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.