Reflective Light-Curves of Ellipsoidally-Shaped Outer Solar-System Objects and Exoplanets

Abstract

Faculty EWI and TNW. BSc Industrial and Applied Mathematics and Applied Physics. In this thesis we consider reflective light-curves of planets, graphs of the intensity of the light originating from the parent star that the planet reflects versus time. In almost all instances, planets are modelled as spheres, like in [1] and [2]. However, planets are better approximated by ellipsoids as proved by Isaac Newton in the Principia [3]. In our own solar system, we can observe that planets are not spheres when we look at for example Jupiter [4] and Haumea [5]. We study the effect on the light-curve of the change from a spherical to an ellipsoidal model. For example, a spherical model of a homogeneous planet at edge-on observation would predict a constant light-intensity during one rotation around its axis due to the symmetry of the model. Note that we assumed here that the planet sits approximately still in the sky during one rotation around its axis. However, an ellipsoidal model of a homogeneous planet at edge-on observation would predict a variable light-intensity. [5] shows a difference between the maximum and the minimum value of the measured light-curve of Haumea of 0.32 magnitudes. This shows that an ellipsoidal model has a significant effect on the light-curve. Furthermore, in contrast to the spherical model, with the light-curves for the ellipsoidal model we can for example calculate the spin of the planet in certain cases, which makes them more interesting. By calculating the light-curve with the ellipsoidal model, we can determine the shape of planets and therefore gain knowledge about the internal structure of planets. Where others, like [5], have calculated the light-curve for an ellipsoidally-shaped planet numerically, we calculate them analytically. We accomplish this with the analytical equation of the light-curve postulated by [6]. We assume that planets have a homogeneous reflecting surface, we assume parallel incident light-rays, we assume Lambertian reflection and we assume that the planet is in a circular orbit around its parent star. We did not calculate the light-curves for non-circular orbits. However, the results can easily be modified to include elliptical Kepler orbits since the light-curve depends linearly on the orbital radius. We consider the applications of a solar system triaxially-shaped planet, like Haumea, a spheroidally-shaped exoplanet and a tidally-locked, triaxially-shaped exoplanet. We considered both edge-on and face-on observation. We confirmed the dimensions given by [5] for the dwarf planet Haumea. We found that for a given tilt of the planet’s rotation axis, there are enough measurable Fourier coefficients to determine the dimensions of the planet in each of our applications. The only exception we found is a spheroidally-shaped exoplanet at edge-on observation without tilt. In that case, we do not have enough information to differentiate between the flattening and the size of the exoplanet.