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Rotation Curves • Spherical mass shell, uniform density Inside shell: no effect Outside shell: acts as if all mass at center • Fcentripital = Fgravity Hwk 2 due Sept 23 Hwk 3 not yet assigned, but due Sept 30 Midterm 1 Wed, Oct 2 dFgravity ρ × d (vol ) ∝ dΩ r2 2 d ( vol ) = dΩ r dr mv 2 GM ( r )m = r r2 1/ 2 GM ( r ) v= r • Inside spherical mass distribution r 4 M ( r ) = ∫ ρ ( r′)4π r 2dr′ = ρ π r 3 0 3 v∝r • Outside spherical mass distribution M ( r ) = const. • Spher. Distr. + exponential disk v ∝ r −1 / 2 The Milky Way’s Rotation Curve Radio data Tangent point method Young star clusters 1 receeding Expected: approaching Rotation curves in other galaxies Spec slit Observed: systemic velocity • Vera Rubin & Kent Ford (late 1970’s) • Image tube spectrograph [CO pg. 917] Density as shown by flat rotation curves • dM(r)/dr ~ constant • • Unbounded mass distribution?? NFW profile ρ (r) = (r) • • • ρ0 (r / a )(1 + r / a )2 Predicted for Cold dark matter (CDM) Actual derived dark matter profiles often slightly different than this What is CDM? Coming later in course. 2 The Milky Way’s Dark Matter “Halo” M31 770 kpc Components of our Galaxy 10 kpc (1010 M☼) Nuclear bulge 1 Disk 7 Halo 0.3 Dark matter 190 out to r ~ 230 kpc Milky Way H I Spectra of Spiral Galaxies 3 Tully-Fisher Relation Rotation Curves • Maximum rotation velocity-Luminosity relation [FIG 25.10] • Tully-Fisher relation • MB = -9.95 log10VMax + 3.15 • MB = -10.2 log10VMax + 2.71 • MB = -11.0 log10VMax + 3.31 Semi-derivation of Tully-Fisher Relation: MB = -9.95 log10Vmax + 3.15 MB = -10.2 log10Vmax + 2.71 MB = -11.0 log10Vmax + 3.31 (Sa) (Sb) (Sc) (Sa) (Sb) (Sc) • Mass interior to outermost R where rotation curve can be measured: Mass = • Assume 2 Vmax R G L = Mass / const. • “Freeman Law” (observed fact ---maybe): L = const. Surf .Bright. = 4πR 2 4 L = const × Vmax Important as a DISTANCE calibrator! • Convert to Absolute B-band magnitudes: L = −10 log10 Vmax + const. M B = M sun − 2.5 log10 Lsun 4 SO FAR: • Galaxy types • Ancient history • Milky Way and spiral galaxy morphology • Nuclear bulge • Disk • Stellar halo • Dark matter halo • Star-forming regions • Chemical enrichment Hwk 2 due Sept 23 Hwk 3 not yet assigned, but due Sept 30 Midterm 1 Wed, Oct 2 Kinematics of spiral galaxies • Rotation curves Î mass distribution (includes sidetrack about measuring distances) • Spiral structure [CO 25.3] • General properties of S, E, Irr galaxies • Midterm 1 (Wed. Oct 2) 5