STAR CLUSTERS Lecture 3 Kinematic Properties Nora Lützgendorf (ESA)
LECTURE 2 1. Star Formation •from gas clouds, fragmentation •Initial mass function (IMF): multiple power laws, changes with time 2. Multiple Stellar populations •Photometric evidence: Multiple sequences in CMD •Spectroscopic evidence: Na-O anti-correlation •Explanations: 1. Polluters + 2nd Generation 2. Polluters •Problems: Mass budget problem (must have lost 90% of their mass??…), and many more… Nora Lützgendorf, KAS16 2 / 51
O utline 1. The Gravitational N-body problem 2. Dynamic Equilibrium 3. Negative Heat Capacity 4. Core Collapse 5. Equipartition of energies 6. Mass Segregation Nora Lützgendorf, KAS16 3 / 51
O utline 1. The Gravitational N-body problem 2. Dynamic Equilibrium 3. Negative Heat Capacity 4. Core Collapse 5. Equipartition of energies 6. Mass Segregation Nora Lützgendorf, KAS16 4 / 51
G ravitation N ~ r j − ~ r i ~ X F i = Gm i m j r i | 3 | ~ r j − ~ j =1 ,j 6 = i m j − 1 m j m j − 2 ~ r i − ~ r j m j +1 m i Nora Lützgendorf, KAS16 5 / 51
G ravitation N ~ r j − ~ r i ¨ X ~ r i = − G m j r i | 3 | ~ r j − ~ j =1 ,j 6 = i m j − 1 m j m j − 2 ~ r i − ~ r j m j +1 m i Nora Lützgendorf, KAS16 6 / 51
G ravitation - N = 2 2 ~ r j − ~ r i ¨ X ~ r i = − G m j | ~ r i | 3 r j − ~ j =1 ,j 6 = i m j ~ r i − ~ r j m i Nora Lützgendorf, KAS16 7 / 51
G ravitation - N = 2 a (1 − e 2 ) r ( θ ) = 1 + 2 cos( θ ) e=2 e=1 e=0.5 e=0 Nora Lützgendorf, KAS16 8 / 51
G ravitation - N = 2 Nora Lützgendorf, KAS16 9 / 51
G ravitation - N = 3 3 ~ r j − ~ r i ¨ X ~ r i = − G m j | ~ r i | 3 r j − ~ j =1 ,j 6 = i m j m j − 1 ~ r i − ~ r j m i Nora Lützgendorf, KAS16 10 / 51
G ravitation - N = 3 Nora Lützgendorf, KAS16 11 / 51
G ravitation - N = 3 Nora Lützgendorf, KAS16 12 / 51
G ravitation - N = 3 C 3 ~ r j − ~ r i ¨ X ~ r i = − G m j H | ~ r i | 3 r j − ~ A j =1 ,j 6 = i O S m j m j − 1 ~ r i ! − ~ r j BUT… m i Nora Lützgendorf, KAS16 13 / 51
G ravitation - N = 3 C 3 ~ r j − ~ r i ¨ X ~ r i = − G m j H | ~ r i | 3 r j − ~ A j =1 ,j 6 = i O S m j m j − 1 ~ r i ! − ~ r j BUT… m i Nora Lützgendorf, KAS16 14 / 51
G ravitation - N = 3 L4 L3 L1 L2 L5 Nora Lützgendorf, KAS16 15 / 51
E xplanations - Problems WIND LISA PATHFINDER HERSCHEL SOHO JWST L 2 GAIA Nora Lützgendorf, KAS16 16 / 51
G ravitation - N > 3 C N ~ r j − ~ r i ¨ X ~ r i = − G m j H r i | 3 | ~ r j − ~ A j =1 ,j 6 = i O m j − 1 S m j m j − 2 ~ r i ! − ~ r j m j +1 m i Nora Lützgendorf, KAS16 17 / 51
O utline 1. The Gravitational N-body problem 2. Dynamic Equilibrium 3. Negative Heat Capacity 4. Core Collapse 5. Equipartition of energies 6. Mass Segregation Nora Lützgendorf, KAS16 18 / 51
D ynamic Equilibrium EQUILIBRIUM Nora Lützgendorf, KAS16 19 / 51
D ynamic Equilibrium COLD (v = small or 0) Nora Lützgendorf, KAS16 20 / 51
D ynamic Equilibrium COLD (v = small or 0) Nora Lützgendorf, KAS16 21 / 51
D ynamic Equilibrium HOT (v = large) Nora Lützgendorf, KAS16 22 / 51
D ynamic Equilibrium HOT (v = large) Nora Lützgendorf, KAS16 23 / 51
D ynamic Equilibrium - Definition EQUILIBRIUM: ‣ No EXPANSION, and no CONTRACTION, even though all particles are in MOTION Nora Lützgendorf, KAS16 24 / 51
V irial Theorem N K = 1 KINETIC ENERGY X v 2 m i ~ i 2 i =1 N N m i m j W = − G 1 POTENTIAL ENERGY X X | ~ r j | r i − ~ 2 i =1 i 6 = j CONSERVATION OF ENERGY E = W + K = const. VIRIAL THEOREM W = − 2 K Nora Lützgendorf, KAS16 25 / 51
O utline 1. The Gravitational N-body problem 2. Dynamic Equilibrium 3. Negative Heat Capacity 4. Core Collapse 5. Equipartition of energies 6. Mass Segregation Nora Lützgendorf, KAS16 26 / 51
“ T emperature” Like in a gas: ‣ Particles move fast system is HOT ‣ Particles move slow system is COLD 1 v 2 = 3 2 m ¯ 2 k B T K = 3 2 Nk B ¯ T Nora Lützgendorf, KAS16 27 / 51
H eat Capacity K = 3 2 Nk B ¯ T VIRIAL THEOREM E = W + K W = − 2 K = − K = − 3 2 Nk B ¯ T C ≡ dE = − 3 2 Nk B d ¯ T Nora Lützgendorf, KAS16 28 / 51
H eat Capacity C = positive GETS HOTTER ENERGY C = negative C ≡ dE T = − 3 2 Nk B d ¯ GETS COLDER Nora Lützgendorf, KAS16 29 / 51
H eat Capacity C = positive GETS COLDER ENERGY C = negative C ≡ dE T = − 3 2 Nk B d ¯ GETS HOTTER Nora Lützgendorf, KAS16 30 / 51
H eat Capacity C = positive C = negative Nora Lützgendorf, KAS16 31 / 51
H eat Capacity V 1 ENERGY V 2 V 2 > V 1 HOTTER COLDER Nora Lützgendorf, KAS16 32 / 51
O utline 1. The Gravitational N-body problem 2. Dynamic Equilibrium 3. Negative Heat Capacity 4. Core Collapse 5. Equipartition of energies 6. Mass Segregation Nora Lützgendorf, KAS16 33 / 51
C ore Collapse Cluster of stars with equal mass: Stars deeper in the potential move faster (hot) Nora Lützgendorf, KAS16 34 / 51
C ore Collapse Cluster of stars with equal mass: Stars deeper in the potential move faster (hot) Nora Lützgendorf, KAS16 35 / 51
C ore Collapse Encounters of fast and slow stars: ENERGY ~ P = M 1 · ~ v 1 + M 2 · ~ v 2 = const. Slow star gets faster, fast star gets slower Nora Lützgendorf, KAS16 36 / 51
C ore Collapse Fast star: looses energy ⇒ sinks deeper in the potential well ⇒ gains speed ⇒ becomes even faster (hotter) Slow star: gains energy ⇒ climbs out of the potential well, ⇒ looses speed ⇒ becomes even slower (colder) Nora Lützgendorf, KAS16 37 / 51
C ore Collapse Nora Lützgendorf, KAS16 38 / 51
C ore Collapse Nora Lützgendorf, KAS16 39 / 51
C ore Collapse Nora Lützgendorf, KAS16 40 / 51
C ore Collapse Nora Lützgendorf, KAS16 41 / 51
C ore Collapse Nora Lützgendorf, KAS16 42 / 51
C ore Collapse M28 M15 Nora Lützgendorf, KAS16 43 / 51
C ore Collapse Surface Brightness M28 Distance Surface Brightness Core Collapsed Distance M15 Nora Lützgendorf, KAS16 44 / 51
O utline 1. The Gravitational N-body problem 2. Dynamic Equilibrium 3. Negative Heat Capacity 4. Core Collapse 5. Equipartition of energies 6. Mass Segregation Nora Lützgendorf, KAS16 45 / 51
E quipartition of Energies Cluster of stars with UN - equal mass: Nora Lützgendorf, KAS16 46 / 51
E quipartition of Energies Encounters of high-mass and low-mass stars: ENERGY ~ P = M 1 · ~ v 1 + M 2 · ~ v 2 = const. Low-mass star gets faster, high-mass star gets slower K i ∼ M i Kinetic energies become more equal 2 v 2 i Nora Lützgendorf, KAS16 48 / 51
E quipartition of Energies When all stars (at radius R) have the same kinetic energy High-mass stars are slow, low-mass stars are fast Anderson & van der Marel, 2010 EQUIPARTITION V ~ 1/sqrt(M) T I T I O N N O E Q U I P A R Nora Lützgendorf, KAS16 49 / 51
O utline 1. The Gravitational N-body problem 2. Dynamic Equilibrium 3. Negative Heat Capacity 4. Core Collapse 5. Equipartition of energies 6. Mass Segregation Nora Lützgendorf, KAS16 50 / 51
M ass Segregation Equipartition of energies: High-mass stars sink to the center Low-mass stars rise to the outskirts Nora Lützgendorf, KAS16 51 / 51
M ass Segregation Mass gradient from center to the outskirts Dynamical Mass Loss log N log M Nora Lützgendorf, KAS16 52 / 51
S ummary - 1 1. The Gravitational N-body problem •N=2: exactly solvable •N=3: approximately solvable •N>3: only numerical solvable 2. Dynamic Equilibrium •No EXPANSION or CONTRACTION of the system 3. Negative Heat Capacity •Remove energy —> hotter •Gain energy —> colder Nora Lützgendorf, KAS16 53 / 51
S ummary - 2 4. Core Collapse •Very condensed core, steep light profile 5. Equipartition of Energies •All the stars (at radius R) have the same kinetic energy •High-mass stars: slow, low-mass stars: fast 6. Mass Segregation •Mass gradient from center to the outskirts Nora Lützgendorf, KAS16 54 / 51
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