Issue |
A&A
Volume 508, Number 2, December III 2009
|
|
---|---|---|
Page(s) | 1011 - 1020 | |
Section | Planets and planetary systems | |
DOI | https://doi.org/10.1051/0004-6361/200912252 | |
Published online | 15 October 2009 |
A&A 508, 1011-1020 (2009)
Transit timing analysis of the exoplanets TrES-1 and TrES-2![[*]](/icons/foot_motif.png)
M. Rabus1,2 - H. J. Deeg1,2 - R. Alonso3,4 - J. A. Belmonte1,2 - J. M. Almenara1,2
1 - Instituto de Astrofísica de Canarias, vía Lactea s/n, 38205 La Laguna, Spain
2 - Departamento de Astrofísica, Universidad de La Laguna, 38205 La Laguna, Tenerife, Spain
3 - Laboratoire d'Astrophysique de Marseille (UMR 6110), Technopôle de Marseille-Étoile, 13388 Marseille, France
4 - Observatoire de Genève, Université de Genève, 1290 Sauverny, Switzerland
Received 1 April 2009 / Accepted 28 August 2009
Abstract
Aims. The aim of this work is a detailed analysis of transit
light curves from TrES-1 and TrES-2, obtained over a period of three to
four years, in order to search for variabilities in observed
mid-transit times and to set constraints on the presence of additional
third bodies.
Methods. Using the IAC 80 cm telescope, we observed
transits of TrES-1 and TrES-2 over several years. Based on these new
data and previously published work, we studied the observed light
curves and searched for variations in the difference between observed
and calculated (based on a fixed ephemeris) transit times. To model
possible transit timing variations, we used polynomials of different
orders, simulated O-C diagrams corresponding to a perturbing third
mass, and we used sinusoidal fits. For each model we calculated the residuals and the false alarm probability (FAP).
Results. For TrES-1, we can exclude planetary companions (>1
)
in the 3:2 and 2:1 MMRs having high FAPs based on our transit
observations from the ground. Likewise, a light time effect caused,
e.g., by a 0.09
mass star at a distance of 7.8 AU is possible. As for TrES-2, we find a better ephemeris of
Epoch
and a good fit for a sine function with a period of 0.2 days,
compatible with a moon around TrES-2 and an amplitude of 57 s, but
it is not a uniquely low
value that would indicate a clear signal. In both cases, TrES-1 and
TrES-2, we are able to put upper constraints on the presence of
additional perturbers masses. We also conclude that any sinusoidal
variations that might be indicative of exomoons need to be confirmed
with higher statistical significance by further observations, noting
that TrES-2 is in the field-of-view of the Kepler Space Telescope.
Key words: planetary systems - methods: N-body simulations - techniques: photometric
1 Introduction
In 1992, the first exoplanets around the pulsar PSR B1257+12 were discovered by detections of anomalies in the pulsation period (Wolszczan & Frail 1992). Similarly, by studying variations in the times of transit, transiting exoplanets give another possibility of finding additional companions, even down to Earth masses. There are different mechanisms causing these variations. For one the gravitational influence of a perturbing body can alter the orbital period of the transiting planet (Agol et al. 2005; Holman & Murray 2005). This effect is particularly sensitive to detections of additional bodies in mean-motion resonances with the transiting planet, or of moons around that planet. For another, a perturbing mass in an orbit larger than the transiting planet can cause the ``star - transiting planet'' system to wobble around the barycentre and alter the observed periodicity, something that is known as the light-time effect (Schneider 2005; Doyle & Deeg 2004; Irwin 1959; Schneider & Doyle 1995). These anomalies are reflected in the times of transit. Using a linear ephemeris and several observations of transits, it is therefore possible to get the difference O-C between the observed transit times and the calculated ones from the ephemeris. From this difference we can learn about perturbations on strict orbital periodicity caused by a possible companion object that has not been detected yet.
Motivated by the possible detection of low-mass companions around transiting planets, we started a long-term campaign to observe transits of the exoplanets TrES-1 (Alonso et al. 2004) and TrES-2 (O'Donovan et al. 2006) over several years. Here we present the results of our observations and a study of the transit timing variations that were found.
Searches for planet-mass objects from the light time effect were first proposed for low-mass eclipsing binaries by Doyle et al. (1998) and have been performed in-depth on the system CM-Draconis, with ambiguous results to date (Morales et al. 2009; Deeg et al. 2008,2000). They were soon followed by several studies of timing effects in transiting planet systems: TrES-1 (Steffen & Agol 2005), HD 189 733 (Miller-Ricci et al. 2008b), HD 209 458 (Agol & Steffen 2007; Miller-Ricci et al. 2008a), GJ 436 (Bean & Seifahrt 2008; Alonso et al. 2008) and CoRoT-1b (Bean 2009). In all cases the authors could only constrain the parameters of a potential second planet.
The first search for variability in the transit times of TrES-1 was conducted by Steffen & Agol (2005). They fitted perturbing planets in different orbits, using transit timing observations from Charbonneau et al. (2005). Steffen & Agol (2005) did not find any strong evidence of a third object. Winn et al. (2007) observed three additional transits of TrES-1; they found that the transit times of these three observations occurred progressively later. However, as mentioned by the authors, these three measurements were not enough to make any firm conclusions. Recently, Raetz et al. (2009b) has reported a transit timing analysis of TrES-1 with the only result of being an improved ephemeris. Previous studies of transit timings for TrES-2 have been performed by Holman et al. (2007) and Raetz et al. (2009a), neither finding evidence of an unseen perturbing planet. Very recently Mislis & Schmitt (2009) analysed transit time durations between 2006 and 2008, indicating a possible change of inclination.
In Sect. 2 we describe the procedure to fit our observations to a transit model and to obtain the observed mid-transit times and their error estimates. These values are then used in Sect. 3 to interpret the O-C diagram by means of different models. Finally we discuss the results in Sect. 4.
2 Light curve fitting
All observations reported here were performed with the IAC 80 cm telescope (IAC-80) at the Teide Observatory, Tenerife. We used the same observing and analysis procedure for both TrES-1 and TrES-2. For further investigation described here, we only used observations where a complete transit light curve was acquired. A detailed description of the analysis leading from the telescope images to the light curves can be found in Rabus et al. (2009), where a study of flux variations during transits caused by starspots or additional transiting planets for TrES-1 was presented. We decorrelated the light curves against the airmass by subtracting a quadratic polynomial fit. Due to a slight defocusing of the telescope during observations, the centroids are not well defined and de-correlation against the target's detector position is not applicable. However, the de-focusing and spread of the flux over several pixels will lower the noise correlated with the target's detector position.
For TrES-1 we obtained eight useful transit observations (Table 2) over a period of three years and for TrES-2 five observations (Table 3)
over a period of two years. In order to measure the observed
mid-transit times, we first created a template of the transit event
from a folding in phase and by binning (6-point bins, with a mean size
of 66 s for TrES-1 and 90 s for TrES-2) of all observations
of the respective transiting planet (Fig. 1). The standard deviations inside the individual bins were 1.63 mmag for TrES-1 and
1.79 mmag
for TrES-2, whereas the standard deviation outside the transit part of
the binned light curve is 0.7 mmag for TrES-1 and 0.8 mmag
for TrES-2.
We used these phased and binned light curves to create a model of the transit light curve, using the formalism from Giménez (2006) and the simplex-downhill fitting algorithm (Press et al. 1992). We fitted for the planetary and stellar radii ratio, k, the sum of the projected radii, rr, and orbital inclination, i, while we kept the eccentricity fixed at zero and the limb darkening coefficient fixed, as obtained by the tables of Claret et al. (1995), assuming quadratic coefficients. The best-fit models are plotted over phased data in Fig. 1 and the best-fit parameters are shown in Table 1. We can see that our light curve model parameters are consistent with the parameters from Southworth (2008) for both TrES-1 and TrES-2.
Table 1: Comparison between best-fit values of this work and parameters from Southworth (2008) for transit light curve models of TrES-1 (upper values) and TrES-2 (lower values).
Table 2: Overview of TrES-1 transits used in this work.
Table 3: Same as Table 2, for TrES-2.
![]() |
Figure 1: Phased and binned light curve of all IAC-80 observations of TrES-1 ( top) and TrES-2 ( bottom) transits with the solid (red) line being the best-fit model light curve. |
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We then shifted in time our model with the fixed best-fit parameters
against each individual observed light curve and calculated the
residuals for each shift. The fit with the minimum
value gave us the observed mid-transit time. We estimated the timing error within a 68% (1-
)
confidence interval of the
values, given by the range where the
residuals increases by 1 over the minimum value. We also calculated the timing precision of each individual light curve,
,
by propagating their respective photometric precision,
,
using the equation (Doyle & Deeg 2004):
where L(ti) is the stellar brightness at ti,



Finally we subtracted the observed mid-transit times from the
calculated ones, obtaining the O-C values. The calculated mid-transit
times for TrES-1 were obtained from the ephemeris
Epoch (Winn et al. 2007) and for TrES-2 we used
Epoch (Holman et al. 2007).
In the following transit timing analysis we included several published mid-transit observations, by Charbonneau et al. (2005), Winn et al. (2007) and Hrudková et al. (2009) for TrES-1 (Table 2), and for TrES-2 by Holman et al. (2007) (Table 3). We considered only O-C times with errors below 60 s, which led to the rejection of some O-C values from Charbonneau et al. (2005). In the O-C residuals of TrES-1 we also removed two outliers at Epoch 255 and 358 that have been identified by Rabus et al. (2009) as transits with possible starspots. Several further transit times of TrES-2 and TrES-1 were recently reported by Raetz et al. (2009b,a), respectively, and for TrES-2 by Mislis & Schmitt (2009), but we did not include these in our study, since they had individual errors and an internal scatter several times larger than the data included in this study. The transit mid-time corresponding to Epoch 0 of TrES-2, reported by O'Donovan et al. (2006) has also been removed due to its high error of >60 s. We obtained for TrES-1 an O-C diagram spanning four years, with 16 points (Fig. 2) and for TrES-2 a diagram spanning two years, with 8 points (Fig. 3).
![]() |
Figure 2:
Difference between calculated (based on ephemeris from Winn et al. 2007) and observed mid-transit times for TrES-1. Filled dots are O-C values obtained with the IAC-80, squares are taken from Charbonneau et al. (2005), triangles from Winn et al. (2007) and asterisk are from Hrudková et al. (2009).
The lines show different polynomial fits, where the solid black line
indicates a linear fit, the long-dashed (red) line a quadratic and the
short-dashed (blue) line a cubic polynomial. The dotted (red) lines
show the fits corresponding to the variation of the quadratic term
within 1- |
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![]() |
Figure 3:
Similar as Fig. 2 (based on ephemeris from Holman et al. 2007) including only polynomials fits and corresponding 1- |
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3 Transit timing analysis
Two mechanisms have been brought forward that may cause deviations of transit times from strict periodicity. For one the light time effect and for another the influence of a third body on the transiting planet's intrinsic periodicity.
3.1 Search for the light time effect
We first consider the light time effect, where the barycentre of the binary system, in our case the system ``star-transiting planet'', is offset against the barycentre formed by an additional third body. This will cause the light to travel a longer or shorter distance to the observer and hence the observer will see a different period, where the intrinsic period is unknown.
For the analysis of the O-C times, we fitted three polynomials of different orders to the O-C values, namely a linear,
,
a parabolic,
,
and a cubic polynomial,
,
where E is the Epoch number and
the fitted polynomial coefficients. We also analysed the case of fixed original ephemeris (
)
and the case of maintaining the original period and fitting only for a constant offset in O-C (
). Table 4 shows the obtained best-fit coefficient for TrES-1 and TrES-2 respectively, and Figs. 2 and 3
show a plot of the respective polynomials. In order to quantify the
improvement of the different fits, we calculated the false alarm
probability (FAP), that is the significance level of the fit quality
improvement and indicates the probability of making a type I error.
Therefore, we first calculated the F-values according to equation:
where




As for the linear coefficients
and
these are without relevant physical meaning, but indicate a slightly different ephemeris than used, where
is an offset of the mid-transit time at epoch 0 and
a
correction to the period of the ephemeris. However, we can give a
physical meaning to the quadratic coefficient. The quadratic term (
), gives the system's acceleration along the line of sight by using the equation (Deeg et al. 2008):
where c is the speed of light and P is the observed period.
In order to estimate the error of the quadratic term, we stepped through different values of ,
fitting for
and
.
The
values which increased the best-fit
residuals by 2.3 gave the 1-
confidence limits (see Table 4); dotted lines in Figs. 2 and 3 show the corresponding fits at these limits. Our best fit parabola model for TrES-1 gave a quadratic term of
s. Solving Eq. (3) we obtained an acceleration of
m s-2 and
s-1. The cubic term
would indicate a constant change in acceleration; however the cubic fit
is also less likely than the quadratic one and will not be further
discussed.
For the quadratic solution of TrES-2 we obtained a best-fit value of
smaller than its error. This high error is consistent with a low
significance of the quadratic solution, therefore we support the linear
case, with a 1-
upper limit for accelerations of
m s-2.
Considering the clear linear trend of O-C times in Figs. 2 and 3, and using the coefficients from the linear fit, we indicate here an improved ephemeris for TrES-1, given by:
where the values in parenthesis give the uncertainty in the last two digits and for TrES-2:
3.2 Search for a perturbation of the intrinsic period
The other cause for transit timing variations could be a perturbation of the intrinsic planet period P' caused by the gravitational influence of a third body on the transiting planet. Regarding such perturbations, there exist no analytical equations that describe the gravitational influence on a transiting planet caused by an undetected third body. Generally, N-body simulations are used, iterating over a large orbital parameter space for a maximum possible mass range, see e.g. Agol et al. (2005); Agol & Steffen (2007); Holman & Murray (2005); Miller-Ricci et al. (2008b); Steffen & Agol (2005); Nesvorný & Morbidelli (2008); Miller-Ricci et al. (2008a).
In order to find the parameter space of third bodies compatible with the observed transit times, we created a numerical 2-dimensional simulation of a three body system by integrating over the equations of motion. We considered the problem in two dimensions, assuming that the orbits of the exoplanetary system and its respective perturber are co-aligned. We further considered the problem in a heliocentric frame, with the star in the centre of the coordinate system, meaning that we neglect the light-time effect in this context. To integrate the equation of motion we used the Burlisch-Stoer algorithm (Press et al. 1992) with a 1 s time step and an accuracy of 10-10.
Table 4:
Comparison of different residuals for TrES-1 (upper values) and TrES-2 (lower values).
We considered the problem for an inner and an outer perturber
separately and neglected the 1:1 mean motion resonance (MMR). Each
simulation started with a perturber 0.005 AU away from the
transiting planet, with an initial zero eccentricity and with a phase
shift towards the transiting planets between 0
and 315
in steps of 45
.
We advanced the simulations in orbital steps of 0.001 AU and used 100 mass steps from 1
to 100
for the perturbing object. We simulated 1000 transits, corresponding to
approximately eight years coverage for TrES-1 and seven years coverage
for TrES-2. We used a mass of 0.61 MJ for TrES-1 and a mass of 0.88
for its host star (Torres et al. 2008), and a mass of 1.2 MJ for TrES-2 and 0.98
for its host star (Sozzetti et al. 2007). After obtaining the simulated O-C diagrams, we applied a Fourier transformation to the synthetic O-C diagrams (Miller-Ricci et al. 2008b,a)
and derived the maximum obtained O-C amplitudes as a function of the
perturber's semi-major axis; showing them for masses of 10, 20 and
40
in Fig. 4, where also the MMRs are indicated by vertical lines. Figures 5 and 6 show some examples of synthetic O-C diagrams for TrES-1.
![]() |
Figure 4:
Maximum O-C amplitudes of a 2 dimensional N-body simulation for the system TrES-1 ( upper two panels) and TrES-2 ( lower two panels)
caused by a perturber during 1000 transits plotted against the
semi-major axes of the perturber. The coloured solid lines correspond
to different perturbing masses, namely 10 |
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![]() |
Figure 5:
Example synthetic O-C diagrams for TrES-1 and a perturber of 30 |
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![]() |
Figure 6: Same as Fig. 5, but for a perturber distance of 0.062 AU (2:1 MMR). |
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To establish potential third body orbital parameters, we fitted for
each perturber distance the observed O-C diagram to all synthetic ones
that had been generated for masses of 1-100
(Fig. 7).
We left three parameters open in the fits: one parameter to shift the
O-C diagrams in Epoch, corresponding to find the best moment for the
first real observation (E=0) within the simulated data, and two
parameters describing linear deviations between simulated and modelled
O-C diagrams, that have no relevant physical meaning.
For the best-fit masses, we see a general trend to higher masses for
perturbers closer to the host star and at larger semi-major axis, and
we can also identify the mean-motion resonances, where the best fit for
a consistent perturber indicates lower masses (vertical lines in
Fig. 7).
At larger orbits for the perturber, the best-fit perturber's mass
increases and it increases more steeply for TrES-2 than for TrES-1.
This is due to TrES-2 having a higher mass than TrES-1 and being closer
to the central star. The high peaks at the 2:3 and 2:1 MMRs for TrES-1 reveal that a >1
planet in that configuration might have been detected in our data set.
Again, in order to quantify the improvement of the simulated O-C fits
in the lower MMRs and the best-fit simulated O-C against the linear
polynomial, we calculate the FAPs. In these cases it is better not to
use Eq. (2) to calculate the F-value,
since this equation is better suited for cases where one expects small
changes for additional parameters, whereas for the simulated O-C
values, we expect big changes. Therefore, we used
to calculate the F-value for the MPFTEST-routine. In Table 4, we show the
and FAPs for some special cases of our simulated O-C values.
For TrES-1 we found that most of the
values for the simulated O-C models were below the one of the linear
fit, but based on the available data, we can exclude the case of 2:3
and 2:1 MMR for planets more massive than 1
with high FAPs. We found a best fit for a perturber with a mass of 2
and a semi-major axis of 0.05 AU, but this peak is not outstandingly low. Similarly, for TrES-2, all
values for the O-C models are below the one from the linear fit. We can identify some low
peaks, with the lowest at 0.049 AU (
)
for a perturbing mass of 18
.
Figure 8 shows for TrES-2 the O-C simulation for the lowest
with the observed O-C residuals overplotted. Given that none of the simulated perturbers indicate a uniquely low
,
we do not find support for any of them.
In order to calculate an upper mass limit above which we can reject perturber masses due to a high increase in ,
we first set a
threshold to 50.2 for TrES-1 and 28.5 for TrES-2. These thresholds correspond to a 90% FAP against the linear fit using the MPFTEST-routine. We then increased the masses for each semi-major axis until the
residuals reached the previously established threshold, which gave us
the upper mass limit. Masses above the threshold can be rejected with a
FAP higher than 90% (Fig. 7).
![]() |
Figure 7:
Masses of third bodies with the lowest |
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3.3 Search for a sinusoidal transit timing variability
In order to search for any sinusoidal periodicity in the data with
periods on the order of the campaign duration and shorter, we fitted
amplitude and phase of a sinusoidal function to the O-C residuals after
subtracting the linear fit with the parameters of Table 4. A sinusoidal transit timing variation might be caused by an exomoon around the transiting planet (Doyle & Deeg 2004; Simon et al. 2007; Sartoretti & Schneider 1999; Kipping 2009). For this fit we stepped through different trial periods between 1500 days (
cycl/d) and 0.1 days (10 cycl/d) for TrES-1 and for TrES-2 between 600 days (
cycl/d)
and 0.1 days (10 cycl/d) in steps of 0.00001, 0.0001, 0.001,
0.01 and 0.1 cycl/d. For each trial period we then logged the
residuals at the best-fitting amplitude and phase.
From Fig. 9, showing the result of the
sine fit, we see that we lack outstanding peaks of low
values for TrES-1 and TrES-2, being it very unlikely that a real
sinusoidal signal has more than one period. However, we also calculated
the FAPs of the best sinusoidal fit against the linear polynomial, as
described in Sect. 3.2 and show the results in Table 4
for comparison. We note that the best sinusoidal fit for TrES-1 with a
period of 16.7 days and an amplitude of 25 s (Fig. 10, upper graph) does not give a significant improvement against the linear polynomial, having a lower
residual but a high FAP, but for TrES-2 we found a good sinusoidal fit
with a FAP of 1% for a period of 0.2 days and an amplitude of
57 s (Fig. 10, lower graph).
4 Discussion
For TrES-1 we obtained eight new O-C values and made use of
10 previously published values of which we removed two
observations with a high probability of containing star spots. The
standard deviation of the O-C values is 33 s and the maximum O-C deviation from the established ephemeris (Winn et al. 2007) is
60 s. Similarly, for TrES-2 we observed five transits and made use of three additional published mid-transit times (Holman et al. 2007).
The transit timing estimation at epoch 0 for TrES-2 had an error
greater than 60 s and was hence removed. After correcting the
ephemeris, the standard deviation of this O-C diagram is
43.9 s and the maximum O-C deviation, is 66.5 s.
![]() |
Figure 8:
Simulated O-C diagram of TrES-2 and the best fit perturber with a mass of 18 |
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In order to search for a light time effect, we fitted different
polynomials. Assuming the validity of the quadratic function for
TrES-1, we can use the quadratic term to estimate a possible
acceleration and its corresponding 1-
error of
s-1,
the negative value indicating a decelerating system. Comparing this
deceleration to the acceleration for the solar system of a few 10-19 s-1 (Zakamska & Tremaine 2005) and CM Draconis of
10-17 s-1 (Deeg et al. 2008),
the acceleration of TrES-1 could be several orders of magnitudes
larger, but is not well constrained given the current data. Following Deeg et al. (2008), the minimum mass of a third body causing a given acceleration in dependence of its lateral distance
is given by
We note that this is independent of the mass of the accelerating system. Using the distance to TrES-1 of 157 pc (Sozzetti et al. 2004) and above value for

This relation allows the identification of nearby objects found in any future high-resolution imaging; that is, any possibly found object has to fulfil Eq. (7) in order to be a potential source of the observed acceleration. For example, for a third object at a distance of 0.05




A second approach has been assuming perturbing objects at nearby orbits. Then for TrES-1, perturbers with a masses >1
at the 2:3 and 2:1 MMRs are very unlikely due to the high
residual with a high FAP against the linear fit. However, none of the simulated perturbers indicated a uniquely low
peak.
Therefore, we do not find support for any perturber, but we established
upper mass limits above which we would obtain a FAP of 90% against the
linear fit. For TrES-2, we obtained some low
peaks; none of them in a low-order MMR. For the best-fit simulated O-C we obtained a perturber of 18
and 0.049 AU. Due to the
residuals below the one from the linear fit, none of the perturbers
could be excluded and again we established upper mass limits for
perturbers in the TrES-2 system.
![]() |
Figure 9:
Results of the |
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![]() |
Figure 10: Best fit sine function (solid line) and observed O-C values with error bars for TrES-1 and TrES-2, phased with period of that fit. This period is 0.06 cycles/day or 16.7 days for TrES-1 and 4.8 cycles/day or 0.21 days for TrES-2. |
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Steffen & Agol (2005) did an
analysis of transit times of TrES-1, but they found no convincing
evidence of a second planet. We also found generally a higher
near to the low-order MMRs. Steffen & Agol (2005)
gave upper mass limits for additional planets in the system TrES-1,
whereas we give additionally the best-fitting mass for any orbital
distance. In general, their upper mass limit for additional planets
decreases closer to the transiting planet, similar to our findings in
Fig. 7. We also found that additional planets with masses above 10
at the MMRs are very unlikely, that is consistent with the results from Steffen & Agol (2005), indicating that at the low-order MMRs and near zero eccentricity, the mass of the additional planet has to be below 10
.
Assuming sinusoidal transit timing variations, we note that we find a good O-C fit for TrES-2 with a FAP of 1.1%. However, since we find several good periods beyond the best one of 0.2 days, we conclude that we need more observations and maybe with higher precision in order to confirm any one of them as a possible exomoon. Whereas for TrES-1, we found no evidence that a sinusoidal function improved the fit significantly with respect to the linear one.
Sartoretti & Schneider (1999), Doyle & Deeg (2004), Simon et al. (2007) and recently Kipping (2009)
discussed the possibility of detections of moons around extrasolar
planets using a timing offset induced by the wobble of the planet
around the planet-moon barycentre. For a given timing offset
the mass of the possible moon
can be estimated using (Sartoretti & Schneider 1999):
where





where



where M* is the mass of the star. Using the radius of the transiting planets, i.e.





We note that Barnes & O'Brien (2002)
gave an analytical expression for an upper mass limit of possible
moons. Generally, the more massive the moon the shorter its lifetime. Lubow et al. (1997) estimated that close-in orbiting planets spin down into synchronous rotation very quickly. Therefore we use the procedure from Barnes & O'Brien (2002)
to estimate the upper mass limit of a moon around TrES-1 and TrES-2. We
obtain an upper mass limit around TrES-1 of the order of 10-6
and around TrES-2 of the order of
10-7
,
similar to one obtained for HD 209 458b by Barnes & O'Brien (2002). This means that moons with masses greater than 10-6
will not have survived until now. This upper mass limit is clearly way
below our detection limit for masses causing transit timing variations
in the system TrES-1 and TrES-2. The closest moon at this upper mass
limit would cause a timing amplitude of the order of 10-6 s, which is not detectable even from space by several orders of magnitude.
Transit duration investigations on TrES-2 performed by Mislis & Schmitt (2009) indicated a decrease of 3 min. between 2006 and 2008 in the duration. However, in our transit timing analysis, we do not find non-linear deviations of O-C times with a similar magnitude. While we can not provide any firm detection, we can put upper mass limits which are consistent with our observations for TrES-1 and TrES-2, respectively. However, due to the gaped data, it might have been possible that we missed important points in the observed O-C diagram, like e.g. a transit timing measurement at the highest amplitude at the mean-motion resonances. We need at least about 3 years of continuous transit observations and with several transit observations per year in order to avoid the missing of windows of high-amplitude O-C deviations from perturbing bodies, with the maximum amplitudes being observable only during a few transit events, see Fig. 6. But even for low-mass perturbers such amplitudes should be easily measurable from the ground at mean-motion resonances. On the other hand, outside the mean-motion resonances, we need higher precision than obtainable from the ground for detections of the small amplitudes caused by a perturbing body of planetary mass, something that may be expected from the current satellite missions in operation. Any transit timing variation in TrES-2 may be expected to be confirmed in the near future by observations done with the Kepler Space Telescope.
AcknowledgementsThis article publishes observations made with the IAC-80 telescope operated by the Instituto de Astrofísica de Canarias in the Observatorio del Teide. We thank the students and staff of the Teide Observatory and Angel de Vicente Garrido for his help with the IAC Supercomputing facility CONDOR. M. R. likes to thank the European Association for Research in Astronomy (EARA) for their support by a EARA - Marie Curie Early Stage Training fellowship. Part of this work was financed by grant ESP2007-65480-002-02 by the Spanish investigation and development ministry. We also thank Tristan Guillot and the anonymous referee for valuable comments that led to a significant improvement of the analysis presented here.
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Footnotes
- ... TrES-2
- Photometric data for TrES-1 and TrES-2 are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/508/1011
All Tables
Table 1: Comparison between best-fit values of this work and parameters from Southworth (2008) for transit light curve models of TrES-1 (upper values) and TrES-2 (lower values).
Table 2: Overview of TrES-1 transits used in this work.
Table 3: Same as Table 2, for TrES-2.
Table 4:
Comparison of different residuals for TrES-1 (upper values) and TrES-2 (lower values).
All Figures
![]() |
Figure 1: Phased and binned light curve of all IAC-80 observations of TrES-1 ( top) and TrES-2 ( bottom) transits with the solid (red) line being the best-fit model light curve. |
Open with DEXTER | |
In the text |
![]() |
Figure 2:
Difference between calculated (based on ephemeris from Winn et al. 2007) and observed mid-transit times for TrES-1. Filled dots are O-C values obtained with the IAC-80, squares are taken from Charbonneau et al. (2005), triangles from Winn et al. (2007) and asterisk are from Hrudková et al. (2009).
The lines show different polynomial fits, where the solid black line
indicates a linear fit, the long-dashed (red) line a quadratic and the
short-dashed (blue) line a cubic polynomial. The dotted (red) lines
show the fits corresponding to the variation of the quadratic term
within 1- |
Open with DEXTER | |
In the text |
![]() |
Figure 3:
Similar as Fig. 2 (based on ephemeris from Holman et al. 2007) including only polynomials fits and corresponding 1- |
Open with DEXTER | |
In the text |
![]() |
Figure 4:
Maximum O-C amplitudes of a 2 dimensional N-body simulation for the system TrES-1 ( upper two panels) and TrES-2 ( lower two panels)
caused by a perturber during 1000 transits plotted against the
semi-major axes of the perturber. The coloured solid lines correspond
to different perturbing masses, namely 10 |
Open with DEXTER | |
In the text |
![]() |
Figure 5:
Example synthetic O-C diagrams for TrES-1 and a perturber of 30 |
Open with DEXTER | |
In the text |
![]() |
Figure 6: Same as Fig. 5, but for a perturber distance of 0.062 AU (2:1 MMR). |
Open with DEXTER | |
In the text |
![]() |
Figure 7:
Masses of third bodies with the lowest |
Open with DEXTER | |
In the text |
![]() |
Figure 8:
Simulated O-C diagram of TrES-2 and the best fit perturber with a mass of 18 |
Open with DEXTER | |
In the text |
![]() |
Figure 9:
Results of the |
Open with DEXTER | |
In the text |
![]() |
Figure 10: Best fit sine function (solid line) and observed O-C values with error bars for TrES-1 and TrES-2, phased with period of that fit. This period is 0.06 cycles/day or 16.7 days for TrES-1 and 4.8 cycles/day or 0.21 days for TrES-2. |
Open with DEXTER | |
In the text |
Copyright ESO 2009
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