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A distinction between physical constitution and perceptual response is plausible and scientifically useful. A division into wholly objective primary qualities and wholly unreal secondary qualities is much harder to defend. These are different interpretations of the primary and secondary qualities distinction.

John Locke attributes extension, solidity, shape, number and motion to bodies as primary qualities. Secondary qualities are powers to produce sensations through the constitution of bodies: a body's power to produce a colour experience is distinct from the experience itself. This distinction need not make the power unreal. Nor should the further resemblance thesis, that ideas of primary qualities resemble their causes, be confused with the more modest claim that bodily constitution explains experience.^[ref]

A modern model of colour perception shows what can be retained. For a nonfluorescent surface in fixed viewing geometry, a simplified cone cell response isHere $L$ is the incident spectral irradiance, $R$ the spectral reflectance, and $S_i$ an effective sensitivity for one of three cone cell classes. Neural processing and adaptation also matter, so these responses are not a complete identification of perceived colour. Nevertheless, the model separates the surface, illumination and observer rather than assigning every experienced variation to a changing surface.

It follows that one surface can produce different responses under different illumination. Conversely, distinct spectral reflectance functions can produce the same three responses for a specified illuminant and observer, a case of metamerism. Three weighted integrals do not in general uniquely determine a whole function. Thus identifying experienced colour with one intrinsic surface property loses both the dependence on the observer and the many-to-one character of the response.

The scientific benefits are concrete. A comparison of paints should specify illumination and observer conditions. A measurement intended to characterize a material can instead estimate its spectral reflectance, preserving information that a single colour match discards. In both cases the distinction tells an investigator what to hold fixed and what to vary. It organizes experiments and explains disagreement without requiring that one observer be hallucinating.

However, perceptual variability alone does not distinguish primary qualities from secondary qualities. George Berkeley points out that apparent extension and shape also change with viewpoint and perceiver.^[ref] A circular disc yields different projected shapes at different viewing angles. The stable object is inferred through a model of that variation; it is not simply copied into experience. Primary qualities therefore have no automatic exemption from the interpretative work involved in perception.

Nor is relationality equivalent to subjectivity. In special relativity, measured length depends on an inertial frame, but relations between such measurements are tightly constrained. Conversely, a disposition to affect a specified perceiver under specified conditions can be an objective physical fact. Classifying a property as relational does not settle whether it is real, measurable or explanatory.

The best distinction is consequently functional and revisable: distinguish the experience to be explained from physical properties and relations used to explain it. That retains the scientific benefits of separating constitution, environment and response. It does not establish an immutable list of primary qualities, a simple resemblance relation between experience and object, or a proof of scientific realism.

Spectral irradiance $F_{\lambda }$ is radiant power crossing unit area per unit wavelength, integrated over the relevant hemisphere with the projected-area factor: $F_{\lambda }=\int I_{\lambda }\cos \theta d\Omega$. Its units are $\mathrm{W}{\mathrm{m}}^{-2}$ per unit wavelength; spectral radiance $I_{\lambda }$ additionally refers to unit solid angle. For an isotropic outgoing hemisphere with Planck function $B_{\lambda }$, the integral is $F_{\lambda }=\pi B_{\lambda }$.

The vertical optical depth above height $z$ is the dimensionless extinction integralHere $z$ increases upward, $n_a$ is absorber number density, ${\sigma }_{\lambda }$ its absorption cross section, ${\rho }_a$ its mass density, and ${\kappa }_{\lambda }$ the absorption coefficient per unit absorber mass. Equivalently, using bulk air density $\rho$ and absorber mass fraction $q_a$, $d{\tau }_{\lambda }/dz=-{\kappa }_{\lambda }{q}_a\rho$. The coefficient may depend on temperature, pressure and wavelength. For purely absorbing gas, transmission along a ray is $e^{-{\tau }_{\mathrm{ray}}}$; scattering, if present, contributes to extinction and requires its own source term.

Use the optical-depth weighting stipulated in the paper. A jump in the Planck function at ${\tau }_1$ gives $dB/d\tau =(B_1-B_0)\delta (\tau -{\tau }_1)$, soThe upper isothermal layer emits $\pi B_0(1-e^{-{\tau }_1})$ and transmits the lower region's blackbody spectral irradiance with factor $e^{-{\tau }_1}$. The lower layer and black surface have the same $B_1$, so their combined radiation entering the upper layer is $\pi B_1$, independent of how its emission is divided between the lower gas and surface.

More generally, integration by parts rewrites the supplied expression asThe terms are attenuated surface emission and the atmosphere's own emission. In the original boundary-plus-gradient form, $\pi B_0$ is the isothermal reference flux and the integral corrects for temperature variation below the top. Optical depths of order unity contribute most strongly. These formulas follow the paper's specified exponential weighting; a full angular radiative transfer calculation uses ray-angle-dependent optical paths rather than identifying every ray with a vertical path.

For the linear Planck function profile, direct integration givesin the optically thick limit. A sudden doubling of absorber abundance doubles optical depth at every physical height while initially leaving the temperature profile fixed. With $\tilde{\tau }=2\tau$, the same profile is $B=B_0+(\alpha /2)\tilde{\tau }$, not $B_0+\alpha \tilde{\tau }$. ThusIf temperature increases downward, $\alpha >0$ and outgoing spectral irradiance decreases immediately because the effective emitting region moves upward into cooler gas. Subsequent thermal adjustment is outside this instantaneous comparison.

For the quadratic Planck function profile,The optically thick limit and the instantaneous doubled-abundance result are thereforeThe new coefficient is $\beta /4$, since $B(\tilde{\tau })=B_0+\beta (\tilde{\tau }/2{)}^2$ at fixed physical temperature. For positive $\beta$, the outgoing spectral irradiance decreases. The factor differs from the linear case because the temperature-dependent source function curves with depth; doubling abundance does not mean holding its old functional coefficients fixed in the new optical-depth coordinate.