Characterizing Kepler's Planets with Transit Timing Variations

Introduction

The Kepler telescope has discovered thousands of planets outside our solar system. The plot below shows the size and orbital periods of Kepler's discoveries:

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The transit method that Kepler uses to discover planets readily yields information about those planets' sizes (from the depth of the transit) and orbital periods (from how frequently transits occur). In the plot above you can see that Kepler has discovered many planets that are intermediate in size between Earth and Neptune (approximately 4x the size of Earth), and with orbits ranging from a few days to ~100 days or so. Given the fact that our Solar System has no analogous planets, its natural to ask: What are these planets like?

To answer these questions, we need to know more than just these planets' sizes and orbital periods. In particular, we need to know their densities in order to distinguish between rocky, terrestrial planets similar to Earth versus planets that are similar to the gas and ice giants of the outer Solar System. Since we already know the sizes of these planets from measuring their transits, we can calculate the planets' densities if we can measure their masses. We'll be looking at a method of determining masses, and therefore measuring densities, of exoplanets based on transit observations.

Kepler's Laws tell us how a single, isolated planet orbits under the influence of its host star's gravity. Such a planet traces out an ellipse in an amount of time, the orbital period, related to the planet's average distance from the star. However, when multiple planets orbit the same star, they experience gravitational forces from each other in addition to the gravity of their host star. These gravitational forces are much weaker than the gravitational force of the star becuase the planets are much less massive. Therefore, Kepler's Laws are still an excellent approximation to the planets' motion, but are not perfect. In particular, the planets' orbital periods are no longer constant-- each successive orbit is slightly longer or shorter relative to the orbit the planet would have in the absence of other planets. In fact, the length of Earth's year can vary by up to a few minutes each year due to interactions with the other planets in the Solar System. Because the gravitational force between two objects depends on those objects' masses, if we can measure the changes in planets' orbits caused by their mutual gravitational interactions, we have an indirect probe of their masses.

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The above plot illustrates this effect for two hypothetical planets. Imagine that we start observing a star with a single transiting planet. Suppose that each time we observe the planet transiting, we record the time measured relative to when we started observing. In this way, we can construct a plot of transit time versus transit number as shown below in the blue points. What does the slope of this line represent? If there are other planets orbiting the same star and interacting gravitiationally with the transiting planet, our plot of transit number versus transit time would look something like the yellow points instead.

Computing Transit Timing Variations:

Use interactive module below to compute transit times. The module solves for the motion of two planets orbiting a star and records the times at which the inner (blue) planet transits the star (as would be observed from the direction indicated by the `Towards Earth' line). Adjust the planet's orbital properties using the sliders provided or enter values by hand after clicking the `+' sign next to each slider. Once you have set the planet parameters, compute transit times using the 'Compute Transit Times' button. After computing transit times, you can copy them your clipboard using the 'Copy Transit Times' button. They can then be pasted into a spreadsheet program such as plot.ly .

Important: Be sure to recompute transit times after adjusting any parameters. The proper transit times are not displayed until the 'Compute Transit Times' button is pressed again.

The 'Copy Transit Times' button does not function properly on the website. Download the stand-alone CDF document for a properly functioning module.

1. Run the simulation with the initial parameters. Copy the transit times to plot.ly grid. Fit a line to the data. What does the slope of this line represent? Does this agree with what you expect based on the the input paramter(s)?

2. Compute a TTV: If the blue planet didn't have a companion, the line that we fit in Part 1 would predict the transit times of the planet perfectly (Why?). However, the gravitational interactions with the outer planet result in transit times similar to the `Perturbed Planet' shown above in the discussion. However, that plot has been greatly exaggerated, and the TTVs are so small that they cannot be seen on the graph produced in Part 1. In order to see the TTVs, we'll subtract off this line. This can be done in plot.ly as follows:

3. Kepler-18 `c' and `d' are a pair of planets slightly larger than Uranus:

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and orbits of approximately 7 and 14 days, respectively. The planet pair exhibits TTVs with amplitudes of ~10 minutes as shown below:
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Astronomers have used their TTVs to determine their masses. Input the observed periods and masses quoted for these planets in the link above. Do the TTVs you compute have a similar amplitude and period? (Note: there is a third planet, Kepler 18-b in the Kepler-18 system. This planet is signifcantly smaller than the other two planets and we can this ingore theis planet for our purposes.)