Stacking transmission spectra of different exoplanets

TL;DR

This work establishes a rigorous framework for stacking exoplanet transmission spectra and shows that, under self-similar abundance profiles, the stacked spectrum is well described by the geometric mean of abundance ratios across planets within JWST-NIRSpec/G395H coverage. It derives how slant optical depth and transit-radius observables scale with harmonic-mean and geometric-mean quantities in isothermal and non-isothermal regimes, introduces representative-planet parameters, and demonstrates, with realistic forward models, when the geometric-mean-abundance interpretation holds. The results indicate stacking can yield population-level atmospheric insights and enable more precise detections in cases of shared chemistry, while warning that large temperature spreads or cross-chemistry transitions (e.g., CO/CH$_4$) bias the interpretation. The findings offer practical guidelines for applying stacking to JWST-era exoplanet atmospheres and for interpreting population trends in chemical abundances and atmospheric structure.

Abstract

In many areas of astronomy, spectra of different objects are co-added or stacked to improve signal-to-noise and reveal population-level characteristics. As the number of exoplanets with measured transmission spectra grows, it becomes important to understand when stacking spectra from different exoplanets is appropriate and what stacked spectra represent physically. Stacking will be particularly valuable for long-period planets, where repeated observations of the same planet are time-consuming. Here, we show that stacked exoplanet transmission spectra are approximately mathematically equivalent to spectra generated from the geometric mean of each planet's abundance ratios. We test this by comparing stacked and geometric mean spectra across grids of forward models over JWST's NIRSpec/G395H wavelength range (2.8-5.2$μ$m). For two dominant species (e.g., H$_2$O and CO$_2$), the geometric mean accurately reflects the stacked spectrum if abundance ratios are self-similar across planets. Introducing a third species (e.g., CH$_4$) makes temperature a critical factor, with stacking becoming inappropriate across the CO/CH$_4$ boundary. Surface gravity exerts only a minor influence when stacking within comparable planetary regimes. We further assess the number of stacked, distinct sub-Neptunes with high-metallicity atmospheres and low-pressure cloud decks required to rule out a flat spectrum at $>5σ$, as a function of both cloud deck pressure and per-planet spectral precision. These results provide guidance on when stacking is useful and on how to interpret stacked exoplanet spectra in the era of population studies of exoplanets.

Figures (14)

• Figure 1: The transmission spectrum, shown as $[R_t-R_t(\lambda=1~\mu{\rm m})]/H$ of three planets with masses of 0.25, 1.25 & 6.25 M$_{\rm J}$. The atmosphere contains two (arbitrary) species with Gaussian cross-sections. The left-panel shows a scenario where the abundance profiles as a function of pressure are fixed in the atmosphere. In this case because the transmission spectra of the different mass planets probe different pressure levels the abundance ratios of the two species are different at the observable pressures. Thus, despite the fact the abundance profiles are identical the different planets produce different transmission spectra. However, in the right-panel the abundances of species 2 are adjusted so that they are identical at the different pressures probed by transmission spectra for the different mass planets. This results in identical transmission spectra. This toy model highlights that different planets will only have identical transmission spectra (in terms of $\Delta R_t/H$) if the extinction profiles are identical in the region probed by transmission spectra, not if the extinction profiles are identical.
• Figure 2: The transmission spectrum, shown as $[R_t-R_t(\lambda=1~\mu{\rm m})]/H$, for 30 planets where the extinction profile with pressure $l$ has been randomly drawn between -0.3 and 3. The stacked spectrum is shown as the solid blue line. A spectrum calculated assuming the $l$ is given by the harmonic mean of all the ${l_j+1}$ is shown as the orange dashed line, perfectly agreeing with the stacked spectrum. This demonstrates that for a given species the stacked spectrum is representative of the harmonic mean of each planet's individual abundance profile.
• Figure 3: Demonstration that for $c=0$, the stacked spectra is given by the natural logarithm of the geometric mean of the opacity ratio.
• Figure 4: Demonstration that varying all the stacked spectra is given by the harmonic mean of the scale height and geometric mean of the opacities.
• Figure 5: Stacking model transmission spectra of identical planets with different H$_2$O and CO$_2$ abundances ($0.01,1,10\times$ solar) but constant H$_2$O/CO$_2$ abundance ratios. Top panel: the individual planets' model spectra are shown as black/grey lines. The stacked spectrum is in blue and the spectrum derived from the geometric mean of the three planets' abundances (labelled 'GMA') is shown in orange. Bottom panel: the difference between the stacked and GMA spectra. This difference is within the uncertainty in the stacked spectrum (shown as the grey shaded region) which assumes WASP-39b-like uncertainties (from Alderson2023) for each of the three planets.