Transiting Exoplanet Monitoring Project (TEMP). V. Transit Follow Up for HAT-P-9b, HAT-P-32b, and HAT-P-36b

A total of six light curves were obtained for the HAT-P-9b system between 2016 January and 2017 February using the Xinglong Schmidt Telescope and Xinglong 60 cm Telescope operated by the National Astronomical Observatories of China. Seven light curves were obtained for the HAT-P-32b system between 2012 November and 2016 January using the same two telescopes. For the HAT-P-36b system, we collected a total of 26 light curves using the aforementioned telescopes as well as the Chungbuk National University Observatory in Jincheon (CbNUO) 60 cm Telescope (Kim et al. 2014), which is operated by CbNUOJ in South Korea. The observations of the HAT-P-36b system span about six years from 2012 March to 2017 April.

The Xinglong Schmidt Telescope (Zhou et al. 1999, 2001) is equipped with a 4K × 4K charge-coupled device (CCD). The field of view (FOV) is 94' × 94', and the pixel scale is 138 pixel⁻¹. The images were windowed down to 512 × 512 pixels to reduce the readout time from 93 s to 12 s. A Johnson/Cousins R-band filter was used for this telescope during our observations.

The Xinglong 60 cm Telescope used three different CCDs over the course of our observing program. Before 2014 November, the telescope was equipped with a 512 × 512 CCD, with a FOV of 17' × 17', a pixel scale of 195 pixel⁻¹, and a standard readout time of 3 s. After that, the telescope used a 1K × 1K CCD, giving a FOV of 17' × 17', a pixel scale of 099 pixel⁻¹, and a readout time of about 23 s. In 2015 October readout problems led to this CCD being replaced by a 2K × 2K CCD with a FOV of 36' × 36', a pixel scale of 106 pixel⁻¹, and a readout time of 6 s. Finally in 2016 June the 1K × 1K CCD was equipped again and used for the rest of the observing program. All of the observations for this telescope also used a Johnson/Cousins R-band filter.

The CCD system of the CbNUO 60 cm Telescope was upgraded in 2012/2013. For the 2012 observations, a 1530 × 1020 SBIG ST-8XE CCD was used, giving a FOV of 27' × 18', a pixel scale of 105 pixel⁻¹, and a readout time of 10 s. The 2013 observations used a 4K × 4K SBIG STX-16083 CCD, which had a FOV of 72' × 72', a pixel scale of 105 pixel⁻¹, and a readout time of 18 s. The 2012 observations were taken with no filter, and the 2013 observations used an R-band filter.

In order to increase the duty cycle of the observations, and reduce the Poisson and scintillation noise, we slightly defocused each telescope following the description in Southworth et al. (2009) and Hinse et al. (2015). The exposure time was set based on the target magnitude and weather conditions to get ideal cadence and enough counts in the CCD linear response regime. Exposure times would be adjusted due to significant weather variations, but we kept it constant during the ingress and egress phases, as it is important for accurate determination of mid-transit times. The telescope times were frequently updated based on GPS time servers. For each exposure, the beginning time was recorded in the image header using the UTC time standard. The summaries of our observations for the HAT-P-9b, HAT-P-32b, and HAT-P-36b systems are listed in Tables 1, 2, and 3, respectively.

Table 1.  Overview of Observations and Data Reduction for HAT-P-9

Date	Time	Telescope	Band	Frames	Exposure	Read	Air Mass	Moon	Comp.	Aperturea	Scatterb
(UTC)	(UTC)				(s)	(s)		Illum.	Stars	(pixels)	(mmag)
2016 Jan 11	$15:38:45\to 22:02:56$	Xinglong 60 cm	R	472	35–50	13	$1.01\to 1.00\to 2.45$	0.03	4	18	2.3
2016 Mar 10	$11:09:20\to 16:19:25$	Xinglong Schmidt	R	275	45–60	12	$1.03\to 1.00\to 1.46$	0.04	4	10	2.5
2016 Nov 08	$16:10:50\to 21:53:45$	Xinglong Schmidt	R	135	130–150	12	$1.47\to 1.00\to 1.06$	0.59	4	12	1.8
2016 Dec 29	$16:43:02\to 21:34:39$	Xinglong Schmidt	R	133	120	12	$1.00\to 1.66$	0.00	4	14	1.5
2017 Jan 06	$12:28:55\to 17:31:00$	Xinglong Schmidt	R	112	120–180	12	$1.43\to 1.00\to 1.03$	0.60	3	10	2.2
2017 Feb 26	$11:39:06\to 17:35:08$	Xinglong Schmidt	R	138	140–160	12	$1.05\to 1.00\to 1.62$	0.00	4	14	2.0

Notes.

^aThis column indicates the aperture diameter used in SExtractor. ^bThis column presents the rms scatter of residuals from the best-fitting model.

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Table 2.  Overview of Observations and Data Reduction for HAT-P-32

Date	Time	Telescope	Band	Frames	Exposure	Read	Air Mass	Moon	Comp.	Aperturea	Scatterb
(UTC)	(UTC)				(s)	(s)		Illum.	Stars	(pixels)	(mmag)
2012 Nov 05	12:02:32 → 19:28:02	Xinglong Schmidt	R	312	30–50	12	$1.21\to 1.01\to 1.43$	0.63	5	17	1.9
2012 Nov 18	11:30:17 → 17:07:22	Xinglong Schmidt	R	236	35	12	$1.17\to 1.01\to 1.16$	0.30	3	17	3.3
2012 Dec 03	12:59:24 → 17:57:58	Xinglong Schmidt	R	226	30	12	$1.01\to 1.51$	0.78	4	17	2.0
2013 Dec 10	12:10:40 → 16:59:22	Xinglong Schmidt	R	173	60	12	$1.02\to 1.01\to 1.39$	0.62	5	13	2.0
2014 Dec 17	11:23:21 → 16:08:56	Xinglong Schmidt	R	336	30–40	12	$1.03\to 1.01\to 1.29$	0.21	3	18	2.7
2015 Jan 01	11:58:10 → 16:27:07	Xinglong Schmidt	R	273	40–60	12	$1.01\to 1.63$	0.88	4	13	3.2
2016 Jan 08	10:27:14 → 15:29:50	Xinglong 60 cm	R	552	25–30	13	$1.01\to 1.46$	0.02	4	20	1.9

Notes.

^aThis column indicates the aperture diameter used in SExtractor. ^bThis column presents the rms scatter of residuals from the best-fitting model.

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Table 3.  Overview of Observations and Data Reduction for HAT-P-36

Date	Time	Telescope	Band	Frames	Exposure	Read	Air Mass	Moon	Comp.	Aperturea	Scatterb
(UTC)	(UTC)				(s)	(s)		Illum.	Stars	(pixels)	(mmag)
2012 Mar 20	13:25:51 → 19:50:38	CbNUO 60 cm	Nc	179	120	10	1.17→1.00→1.34	0.04	5	16	1.8
2012 Mar 28	13:07:46 → 19:01:14	CbNUO 60 cm	Nc	163	120	10	1.15→1.00→1.28	0.30	5	18	1.8
2013 Mar 04	14:09:40 → 20:47:21	CbNUO 60 cm	R	175	120	18	$1.22\to 1.00\to 1.32$	0.53	3	15	2.2
2013 Mar 20	13:26:42 → 19:58:27	CbNUO 60 cm	R	91	240	18	$1.17\to 1.00\to 1.38$	0.59	5	15	1.2
2014 Mar 04	16:32:07 → 21:50:59	Xinglong 60 cm	R	274	60	3	$1.04\to 1.00\to 1.37$	0.15	3	16	3.4
2014 Mar 08	15:48:53 → 21:54:47	Xinglong 60 cm	R	344	60	3	$1.07\to 1.00\to 1.45$	0.52	3	12	3.4
2014 Mar 12	15:32:33 → 21:49:45	Xinglong 60 cm	R	307	60–90	3	$1.07\to 1.00\to 1.50$	0.86	3	9	4.1
2014 Apr 01	11:40:47 → 21:05:00	Xinglong 60 cm	R	460	90	3	$1.47\to 1.00\to 1.69$	0.05	3	8	4.6
2014 Apr 05	11:41:21 → 21:02:57	Xinglong 60 cm	R	463	60–80	3	$1.40\to 1.00\to 1.79$	0.35	3	8	3.2
2014 May 07	12:07:56 → 19:15:03	Xinglong 60 cm	R	385	60	3	$1.05\to 1.00\to 1.93$	0.56	3	8	4.4
2015 Feb 04	16:50:42 → 22:17:22	Xinglong 60 cm	R	238	60	23	$1.18\to 1.00\to 1.14$	0.99	4	12	3.1
2015 Feb 12	17:11:42 → 22:20:31	Xinglong 60 cm	R	225	60	23	$1.08\to 1.00\to 1.21$	0.44	3	17	2.6
2015 Feb 16	16:56:47 → 22:12:53	Xinglong 60 cm	R	230	60	23	$1.08\to 1.00\to 1.23$	0.08	3	17	2.3
2015 Apr 25	11:58:38 → 19:38:56	Xinglong 60 cm	R	335	60	23	$1.13\to 1.00\to 1.75$	0.47	3	12	2.8
2016 Jan 10	16:23:35 → 22:43:20	Xinglong 60 cm	R	559	35	6	$1.68\to 1.00\to 1.04$	0.01	3	18	2.9
2016 Jan 14	15:41:41 → 20:18:11	Xinglong 60 cm	R	299	50	6	$1.87\to 1.02$	0.25	4	16	2.6
2016 Jan 18	15:26:35 → 19:12:18	Xinglong 60 cm	R	88	150	6	$1.86\to 1.06$	0.69	3	25	3.1
2016 Feb 15	13:37:47 → 16:23:57	Xinglong 60 cm	R	65	150	6	$1.85\to 1.15$	0.54	3	26	2.7
2016 Feb 16	18:09:38 → 22:03:27	Xinglong Schmidt	R	80	180	12	$1.02\to 1.00\to 1.21$	0.65	3	14	2.7
2016 Feb 20	17:44:32 → 21:58:28	Xinglong Schmidt	R	108	140	12	$1.02\to 1.00\to 1.24$	0.96	3	13	2.6
2016 Feb 28	17:52:14 → 21:51:40	Xinglong Schmidt	R	75	180	12	$1.00\to 1.30$	0.71	3	15	2.0
2016 Mar 11	16:53:28 → 21:39:41	Xinglong 60 cm	R	675	20	6	$1.01\to 1.00\to 1.45$	0.10	5	20	3.6
2016 Apr 12	14:24:58 → 17:19:11	Xinglong 60 cm	R	226	30	6	$1.05\to 1.00\to 1.04$	0.29	3	12	3.3
2016 May 06	13:34:20 → 16:24:43	Xinglong 60 cm	R	221	40	6	$1.00\to 1.15$	0.00	3	15	5.3
2016 Jun 24	12:39:30 → 16:24:16	Xinglong 60 cm	R	258	25–30	23	$1.10\to 2.13$	0.81	3	15	6.3
2017 Apr 20	13:39:23 → 17:00:24	Xinglong 60 cm	R	263	20	23	$1.03\to 1.00\to 1.10$	0.37	3	14	5.4

Notes.

^aThis column indicates the aperture diameter used in SExtractor. ^bThis column presents the rms scatter of residuals from the best-fitting model. ^cNo band, empty band wheel slot.

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We perform a TEMP data reduction with a homogeneous routine to avoid adding systematic errors. The basic reduction procedure is described in Wang et al. (2017b). There is no exception in this work, so we will not repeat the reduction details for each system, only give a brief description for all of the three systems as follows.

The raw science frames were homogeneously calibrated using a standard procedure including bias and flat corrections. Aperture differential photometry was performed with SExtractor (Bertin & Arnouts 1996). Reference stars were chosen based on a photometric non-variability test. Aperture diameters from 5 to 35 pixels were manually varied to get the best differential light curves. Linear trends caused by variations in weather conditions were removed, and the time stamps were converted from UTC to BJD_TDB following Eastman et al. (2010). The number of comparison stars used and the aperture sizes for each observation for HAT-P-9b, HAT-P-32b, and HAT-P-36b are given in Tables 1, 2, and 3. The final light curves are presented in Tables 4, 5, and 6, respectively.

To refine the system parameters for these three systems, we performed global fits on our new light curves together with published radial-velocity (RV) data using a fast exoplanetary fitting package EXOFAST.¹³ The package calculates the parameter uncertainties through a differential evolution Markov chain Monte Carlo (DE-MCMC) algorithm. See Eastman et al. (2013) for more details about EXOFAST.

The basic processes of these global fits were similar to that described in Wang et al. (2017b). The RV data we used were from Shporer et al. (2009a), Knutson et al. (2014), and Bakos et al. (2012) for the HAT-P-9b, HAT-P-32b, and HAT-P-36b systems, respectively. The light curves we used in the global fits were from our data. We only use those with an rms scatter less than 2.5 mmag. The priors for system parameters in the fits were drawn from Shporer et al. (2009a), Hartman et al. (2011), and Bakos et al. (2012) for the HAT-P-9b, HAT-P-32b, and HAT-P-36b systems, respectively. The priors for the limb-darkening parameters (the linear limb-darkening coefficient u₁ and the quadratic limb-darkening coefficient u₂) for each system were obtained from Claret & Bloemen (2011). During the fits, the limb-darkening parameters and stellar parameters (effective temperature ${T}_{\mathrm{eff}}$, metalicity $\left[\mathrm{Fe}/{\rm{H}}\right]$, and surface gravity $\mathrm{log}({g}_{* })$) were kept fixed and the remaining parameters were set to be freely varied.

In the first stage of the global fit, EXOFAST fit the RV and transit data independently to scale the uncertainties with a reduced ${\chi }_{\mathrm{red}}^{2}=1$ for each best-fitting model. Then it fits both data sets simultaneously with 32 Markov chains. The solution converged when the Gelman–Rubin statistic was below 1.01 and the number of independent draws exceeded 1000 (Eastman et al. 2013). EXOFAST stopped after passing the convergence test six times. The resulting system parameters and their 1σ credible uncertainties for each system are listed in Tables 7–9.

Table 7.  System Parameters for HAT-P-9

Parameter	Units	This Work	Shporer et al. (2009a)	Southworth (2012)
Stellar Parameters:
M*	Mass (${M}_{\odot }$)	${1.281}_{-0.056}^{+0.057}$	1.28 ± 0.13	1.28 ± 0.10
R*	Radius (${R}_{\odot }$)	${1.338}_{-0.060}^{+0.065}$	1.32 ± 0.07	1.339 ± 0.08
L*	Luminosity (${L}_{\odot }$)	${2.62}_{-0.23}^{+0.26}$	⋯	⋯
ρ*	Density (cgs)	${0.755}_{-0.088}^{+0.094}$	⋯	⋯
$\mathrm{log}({g}_{* })$	Surface gravity (cgs)	${4.293}_{-0.035}^{+0.033}$ a	${4.29}_{-0.04}^{+0.03}$	4.293 ± 0.046
${T}_{\mathrm{eff}}$	Effective temperature (K)	6350 ± 150a	6350 ± 150	⋯
$\left[\mathrm{Fe}/{\rm{H}}\right]$	Metalicity	0.12 ± 0.20a	0.12 ± 0.20	⋯
Planetary Parameters:
e	Eccentricity	${0.084}_{-0.047}^{+0.052}$	⋯	⋯
ω*	Argument of periastron (degrees)	${152}_{-39}^{+42}$	⋯	⋯
P	Period (days)	3.92281072 ± 0.00000102b	3.92289 ± 0.00004	3.922814 ± 0.000002
a	Semimajor axis (au)	0.05287 ± 0.00078	0.053 ± 0.002	0.0529 ± 0.0014
MP	Mass (${M}_{{\rm{J}}}$)	${0.749}_{-0.063}^{+0.064}$	0.78 ± 0.09	0.778 ± 0.083
RP	Radius (${R}_{{\rm{J}}}$)	${1.393}_{-0.065}^{+0.067}$	1.40 ± 0.06	1.38 ± 0.10
ρP	Density (cgs)	${0.342}_{-0.048}^{+0.057}$	0.35 ± 0.06	⋯
log(gP)	Surface gravity	${2.979}_{-0.051}^{+0.050}$	⋯	⋯
Teq	Equilibrium temperature (K)	${1540}_{-30}^{+32}$	1530 ± 40	1540 ± 53
Θ	Safronov number	${0.0443}_{-0.0040}^{+0.0042}$	0.046 ± 0.007	0.0463 ± 0.0056
$\langle F\rangle$	Incident flux (${10}^{9}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2}$)	${1.267}_{-0.094}^{+0.10}$	1.3 ± 0.3	⋯
RV Parameters:
$e\cos {\omega }_{* }$	⋯	$-{0.060}_{-0.050}^{+0.046}$	⋯	⋯
$e\sin {\omega }_{* }$	⋯	${0.028}_{-0.041}^{+0.056}$	⋯	⋯
TP	Time of periastron (${\mathrm{BJD}}_{\mathrm{TDB}}$)	${2457811.72}_{-0.37}^{+0.47}$	⋯	⋯
K	RV semi-amplitude (m s−1)	${82.1}_{-6.4}^{+6.5}$	84.7 ± 7.9	⋯
${M}_{P}\sin i$	Minimum mass (${M}_{{\rm{J}}}$)	0.748 ± 0.063	⋯	⋯
${M}_{P}/{M}_{* }$	Mass ratio	${0.000558}_{-0.000044}^{+0.000045}$	⋯	⋯
γ	Systemic velocity (m s−1)	${22666.2}_{-5.3}^{+5.4}$	22665.0 ± 6.0	⋯
$\dot{\gamma }$	RV slope (m s−1 day−1)	$-{0.047}_{-0.050}^{+0.048}$	⋯	⋯
Primary Transit Parameters:
TC	Time of transit (${\mathrm{BJD}}_{\mathrm{TDB}}$)	2455484.913088 ± 0.000386b	⋯	⋯
${R}_{P}/{R}_{* }$	Radius of planet in stellar radii	${0.10696}_{-0.00087}^{+0.00086}$	0.1083 ± 0.0005	⋯
a/R*	Semimajor axis in stellar radii	${8.50}_{-0.35}^{+0.34}$	8.6 ± 0.2	⋯
u1	linear limb-darkening coeff.	0.286c	⋯	⋯
u2	quadratic limb-darkening coeff.	0.320c	⋯	⋯
i	Inclination (degrees)	${86.44}_{-0.36}^{+0.37}$	86.5 ± 0.2	86.10 ± 0.54
b	Impact parameter	${0.510}_{-0.052}^{+0.044}$	0.52 ± 0.03	⋯
δ	Transit depth	${0.01144}_{-0.00019}^{+0.00018}$	⋯	⋯
TFWHM	FWHM duration (days)	${0.12212}_{-0.00068}^{+0.00067}$	⋯	⋯
τ	Ingress/egress duration (days)	0.0178 ± 0.0013	0.019 ± 0.003	⋯
T14	Total duration (days)	0.1400 ± 0.0014	0.143 ± 0.004	⋯
PT	A priori non-grazing transit prob.	${0.1086}_{-0.0079}^{+0.011}$	⋯	⋯
PT, G	A priori transit prob.	${0.1347}_{-0.0098}^{+0.014}$	⋯	⋯
F0	Baseline flux	0.999726 ± 0.000100	⋯	⋯
Secondary Eclipse Parameters:
TS	Time of eclipse (${\mathrm{BJD}}_{\mathrm{TDB}}$)	2457812.95 ± 0.12	⋯	⋯
bS	Impact parameter	${0.540}_{-0.046}^{+0.047}$	⋯	⋯
TS, FWHM	FWHM duration (days)	${0.1262}_{-0.0061}^{+0.0091}$	⋯	⋯
τS	Ingress/egress duration (days)	${0.0194}_{-0.0019}^{+0.0025}$	⋯	⋯
TS,14	Total duration (days)	${0.1457}_{-0.0078}^{+0.011}$	⋯	⋯
PS	A priori non-grazing eclipse prob.	${0.1025}_{-0.0031}^{+0.0033}$	⋯	⋯
PS, G	A priori eclipse prob.	${0.1270}_{-0.0041}^{+0.0043}$	⋯	⋯

Notes. The published system parameters of HAT-P-9 from Shporer et al. (2009a) and Dittmann et al. (2012) are presented for comparison.

^aThese stellar parameters were directly cited from the discovery work (Shporer et al. 2009a). ^bThe P and T_C were obtained from a linear fit based on all of the mid-transit times (see Section 4.2.1). ^cThe limb-darkening parameters were obtained from Claret & Bloemen (2011).

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Table 8.  System Parameters for HAT-P-32

Parameter	Units	This Work	Hartman et al. (2011)	Knutson et al. (2014)
Stellar Parameters:
M*	Mass (${M}_{\odot }$)	${1.132}_{-0.050}^{+0.051}$	${1.176}_{-0.070}^{+0.043}$	⋯
R*	Radius (${R}_{\odot }$)	${1.367}_{-0.030}^{+0.031}$	1.387 ± 0.067	⋯
L*	Luminosity (${L}_{\odot }$)	${2.178}_{-0.096}^{+0.099}$	2.43 ± 0.30	⋯
ρ*	Density (cgs)	0.625 ± 0.014	⋯	⋯
$\mathrm{log}({g}_{* })$	Surface gravity (cgs)	4.22 ± 0.04a	4.22 ± 0.04	⋯
${T}_{\mathrm{eff}}$	Effective temperature (K)	6001 ± 88a	6001 ± 88	⋯
$\left[\mathrm{Fe}/{\rm{H}}\right]$	Metalicity	−0.16 ± 0.08a	−0.16 ± 0.08	⋯
Planetary Parameters:
e	Eccentricity	${0.159}_{-0.028}^{+0.051}$	0.163 ± 0.061	${0.20}_{-0.13}^{+0.19}$
ω*	Argument of periastron (degrees)	${50}_{-18}^{+27}$	52 ± 29	${58}_{-53}^{+28}$
P	Period (days)	2.15000820 ± 0.00000013b	2.150009 ± 0.000001	⋯
a	Semimajor axis (au)	${0.03397}_{-0.00050}^{+0.00051}$	${0.0344}_{-0.007}^{+0.004}$	⋯
MP	Mass (${M}_{{\rm{J}}}$)	${0.68}_{-0.10}^{+0.11}$	0.941 ± 0.166	0.79 ± 0.15
RP	Radius (${R}_{{\rm{J}}}$)	1.980 ± 0.045	2.037 ± 0.099	⋯
ρP	Density (cgs)	${0.108}_{-0.017}^{+0.018}$	${0.14}_{-0.02}^{+0.03}$	⋯
log(gP)	Surface gravity	${2.632}_{-0.072}^{+0.064}$	2.75 ± 0.07	⋯
Teq	Equilibrium temperature (K)	${1835.7}_{-6.9}^{+6.8}$	1888 ± 51	⋯
Θ	Safronov number	${0.0205}_{-0.0032}^{+0.0033}$	0.027 ± 0.004	⋯
$\langle F\rangle$	Incident flux (${10}^{9}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2}$)	${2.505}_{-0.052}^{+0.042}$	2.86 ± 0.31	⋯
RV Parameters:
$e\cos {\omega }_{* }$	⋯	${0.101}_{-0.072}^{+0.077}$	0.099 ± 0.080	${0.076}_{-0.079}^{+0.11}$
$e\sin {\omega }_{* }$	⋯	${0.120}_{-0.015}^{+0.011}$	0.124 ± 0.037	${0.15}_{-0.15}^{+0.19}$
TP	Time of periastron (${\mathrm{BJD}}_{\mathrm{TDB}}$)	${2456237.031}_{-0.062}^{+0.11}$	⋯	⋯
K	RV semi-amplitude (m s−1)	${99}_{-15}^{+16}$	136.1 ± 23.8	${112}_{-21}^{+20}$
${M}_{P}\sin i$	Minimum mass (${M}_{{\rm{J}}}$)	${0.68}_{-0.10}^{+0.11}$	⋯	⋯
${M}_{P}/{M}_{* }$	Mass ratio	${0.000572}_{-0.000088}^{+0.000090}$	⋯	⋯
γ	Systemic velocity (m s−1)	76.1 ± 9.7	⋯	⋯
$\dot{\gamma }$	RV slope (m s−1 day−1)	−0.104 ± 0.015	⋯	−0.097 ± 0.023
Primary Transit Parameters:
TC	Time of transit (${\mathrm{BJD}}_{\mathrm{TDB}}$)	2455867.402743 ± 0.000049b	2454416.14639 ± 0.00009	⋯
RP/R*	Radius of planet in stellar radii	${0.14886}_{-0.00054}^{+0.00056}$	0.1508 ± 0.0004	⋯
a/R*	Semimajor axis in stellar radii	${5.344}_{-0.039}^{+0.040}$	5.32 ± 0.22	⋯
u1	linear limb-darkening coeff.	0.316c	⋯	⋯
u2	quadratic limb-darkening coeff.	0.303c	⋯	⋯
i	Inclination (degrees)	${88.98}_{-0.85}^{+0.68}$	88.7 ± 0.6	⋯
b	Impact parameter	${0.083}_{-0.055}^{+0.070}$	${0.108}_{-0.044}^{+0.043}$	⋯
δ	Transit depth	${0.02216}_{-0.00016}^{+0.00017}$	⋯	⋯
TFWHM	FWHM duration (days)	${0.11284}_{-0.00038}^{+0.00037}$	⋯	⋯
τ	Ingress/egress duration (days)	${0.01712}_{-0.00015}^{+0.00032}$	0.0171 ± 0.0002	⋯
T14	Total duration (days)	${0.13002}_{-0.00049}^{+0.00052}$	0.1292 ± 0.0003	⋯
PT	A priori non-grazing transit prob.	${0.1837}_{-0.0036}^{+0.0037}$	⋯	⋯
PT, G	A priori transit prob.	${0.2479}_{-0.0049}^{+0.0050}$	⋯	⋯
F0	Baseline flux	0.99940 ± 0.00013	⋯	⋯
Secondary Eclipse Parameters:
TS	Time of eclipse (${\mathrm{BJD}}_{\mathrm{TDB}}$)	${2456236.268}_{-0.098}^{+0.10}$	2454417.357 ± 0.109	⋯
bS	Impact parameter	${0.105}_{-0.070}^{+0.086}$	⋯	⋯
TS, FWHM	FWHM duration (days)	${0.1431}_{-0.0044}^{+0.0034}$	⋯	⋯
τS	Ingress/egress duration (days)	${0.02194}_{-0.00055}^{+0.00047}$	0.0221 ± 0.0017	⋯
TS,14	Total duration (days)	${0.1650}_{-0.0048}^{+0.0038}$	0.1653 ± 0.0120	⋯
PS	A priori non-grazing eclipse prob.	${0.1437}_{-0.0021}^{+0.0046}$	⋯	⋯
${P}_{S,G}$	A priori eclipse prob.	${0.1939}_{-0.0029}^{+0.0062}$	⋯	⋯

Notes. The published system parameters of HAT-P-32 from Hartman et al. (2011) and Knutson et al. (2014) are presented for comparison.

^aThese stellar parameters were directly cited from the discovery work (Hartman et al. 2011). ^bThe P and T_C were obtained from a linear fit based on all of the mid-transit times (see Section 4.2.2). ^cThe limb-darkening parameters were obtained from Claret & Bloemen (2011).

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Table 9.  System Parameters for HAT-P-36

Parameter	Units	This Work	Bakos et al. (2012)	Mancini et al. (2015)
Stellar Parameters:
M*	Mass (${M}_{\odot }$)	${1.049}_{-0.046}^{+0.048}$	1.022 ± 0.049	1.030 ± 0.029
R*	Radius (${R}_{\odot }$)	${1.108}_{-0.024}^{+0.025}$	1.096 ± 0.056	1.041 ± 0.013
L*	Luminosity (${L}_{\odot }$)	${1.053}_{-0.046}^{+0.048}$	1.03 ± 0.15	⋯
ρ*	Density (cgs)	${1.089}_{-0.024}^{+0.025}$	⋯	⋯
$\mathrm{log}({g}_{* })$	Surface gravity (cgs)	4.37 ± 0.04a	4.37 ± 0.04	4.416 ± 0.010
${T}_{\mathrm{eff}}$	Effective temperature (K)	5560 ± 100a	5560 ± 100	5620 ± 40
$\left[\mathrm{Fe}/{\rm{H}}\right]$	Metalicity	0.26 ± 0.10a	0.26 ± 0.10	0.25 ± 0.09
Planetary Parameters:
e	Eccentricity	${0.063}_{-0.023}^{+0.021}$	0.063 ± 0.032	⋯
ω*	Argument of periastron (degrees)	${51}_{-19}^{+20}$	95 ± 63	⋯
P	Period (days)	1.32734660 ± 0.00000033b	1.327347 ± 0.000003	1.32734683 ± 0.00000048
a	Semimajor axis (au)	${0.02402}_{-0.00035}^{+0.00036}$	0.0238 ± 0.0004	0.02388 ± 0.00022
MP	Mass (${M}_{{\rm{J}}}$)	${1.925}_{-0.081}^{+0.085}$	1.832 ± 0.099	1.852 ± 0.088
RP	Radius (${R}_{{\rm{J}}}$)	${1.357}_{-0.034}^{+0.035}$	1.264 ± 0.071	1.304 ± 0.021
ρP	Density (cgs)	${0.955}_{-0.054}^{+0.057}$	1.12 ± 0.19	⋯
$\mathrm{log}({g}_{P})$	Surface gravity	3.413 ± 0.017	3.45 ± 0.05	⋯
Teq	Equilibrium temperature (K)	${1820.2}_{-6.8}^{+6.7}$	1823 ± 55	1788 ± 15
Θ	Safronov number	0.0649 ± 0.0026	0.067 ± 0.005	0.0658 ± 0.003
$\langle F\rangle$	Incident flux (${10}^{9}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2}$)	2.479 ± 0.037	2.49 ± 0.30	⋯
RV Parameters:
$e\cos {\omega }_{* }$	⋯	${0.037}_{-0.023}^{+0.022}$	−0.002 ± 0.032	⋯
$e\sin {\omega }_{* }$	⋯	${0.045}_{-0.020}^{+0.021}$	0.051 ± 0.040	⋯
TP	Time of periastron (${\mathrm{BJD}}_{\mathrm{TDB}}$)	${2456698.610}_{-0.064}^{+0.065}$	⋯	⋯
K	RV semi-amplitude (m s−1)	344 ± 11	334.7 ± 14.5	316 ± 39
${M}_{P}\sin i$	Minimum mass (${M}_{{\rm{J}}}$)	${1.919}_{-0.080}^{+0.084}$	⋯	⋯
MP/M*	Mass ratio	0.001754 ± 0.000061	⋯	⋯
γ	Systemic velocity (m s−1)	$-{11.3}_{-7.8}^{+7.9}$	⋯	⋯
$\dot{\gamma }$	RV slope (m s−1 day−1)	$-{1.33}_{-0.67}^{+0.69}$	⋯	⋯
Primary Transit Parameters:
TC	Time of transit (${\mathrm{BJD}}_{\mathrm{TDB}}$)	2456698.735910 ± 0.000149b	2455565.18144 ± 0.00020	2455565.18167 ± 0.00036
RP/R*	Radius of planet in stellar radii	0.1260 ± 0.0011	0.1186 ± 0.0012	⋯
a/R*	Semimajor axis in stellar radii	${4.665}_{-0.034}^{+0.035}$	4.66 ± 0.22	⋯
u1	linear limb-darkening coeff.	0.424c	⋯	⋯
u2	quadratic limb-darkening coeff.	0.250c	⋯	⋯
i	Inclination (degrees)	${85.19}_{-0.55}^{+0.72}$	86.0 ± 1.3	85.86 ± 0.21
b	Impact parameter	${0.373}_{-0.062}^{+0.048}$	${0.312}_{-0.105}^{+0.078}$	⋯
δ	Transit depth	${0.01588}_{-0.00028}^{+0.00029}$	⋯	⋯
TFWHM	FWHM duration (days)	${0.08096}_{-0.00061}^{+0.00062}$	⋯	⋯
τ	Ingress/egress duration (days)	0.01205 ± 0.00062	0.0107 ± 0.0007	⋯
T14	Total duration (days)	${0.09302}_{-0.00075}^{+0.00073}$	0.0923 ± 0.0007	⋯
PT	A priori non-grazing transit prob.	${0.1966}_{-0.0045}^{+0.0049}$	⋯	⋯
PT, G	A priori transit prob.	${0.2533}_{-0.0055}^{+0.0060}$	⋯	⋯
F0	Baseline flux	1.00022 ± 0.00012	⋯	⋯
Secondary Eclipse Parameters:
TS	Time of eclipse (${\mathrm{BJD}}_{\mathrm{TDB}}$)	2456698.103 ± 0.019	2455565.844 ± 0.027	⋯
bS	Impact parameter	${0.408}_{-0.054}^{+0.038}$	⋯	⋯
${T}_{S,{FWHM}}$	FWHM duration (days)	${0.0871}_{-0.0031}^{+0.0038}$	⋯	⋯
τS	Ingress/egress duration (days)	${0.01338}_{-0.00032}^{+0.00034}$	0.0120 ± 0.0015	⋯
TS,14	Total duration (days)	${0.1006}_{-0.0031}^{+0.0036}$	0.1013 ± 0.0071	⋯
PS	A priori non-grazing eclipse prob.	${0.1796}_{-0.0036}^{+0.0035}$	⋯	⋯
PS, G	A priori eclipse prob.	${0.2314}_{-0.0049}^{+0.0048}$	⋯	⋯

Notes. The published system parameters of HAT-P-36 from Bakos et al. (2012) and Mancini et al. (2015) are presented for comparison.

^aThese stellar parameters were directly cited from the discovery work (Bakos et al. 2012). ^bThe P and T_C were obtained from a linear fit based on all of the mid-transit times (see Section 4.2.3). ^cThe limb-darkening parameters were obtained from Claret & Bloemen (2011).

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To obtain precise mid-transit times with reliable uncertainties, and thus to further refine the planetary orbital ephemerides and perform TTV analyses, we separately fit each light curve using the task 9 algorithm (a residual permutation algorithm) in the JKTEBOP¹⁴ code (Southworth et al. 2004; Southworth 2008).

The mid-transit time T_c and light scale factor F₀ were the only two free parameters in these separate fits. All of the others were fixed at the values obtained from the global fits described in Section 3.1. These fixed parameters include the sum of radii R_P/a + R_*/a (where R_* and R_P are the absolute stellar and planetary radii, respectively, and a is the semimajor axis of the planetary orbit), ratio of the radii R_P/R_*, orbital inclination i, mass ratio of the system M_P/M_*, and the combination of orbital eccentricity e and periastron longitude ω_* (presented as $e\cos {\omega }_{* }$ and $e\sin {\omega }_{* }$). The limb-darkening values for each band were fixed at the values from Claret & Bloemen (2011). The mid-transit times with their 1σ credible uncertainties for each system were finally obtained and are listed in Tables 10–12.

Table 10.  Mid-transit Times for HAT-P-9b

Epoch	Telescope	Tc	${\sigma }_{{T}_{{\rm{c}}}}$	O − C
		(BJDTDB)	(s)	(s)
−272	Shporer et al. (2009a)	2454417.90848a	25.92	−6.27
−62	Dittmann et al. (2012)	2455241.69920a	216.00	32.55
−49	Dittmann et al. (2012)	2455292.69620a	77.76	72.35
488	Xinglong 60 cm	2457399.24582	120.43	95.08
503	Xinglong Schmidt	2457458.08549	104.99	−119.67
565	Xinglong Schmidt	2457701.30337	80.85	192.15
578	Xinglong Schmidt	2457752.29752	99.93	−14.40
580	Xinglong Schmidt	2457760.14058	72.42	−235.68
593	Xinglong Schmidt	2457811.14137	85.64	131.65

Note.

^aThese mid-transit times are cited directly from Shporer et al. (2009a) and Dittmann et al. (2012).

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Table 11.  Mid-transit Times for HAT-P-32b

Epoch	Telescope	Tc	${\sigma }_{{T}_{{\rm{c}}}}$	O − C
		(BJDTDB)	(s)	(s)
−697	Hartman et al. (2011)	2454368.84711a	18.66	7.31
−684	Hartman et al. (2011)	2454396.79734a	32.25	18.24
−677	Hartman et al. (2011)	2454411.84695a	13.77	−20.75
−671	Hartman et al. (2011)	2454424.74713a	16.95	−9.02
−664	Hartman et al. (2011)	2454439.79714a	15.79	−13.82
−11	Sada et al. (2012)	2455843.75341b	16.42	65.43
−10	Sada et al. (2012)	2455845.90287b	20.74	18.06
−10	Sada et al. (2012)	2455845.90314b	20.74	41.39
0	Seeliger et al. (2014)	2455867.40301b	63.07	23.07
6	Seeliger et al. (2014)	2455880.30267b	28.51	−10.56
13	Seeliger et al. (2014)	2455895.35297b	13.82	10.40
13	Seeliger et al. (2014)	2455895.35249b	69.12	−31.07
14	Seeliger et al. (2014)	2455897.50328b	28.51	36.48
20	Seeliger et al. (2014)	2455910.40274b	37.15	−14.43
26	Seeliger et al. (2014)	2455923.30295b	26.78	−0.54
35	Seeliger et al. (2014)	2455942.65287b	55.30	−13.83
134	Seeliger et al. (2014)	2456155.50385b	22.46	0.67
135	Seeliger et al. (2014)	2456157.65470b	62.21	73.40
144	Gibson et al. (2013)	2456177.00392b	21.60	−0.37
147	Seeliger et al. (2014)	2456183.45364b	73.44	−26.69
147	Seeliger et al. (2014)	2456183.45361b	42.34	−29.28
148	Seeliger et al. (2014)	2456185.60375b	28.51	−17.90
160	Seeliger et al. (2014)	2456211.40361b	48.38	−38.50
164	Gibson et al. (2013)	2456220.00440b	16.42	26.92
172	Xinglong Schmidt	2456237.20386	24.84	−25.50
178	Xinglong Schmidt	2456250.10439	36.26	16.49
180	Seeliger et al. (2014)	2456254.40404b	19.01	−15.52
185	Xinglong Schmidt	2456265.15466	41.37	34.57
314	Seeliger et al. (2014)	2456542.50538b	27.65	5.27
314	Seeliger et al. (2014)	2456542.50530b	15.55	−1.64
314	Seeliger et al. (2014)	2456542.50522b	44.93	−8.55
328	Seeliger et al. (2014)	2456572.60532b	15.55	−9.84
340	Seeliger et al. (2014)	2456598.40539b	14.69	−12.29
341	Seeliger et al. (2014)	2456600.55546b	14.69	−6.96
354	Seeliger et al. (2014)	2456628.50585b	26.78	17.53
358	Xinglong Schmidt	2456637.10480	20.12	−75.62
367	Seeliger et al. (2014)	2456656.45533b	38.88	−36.62
531	Xinglong Schmidt	2457009.05775	69.24	56.61
538	Xinglong Schmidt	2457024.10743	39.61	23.77
711	Xinglong 60 cm Telescope	2457396.05803	78.22	−46.78

Notes.

^aThese mid-transit times are obtained from our fits based on the light curves in Hartman et al. (2011). ^bThese mid-transit times are cited directly from Sada et al. (2012), Gibson et al. (2013), and Seeliger et al. (2014).

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Table 12.  Mid-transit Times for HAT-P-36b

Epoch	Telescope	Tc	${\sigma }_{{T}_{{\rm{c}}}}$	O − C
		(BJDTDB)	(s)	(s)
−861	Bakos et al. (2012)	2455555.89140a	42.04	78.59
−830	Bakos et al. (2012)	2455597.03726a	86.26	−83.66
−827	Bakos et al. (2012)	2455601.02027a	82.48	−0.56
−821	Bakos et al. (2012)	2455608.98474a	85.21	33.13
−521	CbNUO 60 cm	2456007.18909	60.92	65.33
−515	CbNUO 60 cm	2456015.15110	29.65	−113.51
−258	CbNUO 60 cm	2456356.28178	90.16	111.53
−246	CbNUO 60 cm	2456372.20849	67.82	−13.24
−227	Mancini et al. (2015)	2456397.42892a	25.36	59.61
17	Xinglong 60 cm Telescope	2456721.30085	72.63	4.19
20	Xinglong 60 cm Telescope	2456725.28100	58.38	−159.16
23	Xinglong 60 cm Telescope	2456729.26439	82.17	−42.30
38	Xinglong 60 cm Telescope	2456749.17580	93.30	61.76
41	Xinglong 60 cm Telescope	2456753.15819	73.07	92.29
48	Mancini et al. (2015)	2456762.44829a	36.61	−21.81
51	Mancini et al. (2015)	2456766.43075a	36.58	14.13
65	Xinglong 60 cm Telescope	2456785.01425	110.19	70.05
271	Xinglong 60 cm Telescope	2457058.44614	128.65	−60.08
277	Xinglong 60 cm Telescope	2457066.41202	67.07	95.51
280	Xinglong 60 cm Telescope	2457070.39280	74.15	−13.35
331	Xinglong 60 cm Telescope	2457138.08645	63.37	−102.27
527	Xinglong 60 cm Telescope	2457398.24830	37.56	63.61
530	Xinglong 60 cm Telescope	2457402.22948	35.58	−11.22
533	Xinglong 60 cm Telescope	2457406.21179	79.67	12.21
554	Xinglong 60 cm Telescope	2457434.08580	113.94	−10.64
555	Xinglong Schmidt	2457435.41340	128.99	10.73
558	Xinglong Schmidt	2457439.39401	84.53	−112.34
564	Xinglong Schmidt	2457447.35849	45.65	−77.66
573	Xinglong 60 cm Telescope	2457459.30494	84.96	−49.05
597	Xinglong 60 cm Telescope	2457491.16331	61.79	127.44
615	Xinglong 60 cm Telescope	2457515.05405	88.24	−1.97
652	Xinglong 60 cm Telescope	2457564.16608	189.71	16.13
878	Xinglong 60 cm Telescope	2457864.14544	140.73	−68.05

Note.

^aThese mid-transit times are obtained from our fits based on the published light curves from Bakos et al. (2012) and Mancini et al. (2015).

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4.1.1. HAT-P-9b

The global system parameters we obtained for the HAT-P-9 system, and those from previous studies (Shporer et al. 2009a; Southworth 2012) are listed in Table 7. The best-fitting models for the transit and RV data are plotted in Figures 1 and 2, respectively.

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As shown in Table 7, all of our new measurements of the system parameters of HAT-P-9b show excellent agreement with those in Shporer et al. (2009a) and Southworth (2012), with the high-precision light curves allowing us to place even tighter constraints on the uncertainties.

4.1.2. HAT-P-32b

The refined system parameters for the HAT-P-32b system together with those of previous studies (Hartman et al. 2011; Knutson et al. 2014) are listed in Table 8. The best-fitting models for the transit and RV data are plotted in Figures 3 and 4, respectively.

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As Table 8 shows, the system parameters we found show good agreement with previous work (Hartman et al. 2011; Knutson et al. 2014).

4.1.3. HAT-P-36b

The refined system parameters resulting from our global fit for the HAT-P-36b system together with the previously published results (Bakos et al. 2012; Mancini et al. 2015) are listed in Table 9. The best-fitting models for the transit and RV data are plotted in Figures 5 and 6, respectively.

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As shown in Table 9, the system parameters from our analysis show good agreement with previous studies, except for R_P/R_*. The R_P/R_* of 0.1260 ± 0.0011 we found in the R band is 4.5σ larger than the published one (0.1186 ± 0.0012) in the i band.

To demonstrate that the R_P/R_* discrepancy we found does not arise from differences in the fitting process, we conducted a fit based only on the i-band data from the discovery paper (Bakos et al. 2012). All of the resulting system parameters, including R_P/R_* (0.1192 ± 0.0010), agree with those from the discovery work. The i-band best-fitting model together with our R-band model are plotted in Figure 7 for comparison.

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Multi-band photometry of hot Jupiters can reveal the Rayleigh scattering and absorption features of molecules (e.g., H₂O) and metal (e.g., Na, K, TiO, VO) in their atmospheres (Sing et al. 2016). The transit-depth discrepancy we found between the i and R bands should be useful to infer atmospheric compositions and the cloud/haze properties of HAT-P-36b. The transit-depth discrepancy in different bands has also been found in previous works studying other systems (Mancini et al. 2013; Sing et al. 2015).

4.2.1. HAT-P-9b

To perform a TTV analysis for the HAT-P-9b system, we need an accurate planetary orbital ephemeris. The orbital ephemeris in the discovery work was obtained based on four light curves within a time span of only two months (Shporer et al. 2009a). Dittmann et al. (2012) then derived a new ephemeris with two more observations, extending the time span to two years. In this work, we have added another six light curves, greatly augmenting the time span to a total of eight years.

As described in Section 3.2, we performed a separate fit for each of our six light curves to get accurate mid-transit times (T_c). The best-fitting models are plotted in Figure 8. The obtained mid-transit times together with the published ones from Shporer et al. (2009a) and Dittmann et al. (2012) are listed in Table 10. We then fit the mid-transit times with a linear function as

$$\begin{eqnarray}&&{T}_{{\rm{c}}}(N)={T}_{{\rm{c}}}(0)+N\times P,\end{eqnarray} \tag{ 1 }$$

where P is the planetary orbital period and N represents transit epoch number. T_c(0) is the zero epoch mid-transit time and T_c(N) is the time of epoch N. In this fit, we have chosen T_c(0) to be at the middle of the data time span, which minimized parameter correlations between T_c(0) and P (Shporer et al. 2009b). The best-fitting parameters are

$$\begin{eqnarray}&&{T}_{{\rm{c}}}(0)=2455484.913087\pm 0.000386\,[{\mathrm{BJD}}_{\mathrm{TDB}}],\end{eqnarray} \tag{ 2 }$$

and

$$\begin{eqnarray}&&P=3.92281072\pm 0.00000102[\mathrm{days}].\end{eqnarray} \tag{ 3 }$$

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The P from our analysis agrees with P = 3.92289 ± 0.00004 found in Shporer et al. (2009a) within 2.0σ, and it agrees with P = 3.922814 ± 0.000002 found in Dittmann et al. (2012) within 1.5σ. We believe our updated ephemeris is more precise and reliable due to the extended time span of the observations.

During the fitting process, the errors of the mid-transit times were rescaled to get ${\chi }^{2}/{N}_{\mathrm{dof}}=1$, which provides a more conservative uncertainty for the resulting period. The uncertainties of the times presented in Table 10 as well as the error bars plotted in Figure 9, however, were not rescaled.

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Figure 9 shows the deviations of mid-transit times from our new orbital ephemeris (Equations (1)–(3)), with an rms of 131 s. Though there is no significant TTV anomaly, we can put a upper mass limit on a potential perturber in the HAT-P-9b system (see Section 4.3).

4.2.2. HAT-P-32b

For HAT-P-32b, we used the same technique as in Section 4.2.1 to refine the orbital ephemeris and analyze TTVs. The accurate mid-transit times (T_c) for HAT-P-32b obtained from separately fitting our seven light curves (Figure 10) and the available data (Hartman et al. 2011) are listed in Table 11. Additional mid-transit times cited from Sada et al. (2012), Gibson et al. (2013), and Seeliger et al. (2014) are also listed in Table 11. We then fit these mid-transit times with a linear function (similar to Equations (1)–(3)), resulting in

$$\begin{eqnarray}&&{T}_{{\rm{c}}}(N)=2455867.402743(49)+N\times 2.15000820(13).\end{eqnarray} \tag{ 4 }$$

The quantities in the parentheses represent the uncertainties in the final digit of the preceding number. The orbital ephemeris agrees well with the results from Seeliger et al. (2014).

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Figure 11 shows that the deviations of the mid-transit times from the refined orbital ephemeris are small, with an rms of 31 s. A detailed TTV study for HAT-P-32b has been conducted by Seeliger et al. (2014), who analyzed a total of 29 mid-transit times (shown by green points in Figure 10), and excluded TTVs with amplitudes larger than 1.5 minutes. Our results based on the new data shown by the red points are consistent with their conclusion. An upper mass limit of a potential perturber in the HAT-P-32b system will be presented in Section 4.3.

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4.2.3. HAT-P-36b

We used the exact same technique as that used in Sections 4.2.1 and 4.2.2 to update the orbital ephemeris and to analyze the TTVs for HAT-P-36b. The separate fits were applied on each of our 26 light curves and the published data from Bakos et al. (2012) and Mancini et al. (2015). The best-fitting models for each light curve are plotted in Figure 12. The resulting mid-transit times are listed in Table 12. The updated transit ephemeris is

$$\begin{eqnarray}&&{T}_{{\rm{c}}}(N)=2456698.735910(149)+N\times 1.32734660(33).\end{eqnarray} \tag{ 5 }$$

The quantities in the parentheses represent the uncertainties in the final digit of the preceding number. Our orbital ephemeris agrees well with the result from Bakos et al. (2012) and Mancini et al. (2015).

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Figure 13 shows the deviations of mid-transit times from the new orbital ephemeris for HAT-P-36b, giving an rms of 71 s. Although the follow-up time span and quantity for HAT-P-36b are significantly extended, no obvious TTV signal is detected. Similar to the HAT-P-9b and HAT-P-32b systems, an upper mass limit on a potential perturber in this system is placed in Section 4.3.

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The results from our mid-transit time study (see Section 4.2) allow us to infer an upper mass limit for an additional planet in each system. A perturbing planet will introduce a change in the mid-transit times of a known planet, which can be quantified by the rms scatter around the nominal (unperturbed) linear ephemeris. The TTV effect is amplified for orbital configurations involving mean-motion resonances (Agol et al. 2005; Holman & Murray 2005; Nesvorný & Morbidelli 2008). In principle, this amplification would allow for the detection of low-mass planetary perturbers. A larger perturbation implies a larger scatter around the nominal ephemeris.

The calculation of an upper mass limit is performed numerically via direct orbit integrations. For this task, we have modified the fortran-based MICROFARM¹⁵ package (Goździewski 2003; Goździewski et al. 2008) which utilizes OpenMPI¹⁶ to spawn hundreds of single-task parallel jobs on a suitable super-computing facility. The package's main purpose is the numerical computation of the Mean Exponential Growth factor of Nearby Orbits (MEGNO; Cincotta & Simó 2000; Goździewski et al. 2001; Cincotta et al. 2003) over a grid of initial values of orbital parameters for an n-body problem. The calculation of the rms scatter of TTVs in the present work follows a direct brute-force method, which proved to be robust given the availability of computing power.

Within the framework of the three-body problem, we integrated the orbits of one of our three hot Jupiters and an additional perturbing planet around a central mass. The mid-transit time was calculated iteratively to a high precision from a series of back-and-forth integrations once a transit of the transiting planet was detected. The best-fit radii of both the planet and the host star were accounted for. We then calculated an analytic least-squares regression to the time series of transit numbers and mid-transit times to determine a best-fitting linear ephemeris with an associated rms statistic for the TTVs. The rms statistic was based on a 20-year integration corresponding to 1864 transits for HAT-P-9b, 3398 transits for HAT-P-32b, and 5505 transit events for HAT-P-36b. This procedure was then applied to a grid of masses and semimajor axes of the perturbing planet while fixing all of the other orbital parameters. In this study, we have chosen to start the perturbing planet on a circular orbit that is coplanar with the transiting planet; this implies that Ω₂ = 0° and ω₂ = 0° for the perturbing planet. This setting provides the most conservative estimate of the upper mass limit of a possible perturber (Bean 2009; Fukui et al. 2011; Hoyer et al. 2011, 2012). For the interested readers, we refer to Wang et al. (2018d), which has studied the effects of TTVs on varying initial orbital parameters.

In order to calculate the location of mean-motion resonances, we have used the same code to calculate MEGNO on the same parameter grid. However, this time we integrated each initial grid point for 1000 yr, allowing this study to highlight the location of weak chaotic high-order mean-motion resonances. In short, MEGNO quantitatively measures the degree of stochastic behavior of a nonlinear dynamical system and has been proven useful in the detection of chaotic resonances (Goździewski et al. 2001; Hinse et al. 2010). In addition to the Newtonian equations of motion, the associated variational equations of motion are solved simultaneously, allowing the calculation of MEGNO at each integration time step. The MICROFARM package implements the ODEX¹⁷ extrapolation algorithm to numerically solve the system of first-order differential equations.

Following the definition of MEGNO (denoted as $\langle Y\rangle$; Cincotta & Simó 2000), in a dynamical system that evolves quasi-periodically, the quantity $\langle Y\rangle$ will asymptotically approach 2.0 for $t\to \infty$. In that case, often the orbital elements associated with that orbit are bounded. In case of a chaotic time evolution, the $\langle Y\rangle$ diverges away from 2.0 with orbital parameters exhibiting erratic temporal excursions.

Importantly, MEGNO is unable to prove that a dynamical system is evolving quasi-periodically, meaning that a given system cannot be proven to be stable or bounded for all of the times. The integration of the equations of motion only considers a limited time period. However, once a given initial condition has found to be chaotic, there is no doubt about its erratic nature in the future.

In the following, we will present the results of each system for which we have calculated the scatter of TTVs on a grid of the masses and semimajor axes of a perturbing planet in a circular, coplanar orbit. The results are shown in Figures 14–16 with a resolution of 1024 × 500 pixels. In each of the three cases, we find the usual instability region located in the proximity of the transiting planet with MEGNO color coded as yellow (corresponding to $\langle Y\rangle \gt 5$). The extent of this region coincides with the results presented in Barnes & Greenberg (2006).

The perturbing planet was always started on a circular orbit with the same orbital orientation as the transiting planet. If there is an additional planet in the system, we think it is reasonable for the two planets to share the same orbital plane. The assumption on the orbital shape of the perturber is somewhat arbitrary. We refer the reader to Wang et al. (2018d) for an exploration of the effects of starting the perturbing planet on a higher eccentricity orbit.

For the HAT-P-9b system, the considered initial conditions seem to render the P₂/P₁ = 1/1 co-orbital resonance to be stable/quasi-periodic. In comparison, this is not the case for the other two systems. In each map, we mark the locations of several mean-motion resonances with arrows. By overplotting the rms scatter of the mid-transit times for a certain value, we find that the TTVs are relatively more sensitive at orbital architectures involving mean-motion resonances, confirming the results by Agol et al. (2005) and Holman & Murray (2005). As shown in Figure 14, we find that a perturber of mass (upper limit) around 1 M_⊕ will produce an rms of 131 s when located in the P₂/P₁ = 2:1, 5:2, and 3:1 exterior resonance. For the 1:2 interior resonance, a perturber mass (upper limit) as small as 0.15 M_⊕ could also generate a mid-transit time scatter of 131 s.

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For the HAT-P-32b system, a perturbing planet with an upper mass limit in the range 0.1 to 1 M_⊕ could theoretically cause a mid-transit time scatter of 31 s when located in a 1:4, 1:3, 1:2 (interior) or 2:1, 9:4, 7:3, 5:2, 8:3, 3:1, 7:2, or 4:1 (exterior) orbital resonance. This is seen from Figure 15.

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For the HAT-P-36b system, the result is somewhat more complex, and we refer to Figure 16. For the majority of the aforementioned orbital resonance configurations, a mid-transit time scatter of 71 s is produced by a perturber with an upper mass in the range of 1–10 M_⊕. The exception is the 1:2 interior resonance, for which we find that an upper mass limit of 0.3 M_⊕ can produce the same scatter.

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These observations provide accurate anchors for searches for transit time variations with the ongoing Transiting Exoplanet Survey Satellite (TESS) mission (Ricker et al. 2015), which will complete and follow up almost all transiting hot Jupiters orbiting bright stars (Wang et al. 2019).