‘Tis the season to stare at light curves. While most people were unwrapping presents on December 25th, I was staring at synthetic flux drop-offs and debugging a limb-darkening model. The result is a browser-based simulator for exoplanet transit photometry, including binary eclipses and exomoon scenarios. It does not detect any real exomoons. It does, however, correctly model why detecting them is comically hard.

Source: sebastianspicker/exoplanet-exomoon-simulation

Background

The gift of transits

When a planet crosses in front of its host star, the observed stellar flux drops by a fraction proportional to the ratio of their projected areas:

For Jupiter transiting the Sun, \( \delta \approx 1\% \). For Earth, about 84 ppm. For an exomoon orbiting a Jupiter-sized planet — well, unwrap that calculation yourself:

A Moon-sized exomoon around a Sun-like star contributes roughly 7 ppm of flux variation. The Kepler space telescope’s photometric precision was on the order of 20–30 ppm per 6-hour cadence for bright targets. Ho ho hold on — that signal is buried.

Why stars are not uniformly bright (and why that ruins everything)

A star is not a flat disk of uniform intensity. It is darker at the limb than at the centre — an effect called limb darkening — because the line of sight through the stellar atmosphere is shallower at the edges, sampling cooler, less emissive layers. The quadratic limb-darkening law is:

where \( \mu = \cos\theta \) is the cosine of the angle from disk centre, and \( u_1, u_2 \) are stellar-type-dependent coefficients. This matters for transit modelling because the depth of the light curve dip changes as the planet traverses from limb to centre to limb — the transit is not a flat-bottomed box, it is a rounded trough. Fitting it incorrectly biases \( R_p / R_\star \) and, more critically for exomoon searches, generates false residuals that look suspiciously like a secondary dip.

Exomoon detection: the indirect approach

No exomoon has been unambiguously confirmed as of the time of writing this post (Christmas Day, 2025 — yes, really). The most promising indirect signatures are:

Transit Timing Variations (TTV). The planet–moon system orbits their common barycentre. This causes the planet’s transit to arrive slightly early or late relative to a pure Keplerian ephemeris. The timing offset scales as:

where \( m_m / m_p \) is the moon-to-planet mass ratio, \( a_m \) is the moon’s semi-major axis around the planet, and \( v_p \) is the planet’s orbital velocity. For an Earth-mass moon at 10 planetary radii around a Jupiter-mass planet at 1 AU, this is on the order of minutes — measurable, in principle, with long baselines.

Transit Duration Variations (TDV). The same barycentre wobble modulates the planet’s velocity along the line of sight at transit ingress, changing transit duration. TDV and TTV are 90° out of phase, which lets you solve for both moon mass and orbital radius given enough transits.

Neither signal is clean in practice. Stellar activity, instrument systematics, and other planets in the system all contribute correlated noise at similar timescales. The residuals of the best exomoon candidate to date — Kepler-1625b-i (Teachey & Kipping, 2018) — remain contested. Season’s readings: disputed.

The Simulation

What it actually does

The simulator is a TypeScript application (Vite build, runs in-browser) built around a deterministic, SI-unit physics core. The main pipeline:

UI parameters
  → V4 normalisation
  → runtime creation  (realtime | reference)
  → orbital integrator (Kepler + N-body)
  → geometry & photometry
  → flux decomposition
  → canvas render + plots

Two runtime modes:

• Realtime — fast integrator, interactive rendering, good for exploration
• Reference — high-fidelity integrator, deterministic export, good for sanity-checking against known systems

The photometry layer computes quadratic limb-darkened transit flux, handles binary eclipse geometry (for eclipsing binary configurations), and exposes hooks for phase curves and instrument noise.

The diagnostics layer is the part I find most useful: energy conservation checks across the integration, radial velocity time series, astrometry, and transit timing outputs. If your N-body integrator is drifting, the energy plot tells you immediately.

The repo ships a real-systems.snapshot.json with versioned data from the NASA Exoplanet Archive — so you can load, e.g., TRAPPIST-1 or HD 209458 as a starting configuration.

What it deliberately does not do

The relativistic corrections are approximations. This is not a GR integrator. For the systems it is designed for (short-period planets around Sun-like stars), the relativistic perihelion precession is tiny — Mercury’s 43 arcseconds per century is the canonical example and that is already a demanding target — but for millisecond pulsars or extremely compact binaries, do not trust it.

The atmospheric module exposes hooks but is not a radiative-transfer solver. If you want realistic transmission spectra, point yourself at something like petitRADTRANS and use this for the orbital geometry only.

Discussion

The simulation is educational in intent — hence the built-in didactic mode (black-box exploration → hypothesis → reveal → A/B comparison → rubric scoring). But the physics is not dumbed down: the limb darkening is real, the N-body integrator tracks multi-body gravitational interactions, and the TTV outputs are computed from first principles rather than parameterised fits.

The thing I kept running into while building this is how much of exomoon detection reduces to a residuals-hunting problem. You fit the best planet-only model you can, examine the timing and duration residuals, and look for a coherent signal. The simulator lets you inject a synthetic exomoon of specified mass and orbital radius, generate synthetic light curves with configurable noise, and see what the residuals look like — which is exactly the kind of intuition-building exercise that is tedious to set up from scratch with, say, a raw BATMAN lightcurve model and a custom integrator.

Limitations worth being honest about. The performance budget is real: some effects are profile-gated to keep the interactive mode responsive, which means the reference mode exists specifically for cases where you want the full physics at the cost of speed. For a publication-quality simulation you would want a dedicated N-body code (REBOUND is the obvious choice), not a browser runtime. This is a tool for understanding the problem, not for writing papers about it — which, fitting for a Christmas project, is exactly what I have time for right now.

References

• Teachey, A. & Kipping, D. M. (2018). Evidence for a large exomoon orbiting Kepler-1625b. Science Advances, 4(10). https://arxiv.org/abs/1810.02362

• Kipping, D. M. (2009). Transit timing effects due to an exomoon. MNRAS, 392(1), 181–189. https://arxiv.org/abs/0810.2243

• Mandel, K. & Agol, E. (2002). Analytic light curves for planetary transit searches. ApJL, 580, L171. https://arxiv.org/abs/astro-ph/0210099

• Claret, A. (2000). A new non-linear limb-darkening law for LTE stellar atmosphere models. A&A, 363, 1081–1190.

Merry Christmas. If you came here expecting warmth and cheer, I offer instead a synthetic light curve with a 7 ppm exomoon signal buried in 30 ppm of photon noise. Practically the same thing.

For the physical version of this — a lamp, a ball, and a smartphone measuring real transit light curves in a classroom — see Hunting Exoplanets with Your Phone. For context on where those experiments came from, see The Lab Goes Home.

Changelog

• 2026-03-05: Corrected the description of the limb-darkening variable from “$\mu = \cos\theta$ is the angle from disk centre” to “$\mu = \cos\theta$ is the cosine of the angle from disk centre.” $\theta$ is the angle; $\mu$ is its cosine.
• 2026-03-05: Corrected Claret (2000) page range from 1081–1090 to 1081–1190. The paper contains extensive tables of limb-darkening coefficients spanning 109 pages.

The Gap in the Curriculum

Most physics and astronomy teaching units that address the search for extraterrestrial life focus on exoplanets. The transit method gets visualised, a light curve gets plotted, and the lesson ends with: some exoplanets are in the habitable zone. The end.

What tends to get omitted: moons of exoplanets — exomoons — could equally be candidates for extraterrestrial life, particularly if the exoplanet itself sits in the habitable zone. The moon would then be in the habitable zone too, and a large moon could maintain the atmospheric conditions necessary for liquid water. The possibility is taken seriously in the astrophysics community, and survey data consistently shows that students find the question of life in the universe among the most interesting topics in all of science.

The pedagogical gap is this: the transit method is routinely demonstrated in analogy experiments, but the extension to exomoon detection is almost never treated experimentally, even though it is a natural continuation of the same experiment with only minor modifications. This paper is an attempt to close that gap.

What an Exomoon Signal Looks Like

When only a planet transits a star, the resulting light curve shows a characteristic symmetric dip: flux drops as the planet moves in front of the star, holds at a reduced level during full transit, and recovers as the planet exits. The normalised flux during the flat-bottomed phase is:

where the dip depth $\delta = A_p / A_s$ is determined by the ratio of the planet’s cross-sectional area to the star’s.

When the planet has a moon, the situation is more complex. The light curve is now governed by:

where $A_m$ is the moon’s cross-sectional area and $A_{pm}(t)$ is the time-dependent overlap between the planet’s and moon’s projected disks (the moon is orbiting the planet, so this overlap changes during the transit).

The consequence: additional dips and asymmetries appear in the light curve. The moon can transit slightly before the planet (causing a small flux dip before the main transit begins), or slide in front of the planet during the transit (temporarily reducing the combined occulting area, causing a brief flux recovery in the middle of the dip), or emerge from behind the planet on the exit side (causing a small dip after the main transit ends). The exact signature depends on the relative sizes of planet and moon, their orbital period ratio, and the geometry of the particular transit.

These signatures are small. In real astrophysics, this is why no exomoon has been unambiguously confirmed. In a classroom analogy experiment, the signals are large enough to see clearly — which is exactly what makes the experiment pedagogically useful.

The Experimental Setup

The starting point is a standard transit analogy experiment: a sphere (the planet) on a rod, moved slowly around a lamp (the star) by a slowly rotating motor. A light sensor — an Android smartphone running phyphox, or an Arduino with a suitable sensor — records the illuminance over time. The resulting light curve shows the characteristic symmetric transit dip.

The modification is straightforward: attach a small battery-powered motor to the planet sphere, with a smaller sphere (the moon) on the motor’s arm. The motor we used is a disco ball motor — inexpensive, widely available, and with a rotation speed that works well relative to the transit timescale if you choose the geometry appropriately.

The result is a physical system with two independent circular motions:

• The planet orbiting the star (driven by the main slow-rotation motor)
• The moon orbiting the planet (driven by the disco ball motor)

When this system transits the “star” (the lamp), the light sensor records a compound light curve with the exomoon signatures described above.

One technical note on sensors: High sample rate matters here. The exomoon signatures are brief features on top of the transit dip, and a sensor that samples too slowly will average them out. We found that the TI SensorTag CC2650, despite being a reasonable choice for the basic transit experiment, has a light sensor sample rate of only 1.25 Hz — too slow to resolve exomoon signatures reliably. Android smartphones and Arduinos both achieve adequate sample rates. The Pasco light sensor used in the paper samples at up to 20 Hz and resolves the features clearly.

Reading the Light Curves

The paper presents two distinct light curve types that emerge from the experiment, each with a different exomoon orbital configuration.

Type 1: The moon’s orbital period is short relative to the transit duration. Multiple exomoon signatures appear within a single transit. These include:

• A small dip before the main transit begins (moon transiting alone)
• Asymmetric ingress/egress (moon leading or trailing the planet)
• A brief flux recovery midway through the transit (moon passing behind the planet, reducing the total occluding area)
• A small post-transit dip (moon still in front of the star after the planet has exited)

Type 2: Specific orbital phase alignment where the moon moves directly behind the planet at the moment of maximum occultation. In this case, the deepest point of the transit corresponds to planet alone blocking the star (moon hidden behind planet). As the moon emerges from behind the planet, the total occluded area increases again briefly before both planet and moon exit.

This second case is particularly useful for quantitative analysis: if the orbital geometry is right, students can separately determine the planet’s radius from the secondary dip depth and the combined planet-moon radius from the primary dip depth.

Video + Light Curve Together

The paper recommends recording a video of the experiment simultaneously with the light sensor measurement, from the perspective of the sensor (i.e., looking up at the lamp from below). This technique — which is also central to the transit method paper — is even more valuable here.

Without the video, the exomoon signatures in the light curve are easy to misread as noise or experimental error. With the video, students can advance frame by frame through the moments corresponding to the unusual features and see exactly what the physical system was doing: the moon sliding in front of the planet, the moon emerging from the planet’s shadow, the moon transiting alone at the start or end of the main event.

The cognitive load of interpreting an unfamiliar, complex signal drops substantially when the signal can be correlated frame by frame with a visual record of what produced it.

Differentiation and Extensions

The paper suggests the exomoon experiment as an extension for higher-ability students at the end of a unit on exoplanet detection, not as the entry point. The transit method should come first; the exomoon experiment builds on it.

For students who are comfortable with quantitative analysis, the formula above allows a full treatment: given the measured light curve and a known lamp radius, students can derive both the planet radius and the moon radius from the dip depths at the appropriate moments.

Possible further extensions:

1. Noise floor investigation: systematically vary the moon’s size and determine the smallest moon still detectable. This connects directly to the real astrophysical problem — the reason no exomoon has been confirmed is that the signal is buried in noise.
2. Period ratio effects: vary the transit speed (and thus the effective period ratio between moon and planet) to see how the light curve changes.
3. Sensor comparison: test different sensor types and compare their ability to resolve exomoon signatures. This turns the instrumental limitation into an explicit investigation.

For the deeper theoretical connections — transit timing variations, the David Kipping approach to exomoon detection — see the transit simulation post, which models these effects in a browser-based tool.

For the secondary school curriculum context and the Direct Imaging pre-experiment that typically precedes the transit unit, see Fremde Welten.

References

Küpper, A., & Spicker, S. J. (2023). Ein Analogieexperiment zur Suche nach Exomonden. MNU Journal, 76(5).

Sato, M., & Asada, H. (2009). Effects of mutual transits by extrasolar planet-companion systems on light curves. Publications of the Astronomical Society of Japan, 61(4), L29–L34.

Tusnski, L. R. M., & Valio, A. (2011). Transit model of planets with moon and ring systems. The Astrophysical Journal, 743(1), 97.

Heller, R. (2018). On the detection of extrasolar moons and rings. In H. J. Deeg & J. A. Belmonte (Eds.), Handbook of Exoplanets (pp. 835–851). Springer.

Küpper, A., Spicker, S. J., & Schadschneider, A. (2022). Analogieexperimente zur Transitmethode für den Physik- und Astronomieunterricht in der Sekundarstufe I. Astronomie+Raumfahrt im Unterricht, 59(188), 46–50.

Changelog

• 2025-10-03: Updated the Tusnski & Valio (2011) reference to use article number 97, replacing the previous page range “1–16.”