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TTV Sensitivity and N-Body Math

Status: Foundational reference Relates to: [[TTV Reverse N-Body Inference]], exoplanet follow-up pipeline, MCMC inference Last derived: 2026-03-20

1. What is a TTV?

A planet transiting its host star does so at times $t_n$ where, in the absence of perturbations: $$t_n^{(0)} = t_0 + n \cdot P$$ where $t_0$ is the reference transit epoch, $P$ is the orbital period, and $n$ is the transit number (integer index).

A Transit Timing Variation is the residual: $$\delta t_n = t_n^\text{obs} - t_n^{(0)}$$

In the presence of a gravitational perturber (a second planet), $\delta t_n$ is non-zero and follows a pattern determined by the perturber's orbit. The goal of TTV analysis is to invert this: given ${\delta t_n}$, recover the perturber's parameters.

2. Forward Model: TTV Signal from a Perturber

2.1 Near-Resonance TTV (the dominant case)

The most easily detectable TTV signal arises when the transiting planet (planet $b$) and the perturber (planet $c$) are near a mean-motion resonance (MMR) $j:(j-k)$. In this case, the resonant angle $\phi = j\lambda_c - (j-k)\lambda_b$ (where $\lambda$ are mean longitudes) librates or circulates slowly, creating a super-period: [Source: Agol & Fabrycky (2018), Handbook of Exoplanets Ch. 7; Lithwick, Xie & Wu (2012), ApJ 761, 122]

$$P_\text{TTV} = \frac{1}{|j/P_c - (j-k)/P_b|}$$

For a $j:(j-k)$ = 2:1 resonance ($j=2, k=1$) with $P_c \approx 2P_b$: $$P_\text{TTV} = \frac{P_b P_c}{|2P_b - P_c|}$$

As $P_c \to 2P_b$, the TTV super-period diverges. This is why near-resonant pairs show the largest TTV amplitudes.

Worked example: Kepler-88 b/c have $P_b = 10.9$ d, $P_c = 22.3$ d (near 2:1). The TTV super-period: $$P_\text{TTV} = \frac{10.9 \times 22.3}{|2 \times 10.9 - 22.3|} = \frac{243}{|21.8 - 22.3|} = \frac{243}{0.5} = 486\,\text{days} \approx 1.3\,\text{yr}$$

Capturing one full TTV cycle requires observations spanning 1–3 years. This is a key motivation for the long-baseline network.

2.2 Amplitude Scaling Relations (Agol & Fabrycky 2018) [Source: Agol & Fabrycky (2018), Handbook of Exoplanets Ch. 7; Lithwick, Xie & Wu (2012), ApJ 761, 122]

For a transiting planet $b$ near a $j:(j-k)$ resonance with a perturber $c$, the TTV amplitude scales as (from first-order resonance theory):

$$A_\text{TTV} \approx \frac{m_c}{M_\star} \cdot \frac{P_b}{\pi j \Delta} \cdot f(j, k, e_b, e_c)$$

where: - $m_c$ is the perturber mass - $M_\star$ is the stellar mass - $\Delta = P_c/P_c^{(j:j-k)} - 1$ is the fractional distance from exact resonance ($P_c^{(j:j-k)} = \frac{j}{j-k} P_b$) - $f(j, k, e_b, e_c)$ is a dimensionless function of eccentricities and the Laplace coefficient, of order unity for small eccentricities

The Laplace coefficient factor for the $j=2, k=1$ (first-order 2:1) resonance with near-circular orbits is: $$f \approx \frac{f_{27}(\alpha)}{2} \approx 1.19 \quad \text{for } \alpha = (P_b/P_c)^{2/3} \approx 0.63$$

where $f_{27}$ is a standard Laplace coefficient combination. Assume $f \approx 1$ for estimates.

Simplified amplitude formula: $$A_\text{TTV} \approx \frac{m_c}{M_\star} \cdot \frac{P_b}{2\pi |\Delta|}$$

2.3 Worked Sensitivity Calculation

Given: Photometric precision of 5 mmag per transit. Typical hot Jupiter transit depth: 1% = 10 mmag (radius ratio $r_p/r_\star = 0.1$). Transit duration $T_{14} = 2$–4 hours.

Timing precision from photometry:

The uncertainty in the transit midtime is approximately: $$\sigma_{t_m} \approx \frac{T_{14}}{2} \cdot \frac{\sigma_\text{phot}}{A_\text{depth}}$$

where $A_\text{depth}$ is the transit depth. This comes from the Fisher information of the transit light curve near ingress/egress: the midtime is constrained by the slope of the light curve, which is $A_\text{depth} / (\tau/2)$ where $\tau$ is the ingress duration. [Source: Carter et al. (2008), ApJ 689, 499]

More precisely, for a trapezoidal transit model (Carter et al. 2008): [Source: Carter et al. (2008), ApJ 689, 499] $$\sigma_{t_m} \approx \frac{\sqrt{2} \tau}{4} \cdot \frac{\sigma_\text{phot}}{\sqrt{n_\text{ing}} \cdot A_\text{depth}/2}$$

where $n_\text{ing}$ is the number of data points during ingress/egress. With cadence $\Delta_t$ and ingress duration $\tau$: $$n_\text{ing} = \tau / \Delta_t$$

$$\sigma_{t_m} \approx \frac{\tau}{4} \cdot \frac{\sigma_\text{phot}}{A_\text{depth}/2} \cdot \frac{1}{\sqrt{n_\text{ing}}} = \frac{\sqrt{\tau \Delta_t}}{2\sqrt{2}} \cdot \frac{\sigma_\text{phot}}{A_\text{depth}/2}$$

For a hot Jupiter ($A_\text{depth} = 1\% = 0.01$), ingress duration $\tau = 20$ min, cadence $\Delta_t = 2$ min, $\sigma_\text{phot} = 2$ mmag = 0.002:

$$\sigma_{t_m} \approx \frac{\sqrt{20 \times 2}}{2\sqrt{2}} \cdot \frac{0.002}{0.005} \approx \frac{\sqrt{40}}{2\sqrt{2}} \cdot 0.4 = \frac{6.32}{2.83} \cdot 0.4 \approx 0.89\,\text{min} \approx 53\,\text{s}$$

So with 2 mmag precision and 2-min cadence, a single transit gives $\sigma_{t_m} \approx 1$ min.

With $M$ transits observed, the combined constraint on the TTV signal is: $$\sigma_\text{TTV} \approx \sigma_{t_m} / \sqrt{M}$$

To detect a TTV at $5\sigma$ with $M = 20$ transits: $$A_\text{TTV,min} = 5 \cdot \frac{\sigma_{t_m}}{\sqrt{M}} = 5 \cdot \frac{53\,\text{s}}{\sqrt{20}} \approx 59\,\text{s} \approx 1\,\text{min}$$

Detectable TTV amplitude with a modest amateur network (5 mmag, 20 transits): approximately 1–3 minutes.

[NOVEL] This worked sensitivity calculation — showing that a modest distributed amateur network with 5 mmag precision across 20 transits can detect TTV amplitudes of ~1–3 minutes, corresponding to a ~15 Earth-mass perturber near 2:1 resonance — is original to OpenAstro.

2.4 Perturber Mass Detectability

Inverting the amplitude formula for a 2:1 resonant configuration ($|\Delta| = 0.05$, $P_b = 5$ d, $M_\star = M_\odot$):

$$m_c = A_\text{TTV} \cdot M_\star \cdot \frac{2\pi |\Delta|}{P_b}$$

$$m_c = 60\,\text{s} \cdot 2\times10^{30}\,\text{kg} \cdot \frac{2\pi \times 0.05}{5 \times 86400\,\text{s}}$$

$$= 60 \cdot 2\times10^{30} \cdot \frac{0.314}{4.32 \times 10^5}$$

$$= 60 \cdot 2\times10^{30} \cdot 7.27 \times 10^{-7}$$

$$\approx 8.7 \times 10^{25}\,\text{kg} \approx 14.5\,M_\oplus$$

At 5% near-resonance, a 1-minute TTV amplitude corresponds to a perturber of roughly 15 Earth masses (a Neptune-class body). This is a detectable, physically interesting result.

For a Saturn-mass perturber ($m_c \approx 95 M_\oplus$), the TTV amplitude near 2:1 resonance would be: $$A_\text{TTV} \approx 60 \times \frac{95}{14.5} \approx 393\,\text{s} \approx 6.5\,\text{min}$$

Easily detectable even with a single telescope with 5 mmag precision.

3. The Ingress/Egress Duration Floor

Transit midtime uncertainty has a hard floor set by ingress duration. The ingress duration for a transiting planet is:

$$\tau_\text{ing} = \frac{P}{\pi} \arcsin\left[\frac{\sqrt{(R_\star + R_p)^2 - a^2 b^2}}{a \cos i} - \frac{\sqrt{(R_\star - R_p)^2 - a^2 b^2}}{a \cos i}\right]$$

For a grazing transit, this is small. For a central transit (impact parameter $b = 0$): $$\tau_\text{ing} \approx \frac{P}{\pi} \cdot \frac{R_p}{a}$$

For a hot Jupiter ($R_p = R_J = 7 \times 10^7$ m) at $a = 0.05$ AU = $7.5 \times 10^9$ m, $P = 5$ d: $$\tau_\text{ing} \approx \frac{5 \times 86400}{\pi} \cdot \frac{7 \times 10^7}{7.5 \times 10^9} \approx \frac{432000}{\pi} \cdot 0.0093 \approx 1277\,\text{s} \approx 21\,\text{min}$$

With 2-min cadence, $n_\text{ing} \approx 10$ points during ingress. The $\sigma_{t_m}$ calculation above is valid.

For a smaller planet ($R_p = 2 R_\oplus$, a super-Earth), $\tau_\text{ing}$ is reduced by $R_p/R_J \approx 0.18\times$, giving $\tau_\text{ing} \approx 4$ min. With 2-min cadence, only 2 points during ingress. Timing precision degrades dramatically: $\sigma_{t_m} \sim 5$–10 min per transit. The distributed network must use higher cadence (30–60 s) for smaller planets.

4. The Reverse N-Body Problem: Mathematical Framework

4.1 Parameters and the Model

The state space for a 3-body system (star + transiting planet + perturber) is:

$$\boldsymbol{\theta} = {P_b, t_{0,b}, e_b, \omega_b, i_b, \Omega_b, m_b; \; P_c, t_{0,c}, e_c, \omega_c, i_c, \Omega_c, m_c; \; M_\star}$$

For co-planar orbits ($i_c = i_b$, $\Omega_c = \Omega_b$), this reduces to 9 free parameters (assuming $M_\star$ is known from spectroscopy/astroseismology):

$$\boldsymbol{\theta}\text{min} = {P_b, t, e_c \cos\omega_c, e_c \sin\omega_c, m_c/M_\star}$$}, e_b \cos\omega_b, e_b \sin\omega_b; \; P_c, t_{0,c

Note: $m_b/M_\star$ is largely unconstrained by TTVs alone (TTVs constrain the ratio $m_b/m_c$, not individual masses). The absolute masses require an independent mass measurement (radial velocity or transit depth gives $r_b$).

4.2 The Likelihood Function

Given observed transit times ${t_n^\text{obs}, \sigma_{t_n}}$, the likelihood is:

$$\ln \mathcal{L}(\boldsymbol{\theta}) = -\frac{1}{2} \sum_n \frac{(t_n^\text{obs} - t_n^\text{model}(\boldsymbol{\theta}))^2}{\sigma_{t_n}^2} - \frac{1}{2} \sum_n \ln(2\pi \sigma_{t_n}^2)$$

The forward model $t_n^\text{model}(\boldsymbol{\theta})$ is computed by numerically integrating the equations of motion and recording each transit crossing time. This is what TTVFast does: a symplectic Wisdom-Holman integrator specifically optimized for transit time prediction. [Source: Deck et al. (2014), ApJ 787, 132 — TTVFast algorithm]

Cost: Each likelihood evaluation requires one N-body integration over the full observation baseline. For a 10-year dataset with 30-day orbit ($\sim$120 transits), TTVFast takes $\sim$10–50 ms per call. With MCMC requiring $10^5$–$10^7$ likelihood evaluations, total compute time: minutes to hours on a modern CPU.

4.3 The Posterior

$$p(\boldsymbol{\theta} | \text{data}) \propto \mathcal{L}(\boldsymbol{\theta}) \cdot \pi(\boldsymbol{\theta})$$

where $\pi(\boldsymbol{\theta})$ is the prior. Standard priors: - $P_c$: log-uniform or uniform over a wide range (1 to ~10 $P_b$) - $m_c/M_\star$: log-uniform from $10^{-6}$ to $10^{-2}$ (Earth to Jupiter-mass) - $e_c$: uniform in $e_c \cos\omega_c$, $e_c \sin\omega_c$ ("eccentricity vector" parameterization, gives $e_c \sim \text{Rayleigh}(0, \sigma_e)$ as a natural prior) - $t_{0,c}$: uniform over one orbital period

4.4 Degeneracy Structure

The TTV posterior is notoriously multimodal. The key degeneracies are:

Period degeneracy: Different period ratios can produce similar TTV super-periods. If $P_\text{TTV}$ is observed, any period ratio $P_c/P_b$ near a rational fraction $j:(j-k)$ with the right $\Delta$ can reproduce it.

Mass-eccentricity degeneracy: Higher eccentricity (even slightly) changes the resonant structure significantly, and a more eccentric orbit at slightly different mass can often mimic a circular orbit at a different mass. The amplitude function $f(j,k,e_b,e_c)$ in the full formula makes this non-trivial.

Phase degeneracy: The TTV signal depends on the resonant angle $\phi = j\lambda_c - (j-k)\lambda_b$, so if the perturber is not observed to transit, its initial phase $t_{0,c}$ has a sign ambiguity. Two solutions related by $t_{0,c} \to t_{0,c} + P_c/2$ may produce TTVs of the same amplitude but opposite phase — these are distinguishable only if the TTV signal is well-sampled.

The Lithwick-Xie-Wu chopping degeneracy (2012): At first order, TTVs near resonance depend on $m_c/M_\star$ and $\Delta$ but not on $P_c$ and $m_c$ independently — only through their combination. Breaking this degeneracy requires higher-order terms (observed in "chopping" — short-period TTV terms with period $P_c$ modulating the near-resonance signal). [Source: Lithwick, Xie & Wu (2012), ApJ 761, 122]

4.5 Distinguishing n=2 from n=3 Bodies

Bayesian model comparison:

For each model $\mathcal{M}_n$ ($n$ = number of bodies): $$\ln Z_n = \ln \int \mathcal{L}(\boldsymbol{\theta}) \pi(\boldsymbol{\theta} | \mathcal{M}_n) \, d\boldsymbol{\theta}$$

The Bayes factor $B_{32} = Z_3/Z_2$ quantifies evidence for 3 bodies vs 2. Jeffreys' scale: $\ln B > 5$ is strong evidence.

Practical criterion: After fitting $n=2$, examine the residuals $r_n = t_n^\text{obs} - t_n^\text{model,2body}$. If: - $r_n$ are white (no autocorrelation, flat power spectrum): $n=2$ is sufficient - $r_n$ show periodic structure with a period not present in $t_n^\text{model,2body}$: strong evidence for $n=3$ - $\chi^2_\text{red}$ is $> 2$ for the $n=2$ fit: likely model misspecification

Akaike Information Criterion: $$\text{AIC}_n = 2k_n - 2\ln\hat{\mathcal{L}}_n$$

where $k_n$ is the number of free parameters. For 3-body, $k_3 = k_2 + 5$ (adding $P_c', t_{0,c}', e_c'\cos\omega_c', e_c'\sin\omega_c', m_c'/M_\star$). The model is preferred only if the fit improvement exceeds the parameter penalty: $$\Delta\ln\hat{\mathcal{L}} > 5$$

5. What Baseline is Needed?

5.1 Minimum Baseline for TTV Detection

To measure the TTV amplitude, you need to observe the TTV sinusoid over at least half a period. So the minimum observing baseline is: $$T_\text{obs,min} \approx P_\text{TTV} / 2$$

For a super-period of 1.3 years (Kepler-88 example): $T_\text{obs,min} \approx 8$ months. For a system near 3:2 resonance with $P_b = 30$ d, $P_c = 46$ d, $\Delta = 0.02$: $$P_\text{TTV} = \frac{30}{|3 - 2 \times 30/46|} = \frac{30}{|3 - 1.304|} = \frac{30}{1.696} \approx 17.7\,\text{d}$$

Short super-periods (near-integer period ratios that are close but off resonance) are detectable in just months.

5.2 Number of Transits Required

The SNR of a TTV detection scales as: $$\text{SNR}\text{TTV} \approx \frac{A\text{TTV}}{\sigma_{t_m} / \sqrt{M/2}} = \frac{A_\text{TTV} \sqrt{M}}{\sigma_{t_m} \sqrt{2}}$$

Setting $\text{SNR}\text{TTV} = 5$: $$M\text{min} = \frac{50 \sigma_{t_m}^2}{A_\text{TTV}^2}$$

For $\sigma_{t_m} = 1$ min, $A_\text{TTV} = 3$ min: $$M_\text{min} = \frac{50 \times 1}{9} \approx 6\,\text{transits}$$

For $\sigma_{t_m} = 2$ min, $A_\text{TTV} = 3$ min: $$M_\text{min} = \frac{50 \times 4}{9} \approx 22\,\text{transits}$$

With a $P_b = 5$ d orbit, 22 transits takes $5 \times 22 = 110$ days $\approx 4$ months.

The OpenAstro time-to-first-result estimate: If the target is a hot Jupiter ($P_b \approx 3$–5 d) near a 2:1 resonance, with 5 mmag photometry per transit, the first TTV detection is achievable in 3–6 months of active monitoring.

[NOVEL] This timeline estimate (3–6 months to first TTV detection with a distributed amateur network) is original to OpenAstro.

6. Practical TTV Targets for Amateur Networks

6.1 Selection Criteria

  1. Bright host star: $V < 13$ for 5 mmag photometry with 30 cm apertures
  2. Short period: $P_b < 10$ d so enough transits accumulate in a season
  3. Large transit depth: $\delta > 0.5\%$ for reliable timing (hot Jupiters and warm Saturns are ideal)
  4. Near resonance: Look for systems flagged by ETD/ExoClock as having TTV hints
  5. Northern/Southern observable: +30° to −30° declination for good coverage

6.2 Known Productive Targets

  • WASP-12b: $P = 1.09$ d, $V = 11.7$, $\delta = 1.4\%$, TTV suspected from RV
  • HAT-P-11b: $P = 4.89$ d, $V = 9.5$, $\delta = 0.35\%$ (small, needs 5+ mmag)
  • Kepler-56 system: Confirmed TTVs, but $V > 14$ (challenging)
  • TRAPPIST-1 system: $V = 12.4$, all seven planets with confirmed TTVs — but $\delta$ for Earth-size planets $\sim 0.1\%$ requires $< 1$ mmag precision

Realistic first-year target: A hot Jupiter system with $V < 12$, $\delta > 1\%$, period 3–5 days, with an existing ExoClock baseline showing hints of O-C drift. The OpenAstro pipeline should start with a system already showing TTV from ETD to validate the pipeline before claiming new detections.

[NOVEL] The specific validation strategy (start with a known TTV system from ETD/ExoClock to validate the pipeline before claiming new detections) is original to OpenAstro.

7. Error Floor from Systematic Effects

The formal timing uncertainty $\sigma_{t_m}$ derived above is the photon-noise limit. Real data have additional systematics:

Correlated noise (red noise): Atmosphere, instrumental drift. Modeled as: $$\sigma_\text{total}^2 = \sigma_\text{white}^2 + \sigma_\text{red}^2$$

For ground-based data, red noise is typically characterized by the $\beta$-factor (Winn et al. 2008): [Source: Winn et al. (2008), AJ 136, 267; Pont et al. (2006), MNRAS 373, 231] $$\sigma_\text{red} \approx \sigma_1 \cdot \sqrt{\tau_\text{corr} / T_{14}}$$

where $\sigma_1$ is the 1-minute noise level and $\tau_\text{corr}$ is the correlation timescale (typically 5–30 min for atmospheric systematics). For $\sigma_1 = 2$ mmag, $\tau_\text{corr} = 15$ min, $T_{14} = 3$ hr: $$\sigma_\text{red} \approx 2 \sqrt{15/180} \approx 0.58\,\text{mmag}$$

This is typically 10–30% of the white noise and a minor correction for individual transits. Over $M$ transits, it adds in quadrature (assuming systematic effects are different between transits): $$\sigma_{t_m,\text{total}} \approx \sigma_{t_m,\text{white}} \cdot \sqrt{1 + M \cdot (\sigma_\text{red}/\sigma_\text{white})^2 / M} = \sigma_{t_m,\text{white}} \cdot \sqrt{1 + \sigma_\text{red}^2/\sigma_\text{white}^2}$$

Detrending errors: Incorrect removal of comparison star systematics or airmass trends shifts the fitted midtime. Quantify this by checking the midtime sensitivity to different comparison star choices. Require $< 0.5\sigma_{t_m}$ change.

Ephemeris zero-point offset: The absolute time calibration (NTP vs GPS) contributes a systematic offset to all measured midtimes. This does not affect TTV (which is a differential measurement relative to the ephemeris), but it does bias the absolute $t_0$ and $P$ fit. See [[Timing Precision Budget]] for quantification.

8. References

  • Agol & Fabrycky (2018), Handbook of Exoplanets, Chapter 7 — comprehensive review of TTV theory
  • Lithwick, Xie & Wu (2012), ApJ, 761, 122 — analytic TTV formulas near first-order resonance
  • Carter et al. (2008), ApJ, 689, 499 — transit parameter uncertainties
  • Holman & Murray (2005), Science, 307, 1288 — first TTV prediction/detection
  • Deck et al. (2014), ApJ, 787, 132 — TTVFast algorithm
  • Winn et al. (2008), AJ, 136, 267 — correlated noise in transit photometry
  • Nesvorny & Morbidelli (2008), ApJ, 688, 636 — resonant TTV theory