Substellar Interiors. PHY 688, Lecture 13

Transcription

1 Substellar Interiors PHY 688, Lecture 13

2 Outline Review of previous lecture curve of growth: dependence of absorption line strength on abundance metallicity; subdwarfs Substellar interiors equation of state (EOS) thermeodynamics of hydrogen Feb 23, 2009 PHY 688, Lecture 13 2

3 Previously in PHY 688 Feb 23, 2009 PHY 688, Lecture 13 3

4 Alkali (Na, K) lines in visible spectra of late-l and T dwarfs become saturated! 2005) Feb(Kirkpatrick 23, 2009 PHY 688, Lecture 13 4

5 Lorentzian Line Profile at Increasing τ simulation for the Hα line profile Feb 23, 2009 PHY 688, Lecture 13 5

6 Lorentzian Line Profile at Increasing τ simulation for the Hα line profile saturation at τ > 5 Feb 23, 2009 PHY 688, Lecture 13 6

7 Lorentzian Line Profile at Increasing τ simulation for the Hα line profile Feb 23, 2009 PHY 688, Lecture 13 7

8 Lorentzian vs. Gaussian Line Profiles: Small τ simulation for the Hα line profile Lorentzian Gaussian Feb 23, 2009 PHY 688, Lecture 13 8

9 Lorentzian vs. Gaussian Line Profiles: Large τ simulation for the Hα line profile core more sensitive to Gaussian parts wings more influenced by Lorentzian parts Lorentzian Gaussian Feb 23, 2009 PHY 688, Lecture 13 9

10 Universal Curve of Growth each absorbing species follows a different curve of growth however, ratio of W to Doppler line width Δλ depends upon the product of N and a line s oscillator strength f in the same way for every spectral line useful for determining element abundances " log W # $ % ' &! ( linear W " N flat ( ) log Nf square root W " ln N W " N "# = # v c Feb 23, 2009 PHY 688, Lecture = # c 2kT m

11 Metallicity [m/h] * = log N(m) * / log N(H) * log N(m) Sun / log N(H) Sun [Fe/H] Sun = 0.0 is the solar Fe abundance [Fe/H] = 1.0 is the same as 1/10 solar [m/fe] * = log N(m) * / log N(Fe) * log N(m) Sun / log N(Fe) Sun [Ca/Fe] = +0.3 means twice the number of Ca atoms per Fe atom Sun s metal content is Z 1.6% by mass Feb 23, 2009 PHY 688, Lecture 13 11

12 The Early Universe Had No Heavy Elements (figure credit: N. Wright, UCLA) Feb 23, 2009 PHY 688, Lecture 13 12

13 Metal Enrichment Is Due to Stars H He nuclear fusion neutron capture Fe Li Feb 23, 2009 PHY 688, Lecture 13 13

14 Stellar Populations Young stars (Pop I) are metal-rich [m/h] > 2.0 Milky Way disk, spiral arms Old stars (Pop II) are metal-poor [m/h] < 2.0 Milky Way halo, bulge Ancient stars (Pop III) are expected to have been nearly metal-free (at birth) Feb 23, 2009 PHY 688, Lecture 13 14

15 Cool Metal-Poor Stars M-type metal-poor stars are most numerous M (+L) stars are longest lived dm: normal M dwarfs [m/h] > 1 sdm: M subdwarfs [m/h] ~ 1.3 esdm: M extreme subdwarfs [m/h] ~ 2 i.e., Pop II usdm: M ultra subdwarfs [m/h] < 2 also Pop II Feb 23, 2009 PHY 688, Lecture 13 15

16 Cool Metal-Poor Stars lower atmospheric opacity of subdwarfs reveals the deeper, hotter layers of these stars a dm and an sdm star of the same mass have approximately the same luminosity, but the sdm star can be up to 700 K hotter subdwarfs : dwarfs ~ 1 : 400 (in Milky Way) sd s identified by their high velocities relative to Sun kinematics characteristic of galactic halo stars Feb 23, 2009 PHY 688, Lecture 13 16

17 Subdwarf SEDs signatures of metal deficiencies higher gravity in deeper layers? L * dm esdm usdm models: (Jao et al. 2008) Feb 23, 2009 PHY 688, Lecture 13 17

18 sdm s sdl7 L7 + sdl s enhanced hydrides, CIA H 2 (Burgasser 2008) Feb 23, 2009 PHY 688, Lecture 13 18

19 Outline Review of previous lecture curve of growth: dependence of absorption line strength on abundance metallicity; subdwarfs Substellar interiors equation of state (EOS) thermeodynamics of hydrogen Feb 23, 2009 PHY 688, Lecture 13 19

20 0.05 Msun Brown Dwarf at 5 Gyr (X = 100% H) (1 bar = 10 6 dyn/cm 2 = 0.99 atm) Feb 23, 2009 PHY 688, Lecture 13 20

21 From Lecture 4: Why Are Low-Mass Stars and Brown Dwarfs Fully Convective? radiation photons absorbed by cooler outer layers " dt % efficient in: >1 M Sun $ ' star envelopes # dr & cores of M Sun stars all stellar photospheres convection adiabatic exponent γ = C P /C V " dt % important when radiation inefficient: $ ' interiors of brown dwarfs and <0.3 M Sun stars # dr & cores of >1 M Sun stars envelopes of ~1 M Sun stars in low-mass stars and brown dwarfs: large opacity κ (dt/dr) rad steepens more d.o.f. per gas constituent n = d.o.f / 2; γ = 1 + 1/n (dt/dr) ad becomes shallower Feb 23, 2009 PHY 688, Lecture rad ad = ( 3)*L r 64+r 2,T 3 " = 1( 1 % $ ' T # - & P dp dr

22 Convective Interiors Mean That: entropy (S) is constant throughout ds = dq / T = 0 in convective interior (disregarding radiative atmosphere) equation of state is adiabatic S = 0 holds for a reversible adiabatic process P = Kρ γ (γ = 1+1/n, n = 1.5) brown dwarfs are polytropes of index n = 1.5 Feb 23, 2009 PHY 688, Lecture 13 22

23 From Lecture 4: A Special Solution to the EOS Stars as Polytropes P P(ρ) = Kρ γ, γ = 1+1/n K - constant, n - polytropic index Lane-Emden equations: dimensionless forms of equation of hydrostatic equilibrium solutions: P = P 0 " n +1, # = # 0 " n, T = T 0 " important polytropes: n = 3: normal stars n = 1.5: brown dwarfs, planets, white dwarfs (all are degenerate objects) n = 1: neutron stars n = isothermal proto-stellar clouds 1 d $ " 2 d" " 2 d" ' & ) + # n " % d# (. " * r /a, a * n +1 / 0 ( ) ( ) = 0 K 4+G, c # - dimensionless potential Boundary conditions : ( ) = 0 stellar surface ( ) =1 stellar core # " = " 1 # " = 0 $ d# ' & ) % d" ( " = 0 = 0 stellar core (1-n )/ n / 2 Feb 23, 2009 PHY 688, Lecture 13 23

24 Solutions for Substellar Polytropes n = 1.5: R M 1/3 n = 1.0: R M 0 = const " c # M 2n /(3$n ) P c # M 2(1+n)/(3$n ) (1$n )/(3$n) R # M (Burrows & Liebert 1993) important for M < 4 M Jup objects, in which there are Coulomb corrections to P(ρ) degenerate EOS analytic fit for brown dwarfs and planets (Zapolsky & Salpeter 1969) # R = 2.2 "10 9 % $ M Sun M & ( ' 1/ 3 * # M, 1+ % +, $ M Sun & ( ' )1/ 2 - /./ 4 / 3 cm Feb 23, 2009 PHY 688, Lecture 13 24

25 Sun R M 0.8 Substellar radius changes by <50%; always near 1 R Jup 0.12 R Sun (Burrows & Liebert 1993) Feb 23, 2009 PHY 688, Lecture 13 25

26 10 orders of magnitude change in pressure 3 orders of magnitude change in temperature EOS and Thermodynamics of Hydrogen expect different phases of hydrogen (X = 100% H) Feb 23, 2009 PHY 688, Lecture 13 26

27 Hydrogen phase diagram T Feb 23, 2009 PHY 688, Lecture 13 (Burrows & Liebert 1993) 27

28 Low-Density Regime temperature is sufficient to excite rotational levels of H 2 into equipartition, although not the vibrational levels d.o.f. of H 2 molecules: 3 (spatial motion) + 2 (rotation around short axes) = 5 polytropic index n = d.o.f./2 = 2.5 γ = 1+1/n =1.4 P = Kρ γ ; P = ρkt/µ T ρ γ 1 = ρ 0.4 compare with T ρ 0.67 for n = 1.5 polytrope, as in: high-density interiors of brown dwarfs Jupiter, which does not have sufficient temperature in atmosphere to excite H 2 rotations into equipartition Feb 23, 2009 PHY 688, Lecture 13 28

29 Hydrogen phase diagram T ρ 0.67 T T ρ 0.4 T ρ 0.67 Feb 23, 2009 PHY 688, Lecture 13 (Burrows & Liebert 1993) 29

30 High-Density Regime expect phase change at low T, sufficiently high ρ, P plasma phase transition (PPT) pressure ionization and metallization of H or H+He mixture occurs at ρ ~ 1 gm/cm 3, P = 1 3 Mbar interior is strongly coupled Coulomb plasma bulk of Jupiter (~85%), Saturn (~50%), brown dwarfs (>99.9%) Feb 23, 2009 PHY 688, Lecture 13 30

31 Onset of Coulomb Dominance Γ (dimensionless) ratio of Coulomb energy per ion to kt Ze ionic charge (Z=1 for H) T 6 = T / 10 6 K µ e number of baryons per electron n e = ρn A µ e electron density relevant length scales r s radius of sphere that contains one nucleus on average r e electron spacing parameter Γ << 1 : weekly coupled regime (Debye-Hückel limit) Sun, blood plasma Γ ~ 1 : transition to strong coupling brown dwarf matter, planetary interiors : 1 < Γ < 50 " = Z 2 e 2 r s kt = $ # ' & ) % µ e ( 1 = X + 1 µ e 2 Y = 1+ X 2 $ 3Z ' r s = & ) % 4*n e ( $ 3 ' r e = & ) % 4*n e ( Feb 23, 2009 PHY 688, Lecture / 3 $ 3µ = e Z ' & ) % 4*N A #( $ 3µ = e ' & ) % 4*N A #( Z 5 / 3 T = r s Z 1 3

32 Hydrogen phase diagram T strongly coupled Coulomb plasma Feb 23, 2009 PHY 688, Lecture 13 (Burrows & Liebert 1993) 32