Moons can orbit the TRAPPIST-1 star’s planets but only in a limited region around each planet.

TRAPPIST-1 is an ultra-cool red dwarf star with seven planets. Moons can orbit the TRAPPIST-1 planets but only in a limited region around each planet. planets’ gravity makes this stable zone slightly smaller. Only tiny moons are likely to survive long term — bigger ones would be torn away or fall in over time.

Scientists used the REBOUND N-body code with the IAS-15 integrator which is a high-precision gravitational solver.

A moon can remain in a stable orbit from very close to the planet (just outside the Roche limit) up to about 40–45% of the planet’s Hill radius. The Roche limit sets the innermost safe distance — closer than this, tidal forces would break the moon apart. The Hill radius is the outermost gravitational influence of the planet — beyond this, the star’s gravity dominates.

The gravitational interactions between the TRAPPIST-1 planets slightly reduce the stable region for moons.

The tidal decay equation is used which gives the maximum possible mass of a moon that can survive around a planet for a long time while tidal force is considered.

It shows that moons survive more easily if the planet is massive, compact, and weakly dissipative, and if the moon orbits farther out. In contrast, strong stellar gravity, large planetary radius, and strong tidal dissipation make moon survival harder.

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.

Using the ALMA telescope, researchers observed radio waves, to look at water molecules in the disk. They observed water with a slightly heavier oxygen atom (H₂¹⁸O) and Heavy water (HDO where hydrogen is replaced with deuterium).

Band 7 is a specific range of radio frequencies that the ALMA telescope uses to detect molecular emission lines. Here Band 7 HDO lines observed are much weaker than expected. It reveals that the water emission may originate from a more compact and hotter region having a radius of 53 au from the star or regions hidden by optically thick dust.

Rotational temperature is a measure of the average temperature based on how many molecules are in different "spinning" (rotational) energy states. Rotational diagram is used to determine Rotational temperature (an estimate of the gas excitation temperature) and Column density of molecules.

The below equation describes the dust optical depth at a given observing frequency. Larger optical depth (𝜏) means dust blocks more radiation.

The researchers used atmospheric and radiative-transfer simulations to show that when light emitted from inside an atmosphere is scattered by clouds, it becomes polarized, and this polarization depends strongly on cloud particle size, cloud thickness, and how temperature changes with altitude.

They found that small and large cloud grains produce distinct polarization signatures, thick clouds reduce polarization, and temperature gradients produce wavelength-dependent polarization because different wavelengths come from layers at different temperatures.

The polarization amplitude and the location of polarization peaks are governed by the scattering regime (Rayleigh vs. Mie), which depends on the ratio of particle size to wavelength.

Rayleigh scattering occurs when the cloud particles (or molecules) are much smaller than the wavelength of light. Scattering is strongly wavelength-dependent (∝ 1/λ⁴) → shorter wavelengths scatter more. high linear polarization is produced, especially at ~90° scattering angles.

Mie scattering occurs when particle size is comparable to or larger than the wavelength. Lower net polarization is produced due to multiple scattering directions. It is applied to larger cloud grains (~1–10 μm) in brown dwarf / exoplanet atmospheres.

Optical depth measures how much light is absorbed or scattered as it travels through the star's atmosphere. At low optical depths, insufficient scattering occurs which limits polarization production, whereas at high optical depths, multiple scattering leads to depolarization. Maximum polarization occurs at intermediate optical depths.

Cloud materials observed here: Silicate clouds (MgSiO₃, Mg₂SiO₄) dominate in near-infrared polarization features. Iron and Al₂O₃ contribute mainly at very hot temperatures and shorter wavelengths.

Quantum-mechanical (QM) calculations of the binding-energy (BE) distribution of OCS on ice grains are calculated to know at what temperature OCS sublimates. Binding Energy is the amount of energy needed to pull a molecule (like OCS) off the surface of a dust grain or ice grain.

K dec is(sublimation) rate — how fast OCS leaves the ice

This formula tells researchers at what temperature OCS evaporates off dust grains.

In cold protostellar envelopes (10–20 K), OCS molecules stick to icy dust grains. Molecular line observations are measured to map where OCS is in the gas phase around each protostar.

Radiative-transfer and dust–gas thermal modelling calculations of the protostellar envelope are done to compute a temperature profile T(r) at given luminosity, and OCS sublimation radius is calculated. The density profile is also calculated.
T(r) Dust temperature at distance r from the protostar is calculated as

Here, Bv is plank function, Tv optical depth of OCS line, it is calculated from density profile given below

Researchers found both young stars in binary have very similar luminosities, around 7 times the Sun’s brightness each.

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