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Shoya Kamiaka, Kento Masuda, Yuxin Xue, Yasushi Suto, Tsubasa Nishioka, Risa Murakami, Koichiro Inayama, Madoka Saitoh, Michisuke Tanaka, Atsunori Yonehara, Revisiting a gravity-darkened and precessing planetary system PTFO 8-8695: A spin–orbit non-synchronous case, Publications of the Astronomical Society of Japan, Volume 67, Issue 5, October 2015, 94, https://doi.org/10.1093/pasj/psv063
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Abstract
We reanalyse the time-variable light curves of the transiting planetary system PTFO 8-8695, in which a planet of 3 to 4 Jupiter masses orbits a rapidly rotating pre-main-sequence star. Both the planetary orbital period Porb of 0.448 d and the stellar spin period Ps of less than 0.671 d are unusually short, which makes PTFO 8-8695 an ideal system to check the model of gravity darkening and nodal precession. While the previous analysis of PTFO 8-8695 assumed that the stellar spin and planetary orbital periods are the same, we extend the analysis by discarding the spin–orbit synchronous condition, and find three different classes of solutions roughly corresponding to the nodal precession periods of 199 ± 16, 475 ± 21, and 827 ± 53 d that reproduce the transit light curves observed in 2009 and 2010. We compare the predicted light curves of the three solutions against the photometry data of a few percent accuracy obtained at Koyama Astronomical Observatory in 2014 and 2015, and find that the solution with a precession period of 199 ± 16 d is preferred even though it is preliminary. Future prospects and implications for other transiting systems are briefly discussed.
1 Introduction
PTFO 8-8695 is a pre-main-sequence star (weak-line T-Tauri star) located in the Orion OB1a star-forming region, first discovered by Briceño et al. (2005). Subsequent photometric transit and spectroscopic radial velocity measurements revealed that PTFO 8-8695 harbors one close-in planet (PTFO 8-8695 b) with planetary mass Mp < 5.5 MJ (van Eyken et al. 2012). This is the first transiting exoplanet candidate around a pre-main-sequence star.
van Eyken et al. (2012) discovered that PTFO 8-8695 in 2009 and 2010 exhibited very different transit light curves. Barnes et al. (2013, hereafter B13) found that the peculiar time-variation can be explained by the gravity darkening of the central star and the spin–orbit nodal precession of the system. Indeed they were successful in simultaneously fitting the light curves from 2009 and 2010, and estimated the system parameters including stellar and planetary radii (Rs ∼ 1.0 R⊙, Rp ∼ 1.7 RJ), planetary mass (Mp = 3.0 or 3.6 MJ), and spin–orbit angle (ϕ ∼ 70°; that is, the angle between stellar spin and planetary orbital axes).
In doing so, they assumed that the stellar spin and planetary orbital motion synchronize with each other for simplicity. Given the large spin–orbit misalignment estimated for the PTFO 8-8695 system, however, the synchronous assumption may not be justified from the dynamical point-of-view. Therefore we reanalyse the PTFO 8-8695 transit photometry without adopting the spin–orbit synchronous assumption. As is expected, the permitted parameter space is increased, and we discuss the feasibility of breaking the degeneracy among different solutions with future observations.
The rest of the present paper is organized as follows. Section 2 briefly summarizes the previous work on the PTFO 8-8695 system. We adopt a model of gravity darkening and spin–orbit nodal precession of B13, which is described in section 3. Our data analysis and parameter-fitting procedures are presented in subsections 4.1 and 4.2, respectively, and the resulting constraint on the system parameters from 2009 and 2010 data is shown in subsection 4.3. We also discuss how the long-term photometric monitoring of the system is useful for further constraining the parameter space. A preliminary analysis with the observation performed in 2014–2015 at the Koyama Astronomical Observatory (KAO) is presented in subection 4.4. We discuss the long-term dynamical stability of the system in section 5. The final section is devoted to the summary of the paper and further discussion.
2 Exoplanetary system PTFO 8-8695
The transits of PTFO 8-8695 b were observed at Palomar Observatory during 2009 December 1–2010 January 15 (hereafter, referred to as the 2009 observation), and 2010 December 8–17 (the 2010 observation) as part of the Palomar Transient Factory (PTF) that ran between 2009–2012. The 2009 and 2010 observations comprise 11 and six reliable transit light curves, respectively, with the transit period of 0.448413 d (van Eyken et al. 2012). As already mentioned in the Introduction, the light curves exhibit a significant time variation, and B13 solved this puzzle by taking into account both the rapid rotation of the star and the short transit period of the planet.
Spectroscopic measurement of PTFO 8-8695 implies that the host star is a rapid rotator with a spin period of Ps < 0.671 d (van Eyken et al. 2012). This is consistent with the observational evidence that T-Tauri stars (especially weak-line T-Tauri stars like PTFO 8-8695) tend to rotate more rapidly than main-sequence stars (Edwards et al. 1993). Because of the rapid rotation, the equator of PTFO 8-8695 would be significantly expanded relative to its polar radius.
Because of the stellar surface distortion, PTFO 8-8695 should exhibit significant gravity darkening (von Zeipel 1924; Lucy 1967) and the nodal precession of stellar spin and planetary orbital axes. B13 took account of both effects properly, and are successful in reproducing both 2009 and 2010 light curves for two different stellar masses, Ms = 0.34 and 0.44 M⊙ (Briceño et al. 2005).
The nodal-precession period depends sensitively on both the stellar spin and planetary orbital periods, Ps and Porb. While the latter is well-determined from the transit period (0.448 d), only the upper limit is obtained observationally for the former, Ps < 0.671 d. Since van Eyken et al. (2012) concluded that the stellar spin and the planetary orbital motion most likely synchronize (Ps = Porb), B13 adopted this assumption for simplicity throughout their analysis.
Theoretically speaking, however, the validity of this assumption is not clear. One of the most probable mechanisms for the spin–orbit synchronization is the tidal effect between the star and the planet. In this case, the spin–orbit synchronization and the spin–orbit alignment occur simultaneously. The equilibrium tidal model, for instance, predicts that the effective time-scales for the two processes are almost the same (Lai 2012; Rogers & Lin 2013). Nevertheless the result of B13 indicates the significant spin–orbit misalignment (ϕ ∼ 70°), and indeed they admitted that the truly synchronous rotation might be difficult to achieve in such a misaligned state. This is why we attempt in the present paper to reanalyse the PTFO 8-8695 system, to see how the estimated parameters are sensitive to the synchronous condition.
3 Basic equations
This section summarizes the basic equations that we use in the upcoming analysis, largely following B13. Our model employs 16 parameters in total, and for clarity we list them in table 1.
Parameter . | Symbol . | Fixed value . | Notes* . |
---|---|---|---|
Mean stellar density | ρs | – | a |
Stellar effective temperature at the pole | T pol | 3470 K | b |
Stellar rotation period | P s | – | |
Time of inferior conjunction | t c | 2455543.9402 HJD | c, d |
Stellar inclination at tc | i s | – | e |
Stellar moment of inertia coefficient | C | 0.059 | f |
Limb-darkening parameter | c 1 = u1 + u2 | 0.735 | f |
Limb-darkening parameter | c 2 = u1 − u2 | 0.0 | |
Gravity-darkening parameter | β | 0.25 | f |
Planet-to-star mass ratio | M p/Ms | – | |
Planet-to-star radius ratio | R p/Rs | – | a |
Planetary orbital period | P orb | 0.448413 d | d |
Orbital eccentricity | e cos ω | 0 | g |
Orbital eccentricity | e sin ω | 0 | g |
Planetary orbital inclination at tc | i orb | – | e |
Longitude of the ascending node at tc | Ω | – | e |
Parameter . | Symbol . | Fixed value . | Notes* . |
---|---|---|---|
Mean stellar density | ρs | – | a |
Stellar effective temperature at the pole | T pol | 3470 K | b |
Stellar rotation period | P s | – | |
Time of inferior conjunction | t c | 2455543.9402 HJD | c, d |
Stellar inclination at tc | i s | – | e |
Stellar moment of inertia coefficient | C | 0.059 | f |
Limb-darkening parameter | c 1 = u1 + u2 | 0.735 | f |
Limb-darkening parameter | c 2 = u1 − u2 | 0.0 | |
Gravity-darkening parameter | β | 0.25 | f |
Planet-to-star mass ratio | M p/Ms | – | |
Planet-to-star radius ratio | R p/Rs | – | a |
Planetary orbital period | P orb | 0.448413 d | d |
Orbital eccentricity | e cos ω | 0 | g |
Orbital eccentricity | e sin ω | 0 | g |
Planetary orbital inclination at tc | i orb | – | e |
Longitude of the ascending node at tc | Ω | – | e |
*(a) Specified by stellar equatorial radius Rs, eq;
Parameter . | Symbol . | Fixed value . | Notes* . |
---|---|---|---|
Mean stellar density | ρs | – | a |
Stellar effective temperature at the pole | T pol | 3470 K | b |
Stellar rotation period | P s | – | |
Time of inferior conjunction | t c | 2455543.9402 HJD | c, d |
Stellar inclination at tc | i s | – | e |
Stellar moment of inertia coefficient | C | 0.059 | f |
Limb-darkening parameter | c 1 = u1 + u2 | 0.735 | f |
Limb-darkening parameter | c 2 = u1 − u2 | 0.0 | |
Gravity-darkening parameter | β | 0.25 | f |
Planet-to-star mass ratio | M p/Ms | – | |
Planet-to-star radius ratio | R p/Rs | – | a |
Planetary orbital period | P orb | 0.448413 d | d |
Orbital eccentricity | e cos ω | 0 | g |
Orbital eccentricity | e sin ω | 0 | g |
Planetary orbital inclination at tc | i orb | – | e |
Longitude of the ascending node at tc | Ω | – | e |
Parameter . | Symbol . | Fixed value . | Notes* . |
---|---|---|---|
Mean stellar density | ρs | – | a |
Stellar effective temperature at the pole | T pol | 3470 K | b |
Stellar rotation period | P s | – | |
Time of inferior conjunction | t c | 2455543.9402 HJD | c, d |
Stellar inclination at tc | i s | – | e |
Stellar moment of inertia coefficient | C | 0.059 | f |
Limb-darkening parameter | c 1 = u1 + u2 | 0.735 | f |
Limb-darkening parameter | c 2 = u1 − u2 | 0.0 | |
Gravity-darkening parameter | β | 0.25 | f |
Planet-to-star mass ratio | M p/Ms | – | |
Planet-to-star radius ratio | R p/Rs | – | a |
Planetary orbital period | P orb | 0.448413 d | d |
Orbital eccentricity | e cos ω | 0 | g |
Orbital eccentricity | e sin ω | 0 | g |
Planetary orbital inclination at tc | i orb | – | e |
Longitude of the ascending node at tc | Ω | – | e |
*(a) Specified by stellar equatorial radius Rs, eq;
3.1 Configuration of the system
Figure 1 defines the parameters specifying the geometric configuration of the system. We define the Cartesian coordinates in such a way that the star is located at the origin and the x–y plane coincides with the sky plane, with the positive z-direction pointing toward the observer. The stellar spin vector projected onto the sky plane is defined as the positive y-direction, and the x-direction is set so as to form a right-handed triad.

Schematic illustration for spin–orbit angle ϕ as a function of stellar inclination is, planetary orbital inclination iorb, and longitude of the ascending node Ω. (Color online)
The stellar inclination and planetary orbital inclination, is and iorb, respectively, are measured from the line-of-sight pointing to the observer (positive z-axis). The longitude of the ascending node, Ω, is measured from the positive x-axis to the direction toward the ascending node counterclockwise.
In what follows, S and L denote the stellar spin and planetary orbital angular momentum vectors, respectively. We define the total angular momentum vector as J = S + L. Their unit vectors and norms are denoted as s, l, and j, and S, L, and J, respectively.
3.2 Flux profile of the central star
3.3 Spin–orbit nodal precession
Note that since the equations of motion depend on ρs alone, we cannot evaluate the stellar mass (Ms) and radius (Rs) separately. We also emphasize here that the current analysis does not fix the absolute mass or radius (Ms, Mp, Rs, Rp), but fixes their ratios (Mp/Ms, Rp/Rs) only.
4 Data analysis: methods and results
4.1 Data reduction
The photometric data analysed by B13 are kindly provided to us by Julian van Eyken, and include 11 reliable transits in 2009 and six in 2010 observed at 0.658 μm. The effects of stellar activity, long-term periodical fluctuations (for instance, due to stellar spots), and background noise are already removed and whitened.
We repeat the data reduction ourselves following B13.
Phase-fold. All transit light curves are stacked with the orbital period of 0.448413 d.
Clipping. During the orbital period of 0.448413 d, the transit actually continues for ∼ 0.2 d at best. Thus a major part of the phase-folded light curve corresponds to out-of-transit phase. In order to reduce the number of data points and subsequent computational cost, we focus on ±0.1 d around the transit center.
Binning. All data points are combined into one-minute bins.
Positioning the 2009 and 2010 light curves. B13 found that the transit centers for 2009 and 2010 data are 30861700 and 60848300 s measured from 2009 January 1 UTC. Therefore we position the 2009 and 2010 light curves at those epochs. The mutual separation between the 2009 and 2010 light curves corresponds to 774 transits.
4.2 Parameter fitting procedure
Next we perform the parameter fitting as follows.
Calculate
assuming a set of values for the parameters listed in table 1.Calculate s and l at the observed epochs in 2009 and 2010 by extrapolating those at tc through the precession model prediction.
Evaluate angular parameters (is, iorb, Ω) at the observed epochs in 2009 and 2010.
Compute the model light curves at those epochs using a routine developed by Masuda (2015), and evaluate
between model and phase-folded light curves, where i indicates the variables evaluated at time ti in 2009 and 2010.
The above procedures are iterated so as to find the set of parameters that minimizes χ2 with the Levenberg–Marquardt algorithm (Markwardt 2009). In what follows, we discuss in terms of the reduced chi-square
We see how the minimum
4.3 Results
Figure 2 shows the minimum

Minimum
Figure 2 suggests that slightly better, or at least equally good, fits are obtained outside the synchronous condition (Ps = 0.448413 d), i.e., in the more rapid stellar spin and massive planet regime. Therefore, the synchronous condition over constrains the system parameters, at least unless we have more precise observational data for the stellar spin period, Ps.
It is difficult to see from figure 2, but the resulting solutions are roughly divided into three groups that are different in each precession period (

Precession period dependence of
Best-fitting parameters for three our solutions and two solutions in Barnes et al. (2013).*
Parameter . | Short solution . | Middle solution . | Long solution . | B13 (Ms = 0.34 M⊙) . | B13 (Ms = 0.44 M⊙) . |
---|---|---|---|---|---|
ρs (g cm−3) | 0.32 ± 0.01 | 0.27 ± 0.01 | 0.30 ± 0.01 | 0.43 ± 0.01 | 0.57 ± 0.02 |
P s (d) | 0.390 ± 0.008 | 0.367 ± 0.006 | 0.331 ± 0.008 | 0.448410 | 0.448413 |
i s (°) | 123.9 ± 2.2 | 128.2 ± 1.7 | 131.3 ± 1.5 | 119.4 ± 0.3 | 120.3 ± 1.3 |
M p/Ms | 0.0129 ± 0.0014 | 0.0178 ± 0.0009 | 0.0199 ± 0.0008 | 0.0084 ± 0.0006 | 0.0078 ± 0.0007 |
R p/Rs | 0.169 ± 0.003 | 0.208 ± 0.008 | 0.289 ± 0.016 | 0.159 ± 0.007 | 0.164 ± 0.007 |
i orb (°) | 62.9 ± 0.8 | 55.3 ± 0.5 | 50.9 ± 0.6 | 65.2 ± 1.6 | 69.3 ± 1.3 |
Ω (°) | 133.4 ± 3.3 | 131.2 ± 2.4 | 130.2 ± 2.2 | 136.1 ± 5.2 | 125.5 ± 0.5 |
ϕ (°) | 75.3 ± 2.5 | 85.8 ± 1.8 | 92.3 ± 1.6 | 69 ± 3 | 73.1 ± 0.6 |
198.7 ± 15.6 | 474.6 ± 21.1 | 826.9 ± 53.3 | 292.6 | 581.2 |
Parameter . | Short solution . | Middle solution . | Long solution . | B13 (Ms = 0.34 M⊙) . | B13 (Ms = 0.44 M⊙) . |
---|---|---|---|---|---|
ρs (g cm−3) | 0.32 ± 0.01 | 0.27 ± 0.01 | 0.30 ± 0.01 | 0.43 ± 0.01 | 0.57 ± 0.02 |
P s (d) | 0.390 ± 0.008 | 0.367 ± 0.006 | 0.331 ± 0.008 | 0.448410 | 0.448413 |
i s (°) | 123.9 ± 2.2 | 128.2 ± 1.7 | 131.3 ± 1.5 | 119.4 ± 0.3 | 120.3 ± 1.3 |
M p/Ms | 0.0129 ± 0.0014 | 0.0178 ± 0.0009 | 0.0199 ± 0.0008 | 0.0084 ± 0.0006 | 0.0078 ± 0.0007 |
R p/Rs | 0.169 ± 0.003 | 0.208 ± 0.008 | 0.289 ± 0.016 | 0.159 ± 0.007 | 0.164 ± 0.007 |
i orb (°) | 62.9 ± 0.8 | 55.3 ± 0.5 | 50.9 ± 0.6 | 65.2 ± 1.6 | 69.3 ± 1.3 |
Ω (°) | 133.4 ± 3.3 | 131.2 ± 2.4 | 130.2 ± 2.2 | 136.1 ± 5.2 | 125.5 ± 0.5 |
ϕ (°) | 75.3 ± 2.5 | 85.8 ± 1.8 | 92.3 ± 1.6 | 69 ± 3 | 73.1 ± 0.6 |
198.7 ± 15.6 | 474.6 ± 21.1 | 826.9 ± 53.3 | 292.6 | 581.2 |
*Our angular parameters (is, iorb, Ω) are evaluated at tc = 2455543.9402 HJD (Heliocentric Julian Date) while those of B13 are at tc = 2455536.7680 HJD, the interval of which corresponds to 7.1722 d or 16 transits. Spin–orbit angle ϕ is time-invariant.
Best-fitting parameters for three our solutions and two solutions in Barnes et al. (2013).*
Parameter . | Short solution . | Middle solution . | Long solution . | B13 (Ms = 0.34 M⊙) . | B13 (Ms = 0.44 M⊙) . |
---|---|---|---|---|---|
ρs (g cm−3) | 0.32 ± 0.01 | 0.27 ± 0.01 | 0.30 ± 0.01 | 0.43 ± 0.01 | 0.57 ± 0.02 |
P s (d) | 0.390 ± 0.008 | 0.367 ± 0.006 | 0.331 ± 0.008 | 0.448410 | 0.448413 |
i s (°) | 123.9 ± 2.2 | 128.2 ± 1.7 | 131.3 ± 1.5 | 119.4 ± 0.3 | 120.3 ± 1.3 |
M p/Ms | 0.0129 ± 0.0014 | 0.0178 ± 0.0009 | 0.0199 ± 0.0008 | 0.0084 ± 0.0006 | 0.0078 ± 0.0007 |
R p/Rs | 0.169 ± 0.003 | 0.208 ± 0.008 | 0.289 ± 0.016 | 0.159 ± 0.007 | 0.164 ± 0.007 |
i orb (°) | 62.9 ± 0.8 | 55.3 ± 0.5 | 50.9 ± 0.6 | 65.2 ± 1.6 | 69.3 ± 1.3 |
Ω (°) | 133.4 ± 3.3 | 131.2 ± 2.4 | 130.2 ± 2.2 | 136.1 ± 5.2 | 125.5 ± 0.5 |
ϕ (°) | 75.3 ± 2.5 | 85.8 ± 1.8 | 92.3 ± 1.6 | 69 ± 3 | 73.1 ± 0.6 |
198.7 ± 15.6 | 474.6 ± 21.1 | 826.9 ± 53.3 | 292.6 | 581.2 |
Parameter . | Short solution . | Middle solution . | Long solution . | B13 (Ms = 0.34 M⊙) . | B13 (Ms = 0.44 M⊙) . |
---|---|---|---|---|---|
ρs (g cm−3) | 0.32 ± 0.01 | 0.27 ± 0.01 | 0.30 ± 0.01 | 0.43 ± 0.01 | 0.57 ± 0.02 |
P s (d) | 0.390 ± 0.008 | 0.367 ± 0.006 | 0.331 ± 0.008 | 0.448410 | 0.448413 |
i s (°) | 123.9 ± 2.2 | 128.2 ± 1.7 | 131.3 ± 1.5 | 119.4 ± 0.3 | 120.3 ± 1.3 |
M p/Ms | 0.0129 ± 0.0014 | 0.0178 ± 0.0009 | 0.0199 ± 0.0008 | 0.0084 ± 0.0006 | 0.0078 ± 0.0007 |
R p/Rs | 0.169 ± 0.003 | 0.208 ± 0.008 | 0.289 ± 0.016 | 0.159 ± 0.007 | 0.164 ± 0.007 |
i orb (°) | 62.9 ± 0.8 | 55.3 ± 0.5 | 50.9 ± 0.6 | 65.2 ± 1.6 | 69.3 ± 1.3 |
Ω (°) | 133.4 ± 3.3 | 131.2 ± 2.4 | 130.2 ± 2.2 | 136.1 ± 5.2 | 125.5 ± 0.5 |
ϕ (°) | 75.3 ± 2.5 | 85.8 ± 1.8 | 92.3 ± 1.6 | 69 ± 3 | 73.1 ± 0.6 |
198.7 ± 15.6 | 474.6 ± 21.1 | 826.9 ± 53.3 | 292.6 | 581.2 |
*Our angular parameters (is, iorb, Ω) are evaluated at tc = 2455543.9402 HJD (Heliocentric Julian Date) while those of B13 are at tc = 2455536.7680 HJD, the interval of which corresponds to 7.1722 d or 16 transits. Spin–orbit angle ϕ is time-invariant.
The reason why we have three different solutions can be understood from figure 4, which plots the predicted light curves for the three solutions; the short, middle, and long solutions are plotted as red, green, and blue lines, respectively. Because the transit period is very short, the light curves in the lower panel basically show the evolution of the transit depth for three solutions. The five vertical lines in the lower panel are drawn at three-month intervals, each corresponding to the enlarged graph at the upper panel. By construction, all the three solutions reproduce the 2009 and 2010 data very well (upper-left and upper-right panels), and the significant difference of the light curves shows up only during the unobserved epochs.

Evolution of the transit light curves for three possible solutions in table 2 from 2009 to 2010 observational epochs. Short, middle and long solutions are shown by red, green, and blue lines, respectively. The lower panel illustrates the whole evolution from 2009 (leftmost vertical line) to 2010 (rightmost vertical line). The five vertical lines are drawn at about every three months, each corresponding to the enlarged graph at the upper panel. Enlarged graphs for 2009 (leftmost) and 2010 (rightmost) in the upper panel include the observational data (black points), while the three intermediate graphs show only theoretical calculations of light curves. (Color online)
The examples plotted in the three upper-middle panels illustrate the importance of frequent monitoring of the system in order to specify the system parameters precisely. Indeed some phases without any transit signals are predicted in their whole evolution because planetary orbital inclination significantly deviates from 90°. Thus measurements of the transit depth, even with a relatively low signal-to-noise ratio, greatly help to distinguish among the three solution groups.
4.4 Preliminary comparison with the photometry at Koyama Astronomical Observatory (KAO)
In this subsection, we attempt a preliminary comparison with the transit photometry taken with the Araki telescope at KAO on 2014 November 23 and 27, 2014 December 2 and 23, and 2015 January 10. The photometry has been performed by a dual-band imager, ADLER (Araki telescope DuaL-band imagER, 2048 × 2048 pixels and 12 arcmin2 field of view for each CCD camera). The observation was performed with a 670 nm split dichroic mirror, and the Sloan Digital Sky Survey (SDSS) g′- and i′-band filters. The exposure time was set to 180 s, and two exposures are stacked. The observational data were reduced with standard procedures for photometry, i.e., the dark-subtraction, flat-fielding, and aperture photometry (the daophot task in IRAF). For relative photometry, we chose a star located at (RA, Dec) = (05h25m07
The bottom panel in figure 5 shows the predicted evolution of transit depth for the three solutions over the observed epoch. The five vertical black lines indicate the above five dates, the enlarged views of which are shown in the upper panel. The predicted transit depths at those dates are less than 1% for the short solution and are greater than approximately 10% for the middle and long solutions. The filled circles, on the other hand, indicate the data points after removing the long-term stellar activity. Since there are no clear transit signals beyond the photometric noise level of a few percent, the data seem to prefer the short solution. Although this conclusion is admittedly very premature, it clearly indicates that such fairly crude photometry may be very useful for distinguishing among the three solutions. We plan to conduct the photometric observations again in the winters of 2015 and 2016.

Same as figure 4, but for the observational term of KAO from 2014 November to 2015 January. (Color online)
5 Long-term dynamical stability
So far, the results and discussion are all focused on the short-term dynamics of the system; the nodal precession has an expected time scale of several hundred days. In addition to such a short-term behavior, long-term dynamics also contributes to getting an insight into the nature of the pre-main-sequence star and close-in planet system. When pursuing the longer-term evolution, the tidal effect between the star and close-in planet should be considered. In the PTFO 8-8695 system, particularly, the star and planet are expected to be strongly influenced by the tidal effects because the planetary semimajor axis is very small (a ≤ 2Rs).
We investigate the long-term stability of this system following the equilibrium tidal model by Correia et al. (2011). Note here that two new parameters appear, in order to describe the efficiency of the tidal effect; the Love number k2 and tidal delay time Δt, both of which refer to the fluid property of the central star. The tidal evolution of the system is computed by the numerical integration of equations (19)– (21) in Correia et al. (2011). The efficiency of the tidal effect is also represented by the reduced tidal quality factor Q, which is written in terms of the above parameters as

Tidal evolution of the planetary orbital semimajor axis (top), spin–orbit angle ϕ (middle), and stellar rotational and planetary orbital period (bottom). In the upper panel, the stellar inside (a/Rs ≤ 1.0) is described. In the lower panel, stellar rotational and planetary orbital periods are shown as red and blue lines, respectively. In this calculation we employ the Love number of k2 = 0.028 and tidal delay time of Δt = 0.1 s, both of which are estimated for Sun-like stars. (Color online)
This rapid orbital decay is totally inconsistent with the observational picture that favorably represents the survival of a close-in planet in the PTFO 8-8695 system. In addition, both spin–orbit angle ϕ and orbital period Porb evolve quickly in accordance with the rapid orbital decay (the middle panel and the blue line in the lower panel), which is also incompatible with the significant spin–orbit misalignment or spin–orbit near synchronization obtained by the light-curve analysis. Since k2 = 0.028 and Δt = 0.1 s are estimated for Sun-like stars, these issues suggest that the internal structure or fluid property of PTFO 8-8695 is significantly different from those of the main-sequence stars. Reasonable constraints on k2 and Δt of PTFO 8-8695, therefore, allow us to approach the internal structure and fluid properties of pre-main-sequence stars, which are poorly known today due to the scarcity of the observed samples.
6 Summary and discussion
The spin–orbit angle ϕ plays a key role in investigating the formation mechanism of hot Jupiters (e.g., Crida & Batygin 2014 and references therein), and thus various methods for determining the angular configuration of the planetary system have been developed and utilized. The most popular one is to use the Rossiter–McLaughlin (RM) effect (Rossiter 1924; McLaughlin 1924), which reveals the sky-projected spin–orbit angle through spectroscopy (e.g., Ohta et al. 2005; Hirano et al. 2012). By combining the RM effect and asteroseismology, furthermore, the three-dimensional spin–orbit angle ϕ can be determined (Lund et al. 2014; Benomar et al. 2014).
The stellar gravity-darkening effect provides another methodology of measurement of ϕ through the analysis of photometric transit light curves for rapid rotators (Barnes et al. 2011; Zhou & Huang 2013; Ahlers et al. 2014; Masuda 2015). Advantages of this method are that one can determine the three-dimensional (i.e., not projected) spin–orbit angle directly and that the analysis is possible without the spectroscopic data (i.e., all we need is the photometric transit light curves). When nodal precession occurs, furthermore, more precise constraints on the system are obtained (B13; Philippov & Rafikov 2013; Masuda 2015).
Being a rapid rotator, PTFO 8-8695 is therefore an ideal sample to check the model of gravity darkening and nodal precession. We reanalyse the time-variable transit light curves of this planetary system with the above theoretical model, discarding the spin–orbit synchronous condition assumed in the previous analysis (B13). We find three different groups of solutions that reproduce the observed data, and their precession periods are 199 ± 16, 475 ± 21, and 827 ± 53 d. These solutions are slightly better, or at least equally good compared to those in the previous work. The three solutions reproduce 2009 and 2010 observational photometry very well, whereas the theoretically predicted transit light curves at unobserved epochs are found to be totally different. Difference in transit depth is particularly useful for distinguishing among them by frequently monitoring the system. We also present a preliminary comparison of the light-curve prediction with the additional photometry taken with the Araki telescope at KAO, and find that the solution with the precession period of 199 ± 16 d seems to be preferable. The solution implies ρs = 0.32 ± 0.01 g cm− 3, Ps = 0.390 ± 0.008 d, Mp/Ms = 0.0129 ± 0.0014, Rp/Rs = 0.169 ± 0.003, and ϕ = 75| $_{.}^{\circ}$|3 ± 2| $_{.}^{\circ}$|5. Although this result is still preliminary, more precise constraints on the system configuration are expected to result from our continuous photometric observations in the winters of 2015 and 2016.
The long-term analysis that takes into account the tidal effect between star and planet in the PTFO 8-8695 system suggests that tidal parameters k2 and Δt for pre-main-sequence stars are incompatible with those for main-sequence stars; otherwise, the orbiting planet is engulfed by the central star on the very short time-scale. This fact implies that the internal structure of young stars is likely to be different from that for matured stars, and requires further investigation.
Indeed, gravity darkening becomes more manifest as the star rotates more rapidly. This is in contrast to the RM effect, which becomes harder to measure in the rapid rotators. Therefore gravity darkening provides a complementary methodology to the RM effect in the measurement of ϕ. Since younger stars are known to rotate more rapidly than stars that have matured, gravity darkening is available to unveil the distribution of ϕ for the younger systems in future observations. Since the amplitude of ϕ becomes smaller in the course of the tidal evolution (Barker & Ogilvie 2009; Lai 2012; Xue et al. 2014), it is desirable to preferentially select younger systems which still retain the initial condition. This is yet another reason why the gravity darkening for younger systems is useful for probing the formation scenario of close-in planets.
We are grateful to Julian van Eyken, who kindly provided the photometry data of PTFO 8-8695, and to the referee, Jason W. Barnes, for very constructive comments on the manuscript. We also thank Masahiro Onitsuka for bringing our attention to PTFO 8-8695, and Masahiro Ikoma, Chelsea X. Huang, and Kengo Tomida for fruitful discussion. We are also grateful to Mizuki Isogai, Akira Arai, Naofumi Fujishiro, and Hideyo Kawakita for their technical support on observations. Data analysis was in part carried out on PC cluster at Center for Computational Astrophysics, National Astronomical Observatory of Japan. S.K. acknowledges the receipt of a travel grant from the Hayakawa Satio Fund, the Astronomical Society of Japan. K.M. is supported by JSPS (Japan Society for the Promotion of Science) Research Fellowships for Young Scientists (No. 26-7182) and by the Leading Graduate Course for Frontiers of Mathematical Sciences and Physics. Y.S. and A.Y. gratefully acknowledge the support from Grants-in Aid for Scientific Research by JSPS No. 24340035, and for Young Scientists (B) by JSPS No. 25870893, respectively.
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