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tokens: distance between the points on a zenith angle circle being ∼ R / N μ ) and equally spaced points on the dayside from terminator to disc edge (Fig. 1 ) with the same approximate separation. Our computed disc-averaged radiance was thus calculated as where μ i are the Gaussian quadrature points, Δμ i are the Gaussian weights, φ ij are the azimuth angles (which are different for different μ i ), and ij are the azimuth angle trapezium-rule integration weights. To ensure correct normalization, ∑ j ij = 1.0 and (i.e. ). In addition, we assumed that the exoplanet had a transit with low impact parameter such that northern and southern hemispheres were observed at the same emission angle and we also assumed north/south symmetry such that we only had to integrate over half the disc. Figure 1. Example of disc-averaging quadrature points chosen for N μ = 2, 3, 4, and 5 and phase angle ζ = 45° (where 0° is nightside and 180° is dayside). The daylight side of the planet is coloured light grey, while the nightside is dark grey. The zenith angles of the chosen Gauss–Lobatto quadrature scheme are shown by the dotted white lines. The azimuth points along each zenith angle line (indicated by the asterisk symbols) are chosen to coincide with the terminator for zenith angles that intersect the terminator and be roughly equally spaced with a fractional disc distance of 1.0/ N μ . As we assume north–south symmetry, only points on or above the equator are needed for disc-integration. An example of the chosen quadrature points for our scheme using N μ = 2, 3, 4, or 5 zenith angles is shown in Fig. 1 for a phase of 0.125 (i.e. a phase angle ζ = 45°). Here we can see that as we add more zenith angles, the number of sampled points on the disc is greatly increased. In our scheme if a given zenith angle circle intersects the day–night terminator, one azimuth quadrature point is placed on that terminator and the remaining azimuth points are chosen to be roughly equally spaced on the day and nightside, as noted above. The reason for doing this is that if we were to observe a planet where the reflected starlight is strong, we would expect that contribution to fall rapidly towards the terminator. By making sure we place a quadrature point at the terminator, we can thus ensure a better estimate of the azimuthal average of the radiance at a given zenith angle. Although in this study we have not included reflected starlight in our models, the disc-integration scheme we have defined means that we well be able to seamlessly and accurately incorporate this in future. 2.1 Computation of contribution of reflected starlight Assuming a Lambertian reflecting surface, the scattered radiance at any point on the disc is equal to α F star /π, where α is the reflectivity and F star is the local stellar irradiance (i.e. W m −2 ), calculated as F star = P cos θ/(4π d 2 ), where P is the total power of the star, d is the distance of the planet from the star, and θ is the local stellar zenith