Physics 688, From Brown Dwarfs to Giant Planets
Midterm Examination
April 1, 2009
Duration: 55 minutes
Closed notes. Calculators are necessary. You can answer questions on the exam paper
and diagrams. One-sentence answers suffice for most questions.
1.
(20 points) Hydrogen phase diagram (Burrows & Liebert 1993, RvMP, 65, 301).
X
T
a) (5 points) Explain qualitatively the different states of hydrogen in the interiors of
substellar objects as a function of temperature and pressure. In what hydrogen state is
the bulk of a brown dwarf’s mass
b) (5 points) What is the parameter Γ, and how does it affect the rate of
thermonuclear reactions in the cores of degenerate objects? Why? How does the T(ρ)
equation of state change with substellar mass and age?
c) (10 points) Starting from the polytropic equation of state P(ρ), derive the
numerical values of the exponent α in T(ρ) ∝ ρα for: (1) the interiors and for (2) the
atmospheres of brown dwarfs. Qualitatively, why does the polytropic index not
decrease from the interior to the atmosphere of a giant planet, as it does in higher
mass brown dwarfs?
2.
!
(20 points) Transit radius effect. Burrows et al. (2007, ApJ, 661, 502) note:
“Measuring a transit provides the impact parameter of the planet, not its photospheric
radius. This means that the planetary limb, through which the light from the star that
defines the depth of the transit emerges, is at a slightly larger distance from the
projected planet center than the canonical τ = 2/3 planetary radius.” Here you will
estimate the magnitude of this transit radius effect.
The wavelength-dependent optical depth, τch, along a chord followed by the
stellar beam through the planet’s upper atmosphere, is approximately
2%R &'R ch H
,
" ch ~ #$H
e
H
where κ is the wavelength-dependent opacity, ρ is the mass density of the
photosphere, ΔRch is the excess radius over and above the τ = 2/3 radius (the radius R
of the traditional photosphere), and H ≈ kT / (µgmp) is the atmospheric density scale
height, where k is the Boltzmann constant, µ is the mean molecular weight, g is the
surface gravity, and mp is the proton mass. By definition, and assuming an
exponential atmosphere, τ = κρH = 2/3.
2#R
. (Hint: make the same assumption for τch
H
as for the optical depth level τ of the traditional photosphere.)
a) (5 points) Show that "Rch = H ln
b) (10 points) Estimate ΔRch for a hot Jupiter (e.g., HD 189733b; M = 1.15 MJup, R =
!
1.15 RJup, T = 1120 K) and for a very hot Jupiter (e.g., HD 209458b; M = 0.64 MJup, R
= 1.32 RJup, T ≈ 2000 K). Assume 75% H2 (note: not neutral or ionized H atoms) and
25% He atmospheric mass composition for estimating µ. Use:
RJup = 7.1 ×109 cm
gJup = 2.31×103 cm s–2
k/mp = 8.26×107 erg K–1 g–1.
c) (5 points) What fraction of R is ΔRch in each case? Can the transit radius effect
alone explain the ~10% oversized appearance of HD 290458b?
3.
(15 points) Young and old brown dwarfs.
Na I
TiO
X
X
VO
X
TiO
X
CrH
TiO
FeH
H2O
TiO
TiO
VO
TiO
Y
Cs I
TiO
TiO
FeH
TiO
Rb I
Rb TiO
I
TiO
The above figure from McGovern et al. (2002) is used to qualitatively estimate the
surface gravities (and hence, ages) of two candidate brown dwarfs, σ Ori 47 and σ
Ori 51, detected in projection toward the young (~3 Myr) σ Orionis star forming
region. The insets compare one of the candidates, σ Ori 47, to ultra-cool dwarf
standards (G 196–3B and 2M 0632–01) of similar spectral types and known ages.
a) (5 points) σ Ori 47 and σ Ori 51 span an important spectral type transition. Which
is it: M/L, or L/T? What are the mystery absorbers X and Y? Which one of the two
objects has the later spectral type and is hence cooler? How did you conclude this?
b) (5 points) What are the relative surface gravities and ages (e.g., “low” vs. “high”
and correspondingly “young” vs. “old”) of the two standard objects? How do σ Ori
47 and σ Ori 51 compare? Hence, which one of the latter is likely to be a bona-fide
member of the young σ Orionis star-forming region? On what gravity-sensitive
spectroscopic features did you base your reasoning?
c) (5 points) What does the presence of Li I absorption at 6708 angstroms tell you
about the standard G 196–3B? Given your inference for the age of σ Ori 47 from
point (b) and the lack of detectable Li I absorption in its spectrum, what can you say
about its possible substellar nature? What about σ Ori 51?