Runaway Accretion & KBO Size Distribution Re’em Sari Caltech

Minimum Mass Solar Nebula Surface density (g cm -2 )

The Kuiper Belt ~40 AU from the Sun. Source of the short period comets –Hypothesized before first object found Pluto Charon –Largest members are in a binary! ~700 detected objects Size estimates based on brightness – Typical radius R=100km (for 4% albedo). –SCUBA observation (Varuna): albedo 7  3% Equal mass per logarithmic mass interval (Jewitt) Trujillo et. Al. Quaoar 18m 1:46m 1:21m

Binaries Stars: –mass ratio q~1 –a: uniform distribution over the range of possible scales Moons of planets: –mass ratio q<<1 (PC 10 -1, EM 10 -2, ST&NT ) –a: up to Hill radius Near Earth & Main Belt Asteroids: –Discoveries by ground based radar and optical photometry and spacecraft imaging. Kuiper Belt: –mass ratio q~1 –a: up to Hill radius (Veillet 2002) e=0.8 P=570days

Kuiper Belt Binaries

“Standard” Binary Formation Requires energy dissipation! Standard mechanisms: –Collision –Tidal capture –Three body gravitational interaction Collisions & Tidal capture: –Small initial periapse –Tidal torque can raise periapse: Works perfectly for the Earth-Moon. Limits: angular momentum of the primary & timescale Maximum periapse:

Kuiper Belt Binaries – A Mystery Maximal periapse: Kuiper Belt Binaries – impossible unless 1-e<<1 Objecta [km]ei [deg]TypeQ [arcsec]P [days]DMag Pluto19, PKBO WW3122, CKBO QT CKBO QW CKBO TC36--- PKBO SM SKBO CQ29--- CKBO CF CKBO (Jewitt) R per =a(1-e)~4000km~40R kbo

Geometric Accretion Collision cross section is geometric Scale height h~v/  –n  /h In terms of surface density: –Independent of velocity For MMSN: –Earth (6,400km) 10 8 yr –Jupiter’s core 10 9 yr –Neptune (25,000km) yr –Pluto (1,100km) yr

Gravitational Focusing Larger collision cross section Growth rate depends on Safronov’s number (v esc /v) 2 Safronov: velocity, v, is not a free parameter. –Controlled by interactions between bodies. Form MMSN: geometricrequired limits time (yr)time (yr)on eccentricity –Earth (6,400km) 10 8 yrt<10 8 none –Jupiter’s core 10 9 yrt<10 7 e<0.1 –Neptune (25,000km)10 12 yr10 7 <t< <e<0.4 –Pluto (1,100km) yrt<10 10 e<0.05

Runaway Accretion Without gravitational focusing –Bodies tend to become of equal size With focusing –Few large bodies become larger than their peers. Could eccentricities be excited to required levels ? Is there enough mass in the excited region ? Does oligarchic stage stall accretion ? Most of the Bodies & mass Exponential cutoff Runaway tail m m N(m)

Mass Spectrum ? With focusing Mapping of initial size Almost generally Most of the Bodies & mass Exponential cutoff Runaway tail m m N(m) Does not agree with Kenyon’s simulations! Does not agree with observed KBO size distribution!

Runaway Accretion Without gravitational focusing –Bodies tend to become of equal size With focusing –Few large bodies become larger than their peers. Could eccentricities be excited to required levels ? Is there enough mass in the excited region ? Do oligarchic or orderly stages stall accretion ? Most of the Bodies & mass Exponential cutoff Runaway tail m m N(m)

Interaction with a cold surrounding

Hill sphere –Tidal effects from the Sun –Sets a minimum drift velocity –Sets the maximum binary separation Viscous stirring –Radial and tangential velocity are coupled - eccentricity –Even elastic deflections increase velocity dispersion –Results in much faster heating: temperature doubles in one deflection timescale Disk Effects VrVr VtVt Elastic scat. V r ->V t Initial velocity dispersion Elastic scat. V t ->V r

Solar Angle -   - Angle subtended by Solar radius. –  = at 1AU. –  ~10 -4 at Kuiper belt. Significance: spacing of scales: –Hill radius: R H  -1 R –Hill velocity v H  1/2 v esc Kuiper Belt binaries: –Possible separations 2R<a<R H –From Earth, R H for a 100km body subtends 20’’ R  100km R H  -1 R  10 6 km   AU= cm v H  1 m/s

Simplified Runaway Accretion Setup: –Many small bodies -   0.3 g cm -3 –Few large bodies -    R  100km Processes: –Runaway: large bodies grow by accreting small ones –Small bodies stirred by large bodies –Large bodies: stirring from other large bodies balanced by dynamical friction from small bodies.

Simplified Runaway Accretion Setup: –Many small bodies:   0.3 g cm -2 (from min solar nebula) –Few large bodies:    R  100km (We see them) Processes: , u , v stirring friction accretion stirring =

Simple Solution Consistency: –We assume u>v H & v<v H which requires –We neglected collisions between small bodies Generalization yields the complete velocity spectrum. Mass spectrum more subtle Agrees with detailed simulations (Kenyon) (Kenyon 2002)

Physical Collisions during runaway? Collisions are negligible if Initial body size (GW) –About 1 km for MMSN –Independent of a

Collisions in KB after runaway? Observed velocity dispersion v dis ~ 1 km/s Observed surface density  ~ 3  gr/cm 2 Age of the solar system T ~ 4 Gyr Collisions between bodies of similar size r occur if: Collision of size r on size R:

Collisions After Runaway Bodies above 70 km are safe. The 10 km bodies, can be destroyed by smaller bodies. sufficient rate sufficient energy Bernstein et. al. astro-ph/ suggest 40 km cutoff. Agrees also with Stern (95) calculations. Needs self consistent calculation, e.g. Dohnanyi spectrum.

Binary Formation Binaries cannot form today. –Velocity dispersion is too high (~1 km/s) –No gravitational focusing: Pluto’s escape velocity 1 km/s –Current collision rate: once every yr Formation during runaway accretion –Low velocity dispersion (necessary for focusing) –Kuiper Belt has been frozen since then (by heating!). –Happened before isolation of large objects Laboratory for early stages of runaway accretion.

Binary Formation – Two Channels Setup – just that of runaway accretion: –Large bodies (  - observed) v~0.3v H –Small bodies (  - extrapolation of MSN) u~3v H Stabilization of Transient Binary –requires energy loss L 3 Channel: –Three large bodies share a common Hill sphere –One escapes with energy increase L 2 s Channel –A transient binary looses energy to the sea of small bodies by dynamical friction. v H ~1m/s

L 2 s Channel Initial separation 0.9 Hill Radius Initial separation 1.0 Hill Radius

L 2 s Channel Initial separation 0.9 R H

L 2 s Channel Initial separation 1.0 R H

L 2 s Channel Red: Non capture encounters Blue: Capture encounters Green: Lagrange points L 1 & L 2 Fraction captured is proportional to the drag.

L 3 Channel Initial separation 1.3 Initial separation 1.4 Initial separation 1.5

L 3 Channel Initial separations 1.3 R H

L 3 Channel Initial separations 1.4 R H

L 3 Channel Initial separations 1.5 R H

Binary Formation – Rates L 3 Channel: –Bottleneck: 3 bodies together. –Formation rate per body is (  /  R) 2  -4  =0.3 / My L 2 s Channel: –Bottleneck: 2 bodies together + enough dynamical friction. –Formation rate per large body: (  /  R) 2 (  /  )  -2  = 3 / My L 2 s/L 3 = (  /  ) 3  -2 ~ 10

Binary Evolution = Separation Distribution 3’’ L 2 s or L 3 Create binaries here L 2 s every 3·10 5 yr 0.2’’ Probability a p(a) separation 5% Orbit decays by dynamical friction. With u=3m/s, timescale is 10 3 years Orbit decays by dynamical friction. Larger u therefore longer timescale Contact achieved over 1Myr = growth timescale 300% 20’’

Binary Evolution - Separation L 2 s and L3 initial separation a  R H Binary shrinks due to dynamical friction: –Initially, constant decay rate. –Below r u ~3’’ small body velocity dispersion increases –Below r u ~3’’ decay slows down. Separation distribution: –Constant per logarithmic interval above r u 0.3% –Increases inversely with separation below r u Contact achieved on growth timescale 3’’ 0.3%

Binary Separation Distribution With our parameters: –For a>3’’, a p(a)  0.3% independent of a. –For a<3’’ a p(a)  0.3 (a/3’’) -1 %. –Predicts 5% of KBO are binaries with a>0.2’’. –Compatible with HST survey by Brown. During runaway accretion: –Each large body captures a companion every y. –Lifetime until binary merges 10 6 y.

Binary Coalescence Dominates growth for R  (  /  )  -2 =300km. Produces bodies rotating near breakup. Leaves close binaries light curves holds clues: –30% greater than 0.15Mag –20% greater than 0.4Mag –Jacobi Ellipsoid or Eclipsing binaries or Albedo variation 2000 GN 171 (Sheppard & Jewitt) Small numbers statistics P=8.3h

Predictive Power Separation distribution. Eccentricity distribution. Inclination distribution. Most KBO’s are tight binaries Most are rotating close to breakup. Stable systems of higher multiplicity ? Distribution of the binary mass ratio - q. Requires understanding of velocity anisotropy, and its effect on dynamical friction

Pluto - Charon ? Angular momentum –Marginal for collision scenario Could be produced by L 2 s or L 3 !! –Then decay by dynamical friction to current separation. Inclination distribution may reveal answer –Theoretically and observationally HST program to determine binary orbital elements. –Does binary inclination correlate with separation?