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Extending TESS flare frequency distributions with CHEOPS: Power-law versus lognormal

This study analyzes 5,620 flares from 110 M dwarfs observed by TESS and CHEOPS to demonstrate that while flare frequency distributions based on equivalent duration follow a power law, those based on bolometric energy are better described by a lognormal distribution or a truncated power law, with deviations from a pure power law primarily attributed to observational biases and limited sampling of the most energetic events rather than intrinsic flare generation mechanisms.

Original authors: Julien Poyatos, Octavi Fors, José Maria Gómez Cama

Published 2026-03-25

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CC BY 4.0

Original authors: Julien Poyatos, Octavi Fors, José Maria Gómez Cama

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: Star Tantrums and the "Goldilocks" Zone

Imagine stars, specifically the small, cool ones called M dwarfs, are like toddlers. Sometimes, they throw massive tantrums called flares. These aren't just mood swings; they are violent bursts of radiation that can strip the atmosphere right off a planet orbiting nearby.

For a long time, scientists thought these tantrums followed a simple rule: The Power Law.

The Analogy: Think of a snowball fight. You get a lot of tiny snowballs, fewer medium ones, and very few giant boulders. The rule was: "For every huge boulder, there are 10 medium ones, and 100 tiny ones." It's a predictable, straight line on a graph.

But recently, some scientists started wondering: Is it really that simple? Maybe the distribution of tantrums looks more like a Lognormal curve (a bell curve that's squashed to one side), which would mean the "giant boulders" are actually much rarer than we thought. This matters a lot for habitability: if giant flares are rare, planets might be safer. If they are common, those planets might be sterilized.

The Detective Work: Two Different Cameras

To solve this mystery, the authors (Poyatos, Fors, and Gómez Cama) acted like detectives using two different types of cameras to watch these stars:

TESS (The Wide-Angle Lens): This satellite watches many stars for a long time. It's great at catching the rare, massive tantrums (the boulders) because it has a long observation window. However, its "vision" is a bit blurry; it might miss the tiny, quick sneezes (small flares).
CHEOPS (The High-Magnification Lens): This satellite watches fewer stars but with incredible precision and speed. It's like a high-definition microscope. It catches the tiny, quick sneezes that TESS misses, but it doesn't stay long enough to see the rare, massive tantrums.

The Strategy: They combined the data from both. It's like having a security guard who watches the whole neighborhood for a year (TESS) and a security guard with a super-zoom lens who watches one house for an hour (CHEOPS). Together, they see the whole picture.

The "Complex" Problem: One Big Tantrum or Many Small Ones?

A major issue in the past was decomposition. Sometimes, a star throws a tantrum that looks like one big explosion, but it's actually a chain reaction of smaller explosions happening almost at the same time.

The Analogy: Imagine a row of dominoes falling. To a slow camera, it looks like one long, continuous crash. To a fast camera, you see 50 individual dominoes hitting the floor one by one.
The Fix: The authors used a fast camera (CHEOPS) to break these "complex flares" down into their individual components. They realized that if you count a chain of 10 small flares as one big flare, you mess up the math. You need to count them as 10 separate events.

The Results: What Did They Find?

After cleaning up the data, correcting for the fact that their cameras missed some tiny flares, and breaking down the complex events, they found two very different stories depending on how they measured the energy:

1. The "Raw" Measurement (Equivalent Duration)

When they measured the flare just by how much the light brightened (without worrying about the star's total power), the data fit the Power Law perfectly.

The Takeaway: The raw occurrence of flares is predictable. It's a straight line. The "bell curve" theories were likely wrong because of how the data was being looked at.

2. The "Real" Energy (Bolometric Energy)

When they converted that light into actual total energy (taking into account how bright the star is), the graph changed shape. It didn't look like a straight line anymore; it looked like a Truncated Power Law.

The Analogy: Imagine a staircase. It goes up perfectly straight for a long time, but then, suddenly, the stairs stop. There are no steps above a certain height.
The Break Point: They found a "ceiling" at about $10^{35}$ ergs.
- Below this ceiling: The power law works.
- Above this ceiling: The data drops off sharply.

Why did it drop off?
The authors ran a statistical test (like a "roll of the dice" simulation) and realized this wasn't a physical law of the universe. It was a sampling error.

The Explanation: We just haven't been watching the stars long enough to catch the truly massive, once-in-a-century super-flares. The "ceiling" is just the limit of our current observation time, not a limit of the star's ability to throw a tantrum.

Why Does This Matter?

For Alien Life: If the "ceiling" is real (meaning super-flares are physically impossible), then planets around M dwarfs might be safer than we feared. If the ceiling is just because we haven't watched long enough, then those planets might be getting bombarded more often than we think. The authors suggest that with better data, the "safe zone" might actually be a bit safer than the old "Power Law" models predicted, but we need more time to be sure.
For Future Missions: This study proves that you can't just rely on one telescope. You need the "wide view" (TESS) and the "zoom view" (CHEOPS) working together.
The Future (PLATO): The paper ends by looking forward to the PLATO mission. PLATO will be the ultimate detective: it will watch many stars for many years with high precision. It will finally tell us if that "ceiling" at $10^{35}$ ergs is a real physical wall or just a fog we haven't cleared yet.

Summary in One Sentence

By combining a wide-angle telescope and a high-speed microscope, scientists discovered that while star flares generally follow a predictable pattern, our current view of the biggest flares is cut short by how long we've been watching, not by the laws of physics.

1. Problem Statement

Stellar flares on M dwarfs are critical for assessing exoplanet habitability, as they can erode atmospheres or trigger prebiotic chemistry. Flare Frequency Distributions (FFDs) are traditionally modeled as power laws ( $dN/dE \propto E^{-\alpha}$ ). However, recent studies suggest deviations from this model, proposing lognormal distributions or truncated power laws. These deviations imply different flare generation mechanisms (e.g., sympathetic flares vs. self-organized criticality) and significantly alter predictions for the frequency of high-energy "superflares."

Two main issues complicate FFD analysis:

Observational Biases: Low-energy flares are often missed due to detection thresholds, leading to an apparent flattening of the distribution at the low-energy end.
Metric Conversion: Converting the directly measured Equivalent Duration (ED) to Bolometric Energy ( $E_{bol}$ ) involves multiplying by stellar luminosity. Since stellar luminosities in large samples follow a lognormal distribution, this convolution can distort the resulting $E_{bol}$ -based FFD, making it appear non-power-law even if the underlying ED distribution is a power law.

The study aims to resolve these ambiguities by combining data from TESS (long baseline, lower cadence) and CHEOPS (short baseline, ultra-high cadence/precision) to cover a wider energy range and correct for detection biases.

2. Methodology

The authors analyzed 110 M dwarfs (M0–M5) observed by both missions, detecting 5,620 flares after decomposition. The analysis pipeline included:

Data Processing:
- TESS: Used 20s and 120s cadence PDCSAP light curves.
- CHEOPS: Used 3s cadence "imagettes" photometry.
- Detrending: Applied Savitzky-Golay filters to remove rotational modulation.
Flare Detection & Decomposition:
- Used the AltaiPony package with criteria from Chang et al. (2015).
- Decomposition: Complex flares (multi-peaked) were decomposed into individual components using the Llamaradas Estelares model and Akaike Information Criterion (AIC) optimization. This treated sub-events as individual flares rather than single complex events.
Bias Correction (Injection-Recovery):
- Performed synthetic flare injection-recovery tests (50 iterations for TESS, 200 for CHEOPS) to quantify detection probability and energy recovery ratios as a function of flare energy.
- Applied corrections to real flare counts and energies to account for missed low-energy events and energy underestimation.
Energy Calibration:
- Converted ED to Bolometric Energy ( $E_{bol}$ ) using a 10,000 K blackbody flare spectrum and instrument-specific bolometric correction factors ( $\epsilon_{CHEOPS} = 0.34$ , $\epsilon_{TESS} = 0.18$ ).
Statistical Fitting:
- Constructed FFDs for ED and $E_{bol}$ for TESS, CHEOPS, and a combined sample.
- Fitted Power Law, Lognormal, and Truncated Power Law models.
- Used Vuong's Likelihood Ratio Test (for non-nested models) and Stabilized Kolmogorov-Smirnov (sKS) and Exceedance tests to determine the best fit and distinguish between physical truncation and sampling limits.

3. Key Contributions

Multi-Mission Synergy: Successfully merged TESS and CHEOPS data to extend the dynamic range of FFDs by six orders of magnitude in bolometric energy ( $10^{30}$ to $10^{36}$ erg).
Complex Flare Handling: Demonstrated that decomposing complex flares into components significantly alters FFD slopes, particularly for high-cadence CHEOPS data which resolves more sub-structures than TESS.
Bias Correction Framework: Applied rigorous injection-recovery corrections to both missions, effectively removing the artificial "flattening" of FFDs at low energies previously attributed to intrinsic lognormal behavior.
Metric Distinction: Clearly distinguished between ED-based distributions (intrinsic photometric occurrence) and $E_{bol}$ -based distributions (physical energy), showing how stellar luminosity distributions convolve with power laws to create apparent deviations.

4. Key Results

ED-Based FFDs: Both individual and combined ED-based distributions closely follow a power law. The power-law index ( $\alpha$ ) steepened after bias correction (e.g., combined $\alpha$ increased from $1.85$ to $1.90$), confirming that the low-energy flattening was an observational artifact.
Bolometric Energy ( $E_{bol}$ ) FFDs:
- $E_{bol}$ -based distributions deviate from a pure power law.
- While a lognormal distribution provided a better fit than a simple power law, the Truncated Power Law provided the statistically best fit.
- Break Point: A truncation/break was identified at $1.8 \times 10^{35}$ erg.
Nature of the Truncation:
- sKS and Exceedance Tests: The deviation from a pure power law at high energies ( $>10^{35}$ erg) was attributed to limited sampling rather than an intrinsic physical cutoff. The observed maximum energy lies below the 2.5th percentile of simulated maxima drawn from an infinite power law, suggesting current instruments cannot reliably sample the rarest superflares.
Complex Flares: CHEOPS detected nearly twice as many complex flares (44%) compared to TESS (23%), confirming that high cadence is essential for resolving flare sub-structures.

5. Significance and Implications

Exoplanet Habitability:
- The finding that the low-energy flattening is an artifact supports the use of power laws for low-energy flare statistics.
- The truncation at $1.8 \times 10^{35}$ erg suggests that superflares (which drive atmospheric erosion) may occur less frequently than extrapolations from pure power laws would predict. This slightly relaxes concerns regarding the sterilization of planets orbiting active M dwarfs.
Future Missions (PLATO):
- Current instruments (TESS/CHEOPS) are insufficient to reliably sample flares above $10^{35}$ erg.
- The upcoming PLATO mission, with its longer baselines and high precision, is required to bridge the gap between observational limits and intrinsic flare physics, particularly for the high-energy regime relevant to habitability.
Methodological Impact: The study provides a robust framework for correcting FFDs using injection-recovery and decomposition, which is essential for generating realistic synthetic light curves for future mission planning and sensitivity analysis.

Conclusion: The paper concludes that while ED-based flare occurrences follow a power law, the conversion to bolometric energy introduces lognormal distortions due to stellar luminosity distributions. The apparent deviation from power laws at high energies is primarily a sampling limitation. A truncated power law with a break at $\sim 1.8 \times 10^{35}$ erg best describes the data, implying that the most energetic superflares are rarer than previously thought under pure power-law assumptions.