Predicting the Detection Rates of Transiting Hot Jupiters and Very Hot Jupiters in Wide-field Photometric Surveys
Abstrak
Seven years after the first sighting of a transiting planet around HD209458, it now appears that early predictions of the detection rate of wide-field photometric surveys substantially overestimated the rate at which extrasolar transiting planets have actually been seen. In this paper, we use a $\chi^2$ test to develop a mathematical formalism that allows for a more accurate prediction of the number of transit detections that a given photometric survey should see. We have taken into account such factors as the frequency of gas giants around main sequence stars, the probability of transits occurring, stellar density changes from galactic structure, and the effects of interstellar extinction. We then apply our method to the Trans-Atlantic Exoplanet Survey (TrES), a currently ongoing ground-based search for transiting planets, and the space-based Kepler mission, due to launch in 2008. For both surveys, we offer a prediction for the total number of hot Jupiters and very hot Jupiters that each can expect to detect, as well as the expected distribution of the detections across a number of parameters.
Chapter I
Introduction
1. Background
Since the first detection of a planetary-mass object in 1989 (Latham et al. 1989) and the explicit detection six years later of a Jupiter-sized planet in orbit about 51 Peg (Mayor & Queloz 1995), the number of known extrasolar planets has ballooned to almost 190. By any standard of measure, these discoveries have dramatically altered our conception of the universe. Indeed, until the discovery of the planet around 51 Peg, one could plausibly argue that planet formation was rare, and that our Solar System was one of a few (if not the only) planetary systems in the galaxy. Instead, we are confronted with the prospect that planetary formation is a relatively routine occurrence during the formation of stars; with all that this implies about the chances of finding a world similar to Earth elsewhere in the cosmos.
In addition to confirming the frequency of planetary systems, the properties displayed by the discovered extrasolar planets have forced a rethinking of modern planet formation and dynamical interaction theories. Specifically, the presence of large numbers of jovian planets orbiting very close to their parent stars (so called “hot Jupiters”) have necessitated the creation of new models of planetary migration that allow jovian-worlds formed outside the “snow-line” to spiral in closer to their parent star (Lin et al. 1996; Ward 1997).
To date, there are three reliable ways by which extrasolar planets have been detected. The first is the Radial Velocity (RV) method, which uses the Doppler shift of observed stellar spectra to look for periodic variations in the target star’s radial velocity. By then determining the mass of stellar target (or otherwise estimating it from models), the observed radial velocity curve and velocity semi-amplitude can then be used to directly calculate the inclination-dependent mass (M sin i) of the companion object. While this gives only the minimum mass of these objects, the large number of detections of systems whose unseen companion has a mass on the order of 1MJup sin i statistically ensures that the majority of these are planetary bodies.
The third method by which extrasolar planets have been discovered is the Transit Detection technique, which looks for the periodic dimming of a target star that occurs when an orbiting planet passes in front of the stellar disk. This requires a very specific set of orbital characteristics to yield a transit visible from Earth (the orbital plane has to be aligned to within a few degrees of the line of sight), and therefore transiting planets are expected to be a rarer sight than planets detectable through RV observations. Nevertheless, a transiting extrasolar planet offers the opportunity to not only determine the mass of that planet (assuming that follow-up RV work is feasible) since the i in M sin i is now measurable, but also the planetary radius. This allows for a deeper understanding of not only the composition of extrasolar planets, but also the dynamics of the interiors of jovian worlds. Additionally, and unlike RV surveys, transiting planets should be readably detectable down to 1R© and beyond, even for relatively long periods. NASA’s Kepler mission, which is presently scheduled for launch in the fall of 2008, will consist of a space-based telescope whose primary mission is to search for transiting planets of just this size within the habitable zones of main-sequence stars.
The first transiting planet was discovered around HD209458 in late 1999. It was initially identified as an extremely short-period planetary system by RV measurements in the spring of that year; it was its short (3.5 day) period and the relative brightness of the parent star (mV = 7.65) that prompted photometric observations of HD209458 in the hope that the orbital geometry was sufficient for an observable transit (Charbonneau et al. 2000;
This dearth of results is in stark contrast to the general expectation following the discovery of HD209458b that wide-field photometric transit surveys would discover literally thousands of transiting extrasolar planets (Horne 2003).
The reasons for the low number of transit detections relative to expectations are varied and complex. Partly, this is because the frequency of planets in close orbits about their parent stars (the planets most likely to transit) is much lower than was originally expected.
Recent examinations of the results from the OGLE-III field by Gould et al. (2006b) indicate that the frequency of short-period jovian-worlds is on the order of 1400 , not 1 100 as is often assumed from looking at RV surveys. They point out that most spectroscopic planet searches are usually intentionally biased by the planet-hunting observer towards targeting metal-rich stars, which are expected to have more planets than the average solar-metallicity star.
Furthermore, many of the older estimates for the number of expected detections assumed that all the stars with a photometric precision of less than 1% in a given field would allow for successful detections, provided the orbital geometry yielded a transit (itself a roughly 10% chance). However, not all of the stars in a field are on the main-sequence.
Evolved stars constitute a non-negligible population that are capable of allowing precise photometric measurements, but any planets in orbit around such stars are beyond the detectable reach of present-day photometric surveys.
Also, further reflection reveals that the photometric precision afforded by a star may not be the proper metric by which to judge the detection abilities of a transit survey. After all, one may observe three transits of a star with a precision of 1%, but if one only has three in-transit data points on a 2% deep transit, then the later identification of that event as a transit will be difficult to say the least. Therefore, instead of photometric precision, a better detection metric is the signal-to-noise (S/N) ratio of a transit.
In this thesis, we use the S/N of transits to statistically calculate the number of hot-Jupiters (HJs) and very hot-Jupiters (VHJs) that a given transiting planet survey should be able to detect. We account for factors such as the low frequency of short period jovian planets, variations in stellar density due to galactic structure, and extinction due to interstellar dust. We also realistically account for the number of main-sequence stars that will be in a given field. We restrict ourselves to HJs and VHJs because their short periods (1-5 days) dramatically increase not only their probability of transiting their parent star, but also the probability that they will be observed in transit within a reasonable amount of time by a survey (the so-called “window probability”). It should be noted, however, that our methods are easily expanded to consider a wider set of possible planets. We first describe the mathematical formalism with which we have chosen to address this problem,
and then move on to discuss our specific assumptions and our choices for fixed fiducial parameters. We then offer our own predictions for the detection rate of a ground-based (TrES) and a space-based photometric survey (Kepler), and compare these to either the actual detection rates (for TrES) or the detection rates proposed elsewhere (for Kepler). 2. General Formalism In the most general sense, we can describe the average number of planets that a transit survey should detect as the probability of detecting a transit multiplied by the local stellar mass function, integrated over mass, distance, and the size of the observed field (described Where Pdetectable(M, r) is the probability that a given star of mass M and distance r will be orbited by a Jupiter-sized planet, and that the system will show a detectable transit. Dn dM is the present day mass function, and ½¤ is the local stellar density for the three-dimensional position defined by (r, l, b). We use r2 cos(b) instead of the usual volume element for spherical coordinates, r2 sin(Á), because b is defined opposite to Á: b = 90± occurs at the pole.
2.1. Calculating the Detection Probability Pdetection
The probability that a given star will possess a HJ or VHJ in an orbit such that it shows a detectable transit can be treated as a straight forward expected value calculation. For a given orbital period, there is a probability that a planet with that period will exist around a star and show a transit, as well as a separate distribution (Pperiod) that describes the probability of having that particular period. The expected value may therefore be defined as may be further broken up into two separate probabilities: the chance that a star will have a planet of period p in orbit around it, and the probability that a planet with period p around a star of mass M at a distance r will show an observable transit:
Pdetection(M, r, p) = Pplanet(p)Pobservable(M, r, p) (3) For a given p these two distributions are independent, as the variables governing the physical processes by which planetary systems form are not correlated with those describing the capabilities of a given survey telescope. We therefore can treat each probability separately.
2.1.1. Determining Pobservable
To begin calculating the probability that a planetary system will show detectable transits, we must first select the appropriate statistical test to determine whether or not our notional data shows a discernible transit. In the world of transiting planet searches, many different methods are used either individually or together to test the statistical significance of a possible transit and alert the researcher to its existence. For example, the Hungarian-made Automated Telescope (HAT) survey uses a combination of the Dip Significance Parameter (DSP) and a Box-Least Squares (BLS) power spectrum (amongst others) to test its photometric data. In general though, whatever test one uses, it is ultimately the signal-to-noise ratio of a transit shape in the observed data is what determines whether or not that transit will be identified as an event.
To determine a relation for ¾, we assume Poisson statistics, and find that for a given exposure with an observed number of NS source photons from the target star and NB background photons, e¸ is the efficiency of the overall observing set-up, and can take values from 0 to 1. It accounts for photon losses in places such as the wavelength filter, the mirrors of the telescope, the CCDs, and passage through the atmosphere. FS,¸ is the source photon flux in the observed bandpass in units of ° m2s . texp is the exposure time of each observation, and A is simply the light collecting area of the telescope.
Having previously derived an expression for Pdetect, we need only determine the appropriate formulation for the stellar mass function, dn dM . Using the results of Reid et al.
(2002), who conducted a volume-limited survey of using Hipparcos data out to 25 pc, we adopt. Where knorm is the appropriate normalization constant. Since the stellar mass density in the Solar Neighborhood is approximately 0.030 − 0.034M¯ pc−3, we may define knorm as the solution to the equation 0.032M¯ pc−3 = Z Mmax with an additional factor of M included to ensure the correct units of M¯ pc−3 (integrating equation (26) simply gives N pc−3). In theory, the upper limit of the integral in equation (26), Mmax, should be set to infinity, but this would be computational prohibitive. In practice, we find that a value of Mmax = 7M¯ is sufficient to ensure convergence of the numerical integral. Using this value, we find a normalization of knorm = 0.02124 pc−3 as the solution.
2.3. Integrating Over r, l, and b: Dealing with Galactic Structure
It is quickly evident that while our normalization of the mass integral properly describes the Solar Neighborhood, beyond about 100 pc we must account for variations in stellar density due to galactic structure. Additionally, as mentioned in previous discussion over calculating the observed number of source photons using equation (9), we must also take into consideration the effects of extinction due to interstellar dust.
2.3.1. Galactic Density Model
For the purposes of this project, we chose to adopt the rather straightforward Bahcall disk model of the galaxy (Bahcall & Soneira 1980). The model treats stellar density in the galactic Thin Disk as a double exponential function involving both the distance from the galactic center and height above the galactic plane. Specifically, Where dgc is the distance to the galactic center where ½¤ = 1, in our case the distance of the Solar System from the center (8 kpc). hd,¤ is the scale length of the density changes in the galactic disk, and hz,¤ is the scale height of the Thin Disk. We initially describe the observed volume of space in relation to the observer (r, l, b), but then transform this into distance from the galactic center along the galactic plane (d) and height above the plane itself (z).
We adopt a value of hd,¤ = 2700 pc for the scale length of the disk. For the vertical scale height, we have used the data from Gilmore & Zeilik (2000) to construct a relation for hz,¤ as a function of mass. Unsurprisingly, older and less massive stars have a greater vertical dispersion than younger stars, and the scale height governing their distribution is correspondingly larger. After using our earlier routines for computing the photon luminosity of a star for a given mass to also calculate the absolute magnitude for the same star, we linearly interpolated between the data in Table 19.9 from Gilmore & Zeilik (2000) to arrive at a relation for the vertical scale height of the galactic disk.
That all being said, some discussion is in order over our decision to treat only the Thin Disk, and to do so using the Bahcall model. This choice was motivated by several reasons. We felt that an expanded analysis that also looked at stars in the Thick Disk and Halo would not add a significant number of detections to our final result. Thick Disk and Halo stars being much older and much farther away on average, are extremely difficult to detect transits around in any great quantity. Furthermore, the density of both populations is exceedingly low compared to Thin Disk stars observed out to the distances typically covered by photometric surveys.
Secondly, we decided to use the Bahcall model instead of other more recent, and probably more accurate, models because of speed considerations. While other models take into account features such as density variations from the spiral structure of the galaxy and the Galactic Bulge, integrations over reasonable ranges of r, l and b require upwards of 100,000 numerical iterations. Even if the more thorough models took only 1 10 of a second to produce a solution, these density calculations alone would take almost three hours to complete. Furthermore, because of the distance ranges that transit surveys typically use, the outer edges of the Bulge are too distant to warrant its inclusion (McNamara et al. 2000).
2.3.2. Interstellar Extinction Modeling
Similarly then, we also model the density of interstellar dust as a double exponential:
Which gives the dust density at a distance r along a particular line of sight (l, b) in relation to the density in the Solar Neighborhood, which we define as ½dust(0, l, b) = 1. To calculate the extinction in a given bandpass for a star at a given distance, we then integrate equation (29) over r to arrive at the line-of-sight dust column density, ¿
3. Assumptions
It is important to keep in mind that in the preceding derivation of our mathematical formalism, we made several assumptions about the nature of extrasolar planetary systems, as well as some simplifications to keep the math and computing time more manageable.
4. Results
Using a computer program built upon the preceding formalism, we simulated two different ground- and space-based photometric surveys to see how our predicted results compared with the historical detection rate of an ongoing survey, and the rates predicted in already published literature for a future survey.
To begin, we fixed our two fiducial parameters Rp and Â2 min on reasonable values. Both are shown in Table 2. Our decision to set Rp = 1.1 RJup was motivated from our previous decision to only predict the number of short-period jovian worlds detected by the transit surveys. More specifically, though only ten Jupiter-sized worlds have known radii (Jupiter itself plus the nine known transiting planets), we felt that 1.1 RJup was a reasonable size to expect an extrasolar giant-planet to take.
For the space-based Kepler mission, the Â2 min parameter is expected to be much less than in the case of ground-based surveys such as TrES; both because Kepler is designed to find much smaller Earth-sized worlds, and because space affords a more stable viewing environment as compared to sites inside the Earth’s atmosphere. One cannot set Â2min = 0, as this would imply an ability to detect planets in systems that did not even show transits, so we therefore tested different Â2 min values to see what effect they would have on the predicted detection rates. Based on these tests, we made the slightly arbitrary decision to set Â2 min = 25 for the Kepler simulations.
4.1. TrES
One of the survey programs that we chose to model, and one that is currently ongoing, is the Trans-Atlantic Exoplanet Survey (TrES). TrES is composed of a network of three telescopes sited on the Canary Islands, at Lowell Observatory in Arizona, and on Mt. Palomar, California. All are small, 10 cm aperture, wide-field (6±), CCD-based telescopes that operate in unison to observe the same field of sky nearly continuously over one to two month periods; the specifics are described in more detail by Dunham et al. (2004) and by Brown & Charbonneau (1999). The relevant quantities for our modeling are listed in Table 4.
The throughput of the TrES telescopes is the fraction of photons from a source that reach the CCD detector, and it takes into account such factors as losses due to airmass, poor seeing, filters, and lenses. To determine the effective throughput shown in Table 4, which is much lower than the theoretical throughput of TrES, we varied that parameter until our calculated RMS magnitude residuals matched those of the TrES survey. A comparison of our final calculated RMS residuals in the And0 field to the actual residuals in that field is shown in Fig. 2. Note that the instrumental R magnitudes listed on the TrES plot, which are mostly for calibration purposes, correspond to actual R magnitudes two magnitudes brighter.
For this paper, we treat the three TrES telescopes as one single instrument, due to the strong similarities amongst the telescopes. By taking the total amount of time that the network observes a particular field, we are then able to calculate a predicted detection rate in each of the fields.
We simulated ten different TrES fields using this methodology. The network observation times and the location of each field are shown in Table 5. The data for this table was collected off from the Sleuth observing website.5 The expected detections for each of the fields (and the total number of detections over all simulated fields) are shown in Table 6.
The uncertainty in the total detection prediction is a result of Poisson counting error, as well as uncertainty in the value of Pplanet from Section 2.1.2. The effects of our galactic structure model can clearly be seen in distribution of expected detections. Both the UMa0 and Crb0 fields are substantially above the plane of the galaxy (b ¼ 48±), and thus show a much lower detection rate than those fields directed towards denser regions of the sky.
The potential reasons for this discrepancy between the predicted number of detections and the actual number of detections are many. It may be that the close orbits of HJs and VHJs about their parent stars result in the atmospheres of these planets evaporating away.
The large thermal radiation flux that a Jupiter-sized planet would absorb, only 0.04 AU from a star (a common distance for HJs and VHJs orbiting solar-type stars), would cause the planet’s atmosphere to heat up significantly and therefore expand. This would lead to the eventual result of the atmosphere being so bloated that it could be stripped from the shallower portions of the planet’s gravity well by stellar wind and photon pressure from the parent star. After a period of time, this would leave the now denuded HJ or VHJ as a relatively large, massive core surrounded by a small blanket of gas. To an observer on Earth, such a planet would more difficult to detect in transit, as without an atmosphere, its radius would be substantially below 1.1 RJup. Hubble observations of HD209458 have shown that its planet’s atmosphere is being “evaporated” in just this way (Vidal-Madjar & Etangs 2004), which lends support to this possibility.
Alternatively, it may be that transit surveys suffer from systematic errors that are as yet unknown. Certainly, the accurate reduction of differential photometry is a non-trivial task. Still, without a greater number of planetary detections, it is hard to say whether the actual results of the TrES survey reflect the physical reality of extrasolar planets, biases in photometric surveys, or if they are simply an unlikely fluke of statistics.
4.2. Kepler
The Kepler mission is one of NASA’s Discovery-class missions, and is primarily designed to look for Earth-sized planets orbiting within the habitable zones of other stars. It consists of a single spacecraft that will be launched into an Earth-trailing heliocentric orbit in late 2008. On board, Kepler will have 42 CCDs that will measure the light from the 100,000 main-sequence stars within the telescope’s field of view, and do so with the precision necessary to discriminate transit signals from Earth-like worlds. Table 3 gives a more detailed list of the properties of the Kepler spacecraft.
Beyond looking for small extrasolar Earths, the immense amount of photometry that Kepler is expected to produce over its four year operational lifetime will also allow for the detection of larger jovian worlds transiting other stars. Previous estimates of the number of VHJs and HJs expected to be found by Kepler have predicted the discovery of about 180 close-in extrasolar giant planets (Jenkins & Doyle 2003). Clearly, given that there are only nine currently known transiting planets, this would be an enormous windfall for those scientists studying the properties of transiting planets.
To see how Kepler would fair under our methodology, we used the values in Table 3 to simulate observations of the Kepler field out to a distance of 6kpc. We found that this distance was sufficient to describe the HJ and VHJ detection rate of Kepler, as the number of detections drops to effectively zero beyond 6kpc. One important note is our choice of bandpass. The real Kepler mission will use a filter that covers most of the V- and R-bands.
To simulate this in our program, we select R as the primary band, but increase the CCD throughput parameter beyond 1.0 to 2.0 to compensate for the greater filter width of the actual spacecraft.
While the location of the Kepler field is currently centered on galactic coordinates (76.32, 13.5), at the time of the Jenkins & Doyle (2003) (hereafter referred to as simply JD03) estimate the field was centered much closer to the galactic plane at (76.32, 0.22).
This switch in the position of the field to higher galactic latitudes was done in order to obviate the necessity of sorting through the larger number of giant stars that are found in fields directed along the plane of the galaxy. For the purposes of comparison with the JD03 estimate, we therefore examined the old Kepler field in addition to the new field.
Tables 7 - 9 show the distribution of our predictions by spectral type. To facilitate comparison with JD03, we have included not only the final prediction numbers for both old and new Kepler fields, but also the calculated total number of stars in each field and the calculated number of transiting HJs and VHJs in each field. Interestingly, the calculated number of stars in both fields is approximately 15% lower than the JD03 estimate (for the old field) and the current Kepler input catalog of 100,000 stars (for the new field). After thoroughly checking all of our assumptions and programing, we were not able to account for these slightly lower star counts. The effect, however, is smaller than the other errors that contribute to the uncertainty in our final predictions6, so we decided to proceed with the reduced star counts.
As one can see in Table 9, our final prediction of Ndetect = 58+9.7 −11.4 for the old Kepler field is substantially lower than the JD03 prediction. This is a result of two factors. First, the Gould et al. (2006b) results that we use for Pplanet posit a HJ and VHJ frequency approximately 1 3 of the frequency used in JD03. With our reduced star count, this would imply a detection number of about 50 planets. The additional ten detections come from our 6For example, the fractional uncertainty in Pplanet is twice as large.
More explicit calculation of the transit probabilities of each system; JD03 uses the simple (and slightly low) approximation that 10% of HJs and VHJs transit.
In the new Kepler field, we see that looking away from the plane of the galaxy halves the number of HJs and VHJs detected to Ndetect = 24+6.2 −7.3 Figs. 8 - 11 show plots of our predicted results for Kepler in this new field. One can clearly see in Fig. 9 the mR = 15 magnitude limit of the survey, as well as its predicted saturation point around mR = 9.
While the prediction number of detections in the new Kepler field is lower than the corresponding JD03 prediction would be for the same star count, this is accounted for in the same way as in the old field. Indeed, we find that our predictions are consistent with those of JD03, except for the difference in our assumptions about the frequency of HJs and VHJs. The eventual results from Kepler therefore offer an excellent opportunity to more robustly determine the actual prevalence of short period jovian-worlds.
Postingan
Tidak ada komentar:
Posting Komentar