M.R. Burleigh 2601/Unit 4 DEPARTMENT OF PHYSICS AND ASTRONOMY LIFECYCLES OF STARS Option 2601

M.R. Burleigh 2601/Unit 4 Stellar Physics  Unit 1 - Observational properties of stars  Unit 2 - Stellar Spectra  Unit 3 - The Sun  Unit 4 - Stellar Structure  Unit 5 - Stellar Evolution  Unit 6 - Stars of particular interest

M.R. Burleigh 2601/Unit 4 DEPARTMENT OF PHYSICS AND ASTRONOMY Unit 4 Stellar Structure

M.R. Burleigh 2601/Unit 4 Starbirth

Young Stars

M.R. Burleigh 2601/Unit 4 Globular Clusters

M.R. Burleigh 2601/Unit 4 Star Death

M.R. Burleigh 2601/Unit 4 Star Death

M.R. Burleigh 2601/Unit 4 Star Death

M.R. Burleigh 2601/Unit 4 Star Death

M.R. Burleigh 2601/Unit 4 Stellar Structure  Hydrostatic equilibrium  Equations of state  Energy transport (not derived)  Energy sources  Stellar models  Mass-Luminosity relation  Eddington Limit

M.R. Burleigh 2601/Unit 4 r = R r Centre (r = 0)  (r) P + dPPdr Hydrostatic Equilibrium

M.R. Burleigh 2601/Unit 4 Equation of hydrostatic equilibrium: Only need to know  (r) to determine mass of star radius R Hydrostatic Equilibrium Hydrostatic Equilibrium (1) However:(2)

M.R. Burleigh 2601/Unit 4 E.g. Sun’s central pressure G=6.67x Nm 2 Kg -2, M  =1.989x10 30 kg, R  =6.96x10 8 m   (ave)  =3M  /4  R  3 =1410kgm -3 Surface pressure = 0 Let r = dr = R  and M(r)= M  P c ~G M   (ave)  / R  = 2.7x10 14 Nm -2

M.R. Burleigh 2601/Unit 4 Assume material is a perfect gas Obeys perfect gas law: Number density of particles Boltzmann’s constant (1.381  JK -1 ) Equations of State (3)

M.R. Burleigh 2601/Unit 4 n(r) is dependant on density and composition: m H = 1.67  kg = mass of a hydrogen atom  = mean molecular weight Mass fractions of: H He Metals (all other heavier elements) Equations of State

M.R. Burleigh 2601/Unit 4 In massive stars, radiation pressure also contributes to the total pressure: a =  Jm -3 K -4 = radiation constant (  = ¼ ac) Radiation Pressure

M.R. Burleigh 2601/Unit 4 E.g. Sun’s central temperature Use P c and  estimates, assume  ~ ½ Then T c ~ P c  m H /  (ave)  k ~ 1.2x 10 7 K Gas dissociated into ions & electrons but overall electrically neutral… a plasma

M.R. Burleigh 2601/Unit 4 Energy Transport T(r) depends on how energy is transported from interior  surface Three processes… 1.Conduction – collision of hot energetic atoms with cooler… poor in gases 2.Convection – mass motions of fluids, need steep temp. gradient… happens in some regions of most stars

M.R. Burleigh 2601/Unit 4 Energy Transport Three processes… 1.Conduction 2.Convection 3.Radiation – high energy photons flow outward losing energy by scattering and absorption… opacity sources at high T are i) electron scattering and ii) photoionization

M.R. Burleigh 2601/Unit 4  (r) (opacity) depends only upon N(r), T(r) and  (r) L(r) at the surface is the star’s bolometric luminosity Radiative Transport Equation (4)

M.R. Burleigh 2601/Unit 4 For the Sun  L  ~ 9.5x10 29 /  Joules s -1  However, we do not know  very well  Ranges from <<  << 10 7  Therefore… –10 22 << L  << Joules s -1  Measured value is 3.9x10 26 implies –  ~ 2.4x10 3

M.R. Burleigh 2601/Unit 4 The Virial Theorem  Considers total energy in a star  Gravitational contraction  Gravitational potential energy  kinetic energy  Kinetic energy in bulk  Heat

M.R. Burleigh 2601/Unit 4 Take the equation of hydrostatic equilibrium: and But: The Virial Theorem

M.R. Burleigh 2601/Unit 4 Integrate over the whole star: P,  and r are functions of m Zero at both limits (P(m) = 0 marks the boundary of the star) Twice the thermal (kinetic) energy -2U Gravitational binding energy  Total energy of a star:

M.R. Burleigh 2601/Unit 4 Gravitational contraction   ½  excess must be lost by radiation But, using Virial theorem: 1)Star gets hotter 2)Energy is radiated to space 3)Total energy of the star decreases (becomes more –ve  more tightly bound) Gravitational Contraction

M.R. Burleigh 2601/Unit 4 Stellar Thermonuclear Reactions  Light elements “burn” to form heavier elements  Stellar cores have high enough T and  for nuclear fusion  Work (after 1938) by Hans Bethe and Fred Hoyle

M.R. Burleigh 2601/Unit 4 Stellar Thermonuclear Reactions  Energy release can be calculated from E=mc 2 –e.g. 4 x 1 1 H atoms  1 x 4 2 He atom  4 x x kg = x kg  1 x x kg  E = 4.26x J

M.R. Burleigh 2601/Unit 4 Stellar Thermonuclear Reactions  In the Sun ~10% of its volume is at the T and  required for fusion  Total energy available is… –Energy per reaction x mass/mass in each reaction  E tot = 4.26x x 2x10 29 /6.6916x = 1.27x10 44 J  L  = 3.9x10 26  t ~ 3.3x10 17 s ~ yrs

M.R. Burleigh 2601/Unit 4 CNO cycle Stellar Thermonuclear Reactions Proton – proton chain (PPI, T < 2  10 7 K) 1.44MeV 5.49MeV 12.9MeV

M.R. Burleigh 2601/Unit 4 The PPI Chain

M.R. Burleigh 2601/Unit 4 a b c PPI Chain

M.R. Burleigh 2601/Unit 4 The CNO Cycle

M.R. Burleigh 2601/Unit 4 CNO Cycle

M.R. Burleigh 2601/Unit 4 Triple Alpha High level reactions ~10 8 K Addition of further alphas

M.R. Burleigh 2601/Unit 4 n = number density of particles  = mean molecular weight Hydrostatic equilibrium:(1) Mass equation:(2) Equation of state:(3) Stellar Models: Equations

M.R. Burleigh 2601/Unit 4 Radiation pressure: a = radiation constant =  Jm -3 K -4  = ¼ ac  = opacity ε = rate of energy production (Js -1 kg -1 ) Radiative transport:(4) Energy generation: (5)

M.R. Burleigh 2601/Unit 4 e.g. r = 0 M(r) = 0 L(r) = 0 r = R M(r) = M L(r) = L T(r) = T eff And  (r), P(r)  0 Need to apply boundary conditions to the equations to use them, i.e. fix/know values at certain values of r (centre or surface) Boundary Conditions

M.R. Burleigh 2601/Unit 4 From (1) write dP   P and dr   r Then:  P = P S – P C = 0 - P C SurfaceCentre and  r = R For a perfect gas P   T

M.R. Burleigh 2601/Unit 4 From (4) Also: Substitute  Observed relationship is L  M 3.3 (  is dependant on T and  )

M.R. Burleigh 2601/Unit 4 Hydrostatic equilibrium assumes no net outward motion of material from the star, but the outward flow of radiation imparts a force on the material Momentum of radiation =  T = cross-section of electron-photon scattering = 6.7  m 2 This is opposed by gravitational force =  Force = Eddington Limit

M.R. Burleigh 2601/Unit 4 The forces are equal at the Eddington limit erg s -1 So if L > L E material is expelled

M.R. Burleigh 2601/Unit 4 Stellar Structure  Hydrostatic equilibrium  Equations of state  Energy transport (not derived)  Energy sources  Stellar models  Mass-Luminosity relation  Eddington Limit