12 From the Hayashi Track to the Main Sequence: During collapse, the Virial Theorem says that half the energy goes into radiation and the other half into heat (if n fixed): 1 dE n 3 GM 2 dR L = = − ; − 2 dt 5 n 2R2 dt − dL 1 d2E n 3 GM 2 2 dR 2 d2R = = − + : dt − 2 dt2 5 n 2R2 −R dt dt2 − " # 2 4 Constancy of Teff with L = 4πσR Teff implies dL=L = 2dR=R and d2R=dt2 = 4(dR=dt)2=R. This leads to a solution 1=3 6LoRot − R = Ro 1 + ; GM 2 where Ro and Lo refer to the initial radius and luminosity. Continued collapse to the main sequence involves a change in the polytropic structure from n = 3=2 to n = 3 (the standard model). If we neglect this, however, and use the hydrostatic condition d2I=dt2 = 0 where I MR2 is the moment of iner- tia, one finds that Rd2R=dt2 =/ (dR=dt)2. This implies that dL=L = 3dR=R, which is now p−ositive, and to a solution − 4RoLot R = Ro 1 2 : r − GM This also implies that d ln Teff =d ln R = 5=4 and d ln L=d ln Teff = 12=5. This phase of collapse is known−as the Henyey phase, and results in an increase in L and Teff with stellar shrink- age. The stellar radius, for a one solar mass star, must shrink from Rmin to R , the effective temperature rises from about 12=5 3500 K to 5500 K, so Lmin (3500=5500) 3 L and 4=5 ' ' Rmin (5500=3500) 0:7 R . The slope in the H-R dia- gram of' the Henyey trac'k is 2.4, smaller than that of the main sequence, so the Henyey trajectory will eventually intersect it. 13 Note from the above that Lmin varies with mass less steeply than does Lssm. Since we expect that for solar mass stars that Lmin=Lssm < 1 (although we didn't get this result in the above), in principle as mass is decreased there will be a point at which Lmin = Lssm. For small enough masses, the Henyey phase essentially disappears, and a lower limit to stellar masses occurs. Main Sequence Structure The main sequence is defined as those stars that burn hydro- gen to helium in their cores. Lower mass stars do this via the pp cycle, and higher mass stars via the CNO cycle. The latter is much more temperature sensitive, and leads to convective cores in massive stars. The adiabatic temperature gradient is dT 2 T dP 8π µGρ r = = o (25) dr ad −5 P dr −15 No for a perfect gas near the center. Diffusive transport leads to 2 dT 3 κρ L κoρccr = 3 2 3 ; (26) dr rad −4ac T 4πr ' − acTc where we used the Thomsen opacity (valid at high tempera- 3 tures) and approximated L(r) (4π=3)ρccr . The condition for convective instability is simply' dT=dr > dT=dr , or j jrad j jad 3 3 3 8π µGacTc 4 Tc;6 150 g cm− 1 1 c > 1:2 10 erg g− s− : 15 Noκoρc ' × 14 ρc (27) For comparison, the pp cycle has an energy generation rate of 4 T6 ρ 1 1 pp 17 erg g s ; (28) ' 14 150 g cm 3 − − − 14 so the Sun's core must be radiative. In a massive star, for CNO, we have 17 5 T6 ρ 1 1 4 10 erg g− s− ; (29) CNO ' × 25 150 g cm 3 − these stars have convective cores. If energy generation is very temperature sensitive, approxi- mate the core as having a point-like energy source. Thus, for r > 0, we have L(r) = L. We look for a power-law solution for T outside the convective core: T = brm. With the radiative transport equation, 1 3κ ρ L m = b4 = o c (30) −4 16πac if ρ is constant. The convective core's radius will be determined by dT=dr = dT=dr : j jrad j jad 1=9 4=9 3κoL 15No 1=3 r = ρ− : (31) core 16πac 32πµG c 3 Using the relation ρc M=R we have / 1=9 1=3 5=18 rcore=R L =M M (32) / / using the standard solar model for L(M). Convective cores increase with stellar mass..