Atmospheres on Nonsynchronized Eccentric-tilted Exoplanets. II. Thermal Light Curves

First, we show the thermal light curves of tilted planets in a circular orbit for different τ_rad, θ, and Λ (Figure 3). For circular orbits, one can assume f_sec = 0°. We also plot the light curves of nontilted planets for comparison. Following the convention, the flux peak occurring before the secondary eclipse is referred to a positive peak offset, while the peak occurring after the secondary eclipse is referred to a negative peak offset (Parmentier & Crossfield 2017).

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For nontilted planets, the light curves always show the positive peak offsets when the radiative timescale is short (top row of Figure 3). To better understand how the planet looks at each orbital phase, we show the height fields on the visible hemisphere on top of the light curves (Figure 4). As seen in panel (A) of Figure 4, the positive peak offset is caused by the eastward shift of the hot spot from the substellar point due to a time delay of the height field in response to stellar irradiation. As the radiative timescale increases, the height field is more homogenized in longitude (see Figure 1). Therefore, as the radiative timescale increases, the light-curve amplitude is weaker, and the light curve eventually becomes flat, as shown in the middle and bottom rows of Figure 3. These behaviors are consistent with the synthetic light curves for nonsynchronized planets in a circular orbit in previous studies (Showman et al. 2015; Penn & Vallis 2017, 2018; Rauscher 2017).

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For planets with nonzero obliquities, as shown in Figure 3, the behaviors of the thermal light curves are very complex, depending on the radiative timescale, planetary obliquity, and viewing geometry. These complex behaviors are mainly caused by the geometric effect of the "projected hot spot" and the effect of the "seasonal polar heating" argued in Section 3. The former effect is responsible for light curves in regime (I)—those with short radiative timescales (τ_rad ≪ P_rot). The latter effect is mainly responsible for regime (III)—those with long radiative timescales (τ_rad ≫ P_rot) and large obliquities (≳18°). For planets with very weak seasonality in regimes (II), (IV), and (V), the light curves are almost flat because the height fields are nearly constant throughout the planetary orbit (Figure 3).

In the case of τ_rad = 0.1 day (top row in Figure 3), both the amplitude and peak offset of the light curves appreciably vary with obliquity because of the aforementioned geometrical effect. Interestingly, the dependence of obliquity on amplitude is different for different viewing geometry characterized by Λ. For Λ = ±90° geometry, in which the observer is facing the equator at the secondary eclipse (see Figure 2), the amplitude of the light curve decreases with increasing obliquity (top left panel of Figure 3). On the other hand, for Λ = 0° geometry, in which the solstice occurs at the secondary eclipse and the observer views the polar region then, the amplitude increases with increasing obliquity (top right panel of Figure 3). According to the analytical light curve, the amplitude of the light curve is scaled by ${C}_{{\rm{t}}}(\theta ,\varphi ,{\rm{\Lambda }},\psi )$ (Section 3). In the limit of short radiative timescales, the amplitude factor C_t can be approximated by $C(\theta ,\varphi ,{\rm{\Lambda }})\cos \varphi$, where C is given by Equation (25) in Appendix A.1. The prefactor $C(\theta ,\varphi ,{\rm{\Lambda }})$ for Λ = 90° is given by (Equation (30) in Appendix A.1)

$$\begin{eqnarray}&&C{| }_{{\rm{\Lambda }}={90}^{^\circ }}=\sqrt{{\cos }^{2}\varphi +{\sin }^{2}\varphi \,{\cos }^{2}\theta }.\end{eqnarray} \tag{ 14 }$$

Therefore, the amplitude decreases with increasing obliquity for Λ = 90°, in agreement with Figure 3. This is qualitatively due to the fact that the projected maximum emission flux of the hot spot along the line of sight to the observer will be smaller if the obliquity is higher in this geometry. On the other hand, the prefactor C for Λ = 0° is given by (Equation (31) in Appendix A.1)

$$\begin{eqnarray}&&C{| }_{{\rm{\Lambda }}=0^\circ }=\sqrt{{\left({\sin }^{2}\theta +\cos \varphi {\cos }^{2}\theta \right)}^{2}+{\sin }^{2}\varphi \,{\cos }^{2}\theta }.\end{eqnarray} \tag{ 15 }$$

Here the prefactor C is nearly unity for all obliquities as long as the phase shift of the hot spot is small, while Figure 3 indicates that the amplitude increases with increasing obliquity. This is due to the fact that the actual light curve is obtained from the disk-integrated flux. The hot spot is less smoothed out at the polar region but more smoothed out at the equator, as seen in panels (A) and (C) in Figure 4, which induces the higher disk-integrated flux from the polar region than that from the equator.

For a short radiative timescale (τ_rad ≪ P_rot), the phase (or time) of the peak in the light curve approaches the secondary eclipse as obliquity increases for Λ = ±90° and 0°. According to Equation (9), the phase of the flux peak approaches the secondary eclipse as the obliquity approaches θ = 90°, which is consistent with the trends of the peak offset. This is qualitatively originated from the fact that the equatorial and orbital planes will be more and more misaligned with each other as the obliquity increases. For example, in the Λ = 90° and θ = 90° case (panel (B) in Figure 4), the equatorial plane is essentially perpendicular to the orbital plane; thus, the observer will always see the flux peak from the projected hot spot occurring right at the secondary eclipse. On the other hand, the peak in the light curve moves toward the phase before and after the secondary eclipse for Λ = −45° and 45°, respectively. This phase shift from the secondary eclipse is caused by the flux from the polar region that enhances the total emergent flux around the solstice when the obliquity is high. For example, in the Λ = 45° and θ = 90° case, in which the solstice takes place after the secondary eclipse (panel (D) in Figure 4), the polar region undergoes strong heating around the solstice, and the peak in the light curve also occurs near there. Note that the effect of polar heating is not responsible for Λ = ±90° because the polar region is no longer visible to the observer in that geometry.

In the case of τ_rad = 5 days (middle row of Figure 3), the shapes of the light curves are significantly different from those of τ_rad = 0.1 day. For planets with obliquity smaller than 18° (regime II), the light curves look nearly flat. The phase offset, if there is one, is shifted before the secondary eclipse. In this regime, because of the weak seasonality, the peak offset is still determined by the projected hot spot rather than the seasonal polar heating. On the other hand, the light curves of highly tilted planets exhibit noticeable peaks. This is caused by the polar heating occurring at around the solstice, which produces a strong emergent flux (see Figure 1). Since the polar heating occurs if the obliquity is higher than 18° in this regime (Section 2), a planet with θ > 18° potentially exhibits a flux variation in the light curve. In this regime, since there is a significant time lag in the heating in the polar region (panel (E) in Figure 4), the flux peak occurs after the solstice, as seen in the light curves for Λ = ±45° and 90° (middle and right panels in Figure 3). This can also be known from Equation (10), which shows the phase of the flux peak ${f}_{\mathrm{peak}}\approx {\rm{\Lambda }}+2\pi {\tau }_{\mathrm{rad}}/{P}_{\mathrm{orb}}$. The light curves behave nearly flat in the geometry with Λ = ±90° (left panel in Figure 3), in which the subobserver point is at the equator and the flux from the poles is negligible. For Λ = ±90°, highly tilted planets produce double flux peaks in the light curves. This is caused by the flux from the equatorial region heated at the vernal and autumn equinoxes. The bimodal light curve for this specific geometry can be also seen in previous studies (Gaidos & Williams 2004; Rauscher 2017).

The remarkable feature is that a tilted planet produces a flux peak after the secondary eclipse in some cases, for example, the light curves for Λ = 0 and θ ≥ 30° (middle right panel of Figure 3). This originates from the fact that the phase of the flux peak highly depends on the orbital phase of the solstice when the high latitudes are illuminated. The middle row of Figure 3 shows that the flux peak could occur after the secondary eclipse for θ = 30°, 60°, and 90°. Note that Λ > 0 corresponds to the northern summer solstice occurring after the secondary eclipse (see Figure 2). Because there is also a time lag in the polar heating, the maximum temperature tends to occur after the solstice and secondary eclipse, as seen in Figure 3. These behaviors are in agreement with Rauscher (2017), in which the peaks of the light curves tend to occur after the secondary eclipse. Therefore, one can infer that the obliquity is higher than about 18° if the negative peak offset is found for planets with a long radiative timescale, as classified into regimes (II) and (III). However, it would be difficult to identify the exact value of the obliquity only from the peak offset because the peak offset is controlled by solstice phase Λ and independent of obliquity (Equation (10)).

In the case of τ_rad = 100 days (bottom row of Figure 3), the light curves are almost flat for all obliquities. The amplitudes of the light curves are at least an order of magnitude smaller than the light curves for τ_rad = 5 days. This is simply because, in regimes (IV) and (V), the height fields are controlled by the annual mean insolation and do not change throughout the planet orbit (Section 2).

We also show the light curves for planets with retrograde rotations (i.e., θ > 90°) in Figure 5. For τ_rad = 0.1 day, the peak of the light curve occurs significantly after the secondary eclipse as the obliquity approaches θ = 180°. This is again caused by the aforementioned geometrical effect. The eastward displacement of the hot spot on the equatorial plane is still present for θ > 90° (see Paper I). Because the hot spot projected on the orbital plane effectively shifts westward for the observer when θ > 90° (see Equation (9) and panel (F) of Figure 4), the flux peak occurs after the secondary eclipse. For τ_rad = 5 days, the shape of the light curve is nearly the same between planets with θ (Figure 3) and 180° − θ (Figure 5). This is because the geometrical effect shown in panel (F) of Figure 4 disappears for a planet with a long radiative timescale and a longitudinally homogenized height field (see Figure 1). In this regime, the shape of the light curve is largely determined by the flux from the poles, which only depends on the subobserver latitude ϕ_obs and Λ. Since the subobserver latitude for the case with θ is identical to that with 180° − θ (see Equation (12)), the two planets produce nearly the same light curves for the same Λ.

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We summarize the peak offset calculated from our 2D simulations, as well as the prediction of our analytical theory for tilted planets with various radiative timescales, obliquities, and viewing geometries, in Figure 6. As seen in Figure 6, our analytical theory well captures the general trends of the peak offset of the light curves of tilted planets. For short radiative timescales (left panel of Figure 6), the phase of the flux peak is mainly controlled by the shifted hot spot. The flux peak tends to occur near the secondary eclipse as the obliquity approaches θ = 90° and intrinsically occurs after the secondary eclipse for planets with retrograde rotation (θ > 90°). For long radiative timescales (right panel of Figure 6), the phase of the flux peak is mainly determined by the time lag of the polar heating behind the solstice Λ, but the peak offsets for planets with small obliquities (θ ≲ 18°) are still influenced by the projected hot spot.

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In most cases, our analytical predictions agree very well with the simulation results. The analytical predictions for τ_rad = 0.1 day and Λ = ±45° deviate little from the numerical results. This is probably caused by the fact that the analytical model currently ignores the meridional heat transport for a time evolution of the height field at the pole. The left panel of Figure 6 shows that the light curves without polar flux (dotted lines) underestimate the magnitude of the peak offset, while those with polar flux produce the peak close to the orbital phase of the summer solstice. This means that the flux from the polar region is overestimated. The light curves for τ_rad = 5 days and Λ = ±90° are also completely different from the numerical results. In this specific case, the emergent fluxes from the hot spot and the polar region are not important, and the light curves show double flux peaks (see the left middle panel of Figure 3). But also note that the light curves are almost flat in these cases.

In summary, for planets in circular orbits, nontilted planets always exhibit a positive peak offset, while tilted planets potentially exhibit a negative peak offset if the solstice takes place after the secondary eclipse and/or the planet is retrograde rotating. For hot planets with a short radiative timescale in regime (I), the planets with retrograde rotation produce the flux peak after the secondary eclipse because of the geometrical effect. Therefore, the negative peak offset potentially indicates θ ≥ 90° for planets with a short radiative timescale. For relatively cold planets with a long radiative timescale in regimes (II) and (III), the peak of the light curve occurs after the secondary eclipse because of the polar heating at the solstice. Since the polar heating occurs for θ ≥ 18°, one can use the negative peak offset as a diagnosis of high obliquity θ ≥ 18°. However, it should be pointed out that the negative peak offset can also be produced by other mechanisms, for example, the presence of clouds (Demory et al. 2013; Oreshenko et al. 2016; Parmentier et al. 2016), a westward shift of the hot spot caused by a rotation slower than the orbital motion (Rauscher & Kempton 2014), trapped Rossby waves for slow substellar motion (Penn & Vallis 2017, 2018), and time-varying winds caused by magnetic fields (Rogers 2017). Therefore, one must be cautious when interpreting the negative peak offset seen in the light curves in future observations.

For a planet with a nonzero eccentricity, the shape of the thermal light curve depends not only on eccentricity e, obliquity θ, and Λ but also on the true anomaly of the secondary eclipse f_sec. The situation could be highly complicated if we discuss every possible geometry. In this study, for simplicity, we follow a similar discussion in Kataria et al. (2013), which also examined the light curves for eccentric planets. We only discuss the geometries of f_sec = 0° and ±90°, as shown in Figure 2. Figures 7 and 9 show the light curves of ET planets for a variety of θ and Λ for τ_rad = 0.1 and 5 days. The light curves for τ_rad = 100 days are nearly flat, so we omit them in this section.

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When the radiative timescale is short, the eccentricity effect tends to produce a sharp peak near the periapse, where the planet undergoes intense irradiation from the central star. Generally speaking, eccentricity leads to the shape of the light curve that is narrow around the periapse because the time duration for orbiting around the periapse is very short. For τ_rad = 0.1 day (Figure 7), one can see that the light curves for both tilted and nontilted planets are peaked at around the periapse regardless of the position of the solstice. This is because the magnitude of the height field is primarily determined by the equilibrium value in the limit of the short radiative timescale. If the solstice does not take place at the periapse, the height field will be maximized at the periapse and produce the flux peak near the periapse instead of the solstice in the light curve (Figure 7).

As noted, for eccentric planets, the shape of the light curve depends on the viewing geometry. To better explain the effects of the viewing geometry, we show light curves and height fields on the visible hemisphere at an orbital phase around the secondary eclipse for f_sec = 0° and ±90° in Figure 8. For nontilted planets with a short τ_rad (left columns of Figure 8), the observed flux peak depends on both the viewing geometry and the hot-spot displacement. If the periapse passage occurs before the secondary eclipse (for example, f_sec = 90°; left panel of Figure 2), the strong emergent flux can be observed at around the periapse because the hot spot is displaced toward the observer (bottom left panel of Figure 8). By contrast, if the periapse passage occurs after the secondary eclipse (for example, f_sec = −90°; right panel of Figure 2), the hot spot is displaced away from the observer at the periapse, and the peak flux is smaller than the former case with f_sec = 90° (top left panel of Figure 8). If the secondary eclipse occurs right at the periapse (middle panel of Figure 2), the strong flux peak occurs slightly before the secondary eclipse because of the eastward displacement of the hot spot (middle left panel of Figure 8). These behaviors are seen in the light curves in Figure 7. Our results are also qualitatively consistent with the light curves from the 3D simulations of planets in eccentric orbits in Kataria et al. (2013), which showed that the peak of the light curve is relatively weak for f_sec = −90° (ω = 360° in their context).

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The peak offset is insensitive to the obliquity because the emergent flux is largely controlled by the intense heating at the periapse; however, an amplitude of the light curve is drastically influenced by the obliquity. For example, in the case of f_sec = −90° (top row of Figure 7), a tilted planet exhibits a light curve with a large peak as the obliquity approaches θ = 90°. The reason is that, in contrast to the nontilted planets, where the hot spot is displaced away from the observer in this geometry, the emergent flux is dominated by the flux from the heated pole that is still visible at the periapse (see top right panel of Figure 8). For obliquity θ > 90°, the hot spot is displaced toward the observer at the periapse because of the retrograde rotation (Section 4.1), leading to a large peak in the light curve. On the other hand, in the case of f_sec = 90° (bottom row of Figure 7), the hot spots on highly tilted planets are not displaced toward the observer at the periapse (see bottom right panel of Figure 8). Therefore, the flux peak occurs closer to the secondary eclipse where the equilibrium height field is smaller than that for the periapse. Specifically, for obliquity θ > 90°, the hot spot is displaced away from the observer as in nontilted planets for f_sec = −90°. As a result, for f_sec = 90°, the higher θ case has a smaller peak amplitude in the light curve. Therefore, for eccentric planets with a short radiative timescale in regime (I), it would be difficult to infer the planetary obliquity from the peak offset. Alternatively, a light curve with an abnormally large or small peak might imply a high obliquity of the planet in the geometry where the secondary eclipse occurs before (f_sec = −90°) and after (f_sec = 90°) the periapse passage, respectively.

As the radiative timescale increases, the peak of the light curve lags behind the planet periapse passage, as seen in the case of τ_rad = 5 days (Figure 9). Figure 10 shows the light curves and height fields on the visible hemisphere around the secondary eclipse for τ_rad = 5 days. For nontilted planets, the light curves are always peaked a little behind the periapse passage due to a delayed response of the height field to the stellar heating (left column of Figure 10). In contrast to the cases of τ_rad = 0.1 day, the shape of the light curve depends less on the viewing geometry, since the height field is nearly homogenized in longitude. This is why the shapes of the light curves of nontilted planets at different viewing geometries look roughly similar, although the phase of the flux peak also depends on f_sec.

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For tilted planets, the light curves for θ = 10° and 180° are superposed on those for nontilted planets, while the light curves for θ = 30°, 60°, 90°, 120°, and 150° look different. In addition, light curves for θ = 30° and 60° are almost superposed on those for θ = 150° and 120°, respectively. This behavior is similar to the cases of circular-orbit planets (Section 4.1). This indicates that the shape of the light curve is influenced by the polar heating that occurs for obliquity θ > 18°. However, for Λ = ±90°, the light curves for all obliquities are superposed on each other because the flux from the poles is negligible in this geometry. In other words, the light curves of the tilted planets with equinox at the secondary eclipse are indistinguishable from those of the nontilted planets.

For the ET planets with a long radiative timescale, the peak offset of the light curve from the secondary eclipse is more influenced by the obliquity than the planets with short radiative timescales. For example, in the cases of Λ = 45° (solid lines in the middle column of Figure 9), the phase of the flux peak is shifted toward the phase of the solstice from the periapse passage. As obliquity increases, the peak flux is enhanced if the solstice is close to the periapse, as in the case of f_sec = −90° and 0° (top and middle rows), and weakened if the solstice is far away from the periapse, as in the case of f_sec = 90° (bottom row). Take Λ = 0° as a clearer example (right column); since there is a time delay in the response of the height field to the stellar heating, the flux peak shows a time lag behind the solstice phase (see right column of Figure 10). In summary, when the radiative timescale is as long as classified into regime (II) and (III), the peak offset of the tilted planet is substantially controlled by the solstice phase, while the nontilted planets produce the flux peak around the periapse.⁴

As shown above, eccentricity significantly affects the shape of the light curve. Does it help us to infer the obliquities of eccentric planets? Here we emphasize that the eccentricity e and f_sec are a priori known parameters from observations, while the obliquity θ and f_sol are a priori unknown parameters. If the radiative timescale is significantly short, the flux peak is strongly restricted at around the periapse, where the irradiation is maximum. In that case, the peak offset depends less on obliquity (see Figure 7). Therefore, in terms of peak offset, it seems difficult to distinguish tilted planets from nontilted planets, although obliquity significantly affects the peak amplitude that might offer clues to infer the obliquity. On the other hand, planets with relatively long radiative timescales (in regimes II and III) are better candidates for retrieving the information on the obliquity using the peak offset. This is because the shape of the light curve easily deviates from that of nontilted planets once the obliquity (use 180° − θ for θ > 90°) exceeds 18° and the polar heating occurs.

Specifically, we suggest that the observational geometry in which the secondary eclipse takes place after the periapse (i.e., f_sec > 0°, for example) offers a good opportunity to infer the obliquity. In that geometry, the light curve of a nontilted planet tends to exhibit a positive peak offset (i.e., flux peak before the secondary eclipse) because the flux peak is controlled by the heating at the periapse (Figures 9 and 10), although it would also depend on the radiative timescale. On the other hand, the light curve of a tilted planet potentially exhibits a negative peak offset (i.e., flux peak after the secondary eclipse) because the emergent flux is substantially influenced by the obliquity and the orbital phase of the solstice (the cases of Λ = 0° and 45° in Figure 9, for example). Summarizing, a negative peak offset might indicate a nonzero obliquity (at least, θ > 18°) if the planet is orbiting where the periapse takes place before the secondary eclipse (${f}_{\sec }\gt \,0$).

What radiative timescale is better to infer the obliquity of eccentric planets in that geometry? If the radiative timescale is too short, a tilted planet produces the flux peak before the secondary eclipse. Thus, a longer radiative timescale is preferred. On the other hand, if the radiative timescale is too long, nontilted planets also show the flux peak after the secondary eclipse because of the time delay of the height field response from heating at the periapse. Therefore, to use the negative peak offset as a diagnosis of nonzero obliquity, the radiative timescale should not significantly exceed a critical timescale. Assuming a nontilted planet produces the flux peak after the periapse passage by a time lag of ∼τ_rad, the critical timescale may be given as the duration a planet takes to travel from the periapse to the secondary eclipse. The time duration of planet traveling can be calculated from the Kepler equation, given by Murray & Dermott (1999),

$$\begin{eqnarray}&&\displaystyle \frac{{dE}}{{dt}}=\displaystyle \frac{2\pi }{{P}_{\mathrm{orb}}}\displaystyle \frac{1}{1-e\cos E},\end{eqnarray} \tag{ 16 }$$

where E is the eccentric anomaly associated with the true anomaly f as

$$\begin{eqnarray}&&\tan \displaystyle \frac{f}{2}=\sqrt{\displaystyle \frac{1+e}{1-e}}\tan \displaystyle \frac{E}{2}.\end{eqnarray} \tag{ 17 }$$

Integrating Equation (16) from f = 0 to f_sec with Equation (17), the critical radiative timescale is given by

$$\begin{eqnarray}&&{\tau }_{\mathrm{crit}}=\displaystyle \frac{{P}_{\mathrm{orb}}}{2\pi }\left[2{\tan }^{-1}\left(\sqrt{\displaystyle \frac{1-e}{1+e}}\tan \displaystyle \frac{{f}_{\sec }}{2}\right)-\displaystyle \frac{e\sqrt{1-{e}^{2}}\sin {f}_{\sec }}{1+e\cos {f}_{\sec }}\right].\end{eqnarray} \tag{ 18 }$$

Figure 11 shows the critical radiative timescale normalized by the orbital period (dashed line) and results from the light curves simulated by the shallow-water model (stars and crosses). We find that tilted planets produce a negative peak offset, while nontilted planets produce a positive peak offset for the radiative timescale close to ∼τ_crit. Tilted planets are still distinguishable for a radiative timescale longer than the critical timescale because nontilted planets produce the flux peak after the periapse passage with a time lag smaller than τ_rad (for example, see Figure 9). This is due to the fact that the incoming stellar flux rapidly decreases as a planet moves far away from the periapse. The critical radiative timescale is roughly ∼0.1 P_orb for f_sec = 90°.

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We now attempt to evaluate the temperature range corresponding to the critical radiative timescale, although in a very crude way. The radiative timescale is roughly evaluated as (Showman & Guillot 2002)

$$\begin{eqnarray}&&{\tau }_{\mathrm{rad}}\sim \displaystyle \frac{{P}_{\mathrm{ph}}}{g}\displaystyle \frac{{c}_{{\rm{p}}}}{4\sigma {T}^{3}},\end{eqnarray} \tag{ 19 }$$

where P_ph is the photospheric temperature, c_p is the specific heat of the atmosphere, and σ is the Stefan–Boltzmann constant. If we assume P_orb = 30 days, c_p = 1.3 × 10⁴ J kg⁻¹ K⁻¹, g = 21 m s⁻², and P_ph = 250–670 mbar (Showman et al. 2015; Rauscher 2017), a critical radiative timescale of ∼3 days corresponds to a temperature of ∼600–900 K. This suggests that eccentric Jupiter-size planets with equilibrium temperatures of ∼600–900 K might be good candidates to infer nonzero obliquities in future observations. However, it should be noted that an actual radiative timescale is determined by complex atmospheric properties, such as chemical and thermal structures (e.g., Li et al. 2018). Further studies with realistic radiative transfer will be needed to revisit the estimate here.

For τ_rad ≪ P_rot, the emergent flux is mainly dominated by the flux from the hot spot displaced from the substellar point. Therefore, the emergent flux might be diagnosed by the hot-spot vector r_hs projected onto the subobserver point vector ${{\boldsymbol{r}}}_{\mathrm{obs}}$, i.e., ${{\boldsymbol{r}}}_{\mathrm{hs}}\cdot {{\boldsymbol{r}}}_{\mathrm{obs}}$. The hot-spot vector r_hs is related to the substellar point vector r_ss, which is given by (for derivation, see Dobrovolskis 2009, 2013)

$$\begin{eqnarray}&&{{\boldsymbol{r}}}_{\mathrm{ss}}=\left(\begin{array}{c}(1-\cos \theta )\cos (f-{f}_{\mathrm{sol}})\sin {{\rm{\Omega }}}_{\mathrm{rot}}t-\sin ({{\rm{\Omega }}}_{\mathrm{rot}}t-f+{f}_{\mathrm{sol}})\\ (1-\cos \theta )\cos (f-{f}_{\mathrm{sol}})\cos {{\rm{\Omega }}}_{\mathrm{rot}}t-\cos ({{\rm{\Omega }}}_{\mathrm{rot}}t-f+{f}_{\mathrm{sol}})\\ \cos (f-{f}_{\mathrm{sol}})\sin \theta \end{array}\right).\end{eqnarray} \tag{ 20 }$$

Because the movements of the substellar and subobserver points due to the planetary rotation are the same, one can simplify the problem in a nonrotating framework. Substituting Ω_rot = 0 into Equation (20), the substellar point movement is expressed by

$$\begin{eqnarray}&&{{\boldsymbol{r}}}_{\mathrm{ss}}=\left(\begin{array}{c}\sin (f-{f}_{\mathrm{sol}})\\ -\cos \theta \cos (f-{f}_{\mathrm{sol}})\\ \cos (f-{f}_{\mathrm{sol}})\sin \theta \end{array}\right)=\left(\begin{array}{c}\sin (f-{\rm{\Lambda }})\\ -\cos \theta \cos (f-{\rm{\Lambda }})\\ \cos (f-{\rm{\Lambda }})\sin \theta \end{array}\right),\end{eqnarray} \tag{ 21 }$$

where we assume f_sec = 0 for planets in circular orbits and thus f_sol = Λ. In this context, f expresses the orbital phase from the secondary eclipse. Since the subobserver point is identical to the substellar point at the secondary eclipse (f = 0), the subobserver point is given by

$$\begin{eqnarray}&&{{\boldsymbol{r}}}_{\mathrm{obs}}=\left(\begin{array}{c}-\sin \,{\rm{\Lambda }}\\ -\cos \theta \,\cos \,{\rm{\Lambda }}\\ \cos \,{\rm{\Lambda }}\,\sin \theta \end{array}\right).\end{eqnarray} \tag{ 22 }$$

On the other hand, the hot spot displaced from the substellar point by the phase shift φ (hereafter called the original phase shift) is given by

$$\begin{eqnarray}\begin{array}{rcl}{{\boldsymbol{r}}}_{\mathrm{hs}} & = & \left(\begin{array}{ccc}\cos \varphi & -\sin \varphi & 0\\ \sin \varphi & \cos \varphi & 0\\ 0 & 0 & 1\end{array}\right){{\boldsymbol{r}}}_{\mathrm{ss}}\\ & = & \left(\begin{array}{c}\cos \varphi \sin (f-{\rm{\Lambda }})+\cos \theta \sin \varphi \cos (f-{\rm{\Lambda }})\\ \sin \varphi \sin (f-{\rm{\Lambda }})-\cos \theta \cos \varphi \cos (f-{\rm{\Lambda }})\\ \cos (f-{\rm{\Lambda }})\sin \theta \end{array}\right).\end{array}\end{eqnarray} \tag{ 23 }$$

Now the projection factor ${ \mathcal P }\equiv {{\boldsymbol{r}}}_{\mathrm{obs}}\cdot {{\boldsymbol{r}}}_{\mathrm{hs}}$ is obtained as

$$\begin{eqnarray}\begin{array}{rcl}{ \mathcal P } & = & \cos (f-{\rm{\Lambda }})[({\sin }^{2}\theta +\cos \varphi \,{\cos }^{2}\theta )\cos \,{\rm{\Lambda }}\\ & & -\,\sin \varphi \cos \theta \sin {\rm{\Lambda }}]\\ & & -\,\sin (f-{\rm{\Lambda }})(\cos \varphi \sin \,{\rm{\Lambda }}+\sin \varphi \cos \theta \cos \,{\rm{\Lambda }})\\ & = & C(\theta ,\varphi ,{\rm{\Lambda }})\cos (f-{\rm{\Lambda }}+{\varphi }_{* }),\end{array}\end{eqnarray} \tag{ 24 }$$

where $C(\theta ,\varphi ,{\rm{\Lambda }})$ is the prefactor controlling the light curve amplitude, given by

$$\begin{eqnarray}&&\begin{array}{l}C(\theta ,\varphi ,{\rm{\Lambda }})\,=\,\sqrt{{\left({\sin }^{2}\theta +\cos \varphi {\cos }^{2}\theta \right)}^{2}{\cos }^{2}{\rm{\Lambda }}-({\sin }^{2}\theta +\cos \varphi \,{\cos }^{2}\theta -\cos \varphi )\sin \varphi \cos \theta \sin 2{\rm{\Lambda }}+{\cos }^{2}\varphi \,{\sin }^{2}{\rm{\Lambda }}+{\sin }^{2}\varphi \,{\cos }^{2}\theta }.\end{array}\end{eqnarray} \tag{ 25 }$$

The projection factor ${ \mathcal P }$ is maximized at the orbital phase of $f={\rm{\Lambda }}-{\varphi }_{* }$, where ${\varphi }_{* }$ is given by

$$\begin{eqnarray}&&\begin{array}{l}{\varphi }_{* }\\ =\,{\tan }^{-1}\left[\displaystyle \frac{\cos \varphi \sin \,{\rm{\Lambda }}+\sin \varphi \cos \theta \cos \,{\rm{\Lambda }}}{({\sin }^{2}\theta +\cos \varphi \,{\cos }^{2}\theta )\cos \,{\rm{\Lambda }}-\sin \varphi \cos \theta \sin \,{\rm{\Lambda }}}\right].\end{array}\end{eqnarray} \tag{ 26 }$$

As seen in Equation (26), the peak offset of the tilted planet in regime (I) is determined in a very complex manner. But if one crudely assumes that the phase shift of the hot spot φ is small, which might be valid in the limit of short radiative timescales and weak zonal winds, the problem is greatly simplified. In that case, Equation (26) can be approximated as

$$\begin{eqnarray}\begin{array}{rcl}{\varphi }_{* } & \approx & {\tan }^{-1}\left[\displaystyle \frac{\sin \,{\rm{\Lambda }}+\varphi \cos \theta \cos \,{\rm{\Lambda }}}{\cos \,{\rm{\Lambda }}-\varphi \cos \theta \sin \,{\rm{\Lambda }}}\right]\\ & \approx & {\tan }^{-1}\left[\displaystyle \frac{\sin ({\rm{\Lambda }}+{\tan }^{-1}(\varphi \cos \theta ))}{\cos ({\rm{\Lambda }}+{\tan }^{-1}(\varphi \cos \theta ))}\right]\\ & \approx & {\rm{\Lambda }}+\varphi \cos \theta ,\end{array}\end{eqnarray} \tag{ 27 }$$

where we truncate the terms of φ higher than second order. Equation (27) indicates that, for a small phase shift of the hot spot, the flux peak occurs at $f=-\varphi \cos \theta$, which is just the projection of the original phase shift onto the orbital plane.

Here we show some examples to interpret the light curves for a short radiative timescale in Section 4.1. In the simplest case, the projection factor for a nontilted planet (θ = 0) is given by

$$\begin{eqnarray}&&{ \mathcal P }{| }_{\theta =0}=\cos (f+\varphi ),\end{eqnarray} \tag{ 28 }$$

where ${ \mathcal P }{| }_{\theta ={0}^{^\circ }}$ is maximized at $f=-\varphi$ and thus the flux peak occurs before the secondary eclipse. On the other hand, for θ = 180°, the projection factor is given by

$$\begin{eqnarray}&&{ \mathcal P }{| }_{\theta ={180}^{^\circ }}=\cos (f-\varphi ),\end{eqnarray} \tag{ 29 }$$

where ${ \mathcal P }{| }_{\theta ={180}^{^\circ }}$ is maximized at $f=\varphi$ and thus the flux peak occurs after the secondary eclipse. For another example, the projection factor for planets in the geometry of Λ = 90° is given by

$$\begin{eqnarray}&&{ \mathcal P }{| }_{{\rm{\Lambda }}={90}^{^\circ }}=\sqrt{{\cos }^{2}\varphi +{\sin }^{2}\varphi \,{\cos }^{2}\theta }\cos (f+{\varphi }_{90}),\end{eqnarray} \tag{ 30 }$$

where ${\varphi }_{90}={\tan }^{-1}(\cos \theta \tan \varphi )$ and the flux peak occurs at $f=-{\varphi }_{90}$. On the other hand, the projection factor for Λ = 0° is also given by

$$\begin{eqnarray}&&\begin{array}{l}{ \mathcal P }{| }_{{\rm{\Lambda }}=0^\circ }\\ =\,\sqrt{{\left({\sin }^{2}\theta +\cos \varphi {\cos }^{2}\theta \right)}^{2}+{\sin }^{2}\varphi \,{\cos }^{2}\theta }\cos (f+{\varphi }_{0}),\end{array}\end{eqnarray} \tag{ 31 }$$

where ${\varphi }_{0}={\tan }^{-1}(\cos \theta \sin \varphi /({\sin }^{2}\theta +\cos \varphi \,{\cos }^{2}\theta ))$ and the flux peak occurs at $f=-{\varphi }_{0}$.

The original phase shift of the hot spot φ can be evaluated from linearized shallow-water equations. Without the drag and Coriolis term, a linearized shallow-water equation at the equator can be written as (for the derivation, see Penn & Vallis 2017)

$$\begin{eqnarray}&&\displaystyle \frac{{dh}^{\prime} }{d\lambda ^{\prime} }=-\xi h^{\prime} +\xi {\rm{\Delta }}h\cos \lambda ^{\prime} { \mathcal H }(\cos \lambda ^{\prime} ),\end{eqnarray} \tag{ 32 }$$

where ${ \mathcal H }$ is the Heaviside step function defined as Equation (13), $h^{\prime} =h-H$ is the difference of the height from the mean value, $\lambda ^{\prime} =\lambda +2\pi t/{P}_{\mathrm{rot}}$ is the longitude from the substellar point moving westward, and ξ is the nondimensional parameter given by

$$\begin{eqnarray}&&\xi =\displaystyle \frac{{P}_{\mathrm{rot}}}{2\pi {\tau }_{\mathrm{rad}}}{\left(\displaystyle 1-\frac{{gH}}{{v}_{s}{s}^{2}}\right)}^{-1},\end{eqnarray} \tag{ 33 }$$

where ${v}_{\mathrm{ss}}=2\pi {R}_{{\rm{p}}}/{P}_{\mathrm{rot}}$ is the substellar velocity. Note that the substellar point is assumed to move westward since we have assumed P_rot ≪ P_orb. The analytical solution of Equation (32) for $-\pi /2\lt \lambda \lt \pi /2$ is given by

$$\begin{eqnarray}\begin{array}{rcl}h^{\prime} & = & {\rm{\Delta }}h\displaystyle \frac{\xi }{1+{\xi }^{2}}\left(\Space{0ex}{3.25ex}{0ex}\xi \cos \lambda ^{\prime} +\sin \lambda ^{\prime} \right.\\ & & \left.+\,\displaystyle \frac{{e}^{(3/2)\pi \xi }+{e}^{(1/2)\pi \xi }}{{e}^{2\pi \xi }-1}\exp (-\xi \lambda ^{\prime} )\right),\end{array}\end{eqnarray} \tag{ 34 }$$

where we adopt a periodic boundary condition at $\lambda ^{\prime} =\pm \pi /2$ and ±π following Zhang & Showman (2017). The phase shift of the hot spot φ can be found as a solution of ${dh}^{\prime} /d\lambda ^{\prime} =0$, i.e.,

$$\begin{eqnarray}&&\xi \sin \varphi -\cos \varphi +\xi \displaystyle \frac{{e}^{(3/2)\pi \xi }+{e}^{(1/2)\pi \xi }}{{e}^{2\pi \xi }-1}\exp (-\xi \varphi )=0.\end{eqnarray} \tag{ 35 }$$

The solution of this equation cannot be explicitly shown but can be numerically obtained. In the limit of ξ ≫ 1, the phase shift is expressed by $\varphi \approx {\tan }^{-1}(1/\xi )$.

For planets with long radiative timescales of τ_rad ≫ P_rot, the hot spot is less important for the total emergent flux. When the obliquity is small, as in regime (II), the shape of the light curve is still influenced by the projected hot spot argued in Appendix A.1. On the other hand, if the planetary obliquity is high, as in regime (III), the total flux is largely dominated by the flux from the polar region. To evaluate the flux from the polar region, we construct a simple kinematic model of the height field evolution at the pole. When the meridional heat transport is inefficient, the time evolution of the height field at the pole is expressed by

$$\begin{eqnarray}&&\displaystyle \frac{2\pi }{{P}_{\mathrm{orb}}}\displaystyle \frac{{dh}^{\prime} }{{df}}=\displaystyle \frac{{\rm{\Delta }}h\sin \theta \cos (f-{\rm{\Lambda }}){ \mathcal H }(\cos (f-{\rm{\Lambda }}))-h^{\prime} }{{\tau }_{\mathrm{rad}}}.\end{eqnarray} \tag{ 36 }$$

This equation can be solved in the same way as Equation (32). Again adopting the periodic boundary condition at f = Λ ± π/2 and Λ ± π, we obtain the height field evolution at the pole,

$$\begin{eqnarray}\begin{array}{rcl}{h}_{\mathrm{pole}}^{{\prime} } & = & {\rm{\Delta }}h\displaystyle \frac{\psi \sin \theta }{1+{\psi }^{2}}\left[\Space{0ex}{3.25ex}{0ex}\psi \cos (f-{\rm{\Lambda }})+\sin (f-{\rm{\Lambda }})\right.\\ & & +\,\left.\displaystyle \frac{{e}^{\psi ({\rm{\Lambda }}-3\pi /2)}+{e}^{\psi ({\rm{\Lambda }}-2\pi )}}{1-{e}^{-3\pi \psi /2}}\exp (-\psi f)\right],\end{array}\end{eqnarray} \tag{ 37 }$$

where we define

$$\begin{eqnarray}&&\psi \equiv \displaystyle \frac{{P}_{\mathrm{orb}}}{2\pi {\tau }_{\mathrm{rad}}}.\end{eqnarray} \tag{ 38 }$$

The height field at the pole is maximized when ${{dh}}_{\mathrm{pole}}^{{\prime} }/{df}=0$ is satisfied, i.e.,

$$\begin{eqnarray}&&\begin{array}{l}\psi \sin (f-{\rm{\Lambda }})-\cos (f-{\rm{\Lambda }})\\ \quad +\,\psi \displaystyle \frac{{e}^{\psi ({\rm{\Lambda }}-3\pi /2)}+{e}^{\psi ({\rm{\Lambda }}-2\pi )}}{1-{e}^{-3\pi \psi /2}}\exp (-\psi f)=0.\end{array}\end{eqnarray} \tag{ 39 }$$

Again, one needs to solve this equation numerically. But, in the limit of ψ ≫ 1, the phase of the maximum height field is given by

$$\begin{eqnarray}&&f={\rm{\Lambda }}+{\tan }^{-1}\left(\displaystyle \frac{1}{\psi }\right)\approx {\rm{\Lambda }}+\displaystyle \frac{2\pi {\tau }_{\mathrm{rad}}}{{P}_{\mathrm{orb}}}.\end{eqnarray} \tag{ 40 }$$

This indicates that the flux from the polar region is maximized after the planet passes the solstice. Therefore, the phase of the light curve occurs after the solstice passage, which is in agreement with the light curves in regime (III) in Figure 3. Equation (39) also indicates that the phase of the flux peak from the polar region is independent of the obliquity, which explains why planets with different obliquities produce a flux peak at the same orbital phase for fixed Λ in Figure 3. Rauscher (2017) assumed that the diurnally averaged insolation, and thus the simulated light curves, should fall into this regime. Figure 12 shows the peak offset retrieved from the light curves in Rauscher (2017) and our analytical model (Equation (10)). The theory presented here excellently matches the results of Rauscher (2017), implying that the flux from the polar region indeed controls the shape of the light curves in this regime.

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The light curves for regimes (IV) and (V) can also be understood from the limit of ψ ≪ 1. In the limit of ψ ≪ 1, the height field at the pole is approximated as

$$\begin{eqnarray}&&{h}_{\mathrm{pole}}^{{\prime} }\approx {\rm{\Delta }}h\psi \sin \theta \left[\sin (f-{\rm{\Lambda }})+\displaystyle \frac{4+\psi (4{\rm{\Lambda }}-7\pi )}{3\pi \psi }\right].\end{eqnarray} \tag{ 41 }$$

In this case, the flux peak occurs at f = Λ + π/2, and this is consistent with the light curves for regimes (IV) and (V) in Figure 3, although the light curves are nearly flat because the amplitude scales as ${\rm{\Delta }}h\psi \sin \theta$.

We now attempt to derive a universal formula of the peak offset for tilted planets. The transition from the short-τ_rad limit to the long-τ_rad and high-obliquity limit occurs when the flux from the polar region becomes stronger than the flux from the hot spot shifted from the substellar point. To express the smooth transition of these two limits, we construct a toy model; the height field is excited from the mean value only at the shifted hot spot and the pole. In this context, the height field is expressed as

$$\begin{eqnarray}&&h=H+h^{\prime} ({\boldsymbol{r}})(\delta ({\boldsymbol{r}}-{{\boldsymbol{r}}}_{\mathrm{hs}})+\delta ({\boldsymbol{r}}-{{\boldsymbol{r}}}_{\mathrm{pole}})),\end{eqnarray} \tag{ 42 }$$

where δ is the delta function. Substituting Equation (42) into Equation (11), the diagnosis of the emergent flux is given by

$$\begin{eqnarray}\begin{array}{rcl}F & = & {\displaystyle \int }_{0}^{2\pi }{\displaystyle \int }_{0}^{\pi /2}[H+h({\boldsymbol{r}})(\delta ({\boldsymbol{r}}-{{\boldsymbol{r}}}_{\mathrm{hs}})\\ & & +\,\delta ({\boldsymbol{r}}-{{\boldsymbol{r}}}_{\mathrm{pole}}))]({\boldsymbol{r}}\cdot {{\boldsymbol{r}}}_{\mathrm{obs}})\sin \phi ^{\prime} d\phi ^{\prime} d\lambda ^{\prime} ,\end{array}\end{eqnarray} \tag{ 43 }$$

where ϕ' is the angle from the subobserver latitude (i.e., $\phi ^{\prime} ={\cos }^{-1}({\boldsymbol{r}}\cdot {{\boldsymbol{r}}}_{\mathrm{obs}})$) and λ' is the longitude on the plane perpendicular to the subobserver point vector. Equation (43) yields

$$\begin{eqnarray}\begin{array}{rcl}F & = & \pi H+h^{\prime} \left({{\boldsymbol{r}}}_{\mathrm{hs}}\right)\left({{\boldsymbol{r}}}_{\mathrm{obs}}\cdot {{\boldsymbol{r}}}_{\mathrm{hs}}\right){ \mathcal H }\left({{\boldsymbol{r}}}_{\mathrm{obs}}\cdot {{\boldsymbol{r}}}_{\mathrm{hs}}\right)\\ & & \times \sqrt{1-{\left({{\boldsymbol{r}}}_{\mathrm{obs}}\cdot {{\boldsymbol{r}}}_{\mathrm{hs}}\right)}^{2}}\\ & & +\,h^{\prime} \left({{\boldsymbol{r}}}_{\mathrm{pole}}\right)\left({{\boldsymbol{r}}}_{\mathrm{obs}}\cdot {{\boldsymbol{r}}}_{\mathrm{pole}}\right)\sqrt{1-{\left({{\boldsymbol{r}}}_{\mathrm{obs}}\cdot {{\boldsymbol{r}}}_{\mathrm{pole}}\right)}^{2}}\\ & \approx & \pi H+h^{\prime} \left({{\boldsymbol{r}}}_{\mathrm{hs}}\right){ \mathcal P }{ \mathcal H }({ \mathcal P })+h^{\prime} \left({{\boldsymbol{r}}}_{\mathrm{pole}}\right)\cos \,{\rm{\Lambda }}\,\sin \theta .\end{array}\end{eqnarray} \tag{ 44 }$$

The first, second, and third terms express the emergent flux from the mean height field, shifted hot spot, and pole, respectively. Inserting Equation (35) into Equation (34) with λ' = φ, the height field at the hot spot is given by

$$\begin{eqnarray}&&h^{\prime} ({{\boldsymbol{r}}}_{\mathrm{hs}})={\rm{\Delta }}h\cos \varphi .\end{eqnarray} \tag{ 45 }$$

Therefore, using Equation (24), the second term is expressed as

$$\begin{eqnarray}\begin{array}{rcl}h^{\prime} ({{\boldsymbol{r}}}_{\mathrm{hs}}){ \mathcal P } & = & {\rm{\Delta }}h\cos \varphi \{\cos (f-{\rm{\Lambda }})\\ & & \times \,[({\sin }^{2}\theta +\cos \varphi \,{\cos }^{2}\theta )\cos \,{\rm{\Lambda }}\\ & & -\,\sin \varphi \cos \theta \sin \,{\rm{\Lambda }}]-\sin (f-{\rm{\Lambda }})(\cos \varphi \sin \,{\rm{\Lambda }}\\ & & +\,\sin \varphi \cos \theta \cos \,{\rm{\Lambda }})\},\end{array}\end{eqnarray} \tag{ 46 }$$

where we focus on the orbital phase around the flux peak (${ \mathcal P }\gt 0$) and thus ${ \mathcal H }({ \mathcal P })=1$. On the other hand, from Equation (37), the flux from the pole is expressed by

$$\begin{eqnarray}&&\begin{array}{l}h^{\prime} ({{\boldsymbol{r}}}_{\mathrm{pole}})\cos \,{\rm{\Lambda }}\,\sin \theta ={\rm{\Delta }}h\displaystyle \frac{\psi \cos {\rm{\Lambda }}\,{\sin }^{2}\theta }{1+{\psi }^{2}}\left[\Space{0ex}{3.00ex}{0ex}\psi \cos (f-{\rm{\Lambda }})\right.\\ \quad \left.+\,\sin (f-{\rm{\Lambda }})+\displaystyle \frac{{e}^{\psi ({\rm{\Lambda }}-3\pi /2)}+{e}^{\psi ({\rm{\Lambda }}-2\pi )}}{1-{e}^{-3\pi \psi /2}}\exp (-\psi f)\right].\end{array}\end{eqnarray} \tag{ 47 }$$

Here we only consider the case of $\psi \gg 1$ and truncate the last term in the bracket of Equation (47) because the light curve is nearly flat in the limit of ψ ≪ 1. From Equations (46) and (47), we obtain the total emergent flux as

$$\begin{eqnarray}\begin{array}{rcl}F(f) & = & \pi H+{\rm{\Delta }}h\cos (f-{\rm{\Lambda }})\Space{0ex}{3.50ex}{0ex}[\cos \varphi (({\sin }^{2}\theta \\ & & +\,\cos \varphi \,{\cos }^{2}\theta )\cos {\rm{\Lambda }}\\ & & -\left.\,\sin \varphi \cos \theta \sin {\rm{\Lambda }})+\displaystyle \frac{{\psi }^{2}\cos {\rm{\Lambda }}\,{\sin }^{2}\theta }{1+{\psi }^{2}}\right]\\ & & -\,{\rm{\Delta }}h\sin (f-{\rm{\Lambda }})\Space{0ex}{3.50ex}{0ex}[\cos \varphi (\cos \varphi \sin {\rm{\Lambda }}\\ & & +\,\left.\sin \varphi \cos \theta \cos {\rm{\Lambda }})-\displaystyle \frac{\psi \cos {\rm{\Lambda }}\,{\sin }^{2}\theta }{1+{\psi }^{2}}\right]\\ & = & \pi H+{\rm{\Delta }}{{hC}}_{t}(\theta ,\varphi ,{\rm{\Lambda }},\psi )\cos [f-{\rm{\Lambda }}\\ & & +\,{\varphi }_{\mathrm{peak}}(\theta ,\varphi ,{\rm{\Lambda }},\psi )],\end{array}\end{eqnarray} \tag{ 48 }$$

where ${C}_{{\rm{t}}}(\theta ,\varphi ,{\rm{\Lambda }},\psi )$ is the prefactor controlling the light-curve amplitude, given by

$$\begin{eqnarray}\begin{array}{rcl}{C}_{{\rm{t}}}^{2} & = & {C}^{2}\,{\cos }^{2}\varphi +\displaystyle \frac{{\psi }^{2}\,{\cos }^{2}{\rm{\Lambda }}\,{\sin }^{4}\theta }{1+{\psi }^{2}}\\ & & +\,\displaystyle \frac{2\psi \cos \varphi \cos {\rm{\Lambda }}\,{\sin }^{2}\theta }{1+{\psi }^{2}}\\ & & \times \,[\psi (({\sin }^{2}\theta +\cos \varphi \,{\cos }^{2}\theta )\cos {\rm{\Lambda }}-\sin \varphi \cos \theta \sin {\rm{\Lambda }})\\ & & -\,\left.(\cos \varphi \sin {\rm{\Lambda }}+\sin \varphi \cos \theta \cos {\rm{\Lambda }})\right].\end{array}\end{eqnarray} \tag{ 49 }$$

The total flux is maximized at ${f}_{\mathrm{peak}}={\rm{\Lambda }}-{\varphi }_{\mathrm{peak}}$, where

$$\begin{eqnarray}&&{\varphi }_{\mathrm{peak}}={\tan }^{-1}\left[\displaystyle \frac{(1+{\psi }^{2})(\cos \varphi \sin {\rm{\Lambda }}+\sin \varphi \cos \theta \cos {\rm{\Lambda }})\cos \varphi -\psi \cos {\rm{\Lambda }}\,{\sin }^{2}\theta }{(1+{\psi }^{2})[({\sin }^{2}\theta +\cos \varphi \,{\cos }^{2}\theta )\cos {\rm{\Lambda }}-\sin \varphi \cos \theta \sin {\rm{\Lambda }}]\cos \varphi +{\psi }^{2}\cos {\rm{\Lambda }}\,{\sin }^{2}\theta }\right].\end{eqnarray} \tag{ 50 }$$

In the limit of short radiative timescales (ψ ≫ 1) and small obliquities, Equation (50) returns to Equation (26). On the other hand, in the limit of radiative timescales longer than the planet day (i.e., ξ ≪ 1 and thus $\varphi \to \pi /2$), Equation (50) returns to ${\varphi }_{\mathrm{peak}}\approx -{\tan }^{-1}(1/\psi )$. Consequently, the phase of the flux peak can be evaluated by ${f}_{\mathrm{peak}}={\rm{\Lambda }}-{\varphi }_{\mathrm{peak}}$ with Equation (50) for arbitrary radiative timescales, obliquities, and viewing geometries.