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A comprehensive Rossiter-Mclaughlin Modelling Framework in TLCM: Application to HD 2685 $=$ TOI-135 system

This paper presents an updated Rossiter-McLaughlin modeling framework in the TLCM code, validated on nine known systems, and applies it to new HARPS and TESS observations of the HD 2685 (TOI-135) system to detect an intermediate sky-projected obliquity of approximately 55.6 degrees.

Original authors: Szilárd Csizmadia, Alexis M. S. Smith, J. V. Harre, Gábor G. Balázs

Published 2026-06-03

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CC BY 4.0

Original authors: Szilárd Csizmadia, Alexis M. S. Smith, J. V. Harre, Gábor G. Balázs

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A New Tool for a Cosmic Dance

Imagine a star as a giant, spinning top and a planet as a small marble orbiting it. Usually, the marble spins in the same direction the top spins, like a child running alongside a spinning merry-go-round. But sometimes, the marble runs in a weird direction—maybe even backwards or at a sharp angle.

Scientists want to know: How tilted is the planet's path compared to the star's spin? This tilt is called the "spin-orbit angle."

The authors of this paper have built a new, super-precise computer program (an upgrade to a tool called TLCM) to measure this tilt. They tested this new tool on nine known star systems to make sure it works, and then they used it to study a specific system called HD 2685.

The Main Event: The "Shadow" Effect (Rossiter-McLaughlin)

To measure the tilt, the scientists look for a phenomenon called the Rossiter-McLaughlin (RM) effect. Here is a simple way to visualize it:

The Spinning Star: Imagine the star is a giant, rotating pizza. The side spinning toward us is moving fast (blue-shifted), and the side spinning away is moving slow (red-shifted).
The Planet Passes: When the planet (the marble) crosses in front of the pizza, it blocks a slice of the spinning surface.
The Disturbance:
- If the planet blocks the part spinning toward us, the star suddenly looks like it's moving away a tiny bit (because that blue light is gone).
- If it blocks the part spinning away, the star looks like it's moving toward us.
The Result: As the planet moves across the star, it creates a wobble in the star's measured speed. By watching the shape of this wobble, scientists can figure out if the planet is crossing the star's "equator" (aligned) or cutting across it at a weird angle (misaligned).

What They Did: The "HD 2685" Case Study

The team focused on a system called HD 2685.

The Star: It's a hot, aging star (like a middle-aged human) that is slightly larger and brighter than our Sun.
The Planet: It's a "Hot Jupiter"—a massive gas giant that orbits very close to its star, completing a lap every 4 days.

The Upgrade:
Previous versions of the software used a simple, one-size-fits-all formula to calculate the RM effect. The authors updated the code to be much more flexible.

The Analogy: Imagine trying to measure the shadow of a tree. The old software assumed the tree was a perfect cylinder. The new software can handle trees that are twisted, have irregular leaves, or are shaped like cones. It uses a "numerical integration" method, which basically means it breaks the star's surface into thousands of tiny pixels, calculates the speed of each pixel, and adds them up. This is much more accurate than using a single formula.

The Data:

They used new data from the TESS space telescope (which took photos of the star 7 times over several years, giving them a much clearer picture than the 1 time previous studies had).
They used new data from the HARPS telescope (which measured the star's speed while the planet was passing in front of it).

The Findings: A Tilted Orbit

After running their new, high-precision model on the data, they found:

The Tilt: The planet's orbit is tilted at an angle of about 56 degrees.
What this means: It's not perfectly aligned (0 degrees), and it's not completely upside down (90 degrees). It's in the middle, like a runner on a track who is leaning heavily to one side.
Reliability: They tested the result by changing their assumptions about how fast the star spins. Even when they changed the assumptions, the answer stayed around 56 degrees. This proves the result is solid.

Why Does This Matter?

The paper connects this finding to a bigger mystery in astronomy: Why are some planets tilted and others straight?

The "Hot Star" Rule: Scientists noticed that planets around cool stars (like our Sun) usually line up perfectly. But planets around hot stars (like HD 2685) often have wild tilts.
The Theory: Hot stars have a "radiative" outer layer (like a smooth, slippery shell), while cool stars have a "convective" layer (like boiling soup). The "boiling soup" helps drag the planet into alignment over time. The "smooth shell" doesn't, so the planet keeps its weird tilt.
The Twist: The team also looked at the mass of the planet. There is a theory that very heavy planets (high mass ratio) stay aligned even around hot stars because they formed that way. However, the planet in HD 2685 is not heavy enough to force alignment, so its 56-degree tilt fits the "hot star" rule perfectly.

Summary

The authors built a better calculator for measuring how tilted exoplanets are. They tested it, and it works. They used it to prove that the planet HD 2685 b is tilted at a moderate angle, which fits our current understanding of how hot stars and their planets interact. This adds another piece to the puzzle of how planetary systems form and evolve.

Technical Summary: A Comprehensive Rossiter-McLaughlin Modelling Framework in TLCM: Application to HD 2685 = TOI-135 System

Problem Statement
The Rossiter-McLaughlin (RM) effect is a primary method for measuring the sky-projected spin-orbit angle ( $\lambda$ ) of transiting exoplanets, offering critical insights into planet migration, dynamical excitation, and tidal evolution. While empirical trends suggest a correlation between stellar effective temperature ( $T_{\text{eff}}$ ) and obliquity (with hot stars exhibiting wider obliquity distributions), recent demographic studies indicate that factors such as stellar multiplicity, planet-to-star mass ratios, and primordial disc conditions also play significant roles. To refine these population-level diagnostics, a robust, flexible, and validated modelling framework is required to handle diverse system architectures, including eccentric orbits, multiple radial velocity (RV) offsets, and various limb-darkening laws. Furthermore, specific systems like HD 2685, a hot Jupiter orbiting an evolved, hot host star, require updated analysis with expanded photometric and spectroscopic datasets to constrain their orbital architectures and obliquities.

Methodology
The authors present an updated implementation of the Rossiter-McLaughlin effect within the Transit and Light Curve Modeller (TLCM) code. The key methodological advancements include:

Numerical Integration of RM Anomaly: Unlike previous analytic approaches limited to specific limb-darkening laws (e.g., quadratic), the new framework computes the RM anomaly via direct numerical integration over the occulted portion of the stellar disc. This allows the model to accommodate seven different limb-darkening laws (linear, power-2, quadratic, logarithmic, square-root, Sing-3, and Claret four-parameter) without deriving new analytic integrals for each.
Convective Blueshift (CB) Modelling: The code incorporates an analytic description of convective blueshift, a phenomenon that can induce systematic errors in $\lambda$ measurements. The model uses a temperature-dependent formula derived from Liebing et al. (2021) to account for the net blue-shift of the stellar surface, applying exponential cut-offs for high temperatures and constant values for low temperatures.
Joint Modelling Capabilities: The RM module is coupled with TLCM's existing joint photometric and RV framework. It simultaneously models:
- Transit, occultation, and phase curves (including beaming, reflection, and ellipsoidal effects).
- Eccentric orbits and multiple Keplerian signals.
- Instrumental zero-point offsets and RV jitter.
- Correlated noise using a wavelet-based model, offering a computationally efficient alternative to dense Gaussian processes for long, high-cadence light curves.
Validation: The updated code was validated against nine known systems (including WASP-15, HAT-P-1, HAT-P-7, HAT-P-11, HAT-P-32, etc.) covering aligned, intermediate, polar, and retrograde orbits, as well as systems with eccentric orbits and multiple RV offsets.

Application to HD 2685 = TOI-135
The framework was applied to the HD 2685 system, a single, evolved F-type star ( $T_{\text{eff}} \approx 6801$ K) hosting a $\sim 1.15 M_{\text{Jup}}$ hot Jupiter in a $\sim 4.1$ -day orbit.

Data: The study utilized TESS photometry from seven sectors (1, 27, 28, 67, 68, 94, 95), significantly expanding the dataset from the original discovery (Sector 1 only). New HARPS radial velocity measurements were obtained in-transit and out-of-transit, combined with archival data from CORALIE, CHIRON, and FEROS.
Analysis: A joint fit of the light curves and RV data was performed using Markov Chain Monte Carlo (DE-MCMC) methods. The analysis tested the sensitivity of the derived obliquity to the prior on the projected stellar rotational velocity ( $V \sin I_*$ ).

Key Results

System Parameters: The joint analysis refined the system parameters, yielding a scaled semi-major axis of $a/R_* \approx 7.6$ , a planetary radius of $\approx 1.44 R_{\text{Jup}}$ , and a stellar radius of $\approx 1.57 R_\odot$ . The eccentricity was found to be consistent with zero ( $e \approx 0.004$ ), suggesting a circular orbit, differing slightly from previous estimates.
Rossiter-McLaughlin Detection: A clear RM signal was detected. For the fiducial spectroscopic prior on $V \sin I_*$ ( $15.4 \pm 0.2$ km/s), the sky-projected obliquity was measured as $\lambda = 55.6^{+10.9}_{-11.9}$ degrees.
Robustness: Tests with broader priors on $V \sin I_*$ (including a uniform prior $U(5, 25)$ km/s) showed that while the formal uncertainty on $\lambda$ increases (up to $+13^\circ, -24^\circ$ ), the central value remains stable around $56^\circ$ . This confirms the system possesses an intermediate projected obliquity.
Jitter Calibration: The study re-derived RV jitter values for the different instruments used, finding values of 46 m/s (HARPS), 48 m/s (CORALIE), 69 m/s (CHIRON), and 23.5 m/s (FEROS).

Significance and Claims
The paper claims significance in two primary areas:

Methodological Advancement: The updated TLCM provides a flexible, numerically robust framework for RM modelling that is not restricted by analytic limitations of specific limb-darkening laws. The successful validation across nine diverse systems demonstrates its reliability for population studies.
Population Diagnostics: The measurement of HD 2685b contributes a data point to the sparsely sampled region of the $T_{\text{eff}} - |\lambda|$ diagram corresponding to hot, evolved stars with intermediate obliquities. The result is consistent with the empirical trend that hot stars exhibit a broader range of obliquities compared to cool stars, likely due to the lower efficiency of tidal realignment in stars with radiative envelopes. However, the authors note that the system's low planet-to-star mass ratio ( $\sim 0.0008$ ) aligns with theoretical expectations (Rusznak et al. 2025) for misalignment in low-mass-ratio systems, contrasting with high-mass-ratio systems that tend to remain aligned.

The authors conclude that while their measurement supports existing empirical trends, further observational and theoretical work is required to fully explain the scatter in the spin-orbit angle distribution across different stellar and planetary parameter spaces.