ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion O O O O O B-CelestialObject O B-CelestialObject O O O O O O O O O B-CelestialObjectRegion O O O O O O O O O B-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
tokens: At every rotational phase, a cross-section – or ‘great circle’ – is defined by a west longitude and an east longitude 180° to the east. The first step in calculating the cross-sectional area is finding the distance from the centre of the planet to every point on the projected silhouette. This distance might not necessarily be the elevation at the point on the great circle because the silhouette could be dominated by a taller point that sits on a longitude that isn’t used to define the great circle. In other words, a taller point might be behind or in front of the point on the great circle from our point of view. To account for this projected geometry issue, we first define an ‘α angle’, which is the number of degrees a planet would have to rotate in order for its tallest feature to disappear below the horizon after it appears on the great circle. For example, Everest is the tallest mountain on Earth, and Earth would have to rotate 32 from the time when Everest is on the great circle before the tip of Everest would no longer appear in the planet’s silhouette. Here, l is the latitude of the highest point on the planet and h is the elevation of the highest point. For both the west and east longitude lines used to define the great circle, we step through in latitude, and at every latitude, we find the maximum projected height. The projected height of point A is where d is the actual elevation of point A and β is the angle between A and the point along the great circle. β ranges from −α to +α. The projected heights give us the distances from the centre of the planet to every point along the silhouette. The area of the silhouette can then be found using trapezoidal integration. Once the planet’s cross-sectional area is found, the transit depth can be calculated using equation ( 1 ). 3.5 Defining scatter The projected geometry method described above provides a list of transit depths for every rotational phase, so σ B is easily defined as the std of all of the depths. We use these σ B values to derive the general relationship between B an σ B in Section 3.6 . In the Earth’s case, a transit would last for up to approximately 13 h – more than half a rotational period – and thus we would observe the full range of phases necessary to calculate σ B . However, other transiting planets may not last long enough for us to observe a sufficient fraction of a planet’s rotation. Instead, individual transits will show various ranges of rotational phases, which may or may not overlap depending on the planet’s rotational period. Unfortunately, this means that we might not observe every rotational phase. When this is the case, the σ B we find in our observations may be less accurate than the ‘true value. In this section, we merely describe the two ways to evaluate σ B . The first way is to find the average depth of each individual transit (see Fig. 4 ), and take the std of those averaged depths. The second way is to fold the light curve and take the std of the