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 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 . 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 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[−|z|/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 , show that the disk’s total L LD = 2πI(R = 0)hR. • For the , 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.

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