Problem Set 2 for Astro 322 Read chapter 24.2. (Some of this material isn't in the textbook.) Due Jan. 27th, 5 pm, to the Astro322 drop box on 3rd floor CEB. Problem 1: Assume the Salpeter equation describes stars formed in a cluster with masses between Ml and Mu >> Ml. • Write down and solve the integrals that give (a) the number −3:5 of stars, (b) their total mass, and (c) the total luminosity, assuming L = L (M=M ) . • Explain why the number and mass of stars depend mainly on the mass Ml of the smallest stars, while the luminosity depends on Mu, the mass of the largest stars. • Taking Ml = 0:3M and Mu >> 5M , show that only 2.2% of all stars have M > 5M , while these account for 37% of the mass. • The Pleiades cluster has M ∼ 800M ; show that it has about 700 stars. • Taking Mu = 10M , show that the few stars with M > 5M contribute nearly 80% of the light. Problem 2: Thin disk stars make up 90% of the total stellar density in the midplane, while 10% belong to the thick disk. However, hz in the thin disk is roughly three times smaller than for the thick disk. Show that the surface density of stars per square parsec follows Σ(R, thin disk)∼ 3Σ(R, thick disk). Problem 3: • By integrating the expression n(R; z; S) = n(0; 0;S) exp[−R=hR(S)] exp[−|zj=hz(S)], show that at radius R the number of stars per unit area (surface density) of type S is Σ(R; S) = 2n(0; 0;S)hz(S) exp[−R=hR(S)]. If each has luminosity L(S), the surface bright- ness I(R; S) = L(S)Σ(R; S). • Assuming that hR and hz are the same for all types of 2 star, show that the disk's total L LD = 2πI(R = 0)hR. • For the Milky Way, taking 10 LD = 1:5 × 10 L in the V band and hR = 4 kpc, show that the disk's surface brightness −2 at the Sun's position 8 kpc from the center is ∼ 20L pc . The mass density of stars in the −2 disk is 40-60 M pc , so we have M=LV ∼2-3. • Why could this be larger than M=LV for stars within 100 pc of the Sun? Problem 4: Fig. 1 below plots the numbers of stars at each B − V color with apparent V magnitude 19 < mV < 20, per square degree near the North Galactic Pole. Use Fig. 2 for the stars in the local disk, and the metal-poor globular M92 to represent stars in the halo (Fig. 3). (a) What is the absolute magnitude MV of a disk star at B − V = 0:4? How far away must it be to have mV = 20? In M92, the bluest stars still on the main sequence have B − V ∼ 0:4. Show that, if such a star has apparent magnitude mv = 20, it must be at d ∼ 20 kpc. (b) What absolute magnitudes MV could a disk star have, if it has B −V = 1:5? How far away would that star be at mV = 20? In M92, what is MV for the reddest stars, with B −V = 1:2? How distant must these stars be if mV = 20? (c) Explain why the reddest stars in Fig. 1 are likely to belong to the disk, while the bluest stars belong to the halo. 1 Fig. 1.| Numbers of stars at each B−V color for 19 < mv < 20 toward north Galactic pole. The solid line shows the prediction of a model: thin-disk stars (triangles) are red, halo stars (stars) are blue, thick-disk stars (squares) have intermediate colors. Fig. 2.| CMD from Hipparcos. Fig. 2.| CMDs for the metal-poor globular cluster M92, and metal-rich globular 47 Tuc. 2.