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How Dry is the Brown Dwarf Desert?: Quantifying the Relative Number of Planets, Brown Dwarfs and Stellar Companions around Nearby Sun-like Stars
Daniel Grether1 & Charles H. Lineweaver
1 1,2

arXiv:astro-ph/0412356 v1 15 Dec 2004

Department of Astrophysics, School of Physics, University of New South Wales, Sydney, NSW 2052, Australia

Planetary Science Institute, Research School of Astronomy and Astrophysics & Research School of Earth Sciences, Australian National University, Canberra, ACT, Australia ABSTRACT Sun-like stars have stellar, brown dwarf and planetary companions. To help constrain their formation and migration scenarios, we analyse the close companions (orbital p eriod < 5 years) of nearby Sun-like stars. By using the same sample to extract the relative numb ers of stellar, brown dwarf and planetary companions, we verify the existence of the brown dwarf desert and describ e it quantitatively. With decreasing mass, the companion mass function drops by more than an order of magnitude from 1 M stellar companions to the brown dwarf desert and then rises by approximately half an order of magnitude from brown dwarfs to Jupiter-mass planets. The slop es of the planetary and stellar companion mass functions are of opp osite sign and are incompatible at the 3 sigma level, thus yielding a brown dwarf desert. The minimum numb er of companions p er unit interval in log mass (the driest part of the desert) is at M = 17+14 MJ up . -6 Parab olic fits to the companion mass function suggest that the driest part of the desert scales with host mass and we predict a Jupiter-mass desert and a stellar companion desert for hosts of mass < 0.5 M and > 2 M resp ectively. However, we find no evidence that companion mass scales with host mass in general. Approximately 16% of Sun-like stars have close (P < 5 years) companions more massive than Jupiter: 11%±3% are stellar, 1% ± 1% are brown dwarf and 4% ± 2% are giant planets. The companion mass function in the brown dwarf and stellar mass range, has a different shap e than the initial mass function of individual stars and free-floating brown dwarfs. This suggests either a different sp ectrum of gravitational fragmentation in the formation environment or p ost-formation migratory processes disinclined to leave brown dwarfs in close orbits.

2

1.

Intro duction

The formation of a binary star via molecular cloud fragmentation and collapse, and the formation of a massive planet via accretion around a core in a protoplanetary disk b oth involve the production of a binary system, but are usually recognized as distinct processes (e.g. Heacox 1999;


­2­ Kroupa & Bouvier 2003, see however Boss 2002). The formation of companion brown dwarfs, with masses in b etween the stellar and planetary mass ranges, may have elements of b oth or some new mechanism (Bate 2000; Rice et al. 2003; Jiang, Laughlin & Lin 2004).
> Brown dwarfs are usually defined as b odies massive enough to burn deuterium (M 13 MJ up ), < but not massive enough to burn hydrogen (M 80 MJ up e.g. Burrows 1997). These mass b oundaries, based on the fusion of deuterium and hydrogen, are fundamental to our understanding of what the words "planet", "brown dwarf" and "star" mean. However, since fusion does not turn on in gravitationally collapsing fragments of a molecular cloud until the final masses of the fragments are largely in place, gravitational collapse, fragmentation and accretion should produce a sp ectrum of masses that does not know ab out these deuterium and hydrogen burning b oundaries. Thus, these mass b oundaries should not necessarily corresp ond to transitions in the mode of formation. The physics of gravitational collapse, fragmentation, accretion disk stability and the transfer of angular momentum, should b e resp onsible for the relative abundances of ob jects of different masses, not fusion onset limits.

However, there seems to b e a brown dwarf desert ­ a deficit in the frequency of brown dwarf companions either relative to the frequency of less massive planetary companions (Marcy & Butler 2000) or relative to the frequency of more massive stellar companions to Sun-like hosts (McCarthy & Zuckerman 2004). The goal of this work is (i) to verify that this desert is not a selection effect due to our inablility to detect brown dwarfs and (ii) to quantify the brown dwarf desert more carefully with resp ect to b oth stars and planets. By selecting a single sample of nearby stars as p otential hosts for all typ es of companions, we can b etter control selection effects and more accurately determine the relative numb er of companions more and less massive than brown dwarfs. Various models have b een suggested for the formation of companion stars, brown dwarfs and planets (e.g. Larson 2003, Kroupa & Bouvier 2003, Bate 2000, Matzner & Levin 2004, Boss 2002, Rice et al.2003). All models involve gravitational collapse and a mechanism for the transfer of energy and angular momentum away from the collapsing material. Observations of giant planets in close orbits have challenged the conventional view in which giant planets form b eyond the ice zone and stay there (e.g. Udry 2003). Various typ es of migration have b een prop osed to meet this challenge. The most imp ortant factors in determining the result of the migration is the time of formation and mass of the secondary and its relation to the mass and time evolution of the disk (e.g. Armitage & Bonnell 2002). We may b e able to constrain the ab ove models by quantitative analysis of the brown dwarf desert. For example, if two distinct processes are resp onsible for the formation of stellar and planetary secondaries, we would exp ect well-defined slop es of the mass function in these mass ranges to meet in a sharp brown dwarf valley. If these processes blend in a smooth way, then a parab olic fit to a brown dwarf `b owl' would b e a b etter fit to the data. We examine the mass, and p eriod distributions for companion brown dwarfs and compare them with those of companion stars and planets. The work most similar to our analysis has b een carried


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Fig. 1.-- Our Close Sample. Hertzsprung-Russell diagram for Hipparcos stars closer than 25 p c. Small black dots are Hipparcos stars not b eing monitored for p ossible companions by one of the 8 high precision Doppler surveys considered here (Lineweaver & Grether 2003). Larger blue dots are the subset of Hipparcos stars that are b eing monitored ("Target Stars") but have as yet no known planetary companions. The still larger red dots are the subset of target stars hosting detected planets ("Planet Host Stars") and the green dots are those hosts with larger mass (M2 > 80MJ up ) companions ("Other Host Stars"). Only companions in our less biased sample (P < 5 years and M2 > 10-3 M ) are shown (see Section 2.2). Our Sun is shown as the black cross. The grey parallelogram is the region of Mv - (B - V ) space that contains the highest fraction (as shown by the triangles) of Hipparcos stars that are b eing monitored for exoplanets. This Sun-like region ­ late F to early K typ e main sequence stars ­ contains our Hipparcos Sun-like Stars. The target fraction needs to b e as high as p ossible to minimize selection effects p otentially associated with companion frequency. The target fraction is calculated from the numb er of main sequence stars, i.e., the numb er of stars in each bin b etween the two dotted lines. This plot contains 1509 Hipparcos stars, of which 671 are Doppler target stars. The Sun-like region contains 464 Hipparcos stars, of which 412 are target stars. Thus, the target fraction in the Sun-like grey parallelogram is 90%(= 412/464).


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Fig. 2.-- Our Far Sample. Same as Fig. 1 but for all Hipparcos stars closer than 50 p c. The ma jor reason the target fraction ( 65%, triangles) is lower than in the 25 p c sample ( 90%) is that K stars b ecome too faint to include in many of the high precision Doppler surveys where the apparent magnitude is limited to V < 7.5 (Lineweaver & Grether 2003). This plot contains 6924 Hipparcos stars, of which 2527 are target stars. The grey parallelogram contains 3297 Hipparcos stars, of which 2140 are high precision Doppler target stars (65% 2140/3290). The stars b elow the main sequence and the stars to the right of the M dwarfs are largely due to uncertainties in the Hipparcos parallax or B - V determinations.


­5­ out by Heacox (1999); Zucker & Mazeh (2001) and Mazeh et al. (2003). Heacox (1999) and Zucker & Mazeh (2001) b oth combined the stellar sample of Duquennoy & Mayor (1991) along with the known substellar companions and identified different mass functions for the planetary mass regime b elow 10 MJ up but found similar flat distributions in logarithmic mass for brown dwarf and stellar companions. Heacox (1999) found that the logarithmic mass function in the planetary regime is b est fit by a p ower-law with a slightly negative slop e whereas Zucker & Mazeh (2001) found an approximately flat distribution. Mazeh et al. (2003) looked at a sample of main sequence stars using infrared sp ectroscopy and combined them with the known substellar companions and found that in log mass, the stellar companions reduce in numb er towards the brown dwarf mass range. They identify a flat distribution for planetary mass companions. We discuss the comparison of our results to these in Section 3.1.

2.

Defining a Less Biased Sample of Companions 2.1. Host Sample Selection Effects

High precision Doppler surveys are monitoring Sun-like stars for planetary companions and are necessarily sensitive enough to detect brown dwarfs and stellar companions within the same range of orbital p eriod. However, to compare the relative abundances of stellar, brown dwarf and planetary companions, we cannot select our p otential hosts from a non-overlapping union of the FGK sp ectral typ e target stars of the longest running, high precision Doppler surveys that are b eing monitored for planets (Lineweaver & Grether 2003) b ecause Doppler survey target selection criteria often exclude close binaries (separation < 2") from the target lists, and are not focused on detecting stellar companions. Some stars have also b een left off the target lists b ecause of high stellar chromospheric activity (Fischer et al. 1999). These surveys are biased against finding stellar mass companions. We correct for this bias by identifying the excluded targets and then including in our sample any stellar companions from other Doppler searches found in the literature. We also observe and minimally correct for an asymmetry in our sample b etween the numb er of stellar companions in the northern and southern hemispheres. Our sample selection is illustrated in Figs. 1 and 2. Most Doppler survey target stars come from the Hipparcos catalogue b ecause host stars need to b e b oth bright and have accurate masses for the Doppler method to work. One could imagine that the Hipparcos catalogue would b e biased in favor of binarity since hosts with bright close-orbiting stellar companions would b e over-represented. We have checked for this over-representation by looking at the absolute magnitude dep endence of the frequency of stellar binarity for systems closer than 25 and 50 p c (Fig. 3). We found no significant decrease in the fraction of binaries in the dimmer stellar systems for the 25 p c sample and only a small decrease in the 50 p c sample. Thus, the Hipparcos catalogue provides a good sample of p otential hosts for our analysis, since it (i) contains the Doppler target lists as subsets (ii) is volume-limited for Sun-like stars out to 25 p c (Reid 2002)


­6­ and (iii) it allows us to identify and correct for stars and stellar systems that were excluded. We limit our selection to Sun-like stars (0.5 B - V 1.0) or approximately those with a sp ectral typ e b etween F7 and K3. Following Udry (private communication) and the construction of the Coralie target list, we limit our anaylsis to main sequence stars, or those b etween -0.5 and +2.0 dex (b elow and ab ove) an average main sequence value as defined by 5.4(B - V ) + 2.0 Mv 5.4(B - V ) - 0.5. This sampled region, which we will call our "Sun-like" region of the HR diagram, is shown by the parallelogram in Figs. 1 & 2.

Fig. 3.-- Fraction of stars that are close (P < 5 years) Doppler binaries as a function of absolute magnitude. For the 25 p c Sun-like sample (large dots), 11% of stars are binaries and within the error bars, brighter stars do not app ear to b e over-represented. If we include the extra stars to make the 50 p c Sun-like sample (small dots), the stellar binary fraction is lower and decreases as the systems get fainter. The Hipparcos sample is essentially complete to an absolute visual magnitude of Mv = 8.5 (Reid 2002) within 25 p c of the Sun. Thus the stars in our 25 p c Sun-like sample represent a complete, volume-limited sample. In our sample we make corrections in companion frequency for stars that are not b eing targeted by Doppler surveys as well as corrections for mass and p eriod selection effects in companion detection (see Section 2.2). The result of these corrections is our less-biased distribution of companions to Sun-like stars within 25 p c. We also analyse a much larger sample of stars out to 50 p c to understand the effect of distance on target selection and companion detection. Although less complete, with resp ect to the relative numb er of companions of different masses, this 50 p c sample is similar to the 25 p c sample (Section 3). Stars in our Sun-like region are plotted as a function of distance in Fig. 4. Each histogram bin represents an equal volume spherical shell hence a sample complete in distance would produce


­7­

Fig. 4.-- Distance Dep endence of Sample and Companions. Here we show the numb er of nearby Sun-like stars as a function of distance. Each histogram bin represents the stars in an equal volume spherical shell. Hence, a sample that is complete in distance out to 50 p c would produce a flat histogram (indicated by the horizontal dashed line). The lightest shade of grey represents Hipparcos Sun-like Stars out to 50 p c that fall within the parallelogram of Fig. 2 ("HSS"). The next darker shade of grey represents the numb er of Hipparcos stars that are b eing monitored for planets using the high precision Doppler techniques (8 groups describ ed in (Lineweaver & Grether 2003)). The triangles represent this numb er as a fraction of Hipparcos stars. This fraction needs to b e as large as p ossible to minimize selection effects in the target sample p otentially associated with companion frequency. Also shown (darker grey) are the numb er of Hipparcos stars that have one or more companions in the mass range 10-3 < M /M < 1, and those that host planets (darkest grey). Only those companions in the less biased sample, P < 5 years and M2 > 10-3 M are shown (Section 2.2).


­8­ a flat histogram. This target excluded stars for which ther p ossible correlations b etween their companions that we are fraction needs to b e as large as p ossible to minimize the numb er of e are complicating effects of inhomogenous detection efficiencies and the reasons stellar systems have b een excluded and the prop erties of trying to quantify.

Since nearly all of the high precision Doppler surveys have apparent magnitude limited target lists (often V < 7.5), we investigate the effect this has on the total target fraction as a function of distance. The fraction of stars having an apparent magnitude V brighter (lower) than a given value are shown by the 5 dotted lines for V < 7.5 to V < 9.5. For a survey, magnitude limited to V = 7.5, 80% of the Sun-like Hipparcos stars will b e observable b etween 0 p c and 25 p c. This rapidly drops to only 20% for stars b etween 48 and 50 p c. Thus the ma jor reason why the target fraction drops with increasing distance is that the stars b ecome too faint for the high precision Doppler surveys to monitor. The fact that the target fraction (triangles) lie near the V < 8.5 line indicates that on average V 8.5 is the effective limiting magnitude of the targets monitored by the 8 combined high precision Doppler surveys. In Fig. 1, 52(= 464 - 412) or 11% of Hipparcos stellar systems are not present in any of the Doppler target lists. The triangles in Fig. 1 indicate that the ones left out are spread more or less evenly in B-V space spanned by the grey parallelogram. Similarly in Fig. 2, 1157(= 3297 - 2140) or 35% are not included in any Doppler target list, but the triangles show that more K stars compared to FG stars have not b een selected, again p ointing out that the lower K dwarf stellar brightness is the dominant reason for the lower target fraction, not an effect strongly biased with resp ect to one set of companions over another. In the Sun-like region of Fig. 1 we use the target numb er (412) as the mother p opulation for planets and brown dwarfs and the Hipparcos numb er (464) as the mother p opulation for stars. We then assume that the fraction of these 412 targets that have exoplanet or brown dwarf companions is representative of the fraction of the 464 Hipparcos stars that has exoplanet or brown dwarf companions. Thus we scale up the planetary and brown dwarf companions which have the target sample as their mother p opulation to the Hipparcos sample by 13% (464/412 = 1.13) to achieve the same normalizations for planetary, brown dwarf and stellar companions. Since close-orbitting stellar companions are anti-correlated with close-orbitting sub-stellar companions, the results from the sample of 412 may b e a slight over-estimate of the frequency of sub-stellar companions. However, this over-estimate will b e less than 11% b ecause this is the frequency of close-orbitting stellar secondaries. A non-overlapping sample of the 8 high precision Doppler surveys (Lineweaver & Grether 2003) is used as the exoplanet target list where the Elodie target list was kindly provided by C. Perrier (private communication) and additional information to construct the Coralie target list from the Hipparcos catalogue was obtained from S. Udry (private communication). The Keck and Lick target lists are up dated to those of Wright et al. (2004) from those of Nidever et al. (2002).


­9­

Fig. 5.-- Brown Dwarf Desert in Mass and Period. Estimated companion mass M2 versus orbital p eriod for the companions to Sun-like stars of our two samples: companions with hosts closer than 25 p c (large symb ols) and those with hosts closer than 50 p c, excluding those closer than 25 p c (small symb ols). The companions in the thick solid rectangle are defined by p eriods P < 5 years, < and masses 10-3 < M2 M , and form our less biased sample of companions. The stellar (op en circles), brown dwarf (grey circles) and planetary (filled circles) companions are separated by dashed lines at the hydrogen and deuterium burning onset masses of 80 MJ up and 13 MJ up resp ectively. This plot clearly shows the brown dwarf desert for the P < 5 year companions. Planets are more frequent at larger p eriods (see Fig. 6). The "Detected", "Being Detected" and "Not Detected" regions of the mass-p eriod space show where the high precision Doppler method is currently able to find companions (Lineweaver & Grether 2003). See App endix for discussion of M2 mass estimates.


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Fig. 6.-- Pro jection of Fig. 5 onto the p eriod axis for the 25 p c (dark grey) and 50 p c (light grey) samples. Planets are more clump ed towards higher p eriods. This would b e a selection effect with no significance if the efficiency of finding short p eriod stellar companions with the low precision Doppler technique used to find sp ectroscopic binaries, was much higher than the efficiency of finding exoplanets with high precision sp ectroscopy. Konacki et al. (2004) and Pont et al. (2004) conclude that the fact that the transit photometry method has found planets in sub 2.5 day p eriods (while the Doppler method has found none) is due to higher efficiency for small p eriods and many more target stars and thus that these two observations do not conflict. Thus there seems to b e a real difference in the p eriod distributions of stellar and planetary companions. Although we are dealing with small numb er statistics, the p eriod distribution of brown dwarfs is consistent with b eing midway b etween the p eriod distributions of stellar and planetary companions.


­ 11 ­ 2.2. Companion Detection and Selection Effects

The companions to the ab ove Sun-like sample of host stars have primarily b een detected using the Doppler technique (but not exclusively high precision exoplanet Doppler surveys) with some of the stellar pairs also b eing detected as astrometric or visual binaries. Thus we need to consider the selection effects of the Doppler method in order to define a less biased sample of companions (Lineweaver & Grether 2003). As a consequence of the exoplanet surveys' limited monitoring duration we only select those companions with an orbital p eriod P < 5 years. To reduce the selection effect due to the Doppler sensitivity we also limit our less biased sample to companions of mass M2 > 0.001M . Fig. 5 shows all of the Doppler companions to the Sun-like 25 p c and 50 p c samples within the mass and p eriod range considered here. Our less biased companions are enclosed by the thick solid rectangle. Given a fixed numb er of targets, the "Detected" region should contain all companions that will b e found for this region of mass-p eriod space. The "Being Detected" region should contain some but not all companions that will b e found in this region and the "Not Detected" region contains no companions since the current Doppler surveys are either not sensitive enough or have not b een observing for a long enough duration. To avoid the incomplete "Being Detected" region we limit our sample of companions to M2 > 0.001M . The companions in Fig. 5 all have radial velocity (Doppler) solutions. Some of the companions also have additional photometric, inteferometric, astrometric or visual solutions. The exoplanet Doppler orbits are taken from the Extrasolar Planets Catalog (2004). Only the planet orbiting the star HIP 108859 (HD 209458) has an additional photometric solution but this companion falls outside our less biased region (M2 < MJ up ). For the stellar companion data, the single-lined (SB1) and double-lined (SB2) sp ectroscopic binary orbits are primarily from the 9th Catalogue of Sp ectroscopic Binary Orbits (2004) with additional interferometric, astrometric or visual solutions from the 6th Catalog of Orbits of Visual Binary Stars (Washington Double Star Catalog). Additional binaries and solutions come from Endl et al. (2004); Halbwachs et al. (2000, 2003); Tinney et al. (2001); Jones et al. (2002); Vogt et al. (2002). For the stars closer than 25 p c, 56 have companions in the less biased region (rectangle circumscrib ed by thick line) of Fig. 5. Of these, 17 are exoplanets, 0 are brown dwarfs and 39 are of stellar mass. Of the stellar companions, 25 are SB1s and 14 are SB2s. Of these, 10 SB1s and 7 SB2s have an additional interferometric, astrometric or visual solution resp ectively. For the stars closer than 50 p c, 174 have companions in the less biased region. Of these, 52 are exoplanets, 12 are brown dwarfs and 110 are stars. Of the stellar companions, 64 are SB1s and 46 are SB2s. Of these, 19 SB1s and 11 SB2s have an additional orbital solution along with 1 of the brown dwarfs. If it were complete, the 50 p c sample would have approximately 8 times the numb er of companions as the 25 p c sample, since the 50 p c sample has 8 times the volume of the 25 p c sample. Instead we find that it only has 3 times the numb er of companions, however the relative numb er of planetary and stellar companions remains approximately unchanged in the 50 p c sample compared to the 25


­ 12 ­ p c sample (Section 3). We find an asymmetry in the north/south declination distribution of the Sun-like stars with companions, probably due to undetected or unpublished stellar companions in the south. The numb er of hosts closer than 25 p c with planetary or brown dwarf companions are symmetric in north/south declination to within one sigma Poisson error bars, but many more of the hosts with stellar companions with orbital solutions are in the northern hemisphere (30) compared with the southern (9). We assume that the northern sample of hosts with stellar companions is complete (Halbwachs et al. 2003). We estimate the numb er of missing stellar companions from the south by making a minimal correction up to the one sigma error level b elow the exp ected numb er. Of the 464 Sun-like stars closer than 25 p c, 211 have a southern declination (Dec 0 ) and 253 have a northern declination (Dec > 0 ) and thus 25(25/211 = 30/253) stars in the south should have a stellar companion when fully corrected or 20 if we make a minimal correction. Thus we estimate that we are missing 11(= 20 - 9) stellar companions in the south, 7 of which we assume have b een detected by Jones et al. (2002). Although these 7 stellar companions detected by Jones et al. (2002) have as yet no published orbital solutions, we assume that the stellar companions detected by Jones et al. (2002) have P < 5 years since they have b een observed (as part of the high Doppler precision program at the Anglo-Australian Observatory which started in 1998) for a duration of less than 5 years b efore b eing announced. The additional estimated stellar companions are assumed to have the same mass distribution as the other SB1 stellar companions. We similarly correct the declination asymmetry in the sample of Sun-like stars closer than 50 p c. We find that there should b e, after minimal correction, an additional 49 stars that are stellar companion hosts in the southern hemisphere. 14 of these 49 stellar companions are assumed to have b een detected by Jones et al. (2002). Due to the much larger numb er of stars that are high precision Doppler targets in the south there is also an asymmetry in the numb er of Sun-like stars closer than 50 p c with planetary companions. We estimate that there are an additional 7 stars in the northern hemipshere that are hosts to planetary companions. Table 1: Hipparcos Sample, Doppler Targets and Detected Companions for Near and Far Samples
Sample Hipparcos Number 1508 464 211 253 6924 3297 1647 1650 Doppler Target Number 671 (44%) 412 (89%) 211 (100%) 201 (79%) 2527 (36%) 2140 (65%) 1527 (93%) 613 (37%) Non-Target Number 837 (56%) 52 (11%) 0 (0%) 52 (21%) 4397 (64%) 1157 (35%) 120 (7%) 1037 (63%) Planets


Companions BDs Stars Total SB1 0 0 0 12 6 6 39 (+11) 9 (+11) 30 110 (+49) 26 (+49) 84 25

SB2 14

d < 25 p c (Sun-like) (Dec < 0 ) (Dec 0 ) d < 50 p c (Sun-like) (Dec < 0 ) (Dec 0 )


20 17 9 8 56 52 (+7) 32 20 (+7)

64

46

Correction in brackets.


­ 13 ­ 3. Companion Mass Function

The close companion mass function to Sun-like stars clearly shows a brown dwarf desert for b oth the 25 p c (Fig. 7) and the 50 p c (Fig. 8) samples. The numb ers of b oth the planetary and stellar mass companions decrease toward the brown dwarf mass range. Both plots contain the detected Doppler companions, shown as the grey histogram, within our less biased sample of companions (P < 5 years and M2 > 10-3 M , see Section 2.2). We minimally correct b oth of the less biased samples of companions for unpublished or undetected planetary and stellar companions inferred from the asymmetry in the host declination distribution (Section 2.2). The hashed histograms at large mass show the subset of the stellar companions that are not included in any of the exoplanet Doppler surveys. A large bias against stellar companions would have b een present if we had only included companions found by the exoplanet surveys. For multiple planetary systems, we select the most massive companion to represent the system. We put the few companions (3 in the 25 p c sample, 5 in the 50 p c sample) that have a mass slightly larger than 1 M in the largest mass bin in the companion mass distributions. Fitting straight lines using a weighted least squares method to the 3 bins on the left-hand side (LHS) and right-hand side (RHS) of the brown dwarf region of the mass histograms (Figs. 7 & 8), gives us gradients of -10.8 ± 4.4 (LHS) and 20.0 ± 7.8 (RHS) for the 25 p c sample and -7.2 ± 4.7 (LHS) and 29.5 ± 7.6 (RHS) for the 50 p c sample. Since the slop es have opp osite signs, they form a valley which is the brown dwarf desert. The presence of a valley b etween the negative and p ositive slop ed lines is significant at more than the 3 sigma level. The ratio of the corrected numb er of less biased companions on the LHS to the RHS along with their p oisson error bars is (19 ± 7)/(50 ± 12) = 0.38 ± 0.17 with no companions in the middle 2 bins for the 25 p c sample. For the larger 50 p c sample the corrected less biased LHS/RHS ratio is (59 ± 13)/(159 ± 21) = 0.37 ± 0.10, with 12 ± 5 brown dwarf companions in the middle 2 bins. Thus the LHS and RHS slop es agree to within ab out 1 sigma and so do the LHS/RHS ratios, indicating that the companion mass distribution for the larger 50 p c sample is not significantly different from the more complete 25 p c sample and that the relative fraction of planetary, brown dwarf and stellar companions is approximately the same. To find the driest part of the desert, we fit a weighted least squares parab ola, dN/dlogM = a(logM - logM0 )2 + y0 to the data (solid lines) in Figs. 7 & 8. The minimum occurs at logM0 = -1.79 +0.26 (M0 = 17+14 MJ up ) in the 25 p c sample and at logM0 = -1.76 +0.13 (M0 = 18+7 MJ up ) -0.17 -6 -0.12 -4 in the 50 p c sample and has a p ercentage of companions p er unit logM of y0 = 1.1% ± 0.3% and y0 = 0.8% ± 0.1% resp ectively. The weighted average of b oth samples, the driest part of the desert, is at logM0 = -1.77 ± 0.11 (M0 = 18+5 MJ up ). The driest part of the desert is virtually the same -4 for b oth samples even though we detect a bias in the stellar binarity fraction of the 50 p c sample (Fig. 3). The deep est part of the valley where the straight lines cross is at logM = -1.53 +0..28 and -0 35 logM = -1.22 +0.22 for the 25 and 50 p c samples resp ectively. We have done the analysis with and -0.26 without the minimal declination asymmetry correction. The p osition of the brown dwarf minimum and the slop es seem to b e robust to this correction.


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Fig. 7.-- Brown Dwarf Desert in Close Sample. Histogram of the companions to Sun-like stars closer than 25 p c plotted against mass. The grey histogram is made up of Doppler detected companions in our less biased (P < 5 years and M2 > 10-3 M ) sample. The corrected version of this less biased sample includes 11 extra SB1 stars from an asymmetry in the host declination distribution (Section 2.2). The planetary mass companions are also corrected for the small numb er of Hipparcos Sun-like stars that are not b eing Doppler monitored (13% correction, see Section 2.1). The hatched histogram is the subset of detected companions to hosts that are not included on any of the exoplanet search target lists and hence shows the extent to which the exoplanet target lists are biased against the detection of stellar companions. Since an instrument with sensitivity KS is used for all the companions, there should b e no other substantial biases affecting the relative amplitudes of the stellar companions on the right-hand side (RHS) and the planetary companions on the left-hand side (LHS). A distinct lack of brown dwarf sized companions is noted.


­ 15 ­

Fig. 8.-- Same as Fig. 7 but for the larger 50 p c sample. Fitting straight lines using a weighted least squares fit to the 3 bins on the LHS and RHS, gives us gradients of -7.2 ± 4.7 and 29.5 ± 7.6 resp ectively (dashed lines). Hence the brown dwarf desert is significant at the 4 sigma level. These LHS and RHS slop es agree to within 1 sigma of those in Fig. 7. The ratio of the numb er of companions on the LHS to the RHS is also ab out the same for b oth samples. Hence the relative numb er and distribution of companions is approximately the same as in Fig. 7. Fitting a parab ola as in Fig. 7, the minimum of the brown dwarf desert is at logM0 = -1.76 ± 0.13 (or Mdriest = 18+7 MJ up ) where the fraction of companions p er unit logM is y0 = 0.8% ± 0.1%. 16% -4 of the stars have companions in our less biased region. Of these, 4% ± 2% have companions of planetary mass, 1% ± 1% have brown dwarf companions and 11% ± 3% have companions of stellar mass. To b etter facilitate comparisons b etween the 2 samples, i.e. to compare the LHS and RHS slop es and the shap e of the parab olas which are b oth dep endent on the amplitude of the vertical axis, we scale the vertical axis of the 50 p c sample down to the size of the 25 p c sample. We compare each bin in Fig. 8 with its corresp onding bin in Fig. 7 and scale the vertical axis of Fig. 8 so that the difference in height b etween the bins is on average a minimum. We find that the optimum scaling factor is 0.34.


­ 16 ­ The smaller 25 p c Sun-like sample contains 464 stars with 14.9% ± 4.2% of these having companions in our corrected less biased sample. Of these 15% with companions, 4.1% ± 1.6% are of planetary mass and 10.8% ± 2.6% are of stellar mass. None is of brown dwarf mass. This agrees with previous estimates of stellar binarity such as that found by Halbwachs et al. (2003) of 14% for a sample of G-dwarf companions with a slightly larger p eriod range (P < 10 years). The planet fraction agrees with the fraction 4% ± 1% found in Lineweaver & Grether (2003) when all of the known exoplanets are considered. The 50 p c sample has a large incompleteness due to the lower fraction of monitored stars (Fig. 4) but as shown ab ove, the relative numb er of companion planets, brown dwarfs and stars is approximately the same as for the 25 p c sample. The scaled 50 p c sample has a total companion fraction of 16.6% ± 4.9%, where 4.3% ± 1.6% of the companions are of planetary mass, 0.9% ± 0.6% are of brown dwarf mass and 11.5% ± 2.6% are of stellar mass. Surveys of the multiplicity of nearby Sun-like stars yield the relative numb ers of single, double and multiple star systems. According to Duquennoy & Mayor (1991), 51% of star systems are single stars, 40% are double star systems, 7% are triple and 2% are quadruple or more. Of the 49%(= 40 + 7 + 2) which are stellar binaries or multiple star systems, 11% have stellar companions with p eriods less than 5 years and thus we can infer that the remaining 38% have stellar companions with P > 5 years. Among the 51% without stellar companions, we find that 4% have close (P < 5 years) planetary companions with 1 < M /MJ up < 13, while 1% have close brown dwarfs companions.

3.1.

Comparison with Other Results

Although there are some similarities, the companion mass function found by Heacox (1999); Zucker & Mazeh (2001); Mazeh et al. (2003) is different from that shown in Figs. 7 & 8. Unlike our approach that uses a single sample of stars from which we derive the companion mass function from planets to stars, these authors use two different samples of stars, one to find the planetary companion mass function and another to find the stellar companion mass function, which are then scaled and combined into one mass function. No correction for the north/south asymmetry was made. The different host star prop erties and levels of completeness of the two samples may make this method more prone than our method, to biases in the frequencies of companions. Both Heacox (1999) and Zucker & Mazeh (2001) combined the companions of the stellar mass sample of Duquennoy & Mayor (1991) with the known substellar companions, but identified different mass functions for the planetary mass regime b elow 10 MJ up and similar flat distributions in logarithmic mass for brown dwarf and stellar mass companions. Heacox (1999) found that the logarithmic mass function in the planetary regime is b est fit by a p ower-law (dN/dlogM M ) with index b etween 0 and -1 whereas Zucker & Mazeh (2001) find an approximately flat distribution (p ower-law with index 0). Our work suggests that neither the stellar nor the planetary companion distributions are flat. Rather, they b oth slop e down towards the brown dwarf desert.


­ 17 ­ The work most similar to ours is probably (Mazeh et al. 2003) who looked at a sample of main sequence stars with primaries in the range 0.6 - 0.85 M and P < 3000 days using infrared sp ectroscopy and combined them with the known substellar companions of these main sequence stars and found that in logarithmic mass the stellar companions reduce in numb er towards the brown dwarf mass range. This agrees with our results for the shap e of the stellar mass companion function. However, they identify a flat distribution for the planetary mass companions in contrast to our rising function. Mazeh et al. (2003) found the frequency of stellar and planetary companions (M2 > 1 MJ up ) to b e 15% (for stars b elow 0.7 M ) and 3% resp ectively. This compares with our estimates of 8% (for stars b elow 0.7 M ) and 4%. The larger p eriod range used by Mazeh et al. (2003) can account for the difference in stellar companion fractions. Table 2: Companion Slop es and Companion Desert Mass Minima
Sample d d d d d d < < < < < < 2 5 2 5 2 5 5 0 5 0 5 0 p p p p p p c c c c c c LHS slope -10.8 ± 4.4 -7.2 ± 4.7 & & & & M M M M
1 1 1 1

RHS slope 20.0 ± 7.8 29.5 ± 7.6

Slope Minima log M /M [MJ up ] -1.53 +0.28 31 +28 -0.35 -17 -1.22 +0.22 63 +41 -0.26 -28

< <

1 1 1 1

M M M M



Parabola Minima log M /M [MJ up ] -1.79 +0.26 17 +14 -0.17 -6 -1.76 +0.13 1 8 +7 -0.12 -4 -1.97 +0.55 11 +29 -0.34 -6 -1.95 +0.16 1 2 +5 -0.10 -2 -1.67 +0.38 22 +31 -0.34 -12 -1.64 +0.16 24 +11 -0.16 -7

The Slope Minima are the values of mass where the two (LHS and RHS) straight lines intersect. The Parabola Minima are the values of the mass at the lowest point of the parabola.

Table 3: Companion Fraction Comparison
Sample d < 25 d < 50 d < 25 d < 50 d < 25 d < 50 p p p p p p c c c c c c Total 14.9 ± 16.6 ± 14.9 ± 16.6 ± 14.9 ± 16.6 ± % 4.2 4.9 3.8 4.8 4.2 5.0 Planetary % 4.1 ± 1.6 4.3 ± 1.6 2.9 ± 1.1 2.7 ± 1.3 5.1 ± 1.8 6.0 ± 1.9 Brown 0.0 0.9 0.0 1.3 0.0 1.4 Dwarf % ± 0.0 ± 0.6 ± 0.0 ± 0.7 ± 0.0 ± 0.8 Stellar % 10.8 ± 2.6 11.5 ± 2.6 12.0 ± 2.7 12.6 ± 2.8 9.8 ± 2.4 9.2 ± 2.3

& & & &

M M M M

1 1 1 1

< <

1 1 1 1

M M M M



3.2.

Companion Mass as a Function of Host Mass

We find that the minimum of the companion mass desert is a function of host mass with lower mass hosts having a lower companion mass desert. As seen in Figs. 9 and 10 for the 25 and 50 p c samples resp ectively, lower mass hosts have more stellar companions and fewer giant planet companions while higher mass hosts have fewer stellar companions and more giant planet companions.


­ 18 ­

Fig. 9.-- Same as Fig. 7 but for the 25 p c sample split into companions to lower mass hosts (M1 < 1M ) and companions to higher mass hosts (M1 1M ). The lower mass hosts have 3% planetary, 0% brown dwarf and 12% stellar companions. The higher mass hosts have 5% planetary, 0% brown dwarf and 10% stellar companions. The Doppler method should preferentially find planets around lower mass stars where a greater radial velocity is induced. This is the opp osite of what we observe which suggests that the relative numb er of stellar, brown dwarf and planetary companions is correlated with host mass. To aid comparison, b oth samples are scaled such that they contain the same numb er of companions as the full corrected less biased 25 p c sample of Fig. 7.


­ 19 ­

Fig. 10.-- Same as Fig. 8 but for the 50 p c sample split into companions to lower mass hosts (M1 < 1M ) and companions to higher mass hosts (M1 1M ). Both samples are scaled such that they contain the same numb er of companions as the corrected less biased 50 p c sample of Fig. 8. Also shown are the b est-fit parab olas to the two p opulations. The shap e of the parab ola is approximately the same for b oth ranges of host mass but the miniumum of the parab ola app ears to b e correlated with host mass. The minima of the parab ola are at logM = -1.95+0.16 (M = -0.10 12+5 MJ up ) and logM = -1.64 ± 0.16 (M = 24+11 MJ up ) for the lower and higher mass samples, -2 -7 resp ectively.


­ 20 ­

Fig. 11.-- Companion mass as a function of host mass. The grey histogram of host masses is also shown. We find the average companion mass (triangles) for each of the four host mass bins for stellar, brown dwarf and planetary companions in our Sun-like 50 p c sample (big dots are closer than 25 p c and smaller dots are b etween 25 and 50 p c) to investigate any host-mass/companion-mass correlation. The hashed region of stellar companions is ignored in the analysis of the correlation b etween M1 and M2 to avoid introducing a bias. The companion mass app ears uncorrelated with host mass for either stellar, brown dwarf or planetary companions. As indicated in Fig. 9 there are more companions in the upp er left and lower right. However, the numb er of companions to lower mass hosts M1 < 1M (89) is approximately the same as the numb er of companions to higher mass hosts M1 1M (85). The 4 crosses represent the minimum of the 4 parab olas in the lower and higher mass samples of the 0 pc < d < 25 pc and 25 pc < d < 50 pc samples together with the corresp onding average host mass for these 4 samples. The linear b est fit to these crosses (thick line) shows the driest part of the companion mass distribution as a function of host mass. The 4 crosses are indep endent of each other.


­ 21 ­ The Doppler method should preferentially find planets around lower mass stars where a greater radial velocity is induced. This is the opp osite of what is observed. The Doppler technique is also a function of B - V color (Saar et al. 1998) with the level of systematic errors in the radial velocity measurements, decreasing as we move from high mass to low mass (B - V = 0.5 to B - V = 1.0) through our two samples, p eaking for late K sp ectral typ e stars b efore increasing for the lowest mass M typ e stars again. Hence again finding planets around the lower mass stars (early K sp ectral typ e) in our sample should b e easier. We split the 25 and 50 p c samples into companions to hosts with masses ab ove and b elow 1 M (Figs. 9 and 10) and scale these smaller samples such that they contain the same numb er of companions as the full 25 and 50 p c samples (Figs. 7 and 8). We then find the minimum of the b est-fit parab olas (crosses) and note that the minimum is a function of host mass. The linear b est fit to the 4 crosses (thick line) can b e describ ed by: MH os M
t 3.2±1.0

MD

r iest

20 MJ

up

.

(1)

Thus, for hosts of half a solar mass, the short p er b e for 2MJ up ob jects ­ a "Jupiter Desert", and for 2 minimum would b e for 100 MJ up ob jects ­ a low mass are outside the brown dwarf mass regime as shown in dwarf hosts (0.1 M ) we find that the minimum of the the planetary regime at 0.01 MJ up ( 3ME arth ).

iod (P < 5 years) desert minimum would solar mass hosts, the short p eriod desert "Stellar Desert". Both of these "deserts" Fig. 11. Extrap olating down to small M short p eriod desert would occur well into

We find no evidence against our prop osed stellar desert for higher mass hosts from the small numb er of additional stellar companions to larger mass Hipparcos main sequence hosts from 9th Catalogue of Sp ectroscopic Binary Orbits (2004) closer than 25 p c. The mass ratio q (= M2 /M1 ) distribution should also show a drop in frequency at q < 0.1 for higher mass hosts where we prop ose a low stellar companion mass desert. Halbwachs et al. (2003) observes a higher q fraction at q < 0.4 for the less massive K sp ectral typ e host stars than for the more massive F7-G host stars. Although the driest part of the companion mass desert is a function of host mass we find no evidence for a direct correlation b etween host mass and companion mass. This is shown in Fig. 11 by the distribution of the average mass for stellar, brown dwarf and planetary companion for the 50 p c sample (triangles).

4.

Comparison with the Initial Mass Function

Brown dwarfs found as free-floating ob jects in the solar neighb ourhood and as memb ers of young star clusters (since they are more luminous when young), have b een used to extend the initial mass function (IMF) well into the brown dwarf regime. Comparing the mass function of


­ 22 ­

Fig. 12.-- The mass function of companions to Sun-like stars (lower left) compared to the initial mass function (IMF) of cluster stars (upp er right). Our mass function of the companions to Sunlike stars is shown by the green dots (bigger dots are the d < 25 p c sample, smaller dots are the d < 50 p c sample). The parab ola we fit to the data in Fig. 7 is also shown along with its error. The companion mass function is normalised to the IMF of the cluster stars by scaling the three companion p oints of stellar mass to b e on average 7% for P < 5 years (derived from the stellar multiplicity of Duquennoy & Mayor (1991) discussed in Section 3, combined with our estimate that 11% of Sun-like stars have stellar secondaries). Data for the numb er of stars and brown dwarfs in the Orion Nebula Cluster (ONC) (circles), Pleiades cluster (triangles) and M35 cluster (squares) come from Hillenbrand & Carp enter (2000); Slesnick et al. (2004), Moraux et al. (2003) and Barrado y Navascues et al. (2001) resp ectively and are normalised such that they overlap for masses larger than 1M where a single p ower-law slop e applies. The absolute normalisation is arbitrary. The average p ower-law IMF derived from various values of the slop e of the IMF quoted in the literature (Hillenbrand 2003) is shown as larger red dots along with two thin red lines showing the root-meansquare error. If the turn down in the numb er of brown dwarfs of the IMF is due to a selection effect b ecause it is hard to detect brown dwarfs, then the two distributions are even more different from each other.


­ 23 ­

Fig. 13.-- The initial mass function (IMF) for clusters represented by a series of p ower-law slop es (Hillenbrand 2003). Each p oint represents the p ower-law slop e claimed to apply within the mass range indicated by the horizontal lines. Although the IMF is represented by a series of p ower-laws, the IMF is not a p ower-law for masses less than 1M where the slop e continually changes. The green dots show the slop e of the companion mass function to Sun-like stars b etween the bins of Figs. 7 & 8 with the larger and smaller dots resp ectively. The parab olic fit to the data in Fig. 7 and its associated error are shown by the curve inside the grey region. The p ower-law fit of Lineweaver & Grether (2003) (shown as the green dot with a horizontal line indicating the range over which the slop e applies) is consistent with the parab olic fit. The larger red dots with error bars represent the average p ower-law IMF with a root-mean-square error. and - are the resp ective logarithmic and linear slop es of the mass function. The logarithmic mass p ower-law distribution is dN/dlogM M and the linear mass p ower-law distribution is dN/dM M - where = 1 - . The errors on the parab olic fit get smaller at M 10-3 M and M 1 M since for all parab olas as log(M /M ) tends to ±, tends to 0. This can also b e seen in Fig. 12 where the slop es of the upp er and lower contours b ecome increasingly similar.


­ 24 ­ our sample of close-orbiting companions to Sun-like stars, to the IMF of single stars indicates how the environment of a host affects stellar and brown dwarf formation and/or migration. Here we quantify how different the companion mass function is from the IMF (Halbwachs et al. 2000). The galactic IMF app ears to b e remarkably universal and indep endent of environment and metallicity with the p ossible exception of the substellar mass regime (a weak empirical trend with metallicity is suggested for very low mass stars and brown dwarfs where more metal rich environments may b e producing relatively more low mass ob jects. This is consistent with an extrap olation up in mass from the trend found in exoplanet hosts, Kroupa (2002)). The IMF is often represented as a p ower-law, although this only app ears to b e accurate for stars with masses ab ove 1M (Hillenbrand 2003). The stellar IMF slop e gets flatters towards lower masses and extends smoothly and continously into the substellar mass regime where it app ears to turn over. Free floating brown dwarfs may b e formed either as ejected stellar embryos or from low mass protostellar cores that have lost their accretion envelop es due to photo-evap oration from the chance proximity of a nearby massive star (Kroupa & Bouvier 2003). This hyp othesis may explain their occurence in relatively rich star clusters such as the Orion Nebula cluster and their virtual absence in pre-main sequence stellar groups such as Taurus-Auriga. In Fig. 12 we compare the mass function of companions to Sun-like stars with the IMF of cluster stars. The mass function for companions to Sun-like stars is shown by the green dots from Figs. 7 and 8 (bigger dots are the d < 25 p c sample and smaller dots are the d < 50 p c sample). The parab ola from Fig. 7 and its one sigma confidence region is also shown. The mass function of companions continually reduces in numb er from 1M stars to brown dwarfs whereas the cluster IMF continues to rise nearly until the substellar regime b efore dropping in numb er. The IMF is shown as a series of p ower-law slop es in Fig. 13, which are taken from the literature and kindly provided by Hillenbrand (2003). The IMF is represented by a series of p ower-laws and is clearly not a single p ower-law, as shown by the continual change of the slop e b elow 1M . The larger red dots with error bars represent our b est fit for an average p ower-law IMF with a root-mean-square error. The IMF for young clusters (yellow dots) is statistically indistinguishable from that of older stars (blue dots) and follows the average IMF. In linear mass, b oth the mass function of companions to Sun-like stars and the IMF continually rise ( > 0) with decreasing mass. Thus the brown dwarf desert is apparent only in logarithmic mass.

5.

Summary and Discussion

We analyse the close-orbitting (P < 5 years) planetary, brown dwarf and stellar companions to Sun-like stars to help constrain their formation and migration scenarios. We use the same sample to extract the relative numb ers of planetary, brown dwarf and stellar companions and verify the existence of a brown dwarf desert. Both planetary and stellar companions reduce in numb er towards the brown dwarf mass range. The companion mass function over the range that


­ 25 ­
< we analyse (0.001 < M /M 1.0) is fit b etter by a parab ola than by two lines fit separately to the planetary and stellar data p oints. This suggests that the formation and migration mechanisms for brown dwarfs is a smooth combination of planet and stellar mechanisms.

We find that the minimum of the companion mass desert is a function of host mass with lower mass hosts having a lower mass companion desert. The p eriod distribution of close-orbitting (P < 5 years) companion stars is different from that of the planetary companions with brown dwarfs lying somewhere in b etween. The close-in stellar companions are fairly evenly distributed over logP with planets tending to b e clump ed towards higher p eriods. We compare the companion mass function to the IMF for b odies in the brown dwarf and stellar regime. We find that starting at 1 M and decreasing in mass, stellar companions reduce in numb er b efore plateauing in the brown dwarf regime, while cluster stars increase in numb er b efore reaching a maximum just b efore the brown dwarf regime. This leads to a difference of approximately 1.5 orders of magnitude b etween the much larger numb er of brown dwarfs found in clusters to those found as close-orbitting companions to Sun-like stars. The p eriod distribution of close-orbiting companions may b e more a result of p ost-formation migration and jostling than representive of the relative numb er of companions that are formed at a sp ecific distance from their hosts. The companion mass distribution is more fundamental than the p eriod distribution and should provide b etter constraints on formation models, but our ability to sample the mass distribution is only for P < 5 years. We show in Figs. 9 and 10 that lower mass hosts have more stellar companions and fewer giant planet companions while higher mass hosts have fewer stellar companions but more giant < planet companions. The brown dwarf desert is generally thought to exist at close separations 3 AU (or equivalently P 5 years) (Marcy & Butler 2000) but may disapp ear at wider separations. Gizis et al. (2001) suggests that at very large separations (> 1000 AU) brown dwarf companions may b e more common. However, McCarthy & Zuckerman (2004) in their observation of 280 GKM stars find only 1 brown dwarf b etween 75 and 1200 AU. Our sample is limited to close separations (P < 5 years) b etween the host and companion but we do find a larger prop ortion of brown dwarfs at higher p eriods than at lower ones (Fig. 6). Gizis et al. (2003) rep orts that 15% ± 5% of M/L dwarfs are brown dwarf binaries with separations in the range 1.6 - 16 AU. This falls to 5% ± 3% of M/L dwarfs with separations less than 1.6 AU and none with separations greater than 16 AU. This differs greatly from the brown dwarfs orbiting Sun-like stars but is consistent with our hostcompanion mass relationship, i.e., we exp ect no short p eriod brown dwarf desert arond M or L typ e stars. We find that approximately 16% of Sun-like stars have a close companion more massive than Jupiter. Of these 16%, 11% ± 3% are stellar, 1% ± 1% are brown dwarf and 4% ± 2% are planetary companions. Although Lineweaver & Grether (2003) show that the fraction of Sun-like stars with planets is greater than 25%, this is for target stars that have b een monitored the longest ( 15 years) and at optimum conditions (stars with low-level chromospheric activity or slow rotation)


­ 26 ­ using the high precision Doppler method. When we limit the analysis of Lineweaver & Grether (2003) to planetary companions with p eriods of less than 5 years and masses larger than Jupiter, we find the same 4% that we calculate here. When we split our sample of companions into those with hosts ab ove and b elow 1M , we find that for the lower mass hosts: 12% have stellar, 1% have brown dwarf and 3% have planetary companions and that for the higher mass hosts: 9% have stellar, 1% have brown dwarf and 6% have planetary companions resp ectively. The constraints that we have identified for the companions to Sun-like stars suggest to us that brown dwarfs may form from some combination of planetary and stellar formation processes, i.e., core accretion, disk instability and cloud fragmentation. The fact that there is a close-orbitting brown dwarf desert but no free floating brown dwarf desert suggests that p ost-collapse migration mechanisms may b e resp onsible for this relative dearth of observable brown dwarfs rather than some intrinsic minimum in fragmentation and gravitational collapse in the brown dwarf mass regime (Ida & Lin 2004). Whatever migration mechanism is resp onsible for putting hot Jupiters in close orbits, its effectiveness may dep end on the mass ratio of the ob ject to the disk mass. Since there is evidence that disk mass is correlated to host mass, the migratory mechanism may b e correlated to host mass, as prop osed by Armitage & Bonnell (2002).

6.

Acknowledgements

We would like to thank Christian Perrier for providing us with the Elodie exoplanet target list, Stephane Udry for addititional information on the construction of the Coralie exoplanet target list and Lynne Hillenbrand for sharing her data collected from the the literature on the p ower-law IMF fits to various stellar clusters. This research has made use of the SIMBAD database, op erated at CDS, Strasb ourg, France. This research has made use of the Washington Double Star Catalog maintained at the U.S. Naval Observatory.

7.

App endix: Companion Mass Estimates

The Doppler method for companion detection cannot give us the mass of a companion without some additional astrometric or visual solution for the system or by making certain assumptions ab out the unknown inclination except in the case where a host star and its stellar companion have approximately equal masses and a double-lined solution is available. Thus to find the companion mass M2 that induces a radial velocity K1 in a host star of mass M1 we use 1 2 G 1/3 M2 sin(i) ) 2/3 P (M1 + M2 ) (1 - e2 )1/2

K1 = (

(2)

This equation can b e expressed in terms of the mass function f (m)


­ 27 ­

f (m) =

3 3 P K1 (1 - e2 )3/2 M2 sin3 (i) = (M1 + M2 )2 2 G

(3)

Eq. 3 can then b e expressed in terms of a cubic equation in the mass ratio q = M2 /M1 , where Y = f (m)/M1 . q 3 sin3 (i) - Y q 2 - 2Y q - Y = 0 (4)

For planets (M1 >> M2 ) we can simplify Eq. 2 and directly solve for M2 sin(i) but this is not true for larger mass companions such as brown dwarfs and stars. We use Cox (2000) to relate host mass to sp ectral typ e. When a double-lined solution is available, the companion mass can b e found from q = M2 /M1 = K1 /K2 . For all single-lined Doppler solutions, where the inclination i of a companion's orbit is unknown (no astrometric or visual solution), we assume a random distribution P (i) for the orientation of the inclination with resp ect to our line of sight, P (i)di = sin(i)di (5)

From this we can find probability distributions for sin(i) and sin3 (i). Heacox (1995) and others suggest using either the Richardson-Lucy or Mazeh-Goldb erg algorithms to approximate the inclination distribution. However, Hogeveen (1991) and Trimble (1990) argue that for low numb er statistics, the simple mean method produces similar results to the more complicated methods. We have large bin sizes and small numb er statistics, hence we use this method. The average values of the sin(i) and sin3 (i) distributions assuming a random inclination are < sin(i) >= 0.785 and < sin3 (i) >= 0.589, which are used to estimate the mass for planets and other larger single-lined sp ectroscopic binaries resp ectively. For example, in Fig. 5, of the 174 mass estimates in the 50 p c sample, 46 (26%) come from double-lined Doppler solutions, 20 (12%) come from knowing the inclination (astrometric or visual solution also available for system), 56 (32%) come from assuming < sin(i) >= 0.785 and 52 (30%) from assuming < sin3 (i) >= 0.589.


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A This preprint was prepared with the AAS L TEX macros v5.2.