gure

6.1.

Figure 6.1: Poisson distribution for = 2 .3 and a condence l of 90% respectively.

Increasing the condence limit (shifting C.L. to the right in gure 6.1) always results ina decrease of P (0). The probability of detecting zero ISOs becomes the smaller the higher


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the Condence Limit (C.L.) is. Substituting N with the derived Poisson probability of anon-detection equation 6.2 reads

C.L. = ln (1 l)V

(6.6)

6.2 Determination of V with a MOPS simulationTo determine V a simulation of the PS1 survey was conducted with MOPS (section 4.5)using the developed ISO model described in chapter 5 and all elds that were actuallyobserved by the PS1 survey. It is dened asV (, H ) =

H

T survey

V A

(r, ,,H , t ) dV dt dH (6.7)

where H is the absolute magnitude of ISOs at which the SFD is cut off, r is the positionvector in a heliocentric frame, is the set of orbit element distributions of ISOs, T surveyis the timespan of the PS1 survey, V A is the volume of the valid sphere, is the slopeparameter of the SFD and is the average ISO detection efficiency for objects positioned at

r given the other variables. While T survey , V A and are xed, and H are kept variable,which is why they are explicitly called out. To determine the number density limit as afunction of and H for practical reasons the simulation was conducted only once with aat SFD, all objects having an absolute magnitude of 0. Selected SFDs were then assignedto the output of the simulation as described in section 6.3.

It is apparent that the IEOV of a simulation

V S is approximately identical to

V if all

parameters given in equation 6.7 are approximately identical to the real survey. Consequen-tially, it is a requirement on the simulation that real elds and real efficiencies are admitted.Additionally, it is a requirement on the used ISO model that the normalised distribution of ISOs is identical to the one expected for the solar system in spatial and SFD-domain sinceeach specic depends upon the entire orbit element distribution. This is due to the factthat gravitational focusing distorts the otherwise homogenous distribution of ISOs aroundthe Sun. An effect not negligible since the majority of ISOs is detected at small heliocentricdistances where the number density can be almost 5 times as high as in interstellar spaceas illustrated in gure 5.12. With the requirements fullled

V S (, H )

V (, H ) (6.8)

where the index S denotes for all variables the origin from the PS1 survey simulationconducted by MOPS. V S can be determined from the simulation rearranging equation 6.2:V S (, H ) =

N S (, H )S (, H )

(6.9)

where S is the average density of V S . Due to the fact that the number density of ISOs in the model is much higher than the real number density it can be assumed for an


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6.2. DETERMINATION OF V WITH A MOPS SIMULATION 45accessible sphere of 50AU that

N S (, H ) = N S (, H ) (6.10)

at any given time, where N S is the count of discovered objects in the simulation. Witha known number density distribution of ISOs in the simulation the limit for the real numberdensity distribution can be determined combining equations 6.6, 6.9 and 6.10.

C.L. (, H ) = ln (1 l)N S (, H )

S (, H ) (6.11)

C.L. (, H ) and S (, H ) are average number densities integrated over V and V S , fol-lowing(, H ) =

V

(r, , H ) dV (6.12)

for xed parameters incorporated in V . With the assumption that in a heliocentricframe the number density distribution of ISOs for a given and H depends only on R (dueto gravitational focusing) the equation can be re-written as(, H ) = IS (, H )

V

f (R) dV (6.13)

where IS (, H ) is the number density of ISOs in interstellar space and

f (R) = (R,,H )IS (, H )

(6.14)

Substituting C.L. (, H ) and S (, H ) in equation 6.11 with equation 6.13 gives

IS C.L. (, H ) V

f (R) dV = ln (1 l)N S (, H )

S,IS (, H ) V

f S (R) dV (6.15)

Given the assumption that the orbit distribution is identical for real ISOs and theones in the simulation

f (R) f S (R) (6.16)

the integrals cancel out and the number density limit of ISOs in interstellar space isgiven as

IS C.L. (, H ) = ln (1 l)N S (, H ) S,IS (, H ) (6.17)Since the ISO efficiency is not implemented in MOPS the simulation contains all objects

that were in PS1 elds during the simulation. To get the number count of objects in thesimulation that would have been discovered eventually the probability for each object to be


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found has to be determined with the ISO efficiency given in equation 4.14 and summed uplike

N S (, H ) =i

1 N t

n i =1[ 1( t,nf (V i ) H(D n i 90) ) ] (6.18)

6.3 Assignment of a Size Frequency Distribution (SFD)

The number of objects discovered in the simulation is dependent on the SFD, which is notknown. We use common practice and assign SFDs following an exponential function as givenin equation 2.1 (section 2.1) for all combinations of and H in the ranges = 0 .1...0.8with steps of 0.05 and H = 0 ...20 with steps of full magnitudes.

Assigning a new absolute magnitude to an object also requires a correction of the ap-parent magnitude. Using the apparent magnitude according to Bowell et al. [ 2] it can becalculated with

V = H + 5 log10(R) 2.5log10((1 G)1 + G2) (6.19)where R is the heliocentric distance of the object, its topocentric distance and G the

phase curve of the bodys albedo (not the albedo itself!). G 0 for low-albedo objects andG 1 for high-albedo objects. As described in section 2.2 we use an albedo of p = 0 .04and therefore G is small.

1 = e A 1 tan ( 2 )

B 1

(6.20)

2 = e A 2 tan ( 2 )

B 2

(6.21)

with A1 = 3 .33, B1 = 0 .63, A2 = 1 .87 and B2 = 1 .22 and the phase angle (angle betweenEarth and Sun as seen from the object)

() = arccosr 2 2 R2

2 R (6.22)

where r is the distance between the Earth and Sun. For a xed position equation 6.19can be rewritten as

V = H + const. (6.23)

Since V is computed by MOPS for all objects with H = 0, denoted V 0 we can computethe new apparent magnitude with the simple relation

V (H ) = V 0 + H new (6.24)


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6.4. PRE-COMPUTATION OF DIGEST SCORES 47

where H new is the assigned absolute magnitude. To assign a SFD of the form given inequation 2.1 randomly to the output of the MOPS simulation the equation was invertedand fed with a random variable

H random = log10 X u+ H 0 (6.25)

where X u is a uniformly distributed random variable ranging from 0 to 1. ChoosingH 0 to be H the SFD will automatically be normalised to the number of objects in thesimulation resulting in a cutoff exactly at H as illustrated in gure 6.2. This particularmethod has the effect, that for small values of H more objects are distributed over a smallerrange and that therefore the statistics are better in comparison to distributions up to a largeH .

Figure 6.2: Normalised cumulative distribution of 10,000 randomly generated H -values according toequation 6.25 for the following -H combinations.

blue: = 0 .5, H = 11 green: = 0 .7, H = 11 red: = 0 .5, H = 19 orange: = 0 .7, H = 19.

6.4 Pre-computation of Digest Scores

The digest score is computed automatically by MOPS for each object in the simulation. Itis dependent i.a. on the apparent magnitude (section 4.3) and therefore not valid anymore


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once a new V is assigned to the object. New scores had to be computed 10 times for theentire ISO population (to improve statistics, see section 6.5). The digest score underlies acomplex algorithm which is implemented in a C code and not trivial to reproduce. It wasregarded as the most feasible solution to pre-compute digest scores for all object positionsfor full apparent magnitudes in the range H = 0 ...20 and store them in a look-up table.In the implementation of equation 6.18 the closest pre-computed digest score was used forD N i as approximation for the actual value. The resulting error here is negligible.

6.5 90% Condence Limit (C.L.)

To eventually compute the 90% C.L. for the number density of ISOs a Python script waswritten to loop over all -H combinations and for each of them over all tracklets generatedby the MOPS simulation (section 6.2), assigning them random H values according to themethod described in section 6.3. From the assigned H value the corrected apparent magni-tude V is computed with equation 6.24, which in combination with the tracklet metadataabout the used lter directly yields the corresponding tracklet efficiency (see section 4.4).Furthermore V is used to retrieve the closest match from the pre-computed digest table(section 6.4), which is fed together with the efficiency into equation 6.18. Combined effi-ciencies are computed according to equation 4.13 for tracklets originating from the sameobject (identical S3M ID, see gure 4.1) and subsequently summed up to obtain the to-tal number of objects that would have been discovered by PS1 in the simulation for theparticular -H combination.

To improve statistics the loop over all tracklets is repeated 10 times, assigning differentH values to the tracklets, retaining the same SFD. The resulting numbers for discoveredobjects are averaged and used as input for N S (, H ) in equation 6.17. For S,IS the modeldensity of 0.66AU 3 at 50AU was used as approximation for the model density in interstellar

space (see section 5.3). The result of the additional loops over all -H combinations is a2-dimensional array giving the 90% C.L. for the number density of ISOs depending on SFDslope parameter and SFD cutoff magnitude H , as illustrated in a colour-coded surface plotin gure 6.3. The repetition cycle of 10 was determined empirically so that the computednumber density limits have smooth transitions between different -H combinations. If therepetition cycle were too low the statistics would generate noise in gure 6.3. The sameprocedure was repeated correcting the H -values for the assumption of cometary activity asdescribed in section 2.2. The result is a tighter limit as shown in gure 6.4. Both outputarrays were plotted on the same colour scale for optimal comparison.

To give an idea of the statistical quantier the cumulative number of objects was plottedwhich has been detected in the simulation for each of the H combinations, separatelyfor inert and active objects (gures 6.5 and 6.6). As highlighted previously the statistics arethe better the lower and H . For the C.L. computation in this case the lowest detectednumber of objects was no less than 10. With the C.L. computed the IEOV is directly givenwith equation 6.6. It is illustrated in gures 6.7 and 6.8.


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6.5. 90% CONFIDENCE LIMIT (C.L.) 49

Figure 6.3: 90% condence limit of the number density of ISOs versus slope parameter and limitingabsolute magnitude H of the SFD, without implementation of cometary activity.

Figure 6.4: 90% condence limit of the number density of ISOs versus slope parameter and limitingabsolute magnitude H of the SFD, including the implementation of cometary activity.


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Figure 6.5: Cumulative number of detected ISOs in the PS1 survey simulation versus slope parameter and limiting absolute magnitude H of the SFD, without implementation of cometary activity.

Figure 6.6: Cumulative number of detected ISOs in the PS1 survey simulation versus slope parameter and limiting absolute magnitude H of the SFD, including the implementation of cometary activity.


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6.5. 90% CONFIDENCE LIMIT (C.L.) 51

Figure 6.7: IEOV of the PS1 survey versus slope parameter and limiting absolute magnitude H of theSFD, without implementation of cometary activity.

Figure 6.8: IEOV of the PS1 survey versus slope parameter and limiting absolute magnitude H of theSFD, including the implementation of cometary activity.


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52 CHAPTER 6. OBSERVATIONAL N

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Setting an Observational Upper Limit to the Number Density of Interstellar Objects

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