Effects of Young Clusters

on Forming Solar Systems

WITH: Eva M. Proszkow, Anthony Bloch (Univ. Michigan)Philip C. Myers (CfA), Marco Fatuzzo (Xavier University)David Hollenbach (NASA Ames), Greg Laughlin (UCSC)

Fred C. AdamsUniversity of Michigan

Stars to Planets, Univ. Florida

Most stars form in clusters:

In quantitative detail: What effects does the cluster environment haveon solar system formation?

•N-body Simulations of Clusters

•UV Radiation Fields in Clusters

•Disk Photoevaporation Model •Scattering Encounters

Outline

Cumulative Distribution: Fraction of stars that form in stellar aggregates with N < N as function of N

Lada/LadaPorras

Median point: N=300

Simulations of Embedded Clusters

• Modified NBODY2 Code (S. Aarseth)• Simulate evolution from embedded

stage out to ages of 10 Myr• Cluster evolution depends on the

following:–cluster size– initial stellar and gas profiles–gas disruption history–star formation history–primordial mass segregation– initial dynamical assumptions

• 100 realizations are needed to provide robust statistics for output measures

Virial Ratio Q = |K/W|virial Q = 0.5; cold Q = 0.04Mass Segregation: largest

star at center of cluster

Simulation Parameters

( )300Npc0.1NR

*=

Cluster MembershipN = 100, 300, 1000

Cluster Radius

Initial Stellar DensityGas Distribution

Star Formation Efficiency0.33

Embedded Epoch t = 0–5 Myr

Star Formation t = 0-1 Myr

( )*

*

Rr,

rM2

,1 3

s03

0G ==

+= ξ

πρ

ξξρρ

€

ρ∗∝ r−1

Dynamical Results

•Distribution of closest approaches

•Radial position probability distribution

(given by cluster mass profiles)

I. Evolution of clusters as astrophysical objects

II.Effects of clusters on forming solar systems

Mass Profiles

SubvirialVirial

p

a

a

T 1M)(M

⎟⎟⎠⎞

⎜⎜⎝⎛

+=

ξξξ ξ = r/r0

Simulation p r0 a100 Subvirial

0.69 0.39 2

100 Virial 0.44 0.70 3300 Subvirial

0.79 0.64 2

300 Virial 0.49 1.19 31000 Subvirial

0.82 1.11 2

300 Subvirial

0.59 1.96 3Stellar Gravitational Potential

∫∞ ⎟⎠

⎞⎜⎝⎛

+≡

00 11 duu

p

aψ00

** ψΨ

ρGM T=

Closest Approach Distributions

€

G=G0b

1000AU ⎛ ⎝ ⎜

⎞ ⎠ ⎟γ

Simulation G0 bC (AU)100 Subvirial 0.166 1.50 713100 Virial 0.059

81.43 1430

300 Subvirial 0.0957

1.71 1030

300 Virial 0.0256

1.63 2310

1000 Subvirial

0.0724

1.88 1190

1000 Virial 0.0101

1.77 3650Typical star experiences one close encounter with impact parameter bC during 10 Myr time span

– Photoevaporation of a circumstellar disk

– Radiation from the background cluster often dominates radiation from the parent star (Johnstone et al. 1998; Adams & Myers 2001)

– FUV radiation (6 eV < E < 13.6 eV) is more important in this process than EUV radiation

– FUV flux of G0 = 3000 will truncate a circumstellar disk to rd over 10 Myr, where

Effects of Cluster Radiation on Forming/Young Solar

Systems

⋅

=MMrd

*AU36

Calculation of the Radiation FieldFundamental Assumptions

– Cluster size N = N primaries (ignore binary companions)

– No gas or dust attenuation of FUV radiation– Stellar FUV luminosity is only a function of mass– Meader’s models for stellar luminosity and

temperature Sample IMF → LFUV(N) SampleCluster Sizes

Expected FUVLuminosity in SF

Cluster→

FUV Flux depends on:– Cluster FUV luminosity– Location of disk within

clusterAssume:– FUV point source

located at center of cluster

– Stellar density ρ ~ 1/r

Photoevaporation of Circumstellar Disks

G0 = 1 corresponds to FUV flux 1.6 x 10-3 erg s-1 cm-2

Median 900Peak 1800Mean 16,500

G0 Distribution

Photoevaporation Model

(Adams et al. 2004)

Results from PDR Code

Lots of chemistry and many heating/cooling linesdetermine the temperatureas a function of G, n, A

Solution for Fluid Fields

outer disk edge

sonic surface

Evaporation Time vs FUV Field

-----------------------

(for disks around solar mass stars)

Photoevaporation in Simulated

ClustersRadial Probability Distributions

01)(

rr

NrN

p

a

a

T

=⎟⎟⎠⎞

⎜⎜⎝⎛

+= ξ

ξξ

Simulation reff (pc) G0 mean

rmed (pc)

G0 median

100 Subvirial 0.080 66,500 0.323 359100 Virial 0.112 34,300 0.387 250300 Subvirial 0.126 81,000 0.549 1,550300 Virial 0.181 39,000 0.687 9921000 Subvirial

0.197 109,600 0.955 3,600

1000 Virial 0.348 35,200 1.25 2,060FUV radiation does not evaporate enough disk gas to prevent giant planet formation

for Solar-type stars

Evaporation Time vs Stellar MassEvaporation is much more effective for disks around low-mass stars:Giant planet formation can be compromised

G=3000

Evaporation vs Accretion

Disk accretion aids and abetsthe disk destruction process by draining gas from the inside, while evaporation removes gas from the outside . . .

Solar System ScatteringMany Parameters +Chaotic Behavior

Many Simulations Monte Carlo

Monte Carlo Experiments

• Jupiter only, v = 1 km/s, N=40,000 realizations• 4 giant planets, v = 1 km/s, N=50,000 realizations• KB Objects, v = 1 km/s, N=30,000 realizations • Earth only, v = 40 km/s, N=100,000 realizations • 4 giant planets, v = 40 km/s, Solar mass, N=100,000 realizations• 4 giant planets, v = 1 km/s, varying stellar mass, N=100,000 realizations

Red Dwarf captures the Earth

Sun exits with one red dwarf as a binary companion

Earth exits with the other red dwarfSun and Earth encounter

binary pair of red dwarfs

9000 year interaction

Ecce

ntric

ity e

Semi-major axis aJupiter

Saturn UranusNeptune

Scattering Results for our Solar System

20 AU1601350±≈C

Cross Sections

2.0 M 1.0 M

0.5 M 0.25 M

2/1

*0

−

⋅⎟⎟⎠⎞

⎜⎜⎝⎛

⎟⎟⎠⎞

⎜⎜⎝⎛

≈MMa

C p

AUejσ

Solar System Scattering in Clusters

( )4

*2

00eject AU 1000

)AU/(γγ

π

−

⋅⎟⎟⎠⎞

⎜⎜⎝⎛

⎟⎟⎠⎞

⎜⎜⎝⎛

=MMaC pΓΓ

Ejection Rate per Star (for a given mass)

Integrate over IMF(normalized to cluster

size)∫∫ ⎟⎠

⎞⎜⎝⎛=⎟⎠

⎞⎜⎝⎛ −

dmdNdmNm

dmdNdm where4

γ

Subvirial N=300 ClusterG0 = 0.096, = 1.7 GJ = 0.15 per Myr

1-2 Jupiters are ejected in 10 Myr

Less than number of ejections from internal solar system scattering

(Moorhead & Adams 2005)

Conclusions• N-Body simulations of young embedded clusters

– Distributions of radial positions and closest approaches

– Subvirial clusters more concentrated & longer lived

• Clusters have modest effects on star and planet formation– FUV flux levels are low & leave disks

unperturbed– Disruption of planetary systems rare, bC ~

700-4000 AU – Planet ejection rates via scattering encounters

are low------------------------------------------------------------------

---• Photoevaporation model for external FUV

radiation• Distributions of FUV flux and luminosity • Cross sections for solar system disruption • [Orbit solutions, triaxial effects, spirographic

approx.]

Time Evolution of Parameters

Subvirial Cluster Virial Cluster100 300 1000

• In subvirial clusters, 50-60% of members remain bound at 10 Myr instead of 10-30% for virial clusters • R1/2 is about 70% smaller for subvirial clusters

• Stars in the subvirial clusters fall rapidly towards the center and then expand after t = 5 Myr

Orbits in Cluster Potentials

€

ρ = ρ0

ξ (1+ ξ )3 ⇒ Ψ = Ψ0

1+ ξ

ε ≡ E /Ψ0 and q ≡ j 2 /2Ψ0rs2

ε = ξ1 + ξ 2 + ξ1ξ 2

(ξ1 + ξ 2)(1+ ξ1 + ξ 2 + ξ1ξ 2)

q = (ξ1ξ 2)2

(ξ1 + ξ 2)(1+ ξ1 + ξ 2 + ξ1ξ 2)

Orbits (continued)

€

qmax = 18ε

(1+ 1+ 8ε − 4ε)3

(1+ 1+ 8ε )2

ξ∗ = 1− 4ε + 1+ 8ε4ε

Δθπ

= 12

+ (1+ 4ε)−1/ 4 − 12

⎡ ⎣ ⎢

⎤ ⎦ ⎥1+ log(q /qmax )

6log10 ⎡ ⎣ ⎢

⎤ ⎦ ⎥3.6

limq→qmax

Δθ = π (1+ 8ε)−1/ 4

(effective semi-major axis)

(angular momentum of the circular orbit)

(circular orbits do not close)

Spirographic Orbits!

(Adams & Bloch 2005)

€

x(tp ) = (α −β )cos tp + γ cos[(α −β )tp /β ]

y(tp ) = −(α −β )sin tp + γ sin[(α −β )tp /β ]€

(ε, q)

(ξ1,ξ 2)(α ,β,γ )

Orbital Elements

Allowed Parameter Space

Spirographic approximationis valid over most of the plane

Application to LMC Orbit

Spirographic approximationreproduces the orbital shape with 7 percent accuracy & conserves angular momentumwith 1 percent accuracy.Compare with observationaluncertainties of 10-20 percent.

Triaxial Potentials in Clusters

Box Orbit

Growth of perpendicular coordinate

Where did we come from?

Solar Birth Aggregate

Supernova enrichment requires large N

€

M∗ > 25Mo

Well ordered solar system requires small N

€

FSN = 0.000485

€

ε(Neptune) < 0.1

€

ΔΘ j < 3.5o

Stellar number N

Prob

abilit

y P(

N)

Expected Size of the Stellar Birth Aggregatesurvival supernova

Adams & Laughlin, 2001, Icarus, 150, 151

Constraints on the Solar Birth Aggregate

€

N ≈ 2000 ±1100

€

P ≈ 0.017 (1 out of 60)

(Adams & Laughlin 2001 - updated)

Probability as function of system size N

(Adams & Myers 2001)

(Walsh et al. 2006)

NGC 1333 - cold start

€

(Δv)Z ≈ 0.1 km /s

(Δv)T (Δt)SF ≈ 0.02pc

Probability of Supernovae

€

PSN (N) =1− fnotN

Probability of Supernovae

€

PSN (N) =1− fnotN =1− (1− FSN )N

Probability of Scattering

€

G=n σvScattering rate:

Survival probability:

€

Psurvive ∝ exp − Γdt∫[ ]

Known results provide n, v, t as function of N (e.g. BT87) need to calculate the interaction cross sections

€

σ

Cross Section for Solar System

Disruption

€

σ ≈(400AU)2