
The observed misalignment between stellar rotation axes and protoplanetary disk or planetary orbital planes.Stellar-planetary misalignment refers to the observed discrepancy between the rotation axis of a star and the orbital plane of its surrounding protoplanetary disk or its planets. In the simplest model of solar system formation, the disk and star should spin in the same plane, but observations of many exoplanetary systems show significant tilts, sometimes even retrograde orbits. This misalignment provides crucial clues about the dynamical history of planetary systems, including violent scattering events and gravitational interactions with companion stars.
The first measurements of misalignment were made in the early 2000s using the Rossiter-McLaughlin effect, which detects the spin-orbit alignment of transiting hot Jupiters. The discovery of many misaligned systems, including some with retrograde orbits, challenged the standard formation model.
The misalignment can arise from several mechanisms. One is the gravitational torque from a distant companion star or giant planet that tilts the disk after the star has formed. Another is the Kozai-Lidov mechanism, where a binary companion drives oscillations in the eccentricity and inclination of a planet's orbit, eventually leading to a tilted or even flipped orbit. Additionally, planet-planet scattering during the late stages of formation can eject planets into inclined orbits, leaving the system misaligned.
Measuring stellar obliquity helps astronomers test models of planetary formation and migration, and it is used to infer the presence of unseen companions in exoplanetary systems.
Some exoplanets orbit their star backwards, in the opposite direction to the star's rotation, a direct result of extreme misalignment.
| \Gamma_1 | variable | First adiabatic index — adiabatic index |
| M | variable | Mass of the star — mass kilogram |
| R | variable | Radius of the star — radius metre |
| c | variable | Speed of light — speed metre per second |
| G | constant | Newtonian gravitational constant — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared |
| k | constant | Numerical constant of order unity depending on stellar structure — structure constant = 1 |
| MAE | variable | Mean absolute error — mean absolute error same as dependent variable |
| n | variable | Number of observations — sample size |
| y_i | variable | True value for observation i — true value same as dependent variable |
| \hat{y}_i | variable | Predicted value for observation i — predicted value same as dependent variable |
| \sum_{i=1}^n | operator | Sum over all observations from 1 to n — summation |
| |\cdot| | operator | Takes the absolute value of the difference — absolute value |


















