Assessing the Yarkovsky Effect in the Kuiper Belt - Astronomy ...

Assessing the Yarkovsky Effect in the Kuiper BeltbyDaniel Josef MajaessA THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHEREQUIREMENTSFORTHEDEGREEOFBACHELOR OF SCIENCEwith an honors inAstrophysics(Department of Astronomy and Physics, Dr. Joseph Hahn Supervising Faculty)...................................................................................................................................................................................................................................................................................SAINT MARY’S UNIVERSITYMay 3, 2005c○ Daniel Josef Majaess, 2005

AbstractAssessing the Yarkovsky Effect in the Kuiper Belt, byDaniel Majaess, submittedon May 3, 2005:The Yarkovsky Effect (YE) is a radiation force that while minute, has been shownto help deliver asteroids into near earth crossing orbits (NEOs). The force is dividedinto a seasonal variant due to an object’s yearly revolution and a diurnal componentdue to its daily rotation. We show that the seasonal YE causes an orbital decay ofȧ ∼−1AU/Gyr for a small subset of Kuiper Belt Objects (KBOs) whose diametersrange from 5 − 100m, while the diurnal effect is of little consequence in the KuiperBelt. Moreover, compared to a controlled run, the seasonal YE augments the numberof KBOs that diffuse into the inner Solar System by a factor of ∼ 100. It was thoughtthat the YE could enhance the inward transport of KBOs where they may be detectedas Jupiter Family Comets (JFCs), but the bodies considered here are 1-2 orders ofmagnitude to small.

ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 The Yarkovsky Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1 An Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 The Seasonal YE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 The Diurnal YE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 The N-body Integrator, Mercury6 . . . . . . . . . . . . . . . . . . . . 215 Discussion and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.1 The Delivery Model: YE → Resonance → Ejection. . . . . 235.2 The YE: changing the KB initial & final states. . . . . . . . . . . . . 24

5.3 The YE: increasing the inward flux of KBOs. . . . . . . . . . . . . . . 276 Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31A YE Subroutine for Mercury6 . . . . . . . . . . . . . . . . . . . . . . . 32B IDL Subroutine: Calculating E(x) and d(x) . . . . . . . . . . . . . 35C Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

List of Figures2.1 A depiction of the seasonal Yarkovsky effect . . . . . . . . . . . . . . 92.2 A contour plot of |ȧ| as a function of x and θ. . . . . . . . . . . . . . 132.3 ȧ for varying thermal conductivity parameters: K θ=3 =2.2 × 10 4 ergKcms(solid), K ice =5.2 × 10 4ergKcms(dashed). . . . . . . . . . . . . . . . . . 153.1 Ejection of a KBO due to eccentricity pumping . . . . . . . . . . . . 193.2 An anology of Kirkwood Gaps in the Kuiper Belt . . . . . . . . . . . 204.1 The fictitious force created to mimic the YE . . . . . . . . . . . . . . 225.1 The delivery model in action: YE → resonance → ejection . . . . . . 235.2 The population layout for the KB at the initial and final states . . . . 255.3 The eccentricity layout for the KB at the final state (t =4.5Gyrs).The top graph is the simulation evolved with the YE, and the bottomis the control (same simulation without YE). . . . . . . . . . . . . . . 265.4 The inward flux of KBOs over time . . . . . . . . . . . . . . . . . . . 27C.1 Increased eccentircity results in a decreased perihelion . . . . . . . . . 36

List of Tables2.1 Definitions for some of the chemical and geological variables . . . . . 122.2 Geological parameters used to describe our KBO . . . . . . . . . . . . 14C.1 KB resonances and their corresponding semi-major axes . . . . . . . 37

Chapter 1IntroductionThe Kuiper Belt (KB) is a vast swarm of comets that extends beyond Neptune from35 on upwards to 50 AU. These bodies orbit the sun and are considered to be abelt much like the main asteroid field between Mars and Jupiter. Unlike their rockyasteroid counterparts, the KB constituents are comets that are made of ices andvolatile material leftover from the primordial solar system. In this project, the KBis of main interest because it is generally believed that it is a reservoir for JupiterFamily comets (JFCs) (Levison, 1996). If this is true, an immediate question arisesregarding the history of these comets, mainly what mechanism caused their inwardtransport from the KB? Conventional thinking suggests that KBOs get ejected fromthe belt via resonances, however recent HST observations show that there are notenough small KBOs to account for the observed abundance of JFCs (Bernstein et al.,2004).Recent work on the main asteroid belt has shown that a force entitled the YarkovskyEffect (YE) plays an important role in the delivery of meteorites and earth crossingobjects (Bottke et al., 2000). The goal of this project is to see whether this effect—when coupled with resonances—can enhance the diffusion of KBOs where they’reobserved as JFCs.

Chapter 2The Yarkovsky Effect2.1 An IntroductionThe YE is divided into two main components due to a body’s daily rotation and itsannual revolution around the sun. The diurnal YE is the daily variant and it arisesfrom the temperature differential between the surface exposed to the sun and thedarkside. Assuming a blackbody relation, the flux of photons from the warm side isgreater than that of the dark by Ratio =( TwarmT dark) 4 . Hence thermal energy is reradiatedasymmetrically and because photons carry momentum, the body will experience athrust which depending on its rotation may cause an outwards or inwards orbitaldrift. Similarly, the seasonal component arises from the uneven heating experiencedby a body’s poles, where one may spend half the orbital period in sunlight and theother in the dark—creating a latitudinal temperature differential. Contrary to thediurnal component, the net force due to the seasonal effect always causes orbital decay(Figure 2.1).Incidentally these ideas were thought of almost a century ago by Ivan Yarkovsky,a civil engineer who studied various problems in his spare time. Yarkovsky theorizedthat a radiation force, while minute, could affect the dynamical evolution of smallasteroids and particles in space. Some forty years after seeing Ivan’s theories in a

ochure, astronomer Ernst Opik developed the first paper on it in 1951 (Vokrouhlicky,1998).Figure 2.1: A depiction of the seasonal Yarkovsky effect: the magnitude of theforce is characterized by the size of the arrow which is given for several points alonga body’s revolution around the sun, note that the spin axis is lying in the orbitalplane. Consider the body starting at the top of the diagram, where incident sunlightstrikes the south pole and is absorbed. The body retains this energy for a time Δtand then reradiates it with a component in the direction of orbital motion. This actslike a photon rocket and the recoil reduces the body’s orbital velocity, always causingthe orbit to decay, ie. the body slowly spirals into the sun. Figure from (Rubincam,1998)2.2 The Seasonal YEThe rate at which a body’s semi-major axis (SMA) decays due to the seasonal YE isoutlined in (Vokrouhlický, 1999):

ȧ = dadt = 4α 9Φ E(x)sinδ(x)sin 2 γ (2.1)w rev 1+θ/xwhere α =1− A, is the body’s reflectivity since A (the albedo), is the fraction ofsunlight absorbed. The angle between a body’s equator and its orbital plane is theobliquity γ, andw rev is the body’s mean orbital angular velocity in its path aroundthe sun:w rev =√GM∗a 3 (2.2)where G is the gravitational constant and M ∗ is the mass of the sun with a beingthe semi-major axis (ie, the body’s average distance from the sun). The radiationpressure coefficient Φ is given by:Φ= πR2 F ∗mc(2.3)where m and R are the mass and radius respectively. The incoming solar radiationor solar flux F ∗ is given by:F ∗ =L ∗4πa 2 (2.4)where L ∗ is the amount of energy radiated by the sun per second and the massof a spherical body is m = ρ × 4 3 πR3 , rho and R are the body’s density and radiusrespectively. Eqn(2.3) can now be rewritten as:

Φ= πR2 F ∗mc= πr24π3 ρR3 cL ∗4πa 2 =3L ∗16πρa 2 Rc(2.5)Let’s recast Eqn(2.1) using:f θ (x) = E(x)sinδ(x)1+θ/x(2.6)ȧ = 4α 9Φw revf θ (x)sin 2 γ (2.7)We can characterize the size of our body by a dimensionless parameter x, andits geological and thermal parameters by θ.The numerator E(x)sinδ(x), is a setof complicated functions of x & θ and I refer the reader to Vokrouhlický (1999) forthe details; Appendix (B) provides an IDL subroutine to solve these equations. Thethermal parameter θ is a function of the body’s thermal conductivity K, the specificheat at constant pressure C P , Boltzmann’s constant σ, the thermal emissivity ɛ ,andthe temperature of the body T :θ =√ρCp KwɛσT 3 (2.8)When dealing with the seasonal YE we set w = w rev , and for the diurnal YE weset w = w rot , the rotational velocity. The body’s dimensionless size parameter x is

C pKlɛThe heat capacity is the amount of energy requiredto increase a material’s temperature by ΔT .The thermal conductivity is the rate at which energytravels through a material.The thermal depth is the distance an energy waveis able to penetrate through a body’s surface.The thermal emissivity is a measure of a material’sability to absorb and radiate energy.Table 2.1: Definitions for some of the chemical and geological parameters used todescribe a ‘body’given in terms of the thermal depth l and R:x =√2Rl(2.9)where√Kl =ρC p n(2.10)The temperature of the sunward-facing side of a rotating body is given in (Vokrouhlický,1999) as :T 4 =L ∗16πa 2 σ(2.11)Finding θ and x Which Optimize 〈ȧ〉We anticipate that the YE will be very weak in the KB due to the great distancefrom the sun. Thus for demonstration purposes we only need to consider those bodiesthat suffer the maximum orbital decay—this will allow us to determine if the YE isof any consequence in the KB. We can further simplify Eqn(2.7) by introducing the

variable Γ:Γ= 4α 9Φw revsin 2 γ (2.12)ȧ = 4α 9Φw revf θ (x)sin 2 γ = f θ (x)Γ (2.13)Treating Γ as a constant, Figure (2.2) shows a contour plot of |a(x, θ)| whichsuggests that bodies with a value of x & θ ≈ 3 have the maximum orbital decay(strongest YE).Figure 2.2: A contour plot of |ȧ| as a function of x and θ. Notethat|ȧ| has a localmax at x ≈ 3andθ ≈ 3.

C p The Heat Capacity of Ice 6.1 × 10 6 ergsK·gmρ Density of Ice ρ ≈ 2 gcm 3K The Thermal Conductivity of Ice at T = 250K K=5.2×10 4 ergK·cm·sɛ Thermal Emissivity assuming the body is ’dark’ and small ɛ =1−10 rad*n Angular Revolution With a SMA =40AU n=7.9×10s*T Temperature of the Body With a SMA =40AU T≈44K=-229 ◦ Ca Semi-major axis a =40AUTable 2.2: Geological parameters used to describe our KBO, variables marked with‘*’ have been calculated using the equations above.Finding the Radius, Thermal Depth, and Thermal Conductivity WhichMaximizes the Orbital Decay due to the YETable (2.2) indicates the values of K, C p & ρ used to describe a KBO made of ice(Lide, 1999). Solving Eqn(2.8) we find that θ = 5, which is comparable to our optimalvalue of θ = 3 (Figure 2.2). Adapting x & θ ≈ 3, we can recursively determine thecorresponding thermal conductivity, thermal depth, and size. The thermal conductivitythat optimizes the YE is K θ=3 =2.2 × 10 4ergKcms, solving Eqn(2.10) we obtain athermal depth of l θ=3 ≈ 15m. Eqn(2.9) is re-arranged to find a radius of R θ=3 ≈ 30m.Semi-Major Axis and Eccentricity EvolutionNow that all the variables and equations are known we can solve the constant Γ inEqn(2.13) and obtain a value for the optimal decay rate:ȧ = −0.75 AUGyr≈−1AUGyr(2.14)Lastly, due to the YE a body’s eccentricity evolves according to Eqn(1) in (Ru-

Figure 2.3: ȧ for varying thermal conductivity parameters: K θ=3 =2.2 × 10 4(solid), K ice =5.2 × 10 4 erg (dashed).Kcmsbincam, 1998) :ergKcmse(t) =e 0[ a(t)a 0] 0.72(2.15)where e 0 and a 0 are the eccentricity and semi-major axis evaluated at time t = t 0 .We can differentiate Eqn(2.15) to obtain:ė = dedt = −0.72eȧ a(2.16)

2.2.1 SummaryFigure (2.3) shows that an optimal SMA decay of ȧ ∼−1.0AU/Gyr occurs for aKBO of radius R ≈ 30m. With the present age of our solar system at ≈ 4.5Gyrs,a comet of that size would take 15 Gyrs to arrive into the inner solar system fromthe KB. Thus we can conclude that the seasonal YE is much to weak on its own tocause the diffusion of comets from the KB into the inner solar system. However, thefollowing simulations in Chapter 5 demonstrate that the YE can play a role in anoverall delivery system by replenishing local resonances with KBOs.2.3 The Diurnal YEThe rate at which a body’s SMA decays due to the diurnal YE is outlined in (Vokrouhlicky,1998), Eqn(35):ȧ = dadt = −8α 9Φw revf θ (x)cosγ (2.17)dadt = −f θ(x) 8α 9Φw revcosγ = f θ (x)Γ 2 (2.18)Notice that the equation differs from the seasonal YE (Eqn 2.7) by a constant, sothe diurnal variant is also strongest when x & θ ≈ 3. Using Table (2.2), let’s calculatethe thermal parameter due to a body’s rotational motion:θ =√ρCp Kw rotɛσT 3 (2.19)Sheppard and Jewitt (2002) suggest that KBOs of diameter >200km have or-

ital periods of approximately 8 hours, so the rotational velocity becomes w rot =2πP rot(8hours)=2× 10−4 radsec. Plugging this into the equation above, θ ≈ 2000 which ismuch bigger than our optimal value of θ = 3. Thus we can conclude that the orbitaldrift caused by the diurnal YE is too small to be of any consequence in the KB.

Chapter 3ResonancesA resonance is a site in the Solar System where the orbital periods of two bodies arecommensurate (ratios of integers). For example, a KBO that is near a 5:2 resonancewill orbit the sun twice for every five of Neptune’s. More importantly, KBOs nearthese resonances will experience periodic perturbations from the giant planet whichcan lead to an increase in eccentricity and a more elongated orbit. With increasingeccentricity the body’s perihelia 1 decreases and this can cause the KBO’s orbit tocross Neptune’s. 2A close-approach may result in the particle being gravitationallyscattered by the planet; an example is shown in Figure (3.1).Astronomers who study the main belt are fortunate in that they can directlyobserve the strength of the resonance phenomena through Kirkwood Gaps, areas inthe main belt devoid of asteroids due to resonances with Jupiter. However, due tosmall comet sizes and their distance away from earth, the KB remains just outside thelimit of acquiring solid observational data. As a result, these ‘telescopic effects’ hinderus from knowing the true KB layout and so we must depend heavily on computersimulations and mathematical extrapolation (Grundy et al., 2002). Figure (3.2) showsa numerically generated KB with 10 4 particles evenly distributed between 35 & 701 The point in the orbit nearest to the sun.2 Increase in eccentricity → decrease in perihelia → overlapping orbits, see Appendix (C), Figure(C.1)

AU—this simulation exhibits gaps devoid of KBOs between (5:4 & 4:3) and (3:2 &5:3) resonances 3 , a direct analogy of Kirkwood Gaps.Figure 3.1: Ejection of a KBO due to eccentricity pumping: a particle starts in the4:3 resonance where its eccentricity is pumped up by Neptune until it crosses thatplanet’s orbit. A close-approach results in gravitational scattering and the particle isthrown about the solar system. The horizontal dashed line indicates the eccentricitya particle must have in order to cross Neptune’s orbit: e>1 − r p /a where (r p )isNeptune’s perihelion distance.3 For a list of strong KB resonances and their corresponding semi-major axes see Appendix (C),Table (C).

Chapter 4The N-body Integrator, Mercury6The n-body integrations in this project were performed using the Mercury6 code(Chambers, 1999). The simulations used the hybrid symplectic/Bulirsch-Stoer integratoralong with an additional subroutine used to mimic the YE (see Appendix (A)).Mercury6 allows the user to add an additional acceleration, f: ⃗⃗f =[0.98 (⃗v − ⃗v c)+ 1 ]an 2 ˆθ 〈ȧ〉 n (4.1)where ⃗v is the velocity of a particle, ⃗v cis its velocity in a circular orbit, theacceleration of the particle is in the polar direction ˆθ, and recall that the rate of theorbital decay due to the seasonal YE is given by 〈ȧ〉 ∼−1AU/Gyr. By inserting thisacceleration f ⃗ into Gauss’ eqns (Murray and Dermott, 1999), one can show that theparticle’s semi-major axis a(t) and eccentricity e(t) will decay at the desired rates,Eqns (2.14) and (2.16). Moreover, Figure 4.1 supports that this is indeed the case.Having the code equipped with the YE, we can go ahead with our simulations inwhich the following is assumed:• 10 4 particles will be evenly distributed between 35-75AU and evolved for 4.5Gyrs—the current age of our solar system.• The YE shall cause a constant SMA decay of ȧ ∼−1AU/Gyr for all particles

Figure 4.1: Using the fictitious force created to mimic the YE Eqn(4.1), a particle’sorbit decays at the desired rates a(t) Eqn(2.14), and e(t) Eqn(2.16). The jagged linesare a result of perturbations on the body due to the force of gravity with the Sun andthe four giant planets.but they shall have varying ė according to Eqn(2.16), which implies a size ofR ∼ 30m• These particles are massless which is justified since we are dealing with smallsized bodies. The gravitational force felt by a given particle is due to the sunand the four giant planets: Jupiter, Saturn, Uranus, and Neptune.• Solar system formation models require KBOs to be be dynamically cold—meaning the particles shall start with very low eccentricities ē =1.0 × 10 −3 .• Collisions among the particles were ignored, see the discussion in Chapter 6.

Chapter 5Discussion and Results5.1 The Delivery Model: YE → Resonance → Ejection.Following the role played by the YE in the asteroid belt, we will consider whetherthe force can deliver KBOs into nearby resonances where eccentricity pumping maycause a close-approach with Neptune and result in a KBO being scattered (Figure5.1).Figure 5.1: The delivery model in action: the top graph shows a particle startingat ≈46AU being transported to the 5:3 resonance by the YE. Eccentricity pumpingleads it to exceed the threshold required for scattering by Neptune and it is thrownabout the solar system. Similarly, the bottom graph depicts a particle starting at≈44AU and being dragged by the YE into the 5:3 resonance where it shares the samefate as the particle above.

5.2 The YE: changing the KB initial & finalstates.Examining the consequence of the YE on the KB is accomplished by comparing theKB layout at t =4.5Gyrs to a control, which is the same simulation except with theYE turned off. According to Figure (5.2) & (5.3), the YE has completely eroded theKB of these small R ∼ 30m bodies between the 5:4 & 5:3 resonances and then againbetween the 5:3 & 3:2—a distance of ∼ 7AU for the former! The YE has helpedthe 2:1 resonance trap over 1200 particles (Figure 5.2), almost four times the amountwithout the YE. Resonance trapping occurs when the energy and angular momentumtransferred to the particle from the giant planet are balanced by that which the YEhas imposed, effectively initiating an equilibrium so the particle remains ‘trapped’ inthat resonance. According to Figure (5.3), particles in every major resonance havehad their eccentricites pumped, the YE facilitates this process by dragging particleseven closer to the resonance’s center. Moreover, significant pumping has occurred atthe 5:3 & 7:4 resonances and particles that passed through the 2:1 and the weak 3:1resonance have had their orbits perturbed.

Figure 5.2: Population statistics for the KB at the initial and final states: thedashed line indicates t = 0, and the solid t =4.5Gyrs. The top graph is the simulationevolved with the YE, and the bottom is the control (same simulation without YE).

Figure 5.3: The eccentricity layout for the KB at the final state (t =4.5Gyrs).The top graph is the simulation evolved with the YE, and the bottom is the control(same simulation without YE).

5.3 The YE: increasing the inward flux of KBOs.Figure (5.2) demonstrates that the KB’s inner edge is almost completely depletedof these small bodies. We examine the possibility that these missing particles wereejected into planet-crossing orbits within the inner solar system.Figure 5.4: The inward flux of KBOs over time: at the final iteration of t =4.5Gyrsthe YE contributes to an increase of ∼ 100 times the incoming KBOs.Figure (5.4) is a plot of the number of simulated particles having a semi-majoraxis less than Neptune’s at time t =4.5Gyrs. Again, the solid line represents thesimulation having the YE turned on and the dashed line is the control with YE off.Initially both runs have an equal amount of incoming bodies due to the fact thatthey have the same initial KBO distribution.Hence, particles starting near or inresonances may get ejected in both.However, without the YE the flux decreases

at later times because there is no mechanism to deliver particles from their startingposition into a strong resonance. Essentially, the YE gives particles who don’t start inor near a resonance the possibility of getting there due to the imposed semi-major axisdecay rate of ȧ ∼−1AU/Gyr. Thus the effect will continually replenish resonanceswith KBOs and allow them the opportunity to be ejected. As a result, it appearsthat the Yarkovsky enabled run ejects ∼ 100 times the number of KBOs into thesolar system at the final iteration (t =4.5Gyrs).

Chapter 6CollisionsIf the collision timescale in the KB is much less than that of the YE (T YE ≈ 10 9 yrs),KBOs would continue to fragment until they are significantly smaller than the optimalradius of 30m—thus making the YE negligible. Let’s go ahead and obtain a roughapproximation for the collision timescale.Using an estimate for the mass of the KB, M KB =0.02M ⊕ (Hahn and Malhotra,2005), we can derive the total number of KBOs assuming that they’re all of equalsize and composition:N = M kbm kbo=M kb4π3 ρR3 kbo(6.1)where m kbo is the mass of a single KBO and R kbo is its radius. Assuming the KBis a disk with an outer radius of 50AU and an inner of 30, the surface area is givenby:A kb = π(Rout 2 − Rin) 2 =10 30 cm 2 (6.2)depth:The fraction of the belt that is occupied by KBOs can be described by the opticalτ = N × A kboA kb= 3M kb4ρR kbo A kb(6.3)where A kbo = πRkbo 2 , is just the cross-sectional area of a KBO. Finally, the collision

timescale is related to orbital period P orb and the optical depth τ:T c = P orbτ(6.4)where the orbital period is simply2πw rev=250 yrs, and recall w rev is given byEqn(2.2).Since a KBO is in an inclined orbit, it penetrates the KB plane twiceper period, so the probability of a collision per orbit is P c =2τ/P orb . The collisiontimescale is thus:T c = P −1c= P orb2τ= 2P orbρR kbo A kb3M kb(6.5)Of course our results will depend upon the details of the KBO size distributionwhich our simple calculation neglects.If we assume the KB’s optical depth τ isdominated by 1 km sized bodies R kbo ∼ 1km, thenT c =10 11 yrs. Thissuggests thatcollisions can be neglected since our collision timescale is even much larger than theage of the solar system.

Chapter 7ConclusionsWe have shown that the orbital drift due to the diurnal YE is too small to be ofany consequence in the KB. However, the seasonal variant causes a maximum orbitdecay of ȧ ∼−1AU/Gyr for a small subset of KBOs whose physical properties mustconspire such that their thermal and size parameters, θ and x, are both near 3.These comets correspond to having a diameter of ∼60m and a thermal conductivityof K θ=3 =2.2 × 10 4ergKcms, which is approximately half that of ice water.By continually replenishing resonances with KBOs, the seasonal YE significantlyenhances their transport into the inner solar system. Unfortunately, these YarkovskydrivenKBOs are 1-2 orders of magnitude too small to be considered the progenitorsof the JFCs.

Appendix AYE Subroutine for Mercury6c%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%cc MFO_USER.FOR (ErikSoft 2 March 2001)c (YE Subroutine Added Summer 2004)cc%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%cc Author: John E. Chambersc Author: Daniel Majaessc Physics Derived By: Prof. Josef Hahnc Applies the YE.cc If using with the symplectic algorithm MAL_MVS, the force should bec small compared with the force from the central object.c If using with the conservative Bulirsch-Stoer algorithm MAL_BS2, thec force should not be a function of the velocities.cc N.B. All coordinates and velocities must be with respect to centralc bodyc ===c----------csubroutine mfo_user (time,nbod,nbig,m,x,v,a,params)cimplicit noneinclude ’mercury.inc’cc Input/Outputinteger nbod, nbigreal*8 time,jcen(3),m(nbod),x(3,nbod),v(3,nbod),a(3,nbod)real*8 params(3,nbod)cc Localinteger j,k

eal*8 tau,da,sigma,t_factor,r,v2,one_over_a,factor,dadtreal*8 r_1,a_s,v_2,n,a_dot,b1,b2,b3cc--------cdo j = 2, nbodc zero the accelerationsa(1,j) = 0.d0a(2,j) = 0.d0a(3,j) = 0.d0c migration parameterstau=params(1,j)da=params(2,j)sigma=params(3,j)c Compute the planet’s accelerations.if (tau .gt. 0.d0) thent_factor=0.5d0*exp(-time/tau)/taur=x(1,j)**2+x(2,j)**2+x(3,j)**2r=sqrt(r)v2=v(1,j)**2+v(2,j)**2+v(3,j)**2one_over_a=2.d0/r-v2/K2factor=t_factor*da*one_over_aa(1,j)=factor*v(1,j)a(2,j)=factor*v(2,j)a(3,j)=factor*v(3,j)end ifa_dot=params(3,j)if (abs(a_dot) .gt. 0.d0) thenccccDeclare EquationsVelocity Vectorv_2=(v(1,j)**2+v(2,j)**2+v(3,j)**2)Radius Vectorr_1=sqrt(x(1,j)**2+x(2,j)**2+x(3,j)**2)SMAa_s=1.d0/(2.d0/r_1-v_2/K2)

cN, mean motionn=sqrt(K2/abs(a_s)**3)c Calculate some terms for efficiencyb1=0.98*a_dot/a_sb2=0.5*a_dot*n/r_1b3=a_s*n/r_1c&&The acceleration Componentsa(1,j)=b1*(v(1,j)+x(2,j)*b3)-x(2,j)*b2a(2,j)=b1*(v(2,j)-x(1,j)*b3)+x(1,j)*b2a(3,j)=0.0d0end ifend doccc-----------creturnendcc%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Appendix BIDL Subroutine: Calculating E(x)and d(x);Created by Prof. Josef Hahn.;calculate the E and delta functions used in the formula for the YE;described in Vokrouhlicky (1999), A&A, v344, p362. The inputs are the;dimensionless radius x (which can be an array) and theta (a number),;while the outputs are E(x) and delta(x)pro Ed,x,theta,E,deltachi=theta/xfactor=chi/(1.0+chi);A,B,C,D functionsA=-(x+2.0)-exp(x)*( (x-2.0)*cos(x) - x*sin(x) )B=-x-exp(x)*( x*cos(x) + (x-2.0)*sin(x) )C=A+factor*(3.0*(x+2.0)+exp(x)*(3.0*(x-2.0)*cos(x) + x*(x-3.0)*sin(x)))D=B+factor*(x*(x+3.0)-exp(x)*(x*(x-3.0)*cos(x) - 3.0*(x-2.0)*sin(x)));E*exp(i*delta)num=dcomplex(A,B)den=dcomplex(C,D)Ed=num/denEd_real=real_part(Ed)Ed_imag=imaginary(Ed);E and deltaE=sqrt(Ed_real^2+Ed_imag^2)delta=atan(Ed_imag,Ed_real)returnend

Appendix CMiscellaneousFigure C.1: Consider the orbits of two bodies with different semi-major axes (SMA)and masses, the larger’s orbit in solid and the crosshair indicates the location of thesun. With increasing eccentricity, the smaller body’s perihelion decreases and so itapproaches the sun and the other body. With a high enough eccentricity a closeencounter will most likely result in the smaller body being scattered.

Resonance Semi-major axis (AU)5:4 34.94:3 36.53:2 39.55:3 42.37:4 43.72:1 47.83:1 62.6Table C.1: KB resonances and their corresponding semi-major axes

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AcknowledgementsThese simulations were performed on the McKenzie computer cluster at the CanadianInstitute for Theoretical Astrophysics (CITA) at the University of Toronto; thosemachines are funded by the Canada Foundation for Innovation (CFI) and the OntarioInnovation Trust (OIT). Special credit goes out to the administrators Robin Humbleand Chris Loken.I would like to thank my fellow students and future astronomers Joel Tanner, JonSavoy, Adam Chaffey, and Chris Geroux.Lastly, I would like to acknowledge theefforts of Prof.Joseph Hahn who showed great patience towards me and providedcountless hours to the success of this project; and Prof. David Clarke for giving meencouragement when it was most needed.