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Supermassive black hole mass inference with the optical flares of tidal disruption events

This paper introduces TDEFLARE, a publicly available code that rapidly and reliably estimates supermassive black hole masses from early optical/UV tidal disruption event flares using empirical relationships with thermal X-ray and UV plateau emissions, thereby enabling the analysis of 89 events to derive the intrinsic black hole mass function while accounting for Malmquist-Hills bias.

Original authors: Andrew Mummery

Published 2026-01-22

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Andrew Mummery

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: Weighing Invisible Giants

Imagine a supermassive black hole as a giant, invisible monster sitting at the center of a galaxy. We can't see the monster itself, but sometimes, a star wanders too close and gets ripped apart by the monster's gravity. This event is called a Tidal Disruption Event (TDE). As the star is shredded, it creates a massive, bright flash of light (an optical flare) that we can see from Earth.

The main goal of this paper is to answer a simple question: How heavy is the monster?

Knowing the weight of these black holes helps astronomers understand how galaxies grow and evolve. However, weighing them is tricky. Usually, you need to watch a TDE for years to get a good measurement, but by then, the light has faded, and we often lose the data.

The Problem: Waiting Too Long

Think of a TDE like a firework.

The Explosion (Early Phase): It's bright, chaotic, and happens quickly.
The Glow (Late Phase): After the explosion, there is a steady, dim glow that lasts for years.

Scientists have known for a while that if you wait for the "glow" (the late-time data) and combine it with X-ray data, you can weigh the black hole very accurately. But waiting years is expensive and difficult. By the time you get that data, you might have missed the chance to study thousands of other fireworks that are too far away or too faint to wait for.

The Solution: A "Quick-Weigh" Tool

The author, Andrew Mummery, has created a new tool called TDEFLARE. Think of this tool as a smart scale that can guess your weight just by looking at how you jumped, without needing to wait for you to land.

Instead of waiting for the long, steady glow, TDEFLARE looks at the initial "explosion" (the early optical flare) and uses some clever math to estimate the black hole's mass immediately.

How It Works: The "Recipe"

The paper argues that while the physics of the initial explosion is complicated, the result follows a simple pattern. The author found three "recipes" (scaling relationships) that link the brightness of the flare to the weight of the black hole:

The Peak Brightness: How bright the flare gets at its highest point.
The Total Energy: How much total light was released as the flare faded.
The Plateau: If we do have late data, the steady glow level.

The Analogy: Imagine you are trying to guess the size of a car engine just by listening to the sound it makes when you start it. You don't need to drive the car for 100 miles to know the engine size; the initial "roar" tells you enough. TDEFLARE listens to the "roar" of the black hole.

Key Findings (What the Tool Can Do)

The paper tests this tool on real data and makes four main claims:

It's Accurate: When the author compares TDEFLARE's results to much more complex, physics-heavy computer models (which take days to run), the results are almost identical. TDEFLARE gets the weight right in seconds.
It Works for Partial Eaters: Sometimes the black hole only takes a bite out of the star and spits the rest out (a "partial disruption"). TDEFLARE works for these too, not just the ones where the star is completely destroyed.
It Doesn't Need to Wait: You can get a reliable weight estimate using only the data from the first few weeks after the flare. You don't need to wait years for the "glow."
It Fixes a "Bias" in Our Data: The paper points out a sneaky trick nature plays on us.
- The "Hills Bias": Very heavy black holes tend to swallow stars whole without making a flash. So, the ones we see are usually the slightly lighter ones that let some light escape.
- The "Malmquist Bias": In a survey, we are more likely to see the brightest (and therefore heaviest) black holes because they are easier to spot.
- The Result: These two biases cancel each other out in a weird way, making the relationship between black hole mass and galaxy speed look "flatter" than it really is. TDEFLARE helps us see through this distortion.

The "Malmquist-Hills" Bias Explained Simply

Imagine you are trying to guess the average height of people in a room, but you can only see people who are jumping.

The Hills Effect: Very tall people (super heavy black holes) might be so tall they trip and fall before they can jump. You only see the slightly shorter people jumping.
The Malmquist Effect: You can only see people jumping if they jump high enough to be seen from your window. Short people who jump low are invisible. You mostly see the tall jumpers.

If you combine these, your sample of "jumpers" looks weirdly different from the actual crowd in the room. TDEFLARE helps astronomers correct for this so they don't draw the wrong conclusions about how black holes and galaxies are related.

The Bottom Line

The paper introduces TDEFLARE, a fast, easy-to-use code that lets astronomers weigh supermassive black holes using only the early, bright flash of a tidal disruption event.

Why it matters: As new telescopes (like the Rubin/LSST) start finding thousands of these events, we won't have time to wait years for data on each one. TDEFLARE allows us to weigh them all quickly.
The Caveat: The author is clear: TDEFLARE is a "shortcut." If you have years of data and X-ray observations, you should still use the complex, physics-heavy models because they tell you more about how the system works. But for a quick, reliable weight check on a massive population of black holes, TDEFLARE is the best tool available.

In short: TDEFLARE is a "quick-weigh" scale for black holes that works by listening to the initial explosion, saving astronomers years of waiting time while still giving them the right answer.

Technical Summary: Supermassive Black Hole Mass Inference with Tidal Disruption Event Optical Flares

Problem Statement
Tidal Disruption Events (TDEs) offer a unique probe of the quiescent supermassive black hole (SMBH) population, particularly for intermediate-mass black holes and low-mass scaling relationships. However, the vast majority of TDEs discovered by current and future optical surveys (e.g., Rubin/LSST) will be observed primarily in optical-UV wavelengths. While late-time UV plateaus and X-ray emission allow for robust mass inference via accretion disk physics, securing such data is observationally expensive, requiring long baselines and often UV coverage. Conversely, early-time optical/UV flare models are theoretically complex and often fail to reproduce known galactic scaling relationships at statistically significant levels. There is a need for a rapid, reliable method to infer SMBH masses using only the early-time optical/UV data typically available in large survey samples.

Methodology
The paper introduces TDEFLARE, a code designed to infer SMBH masses ( $M_\bullet$ ) from the optical/UV light curves of TDEs using empirically calibrated scaling relationships. The methodology proceeds in three stages:

Phenomenological Light Curve Modeling:
The code fits multi-wavelength TDE light curves using a simple phenomenological model consisting of:
- A Gaussian rise ( $L_{rise}$ ).
- An exponential decay ( $L_{decay}$ ) characterized by a peak spectral luminosity ( $L_{pk}$ ) and a radiated energy ( $E_g$ ).
- A constant late-time "plateau" luminosity ( $L_P$ ).
  The model assumes a blackbody spectral energy distribution (SED) to utilize multi-wavelength data, though the core parameters ( $L_{pk}, E_g, L_P$ ) are extracted from the g-band (rest-frame, $\nu = 6 \times 10^{14}$ Hz).
Empirical Scaling Relationships:
The extracted parameters are mapped to black hole mass using three scaling relationships calibrated against black hole masses derived from first-principles accretion disk theory (which agree with $M_\bullet-\sigma$ and $M_\bullet-M_{bulge}$ relations):
- Plateau Luminosity ( $L_P$ ): Derived from relativistic accretion theory (Equation 1).
- Peak Luminosity ( $L_{pk}$ ): Empirically calibrated (Equation 2).
- Radiated Energy ( $E_g$ ): Empirically calibrated (Equation 3).
  The code accounts for intrinsic scatter in these relationships ( $\epsilon \approx 0.38 - 0.53$ dex).
Statistical Inference and Hills Mass Correction:
- Conflation: The code combines the probability density functions (PDFs) derived from independent observables (specifically $E_g$ and $L_P$ ) using "conflation" (multiplying PDFs and renormalizing) to minimize information loss and produce a tighter mass posterior.
- Hills Mass Suppression: For high-mass systems, the code incorporates the Hills mass ( $M_{Hills}$ ), the limit above which a star is swallowed whole rather than disrupted. Using a Bayesian framework, the code marginalizes over stellar mass, black hole spin, and orbital inclination, effectively truncating the mass posterior at $M_{Hills}$ .

Key Contributions and Results
The paper demonstrates the efficacy of TDEFLARE through several specific results:

Consistency with Physical Models: When applied to TDEs with extensive data (e.g., AT2019dsg, eJ2344), TDEFLARE produces mass posteriors consistent (at $1\sigma$ ) with significantly more complex, physically motivated disk codes (FitTeD and kerrSED) that include X-ray data and full relativistic disk physics.
Recovery of Scaling Relationships: Applied to a sample of $N=89$ TDEs, the code reproduces galactic scaling relationships ( $M_\bullet-\sigma$ , $M_\bullet-M_{bulge}$ , $M_\bullet-M_{gal}$ ) with high statistical significance ( $>5\sigma$ ).
Robustness to Data Availability: The method yields reliable mass estimates even without late-time plateau data, utilizing only early-time peak and decay data. This is crucial for high-redshift or short-baseline observations.
Partial and Repeating TDEs: The code successfully infers consistent masses for partial TDEs and repeating flares (e.g., AT2022dbl), confirming its applicability beyond full disruptions.
Intrinsic Mass Function: Using the TDEFLARE constraints, the authors derive the intrinsic black hole mass function of the TDE population. The distribution peaks at $\sim 10^6 M_\odot$ , decays as a power law up to $10^8 M_\odot$ , and drops super-exponentially thereafter.
Malmquist-Hills Bias: The paper identifies and quantifies a dual bias affecting TDE scaling relationships:
- Hills Bias: At high masses, only lower-mass black holes (relative to the scaling relation) produce observable TDEs because higher masses swallow stars whole.
- Malmquist Bias: At low masses, flux-limited surveys preferentially detect the most luminous (and thus most massive) TDEs.
- Result: These combined biases artificially flatten the observed $M_\bullet-\sigma$ relationship. The authors caution that observed flattening in TDE samples does not necessarily imply a flattening of the intrinsic galactic scaling relation.

Significance and Claims
The authors position TDEFLARE as a practical, population-level tool for the upcoming Rubin/LSST era, where thousands of optically bright TDEs will be discovered. Its primary significance lies in its ability to:

Provide rapid mass constraints (seconds on a laptop) compared to computationally expensive physical disk models.
Utilize well-understood disk physics to calibrate empirical relationships for the poorly understood early-time flare phase.
Enable the study of SMBH demographics and galaxy correlations without waiting for late-time data.

The paper explicitly states that TDEFLARE is not a replacement for physical models when high-quality, multi-wavelength, late-time data is available, as physical models offer superior interpretability (e.g., constraining spin, accretion rates, and disk mass). However, for the vast majority of survey-discovered TDEs where only early optical/UV data exists, TDEFLARE offers a reliable, empirically grounded framework for mass inference. The authors emphasize that the code does not solve the fundamental theoretical problem of what powers the early flare but provides a robust empirical bridge to black hole mass constraints.