12 The Initial Mass Function: Observations

Suggested background reading: As we continue to march downward in size scale, we now turn • Offner, S. S. R., et al. 2014, in “Proto- from the way gas clouds break up into clusters to the way clusters stars and Planets VI", ed. H. Beuther break up into individual stars. This is the subject of the initial mass et al., pp. 53-75 function (IMF), the distribution of stellar masses at formation. The Suggested literature: IMF is perhaps the single most important distribution in stellar • van Dokkum, P. G., & Conroy, C. 2010, Nature, 468, 940 and galactic astrophysics. Almost all inferences that go from light • da Rio, N., et al. 2012, ApJ, 748, 14 to physical properties for unresolved stellar populations rely on an assumed form of the IMF, as do almost all models of galaxy formation and the ISM.

12.1 Resolved Stellar Populations

There are two major strategies for determining the IMF from obser- vations. One is to use direct star counts in regions where we can resolve individual stars. The other is to use integrated light from more distant regions where we cannot.

12.1.1 Field Stars

The first attempts to measure the IMF were by Salpeter (1955),1 using 1 This has to be one of the most cited stars in the Solar neighborhood, and the use of Solar neighborhood papers in all of astrophysics – nearly 5,000 citations as of this writing. stars remains one of the main strategies for measuring the IMF today. Suppose that we want to measure the IMF of the field stars within some volume or angular region around the Sun. What steps must we carry out?

Constructing the Luminosity Function The first step is to construct a luminosity function for the stars in our survey volume in one or more photometric bands. This by itself is a non-trivial task, because we require absolute luminosities, which means we require distances. If we are carrying out a volume-limited instead of a flux-limited survey, 192 notes on star formation 2684 BOCHANSKI ET AL. Vol. 139

3.0 3.0

2.5 2.5 we also require distances to determine if the target stars2.0 are within 2.0 r i - -

1.5 r 1.5 our survey volume. g The most accurate distances available are from parallax,1.0 but this 1.0 presents a challenge. To measure the IMF, we require a0.5 sample of 0.5 0.0 0.0 stars that extends down to the lowest masses we wish to-0.5 measure. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 r - i i - z Figure 5. Color–color diagrams of the final photometric sample with the 5 Gyr isochrones of Baraffe et al. (1998,reddashedline)andGirardietal.(2004,yellow As one proceeds to lower masses, the starsdashed very line) overplotted. rapidly The contours representbecome 0.2% of our entire sample, with contours increasing every 10 stars per 0.05 color–color bin. Note that the model predictions fail by nearly 1 mag in some locations of the stellar locus. dimmer, and as they become dimmer it becomes(A color version of this harder figure is available and in the online harder journal.)

5 to obtain accurate parallax distances. For 0.1 M stars,Mismatched typical and Counted in SDSS ⇠ 16 Stars within 4x4x4 kpc cube absolute V band magnitudes are M 14, and parallax catalogs V ⇠ at such magnitudes are only complete out to 5 10 pc. A survey 18⇠ of this volume only contains 200 300 stars and brown dwarfs, ⇠ and this sample size presents a fundamentalr limit on how well 10 the IMF can be measured. If one reduces the20 mass range being r studied, parallax catalogs can go out somewhat further, but then one M is trading off sample size against the mass range22 that the study can 15 probe. Hopefully Gaia will improve this situation-2 -1 significantly. 0 1 2 3 4 5 r - z For these reasons, more recent studiesFigure have 6. Hess diagramtended for objectsto identified rely as stars onin the SDSS less pipeline, but as galaxies with high-resolution ACS imaging in the COSMOS footprint (red This Study filled circles). The black points show 0.02% of the final stellar sample used in D. A. Golimowski et al. 2010, in prep. the present analysis. Note that galaxy contamination is the most significant at West et al. 2005 accurate spectroscopic or photometric distances. These introduce Hawley et al. 2002 faint, blue colors. These colors and magnitudes are not probed by our analysis, Juric et al. 2008 since these objects lie beyond our 4 4 4 kpc distance cut. Sesar et al. 2008 × × Baraffe et al. 1998 significant uncertainties in the luminosity(A color function, version of this figure but is available they in the online are journal.) 20 0.5 1.0 1.5 2.0 2.5 3.0 stars, clusters, etc.), and mathematical relations are fitted to their more than compensated for by the vastly larger number of stars r - i color (or spectral type)—absolute magnitude locus. Thus, the color of a star can be used to6 estimate its absolute magnitude, Figure 7. Mr vs. r i CMD. The parallax stars from the nearby star sample are available, which in the most recent studies can be > 10 . The general Figureshown as filled12 circles,−.1: and Color-magnitude the best-fit line from Table 4 is the diagram solid red line. and in turn, its distance, by the well-known distance modulus Other existing parallax relations are plotted for comparison: West et al. (2005, (m M): forpurple stars dash-dotted with line), Juri well-measuredcetal.(´ 2008, their “bright” relation; parallax green dash- procedure for photometric distances is to construct− color-magnitude dotted line), Sesar et al. (2008,yellowdash-dottedline),andD.A.Golimowski mλ,1 Mλ,1(mλ,1 mλ,2) 5logd 5, (1) distances.et al. (2010, in preparation, The solid filters blue line). used The original are West the et al. SDSS (2005) − − = − relations have been transformed using the data from their Table 1. In addition, (CMD) diagrams in one or more colors for Solar neighborhood stars the 5 Gyr isochrone from the Baraffe et al. (1998) models appears as the dashed where d is the distance, mλ, is the apparent magnitude in one r and i. Credit: Bochanski et al. (2010), 1 line. filter, and mλ,1 mλ,2 is the color from two filters, which is used − ©AAS.(A color version Reproduced of this figure is available with in the online permission. journal.) using the limited sample of stars with measuredto calculate the absolute parallax magnitude, distances,Mλ,1. There have been multiple photometric parallax relations,10 perhaps aided by theoretical models. Figureas shown12 in Figure.1 shows7,constructedforlow-massstarsobserved an example by SDSS (Hawley et al. 2002;Williamsetal.2002; West et al. 2005;Juricetal.´ 2008;Sesaretal.2008;D.A.Golimowskietal. 2010, in preparation). There is a spread among the relations, of such a CMD. Each observed star with an10 Photometric unknown parallax relations distance are often referred to as is color–magnitude then seen in Figure 8,whicharevalidoverdifferentcolorranges. relations. We use both names interchangeably throughout this manuscript. Additional photometry in ugrizJHK of a large sample of nearby assigned an absolute magnitude based on its color and the CMD. s The absolute magnitude plus the observed magnitude also gives a distance. The spectroscopic parallax method is analogous, except that one uses spectral type - magnitude diagrams (STMD) in place of color-magnitude ones to assign absolute magnitudes. This can be more accurate, but requires at least low resolution spectroscopy instead of simply photometry.

Bias Correction Once that procedure is done, one has in hand an absolute luminosity function, either over a defined volume or (more- commonly) a defined absolute magnitude limit. The next step is to correct it for a series of biases. We will not go into the technical details of how the corrections are made, but it is worth going through the list just to understand the issues, and why this is not a trivial task. Metallicity bias: the reference CMDs or STMDs used to assign abso- lute magnitudes are constructed from samples very close to the Sun with parallax distances. However, there is a known negative metallic- the initial mass function: observations 193

ity gradient with height above the galactic plane, so a survey going out to larger distances will have a lower average metallicity than the reference sample. This matters because stars with lower metallicity have higher effective temperature and earlier spectral type than stars of the same mass with lower metallicity. (They have slightly higher absolute luminosity as well, but this is a smaller effect.) As a result, if the CMD or spectral type-magnitude diagram used to assign absolute magnitudes is constructed for Solar metallicity stars, but the star being observed is sub-Solar, then we will tend to assign too high an absolute luminosity based on the color, and, when comparing with the observed luminosity, too large a distance. We can correct for this bias if we know the vertical metallicity gradient of the galaxy. Extinction bias: the reference CMDs / STMDs are constructed for nearby stars, which are systematically less extincted than more distant stars because their light travels through less of the dusty Galactic disk. Dust extinction reddens starlight, which causes the more distant stars to be assigned artificially red colors, and thus artifi- cially low magnitudes. This in turn causes their absolute magnitudes and distances to be underestimated, moving stars from their true luminosities to lower values. These effects can be mitigated with knowledge of the shape of the dust extinction curve and estimates of how much extinction there is likely to be as a function of distance. Malmquist bias: there is some scatter in the magnitudes of stars at fixed color, both due to the intrinsic physical width of the main sequence (e.g., due to varying metallicity, age, stellar rotation) and due to measurement error. Thus at fixed color, magnitudes can scatter up or down. Consider how this affects stars that are near the distance or magnitude limit for the survey: stars whose true magnitude should place them just outside the survey volume or flux limit will artificially scatter into the survey if they scatter up but not if they scatter down, and those whose true magnitude should place them within the survey will be removed if they scatter to lower magnitude. This asymmetry means that, for stars near the distance or magnitude cutoff of the survey, the errors are not symmetric; they are much more likely to be in the direction of positive than negative flux. This effect is known as Malmquist bias. It can be corrected to the extent that one has a good idea of the size of the scatter in magnitude and understands the survey selection. Binarity: many stars are members of binary systems, and all but the most distant of these will be unresolved in the observations and will be mistaken for a single star. This has a number of subtle effects, which we can think of in two limiting cases. If the binary is far from equal mass, say q = M /M 0.3 or less, then the secondary star 2 1 ⇠ contributes little light, and the system colors and absolute magnitude 194 notes on star formation

will not be that different from those of an isolated primary of the same mass. Thus the main effect is that we correctly include the primary in our survey, but we miss the secondary entirely, and therefore undercount the number of low luminosity stars. On the other hand, if the mass ratio q 1, then the main effect is that ⇠ the color stays about the same, butNo. 6, using2010 our CMD THE LUMINOSITY we assign AND the MASS FUNCTIONS OF LOW-MASS STARS. II. 2689

luminosity of a single star when the true luminosityTable 5 is actually twice 0.010 Measured Galactic Structure that. We therefore underestimate the distance, and artificially scatter Property Raw Value Uncertainty things into the survey (if it is volumeZo limited),thin or255 out pc of the 12 survey pc (if 0.008 Ro,thin 2200 pc 65 pc

Zo,thick 1360 pc 300 pc ) it is luminosity-limited). At intermediate mass ratios, we get a little -3 Ro,thick 4100 pc 740 pc pc 0.006 of both effects. f 0.97 0.006 -1

The main means of correcting for this is, if weTable have 6 a reasonable 0.004

Bias-corrected Galactic Structure (0.5 mag Φ estimate of the binary fraction and massProperty ratio Corrected distribution, Value Uncertainty to guess a true luminosity function, determineZo,thin which stars300 pc are binaries, 15 pc add 0.002 Ro,thin 3100 pc 100 pc them together as they would be addedZo,thick in the observation,2100 pc 700filter pc the Ro,thick 3700 pc 800 pc f 0.96 0.02 0.000 resulting catalog through the survey selection, and compare to the 8 10 12 14 16 2692 BOCHANSKI ET AL.M Vol. 139 observed luminosity function. This procedure is then repeated, ad- r Figure 15.0.008Raw r-band LF for the stellar sample, using the (Mr ,r z)CMR. those from the observationsSystem and analysis, e.g30%., Malmquist bias. Single Star − 40% Note the smooth behavior, with a peak near Mr 11, corresponding to a spectral justing the guessed luminosity function,The systematic0.008 differencesuntilSingle the manifestedStar simulated in different50% observed CMRs, which type of M4. The errorSystem bars (many of which∼ are smaller than the points) are Covey vary according to stellar metallicity, interstellar extinction, and the formal∼ uncertainties from fitting the local densities in each 0.5 mag absolute luminosity function matches thecolor, actually are isolated observed and discussed one. in Sections 5.1 and 5.2,and magnitude slice in stellar density. the results are used in Section 5.3 to estimate the systematic 0.006 ) )

-3 0.006 Once all these bias correctionsuncertainties are made, in the LF the and GS. result is a corrected To-3 test the systematic effects of metallicity on this study, the pc Malmquist bias (Section 5.4)andunresolvedbinarity ([Fe/H], pc M )relationfromIvezicetal.(´ 2008) was adopted. -1 ∆ r luminosity function that (should)(Section faithfully5.5)werequantifiedusingMonteCarlo(MC)models. reproduce the actual We note-1 that this relation is appropriate for more luminous F Each model was populated with synthetic stars that were con- and G stars,0.004 near the main-sequence turnoff, but should give us 0.004 luminosity function in the surveysistent volume. with the observed Figure GS and12 LF..2 Theshows mock stellar an catalog aroughestimateforthemagnitudeoffset.TheadoptedGalactic (0.5 mag was analyzed with the same pipeline as the actual observations metallicity (0.5 mag gradient is Φ example of raw and corrected luminosityand the differences functions. between the input and “observed” GS and Φ 4 LF were used to correct the observed values. 0.002[Fe/H] 0.0958 2.77 10− Z . (9) 0.002 = − − × | | 5.1. Systematic CMRs: Metallicity At small Galactic heights (Z . 100 pc), this linear gradient The Mass-Magnitude Relation The next step is to convert the lumi- produces a metallicity of about [Fe/H] 0.1, appropriate for A star0.000 with low metallicity will have a higher luminosity nearby, local0.000 stars (Allende Prieto et al.=2004− ). At a height of nosity function into a mass function,and temperature which compared8 requires 10 to its solar-metallicity 12 knowledge 14 counterpart 16 of 2kpc(themaximumheightprobedbythisstudy),themetal-8 10 12 14 16 of the same mass, as first describedM by Sandage & Eggen ∼ r licity is [Fe/H] 0.65, consistentM withr measured distributions (1959). However, at a fixed color, stars with lower metallicities ∼− the mass-magnitude relation (MMR)Figure 20. System in whatever and single-star M photometricr LFs for our four different band binary Figure(Ivezic 21.´ etSingle-star al. 2008 (red). The filled actual circles) metallicity and system(black distribution filled circles) is prob- LFs. prescriptions.have fainter The absolute spread between magnitudes. prescriptions Failing in each to bin account is used to calculatefor this Noteably thatFigure more the major complex,12 differences.2: but Luminosity given between the our uncertainties system function and single-star associated for LFs occur with at we have used for our luminositytheeffect final function. uncertaintyartificially in brightens the This systemlow-metallicity and must single-star be LFs. stars, determined increasing their lowthe luminosities, effectsMilky of metallicityWay since low-mass stars on stars Mbefore dwarfs, can be companions (top) adopting and to a stars more af- of complex any higher (Aestimated color version distance. of this figure This is available inflates in densities the online journal.) at large distances, mass,description including is masses not justified. above those The sampled correction here. to the absolute mag- increasing the observed scale heights (e.g., King et al. 1990). (A colorter version (bottom) of this figure bias is available correction. in the online Credit: journal.) by either theoretical modelling, empirical calibration, or both. Par- nitude, ∆Mr ,measuredfromFandGstarsinclustersofknown Quantifying the effects of metallicity on low-mass dwarfs metallicityBochanski and distance et al. (Ivezi (2010cetal.´ ), ©2008AAS.)isgivenby Repro- ticularly at the low mass end, thecolorsis complicated theoretical and brightnesses by multiple aremodels calculated factors. tend for First, each direct to system. have metallicity Scatter is introduced in color and absolute magnitude, as described highlightsduced the with need for permission. additional investigations into2 the thick measurements of these cool stars are difficult (e.g., Woolf & ∆Mr 0.10920 1.11[Fe/H] 0.18[Fe/H] . (10) inWallerstein Section 5.42006.Thestellarcatalogisanalyzedwiththesame;Johnson&Apps2009), as current models do disk and suggests= − that future− investigations− should be presented significant uncertainties arising from complex atmospheric chemistry in terms of stellar mass contained in the thick disk, not scale pipelinenot accurately as the data, reproduce and the their output complex model spectral LF is features. compared Cur- to Substituting Equation (9)intoEquation(10)yieldsaquadratic the observed LF. The input PSLF is then tweaked according to height and normalization. that affects the optical and evenrently, near-infrared measurements of colors. metallicity-sensitive For empirical molecular band equation for ∆Mr in Galactic height.X. Delfosse After initially et al.: Accurate assigning masses of very low mass stars. IV 219 theheads differences (CaH and between TiO) are the used observed to estimate system the LF metallicity and the model of M absoluteThe iterative magnitudes process and described distances with above the accounts CMRs appropriate for binary calibrations, the data are only assystemdwarfs good LF. at the This as1dexlevel(seeGizis loop the is empiricalrepeated until1997 the mass artificial;Lepine´ deter- etsystem al. 2003 LF; starsfor nearby in the stars, sample each and star’s allows estimated us to compare height above the system the plane, LF matchesBurgasser the & observed Kirkpatrick∼ system2006 LF.;Westetal. Note that the2008 GS), butparameters detailed andZ ,wascomputed.Thisisrelatedtothestar’sactualheight, single-star LF in Figure 20.MostobservedLFsaresystem LFs,ini except for the local volume-limited surveys. However, most minations, which must come fromaremeasurements also orbit adjusted aremodelling. during only available this process, for This a few and stars.requires the bias-corrected The effects the of Ztrue,throughthefollowingequation: valuesmetallicity are given on the in Tableabsolute6.Thethindiskscaleheight,which magnitudes of low-mass stars are theoretical investigations into the IMF predict the form of the ∆Mr (Ztrue) has a strong effect on the derived LF, is in very good agreement single-star MF. Note that for all binary− prescriptions, the largest usual schemes for measuring stellarpoorly constrained. masses Accurate from parallaxes orbits, for e.g., nearby binaries subdwarfs Ztrue Zini10 5 . (11) withdo exist previous (Monet values. et al. As1992 the;Reid measured1997;thin Burgasser disk scale et al. height2008), differences between the two= LFs are seen at the faintest Mr,since that are both spectroscopic andincreases,but eclipsing measurements the density and of gradientstheir thus precise decrease, have metal abundances knownand a smaller are in- local diffi- theA lowest-luminosity star’s true height above stars the are plane most was easily calculated hidden by in finding binary systems. densitycult given is needed the extreme to explain complexity distant of calculating structures. the This opacity change of the root of this nonlinear equation. Since ∆Mr is a positive value, clinations, or visual binaries withisthe most molecular measured pronounced absorption at radial the bands bright that velocities. end, dominate where the the spectra majority Figure of of M the actual distance from the Galactic plane, Ztrue,issmaller thedwarfs. stars Observations are many thin of disk clusters scale with heights known away metallicities from the could Sun than the initial6. RESULTS: estimate, LUMINOSITYZini.Asexplainedabove,thiseffect FUNCTION 12.3 shows an example empirical(see MMR. Figure 19). The preferred model thin disk and thick disk mitigate this problem (Clem et al. 2008;Anetal.2008), but becomesThe final important adopted atsystem larger and distances, single-star movingM LFs stars are inward presented and scalethere lengths are no were comprehensive found to be observations similar. This in is the mostugriz likelysystem due decreasing the density gradient. Thus, if metallicityr effects are in Figure 21.TheLFswerecorrectedforunresolvedbinarity As with the luminosity function,tothat the probe limited there the radiallower are main extent a sequence. number of the survey of compared possible to their neglected, the scale heights and lengths are overestimated. and Malmquist bias. The uncertainty in each bin is computed typical scale lengths. Upcoming IR surveys of disk stars, such from the spread due to CMR differences, binary prescriptions, biases, because the stars are notas uniformAPOGEE (Allende in either Prieto et al. age2008 or), should metallicity, provide more Figure 12.3: Empirically-measured and Malmquist corrections. The mean LFs and uncertainties are accurate estimates of these parameters. and as a result there is no true single MMR. This would only intro- listedmass-magnitude in Tables 7 and 8.Thedifferencesbetweenthesingle relationship in V band. SDSS observations form a sensitive probe of the thin disk and systemCredit: LFs Delfosse are discussed et belowal., A&A, and compared364, 217 to previous, and thick disk scale heights, since the survey focused mainly on duce a random error if the age and metallicity distribution of the studies in both M and M . the northern Galactic cap. Our estimates suggest a larger thick 2000, reproducedr J with permission © sample used to construct the MMRdisk scale were height the and smaller same thick as disk that fraction in the than recentIMF ESO.6.1. Single-star versus System Luminosity Functions studies (e.g., Siegel et al. 2002;Juricetal.´ 2008). However, survey. However, there is no reasonthese two to parameters believe are that highly thisdegenerate is actually (see Figure 1 of Figure 21 demonstrates a clear difference between the single- Siegel et al. 2002). In particular, the differences between our star LF and the system LF. The single-star LF rises above investigation and the Juricetal.(´ 2008) study highlight the sen- the system LF near the peak at Mr 11 (or a spectral sitivity of these parameters to the assumed CMR and density type M4) and maintains a density about∼ twice that of the profiles, as they included a halo in their study and we did not. system∼ LF.13 This implies that lower-luminosity stars are easily The Juricetal.(´ 2008)studysampledlargerdistancesthanour hidden in binary systems, but isolated low-luminosity systems work, which may affect the resulting Galactic parameters. How- Fig. 1. V, J, H and K band M/L relations. The circles are data from Henry & McCarthy (1993), Torres et al. (1999), Henry et al. (1999), Benedict ever, the smaller normalization found in our study is in agree- 13 Weet note al. (2000)that the and differences Metcalfe between et al. (1996). our system The triangles and single-star represent LFs our recent mesurements (Segransan´ et al. 2000, in prep.; and Forveille et al. ment with recent results from a kinematic analysis of nearby disagree1999). considerably The masses with and those luminosities reported used by Covey in this etfigure al. ( are2008 also). These listed in Table 3. The two curves represent the piecewise linear relation of Henry MdwarfswithSDSSspectroscopy(J.S.Pinedaetal.2010,in differences& McCarthy were investigated, (1993; dotted and line) the and Covey our etpolynomial al. (2008) fit binary (solid corrections line). were found to be erroneous, with companion stars sampled from the MF preparation). They find a relative normalization of 5%, simi- convolved with the full sample volume, which is inappropriate for companion lar to the present investigation. The discrepancy in scale∼ height stars. Thethus authors in principle regret be the needed error. to derive V band magnitude differ- ences. A comparison with the direct measurements of Henry et al. (1999) for the sources in common shows maximum relative errors of 10% from neglecting this transformation: a 0.5 mag- nitude contrast⇠ is in error by at most 0.05 magnitude, and a 2 magnitudes one by at most 0.2 magnitude. We have therefore adopted the larger of 0.05 magnitude and 10% of the magnitude difference as a conservative estimate of the standard error for these spectroscopic magnitude differences.

3. Visible and infrared mass/Luminosity relations The masses are listed in Table 3, with the individual absolute magnitudes derived from Table 1 and Table 2 for the four photo- metric bands (V, J, H and K) which have significant numbers of Fig. 2. mass-colour (V-K) relation for M dwarfs. The three curves are measurements. Fig. 1 shows the M/L relations for these 4 pho- 5 Gyr theoretical isochrones from Baraffe et al. (1998) for two metal- tometric bands. As can be seen immediately in Fig. 1, 20 stars licities and our polynomial fit. The Siess et al. (2000) model are rep- define the V and K relations, while the J and H ones⇠ still have resented for 5 Gyr and solar metallicity with asterisks. smaller numbers of stars. A number of systems still lack mag- nitude difference measurements in those two bands. dispersions to be generally useful, and is provided here mostly Fig. 2 presents the relation between stellar mass and the V-K for illustration, and as a warning to potential users of similar colour index. This relation probably has too large an intrinsic relations. the initial mass function: observations 195

the case. The selection function used to determine the empirical mass-magnitude sample is complex and poorly characterized, but it is certainly biased towards systems closer to the Sun, for example. Strategies to mitigate this are similar to those used to mitigate the corresponding biases in the luminosity function. Once the mass-magnitude relationship and any bias corrections have been applied, the result is a measure of the field IMF. The results appear to be well-fit by a lognormal distribution or a broken powerlaw, along the lines of the Chabrier (2005) and Kroupa (2002) IMFs introduced in Chapter 2.

Age Correction The strategy we have just described works fine for stars up to 0.7 M in mass. However, it fails with higher ⇠ mass stars, for one obvious reason: stars with masses larger than this can evolve off the main sequence on timescales comparable to the mean stellar age in the Solar neighborhood. Thus the quantity we measure from this procedure is the present-day mass function (PDMF), not the IMF. Even that is somewhat complicated because stars’ luminosities start to evolve non-negligibly even before they leave the main sequence, so there are potential errors in assigning masses based on a MMR calibrated from younger stars. One option in this case is simply to give up and not say anything about the IMF at higher masses. However, there is another option, which is to try to correct for the bias introduced by stellar evolution. Suppose that we think we know both the star formation history of the region we are sampling, M˙ (t), and the initial mass-dependent ⇤ main-sequence stellar lifetime, tMS(m). Let dn/dm be the IMF. In this case, the total number of stars formed over the full lifetime of the galaxy in a mass bin from m to m + dm is

dn dn 0 form = dt M˙ (t) (12.1) dm dm • ⇤ Z where t = 0 represents the present. In contrast, the number of stars per unit mass still on the main sequence is

dn dn 0 MS = dt M˙ (t) (12.2) dm dm t (m) ⇤ Z MS

Thus if we measure the main sequence mass distribution dnMS/dm, we can correct it to the IMF just by multiplying:

0 ˙ dn dnMS • dt M (t) µ ⇤ . (12.3) dm dm 0 ˙ Rt (m) dt M (t) MS ⇤ This simply reduces to scaling theR number of observed stars by the fraction of stars in that mass bin that are still alive today. 196 notes on star formation

Obviously this correction is only as good as our knowledge of the star formation history, and it becomes increasingly uncertain as the correction factor becomes larger. Thus attempts to measure the IMF from the Galactic field even with age correction are generally limited to masses of no more than a few M .

12.1.2 Young Clusters

To measure the IMF for more massive stars requires a different technique: surveys of young star clusters. The overall outline of the technique is essentially the same as for the field: construct a luminosity function, correct for biases, then use a mass-magnitude relation to convert to a mass function. However, compared to the field, studying a single cluster offers numerous advantages:

• If the population is young enough, then even the most massive stars will remain on the main sequence, so there is no need to worry about correcting from the PDMF to the IMF. Even for some- what older clusters, one can probe to higher masses than would be possible with the 5 10 Gyr old field population. ⇠ • The stellar population is generally uniform in metallicity or very close to it, so there are no metallicity biases.

• The entire stellar population is at roughly the same distance, so there are no Malmquist or extinction biases. Moreover, in some cases the distance to the cluster is known to better than 10% from radio parallax – some young stars flare in the radio, and with radio interferometry it is possible to obtain parallax measurements at much larger distances than would be possible for the same stars in the optical.

• Low-mass stars and brown dwarfs are significantly more luminous at young ages, and so the same magnitude limit will correspond to a much lower mass limit, making it much easier to probe into the brown dwarf regime.

These advantages also come with some significant costs.

• The statistics are generally much worse than for the field. The most populous young cluster that is close enough for us to resolve individual stars down to the hydrogen burning limit is the Orion Nebula Cluster, and it contains only 103 104 stars, as compared ⇠ to 106 for the largest field surveys. ⇠ • The MMR that is required to convert an observed magnitude into a mass is much more complex in a young cluster, because a significant fraction of the stars may be pre-main sequence. For the initial mass function: observations 197

such stars, the magnitude is a function not just of the mass but also the age, and one must fit both simultaneously, and with significant theoretical uncertainty. We will discuss this issue further in Chapter 17. How much of a problem this is depends on the cluster age – for a 100 Myr-old cluster like the Pleiades, all the stars have reached the main sequence, while for a 1 2 ⇠ Myr-old cluster like Orion, almost none have. However, there is an obvious tradeoff here: in a Pleiades-aged cluster, the correction for stars leaving the main sequence is significant, while for an Orion-aged cluster it is negligible.

• For the youngest clusters, there is usually significant dust in the vicinity of the stars, which introduces extinction and reddening that is not the same from star to star. This introduces scatter, and also potentially bias because the extinction may vary with position, and there is a systematic correlation between position and mass (see next point).

• Mass segregation can be a problem. In young clusters, the most massive stars are generally found closer to the center – whether this is a result of primordial mass segregation (the stars formed there) or dynamical mass segregation (they formed elsewhere but sank to the center), the result is the same. Conversely, low mass stars are preferentially on the cluster outskirts. This means that studies must be extremely careful to measure the IMF over the full cluster, not just its outskirts or core; this can be hard in the cluster center due to problems with crowding. Moreover, if the extinction is not spatially uniform, more massive stars toward the cluster center are likely to suffer systematically more extinction that low-mass ones.

• Dynamical effects can also be a problem. A non-trivial fraction of O and B stars are observed to be moving with very high spatial velocities, above 50 km s 1. These are known as runaways. They ⇠ are likely created by close encounters between massive stars in the core of a newly-formed cluster that lead to some stars being ejected at speeds comparable to the orbital velocities in the en- counter. Regardless of the cause, the fact that this happens means that, depending on its age and how many ejections occurred, the cluster may be missing some of its massive stars. Conversely, be- cause low-mass stars are further from the center, if there is any tidal stripping, that will preferentially remove low-mass stars.

• Binary correction is harder for young stars because the binary fraction as a function of mass is much less well known for young clusters than it is for field stars. 198 notes on star formation

Probably the best case for studying a very young cluster is the Orion Nebula Cluster, which is 415 pc from the Sun. Its distance is known to a few percent from radio interferometry (Sandstrom et al., 2007; Menten et al., 2007; Kim et al., 2008). It contains several thousand stars, providing relatively good statistics, and it is young enough that all the stars are still on the main sequence. It is close enough that we can resolve all the stars down to the brown dwarf limit, and even beyond. However, the ONC’s most massive star is only 38 M , so to study the IMF at even higher masses requires the use of more distant clusters within which we cannot resolve down to low masses. For somewhat older clusters, the best case is almost certainly the Pleiades, which has an age of about 120 Myr. It obviously has no very massive stars left, but there are still 10 M stars present, and it is ⇠ also close and very well-studied. The IMF inferred for the Pleiades appears to be consistent with that measured in the ONC.

12.1.3 Globular Clusters

A final method for studying the IMF is to look at globular clusters. Compared to young clusters, globular clusters lack the massive stars because they are old, and suffer somewhat more from confusion problems due to their larger distances. Otherwise they are quite similar in terms of methodological advantages and disadvantages. The main reason for investigating globular clusters is that they provide us with the ability to measure the IMF in an environment as different as possible from that of young clusters forming in the disk of the Milky Way today. The stars in globular clusters are ancient and metal poor, and they provide the only means of accessing that population without resorting to integrated light measurements. They are therefore a crucial bridge to the integrated light methods we will discuss shortly. The major challenge for globular clusters is that all the dynamical effects are much worse, due to the longer time that the clusters have had to evolve. Over long times, globular clusters systematically lose low-mass stars due to tidal shocking and a phenomenon known as two-body evaporation, whereby the cluster attempts to relax to a Maxwellian velocity distribution, but, due to the fact that the cluster is sitting in a tidal potential, the tail of that distribution keeps escaping. This alters the IMF. There can also be stellar collisions, which obviously move low mass stars into higher mass bins. Accounting for all these effects is a major challenge, and the usual method is to adopt a proposed IMF and then try to simulate the effects of dynamical evolution over the past 13 Gyr in order to ⇠