Fundamental (Sub)stellar Parameters:
Masses and Radii
PHY 688, Lecture 10
Outline
• Review of previous lecture
– brown dwarf effective temperatures
– finding cool brown dwarfs
– current problem: what are the coolest brown dwarfs?
• (Sub)stellar masses and radii
– binary stars
• Current astrophysics problem:
– dynamical mass and radius determination of brown
dwarf binaries
Feb 16, 2009
PHY 688, Lecture 10
2
Previously in PHY 688…
Feb 16, 2009
PHY 688, Lecture 10
3
(Sub)stellar Effective Temperature
% d2
Teff = '' 2
& R*
•
!
•
•
Feb 16, 2009
(Cushing
et al. 2006; Marley & Leggett 2008)
PHY 688, Lecture 10
#
14
(
f " d"
*
$ *)
fλ – recorded
spectrum
d – trigonometric
parallax distance
R* – (sub)stellar
radius (for brown
dwarfs R* ≈ 0.1 RSun)
4
Teff’s of L and T Dwarfs
Feb 16, 2009
PHY 688, Lecture 10
(Kirkpatrick 2005)
5
Teff’s without Spectra
• estimate SpT
from color(s):
e.g., z – J
• use SpT-specific
bolometric
correction (BC)
to get total flux
z
• get Teff if distance d is known
J
Feb 16, 2009
(Cushing
et al. 2006; Marley & Leggett 2008)
PHY 688, Lecture 10
6
Bolometric Corrections
BCλ(SpT) = Mbol(SpT) – Mλ
•
Mbol(SpT) is known from the few
objects of given SpT with nearcomplete SEDs
BC(Sun) ≡ 0 mag
at all wavelengths λ
Feb 16, 2009
PHY 688, Lecture 10
(Golimowski et al. 2004)
7
Searching for Cool Brown Dwarfs:
T’s and Y’s
Y?
L
T0–T4
T5–T8
SDSS
Y?
Feb 16, 2009
PHY 688, Lecture 10
(Fan et al. 2001)
8
Possible NH3 at T8.5–T9
The First Y0’s?
KS (2.15 µm)
(T8)
(T8)
CFBDS 0059
(T8.5)
New T9 dwarf from CFBDS
(T9)
Teff ≈ 600 K: coolest brown dwarf to date
Feb 16, 2009
PHY 688,
Lecture
(Delorme
et al.
2008)10
9
• UKDISS
Large-Area Sky Surveys
of the Near Future
UKIRT
– 0.95–2.4 µm, 18% sky
– K = 18.4 mag
• cf. K = 14.3 for 2MASS
– UKIRT (3 m, ground-based)
– 2005–2011
• WISE
–
–
–
–
3–25 µm, 99% sky
0.4 m, space-based
November 2009 launch
8-month lifetime
LSST
• PanSTARRS / LSST
– visible, ~75% sky, synoptic
– 4 x 1.8 m / 8.4 m ground-based
– 2009–? / 2015
Feb 16, 2009
PHY 688, Lecture 10
10
Outline
• Review of previous lecture
– brown dwarf effective temperatures
– finding cool brown dwarfs
– current problem: what are the coolest brown dwarfs?
• (Sub)stellar masses and radii
– binary stars
• Current astrophysics problem:
– dynamical mass and radius determination of brown
dwarf binaries
Feb 16, 2009
PHY 688, Lecture 10
11
Mass
• most fundamental
of stellar parameters
– L ∝ M3.8
– τMS ≈
1010 yr (M/MSun)–2.8
• impossible to
measure for isolated
stars
Feb 16, 2009
PHY 688, Lecture 10
12
Dynamical Masses:
Binary Stars to the Rescue
• ~ 1/3 of stars are binaries
• ~ 50% of Sun-like (~ 1 MSun) stars are binaries
Feb 16, 2009
PHY 688, Lecture 10
13
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 16, 2009
PHY 688, Lecture 10
14
Visual
Binary
• GJ 569Bab
binary
brown dwarf
• a > 5–10AU
(Lane et al. 2001)
Feb 16, 2009
PHY 688, Lecture 10
15
First Dynamical Mass of T Dwarf
(Visual) Binary: 2MASS J1534–2952AB
Feb 16, 2009
PHY 688, Lecture 10
(Liu et al., 2008)
16
Astrometric Binary
Feb 16, 2009
PHY 688, Lecture 10
17
Astrometric Binary: Sirius AB
• Sirius A:
– nearby luminous B star
– brightest star in the sky
• ~1 MSun white dwarf
companion first
inferred from its large
astrometric effect on
primary
• now also a visual
binary
Feb 16, 2009
B
Hubble Space Telescope image
PHY 688, Lecture 10
18
Astrometric Binary: GJ 802AB
• unseen
brown dwarf
companion
• a > 0.5–2AU
(Pravdo et al. 2005)
Feb 16, 2009
PHY 688, Lecture 10
19
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 16, 2009
PHY 688, Lecture 10
20
Radial Velocity vs. Time for Doublelined SB in a Circular Orbit
Feb 16, 2009
PHY 688, Lecture 10
21
Radial Velocity vs. Time for Doublelined SB in Elliptical Orbit (e = 0.4)
Feb 16, 2009
PHY 688, Lecture 10
22
Spectroscopic Binary: 51 Peg Ab
(SB1)
• first planet detected
around a mainsequence star
– primary SpT: G2 V
• Mp sin i = 0.47 MJup
(Mayor & Queloz 1995)
Feb 16, 2009
PHY 688, Lecture 10
23
Visual + Spectroscopic
Binary Brown Dwarf: Gl 569Bab
•
•
Feb 16, 2009
first BD dynamical mass
Mtot = 0.l25 ± 0.007 MSun
PHY 688, Lecture 10 (Lane et al. 2001; Simon et al. 2006) 24
Totally Eclipsing Binaries
Feb 16, 2009
PHY 688, Lecture 10
25
Totally Eclipsing Binaries
ta – start of secondary ingress
tb – end of secondary ingress
tc – start of secondary egress
td – end of secondary egress
Feb 16, 2009
PHY 688, Lecture 10
26
Dynamical Mass Determination
– If orbital major axes (relative to center of mass) or radial
velocity amplitudes are known, so is the ratio of masses:
m1 a2 v2 r
=
=
m2 a1 v1r
– If the period, P, and the sum of major axis lengths,
a = a1 + a2 , are known, Kepler’s third law can give
masses separately:
12
2
#
&
4"
P =%
a3 (
$ G(m1 + m2 ) '
Feb 16, 2009
PHY 688, Lecture 10
27
Dynamical Mass Determination
– If only the two radial velocities are known (SB2), the
sum of masses (from Kepler’s third law) is:
3
P ! v1r + v2 r $
m1 + m2 =
#
&
2'G " sin i %
– If only one radial velocity is known (SB1), a useful
quantity is the mass function:
3
3
v1 P ( m2 sin i)
f (m1,m2 ) =
=
2"G ( m1 + m2 ) 2
– If orientation angle of orbit, i, is known, this allows
separate determination of the mass(es)
!
Feb 16, 2009
PHY 688, Lecture 10
28
Other Uses for Totally Eclipsing
Binary Systems: Radii and Teff’s
• Duration of eclipses and shape of light curve can be used to
determine radii of stars:
v1 + v2
=
(t2 ! t1 )
2
v1 + v2
Rl =
(t3 ! t1 )
2
(radius of R
s
secondary)
(radius of
primary)
t1 – start of secondary ingress
t2 – end of secondary ingress
t3 – start of secondary egress
Relative depth of primary (deepest) and secondary brightness
minima of eclipses can be used to determine the ratio of effective
temperatures of the stars:
F0 ! Fprimary
F0 ! Fsecondary
Feb 16, 2009
" Te ,s
=$
$ Te ,l
&
PHY 688, Lecture 10
#
%%
'
4
.
29
Luminosity-Mass Relation for Stars
with Well-determined Orbits
Feb 16, 2009
PHY 688, Lecture 10
(Popper 1980)
30
Radius-Mass Relation for Binary
Stars with Well-Determined Orbits
100
Radius (R e )
10
1
Detached binaries
Semidetached/contact binaries
0.1
0.1
Feb 16, 2009
1
10
Mass ( M e )
PHY 688, Lecture 10
100
(Malkov 1993)
31
Temperature-Mass Relation for Stars
with Well-determined Orbits
50000
Detached binaries
Temperature (K)
40000
Semidetached/contact binaries
30000
20000
10000
0
0.1
1
10
Mass ( M e )
Feb 16, 2009
PHY 688, Lecture 10
100
(Malkov 1993)
32
Binary Separations
Feb 16, 2009
PHY 688, Lecture 10
(Reid & Metchev 2007)
33
Binary Mass Ratio Distribution
Feb 16, 2009
PHY 688, Lecture 10
(Reid & Metchev 2007)
34
Outline
• Review of previous lecture
– brown dwarf effective temperatures
– finding cool brown dwarfs
– current problem: what are the coolest brown dwarfs?
• (Sub)stellar masses and radii
– binary stars
• Current astrophysics problem:
– dynamical mass and radius determination of brown
dwarf binaries
Feb 16, 2009
PHY 688, Lecture 10
35
First Determination of
Substellar Radii
Feb 16, 2009
PHY 688, Lecture 10
(Stassun et al., 2005)
36
First Determination of
Substellar Radii
Feb 16, 2009
PHY 688, Lecture 10
(Stassun et al., 2005)
37
First Determination of
Substellar Radii
Feb 16, 2009
PHY 688, Lecture 10
(Stassun et al., 2005)
38