Rolf Kudritzki SS 2015
11. Virial Theorem Distances of
Spiral and Elliptical Galaxies
Virial Theorem
2
vrot
2
M (R0 )
= ↵s G
R0
(s1)
spiral galaxies
M (Ref f )
= ↵e G
Ref f
(e1)
elliptical galaxies
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
1
Rolf Kudritzki SS 2015
radial surface brightness distribution
r
R0
I(r) = I0 e
I(r) = Ief f e
⇣
7.67 ( Rerf f
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⇡
L = 2⇡
Z
Z
L
= 2⇡
2
1
0
1
0
Z
I(r)rdr = 2⇡R02 I0
2
I(r)rdr = 7.22⇡Ref
f Ief f
Ref f
I(r)rdr
0
(e3a)
(s3)
spiral galaxies
(e3)
elliptical galaxies
(see next page)
for elliptical galaxies
explains “half-light radius”
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Rolf Kudritzki SS 2015
elliptical galaxies
calculation of integrated
luminosity form surface
brightness profile
3
spiral galaxies
Rolf Kudritzki SS 2015
combining (s1) and (s3) we get
or
2
M
L = 2⇡I0 ↵s2 G2 4
vrot
4
vrot
=
2 2 M 2
2⇡I0 ↵s G ( ) L
L
(s4)
basis for Tully-Fisher relationship (Tully & Fisher, 1977, A&A 54, 661)
MB,V,I.. = n · log(vrot ) + CT F
(s5)
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 !!!!
4
Rolf Kudritzki SS 2015
Lesson from the past: why the TFR
gave H0 in the mid 80’s 20 years ago
examples of TF
calibration
(see Brent Tully’s
chapter 8)
M31
M81
NGC 2403
M33
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
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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)
!
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Rolf Kudritzki SS 2015
elliptical galaxies
combining (e1) and (e3) we obtain similar as for spirals
4
or
=
2 2 M 2
7.22⇡Ief f ↵e G ( ) L
L
MB,V,I.. = n · log( ) + CF J
(e4)
(e5)
Faber & Jackson,
1976, ApJ 204, 668
Faber – Jackson Relationship for elliptical galaxies
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Rolf Kudritzki SS 2015
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, 1275
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Rolf Kudritzki SS 2015
Dressler et al., 1987, Bender et al., 1992, ApJ 399, 462 à
slight dependence of M/L on M
combination of (e1) and (e4) à
(e1), (e6) and (e7) à
M
⇠ M , ⇡ 0.2
L
M 1 1
Ref f ⇠ ( ) Ief f
L
Ref f ⇠ (
2
Ref f )
2
2
(e6)
(e7)
(e8)
Ief1f
solving (e8) for Reff à
Ref f ⇠
2 11+
1
1+
Ief f
(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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Rolf Kudritzki SS 2015
fundamental plane relation in Coma cluster
μ is surface
brightness in
magnitudes
~ -2.5 log Ieff
Allanson et al., 2009, ApJ 702, 1275
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Rolf Kudritzki SS 2015
with
à
⇥ef f = 2
Ref f
d
⇥ef f ⇠
2 11+
angular half-light diameter
1
1+
Ief f
1
d
(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
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
1
Dn ⇠
d
2 11+
1
1+
Ief f
+0.8
1
⇡
d
2 11+
(e12)
The modified relation (e12) is then called the Dn – σ relation.
The method has been successfully applied by the HST Key Project.
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Rolf Kudritzki SS 2015
fundamental plane
distance removed
angular diameter
distance removed
Kelson et al., 2000, ApJ 529, 768
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Rolf Kudritzki SS 2015
relation between Dn, Reff, Ieff
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)
(A1)
2
2
L(r) = a⇡Ref
I
F
(x)
=
⇡r
I(r)
f ef f
where a=7.22 and F (x) =
1
7!
Z
x
e
x 7
x dx
x = 7.67
0
✓
r
Ref f
◆ 14
note that F(∞) = 1 and F(7.67) = ½. A good approximation for F(x) in the
range 4 < x < 8 is
1 ⇣ x ⌘3
see plot next page
F (x) ⇡
2
✓
From (A1) we then obtain
7.67
◆2
r
Ref f
or
r
Ref f
=
a
Ief f
I(r)
=a
Ief f
I(r)
✓
r
Ref f
◆ 34
! 45
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Rolf Kudritzki SS 2015
1 ⇣ x ⌘3
F (x) ⇡
2 7.67
1
F (x) =
7!
Z
x
e
x 7
x dx
0
14