Rolf Kudritzki SS 2015 11. Virial Theorem Distances Rolf Kudritzki SS 2015 2 radial surface...

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Transcript of Rolf Kudritzki SS 2015 11. Virial Theorem Distances Rolf Kudritzki SS 2015 2 radial surface...

  • Rolf Kudritzki SS 2015

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    11. Virial Theorem Distances of Spiral and Elliptical Galaxies

    Virial Theorem

    vrot rotational velocity of spiral disks, ~ constant beyond disk scale length R0 disk scale length σ velocity dispersion of stars in elliptical galaxies Reff “half light radius” of ellipticals (see below) M galaxy mass (visible and dark matter) confined within R0 and Reff αs and αe form factors of order 1 to 10 resulting from mass distribution

    v2 rot

    = ↵ s

    G M(R0)

    R0 (s1) spiral galaxies

    (e1) elliptical galaxies � 2 = ↵eG

    M(Reff )

    Reff

  • Rolf Kudritzki SS 2015

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    radial surface brightness distribution

    I(r) = I0e � rR0

    I(r) = Ieffe �7.67

    ⇣ ( rReff )

    1 4 �1

    ⌘ (s2) spiral galaxies

    (e2) elliptical galaxies (de Vaucouleurs, 1941)

    galaxy luminosity L = LB,V,I,… depending in which pass band we observe

    L = 2⇡

    Z 1

    0 I(r)rdr = 2⇡R20I0 (s3) spiral galaxies

    L = 2⇡

    Z 1

    0 I(r)rdr = 7.22⇡R2effIeff (e3) elliptical galaxies (see next page)

    L

    2 = 2⇡

    Z Reff

    0 I(r)rdr (e3a) for elliptical galaxies

    explains “half-light radius”

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    elliptical galaxies calculation of integrated luminosity form surface brightness profile

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    spiral galaxies

    combining (s1) and (s3) we get

    basis for Tully-Fisher relationship (Tully & Fisher, 1977, A&A 54, 661)

    or

    calibrate cTF and n from vrot measurements of ISM HI 21cm emission and integrated LB,V,I,…luminosities !!!! Note that for TF to work M/L needs to be similar in all spirals !!!!

    L = 2⇡I0↵ 2 s

    G2 M2

    v4 rot

    v4 rot

    = 2⇡I0↵ 2 s

    G2( M

    L )2L

    M

    B,V,I..

    = n · log(v rot

    ) + C TF

    (s4)

    (s5)

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    Lesson from the past: why the TFR gave H0 in the mid 80’s 20 years ago

    M33 NGC 2403

    M81 M31

    A fit of the same slope fixed to the 4 galaxies in red would lead to a zero point 9% fainter ==> H0 9% (7 km/s/Mpc) larger

    examples of TF calibration (see Brent Tully’s chapter 8)

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    CF2: ! I & [3.6] band Luminosity - HI Linewidth Calibration

    Tully & Courtois 2012, ApJ, 749: 78 (I band)!

    Sorce et al. 2013, ApJ, 765: 94 (Spitzer mid-IR)! 3.6µm!

    0.8µm

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    elliptical galaxies

    combining (e1) and (e3) we obtain similar as for spirals

    �4 = 7.22⇡Ieff↵ 2 eG

    2( M

    L )2L

    MB,V,I.. = n · log(�) + CFJ or

    (e4)

    (e5) Faber & Jackson, 1976, ApJ 204, 668

    Faber – Jackson Relationship for elliptical galaxies

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    FJ relation in Coma cluster

    Allanson et al., 2009, ApJ 702, 1275

    scatter ~ 0.6 mag larger than uncertainty of individual measurements à  dependence on additional parameter (Dressler et al., 1987, ApJ 313,59)

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    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 Reff ⇠ ( M

    L )�1I�1eff�

    2 (e7)

    (e1), (e6) and (e7) à Reff ⇠ (�2Reff )���2I�1eff (e8)

    solving (e8) for Reff à

    Reff ⇠ �2 1�� 1+� I

    � 11+� 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.

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    fundamental plane relation in Coma cluster

    Allanson et al., 2009, ApJ 702, 1275

    μ is surface brightness in magnitudes ~ -2.5 log Ieff

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    ⇥eff ⇠ �2 1�� 1+� I

    � 11+� eff

    1

    d

    ⇥eff = 2 Reff d

    with

    (e11)

    angular half-light diameter

    à

    (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 Dn = angular diameter within which the total mean surface brightness (in mag) has a certain value, for instance 20.75 mag (B-Band). Since Dn ~ ΘeffIeff0.8 (see below), this removes the Ieff dependence (e12) The modified relation (e12) is then called the Dn – σ relation. The method has been successfully applied by the HST Key Project.

    Dn ⇠ 1

    d �2

    1�� 1+� I

    � 11+�+0.8 eff ⇡

    1

    d �2

    1�� 1+�

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    fundamental plane distance removed

    angular diameter distance removed

    Kelson et al., 2000, ApJ 529, 768

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    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⇡R2effIeffF (x) = ⇡r 2 I(r)

    relation between Dn, Reff, Ieff

    where a=7.22 and F (x) =

    1

    7!

    Z x

    0 e

    �x x

    7 dx

    x = 7.67

    ✓ r

    Reff

    ◆ 1 4

    note that F(∞) = 1 and F(7.67) = ½. A good approximation for F(x) in the range 4 < x < 8 is

    see plot next page F (x) ⇡ 1

    2

    ⇣ x

    7.67

    ⌘3

    (A1)

    From (A1) we then obtain ✓ r Reff

    ◆2 = a

    Ieff

    I(r)

    ✓ r

    Reff

    ◆ 3 4

    or r

    Reff =

    a Ieff

    I(r)

    ! 4 5

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    F (x) ⇡ 1 2

    ⇣ x

    7.67

    ⌘3

    F (x) = 1

    7!

    Z x

    0 e

    �x x

    7 dx