Simulating Transits of Large Objects at the L₁ Lagrange Point for the 2018 Feature Film Clara

The motion picture Clara (Sherman 2018) tells the story of an exoplanetary astronomer who makes a momentous discovery. As part of the production process in early 2017, we served as scientific advisors and were tasked with producing simulated light curves to the art department for large objects located at the L₁ Lagrange point of extrasolar planetary systems. These were of particular interest to the filmmakers as the L₁ point of a star and planet system is inherently unstable. Therefore, any object located there must have a means of performing active station-keeping, suggesting an artificial origin as described by Gaidos (2017), though, at the time of production, simulated light curves resulting from objects at L₁ or nearby had not been published.

The position of the L₁ point in a star–planet system may be approximated as the edge of the planet's Hill sphere along a line linking the centers of the planet and the star at a distance, d, from the planet (de Pater & Lissauer 2010). There are two potential geometries that can cause modifications of the planet's primary transit. (1) An object precisely at the L₁ point is visible against the star near the beginning and end of the transit, but is hidden behind the planet at near mid-transit, producing a "W" shaped transit (Gaidos 2017) or (2) an object orbiting the L₁ point offset to one side of the planet producing an unbalanced light curve, much as would be expected for an exomoon (Simon et al. 2007).

For case (2), many transits must be examined before the object can be conclusively be determined to orbit the L₁ point and not the planet itself. However, for case (1), a single transit is diagnostic (see Figures 1(b), (e)). Considering similar triangles, the separation between the centers of the planet and the L₁ object, ΔR, projected against the stellar disc can be determined at the stellar limb, as a fraction of the radius of the stellar disc, R:

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}R}{R}\cong \left(1-{\left(\displaystyle \frac{{M}_{\mathrm{planet}}}{3{M}_{\mathrm{star}}}\right)}^{1/3}\right).\end{eqnarray} \tag{ 1 }$$

Considering typical mass–radius relationships across a variety of planetary compositions (Fortney et al. 2007), radius typically increases somewhat less rapidly than the cube root of planetary mass due to internal compression. As such, the effect on the transit light curve is maximized for massive planets. A representative case is shown in Figure 1 for the star Gliese 832 and a simulated terrestrial planet. Here the disc of the star is represented by an image of the solar disc (captured by Geoff Elston on 2013 October 27) and noise has been artificially added.

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An important question not addressed by our work is whether or not large objects at or near L₁ are expected. This location is advantageous for spacecraft that seek to observe the Sun or to continuously monitor the full illuminated disc of the Earth. However, the most likely purpose of a large object at this location would be a shade designed to maintain a constant planetary temperature as the parent star ages and its luminosity increases (Gaidos 2017). A shade at L₁ would produce a constant annular eclipse as viewed from the planetary surface. For an earth-like planet orbiting a Sun-like star, a reduction in solar flux by 10%, the amount by which solar luminosity is expected to increase over the next billion years would require a shade 2200 km in diameter. This is the same order of magnitude as the planetary radius, suggesting that if such artifacts exist they would produce a maximal "W" signature.