Midterm Review

PHY 688, Lecture 27 Mar 30, 2009 Outline • Course administration – final presentations: select topics today – problem set 3

• Midterm exam – next Wednesday, April 1, 10:40–11:35am – review

• Review of previous lecture – search techniques; direct imaging

Mar 30, 2009 PHY 688, Lecture 27 2 Example Presentation Topics

• Nearby young brown dwarfs (Josh Schlieder) • Y dwarfs: empirical and theoretical expectations • The 2MASS J0535–0546 eclipsing binary: why is the less massive component hotter? • The substellar initial mass function: comparison among star forming regions and implications for the formation of brown dwarfs • Non-equilibrium chemistry in substellar atmospheres • Magnetic activity and rotation in low-mas stars and brown dwarfs • Nemesis: Sun's hypothetical binary companion • The origin of hot Jupiters • The Kozai mechanism and the eccentricities of exoplanet orbits • Searching for exo-earths through gravitational microlensing • formation through core • Planet formation through gravitational disk instability • Planet migration and the architecture of planetary systems • Extremely high contrast imaging of : present and future • Dynamical signatures of unseen in optically thin circumstellar debris disks

Mar 30, 2009 PHY 688, Lecture 27 3 Outline • Course administration – final presentations: select topics today – problem set 3

• Midterm exam – next Wednesday, April 1, 10:40–11:35am – review

• Review of previous lecture – exoplanet search techniques; direct imaging

Mar 30, 2009 PHY 688, Lecture 27 4 Review Topics

• binary stars: dynamical masses • stellar and substellar interiors; evolution • nuclear fusion in the cores low-mass objects • spectral energy distributions, dominant atmospheric absorbers, spectral classification • stellar and substellar photospheres – effective temperature, surface gravity, metallicity, evolution • dust and clouds in substellar atmospheres • hot Jupiters: – temperature structure and dynamics of atmospheres – radii; Rossiter-McLaughlin effect

Mar 30, 2009 PHY 688, Lecture 27 5 Dynamical Masses

• Resolved visual binaries: see stars separately, measure orbital axes and speeds directly. • Astrometric binaries: only brighter member seen, with periodic wobble in the track of its proper motion. • Spectroscopic binaries: unresolved (relatively close) binaries told apart by periodically oscillating Doppler shifts in spectral lines. Periods = days to years. – Eclipsing binaries: orbits seen nearly edge on, so that the stars actually eclipse one another. (Most useful.)

Mar 30, 2009 PHY 688, Lecture 27 6 Dynamical Mass Determination

– If orbital major axes (relative to center of mass) or amplitudes are known, so is the ratio of masses: m a v 1 = 2 = 2r m2 a1 v1r

– If the period, P, and the sum of major axis lengths, a = a 1 + a 2 , are known, Kepler’s third law can give masses separately: 2 1 2 # 4" 3& P = % a ( $ G(m1 + m2 ) ' Mar 30, 2009 PHY 688, Lecture 27 7

! Internal Equilibrium Equations dP GM # • hydrostatic equilibrium = " r dr r2

• mass continuity dM 2 r = 4$r # dr

• conservation of energy dLr 2 – energy emitted in neutrinos = 4$r #(% "%& ) εν dr

• temperature continuity – depends on mode of energy transport ! Mar 30, 2009 PHY 688, Lecture 27 8 Modes of Energy Transport and the Temperature Continuity Equation • radiation – photons absorbed by cooler outer layers dT 3#$Lr – efficient in: = " 2 3 • >1 MSun star envelopes dr 64%r &T • cores of 0.3–1 MSun stars • all stellar photospheres • convection ( + – adiabatic exponent = C /C dT 1 T dP γ P V = *1 " - – important when radiation inefficient: dr ) ' , P dr • interiors of brown dwarfs and <0.3 MSun stars • cores of >1 MSun stars • envelopes of ~1 MSun stars

!

Mar 30, 2009 PHY 688, Lecture 27 9 HydrogenHydrogen phasephase diagram diagram Evolution is towards: • lower entropy S 0.67 ρ • higher degeneracy η T ∝

T

0.4 T ∝ ρ

0.67 ρ T ∝

Mar 30, 2009 PHY 688, Lecture 27 (Burrows & Liebert 1993) 10 A Brown Dwarf’s and Jupiter’s Interiors

(1 bar)

0.05 M Sun 9 brown dwarf (4×10 bar)

(Guillot 2006) Mar 30, 2009 PHY 688, Lecture 27 11 Radius vs. Mass: Comparison with Known Planets • for polytropes 1#n R " M 3#n

• n = 1.5 for brown dwarfs ! • n = 0.5–1.0 for 0.1–1 MJup planets • (n = 0: uniform density) H2O planet

SiO4 planet • icy/rocky cores in olivine (Mg,Fe)2 Neptune, Uranus?

Mar 30, 2009 PHY 688, Lecture 27 (Guillot 2006) 12 p-p Chain Reaction Rate in Low-Mass Stars Is Decided by the First Step (p+p)

take into account Coulomb coupling

Mar 30, 2009 PHY 688, Lecture 27 13 Minimum Main Sequence Mass

$ 10"2 g cm"2 ' I(*) MMMSM = 0.0865MSun& ) % # R ( I(*min ) 1.509 (* + +) I(*) = *1.325

• MMMSM depends on:

– opacity κR (i.e., metallicity Z) ! – He content Y (through α(Y))

• MMMSM = 0.075 MSun at solar Y (25%), Z (1.6%) – lower for higher Y, Z

Mar 30, 2009 PHY 688, Lecture 27 14 Li and D: Depleted within Few 100Myr

50% D depletion 50% Li depletion

stars brown dwarfs “planets”

Mar 30, 2009 PHY 688, Lecture 27 (Burrows et al. 2001) 15 The Optical to IR SEDs of UCDs

(CushingMar 30, 2009et al. 2006; Marley & Leggett 2008) PHY 688, Lecture 27 16 SEDs: Near-IR Wavelengths

• reddish L dwarf colors due J H K to dust in the visible atmosphere

• neutral T dwarf colors due to dust-free atmosphere, molecular opacity

CIA H2 • 2MASS J – Ks colors: M5 : ~0.9 mag L5 : ~1.7 mag T6 : ~0 mag

Mar 30, 2009 PHY 688, Lecture 27 17 Near-IR CMD of Stars and Brown Dwarfs

Mar 30, 2009 PHY 688, Lecture 27 (Kirkpatrick 2005) 18 Luminosity (i.e., Surface Gravity) Effects at A0

(figure: D. Gray) Mar 30, 2009 PHY 688, Lecture 27 19 From Lecture 5: Line Profiles # /2$ • Natural line width (Lorentzian [a.k.a., Cauchy] profile) I I " = 0 2 2 – Heisenberg uncertainty principle: ∆ν =∆E/h (" %" 0 ) + # /4 • Collisional broadening (Lorentzian profile) # & Lorentzian FWHM – collisions interrupt photon emission process #E i + #E f 1 1 –9 – ∆t < ∆t ~ 10 s " natural = = + coll emission h /2$ #t #t – dependent on T, ρ i f • Pressure broadening (~ Lorentzian profile) ! " collisional = 2 #tcoll – ∆t > ∆t &n interaction emission " pressure % r ; n = 2,3,4,6 – nearby particles shift energy levels of emitting particle • Stark effect (n = 2, 4) 2 (" %" 0 ) • van der Waals force (n = 6) cool stars % 1 2$ 2 • dipole coupling between pairs of same species (n = 3) I = e ! " 2#$ – dependent mostly on ρ, less on T • Thermal Doppler broadening (Gaussian profile) $ & Gaussian FWHM – emitting particles have a Maxwellian distribution of velocities kT "thermal = # 0 • Rotational Doppler broadening (Gaussian profile) mc 2 – radiation emitted from a spatially unresolved rotating! body "rotational = 2# 0u/c •MarComposite 30, 2009 line profile: LorentzianPHY + 688, Gaussian Lecture 27 = Voigt profile 20

! Gravity-Sensitive Features in UCDs

Mar 30, 2009 PHY 688, Lecture 27 (McGovern et al. 2004) 21 Gravity in UCDs Key species: • neutral alkali elements (Na, K) – weaker at low g • hydrides – CaH weaker at low g – FeH unchanged • oxides – VO, CO, TiO stronger at low g

– H2O ~ unchanged

log g and Teff are measurable properties Wavelength (µm) Mar 30, 2009 (Kirkpatrick et al. 2006) PHY 688, Lecture 27 22 Curve of Growth: Dependence of Line Equivalent Width W on Column Density N • N ≡ integral of number density of absorbing atoms or molecules along line of sight [cm-2] – for small N, W ∝ N • linear part of the curve of growth – for larger N, W " ln N • after the Gaussian core bottoms out • flat part of the curve of growth – for even larger N, W " N • !after the absorption by the Lorentzian wings becomes strong • square root part of the curve of growth • There is a different curve of growth, W(N), for each spectral line!

Mar 30, 2009 PHY 688, Lecture 27 23 Universal Curve of Growth

• the 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 (e.g. Unsöld 1955).

1 v "# = # flat square c " W # root log $ % 0 W " N # 2kT ' &! ( W " ln N = c m linear 1 W " N ! ! 1 0 1 2 3 4 ! ! log (Nf ) Mar 30, 2009 PHY 688, Lecture 27 24 Curve of Growth: Determining Abundances • Measure W for a lot of lines (each with distinct, known f) of a bunch of atomic or ionic species. • Plot W/∆λ against xNf where: – N is the column density of one species – x is the relative abundance of the atomic species that gives rise to the line (ratio of number density of that species to the number density of the first species), • Adjust x, N, and ∆λ until the points fit the universal curve of growth. • Then one knows these three quantities for each species.

Mar 30, 2009 PHY 688, Lecture 27 25 Subdwarf SEDs

• signatures of metal deficiencies dM esdM • higher gravity in deeper layers?usdM

(Jao et al. 2008)

Mar 30, 2009 PHY 688, Lecture 27 26 sdL7 sdM’s L7 sdL’s

enhanced hydrides,

H2

Mar 30, 2009 PHY 688, Lecture 27 27 Simple Chemical Picture of Atmospheric Cooling for MLT’s • As gas temperature of a (brown) dwarf drops, atoms: – first favor an ionized state • e.g., Ca II, Fe II in Sun – then favor a neutral state • e.g., Na I, K I in M/L/T dwarfs – then form molecules

• e.g, H2O, TiO, FeH, CH4 in M/L/T dwarfs – then condense into a solid or liquid

• e.g., Mg2SiO4, Al2O3 in L/T dwarfs • dust clouds • More refractory elements tend to condense first • Exact sequence of molecule and condensate formation depends on – gas pressure – metallicity – turbulent mixing from warmer or colder layers, etc

Mar 30, 2009 PHY 688, Lecture 27 28 Dust Cloud Chemistry

(Burrows et al. 2001)

Mar 30, 2009 PHY 688, Lecture 27 29 Cloud Level: Balance of Turbulent Mixing and Sedimentation • Cloud condensates will settle under gravity to a level where there is enough upward convective (turbulent) motion to keep them afloat. • Level and vertical extent of clouds depend on – droplet size (i.e., mass) – convective velocity, mixing efficiency 4 3$ '1 3 #qt H $ L ' RF "K " f rain w*qc = 0, K = & ) & ) #z 3 % H ( % µ*ac p ( • K – vertical eddy diffusion coefficient (~105–109 cm2 s–1) – H = RT/µg – atmospheric scale height (~10 km); L – turbulent mixing length (~H); R – –1 universal gas constant; µ – atmospheric molecular weight (2.2 g mol assumed); ρa – 4 atmospheric density; cp – specific heat of atmosphere at constant pressure (ideal gas); F = σTeff ! • qc – condensate mixing ratio (mole of condensate per mole of atmosphere) • qt = qc+qv – total mixing ratio (condensate + vapor) –1 • w* = K/L – convective velocity scale (~1 m s ) • frain – sedimentation efficiency (~2–6 in bulk of cumulus clouds on Earth)

– ratio of mass-weighted droplet sedimentation velocity to w*

Mar 30, 2009 PHY 688, Lecture 27 (Ackerman & Marley 2001) 30 Condensate Clouds (AM01 Baseline Models)

L dwarf T dwarf

Mar 30, 2009 PHY 688, Lecture 27 (Ackerman & Marley 2001) 31 Emergent Flux Depends on Wavelength and Cloud Level

τcloud < 0.5; hcloud > hphotosphere

τcloud > 1; hcloud ~ hphotosphere silicate cloud

(frain = 3)

τcloud > 1; hcloud < hphotosphere

Mar 30, 2009 PHY 688, Lecture 27 (Ackerman & Marley 2001) 32 Modeling M L and T Dwarfs • Models that incorporate suspended L dust (DUSTY) successfully reproduce L dwarf colors T • Late T dwarfs well fit by dust-free photospheres (e.g., COND models: dust removed upon DUSTY models (dust remains formation) suspended)

• Transition can be explained by sedimentation of silicate clouds COND models below visible photosphere (dust is removed)

Mar 30, 2009 PHY 688, Lecture 27 (Baraffe et al. 2003)33 The L/T Transition Problem

• photospheres turn blue in the near-IR unusually quickly • clouds sink comparatively slowly – need to be “rained out” (sedimented) faster

• reddest L dwarfs require

inefficient sedimentation ∞ → (frain < 3) f rain • early T dwarfs require frain > 3 • late T’s require no visible clouds ∞) = 3 (frain → frain

Mar 30, 2009 PHY 688, Lecture 27 (Burgasser et al. 2002) 34 What Is the Weather on an Early T Dwarf?

• partly cloudy?

• uniformly hazy?

• raining “cats and dogs”? – i.e., silicates and iron

Mar 30, 2009 PHY 688, Lecture 27 35 Detecting Thermal Emission From Planet’s “Day” Side: Secondary Eclipse

Secondary Eclipse See thermal radiation from planet disappear and reappear

Primary Eclipse See radiation from star transmitted Through the planet’s atmosphere

Mar 30, 2009 PHY 688, Lecture 27 36 Effect of Irradiation

• balance between internal flux and flux incident from star 4 4 4 Teff = Tint + W T* • W – dimensionless “dilution” factor ~ 10–3

• incident light penetrates to depth τpen, such that # & 4 T* " pen = W % ( )1 $ Tint '

• for τ < τpen, Teff is governed by irradiation and is constant – isothermal, radiative region ! • for τ > τpen, Teff ≈ Tint, and rises monotonically with τ

Mar 30, 2009 PHY 688, Lecture 27 37 P-T Profiles of Hot Jupiters AU

• isothermal regions are radiative

Mar 30, 2009 PHY 688, Lecture 27 (Fortney et al. 2007) 38 Hot and Very Hot Jupiters: pL vs. pM Planets

• distinction: – based on lack or presence of high-level TiO/VO associated with a stratosphere – cf. L vs. M stellar spectral types • transition at around 0.04–0.05 AU equivalent separation from the Sun • note dependences on: – observed planetary hemisphere (Fortney et al. 2008) – orbital phase for planets on very eccentric orbits • HD 17156b, HD 80606b, HD 147506b Mar 30, 2009 PHY 688, Lecture 27 39 Winds: U Cooling vs. Advection • advection time scale tadvec = Rp/U

– Rp – planet radius – U – wind speed • balance of cooling vs. advection decides wind speed U

"Tday – night ~ 1# e#tadvec / trad "Trad • winds of several km/sec (~ sound speed) expected from 2D and 3D ! dynamical models Mar 30, 2009 PHY 688, Lecture 27 (Fortney et al. 2008) 40 Radii of Very Hot Jupiters • some large radii cannot be explained by coreless planet models with high-altitude stratospheres: – extra internal power source? • stratospheric heat trap • tidal heating • damping or orbital eccentricity and apparent resetting of planet age? – host stars are giga-years old – preferential evaporation of neutral helium? (Fortney et al. 2007)

Mar 30, 2009 PHY 688, Lecture 27 41 Rossiter-McLaughlin Effect

• first detected in eclipsing binary stars – as in bottom panel

• effective Doppler shift of (absorption) line changes depending on the part of the host star that is occulted

Mar 30, 2009 PHY 688, Lecture 27 (Gaudi & Winn 2007) 42 RM Effect Geometry

• seek to measure angle λ between projected stellar rotation axis and planetary orbital axis

• note: ΩS sin IS here is the same as Vs sin Is and V sin Is in the following slides: the (Ohta et al. 2005) projected stellar spin rate

Mar 30, 2009 PHY 688, Lecture 27 43