Emission Spectroscopy

Astro 497, Week 10, Friday

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–- Credit: Fig 1 from Winn chapter of Exoplanets textbook

–- Kreidberg, chapter from Handbook of Exoplanets

Example Emission Spectrum

Start with timeseries

–- Mansfield et al. CC BY 4.0

Reduce to single emission spectrum

–- Mansfield et al. CC BY 4.0

What sets the scale for emission spectroscopy signal?

Increase in observed flux outside of transit

$$f_{\mathrm{out-of-occultation}} \simeq \pi R_\star^2 B_\lambda(T_\star) + \pi R_p^2 B_\lambda(T_{pl})$$

$$f_{\mathrm{during-occultation}} \simeq \pi R_\star^2 B_\lambda(T_\star)$$

Measurement wavelength: $\lambda$
Surface temperature of star ($T_\star$) and planet ($T_pl$)
Radius of Planet: $R_p$ (where optically think at all wavelengths observed)
Radius of Star: $R_\star$

Planck function for Blackbody radiation

$$B_\lambda(T) = \frac{2hc^2}{\lambda^3} \frac{1}{e^{hc/(\lambda k_B T)}-1}$$

If both star and planet have uniform surface brightness, then

$$\delta_{occ}(\lambda) = \frac{\pi R_p^2}{\pi R_\star^2} \frac{B_\lambda(T_p)}{B_\lambda(T_\star)}$$

Then integrate $\delta_{occ}(\lambda)$ over spectral bandpass observed.

function Bλ(T::Real; λ::AbstractArray{T1} = range(0.5, stop=10, length=1000) ) where { T1 <: Real }
    h = 6.62607015e−34
    c =	299792458
    k = 1.380649e−23
    (2*h*c^2)./(λ.^5 .*(exp.((h*c/(k*T))./λ).-1))
end

Bλ (generic function with 1 method)

let
    λ = range(0.5, stop=10, length=1000)
    plt = plot(xlabel="λ (μm)", ylabel="Spectral Energy Density", widen=false)
    if logscalex plot!(plt,xscale=:log10) end
    if logscaley plot!(plt,yscale=:log10) end	
    plot!(plt,λ,Bλ(1000; λ=λ.*1e-6), label="T = 1000 K")
    plot!(plt,λ,Bλ(1200; λ=λ.*1e-6), label="T = 1200 K")
    plot!(plt,λ,Bλ(1400; λ=λ.*1e-6), label="T = 1400 K")
    plot!(plt,λ,Bλ(1600; λ=λ.*1e-6), label="T = 1600 K")
end

Log scale: X Y

Rayleigh-Jeans limit

If $\lambda \gg \frac{hc}{k_B T}$, then

$$B_\lambda(T) \rightarrow \frac{2 k_B T}{\lambda^2}$$

and

$$\delta_{occ}(\lambda) \rightarrow \left(\frac{R_p}{R_\star}\right)^2 \frac{T_p}{T_\star}$$

Representative values

Planet	$\delta$	$T$	$\Delta\delta_{occ}$
		(K)
Hot-Jupiter	$\simeq 10^{-2}$	$\simeq 1300$	$\simeq 2\times 10^{-3}$
Earth	$\simeq 10^{-4}$	273	$\simeq 5 \times 10^{-6}$

What could we detect?

–- Mansfield et al. CC BY 4.0

question(md"How significant is the error when approximating blackbody temperature with brightness temperature?")

Question

How significant is the error when approximating blackbody temperature with brightness temperature?

–- Credit: Natasha Batalha & PICASO manual

question(md"In a lot of the emission spectra, the wavelength range is 1-10 microns. 
Are there any important elements that can be detected that do not lie in this range?")

Question

In a lot of the emission spectra, the wavelength range is 1-10 microns. Are there any important elements that can be detected that do not lie in this range?

Comparison to models

–- Mansfield et al. CC BY 4.0

#md"""#### Things to look for"""
question(md"Is occultation spectroscopy a good way to find out the type of planet we are looking at?")

Question

Is occultation spectroscopy a good way to find out the type of planet we are looking at?

question(md"Does the composition of clouds impact what sort of reflected light is seen?")

Question

Does the composition of clouds impact what sort of reflected light is seen?

How to prioritize planets for detailed atmospheric characterization?

Emission spectroscopy metric

$$\begin{eqnarray} \mathrm{ESM} & = & 4.29 \times 10^6 \times \left(\frac{R_p}{R_\oplus}\right)^2 \left(\frac{B_{7.5}(T_{day})}{B_{7.5}(T_\star)}\right) \times 10^{-m_K/5} \\ \end{eqnarray}$$

Planck's function for temperature $T$ evaluated at 7.5μm ($B_{7.5}(T))$
Planet's dayside temperature ($T_{day}$)
Apparent magnitude of host star in K band ($m_K$), since that's what's commonly available
Proportional to SNR for emissions spectroscopy measurement at mid-IR wavelengths
MIRI LRS bandpass is ~5–10 μm
Assumes blackbody emission model for star and planet
Planet emission can differ by over an order of magnitude due to molecular absoprtion bands (and opacity low windows between them)
Sometimes approximate $T_{day} \simeq 1.1 T_{eq}$ based on models that must assume heat redistristribution factor (0.53), albedo (0.3), etc.

–- Kempton et al. (2018)

question(md"""
Are there specific types of stars for which occultation spectroscopy is better suited/more successful? 
""")

Question

Are there specific types of stars for which occultation spectroscopy is better suited/more successful?

Example/Reference case

GJ 1132b

$$R_p/R_\star = 0.0455$$

$$a/R_\star = 16.54$$

$$T_\star = 3270\;\mathrm{K}$$

$$ESM = 7.5$$

–- Mansfield et al. CC BY 4.0

question(md"""Are there any drawbacks to emission spectroscopy?""")

Question

Are there any drawbacks to emission spectroscopy?

question(md""" "Occultation spectroscopy and transit spectroscopy thereby provide different and complementary information about the planetary atmosphere."

What exact different information can these two spectroscopy  provide?""")

Question

"Occultation spectroscopy and transit spectroscopy thereby provide different and complementary information about the planetary atmosphere."

What exact different information can these two spectroscopy provide?

Reading Questions

Setup & Helper Code

ChooseDisplayMode()

Full Width Mode Present Mode

using PlutoUI, PlutoTeachingTools

using Plots, ColorSchemes

question(str; invite="Question") = Markdown.MD(Markdown.Admonition("tip", invite, [str]))

question (generic function with 1 method)

Built with Julia 1.8.2 and

ColorSchemes 3.19.0
Plots 1.35.5
PlutoTeachingTools 0.2.3
PlutoUI 0.7.44

To run this tutorial locally, download this file and open it with Pluto.jl.