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ASTR2050 Spring 2005 Lecture 11am 22 March 2005 In this class we will cover “Clusters of Stars”: • Types of clusters • Distances to clusters • Cluster dynamics: The Virial Theorem • HR diagrams: Ages and populations 1 Types of Clusters Assumption: All stars in a cluster are (about) the same age • “Galactic” or “Open” clusters • In the Galactic disk • ≤1000 stars and ≤10pc across • Sometimes associated with gas/dust clouds • “Globular” clusters • Appear grouped about the Galactic center ➽ Find center of the Galaxy at 10kpc 4 to 106 stars and 20 to 100 pc across 10 • • No gas/dust clouds 2 Examples of Open Clusters M6 “Butterfly Cluster” M37 Age ≈50 Million years Age ≈300 Million years See Kutner Figure 13.1 3 Examples of Globular Clusters M5 M80 Age ≈13 Billion years Age ≈15 Billion years See Kutner Figure 13.2 4 Distances to Clusters • Main Sequence matching • Recall 18 Feb studio class (“Pleiades”) • OK for all clusters (but, later this class...) • Cepheids and RR Lyrae stars • Useful if you can find them • Globular clusters have many RR Lyrae • Star motion in “moving” clusters • Accurate method, but... • Clusters must be nearby 5 Distances to Moving Clusters Kutner Fig 13.3 vr A ) vT v Distance d is determined from proper angular motion µ which arises from tangential speed vT : vT (km/sec) µ(rad/sec) = d(km) Li ne of Sig ht Star The Doppler shift is used to measure the radial velocity vr. A ) Observer 2 2 2 =vr + vT v tan A=vT /vr or for d(pc) and µ(arcsec/year) d = vT /4.74µ = vr tan A/4.74µ So how can we measure A? 6 Convergent Point A cluster of stars will appear to move towards a “convergent point”. This makes it possible to determine the angle A. Example: The Hyades Identify the “convergent point” using a large number of moving stars. A 7 Cluster Dynamics and The Virial Theorem Consider the motion of a mass in a circular orbit: 1 2 1 mv2 1 GMm 1 GMm 1 K= mv = R = R 2 = =− U 2 2 R 2 R 2 R 2 So, the total energy is E = K +U = U/2 < 0 For a collection of objects (like a cluster of stars) that are moving in equilibrium, this relationship still holds: !K" = −!U"/2 and E = !K" + !U" = !U"/2 This powerful result is known as “the virial theorem”. See Kutner section 13.3.1 for the derivation 8 Example: Why don’t clusters fly apart? See Kutner Section 13.3.2 2 3 GM Assume the cluster is a sphere of mass M and radius R. Then U = − 5 R (15 Feb 10am) If the cluster has N stars each # of mass $ m, then N K=! i=1 ! " 1 2 1 1 N 2 m!vi = (mN) !vi ! 2 2 N i=1 1 2 = M!v " 2 so that !v " = 2K/M = −U/M = (3/5)GM/R But 2 1 2 GMm GM 2 2 mvesc = ⇒ vesc = 2 > "v # 2 R R Also: Homework problem on “Virial Mass”. (We’ll use this later.) 9 HR Diagrams: Ages and Populations A young cluster An old cluster 10 Globular clusters are different M3 12 Horizontal branch 14 16 V 18 RR Lyrae Blue stragglers Turnoff 20 22 -0.5 0 Sun would be still on the main sequence, but globular clusters are much older !? 11 0.5 B-V 1 1.5 ☛ Stellar Populations Globular clusters are nearly pure H and He (no “metals”) and stellar evolution details are sensitive to composition. Galactic clusters are more typical of stars in the galactic disk, including “metals”. We call these Population I stars. Low metallicity stars, like those found in globular clusters, are called Population II stars. We think of Population I stars as “young”, and made up of recycled materials. Population II stars are “old”, and might be made from primordial materials from the birth of the universe. 12