Today in Astronomy 142: stellar structure Observations of stars which we seek to explain theoretically. Principles of stellar structure Hydrostatic equilibrium • Crude stellar interior models • Pressure, density and temperature in the center (where the luminosity is produced) Opacity of the Sun: diffusion of light from center to surface. Rotational structure of the Sun: the different colors indicate flows slower (blue) and faster (red) than average. (SOHO/NASA). 29 January 2013 Astronomy 142, Spring 2013 1 Data on eclipsing binary stars The most useful, ongoing, big compendium of detached, main-sequence, double-line eclipsing binary data is by Oleg Malkov and collaborators (1993, 1997, 2006, 2007). Much of the data come from many decades of work by Dan Popper. Those objects for which orbital velocities have been measured comprise the fundamental reference data on the dependences of stellar parameters – temperature, radius, luminosity on their masses. A strong trend emerges in plots of mass vs. any other quantity, involving all stars which are not outliers in the mass vs. radius plot. This trend, in whatever graph it appears, is called the main sequence, and its members are called the dwarf stars. 24 January 2013 Leftovers Astronomy 142, Spring 2013 2 Radii of eclipsing binary stars 100 Giants 10 ) R Radius ( Radius 1 Detached binaries Semidetached/contact binaries 0.1 0.1 1 10 100 Mass (M) 24 January 2013 Leftovers Astronomy 142, Spring 2013 3 Luminosities of eclipsing binary stars 1000000 10000 ) L 100 1 Luminosity ( Luminosity 0.01 Detached binaries Semidetached/contact binaries 0.0001 0.1 1 10 100 Mass (M) 24 January 2013 Leftovers Astronomy 142, Spring 2013 4 Effective temperatures of eclipsing binary stars 50000 Detached binaries 40000 Semidetached/contact binaries 30000 20000 Temperature (K) Temperature 10000 0 0.1 1 10 100 Mass (M) 24 January 2013 Leftovers Astronomy 142, Spring 2013 5 The H-R diagram We can’t, of course, measure the masses of single stars, or binary stars in orbits of unknown orientation. Unfortunately this includes more than 99% of the stars in the sky. We can usually measure luminosity (from bolometric magnitude) and temperature (from colors) though. The plot of this relation is called the Hertzsprung-Russell diagram. The H-R diagram for detached eclipsing binaries is very similar to that of stars in general: it corresponds to what is obviously the main sequence in other samples of stars. This correspondence allows us to infer the masses of single or non-eclipsing multiple stars. 24 January 2013 Leftovers Astronomy 142, Spring 2013 6 H-R diagram for eclipsing binary stars Usually the T axis is plotted backwards in H-R diagrams. 1000000 Detached binaries Semidetached/contact binaries 10000 ) L 100 Giants 1 Luminosity ( Luminosity 0.01 0.0001 10000 1000 Effective temperature (K) 24 January 2013 Leftovers Astronomy 142, Spring 2013 7 Eclipsing binaries vs. nearby stars Stars within 25 pc: Gliese & Jahreiss 1991. 1000000 Stars within 25 pc of the Sun 10000 Detached binaries ) L 100 1 Luminosity ( Luminosity 0.01 0.0001 10000 1000 Effective temperature (K) 24 January 2013 Leftovers Astronomy 142, Spring 2013 8 Eclipsing binaries vs. young stars in clusters X-ray selected Pleiades: Stauffer et al. 1994. 1000000 The Pleiades young open cluster 10000 Detached binaries ) L 100 1 Luminosity ( Luminosity 0.01 0.0001 10000 1000 Effective temperature (K) 24 January 2013 Leftovers Astronomy 142, Spring 2013 9 Eclipsing binaries vs. bright or very nearby stars Bright/nearby: Allen’s Astrophysical Quantities (2000). 1000000 10000 ) L 100 1 Nearest and Luminosity ( Luminosity brightest stars 0.01 Detached binaries 0.0001 10000 1000 Effective temperature (K) 24 January 2013 Leftovers Astronomy 142, Spring 2013 10 Theoretical principles of stellar structure Vogt-Russell “theorem:” the mass and composition of a star uniquely determine its radius, luminosity, internal structure, and subsequent evolution. Stars are spherical, to good approximation. Stable stars are in hydrostatic equilibrium: the weight of each infinitesimal piece of the star’s interior is balanced by the force from the pressure difference across the piece. • Most of the time the pressure is gas pressure, and is described well in terms of density and temperature by the ideal gas law, PV = NkT. • However, in very hot stars or giant stars, the pressure exerted by light – radiation pressure – can dominate. 29 January 2013 Astronomy 142, Spring 2013 11 Principles of stellar structure (continued) Energy is transported from the inside to the outside, most of the time in the form of light. • The interiors of stars are opaque. Photons are absorbed and reemitted many many times on their way from the center to the surface: a random-walk process called diffusion. • The opacity depends upon the density, temperature and chemical composition. • Most stars have regions in their interiors in which the radial variations of temperature and pressure are such that hot bubbles of gas can “boil” up toward the surface. This process, called convection, is a very efficient energy-transport mechanism and can be more important than light diffusion in many cases. 29 January 2013 Astronomy 142, Spring 2013 12 The equations of stellar structure Hydrostatic equilibrium: Equations of state dP GM ρ = − r Pressure: 2 dr r PP= (ρ , T ,composition) in general Mass conservation: ρσkT4 T 4 dM 2 = + throughout most normal stars r = 4πρr µ 3c dr Energy generation: Opacity: κ κρ dL = ( ,T ,composition) in general r = 4πr2 ρε dr Energy generation: ε= ερ ( ,T , composition) in general Energy transport: dT 3 κρ L = − r Boundary conditions: dr 16σ 22π rr4 Mr → 0 Adiabatic temperature gradient: as r → 0 Lr → 0 dT 1 µ GM =−−1 r T → 0 2 dr rad γ k r P → 0 as r → Rstar Convection occurs if: ρ → 0 T dP γ < Mr , Lr : mass or luminosity contained P dT γ − 1 within radius r. 29 January 2013 Astronomy 142, Spring 2013 13 The equations of stellar structure (continued) There are no analytical solutions to the equations; usually stellar-interior models are generated by computer solution. Does that mean we get to go home now? No. Progress can be made by assuming a formula for one of the parameters, and solving for the rest. In this manner we can learn the workings of some of these differential equations, and establish some of the scaling relations useful in understanding the shapes of the empirical RM ( ) , T ee ( M ) , LM ( ) , and LT ( ) results. First it will be useful to derive the stellar-structure equation we’ll use most, the equation of hydrostatic equilibrium. 29 January 2013 Astronomy 142, Spring 2013 14 Hydrostatic equilibrium In AST 111 we applied hydrostatic equilibrium to the structure of planetary atmospheres. Because atmospheres are thin compared to the radii of planets, it was sufficient there to approximate atmo- spheres as plane parallel slabs with constant gravitational acceleration. Thus we got a Cartesian one-dimensional differential equation, which we solved in various cases: dP = −ρ g . dz In stellar interiors we still get a one-D differential equation (because stars are spherically symmetric), but we can’t ignore the spherical shape or the radial dependence of gravitational acceleration. 29 January 2013 Astronomy 142, Spring 2013 15 Hydrostatic equilibrium (continued) Consider a spherical shell of radius r and thickness ∆r r within a star with mass density ∆m (mass per unit volume) ρ(r). Its M weight is r ∆r GM∆ m ∆=−F r r r2 GM =−∆r 4πρrrr2 ( ) 2 R r M =−∆4.πρGMr ( r) r Mr : mass contained (minus sign: force points inward) within radius r 29 January 2013 Astronomy 142, Spring 2013 16 Hydrostatic equilibrium (continued) If the star is in hydrostatic equilibrium the weight is balanced by the pressure difference across the shell: 2 −∆F = Pr( ) 44ππ r2 − Pr( +∆ r) ( r +∆ r) (∆rr ) 22dP (to first order ≅Pr( ) 44ππ r − Pr( ) +∆ r r dr in pressure) dP =−∆4,π rr2 dr dP∆ F 4πρGM( r)∆ r GM ρ( r) so ==−=rr−. dr 44ππrr22∆∆ rr r 2 29 January 2013 Astronomy 142, Spring 2013 17 Crude stellar models (Not even complex enough to be called “simple”…) One can learn a surprisingly large amount about stellar interiors by starting with a formula for the mass density, ρ(r), and solving the equations of hydrostatic equilibrium and the equation of state self consistently for pressure and temperature. One learns most, of course, if the formula for density resembles the real thing. In AST 142 we will consider problems in which density of polynomial or exponential form to be within bounds. That is, all forms in which the hydrostatic equilibrium equation can be integrated straightforwardly. 29 January 2013 Astronomy 142, Spring 2013 18 Crudest approximation: the “uniform Sun” Since we know its distance (from radar), mass (from Earth’s orbital period) and radius (from angular size and known distance): r =1 AU = 1.4960 × 1013 cm 33 M =1.98843 × 10 gm = × 10 R 6.9599 10 cm we know the average mass density (mass per unit volume): MM3 ρ = = = 1.41 g cm-3 3 V 4π R = only 26% of Earth's. What would the pressure and temperature be, if the Sun had uniform density? 29 January 2013 Astronomy 142, Spring 2013 19 The “uniform Sun” (continued) 3 Let us therefore assume that ρρ ( r ) = = 3 MR 4, π and integrate away: 2 dP GM ρ G r3 33M GM =−=−r Mr =− dr 22 33 6 r rRR 44ππ R R 0 3GM2 dP = − rdr Surface has P = 0, if it doesn't mov. e ∫∫π 6 P(00) 4 R 22 2 3GM R 3 GM −=−PP(0) ⇒( 0.) = 6428π 4π RR 15 -2 9 PC ==P(0) 1.34×= 10 dyne cm 1.3× 10 atmospheres.