Midterm Review

PHY 688, Lecture 27Mar 30, 2009

Mar 30, 2009 PHY 688, Lecture 27 2

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 3

Example Presentation Topics• Nearby young brown dwarfs (Josh Schlieder)• Y dwarfs: empirical and theoretical expectations• The 2MASS J0535–0546 brown dwarf 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• Planet formation through core accretion• Planet formation through gravitational disk instability• Planet migration and the architecture of planetary systems• Extremely high contrast imaging of exoplanets: present and future• Dynamical signatures of unseen planets in optically thin circumstellar debris disks

Mar 30, 2009 PHY 688, Lecture 27 4

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 5

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 6

Binary Star Dynamical Masses

• Resolved visual binaries: see stars separately, measureorbital axes and speeds directly.

• Astrometric binaries: only brighter member seen, withperiodic wobble in the track of its proper motion.

• Spectroscopic binaries: unresolved (relatively close)binaries told apart by periodically oscillating Dopplershifts 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 7

Dynamical Mass Determination

– If orbital major axes (relative to center of mass) or radialvelocity amplitudes are known, so is the ratio of masses:

– If the period, P, and the sum of major axis lengths, are known, Kepler’s third law can givemasses separately:

m

m

a

a

v

v

r

r

1

2

2

1

2

1

= =

!

P =4" 2

G(m1

+ m2)a3

#

$ %

&

' (

1 2

1 2 ,a a a= +

Mar 30, 2009 PHY 688, Lecture 27 8

Internal Equilibrium Equations

• hydrostatic equilibrium

• mass continuity

• conservation of energyεν – energy emitted in neutrinos

• temperature continuity– depends on mode of energy transport

!

dP

dr= "

GMr#

r2

dMr

dr= 4$r2#

dLr

dr= 4$r2#(% "%& )

Mar 30, 2009 PHY 688, Lecture 27 9

Modes of Energy Transport and theTemperature Continuity Equation

• radiation– photons absorbed by cooler outer layers– efficient in:

• >1 MSun star envelopes• cores of 0.3–1 MSun stars• all stellar photospheres

• convection– adiabatic exponent γ = CP/CV– important when radiation inefficient:

• interiors of brown dwarfs and <0.3 MSun stars• cores of >1 MSun stars• envelopes of ~1 MSun stars

!

dT

dr= "

3#$Lr

64%r2&T 3

dT

dr= 1"

1

'

(

) *

+

, - T

P

dP

dr

Mar 30, 2009 PHY 688, Lecture 27 10(Burrows & Liebert 1993)

T

Hydrogen phase diagram

T ∝ ρ0.67

T ∝ ρ0.67

T ∝ ρ0.4

Hydrogen phase diagram

Evolution is towards:• lower entropy S• higher degeneracy η

Mar 30, 2009 PHY 688, Lecture 27 11

A Brown Dwarf’s and Jupiter’sInteriors

(1 bar)

(4×109 bar)0.05 MSunbrown dwarf

(Guillot 2006)

Mar 30, 2009 PHY 688, Lecture 27 12

Radius vs. Mass:Comparison with Known Planets

• for polytropes

• n = 1.5 for browndwarfs

• n = 0.5–1.0 for0.1–1 MJup planets

• (n = 0: uniformdensity)

• icy/rocky cores inNeptune, Uranus?

(Guillot 2006)

!

R"M

1#n

3#n

olivine (Mg,Fe)2SiO4 planetH2O planet

Mar 30, 2009 PHY 688, Lecture 27 13

p-p Chain Reaction Rate in Low-MassStars Is Decided by the First Step (p+p)

take into accountCoulomb coupling

Mar 30, 2009 PHY 688, Lecture 27 14

Minimum Main Sequence Mass

• 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

!

MMMSM = 0.0865MSun

10"2g cm"2

#R

$

% &

'

( ) I *( )I *min( )

I *( ) =* ++( )

1.509

*1.325

Mar 30, 2009 PHY 688, Lecture 27 15

starsbrown dwarfs“planets”

(Burrows et al. 2001)

Li and D: Depleted within Few 100Myr

50% Li depletion50% D depletion

Mar 30, 2009 PHY 688, Lecture 27 16

The Optical to IR SEDs of UCDs

(Cushing et al. 2006; Marley & Leggett 2008)

Mar 30, 2009 PHY 688, Lecture 27 17

CIA H2

SEDs: Near-IR Wavelengths• reddish L dwarf colors due

to dust in the visibleatmosphere

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

• 2MASS J – Ks colors:M5 : ~0.9 magL5 : ~1.7 magT6 : ~0 mag

J H K

Mar 30, 2009 PHY 688, Lecture 27 18

Near-IR CMD of Stars and Brown Dwarfs

(Kirkpatrick 2005)

Mar 30, 2009 PHY 688, Lecture 27 19

Luminosity (i.e., Surface Gravity)Effects at A0

(figure: D. Gray)

Mar 30, 2009 PHY 688, Lecture 27 20

From Lecture 5: Line Profiles• Natural line width (Lorentzian [a.k.a., Cauchy] profile)

– Heisenberg uncertainty principle: ∆ν =∆E/h• Collisional broadening (Lorentzian profile)

– collisions interrupt photon emission process– ∆tcoll < ∆temission ~ 10–9 s– dependent on T, ρ

• Pressure broadening (~ Lorentzian profile)– ∆tinteraction > ∆temission– nearby particles shift energy levels of emitting particle

• Stark effect (n = 2, 4)• van der Waals force (n = 6)• dipole coupling between pairs of same species (n = 3)

– dependent mostly on ρ, less on T• Thermal Doppler broadening (Gaussian profile)

– emitting particles have a Maxwellian distribution of velocities• Rotational Doppler broadening (Gaussian profile)

– radiation emitted from a spatially unresolved rotating body• Composite line profile: Lorentzian + Gaussian = Voigt profile

!

I" =1

2#$e

%" %"

0( )2

2$2

$ &Gaussian FWHM

!

"thermal

= #0

kT

mc2

"rotational

= 2#0u /c

!

" natural =#Ei + #E f

h /2$=1

#ti+1

#t f

" collisional = 2 #tcoll

" pressure % r&n; n = 2,3,4,6

!

I" = I0

# /2$

" %"0( )

2

+ # 2/4

# & Lorentzian FWHM

cool stars

Mar 30, 2009 PHY 688, Lecture 27 21

Gravity-Sensitive Features in UCDs

(McGovern et al. 2004)

Mar 30, 2009 PHY 688, Lecture 27 22

Gravity inUCDs

(Kirkpatrick et al. 2006)Wavelength (µm)

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

Mar 30, 2009 PHY 688, Lecture 27 23

Curve of Growth: Dependence of LineEquivalent 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,

• after the Gaussian core bottoms out• flat part of the curve of growth

– for even larger 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 eachspectral line

!

W " ln N

!

W " N

Mar 30, 2009 PHY 688, Lecture 27 24

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 thesame way for every spectral line (e.g. Unsöld 1955).

1 0 1 2 3 4

1

0

1

logW

!

" #$ %&' (

( )log Nf

linear

flat square

root

!

W "N

!

W " ln N

!

W " N

!

"# = #v

c

=#

c

2kT

m

Mar 30, 2009 PHY 688, Lecture 27 25

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 numberdensity of the first species),

• Adjust x, N, and ∆λ until the points fit the universal curveof growth.• Then one knows these three quantities for each species.

Mar 30, 2009 PHY 688, Lecture 27 26

Subdwarf SEDs

• signatures of metal deficiencies• higher gravity in deeper layers?

dMesdMusdM

(Jao et al. 2008)

Mar 30, 2009 PHY 688, Lecture 27 27

enhancedhydrides,H2

sdM’ssdL’s

sdL7L7

Mar 30, 2009 PHY 688, Lecture 27 28

Simple Chemical Picture ofAtmospheric 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 29

(Burrows et al. 2001)

Dust CloudChemistry

Mar 30, 2009 PHY 688, Lecture 27 30

Cloud Level: Balance of TurbulentMixing and Sedimentation

• Cloud condensates will settle under gravity to a level where there is enoughupward convective (turbulent) motion to keep them afloat.

• Level and vertical extent of clouds depend on– droplet size (i.e., mass)– convective velocity, mixing efficiency

• K – vertical eddy diffusion coefficient (~105–109 cm2 s–1)– H = RT/µg – atmospheric scale height (~10 km); L – turbulent mixing length (~H); R –

universal gas constant; µ – atmospheric molecular weight (2.2 g mol–1 assumed); ρa –atmospheric density; cp – specific heat of atmosphere at constant pressure (ideal gas); F = σTeff

4

• qc – condensate mixing ratio (mole of condensate per mole of atmosphere)• qt = qc+qv – total mixing ratio (condensate + vapor)• w* = K/L – convective velocity scale (~1 m s–1)• frain – sedimentation efficiency (~2–6 in bulk of cumulus clouds on Earth)

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

(Ackerman & Marley 2001)

!

"K#qt#z

" frainw*qc = 0, K =

H

3

L

H

$

% &

'

( )

4 3RF

µ*acp

$

% & &

'

( ) )

1 3

Mar 30, 2009 PHY 688, Lecture 27 31

Condensate Clouds(AM01 Baseline Models)

(Ackerman & Marley 2001)

L dwarf T dwarf giantplanet

Mar 30, 2009 PHY 688, Lecture 27 32

Emergent Flux Depends on Wavelengthand Cloud Level

(Ackerman & Marley 2001)

τcloud < 0.5; hcloud > hphotosphere

τcloud > 1; hcloud ~ hphotosphere

silicate cloud(frain = 3)

τcloud > 1; hcloud < hphotosphere

Mar 30, 2009 PHY 688, Lecture 27 33

ModelingL and T Dwarfs

• Models that incorporate suspendeddust (DUSTY) successfullyreproduce L dwarf colors

• Late T dwarfs well fit by dust-freephotospheres (e.g., CONDmodels: dust removed uponformation)

• Transition can be explained bysedimentation of silicate cloudsbelow visible photosphere

(Baraffe et al. 2003)

DUSTY models(dust remainssuspended)

COND models(dust is removed)

L

M

T

Mar 30, 2009 PHY 688, Lecture 27 34

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 requireinefficient sedimentation(frain < 3)

• early T dwarfs require frain > 3• late T’s require no visible clouds

(frain → ∞)

(Burgasser et al. 2002)

frain = 3

f rain →

∞

Mar 30, 2009 PHY 688, Lecture 27 35

What Is the Weather onan Early T Dwarf?

• partly cloudy?

• uniformly hazy?

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

Mar 30, 2009 PHY 688, Lecture 27 36

Detecting Thermal Emission FromPlanet’s “Day” Side: Secondary Eclipse

Primary Eclipse

Secondary Eclipse

See thermal radiation from planet disappear and

reappear

See radiation from star transmittedThrough the planet’s atmosphere

Mar 30, 2009 PHY 688, Lecture 27 37

Effect of Irradiation

• balance between internal flux and flux incident from starTeff

4 = Tint4 + W T*

4

• W – dimensionless “dilution” factor ~ 10–3

• incident light penetrates to depth τpen, such that

• for τ < τpen, Teff is governed by irradiation and is constant– isothermal, radiative region

• for τ > τpen, Teff ≈ Tint, and rises monotonically with τ!

" pen =WT*

Tint

#

$ %

&

' (

4

)1

Mar 30, 2009 PHY 688, Lecture 27 38

P-T Profiles of Hot Jupiters

• isothermal regions are radiative

(Fortney et al. 2007)

AU

Mar 30, 2009 PHY 688, Lecture 27 39

Hot andVery Hot Jupiters:pL vs. pM Planets• distinction:

– based on lack or presenceof high-level TiO/VOassociated with astratosphere

– cf. L vs. M stellar spectraltypes

• transition at around0.04–0.05 AU equivalentseparation from the Sun

• note dependences on:– observed planetary

hemisphere– orbital phase for planets on

very eccentric orbits• HD 17156b, HD 80606b,

HD 147506b

(Fortney et al. 2008)

Mar 30, 2009 PHY 688, Lecture 27 40

Winds:Cooling vs.Advection

• advection time scaletadvec = Rp/U– Rp – planet radius– U – wind speed

• balance of cooling vs.advection decides windspeed U

• winds of several km/sec(~ sound speed) expectedfrom 2D and 3Ddynamical models

(Fortney et al. 2008)

U

!

"Tday–night

"Trad~ 1# e# tadvec / trad

Mar 30, 2009 PHY 688, Lecture 27 41

Radii of VeryHot Jupiters

• some large radii cannot beexplained by coreless planetmodels with high-altitudestratospheres:– 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 neutralhelium? (Fortney et al. 2007)

Mar 30, 2009 PHY 688, Lecture 27 42

Rossiter-McLaughlin Effect• first detected in

eclipsing binary stars– as in bottom panel

• effective Doppler shiftof (absorption) linechanges depending onthe part of the host starthat is occulted

(Gaudi & Winn 2007)

Mar 30, 2009 PHY 688, Lecture 27 43

RM Effect Geometry

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

(Ohta et al. 2005)

• note: ΩS sin IS here is the same as Vs sin Isand V sin Is in the following slides: theprojected stellar spin rate