An introduction to exoplanets

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3.4  Do it yourself: planet size measurement

You’ve now learned everything you need to know to interpret some transits.

Activity 3  When are exoplanet transits seen?

The interactive application below allows you to simulate a transit light curve. The sliders at the top allow you to choose the radii of the star and planet, the orbital period and the orbital inclination. You can also use the arrow keys on the keyboard to amend the values.

What does the term ‘orbital period’ mean?

It is the time taken for a planet (and its star) to complete one orbit around the centre of mass.

What does the term ‘orbital inclination’ mean?

It is the angle from which the system is viewed. An orbital inclination of exactly 90˚ would mean the orbits were seen exactly edge-on, and the planet would transit across the centre of the star from the observer’s point of view.

Below the sliders is a graph that shows the brightness of the star – a light curve. The vertical axis is brightness, with a value of 1 indicating the normal brightness of the star if looked at through a telescope with nothing obscuring the view. The horizontal axis is time in hours. The moment at which the planet passes closest to us is chosen to be time zero on this graph.

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Interactive feature not available in single page view (see it in standard view).

Reading from the graph, what is the measured brightness of the star at t = 0? You may find that a transparent ruler is helpful here. ‘(Make sure the sliders are set to Rstar = 1.27 RSun; Rplanet = 1.75 RJ; Porb = 10 days; i = 90°.)

0.98; you may have got a slightly different answer depending on how accurately you measured from the screen.

What is the value for the transit depth (displayed in the text on the right-hand side above the graph)?

2.00%

How does this value of 2% for the transit depth relate to the measured brightness of the star at mid-transit?

The measured brightness of the star is 0.98, which is 2% less than the value of exactly 1 that is the normal brightness of the star. The transit depth tells us how much light is blocked by the planet, and is expressed as a percentage of light blocked from the star.

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The interactive application has been repeated so the sliders can be adjusted as you work through the next questions.
Interactive feature not available in single page view (see it in standard view).

Adjust the value of the orbital inclination, slowly reducing it from i = 90° to i = 85.5°. Watch carefully how the graph changes.

Describe the graph for i = 85.5°.

The graph is almost entirely a straight line of brightness equal to exactly 1. There is just a tiny dip centred on t = 0.

What is the transit depth for these settings? (Rstar = 1.27 RSun; Rplanet = 1.75 RJ; Porb = 10 days; i = 85.5°.)

0.15% according to the text shown in the application. It would be impossible to measure the graph with this precision.

Can you explain the changes in the graph as you varied the orbital inclination?

Hint: it might help to look back at Figure 12, and imagine what would happen to your three-dimensional view if your viewpoint got higher. A higher viewpoint is what a decrease in the angle i means.

For i = 90° exactly, the planet crosses the middle of the star’s circular cross-section, passing through the centre of the circle. As i is decreased, your viewpoint gets higher, and the planet appears to cross the star below the middle. The planet is in front of your view of the star for a shorter time for i < 90°; as shown in Figure 12, the planet’s path across the star is slightly shorter when it appears below the centre. As the planet’s path across the star gets lower as a result of our viewpoint getting higher, it also gets shorter. Finally, the planet will just graze the edge of the star. Now, the area covering the star is less than a full circle so the transit depth decreases. The light curve also changes shape: the dip is no longer flat-bottomed because the amount of light being blocked is continuously changing. It is this grazing transit that is shown by the graph for orbital inclinations between about i = 86.4° and i = 85.5°.

Can you predict what the graph will show when the orbital inclination decreases from i = 85° down to i = 0°? Explain your reasoning.

For smaller values of i our viewpoint will be so high that the planet will never appear to pass in front of the star. The full brightness of the star should be visible at all times. The graph will be a straight line with measured brightness of exactly 1 throughout.

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