Stellar Populations - Lecture IV RUSSELL SMITH Rochester 333 [email protected] http://astro.dur.ac.uk/~rjsmith/stellarpops.html Introduction to Astronomy & Astrophysics: Stellar Populations 2011/12 Lecture IV Homework** STELLAR POPULATIONS problem... ASSIGNMENT: MASS, LUMINOSITY AND THE IMF ** 1.) Assume a galaxy forms its stars in a single rapid burst, with a Salpeter initial mass function extending from 0.1 to 100 Msun. After 10 Gyrs have passed, divide the surviving stars into three groups: lower main-sequence upper main sequence (MS stars with mass > 0.75 Msun), lower main sequence (MS stars with mass < 0.75 Msun), and red giants. Calculate the fractional contribution of each group to the a) total *number* of surviving stars. b) total *mass* in surviving stars. c) total *luminosity* of the galaxy. Hints: *Assume the MS lifetime-vs-mass relation and RGB lifetimes as given -2.5in Lecture I. Ignore subsequentMS lifetime-mass evolutionary relationship:stages. TMS ~ 10 Gyr x (M/Msun) *For MS stars, assume luminosity-mass relation of (L/Lsun) = (M/Msun)^3.5. *Assume average➔ Main-sequence RGB luminosity isturn-off ~100 times mass higher at age than 10 the Gyr turn-off is Mluminosity.TO = 1 Msun Stars with mass ~1-2 Msun have RGB lifetimes ~1 Gyr 2.) It has sometimes been proposed that elliptical galaxies have a "bottom-heavy" initial mass function, e.g.➔ a power-lawMost massive with slope surviving -3.5 instead star ofmust the Salpeter have had -2.35. MS Calculate lifetime the ~9Gyr total luminosity contributions for the same three mass ranges for the bottom-heavy case. ➔ Corresponds to a mass of Mup =1.043 Msun 3.) Some absorption lines in the spectra of stars depend explicitly on the mass of the star, as well as on temperature and metallicity. Introduction to Astronomy & Astrophysics: Stellar Populations 2011/12 Lecture IV Assume that for one of these features, we define a Lick-like line-index with a "feature band" 25 Angstroms in width. In stars, this feature has an equivalent width of ~5 Angstroms for all stars of mass below 0.75 Msun, but is essentially absent from all stars of higher mass. We want to exploit this feature to constrain the dwarf-star content of a distant galaxy, from the integrated spectrum. Qualitatively, how will the measured index depend on the slope of the initial mass function? What signal-to-noise ratio (per angstrom) is required in the spectrum to distinguish between the Salpeter and bottom-heavy scenarios at 3-sigma significance? TO COMPLETE BY FEB / 7 ** STELLAR POPULATIONS ASSIGNMENT: MASS, LUMINOSITY AND THE IMF ** 1.) Assume a galaxy forms its stars in a single rapid burst, with a Salpeter initial mass function extending from 0.1 to 100 Msun. After 10 Gyrs have passed, divide the surviving stars into three groups: lower main-sequence upper main sequence (MS stars with mass > 0.75 Msun), lower main sequence (MS stars with mass < Homework problem...0.75 Msun), and red giants. Calculate the fractional contribution of each group to the a) total *number* of surviving stars. b) total *mass* in surviving stars. c) total *luminosity* of the galaxy. Hints: Fractional contribution*Assume of the lower-MS MS lifetime-vs-mass to total number relation and of RGB surviving lifetimes stars: as given in Lecture I. Ignore subsequent evolutionary stages. *For MS stars, assume luminosity-mass relation of (L/Lsun) = (M/Msun)^3.5. Mcut *Assume average RGB luminosity is ~100Where: times higher than the turn-off luminosity. N(m) dm ∫ -2.35 N(m) = N0 m (Salpeter IMF) 2.)M It has sometimes been proposed that elliptical galaxies have a "bottom-heavy" initial mass function,lo e.g. a power-law with slope -3.5 instead of the Salpeter -2.35. Calculate the total luminosity FLMS = contributions for the same three mass rangesMlo =for 0.1 the Mbottom-heavysun case. Mup Mcut = 0.75 Msun 3.)∫ SomeN(m) absorption dm lines in the spectra of stars depend explicitly on the mass of the star, as well as on temperature and metallicity. Mup = 1.043 Msun Mlo Assume that for one of these features, we define a Lick-like line-index with a "feature band" 25 Angstroms in width. In stars, this feature has an equivalent width of ~5 Angstroms for all stars of mass below 0.75 Msun, but is essentially absent from all stars of higher mass. ➔ FLMS = 0.975. Change limits to get FUMS = 0.022; FRGB = 0.0025 We want to exploit this feature to constrain the dwarf-star content of a distant galaxy, from the Lower MS dominateintegrated the total spectrum. number Qualitatively, of stars how will the measured index depend on the slope of the initial Introduction to Astronomy & Astrophysics:mass function? Stellar Populations 2011/12 Lecture IV What signal-to-noise ratio (per angstrom) is required in the spectrum to distinguish between the Salpeter and bottom-heavy scenarios at 3-sigma significance? TO COMPLETE BY FEB / 7 ** STELLAR POPULATIONS ASSIGNMENT: MASS, LUMINOSITY AND THE IMF ** 1.) Assume a galaxy forms its stars in a single rapid burst, with a Salpeter initial mass function extending from 0.1 to 100 Msun. After 10 Gyrs have passed, divide the surviving stars into three groups: lower main-sequence upper main sequence (MS stars with mass > 0.75 Msun), lower main sequence (MS stars with mass < Homework problem...0.75 Msun), and red giants. Calculate the fractional contribution of each group to the a) total *number* of surviving stars. b) total *mass* in surviving stars. c) total *luminosity* of the galaxy. Hints: Fractional contribution*Assume of the lower-MS MS lifetime-vs-mass to total mass relation of and surviving RGB lifetimes stars: as given in Lecture I. Ignore subsequent evolutionary stages. *For MS stars, assume luminosity-mass relation of (L/Lsun) = (M/Msun)^3.5. Mcut *Assume average RGB luminosity is ~100Where: times higher than the turn-off luminosity. N(m) m dm ∫ -2.35 N(m) = N0 m (Salpeter IMF) 2.)M It has sometimes been proposed that elliptical galaxies have a "bottom-heavy" initial mass function,lo e.g. a power-law with slope -3.5 instead of the Salpeter -2.35. Calculate the total luminosity FLMS = contributions for the same three mass rangesMlo =for 0.1 the Mbottom-heavysun case. Mup Mcut = 0.75 Msun 3.)∫ SomeN(m) absorption m dm lines in the spectra of stars depend explicitly on the mass of the star, as well as on temperature and metallicity. Mup = 1.043 Msun Mlo Assume that for one of these features, we define a Lick-like line-index with a "feature band" 25 Angstroms in width. In stars, this feature has an equivalent width of ~5 Angstroms for all stars of mass below 0.75 Msun, but is essentially absent from all stars of higher mass. ➔ FLMS = 0.904. Change limits to get FUMS = 0.085; FRGB = 0.012 We want to exploit this feature to constrain the dwarf-star content of a distant galaxy, from the Lower MS also dominateintegrated the spectrum. total massQualitatively, of stars how will the measured index depend on the slope of the initial Introduction to Astronomy & Astrophysics:mass function? Stellar Populations 2011/12 Lecture IV What signal-to-noise ratio (per angstrom) is required in the spectrum to distinguish between the Salpeter and bottom-heavy scenarios at 3-sigma significance? TO COMPLETE BY FEB / 7 ** STELLAR POPULATIONS ASSIGNMENT: MASS, LUMINOSITY AND THE IMF ** 1.) Assume a galaxy forms its stars in a single rapid burst, with a Salpeter initial mass function extending from 0.1 to 100 Msun. After 10 Gyrs have passed, divide the surviving stars into three groups: lower main-sequence upper main sequence (MS stars with mass > 0.75 Msun), lower main sequence (MS stars with mass < Homework problem...0.75 Msun), and red giants. Calculate the fractional contribution of each group to the a) total *number* of surviving stars. b) total *mass* in surviving stars. c) total *luminosity* of the galaxy. Hints: Fractional contribution*Assume of lower-MS the MS lifetime-vs-mass to total luminosity relation and of RGB surviving lifetimes stars: as given in Lecture I. Ignore subsequent evolutionary stages. *For MS stars, assume luminosity-mass relation of (L/Lsun) = (M/Msun)^3.5. *AssumeMcut average RGB luminosity is ~100 timesWhere: higher than the turn-off luminosity. 3.5 ∫ N(m) LMS(m) dm LMS(m) = Lsun (M/Msun) 2.) It has sometimes been proposed that elliptical(luminosity-mass galaxies have a "bottom-heavy" relation initial mass Mlo function, e.g. a power-law with slope -3.5 insteadfor of theMS) Salpeter -2.35. Calculate the total luminosity FLMS = contributions for the same three mass ranges for the bottom-heavy case. MTO Mup LRGB = 100 LTO = 100 Lsun ∫ N(m)3.) Some LMS(m) absorption dm + lines ∫ N(m) in the spectraLRGB dm of stars depend explicitly on the mass of the star, as well as on temperature and metallicity. Mlo MTO Assume that for one of these features, we define a Lick-like line-index with a "feature band" 25 Angstroms in width. In stars, this feature has an equivalent width of ~5 Angstroms for all stars of mass below 0.75 Msun, but is essentially absent from all stars of higher mass. ➔ FLMS = 0.054. Change limits to get FUMS = 0.047; FRGB = 0.899 We want to exploit this feature to constrain the dwarf-star content of a distant galaxy, from the RGB dominates the integratedtotal light spectrum. output. Qualitatively, how will the measured index depend on the slope of the initial Introduction to Astronomy & Astrophysics:mass function? Stellar Populations 2011/12 Lecture IV What signal-to-noise ratio (per angstrom) is required in the spectrum to distinguish between the Salpeter and bottom-heavy scenarios at 3-sigma significance? TO COMPLETE BY FEB / 7 ** STELLAR POPULATIONS ASSIGNMENT: MASS, LUMINOSITY AND THE IMF ** 1.) Assume a galaxy forms its stars in a single rapid burst, with a Salpeter initial mass function extending from 0.1 to 100 Msun.