In the <annulus model for transmission spectroscopy>, an added aerosol <optical depth> changes the signal to $A[1-e^{-(\tau_{\rm gas}+\tau_{\rm aerosol})}]$. Large particles can supply nearly wavelength-independent extinction. An opaque high <exoplanet cloud deck> then masks deeper gas, flattens the optical <exoplanet transmission spectrum>, and weakens atomic or molecular features.

Small particles can instead produce a rising transit radius toward short wavelengths. In an isothermal atmosphere, the <slant optical depth of an isothermal atmosphere> is
$$
\tau_\lambda(z)\simeq\sigma_\lambda n_0e^{-z/H}\sqrt{2\pi R_pH}.
$$
Taking the effective radius near $\tau_\lambda\sim1$ gives $z(\lambda)=H\log\sigma_\lambda+\mathrm{constant}$. For <Rayleigh scattering>, $\sigma_\lambda\propto\lambda^{-4}$, so the <scattering slope of a transmission spectrum> is
$$
\boxed{\frac{dR_p}{d\log\lambda}=-4H.}
$$
A steeper-than-expected optical slope or suppressed gas features can therefore indicate <atmospheric haze> or <exoplanet clouds>. These signatures are not unique: high mean molecular mass reduces $H$, gas itself can produce <Rayleigh scattering>, and stellar surface heterogeneity can mimic slopes. Consistent optical and infrared features help distinguish these explanations.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 315 / 1 / c / Solution

For an isothermal thin atmosphere with <number density> $n(z)=n_0e^{-z/H}$ and extinction cross-section $\sigma_\lambda$, a tangent ray near radius $R_p+z$ has height $z+\ell^2/(2R_p)$ at distance $\ell$ along the ray. The resulting <Gaussian integral> gives slant <optical depth>
$$
\tau_\lambda(z)\simeq\sigma_\lambda n(z)\sqrt{2\pi R_pH}.
$$
The effective transit height is near $\tau_\lambda\sim1$, so $z(\lambda)=H\log\sigma_\lambda+\mathrm{constant}$ if composition is constant. This explains the <scattering slope of a transmission spectrum>.

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 Incoming links: Slant optical depth of an isothermal atmosphere