11. Virial Theorem Distances of Spiral and Elliptical Galaxies

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1 Virial Theorem 11. Virial Theorem Distances of Spiral and Elliptical Galaxies v 2 rot = s G M(R 0) R 0 2 = e G M(R eff ) R eff (s1) spiral galaxies (e1) elliptical galaxies v rot rotational velocity of spiral disks, ~ constant beyond disk scale length R 0 disk scale length σ velocity dispersion of stars in elliptical galaxies R eff half light radius of ellipticals (see below) M galaxy mass (visible and dark matter) confined within R 0 and R eff α s and α e form factors of order 1 to 10 resulting from mass distribution 1

2 radial surface brightness distribution I(r) =I 0 e r R 0 I(r) =I eff e 7.67 ( r Reff ) (s2) spiral galaxies (e2) elliptical galaxies (de Vaucouleurs, 1941) galaxy luminosity L = L B,V,I, depending in which pass band we observe Z 1 L =2 I(r)rdr =2 R0I 2 0 (s3) spiral galaxies Z 0 1 L =2 I(r)rdr =7.22 Reff 2 I eff (e3) elliptical galaxies (see next page) L 2 =2 0 Z Reff 0 I(r)rdr (e3a) for elliptical galaxies explains half-light radius 2

3 Rolf Kudritzki SS 2015 elliptical galaxies calculation of integrated luminosity from surface brightness profile 3

4 spiral galaxies combining (s1) and (s3) we get or L =2 I 0 2 sg 2 M 2 v 4 rot v 4 rot =2 I 0 2 sg 2 ( M L )2 L (s4) basis for Tully-Fisher relationship (Tully & Fisher, 1977, A&A 54, 661) M B,V,I.. = n log(v rot )+C TF (s5) calibrate c TF and n from v rot measurements of ISM HI 21cm emission and integrated L B,V,I, luminosities!!!! Note that for TF to work M/L needs to be similar in all spirals!!!! 4

5 Rolf Kudritzki SS 2015 Lesson from the past: why the TFR gave H 0 in the mid 80 s 20 years ago examples of TF calibration (see Brent Tully s chapter 8) M81 M31 NGC 2403 M33 A fit of the same slope fixed to the 4 galaxies in red would lead to a zero point 9% fainter ==> H 0 9% (7 km/s/mpc) larger 5

6 Rolf Kudritzki SS 2015 CF2:! I & [3.6] band Luminosity - HI Linewidth Calibration Tully & Courtois 2012, ApJ, 749: 78 (I band)! 0.8µm 3.6µm! Sorce et al. 2013, ApJ, 765: 94 (Spitzer mid-ir)! 6

7 elliptical galaxies combining (e1) and (e3) we obtain similar as for spirals 4 =7.22 I eff 2 eg 2 ( M L )2 L (e4) or M B,V,I.. = n log( )+C FJ (e5) Faber & Jackson, 1976, ApJ 204, 668 Faber Jackson Relationship for elliptical galaxies 7

8 FJ relation in Coma cluster scatter ~ 0.6 mag larger than uncertainty of individual measurements à dependence on additional parameter (Dressler et al., 1987, ApJ 313,59) Allanson et al., 2009, ApJ 702,

9 Dressler et al., 1987, Bender et al., 1992, ApJ 399, 462 à slight dependence of M/L on M M L M, 0.2 (e6) combination of (e1) and (e4) à R eff ( M L ) 1 I 1 eff 2 (e7) (e1), (e6) and (e7) à R eff ( 2 R eff ) 2 I 1 eff (e8) solving (e8) for R eff à R eff I 1 1+ eff (e9) this is the fundamental plane relationship of elliptical galaxies, which relates size with velocity dispersion and surface brightness. One can use this for distances, because it provides an absolute length, which then leads distance through the angular size. 9

10 fundamental plane relation in Coma cluster μ is surface brightness in magnitudes ~ -2.5 log I eff Allanson et al., 2009, ApJ 702,

11 with à eff =2 R eff d eff I 1 1+ eff 1 d angular half-light diameter (e11) (e11) allows to determine distance through a measurement of - angular diameter and surface brightness through photometry - velocity dispersion through spectroscopy and line width Following Dressler et al., 1987, one frequently uses the quantity D n = angular diameter within which the total mean surface brightness (in mag) has a certain value, for instance mag (B-Band). Since D n ~ Θ eff I eff 0.8 (see below), this removes the I eff dependence D n 1 d I eff 1 d (e12) The modified relation (e12) is then called the D n σ relation. The method has been successfully applied by the HST Key Project. 11

12 fundamental plane distance removed angular diameter distance removed Kelson et al., 2000, ApJ 529,

13 relation between D n, R eff, I eff From (e2) we can calculate the integrated luminosity L(r) at radius r and the mean surface brightness at that radius (see also page 3) L(r) =a R 2 eff I eff F (x) = r 2 I(r) where a=7.22 and F (x) = 1 7! Z x 0 e x x 7 dx x =7.67 r R eff note that F( ) = 1 and F(7.67) = ½. A good approximation for F(x) in the range 4 < x < 8 is x 3 see plot next page From (A1) we then obtain F (x) r = a I eff R eff I(r) (A1) r R eff or! 4 r = a I 5 eff R eff I(r) 13

14 F (x) 1 2 x F (x) = 1 7! Z x 0 e x x 7 dx 14