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Astro 6590: Galaxies and the Universe Lecture 7A: HW#1&2 Finishing up a few items Oort’s constants Force Law in the galactic plane Then… RG HW#3 due next Wed, 24 Sep HW #1 1 dΦ 1 dA Φ dm m where A(m)≡ dN/dM so A dm = 1 dA M - Mo for a Gaussian LF: =m σ2 A dm dlnA and for a homogeneously distributed so <M>m –Mo = - σ2 dm population dlnA dm = 0.6 ln 10 = 1.38 so <M>m = Mo – 1.38 σ2 => independent of m. 1. Use B&M 3.6.1 Thus, the mean absolute magnitude of a magnitude limited smaple is smaller (brighter) than that of a volume limited sample by 1.38 σ2 2. This is B&M 3.5 ; it follows from the previous result; For a randomly distributed population, the distribution of galaxies in M in a magnitude-limited sample is independent of the survey’s limiting magnitude. 1 HW#1 (cont) 3.1 Straightforward 3.2 Straightforward: for α < -2 3.3 If the 2nd cluster is 5X more distant, we lose 52 in flux. If the survey is good to mlim & for first cluster is complete to M* + 4=> Mlim = mlim+ 5 – 5 log d = M* + 4 for the 2nd cluster Mlim = mlim+ 5 – 5 log (5d) = M* + 4 – 5log5 3.4 For α = -1.1; what is ratio of “missing” Lint? Define f as: L ∫0lim Φ(L) LdL f≡ = fraction of L lost per cluster ∞ ∫0 Φ(L) LdL Nearby cluster loses f=0.037; one at 5d loses 0.517, so the ratio is 13.97. (Some people defined the measure in different ways.) 3.5 Fraction which lies below L* increases as faint end steepens 3.6 An exercise in reading recent literature and integrating… HW #2 1. (m-M) = 5 log d -5 Add extinction of 1 mag/kpc (m-M) = 5 log d – 5 + A => A = 10-3 d Off by 2X at (m-M)=12.4, dtrue ~ 1.5 kpc Re 2. Ltot = 2 ∫ 0 I(R) 2πR dR => straightforward j(r) r dr 2 2 √r – R ∞ 1 dI dR and j(r) = ∫ π r dR √R2 – r2 ∞ 3. Following B&T: I(R) = 2 ∫ R 4. Following B&M: dN/dM = No(M) ζ(M) where ζ (M) is your favorite IMF. Note that stars of M ≥ 2 M have τMS < 109yrs and will die. Assume τMS ~ 1010(M/M)-2.5. Low mass stars dominate because (1) more are formed and (2) they live longer. Assume L∝Mx; Mtot = ∫ M dN/dM dM ~ 1.5 x 109 M answer depends Ltot = ∫ dN/dM L dM ~ 1.3 x 1010 M on x and IMF 2 Local Standard of Rest (LSR) Hipparcos results Uo = 10.00 ± 0.36 km/s => radially inwards Vo = 5.25 ± 0.62 km/s => in direction of Galactic rotation Wo = 7.17 ± 0.38 km/s => vertically upwards Motions of stars near the Sun • Hipparcos catalog • Need to account for variations with stellar type • See Dehnen & Binney 1998 3 Geometry of solar motion From Mihalas & Binney R sin α = R cos l – d Vr = Θ cos α - Θ sin l = (ω − ω) R sin l Vt =(ω − ω) R cos l -ωd where ω = Θ/R Galactic Rotation Relations In the solar neighborhood, d << R and |R-R|<< R Vobs = (ω−ω) R sin l Estimate the variation of (ω−ω) by a first order Taylor expansion: (ω−ω) ≈ dω (R - R) dR R dω d Θ = dR dR R dω = dR R = 1 dΘ Θ R dR R2 so that 1 dΘ Θ R dR R R2 4 Galactic Rotation Relations dω 1 dΘ Θ - = dR R R dR R R2 To first order, then dΘ - Θ Vr = dR R R (R - R) sin l For d << R => Θ R Vr ≈ - R - R ≈ d cos l, so dΘ dR d sin l cos l R Also, sin l cos l = ½ sin 2l, so we can define Vr = A d sin 2 l Θ R A ≡ ½ and dΘ dR Oort’s Constant A R Oort’s Constants Θ R A ≡ ½ Vr = A d sin 2 l dΘ dR R Oort’s Constant A We can show similarly (B&M 637-641) for proper motions: proper motion μ = Vt / 4.74 d where d in pc, μ in arcsec/yr Vt = d (A cos 2 l + B) where B ≡-½ dΘ dR Θ + R Notice the identities: ω = dΘ dR Θ = A-B R = - (A + B) R A ≡ - ½ R dω dR Oort’s Constant B R A and B are known as Oort’s constants A = 14.82 ± 0.84 km/s kpc-1 B = 12.37 ± 0.64 km/s kpc-1 R Feast&Whiteoak 1997 5 Oort’s constants Hipparcos π and μ for 20000 single stars at d>100pc as fn(SpecType). 3 contributions to local kinematics: • Motion of Sun relative to avg motion of *s in S.N. • Differential velocity field due to galactic rotation around Sun (shear) • Peculiar velocities Oort’s constants F. Mignard • Derives Solar Motion: u, v, w, V, (l,b)apex wrt different stellar groups => differences in radial (u) direction than orthoradial (v) for early vs late types. Same with A, less with B. Sun is moving upward (+z) from the GP 7.2±0.2 km/s 6 Force law ⊥ Galactic plane Kz = - ∂Φ ∂z Kapteyn 1922; Oort 1932 Force law ⊥ Galactic plane 7 Force law ⊥ Galactic plane Force law ⊥ Galactic plane 8 Force Law ⊥ Galactic plane Force Law ⊥ Galactic plane 9 Force Law in the galactic plane Bianaymé et al. 2006 A&A 446, 933 • 203 nearby “red clump” stars in Hipparcos catalog (so distances known) • NGP K-giants Force Law in the galactic plane Bianaymé et al. 2006 A&A 446, 933 Try to fit including DM component ρeff => ρeff = 0 works too. Surface mass density w/in |z| from the GP (i.e. disk) Σ0.8kpc = 57-66 Mpc-2 Σ1.1kpc = 57-79 Mpc-2 Σstars = ~40 Mpc-2 Don’t need much ΣISM = ~13 Mpc-2 DM in disk P (z-oscill) = 42±2 Myr Dinosaur extinctions => P=26 Myr (Raup&Sepkoski 1986) 10 On-going surveys RAVE: radial velocities for 1,000,000 stars to < 5 km/s uses 6dF spectrograph on 1.2m UK Schmidt telescope at AAO Programs under SDSS: (APO 2.5m telescope) SEGUE-1: 240,000 stars to 4 km/s SEGUE-2: 350,000 stars to 4 km/s APOGEE: higher res IR spectra for 100,000 stars => metallicity => Powerful probes of structure and history of MW 11