The τ ≈ 1 sweet spot: notes for instructors
These notes go with the τ ≈ 1 sweet-spot cartoon. The notes cover what the cartoon deliberately simplifies, the physics needed to answer the questions a careful student is likely to ask, and some discussion questions.
Background: Section III.C of N. Jeevanjee, “Climate sensitivity from radiative-convective equilibrium: A chalkboard approach,” Am. J. Phys. 91, 731–745 (2023), https://doi.org/10.1119/5.0135727.
The idea
At a single wavelength, how much a layer sends to space is the product of two competing factors. Deeper layers contain more absorbers, and by Kirchhoff’s law they therefore emit more. But a photon from deeper down must cross more absorbers to get out. High up, nearly every photon escapes, but there are few emitters. Deep down there are many emitters, but almost nothing escapes. The product peaks where the optical depth to space is \tau \approx 1.
What the cartoon simplifies
1. The sweet spot exists only per unit height. In the paper’s model for CO₂ (its Eq. 20), \tau \propto p^2: the p^2 comes from the absorber column above (one factor of p) and pressure broadening of the absorption lines (another factor of p). With p \propto e^{-z/H} this gives \tau \propto e^{-2z/H}, so every H/2 \approx 3.75\;\text{km} of height changes \tau by a factor of e. A slab of thickness dz therefore sends to space an amount proportional to
\left|\frac{d\tau}{dz}\right| e^{-\tau} \;\propto\; \tau\, e^{-\tau},
which peaks at \tau = 1. If you instead slice the column into layers of equal \Delta\tau, each slice sends e^{-\tau}\Delta\tau, which is largest at the top: no interior peak. The cartoon’s comparison is fair because the thin layer is drawn at least as tall as the \tau \approx 1 layer.
The thin region physically has no top, since \tau \to 0 only as p \to 0. Truncating it at the dashed top-of-atmosphere (TOA) line is harmless. The whole region above the level where the optical depth is \tau contributes a fraction 1 - e^{-\tau} of all the photons that reach space, which is finite however tall the region is.
2. At \tau = 1 itself, most photons are absorbed. A photon emitted straight up at \tau = 1 escapes with probability e^{-1} \approx 37\%. The cartoon’s 5 of 8 is an average over the whole shaded layer. You can read the layer’s optical depth off the figure using the paper’s interpretation of \tau (its Eq. 17) as total absorber cross-section divided by column area. Each glyph’s width is about 0.099 of the column width, so:
| Level | \tau from glyph widths |
|---|---|
| top of shaded layer | 0.20 |
| bottom of shaded layer | 0.99 |
| surface | 3.7 |
The average escape probability across \tau = 0.20–0.99 is (e^{-0.20} - e^{-0.99})/0.79 \approx 57\%, consistent with 5 of 8.
3. The real peak is broad, and the top of the thick region contributes. By the same e^{-\tau} weighting, the expected number of photons reaching space from each region is:
| Region | Expected | Drawn |
|---|---|---|
| thin | 1.8 | 2 |
| \tau \approx 1 | 4.5 | 5 |
| thick | 3.5 | 0 |
The \tau \approx 1 layer still sends the most. Its lead over the thin layer is drawn about right, 5 to 2 against an expected 4.5 to 1.8. The \tau \approx 1 layer’s lead over the thick region is much smaller than drawn: an expected 4.5 to 3.5, where the drawing shows 5 to 0. The “every photon absorbed” rule describes the deep part of the thick region; its top, just below \tau = 1, sends a sizable share to space. More generally, \tau e^{-\tau} falls to half its peak at \tau \approx 0.23 and \tau \approx 2.7. The span between those two levels is about 1.2H \approx 9\;\text{km}, most of the troposphere.
4. Emission is isotropic. Molecules emit in all directions. The cartoon shows only the upward photon from each.
5. Absorbed energy is re-emitted at the local temperature. An absorbed photon’s energy is thermalized by collisions, and the absorbing layer re-emits at its own temperature. So each photon that reaches space was emitted at the temperature of the layer it left, whatever path the energy took before that. Re-emission at the local temperature is what ties the \tau \approx 1 level to the emission temperature T_\text{em}.
6. The color-to-temperature order assumes temperature falls with height. That holds in the troposphere. The paper’s model has an isothermal stratosphere. In the real stratosphere temperature rises with height, so near the center of the CO₂ band, at 15 μm, where \tau \approx 1 lies in the stratosphere, the optically thin layer is warmer than the \tau \approx 1 layer. That is the narrow emission spike at the band center in observed outgoing spectra.
7. Not to scale. The molecule counts (2, 8 and 27) and the layer heights are illustrative. Each glyph is best read as a unit of absorbing cross-section rather than a fixed number of molecules, because absorption per molecule grows with pressure. If \tau grew by the same factor across every layer as tall as the \tau \approx 1 layer, as \tau \propto e^{-2z/H} implies, the cartoon’s 2 and 8 glyphs would set that factor at 4, and the thick region, two such layers tall, would need 32 + 128 = 160 glyphs, six times the 27 drawn.
From photon counts to W/m²
The cartoon counts photons, but OLR weights each emitter by the Planck function B at its temperature: at one wavelength every photon has the same energy, and a warmer emitter sends out more of them. In the troposphere, deeper layers are warmer, so Planck weighting favors depth. It strengthens the \tau \approx 1 layer’s lead over the thin layer and narrows its lead over the thick layer.
The numbers below are my own estimates for an idealized all-troposphere column; the paper does not give them. The estimates assume a wavelength of 14 μm, T \approx 227\;\text{K}, a lapse rate of 6.5\;\text{K/km} and H = 7.5\;\text{km}.
- How fast B grows with depth: B rises about 2% per Kelvin, so it grows by a factor of e every \approx 7.6\;\text{km} of descent, or about 1.6\times per H/2 layer.
- Why depth still loses: transmittance e^{-\tau}, with \tau \propto e^{-2z/H}, falls double-exponentially with depth. A modest exponential increase in B cannot offset that.
Relative to the \tau \approx 1 layer, for equal-height layers each a factor of e more opaque than the one above, with the energy computed from the Planck function:
| Layer | \tau range | Photons to space | W/m² to space |
|---|---|---|---|
| thin neighbor | 0.22–0.61 | 0.72 | 0.43 |
| \tau \approx 1 | 0.61–1.65 | 1 | 1 |
| thick neighbor | 1.65–4.48 | 0.51 | 0.76 |
With the approximation B \propto \tau^\beta, \beta \approx 0.5, the energy reaching space per unit height goes as \tau^{1+\beta} e^{-\tau}. That peaks at \tau \approx 1.5, about 1.5\;\text{km} below the \tau = 1 level. The contribution has a long tail upward on the cold side. So the level whose temperature reproduces the OLR, \pi B(T_\text{em}), sits higher, at \tau_\text{eff} = [\Gamma(1+\beta)]^{1/\beta} \approx 0.78, where \Gamma is the gamma function (the paper uses \Gamma for the lapse rate). The exact Planck function gives a peak at \tau \approx 1.46 and \tau_\text{eff} \approx 0.72. The paper’s \tau = 1 lies between the peak and \tau_\text{eff}. In this column the \tau = 1 level is at 227 K and T_\text{em} is 219 K, so placing all emission at \tau = 1 overestimates OLR at this wavelength by about 20%.
Discussion questions
- Only some photons from the \tau \approx 1 layer escape, yet that layer sends the most to space. Why? It has many more emitters than the thin layer. Its escape fraction is lower, but not low enough to cancel that advantage.
- Why compare layers of equal height rather than layers of equal optical thickness \Delta\tau? See note 1. With equal-\Delta\tau slices the topmost slice sends the most, so there is no sweet spot. Slices holding equal masses of CO₂ give a third answer: with \tau \propto p^2, each slice’s emission to space goes as \tau^{1/2} e^{-\tau}, which peaks at \tau = 0.5.
- What happens to this picture if the CO₂ concentration doubles? Every level’s \tau doubles, so the \tau = 1 level rises. Where temperature falls with height it is now colder and emits less, which is the route to radiative forcing.
- Why does the emitting layer’s temperature matter for OLR in W/m², not just photon counts? See the section on Planck weighting.
- At a wavelength where CO₂ absorbs more weakly, where does the sweet spot sit? Lower and warmer. In the limit of very weak absorption it reaches the surface, and the atmosphere becomes transparent at that wavelength.
Caption, alt text and credit line
Plain-text versions for pasting into a web page, slide notes or a learning management system. They use Unicode τ so they survive outside LaTeX.
Caption:
Why emission to space, per unit height, peaks near optical depth τ ≈ 1 (CO₂, one wavelength). Each molecule emits one photon straight up; the photon escapes unless a molecule lies directly above it. High up (τ ≪ 1, blue), every photon escapes but there are few emitters. Deep down (τ ≫ 1, red), there are many emitters but every photon is absorbed. In between (τ ≈ 1, purple), emitters are plentiful and many of their photons still escape, so this layer sends the most photons to space. Colors mark the emitting layer, from coldest (blue) to warmest (red).
The caption’s “many” is deliberate. A photon emitted straight up from exactly τ = 1 escapes with probability e^{-1} \approx 37\%; the cartoon’s 5 of 8 is an average over a layer spanning roughly τ ≈ 0.2–1 (see note 2 above).
Short caption:
Emission to space per unit height peaks near τ ≈ 1: enough emitters, and enough of their photons still escape.
Alt text:
Cartoon of an atmospheric column of CO₂ molecules in three layers, with more molecules lower down. Arrows show each molecule emitting a photon upward. Both photons from the sparse top layer cross the top of the atmosphere; five of eight from the shaded middle layer, labeled “τ ≈ 1, sweet spot”, also escape; every photon from the dense bottom layer is absorbed by a molecule above it.
Credit line:
Figure by Spencer A. Hill, adapted from Fig. 3 of N. Jeevanjee, “Climate sensitivity from radiative-convective equilibrium: A chalkboard approach,” Am. J. Phys. 91, 731–745 (2023), https://doi.org/10.1119/5.0135727 (CC BY 4.0).