---
title: "Why a timing wave can leave the mass uncertain"
description: "Derive the mass–free-eccentricity degeneracy, understand transit aliasing, and construct a normalized likelihood with explicit priors."
canonical_url: "https://sandboxphysics.com/stories/transit-timing/mass-and-eccentricity/"
language: "en"
page_type: "story"
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  zh-CN: "https://sandboxphysics.com/zh/stories/transit-timing/mass-and-eccentricity/"
last_modified: "2026-10-28"
---

[Transit timing, from an orbit to a mass](https://sandboxphysics.com/stories/transit-timing/)

L3 · Transit timing variations

# Why a timing wave can leave the mass uncertain

Sandbox Physics · 2026-10-28

A large timing signal can be easy to measure and difficult to interpret. Its slow amplitude depends on a planet's mass and on an eccentricity combination. The shorter conjunction component sometimes supplies the missing independent information.

## Before you begin

The two preceding chapters supply the orbit and fitted clock. Complex numbers are introduced as a compact way to store the size and direction of an eccentricity vector; Gaussian errors and conditional probability are defined here.

## Convert an angular displacement into a time

Near a nominal crossing, a small angular displacement shifts the event along the clock. Expand the angular position to first order in time: the unperturbed angular rate multiplies the timing shift. A positive angular lead therefore corresponds to a negative timing delay.

A perturbation changes both the mean longitude and the eccentric displacement around it. That second contribution matters even when the eccentricity is small. Deriving a timing formula from a change in mean longitude alone would miss part of the mass information.

$$
0\simeq n\,\delta t+\delta\theta,\qquad \delta t=-\frac{\delta\theta}{n}
$$

The angular perturbation is measured relative to a reference crossing, with longitude zero along the line of sight. This linearization requires a small displacement.

Reading: Lithwick · Deck & Agol

## Separate a driven eccentricity from a free one

An eccentricity has a magnitude and a direction toward periapse. Store those two pieces as the real and imaginary parts of a complex number. Multiplication by a complex exponential rotates a vector, so this notation follows the direction without repeatedly writing two trigonometric components.

Near a first-order resonance, the eccentricity vector contains a part driven by the nearby resonance and a freely chosen part fixed by the initial orbit. The driven part rotates with the slow conjunction longitude. The free part evolves only on the longer secular timescale in this approximation.

The leading slow timing signal sees a weighted sum of the two free vectors. It does not measure four separate real eccentricity components. This is an information limit of the formula, even before measurement noise is added.

$$
z=e\,e^{i\varpi}=z_{\rm free}+z_{\rm forced},\qquad Z_{\rm free}=f z_{1,\rm free}+g z_{2,\rm free}
$$

The imaginary unit records the second component; periapse longitude fixes direction. The coefficients f and g come from the gravitational disturbing function. Their near-two-to-one leading values are about negative 1.190 and positive 0.4284.

Reading: Lithwick · Deck & Agol

## Read the degeneracy in the amplitudes

Define the fractional offset from two-to-one using the two mean periods. The complex amplitude stores both the size and phase of the slow timing oscillation. Its mass factor multiplies a bracket containing the free eccentricity. Changing one can compensate a change in the other.

Both planets' timing series help, but they still share the same eccentricity combination. The shape of the allowed mass region can be a curved ridge. Fixing free eccentricity to zero picks a slice through that ridge; it does not demonstrate that the data measured zero eccentricity.

$$
\Delta=\frac{P_2}{2P_1}-1,\quad \phi=2\lambda_2-\lambda_1,\quad \delta t_i=\operatorname{Im}(V_i e^{i\phi})
$$

Periods and transit epochs determine the phase convention. The tutorial jointly varies those ephemerides and carries their uncertainty into the inference.

$$
V_1\simeq\frac{P_1\mu_2}{\pi\,2^{2/3}\Delta}\left(-f-\frac{3Z_{\rm free}^{*}}{2\Delta}\right),\qquad V_2\simeq\frac{P_2\mu_1}{2\pi\Delta}\left(-g+\frac{3Z_{\rm free}^{*}}{2\Delta}\right)
$$

These are leading near-resonant expressions: mu is planet mass divided by stellar mass, and the star denotes complex conjugation. Near exact resonance this expansion fails; physical event times require dynamics that resolves the resonance.

Reading: Lithwick · Deck & Agol

## Find a component that depends differently on mass

At low eccentricity, repeated conjunctions generate harmonics of the synodic angle. Their amplitudes scale with the perturbing mass and functions of orbital spacing. A resolved, nonaliased component can constrain that mass without the same free-eccentricity factor.

The observation is stroboscopic: one transit samples the orbit once each revolution. Near two-to-one, the inner second synodic harmonic falls at the slow timing frequency in the sampled data. The outer first harmonic does too. Those components already contain the zero-free-eccentricity resonant contribution. Adding complete resonant and chopping formulae would count that contribution twice.

The tutorial follows Deck and Agol's decomposition: compute the zero-free-eccentricity chopping terms, then add only the free-eccentricity correction at the slow frequency. The inner first synodic harmonic supplies the cleanest additional mass information in this case. The approximation still needs comparison with actual integrated transit times.

$$
\delta t_{1,\rm chop}=\frac{P_1}{2\pi}\mu_2\sum_{j\ge1}f_1^{(j)}(\alpha)\sin[j(\lambda_1-\lambda_2)],\qquad \alpha=a_1/a_2
$$

The dimensionless coefficients are evaluated from converged Laplace-coefficient integrals. Close spacing, large eccentricity or actual resonance can invalidate the harmonic interpretation.

Reading: Deck & Agol · Lithwick

## Measure a small component in a complete orbit

Use the saved default: twenty-Earth-mass companion, initial ratio 2.04, circular inner orbit, initial companion longitude one radian, and a two-hundred-initial-period baseline. Fit several harmonics at the companion's mean frequency to the 199 inner events while also fitting the linear clock. The first harmonic has a semiamplitude about 0.443 minutes, much smaller than the full 29.5-minute residual range.

The independently calculated leading chopping coefficient predicts about 0.456 minutes. The 2.67-percent difference includes finite mass, finite observing baseline and higher-order eccentricity effects. Fits using four, eight and twelve harmonics give a stable first component. This is a quantified approximation check on actual synthetic Newtonian events, not a new measured planet mass.

| Saved synthetic quantity | Result |
| --- | --- |
| Inner events over 200 initial periods | 199 |
| Full timing peak-to-peak range | 29.5 minutes |
| Fitted first-harmonic semiamplitude | 0.443 minutes |
| Leading perturbative prediction | 0.456 minutes |

Reading: Deck & Agol · Deck et al.

## Work a small inverse problem

Consider an arithmetic model for a measured slow amplitude: mass in Earth units multiplies one half plus ten times a dimensionless eccentricity combination, giving minutes. A ten-minute amplitude is reproduced by mass 10 with eccentricity combination 0.05, mass 20 with zero, or mass 40 with negative 0.025. These are three points on one continuous degeneracy.

Now suppose an independent component has one tenth of a minute per Earth mass and is measured as two minutes with a one-fifth-minute Gaussian error. Conditional on that exact coefficient, it gives mass 20 with a two-Earth-mass standard error. Substituting back into the slow amplitude constrains the eccentricity combination.

This exercise illustrates the geometry of information. Its coefficients are invented arithmetic values, not a calibration for Kepler-289. In a physical fit, uncertain periods, stellar mass, harmonic coefficients and model adequacy also affect the result.

$$
A=m(0.5+10z)\;\mathrm{min},\qquad B=0.1m\;\mathrm{min}
$$

Here m is the numerical mass in Earth units and z is a real teaching parameter. The two observations constrain different directions in the mass–eccentricity plane.

Reading: Lithwick · Deck & Agol

## Put uncertainty into the likelihood

A likelihood assigns a probability density to the measured times for a chosen model. For independent Gaussian errors, each squared residual is divided by its variance. If additional white timing scatter is fitted, its variance adds to the quoted variance. The logarithm of that total variance is essential: otherwise arbitrarily large scatter could improve the fit for free.

A posterior combines that likelihood with proper, finite probability distributions for the parameters. Those priors state what masses and eccentricities were allowed before using this particular timing series. Marginalization integrates over unwanted parameters; replacing them by their best values gives a profile likelihood, a different object.

Our real-data calculation jointly samples two masses, two components of the free vector, four ephemeris parameters and one positive jitter. It reports the priors, independent ensemble diagnostics and a broader-eccentricity-prior rerun. Large jitter or a new poorly explored mode is a reason to question the reduced model and the robustness of its mass interval.

$$
\ln L=-\frac12\sum_n\left[\frac{(t_n-t_{n,\rm model})^2}{\sigma_n^2+s^2}+\ln\!\bigl(2\pi(\sigma_n^2+s^2)\bigr)\right],\qquad p(\vartheta\mid t)\propto L\,p(\vartheta)
$$

The additional scatter s shares the time unit of the supplied errors. This diagonal likelihood omits unpublished timing covariance and is explicitly conditional on the approximate two-planet model.

Reading: Greklek-McKeon et al. · Deck & Agol

## Try the observation

### [Separate orbital sensitivity from inference](https://sandboxphysics.com/labs/transit-timing-variations/)

1. Compare the eccentric preset with the circular default before changing mass.
2. Change spacing and inspect which parts of the full timing pattern move; do not equate a large residual with a unique mass.
3. Continue to the measured-time tutorial and compare the conditional posterior with its residual diagnostics.

## Exercises and explained answers

A fit fixes all free eccentricities to zero and returns a narrow mass interval. What question should you ask first?

Ask whether the data constrain eccentricity independently. Fixing it removes a degeneracy by assumption. Compare a jointly varied free vector and an explicit prior-sensitivity calculation before interpreting the interval.

Why is adding the full two-to-one resonant formula to every chopping harmonic wrong?

Transit sampling aliases the inner second and outer first synodic harmonics with the slow frequency. Their zero-free-eccentricity contribution is already present in the chopping sum. Add only the free-eccentricity correction, using one consistent longitude convention.

A model's quoted timing errors are small, but it requires fifteen minutes of extra jitter. Does a narrow conditional mass interval settle the planet's mass?

It describes the chosen approximate model with an added scatter term. The large discrepancy demands a richer dynamical/photometric model or an explanation of the extra scatter. Comparable physical accuracy requires a model that accounts for the observed residuals.

[Previous: Turn stellar crossings into timing residuals](https://sandboxphysics.com/stories/transit-timing/crossings-and-residuals/) [Next: Reanalyze twenty-five measured Kepler-289 transit times](https://sandboxphysics.com/stories/transit-timing/kepler289-timing-tutorial/)

## Sources and calculation scope

1. [Lithwick, Xie & Wu, Extracting Planet Mass and Eccentricity From TTV Data (2012)](https://arxiv.org/abs/1207.4192v2)

   Pinned v2, Sections 2.1–2.3, Equations 1–10 and Appendix B. The free-eccentricity vector and the near-resonant validity conditions.

2. [Deck & Agol, Measurement of Planet Masses with Synodic Chopping (2015)](https://arxiv.org/abs/1411.0004v1)

   Pinned v1, Sections 2–5.1, Equations 10–15 and Appendix A. The transit sampling aliases some harmonics; the PH3 application explicitly compares mass constraints with and without chopping.

3. [Deck et al., TTVFast (2014)](https://arxiv.org/abs/1403.1895v1)

   Pinned v1, Sections 2.1.3, 2.2–2.3 and 3. Mid-transit is a foreground minimum of astrocentric sky separation. This teaching calculation reuses the site's RK4 kernel and independently checks IAS15; it does not port the GPL TTVFast code.

4. [Greklek-McKeon et al., Kepler-289 Densities and Transit Timing (2023)](https://arxiv.org/abs/2208.00022v1)

   Pinned v1, Sections 2–3, Table 1 and Figures 2–4, 9–10. CC BY 4.0. The author analysis jointly fits transit shapes, then fits the measured times with a three-planet dynamical model.