Read transit duration as an orbital speedometer

Begin with a measured chord and period

A planet can take a short time to cross because the chord is short or because it moves quickly. The duration alone cannot separate those possibilities. Ingress and the fully contained interval help constrain the path, while an independent stellar density supplies an orbital scale for the measured period.

Section 2 of the photoeccentric paper groups those observables into a speed factor. Its approximation treats the speed as nearly constant across a short transit. It is therefore different from solving an arbitrary long segment of an eccentric orbit at one fixed speed.

Define eccentricity $e$ and argument of periapse $\omega$ in the paper’s convention: a transit at periapse has $\omega=90^\circ$ , and one at apoapse has $\omega=-90^\circ$ . The speed relative to the circular expectation is:

g(e,\omega)=\frac{1+e\sin\omega}{\sqrt{1-e^2}}

A larger

g

gives a faster crossing and shorter duration for the same chord. Keeping the orientation convention matters for the sign of the sine.

Reading: Dawson & Johnson 2012, Photoeccentric Effect I, v3 · Kipping 2010, Finite Integration Time, arXiv v1

Follow the density relation one step at a time

The circular duration fit interprets a faster crossing as a larger scaled orbital axis at the same period. In the short-transit approximation this scale changes by $g$ . Kepler’s inferred density contains the cube of that scale, giving the paper’s equation 6:

A discrepancy is informative only after the light curve has been modeled appropriately. Unaccounted exposure averaging, an incorrect limb profile, blending or a wrong host can alter the fitted shape and orbital scale. Our real TESS tutorial therefore reports a conditional circular density; it does not convert that one value into a measured eccentricity.

For example, a fitted circular density 3.375 times an independently known stellar density gives $g=1.5$ . This says that the short-transit speed factor exceeds the circular expectation. It still leaves two orbital parameters in one equation.

\rho_{\rm circ}=g^3\rho_\star,\qquad g=\left(\frac{\rho_{\rm circ}}{\rho_\star}\right)^{1/3}

The first density comes from a circularly interpreted transit; the second is the independent stellar density. The relation retains the paper’s stated orbital and negligible-mass approximations.

Reading: Dawson & Johnson 2012, Photoeccentric Effect I, v3 · Mandel & Agol 2002, Analytic Lightcurves, arXiv v1

Work two orbits with the same factor

At periapse the factor becomes $\sqrt{(1+e)/(1-e)}$ . Solving that expression for the least eccentric orbit that can reach a specified $g>1$ gives:

For $g=1.5$ , the minimum is $e=0.3846$ , attained at periapse. Another allowed choice is $e=0.6$ : its square-root denominator is 0.8, so $1+0.6\sin\omega=1.2$ . Thus $\sin\omega=1/3$ , with orientations near 19.5 or 160.5 degrees. Both give the same speed factor as the less eccentric periapse crossing.

Even $g=1$ does not algebraically force a circular orbit. At nonzero eccentricity an appropriate orientation can reproduce that factor. The next step assigns probability weights to these allowed orbits.

e_{\min}=\frac{g^2-1}{g^2+1}

This is the low-eccentricity edge compatible with the approximate speed factor. Probability weights enter in the next section.

Reading: Dawson & Johnson 2012, Photoeccentric Effect I, v3

Read the contours in Figure 2

Figure 2 plots simulated eccentricity–orientation posteriors for specified speed factors and uncertainty. Read the axes first, then the contour levels and the two orbital-scale rows. These are illustrative posterior calculations under the paper’s assumptions, not repeated measurements of one planet.

The curved regions trace the relation between eccentricity and orientation. Their enclosed probability also depends on the priors and on which systems can transit. A thin geometric curve from the equation would not contain those probability weights.

For a speed factor consistent with one, the paper can prefer low eccentricity after weighting the allowed configurations. That is compatible with the algebraic existence of some higher-eccentricity orientations. Existence, relative probability and a measured confidence interval are different outputs.

Reading: Dawson & Johnson 2012, Photoeccentric Effect I, v3

Change the stellar information in Figure 3

Figure 3 changes the independent stellar-density uncertainty. The cube root weakens its influence on the speed factor: if the circular density were known much more precisely, a small fractional stellar-density error contributes about one third as much fractional error in $g$ . The full joint uncertainty need not be symmetric after transforming into eccentricity.

Read Section 3.2 beside the contours. Its Bayesian construction accounts for additional transit parameters and the requirement that the orbit actually produces the observed transit. A simple least-squares curvature at one optimum cannot describe a curved or boundary-limited posterior by itself.

For a new application, retain the stellar-density source, exposure model, limb and dilution assumptions, eccentricity/orientation priors and transit-selection treatment. State which quantity was directly constrained: a duration, a circular density, a speed factor or a posterior over orbital elements. That sequence makes a strong photoeccentric inference inspectable without turning every short transit into a uniquely determined eccentric orbit.

Reading: Dawson & Johnson 2012, Photoeccentric Effect I, v3 · Kipping 2010, Finite Integration Time, arXiv v1

Read transit duration as an orbital speedometer

Begin with a measured chord and period

Follow the density relation one step at a time

Work two orbits with the same factor

Read the contours in Figure 2

Change the stellar information in Figure 3

Exercises and explanations

Try the prediction

Distinguish a short chord from a fast crossing

Sources and calculation scope