CONSTRAINTS ON LONG-PERIOD PLANETS FROM AN L'- AND M-BAND SURVEY OF NEARBY SUN-LIKE STARS: MODELING RESULTS^*

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 than 10 years (Cumming et al. 2008; Butler et al. 2006; Fischer & Valenti 2005) due to the limited temporal baseline of the observations, and the need to observe for a complete orbital period to confirm the properties of a planet with confidence. The masses of discovered planets range from just a few Earth masses (Bouchy et al. 2009) up to around 20 Jupiter masses (M_Jup). We note that a 20 M_Jup object would be considered by many to be a brown dwarf rather than a planet, but that there is no broad consensus on how to define the upper mass limit for planets. For a good overview of 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 planet populations. Several groups have fit approximate power-law distributions in mass and semimajor axis 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 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-period extrasolar planets. Nor can we see how our own solar system fits into the big picture of planet formation in the galaxy without a good census of planets in Jupiter- and Saturn-like orbits around other 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 planets with periods longer than 10 years, but even as RV temporal baselines increase, long-period planets will remain harder to detect due to their slow orbital velocities. The amplitude of a star's astrometric wobble increases with the radius of its planet's orbit, but decades-long observing programs are still needed to find long-period planets. Direct imaging is the only method that allows us to characterize 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 technology is adaptive optics (AO), which allows 6–10 m ground-based telescopes to obtain diffraction-limited IR images several times sharper than those from the Hubble Space Telescope (HST), despite Earth's turbulent atmosphere. Theoretical models of giant planets indicate that such telescopes should be capable of detecting self-luminous giant planets in large orbits around young, nearby stars. The stars should be young because the glow of giant planets comes from gravitational potential energy converted to heat in their formation and subsequent contraction: lacking any internal fusion, they cool and become fainter as they age.

Several groups have published the results of AO imaging surveys for extrasolar planets around F, 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; Lafrenière et al. 2007; and Chauvin et al. 2010). Of these, most have used wavelengths in the 1.5–2.2 μm range, corresponding to the astronomical H and K_S filters (Masciadri et al. 2005; Biller et al. 2007; Lafrenière et al. 2007; Chauvin et al. 2010). They have targeted mainly very young stars. Because young stars are rare, the median distance to stars in each 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, with proximity prioritized more than youth in the sample selection. The median distance to our survey targets 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 in older systems, but which would be undetectable around all but the nearest stars. More information 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, including extensive tests in which fake planets were randomly placed in the raw data and then recovered by an experimenter who knew neither their positions nor their number. Such tests are essential for establishing the true relationship between source significance (i.e., 5σ, 10σ, etc.) and survey completeness.

Our survey places constraints on a more mature population of planets than those that have focused on very young stars, and confirms that a paucity of giant planets at large separations from Sun-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 the normalization of the power laws. In Section 3, we present the constraints our survey places on the distribution of extrasolar giant planets, based on extensive Monte Carlo simulations. In Section 4, we discuss the promising future of planet-search observations in the L' and especially the M band, and in Section 5 we conclude.

Nearly 400 RV planets are known; see Butler et al. (2006) for a useful, conservative listing of confirmed extrasolar planets as of 2006, or http://exoplanet.eu/catalog-RV.php for a frequently updated catalog of all confirmed and many suspected extrasolar planet discoveries.

The number of RV planets is sufficient for meaningful statistical analysis of how extrasolar planets are distributed in terms of their masses and orbital semimajor axes. The lowest-mass planets and those with the longest orbital periods are generally rejected from such analyses to reduce bias from completeness effects, but there remains a considerable range (2–2000 days in period 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 the shortest-period planets, or "hot Jupiters," represent a separate population, a "pileup" of planets in very close-in orbits that does not follow the same statistical distribution as planets in more distant orbits (Cumming et al. 2008). The hot Jupiters are therefore often excluded from statistical fits to the overall populations of extrasolar planets, or at least from the fits to the semimajor axis distribution.

Cumming et al. (2008) characterize the distribution of RV planets detected in the Keck Planet Search with an equation of the form

$$\begin{equation} dN =C_0 M^{\alpha _L} P^{\beta _L} d\ln (M) d\ln (P), \end{equation} \tag{ 1 }$$

where M is the mass of the planet, P is the orbital period, and C₀ is a normalization constant. They state that 10.5% of solar-type stars have a planet with mass between 0.3 and 10 M_Jup and period between 2 and 2000 days, which can be used to derive a value for C₀ given values for the power-law exponents α_L and β_L. They find that the best-fit values for these are α_L = −0.31 ±  0.2 and β_L = 0.26 ±  0.1, where the _L subscript is our notation to make clear that these 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 ordinary differentials, and to give it in terms of orbital semimajor axis a rather than orbital period P:

$$\begin{equation} dN =C_0 M^{\alpha } a^{\beta } dM da, \end{equation} \tag{ 2 }$$

where C₀, of course, will not generally have the same value for Equations (1) and (2). Manipulating the two equations and using Kepler's third law make it clear that

$$\begin{equation} \alpha = \alpha _L - 1 \end{equation} \tag{ 3 }$$

and

$$\begin{equation} \beta = \frac{3}{2}\beta _L - 1. \end{equation} \tag{ 4 }$$

The Cumming et al. (2008) exponents produce α = −1.31 ± 0.2 and β = −0.61 ± 0.15 when translated into our form. The mass power law is well behaved, but the integral of the semimajor axis power law does not converge as a → ∞, so an outer truncation radius is an important parameter of the semimajor axis distribution.

Butler et al. (2006) present the 2006 Catalog of Nearby Exoplanets, a carefully described heterogeneous sample of planets detected by several different RV search programs. With appropriate caution, Butler et al. (2006) refrain from quoting confident power-law slopes based on the combined discoveries of many different surveys with different detection limits and completeness biases (in contrast, 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 underlying distribution. 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 additional input, we conclude that the true value of the mass power-law slope α is probably between −1.1 and −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 radius of the semimajor axis distribution cannot be constrained by the RV results: surveys like ours 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 to find C₀. We elect not to use the Cumming et al. (2008) value (10.5% of stars having a planet with mass between 0.3 and 10 M_Jup and period between 2 and 2000 days), because this range includes the 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 four years have been detected around these stars. Forty-seven of these stars are marked in their Table 1 as having RV planets. 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 their Table 1. The stars left out are HD 18445 and HD 225261. We cannot find any record of these stars having planets, and therefore as far as we can tell they are typos.

Since all planets with masses above 1 M_Jup and periods less than four years orbiting stars in the Fischer & Valenti (2005) list of 850 may be relied upon to have been discovered, we may pick any sub-intervals in this range of mass and period and divide the number of planets falling into these intervals by 850 to obtain our normalization. We selected the range 1–13 M_Jup in mass and 0.3–2.5 AU 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 thus the 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 stars having 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 slight difference 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 very different constraints if we had used the Cumming et al. (2008) normalization in our Monte Carlo simulations.

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 and masses greater than 1 M_Jup. This is close to our value of 3.29% for a similar range. Lafrenière et al. (2007) and Nielsen et al. (2008) fixed α and β in their simulations and let the normalization be a 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 distribution of known RV planets:

$$\begin{equation} P(\epsilon) = \epsilon e^{-\epsilon ^2/(2\sigma ^2)}. \end{equation} \tag{ 5 }$$

where P() is the probability of a given extrasolar planet having orbital eccentricity , e is the root of the natural logarithm, and σ = 0.3. We find that this mathematical form provides an excellent 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 the Monte Carlo simulations we describe in Section 3.

3.1. Theoretical Spectra

3.2. Introducing the Monte Carlo Simulations

3.3. A Detailed Look at a Monte Carlo Simulation

To evaluate the significance of our survey and provide some guidance for future work, we have analyzed in detail a single Monte Carlo simulation. We chose the Cumming et al. (2008) best-fit values of α = −1.31 and β = −0.61, 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, we normalized the planet distributions so that each star had a 3.29% probability of having a planet with semimajor axis between 0.3 and 2.5 AU and mass between 1 and 13 M_Jup. The simulation consisted of 50,000 realizations of our survey with these parameters. In all, 505,884 planets were simulated, of which 51,879 were detected.

In 38% of the 50,000 realizations, our survey found zero planets, while 37% of the time it found one and 25% of the time it found two or more. The planet distribution we considered in this simulation cannot be ruled out by our survey, since a null result such as we actually obtained turns out not to be very improbable.

The large number of survey realizations in our simulation allows the calculation of precise statistics for potentially detectable planets. The median mass of detected planets in our simulation was 11.36 M_Jup, the median semimajor axis was 43.5 AU, the median angular separation was 2.86 arcsec, and the median significance was 21.4σ. This last number is interesting because it suggests that, for our survey, any real planet detected was likely to appear at high significance, obvious even on a preliminary, "quick-look" reduction of the data. This suggests that performing such reductions at the telescope should be a high priority, to allow immediate confirmation and follow up if a candidate is seen. Figure 1 presents as a histogram the significance of all planets detected in this Monte Carlo simulation.

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We suspected that there would be a detection bias toward very eccentric planets, because these would spend most of their orbits near apastron, where they would be easier to detect. This bias did not appear at any measurable level in our simulation. However, there was a weak but clear bias toward planets in low-inclination orbits, which, of course, spend more of their time at large separations 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 the number would imply if the best two or three stars had most of the probability of hosting detectable planets. Table 3 gives the percentage of planets detected around each star in our sample based on our detailed Monte Carlo simulation. Due to poor data quality, binary orbit constraints, or other issues, a few stars had zero probability of detected planets given the distribution used here. In general, however, the likelihood of hosting detectable planets is fairly well distributed.

Table 3. Percentage of Detected Planets Found Around Each Star

Star Name	Percentage of Total	Median	Median	Median
	Detected Planets	Mass (MJup)	Semimajor Axis (AU)	Separation (arcsec)
GJ 117	6.07	7.66	39.36	3.64
Eri	5.83	6.98	18.26	4.35
HD 29391	5.80	8.14	49.13	2.71
GJ 519	4.74	10.44	40.51	3.28
GJ 625	4.67	9.72	29.18	3.48
GJ 5	4.45	9.60	53.42	3.08
BD+60 1417	3.95	11.58	44.48	2.05
GJ 355	3.81	9.71	53.91	2.34
GJ 354.1 A	3.67	9.58	60.12	2.64
GJ 159	3.57	9.73	57.95	2.71
GJ 349	3.35	11.38	44.40	3.17
61 Cyg B	3.29	11.32	19.53	4.08
GJ 879	3.03	11.18	36.84	3.69
GJ 564	2.94	10.67	56.80	2.70
GJ 410	2.93	12.78	41.83	3.03
GJ 450	2.89	12.90	38.72	3.66
GJ 3860	2.68	12.70	49.72	2.69
HD 78141	2.58	12.47	57.00	2.24
BD+20 1790	2.51	12.14	58.33	2.02
GJ 278 C	2.20	12.68	54.56	3.04
GJ 311	2.19	12.55	52.07	3.20
HD 113449	2.17	12.52	59.31	2.29
GJ 211	2.10	13.59	50.51	3.30
BD+48 3686	2.08	12.56	55.05	2.01
GJ 282 A	2.05	13.39	49.85	2.99
GJ 216 A	2.03	12.71	42.98	4.21
61 Cyg A	1.97	13.70	20.94	4.54
HD 1405	1.54	13.13	66.34	2.04
HD 220140 A	1.54	11.73	36.85	1.73
HD 96064 A	1.49	12.63	46.64	1.75
HD 139813	1.43	14.33	59.71	2.37
GJ 380	0.92	15.76	25.31	4.21
GJ 896 A	0.61	12.43	6.47	0.98
GJ 860 A	0.38	11.58	53.26	6.62
τ Ceti	0.38	17.19	25.49	5.52
GJ 896 B	0.34	11.40	6.78	1.14
ξ Boo B	0.32	12.07	8.25	1.36
HD 220140 B	0.28	12.04	25.92	1.37
ξ Boo A	0.24	12.89	8.72	1.50
GJ 659 B	0.21	17.71	62.54	2.81
GJ 166 B	0.17	16.12	6.19	1.34
GJ 684 A	0.17	14.93	85.98	4.87
HD 96064 B	0.13	14.43	38.55	1.60
GJ 505 B	0.12	15.94	17.11	1.61
GJ 166 C	0.10	15.56	6.43	1.52
GJ 505 A	0.07	16.32	18.08	1.75
GJ 702 A	0.02	15.90	6.21	1.50
GJ 684 B	None	NA	NA	NA
GJ 860 B	None	NA	NA	NA
GJ 702 B	None	NA	NA	NA
HD 77407 A	None	NA	NA	NA
GJ 659 A	None	NA	NA	NA
GJ 3876	None	NA	NA	NA
HD 77407 B	None	NA	NA	NA

Notes. This table applies to our detailed Monte Carlo simulation with 50,000 survey realizations run using α = −1.31, β = −0.61, and semimajor axis truncation radius 100 AU. Of all the simulated planets that were detected, we present here the percentage that were found around each given star, and the median mass, semimajor axis, and projected separation for simulated planets found around each star. The table thus indicates around which stars our survey had the highest likelihood of detecting a planet. Many stars with poor likelihood are binaries, with few stable planetary orbits possible.

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In Table 4, we give the details of planetary orbital constraints used in our Monte Carlo simulations 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 limited in two ways: the apastron could not be too far out or the orbit would be rendered unstable by proximity to HD 96064 A—but the periastron also could not be too far in, or the binary orbit of HD 96064 Ba and HD 96064 Bb would render it unstable. Planets individually orbiting HD 96064 Ba or HD 96064 Bb were not considered in our survey, since to be stable the planets would have to be far too close in for us to detect them. The constraints described in Table 4 account for most of the stars in Table 3 with few or no detections reported.

Table 4. Constraints on Simulated Planet Orbits Around Binary Stars

Star Name	Separation	Constraints on	Constraints on	Constraints on
		Circumprimary	Circumsecondary	Circumbinary
	(arcsec)	Apastron (arcsec, AU)	Apastron (arcsec, AU)	Periastron (arcsec, AU)
HD 220140 AB	10.828	<3.61, 71.3	<2.17, 42.8	No stable orbits
HD 96064 AB	11.628	<3.88, 95.6	<2.33, 57.3	No stable orbits
HD 96064 Bab	0.217	No stable orbits	No stable orbits	>0.65, 16.1
GJ 896 AB	5.366	<1.79, 11.8	<1.79, 11.8	No stable orbits
GJ 860 AB	2.386	<0.79, 3.17	<0.60, 2.41	>7.15, 28.7
ξ Boo AB	6.345	<2.12, 14.2	<2.12, 14.2	No stable orbits
GJ 166 BC	8.781	<2.20, 10.6	<2.20, 10.6	No stable orbits
GJ 684 AB	1.344	<0.45, 6.34	<0.27, 3.80	>4.03, 56.8
GJ 505 AB	7.512	<2.50, 29.8	<2.50, 29.8	No stable orbits
GJ 702 A	5.160	<1.76, 8.85	<1.32, 6.64	>15.9, 79.7
HD 77407 AB	1.698	<0.57, 17.2	<0.34, 10.23	>5.11, 153.7

Notes. Planets orbiting the primary in a binary star were considered to be de-stabilized by the gravity of the secondary if their apastron distance from the primary was too large. Similarly, planets orbiting the secondary had to have small enough apastron distances to avoid being de-stabilized by the primary. Circumbinary planets had to have a large enough periastron distance to avoid be de-stabilized by the differing gravitation of the two components of the binary. Note that HD 96064B is itself a tight binary star, so planets orbiting it had both a minimum periastron and a maximum apastron.

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A final question our detailed simulation can address is how important the M-band observations were to the survey results. In Table 5, we show that when M-band observations were made, they did substantially increase the number of simulated planets detected.

Table 5. Importance of the M-band Data

Star Name	Total Simulated	Two-band	L'-only	M-only
	Detections	Detections (%)	Detections (%)	Detections (%)
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

Notes. The usefulness of M-band observations based on our detailed Monte Carlo simulation. When M-band observations were made on a given star, they did substantially increase the number of simulated planets detected around that star.

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3.4. Monte Carlo Simulations: Constraining the Power Laws

The planet distribution we used in the single Monte Carlo simulation described above could not be ruled out by our survey. To find out what distributions could be ruled out, we performed Monte Carlo simulations assuming a large number of different possible distributions, parameterized by 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 statistics of Fischer & Valenti (2005): any given star had 3.29% probability of hosting a planet with mass between 1 and 13 M_Jup and semimajor axis between 0.3 and 2.5 AU. The mass range for simulated planets was 1–20 M_Jup.

We tested three different values of α: −1.1, −1.31, and −1.51, roughly corresponding to the most optimistic 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. In contrast to the extensive results described in Section 3.3, the only data saved for these simulations were the probability of finding zero planets. Since we did in fact obtain a null result, distributions for 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 our three different values of α. Figure 2 presents constraints based on α = −1.31, the best-fit value from RV statistics, while Figure 3 compares the optimistic case α = −1.1 and the pessimistic case α = −1.51. Each pixel in these figures represents a Monte Carlo simulation involving 15,000 realizations of our survey; generating the figures took several tens of hours on a fast PC. Contours are overlaid at selected probability levels. Regions within the 1%, 5%, and 10% contours can, of course, be ruled out at the 99%, 95%, and 90% confidence levels, respectively. For example, we find that the most optimistic power laws allowed by the Cumming et al. (2008) RV statistics, α = −1.1 and β = −0.46, are ruled out with 90% confidence if R_trunc is 110 AU or greater. Similarly, α = −1.51 and β = −0.3, truncated at 100 AU, are ruled out. Though β = 0.0 is not physically plausible, previous work has sometimes used it as an example: for α = −1.31, we rule out β = 0.0 unless R_trunc is less than 38 AU.

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3.5. Model-independent Constraints

It is also possible to place constraints on the distribution of planets without assuming a power law or any other particular model for the statistics of planetary masses and orbits. Note well that by "model independent" in this context, we mean independent only of models for the statistical distributions of planets in terms of M and a—not independent of models of planetary spectra such as those we obtain from Burrows et al. (2003). The latter are our only means of converting from planetary 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 simulations on a grid of planet mass and orbital semimajor axis. For each grid point, we seek to 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 and semimajor axis a is no more than P(M, a). We determine P(M, a) by a search: first a guess is made, and a Monte Carlo simulation assuming this probability is performed. If more than 10% of the realizations of our survey turn up a null result, the guessed probability is too low; if less than 10% turn up a null result, the probability is too high. It is adjusted in steps of ever-decreasing size until the correct value is reached.

Figure 4 shows the 90% confidence upper limit on P(M, a) as a function of mass M and semimajor 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) is less than 8%, 10%, 25%, 50%, and 75%, with 90% confidence. Note that P(M, a), the value constrained by our simulations, is a probability rather than a fixed fraction. The probability is the more scientifically interesting number, but is harder to constrain. For example, if 3.7% is the fraction of the actual stars in our sample that have planets with easy-to-detect properties, there are two such planets represented in our 54 star survey. However, if the probability of a star like those in our sample having such a planet is 3.7%, there is still a nonzero probability (13% in this case) that no star in our sample actually has such a planet.

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The results presented in Figure 4 can be interpreted as model-independent constraints on planet populations. For example, with 90% confidence we find that less than 50% of stars with properties like those in our survey have a 5 M_Jup or more massive planet in an orbit with a semimajor axis between 30 and 94 AU. Less than 25% of stars like those in our survey have a 7 M_Jup or more massive planet between 25 and 100 AU, less than 15% have a 10 M_Jup or more massive planet between 22 and 100 AU, and less than 12% have a 15 M_Jup or more massive planet/brown dwarf between 15 and 100 AU. Going to the most massive objects considered in our simulations, we can set limits ranging inward past 10 AU: we find that less than 25% of stars like those surveyed have a 20 M_Jup object orbiting between 8 and 100 AU. These constraints hold independently of how planets are distributed in terms of their masses and semimajor axes.

HR 8799 appears to have a remarkable system of three massive planets, seen at projected distances 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 planetary system. For purposes of this simulation, we adopted the masses above, and set the planets' orbital radii equal to their projected separations. Our 8.1% limit represents a step toward determining whether or not systems of massive planets in wide orbits are more common around more massive stars such as HR 8799 than F, G, K stars such as those we have surveyed.

3.6. Our Survey in the Big Picture

In the L' and M bands, the sky brightness is much worse than at shorter wavelengths. However, models (e.g., Burrows et al. 2003) predict that in the L' and M bands, planets fade less severely with increasing age (or, equivalently, decreasing T_eff). Also, planet/star flux ratios are more 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 a more limiting factor than the sky brightness. In Heinze et al. (2008), we have shown that M-band observations tend to do better than those at shorter wavelengths at small separations from bright stars.

The L' and M bands are most useful, however, for detecting the lowest-temperature planets, which have the reddest H − L' and H − M colors. Such very low temperature planets can only be detected around the nearest stars, so it is for very nearby stars that L'- and M-band observations are most useful. For distant stars, around which only relatively 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 in the same wavelength regime. While the K_S band has been used extensively to search for planets around young stars (Masciadri et al. 2005; Chauvin et al. 2010), our comparison here will focus on the H-band regime. Models indicate that it offers better sensitivity than the K_S band except for planets younger than 100 Myr (Burrows et al. 2003; Baraffe et al. 2003), and most of the stars we will 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 Lafrenière et al. (2007), in part because of their optimized narrowband filter. They attained an effective background-limited point-source sensitivity of about H = 23.0. Based on the models of Burrows et al. (2003), Lafrenière et al. (2007) would have set better planetary mass limits than our observations around all of our own survey targets except the very nearest objects, such as 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 detection exposures are attempted, the sensitivity at all wavelengths will increase. This means that low-temperature planets, with their red IR colors, will be detectable at larger distances, and the utility of the L' and especially the M bands will increase. In Figure 5, we show the minimum detectable planet mass for hypothetical stars at 10 and 25 pc distance as a function of the increase over current sensitivity in the H, L', and M bands, and in Figure 6 we present the same comparison for a star at 5 pc. We have taken current sensitivity to be H = 23.0 (i.e., Lafrenière et al. 2007), L' = 16.5, and M = 13.5 (i.e., the present work, scaled to an 8 m telescope such as Lafrenière et al. 2007 used). These are background limits, not applicable close to bright stars. Based on Heinze et al. (2008), we believe the 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 observations with next-generation extreme AO systems such as the Gemini Planet Imager (GPI) and SPHERE will offer improved performance 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 compare background-limited performance only.

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The suppression 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 and 6. Based on Burrows et al. (2003), the entire mass range covered by both figures corresponds to planets with T_eff below 500 K, except for planets with masses above 6.5 M_Jup in the left panel of Figure 5 (25 pc distance, 300 Myr age). This upper section of the 25 pc, 300 Myr panel is irrelevant to the important implications of the figure. According to Hubeny & Burrows (2007), there is no suppression of the M band for effective temperatures below 500 K.

We have deliberately chosen the characteristics of the hypothetical stars in Figures 5 and 6 to be less good than the best available planet-search candidates, so that in each case stars closer and/or younger than the example actually exist. Using the very youngest stars would also have resulted 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 only secondary usefulness since either the H band or the M band always offers sensitivity to lower-mass planets. Second, Figure 6 shows that with a relatively minor increase of 1 mag in sensitivity, the M band will be sensitive to lower-mass planets around all stars within 5 pc than can be detected with H-band observations, even if the H-band sensitivity increases the same amount. Third, Figure 5 shows that the advantage of the M band decreases with increasing distance, but that as larger telescopes and longer exposures increase sensitivities to 2.5 mag above present levels, the M band will 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, the H band will likely 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 searches around stars at a distance of 10 pc or less.

Interestingly, the conclusions of Figures 5 and 6 are essentially independent of age: extensive calculations by Heinze (2007) showed that the relative usefulness of different wavelengths had only a weak dependence on age, for stars at a fixed distance—and even this weak age dependence could change 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 the same, the L'-, M-, and H-band curves will slide up or down but remain essentially fixed in their relative positions. For example, given a 3 mag increase in sensitivity at both wavelengths, M-band observations will detect lower-mass planets than H-band ones around a star at 10 pc, whether the stellar age is 5 Gyr, 1 Gyr, or 100 Myr. This is to be expected, since if one dials down the age of a given hypothetical star system, the T_eff (and therefore IR color) of the faintest detectable planets will remain about the same, though their masses will decrease.

Again, Figures 5 and 6 apply only to background-limited sensitivity. However, given the much more favorable planet/star flux ratios in the M band relative to H, we would expect the longer-wavelength observations to remain equally competitive closer to the star. Advances in M-band coronography will likely parallel the development of H-band extreme AO systems such as GPI and SPHERE. Though at present they are surpassed in sensitivity by H-regime observations for all but the nearest stars, the L' and especially the M bands hold considerable promise for the future.

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 sensitivity tests involving fake planets inserted into our raw data (reported in detail in Heinze et al. 2010), we established a definitive significance versus completeness relation for planets in our data, which we then 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 we surveyed, but we did not detect any planets. Based on this null result, we place constraints on the power laws that may describe the distribution of extrasolar planets in mass and semimajor axis. We also place constraints on planet abundances independent of the distributions. If the distribution of planets is a power law with dN ∝ M^αa^βdMda, the work of Cumming et al. (2008) and Butler et al. (2006) indicates that the most optimistic (i.e., planet-rich) case permitted by the statistics of known RV planets corresponds to about α = −1.1 and β = −0.46. Normalizing the distribution to be consistent with RV statistics, we find that these values of α and β are ruled out at the 90% confidence level, unless the semimajor axis distribution is truncated at a radius R_trunc less than 110 AU. Though β = 0.0 is not physically plausible, previous work has sometimes used it as an example: for α = −1.31, corresponding to the best-fit value from Cumming et al. (2008), we rule out β = 0.0 unless 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 planet orbiting between 30 and 94 AU, no more than 15% have a 10 M_Jup planet orbiting between 22 and 100 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) and Biller et al. (2007), but less tight than those of Nielsen et al. (2008) and especially Lafrenière et al. (2007). The recent work of Nielsen & Close (2009) and Chauvin et al. (2010) also placed tighter constraints 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 mature population of planets. Also, we have confirmed that a paucity of giant planets at large separations from Sun-like stars is robustly observed at a wide range of wavelengths.

The best current H-regime observations, those of Lafrenière et al. (2007), would attain sensitivity to lower-mass planets than did our L'- and M-band observations for all of our survey targets except those lying within 4 pc of the Sun. However, as larger telescopes are built and longer exposures are attempted, the sensitivity of M-band observations may be expected to increase at least as fast as that of H-band observations (in part because M-band detectors are currently a less mature technology). As shown in Figures 5 and 6, a modest increase from current sensitivity levels, even if paralleled by an equal increase in H-band sensitivity, would render the M band the wavelength of choice for extrasolar planet searches around a large number of nearby stars.

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 NAG W-2166. The images of these surveys are based on photographic data obtained using the Oschin Schmidt Telescope on Palomar Mountain and the UK Schmidt Telescope.

Facilities: MMT - MMT at Fred Lawrence Whipple Observatory, SO:Kuiper - Steward Observatory's 1.5 meter Kuiper Telescope