Handout 19: Evolution of stars on the Hertzsprung-Russell Diagram

<strong>Handout</strong> <strong>19</strong>: <strong>Evolution</strong> <strong>of</strong> <strong>stars</strong> onthe Hertzsprung-Russell Diagram• From H.W. #7:• If the T-gradient required by rad’n xfer in order totransport the L outward is steeper than this (dlnT/dlnP) rad > (dlnT/dlnP) adiabatic• Convectively unstableAdiabatic T gradient is : Energy carried by convection rather than rad’n• Only slightly superadiabatic gradient is needed• (dlnT/dlnP) actual~ (dlnT/dlnP) adiabaticγdln( T)dln( P)c Pso dln( T)c V dln( P)γ − 1γc P − c Vc P

Uniform density stellar model isconvectively unstable• P = nkT number density n = constant dT/dP = 1/nk (dlnT/dlnP) = (P/T)(dT/dP) = 1 1 > 0.4, model is convectively unstable everywhere!• Balloon moved from 1 to 2’ (figure) adiabatically i.e. no heat added• Hotter than surroundings (i.e. 2)• Less dense will rise further (convection)• Carries heat energy to higher levels!

Figure: Unstable T-gradientModel,slope=1lnT12’Adiabatic,slope=0.42lnP

Uniform stellar modelM sunρ := constant43 ⋅π⋅ R sun 3gr ():=G⋅ρ4⋅ ⋅π⋅r 33r 2dPdr−g⋅ρ−43 ⋅π⋅G⋅ ρ2 ⋅rPR ( sun ) 0Pr ()R⌠ sun⎮⎮⌡r−4r3 ⋅π⋅G⋅ ρ2 ⋅rdPr ()10 + 4µ := µ = 0.609 n :=20 + 3Tr ()23 ⋅π⋅G⋅ ρ2 2:= ⋅⎛⎝ R sun − r 2ρµ ⋅m H1i:= ⋅Pr()i:= 0..10 r := ⋅nk ⋅i 10 R sun1 .10 7⎞⎠( )Tr iK5 .10 600 0.5 1r iR sun

Convection continued• The maximum adiabatic gradient is 0.4Any additional heat capacity will reduce this• Reducing the gradient toward zero• Forcing convection Zero gradient means no radiative fluxSources <strong>of</strong> heat capacity H ionization 6-10,000 K He ionization 20-50,000 K• 13.6 eV needed to ionize• Plus (3/2)kT = 1.5 eV for the K.E. <strong>of</strong> the electronAdiabatic gradient reduced to 0.1

Convective zones in MainSequence <strong>stars</strong>• H, He ionization create convective zone justbelow the photosphere for sun The convection generates magnetic fields• Leading to sunspots, flares, active chromosphere, and x-rays Extends down to 0.8 <strong>of</strong> solar radius• Hotter <strong>stars</strong>, convection zone shrinks Above 8000 K (F-<strong>stars</strong>) fully radiative• cooler <strong>stars</strong>, the convection zone deepens Stars below 0.3 solar masses fully convective• Above 0.3 solar masses, Luminosity determinedby radiation

Convection crucial in early stellarevolution• Figure 9.10 shows interior κ ~ T -3.5But ~ M/R• Virial theoremIncreasingly large |dT/dr| for larger <strong>stars</strong>Convection sets in• Stars < 5 solar mass start fully convectiveSun at 30 solar radii was “fully convective”Major blockage to luminosity is photosphere• L = 4πR 2 σT(τ=2/3) 4

Fully convective <strong>stars</strong> nearlyconstant effective T, H- opacity• The dominant opacity at low T’s is H- Free-free and free-bound• Very strong T-dependence (Fig. 9.10) H- needs electrons Electrons come from ionized metals Saha dictatesκ κ 0 ⋅P 0.7 ⋅T 5.3• Luminosity determined by transition betweenconvection and radiative photosphere i.e. T e = T at τ = 2/3 levelκ 0 ⋅P a ⋅Tb

Solve for P at τ = 2/3• As in homework, solve for P(τ=2/3) κ not constant, need to do integral Weak dependence <strong>of</strong> T on τ,• T can be taken out <strong>of</strong> integral• Solve for P (see next slide):dr⎡⎢⎣2P ph3 ⋅(a + 1)⋅gbκ 0 ⋅T e⎤⎥⎦1a+1

Photospheric P (i.e. τ = 2/3)23τR⌠⎮⌡r photosphereκρ ⋅drRb ⌠κ 0 ⋅T e ⋅⎮⌡r photosphereP a ⋅ρdrHydrostatic equil. implies23bκ 0 ⋅T e ⋅⎛⎜⎝−1g⎞⎠R⌠⋅⎮⎮⌡r photosphereρP a dP⋅dr−1 dP⋅g drdraκ 0 ⋅T e ⋅⎛⎜⎝g = const.−1g⎞⎠1a+1⋅ ⋅−⎛P photospherea + 1⎝⎞⎠sincedPdr ⋅drdP and Pr ( R) 01⎡2so P ph3 ⋅(a + 1)g⋅⎢bκ 0 ⋅T e⎣⎤⎥⎦a+1

Integrate adiabatic gradient fromcenter <strong>of</strong> star to photospheredln( T)dln( P)γ − 1γEq. (10.75)γ ⌠⋅⎮γ − 1 ⌡T cT e1 dln( T)⌠⎮⌡P cP photosphere1 dln( P)γ⋅( ln( T c ) − ln( T e ))( ln( P c ) − ln( P ph ))γ − 1exponentiate, and solve for P at the tau = 2/3 levelγP cγ−1K'P ph K' ⋅T ewhereγγ−1T c⎡⎢⎣2solve for eff. T T e3 ⋅(1 + a)⋅GM ⋅κ 0 R 2 ⋅− ( 1+a)⋅K'⎤⎥⎦b+γ1γ−1⋅( 1+a)

Vertical, Hayashi tracks on the H-Rdiagram• Insert appropriate values Note: T c ~ M/R, P c ~ (M/R 3 )T c T e ~ R 0.06• L ~ R 2 T e4~ T e36,• T e ~ L 0.03• Tracks nearly vertical on H-R diagram (draw)• Hayashi figured this out in <strong>19</strong>61γγ − 12.5 K' const ⋅M − 0.5 ⋅R − 1.5( )T e const R − 2 R 1.7 ⋅ 9.551.5⋅ ⋅const ⋅R0.061