Ay 20 - Fall 2004 - Lecture 17Stellar Luminosity and MassFunctions* * * * *History and Formation ofOur GalaxyStellar Luminosity and Mass Functions• Basic statistical descriptors of stellar populations:probability distribution for stellar luminosities (a functionof the bandpass) and masses• Most important: stellar Initial Mass Function (IMF) =mass function at the formation time• Key in understanding and modeling star formation andgalaxy evolution• Needed in order to estimate stellar (baryonic?) masscontent of galaxies (and other stellar systems) from theobserved luminosities• Very, very hard to do - depends on having lots of veryreliable, well-understood data and calibrationsDetermining the IMF is a tricky business…• Observed star counts– Understand your selection effects, completeness– Get the distances– Estimate the extinction– Correct for unresolved binaries• Get the Present-Day Luminosity Function (PDLF)– Assume the appropriate mass-luminosity relation– It is a function of metallicity, bandpass, …– Theoretical models tested by observations• Get the Present-Day Mass Function (PDMF)– Assume some evolutionary tracks, correct for the evolved stars(also a function of metallicity, …)– Assume some star formation history• Get the Initial Mass Function (IMF)!(From P. Armitage)Luminosity functionsSuppose we measure the distance and apparent magnitudem of all stars within some limiting distance dmax (a `volumelimited sample’ - easier in theory than in practice!).Convert from apparent magnitude to absolute magnitude M using the known distance d to each star and the definition:€ M = m − 5log10d10 pc      absolute magnitude in somewaveband, e.g. in the visual MVFinally count the number of stars with M between (M-0.5) and(M+0.5), and divide by the volume surveyed. Givesthe luminosity function.€ 43πdmax3More formally:€ Φ M( )ΔMnumber density (stars per pc3) of stars = with absolute magnitude M betweenM and M+ΔMluminosity functionIdentical concept applies to galaxies (though typically measurenumbers of galaxies per Mpc3 rather than per pc3).Can be hard to measure Φ(M):• for very low mass stars (M large), which are dim unlessvery close to the Sun• for massive stars (M small), which are rareLuminosity function is the basic observable for studying a population of stars.(From P. Armitage)Local luminosity function (stars with d < 20 pc) for the Milky Way measured by Kroupa, Tout & Gilmore (1993):faint starsbright stars(From P. Armitage)Initial Mass Function (IMF)Starting from the observed luminosity function, possible to derive an estimate for the Initial Mass Function (IMF). To define the IMF, imagine that we form a large number of stars.Then:€ ξ(M )ΔMthe number of stars that have beenborn with initial masses between M and M+ΔM (careful not to confusemass and absolute magnitude here)=this is the Initial Mass Function or IMFThe IMF is a more fundamental theoretical quantity which isobviously related to the star formation process. Note that the IMF only gives the distribution of stellar masses immediatelyafter stars have formed - it is not the mass distribution in, say,the Galactic disk today.(From P. Armitage)In practice: several obstacles to getting the IMF from theluminosity function:Convert from absolute magnitude to massNeed stellar structure theory, calibrated by observations ofeclipsing binaries.(From P. Armitage)The Mass-Luminosity RelationL ~ M 2.7L ~ M 1.6L ~ M 3.1L ~ M 4.7Use it to convert stellar luminosities into massesAnd, of course, it is a function of bandpass …Massive stars have short lifetimesSuppose we observe the luminosity function of an old cluster.There are no very luminous main sequence stars. But this doesnot mean that the IMF of the cluster had zero massive stars,only that such stars have ended their main sequence lifetimes.More generally, we need to allow for the differing lifetimes ofdifferent stars in deriving the IMF. If we assume that the star formation rate in the disk has been constant with time, meanswe need to weight number of massive stars by 1 / τms, whereτms is the main sequence lifetime.Massive stars are doubly rare - few are formed plus they don’t live as long as low mass stars…(From P. Armitage)Mass lossFor massive stars, mass loss in stellar winds means that thepresent mass is smaller than the initial mass.These difficulties mean that although the local IMF is welldetermined for masses between ~0.5 Msun and ~50 Msun:• not well determined at the very low mass end (mainlybecause the relation between luminosity and massis not so well known)• for very massive stars - simply too rare• in other galaxies, especially in the distant Universe(From P. Armitage)Salpeter Mass FunctionThe Initial Mass Function for stars in the Solar neighborhoodwas determined by Salpeter in 1955. He obtained:€ ξ(M ) =ξ0M−2.35constant which sets the local stellar densitySalpeter IMFUsing the definition of the IMF, the number of stars that form with masses between M and M + ΔM is:€ ξ(M )ΔMTo determine the total number of stars formed with massesbetween M1 and M2, integrate the IMF between these limits:€ N =ξ(M)dM =ξ0M−2.35dMM1M2∫M1M2∫=ξ0M−1.35−1.35      M1M2=ξ01.35M1−1.35− M2−1.35[ ](From P. Armitage)Can similarly work out the total mass in stars born with massM1 < M < M2:€ M*= Mξ(M)dMM1M2∫Properties of the Salpeter IMF:• most of the stars (by number) are low mass stars• most of the mass in stars resides in low mass stars• following a burst of star formation, most of the luminosity comes from high mass starsSalpeter IMF must fail at low masses, since if we extrapolateto arbitrarily low masses the total mass in stars tends to infinity!Observations suggest that the Salpeter form is valid for roughlyM > 0.5 Msun, and that the IMF `flattens’ at lower masses. Theexact form of the low mass IMF remains uncertain.(From P. Armitage)What is the origin of the IMF?Most important unsolved problem in star formation. Many theories but no consensus.Observationally, known that dense cores in molecular cloudshave a power-law mass function rather similar to the IMF. Sothe IMF may be determined in part by how such cores form from turbulent molecular gas.Is the IMF `universal’?i.e. is ξ(M) the same function everywhere?Most theorists say no. Predict that fragmentation is easier if the gas can cool, so primordial gas without any