Detecting and Characterizing Planetary Systems with Transit Timing

TL;DR

This paper introduces a new transit timing variation (TTV) method for detecting low-mass exoplanets, especially in resonant systems, using a network of small ground-based telescopes to improve detection capabilities.

Contribution

It presents a novel TTV detection technique and proposes a modest telescope network to enhance the observation of small, Earth-like exoplanets.

Findings

TTV method can detect planets smaller than Earth.

Small ground-based telescopes can achieve necessary precision.

Network of telescopes can significantly improve detection efficiency.

Abstract

In the coming decades, research in extrasolar planets aims to advance two goals: 1) detecting and characterizing low-mass planets increasingly similar to the Earth, and 2) improving our understanding of planet formation. We present a new planet detection method that is capable of making large advances towards both of these objectives and describe a modest network of telescopes that is able to make the requisite observations. In a system where a known planet transits its host star, a second planet in that system will cause the time between transits to vary. These transit timing variations can be used to infer the orbital elements and mass of the perturbing planet even if it has a mass that is smaller than the mass of the Earth. This detection technique complements other techniques because it is most sensitive in mean-motion resonances where, due to degeneracies, other techniques have reduced sensitivity. Small ground-based observatories have already exceeded the photometric precision necessary to detect sub-Earth mass planets. However, TTV planet searches are currently limited by the relatively small number of high-precision transit data and insufficient observing time on existing telescopes. These issues will be compounded as the number of known transiting planets suitable for TTV study will increase substantially in the near future. A relatively modest investment in a ground-based network of small ($\sim 0.5 {\rm m}$ telescopes could provide the needed coverage and so dramatically increase the effectiveness of transit timing observations.

Full text

**Detecting and Characterizing Planetary Systems with Transit Timing

**Jason H. Steffen, B. Scott Gaudi, Eric B. Ford, Eric Agol, Mathew J. Holman

1. Abstract

In the coming decades, research in extrasolar planets aims to advance two goals: 1) detecting and characterizing low-mass planets increasingly similar to the Earth, and 2) improving our understanding of planet formation. We present a new planet detection method that is capable of making large advances towards both of these objectives and describe a modest network of telescopes that is able to make the requisite observations. In a system where a known planet transits its host star, a second planet in that system will cause the time between transits to vary. These transit timing variations can be used to infer the orbital elements and mass of the perturbing planet even if it has a mass that is smaller than the mass of the Earth. This detection technique complements other techniques because it is most sensitive in mean-motion resonances where, due to degeneracies, other techniques have reduced sensitivity. Small ground-based observatories have already exceeded the photometric precision necessary to detect sub-Earth mass planets. However, TTV planet searches are currently limited by the relatively small number of high-precision transit data and insufficient observing time on existing telescopes. These issues will be compounded as the number of known transiting planets suitable for TTV study will increase substantially in the near future. A relatively modest investment in a ground-based network of small ($\sim 0.5~{}{\rm m}$ telescopes could provide the needed coverage and so dramatically increase the effectiveness of transit timing observations.

2. Introduction: Planetary Transits

After many years of effort, the transit surveys for planets have recently come to fruition (see Charbonneau et al. 2007 for a review). To date 9 transiting planets orbiting bright ($V\lesssim 12$) stars are known, most of these were discovered through these transiting planet surveys (Alonso et al. 2004, McCullough et al. 2006, O’Donovan et al. 2006, Bakos et al. 2007, Collier-Cameron et al. 2007). Beyond the existing ground based transit surveys, the recent launch of ESA’s CoRoT satellite (Baglin et al. 2002) and the planned launch of NASA’s Kepler mission (Borucki et al. 2003) are expected to discover many additional transiting systems.

Once a star has been discovered to host a transiting planet, there is the potential for many exciting follow-up observations (see Charbonneau et al. 2007). In this white paper, we focus on the potential for transit timing observations to search for additional planets. If a star harbors a single transiting planet, then the transit times would be strictly periodic. On the other hand, if a star harbors more than one planet, then the planets’ mutual gravitational perturbations will cause their orbits to deviate from simple Keplerian orbits and hence from strictly periodic times of transit. These transit timing variations (TTV) can be used to infer the orbital elements of the perturbing planet (Agol et al. 2005, Holman & Murray 2005). Since very small changes in a planet’s orbital elements can be measured by timing transits, this represents a very powerful method for searching for low-mass planets. The sensitivity of the TTV method is further enhanced if the perturbing planet is near a mean-motion resonance (MMR) where the timing variation depends on the planet-planet mass ratio rather than planet-star mass ratio. For example, for a transiting, Jupiter mass planet on a 3-day orbit, an Earth mass planet in the 2:1 resonance will cause periodic variations in the transit times that have an amplitude greater than one minute (See Figure 1). This should be compared to the (best) transit timing precision of $\sim$10s that has been demonstrated using small (1.2m) ground-based observations (Holman et al. 2006). Thus, the TTV technique is already capable of probing for planets with masses even less than the mass of the Earth (Agol & Steffen 2007). Further, the enhanced sensitivity of the TTV method resonant planets is particularly interesting to theorists, since other dynamical detection techniques often have a reduced sensitivity to such planets due to potential degeneracies between orbital parameters.

The TTV technique was first applied to transit measurements of the TrES-1 planetary system. In this case, a set of 11 ground-based observations were able to probe for terrestrial mass planets near several interesting MMR’s (Steffen & Agol 2005). In a similar study, 13 Hubble Space Telescope observations of the HD 209458 system were sensitive to planets with masses approaching that of Mars (Agol & Steffen 2007). We note that the ground based observations (using KeplerCAM on the 1.2m Fred Whipple Observatory) of XO-1 (Holman et al. 2006) are of higher precision than some of the HST observations of the HD 209458 system—indicating the great potential for high quality transit timing observations, even with modest ground-based telescopes. The highest transit-timing precision to date, $\sim$ 6 seconds, has been recently obtained for HD 189733b with the Spitzer Space Telescope (Knutson et al. 2007); however, all other known transiting planets would have a lower timing precision due to the faintness of their host stars in the Spitzer bandpass.

3. Scientific Justification for TTV Observations

Analyzing the variations of transit times in planetary systems can address many important questions regarding planetary populations, characteristics, planet formation, and evolution theories. We address several of these questions here.

Planet Detection:

$\bullet$ The TTV method can increase the scientific value of other transit search programs such as NASA’s Kepler mission. It provides the capability to discover additional planets around stars with a transiting planet, even if the second planet does not transit the star. This could be particularly useful for detecting terrestrial-mass planets in the habitable zone where such a planet has a reduced geometric probability of transiting. This is accomplished by analyzing the transits of a planet with an orbit that lies interior to the habitable zone, where planets are more likely to be transiting. For a conservative example, if a Jupiter mass planet were on a 130-day orbit, high quality ground-based observations of the transit times could identify a terrestrial mass perturbing planet on a non-resonant, low eccentricity orbit ($\sim 0.05$) at 1 AU (see Figure 1). From this conservative starting point, the noise remains constant while the TTV signal scales linearly with the mass of the perturber, the eccentricity of the perturber, and the period of the transiting planet. The signal also grows sharply as the system approaches one of several MMR’s.

$\bullet$ Transit timing can be quite sensitive to low-mass planets. Thus, the TTV method can test theoretical models of planet formation and migration predict that terrestrial mass planets should be common around stars with a short-period giant planet (e.g., Narayan et al. 2005).

$\bullet$ Although it is possible to search for additional planets in each system by directly observing their transits (e.g., Brown et al 2001), this generally requires continuous, precise photometry outside of transit, and so a substantial commitment of observation time. Since the TTV method only requires photometric observations near the times of transit, it provides an efficient tool to search for additional companions.

Planet Formation Theory:

$\bullet$ Transit timing observations are particularly valuable because they are extremely sensitive to planets near MMR. The core accretion model of planet formation predicts that terrestrial mass planets will often be caught in MMR with a gas giant planet as that planet migrates inward (Zhou et al. 2005, Thommes 2005, Cresswell & Nelson 2006, Terquem & Papaloizou 2007, Fogg & Nelson 2007, Mandell et al. 2007). This occurs either by: 1) the migrating planet sweeping smaller planets into interior resonances or 2) exterior planets interacting with the gas disk and rapidly spiraling toward the giant and becoming trapped in exterior resonances. Thus, if the core-accretion model is correct, we expect that many stars with transiting planets may also have lower-mass planets in interior and/or exterior resonances. Further, the details of the trapping in different resonances depend on the migration timescale, the disk lifetime, and the eccentricities and inclinations of both bodies at the time the bodies approach resonance. Thus, studying the frequency of planets in various resonances can provide constraints on the migration process. (This is similar to the constraints on the timescale and extent of migration for Neptune based on the resonant Kuiper belt objects).

$\bullet$ Provided that sufficient TTV observations are made, a null result would also be significant. While the core-accretion model predicts that trapped terrestrial planets should be common, the gravitational instability model of planet formation does not anticipate the presence of such planets in any quantity. Thus, the discovery of small objects in MMR with transiting planets—precisely the regime where the TTV technique is most sensitive—would support the former theory (Zhou et al. 2005). The lack of such planets would support the latter or place strong constraints on parameters of migration theory, particularly if the period of the transiting planet were a few tens of days (tidal effects may affect systems with shorter orbits which we discuss next).

$\bullet$ The TTV method could measure the ubiquity of closely packed planetary systems (e.g. Juric & Tremaine 2007) and/or study the dynamical properties of systems with strongly interacting planets. Small planets on interior resonant orbits with a hot Jupiter (or the lack thereof) would constrain models of tidal interactions that may cause the orbit of the inner planet to decay (D. Fabrycky, private communication).

Characterization of Planetary Systems:

$\bullet$ From TTV analyses one may be able to identify the mutual inclination of planetary systems (Miralda-Escudé 2002). That quantity has implications for allowed mechanisms for the growth of eccentricities in planetary systems (e.g., Chatterjee et al. 2007). It can also provide a determination of the mass of a non-transiting perturbing planet, something that is very difficult to identify with other planet detection techniques.

$\bullet$ For systems where multiple planets transit there can be sufficient information to determine the absolute masses and radii of the two planets and the host star independent of stellar models, as with double-lined eclipsing binaries. This is crucial for determining when terrestrial-sized planets have terrestrial-mass (which would otherwise be challenging or impossible with the radial-velocity technique) and allows for a measurement of their densities.

$\bullet$ Knowledge of the mass and orbit of additional planets can help interpret observations of the transiting planet. For example, it has been proposed that planets with unexpectedly large radii and being heated by a combination of tidal dissipation in the planet and orbital perturbations from other planets (e.g., Bodenheimer et al. 2001).

4. Requirements for a Successful Transit Timing Program

Transit Observations:

Searching for planets with the TTV method requires precise photometric observations during many transits. For a photometric precision $\sigma_{ph}$, the central time $T_{c}$ of a given transit can be measured with an accuracy (Ford & Gaudi 2006, Holman & Murray 2005): $\sigma_{T_{c}}\simeq[t_{e}/(2\Gamma)]^{1/2}\sigma_{ph}\rho^{-2},$ where $t_{e}$ is the duration of ingress/egress, $\Gamma$ is the rate at which observations are taken, and $\rho$ is the ratio of the planet radius to stellar radius (this equation ignores limb-darkening). For typical parameters and millimagnitude photometry ($\sigma_{ph}\sim 10^{-3}$), $T_{c}$ can be measured to tens of seconds (e.g., Brown et al. 2001; Holman et al. 2006). Such precision enables the detection of Earth-mass companions to transiting 3-day period gas giants for optimal (i.e. resonant) configurations (Steffen & Agol 2005). For longer period transiting planets, the sensitivity is correspondingly better, scaling as $m_{2}\propto P_{1}^{-1}$, where $1,2$ label the transiting and perturbing planets.

Achieving millimagnitude photometry on stars of $V\sim 11-13$ magnitude is generally not a problem of a sufficient photon rate, even for telescopes as small as $\sim 20^{\prime\prime}$. Rather the difficulties are due to the limited dynamic range, the scarcity of comparison stars of similar magnitude, and systematic errors in the photometry, such as correlated noise (Pont et al. 2004). In this sense, large-format, high-quality, monolithic cameras on relatively small aperture telescopes are ideal. As stated, the most impressive results on follow-up of bright transiting planets have come from the Transit Light Curve (TLC) project using KeplerCam. This camera has a single 4K $\times$ 4K Fairchild 486 CCD with a $23^{\prime}\times 23^{\prime}$ field of view. Using 2 $\times$ 2 binning, the readout time for this detector is 11.5 s. The TLC project has demonstrated the ability to achieve $\sim 0.15\%$ photometry in the $z$-band (where limb darkening is minimized), with 30 second exposures on the bright $(V=11.79)$ transit host star TrES-1 (Winn et al. 2006). They furthermore demonstrated that the uncertainties were essentially uncorrelated, such that by binning on $\sim 30$ minute timescales, they achieved RMS residuals of $\sim 10^{-4}$. The times of individual transits were measured to a precision of $\sim 10$ seconds.

Currently, there are 9 stars with transiting planets that are sufficiently bright for precise transit timing measurements. In principle all of these could be searched for transit timing signals using existing observatories. In practice, there are several barriers. While transit timing can be performed with a relatively modest aperture, it does require a high-quality, large-format camera. Relatively few small observatories are outfitted with such a detector. Second, the observations must occur at a specific time, so scheduling can be difficult, particularly for observatories that scheduled by the night and/or are shared by multiple institutions. Third, the transit of a typical hot Jupiter lasts less than two hours. This also makes it difficult to effectively use an observatory with a per-night scheduling system. Finally, weather and day-night aliasing means that it is difficult to observe many consecutive transits from a single site. These difficulties will be exacerbated when ongoing transit search programs produce more transiting planets in the near future. Clearly, a proper TTV study of a significant fraction of the known transiting planets over the next decade will require additional observatories.

The observational needs of TTV observations of current and future systems could best be met by a network of longitudinally distributed telescopes that can monitor the skies continuously. These telescopes need not be particularly large and could be fully automated. The entire network could be as small as four telescopes and as large or larger than 10 (depending upon the needs and the number of discovered transiting systems). Here we present a strawman proposal for such a network, consisting of a number of 20” dedicated telescopes. This network would accomplish the majority of the science goals outlined above, and at a low cost relative to other major exoplanet initiatives.

Each telescope, which can be bought off-the-shelf from a number of manufacturers, can be equipped with a large-format camera (also off-the-shelf). To this end, we consider a 20” roboticized telescope with a 2K $\times$ 2K $z$-band optimized detector with a $\sim 26^{\prime}\times 26^{\prime}$ FOV. This setup can achieve a photon collection rate that is within a factor of $\sim 2$ of the FLWO 1.2m telescope in the $z$-band. A filter that is more broad may achieve a higher photon rate. Scaling from the results of (Winn et al. 2006), this setup should achieve $\sim 1.5~{}{\rm mmag}$ RMS photometry in the $z$-band in one minute samples on a $V\sim 12$ star. As stated above, this will also allow to measure transit times to tens of seconds (Winn et al. 2006), and so sensitivity to resonant Earth-mass planets. Based on actual price quotes, we estimate the total cost for hardware for each such telescope to be roughly $100k, along with additional costs for site fees and computing resources. For a six-year science program the total cost of the equipment, its operation, and maintenance is approximately$250k per telescope.

A typical planet on a 3-day orbit is in transit for roughly 3% of its orbit. Thus, of the $\sim 10$ known planets there is one in transit nearly 1/3 of the time. If we assume that a planetary system is only visible during six months then once $\sim 70$ systems are discovered we expect a planet to be in transit at all times. If the Kepler and CoRoT missions identify even a moderate fraction of their anticipated yield, then additional telescopes (perhaps located in pairs to mitigate against coincidence losses or separated by a few hundred miles to mitigate against weather) would be required to search most of the expected discoveries of stars hosting transiting planets. A network of small telescopes could easily and rapidly be expanded and optimized to meet the increasing demands of new discoveries. This capability represents a significant advantage for the choice of small, low-cost telescopes.

An even larger potential resource for ground-based transit timing studies is a network of 0.4 meter and 1 meter telescopes is being built with funding from Wayne Rosing to study extragalactic transients and extrasolar planets, with microlensing and transits, called Las Cumbres Observatory Global Network (LCOGT, Brown et al. 2006). By the end of 2007 the first of 30 0.4 meter telescopes will be commissioned, while by the end of 2008 the first of 30 1-meter telescopes will be commissioned which will utilize the same CCDs as KeplerCam with a 4-second readout. Thus, the capability of each of these 60 telescopes will be very similar to the strawman telescope proposal described here, while it is expected that about 15-20% of the LCOGT observing time will be devoted to planetary transit observations. These telescopes will be distributed longitudinally which is ideal for complete transit lightcurves. In addition to these telescopes, there are currently two two-meter robotically controlled telescopes (one in Hawaii and one in Australia) with fast readouts that are part of the LCOGT network. These telescopes are already in operation and will, in principle, allow high precision photometry of fainter transiting planet systems (Tim Brown, priv. comm.).

Transit Timing Analysis:

Inferring the orbital elements and mass of a perturbing planet via TTV is generally more complex than other planet detection schemes. In transit timing data, the signal is a combination of several effects including the reflex motion of the star, the mutual gravitational interaction between the planets, the changing light travel time, or changing tidal field of a distant companion (e.g., Borkovits et al. 2003; Heyl & Gladman 2006), though typically only one effect dominates. Unlike other dynamical detection techniques, the salient characteristic of the TTV approach is the deviations from Keplerian orbits. This requires high accuracy n-body simulations of each model in order to calculate its TTV signature (Holman & Murray 2005; Agol et al. 2005). Given the computational requirements of n-body integrations, practical algorithms must explore a high-dimensional parameter space efficiently. While challenging, preliminary tests involving simulated data show that a large fraction of systems can be correctly identified with appropriate analysis techniques (Steffen & Agol 2006) though much more development of these techniques remains.

3. Conclusions

High precision measurements of the transit times of known of transiting planets provide a method to search for additional low-mass planets in these systems, with sensitivity to (sub) Earth-mass planets and planets near mean-motion resonances. Thus a serious transit timing campaign would provide constraints for planet formation theory, including migration history, tidal interactions, and the frequency of low-mass planets in resonances. We expect that the telescopes currently making transit timing observations will soon be overwhelmed by detections from both ground and space-based transit searches. A network of small, inexpensive telescopes could rapidly be deployed to search for Earth-mass planets around solar-type stars and to address many outstanding scientific questions about planet formation. Such a network would require a relatively modest investment, and would be relatively easy to scale-up as additional transit searches discover more transiting planets.

**References:

Agol E. et al., 2005, MNRAS, 359, 567

Agol E., & Steffen, J. 2007, MNRAS, 374, 941

Alonso R., et al. 2004, ApJ, 613, L153

Baglin A., & COROT Team 2002, ESA SP-485: Stellar Structure & Habitable Planet Finding, 17

Bakos G. A., et al. 2006, ApJ, 656, 552

Bodenheimer P. et al., 2001, ApJ, 548, 466

Borkovits T. et al., 2003, A&A, 398, 1091

Borucki W., et al. 2003, Proc. SPIE, 4854, 129

Brown T. M., et al. 2001, ApJ, 552, 699

Brown, T. et al., 2006, BAAS 208, #56.05

Charbonneau D. et al. 2007, PPV, 701

Chatterjee S. et al., 2007, astro-ph/0703166

Collier Cameron A., et al. 2007, MNRAS, accepted

Cresswell P. & Nelson, R. P., 2006, A&A, 450, 833

Fogg M. J., & Nelson, R. P. 2007, A&A, 461, 1195

Heyl J. & Gladman, B., 2006, astro-ph/0610267

Ford E.B., & Gaudi, B.S. 2006, ApJ, 652, L137

Holman M. & Murray, N. 2005, Science, 307, 1288

Holman M. J. et al., 2006, ApJ, 652, 1715

Juric M. & Tremaine, S., 2007, astro-ph/0703160

Knutson H. et al.  2007, Nature, in press

Mandell A. M. et al., 2007, astro-ph/0701048

McCullough P. R., et al. 2006, ApJ, 648, 1228

Miralda-Escudé J. 2002, ApJ, 564, 1019

Narayan R. et al., 2005, ApJ, 620, 1002

O’Donovan F. T., et al. 2006, ApJ, 651, L61 713, 185

Pont F. et al. 2006, MNRAS, 373, 231

Steffen J. H., & Agol, E. 2005, MNRAS, 364, L96

Steffen J. H., & Agol, E. 2007, astro-ph/0612442

Terquem C. & Papaloizou, J., 2007, ApJ, 654, 1110

Thommes E. W. 2005, ApJ, 626, 1033

Winn J.N. et al., 2006, ApJ, accepted

Wright J. T. 2005, PASP, 117, 657

Zhou J. et al., 2005, ApJ, 631, L85**