Copyright © Michael Richmond.
This work is licensed under a Creative Commons License.
One of the most exciting developments in exoplanet science over the past decade has been the search for evidence of, and analysis of, exoplanetary atmospheres. As usual, we humans tend to find a new world more interesting if there's a chance that we might be able (eventually) to visit it comfortably. In addition, astronomers looking for signs of alien life on other planets naturally suspect that planets with atmospheres are the best places to look; and planets with Earth-like atmospheres might be best of all.
Let's look into some of the basic physical principles that place limits on our ability to find and characterize the atmospheres around other planets.
Contents
Should we use optical telescopes, or radio telescopes -- or X-ray telescopes? Which instruments are most likely to reveal the existence of gaseous atmospheres around exoplanets? One can answer this question in several ways:
The best choice, of course, will be some wavelength regime in which there are plenty of strong markers of atmospheric gases, AND the background is relatively low, AND in which we can observe efficiently and without great expense. Let's consider each of these constraints in turn.
So, the first criterion we can choose is simple: pick a wavelength at which the star emits a lot of light. If we can detect many photons, we can achieve high signal-to-noise ratios, allowing us to pick out very weak signals. For typical main-sequence stars, this limits us to the near-UV, visible, and near-IR.
A space-based telescope can make measurements at any of these wavelengths. If we wish to search from the ground, however, we are limited to those wavelengths which are transmitted by the Earth's atmosphere.
Yes, yes, observing from the ground means that we'll also see absorption due to gases in the Earth's atmosphere, which can complicate the search for exoplanetary features. As we shall see, there are some ways to avoid confusing terrestrial and extra-terrestrial lines.
The graph below shows the transmission of light through the Earth's atmosphere in the UV-visible-IR region.
One can see that our choices for ground-based observations include the entire visible range (from about 0.4 to 0.8 microns), plus a series of windows in the near-infrared. The strong absorption bands separating the windows are largely due to water vapor, so using a telescope in a dry site (at high altitude or in the Antarctic) can yield better results.
At wavelengths beyond 5 microns or so, thermal emission from the atmosphere, the telescope, and the surrounding landscape starts to become a major source of noise.
By this criterion, our choices are UV-visible-IR for space-based telescopes, but limited to certain windows within the visible and near-IR (0.4 to 5 microns or so) for ground-based telescopes.
The higher the level of these background sources, the harder it will be for us to separate and measure the light from the objects of interest. Therefore, we ought to plan our observations in a region of the spectrum where the background levels are relatively low.
In order to illustrate the issue, let's consider three passbands in the visible and IR: B, K, and N. I'll choose numbers from Leinert et al., A&AS, 127, 1 (1998) and Expected NIR sky background at the TAO site.
To compare with the levels at other wavelengths, we can convert this to the number of photons per second per square arcsecond per square meter of collecting area. In those units, the B-band background is roughly
For ground-based observations, the results are clear: one must avoid thermal emission by working in the optical or near-IR regime.
Space-based telescopes don't have to deal with the emission of light from the Earth's atmosphere or surface, but they aren't immune from thermal radiation. Dust particles in the solar system, which are concentated toward the plane of planetary orbits, absorb light from the Sun and re-emit it at longer wavelengths. This zodical light creates a background which rises at longer IR wavelengths, but to a lesser extent.
Using values from Leinert et al., A&AS, 127, 1 (1998), we find that in space,
| band | B | K | N |
| photon/s · m2 · □'' | 10 | 26 | 2600 |
Whether working from the ground or space, one should -- if possible -- avoid the longer infrared wavelengths in order to keep the background levels from overwhelming the signals from exoplanetary atmospheres.
The difference between the n = 1 and n = 2 levels is 10.2 eV, so if a photon with this energy (corresponding to a wavelength of 1216 Angstroms) passes an atom in the ground state, the atom may absorb the photon and enter a higher energy level.
Q: In which portion of the electromagnetic spectrum
do we find a photon with energy of 10.2 eV?
However, there are two reasons that electronic transitions are not particularly good choices for exoplanets. First, at the temperatures expected in most planetary atmospheres (≤ 1500 K), most species will be in molecular, not atomic, form. Second, the energies required to raise most common atoms (H, He, C, O, etc.) out of their ground states are considerably larger than the typical kinetic energies of particles in planetary atmospheres. So, for the most part, we can ignore these lines.
In this case, the force constant has a value of roughly k = 1900 N/m. Treating the molecule as a simple harmonic oscillator, one can compute the energy of its vibrational states, and therefore the energy of the transitions between them. For CO, the energy required to raise the molecule from the ground state to the first excited vibrational state is about 0.26 eV, corresponding to a wavelength of about 4.7 microns. Other simple molecules of interest to exoplanet researchers, such as H2O or NH3, have vibrational states with similar transition energies.
Now, the typical kinetic energy of particles in a gas is of order
At temperatures of a few hundred to a thousand Kelvin, this kinetic energy is somewhat smaller than the energy of the first excited vibrational state. We may therefore expect many of the molecules to be in the ground state, ready to absorb near-IR photons and imprint an absorption line into the spectrum of the host star.
If we consider all these factors -- the wavelengths at which we can observe celestial sources clearly, at which the background levels are low, and at which the constituents of exoplanetary atmospheres are most likely to absorb light -- the conclusion is clear: our best choice is to look in the near-infrared, at wavelengths between, perhaps, 1 and 5 microns.
Looking for evidence for absoption of light by the atmosphere of an exoplanet is a complex process. It will help if we break it down into a series of steps, starting with a much simpler procedure and working our way up to the final result.
So, let's begin with a problem which is easier to describe and to understand: detecting the transit of a planet in front of its host star. Keep in mind that in order to be SURE that we've seen an event, we need to verify that two things are true:
The proper calculation of the uncertainty in a photometric (or spectroscopic) measurement is, itself, a rather complicated issue. I'll take a very simple approach, and assume that the only source of noise in the measurement is Poisson noise ("shot noise") in the number of photons collected from the host star. In that case, if we collect N photons in a measurement, then the uncertainty in that number is √N, and the fractional uncertainty is
For this example, let's consider a rather favorable system to study: a Sun-like star at a distance of just d = 10 pc. We will observe this star in the near-IR; let's pick the H-band, centered at λ0 = 1.66 microns and with a width of Δλ = 0.251 microns (according to Cohen, Wheaton, and Megaeath, AJ, 126, 1090 (2003)). The apparent magnitude of this star at a distance of 10 pc will be, by definition, the absolute magnitude of the Sun; according to the tables in Willmer, C. N. A., ApJS 236, 47 (2018), this is mH = 3.3. Going through the calculations, we find that the flux of photons from this star will be
Let's adopt a relatively modest telescope as our instrument one with a diameter D = 1 meter = 100 cm. The collecting area is just π (D/2)2. If we observe this star under ordinary circumstances, when no planet is blocking any of its light,
and take a single 1-second exposure, how many photons can we expect to collect?
Q: How many photons will we collect in 1 second? Q: What will the fractional uncertainty of our measurement be?
Now, suppose that an airless, solid planet passes in front of the star. During the transit, it will block a small amount of the star's light.
Consider two cases: an Earth-like planet, or a Jupiter-like planet. In each case, how much light will the planet block? Will we be able to detect the small decrease in the star's brightness?
Stellar radius Rs = 6.96 x 108 meters.
Earth Jupiter
----------------------------------------------------------
radius (m) 6.37 x 106 7.15 x 107
fraction blocked
is dip significant?
----------------------------------------------------------
Hmmm. In this case, the numbers show that we should be able to detect the transit of a Jupiter-sized planet easily: the fraction of light it blocks is much larger than the fractional uncertainty in a measurement with a one-second exposure.
But that's not true for the Earth-sized planet. In that case, the fractional uncertainty is LARGER than the size of the decrease in brightness.
Fortunately, there's a simple change we can make to our experiment which will allow us to detect the planet: we can increase the exposure time, collecting more photons, increasing the precision of the measurement. For example, if we increase the exposure time from 1 second to 100 seconds, then the number of photons we collect rises by a factor of 100, and the fractional uncertainty decreases by a factor of √(100) = 10. With the longer exposure, the fractional uncertainty would shrink to about 1 x 10-5, considerably smaller than the change in the star's brightness.
An airless planet crossing the disk of a star is simple -- and boring. Let's add an atmosphere, of height h.
If the atmosphere is opaque, then the amount of light blocked during a transit is now larger:
If the atmospheric thickness is considerably smaller than the solid body of the planet, then one can approximate the fractional increase in the depth of the transit as
Stellar radius Rs = 6.96 x 108 meters.
Earth
----------------------------------------------------------
radius (m) 6.37 x 106
atmospheric height h (m) 8.0 x 103
fraction blocked
if atmosphere opaque
----------------------------------------------------------
Note that in this case, adding an atmosphere causes only a very small increase in the depth of the transit.
Now, let's see how we can use an absorption line to reveal the presence of an atmosphere, using a photometric approach. Suppose that a planet does have an atmosphere, and it contains some gas with a strong absorption line at a wavelength λ0. If we make measurements of the host star during a transit through three filters at slightly different wavelengths,
then our instruments will record a smaller depth of transit through the two "outer" filters, compared to the measurement through the central filter.
To continue our earlier example, suppose that we observe an Earth-like planet with an Earth-like atmosphere transiting a Sun-like star, using a telescope of diameter D = 1 m. Let's make measurements at wavelengths within the H-band, choosing narrow filters which are centered on some wavelength at which the atmosphere has a strong absorption line. For example, suppose that the material absorbs at λ0 = 1.660 microns, and the line extends across the entire (unrealistically wide) 100-Å span from 1.655 to 1.665 microns. We choose filters of the same width, Δλ = 0.010 micron = 100 Å, one at shorter and one at longer wavelengths:
off-line on-line off-line
band (microns) 1.630 - 1.640 1.655 - 1.665 1.680 - 1.690
-------------------------------------------------------------------------
star (photon/s) 3.51 x 106 3.51 x 106 3.51 x 106
frac blocked 8.376 x 10-5 8.398 x 10-5 8.376 x 10-5
measure (photon/s) 3,509,706 3,509,705.2 3,509,706
-------------------------------------------------------------------------
The difference between the measurement inside the line and outside the line is small -- very small. For these parameters, we expect to see a difference of just about 1 fewer photon per second collected through the "on-line" filter than the "off-line" filters.
If we expose for just a single second, then we collect roughly 3.5 million photons through each of the filters. The uncertainty in the measurement throigh each filter is (due to Poisson noise only) the square root of that number, or about ± 1870 photons.
Yikes! There's no way we can detect a change of just 1 photon when the uncertainty in each measurement is almost 2000 photons. This particular experiment will utterly fail to detect the presence of an atmosphere on the planet, if it exists.
But if the planet is larger, and has a more extensive atmosphere, perhaps there is a chance. Let's pick a real system -- HD 209458 -- which has a star roughly the size of the Sun, but a much larger planet; and we'll assign this planet a very thick atmosphere, so that it blocks a much larger amount of light. The estimate of h in the table below is based on the method of Sing et al., Nature, 529, 59 (2016).
Stellar radius Rs = 8.35 x 108 meters.
HD 209458 b
----------------------------------------------------------
radius (m) 9.937 x 107
atmospheric height h (m) 5.7 x 105
fraction blocked
if atmosphere transparent 0.01416
fraction blocked
if atmosphere opaque 0.01433
difference in fraction 0.00016
blocked
----------------------------------------------------------
Aha! The fractional difference between the on-line and off-line measurement is much larger. That means that we don't have to collect trillions of photons in order to detect reliably the difference between the on- and off-filter values, and so prove the presence of an atmosphere.
In the general case of a star with radius RS, planet with solid body radius RP, and planetary atmosphere of height h, one can derive an expression for the change in depth of transit between filters which do and do not contain the absorption line. As long as the atmosphere is considerably smaller than the solid body of the planet, h << RP,
Even for favorable cases like the one involving HD 209458, it is necessary to collect a large number of photons in order for the Poisson noise in the measurement to be smaller than the fractional change between in-line and out-of-line values. Since the fractional error due to Poisson noise is proportional to the square root of the number of photons collected, we can make a rough estimate of the number of photons N required to detect some particular change-in-depth-of-transit:
If some particular combination of observing parameters does not yield a large enough number of photons, one can modify the setup in several ways to increase the signal:
There's another complication in the detection of planetary atmospheres which involves several factors, but boils down the fact that the observed wavelength of the absorption lines will change significantly over the course of the transit.
Why is this important? Remember, the key to detecting an atmosphere is detecting the CONTRAST between the brightness of the host star at wavelengths inside the line and outside the line. In order to maximize the difference between the in-line and out-of-line measurements, we should choose our filter bandwidths carefully.
Consider an absorption feature at 1.66 microns = 16,000 Å, with a width of about 4 Å. If we choose filters much wider than the line width, we'll include lots of light from outside the line itself in the "in-line" filter.
On the other hand, if our filters are too narrow, the "off-line" measurements won't include as many photons as they might, leading to larger Poisson noise.
To optimize the detection of the absorption line, our filters should have bandpasses roughly equal to the width of the line.
Just how wide are typical absorption lines in exoplanetary atmospheres? This is a complicated question, as there are a number of factors which influence the shapes of the lines.
For general purposes, I'll adopt a generic line width of Δλ = 0.3 Å for the following discussion, consistent with measurements of O2 molecular lines in the Earth's atmosphere ( van der Riet Wooley, ApJ 73, 185 (1931) ).
Okay, having adopted this line width, one might guess that measurements would be straightforward: choose a pair of narrow-band filters or spectroscopic regions roughly 0.3 Å wide, on-line and off-line, then make a series of exposures over the course of many hours, extending from before the transit (to establish a baseline) to some time after the transit (to check the baseline). Simple, right?
Wrong!
Unfortunately, there's a complication: the wavelength of the absorption line will change significantly over the course of the transit. Consider the diagram below, which shows the position of the planet in its orbit at the start, middle, and end of the transit.
To make the effect of the orbital motion more easily seen, I've exaggerated the size of the star relative to the orbit in the following diagrams.
At the start of the transit, the planet is moving toward the observer.
Q: Write an expression for the radial velocity
in terms of v and θ.
Okay, but just what is the angle θ at the start (and end) of the transit?
The magnitude of the radial velocity at ingress (or egress) can be expressed as
That radial velocity, in turn, generates a Doppler shift which can change the observed wavelength of the line itself.
In some cases, this shift can be considerably larger than the width of the line. For example, in the case of HD 209458b,
RS = 8.35 x 108 m
rorb = 6.96 x 109 m
v0 = 1.43 x 105 m/s
Q: For an absorption feature with rest wavelength λ0 = 1.660 microns = 16,600 Å
what is the shift in wavelength from ingress to egress?
Recall that the typical width of absorption lines in an exoplanet's atmosphere may be 0.2 or 0.3 Å. This Doppler shift due to orbital motion can cause the line to move by many times its width, complicating the analysis of observations.
Blain et al., arXiv 2408.13536 (2024) made a model of the spectrum they expected to observe over the course of a transit by HD 209458b. One panel from Figure 5 of their paper (slightly modified) is shown below. In this two-dimensional graph, wavelength runs left to right, in the usual manner, but the vertical direction indicates time, running from bottom to top. The purple section of the graph corresponds to the duration of the transit. In that section, features due to the photosphere of the star are shown in white; they remain at the same wavelength at all times. But the features caused by absorption in the exoplanet's atmosphere, shown in dark purple, shift from shorter to longer wavelengths over the course of the transit.
A heaily modified version of Figure 5 from
Blain et al., arXiv 2408.13536 (2024)
To summarize, one who tries to study the atmosphere of an exoplanet faces quite a few challenges:
And, of course, one more very important criterion
This last factor can place a stringent requirement on the size of the telescope used to perform this experiment. How large? Let's continue to use the observations of HD 209458b by Blain et al., arXiv 2408.13536 (2024) as an example.
The contrast in transit depth between in-line and out-of-line measurements is roughly 1.6 x 10-4, which implies that we must collect at least N = 3.8 x 107 photons in order to detect the signal reliably.
If we choose a bandpass of width Δλ = 0.3 Å to match the expected size of atmospheric features, then we find for a star of apparent magnitude mH = 6.366 the flux of photons inside the bandpass will be roughly
photons
flux f = 0.08 ------------
s * sq.cm.
Q: How long will it take to collect N = 3.8 x 107 photons
with a telescope of diameter 1 m? 10 m?
D = 1 m D = 10 m
-------------------------------------------------------------------
Area (sq.cm.)
Rate (photon/s)
Exptime for N (s)
-------------------------------------------------------------------
It is clear that hunting for exoplanet atmospheres is a job for very large telescopes ... but given these numbers, it seems that with the largest current telescopes (diameters greater than 5 or 8 meters), it ought to be possible to make several measurements over the course of a transit which would detect a single line cleanly.
In fact, the calculations above do not tell the whole story: they have been simplified in several ways and avoid dealing with some issues that arise in the real world:
So, in fact, when one runs through these calculations using reasonable values for the parameters, rather than ideal ones, one discovers that even our largest telescopes would have difficulty detecting a single absorption line with any confidence in any known exoplanet system.
Is the whole endeavor hopeless? Are astronomers just wasting their time?
Or is there something else we can do?
Fortunately, scientists have devised clever tricks that can help us to detect the presence of exoplanetary atmospheric features in the spectrum of a host star. Once again, let's turn to Blain et al.'s study of HD 209458b.
To start, the authors use a very large telescope at a very good ground-based site: one of the 8.2-meter VLT units in the Atacama Desert of Chile. They choose a system with a relatively bright host star (mV = 7.65, mH = 6.37) in order to boost the signal. The exoplanet is known to be a hot Jupiter, which suggests that its atmosphere ought to be massive and extended.
The spectrograph in this study, CRIRES+, provides very high spectral resolution. Operating with a narrow slit, they worked at roughly R = 100,000, allowing them to resolve features in the H-band as narrow as 0.16 Å. That's a good match to the expected widths of absorption lines produced by the planet's atmosphere.
Now, here's the real key: rather than attempting to measure just one line at a time, the authors effectively measure tens or hundreds of lines simultaneously, greatly increasing their ability to detect faint signals. The method goes like this:
Now, for each observed spectrum, taken at time t,
If the model of the planetary atmosphere was a good match to the ACTUAL planetary atmosphere, then each of the cross-correlations in the step "C" above should yield some positive result. Yes, due to random noise, some of the spectra might produce correlations of zero, or even negative values; but note that the team acquired several hundred spectra each night. Their final result, moreover, was based on spectra gathered on four good nights, each covering one transit.
As a check on the procedure, one can also run through the same steps, but make one crucial change: in step "A", rather than shifting the planet's model spectrum by the expected radial velocity, one can shift it by some other, incorrect, velocity. In that case, the shifted model features should NOT line with any observed features, and the cross-correlation should yield a result of negligible or negative significance.
An example of this check is shown in the Figure A.2 from Blain et al., below. the Doppler shift applied to the planetary spectrum is varied along the horizontal axis, time during the night runs along the vertical axis (from bottom to top), and the colors show the value of the cross-correlation. Note that the positive (blue) correlations only appear when the planetary model has been shifted by an amount which varies over the course of the transit -- just as we would expect from its orbital motion.
Figure A.2 from
Blain et al., arXiv 2408.13536 (2024)
Of course, this entire method of analysis depends crucially on step "2" in the list above: creating a model spectrum of the planetary atmosphere. If our model includes H2O, but the real atmosphere does not, then no amount of shifting will yield a significantly positive correlation.
Thus, this technique involves one more layer of iteration:
Interpreting the results can be difficult. In Figure 7 from Blain et al., the authors present the results for several possible components of the atmosphere in HD 209458b. In these graphs, blue regions denote positive, and red regions negative, correlations. The axes have been chosen so that real signals from the planet ought to appear near the location where the dashed white lines cross.
Figure 7 from
Blain et al., arXiv 2408.13536 (2024)
We (and Blain et al.) can conclude that H2O is definitely present in the planet's atmosphere, and that H2S might be present, but that there is no evidence for a detection of CO or HCN.
Copyright © Michael Richmond.
This work is licensed under a Creative Commons License.