Practice Final Exam Problems
Astro 497, Week 15, Wednesday
Transmission Spectroscopy
Q1a: Estimate the relative magnitude of the instantenous transmission spectroscopy signal for a hot-Jupiter orbiting a sun-like Star with orbital period of 4 days relative to a planet of the same size and composition and atmospheric structure with an orbital period of 8 days. Assume that the host stars have the same radius and effective temperature.
What sets the scale for transmission spectroscopy signals?
Which of those termswould be different for these two planets?
$$\Delta\delta(\lambda) = \frac{\pi (R_p + N_H(\lambda) H)^2}{\pi R_\star^2} - \frac{\pi R_p^2}{\pi R_\star^2} \simeq 2 N_H \delta \left(\frac{H}{R_p}\right)$$
$$H = \frac{k_B T}{\mu_m g}$$
$$H = \frac{k_B T}{\mu_m g}$$
The atmospheric scale height will differ due to different temperatures.
The temperatures will differ due to different incidenct flux and equilibrium temperatures.
$$\Delta\delta(\lambda) \propto H \propto T$$
$$\Delta\delta(\lambda) \propto H \propto T$$
$$T_{eq} \equiv T_{\star,eff} \sqrt{\frac{R_\star}{a}} \left(2^{-1/4}\right)$$
(assuming zero albedo, instantenous heat transport)
(assuming zero albedo, instantenous heat transport)
Convert from semi-major axis to orbital period
$$P^2 \propto a^3$$
$$\Delta\delta(\lambda) \propto T_{eq} \propto a^{-1/2} \propto P^{-1/3}$$
$$\frac{\Delta\delta(\lambda)_{HJ @ P=4d}}{\Delta\delta(\lambda)_{HJ @ P=8d}} = (8d/4d)^{1/3} = 2^{1/3} \simeq 1.26$$
$$P^2 \propto a^3$$
$$\Delta\delta(\lambda) \propto T_{eq} \propto a^{-1/2} \propto P^{-1/3}$$
$$\frac{\Delta\delta(\lambda)_{HJ @ P=4d}}{\Delta\delta(\lambda)_{HJ @ P=8d}} = (8d/4d)^{1/3} = 2^{1/3} \simeq 1.26$$
Representative values: Transmission Spectroscopy
| Planet | $\delta$ | $T$ | $g$ | $\mu_m$ | $\Delta\delta$ |
|---|---|---|---|---|---|
| (K) | (m/s) | (amu) | |||
| Hot-Jupiter | $\simeq 10^{-2}$ | $\simeq 1300$ | $\simeq 25$ | $\simeq 2$ | $\simeq 10^{-4}$ |
| Earth | $\simeq 10^{-4}$ | 273 | 10 | 28 | $\simeq 10^{-6}$ |
Occultation Spectroscopy
Q2a: Imaging you're trying to plan occultation spectroscopy of one of two hot-Jupiters (both with radius 1 $R_{\mathrm Jup}$) using JWST's mid-infrared instrument (MIRI) in its low-resolution spectrograph mode. Consider the same two planets from Q1, both orbiting a sun-like star (with the same radius and effective temperature). Estimate the relative magnitude of the instantenous occultation spectroscopy signal at 12μm for the planet with an orbital period of 4 days and relative to the planet with an orbital period of 8 days.
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.
$$\delta_{occ}(\lambda) = \frac{\pi R_p^2}{\pi R_\star^2} \frac{B_\lambda(T_p)}{B_\lambda(T_\star)}$$
Then integrate $\delta_{occ}(\lambda)$ over spectral bandpass observed.
Planck function for Blackbody radiation:
$$B_\lambda(T) = \frac{2hc^2}{\lambda^3} \frac{1}{e^{hc/(\lambda k_B T)}-1}$$
Do we need to use the full Planck function? Or are we in a regime that can be well-approximated more simply?
$$B_\lambda(T) = \frac{2hc^2}{\lambda^3} \frac{1}{e^{hc/(\lambda k_B T)}-1}$$
Do we need to use the full Planck function? Or are we in a regime that can be well-approximated more simply?
$$T_{eq} \equiv T_{\star,eff} \sqrt{\frac{R_\star}{a}} \left(2^{-1/4}\right)$$
(assuming zero albedo & instantenous heat transport)
(assuming zero albedo & instantenous heat transport)
begin
Teff = 5770 # K
R_sol_in_AU = 0.00465047 # AU
a_8days = (8/365.25)^(2/3) # AU
Teff_8d = Teff * sqrt(R_sol_in_AU/a_8days) / 2^(1/4) # K
a_4days = (4/365.25)^(2/3) # AU
Teff_4d = Teff * sqrt(R_sol_in_AU/a_4days) / 2^(1/4) # K
Teff_4d, Teff_8d
end(1489.9675689862063, 1182.5880432036022)
Bλ (generic function with 1 method)
let
h = 6.62607015e-34 # m^2 kg / s
c = 299_792_458 # m / s
k_B = 1.380649e−23 # m^2 kg s^-2 K^-1
λ_min = 12e-6 # m
λ_min/(h*c/(k_B*Teff_8d))
end0.9863278135983548
Rayleigh-Jeans limit
If $\lambda \gg \frac{hc}{k_B T}$, then
$$B_\lambda(T) \rightarrow \frac{2 k_B T}{\lambda^2}$$
and
$$\delta_{occ}(\lambda) \rightarrow \left(\frac{R_p}{R_\star}\right)^2 \frac{T_p}{T_\star}$$
$$\Delta\delta(\lambda) \propto T_{eq} \propto a^{-1/2}$$
Convert from semi-major axis to orbital period
$$P^2 \propto a^3$$
$$\Delta\delta(\lambda) \propto P^{-1/3}$$
$$\frac{\Delta\delta(\lambda)_{HJ @ P=4d}}{\Delta\delta(\lambda)_{HJ @ P=8d}} = \left(\frac{8\mathrm{d}}{4\mathrm{d}}\right)^{\frac{1}{3}} \simeq 1.26$$
Convert from semi-major axis to orbital period
$$P^2 \propto a^3$$
$$\Delta\delta(\lambda) \propto P^{-1/3}$$
$$\frac{\Delta\delta(\lambda)_{HJ @ P=4d}}{\Delta\delta(\lambda)_{HJ @ P=8d}} = \left(\frac{8\mathrm{d}}{4\mathrm{d}}\right)^{\frac{1}{3}} \simeq 1.26$$
What if we didnn't use Rayleigh-Jean's limit?
Bλ(Teff_4d, λ=[12e-6]) ./ Bλ(Teff_8d, λ=[12e-6])1-element Vector{Float64}:
1.4208573577725372
Q2b: If one also were to account for the effects of tides on the two planets, would you expect the estimate above to over or underestaimte the true ratio of emission spectroscopy signals (for 4d planet relative to 8d planet)?
What would we expect to be different about the two planets due to tides?
P = 4d planet:
The $P$ = 4d planet is likely to have its rotation period synchronized with its orbital period.
Therefore, one side is constantly illuminated and the other size is always experiencing night.
If heat transport is inefficient, then there could be a substantial difference in the temperature on the day and night sides.
P = 8d planet
The $P$ = 8d planet is not expected to be have its rotation period synchronized with its orbital period.
Even if heat transport is inefficient, the day-night cycle will suppress longitudinal temperature variations
Consider two extrems:
Entire planet at one temperature (limit for 8d planet)
The dark side of the 4d planet is at 0K and radiates no heat.
If only half of the planet's surface area is avaliable to radiate heat, then equilibrium temperature will increase to maintain balance with incoming radiation.
This effect causes an observer to measure more thermal emission from the 4d planet that without tides.
Ignoring tides would cause us to underestimate the ratio of the thermal emission from the 4d planet relative to the 8d planet.
Entire planet at one temperature (limit for 8d planet)
The dark side of the 4d planet is at 0K and radiates no heat.
If only half of the planet's surface area is avaliable to radiate heat, then equilibrium temperature will increase to maintain balance with incoming radiation.
This effect causes an observer to measure more thermal emission from the 4d planet that without tides.
Ignoring tides would cause us to underestimate the ratio of the thermal emission from the 4d planet relative to the 8d planet.
Q2c: Provide an upper bound on how strong this effect could be.
Since $P_{bb} = A \sigma T^4$, the most the temperature could increase is by a factor of $2^{1/4} \simeq 1.19$. Therefore, we can place a bound on the transit depth ratio allowing for effects of tidal circularization of 20% higher than our previous calculation.
Representative values: Occultations
| Planet | $\delta$ | $T$ | $\Delta\delta_{occ}$ |
|---|---|---|---|
| (K) | |||
| Hot-Jupiter | $\simeq 10^{-2}$ | $\simeq 1300$ | $\simeq 2\times 10^{-3}$ |
| Earth | $\simeq 10^{-4}$ | 273 | $\simeq 5 \times 10^{-6}$ |
Atmospheric Scale Height
Consider an Earth-mass, Earth-radius planet in the habitable zome of a sun-like star with unknown atmospheric composition. Imagine you wnat to planning observations ot search for evidence of water in its atmosphere.
Q3a: Evaluate the ratio of the atmospheric scale height for a planet with an atmosphere of pure molecular hydrogen to the atmospheric scale height for a planet with an atmosphere of pure water vapor.
$$H = \frac{k_B T}{\mu_m g}$$
What variables affect the result?
What values to use?
$$\mu_{H_2} \simeq 2$$
$$\mu_{H_2O} \simeq 18$$
$$\mu_{H_2O} \simeq 18$$
$$ H_{H_2} / H_{H_2O} \simeq \frac{18}{2} \simeq 9$$
Q3b: Evaluate the ratio of the atmospheric scale height for the same planet with an atmosphere of pure hydrogen to atmospheric scale height for the same planet with an atmosphere of 79% molecular hydrogen 20% helium and 1% water vapor.
Need to compute μ_mix = 0.79 * 2 + 0.2 * 4 + 0.01*18 ≃ 2.56
$$\frac{H_{H_2}}{H_{\mathrm{mix}}} \simeq \frac{μ_{\mathrm{mix}}}{\mu_{H_2}} \simeq 1.3$$
Disk lifetimes
Q4: Imagine you are reviewing a proposal to search for protoplanetary disks around sun-like stars in the Hyades star cluster, a cluster with ~300 stars with an age of ≃ 600Myr. Identify the main challenge to the success of this proposal. Would you recommend awarding this proposal telescope time? Why or why not?
$$f_{\mathrm{disk}} = \exp(-t/\tau_{\mathrm{disk}})$$
$$\tau_{\mathrm{disk}} = 2.5 \; \mathrm{Myr}$$
$$\tau_{\mathrm{disk}} = 2.5 \; \mathrm{Myr}$$
exp(-600/2.5)5.8792826982452694e-105
Future Transit Surveys
Q5: Evaluate the relative photometric precision of photometry of the host star in the Kepler-11 system for the PLATO mission relative to the Kepler mission. Kepler-11 has an apparent magnitude of 14. You may assume that Kepler-11 is not observered by PLATO's fast cameras.
Precission is/will be photon limited for 14th magnitude targets.
Kepler: 0.95m aperture
PLATO: 26x 120mm cameras: 4 groups of 6 cameras + 2 fast cameras
$$A_{Kepler} = π 0.95^2$$
$$A_{Plato} = 6* π 0.12^2$$
$$A_{ratio} = A_{Plato} / A_{Kepler}$$
$$A_{Plato} = 6* π 0.12^2$$
$$A_{ratio} = A_{Plato} / A_{Kepler}$$
$$\frac{\sigma_{Plato}}{\sigma_{Kepler}} = \left(\frac{A_{Kepler}}{A_{Plato}}\right)^{1/2} \simeq 3.2$$
3.2319656328389152
Setup
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