Modelling the Light Echo in Active Galaxy BL Lacertae Using High-Cadence Light Curve Data

Purpose

The study of active galactic nuclei (AGN) is key to understanding the origin and evolution of the universe, as it consists of galaxies whose dynamics are determined by their nuclei. Conditions near supermassive black holes share similarities with the early universe, such as extreme density, high temperature, and intense gravity [1]. As a result, the processes that occur near supermassive black holes can provide insight into the transformations that occurred following the Big Bang.

The Challenge

Located 900 million light-years from Earth, AGN BL Lacertae (Figure 1) is a blazar with a relativistic jet pointed directly at Earth, overwhelming the light from its host galaxy and making stars and emission lines nearly impossible to detect [2]. As a result, its nucleus cannot be observed directly, and methods that rely on such observations cannot be applied to estimate the spatial dimensions of its nucleus.

Proposed Solution

The physical parameters of an active galaxy can be estimated based on a single high-cadence light curve. This is because the light emitted from the corona gets reflected by baryonic matter coupled to dark matter (Figure 2) and reaches Earth at different times depending on the light travel distance. Therefore, at any given moment, we see a sum of signals emitted during a time interval. Light echo length, the inner radius of dark matter, and orbital velocity can be predicted using regression analysis of the light curve.

Light Curve Data Source

Time Series Data from Transiting Exoplanet Survey Satellite (TESS) (Figure 3):

Regularly sampled light curve (Figure 4) with an exposure time of 20 s

Observed over 27 days (February 26th - March 24th, 2024)

Launched by NASA in 2018, TESS is designed to find exoplanets orbiting bright host stars relatively close to Earth. It is equipped with four wide-field cameras and observes in the optical wavelength range 600-1000 nm, covering red visible to near-infrared light [3].

Method 1 - ML Models

Three types of machine learning models were trained: linear regression, random forest, and multilayer perceptron (MLP). The features flux, flux_lag1, flux_lag2, flux_lag5, flux_lag10, pct_change, mean_5, and std_5 were used as inputs for the models (Figure 5). Model performance was evaluated using the coefficient of determination. R² values were computed and plotted for k = [0, 10, 20,…, 500], where k represents the number of 20-second time steps ahead the models predict. The largest value of k for which R² > 0.9 was selected, resulting in k = 75 steps for linear regression. The maximum k values for random forest and MLP are k = 60 and 70, respectively.  For random forest and MLP models, the tree depth and hidden layer sizes were chosen to maximize R².

Method 2 - Time Series Models

Box-Jenkins models [4] were used to estimate the direct light echo length:

Autocorrelation function (ACF): Correlation of time series with a lagged version of itself.

Partial Autocorrelation function (PACF): Additional correlation explained by each successive lagged term, excluding the correlations due to multiple reflections of light from the corona.

∴ Direct light echo length = time taken for PACF to reach zero.

Physical Parameters of BL Lacertae

The light echo length, ISCO and corona radius, and orbital velocity of baryonic matter and dark matter at ISCO have been estimated using machine learning models and the partial autocorrelation function of a high-cadence light curve from BL Lacertae. By comparing the ISCO radius to the Schwarzschild radius [5], it was established that BL Lacertae is a prograde Kerr black hole, meaning it is spinning in the same direction as its accretion disk [6]. The orbital velocity of baryonic and dark matter surrounding the AGN at ISCO was calculated to be 0.77 times the speed of light.

Statistical Analysis

Unlike ACF and ML prediction horizons, PACF is independent of the light curve length. This is because ACF decreases slowly (Figure 6), and the time it takes for it to reach zero is highly variable, depending on the interval of the light curve observed. Such variability is characteristic of non-stationary time series. Machine learning models are able to predict as long as ACF is greater than zero, meaning their estimates of the light echo length are also highly variable (Figure 7).

The light curve from BL Lacertae can be modelled as a sum of a non-stationary random walk (red noise) component and a stationary white-noise component. The random walk component is what causes the slow decay of ACF and a high variability of the time it takes for it to reach zero. A physical explanation for the sum of these two components could be the superposition of direct light echoes reaching the Earth after a single reflection off the accretion disk and secondary echoes arising from multiple reflections, which can circle the black hole multiple times. Only the direct reflections can inform us of the size of the AGN, while the secondary echoes must be excluded.

PACF registers correlations between two points in time that are not implied by correlations with points in between. Thus, PCF vanishes when direct echo is no longer received, which happens long before the secondary echoes disappear (Figure 8). The sum of a random walk and white noise can be equivalently represented as an ARIMA(1,1,1) process, which enables the use of standard libraries to fit the process to the time series data and calculate the time it takes PACF to reach zero analytically. The resulting estimate is stable with respect to the selection of different intervals or cadences of the light curve, which is required for any estimate of a physical property of a celestial object.

Implication of Results

By studying AGN BL Lacertae, I have used autoregressive models to measure the physical parameters of the blazar, given a single light curve. As a result, the structure of that curve can now be understood and linked to the physical properties of the celestial object emitting it.

Surprisingly, similar processes occur on Earth, such as bats using ultrasonic echolocation to detect objects and track motion in complete darkness. This means that even when an object cannot be viewed directly, its structure can still be determined through the signals it emits or reflects. For active galactic nuclei, this provides a method to estimate spatial dimensions and better understand how matter behaves under extreme gravity. More broadly, this approach expands how astronomers study distant systems by being able to extract physical meaning from light curve data and apply similar methods across both astrophysics and other fields of science.

Next Steps

Expand on the physical properties of the AGN behind the first-order ARIMA model.

Extend research to other BL Lac objects and surveys.

Select objects with a larger broad-line emission region and compare the proposed method to spectral methods, such as reverberation mapping.

Extend and apply the proposed method to unevenly sampled light curve data from other telescopes.

Determine the minimum sample cadence required for the proposed method.

Apply the proposed method to AGN’s with various masses and spins.

Acknowledgments

I would like to thank Tony Rodriguez (Harvard University) and Victoria Lloyd (Stony Brook University) for their guidance and for providing access to NASA’s TESS time series data. I would also like to acknowledge the Waterloo-Wellington Science & Engineering Fair team for their valuable feedback and my father for his encouragement and support.

Reference List

[1] “Accreting Supermassive Black Holes in the Early Universe.” Center for Astrophysics Harvard & Smithsonian, 24 October 2014, https://www.cfa.harvard.edu/news/accreting-supermassive-black-holes-early-universe.

[2] Ruina, Arshia. “Exceptional flare tests blazar emission models.” CERN Courier, 8 July 2025, https://cerncourier.com/exceptional-flare-tests-blazar-emission-models/.

[3] TESS Science Operations Center. “TESS Mission.” MIT TESS, 2018.

[4] Box, George E. P., and Gwilym M. Jenkins. Time Series Analysis: Forecasting and Control. 3rd ed., Prentice Hall, 1994.

[5] Yiting, Cen, and Song Yong. “Upper Bound on the Radius of the Innermost Stable Circular Orbit of Black Holes.” Chengdu University of Technology, 2025.

[6] Teukolsky, Saul A. “The Kerr Metric.” Center for Radiophysics and Space Research, 2015.

Cover slide of “WHY”

NASA. NASA’s Webb Delivers Deepest Infrared Image of Universe Yet. 12 July 2022, nasa.gov/image-article/nasas-webb-delivers-deepest-infrared-image-of-universe-yet/.

Cover slide of “HOW” & Figure 3

NASA. TESS Spacecraft Diagram. 2018. NASA’s Transiting Exoplanet Survey Satellite, MIT, tess.mit.edu/tess-spacecraft/.

Cover slide of “SO WHAT”

Pan-STARRS. BL Lacertae. Aladin Lite, CDS, Strasbourg Observatory, aladin.cds.unistra.fr/AladinLite/?target=BL%20Lacertae.

Cover slide of “WHAT’S NEXT”

Winters, Dan. The Artemis I mission lifts off from NASA's Kennedy Space Center. National Geographic, November 16, 2022, https://www.nationalgeographic.com/magazine/article/nasas-artemis-i-mission-launches-new-era-of-lunar-exploration.

Cover slide of “THANKS”

NASA. ESA astronaut Luca Parmitano during a spacewalk outside the International Space Station. BBC Sky at Night Magazine, 22 May 2023, skyatnightmagazine.com/space-missions/spacewalks.

Cover slide of “REFERENCES”

Gregersen, Erik. "Telescopes: Seeing Stars". Encyclopedia Britannica, 13 Jun. 2025, https://www.britannica.com/story/telescopes-seeing-stars. Accessed 30 April 2026.

Figure 1.

King, Bob. Cygnus and Lacerta star chart showing Deneb and BL Lac. Sky & Telescope, 17 Nov. 2022, skyandtelescope.org/astronomy-news/observing-news/sneak-peek-at-two-promising-ztf-comets/.

Figure 2.

Wilkins, Dan. Diagram of a black hole showing the corona, accretion disc, and X-ray emission, danwilkins.net/research.