Constraints on Long-Period Planets from an L' and M band Survey of Nearby Sun-Like Stars: Modeling Results
A. N. Heinze, Philip M. Hinz, Matthew Kenworthy, Michael Meyer, Suresh Sivanandam, Douglas Miller
aa r X i v : . [ a s t r o - ph . E P ] M a r To be submitted to ApJ.
Constraints on Long-Period Planets from an L ′ and M band Survey of NearbySun-Like Stars: Modeling Results A. N. Heinze
Steward Observatory, University of Arizona, 933 N Cherry Avenue, Tucson, AZ 85721 [email protected]
Philip M. Hinz
Steward Observatory, University of Arizona, 933 N Cherry Avenue, Tucson, AZ 85721 [email protected]
Matthew Kenworthy
Steward Observatory, University of Arizona, 933 N Cherry Avenue, Tucson, AZ 85721 [email protected]
Michael Meyer
Department of Physics, Swiss Federal Institute of Technology (ETH-Zurich), ETH H´onggerberg,CH-8093 Zurich, Switzerland [email protected]
Suresh Sivanandam
Steward Observatory, University of Arizona, 933 N Cherry Avenue, Tucson, AZ 85721 [email protected]
Douglas Miller
Steward Observatory, University of Arizona, 933 N Cherry Avenue, Tucson, AZ 85721 [email protected]
ABSTRACT Observations reported here were obtained at the MMT Observatory, a joint facility of the University of Arizonaand the Smithsonian Institution. L ′ and M band Adaptive Optics (AO) extrasolar planetimaging survey of 54 nearby, sunlike stars using the Clio camera at the MMT. Oursurvey concentrates more strongly than all others to date on very nearby F, G, and Kstars, in that we have prioritized proximity higher than youth. Our survey is also thefirst to include extensive observations in the M band, which supplemented the primary L ′ observations. These longer wavelength bands are most useful for very nearby systemsin which low temperature planets with red IR colors (i.e. H − L ′ , H − M ) could bedetected. The survey detected no planets, but set interesting limits on planets andbrown dwarfs in the star systems we investigated. We have interpreted our null resultby means of extensive Monte Carlo simulations, and constrained the distributions ofextrasolar planets in mass M and semimajor axis a . If planets are distributed accordingto a power law with dN ∝ M α a β dM da , normalized to be consistent with radial velocitystatistics, we find that a distribution with α = − . β = − .
46, truncated at 110AU, is ruled out at the 90% confidence level. These particular values of α and β aresignificant because they represent the most planet-rich case consistent with currentstatistics from radial velocity observations. With 90% confidence no more than 8.1% ofstars like those in our survey have systems with three widely spaced, massive planetslike the A-star HR 8799. Our observations show that giant planets in long-period orbitsaround sun-like stars are rare, confirming the results of shorter-wavelength surveys, andincreasing the robustness of the conclusion. Subject headings: planetary systems, techniques: IR imaging, intrumentation: adaptiveoptics, astrometry, binary stars
1. Introduction
Nearly 400 extrasolar planets have now been discovered using the radial velocity (RV) method.RV surveys currently have good statistical completeness only for planets with periods of less thanten years (Cumming et al. 2008; Butler et al. 2006; Fischer & Valenti 2005), due to the limitedtemporal baseline of the observations, and the need to observe for a complete orbital period toconfirm the properties of a planet with confidence. The masses of discovered planets range fromjust a few Earth masses (Bouchy et al. 2009) up to around 20 Jupiter masses (M
Jup ). We note thata 20 M
Jup object would be considered by many to be a brown dwarf rather than a planet, but thatthere is no broad consensus on how to define the upper mass limit for planets. For a good overviewof RV planets to date, see Butler et al. (2006) or http://exoplanet.eu/catalog-RV.php .The large number of RV planets makes it possible to examine the statistics of extrasolar planetpopulations. Several groups have fit approximate power law distributions in mass and semimajoraxis to the set of known extrasolar planets (see for example Cumming et al. (2008)). Necessarily,however, these power laws are not subject to observational constraints at orbital periods longer 3 –than 10 years – and it is at these orbital periods that we find giant planets in our own solar system.We cannot obtain a good understanding of planets in general without information on long periodextrasolar planets. Nor can we see how our own solar system fits into the big picture of planetformation in the galaxy without a good census of planets in Jupiter- and Saturn-like orbits aroundother stars.Repeatable detections of extrasolar planets (as opposed to one-time microlensing detections)have so far been made by transit detection (e.g. Charbonneau et al. (2000)), by RV variations(Mayor & Queloz 1995), by astrometric wobble (Benedict et al. 2006), or by direct imaging (Marois et al.2008). Of these methods, transits are efficient only for detecting close-in planets. As noted above,precision RV observations have not been going on long enough to detect more than a few planetswith periods longer than ten years, but even as RV temporal baselines increase, long period planetswill remain harder to detect due to their slow orbital velocities. The amplitude of a star’s astro-metric wobble increases with the radius of its planet’s orbit, but decades-long observing programsare still needed to find long-period planets. Direct imaging is the only method that allows us tocharacterize long-period extrasolar planets on a timescale of months rather than years or decades.Direct imaging of extrasolar planets is technologically possible at present only in the infrared,based on the planets’ own thermal luminosity, not on reflected starlight. The enabling technologyis adaptive optics (AO), which allows 6-10m ground-based telescopes to obtain diffraction lim-ited IR images several times sharper than those from HST, despite Earth’s turbulent atmosphere.Theoretical models of giant planets indicate that such telescopes should be capable of detectingself-luminous giant planets in large orbits around young, nearby stars. The stars should be youngbecause the glow of giant planets comes from gravitational potential energy converted to heat intheir formation and subsequent contraction: lacking any internal fusion, they cool and becomefainter as they age.Several groups have published the results of AO imaging surveys for extrasolar planets aroundF, G, K, or M stars in the last five years (see for example Masciadri et al. (2005); Kasper et al.(2007); Biller et al. (2007); Lafreni`ere et al. (2007); and Chauvin et al. (2010)). Of these, mosthave used wavelengths in the 1.5-2.2 µ m range, corresponding to the astronomical H and K S filters(Masciadri et al. 2005; Biller et al. 2007; Lafreni`ere et al. 2007; Chauvin et al. 2010). They havetargeted mainly very young stars. Because young stars are rare, the median distance to stars ineach of these surveys has been more than 20 pc.In contrast to those above, our survey concentrates on very nearby F, G, and K stars, withproximity prioritized more than youth in the sample selection. The median distance to our surveytargets is only 11.2 pc. Ours is also the first survey to include extensive observations in the M band,and only the second to search solar-type stars in the L ′ band (the first was Kasper et al. (2007)).The distinctive focus on older, very nearby stars for a survey using longer wavelengths is natural:longer wavelengths are optimal for lower temperature planets which are most likely to be found inolder systems, but which would be undetectable around all but the nearest stars. More information 4 –on our sample selection, observations, and data analysis can be found in our Observations paper,Heinze et al. (2010), which also details our careful evaluation of our survey’s sensitivity, includingextensive tests in which fake planets were randomly placed in the raw data and then recoveredby an experimenter who knew neither their positions nor their number. Such tests are essentialfor establishing the true relationship between source significance (i.e. 5 σ , 10 σ , etc.) and surveycompleteness.Our survey places constraints on a more mature population of planets than those that havefocused on very young stars, and confirms that a paucity of giant planets at large separations fromsun-like stars is robustly observed at a wide range of wavelengths.In Section 2, we review power law fits to the distribution of known RV planets, including thenormalization of the power laws. In Section 3, we present the constraints our survey places on thedistribution of extrasolar giant planets, based on extensive Monte Carlo simulations. In Section 4we discuss the promising future of planet-search observations in the L ′ and especially the M band,and in Section 5 we conclude.
2. Statistical Distributions from RV Planets
Nearly 400 RV planets are known. See Butler et al. (2006) for a useful, conservative list-ing of confirmed extrasolar planets as of 2006, or http://exoplanet.eu/catalog-RV.php for afrequently-updated catalog of all confirmed and many suspected extrasolar planet discoveries.The number of RV planets is sufficient for meaningful statistical analysis of how extrasolarplanets are distributed in terms of their masses and orbital semimajor axes. The lowest massplanets and those with the longest orbital periods are generally rejected from such analyses toreduce bias from completeness effects, but there remains a considerable range (2-2000 days inperiod, or roughly 0.03-3.1 AU in semimajor axis for solar-type stars; and 0.3-20 M
Jup in mass)where RV searches have good completeness (Cumming et al. 2008). There is evidence that theshortest period planets, or ‘hot Jupiters,’ represent a separate population, a ‘pileup’ of planets invery close-in orbits that does not follow the same statistical distribution as planets in more distantorbits (Cumming et al. 2008). The hot Jupiters are therefore often excluded from statistical fitsto the overall populations of extrasolar planets, or at least from the fits to the semimajor axisdistribution.Cumming et al. (2008) characterize the distribution of RV planets detected in the Keck PlanetSearch with an equation of the form dN = C M α L P β L d ln( M ) d ln( P ) . (1)where M is the mass of the planet, P is the orbital period, and C is a normalization constant. 5 –They state that 10.5% of solar-type stars have a planet with mass between 0.3 and 10 M Jup andperiod between 2 and 2000 days, which information can be used to derive a value for C givenvalues for the power law exponents α L and β L . They find that the best-fit values for these are α L = − . ± . β L = 0 . ± .
1, where the L subscript is our notation to make clear thatthese are the exponents for the form using logarithmic differentials.In common with a number of other groups, we choose to represent the power law with ordinarydifferentials, and to give it in terms of orbital semimajor axis a rather than orbital period P : dN = C M α a β dM da. (2)Where C , of course, will not generally have the same value for Equations 1 and 2. Manipu-lating the two equations and using Kepler’s Third Law makes it clear that α = α L − . (3)and β = 32 β L − . (4)The Cumming et al. (2008) exponents produce α = − . ± . β = − . ± .
15 whentranslated into our form. The mass power law is well behaved, but the integral of the semimajor axispower law does not converge as a → ∞ , so an outer truncation radius is an important parameterof the semimajor axis distribution.Butler et al. (2006) presents the 2006 Catalog of Nearby Exoplanets, a carefully describedheterogenous sample of planets detected by several different RV search programs. With appropriatecaution, Butler et al. (2006) refrain from quoting confident power law slopes based on the combineddiscoveries of many different surveys with different detection limits and completeness biases (incontrast, the Cumming et al. (2008) analysis was restricted to stars in the Keck Planet Search,which were uniformly observed up to a given minimum baseline and velocity precision). Butler et al.(2006) do tentatively adopt a power law with the form of Equation 2 for mass only, and state that α appears to be about -1.1 (or -1.16, to give the exact result of a formal fit to their list of exoplanets).However they caution that due to their heterogeneous list of planets discovered by different surveys,this power law should be taken more as a descriptor of the known planets than of the underlyingdistribution. They do not quote a value for the semimajor axis power law slope β .Based mostly on Cumming et al. (2008), but considering Butler et al. (2006) as helpful addi-tional input, we conclude that the true value of the mass power law slope α is probably between -1.1and -1.51, with -1.31 as a good working model. The value of the semimajor axis power law slope β is probably between -0.46 and -0.76, with -0.61 as a current best guess. The outer truncation 6 –radius of the semimajor axis distribution cannot be constrained by the RV results: surveys likeours exist, in part, to constrain this interesting number.The only other result we need from the RV searches is a normalization that will allow us tofind C . We elect not to use the Cumming et al. (2008) value (10.5% of stars having a planet withmass between 0.3 and 10 M Jup and period between 2 and 2000 days), because this range includesthe hot Jupiters, a separate population.We take our normalization instead from the Carnegie Planet Sample, as described in Fischer & Valenti(2005). Their Table 1 (online only) lists 850 stars that have been thoroughly investigated with RV.They state that all planets with mass at least 1 M
Jup and orbital period less than 4 years havebeen detected around these stars. Forty-seven of these stars are marked in Table 1 as having RVplanets. Table 2 from Fischer & Valenti (2005) gives the measured properties of 124 RV planets,including those orbiting 45 of the 47 stars listed as planet-bearing in Table 1. The stars left outare HD 18445 and and HD 225261. We cannot find any record of these stars having planets, andtherefore as far as we can tell they are typos in Table 1.Since all planets with masses above 1 M
Jup and periods less than 4 years orbiting stars in theFischer & Valenti (2005) list of 850 may be relied upon to have been discovered, we may pick anysub-intervals in this range of mass and period, and divide the number of planets falling into theseintervals by 850 to obtain our normalization. We selected the range 1-13 M
Jup in mass, and 0.3-2.5AU in semimajor axis. Twenty-eight stars, or 3.29% of the 850 in the Fischer & Valenti (2005) list,have one or more planets in this range. Our inner limit of 0.3 AU excludes the hot Jupiters, and thusthe 3.29% value provides our final normalization. We note that if we adopt the Cumming et al.(2008) best-fit power laws, and use the 3.29% normalization to predict the percentage of starshaving planets with masses between 0.3 and 10 M
Jup and orbital periods between 2 and 2000 days,we find a value of 9.3%, which is close to the Cumming et al. (2008) value of 10.5%. The slightdifference is probably not significant, but might be viewed as upward bias in the Cumming et al.(2008) value due to the inclusion of the hot Jupiters. In any case we would not have obtained verydifferent constraints if we had used the Cumming et al. (2008) normalization in our Monte Carlosimulations.For comparison, among the other papers reporting Monte Carlo simulations similar to ours,Kasper et al. (2007) used a normalization of 3% for planets with semimajor axes of 1-3 AU andmasses greater than 1 M
Jup . This is close to our value of 3.29% for a similar range. Lafreni`ere et al.(2007) and Nielsen et al. (2008) fixed α and β in their simulations, and let the normalization bea free parameter. Chauvin et al. (2010) obtained their normalization from Cumming et al. (2008),and Nielsen & Close (2009) obtained theirs from Fischer & Valenti (2005).Juric & Tremaine (2008) provide a helpful mathematical description of the eccentricity dis-tribution of known RV planets: P ( ǫ ) = ǫe − ǫ / (2 σ ) . (5) 7 –where P ( ǫ ) is the probability of a given extrasolar planet’s having orbital eccentricity ǫ , e isthe root of the natural logarithm, and σ = 0 .
3. We find that this mathematical form provides anexcellent fit to the distribution of real exoplanet eccentricities from Table 2 of Fischer & Valenti(2005), so we have used it as our probability distribution to generate random eccentricities for theMonte Carlo simulations we describe in Section 3 below.
3. Constraints on the Distribution of Planets3.1. Theoretical Spectra
Burrows et al. (2003) present high resolution, flux-calibrated theoretical spectra of giant plan-ets or brown dwarfs for ages ranging from 0.1-5.0 Gyr and masses from 1 to 20 M
Jup (these areavailable for download from ). We have integratedthese spectra to give absolute magnitudes in the L ′ and M filters used in Clio (see Tables 1 and2), and have found that the results can be reasonably interpolated to give the L ′ or M band mag-nitudes for all planets of interest for our survey. Baraffe et al. (2003) also present models of giantplanets and brown dwarfs, pre-integrated into magnitudes in the popular infrared bands. Thesemodels predict slightly better sensitivity to low mass planets in the L ′ band and slightly poorersensitivity in the M band, relative to the Burrows et al. (2003) models. We cannot say if the differ-ence is due to the slightly different filter sets used (MKO for Clio vs. Johnson-Glass and Johnsonfor Baraffe et al. (2003)), or if it is intrinsic to the different model spectra used in Burrows et al.(2003) and Baraffe et al. (2003). We have chosen to use the Burrows et al. (2003) models exclu-sively herein, to avoid any errors due to the slight filter differences. Since the Burrows et al. (2003)models predict poorer sensitivity in the L ′ band, in which the majority of our survey was conducted,our decision to use them is conservative. In common with several other surveys (Kasper et al. 2007; Biller et al. 2007; Lafreni`ere et al.2007; Chauvin et al. 2010) we have used our survey null result to set upper limits on planet popula-tions via Monte Carlo simulations. In these simulations, we input our sensitivity data in the form oftabular files giving the sensitivity in apparent magnitudes as a function of separation in arcsecondsfor each star. Various features of our images could cause the sensitivity at a given separation to varysomewhat with position angle: to quantify this, our tabular files give ten different values at eachseparation, corresponding to ten different percentiles ranging from the worst to the best sensitivityattained at that separation. These files are described in detail in Heinze et al. (2010), and areavailable for download from
The Monte Carlo simulations described below allow us to use the observed sensitivity to plan- 8 –Table 1. L ′ Band Absolute Mags from Burrows et al. (2003)Planet Mass Mag at Mag at Mag at Mag at Mag atin M
Jup a a b b b b a No models for these very faint planets appear in Burrows et al. (2003). We have inserted ad hoc values to smooththe interpolations. Any effect of the interpolated magnitudes for planets we could actually detect is negligible. b No models for these bright, hot planets appear in Burrows et al. (2003), which focuses on cooler objects. We haveadded values from Baraffe et al. (2003) and then adjusted them to slightly fainter values to ensure smooth interpolations.
Table 2. M Band Absolute Mags from Burrows et al. (2003)Planet Mass Mag at Mag at Mag at Mag at Mag atin M
Jup a a b b b b a No models for these very faint planets appear in Burrows et al. (2003). We have inserted ad hoc values to smooththe interpolations. Any effect of the interpolated magnitudes for planets we could actually detect is negligible. b No models for these bright, hot planets appear in Burrows et al. (2003), which focuses on cooler objects. Wehave added values from Baraffe et al. (2003) and then adjusted them to slightly fainter values to ensure smoothinterpolations. α and β , anda given outer truncation value R trunc for the semimajor axis distribution. Using the normalizationdescribed in Section 2, the probability P plan of any given star having a planet between 1 and 20M Jup is then calculated from the input α , β , and R trunc . In each realization of our survey, eachstar is randomly assigned a number of planets, based on Poisson statistics with mean P plan . Inmost cases P plan <<
1, so the most likely number of planets is zero. If the star turns out to haveone or more planets, the mass and semimajor axis of each are randomly selected from the inputpower law distributions. The eccentricity is randomly selected from the Juric & Tremaine (2008)distribution, and an inclination is randomly selected from the distribution P ( i ) ∝ sin( i ). If the staris a binary, the planet may be dropped from the simulation at this point if the orbit seems likelyto be unstable. In general, we consider circumstellar planets to be stable as long as their apastrondistance is less than 1 / σ sources, 46% of 7 σ sources, and 16% of 5 σ sources, where σ is a measure of thePSF-scale noise in a given region of the image (see Heinze et al. (2010) for details). This secondand final random choice in our Monte Carlo simulations is therefore arranged to ensure that a ran-domly selected 16% of planets with 5-7 σ significance, and 46% of planets with 7-10 σ significance,are recorded in the simulation as detected objects. Although we have 97% completeness at 10 σ ,we choose to consider 100% of simulated planets with 10 σ or greater significance to be detected,because at only slightly above 10 σ the true completeness certainly becomes 100% for all practicalpurposes. Note that we have conservatively allowed the detection probabilities to increase stepwise, 10 –rather than in a continuous curve, from 5 to 10 σ : that is, in our Monte Carlo simulations, planetswith 5-7 σ significance are detected at the 5 σ rate from our blind sensitivity tests, while those with7-10 σ significance are detected at the 7 σ rate.The low completeness (16%) at 5 σ , as determined from our blind sensitivity tests using fakeplanets, may seem surprising. In these tests we distinguished between planets that were suggestedby a concentration of unusually bright pixels (‘Noticed’), or else confidently identified as real sources(‘Confirmed’). Many more planets were noticed than were confirmed: for noticed planets, therates are 100% at 10 σ , 86% at 7 σ , and 56% at 5 σ . However, very many false positives were alsonoticed, so sources that are merely noticed but not confirmed do not represent usable detections.The completeness levels we used in our Monte Carlo simulations (16% at 5 σ and 46% at 7 σ )refer to confirmed sources. No false positives were confirmed in any of our blind tests. Followupobservations of suspected sources are costly in terms of telescope time, so a detection strategy witha low false-positive rate is important.Though sensitivity estimators (and therefore the exact meaning of 5 σ ) differ among planetimaging surveys, ours was quite conservative, as is explained in Heinze et al. (2010). The lowcompleteness we find at 5 σ , which has often been taken as a high-completeness sensitivity limit,should serve as a warning to future workers in this field, and an encouragement to establish adefinitive significance-completeness relation through blind sensitivity tests as we have done.Note that our blind sensitivity tests, covered in Heinze et al. (2010), are completely distinctfrom the Monte Carlo simulations covered herein. The blind tests involved inserting a little over ahundred fake planets into our raw image data to establish our point-source sensitivity. In our MonteCarlo work we simulated the orbits, masses, and brightnesses of millions of planets, and comparedthem to our previously-established sensitivity limits to see which planets our survey could havedetected. To evaluate the significance of our survey and provide some guidance for future work, we haveanalyzed in detail a single Monte Carlo simulation. We chose the Cumming et al. (2008) best fitvalues of α = − .
31 and β = − .
61, with the semimajor axis truncation radius set to 100 AU.Planets could range in mass from 1 to 20 M
Jup . As described in Section 2 above, we normalized theplanet distributions so that each star had a 3.29% probability of having a planet with semimajoraxis between 0.3 and 2.5 AU and mass between 1 and 13 M
Jup . The simulation consisted of 50,000realizations of our survey with these parameters. In all, 505,884 planets were simulated, of which51,879 were detected.In 38% of the 50,000 realizations, our survey found zero planets, while 37% of the time itfound one, and 25% of the time it found two or more. The planet distribution we considered inthis simulation cannot be ruled out by our survey, since a null result such as we actually obtained 11 –turns out not to be very improbable.The large number of survey realizations in our simulation allows the calculation of precisestatistics for potentially detectable planets. The median mass of detected planets in our simulationwas 11.36 M
Jup , the median semimajor axis was 43.5 AU, the median angular separation was 2.86arcsec, and the median significance was 21.4 σ . This last number is interesting because it suggeststhat, for our survey, any real planet detected was likely to appear at high significance, obviouseven on a preliminary, ‘quick-look’ reduction of the data. This suggests that performing suchreductions at the telescope should be a high priority, to allow immediate confirmation and followupif a candidate is seen. Figure 1 presents as a histogram the significance of all planets detected inthis Monte Carlo simulation.We suspected that there would be a detection bias toward very eccentric planets, because thesewould spend most of their orbits near apastron, where they would be easier to detect. This biasdid not appear at any measurable level in our simulation. However, there was a weak but clearbias toward planets in low-inclination orbits, which, of course, spend more of their time at largeseparations from their stars than do planets with nearly edge-on orbits.A concern with any planet imaging survey is how strongly the results hinge on the best (i.e.nearest and youngest) few stars. A survey of 54 stars may have far less statistical power than thenumber would imply if the best two or three stars had most of the probabilty of hosting detectableplanets. Table 3 gives the percentage of planets detected around each star in our sample based onour detailed Monte Carlo simulation. Due to poor data quality, binary orbit constraints, or otherissues, a few stars had zero probability of detected planets given the distribution used here. Ingeneral, however, the likelihood of hosting detectable planets is fairly well distributed.In Table 4, we give the details of planetary orbital constraints used in our Monte Carlo simula-tions for each binary star we observed, complete with the separations we measured for the binaries.Note that HD 96064 B is a close binary star in its own right, so planets orbiting it were limitedin two ways: the apastron could not be too far out, or the orbit would be rendered unstable byproximity to HD 96064 A – but the periastron also could not be too far in, or the binary orbit ofHD 96064 Ba and HD 96064 Bb would render it unstable. Planets individually orbiting HD 96064Ba or HD 96064 Bb were not considered in our survey, since to be stable the planets would haveto be far too close-in for us to detect them. The constraints described in Table 4 account for mostof the stars in Table 3 with few or no detections reported.A final question our detailed simulation can address is how important the M band observationswere to the survey results. In Table 5, we show that when M band observations were made, theydid substantially increase the number of simulated planets detected. 12 –Fig. 1.— Histogram of detection significance for the 51,879 simulated planets detected in 50,000realizations of our survey with the Cumming et al. (2008) distribution ( α = − . β = − . σ , but some 5-7 σ planets are still detected. The relatively high median significance of 21.4 σ suggests any detectedplanet would most likely be quite obvious – a good argument for doing ‘quick-look’ data reductionsas soon as possible at the telescope. 13 –Table 3. Percentage of Detected Planets Found Around Each Star% of Total Median Median MedianStar Name Detected Planets Mass Semimajor Axis SeparationGJ 117 6.07 7.66 M Jup ǫ Eri 5.83 6.98 M
Jup
Jup
Jup
Jup
Jup
Jup
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The planet distribution we used in the single Monte Carlo simulation described above couldnot be ruled out by our survey. To find out what distributions could be ruled out, we performedMonte Carlo simulations assuming a large number of different possible distributions, parametrizedby the two power law slopes α and β , and by the outer semimajor axis truncation radius R trunc .Regardless of the values of α and β , each simulation was normalized to match the RV statisticsof Fischer & Valenti (2005): any given star had 3.29% probability of hosting a planet with massbetween 1 and 13 M Jup and semimajor axis between 0.3 and 2.5 AU. The mass range for simulatedplanets was 1-20 M
Jup .We tested three different values of α : -1.1, -1.31, and -1.51, roughly corresponding to the mostoptimistic permitted, the best fit, and the most pessimistic permitted values from Cumming et al.(2008). For each value of α , we ran simulations spanning a wide grid in terms of β and R trunc . Inconstrast to the extensive results described in Section 3.3, the only data saved for these simulationswas the probability of finding zero planets. Since we did in fact obtain a null result, distributionsfor which the probability of this was sufficiently low can be ruled out.Figures 2 and 3 show the probability of a null result as a function of β and R trunc for ourthree different values of α . Figure 2 presents constraints based on α = − .
31, the best-fit valuefrom RV statistics, while Figure 3 compares the optimistic case α = − . α = − .
51. Each pixel in these figures represents a Monte Carlo simulation involving 15,000realizations of our survey; generating the figures took several tens of hours on a fast PC. Contoursare overlaid at selected probability levels. Regions within the 1%, 5%, and 10% contours can, ofcourse, be ruled out at the 99%, 95%, and 90% confidence levels respectively. For example, we findthat the most optimistic power laws allowed by the Cumming et al. (2008) RV statistics, α = − . β = − .
46, are ruled out with 90% confidence if R trunc is 110 AU or greater. Similarly, α = − .
51 and β = − .
3, truncated at 100 AU, is ruled out. Though β = 0 . α = − .
31, we rule out β = 0 . R trunc is less than 38 AU. It is also possible to place constraints on the distribution of planets without assuming a powerlaw or any other particular model for the statistics of planetary masses and orbits. Note well thatby “model-independent” in this context, we mean independent only of models for the statisticaldistributions of planets in terms of M and a – not independent of models of planetary spectra suchthose we obtain from Burrows et al. (2003). The latter are our only means of converting fromplanetary mass and age to detectable flux, and as such they remain indispensable.To place our model-independent constraints, we performed an additional series of Monte Carlo 15 –Table 3—Continued% of Total Median Median MedianStar Name Detected Planets Mass Semimajor Axis SeparationGJ 896 A 0.61 12.43 M Jup
Jup τ Ceti 0.38 17.19 M
Jup
Jup ξ Boo B 0.32 12.07 M
Jup
Jup ξ Boo A 0.24 12.89 M
Jup
Jup
Jup
Jup
Jup
Jup
Jup
Jup
Jup α = − . β = − .
61, and semimajor axis trunca-tion radius 100 AU. Of all the simulated planets that were detected, we presenthere the percentage that were found around each given star, and the median mass,semimajor axis, and projected separation for simulated planets found around eachstar. The table thus indicates around which stars our survey had the highest like-lihood of detecting a planet. Many stars with poor likelihood are binaries, withfew stable planetary orbits possible. 16 –Table 4. Constraints on Simulated Planet Orbits Around Binary Starsconstraints on constraints on constraints onseparation circumprimary circumsecondary circumbinaryStar Name (arcsec) apastron apastron periastronHD 220140 AB 10.828 < < < < > < < < < > ξ Boo AB 6.345 < < < < < < > < < < < > < < > β , where dnda ∝ a β , and the outer truncation radius of the semimajor axisdistribution. Here, the slope of the mass distribution α has been taken as -1.31, where dndM ∝ M α .Since we found no planets, distributions that lead to a probability P of finding no planets are ruledout at the 1 − P confidence level: for example, the region above and to the right of the 0.1 contouris ruled out at the 90% confidence level 18 –Table 5. Importance of the M Band DataTotal simulated 2-band L ′ -only M -onlyStar Name detections detections detections detections ǫ Eri 2850 46.98% 8.28% 44.74%61 Cyg B 1610 52.73% 1.55% 45.71%61 Cyg A 965 63.01% 22.80% 14.20% ξ Boo B 157 61.15% 18.47% 20.38% ξ Boo A 115 60.00% 18.26% 21.74%GJ 702 A 9 22.22% 0.00% 77.78%Note. — The usefulness of M band observations, based on ourdetailed Monte Carlo simulation. When M band observations weremade of a given star, they did substantially increase the number ofsimulated planets detected around that star.Fig. 3.— Probability of our survey detecting zero planets, as a function of the power law slope ofthe semimajor axis distribution β , where dnda ∝ a β , and the outer truncation radius of the semimajoraxis distribution. Here, the slope of the mass distribution α has been taken as -1.1 (left) and -1.51(right), where dndM ∝ M α . Since we found no planets, distributions that lead to a probability P offinding no planets are ruled out at the 1 − P confidence level: for example, the regions above andto the right of the 0.1 contours are ruled out at the 90% confidence level 19 –simulations on a grid of planet mass and orbital semimajor axis. For each grid point we seekto determine a number P ( M, a ) such that, with some specified level of confidence (e.g., 90%),the probability of a star like those in our sample having a planet with the specified mass M andsemimajor axis a is no more than P ( M, a ). We determine P ( M, a ) by a search: first a guess ismade, and a Monte Carlo simulation assuming this probability is performed. If more than 10% ofthe realizations of our survey turn up a null result, the guessed probability is too low; if less than10% turn up a null result, the probability is too high. It is adjusted in steps of ever-decreasing sizeuntil the correct value is reached.Figure 4 shows the 90% confidence upper limit on P ( M, a ) as a function of mass M andsemimajor axis a . Each pixel represents thousands of realizations of our survey, with P ( M, a )finely adjusted to reach the correct value. Contours are overplotted showing where P ( M, a ) isless than 8%, 10%, 25%, 50%, and 75%, with 90% confidence. Note that P ( M, a ), the valueconstrained by our simulations, is a probability rather than a fixed fraction. The probability isthe more scientifically interesting number, but is harder to constrain. For example, if 3.7% is thefraction of the actual stars in our sample that have planets with easy-to-detect properties, thereare 2 such planets represented in our 54-star survey. However, if the probability of a star like thosein our sample having such a planet is 3.7%, there is still a nonzero probablity (13% in this case)that no star in our sample actually has such a planet.The results presented in Figure 4 can be interpreted as model-independent constraints on planetpopulations. For example, with 90% confidence we find that less than 50% of stars with propertieslike those in our survey have a 5 M
Jup or more massive planet in an orbit with a semimajor axisbetween 30 and 94 AU. Less than 25% of stars like those in our survey have a 7 M
Jup or moremassive planet between 25 and 100 AU, less than 15% have a 10 M
Jup or more massive planetbetween 22 and 100 AU, and less than 12% have a 15 M
Jup or more massive planet/brown dwarfbetween 15 and 100 AU. Going to the most massive objects considered in our simulations, we canset limits ranging inward past 10 AU: we find that less than 25% of stars like those surveyed havea 20 M
Jup object orbiting between 8 and 100 AU. These constraints hold independently of howplanets are distributed in terms of their masses and semimajor axes.HR 8799 appears to have a remarkable system of three massive planets, seen at projecteddistances of 24, 38, and 68 AU, with masses of roughly 10, 10, and 7 M
Jup , respectively (Marois et al.2008). Using a Monte Carlo simulation like those used to create Figure 4, we find with 90%confidence that less than 8.1% of stars like those in our survey have a clone of the HR 8799 planetarysystem. For purposes of this simulation we adopted the masses above, and set the planets’ orbitalradii equal to their projected separations. Our 8.1% limit represents a step toward determiningwhether or not systems of massive planets in wide orbits are more common around more massivestars such as HR 8799 than FGK stars such as those we have surveyed. 20 –Fig. 4.— 90% confidence level upper limits on the probability P ( M, a ) that a star like those in oursurvey will have a planet of mass M and semimajor axis a . This plot shows, for example, that oursurvey constrains the abundance of 10 M Jup or more massive planets with orbital semimajor axesbetween 22 and 100 AU to be less than 15% around sun-like stars. The abundance of 5 M
Jup ormore massive planets between 25 and 94 AU is constrained to be less than 50%. The latter rangedoes not extend all the way to 100 AU because our sensitivity to planets in very distant orbitsdecreases somewhat due to the possibility of their lying beyond our field of view. 21 –
The surveys of Kasper et al. (2007) and Biller et al. (2007), have set constraints on the dis-tributions of extrasolar planets similar to those we present herein, while Nielsen et al. (2008)and especially Lafreni`ere et al. (2007) have set stronger constraints. More recent analyses byNielsen & Close (2009) and Chauvin et al. (2010) also provide constraints on the planetary dis-tribution. For example, Nielsen & Close (2009) provide a 68% confidence that the Cumming et al.(2008) distribution can be excluded for a truncation radius of 28 AU. However, if different modelsare used this number jumps to 83 AU. Chauvin et al. indicate a similar limit from analyzing theirresults using Baraffe et al. 2003 models. For the standard parameters they indicate a maximumpermitted truncation radius of approximately 35 AU. In this context, the results presented hereprovide looser constraints on the planet distribution, but provide an independent check on themodel-dependent systematic errors which may exist with shorter wavelength data, due to incorrectmodel brightness estimates or age determination.Theoretical spectra of self-luminous extrasolar planets are very poorly constrained observation-ally. The recent detections of possible planets around HR 8799 (Marois et al. 2008), Fomalhaut(Kalas et al. 2008), and β Pic (Lagrange et al. 2009) are either single-band ( β Pic) or only be-ginning to be evaluated at multiple wavelengths (HR 8799, Fomalhaut). The candidate planetsorbiting HR 8799 and β Pic are hotter than we would expect to find orbiting middle-aged starssuch as those in our survey, while HST photometry of Fomalhaut b suggests much of its brightnessis starlight reflected from a circumplanetary dust disk. Our survey, and other exoplanet surveys,must therefore be interpreted using models of planetary spectra that are not yet well-tested againstobservations.Such models predict brightnesses in the H band, and particularly in narrow spectral win-dows within the H band, that are enormously in excess of black body fluxes. The constraintsset by Masciadri et al. (2005); Biller et al. (2007); Lafreni`ere et al. (2007); Nielsen et al. (2008);Nielsen & Close (2009); and Chauvin et al. (2010) depend on the accuracy of these predictionsof remarkable brightness in the H band. The L ′ and M bands that we have used are nearerthe blackbody peaks of low-temperature self-luminous planets, and might be expected to be morereliable.However, Leggett et al. (2007) and Reid & Cruz (2002) suggest that the M band brightnessat least of hotter extrasolar planets will be less than predicted by Burrows et al. (2003) due toabove-equilibrium concentrations of CO from convective mixing. Hubeny & Burrows (2007) presentnew models indicating the effect is present for planets with T eff ranging from 600 to 1800K. Themaximum M band flux supression is about 40%, and flux supression disappears completely for T eff below 500K. Based on Burrows et al. (2003), this T eff value corresponds to planets of about3.5, 6.5, 12, and 15 M Jup at ages of 100 Myr, 300 Myr, 1 Gyr, and 2 Gyr, respectively. In manycases our M band observations were sensitive to planets at lower masses than these values, andtherefore T eff lower than 500K, implying that the CO flux supression would have no effect on our 22 –mass limits. In other cases our M band sensitivity did not extend so low. However, given that M band observations formed a relatively small part of our survey, and CO supression would affectonly a fraction even of them, the total effect on the statistical conclusions of our survey should beentirely negligible.Theoretical spectra such as those of Burrows et al. (2003) may or may not be more reliable inthe L ′ and M bands than at shorter wavelengths. However, so long as the models remain poorlyconstrained by observations at every wavelength, conclusions based on observations at multiplewavelengths will be more secure. Our survey, with that of Kasper et al. (2007), has diversifiedplanet imaging surveys across a broader range of wavelengths.In another sense our survey differs even from that of Kasper et al. (2007): we have investigatedolder stars. This is significant because planetary systems up to ages of several hundred Myr may stillbe undergoing substantial dynamical evolution due to planet-planet interactions (Juric & Tremaine2008; Gomes et al. 2005). Our survey did not necessarily probe the same planet population as, forexample, those of Kasper et al. (2007) and Chauvin et al. (2010).Finally, theoretical models of older planets are likely more reliable than for younger ones,as these planets are further from their unknown starting conditions and moving toward a well-understood, stable configuration such as that of Jupiter. It has been suggested by Marley et al.(2007) and Fortney et al. (2008) that theoretical planet models such as those of Burrows et al.(2003) and Baraffe et al. (2003) may overpredict the brightness of young ( <
100 Myr) planets byorders of magnitude, while for older planets the models are more accurate.We have focused on nearby, mature star systems, and have conservatively handled the agesof stars. This makes our survey uniquely able to confirm that the rarity of giant planets at largeseparations around solar-type stars, first noticed in surveys strongly weighted toward young stars,persists at older system ages. It is not an artifact of model inaccuracy at young ages due to unknowninitial conditions.
4. The Future of the L ′ and M Bands
In the L ′ and M bands, the sky brightness is much worse than at shorter wavelengths. How-ever, models (e.g., Burrows et al. (2003)) predict that in the L ′ and M bands, planets fade lessseverely with increasing age (or, equivalently, decreasing T eff ). Also, planet/star flux ratios aremore favorable in the L ′ and M bands than at shorter wavelengths such as the H and K S bands.It makes sense to use the L ′ and M bands on bright stars, where the planet/star flux ratio is amore limiting factor than the sky brightness. In Heinze et al. (2008), we have shown that M bandobservations tend to do better than those at shorter wavelengths at small separations from brightstars.The L ′ and M bands are most useful, however, for detecting the lowest temperature planets, 23 –which have the reddest H − L ′ and H − M colors. Such very low temperature planets can only bedetected around the nearest stars, so it is for very nearby stars that L ′ and M band observationsare most useful. For distant stars, around which only relativly high T eff planets can be detected,the H and K S bands are much better. We will now quantitatively describe the advantage of L ′ and M band observations over shorter wavelengths for planet-search observations of nearby stars.Most AO planet searches to date have used the H and K S bands, or specialized filters inthe same wavelength regime. While the K S band has been used extensively to search for planetsaround young stars (Masciadri et al. 2005; Chauvin et al. 2010), our comparison here will focuson the H band regime. Models indicate it offers better sensitivity than the K S band except forplanets younger than 100 Myr (Burrows et al. 2003; Baraffe et al. 2003), and most of the stars wewill suggest the L ′ and the M bands are useful for will be older than this. The most sensitive H -regime planet search observations made to date are those of Lafreni`ere et al. (2007), in part becauseof their optimized narrow-band filter. They attained an effective background-limited point-sourcesensitivity of about H = 23 .
0. Based on the models of Burrows et al. (2003), Lafreni`ere et al.(2007) would have set better planetary mass limits than our observations around all of our ownsurvey targets except the very nearest objects, such as ǫ Eri and 61 Cyg. Thus, at present, the H -regime delivers far better planet detection prospects than the L ′ and M bands for most stars.However, as detector technology improves, larger telescopes are built, and longer planet de-tection exposures are attempted, the sensitivity at all wavelengths will increase. This means thatlow-temperature planets, with their red IR colors, will be detectable at larger distances, and theutility of the L ′ and especially the M bands will increase. In Figure 5 we show the minimumdetectable planet mass for hypothetical stars at 10 and 25 pc distance as a function of the increaseover current sensitivity in the H , L ′ , and M bands, and in Figure 6 we present the same compar-ison for a star at 5 pc. We have taken current sensitivity to be H = 23 . L ′ = 16 .
5, and M = 13 . L ′ and M bands will do even better relative to H closer to the star where observations are no longer background limited. Of course H band obser-vations with next-generation extreme AO systems such as GPI and SPHERE will offer improvedperformance close to the star, but advances in M -band AO coronography (e.g. Kenworthy et al.(2007)), will also improve the longer-wavelength results. In any case, Figures 5 and 6 comparebackground-limited performance only.The supression of flux in the M band due to elevated levels of CO (Leggett et al. 2007;Reid & Cruz 2002) does not apply to planets at the low temperatures relevant for Figures 5 and6. Based on Burrows et al. (2003), the entire mass range covered by both Figures corresponds toplanets with T eff below 500K, except for planets with masses above 6.5 M Jup in the left panel ofFigure 5 (25 pc distance, 300 Myr age). This upper section of the 25 pc, 300 Myr panel is irrelevantto the important implications of the figure. According to Hubeny & Burrows (2007), there is nosupression of the M band for effective temperatures below 500K. 24 –Fig. 5.— Minimum detectable planet mass in units of M Jup for stars at 25pc (left) and 10pc (right),in the H , L ′ , and M bands, as a function of increase over current sensitivity. We have taken currentsensitivities to be H = 23 . L ′ = 16 .
5, and M = 13 .
5. While the H band will likely remain thewavelength of choice for planet search observations of stars at 25 pc and beyond, an increase ofonly 2.4 mag over current sensitivities, even though paralleled by an equal increase in H bandsensitivity, will render the M band more sensitive than H for planets around all stars nearer than10 pc. The relative effectiveness of different wavelengths depends sensitively on the distance to astar system, but it is essentially independent of the stellar age, as explained in the text. 25 –Fig. 6.— Minimum detectable planet mass in units of M Jup for stars at 5pc, in the H , L ′ , and M bands, as a function of increase over current sensitivity. We have taken current sensitivities to be H = 23 . L ′ = 16 .
5, and M = 13 .
5. Given only a 1 magnitude increase in M band sensitivity,paralleled by an equal increase at H band, the M band would be the best wavelength for planetsearch observations around all stars nearer than 5 pc. While the sensitivity increases requiredto render M preferable in Figure 5 require substantial improvements to existing instruments andtelescopes, the 1 mag increase required at 5 pc could be obtained by simply increasing the exposuretime. As with Figure 5, this result concerning the relative effectiveness of different wavelengths isindependent of stellar age, to first order. 26 –We have deliberately chosen the characteristics of the hypothetical stars in Figures 5 and 6to be less good than the best available planet search candidates, so that in each case stars closerand/or younger than the example actually exist. Using the very youngest stars would also haveresulted in sensitivities better than 1 M Jup , a mass regime not covered by the Burrows et al. (2003)models used in the Figures.Figures 5 and 6 illustrate three very important points. First, the L ′ band appears to have onlysecondary usefulness since either the H band or the M band always offers sensitivity to lower-massplanets. Second, Figure 6 shows that with a relatively minor increase of 1 magnitude in sensitivity,the M band will be sensitive to lower-mass planets around all stars within 5 pc than can be detectedwith H band observations, even if the H band sensitivity increases the same amount. Third, Figure5 shows that the advantage of the M band decreases with increasing distance, but that as largertelescopes and longer exposures increase sensitivities to 2.5 mag above present levels, the M bandwill be superior to H out to 10 pc. With an increase of 4 mag, the M band would surpass H out to 25 pc – but as such a large sensitivity increase would be difficult to achieve, H band willlikely remain the primary wavelength for stars at 25 pc and beyond. For stars closer than 10 pc,however, the M band already offers excellent sensitivity that has barely been exploited so far.Given reasonable sensitivity increases, M should become the primary band for planet searchesaround stars at a distance of 10 pc or less.Interestingly, the conclusions of Figures 5 and 6 are essentially independent of age: extensivecalculations by Heinze (2007) showed that the relative usefulness of different wavelengths had onlya weak dependence on age, for stars at a fixed distance – and even this weak age dependence couldchange sign on switching from the models of Burrows et al. (2003) to those of Baraffe et al. (2003).This means that if we change the ages of the stars in Figures 5 and 6 but leave the distances thesame, the L ′ , M , and H band curves will slide up or down but remain essentially fixed in theirrelative positions. For example, given a 3 magnitude increase in sensitivity at both wavelengths, M band observations will detect lower mass planets than H -band ones around a star at 10 pc, whetherthe stellar age is 5 Gyr, 1 Gyr, or 100 Myr. This is to be expected, since if one dials down theage of a given hypothetical star system, the T eff (and therefore IR color) of the faintest detectableplanets will remain about the same, though their masses will decrease.Again, Figures 5 and 6 apply only to background-limited sensitivity. However, given the muchmore favorable planet/star flux ratios in the M band relative to H , we would expect the longerwavelength observations to remain equally competitive closer to the star. Advances in M bandcoronography will likely parallel the development of H band extreme AO systems such as GPI andSPHERE. Though at present they are surpassed in sensitivity by H -regime observations for all butthe nearest stars, the L ′ and especially the M bands hold considerable promise for the future. 27 –
5. Conclusion
We have surveyed unusually nearby, mature star systems for extrasolar planets in the L ′ and M bands using the Clio camera with the MMT AO system. By extensive use of blind sensitivitytests involving fake planets inserted into our raw data (reported in detail in Heinze et al. (2010)),we established a definitive significance vs. completeness relation for planets in our data, which wethen used in Monte Carlo simulations to constrain planet distributions.We set interesting limits on the masses of planets and brown dwarfs in the star systems wesurveyed, but we did not detect any planets. Based on this null result, we place constraints onthe power laws that may describe the distribution of extrasolar planets in mass and semimajoraxis. We also place constraints on planet abundances independent of the distributions. If thedistribution of planets is a power law with dN ∝ M α a β dM da , the work of Cumming et al. (2008)and Butler et al. (2006) indicates that the most optimistic (i.e. planet-rich) case permitted by thestatistics of known RV planets correponds to about α = − . β = − .
46. Normalizing thedistribution to be consistent with RV statistics, we find that these values of α and β are ruled outat the 90% confidence level, unless the semimajor axis distribution is truncated at a radius R trunc less than 110 AU. Though β = 0 . α = − .
31, corresponding to the best-fit value from Cumming et al. (2008), werule out β = 0 . R trunc is less than 38 AU. Independent of distribution models, with 90%confidence no more than 50% of stars like those in our survey have a 5 M Jup or more massive planetorbiting between 30 and 94 AU, no more than 15% have a 10 M
Jup planet orbiting between 22 and100 AU, and no more than 25% have a 20 M
Jup object orbiting between 8 and 100 AU.Our constraints on planet abundances are similar to those placed by Kasper et al. (2007) andBiller et al. (2007), but less tight than those of Nielsen et al. (2008) and especially Lafreni`ere et al.(2007), The recent work of Nielsen & Close (2009) and Chauvin et al. (2010) also placed tighterconstraints on exoplanet distributions than our survey. However, we have surveyed a more nearby,older set of stars than any previous survey, and have therefore placed constraints on a more maturepopulation of planets. Also, we have confirmed that a paucity of giant planets at large separationsfrom sun-like stars is robustly observed at a wide range of wavelengths.The best current H regime observations, those of Lafreni`ere et al. (2007), would attain sensi-tivity to lower mass planets than did our L ′ and M band observations for all of our survey targetsexcept those lying within 4 pc of the Sun. However, as larger telescopes are built and longer expo-sures are attempted, the sensitivity of M band observations may be expected to increase at least asfast as that of H band observations (in part because M band detectors are currently a less maturetechnology). As shown in Figures 5 and 6, a modest increase from current sensitivity levels, evenif paralleled by an equal increase in H band sensitivity, would render the M band the wavelengthof choice for extrasolar planet searches around a large number of nearby stars. 28 –
6. Acknowledgements
This research has made use of the SIMBAD online database, operated at CDS, Strasbourg,France, and the VizieR online database (see Ochsenbein et al (2000)).We have also made extensive use of information and code from Press et al. (1992).We have used digitized images from the Palomar Sky Survey (available from http://stdatu.stsci.edu/cgi-bin/dss_form ),which were produced at the Space Telescope Science Institute under U.S. Government grant NAGW-2166. The images of these surveys are based on photographic data obtained using the OschinSchmidt Telescope on Palomar Mountain and the UK Schmidt Telescope.Facilities: MMT, SO: Kuiper
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