Atoms and ions in the atmosphere absorb light at specific wavelengths, leaving dark absorption features in the stellar spectrum. By measuring how the planet appears slightly larger at some wavelengths, scientists build a transmission spectrum. It tells us how the effective size of the planet changes with wavelength due to absorption in the atmosphere.
F(λ,t) is the observed flux,
F0(λ) is the baseline (no transit) flux,
Rp(λ) is the apparent planetary radius at that wavelength,
R∗ is the star’s radius
1D hydrodynamic atmospheric escape (Parker wind) model combined with non-local thermodynamic equilibrium (NLTE) radiative transfer is used here. KELT-9b is so hot that its upper atmosphere behaves like a flowing gas, not a static layer. Gravity and pressure compete, causing gas to expand outward and escape. Parker wind model describes a steady, pressure-driven outflow originally developed for the solar wind. It gives velocity, density, and pressure as functions of radius. Here blueshifted absorption and large mass-loss rate (~10¹² g/s) occur.
Roche lobe is used to detect atmospheric material around KELT-9b is gravitationally bound or escaping. Using the planet–star mass ratio and orbital separation, the Roche lobe radius is calculated and compared with the effective radius from Mg II and Fe II absorption in the transmission spectrum. The Roche lobe radius is small, the planet’s close orbit also enhances mass loss and atmospheric escape.
A logistic sigmoid function is used as a smooth mathematical tool to model how ion absorption gradually appears and disappears with wavelength or velocity in the transmission spectrum.
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