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Sandbox Physics
Transit timing, from an orbit to a mass

L1 · Transit timing variations

Build the orbit before building the clock

Sandbox Physics ·

A transit time is an event on an orbit. To calculate that event reliably, begin with a moving star and planets, define the observer, and establish the repeating schedule that appears when the companion's mass vanishes.

Put the observer on a definite side

Draw the orbital plane horizontally. Set the observer very far along its positive horizontal axis, looking toward the star. The vertical coordinate is then the direction across the stellar image. A planet on the observer's side can block the star; a planet on the far side cannot produce that transit.

Keep two coordinate systems distinct. Barycentric positions are measured from the center of mass of all three bodies. Astrocentric positions subtract the star's location from the planet's. Newton's equations are integrated in the first system; the telescope image uses the second. The star's small wobble belongs in this subtraction.

rp⋆=rp−r⋆,vp⋆=vp−v⋆\boldsymbol r_{p\star}=\boldsymbol r_p-\boldsymbol r_\star,\qquad \boldsymbol v_{p\star}=\boldsymbol v_p-\boldsymbol v_\star
Both positions have length units. Both velocities have length per time units. A relative velocity is the derivative of the relative position.

Reading: Deck et al. · Lithwick

Every body responds to every other body

For each body, add the acceleration caused by every other mass. The displacement vector points toward the attracting body; dividing it by distance cubed leaves the familiar inverse-square strength. There is no gravitational softening in this model.

Pairwise forces are equal and opposite. Their sum therefore preserves the system's total momentum, and central pairwise forces preserve total angular momentum. The kinetic energy plus all pair potentials is also constant for this isolated Newtonian system. Individual planets can exchange energy even while the total stays fixed.

r¨i=G∑j≠imjrj−ri∣rj−ri∣3\ddot{\boldsymbol r}_i=G\sum_{j\ne i}m_j\frac{\boldsymbol r_j-\boldsymbol r_i}{|\boldsymbol r_j-\boldsymbol r_i|^3}
Mass times this acceleration gives force. In SI units the gravitational constant has units cubic meters per kilogram per square second.
E=∑imivi22−∑i<jGmimjrij,Lz=∑imi(xivy,i−yivx,i)E=\sum_i\frac{m_i v_i^2}{2}-\sum_{i<j}\frac{Gm_im_j}{r_{ij}},\qquad L_z=\sum_i m_i(x_i v_{y,i}-y_i v_{x,i})
Energy and angular momentum are independent numerical checks. Timing accuracy needs its own event test even when these invariant errors are small.

Reading: Deck et al. · Deck & Agol

An isolated planet supplies an exact null

Remove the outer planet's gravitational mass. The star and inner planet still orbit their common center of mass. Subtract their two accelerations: the relative orbit feels the sum of their masses. A circular orbit balances this relative gravity against the acceleration needed to turn at constant speed.

The same period relation holds for a bound ellipse when its radius is replaced by semimajor axis. Eccentricity changes the speed around the orbit and the time spent near different parts, but the planet returns to the same viewing configuration once per period. Repeated transits therefore lie on an exactly linear ephemeris in geometric coordinate time.

n2a3=G(M⋆+m1),P=2πnn^2a^3=G(M_\star+m_1),\qquad P=\frac{2\pi}{n}
The angular frequency is radians per time. Omitting the planet's mass is often a small approximation; the numerical null here retains it.

Reading: Deck et al. · Lithwick

Choose units without changing the clock later

Take the initial inner semimajor axis as the length unit and stellar mass as the mass unit. Setting the numerical gravitational constant to one fixes a time unit. An inner orbit of unit semimajor axis lasts about six point two eight such units, not one unit.

The teaching star has one adopted solar mass, the inner mass ratio is thirty millionths, and the physical initial inner period is twelve days. These choices imply an initial semimajor axis of about one tenth of an astronomical unit. We label the long calculation by the initial period throughout; a perturbation of the later orbit cannot quietly redefine elapsed time.

t0=a1,03/(GM⋆),P1,0/t0=2π1+m1/M⋆,tdays=12 tP1,0/t0t_0=\sqrt{a_{1,0}^3/(GM_\star)},\quad P_{1,0}/t_0=\frac{2\pi}{\sqrt{1+m_1/M_\star}},\quad t_{\rm days}=\frac{12\,t}{P_{1,0}/t_0}
In the last expression t is the numerical time in code units. The twelve-day period is an adopted teaching scale; the real Kepler-289 tutorial uses its original measured days.

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

Work a two-body timing check

Start the inner planet at periapse on the observer's side. For a circular orbit this is simply a chosen point on the circle. The first future transit occurs one period later. A finite eccentricity leaves that return interval unchanged.

Integrate for two hundred initial periods with the companion mass set to zero. Compare each computed interior event to its integer multiple of the analytic period before fitting any clock. The maximum absolute timing difference at the default numerical resolution is below one tenth of a second. After subtracting a fitted clock, the residual peak-to-peak range is below two hundredths of a second. Those are numerical residuals around a zero-valued physical prediction.

The endpoint deserves care. A root exactly on the requested final time can fall just inside or outside a floating-point record. The independent reference compares roots at least one thousandth of an initial period before that endpoint; the convention is saved with the checks.

Adopted inputValueExpected consequence
Inner mass ratio0.000030The star also moves
Initial physical period12 daysOne code period has a fixed day conversion
Companion gravitational massZeroExactly periodic geometric crossings
Numerical baseline200 initial periodsNumerical timing drift must be checked

Reading: Deck et al. · Lithwick

Add a companion without adding a timing curve

Give the second planet a nonzero mass and integrate exactly the same force law. Its repeated conjunctions change the inner planet's position and velocity. The timing variation must emerge when those positions cross the observer's direction. Adding a chosen sine wave to event times would skip the physical question.

For the default twenty-Earth-mass companion, initial period ratio 2.04 and circular inner orbit, the saved two-hundred-period calculation gives about 29.5 minutes peak-to-peak for the inner timing residuals. That number belongs to the stated synthetic initial conditions. The next chapter explains how the events and fitted clock produce it.

Reading: Deck et al. · Deck & Agol

Try the observation

Recover the isolated clock

  1. Open An isolated clock and predict the physical timing variation.
  2. Compare its small numerical residual with Near two-to-one; keep the same day scale.
  3. Read the model notice and identify the coordinate time that is being measured.

Exercises and explained answers

Why does making the coordinate origin fixed on the star require care?

The star accelerates. A frame that follows it is not inertial, so the relative planetary equations need the corresponding indirect acceleration. Integrating all three inertial bodies and subtracting the star afterward retains that effect directly.

If the adopted physical inner period doubles while every dimensionless parameter stays fixed, what happens to the timing residuals in minutes?

The numerical orbit and dimensionless event series stay identical. The time unit doubles, so every residual in minutes doubles. With stellar mass fixed, the physical length scale also grows by two to the two-thirds power.

An eccentric isolated planet leaves a slowly curved timing residual. Which check comes first?

Refine the integration and event search, then compare directly to integer analytic Kepler periods. The exact isolated eccentric orbit repeats its geometric schedule; a curved residual can come from accumulated numerical phase error.

Sources and calculation scope

  1. Lithwick, Xie & Wu, Extracting Planet Mass and Eccentricity From TTV Data (2012)

    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)

    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)

    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)

    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.

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