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A comprehensive comparison of the Sun to other stars: searching for self-selection effects
Jos´ A. Robles1 , Charles H. Lineweaver1 , Daniel Grether2 , Chris Flynn3 , Chas A. Egan2,4 , e Michael B. Pracy4 , Johan Holmberg5 and Esko Gardner3

arXiv:0805.2962v1 [astro-ph] 19 May 2008

ABSTRACT
If the origin of life and the evolution of observers on a planet is favoured by atypical prop erties of a planet's host star, we would exp ect our Sun to b e atypical with resp ect to such prop erties. The Sun has b een describ ed by previous studies as b oth typical and atypical. In an effort to reduce this ambiguity and quantify how typical the Sun is, we identify eleven maximally-indep endent prop erties that have plausible correlations with habitability, and that have b een observed by, or can b e derived from, sufficiently large, currently available and representative stellar surveys. By comparing solar values for the eleven prop erties, to the resultant stellar distributions, we make the most comprehensive comparison of the Sun to other stars. The two most atypical prop erties of the Sun are its mass and orbit. The Sun is more massive than 95 ± 2% of nearby stars and its orbit around the Galaxy is less eccentric than 93 ± 1% of FGK stars within 40 parsecs. Despite these apparently atypical prop erties, a 2 -analysis of the Sun's values for eleven prop erties, taken together, yields a solar 2 = 8.39 ± 0.96. If a star is chosen at random, the probability that it will have a lower value ( b e more typical) than the Sun, with resp ect to the eleven prop erties analysed here, is only 29 ± 11%. These values quantify, and are consistent with, the idea that the Sun is a typical star. If we have sampled all reasonable prop erties associated with habitability, our result suggests that there are no sp ecial requirements for a star to host a planet with life.
Subject headings: Sun: fundamental parameters -- Sun: general -- stars: fundamental parameters -- stars: statistics

1.
Planetary Science Institute, Research School of Astronomy & Astrophysics and Research School of Earth Sciences, The Australian National University, Canb erra Australia; josan@mso.anu.edu.au. 2 University of New South Wales, Sydney, Australia. 3 Tuorla Observatory, University of Turku, Finland. 4 Research School of Astronomy & Astrophysics, The Australian National University, Canb erra, Australia. 5 Max Planck Institute for Astronomy, Heidelb erg, Germany.
1

INTRODUCTION

If the prop erties of the Sun are consistent with the idea that the Sun was randomly selected from all stars, this would indicate that life needs nothing sp ecial from its host star and would supp ort the idea that life may b e common in the universe. More particularly, if there is nothing sp ecial ab out the Sun, we have little reason to limit our life-hunting efforts to planets orbiting Sun-like stars. As

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an example of the typ e of anthropic reasoning we are using, consider the following situation. Supp ose uranium (a low abundance element in the Solar System and in the universe) was central to the biochemistry of life on Earth. Further, supp ose that a comparison of our Sun to other stars showed that the Sun had more uranium than any other star. How should we interpret this fact? The most reasonable way to proceed would b e to try to evaluate the probability that such a coincidence happ ened by chance and to determine whether we are justified in reading some imp ortance into it. Although a correlation does not necessarily imply cause, we think that a correlation b etween the Sun's anomalous feature and life's fundamental chemistry would b e giving us imp ortant clues ab out the conditions necessary for life. Sp ecifically, the search for life around other stars as envisioned by the NASA's Terrestrial Planet Finder or ESA's Darwin Pro ject and as currently underway with SETI, would change strategy to focus on the most uranium-rich stars. Another example: supp ose the Sun had the highest [Fe/H] of all the stars that had ever b een observed. Then high [Fe/H] would b e strongly implicated as a precondition for our existence, p ossibly by playing a crucial role in terrestrial planet formation. These are exaggerated examples of the more subtle correlations that a detailed and comprehensive comparison of the Sun with other stars could reveal. Whether the Sun is a typical or atypical star with resp ect to one or a few properties has b een addressed in previous studies. Using an approach similar to ours (comparing solar to stellar prop erties from particular samples), some studies have suggested that the Sun is a typical star (Gustafsson 1998; Allende Prieto 2006), while other studies have suggested that the Sun is an atypical star (Gonzalez 1999a,b; Gonzalez et al. 2001). This apparent disagreement arises from three problems: 2

i) the language used to describ e whether the Sun is, or is not typical, is often confusingly qualitative. For example, rep orting the Sun as "metal-rich", can mean that the Sun is significantly more metal-rich than other stars (e.g. more metal-rich than 80% of other stars) or it can mean that the Sun is insignificantly metal-rich (e.g. more metal-rich than 51% of other stars). ii) selection effects: the stellar samples chosen for the comparison can b e biased with resp ect to the prop erty of interest. iii) the inclusion (or exclusion) of stellar prop erties for which it is susp ected or known that the Sun is atypical, will make the Sun app ear more atypical (or typical). In this pap er we address problem i by using only quantitative measures when comparing the Sun's prop erties to other stars. Our main interest is to move b eyond the qualitative assessment of the Sun as either typical or atypical, and obtain a more precise quantification of the degree of the Sun's (a)typicality. In other words, we want to answer the question `How typical is the Sun?' rather than `Is the Sun typical or not?' There are at least two ways to quantify how typical the Sun is. This can b e done for individual parameters by determining how many stars have values b elow or ab ove the solar value (Table 3). This can also b e done by a joint analysis of multiple parameters (Table 2). If there are several subtle factors that have some influence over habitability, a quantitative joint analysis of the Sun's prop erties may allow us to identify these factors without invoking largely sp eculative arguments linking sp ecific prop erties to habitability. With resp ect to problem ii, most previous analyses have compared the Sun to subsets of Sun-like stars selected to b e Sun-like with


resp ect to one or more parameters. In such analyses, the Sun will app ear typical with resp ect to any parameter(s) correlated with one of the pre-selected Sun-like parameters. For example, elemental abundances [X/H] are correlated with metallicity1 [Fe/H]. The sample of Edvardsson et al. (1993a) was selected to have a wide range of [Fe/H]. This produced a metallicity distribution unrepresentative of stars in general. Recognizing this, Edvardsson et al. (1993a) conditioned on solar metallicity, [Fe/H] 0 and then compared solar abundances for 12 elements to the abundances in a group of nearby stars with solar iron abundance, solar age and solar galactocentric radius. They found the Sun to b e "a quite typical star for its metallicity, age and galactic orbit". Similarly, Gustafsson (1998), after comparing various prop erties of the Sun to solar-typ e stars (stars of similar mass and age), concluded that the Sun seems very normal for its mass and age; "The Sun, to a remarkable degree, is solar typ e". The stellar samples we use for comparison with the Sun are, in our judgement, the least-biased samples currently available for such a comparison. To address problem iii, in Section 2 we compare the Sun to other stars using a large numb er (eleven) of maximally-indep endent prop erties with plausible correlations with habitability. These prop erties can b e observed or derived for a sufficiently large, representative stellar sample (Table 1). Any prop erty of the Sun or its environment which must b e sp ecial to allow habitability would show up in our analysis. However, in contrast to previous analyses which have looked for solar anomalies with resp ect to individual prop erties, we p erform a joint analysis that enables us to quantify how typical the solar values are, taken as a group. In Section 3, the differ1

ences b etween the solar values and the stellar samples' medians are used to p erform first a simple and then an improved version of a 2 analysis to estimate whether the solar values are characteristic of a star selected at random from the stellar samples. The results of our joint analysis are presented in Figure 13 of Section 4. We find that the solar values, taken as a group, are consistent with the Sun b eing a random star. However, there are imp ortant caveats to this interpretation associated with the compromise b etween the numb er of properties analyzed, and their plausibility of b eing correlated with habitability. In Sections 5 and 6 we discuss these caveats and summarize. We discuss the levels of correlation b etween our eleven prop erties in App endix A. 2. Stellar Samples and Solar Values

Metallicity: [Fe/H] is the fractional abundance of Fe relative to hydrogen, compared to the same ratio in the Sun: [Fe/H] log(Fe/H) - log(Fe/H)

We are looking for a signal associated with a prerequisite for, or a prop erty that favors, the origin and evolution of life (see Gustafsson 1998 for a brief discussion of this idea). If we indiscriminately include many prop erties with little or no plausible correlation with habitability, we run the risk of diluting any p otential signal. If we choose only a few prop erties based on previous knowledge that the Sun is anomalous with resp ect to those prop erties, we are making a useful quantification but we are unable to address problem iii. We choose a middle ground and try to identify as many prop erties as we can that have some plausible association with habitability. This strategy is most sensitive if a few unknown stellar prop erties (among the ones b eing tested) contribute to the habitability of a terrestrial planet in orbit around a star. An optimal quantitative comparison of the Sun to other stars would require an unbiased, large representative stellar sample from which indep endent distributions, for as many properties as desired, could b e compared. Such

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Table 1: Samples used to produce the stellar distributions plotted in Figures 1­10.
Fig. 1 2 3 4A 4B 5 6 7 8 9 10 Prop erty Mass [M ] Age [Gyr] [Fe/H] [C/O] [Mg/Si] v sin i [km s-1 ] e Zmax [kp c] RGal [kp c] Mgal [M ]k Mgroup [M ]k Range 0.08 ­ 2 0 ­ 15 -1.20 ­ +0.46 -0.22 ­ +0.32 -0.18 ­ +0.14 0 ­ 36 0­1 0 ­ 9.60 0 ­ 30 107 ­ 1012 109 ­ 1013 Median µ1/2 0.33 5.4 -0.08 0.07 0.01 2.51 0.10 0.14 4.9 1010.2 1011.1
68

0.37 3.25 0.20 0.09 0.04 1.27 0.05 0.10 5.03 0.47 0.47

Solar # stars Value 1 125 4.9+3.1 a 552 -2.7 0 453 0 256 0 231 1.28d 276 0.036 ± 0.002f 1,987 0.104 ± 0.006h 1,987 7.62 ± 0.32i -- 1010.55±0.16 -- 1010.91±0.07 --

Sp ectral typ e A1­M7 F8­K2 F7­K3 FG FG F8­K2 A5­K2 A5­K2 -- -- --

dmax [p c] 7.1 200 25 150 150 80 40 40 50,000 107 107

Source Henry 2006 (RECONS) Rocha-Pinto et al. 2000b Grether & Lineweaver 2007 b G99, R03, BF06 c R03, B05 e Valenti & Fischer 2005 g Nordstr¨m et al. 2004 o g Nordstr¨m et al. 2004 o j BS80, G96, E05 l D94, CB99, L00, BJ01, J03 Eke et al. 2004

Characteristic width of distribution in the direction of the solar value. a Wright et al. (2004), (see footnote in Sec. 2.2). b G99: Gustafsson et al. 1999, R03: Reddy et al. 2003, BF06: Bensby & Feltzing 2006. c R03: Reddy et al. 2003, B05: Bensby et al. 2005. d Solar rotational velocity corrected for random inclination (see Sec. 2.5). e Sub-set of stars within the mass range: 0.9 M M 1.1 M . f Calculated using the solar galactic motion (Dehnen & Binney 1998) and the Galactic p otential (see Sec. 2.6). g Sub-set of volume complete A5­K2 stars within 40 p c. h Integrated solar orbit in the Galactic p otential of Flynn et al. (1996) (see Sec. 2.6). i Eisenhauer et al. 2005. j BS80: Bahcall & Soneira 1980, G96: Gould et al. 1996, E05: Eisenhauer et al. 2005. k Stellar mass, not total baryonic mass, nor total mass. l D94: Driver et al. 1994, CB99: Courteau & van den Bergh 1999, L00: Loveday 2000, BJ01: Bell & de Jong 2001, J03: Jarrett et al. 2003.


a distribution for each prop erty of interest would allow a straightforward analysis and outcome: the Sun is within the n% of stars around the centroid of the N -dimensional distribution. However, observational and sample selection effects prevent the assembly of such an ideal stellar sample. In this study, we compare the Sun to other stars with resp ect to the following eleven basic physical prop erties: (1) mass, (2) age, (3) metallicity [Fe/H], (4) carb on-to-oxygen ratio [C/O], (5) magnesium-to-silicon ratio [Mg/Si], (6) rotational velocity v sin i, (7) eccentricity of the star's galactic orbit e, (8) maximum height to which the star rises ab ove the galactic plane Zmax , (9) mean galactocentric radius RGal , (10) the mass of the star's host galaxy Mgal , (11) the mass of the star's host group of galaxies Mgroup . These eleven prop erties span a wide range of stellar and galactic factors that may b e associated with habitability. We briefly discuss how each parameter might have a plausible correlation with habitability. For each prop erty we have tried to assemble a large, representative sample of stars whose selection criteria is minimally biased with resp ect to that prop erty. For each prop erty the p ercentage of stars with values lower and higher than the solar value are computed. For prop erties (9), (10) and (11), the uncertainties in the p ercentages are determined from the uncertainties of the distributions. For the rest of the prop erties, nominal uncertainties , on the p ercentages were calculated assuming a binomial distribution (e.g. Meyer 1975): = (nlow â nhigh /Ntot )1/2 where nlow (nhigh ) is the fraction of stars with a lower (higher) value than the Sun and Ntot is the total numb er of stars in the sample. The solar value is indicated with the symb ol " " in all figures. We compare the Sun and its environment to other stars and their environments. The analysis of these larger environmental con5

texts provides information ab out prop erties that otherwise could not b e directly measured. For example, supp ose the metallicity of the Sun were normal with resp ect to stars in the solar neighb orhood but that these stars as a group, had an anomalously high metallicity with resp ect to the average metallicity of stars in the Universe. This fact would strongly suggest that habitability is associated with high metallicity, but our comparison with only local stars would not pick this up. In the absence of an [Fe/H] distribution for all stars in the Universe, we use galactic mass as a convenient proxy for any such prop erty that correlates with galaxy mass. 2.1. Mass

Mass is probably the single most imp ortant characteristic of a star. For a main sequence star, mass determines luminosity, effective temp erature, main sequence life-time and the dimensions, UV insolation and temp oral stability of the circumstellar habitable zone (Kasting et al. 1993). Low mass stars are intrinsically dim. Thus a complete sample of stars can only b e obtained out to a distance of 7 parsecs ( 23 lightyears). Figure 1 compares the mass of the Sun to the stellar mass distribution of the 125 nearest main sequence stars within 7.1 p c, as compiled by the RECONS consortium (Henry 2006). Over-plotted is the stellar Initial Mass Function (IMF) (Kroupa 2002, Eqs. 4 & 5, Table 1) normalised to 125 stars more massive than the brown dwarf limit of 0.08 M . Since the IMF app ears to b e fairly universal (Kroupa & Weidner 2005), these nearby comparison stars are representative of a much larger sample of stars. There is good agreement b etween the histogram and the IMF -- the Sun is more massive than 95 ± 2% of the nearest stars, and more massive than 94±2% of the stars in the Kroupa (2002) IMF.


Fig. 1.-- Mass histogram of the 125 nearest stars (Henry 2006, RECONS). The median (µ1/2 = 0.33 M ) of the distribution is indicated by the vertical grey line. The 68% and 95% bands around the median are indicated resp ectively by the vertical dark grey and light grey bands. We also use these conventions in Figs. 2­11. The solid curve and hashed area around it represents the Initial Mass Function (IMF) and its associated uncertainty (Kroupa 2002). The Sun, indicated by " ", is more massive than 95 ± 2% of these stars.

Fourteen brown dwarfs and nine white dwarfs within 7.1 p c were not included in this sample. Including them yields 94% -- the same result obtained from the IMF. Our 95% ± 2% result should b e compared with the 91% rep orted by Gonzalez (1999b). The Sun's mass is the most anomalous of the prop erties studied here. 2.2. Age
2

(Carter 1983). Accurate estimation of stellar ages is difficult. For large stellar surveys (> a few hundred stars), the most commonly used age indicators are based on isochrone fitting and/or chromospheric activity (RHK index). Rocha-Pinto et al. (2000b) have estimated a Star Formation Rate (SFR), or equivalently, an age distribution for the local Galactic disk from chromospheric ages of 552 latetyp e (F8­K2) dwarf stars in the mass range 0.8 M M 1.4 M at distances d 200 p c (Rocha-Pinto et al. 2000a). They applied scale-height corrections, stellar evolution corrections and volume incompleteness corrections that converted the observed age distribution into the total numb er of stars b orn at any given time. Hernandez et al. (2000) and Bertelli & Nasi (2001) have made estimates of the star formation rate in the solar neighb orhood and favour a smoother distribution (fewer bursts) than Rocha-Pinto et al. (2000b). In Figure 2 we compare the chromospheric age of the Sun ( = 4.9 ± 3.0 Gyr, Wright et al. 2004) 2 to the stellar age distribution representing the Galactic SFR (Rocha-Pinto et al. 2000b). The median of this distribution is 5.4 Gyr. The Sun is younger than 53 ± 2% of the stars in the thin disk of our Galaxy. Over-plotted is the cosmic SFR derived by Hopkins & Beacom (2006). According to this distribution with a median µ1/2 = 9.15 Gyrs, the Sun was b orn after 86 ± 5% of the stars that have ever b een b orn. The Galactic and cosmic SFRs are different b ecause the cosmic SFR was dominated by bulges and elliptical galaxies in which the largest fraction of stellar mass in the Universe resides. Bulges and elliptical galaxies (earlyTo ensure that the Sun's age is determined in the same way as the stellar ages to which it is b eing compared, we adopt the chromospheric solar age = 4.9 ± 3.0 Gyr over the more accurate meteoritic age = 4.57 ± 0.002 Gyr (All`gre et al. 1995). e

If the evolution of observers like ourselves takes on average many billions of years, we might exp ect the Sun to b e anomalously old

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typ e galaxies) formed their stars early and quickly and then ran out of gas. The disks of spiral galaxies, like our Milky Way, seem to have undergone irregular bursts of star formation over a longer p eriod of time as they interacted with their satellite galaxies. The volume limited (dmax = 40 p c) subset from Nordstrom et al. (2004) contains ¨ isochrone ages for 1126 A5­K2 stars. The median of this sub-set is 5.9 Gyr and the Sun is younger than 55 ± 2% of the stars. The similarity of this isochrone age result to the chromospheric age result is not obvious since the agreement b etween these two age techniques is rather p oor. This mismatch can b e seen in Fig. 15D, Reid et al. (2007), and in Fig. 8 of Feltzing et al. (2001).

2.3.

Metallicity

Fig. 2.-- The Galactic stellar age distribution (median µ1/2 = 5.4 Gyr) from Rocha-Pinto et al. (2000b). The Sun is younger than 53 ± 2% of the stars in the disk of our Galaxy. The grey curve is the cosmic Star Formation Rate (SFR) with its associated uncertainty (Hopkins & Beacom 2006), according to which the Sun is younger than 86 ± 5% of the stars in the Universe.

Iron is the most frequently measured element in nearby stars. Metallicity [Fe/H], is known to b e a proxy for the fraction of a star's mass that is not hydrogen or helium. In the Sun and p ossibly in the Universe, the dominant contributors to this mass fraction in order of abundance are: O(44%), C(18%), Fe(10%), Ne(8%), Si(6%), Mg(5%), N(5%), S(3%) (Asplund et al. 2005; Truran & Heger 2005). The corresp onding abundances by numb er are: O(48%), C(26%), Ne(7%), N(6%), Mg (4%), Si(4%), Fe(3%), S(2%). Imp ortantly for this analysis, this short list contains the dominant elements in the comp osition of terrestrial planets (O, Fe, Si and Mg) and life (C, O, N and S). Over the last few decades, much effort has gone into determining abundances in nearby stars for a wide range of elements. Stellar elemental abundances for element X are usually normalised to the solar abundance of the same element using a logarithmic abundance scale: [X/H] log(X/H) - log(X/H) . Hence all solar elemental abundances [X/H] , are defined as zero. Sp ectroscopic abundance analyses are usually made differential relative to the Sun by analysing the solar sp ectrum (reflected by the Moon, asteroids or the telescop e dome) in the same way as the sp ectrum of other stars. In this approach, biases introduced by the assumption of local thermodynamic equilibrium (LTE), largely cancel out for Sun-like stars (Edvardsson et al. 1993b). A comparison b etween solar and stellar iron abundances is a common feature of most abundance surveys and most have concluded that the Sun is metal-rich compared to other stars (Gustafsson 1998; Gonzalez 1999a,b). However, for our purp oses, the appropriateness of these comparisons dep ends on the selection criteria of the stellar sample to which the Sun has b een compared. Stellar metallicity analyses such 7


as Edvardsson et al. (1993a); Reddy et al. (2003); Nordstrom et al. (2004); Valenti & Fischer ¨ (2005) have stellar samples selected with different purp oses in mind, e.g., Edvardsson et al. (1993a) aimed to constrain the chemical evolution of the Galaxy and their sample is biased towards low metallicity (average [Fe/H]= -0.25). The sample of Valenti & Fischer 2005 (average [Fe/H]= -0.01), was selected as a planet candidate list and contains some bias towards high metallicity (see Grether & Lineweaver 2007). To assess how typical the Sun is, Gustafsson (1998) limited the sample of Edvardsson et al. (1993a) to stars with galactocentric radii within 0.5 kp c of the solar galactocentric radius, and to ages b etween 4 and 6 Gyrs. The distribution of stars given by this criteria has an average [Fe/H]= -0.09. Grether & Lineweaver (2006, 2007) compiled a sample of 453 Sun-like stars within 25 p c. These stars were selected from the Hipparcos catalogue, which is essentially complete to 25 p c for stars within the sp ectral typ e range F7­K3 and absolute magnitude of MV = 8.5 (Reid 2002). Metallicities for this sample were assembled from a wide range of sp ectroscopic and photometric surveys. In Figure 3, we compare the Sun to the Grether & Lineweaver (2007) sample, which has a median [Fe/H]= -0.08. To our knowledge this is the most complete and leastbiased stellar sp ectroscopic metallicity distribution. The Sun is more metal-rich than 65 ± 2% of these stars. This result should b e compared with Favata et al. (1997) who constructed a volumelimited (dmax = 25 p c) sample of 91 G and K dwarfs ranging in color index (B - V ) b etween 0.5 - 0.8 (Favata et al. 1996). Their distribution has a median [Fe/H]= -0.05 and compared to this sample, the Sun is more metal rich than 56 ± 5% of the stars. Fuhrmann (2008) compared the Sun to a volume complete (dmax = 25 p c) sample of ab out 185 thin-disk mid-F-typ e to early K-typ e stars 8

down to MV = 6.0. He finds a mean [Fe/H] = -0.02 ± 0.18. This mean [Fe/H] is lowered by 0.01 dex if the 43 double-lined sp ectroscopic binaries in his sample are included. His results are consistent with ours.

Fig. 3.-- Stellar metallicity the 453 FGK Hipparcos stars (Grether & Lineweaver 2007). µ1/2 = -0.08. The Sun is m than 65 ± 2% of the stars.

histogram of within 25 p c The median ore metal-rich

2.4.

Elemental ratios [C/O] and [Mg/Si]

The elemental abundance ratios of a host star have a ma jor impact on its protoplanetary disk chemistry and the chemical comp ositions of its planets. Oxygen and carb on make up 62% of the Solar System's non-hydrogen-non-helium mass content (Z = 0.0122, Asplund et al. 2005). Carb on and oxygen abundances are among the hardest to determine. This is due to high temp erature sensitivity and non-LTE effects in their p ermitted lines (e.g. C I 6588, O I 7773), and to the presence of blends in the forbidden lines ([C I] 8727, [O I]


6300). See Allende Prieto et al. (2001) and Bensby & Feltzing (2006) for details on C and O abundance derivations. Carb on pairs up with oxygen to form carb on monoxide. In stars with a C/O ratio larger than one, most of the oxygen condenses into CO which is largely driven out of the incipient circumstellar habitable zone by the stellar wind. In this oxygen-depleted scenario, planets formed within the snowline are formed in reducing environments and are mostly comp osed of carb on comp ounds, e.g. silicon carbide (Kuchner & Seager 2005). Thus, the C/O ratio could b e strongly associated with habitability. As most heavy element abundances relative to hydrogen (e.g. [O/H], [C/H], [N/H]) are correlated with [Fe/H], they were not included in our analysis. After the overall level of metallicity (represented by [Fe/H]), and after the ratio of the two most abundant metals, [C/O], the magnesium to silicon ratio [Mg/Si] is the most imp ortant ratio of the next most abundant elements (excluding the noble gas Ne). For example [Mg/Si] sets the ratio of olivine to pyroxene which determines the ability of a silicate mantle to retain water (Hugh O'Neill, private communication). Stellar elemental abundance ratios are defined as [X1 /X2 ] = [X1 /H] - [X2 /H] . Hence, systematic errors associated with the determination of absolute solar abundances cancel for abundances relative-to-solar. We compile [C/O] and [Mg/Si] ratios from samples with the largest numb er of stars and highest signal-to-noise stellar sp ectra: [C/O]: 256 stars from Gustafsson et al. 1999; Reddy et al. 2003; Bensby & Feltzing 2006 [Mg/Si]: 231 stars from Reddy et al. 2003; Bensby et al. 2005 Due to their selection criteria, these samples are biased towards low metallicity and 9

therefore cannot b e used to create a representative [Fe/H] distribution. Because a correlation exists b etween the [C/O] and [Mg/Si] ratios and [Fe/H] (e.g. Gustafsson et al. 1999), the samples we use have a relatively narrow range of [Fe/H] to reduce the influence of the correlation. Therefore, these small correlations can b e neglected in this study -- see the b ottom panels of Fig. 4 where [Fe/H] versus [C/O] as well as [Fe/H] versus [Mg/Si] are plotted. The top panels show the corresp onding stellar distribution histograms. The Sun's [C/O] ratio is lower than 81 ± 3% of the stars. This is consistent with Gonzalez (1999b) who suggested -- based on data from Edvardsson et al. (1993a) and Gustafsson et al. (1999) -- that the Sun has a low [C/O] ratio relative to Sun-like stars at similar galactocentric radii. See however, Ram´rez et al. (2007) who find that the Sun i is oxygen p oor compared to solar metallicity stars. The Sun's [Mg/Si] ratio is lower than 66 ± 3% of the stars. The [C/O] and [Mg/Si] ratios are also largely indep endent of each other (see Fig. 14 in App endix A). 2.5. Rotational velocity

Stellar rotational velocities are related to the sp ecific angular momentum of a protoplanetary disk and p ossibly to the magnetic field strength of the star during planet formation, and to protoplanetary disk turbulence and mixing. An unusually low stellar rotational velocity may b e associated with the presence of planets (Soderblom 1983). One or several of these factors could b e related to habitability. There is a known correlation b etween mass and v sin i at higher stellar masses (e.g. see Fig. 18.21 of Gray 2005, p. 485). In order to minimise the effect of this correlation (and maximize indep endence b etween parameters),


Fig. 4.-- A: Comparison of the Sun's carb on-to-oxygen ratio ([C/O] 0) to the [C/O] ratios of 256 stars compiled from Gustafsson et al. (1999); Reddy et al. (2003) and Bensby & Feltzing (2006). The Sun's [C/O] ratio is lower than 81 ± 3% of the stars in this sample which has a median µ1/2 = 0.07. B: Comparison of the Sun's magnesium-to-silicon ratio ([Mg/Si] 0), to [Mg/Si] values from 231 stars from Reddy et al. (2003) and Bensby et al. (2005). The Sun's [Mg/Si] ratio is lower than 66 ± 3% of the stars in this sample with median µ1/2 = 0.01. The b ottom panels C & D show the small correlations of these distributions with [Fe/H]. These small correlations can b e neglected for this study.

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we assembled a sample containing 276 stars within the mass range 0.9­1.1 M (F8­K2) from Valenti & Fischer (2005). The selection criteria of the Valenti & Fischer (2005) stars introduces some bias against more active stars. We compared the high v sin i tail of our Valenti & Fischer (2005) sample with the high v sin i tail of a sub-sample from Nordstrom et al. (2004). We estimate ¨ that for our Valenti & Fischer (2005) sample, the bias introduced by the selection criteria is lower than 5%. The v sin i values in Valenti & Fischer (2005) are obtained by fixing the macroturbulence for the stars of a given color, without modeling the stars individually. If the macroturbulence value was underestimated for T > 5800 K, the resulting v sin i values (esp ecially when v sin i is near zero) would b e overestimated (Sec. 4 of Valenti & Fischer 2005). The inclination of the stellar rotational axis to the line of sight is usually unknown so the observable is v sin i. Using the solar sp ectrum reflected by the asteroid Vesta, Valenti & Fischer (2005) derived a solar v sin i = 1.63 km s-1 . For the purp oses of this analysis we use the mean value that would b e derived for the Sun, when viewed from a random inclination: v sin i = 1.63( /4) km s-1 1.28 km s - 1 . The Sun rotates more slowly than 83 ± 7% of the stars in our Valenti & Fischer (2005) sample (Fig. 5). This is in agreement with Soderblom (1983, 1985) who rep orted that the Sun is within one standard deviation of stars of its mass and age. 2.6. Galactic orbital parameters

Fig. 5.-- Rotational velocity histogram for 276 F8­K2 (0.9 M 1.1 M ) stars (Valenti & Fischer 2005). The Sun (v sin i = 1.28 km s-1 ) rotates more slowly than 83 ± 7% of the stars. There is one star to the right of the plot with v sin i = 36 km s-1 .

The Galactic velocity comp onents of a star (U ,V ,W ) with resp ect to the local standard of rest (LSR) may b e used to compute a star's orbit in the Galaxy. How typical or atypical is the solar orbit compared to the orbits of other nearby stars in the Galaxy? The orbit may b e 11

related to habitability b ecause more eccentric orbits bring a star closer to the Galactic center where there is a larger danger to life from sup ernovae explosions, cosmic gamma and Xray radiation and any factors associated with higher stellar densities (Gonzalez et al. 2001; Lineweaver et al. 2004). For a standard model of the Galactic p otential, Nordstrom et al. (2004) computed or¨ bital paramters for the Sun, and for a large sample ( 16700) of A5­K2 stars. Their adopted comp onents of the solar velocity relative to the local standard of rest were (U, V , W ) = (10.0± 0.4, 5.25 ± 0.62, 7.17 ± 0.38) km s-1 (Dehnen & Binney 1998). For each of the 1,987 stars within 40 p c in the Nordstrom et al. (2004) catalog, an in¨ ner and outer radii Rmin and Rmax were computed. This yielded the orbital eccentricity e (Rmax - Rmin )/(Rmin + Rmax ). The solar


eccentricity was computed using the comp onents of the solar motion (Dehnen & Binney 1998) relative to the local standard of rest in the Galactic p otential of Flynn et al. (1996). The b ottom panel of Figure 6 shows the correlation b etween Galactic orbital eccentricity e and the magnitude of the galactic orbital velocities with resp ect to the local standard of rest: vLSR (U 2 +V 2 +W 2 )1/2 . Eccentricity e and vLSR are strongly correlated. We include e, not vLSR , in the analysis since e is less correlated with the maximum height ab ove the Galactic plane Zmax , than is vLSR . This is shown in Fig. 16 in App endix A. The Sun's eccentricity was determined with the same relation as the stellar eccentricities. The uncertainty in our estimate of solar eccentricity came from propagating the uncertainty in the adopted solar motion. We find e = 0.036 ± 0.002 (consistent with the e = 0.043 ± 0.016 found by Metzger et al. 1998). The Sun has a more circular orbit than 93 ± 1% of the A5­K2 stars within 40 p c (with median eccentricity µ1/2 = 0.1). This is the second most anomalous of the eleven solar prop erties we consider here. The frequency of the passage of a star through the thin disk could b e associated with Galactic gravitational tidal p erturbations of Oort cloud ob jects that might increase the impact rate on p otentially habitable planets. This is correlated with the maximum height, Zmax , to which the stars rise ab ove the Galactic plane. Figure 7 shows the stellar distribution of Zmax for the stars shown in Figure 6. We find that 59 ± 3% of the A5­K2 stars within 40 p c of the Sun reach higher ab ove the Galactic plane than the Sun does (Zmax, = 0.104 ± 0.006 kp c). The solar Zmax, was derived by integrating the solar orbit in the Galactic p otential. The uncertainty on W , produces the uncertainty on Zmax, and hence the ±3% uncertainty on 59%. Our results for eccentricity and Zmax are consistent 12

Fig. 6.-- Top panel: eccentricity distribution for the 1,987 stars at d 40 p c from Nordstrom et al. (2004). The Sun has a more ¨ circular orbit than 93 ± 1% of the A5­K2 stars within 40 p c. After mass, eccentricity is the second most anomalous parameter. Bottom panel: Correlation b etween vLSR and eccentricity for the same stars presented in the top panel. Since these prop erties are highly correlated we select only one for the analysis. The large grey p oint with error bars represents the median and the 68% widths of the two onedimensional distributions. As in Fig. 4 the contours corresp ond to 38%, 68%, 82% and 95%.


+ halo) comp onents. Using the current Solar distance from the center (R0 = 7.62 ± 0.32 kp c, Eisenhauer et al. 2005) and a disk scale length h = 3.0 ± 0.4 kp c (Gould et al. 1996), we estimate that the Sun lies farther from the Galactic center than 72+8 % of the stars in the -5 Galaxy. The uncertainty on the result comes from the 68% b ounds of the total distribution, which come from the scale length uncertainty (±0.4 kp c).

Fig. 7.-- The distribution of maximum heights ab ove the Galactic plane for the Nordstrom et al. (2004) sample. 59 ± 3% ¨ of nearby A5­K2 stars (dmax =40 p c) reach higher ab ove the Galactic plane than the Sun reaches. There are 22 stars evenly distributed over Zmax b etween 1.5 and 9.6 kp c. Their exclusion from the comparison reduces the 59% result by less than 1%.

with those obtained using Hogg et al. (2005) LSR values: (U, V , W ) = (10.1 ± 0.5, 4.0 ± 0.8, 6.7 ± 0.2). Using the Hogg et al. LSR values, 92 ± 1% of A5­K2 stars within 40 p c have higher eccentricities than the Sun and 62 ± 4% of A5­K2 stars within 40 p c have larger Zmax values. How does the Sun's distance from the center of the Milky Way compare to the distances of other stars from the center of the Milky Way? In Fig. 8 we show the distribution of the mean radial distances of stars from the Galactic center, based on the star count model of Bahcall & Soneira (1980). To represent the entire Galactic stellar p opulation we include the disk (thin + thick) and spheroidal (bulge

Fig. 8.-- Mean stellar galactocentric radius distribution dN /dRGal . The solid curve represents the sum of the disk (dashed line) and spheroidal (dotted line) stellar comp onents. The 68% uncertainty of the total distribution is shown by the cross-hatched area. The Sun is farther from the Galactic center than 72+8 % -5 of the stars in the Galaxy.

2.7.

Host galaxy mass

The mass of a star's host galaxy may b e correlated with parameters that have an influence on habitability. For example, galaxy mass affects the overall metallicity distribution that a star would find around itself -- an 13


effect that would not show up in Fig. 3, which only shows the local metallicity distribution. The Milky Way is more massive than 99% of all galaxies -- the precise fraction dep ends on the lower mass-limit chosen for an ob ject to b e classified as a galaxy, and the b ehaviour of the low-mass end of the galaxy mass function (Silk 2007). We are referring here to the stellar mass, not the total baryonic mass or the total mass. Despite the Milky Way's large mass compared to other galaxies, if most stars in the Universe resided in even more massive galaxies, the Milky Way would b e a rather low mass galaxy for a star to b elong to. To estimate the fraction of all stars in galaxies of a given mass, we first estimate the distribution of galaxy masses by taking the K-band luminosity function of Loveday (2000) (K-band most closely reflects stellar mass since it is less sensitive than other bands to differences in stellar p opulations) and weighting it by luminosity. We convert this to stellar mass assuming a constant stellar-mass-to-light ratio of 0.5 (Bell & de Jong 2001). This function, plotted in Fig. 9, shows the amount of stellar mass contributed by galaxies of a given mass -- or assuming identical stellar p opulations -- the fraction of stars residing in galaxies of a given stellar mass. We estimate the K-band luminosity of the Milky Way by converting the published Vband magnitude of Courteau & van den Bergh (1999) to the K-band assuming the mean color of an Sb c spiral galaxy from the 2 MASS Large Galaxy Atlas (Jarrett et al. 2003) and applying the color conversion from (Driver et al. 1994). We then convert this to stellar mass using the same stellar-mass-to-light ratio used ab ove, i.e., 0.5. In this way we estimate the stellar-mass content of the Milky Way to b e 1010.55±0.16 = 3.6+1.5 â 1010 M (see -1.1 also Flynn et al. 2006). Comparing this to the stellar masses of other galaxies (Fig. 9), 14

Fig. 9.-- Fraction of all stars that live in galaxies of a given mass, dN /dM (solid curve). The mass of the Sun's galaxy is indicated by the " ". This distribution represents the amount of stellar mass contributed by galaxies of a given mass. Approximately 77+11 % of stars live in galaxies less massive -14 than ours. The cross-hatched band shows the 1 uncertainty associated with the uncertainty in the two Schechter function parameters, and L (Loveday 2000; Schechter 1976). The dashed line shows the unweighted luminosity function (the numb er of galaxies p er luminosity interval dNgals /dM ) according to which the Milky Way is more massive than 99% of galaxies.

we find that 77+11 % of stars reside in galaxies -14 less massive than the Milky Way. 2.8. Host group mass

The mass of a star's host galactic group or galactic cluster may b e correlated with parameters that have an influence on habitability. For example, group mass is correlated with the density of the galactic en-


vironment (numb er of galaxies/Mp c3 ) which could, like galactocentric radius, b e associated with the dangers of high stellar densities: "The presence of a giant elliptical at a distance of 50 kp c would have disrupted the Milky Way Galaxy, so that human b eings (and hence astronomers) probably would not have come into existence." (van den Bergh 2000). Our Local Group of galaxies seems rather typical (van den Bergh 2000) but we would like to quantify this. Proceeding similarly to our analysis of galaxy mass in Sect. 2.7, we ask: What fraction of stars live in galactic groups less massive than our Local Group? Figure 10 shows the luminosityweighted ( stellar-mass-weighted) numb er density of galactic groups. The numb er distribution and luminosity distribution of galactic groups is taken from the Two-degree Field Galaxy Redshift Survey Percolation-Inferred Galaxy Group (2PIGG) catalogue (Eke et al. 2004). It spans the range from weak groups to rich galaxy clusters. We estimated the stellar masses of the 2PIGG groups and Local Group galaxies (Courteau & van den Bergh 1999) by converting from the B-band assuming a constant stellar-mass-to-light ratio of 1.5 (Bell & de Jong 2001). This gives an estimated stellar mass of the local group of 1010.91±0.07 = 8.1+1.4 â -1.2 1010 M . Figure 10 indicates that our Local Group is a typical galactic grouping for a star to b e part of. Approximately 58 ± 5% of stars live in galactic groups more massive than our Local Group. With resp ect to the mass of its galaxy and the mass of its galactic group, the Sun is a fairly typical star in the Universe. 3. 3.1. Joint Analysis of 11 Solar Prop erties Solar 2 -analysis

Fig. 10.-- The dashed histogram shows the luminosity function of galactic groups (numb er of groups p er interval of B-band luminosity). The solid histogram shows the luminosity-weighted group luminosity function (approximately the fraction of stars which inhabit a group of given stellar mass). The horizontal axis has b een converted to stellar mass assuming a constant B-band stellarmass-to-light ratio of 1.5 (Bell & de Jong 2001). The " " shows the estimated mass of the Local Group (Courteau & van den Bergh 1999) and lies just b elow the median (vertical grey line).

noise, i.e., are they consistent with the values of a star selected at random from our stellar distributions. We take a 2 approach to answering this question. First we estimate the solar 2 , by adding in quadrature, for all eleven prop erties, the differences b etween the solar values and the median stellar values. We find:
N =11

We would like to know if the solar properties, taken as a group, are consistent with 15

2 =
i=1

(x

,i

- µ1/2,i )2 = 7.88+ - 2 68,i

0.08 0.30

(1)


where i is the prop erty index, N = 11 is the numb er of prop erties we are considering, µ1/2,i is the median of the ith stellar distribution and 68,i is the difference b etween the median and the upp er or lower 68% zone, dep ending on whether the solar value x ,i is ab ove or b elow the median. The uncertainty on 2 is obtained using the uncertainties of x ,i . Equation (1) can b e improved up on by taking into account: i) the non-Gaussian shap es of the stellar distributions and ii) the larger uncertainties of the medians of smaller samples (our smallest sample is 100 stars). We employ a b ootstrap analysis (Efron 1979) to randomly resample data (with replacement) and derive a more accurate estimate of 2 . Because the b ootstrap is a nonparametric method, the distributions need not b e Gaussian. We obtain 2 = 8.39 ± 0.96. Figure 11 shows the resulting solar chi-squared distribution. The median of this distribution is our adopted solar chi-squared value. Dividing our adopted solar chi-squared by the numb er of degrees of freedom gives our adopted reduced solar chi-squared value: 2 /11 = 0.76 ± 0.09 (2)

Fig. 11.-- Bootstrapp ed solar chi-squared distribution. The median of the distribution (white " ") is 2 = 8.39 ± 0.96. This should b e compared to the solar 2 value from Eq. 1: 7.88+0.08 which is over-plotted (grey " " -0.30 on dotted line).

The standard conversion of this into a probability of finding a lower chi-squared value (assuming normally distributed indep endent variables) yields: P (< 2 = 8.39|N = 11) = 0.32 ± 0.09. (3) 3.2. Estimate of P (< 2 )

The histogram shown in Figure 12 is the resulting Monte Carlo stellar 2 distribution. Three standard chi-squared distributions have b een over-plotted for comparison (N = 10, 11, 12). The probability of finding a star with chi-squared lower than or equal to solar is: PMC ( 2 = 8.39|N = 11) = 0.29 ± 0.11 (4) The Monte Carloed 2 distribution has a similar shap e to the standard chi-squared distribution function for N = 11, and thus b oth yield similar probabilities: PMC ( 2 ) = 0.29 P ( 2 ) = 0.32 (Eqs. 3 and 4). The more appropriate Monte Carlo distribution has a longer tail, produced by the longer sup erGaussian tails of the stellar distributions. Table 2 summarizes our analysis for the Solar 2 values and the probabilities P (< 2 ). 16

To quantify how typical the Sun is with resp ect to our 11 prop erties, we compare the solar 2 (= 8.39) to the distribution of 2 values obtained from the other stars in the samples. We p erform a Monte Carlo simulation (Metrop olis & Ulam 1949) to calculate an estimate of each star's chi-squared value (2 ).


Our simple 2 = 7.88 estimate increased to 8.39 and the uncertainty increased by a factor of 3 after non-Gaussian and sample size effects were included as additional sources of uncertainty. Our improved analysis yields PMC ( 2 ), with a longer tail and brings the probability down from 0.32 ± 0.09 to 0.29 ± 0.11. If this value were close to 1, almost all other stars would have lower chisquared values and we would have good reason to susp ect that the Sun is not a typical star. However, this preliminary low value of 0.29 indicates that if a star is chosen at random, the probability that it will b e more typical ( have a lower 2 value) than the Sun (with resp ect to the eleven prop erties analysed here), is only 29 ± 11%. The details of our improved estimates of 2 and P (< 2 ) can b e found in the App endix B. 4. Results

Figure 13 shows four different representations of our results. Panel (A) compares the solar values to each stellar distribution's median and 68% and 95% zones. The Sun lies b eyond the 68% zone for three prop erties: mass (95%), eccentricity (93%) and rotational velocity (88%). No solar prop erty lies b eyond the 95% zone. The histogram in panel (B) is the distribution of solar values in units of standard deviations: zi = x
,i

Fig. 12.-- Stellar chi-squared distribution from our Monte Carlo simulation. PMC (< 2 = 8.39) = 0.29 ± 0.11 (represented by the grey shade) is calculated integrating from 2 = 0 to 2 = 2 . For comparison, three 2 distribution-curves are over-plotted with 10, 11, and 12 degrees of freedom. The standard probability from the N = 11 curve yields: P (< 2 = 8.39|N = 11) = 0.32 ± 0.09.

-µ 68,i

1/2,i

(5)

For each stellar prop erty i, the Sun has a larger value than ni % of the stars. If the Sun were a randomly selected star, we would exp ect the p ercentages ni % to b e scattered roughly evenly b etween 0% and 100%. When the ni % values are lined up in decreasing order (panel C), we exp ect them to b e near the line given by: n
i,expected

%= 1-

(i - 1/2) â 100% (6) N 17

and plotted in Panel C. Any anomalies would show up as p oints ` ' significantly distant from the line. Panel (D) compares the p ercentages ni % of stars having sub-solar values (shown in Panel C) with the solar values expressed in units of standard deviations from each distribution's median (shown in Panel B). If the stellar distributions were p erfect Gaussians, the translation from zi to ni would b e given by the cumulative Gaussian distribution (black line in Panel D). That the p oints lie along this line demonstrates that the approximation of our distributions as Gaussians is reasonable. Table 3 lists p ercentages ni % of stars for each prop erty (as shown in Fig. 13). In the


Table 2 Summary of and P (< 2 ) results.
2

Analysis simple improved



2

2 /11
+ 0.72-0.01 0.03 0.76 ± 0.09 (Eq. 2)

P (< 2 |N = 11) 0.28+0.01 (Eq. B1) -0.03 0.32 ± 0.09 (Eq. 3)

PMC (< 2 |N = 11) -- 0.29 ± 0.11 (Eq. 4)

7.88+0.08 (Eq. 1) -0.30 8.39 ± 0.96

lower half of the table we list prop erties not included in this analysis b ecause of correlations with prop erties that are included. Individual stellar uncertainties make the observed characteristic widths (68 , column 5 of Table 1) larger than the widths of the intrinsic distributions. This broadening effect makes the Sun app ear more typical than it really is when 68 and the individual stellar uncertainties ( ) are of similar size and the individual stellar uncertainties are much larger than the solar uncertainty ( ). We estimate that our results are not significantly affected by this broadening effect. Our resulting probability of finding a star with a 2 lower or equal to the solar value of 29 ± 11% (Eq. 4), is consistent with the probability we would obtain if stellar multiplicity were included in our study. Using the volume limited sample used for stellar mass in Section 2.1 (125 A1­M7 stars within 7.1 p c) the probability that a randomly selected star will b e single is 52.8 ± 4.5%, which means that half of stars are single while the other half have one or more companions. Including this in our b ootstrap analysis and Monte Carlo simulations (see App endix B.1) marginally increases the probability in Eq. 4 to 33 ± 11%. If the multiplicity data for 246 G dwarfs from Duquennoy & Mayor (1991) is used instead -- the probability that a randomly selected G dwarf will b e single is 37.8 ± 2.9% -- then the probability in Eq. 4 would increase to 18

34 ± 11%. The inclusion of stellar multiplicity marginally increases our rep orted probability. In Figures 6 and 7 of Radick et al. (1998), the Sun's short-term variability as a function of average chromospheric activity, app ears 1 low, compared to a distribution of 35 F3­K7 Sun-like stars (Lockwood et al. 1997). Lockwood et al. (2007) suggest that the Sun's small total irradiance variation compared to stars with similar mean chromospheric activity, may b e due to their limited sample and the lack of solar observations out of the Sun's equatorial plane. We do not include short or long term variability (chromospheric or photometric) in Table 3 b ecause of the small size of the Lockwood et al. (2007) sample. We also do not include the chromospheric index RHK (see Table 3, b ottom panel) as one of our 11 prop erties b ecause of its correlation with the chromospheric ages of our sample.


Fig. 13.-- Various representations of our main results. A: Solar values of eleven prop erties compared to the distribution for each prop erty Each distribution's median value is indicated by a small filled circle. The dark and light grey shades represent the 68% and 95% zones resp ectively. B: Histogram of the numb er of prop erties as a function of the numb er of standard deviations the solar value is from the median of that prop erty. The grey curve is a Gaussian probability distribution normalised to 11 parameters. C: Percentage ni % of stars with sub-solar values as a function of prop erty. The average signal exp ected from a random star is shown by the solid line (see Sec. 4). D: Percentage ni % of stars with sub-solar values as a function of the numb er of standard deviations the solar value is from the median of that prop erty. The solid curve is a cumulative Gaussian distribution -- if every sample were a Gaussian distribution, every solar dot would sit exactly on the line. Just as in (C), the dashed lines encompass the 68% and 95% zones. Similar to the results from Figure 12, these four panels indicate that the Sun is a typical star.

19


Table 3 Summary of How the Sun Compares to Other Stars (see Fig. 13)
Parameter Mass Age [Fe/H] [C/O] [Mg/Si] v sin i e Zmax RGal Mgal Mgroup Fig. 1 2 3 4A 4B 5 6 7 8 9 10 ni % 95 ± 2% 53 ± 2% 65 ± 2% 81 ± 3% 66 ± 3% 83 ± 7% 93 ± 1% 59 ± 3% 72+8 % -5 77+11 % -14 58 ± 5% of of of of of of of of of of of nearby stars in nearby nearby nearby nearby nearby nearby stars in stars in stars in Level of Anomaly stars are less massive than the Sun. the thin disk of the Galaxy are older than the Sun. stars are more iron-p oor than the Sun. stars have a higher C/O ratio than the Sun. stars have a higher Mg/Si ratio than the Sun. Sun-like-mass stars rotate faster than the Sun. stars have larger galactic orbital eccentricities than the Sun. stars reach farther from the Galactic plane than the Sun. the Galaxy are closer to the galactic center than the Sun. the Universe are in galaxies less massive than the Milky Way. the Universe are in groups more massive than the local group.

Prop erties not included in the analysis b ecause they are correlated with the selected 11 parameters Mass: IMFStellar Age: SFRCosmic Agea [Fe/H]b v sin ic log RHK d [O/Fe] Rmin vLSR |U | |V | |W |
a b c

1 2 -- -- -- -- -- -- -- -- -- --

94 86 55 56 92 51 75 91 93 75 82 58

± ± ± ± ± ± ± ± ± ± ± ±

2% 5% 2% 5% 5% 2% 3% 1% 1% 1% 1% 1%

of of of of of of of of of of of of

nearby stars in nearby nearby nearby nearby nearby nearby nearby nearby nearby nearby

stars are more massive than the Sun. the Universe are older than the Sun. Sun-like-mass stars are older than the Sun. stars are more iron-p oor than the Sun. Sun-like-mass stars rotate faster than the Sun. FGKM stars are more chromospherically active. stars have a lower O/Fe ratio than the Sun. stars get closer to the Galactic center. stars have smaller velocity with resp ect to the LSR. stars have larger absolute radial velocity. stars have larger absolute tangential velocity. stars have larger absolute vertical velocity.

1126 stars (A5­K2) from Nordstr¨m et al. (2004). o 91 stars (GK) from Favata et al. (1997).

590 stars (F8­K2) from Nordstr¨m et al. (2004). o 866 stars (FGKM) from Wright et al. (2004).

d

20


5.

Discussion and Interpretation

The probability PMC ( 2 ) = 0.29 ± 0.11 classifies the Sun as a typical star. How robust is this result? The probability of finding a star with a chi-squared lower than or equal to 2 , dep ends on the prop erties selected for the analysis (see problem iii of Section I). For example, if we had chosen to consider only mass and eccentricity data, this analysis would yield PMC (2 2 ) = 0.94 ± 0.4, i.e., the Sun would app ear mildly ( 2 ) anomalous. If on the other hand, we had chosen to remove mass and eccentricity from the analysis, we would obtain PMC (2 2 ) = 0.07 ± 0.04, which is anomalously low. The most common cause of such a result is the over-estimation of error bars. The next most common cause is the preselection of prop erties known to have ni % 50%. Gustafsson (1998) discussed the atypically large solar mass, and prop osed an anthropic explanation -- the Sun's high mass is probably related to our own existence. He suggested that the solar mass could hardly have b een greater than 1.3 M since the main sequence lifetime of a 1.3 M star is 5 billion years (Clayton 1983). He also discussed how the dep endence of the width of the circumstellar habitable zone on the host star's mass probably favours host stars within the mass range 0.8­1.3 M . Our prop erty selection criteria is to have the largest numb er of maximally indep endent prop erties that have a plausible correlation with habitability and, ones for which a representative stellar sample could b e assembled. Our joint analysis does not weight any parameter more heavily than any other. If the only prop erties associated with habitability are mass and eccentricity then we have diluted a 2 signal that would b e consistent with Gustafsson's prop osed anthropic explanation. Our analysis p oints in another direction. If

mass and eccentricity were the only prop erties associated with habitability, then the solar values for the remaining 9 prop erties would b e consistent with noise. However, a joint analysis of just the remaining 9 prop erties produces a 2 ,9 = 3.6 ± 0.4 and the anomalously low probability: P ( 2 ,9 ) = 0.07 ± 0.04, which suggests that the 9 prop erties are unlikely to b e the prop erties of a star selected at random with resp ect to these prop erties. The 2 fit of the 11 p oints in Panel C of Fig. 13 to the diagonal line yields a fit that is substantially b etter then the fit of the remaining 9 prop erties to Eq. 7 with N = 9. In other words, the joint analyis suggests that although mass and eccentricity are the most anomalous solar prop erties, it is unlikely that they are associated with habitability, b ecause without them, it is unlikely that the remaining solar prop erties are just noise. Thus, the Sun, despite its mildly ( 2 ) anomalous mass and eccentricity, can b e considered a typical, randomly selected star. There may b e stellar prop erties crucial for life that were not tested here. If we have left out the most imp ortant prop erties, with resp ect to which the Sun is atypical, then our Sun-is-typical conclusion will not b e valid. If we have sampled all prop erties associated with habitability, our Sun-is-typical result suggests that there are no sp ecial requirements on a star for it to b e able to host a planet with life. 6. Conclusions

We have compared the Sun to representative stellar samples for eleven prop erties. Our main results are: · Stellar tricity ties. T 2% of orbital 21 mass and Galactic orbital eccenare the most anomalous prop erhe Sun is more massive than 95 ± nearby stars and has a Galactic eccentricity lower than 93 ± 1%


FGK stars within 40 p c. · Our joint b ootstrap analysis yields a solar chi-squared 2 = 8.39 ± 0.96 and a solar reduced chi-squared 2 /11 = 0.76 ± 0.09. The probability of finding a star with a chi-squared lower than or equal to solar PMC ( 2 = 8.39 ± 0.96) = 0.29 ± 0.11. To our knowledge, this is the most comprehensive and quantitative comparison of the Sun with other stars. We find that taking all eleven prop erties together, the Sun is a typical star. This finding is largely in agreement with Gustafsson (1998), however our results undermine the prop osition that an anthropic explanation is needed for the comparatively large mass of the Sun. Further work could encompass the inclusion of other prop erties p otentially associated with habitability. Another improvement would come when larger stellar samples b ecome available for which all prop erties could b e derived, instead of using different samples for different prop erties as was done here. In addition, research in the molecular evolution that led to the origin of life may, in the future, b e able to provide more clues as to which stellar prop erties might b e associated with our existence on Earth, orbiting the Sun. Acknowledgments: We would like to thank Charles Jenkins for clarifying discussions of statistics, particularly on how to include stellar multiplicity, and Martin Asplund and Jorge Mel´ndez for discussions of elemental e abundances. JAR acknowledges an RSAA PhD research scholarship. MP acknowledges the financial supp ort of the Australian Research Council. EG acknowledges the financial supp ort of the Finnish Cultural Foundation.

22


A.

Prop erty-correlations

The 2 formalism and the use of the 2 -distribution to obtain P (< 2 |N ), -- improved using Monte-Carlo simulations in Section 3.2 to obtain PMC ( 2 ) -- assumes that each parameter is indep endent of the others. In selecting our 11 prop erties we have selected prop erties which are maximally indep endent based on plotting prop erty 1 vs prop erty 2 for the same stars. We show seven such plots in this App endix. If there are correlations b etween the analysed prop erties, then the numb er of degrees of freedom N could drop from 11 to 10.5 (see Fig. 12). Some prop erties have b een excluded from the analysis due to a correlation with another prop erty in the analysis. A.1. Elemental ratios

Fig. 14.-- Carb on to oxygen ratio [C/O] versus magnesium to silicon ratio [Mg/Si] of 176 FG stars with abundances for these elements (Reddy et al. 2003). In Figure 4 (b ottom panels) we showed that the [C/O] and [Mg/Si] distributions are largely indep endent of [Fe/H]. Here we show that these distributions are also largely indep endent of each other. Note that in this comparison we only use the data from Reddy et al. (2003), since it is the largest available sample with C, O, Mg and Si abundances.

A.2.

Mass, age and rotational velocity

In Figure 15 we show four correlation plots for mass, chromospheric age, rotational velocity and v sin i. We use the stars common to b oth Wright et al. (2004) and Valenti & Fischer (2005) for which these observables are available.

23


Fig. 15.-- Correlation plots b etween various prop erties. For all four panels we use the stars common to b oth Wright et al. (2004) and Valenti & Fischer (2005). Panel (A): mass vs rotational velocity v sin i for 713 FGK stars. This panel shows the degree of correlation b etween mass and v sin i. See Gray (2005) for a stronger correlation b etween these two variables when a larger mass range and more active stars are kept in the sample. To minimize the effect of this correlation on our analysis, we restrict the range of mass in Fig. 5 to 0.9 to 1.1M . Panel (B): chromospheric age versus v sin i for 641 FGK stars. The lack of correlation b etween chromospheric determined ages and rotational velocities is shown. Panel (C): no strong correlation b etween mass and chromospheric age for 639 FGK stars. Panel (D): the ages of 637 stars determined by the chromospheric method versus their ages from the isochrone method.

24


A.3.

Galactic orbital parameters

The Galactic orbital eccentricity (e) and the magnitude of the galactic orbital velocities with resp ect to the local standard of rest (vLSR ) are strongly correlated (see Fig. 6 in Sec. 2.6). We selected e instead of vLSR b ecause of its near indep endence of the maximum height ab ove the galactic plane (Zmax ).

Fig. 16.-- Left panel: Galactic orbital eccentricity e versus Zmax for 1987 FGK stars within 40 p c (Nordstrom et al. 2004). The orbital eccentricity is not correlated with Zmax . Right panel: ¨ vLSR versus Zmax for the same stars. Because vLSR is more strongly correlated with Zmax than eccentricity, eccentricity has b een selected for the joint analysis instead of vLSR . As in Fig. 4, the contours corresp ond to 38%, 68%, 82% and 95%.

B.

Improved Estimates of

2

and P (< 2 )

In Section 3.2, with 11 degrees of freedom, the reduced chi-squared from Equation 4 is 2 / 11 = 0.72+0.01 . Since 2 / 11 < 1, the Sun's prop erties are consistent with the Sun b eing a randomly -0.03 selected star. To improve on this preliminary analysis (but with a similar conclusion), as mentioned in Section 3.2, we employ a b ootstrap analysis (Efron 1979) to randomly resample data (with replacement) and derive a more accurate estimate of 2 . Because the b ootstrap is a non-parametric method, the distributions need not b e Gaussian. For every iteration, each parameter's stellar distribution is randomly resampled and a 2 value is calculated using Eq. (1). The uncertainties ,i of the solar values x ,i are also included in the b ootstrap method: for every iteration, the Solar value for each parameter is replaced in Eq. (1) by a randomly selected value from a normal distribution with median µ1/2,i = x ,i and standard 25


deviation ,i . The process was iterated 100,000 times, although the resulting distribution varies very little once the numb er of iterations reaches 10, 000. The median of this distribution and the error on the median yields our improved value for the reduced 2 (Fig. 11). The uncertainty of the median of each re-sampled distribution varies inversely prop ortionally to the square root of the numb er of stars in the distribution, µ1/2,i 1/ N ,i . In other words, median values are less certain for smaller samples and this uncertainty is included in our improved estimate of 2 , and its uncertainty. We find the probability of finding a star with a 2 value lower than the solar 2 , for N = 11 degrees of freedom in the standard way (Press et al. 1992) and obtain: P (< 2 = 7.88+ -
0.08 0.30

|11) = 0.28+ -

0.01 0.03

(B1)

To improve our estimate of the probability of finding a star with lower chi-squared value than the Sun, we p erform a Monte Carlo simulation (Metrop olis & Ulam 1949) to calculate an estimate of each star's chi-squared value (2 ). For every iteration, we randomly select a star from each stellar distribution. We then calculate its 2 value by replacing the solar value x ,i with that star's value x ,i in Eq. (1). This process was rep eated 100,000 times to create our Monte Carloed stellar chi-squared distribution. Stars were randomly selected with replacement, thus the simulated 2 distribution accounts for small numb er statistics and non-Gaussian distributions. The probability of finding a star with chi-squared lower than or equal to solar is PMC = 0.29 ± 0.11. The results of our analysis for the Solar 2 values and the probabilities P (< 2 ) are summarized in Table 2 B.1. Addition of a discrete parameter

In Section 4 we discuss the addition of stellar multiplicity to our analysis. Since stellar multiplicity cannot easily b e approximated by a one-sided Gaussian (particularly b ecause the Sun is on the edge of the distribution, i.e., it is of multiplicity one), we modified our Monte Carlo procedure to include this discrete parameter. The likelihood of observing a particular 2 for the 11 parameters is 11 1 2 . (B2) exp - i 2
i=1

We take the probability p(1) of a star b eing a single star, to b e 53.8 ± 4.5%, obtained from our sample of nearby stars (Sec. 2.1). The likelihood L of observing a particular 2 and p(1) is the product 11 1 L = p(1) exp - (B3) 2 . i 2
i=1

Taking logarithms we can then compute the distribution of the statistic S , where S = ln p(1) - 1 2
11

2 . i
i=1

(B4)

The distribution of S allows us to obtain the results for multiplicity rep orted at the end of Section 4. 26


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