Section 2.1 showed the effects of stellar activity and transit events have on the observed light curve of a star. In this section a model for simulating such events is developed. Equation (2.4) shows how the radius ratio between the planet and star determines the dip in flux caused by the planet transiting over the stellar disc. This ratio is used to calculate the length scale which shall be used throughout the simulation. A value forRp, in terms of pixels is chosen to allow conversion between physical lengths and computational lengths. This value can either be made smaller to increase the speed of simulation with the consequence of decreasing the accuracy of the resultant light curve or made larger to increase accuracy. From this choice, a value ofR?is then calculated by re-arranging Equation (2.4).
2.2.1
Stellar Disc
The stellar disc is created by first generating a two dimensional array where each element is the radial distance from the centre of the array outwards. The array is then normalised by
−0.10
−0.05
0.00
0.05
0.10
Phase
0.98
0.99
1.00
No
rma
lised flux
Figure 2.6: Simulated transit light curve of a hot Jupiter (RP = 0.1R?), on an aligned orbit (ϕ=0), with an impact parameter (b=−0.2). The corresponding image sequence is shown above the light curve. The bottom of the transit is not flat; rather, the star has been limb-darkened resulting in the transit bottom being curved.
dividing by R? and any values greater than 1 are set to 0 to set the boundary of the stellar disc.
As discussed in Section 2.1.3, limb-darkening changes both the ingress and egress shape of the transit, but also the shape of the transit bottom. Here the quadratic limb-darkening law of Mandel & Agol (2002) for transit light curves is used. The intensity at a given radius,
µ=cosθ=p1−r2, where 0≤r≤1 is the normalised radius is given by
I(µ) I(0) =1− 4 X n=1 an 1−µn/2. (2.8) The coefficients,andetermine the strength of the limb-darkening and are related to the phys-
ical properties of the star. These include: The stellar effective temperature; the surface gravity; stellar metallicity; and also the observing band. They are traditionally found through lookup tables, such as the tables of Sing (2010) for theCoRoT andKeplertransit surveys. Once the star has been limb-darkened, the entire array is divided by the total value to normalise the total unocculted flux.
2.2.2
The Planet and Transit Light Curve
As the planet transits over the stellar disc it will block star light causing a dip in the light curve. To simulate a planet, a separate array is generated that contains flux altering values that can be multiplied onto the stellar disc to create the light curve. In this array, a value of 0 will result in the stellar flux being entirely blocked, whereas a value of 1 does not alter the background star light. The array is generated with default values of unity so that the star light is unchanged.
To simulate the planet, a circular array is created where valuespx2+ y2 <RP =0 and
values ofpx2+y2>R
P =1. This means that the planet will be entirely dark as it transits
over the stellar disc. The path the planet will take over the stellar disc is determined by the impact parameter, b (see Section 2.1). The transit path is also dependent on the obliquityϕ which determines whether the orbital plane of the planet and the rotation axis of the star are aligned (ϕ =0) or misaligned (ϕ 6=0). Accounting for these parameters, an array of(x,y)
coordinates is created that determines the position of the planet at each time step of the transit. For each set of(x,y)values the amount of flux occulted by the planet over its current location on the stellar disc is calculated and the light curve is created. Figure 2.6 shows images of a transit of a hot Jupiter in an aligned orbit with an impact parameter, b=−0.2. The resultant light curve is shown below the image sequence.
2.2.3
Testing the Effects of Limb-Darkening
The limb-darkening law is given by Equation (2.8). By changing the limb-darkening coeffi- cients the consequences of limb-darkening on the resultant transit light curve can be explored. Figure 2 of Mandel & Agol (2002) show analytic light curves for transits of a planet with vari- ous limb-darkening coefficients. Shown in Figure 2.7 is their plot reproduced using the code developed here. In every case,RP = 0.1R? and the impact parameter, b=0. The solid line
corresponds to no limb-darkening, i.e. an = 0,{n= 1, 2, 3, 4}. In this case, as expected the bottom of the transit occurs at a flux value of 0.99. The remaining for light curves are for
an =1,{n=1, 2, 3, 4}and all other coefficients am = 0,m6=n. In these cases, the depth of the transit increases and also the ingress and egress times increase, resulting in the base of the transit becoming curved rather than flat. The reason for the increase in the ingress and egress time becomes clear when the corresponding stellar disc images are considered. The
Phase 0.980 0.985 0.990 0.995 1.000 Normalised Flux 0.05 0.04 0.03 0.02 0.01 0.00 {0, 0, 0, 0} {1, 0, 0, 0} {0,1, 0, 0} {0, 0, 1, 0} {0, 0, 0, 1}
Figure 2.7: Images and corresponding light curves of a hot Jupiter (RP = 0.1R?) on an aligned orbit (ϕ = 0) with an impact parameter (b = 0). For each case the limb- darkening has been changed. The limb-darkening coefficients are shown above the image as {a1,a2,a3,a4} The first image is for a star with no limb-darkening (solid-line), i.e. an = 0,{n = 1, 2, 3, 4}. The remaining light curves are for an = 1,{n= 1, 2, 3, 4} and all other coefficients am = 0,m6= n. This plot has been reproduced from Mandel & Agol
(2002) but using the code presented in this chapter rather than their analytic expressions.
top panel of Figure 2.5 clearly shows the edge of the stellar disc becomes less clearly defined as the strength of the limb-darkening is increased.
2.2.4
The Impact Parameter
As already discussed in Section 2.1.2, the impact parameter changes the trajectory of the transit path over the stellar disc. This has the consequence of altering the transit timing (Equation (2.6)) and also depth of the transit. Figure 2.8 shows multiple light curves for impact parameters, b=0, 0.25, 0.5, 0.75, 1. The light curves show how the impact parameter changes the resultant light curve. As expected, changing the impact parameter alters the shape of the light curve. Firstly, the depth of the transit decreases as the impact parameter increases. This is to be expected because the star is brightest at disc centre and so if the planet does not transit over the centre of the disc it will occult less star light. As the impact parameter increases, the transit duration decreases. Again, this is intuitive because the planet
−0.10 −0.05 0.00 0.05 0.10 Phase 0.94 0.96 0.98 1.00 Normalised flux b = 0.00 b = 0.25 b = 0.50 b = 0.75 b = 1.00
Figure 2.8:Mid transit images and corresponding light curves for a hot Jupiter with various impact parameters. The impact parameters used here are b=0, 0.25, 0.5, 0.75, 1respect- ively. The light curves are offset for clarity.
will transit over the whole diameter of the stellar disc if it is transiting over the centre of the disc making the transit duration as long as possible; whereas, the transit chord intersects less of the stellar disc whenb6=0 (see Figure 2.4). For larger impact parameters, the shape of the transit becomes move V shaped rather than the U shape of a planetary transit. This means that transits of planets with large impact parameters may become confused with transiting grazing binary stars.