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 Midterm #2 — Average: 69
— Median: 69.5
— Max: 96 Midterm #1 Midterm #2 00 20 40 60 100 00 AST 203 (Spring 2011) Star Clusters (Kulner, Ch. 13) AST 203 (Spring 2011) Last Time virial theorem: for a cluster in equilibrium: <K> = g an The total energy of the system is E = K + U so E=§<U> Potential energy is negative—the system is bound. We can use the virial theorem to estimate the mass of a star
cluster. We will see it again with molecular clouds and galaxy clusters. AsT 203 (spring 2011) Cluster Escape Velocity What is the escape velocity of a star at the edge of the cluster? A star at the edge of the cluster has a potential energy of GMm
U:__
R Its kinetic energy is and to escape, it needs K>U —> v>vezil2GTM Note that this is about 2x higher than the RMS speed. AST 203 (Spring 2011) Globular ClusterGl 7 7 ' HSTFWFPC2
in Galaxy M31 PR69611  ST Sci 0P0  April 24, 1996 Michael Rich. Kenneth Mighell. and James D. Neill (Columbia Universily).
Wendy Freedman {Carnegie Observatories) and NASA AST 203 (Spring 2011) Relaxation Time Relaxation: over time, stars equilibrate due gravitational
encounters These encounters exchange energy and momentum. The average velocity after relaxation is the RMS velocity we get
from the virial theorem. Some stars will be slower, some faster
Some stars may have velocities in excess of the escape velocity. Relaxation time: time it takes to reach equilibrium configuration. AST 203 (Spring 2011) Relaxation Time Two stars “collide” when their kinetic energy is equal to the potential energy between them
1 2 07712 —mv :
2 r This defines the radius of the interaction. A star moving through the cluster sweeps out a cylinder. We define the relaxation time as the time until one collision: 2 O
nV : n(7rr vtrel) : 1
1 v3
trel : Z — T
TMTT‘QU 47TG2m2n O AST 203 (Spring 2011 :1 Relaxation Time The number density of stars in the cluster is
N (M/m) _ 3M “2—: (47r/3)R3 (47r/3)R3 — 47TmR3 so the relaxation time becomes
U3R3 U4R2 R tre = — = ——
1 3G2mM 3G2mM v The virial theorem tells us that Note that this is Just saying that we M R N R / need to cross the m U U cluster many
times to relax. therefore AST 203 (Spring 2011) Relaxation Time A more accurate calculation would take into account the
interactions of farther stars, resulting in a shorter relaxation time: 1 N R
trel N — 121n(N/2) I For a typical cluster, N = 106, v = 1.6 x 106 cm s'1, R = 5 pc, we
find trel N 2 X 109 yr Evaporation time: time in which a large # of stars leave the
cluster. This is typically 100 times larger than the relaxation time. AST 203 (Spring 2011) Virial Mass Relaxation time << age of the universe—globular clusters are
virialized. If we assume that a cluster obeys the Virial theorem, then by
measuring the average velocities, we can determine the mass. Again, starting with the virial theorem: 1
_ <K> : —§ <U>
Ifwe take
3GJW2 1 2
U 2 __ (K) : —M t)
then < > 5 R 2 < >
_ 5 <v2>R
M _ E G
AST 203 (Spring 2011)
Virial Mass The velocity we measure is not the full average, but rather the
average along the line of sight. The total velocity is written as 212 = of, + v: + 1): so
<02) = <03) + <93) + <03) On average, the velocities in all directions are the same, <v2> = 3 (vi>
and because of the random orientations, we can take this
component to be the line of sight velocity, so 5<’U,,2.>R
M— G AST 203 (Spring 2011) HertzprungRussell
Diagram
Horizontal axis: spectral Class, B — VI or T man] I
(increasing to left). EIIIIIlEll. EIIIIIIDl. mm. 300le I Em lEIIII DEM] Vertical axis: Luminosity or
absolute magnitude. main sequence: diagonal
line running through all the
spectral classes. Some TL combinations
not realized in nature. Wide range in L for stars of
the same [mum Low L population: white [Minimum dwarfs. AST 203 (Spring 2011) if: E [biogKEEN: 3v Cluster HR Diagram Stars in a cluster form at roughly
the same time (Bipedmwo Massive stars evolve off main
sequence first. Stars leave the main sequence at the
turnoff point. a A r c x M
Spearl 1m Allows us to estimate cluster age. Globular cluster M3 H_R diagram These observations tell us that globular clusters are much older
than open clusters. AST 203 (Spring 2011) Blue Stragglers Stars in a globular should
be old Blue stragglers: mergers? ﬁlliélée AST 203 (Spring 2011) NASA mIdTlm Hubble Helltag lST llAUHAl  Hubbl PSCDUQWFPCZ I Sal P Blue Stragglers Ground Blue Stragglers in Globular Cluster 47 Tucanae HST  WFPC2 PHCQ?35 ' C‘CtOt‘B!‘ 2319"?  ST SCI 0P0
Fl. Saffer {Villanoua Univer‘sny}. D. Zurelr. {ST Sci} and NASA AST 203 (Spring 2011) There's more than one way to make a The Collision Model The Slow Coalescence Model \ Err«2:11 ridline Ruining. \«nry slowly http://hubblesite.org/newscenter/archive/releases/1997/35/image/d/
STScI AST 203 (Spring 2011) Cluster HR Diagram Cluster members are at (roughly)
the same distance mo Construct a cluster HR diagram l 0— Using apparent magnitude and T.  _‘ relative apparent brightness Main sequence is main _ _
seq uence "3: I one . Slide a cluster HR diagram up/ “ down over another cluster's. moo ‘ ' ao'oo surface temperature (Kelvm) (from Bennett et al.) Determine m — M —> the distance ”””””” to the cluster. This is called mainsequence fitting. AST 203 (Spring 2011) HertzprungRussell
Diagram Horizontal axis: spectral
class, B — V, or T
(increasing to left). Vertical axis: Luminosity or
absolute magnitude. main sequence: diagonal
line running through all the
spectral classes. Some TL combinations
not realized in nature. Wide range in L for stars of
the same T. Low L population: white dwa rfs.
AST 203 (Spring 2011) iiIIIII EiiIiiIi 1] i:ii:ii:i 1 [Hill] [I i:ii:i‘i i] i:ii:ii:i‘i [I [Hill] [ii +1 i]
Colour {B EiiIiiIiiIii' [Te ...
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 Simon,M
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