Fundamental (Sub)stellar Parameters:
Surface Gravity
PHY 688, Lecture 11
Outline
• Review of previous lecture
– binary stars and brown dwarfs
– (sub)stellar dynamical masses and radii
• Surface gravity
– stars, brown dwarfs, and giant planets
– determining model-dependent masses
• Curve of growth for absorption lines
– determining photospheric abundances
Feb 18, 2009
PHY 688, Lecture 11
2
Previously in PHY 688…
Feb 18, 2009
PHY 688, Lecture 11
3
Mass
• most fundamental
of stellar parameters
– L ∝ M3.8
– τMS ≈
1010 yr (M/MSun)–2.8
• impossible to
measure for isolated
stars
Feb 18, 2009
PHY 688, Lecture 11
4
Dynamical Masses:
Binary Stars to the Rescue
• 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.)
Feb 18, 2009
PHY 688, Lecture 11
5
Visual Binary: GJ 569Bab
• first with a dynamical
mass
• measure: P, a, i
(+ a1, a2 if
independent
astrometric reference
exists)
• determine: Mtot
(+ M1, M2)
Feb 18, 2009
(Lane
etLecture
al. 2001)
PHY 688,
11
• a > 5–10AU
6
Astrometric Binary: GJ 802AB
•
unseen brown
dwarf com-panion;
first and only to be
discovered
astrometrically
•
measure: P, a1, i
(using independent
astrometric
reference)
•
determine: M1
(a2, M2 can be
constrained from
resolved imaging)
•
a > 0.5–2AU
(Pravdo et al. 2005)
Feb 18, 2009
PHY 688, Lecture 11
7
Spectroscopic
Binary
(a)
• double-lined (SB2)
– spectra of both stars visible
(d)
(a)
(b)
(b)
(c)
(c)
(d)
(d)
• single-lined (SB1)
– only spectrum of brighter star visible
Feb 18, 2009
PHY 688, Lecture 11
8
Radial Velocity vs. Time for an SB2
in a Circular Orbit
• measure: P, v1, v2
• determine: a1 sin i, a2 sin i, M1 sin i, M2 sin i
Feb 18, 2009
PHY 688, Lecture 11
9
SB1 Spectroscopic Binary: 51 Peg Ab
• first planet detected
around a mainsequence star
– primary SpT: G2 V
• Mp sin i = 0.47 MJup
• 0 AU < a < 10 AU
(Mayor & Queloz 1995)
• measure: P, v1
• determine: a sin i, M2 sin i (if M1 approximately known)
Feb 18, 2009
PHY 688, Lecture 11
10
Totally Eclipsing Binaries
(Are Also SB1’s or SB2’s)
ta – start of secondary ingress
tb – end of secondary ingress
tc – start of secondary egress
td – end of secondary egress
• measure: P, v1, i, ∆F1, ∆F2 (+ v2 if SB2)
• determine: a, M1, M2, R1, R2, ratio Teff,1/Teff,2
M1,
Feb 18,–2009
M2 determined exactlyPHY
if SB2;
otherwise,
only ratio is known
688, Lecture
11
11
First Determination of Substellar
Radii: 2MASS 0535–0546 A/B
Feb 18, 2009
PHY 688, Lecture 11
(Stassun et al., 2005)
12
Luminosity-Mass Relation for Stars
with Well-determined Orbits
similar relations
for radius and
Teff dependence
on mass
Feb 18, 2009
PHY 688, Lecture 11
(Popper 1980)
13
Outline
• Review of previous lecture
– binary stars and brown dwarfs
– (sub)stellar dynamical masses and radii
• Surface gravity
– stars, brown dwarfs, and giant planets
– determining model-dependent masses
• Curve of growth for absorption lines
– determining photospheric abundances
Feb 18, 2009
PHY 688, Lecture 11
14
Given Masses and Radii, Estimate
Densities, Surface Gravities
• Sun
MSun = 2.0 "10 33 g
RSun = 7.0 "1010 cm
#Sun = 1.4 g/cm3
log g = GM /R 2 = 4.44 [cgs]
image credit: SOHO (ESA + NASA)
Feb 18, 2009
PHY 688, Lecture 11
15
Given Masses and Radii, Estimate
Densities, Surface Gravities
• Betelgeuse (M2 I)
M " 10MSun
R " 1000RSun
# " 10$8 #Sun
" 1.4 %10$8 g/cm3
log g " $0.6
Feb 18, 2009
PHY 688, Lecture 11
16
Given Masses and Radii, Estimate
Densities, Surface Gravities
• Sirius B (white dwarf)
M " 0.6MSun
R " 0.01RSun
# " 6 $10 5 #Sun
" 8 $10 5 g/cm3
log g " 8
B
credit: Hubble Space Telescope (NASA)
Feb 18, 2009
PHY 688, Lecture 11
17
Given Masses and Radii, Estimate
Densities, Surface Gravities
• Gl 229B (T6.5)
M " 0.03MSun
R " 0.1RSun
# " 30 #Sun
" 40 g/cm3
log g " 5
Feb 18, 2009
PHY 688, Lecture 11
18
Given Masses and Radii, Estimate
Densities, Surface Gravities
• 2MASS 0535–0546B
– secondary of first eclipsing substellar binary
M = 0.034 MSun
R = 0.51RSun
" = 0.26 "Sun
= 0.36 g/cm3
log g = 3.6
Feb 18, 2009
PHY 688, Lecture 11
19
Given Masses and Radii, Estimate
Densities, Surface Gravities
• Jupiter
M = 0.95 "10#3 MSun
R = 0.10RSun
$ = 0.88 $Sun
= 1.25 g/cm3
log g = 3.4
Feb 18, 2009
PHY 688, Lecture 11
20
2M 0535–05A
(0.054 MSun)
At Constant Mass Younger Brown
Dwarfs Have Lower Gravities
2MASS 0535–0546B (0.034 MSun)
stars
brown dwarfs
“planets”
Gl 229B
(~0.03 MSun)
Feb 18, 2009
PHY 688, Lecture 11
(Burrows et al. 2001) 21
At Constant Teff Younger Brown Dwarfs
Are Less Massive, Have Lower Gravities
stars
brown dwarfs
“planets”
2MASS 0535–0546 A/B
M
13
10
M
Ju
p
M
Ju
p
5M
Jup
1M
stars
brown dwarfs
“planets”
Gl 229B
Jup
Jupiter
Feb 18, 2009
(Burrows et al. 2001)
PHY 688, Lecture 11
22
At Constant Teff, Younger Brown Dwarfs
Have Lower
Gravities
Gl 229B
2MASS 0535–0546 A/B
Jupiter
log g vs. Teff for brown
dwarfs and planets
Feb 18, 2009
PHY 688, Lecture 11
(Burrows et al. 1997)
23
Luminosity (i.e., Surface Gravity)
Effects at A0
(figure: D. Gray)
Feb 18, 2009
PHY 688, Lecture 11
24
From Lecture 5: Line Profiles
•
Natural line width (Lorentzian [a.k.a., Cauchy] profile)
I" = I0
– Heisenberg uncertainty principle: ∆ν =∆E/h
•
Collisional broadening (Lorentzian profile)
# & Lorentzian FWHM
– collisions interrupt photon emission process
– ∆tcoll < ∆temission ~ 10–9 s
– dependent on T, ρ
•
Pressure broadening (~ Lorentzian profile)
#E i + #E f
1
1
" natural =
=
+
h /2$
#t i #t f
!
" collisional = 2 #t coll
– ∆tinteraction > ∆temission
" pressure % r
– nearby particles shift energy levels of emitting particle
• Stark effect (n = 2, 4)
• van der Waals force (n = 6)
cool stars
• dipole coupling between pairs of same species
!(n = 3)
&n
; n = 2,3,4,6
(" % " 0 ) 2
%
1
2
I" =
e 2$
2# $
$ & Gaussian FWHM
– dependent mostly on ρ, less on T
•
# /2$
2
(" % " 0 ) + # 2 /4
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
•FebComposite
line profile: Lorentzian
Gaussian
18, 2009
PHY+688,
Lecture 11= Voigt profile
!
kT
mc 2
"rotational = 2# 0 u /c
"thermal = # 0
25
Feb 18, 2009
PHY 688, Lecture 11
(Kleinmann & Hall 1986) 26
Gravity-Sensitive Features in UCDs
Feb 18, 2009
PHY 688, Lecture 11
(McGovern et al. 2004) 27
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
Feb 18, 2009
(Kirkpatrick et al. 2006) PHY 688, Lecture 11
Wavelength (µm)
28
Example: HR8799bcd – Do the
“Planets” Have Planetary Masses?
Keck AO image of the HR 8799bcd planetary system
(Marois et al. 2008, Science)
Feb 18, 2009
PHY 688, Lecture 11
29
Masses of HR8799bcd
Gl 229B
2MASS 0535–0546 A/B
Jupiter
Can use log g and Teff
to infer substellar mass
Feb 18, 2009
PHY 688, Lecture 11
(Burrows et al. 1997)
30