Effects of Young Clusters
on Nascent Solar Systems
Fred C. Adams
University of Michigan
Eva M. Proszkow (University of Michigan)
----------------------------------------------------Philip C. Myers (Harvard Smithsonian CfA)
Marco Fatuzzo (Xavier University)
David Hollenbach (NASA Ames)
Greg Laughlin (UC Santa Cruz)
Anthony Bloch (University of Michigan)
Most stars form in clusters:
What exactly are the effects
of the cluster environment
on the coupled processes of
star and planet formation?
Outline
• N-body Simulations of Young Clusters
• UV Radiation Fields in Young Clusters
• Photoevaporation Models for Disks
• Orbit Solutions in Cluster Potentials
• Scattering Encounters in Young Clusters
• NGC 1333 Case Study
• Constraints on Solar Birth Aggregate
Introduction
• Environmental effects on planet
formation:
– disruptive FUV radiation
– scattering interactions between
star-disk systems
• Large (>1000 stars), dense clusters
disrupt disk and planet formation
• Small (<100 stars) clusters don’t
• Intermediate sized clusters
(100 < N < 1000 members)
• Virial and subvirial initial
conditions (NGC 1333, 2068, 2264)
(Lada & Lada 2003;
Porras et al. 2003)
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
Simulation Parameters
Virial Ratio Q = |K/W|
virial Q = 0.5; cold Q = 0.04
Mass Segregation: largest star
at center of cluster
Cluster Membership
N = 100, 300, 1000
Cluster Radius
N
R* N   1.0 pc
  r 1
Initial Stellar Density
Gas Distribution
G 
0
 1   
3
,

0 
2M*
 rs
300
3
,
 
r
R*
Star Formation Efficiency 0.33
Embedded Epoch t = 0–5 Myr
Star Formation t = 0-1 Myr
Fundamental Plane for Clusters
(from Megeath, Allen et al. PPV)
NGC 1333 - cold start
(v) Z  0.1 km/s
(v)T (t) SF  0.02 pc

(Walsh et al. 2006)
Collection of SPITZER Clusters
Results from Simulations
•Evolution of cluster parameters:
– bound cluster fraction fb
– virial ratio Q
– velocity isotropy parameter b = 1 - vq2/vr2
– half-mass radius R1/2
•Distribution of closest approaches
•Radial position probability distribution
•Cluster mass profiles
Time Evolution of Parameters
Subvirial Cluster
100
Virial Cluster
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
Mass Profiles
Subvirial
Virial
M( )   a 

 
a 
MT
1  
Simulation
p
 = r/r0
p
r0
a
100 Subvirial
0.69
0.39
2
100 Virial
0.44
0.70
3
300 Subvirial
0.79
0.64
2
300 Virial
0.49
1.19
3
1000 Subvirial
0.82
1.11
2
300 Subvirial
0.59
1.96
3
Stellar Gravitational Potential
GM T *
* 
0
r0
0  

0
p
 1 
du

a 
1 u 
Closest Approach Distributions
 b 
  0

1000AU 
0

bC (AU)
100 Subvirial
0.166
1.50
713
100 Virial
0.0598
1.43
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
3650
Simulation

Typical star experiences
one close encounter with
impact parameter bC
during 10 Myr time span
Effects of Cluster Radiation on
Forming/Young Solar Systems
– 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
M*
rd  36 AU
M
Calculation of the Radiation Field
Fundamental 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)
Sample
→
Cluster Sizes
Expected FUV
Luminosity in SF
Cluster
Photoevaporation of Circumstellar Disks
FUV Flux depends on:
– Cluster FUV luminosity
– Location of disk within
cluster
Assume:
– FUV point source located
at center of cluster
– Stellar density  ~ 1/r
G0 Distribution
Median
900
Peak
1800
Mean
16,500
G0 = 1 corresponds to FUV flux
1.6 x 10-3 erg s-1 cm-2
Photoevaporation Model
(Adams et al. 2004)
Results from PDR Code
Solution for Fluid Fields
Evaporation Time vs FUV Field
Photoevaporation
in Simulated Clusters
Radial Probability Distributions
Simulation
N (r )   a
 
a
NT
1 



p

reff (pc)
G0 mean
rmed (pc)
G0 median
100 Subvirial
0.080
66,500
0.323
359
100 Virial
0.112
34,300
0.387
250
300 Subvirial
0.126
81,000
0.549
1,550
300 Virial
0.181
39,000
0.687
992
1000 Subvirial
0.197
109,600
0.955
3,600
1000 Virial
0.348
35,200
1.25
2,060
r
r0
FUV radiation does not evaporate enough disk gas to
prevent giant planet formation for Solar-type stars
Evaporation Time vs Stellar Mass
Evaporation
is much more
effective for
disks around
M-type stars:
Giant planet
formation is
compromised
Evaporation vs Accretion
Orbits in Cluster Potentials
0
0


3
 (1  )
1 
  E /0 and q  j /2 r
2
2
0 s
1   2  1 2

(1   2 )(1 1   2  1 2 )
(1 2 )
q
(1   2 )(1 1   2  1 2 )
2
Orbits (continued)
qmax
1 (1 1 8  4)

2
8 (1 1 8 )
1 4  1 8
 
4
3
(angular momentum
of the circular orbit)
(effective semi-major axis)
1 
1 log( q /qmax )
1/ 4
  (1 4)  1

 2 
2
6log10 
3.6
q
lim q   (1 8)
1/ 4
q q max
(circular orbits do not close)
Spirographic Orbits!
Orbital Elements
(, q)
(1, 2 )
( , b,  )
x(t p )  (  b )cos t p   cos[(  b )t p / b ]
y(t p )  (  b )sin t p 
 sin[(   b )t p / b ]
(Adams & Bloch 2005)
Allowed Parameter Space
Application to LMC Orbit
Spirographic approximation
reproduces the orbital shape
with 7 percent accuracy &
conserves angular momentum
with 1 percent accuracy.
Compare with observational
uncertainties of 10-20 percent.
Solar System Scattering
Many 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 saves the Earth
red dwarf
sun
earth
Red Dwarf captures the Earth
Sun exits with one red dwarf
as a binary companion
9000 year
interaction
Sun and Earth encounter
binary pair of red dwarfs
Earth exits with the
other red dwarf
Scattering Results for our Solar System
Jupiter
Saturn
Uranus
Neptune
Semi-major axis a
Cross Sections
2.0 M
0.5 M
 a p  M * 

 ej  C 0  
 AU  M  
1 / 2
C 0  1350  160AU
2
1.0 M
0.25 M
Solar System Scattering in Clusters
Ejection Rate per Star
(for a given mass)
 eject
 C 0 (a p / AU ) 

  0 
  1000 AU  

2
 M* 


 M 

4
Integrate over IMF
(normalized to cluster size)
 dN   4
 dN 
 dm dm m where N   dm dm 
Subvirial N=300 Cluster
0 = 0.096,  = 1.7
J = 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)
NGC 1333: A Case Study
NGC 1333: A Case Study
• Observations of NGC 1333 provide
position, line of sight velocities, and
mass estimates for 93 N2H+ clumps
(Walsh et al. 2005)
• Reconstruct positions and velocities
of the clumps
• Simulate the evolution of this cluster
for 10 Myr
NGC 1333 is very much like our ‘100 Subvirial’ cluster, except…
• more centrally condensed
– R1/2 is about 2 times smaller
– 69% of cluster remains bound instead of 54%
• more interactive
– G0 is 5 times larger
– bC ≈ 238 AU → disk truncation to rd ~ 80 AU
Where did we come from?
Solar Birth Aggregate
Supernova enrichment
requires large N
Well ordered solar system
requires small N
M  25Mo
(Neptune)  0.1
FSN  0.000485
 j  3.5

o

Probability of Supernovae
PSN (N)  1 f
N
not
Probability of Supernovae
PSN (N)  1 f
N
not
 1 (1 FSN )
N
Probability of Scattering
Scattering rate:
Survival probability:

  n v

Psurvive  exp   dt

Known results provide n, v, t as function of N (BT87)
need to calculate the interaction cross sections 

Basic Cluster Considerations
Velocity:
v 1 km/s
3
n

N
/4R
Density and time:
tR  (R /v) N /(10ln N)

(nvt) 0  N 2 /40R 2 ln N  N 

Cross Section for
Solar System Disruption
  (400AU)
2
Expected Size of the Stellar Birth Aggregate
supernova
Probability P(N)
survival
Stellar number N
Adams & Laughlin, 2001, Icarus, 150, 151
Cumulative Distribution: Fraction of stars that form
in stellar aggregates with N < N as function of N
Porras
Lada/Lada
Median point: N=300
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)
Conclusions
• NBody simulations of young embedded clusters
– Distributions of radial position and closest approach
– Subvirial clusters are more centrally concentrated, retain more
stars, and have more radial velocities (compare with virial)
• Clusters have modest effects on star and planet formation
– most interactive system, NGC 1333, truncates disks to rd ~ 80 AU
– FUV flux levels are low enough to leave disks unperturbed
– disruption of planetary systems is small effect, bC ~ 700-4000 AU
and disruption requires at least 250 AU approach
– ejection rates for encounters with passing cluster members are
lower than ejection rates from planets within the system
• Photoevaporation models from external FUV radiation
• Central concentration and mass segregation can play
important role in increasing the interaction rate
Conclusions (continued)
• Spirographic approximation scheme for orbits
– high accuracy and analytic results
– new orbital elements
• Half a million Monte Carlo scattering simulations
provides us with cross sections for solar system disruption
• Constraints on the birth aggregate of the solar system
– Solar system formed in clusters with N = 2000 +/- 1000
– A priori odds of solar system conditions about 2 percent
alf a million monte carlo scattering simulations)
Future Code Development
–
–
–
–
–
–
Mass segregation (primordial/evolutionary)
Non-spherical star and gas distributions
Primordial filamentary structure
Exponential decay of gas potential
Close encounter binary formation
Primordial binaries
Parameter Space Study
–
–
–
–
–
Stellar membership, N & Virial ratio
Gas potentials (mass, shape) & expulsion
Stellar densities
IMF
Star formation epoch