### HW11

```HW 11
Astronomy 4810/5010, Fall 2014
Due Tuesday Nov 11
1) (U and G) An analytic model of line and continuum flux from a stellar atmosphere. Consider
a stellar atmosphere with density profile ρ(z) = ρ0 e−z/H , where z is altitude above some reference
point, H is the pressure scale height, and ρ0 is the density at the reference point. Let the opacity
be independent of height, but depend on frequency as
κν
= κc + κl e−(ν−ν0 )
2 /∆ν 2
.
(1)
Here κc is the (constant) continuum opacity, and the second term represents a line with maximum
opacity κl ≫ κc , centered at ν = ν0 , and with width ∆ν ≪ ν0 . Let the temperature profile in the
atmosphere be linear in altitude with the form
T (z) = T0 +
dT
z,
dz
(2)
where T0 is the temperature at z = 0, and dT /dz is the temperature gradient.
The (“Eddington-Barbier”) model for the emitted intensity is Iν ≃ Bν (Tν ), where Tν is the temperature at the altitude where frequency ν has τν = 1. Since the opacity is different at different
frequencies, they will have τν = 1 at different altitudes, where the temperature is different, leading
to different values of intensity Bν .
a) Compute the optical depth τν (z) by integrating from altitude z out to infinity.
b) Solve for the altitude zν where τν = 1.
c) Plug the result from part (b) in to find Tν = T (zν ).
d) Find the values of the “continuum altitude” and “continuum temperature” where the
continuum flux is emitted from (assume the frequency is far enough from line center that line
opacity is negligible to do this part).
e) Find how much the temperature drops at the center of the line as compared to the continuum. Be sure to evaluate this part using κl ≫ κc .
f) Taylor expand the blackbody intensity in terms of temperature, and then use the result from
part (e) to derive the fraction decrease in intensity, δIν /Iν , at line center as compared to the
continuum. Express your answer in terms of the three quantities κl /κc ≫ 1, ∇ = d ln T /d ln P
the temperature gradient, and hν/kT .
g) For a given ∇ and κl /κc , is δIν /Iν largest on the Rayleigh-Jeans or Wein tails?
2) (U and G) Probing the atmospheres of extrasolar planets using transmission spectroscopy. A
star of radius Rs a distance d from Earth has a planet orbiting around it. The orbit is edge on, and
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we see the planet pass in front of the star (“transit”). Let each part of the star emit with intensity
Iν (0). Our goal is to understand the drop in flux at each wavelength when the planet is directly in
front, and centered on, the star.
a) Compute the specific flux, measured at Earth, due to the star alone, when the planet is
not blocking any light.
b) Now put the planet in front of the star. Let’s first assume that the planet is an opaque
disk of radius Rp at all frequencies. Compute δFν /Fν , the fractional drop in flux due to the
planet blocking the star. If Rp = Rs /10, what is the size of this drop in flux?
c) Now consider a thin atmosphere of extent z outside the opaque disk of the planet. That
is, the atmosphere extends from radius r = Rp to radius r = Rp + z, where z ≪ Rp . If
this atmosphere was opaque at all frequencies, what would be the additional drop in flux, on
top of that from the opaque disk of the planet. How big is the signal from the atmosphere
compared to the planet, if z = Rp /10?
d) Now consider that the atmosphere is not perfectly opaque. Let the density profile be
ρ(z) = ρ0 e−z/H , where z is altitude, H is the pressure scale height, and ρ0 is the density at
the reference point. Let the opacity κν depend on frequency, but be independent of altitude.
Let s denote the distance along the trajectory, with s =
√ 0 at the closest point. For impact
parameter b, the radius along the trajectory is then r = b2 + s2 ≃ b + s2 /2b, and the density
2
profile along the trajectory is is ρ = ρ(b)e−s /2bH . Integrate over −∞ ≤ s ≤ ∞ to find the
optical depth τν (b) for the trajectory at impact parameter b. Express your answer in terms
of ρ(b), κν , b and H.
e) Given the optical depth in part (d), the drop in intensity along a sightline with impact
parameter b is Iν (0)(1 − e−τν (b) ). We now want to integrate over all sightlines through the
atmosphere in order to find out how much flux was taken away. We will use the plane parallel
limit in which the integral over perpendicular area 2πbdb ≃ 2πRp dz, and the density at the
impact parameter b = Rp + z is ρ(b) = ρ0 e−z/H . In the plane parallel limit (H ≪ Rp ), the
atmosphere extends from z = 0 to z = ∞. The drop in flux is then
Z ∞ 2πRp
−τν (z)
δFν ⌋atmosphere =
dz
1
−
e
.
(3)
I
(0)
ν
d2
0
Compute this integral, and by comparing to part (c), find the effective altitude zν for each
frequency ν. Express your answer in terms of H, ρ0 , κν and Rp . Note: you may find an
integral of the form
Z ∞ −u
du 1 − e−βe
≃ γ + ln(β) + ...,
(4)
0
where γ ≃ 0.577 is Euler’s constant and this formula is valid for β ≫ 1.
f) If the continuum opacity is κc and the line opacity is κl ≫ κc , compute zl −zc , the difference
in altitude (radius) blocked at each frequency.
2
```