The Pan-STARRS Synthetic Solar System Model:

A tool for testing and efficiency determination of the

Moving Object Processing System

Tommy Grav

Department of Physics and Astronomy, Johns Hopkins University

tgrav@pha.jhu.edu

Robert Jedicke

Institute for Astronomy, University of Hawaii

jedicke@ifa.hawaii.edu

Larry Denneau

Institute for Astronomy, University of Hawaii

denneau@ifa.hawaii.edu

Steve Chesley

Jet Propulsion Laboratory, California Institute of Technology

steve.chesley@jpl.nasa.gov

Matthew J. Holman Harvard-Smithsonian Center for Astrophysics

mholman@cfa.harvard.edu

Timothy B. Spahr

Harvard-Smithsonian Center for Astrophysics

tspahr@cfa.harvard.edu

March 27, 2008

Submitted to Icarus.

Manuscript: 76 pages, with 2 tables and 23 figures.

Corresponding author:

Tommy Grav

Department of Physics and Astronomy

Johns Hopkins University

3400 N. Charles St.

Baltimore, MD 21218

Phone: (410) 516-7683

Email: tgrav@pha.jhu.edu

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Abstract

We present here the Pan-STARRS Moving Object Processing System (MOPS)

Synthetic Solar System Model (S3M), the first ever attempt at building a com-

prehensive flux-limited model of the small bodies in the solar system. The

model is made up of synthetic populations of near-Earth objects (NEOs with a

sub-population of Earth impactors), the main belt asteroids (MBAs), Trojans

of all planets from Venus through Neptune, Centaurs, trans-neptunian objects

(classical, resonant and scattered TNOs), long period comets (LPCs), and in-

terstellar comets (ICs). All of these populations are complete to a minimum

of V = 24.5, corresponding to approximately the expected limiting magnitude

for Pan-STARRS’sability to detect moving objects. The only exception to this

rule are the NEOs, which are complete to H = 25 (corresponding to objects of

about 50 meter in diameter).

The S3M provides an invaluable tool in the design and testing of the MOPS

software, and will also be used in the monitoring of the upcoming Pan-STARRS

all-sky survey, scheduled to start science operations in the fall of 2008.

1 Introduction

The first asteroid was discovered over two centuries ago, and since then as-

tronomers have spent untold hours trying to catalog as many of these wander-

ing stars as possible. By the turn of the 20th century slightly more than 330

asteroids were known, mostly discovered by Max Wolf. He pioneered the use of

astrophotography, a technique in which the asteroids appeared as short streaks

on photographic plates with long exposures. While this dramatically increased

the number of discoveries, only a few thousand asteroids had well established

orbits at the end of the 20th century, as most astronomers had little interest in

these vermin of the sky. It was the increased interest in the potential hazardous

asteroids (PHA) that launched another significant increase in the discovery rate

of the asteroid population of the Solar System. The PHAs have orbits which

frequently take them close to the Earth and in time could collide with it. Public

awareness of this asteroid threat resulted in a mandate from the US Congress

to the National Aeronautics and Space Administration (NASA) to search, find

and catalog 90 percent of Near-Earth Objects (NEO) with diameters larger

than 1 kilometer (Morrison, 1992). The current estimate is that this goal will

be achieved some time after the 2008 deadline set forth by Congress and NASA

(Jedicke et al., 2003). Using wide-field charge-coupled devices (CCDs) and au-

tomated reduction pipelines, a number of ground based survey telescopes (like

Spacewatch, the Near-Earth Asteroid Tracking, the Lincoln Near-Earth Aster-

oid Research and the Catalina Sky Survey) have increased the discovery rate

to over 5,000 new asteroids per month (Stokes et al., 2002). Currently, as of

the start of 2008, over 170,000 asteroids have well-defined orbits and have been

numbered (only about 14,000 of these have been named).

These wide-field surveys have given a wealth of new information on the

minor planets of the solar system, but the surveys have either been limited by

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the depth they can reach, or in the area they can cover. In 2001 the astronomy

community of the US issued a decadal review (Astronomy and Astrophysics

Survey Committee et al., 2001) that recommended the implementation of a

large synoptic survey telescope to survey the visible sky every week to a fainter

limit than reached by any existing survey. The goal of this telescope would be

to catalog over 90 percent of the NEOs larger than 300 meters in diameter and

thus assess the threat these objects could pose to Earth’s population. Such a

telescope would also discover an order of magnitude more asteroids for most

other solar system populations than is currently known, as well as perform an

almost unlimited number of projects ranging from studies of the local solar

neighborhood to cosmology. With this recommendation a new paradigm has

come upon the astronomy and astrophysical community, where all-sky surveys

will be able to penetrate significantly deeper and cover significantly larger areas

than before.

The Panoramic Survey Telescope and Rapid Response System (Pan-STARRS;

Hodapp et al., 2004) is one of several different implementations of this next

generation of all-sky surveys. Pan-STARRS is a two-step project: The first

step, a 1.8-meter telescope, called Pan-STARRS 1 (PS1), has been built on

Haleakela, Hawaii. It is to be operated by the Pan-STARRS 1 Science Consor-

tium (www.ps1sc.org) and will begin its all-sky survey during the fall of 2008.

It features a giga-pixel camera (GPC1; with 1.44 billion pixels) with a field of

view of ∼ 7 sq. degrees, which gives a pixel-scale of 0.3 arcseconds. The camera

uses a new CCD technology called Orthogonal Transfer Arrays (OTA; Tonry

et al., 1997, 2006; Burke et al., 2007) that allow charges to move in both spa-

tial directions in real time to compensate for the image motion caused by the

atmosphere and tracking inaccuracies of the telescope. The OTA-technology

thus provides a tip-tilt correction in the electronics rather than through manip-

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ulation of the secondary mirror, allowing for sub-arcsecond seeing over the full

field of view, rather than just inside the isoplanatic angle around the center of

the field. About 60 percent of the available observing time will be dedicated to

observing almost the entire visible sky using the Sloan grizy-filters in what is

called the 3π survey. This survey will cover ∼ 10, 000 sq. degrees each week,

reach a limiting magnitude of ∼ 22.7 in the three main filters (gri), and the

cadence of the observations are designed with the discovery of moving objects

in mind. This gives the system the ability to detect, track and catalog an un-

precedented set of solar system bodies, from the fast moving near-Earth objects

(NEOs) to the slow moving distant trans-neptunian populations (TNOs). The

second part of the project is designated Pan-STARRS 4 (PS4) and consists of

four telescopes of the same size as PS1, each equipped with its own giga-pixel

camera with the same field of view as GPC1. This collection of telescopes will

simultaneously observe the same field of the sky, and coadding of the images

will make it possible to reach a limiting magnitude of ∼ 24.5 in the three main

Sloan filters (gri). PS4 will thus perform the same survey as PS1, but with a

deeper limiting magnitude and a project lifetime two to three times as long.

Each image taken with the GPC1 requires ∼ 2 gigabytes of storage and with

an exposure time of ∼ 30 seconds the survey will accumulate ∼ 1 Terabyte of

data per night. In order to quickly handle and reduce this enormous amount

of data a state-of-the-art software pipeline has been created. First the images

are processed by the Image Processing Pipeline (IPP; Magnier, 2006), which

detect any transient source available in the images through the use of difference

imaging. These transient detections are then handed to a number of science

clients looking for moving objects, supernovae, gamma-ray bursts and other

transient phenomena. The science client in charge of handling moving solar

system objects is the Moving Object Processing System (MOPS; Kubica et al.,

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2007). This fully automatic pipeline is designed to discover, link, track and

catalog 90 percent of the potentially hazardous objects (PHOs) and 80 percent

of all other known populations of small objects in the solar system, if they have

a minimum of two detections on at least three nights during a lunation (note

that a MOPS lunation runs from full moon to full moon, rather than the more

common definition from new moon to new moon). It is expected that the survey

will produce on the order of∼ 5 million new solar system objects, thus increasing

the number of known moving objects by at least an order of magnitude. This

will provide an unprecedented opportunity to understand not only the current

dynamical state of the solar system, but also will provide a window into its

formation and evolution. The Pan-STARRS 1 survey is expected to increase

the number of known NEOs, jovian Trojans and trans-neptunian objects to

rival the number of currently known main belt asteroids (MBAs), finally making

it possible to understand these populations to the same degree that we today

understand the main belt.

One of the major difficulties encountered in interpreting the data collected

during a survey of minor planets is understanding how selection effects have

contributed in generating the sample. This bias is made up of a complex set of

factors dependent on, among others, the physical and dynamical properties of

the asteroids, the characteristics of the detector and telescope, the abilities of

the software, and the decisions of the observers. In order to determine the real

underlying population of objects observed during the survey, a correction must

be determined to remove this bias from the raw survey results. An overview

of the main selection effects and the mathematical basis used in correcting for

them can be found in Jedicke et al. (2002). It is clear that a better under-

standing of the dynamical and physical distributions of the many populations

of minor planets in the solar system can only be achieved with proper debiasing

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techniques. In order to understand the selection effects of the Pan-STARRS

survey and determine any corrections, we have created a model of the minor

planets in the solar system called the PS Synthetic Solar System Model (S3M).

This model will be used to generate synthetic detections in all images taken dur-

ing the surveys. These detections will be processed along with the detections

handed to the MOPS by the IPP. They can thus be used to gauge the pipeline’s

ability to link detections, determine orbits and track the objects throughout the

lifetime of the survey. This provides us with not only a way to monitor that

the pipeline is functioning properly, but it also provides us an important tool in

debiasing the surveys.

The S3M also provides an important tool in developing and testing the

software needed to automatically process the volume moving object detections

that is expected with Pan-STARRS (or other surveys like LSST). By creating a

synthetic survey simulation that mimics the expected survey pattern of the Pan-

STARRS project, detections of the S3M objects can be generated for each field

observed. These detections can then be used to test the efficiency of the MOPS

in linking, orbit determination and tracking the different populations of minor

planets under a number of varying circumstances (like astrometric accuracy,

limiting magnitudes, noise in the detection set, and so forth). Thus a tested

and properly tuned pipeline can be created, ready to handle the enormous data

volume expected in future surveys.

This work presents, to our knowledge, the only attempt at a comprehensive

model of every solar system object visible to a limiting magnitude of V ∼ 24.5.

We are aware of only one other attempt to derive a model describing the orbital

elements of every object in a population. Tedesco et al. (2005) created a model

for the main belt asteroids, named the Statistical Asteroid Model (SAM). The

model presented in this paper is much more extensive than SAM, which was

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limited to the MBAs, given that we include models for nearly every known

population in the solar system. We present in this paper the description of the

procedures used in creating the current version of the S3M. While it has some

limitations and known deficiencies, for which work is underway to correct, we

are publishing this version to document the results used in in studies already

published (Durech et al., 2005; Kubica et al., 2007; Milani et al., 2006, 2008),

as well as to document the underlaying data sets used in the development and

testing of the MOPS software.

2 The MOPS Synthetic Solar System Model

One of the main scientific goals of the Pan-STARRS surveys is to identify and

track 90 percent of the potential hazardous asteroids (PHAs) larger than 300

meters in diameter within its full operational lifetime. As this goal is achieved,

the survey will also find and track several millions of other solar system aster-

oids, increasing the number of known moving objects by at least an order of

magnitude. Currently, there are about a quarter of a million known asteroids

and comets in the solar system, ranging from the near-Earth objects (NEOs)

that frequently cross the orbit of our own Earth, to the distant trans-neptunian

objects (TNOs) that orbit in the outer reaches beyond Neptune. Some popu-

lations, like the main belt asteroids (MBAs) have been extensively tracked and

cataloged, while others have only a few hundred known members (like the NEOs,

Centaurs, and short period comets). In this paper the most recent knowledge of

the known objects in the solar system and the dynamical and physical processes

that shaped their orbits are used to generate a set of synthetic populations.

These populations are modeled as accurately as possible to help in testing the

pipeline, as well as to determine the efficiency of the survey in detecting and

tracking the different populations.

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Each object in the Synthetic Solar System Model (S3M) consists of a set of

orbital elements (in this case represented by perihelion distance, q, eccentricity,

e, inclination, i, longitude of ascending node, Ω, argument of perihelion, ω,

and time of perihelion passage, T ) and an absolute magnitude, H. It should be

noted that we have made no attempts to add physical parameters in this version

of the S3M. To simplify the testing of the MOPS we assume that all synthetic

objects have no rotational light curves, follow a phase brightening according to

G = 0.15, and have similar broadband colors and albedos. Physical properties

will be added to a future version of our S3M and are considered to be beyond

the scope of this paper.

The synthetic populations generated in this model are complete to at least

V ∼ 24.5, except for the NEOs, which due to their close approaches to the

Earth have been made complete to H ∼ 25. In order to convert the apparent

magnitude limit to the corresponding absolute magnitude, HV , we have used

the equation:

HV = V − 5 log(∆r)− P (α)− δV, (1)

where V is the apparent visual magnitude, ∆ is the geocentric distance, r is

the heliocentric distance, P (α) is the phase light curve function, α is the phase

angle and δV is variations in brightness caused by rotational effects of the body

itself. To generate limiting absolute magnitudes we assume that P (α) = 0 and

δV = 0. When necessary, transformation from absolute magnitudes, HV , to

diameters in kilometers, D, can be done using (Bowell et al., 1989)

logD = 3.130− 0.5logA− 0.2HV (2)

where A is the albedo.

The S3M presented here has been created using a large number of differ-

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ent methods and it is important to note that one should in no way expect at

this time that the entire S3M has to be created from a single model. Such a

task may not even be possible within our lifetimes, although it is hoped that

Pan-STARRS and MOPS will contribute significantly to the creation of such

a unifying model. Our approach to the modeling of the various solar system

populations is presented in the following section, roughly in order of heliocentric

distance.

2.1 The Near-Earth Objects (NEOs)

The near-Earth objects (NEOs) are objects whose orbits may take them close

to the Earth. Usually the qualifying criterion is a perihelion distance of q < 1.3.

What makes these objects so interesting is of course the realization that through

perturbations they could evolve onto orbits that would take them on a path

towards a collision with the Earth. Currently there are a little over 5, 000 known

NEOs, of which only ∼ 800 have orbits sufficiently well known to have been

numbered by the Minor Planet Center (MPC). Only these numbered objects

have accurate enough orbits to allow for predictions of future encounters with

the Earth. It is estimated that PS1 will be able to find ∼ 5, 000 new NEOs

during its three years of operations, most of which will have orbits accurate

enough to predict their orbital evolution for the next century.

The position of the known NEOs, as seen from above the ecliptic, is shown

in Figure 1. The significant clustering of objects close to the Earth is caused by

observational bias, since short distances makes it possible to see the more nu-

merous small NEOs. This shows the need to derive an unbiased NEO population

to properly test the MOPS.

The synthetic NEO population in the S3M is based on the orbital element

and absolute magnitude frequency results of Bottke et al. (2002). They created

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a set of theoretical time-residence distributions for objects in the phase space

occupied by NEOs. These distributions derived dynamically from each of five

different source populations for the NEOs: 1) the ν6 secular resonance; 2) the 3:1

mean-motion resonance (MMR) with Jupiter; 3) the intermediate Mars crossing

(IMC) region, where numerous small resonances exist; 4) the outer main belt

(OMB); and 5) the Jupiter family comet population (JFCs). Test objects were

started in reasonable locations within each region and numerically evolved until

they either were ejected from the solar system or they hit a planet or the Sun.

The time-residence distribution, Rk(a, e, i), for source region k is simply the

fraction of time that objects from each region spent in a bin located at semi-

major axis a, eccentricity e and inclination i.

They then developed a simulation of the Spacewatch survey to determine the

bias, B(a, e, i,H), for detection of objects with orbital elements (a, e, i) and also

as a function of the absolute magnitude H. Letting n(a, e, i,H) represent the

number of objects actually detected by Spacewatch as a function of the orbital

elements and absolute magnitude, and assuming that the true H-distribution of

NEOs is proportional to 10αH , the number predicted for each bin by the Bottke

model is:

n′(a, e, i,H) = B(a, e, i,H)N010αH∑k

akRk(a, e, i) (3)

where ak are weighting coefficients for the contribution of each source region to

the NEOs (note that the sum of all ak is 1). It is assumed that the absolute

magnitude distribution is independent of the (a, e, i) and there is currently little

reason to believe that this is incorrect.

Bottke et al. (2002) fitted this equation for the parameters α and ak to

the actual distribution of objects observed by Spacewatch, n(a, e, i,H), and

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produced an unbiased model MNEO for the near-Earth object population:

MNEO(a, e, i,H) = N010αH∑k

akRk(a, e, i) (4)

Due to the lack of Spacewatch data and their inability to properly model

the discovery bias for objects with H > 24 for that project, Bottke et al. (2002)

could not estimate the size of the population below that limit. We simply ex-

trapolate these results out to our model limit of H > 25. Note that Rabinowitz

et al. (2000) found a much steeper size distribution beyond H > 24, based on

direct debiasing of observational data from Spacewatch and NEAT (without any

theoretical modeling). Their result is applicable to H = 31 and it is in agree-

ment with the Bottke et al. (2002) result for H < 24. We elect to disregard this

result as we are only extending the Bottke model by one magnitude and the

difference in result is quite limited. Should we go on to extend our synthetic

population to fainter absolute magnitudes in the future, it would be prudent to

revisit this issue.

Thus, the Pan-STARRS near-Earth object model population adopts the Bot-

tke et al. (2002) model for orbital elements regardless of absolute magnitude:

N(H) ∼ 100.35H (5)

The absolute magnitude model is chosen such that the cumulative number of

objects brighter than H = 18 is 960, which matches the Bottke et al. (2002)

model and other contemporary estimates (Stuart and Binzel, 2004).

Figure 2 shows our synthetic distribution of NEOs based on the Bottke et al.

(2002) model as well as the known distribution of NEOs (as of December 2007;

Spahr, personal communication). It can be seen from this figure that the model

is in fair agreement with the observed population. The differences in eccentricity

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and inclination can easily be explained as effects caused by biases in the observed

sample. Because high eccentricity objects spend more time at larger distances

from the Sun, they are less likely to be observed by current surveys. This leads

to an excess of low eccentricity objects in the known sample of NEOs. The

difference in inclination distribution is due to the fact that most surveys are

constrained to observing close to the ecliptic, which causes the apparent excess

of known NEOs with low inclination. Perhaps the biggest problem with our

synthetic model is the lack of the small NEOs as can be seen in the absolute

magnitude panel of Figure 2. Current surveys have found several hundred NEOs

with H > 25, objects that do not currently exist in our synthetic population.

However, there is no reason to believe that there is any difference in the orbital

element distribution of large and small NEOs, the current synthetic population

will provide an adequate test of the capability of MOPS to find and catalog

NEOs.

2.1.1 Earth Impactors

One of the main scientific goals of the Pan-STARRS project and other next

generation all-sky surveys is the discovery of asteroids that have the potential

to threaten the Earth. But discovery is only half of the challenge. One must also

be able to rapidly recognize the object as a potential threat in order to assure

adequate tracking and, in the event that an impact is confirmed, sufficient time

for a suitable mitigation effort. To this end, after the discovery of a new Earth-

crossing asteroid, the trajectory and associated uncertainty region of the object

are routinely searched for future possibilities of impact (Milani et al., 2005).

It should be noted here that Earth impactors are a subset of the larger

near-Earth population, but there is no certainty that any of these will provide

a possible impact for the next century. In order to predict the performance of

the survey in detecting and recognizing actual impactors, we have decided to

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add to our solar system model a great number of simulated Earth impactors

with impact dates that span from outset of the survey to many decades into the

future. This will allow for the understanding of both the short- and long-term

warning capability provided by the survey.

To make the analysis more realistic, we want our synthetic impactors to be

on orbits that are consistent with the orbital distribution of the actual impact-

ing population. To this end we have derived 10,006 impacting orbits sampled

according to the presumed distribution of impactors. The impacts with Earth

are uniformly distributed from 2010 to 2110. During these 100 years, the Earth

is struck about twice per week on average by this synthetic impactor population.

Impacting asteroids have orbits that are systematically different from those

of the near-Earth population in general. This is because their orbits must pro-

vide both a possibility and a propensity for impact. Our approach for deriving

impactors was first described by Chesley and Spahr (2004) and consists of three

steps. First we begin with the debiased near-Earth population model developed

by Bottke et al. (2002) and described above. We use a dataset of 106 NEO orbits

consisting of semimajor axis a, eccentricity e and inclination i. Objects were

randomly drawn from this population and then slightly fuzzed by adding random

offsets uniformly distributed over ±0.02 AU in perihelion distance q, ±0.02 in

e and ±0.1 in i, always minding that q > 0, 0 < e < 1 and 0 < i < 180. The

argument of perihelion ω and longitude of ascending node Ω were selected with

uniform distribution. This step yielded an oriented NEA orbit: (a, e, i, ω,Ω).

The next step is to determine whether the NEO orbit obtained in the pre-

vious step is a potential impactor. Specifically, the MOID of the object must

be less than its Earth capture cross-section, both terms that will now be de-

fined. The minimum separation between the orbits of the asteroid and Earth

is known as the Minimum Orbital Intersection Distance, or MOID. Here orbit

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is used to refer simply to an osculating ellipse in space, without regard to the

position of the asteroid or Earth along its orbit. The term Earth capture cross-

section, or b⊕, alludes to the fact that the Earth’s gravity can perturb what

otherwise would be a near miss into a collision. The extent of the Earth capture

cross-section, b⊕, is the maximum unperturbed encounter distance that will lead

to impact after the focusing effect of the Earth’s gravitational perturbation is

applied. In other words, it is the impact zone around the Earth, and is larger

than the radius of the Earth by an amount that depends on the unperturbed

encounter velocity v∞ according to

b⊕ = r⊕

√1 +

v2e

v∞(6)

where r⊕ is the Earth’s radius and v2e = 2GM⊕/r⊕ is the square of the Earth’s

escape velocity. If MOID < b⊕ then a collision is possible and the object is

logged as a potential impactor. Otherwise, the object is discarded and a new

NEO orbit is drawn.

In the final step we sample from among the potential impactors according to

their propensity for impact. This propensity is quantified by F , the fraction of

time that a potential impactor can be found within a distance b⊕ of the Earth’s

orbit. In other words, a torus of radius b⊕ is formed around the Earth’s orbit and

F is the fraction of time that the object resides within the torus. Objects with

high-crossing velocities, due generally to some combination of high eccentricity

and high inclination, will tend to spend little time in a capture torus of smaller

volume (due to the smaller b⊕), and are correspondingly unlikely to impact.

Conversely, low-e and low-i objects will cross the torus at less oblique angles

and with lower relative velocities, increasing the chance that the Earth will

be in the way for an impact. Moreover, the lower v∞ objects have higher b⊕

and therefore a corresponding larger volume torus around the Earth’s orbit.

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Figure 3 can be used to compare the orbital distributions of the potential and

sampled impactors. If the potential impactor is selected through an F -weighted

sampling, then we log it as a simulated impactor. The 10,000 sampled impactors

were obtained from among approximately 1.3× 105 potential impactors, which

were in turn derived from about 5.1× 108 NEA samples.

The steps taken in selecting the simulated impactors used very simple two-

and three-body dynamical models. The final step consists of refining the im-

pactor orbits to ensure that they indeed impact with a high-fidelity dynamical

model, including relativistic effects and the perturbations of all eight planets

and the Earth’s moon, as well as Ceres, Pallas, Vesta and Pluto. In the process,

the asteroids are targeted to a randomly selected point on the disk formed by

the Earth capture cross-section. As an example, Figure 4, depicts the impactor

locations over North America and Hawaii.

2.2 The Main Belt Asteroids

The main belt asteroids (MBAs) are roughly identified as objects with orbits

between Mars and Jupiter. They are believed to have formed there in the

early primordial solar nebula, with the gravitational perturbations from the

neighboring planets preventing them from accreting into a planet. Most of the

mass has since been lost, leading to the 4 largest objects ((1) Ceres, (2) Pallas,

(4) Vesta and (10) Hygiea) containing more than half the mass. The orbits of

the MBAs are strongly sculpted by mean-motion resonances with Jupiter, with

the Kirkwood Gaps being the most visible result. Another important feature

is the asteroid families created when larger asteroids break up due to impacts

from smaller asteroids.

In this paper we define the MBAs as those objects that have semi-major axis

1.8AU ≤ a ≤ 4.1AU and a perihelion distance q > 1.666AU. There are currently

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∼ 334, 000 known main belt asteroids, of which ∼ 297, 000 have multi-opposition

orbital arcs. The orbital distributions of the known main belt asteroids are

shown in Figure 5, where the Kirkwood Gaps at 2.5, 2.82, 2.95 and 3.2AU, the

Hungaria family (a ∼ 1.9AU, i ∼ 20), the Phocaea family (a ∼ 2.25− 2.5AU,

i ∼ 25) and the Hilda family (a ∼ 3.95AU) are easily identified. The absolute

magnitude frequency distribution of the MBAs has several features and is shown

in Figure 6. The hump seen for H < 14.5 is believed to be real and a consequence

of collisional evolution among the MBAs (Davis et al., 2002). The sharp drop for

H > 15.5 is most likely due to the fact that asteroids in this absolute magnitude

range are, usually, too faint for contemporary astroid surveys to detect.

The current sample of known MBAs is dramatically affected by observational

biases that skew the distribution towards small distances in semi-major axis

compared to the underlying actual distribution. This is caused by two main

effects: 1) the asteroids on the inner edge of the belt are closer to the Sun and

the Earth and are therefore much brighter than asteroids with the same size

in the outer belt; and 2) the inner edge of the belt is also dominated by the

brighter S-complex asteroids, which in general have higher albedos than the C-

complex asteroids that dominate the outer belt. Similar observational effects are

also major factors in the observed distributions of eccentricity and inclination.

Examples of this are the Hungaria and Phocaea families, which both represent

major groups in the set of known observed MBAs due to these observational

biases, but more than likely they are relatively minor sub-populations in the

actual distribution of MBAs. The three angular orbital elements (longitude of

ascending node, Ω, argument of perihelion, ω and mean anomaly, M) also have a

number of interlinked dependencies that are due both to real and observational

selection effects. The distribution of the longitude of the ascending node can be

fit to a sinusoidal function with a phase shift of ∼ 10. The distribution of the

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argument of perihelion is more or less random, although a slightly sloping linear

fit, with increasing number of objects at increasing argument of perihelion,

can be found at the 1.5σ level. The shape of the distribution of longitude

of perihelion (ω = Ω + ω), and thus of the longitude of ascending node and

argument of perihelion, is driven by perturbations by Jupiter.

It is currently believed that the sample of known MBAs is nearly complete for

H < 14.5 (which for a Hilda-type MBA would correspond to a visual apparent

magnitude of V ∼ 20), based on the lack of new discoveries of asteroids in that

range (Jedicke et al., 2002). This sub-sample contains ∼ 44, 000 objects and

thus makes up about 15% of the known MBAs with multi-opposition orbits.

Assuming that there is no or little dependence of the orbital distribution of

MBAs with their absolute magnitude, the distribution of orbital elements for

objects with H < 14.5 should be representative of the unbiased underlaying

distribution of orbital elements for the MBAs (see Figure 6). It should be noted

that some recent work by Yoshida et al. (2003) has shown hints of a correlation

between size and orbital distribution. They claim that there is a depletion of

sub-kilometer asteroids in the outer parts of the belt compared to the inner.

They further claim that there are more small asteroids in the outer belt than

larger asteroids.

The Pan-STARRS synthetic model of the MBAs should reproduce the main

characteristics of the known unbiased MBAs and this distribution is exceedingly

complicated, as evidenced by the description above. In order to reduce the

number of objects in the model, we elected to limit ourselves to those objects

that are detectable by Pan-STARRS, that is those that can achieve an apparent

magnitude of V < 24.5.

In order to generate each member of our synthetic population of the MBAs

that re-creates all its interdependencies we used the following prodecure: 1) a

16

known MBA with H < 14.5 was selected randomly; 2) each orbital element (x)

was randomly smeared to a new value in the range [x− dx, x+ dx] where dx for

each orbital element is specified in Table 1. These values where chosen such that

the basic shape of each distribution was preserved. It should be noted that this

smearing of the orbital elements does to a certain degree smear out the Kirkwood

Gaps and other resonances, but this has little or no effect on the modeling

the performance of the Pan-STARRS telescope or the MOPS software; 3) the

maximum absolute magnitude (Hmax) at which this object might be visible to

Pan-STARRS assuming a limiting magnitude of V = 24.5 was determined; 4)

the total number of objects among the MBAs with N(H < Hmax) according

to the cumulative distribution given in Figure 8 was determined; 5) an absolute

magnitude for this synthetic object with H < Hmax that follows the cumulative

distribution determined in the previous step was randomly generated. Since

we expect that there are on the order of 107 within reach of Pan-STARRS

the procedure was repeated 107 times in order to generate the full synthetic

population of MBAs. While a smaller sample of MBAs would be enough for

the project to test the ability of MOPS to link and determine orbits, the large

sample of 107 objects ensures that MOPS can in fact handle the volume of

detections and orbits expected with the full Pan-STARRS system. Since the

MBAs carries a signature of interaction with Jupiter the epoch of these orbits is

important and is equal to the epoch for which the population of known MBAs

with H < 14.5 was extracted.

In Figures 7 and 8 our synthetic MBA population is compared to the bright

(H < 14.5) and entire known populations. We see that the model does in fact

retain the most important features of the MBAs. While our smearing of the

orbital elements in the generation of our synthetic population does smear out

the Kirkwood Gaps (as well as other resonant features) they are still fairly easily

17

identified. Furthermore, the most important families, such as the Hungarias and

the Hildas, have also been preserved. Finally, the synthetic population seems to

be more properly debiased, with a larger number of objects in the outer main

belt compared to the known sample.

2.3 The Jovian Trojans

A large number of asteroids share the orbit of the planet Jupiter, having been

captured into its 1:1 mean-motion resonance. These asteroids are found to orbit

one of the two Langrangian points of stability, L4 and L5, found ±60 along the

planets orbit, when seen in a heliocentric coordinate system that rotates with

Jupiter. The first of these asteroids was discovered just over a century ago by

Max Wolf, who named it (588) Achilles after the hero of the epic poem The

Iliad. Subsequent asteroids in similar orbits were also given names from the

poem and the group as a whole became known as the Trojans.

As of December 2007 a little over 2, 000 Trojans are known, of which about

half have reliable orbits and have been numbered. The total population is pre-

sumed to be significantly larger, perhaps on the same order as the main belt

asteroids (Shoemaker et al., 1989). The position of the Trojans relative to

Jupiter librate around the L4 (leading swarm) and L5 (trailing swarm), with

periods on the order of a few hundred years. They have small eccentricities

(e < 0.2) and most have moderate inclinations (i < 20, although a few known

Trojans have inclinations larger than 30). It has further been found that there

are significantly more asteroids in the L4 swarm compared to L5, with a ratio

of NL4/NL5 = 1.6± 0.1 (Szabo et al., 2007). This effect is attributed to obser-

vational bias by some, while others argue that it is a real effect inherent in the

underlying population.

We base our synthetic population of Trojans on the published results of the

18

Sloan Digital Sky Survey (SDSS), which is the largest well-calibrated survey

of these objects to date. Using the third release of the SDSS Moving Object

Catalog, Szabo et al. (2007) studied about 860 unique candidate Trojans. This

represents a complete sample with a limiting magnitude of V = 21.4 (r = 21.2,

corresponding to an absolute brightness of H = 13.8 or about 10 km in diameter

assuming an albedo of 0.04). Out of the 860 objects only 313 have known

orbits and will thus form the basis of our distribution of orbital elements. It is

presumed that the known Trojans are complete to H ∼ 12, so this magnitude

limit is used to determine the differential absolute magnitude distribution, given

by:

log(N) = 2.9 + α(H − 12) (7)

with α = 0.44 ± 0.05. Jewitt et al. (2000) and Yoshida and Nakamura (2005)

arrive at similar results in their surveys. Using the limiting magnitude of Pan-

STARRS, V ∼ 24.5, the average distance of r ∼ 5.2 and ∆ ∼ 4.2 gives a

Hlim ∼ 17.8, corresponding to a diameter of ∼ 2 kilometers when assuming an

albedo of 0.04. This yields an absolute number of about 280, 000 objects bright

enough to be observed by Pan-STARRS. This represents the second largest

single population in the MOPS Synthetic Solar System Model, behind the 10

million objects in the main belt model population.

In order to generate this large population from the sample of 313 SSDS

objects with known orbits, we decided to generate 10, 000 clones. Each clone

was generated by randomly selecting one of the SSDS objects, then shifting the

orbital elements randomly inside some range [x−dx, x+dx] with shifts given in

Table 1. These clones where then integrated for 1 billion years using a standard

Wisdom-Holman symplectic integrator (Wisdom and Holman, 1991). Almost

all of the clones, 9, 389 objects, survived the entire integration as Trojans. We

extract 180 outputs for these surviving clones, where each output were 5 mil-

19

lion years apart. This is then used to generate a normalized time-residence

distribution of the Trojans in (a, e, i) and (Ω,M,L) phase space, where L is

the mean longitude of an object with respect to Jupiter. For each generated

synthetic object we randomly select a bin in the (a, e, i) phase space, weighted

by the time-residence density. Similarly a bin in the (Ω,M,L) phase space was

randomly selected, again weighted by the time-residence density. From this the

exact orbital elements are chosen randomly within the bins. Note that due to

this population’s strong dependence on the mean longitude of Jupiter, Ljup,

the angular elements are given using the mean longitude of objects with re-

spect to the planet L, rather than the usual argument of perihelion. In order

to determine the argument of perihelion for each object ω = L−Ω−M can be

used. An absolute magnitude is then chosen using Equation 7. This procedure

is repeated until 280, 000 Trojans have been generated. While this procedure

was used we insured that the L4 and L5 swarms contained the appropriate 1.6:1

ratio of objects by counting up the number of selected objects in each swarm

and discarding objects from a swarm after it was filled up.

From Figure 10 and 11 it is seen that our synthetic population is a fair

approximation to the distribution of the known Trojans. The biggest difference

is seen in the inclination distribution, where our synthetic population contains

more high inclination objects than the current set of known Trojans. This is

most likely due to the fact that the known population has a heavy bias against

the detection of high inclination objects, since most deep surveys are performed

close to the ecliptic to maximize the number of objects expected (Jewitt et al.,

2000; Yoshida and Nakamura, 2005). However, if we look at the sample of

known Trojans with H < 12, which we believe to represent a complete sample,

it is seen that our inclination distribution is in very good agreement (see Figure

12).

20

2.3.1 Trojans of Other Planets

Besides Jupiter, currently only Mars and Neptune have populations known to

be in their 1:1 mean-motion resonances, both with very limited samples. The

Minor Planet Center recognizes four martian trojans, one in the leading L4

swarm and three in the trailing L5 swarm. Additional trojans of Mars have

been suggested (Connors et al., 2005), but the strong perturbations of the other

planets makes it difficult to confirm these until better orbits can be determined.

For the neptunian trojans the MPC lists five known objects, all in the leading

L4 swarm (Sheppard and Trujillo, 2006). We thus avoid trying to infer these

populations based on the limited current knowledge about them. Instead, we

elect to use our knowledge of the larger sample of jovian trojans described above.

For each planet in the solar system we use the same generation scheme as for our

synthetic population of jovian trojans, then scale and rotate the orbits such that

their semi-major axis and mean longitude is appropriate for their new parent

planet. It should be noted that this of course leads to a number of potentially

unstable objects that quickly can be perturbed away from their trojan orbits.

While this can be viewed as a problem, it actually helps the testing of the MOPS.

One of the science goals of the Pan-STARRS project is to identify trojans of any

planet in the solar system, even though these may only be short-time stable.

The inclusion of a model for trojan populations will insure that MOPS is able

to detect and track these objects. It will also help create the necessary tools to

identifying these trojans among the hundreds of thousands of objects detected

by the pipeline and subsequently for analyzing their stability and dynamical

behavior.

To the impact these hypothetical populations have on the processing of the

S3M while still containing a large enough sample to get accurate statistical

results, we have elected to limit the number of objects in these populations to

21

10, 000 for the terrestrial planets (Mercury, Venus, Earth and Mars) and 20, 000

for the outer planets (Saturn, Uranus and Neptune). It should be noted that

our synthetic population of resonant trans-neptunian objects (see Section 2.5)

also contains a small number of additional neptunian trojans.

2.4 The Centaurs

The Centaurs are a population of icy bodies that orbit between Jupiter and

Neptune, frequently crossing the orbits of the giant planets. The first centaur,

(2060) Chiron, was discovered in 1977 (Kowal et al., 1979). Initially classified

as an asteroid, it was later identified as a comet and is one of the few objects

that also has a cometary designation, 95P/Chiron. Since then, between 50 and

100 Centaurs have been discovered, depending on the definitions used. The

Centaurs are probably the least well-studied population of small bodies in the

solar system. Recent research suggests that they have been ejected from the

trans-neptunian (Levison and Duncan, 1997) and Oort cloud (Emel’Yanenko

et al., 2005) populations, and are currently on extremely chaotic orbits that are

heavily influenced by the giant planets. The dynamical lifetime of objects in

this region of the solar system is on the order of a few million years as they get

kicked into the inner solar system, ejected from the solar system, or collide with

a planet (Horner et al., 2004a,b).

In this paper we define the Centaurs as objects that have 5.5 ≤ a ≤ 35AU

and q > 5.2AU and our model is based on the work of Jedicke and Herron (1997).

They used the results of Duncan et al. (1995) to generate a theoretical centaur

model. Duncan et al. (1995) studied the dynamical structure of the Kuiper Belt

and produced a number of objects that were perturbed giant planet crossing

orbits. These were integrated until they were either ejected from the solar

system or suffered a close approach to a planet. This time-residence distribution

22

of the Centaurs in (a, e, i) space provides the steady-state distribution of objects

in the giant planet crossing region of the solar system.

Jedicke and Herron (1997) were able to fit the dynamically derived a, e and i

distribution for the Centaurs with these relatively simple independent analytical

expressions:

n(a) ∝ exp[−1

2(a− a)2

σ2a

](8)

n(e) ∝ exp[−1

2e2

e20

](9)

n(i) ∝ (i− i) exp[−1

2(i− i)2

i20

](10)

where a = 32AU, σa = 6.9AU, e0 = 0.21, i = −2.7 and i0 = 15. The

resulting normalized distributions are shown in Figures 14 and 15 together with

the distribution of known Centaurs. Jedicke and Herron (1997) only considered

objects to a < 30AU but we have extended the limit to 35AU in this work. Note

that the distribution does not look anything like the theoretical model due to

strong observational selection effects.

The size-frequency distribution or, nearly equivalently, the absolute mag-

nitude distribution of the Centaurs is not well measured. If the differential

absolute magnitude distribution is modeled as n(H) ∝ 10αH , Jedicke and

Herron (1997) used a single centaur found by Spacewatch to measure α =

0.61+0.70−0.40(stat) ± 0.08(sys). A few years later Sheppard et al. (2000) used a

Monte Carlo simulation to determine that the centaur size-frequency distribu-

tion was consistent with α = 0.60. In this work we will assume that α = 0.61.

The distribution used in Jedicke and Herron (1997) was normalized using the

claim that there are about three objects larger than Chiron (the largest centaur

at the time). Chiron has since been modified to have H = 6.5, rather than

the original claim of H = 6.0, and there is only one other centaur known with

23

H = 6.4. We thus assume in this work that there are only two objects larger

than H = 6.5 and normalize the absolute magnitude distribution accordingly,

resulting in a set of 60, 227 synthetic objects.

Our population of synthetic centaurs is now generated by: 1) binning the

normalized (a, e) distribution defined by Equations 8 and 9 into a grid of 1, 000

by 1, 000 bins; 2) the fraction, f(a, e), of the centaur population in each bin is

calculated; 3) the maximum possible absolute magnitude, Hmax, that an object

at this perihelion, q = a(1− e), can have and still be above limiting magnitude

of Pan-STARRS (assumed to be V = 24.5) is determined; 4) the number of

centaurs, N(a, e,Hmax), with H < Hmax in this (a, e) bin is determined.

We then use this information such that for each of the N(a, e,Hmax) cen-

taurs: a) a semi-major axis and eccentricity is generated randomly within the

current bin’s range of a and e; b) an inclination according to Equation 10 is

generated; c) an absolute magnitude following the distribution described above

is picked; d) the three orbital angular elements (Ω, ω,M) in the range [0, 360]

are randomly generated. Since the orientation of the orbits are random, the

epoch of this synthetic centaur population is unimportant.

In Figures 14 and 15 the synthetic population is compared to the current 48

known Centaurs with 5.2 < q < 27AU and 5.5 < a < 35AU. The two popula-

tions are in fair agreement, with the perihelion distance, eccentricity and mean

anomaly distributions having the largest discrepancies. The apparent lack of

objects with low perihelion distances in the known population might seem puz-

zling, as low perihelion objects come closer to Earth and are therefore brighter

and easier to detect. However, from our model it is clear that most low perihe-

lion distance objects also have large eccentricities (and hence, larger semi-major

axes). This means that most low perihelion distance objects spend a majority

of their orbital period at distances where they are either too faint or move too

24

slowly to be detected by current large-field surveys (this is also the reason be-

hind the apparent lack of high eccentricity objects in the known population).

These two effects apparently work to cancel each other out, leading to a rather

flat discovery distribution in perihelion distance (as seen in the known popula-

tion). The distribution of mean anomaly seen in the known population is also

easily explained as most Centaurs, as explained above, are only bright enough

to be detected when they are close to their perihelion distance, that is, when

the mean anomaly is close to 0. Since all Centaurs have orbital periods much

longer than the few recent decades during which they have been discovered, this

feature has remained. Since we make no attempt at removing any Centaurs that

might currently be too distant and faint to be seen during the lifetime of the

Pan-STARRS project, the mean anomaly is evenly distributed in our synthetic

population as expected.

2.5 The Trans-Neptunian Populations

Since the discovery of the population’s first member (15760) 1992 QB1 (Je-

witt and Luu, 1993) in 1992, study of the trans-neptunian objects (TNOs) has

revealed a dynamically complex population. A treasure trove of interesting phe-

nomena and a number of constraints on planet formation and the early evolution

of the solar system has emerged. The division of the TNOs into several distinct

sub-populations is still hotly debated (Gladman et al., 2001). In this work we

elect to simplify the distribution by dividing it into three main components:

the classical Kuiper Belt objects (CKBOs); the resonant Kuiper Belt objects

(RKBOs); and the scattered disk objects (SDOs). We define the CKBOs to be

non-resonant objects with average semi-major axes in the range 29 < a < 48AU,

regardless of eccentricity and inclination (anything closer to the Sun than this

range of semi-major axes is considered a centaur by our over-simplified defini-

25

tion). The RKBOs consist of those objects in major mean-motion resonances

with Neptune (from the trojan population in the 1:1 mean-motion resonance

out to the 3:1 mean-motion resonance). SDOs are considered to be any object

with an average semi-major axis beyond 48AU. It should be noted that this di-

vision of sub-populations is much coarser than that of other authors (Gladman,

2002; Gladman and Marsden, 2008), but gives us the ability to generate a rough

representation of the outer solar system, which can be used in the simulation

and efficiency determination of the Pan-STARRS surveys.

Our model for the trans-neptunian populations is based on the Nice model

(Morbidelli et al., 2007; Levison et al., 2007). It assumes that the giant planets

begin on nearly circular and coplanar orbits, in a compact configuration between

∼ 5.5 and ∼ 14AU, with orbital separations much smaller than today’s solar

system. A key part of the model is that Saturn is assumed to be inside the

2:1 mean-motion resonance with Jupiter and crosses this MMR after ∼ 0.35 −

1.1Gyrs as it migrates outwards (Gomes et al., 2005). Outside this compact

zone of planets is a planetesimal disk with an outer edge at ∼ 34 AU and a

total mass of ∼ 35Mearth. These planetesimals are assumed to be on orbits

with dynamical lifetimes of at least 3Myr, which is long enough to survive the

supposed lifetime of the gas disk. The planetesimals at the inner edge of this disk

evolve onto Neptune-crossing orbits during the first few million years, causing

the planets to migrate at a very slow rate as planetesimals on unstable orbits

are ejected from the system. When Jupiter and Saturn cross their mutual 2:1

mean-motion resonance, their eccentricities are slightly excited. This causes

significant perturbations on the orbits of Uranus and Neptune, which both have

their eccentricities driven up (e ∼ 0.3 − 0.4), such that their orbits essentially

become chaotic and begin to approach one another. With such eccentricities the

two planets penetrate deep into the planetesimal disk, which is destabilized and

26

begins scattering planetesimals throughout the solar system. The eccentricities

of the planets are subsequently dampened during the next few million years, due

to dynamical friction caused by the planetesimals. The planets thus decouple

from each other, leaving the configuration found in the solar system today, with

a significantly depleted planetesimal disk.

The Nice model reproduces several of the key aspects known about the for-

mation of the outer solar system. The orbital distribution of the planets in the

outer solar system is recreated (Tsiganis et al., 2005) and the existence of the

jovian and neptunian trojan populations are explained (Morbidelli et al., 2005;

Tsiganis et al., 2005). Even the origin of the late heavy bombardment in the

inner solar system is explained. The Nice model also reproduces many of the

main features observed in the trans-neptunian population. It creates both the

resonant and non-resonant populations and it retains the a−e distribution seen

in the classical belt. Further, it maintains the lack of an outer edge outside the

2:1 MMR with Neptune and explains the apparent mass deficit. However, some

problems with the Nice model still exist and they are important with respect to

our generation of a synthetic population: 1) all the mean-motion resonances are

noticeably overpopulated compared to the classical belt; 2) the population of

objects with 30 < q < 40 AU is significantly overpopulated; 3) the eccentricities

of the classical Kuiper Belt objects are slightly high; and 4) the classical belt

are significantly lacking high inclination objects (i > 25).

The numerical simulations behind the Nice model is described in Levison

et al. (2007) and Morbidelli et al. (2007) and we use the 2, 055 surviving particles

from their simulation B in generating our synthetic population for the Kuiper

Belt. Since these simulations end shortly after the migration of the planets has

ended, we perform long-term integrations using a standard Wisdom-Holman

symplectic integrator (Wisdom and Holman, 1991) to determine which of the

27

objects remained long-term stable. This removed 919 planetesimals that suffer

close encounters with one of the giant planets. For each of the remaining 1, 136

long term stable objects we use our integrations to determine what class each

object belongs in. We call an object resonant if it spends more than 50%

of its time inside one of the main mean-motion resonances in the TNO region.

Altogether we found that 622 of the surviving objects were resonant, with a little

bit more than one fourth (180 objects) being in the 3:2 mean-motion resonance

(these objects are also called the plutinos). In all we found 27 populated mean-

motion resonances where, besides the Plutinos, the 5:3 (with 77 objecs), 1:1

(neptunian trojans with 65 objects), 2:1 (the twotinos with 59 objects) and 4:3

(with 51 objects) are the heaviest populated resonances (see the bottom panel

of Figure 17). Together these objects account for almost 70% of the stable

resonant population coming out of the Nice model. Of the remaining stable

objects, 347 were found to be classical TNOs and another 167 were determined

to be scattered disk objects (with average a > 48AU).

We then extract the orbital parameters for each of the 1,000 output steps of

the integrations for each of the surviving objects (each of these outputs were 0.1

million years apart). Since the demographics of the trans-neptunian population

are intimately connected to the position of Neptune, each of the orbits was

rotated and scaled such that Neptune had approximately the same position for

all of the epochs. This gives a set of 1,137,136 orbits that we are used in building

our synthetic population of TNOs, but first we need to determine the number

of objects in each of the synthetic sub-populations.

For the trans-neptunian populations the differential magnitude distribution

is given by a power-law relation

Σ(R) = 10α(mR−m0) (11)

28

where we use α = 0.63 ± 0.06 and m0 = 23.04 ± 0.1 (Trujillo et al., 2001).

These numbers are in excellent agreement with other independent attempts at

deriving the differential magnitude distribution of the trans-neptunian popula-

tion (Bernstein et al., 2004; Fraser et al., 2007). Our model is created to be

complete to a V magnitude of 24.5, so we need to convert this distribution to

that magnitude band. Trans-neptunian objects vary in color from the blue ob-

jects, with an average V − R ∼ 0.39, to the very red objects, with an average

of V − R ∼ 0.71 (Barucci et al., 2005). Using the data collected by Barucci

et al. (2005), we derive an average V − R color for the full TNO sample to be

V −R = 0.59 and we use this value to convert R magnitudes to V magnitudes

for the generation of our synthetic TNO population.

We know from the Nice model that the number of resonant objects is heav-

ily overrepresented (Morbidelli et al., 2007), so we need to determine what the

proper ratio of classical to resonant objects is. Trujillo et al. (2001) used maxi-

mum likelihood simulations from which they estimated the number of classical

KBOs to be NCKBOs(D > 100km) = 3.8+2.0−1.5×104. They further estimated the

total number of plutinos larger than 100km, which is quite small compared to

the classical KBOs, to be N(D > 100km) = 1, 400. Similarly they estimated the

number of objects in the 2:1 mean-motion resonance to be ∼ 1.7 times that of

the Plutinos. Trujillo et al. (2000) estimated the total population of SDOs to be

NSDOs(D > 100km) = 3.1+1.9−1.3 × 104, which is approximately the same number

as that for the classical KBOs derived in Trujillo et al. (2001). We can now use

this, together with the appropriate magnitude frequency distribution, to derive

the absolute number of objects in our synthetic model, while maintaining an

approximate ratio of CKBOs:RKBOs:SDOs at 1:0.1:0.8.

The next step is to determine the minimum size of a trans-neptunian object

that can be detected by Pan-STARRS. If we define the heliocentric distance of

29

25AU as the closest a TNO will come, the full PS survey will be able to detect

and track objects as small as 50km (or HV ∼ 10.6 assuming an albedo of 0.04).

The differential size distribution is given by

n(r)dr = N0r−qdr (12)

where q = 5α + 1, or q = 4.2 ± 0.3 using the Trujillo et al. (2001) result.

Given that N(D > 100km) = 38, 000 and q = 4.2, we estimate that there are

∼ 350, 000 classical TNOs with D > 50km. Adding the resonant and scattered

disk objects in the ratio mentioned above, altogether we have to generate ∼

665, 000 synthetic objects.

For each of these synthetic object we now chose randomly from the appro-

priate build set generated by our integration of the Nice Model objects. We

keep the semi-major axis, but shift the eccentricity by a small value randomly

selected in the range [−0.025,+0.025]. Since we know that the inclination dis-

tribution of the Nice model has a deficit of high inclination objects, we shift the

inclination by δi randomly selected in the range [−20 sin i,+20 sin i], where

i is the inclination of the build object selected (this range of shifts was found

through trial and error). This ensures that the inclination distribution gets

widened, but keeps the low inclinations from being shifted too much. Thus the

low inclination core of the classical belt is kept intact. For the classical and

scattered disk objects we randomly select the orbital element angles (longitude

of the ascending node, argument of perihelion and mean anomaly), while for

the resonant objects we have to insure that the appropriate resonance angle is

kept intact.

The object is then given a radius according to the distribution given in

Equation 12, from which the absolute magnitude is computed using Equation 2

and assuming an albedo of 0.04. The current position and apparent magnitude

30

of the object is then calculated. Since most TNOs have minimal changes in

their heliocentric and observer distance over the lifetime of the Pan-STARRS

surveys, we can assume that the visual apparent magnitude stays roughly the

same throughout (especially under the assumption that phase correction and

rotational light curve variations are neglible; see beginning of Section 2). If the

apparent magnitude is brighter than V = 24.5 (or R = 23.91 using a V −R color

of 0.59) the synthetic object is saved; otherwise it is discarded. This results in a

synthetic model consisting of 42, 500 CKBOs, 5, 885 RKBOs and 10, 907 SDOs.

Figure 16 shows the distribution of the semi-major axis, eccentricity and

inclination of the known multi-opposition TNOs (taken from the Minor Planet

Center) and that of our synthetic population of classical and resonant objects. It

is clear from the figure that the two are in reasonable good agreement and that

most of the features seen in the current observed population of TNOs have been

retained. The 1:1, 3:2 and 2:1 MMRs are easily identified in both the known

and synthetic populations. The edge of the classical Kuiper Belt has also been

retained at the proper distance, just outside the 2:1 MMR. Figure 17 shows the

semi-major axis distributions of the resonant and classical synthetic population

compared to the known population of TNOs. While the main features are

retained, it is clear that our synthetic population has an overpopulation of

objects in the 30 < a < 40AU range. This is due to the constraints of and

clearly one of the main deficiencies of the Nice model, but given the success of

the model in retaining other features of the TNO populations, it is one that we

will have to live with for the moment.

Figure 18 shows the eccentricity distribution of our synthetic classical and

resonant populations compared to the current known population of TNOs. It

shows that the dual gaussian representation of the known population can be

explained as a combination of the eccentricity distributions of the classical and

31

resonant populations. However, adding together the distribution of the syn-

thetic CKBOs and RKBOs does not adequately reproduce that of the known

population. This is due to the small ratio of RKBOs to CKBOs found above,

and puts these findings in question. Future updates of the S3M will look at this

problem closer.

Figure 18 also shows the inclination distribution of our synthetic population,

compared to that of the known multi-opposition population. It is seen that

shifting the inclination of each object in the range ±20 sin i goes a long way

towards deriving an inclination distribution that is similar to that of the known

population. It can be argued that the synthetic model still lacks high inclination

objects, but using larger shifts smears out the low inclination core too much. Our

synthetic population is also exhibiting some clumping in inclination (see bottom

left panel in Figure 16), an artifact of the way we generate the population. This

could be remedied by increasing the number of stable planetesimals that survive

the Nice model integrations, but expanding that model itself is beyond the scope

of this work.

The resulting scattered disc population is shown in Figures 19 and 20, and

is seen to be a remarkably good representation for all but the inclination distri-

bution. The relative lack of high inclination (and also the relative lack of high

eccentricity SDOs) is most likely due to the Nice model’s failure to provide any

method for the trans-neptunian populations to excite their objects, beyond the

method of resonance pumping. Note also that while our synthetic population is a

fair representation of the scattered disk population, it does not adequately model

the high perihelion objects, like (90377) Sedna (a = 484, e = 0.84, i = 12).

While our synthetic TNO population has some problems, it will clearly be a

valuable tool in the testing and calibration of the Pan-STARRS surveys. With

an expected ∼ 10, 000 new TNOs to be discovered by Pan-STARRS, an order

32

of magnitude more than presently known, the analysis of how the observational

results of this synthetic population compares with the observed real population

will be extremely important in testing the validity of the Nice model and provide

new constraints for future models, whether they are based on the Nice model or

not.

2.6 The Cometary Population

As icy bodies enter the inner solar system the heat of the sun free trapped

volatiles, which sublimates into gases. This generates the coma and tails that

make comets some of the most interesting and visually stunning objects in the

solar system. There are two main types of cometary populations in the solar

system, the ecliptic periodic comets and the isotropic long-period comets. The

ecliptic, period comets, also known as the short-period comets, have low in-

clinations and fairly short periods (less than ∼ 200 years). They are believed

to originate among the trans-neptunian objects, created when objects are per-

turbed by the giant planets into the inner solar system. The ecliptic comets can

be divided into two main sub-populations, the Jupiter family and Halley family

of comets. In addition, a handful of main belt objects, centaurs and unusual

objects have been found to be cometary. The long-period comets, on the other

hand, are isotropic, more or less evenly distributed in the sky. They are thought

to be Oort cloud objects perturbed into the solar system by passing stars, galac-

tic tides or collisions among the Oort cloud objects. While they are technically

periodic (with periods of more than ∼ 1, 000 years), they are weakly bound and

can be ejected from the solar system through planetary perturbations.

In this version of the S3M we only include the long period comets and a

population of interstellar comets. A more comprehensive study of the demo-

graphics of the short period comets is under way using an approach similar to

33

that used by Francis (2005) for the long period comets (see below). The results

of this study will be published in a separate paper.

Note that we make no attempt at modeling the brightening of the comets due

to outgassing. Numerous attempts have been undertaken to study this behav-

ior, but the lack of a standard in the way magnitudes are being reported makes

it nearly impossible to use existing data to reach firm conclusions. Our primary

goal with the S3M is to make sure that the pipeline is able to determine the

dynamical quantities of the different solar system populations, making us less

concerned at this point about the magnitude evolution of these objects. One of

the science goals of the Pan-STARRS project is to derive a standard method for

determining and reporting of magnitude measurements of the cometary popula-

tion, with the goal of understanding how comets brighten and fade throughout

their orbits in mind.

2.6.1 The Long-Period Comets

Our creation of a population of long-period comets is based on work done by

Francis (2005), where he derived the absolute magnitude and perihelion dis-

tributions of this population using data from the Lincoln Near-Earth Asteroid

Research (LINEAR; Stokes et al., 2000) survey.

The absolute magnitude distribution was determined by Everhart (1967) by

using a broken power-law, with the break happening at Hb ∼ 6. Hughes (2001)

also saw a break at similar absolute magnitudes, but was unable to determine

whether it was a real feature or simply the result of observational bias caused by

increasing incompleteness at fainter magnitudes. Francis (2005) used a broken

power-law on the form:

dn

dH∝

b(H−Hb) H < Hb

f (H−Hb) H > Hb

(13)

34

where he found f = 1.03 ± 0.09 with Hb = 6.5. The fit was unable to provide

a useful constraint on the bright end slope b, so the value of 2.2 was adopted

from Hughes (2001).

However, Francis (2005) adopted a perihelion distribution that was different

than that found by Everhart (1967) and Hughes (2001): one that rises for

q < 2 and follows a power-law beyond that. This preserved the Everhart (1967)

observation that there is a drop in comet numbers with q < 1, but allowed for a

more shallow distribution at larger perihelion distances such that the LINEAR

observations could be recreated. The equation chosen was:

dn

dq∝

1 +√q q < 2AU

2.41× ( q2 )γ q > 2AU(14)

where γ was found to be −0.27± 0.3.

The total number of long-period comets in our synthetic sample is normalized

such that there are 0.53 comets with H < 6.5 and perihelion 0.5 < q < 1.5 per

year (Hughes, 2001), resulting in a total of 135 objects per year with H < 24.5

and q < 15AU. In order to generate a synthetic population for the full Pan-

STARRS project, a time frame of 14 years was used, based on the 10-year

nominal survey lifetime of Pan-STARRS plus a two-year buffer on either side to

include objects detected both outbound and inbound. We thus have to generate

a total synthetic population of 1, 880 long-period comets.

To generate our sample of synthetic long-period comets we use Equations

13 and 14 to determine the absolute magnitude and perihelion of each object.

The distribution of semi-major axes, a, is also adopted from Francis (2005) and

chosen such that 40 percent of the objects are considered to be dynamically

new and are given a = 20, 000AU. The remainder are given a value of a that is

35

chosen uniformly and randomly in the 10, 000 − 20, 000AU range. The time of

perihelion passage was chosen randomly during the 14-year timespan used above,

and the inclination and direction of perihelion passage were chosen randomly to

give a uniform distribution on the celestial sphere. The distribution of absolute

magnitude and perihelion distance for our synthetic population is shown in

Figures 21 and 22, together with the associated analytical expression.

2.6.2 Interstellar Comets

It is commonly believed that a significant portion (> 99%) of the comets and

other icy bodies that accreted in the outer solar system is ejected as the giant

planets formed and migrated to their current orbits. This should create a large

population of interstellar comets (ISCs), but currently no such object has been

found. Using this fact, Whipple (1975) determined an upper limit on their

number-density as ∼ 1013 pc−3. Using the failure of the Spacewatch survey to

identify an ISC, Meinke et al. (2004) assumed a number-density distribution on

the form N = N010α(H −H0), where N0 is the number-density at H0 = 19.1,

which corresponds to a comet with a diameter of ∼ 1km when using an albedo

of 0.04. Using a power-law index of α = 0.5, corresponding to the expected

slope of accreting planetesimals, they determined an upper limit for the ISCs of

∼ 1014 pc−3.

Since it is believed that comets coming from interstellar space have yet to

be observed, they represent a tantalizing prospect for a breakthrough discovery

for the Pan-STARRS project. But in order for these objects to be found in the

vast data set collected by the project, it is important to test the MOPS ability

to find, identify and track these unusual moving objects. Since an ISC would

come into the solar system from the local solar neighborhood, it should have

an encounter velocity similar to that of nearby stars, 10 − 40 km/s (Kresak,

1992; Dehnen and Binney, 1998). We thus assume that the ISCs would have

36

a velocity distribution at infinity given by a gaussian distribution with a mean

value of 25 km/s and a standard deviation of 5 km/s.

Our synthetic sample was created in the following fashion. First we select a

random time in the interval September 30., 2006, to September 30., 2015, which

covers most of the lifetime of the project. This time represents the epoch of

the orbital state, consisting of position and velocity, to be generated. We select

a random position within a sphere of radius 50AU centered on the barycenter

of the solar system. The direction of the velocity component of the object is

selected randomly on a unit sphere centered on the position already generated.

The magnitude of this velocity component is then chosen such that the velocity

at infinity is distributed according to the distribution described above. The po-

sition and velocity at the given epoch is then converted into hyperbolic orbital

elements and the object is given a random apparent magnitude in the range

20 ≤ m ≤ 24.5, which ensures that the hyperbolic comets are within the ob-

servational limiting magnitude of Pan-STARRS. The absolute magnitude, H,

was calculated from Equation 1 using the position and the apparent magnitude.

The resulting distributions can be seen in the bottom panel of Figure 23.

Since the interstellar comets are isotropic and we want a sky density of

roughly one per PS field we can calculate the total number of possible observa-

tions by multiplying the number of PS fields needed to cover the entire sky with

the number of days in the time range used to generate the epochs above. Each

object was then integrate both forward and backwards to the edges of this time

range and the number of nights each object had an apparent magnitude less

than V = 24.5 was counted and added together. A total of 8, 304 objects was

needed to reach the required density. This is a large enough sample to ensure

statistical significance for testing and efficiency determination.

37

3 Summary

We have created an extensive synthetic solar system model covering every ma-

jor known population of minor planets. With around 11 million objects this

model is to our knowledge the most extensive attempt at generating a synthetic

population of minor planets in the solar system.

The synthetic model presented here allows us to simulate the expected per-

formance of both the Pan-STARRS 1 and Pan-STARRS 4 surveys by generating

synthetic detections that can be fed into to the Moving Object Processing Sys-

tem (MOPS). The model therefore provides a valuable tool in the testing of this

new and innovative software in preparation for science operations expected to

start in fall of 2008. Once science operations have begun the synthetic solar sys-

tem will be used to generate synthetic detections for each field observed (similar

to the test simulations). These detections will be handed to the MOPS pipeline

together with the real detections found in each field. This allows us to determine

the observing biases, as well as derive the efficiency of each component and of

the system as a whole. The results can then be used in an iterative process

to debias the survey and provide corrections to our synthetic model, which can

then be fed into the pipeline again and reprocessed, deriving new efficiencies

and new refinements to the model.

It is our belief that the Synthetic Solar System Model can be used for a

multitude of other projects and purposes beyond the testing and monitoring of

the Pan-STARRS survey. The Large Synoptic Survey Telescope (LSST) has ex-

pressed interest in an extended version of the S3M going to V ∼ 26 (Pierfederici,

personal communications) and other surveys might use it to model their current

or expected performance in a similar fashion to the Pan-STARRS project. It

is also our belief that the S3M can provide a valuable tool in determining fore-

ground contamination of asteroids in non-solar system surveys, especially for

38

future space craft missions like the Wide-field Infrared Survey Explorer (WISE;

Mainzer et al., 2005). The S3M is public domain and can be downloaded from

www.mac.com/tgrav/iWeb/Astronomy/.

It should be noted that there are several improvements that can be done

to the version of the S3M presented here, and development and upgrades will

continued into the future. Work on addition of the ecliptic periodic comets (the

Jupiter and Halley family comets) are already underway and will be presented in

a separate paper. Extending the model to include physical parameters is also a

high priority to increase the usability of the model to projects outside the visible

wavelengths. Some work has already been done to test the PS observational

cadences for light curve determination and shape inversion (Durech et al., 2005).

Extending the model to fainter absolute magnitudes is also currently a high

priority to allow for determining the performance of future ground based surveys

(like LSST) and space missions (like WISE).

4 Acknowledgments

The design and construction of the Panoramic Survey Telescope and Rapid

Response System by the University of Hawaii Institute for Astronomy is funded

by the United States Air Force Research Laboratory (AFRL, Albuquerque, NM)

through grant number F29601-02-1-0268. This work was conducted in part

at the Jet Propulsion Laboratory, California Institute of Technology, under a

contract with the National Aeronautics and Space Administration.

39

Tables

Table 1: The range over which the orbital elements can be modified during the

generation of the main belt and trojan synthetic populations.

Table 2: Shown here are the number of known objects for each known or

anticipated population described in this paper. Also shown is the expected

number found during the PS1 and PS4 projects and the number of objects in

the S3M.

40

Table 1:

Orbit element MBAs Trojansa 0.01 AU 0.02 AUe 0.01 0.04i 0.5 5

Ω 1 2

ω 1 2

M 1 2

Table 2: Pan-STARRS solar system

Object Type Known now 1 year PS1 1 year PS 4 MOPS S3MNear Earth Objects ∼ 5, 000 ∼ 5, 000 ∼ 25, 000 268,896Earth Impactors 0 Unknown Unknown 10,006Main Belt Asteroids ∼ 300, 000 ∼ 1, 000, 000 ∼ 5, 000, 000 10,000,000Jovian Trojans ∼ 2, 000 ∼ 20, 000 ∼ 100, 000 280,000Centaurs ∼ 75 ∼ 300 ∼ 1000 60,278Kuiper Belt Objects ∼ 1000 ∼ 3, 000 ∼ 10, 000 42,709Neptunian Trojans 6 ∼ 50 ∼ 200 ∼ 20, 000Scattered Disc Objects ∼ 100 ∼ 500 ∼ 2, 000 10,952Jupiter Family Comets ∼ 300 ∼ 500 ∼ 5, 000 Work in progressLong Period Comets ∼ 1, 000 ∼ 1, 000 ∼ 5, 000 9,400Hyperbolic Comets 0 Unknown Unknown 8300

41

Figures

Figure 1: The position as seen from above the ecliptic of the known near-Earth

objects at julian date 2454400.5 and the synthetic near-Earth objects at julian

date 2454466.5. The positions and orbits of Mercury, Venus, Earth, Venus and

Jupiter are also shown.

Figure 2: The Bottke et al. (2002, ; solid line) and known (as of 2004 Nov

28; dotted line) distributions in semi-major axis, eccentricity, inclination and

absolute magnitude. Only objects with H < 18 are included in the plots of the

orbital elements.

Figure 3: Compared to the Potential Impactors, the F -weighted Impactor

Population shows an increased presence of shallow crossing orbits (q and/or Q

near 1 AU) and low inclination, both of which lead to lower V∞.

Figure 4: Distribution of impactors over North America and Hawaii.

Figure 5: The semi-major axis vs eccentricity distributions (top) and semi-

major axis vs inclination distribution (bottom) of all the known MBAs. The

Kirkwood gaps at 2.5, 2.82, 2.95 and 3.2AU are easily identified, as are the

Hungarias (a ∼ 1.9AU, i ∼ 20), the Phocaea (a ∼ 2.25 − 2.5AU, i ∼ 25)

and the Hildas (a ∼ 3.95AU). The division of the main belt into the inner

(a ∼ 2.1 − 2.5AU), middle (a ∼ 2.5 − 2.8AU) and outer (a ∼ 2.8 − 3.2AU)

regions as demarcated by secular and mean-motion resonances is also seen.

Figure 6: (Top) The histogram shows the absolute magnitude distribution of

the known main belt asteroids as of Dec. 2007. The debiased differential (solid

line) and cumulative (dotted line) distribution are also shown. The vertical

dotted time gives the magnitude limit for which the sample is believed to be

42

complete. (Bottom) The semi-major axis distribution of all known main belt

asteroids (solid histogram) and all known asteroids with H < 14.5 (grey line) is

shown.

Figure 7: The normalized distributions of semi-major axis, eccentricity, incli-

nation and absolute magnitude for the known MBAs with H < 14.5 (dotted

line) and our synthetic (1/30th sample; solid line) with H < 14.5.

Figure 8: The normalized distributions of semi-major axis, eccentricity and

inclination for the known MBAs (dotted line) and our synthetic (1/30th sample;

solid line). For the absolute magnitude distributions the actual distributions for

the known MBAs and our synthetic population (again a 1/30th sample).

Figure 9: The position as seen from above the ecliptic of the known jovian

trojans for julian date 2454400.5 and our synthetic population of jovian trojans

for julian date 2454466.5. The positions and orbits of Mercury, Venus, Earth,

Venus, Jupiter and Saturn are shown.

Figure 10: The synthetic (black line) and known (grey line) populations of the

jovian Trojan population.

Figure 11: The synthetic (black line) and known (grey line) populations of

the jovian Trojan population. The dotted line in the lower right panel gives the

mean longitude of Jupiter at the epoch of the orbital elements plotted.

Figure 12:The full synthetic (black line) and known (grey line) populations

with H < 12 of the jovian Trojan population.

Figure 13: The position as seen from above the ecliptic of the known (triangles)

and synthetic Centaurs for julian date 2454466. The position and orbit of the

planets are also shown.

43

Figure 14: The fractional distributions of the perihelion distance, semi-major

axis, eccentricity and inclination for the synthetic MOPS population (solid line)

and the known population (as per Dec. 2008; dotted line) of Centaurs.

Figure 15: The fractional distributions of the longitude of ascending node, ar-

gument of perihelion and mean anomaly, together with the number distribution

of absolute magnitudes for the synthetic MOPS population (solid line) and the

known population (as per Dec. 2008; dotted line) of Centaurs.

Figure 16: The semi-major axis versus eccentricity distribution of the known

(top left panel) and our synthetic classical and resonant (top right panel) pop-

ulations are shown. The bottom two panels show the semi-major axis versus

inclination distribution of the known (left panel) and our synthetic classical and

resonant (right panel).

Figure 17: Top panel): Comparison of the distributions of the semi-major

axes of the known (grey line) and synthetic (black line) classical and resonant

populations. Middle panel): The distribution of the semi-major axes of our

synthetic classic population. Bottom panel) The distribution of the semi-major

axes of our synthetic resonant populations. In all three panels the dotted line

gives the location from left to right of the 1:1, 5:4, 4:3, 3:2, 5:3 and 2:1 mean-

motion resonances.

Figure 18: Top panel): Comparison of the distributions of the eccentricities

(top) and inclination (bottom) of the known (grey line) and synthetic classical

(black line) and resonant (dotted line) populations.

Figure 19: The semi-major axis versus eccentricity (top panels) and semi-

major axis versus inclination (bottom panels) distributions of our synthetic (left

panels) and the known (right panels) populations of scattered disk objects.

44

Figure 20: Top left): Comparison of the distributions of the perihelion distance

of the known (grey line) and synthetic (black line) scattered disk populations.

Top right): The same as above, but for semi-major axis. Bottom left): The

same as above, but for eccentricity. Bottom right): Same as above, but for

inclination.

Figure 21: The absolute magnitude, H, distribution of our synthetic long

period comets population (solid line histogram), and Equation 13 (dotted line).

Figure 22: The perihelion distance, q, distribution of our long period comet

synthetic population (solid line histogram), and Equation 13 (dotted line).

Figure 23: The perihelion (top), eccentricity (upper middle), inclination (lower

middle) and absolute magnitude (bottom) distributions of our synthetic hyper-

bolic comet population.

45

Figure 1:

46

Figure 2:

47

Figure 3:

48

Figure 4:

49

Figure 5:

50

Figure 6:

51

Figure 7:

52

Figure 8:

53

Figure 9:

54

Figure 10:

55

Figure 11:

56

Figure 12:

57

Figure 13:

58

Figure 14:

59

Figure 15:

60

Figure 16:

61

Figure 17:

62

Figure 18:

63

Figure 19:

64

Figure 20:

65

Figure 21:

66

Figure 22:

67

Figure 23:

68

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