r. capuzzo-dolcetta dept. of physics, univ. of roma la sapienza (roma, italy) r. capuzzo dolcetta...
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R. Capuzzo-Dolcetta
Dept. of Physics, Univ. of Roma “La Sapienza” (Roma, Italy)
R. Capuzzo DolcettaSapienza, Univ. di Roma SAIt 2009, Pisa, 7/5/2009
Supercalcolo e dinamica di sistemi stellari
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Meccanica celesteDinamica stellare Grande scala,
cosmologia
Supercalcolo e dinamica di sistemi stellari
N ≤ 10N ≤ 1012
N → ∞
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Supercalcolo e dinamica di sistemi stellari
La gravità terrestre
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La gravità celeste
Galassia ellittica
Ammasso globulare
Supercalcolo e dinamica di sistemi stellari
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~10-8 ~10-2 ~10-2
lago di Garda
30 pc = 90 al = 6x106 UA
AG: M 13 Ammasso di galassie
1 Mpc =30 Mal = 2 GUA
50 km
auto grav/ext grav
Peculiarità dell’ astrofisica è il ruolo dell’auto-gravità (self-gravity)
Supercalcolo e dinamica di sistemi stellari
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1) divergenza UV ( )
2) divergenza IR (Uij non si annulla mai)
I sistemi auto-gravitanti sono difficili da studiare per la doppia divergenza di Uij1/rij
ij
rU
ij 0lim 0 t
)( 2NO
Supercalcolo e dinamica di sistemi stellari
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Il problema gravitazionale classico degli N corpi (sistema secco)
0
0
13
)0(
)0(
||
ii
ii
N
ijj
jii
ji
mG
rr
rr
rrrr
r
j
Indipendentemente da N, ci sono 10 integrali primi
Soluzioni analitiche solo per N=2.
Supercalcolo e dinamica di sistemi stellari
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Il sistema:
● è di complessità O(N2); ● è lontano dalla linearità;
● ha pochi vincoli nello spazio delle fasi.
Il premio Oscar (re di Svezia):
Dato un sistema di punti di massa che si attraggono secondo la legge di Newton, nell’ipotesi di non avere collisioni,trovare per le coordinate un’espressione in serie di una funzione nota del tempo convergente uniformemente.
Supercalcolo e dinamica di sistemi stellari
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● Il premio fu vinto da H. Poincarè, con un articolo che portò alla teoria del caos. Piccole differenze nelle c.i. portano a grandi differenze nell’evoluzione secolare degli
N corpi.
● La soluzione per N=3 del problema del bando venne nel 1912 da K. Sundman che dimostrò l’esistenza di sviluppo in serie di potenze di t1/3.
● Il risultato di Sundman generalizzato a ogni N nel 1991 da Q. Wang.
Supercalcolo e dinamica di sistemi stellari
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).,...,2,1(,)0(
,)0(
,4
,||
0
0
2
113
Ni
GU
Um
G
ii
ii
ee
N
je
N
ijj
jii
ji
rr
rr
rrrr
r
j
Supercalcolo e dinamica di sistemi stellari
I sistemi astrofisici non sono isolati, né secchi
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N
j
jmGU
13* ||
)( jj
rrrr
r
.)T,,p(f
,GU
),N,...,i)(UU(
,p
),UU(p
g
*g
*g
0
4
1dt
d
dt
dudt
d
,dt
d
2
2
2
ir
v
v
v eq. di continuità g
eq. del moto del gas g+
eq. dell’energia g
eq. del moto stelle g+
eq. di Poisson geq. di stato g
I sistemi astrofisici reali non sono semplici N corpi…Una fase condensata (s) è immersa in una diluita (g)
forza di pressione force p (short-range) forza di gravità force U (long-range)
Supercalcolo e dinamica di sistemi stellari
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fluttuazioni su piccola scala di p(r) introducono grandi
fluttuazioni of p
Sistemi astrofisici 3D auto-gravitantisono ben rappresentabili lagrangianamente (sistemi di particelle: =N corpi, g=SPH)
…tuttavia...
la forza di volume richiede (NSPH+N*)2
valutazioni
Basso costo computaz.; bassa precisione
Alto costo computaz; alta precisione
Supercalcolo e dinamica di sistemi stellari
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small scale fluctuations of p(r) introduce large
fluctuations of p
3D self-gravitating astrophysical systems may be suitably simulated in a Lagrangian
way(particle systems: =N bodies, g=SPH)
…nevertheless... the body force requires (NSPH+N*)2 valutations
Low computational cost low precision
High computational cost high precision
Supercalcolo e dinamica di sistemi stellari
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Profiling in una simulazione tipicaProfiling in una simulazione tipica
task tempo di Cpu
valutazione delle forze gravitazionali, N2 60%
val. delle quantità fluido-dinamiche, n2 25%
integraz. Temporale, N 15%
Supercalcolo e dinamica di sistemi stellari
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222 )()()(|| jijijii zzyyxx j
rr
Si usano vari algoritmi: Erone, Bombelli, Newton, dispendiosi computazionalmente…
la distanza euclidea…
è uno dei problemi…
Supercalcolo e dinamica di sistemi stellari
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flopNN
flopN
n f 352
)1(35
2
N=1000 nf =1.5107 flopt = 1.8/100 sec
N=105 nf =1.51011 flop
t = 180 sec =3 min
N=1011 nf =1.51023 flop
t = 1.81014sec = 5.7 Myr!
nf = n. di op. per passo temporale
Problema 1: valutazione della forza Fij=Uij
35 flop
v/fnt
con un PE da v=1 Gflop/sec,tij =3.510-8sec
Supercalcolo e dinamica di sistemi stellari
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L’età di un a. globulare (~12 Gyr) is 2105 tcross 200 trel 720 anni di simulazione!
Coarse-grain: rilassamento violento tcross
Fine-grain: rilassamento“collisionale” trel
crosse
rel tNlog
Nt
10
1
Problema 2 : lunghezza delle simulazioni
10 tcross
1000 tcross
4x108 tcross
ammasso aperto
ammasso globulare
galassia
Supercalcolo e dinamica di sistemi stellari
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Un’approccio economico all’HPC: GPUsQuasi 1 Tflop/sec per 1250 euro
TESLA C 1060240 cores, 4 Gb memory,1.3Ghz per core. 936 Gflop/sec
FIRESTREAM 9170320 cores, 2 Gb memory,750Mhz, 1.2 Tflop/sec
Supercalcolo e dinamica di sistemi stellari
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2 quadcore Xeon da 2 Ghz
+ =
Potenza: ~ 12 Gflops (CPU) ~ 2 Tflops (GPU)
Costo:~ 7000 euro~ 1000 W
Supercalcolo e dinamica di sistemi stellari
2 TESLA C1060
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High performance... High performance...
1) divergenza UV ( )
2) divergenza IR (Uij never vanishes)
Problema a scale spazio-temporali multiple
I sistemi autogravitanti sono difficili da studiare acausa della doppia divergenza di Uij1/rij
ij
rUlim
ij 0
Impossibile usare metodi perturbativi
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Dinamica di ammassi globulari...Dinamica di ammassi globulari...
Da pochi corpi (N10) a molti corpi (N1011) passando per un ... numero intermedio di corpi (N106)
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Stellar system
Binaries, triple, Plan. systems
Open clusters
Globular clusters, galactic nuclei
Galaxies, Galaxy clusters
N 2,3, 10 10,000 105109 109
Regime Deterministic Collisional Secularly collisional
Collisionless
Time-scales tcross t trel tcross<t tcrosstrel<t tcross< t trel
Gravity Newtonian Newtonian Newtonian,general relativity
Newtonian, gen.relativity
Technique Analytic,Perturbative, Direct N-body
Gas+DirectN-body
Fokker-Planck,Direct N-body
Tree-codes,PM, P3M
t = age of the system, trel= relax. time, tcross= orb.time
Fluid (collision-dominated): trel<< tcross<t
Few body Intermediate N Many body
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High performance... High performance...
La molteplicità dei tempi scala richiede passi temporali individuali
AA, AG, nuclei gal.: Intermed. N body prob. (102109)
Sistemi auto-gravitanti: da pochi a tanti corpi
treltcross<età collisionale; tcross<<trel<età sec. collisionale
tcostante
tvariabile
sbagliato!
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Dinamica di ammassi globulari...Dinamica di ammassi globulari...
Profiling in a typical simulationProfiling in a typical simulation
Pro Cpu time (%)
Gravitational force evaluation
80
Communications
time integration,
20
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Dinamica di ammassi globulari.... Dinamica di ammassi globulari....
tCPU= nstep tstep
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Quasi-circular GC orbit
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High performance...High performance...
clumps!
S-shape
GC tidal tails
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Morphology of GC tidal tails: the S-shape
Palomar 5 (Odenkirchen et al. 2003)
High performance...High performance...
Simulation
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Density profiles
r -3
r -1.6
simulation
Palomar 5
High performance...High performance...
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Morphology of tidal tails: the S-shape
GC
y’
x’
Galaxy centre •
High performance...High performance...
'''' iiiGCii rωrωrωωrrr 2
ir GCr 'irωω 'irω 2 'irω
Planar, clockwise motion
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•Clumps in the tails are not bound structures;
•stars slow down their motion in the clump for a while and then move to the outer part of the tail;
•clumps are symmetrical in the tails;
• clumps are associated with the region where the inner S-shape profile of the tail stretches along the cluster orbit.
clumps
High performance...High performance...
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Dinamica di ammassi globulari...Dinamica di ammassi globulari...
back
• • Simulazione N-corpi ad alta risoluzione
• Ogni AG ha N=250,000 stelle
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Dinamica di ammassi globulari...Dinamica di ammassi globulari...
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Dinamica di ammassi globulari.... Dinamica di ammassi globulari....
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High performance... High performance...
Coarse grain t. scale: tcross Rhm/vvir treg = tcross= = 6104 yr
i
j
rij
Fine grain t. scale: trel
t=min{treg, tij} very small
down to 1 yrijijij vrt /
(back)
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High performance... High performance...
~10-8 ~10-2 ~10-2
Garda lake
30 pc = 90 ly = 6x106 AU
GC: M 13 Galaxy cluster
1 Mpc = 30 Mly = 2 GAU
50 km
self grav/ext grav
Peculiarity of astrophysical simulations is the role of self-gravity
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Dinamica di ammassi globulari...Dinamica di ammassi globulari...
• HST + large ground telescope provide data on GCS distribution mainly in early type galaxies (e.g. Forbes et al. 1996,1998a,1998b; Harris et al. 2000, 2004,2006).
• Growing evidence of presence of very massive (>107 M) YOUNG star clusters in Antennae (Fritze-v. Alvensleben 1999), MCs, M33, Fornax dSph (de Grijs et al. 2005), M31 (Fusi Pecci et al. 2005) as well as OLD (Harris & Pudritz 1994) in M87 and Virgo ellipticals.
• Harris et al. (2006) indicate how up to a 40% of the total mass in GCS of brightest cluster galaxies is contributed by massive (p.d. mass > 1.5 106 M), in good agreement with recent theoretical results by Kravtsov & Gnedin (2005).
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High performance... .High performance... .
Self-gravitating systems: from small to large N
• Solar system stabilityProblem first tackled by Laplace.Why supercomputing? To get superprecision!It depends on resonances, difficult to treat (tides favour resonances).Neptune and Pluto are in a 3:2 resonance.(this is a numerical result by Cohen and Hubbard, 1965, US Naval Weapons Lab.).
Planetary systems: a Few body problem (N<10)
torb << age
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High performance... High performance...
It is just by mean of the next generation of supercomputers that the results by Sussman & Wisdom(1987), Laskar (1989) and Sussman & Wisdom (1992) suggesting:
• the solar system is a chaotic system could be confirmed
The Digital Orrery
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High performance... High performance...
After a violent relaxation phase tcross a metastable configuration is reached fluctuations over the mean field are negligible galaxies are (now) collisionless systems where stars move in a general potential. But, how the metastable configuration was achieved? Why spiral, elliptical, irregular galaxies? Many body dynamics to integrate over a relatively short time.
Galaxies: a Large N body problem (10111012)
Self-gravitating systems: from small to large N
tcross< age << trel collisionless
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High performance... High performance...
The multiplicity of time scales requires individual time stepping
OC and GC: an Intermediate N body problem (102107)
Self-gravitating systems: from small to large N
ttreltcross<age collisional; tcross<<ttrel<age sec. collisional
constant t
variable t
wrong!
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Profiling in a typical simulationProfiling in a typical simulation
Procedura Cpu time (%)
Gravitational force evaluation
80
time integration, communications
20
High performance... High performance...
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High performance...High performance...
tCPU= nstep tstep
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Dinamica di ammassi globulari...Dinamica di ammassi globulari...
M 87M 87
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Dinamica di ammassi globulari...Dinamica di ammassi globulari...
Ammassi globulari nella GalassiaAmmassi globulari nella Galassia
150-200 oggettiprivi di gas età = 13 Gyr0.00 < e < 0.27800 < M (M) < 2.5×106
1000 < N < few ×106
0.50 < c=Log rt/rc < 2.504.90 < Log tr,c< 10.16-1.12 < Log
Gli AG sono i più grandi sistemi di N corpi studiabili 1:1
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Binaries... N=2
Solar and stellar systems are composed by N=2 up to N=1012 stars, often embedded in a gaseous cloud...Multi-phase gravitational N-body problem...
Solar system ... N=10
Supercalcolo e dinamica di sistemi stellari
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Small open clusters N=50
Large open clusters N=1000
15 ly
embedded in their mother cloud...like M16
Supercalcolo e dinamica di sistemi stellari
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Globular clusters 104 N 106
M 13, in Hercules
M 5
30 pc = 90 ly = 6x106 AU
Supercalcolo e dinamica di sistemi stellari
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M 15
1 pc = 3 ly
Supercalcolo e dinamica di sistemi stellari
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M 87a giant elliptical
N = 21011
N = 1012
Andromeda
160,000 ly
Supercalcolo e dinamica di sistemi stellari
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Stellar system
Binaries, triple, Plan. systems
Open clusters
Globular clusters, galactic nuclei
Galaxies, Galaxy clusters
N 2,3, 10 10,000 105109 109
Regime Deterministic Collisional Secularly collisional
Collisionless
Time-scales tcross << t trel tcross<t tcross < trel<t tcross< t << trel
Gravity Newtonian Newtonian Newtonian,general relativity
Newtonian, gen.relativity
Technique Analytic,Perturbative, Direct N-body
Gas+DirectN-body
Fokker-Planck,Direct N-body
Tree-codes,PM, P3M
t = age of the system, trel= relax. time, tcross= orb.time
Fluid (collision-dominated): trel<< tcross<t
Few body Intermediate N Many body
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or (partially...)to dedicated (non programmable) computational architectures like the japanese
Solutions:
Resort to grid methods, like P3M methods (Poisson’s eq.on a grid via FFT and a local direct summation)
orResort to multipole expansions tree algorithms
Supercalcolo e dinamica di sistemi stellari
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TABLE-1 Low-Precision machines Machine Year Peak speedGRAPE-1 1989 240 Mflop/s,GRAPE-3 1991 15 Gflop/sGRAPE-5 1998/9 ~ 1Tflop/s
TABLE-2 High-Precision machines Machine Year Peak speed GRAPE-2 1990 40Mflop/sHARP-1 1993 180 Mflop/sGRAPE-4 1995 1 Tflop/sGRAPE-6 2002 48(64) Tflop/sGRAPE-DR 2008 2 Pflop/s
GRAPE 6
Like a graphics accelerator speeding up graphics calculations on a workstation,the GRAPE acts as a Newtonian force accelerator, in the form of an attached piece of hardware.
Supercalcolo e dinamica di sistemi stellari
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N
j
jmGU
13* ||
)( jj
rrrr
r
.)T,,p(f
,GU
),N,...,i)(UU(
,p
),UU(p
g
*g
*g
0
4
1dt
d
dt
dudt
d
,dt
d
2
2
2
ir
v
v
v continuity eq. g
gas motion eq. g+
energy eq. g
stellar motion eq. g+
Poisson’s eq. geq. of state g
Real astrophysical systems are not simple N-bodies...a condensed phase (s) in a dilute medium (g)
pressure force p (short-range) gravity force U (long-range)
Supercalcolo e dinamica di sistemi stellari