Effect of eccentricity
Simultaneous fitting two transits of an eclipsing binary is not evident. It is not allways clear how the various parameters interact. For SPH10003247 I used this approach:
- start fitting the secondary (shallowest) transit by changing the relative luminosity of the stars. A luminosity ratio of 0.55 matches the secondary transit. And also the primary transit can be matched (black lines in figure). A further increase (say 0.52) of the luminosity difference will make the secondary transit to shallow. A decrease (0.6) will make the primary transit not deep enough.
- A shallow secondary transit can also be produced by having a size difference between the two stars. Increasing the size of the secondary star to 1.26, will produce the right secondary depth. However this will not produce the right depth of the primary transit. The advantage of a size ratio is also that we will have a flat transit as we have in the data.
- Using a luminosity difference seems to be the preferred explanation. The next thing we need to explain is the shape of the bottom of the transits. Unfortunately any change to the size ratio will make the depths shallower, so a correction to the luminosity ratio must be applied as well. Thus we need to increase the size of the secondary star whilst decreasing the luminosity ratio. Unfortunately at a size ratio of 1.15 and brightness ratio of 0.72, the primary transit is no longer deep enough.
- It is also possible to move in the other direction: decreasing size ratio and increasing the luminosity ratio. It is not possible to go below a luminosity ratio of 0.52. At this ratio the secondary star is fully occulted by the primary, and thus the depth of the secondary transit defines this luminosity ratio.
- In order to explain the transits, I need to work with a third parameter: the linear limb brightening coefficient. Standard this is set at 0.6, but in practive it can vary wildly depending on the star. By decreasing this factor to 0, I can increase the lumiosity ratio to 0.46 and decrease the size ratio to 0.88. This still does not explain the observations: the secondary transit is to narrow and the primary transit is to shallow. The size ratio must be as small has 0.77 to explain the width of the secondary transit (red lines in figure).
Thus my current model does not work to explain these observations. I need to have other parameters that I can tune. There seems to be one assumption to many in my model.
This eclipsing binary has an eccenctric orbit as can be seen from the centers of the transits (0.0985 and 0.7561). Thus my assumption of a circular orbit does not hold. There might be a difference in apparent radii due to the orbit. A change from 0.95 to 0.77 is then needed, to explain the observations.
The difference in transit timings reveals at least a minimal eccentricity of 0.25, which corresponds to a relative distance variation between the two stars of 1.6. This might thus explain why my circular orbit does not work. So I need to refine my model to incorporate eccentricity. (Literature used Russell (1912) and Sterne (1939).
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