Astro 160: The Physics of Stars

d) Combine your previous results to show that F c,1 ≈ F c,2 .From (a) we know thatusing this and our solution we findF c,1 ≈ 1 2 ρv3 cF c,2 ≈ ρ ∆E m v c12 ρv3 c ≈ ρ ∆E m v c ⇒ 1 2 v2 c ≈ ∆E mProblem # 2Estimate the convective velocity v c and the dimensionless entropy gradient (ds/dr)(H/c p ) in the convectionzones of 0.1 and 10 M sun stars. Assume that the material undergoing convection is at about themean density of the star and that gas pressure dominates. You can either use a scaling argument to estimatethe density, temperature, luminosity, etc. of such stars or look up in a book (e.g., Carrol & Ostlie) anyproperties of 0.1 and 10M? stars that you need to make your estimate (e.g., radius and luminosity). Butyou can’t just look up v c and (ds/dr)(H/c p ).From class we know thatbut we know thatthuswhich reduces toF c = ρα 3 Cs3 H ds∣C p dr∣3/2F c ≈whereCs 2 = kTm pand for the convective velocity we findL4πR 2∣ ∣∣∣ H dsv c = C s C p dr∣∣ ( ) ∣∣∣ H dsL 2/3C p dr∣ = 14πR 2 ρC 2 s( ) H dsLR 2/3∣C p dr∣ = m p3M ∗ k b T∣ ∣∣∣ H dsv c = C s C p dr∣1/2ρ = 3M4πR 3=( ) LR 1/33Mfrom Carrol and Ostley we find that for 10M sun and .1M sun we find that the radius, and luminosity areapproximatelyM ≈ 10M sun R ≈ 6R sun L ≈ 5700L sun1/2M ≈ 0.1M sun R ≈ 0.2R sun L ≈ .0034L sungiven these values we find∣ ∣∣∣ H dsM = 10M sun C p dr∣ ≈ 3.61 × 10−6 v∣ ∣∣∣ H dsM = 0.1M sun C p dr∣ ≈ 5.8 × 10−10 v14c ≈ 5.3 × 10 4 cm/sc ≈ 698 cm/s