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Galaxies II – Dr Martin Hendry
10 lectures to A3/A4, beginning January 2008
Galaxies II – Dr Martin Hendry
10 lectures to A3/A4, beginning January 2006
Course Topics
1. Galaxy Kinematics
o
o
o
o
o
Spectroscopy and the LOSVD
Measuring mean velocities and velocity dispersions
Rotation curves of disk systems
Evidence for dark matter halos
The Tully-Fisher and Fundamental Plane relations
Galaxies II – Dr Martin Hendry
10 lectures to A3/A4, beginning January 2006
Course Topics
1. Galaxy Kinematics
o
o
o
o
o
Spectroscopy and the LOSVD
Measuring mean velocities and velocity dispersions
Rotation curves of disk systems
Evidence for dark matter halos
The Tully-Fisher and Fundamental Plane relations
2. Abnormal and Active Galaxies
o
o
o
o
o
Starburst galaxies
Galaxies with AGN
The unified model of AGN
Radio lobes and jets
Evidence for supermassive black holes
Galaxies II – Dr Martin Hendry
10 lectures to A3/A4, beginning January 2006
Course Topics
3. Galaxy Formation and Evolution
o
o
o
o
Galaxy mergers and interactions
Polar rings, dust lanes and tidal tails
Star formation in ellipticals and spirals
Chemical evolution models
Galaxies II – Dr Martin Hendry
10 lectures to A3/A4, beginning January 2006
Course Topics
3. Galaxy Formation and Evolution
o
o
o
o
Galaxy mergers and interactions
Polar rings, dust lanes and tidal tails
Star formation in ellipticals and spirals
Chemical evolution models
4. Galaxies and Cosmology
o
o
o
o
Hierarchical clustering theories
Galaxy clusters as cosmological probes
Proto-galaxies and the Lyman-alpha forest
Re-ionisation of the early Universe
Some Relevant Textbooks
(Not required for purchase, but useful for consultation)
o An Introduction to Modern Astrophysics,
B.W. Carroll & D.A. Ostlie (Addison-Wesley)
o Galactic Astronomy,
J. Binney & M. Merrifield
(Princeton UP)
o Galactic Dynamics,
J. Binney & S. Tremaine
(Princeton UP)
o Galaxies and the Universe,
L. Sparke & J.S. Gallagher (Cambridge UP)
1. Kinematics of Galaxies
The key to probing large-scale motions within galaxies
is spectroscopy
Radiation emitted from gas (e.g. stars, nebulae) moving
radially is Doppler shifted
1. Kinematics of Galaxies
The key to probing large-scale motions within galaxies
is spectroscopy
Radiation emitted from gas (e.g. stars, nebulae) moving
radially is Doppler shifted
Radial velocity
(can be +ve or –ve)
Change in
wavelength
(can be +ve
or –ve)
(1.1)

v
z

0 c
Wavelength of light as
measured in the laboratory
Speed of light
(Formula OK if v << c)
1. Kinematics of Galaxies
The key to probing large-scale motions within galaxies
is spectroscopy
Radiation emitted from gas (e.g. stars, nebulae) moving
radially is Doppler shifted
Radial velocity
(can be +ve or –ve)
Change in
wavelength
(can be +ve
or –ve)
(1.1)

v
z

0 c
Wavelength of light as
measured in the laboratory
Speed of light
(Formula OK if v << c)
Analysis of individual spectral
lines can allow measurement of
line of sight velocity
Fine for individual stars (e.g.
spectroscopic binaries – recall
A1Y stellar astrophysics)
Spectroscopic Binaries
Orbits, from above
B
A
A
B
A
B
B
A
To Earth
Spectral lines
0
B
0
A
A+B
0
A
0
B
A+B
When we collect light from some
small projected area of a galaxy,
its spectrum is the sum of spectra
from stars and gas along that line
of sight – all with different line of
sight velocities.
This ‘smears out’ individual
spectral lines
When we collect light from some
small projected area of a galaxy,
its spectrum is the sum of spectra
from stars and gas along that line
of sight – all with different line of
sight velocities.
This ‘smears out’ individual
spectral lines
(Not really a problem for determining cosmological redshifts for
distant galaxies, since broadening of spectral lines across galaxy is
a small effect compared with the radial velocity of entire galaxy.
See e.g. Lyman Ha line: SDSS)
When we collect light from some
small projected area of a galaxy,
its spectrum is the sum of spectra
from stars and gas along that line
of sight – all with different line of
sight velocities.
This ‘smears out’ individual
spectral lines
We define the Line of Sight Velocity Distribution (LOSVD) via:
F ( v LOS ) d v LOS 
Fraction of stars contributing to
spectrum with radial velocities
between v LOS and v LOS  d v LOS
(1.2)
It is useful to define the observed spectrum not in terms of
wavelength or frequency, but spectral velocity, u , via
u  c ln 
(1.3)
It is useful to define the observed spectrum not in terms of
wavelength or frequency, but spectral velocity, u , via
u  c ln 
Hence, a Doppler shift of

u 
(1.3)
corresponds to
c 

 v LOS
(1.4)
It is useful to define the observed spectrum not in terms of
wavelength or frequency, but spectral velocity, u , via
u  c ln 
Hence, a Doppler shift of

u 
corresponds to
c 

Light observed at spectral velocity
spectral velocity
u  v LOS
(1.3)
 v LOS
u
was emitted at
(1.4)
Suppose that all stars have intrinsically identical spectra,
S (u )
S (u ) measures the (relative) intensity
u
Intensity received from a star with line
of sight velocity
v LOS
is
S (u  v LOS )
Relative intensity (arbitrary units)
of radiation at spectral velocity
Wavelength (Angstroms)
Suppose that all stars have intrinsically identical spectra,
S (u )
S (u ) measures the (relative) intensity
u
Intensity received from a star with line
of sight velocity

v LOS
is
S (u  v LOS )
Relative intensity (arbitrary units)
of radiation at spectral velocity
Observed composite spectrum:

G (u ) 
 F v  S u  v  d v
LOS
LOS
LOS
Wavelength (Angstroms)

(1.5)
Suppose that all stars have intrinsically identical spectra,
S (u ) measures the (relative) intensity
of radiation at spectral velocity
u
Intensity received from a star with line
of sight velocity

v LOS
is
S (u  v LOS )
Observed composite spectrum:

G (u ) 
 F v  S u  v  d v
LOS
LOS
LOS

(1.5)
S (u )
Suppose that all stars have intrinsically identical spectra,
S (u )
S (u ) measures the (relative) intensity
of radiation at spectral velocity
u
Intensity received from a star with line
of sight velocity

v LOS
is
S (u  v LOS )
Observed composite spectrum:

G (u ) 


F v LOS  S u  v LOS  d v LOS
Galaxy spectrum is smoothed
version of stellar spectrum –
‘smeared out’ by LOSVD
(1.5)
Of course, stars don’t all have identical spectra,
S (u ) replaced by (local) average spectrum Sav (u, v LOS )
which depends on :
o age
o metallicity
o galaxy environment
Spectral Synthesis
(See Section 3)
Of course, stars don’t all have identical spectra,
S (u ) replaced by (local) average spectrum Sav (u, v LOS )
which depends on :
o age
o metallicity
o galaxy environment
Spectral Synthesis
(See Section 3)

G (u ) 
 F v  S u  v
LOS

av
LOS
, v LOS  d v LOS
(1.6)
Of course, stars don’t all have identical spectra,
S (u ) replaced by (local) average spectrum Sav (u, v LOS )
which depends on :
o age
o metallicity
o galaxy environment
Spectral Synthesis
(See Section 3)

G (u ) 
 F v  S u  v
LOS

av
LOS
, v LOS  d v LOS
We consider here only
the simpler case where
S (u ) is the same
throughout the galaxy
Generally the slowly varying continuum component of the
spectrum is removed first – i.e. we write:
S (u)  Scont (u)  Sline(u) (1.7)
Emission:
Sline(u)  0
Absorption:
Sline (u )  0
Generally the slowly varying continuum component of the
spectrum is removed first – i.e. we write:
S (u)  Scont (u)  Sline(u) (1.7)
Emission:
Sline(u)  0
Absorption:
Sline (u )  0
so that

Gline (u ) 
 F v  S u  v  d v
LOS

line
LOS
LOS
(1.8)
Generally the slowly varying continuum component of the
spectrum is removed first – i.e. we write:
S (u)  Scont (u)  Sline(u) (1.7)
Emission:
Sline(u)  0
Absorption:
Sline (u )  0
so that

Gline (u ) 
 F v  S u  v  d v
LOS
line
LOS
Relative intensity (arbitrary units)

Wavelength (Angstroms)
LOS
(1.8)
Equation (1.8) is an example of an integral equation , where the
function we can observe (the galaxy spectrum) is related to the
integral of the function we wish to determine (the LOSVD).

Gline (u ) 
 F v  S u  v  d v
LOS
line
LOS
LOS

Observed galaxy spectrum
LOSVD
‘Template’ stellar spectra
Equation (1.8) is an example of an integral equation , where the
function we can observe (the galaxy spectrum) is related to the
integral of the function we wish to determine (the LOSVD).

Gline (u ) 
 F v  S u  v  d v
LOS
line
LOS
LOS

Observed galaxy spectrum
LOSVD
‘Template’ stellar spectra
It is a particular type of integral equation: a convolution

g ( y) 
 f x  s y  x  dx

‘Data’ function
‘Source’ function
‘Kernel’ function
(1.9)
We want to estimate the source function, F v LOS  , given the
observed galaxy spectrum, G (u ) , and using a kernel function,
S (u ) , computed from e.g. a stellar spectral synthesis model.
How can we extract
F v LOS  from inside the integral?…
We want to estimate the source function, F v LOS  , given the
observed galaxy spectrum, G (u ) , and using a kernel function,
S (u ) , computed from e.g. a stellar spectral synthesis model.
How can we extract
F v LOS  from inside the integral?…
Fourier Convolution Theorem

Consider a convolution equation of the form
g ( y) 
 f x  s y  x  dx

The Fourier transforms of the functions f , g and
g~ (k ) 
~ ~
f (k ) s (k )
(1.10)
s satisfy the relation
Here
~
f (k ) 

 f ( x) e
ikx
dx

For proof, see Examples 1
In the context of our problem:
And
F v LOS 
~
~
~
G (k )  F (k ) S (k )
~
~ 1  G(k ) 
 F ~ 
 S (k ) 
(1.12)
Inverse Fourier transform
Hence, we can in principle invert the integral equation and
reconstruct the LOSVD, F v LOS 
(1.11)
In the context of our problem:
And
F v LOS 
~
~
~
G (k )  F (k ) S (k )
~
~ 1  G(k ) 
 F ~ 
 S (k ) 
(1.12)
Inverse Fourier transform
Hence, we can in principle invert the integral equation and
reconstruct the LOSVD, F v LOS 
In practice, this method is vulnerable to noise on the observed
galaxy spectrum, G (u ) , and uncertainties in the kernel S (u ) .
Need to filter out high frequency (k) noise
(1.11)
Filter, denoting range of
wavenumbers which give
reliable inversion
Ratio of two small
quantities: very noisy
If we cannot easily reconstruct the complete LOSVD F LOS  ,
we can at least constrain some of the simplest properties of
this function

v LOS 
Mean value
v
LOS
F v LOS  d v LOS
(1.13)

Variance
2
 LOS


2


v

v
 LOS LOS F v LOS  d v LOS

Velocity dispersion
 LOS 
2
 LOS
(1.15)
(1.14)
The Cross-Correlation Function Method
This is a common method for estimating
v LOS and  LOS .
Pioneered by e.g. Tonry & Davis (1979)
We define:

CCF ( v LOS ) 
 G(u) S u  v  du
LOS

(We use continuum-subtracted galaxy and template spectra)
(1.16)
The Cross-Correlation Function Method

CCF ( v LOS ) 
 G(u) S u  v  du
LOS

For a random value of
v LOS the product G(u) S u  v LOS 
fluctuates between +ve and –ve values

G (u )
S u  v LOS 
CCF (v LOS )
is small
+ve
-ve
+ve
-ve
The Cross-Correlation Function Method

CCF ( v LOS ) 
 G(u) S u  v  du
LOS

For a random value of
v LOS the product G(u) S u  v LOS 
fluctuates between +ve and –ve values

G (u )
S u  v LOS 
CCF (v LOS )
is small
+ve
-ve
+ve
-ve
The Cross-Correlation Function Method

CCF ( v LOS ) 
 G(u) S u  v  du
LOS

For a random value of
v LOS the product G(u) S u  v LOS 
fluctuates between +ve and –ve values

G (u )
S u  v LOS 
CCF (v LOS )
is small
+ve
-ve
+ve
-ve
The Cross-Correlation Function Method

CCF ( v LOS ) 
 G(u) S u  v  du
LOS

For a random value of
v LOS the product G(u) S u  v LOS 
fluctuates between +ve and –ve values

G (u )
S u  v LOS 
CCF (v LOS )
is small
+ve
-ve
+ve
-ve
The Cross-Correlation Function Method

CCF ( v LOS ) 
 G(u) S u  v  du
LOS

When
v LOS  v LOS emission and absorption features line up,
and the product

G (u )
S u  v LOS 
G(u ) S u  v LOS  is large everywhere
CCF (v LOS )
is large and positive
+ve
-ve
+ve
-ve
The Cross-Correlation Function Method

CCF ( v LOS ) 
 G(u) S u  v  du
LOS

We estimate
v LOS by
finding the maximum of
the cross-correlation
function.
v LOS
The Cross-Correlation Function Method

CCF ( v LOS ) 
 G(u) S u  v  du
LOS

We estimate
v LOS by
finding the maximum of
the cross-correlation
function.
Width of CCF peak allows
estimation of
Advantages:
 LOS
Fast, objective,
automatic
v LOS
What do we learn from the LOSVD?…
In the Milky Way, analysis of HI 21cm radio emission, has
revealed the spiral structure of the Galaxy
(See A1Y Cosmology and A2 Theoretical Astrophysics)
What do we learn from the LOSVD?…
In the Milky Way, analysis of HI 21cm radio emission, has
revealed the spiral structure of the Galaxy
(See A1Y Cosmology and A2 Theoretical Astrophysics)
What do we learn from the LOSVD?…
In the Milky Way, analysis of HI 21cm radio emission, has
revealed the spiral structure of the Galaxy
Can also probe spiral structure from spectra of HII regions
HII region
=
ISM region surrounding hot young
stars (O and B) in which hydrogen
is ionised.
These trace out spiral arms, where young stars are being born
Examples:
Orion Nebula, Great Nebula in Carina
What do we learn from the LOSVD?…
In the Milky Way, analysis of HI 21cm radio emission, has
revealed the spiral structure of the Galaxy
Can also probe spiral structure from spectra of HII regions
Other MW tracers include:
CO in molecular clouds
H2O masers
Cepheids, RR Lyraes
Globular Clusters
What do we learn from the LOSVD?…
We can construct a rotation curve : a graph of rotation speed
versus distance from the centre of the galaxy.
Milky Way Rotation Curve
Inside 1 kpc
vr  r

‘rigid-body’ rotation
This is consistent with a spherical matter distribution, of
constant matter density
Consider a mass, m , at distance r
Galaxy.
from the centre of the
Equating circular acceleration and gravitational force:
m v2
r
GM r m

r2
Mass interior to radius r
(1.17)
Inside 1 kpc
vr  r

‘rigid-body’ rotation
This is consistent with a spherical matter distribution, of
constant matter density
Consider a mass, m , at distance r
Galaxy.
from the centre of the
Equating circular acceleration and gravitational force:
m v2
r
GM r m

r2
Mass interior to radius r
(1.17)
Equating circular acceleration and gravitational force:
2
v r
3
Mr 
r
G
This is consistent with
Mr   r 
4
3
3
(1.18)
for constant

Equating circular acceleration and gravitational force:
2
v r
3
Mr 
r
G
This is consistent with
Mr   r 
4
3
3
(1.18)
for constant

At large radii (well beyond the limit of the optical disk) the
Milky Way’s rotation curve is flat

Evidence for a halo of dark matter around the Galaxy
In Our Solar System:
Orbital velocity (km/s)
60
50
40
30
20
10
0
0
10
20
30
40
Distance from the Sun (AU)
50
Orbital velocity (km/s)
60
50
vr
40
1 / 2
30
20
10
0
0
10
20
30
40
50
Distance from the Sun (AU)
M r constant for all r  RSun
 v r  constant
2

vr
1 / 2
(1.19)
Same argument gives
v  r 1/ 2 in outer
regions of the Galaxy,
if only a roughly
spherical distribution
of luminous matter
contributes to the
rotation curve.
Observed rotation curve
Instead rotation curve
is flat.
Same behaviour seen
for external galaxies
Rotation curve predicted
from luminous matter
From the Mathewson et al ‘Mark III’ Spirals survey
Outer regions:
vr   const.
This is consistent with a roughly spherical distribution of dark
matter , with density   r 2
Consider a mass, m , at distance r
Galaxy.
from the centre of the
Equating circular acceleration and gravitational force:
m v2
r
GM r m

r2
Mass interior to radius r
(1.17)
Outer regions:
vr   const.
This is consistent with a roughly spherical distribution of dark
matter , with density   r 2
Mr  r

dM r
 const.
dr
but…
(1.18)
dM r
 4 r 2  (r )
dr
for a spherical distribution

 (r )  r 2
as required
(1.19)
Points to note…
Evidence from e.g. HI rotation curves and the motions of satellite
galaxies suggests that halos typically extend to at least 100 kpc.
Points to note…
Evidence from e.g. HI rotation curves and the motions of satellite
galaxies suggests that halos typically extend to at least 100 kpc.
We cannot have  (r )  r to arbitrary radii, however, if the
halo mass is to remain finite.
2
Points to note…
Evidence from e.g. HI rotation curves and the motions of satellite
galaxies suggests that halos typically extend to at least 100 kpc.
We cannot have  (r )  r to arbitrary radii, however, if the
halo mass is to remain finite.
2
In any case, mass distribution of neighbouring halos may overlap:
Galaxies which appear as separate luminous objects may
have formed from a single dark matter halo – the result
of an earlier halo merger
Link between galaxy formation and cosmology – see later!
Points to note…
In order to match the rigid-body rotation of e.g. the Milky
Way in its central region, we need to modify the halo density
at small radii:
The parametric form
C0
 (r )  2 2
a r
has the correct properties
For the Milky Way:
(but see later)
C0  4.6 108 M Sun kpc -1
a  2.8 kpc
(1.20)
So what is the Dark Matter?…
(Revision of A1Y Cosmology)
Simplest candidates: Baryonic Dark Matter:
Can constrain mass and
distribution of MACHOs via
gravitational microlensing
White
dwarfs
Brown
dwarfs
Detecting MACHOs with
Gravitational Microlensing
Large Magellanic Cloud
MACHO’s gravity focuses
the light of the background
star on the Earth
A MACHO
So the background star
briefly appears brighter
Lightcurve of a microlensing event
The shape of the
curve tells about
the mass and
position of the
dark matter which
does the lensing
Time
Lightcurve of a microlensing event
The shape of the
curve tells about
the mass and
position of the
dark matter which
does the lensing
Time
Results indicate not nearly enough MACHOs to explain rotation curves
So what is the Dark Matter?…
(Revision of A1Y Cosmology)
Simplest candidates: Baryonic Dark Matter:
Can constrain mass and
distribution of MACHOs via
gravitational microlensing
Can also measure X-ray
emission from galaxy clusters:
baryonic cold gas
White
dwarfs
Brown
dwarfs
Cluster baryons from X-ray maps
Optical
EM   b
X-ray
2
So what is the Dark Matter?…
(Revision of A1Y Cosmology)
Simplest candidates: Baryonic Dark Matter:
Can constrain mass and
distribution of MACHOs via
gravitational microlensing
Can also measure X-ray
emission from galaxy clusters:
baryonic cold gas
Brown
dwarfs
White
dwarfs
Again, not enough
baryons to explain
motion of galaxies
in clusters!
But nucleosynthesis tells us, in any case, that most of the dark
matter must be non-baryonic
Isotopes of hydrogen
+
+
+
Hydrogen
Deuterium
Tritium
(1 proton)
(1 proton + 1 neutron)
(1 proton + 2 neutrons)
But nucleosynthesis tells us, in any case, that most of the dark
matter must be non-baryonic
But nucleosynthesis tells us, in any case, that most of the dark
matter must be non-baryonic
If the dark matter has to be non-baryonic, what is it?…
Hot dark matter?
(e.g. massive neutrinos)
Neutrinos are now measured to have non-zero rest
mass, but they’re not massive enough to account for
galaxy and cluster dark masses.
Also, they would smear out early structure in the
Universe (see later)
If the dark matter has to be non-baryonic, what is it?…
Hot dark matter?
(e.g. massive neutrinos)
Neutrinos are now measured to have non-zero rest
mass, but they’re not massive enough to account for
galaxy and cluster dark masses.
Also, they would smear out early structure in the
Universe (see later)
Cold dark matter
WIMPs:
axions?
neutralinos?
Haven’t found anything yet. Watch this space!!
The Tully Fisher Relation for Spirals
In A1Y cosmology we considered the Tully Fisher relation for
spiral galaxies, which can be used to estimate galaxy distances.
The relation was first measured empirically, using HI rotation
velocities, by Brent Tully and Richard Fisher in 1977
To Earth
The Tully Fisher Relation for Spirals
In A1Y cosmology we considered the Tully Fisher relation for
spiral galaxies, which can be used to estimate galaxy distances.
The relation was first measured empirically, using HI rotation
velocities, by Brent Tully and Richard Fisher in 1977
To Earth
The Tully Fisher Relation for Spirals
In A1Y cosmology we considered the Tully Fisher relation for
spiral galaxies, which can be used to estimate galaxy distances.
To Earth
HI flux density (Jy)
The relation was first measured empirically, using HI rotation
velocities, by Brent Tully and Richard Fisher in 1977
1000
velocity
1500
kms -1
The Tully Fisher Relation for Spirals
In A1Y cosmology we considered the Tully Fisher relation for
spiral galaxies, which can be used to estimate galaxy distances.
To Earth
HI flux density (Jy)
The relation was first measured empirically, using HI rotation
velocities, by Brent Tully and Richard Fisher in 1977
1000
 Vmax 
 I  - 7.68 log 10 
  4.79
 sin i 
velocity
1500
kms -1
The Tully Fisher Relation for Spirals
In A1Y cosmology we considered the Tully Fisher relation for
spiral galaxies, which can be used to estimate galaxy distances.
To Earth
Absolute
magnitude
HI flux density (Jy)
The relation was first measured empirically, using HI rotation
velocities, by Brent Tully and Richard Fisher in 1977
1000
 Vmax 
 I  - 7.68 log 10 
  4.79
 sin i 
velocity
1500
kms -1
If disk is inclined to the line of sight,
we see only a component of Vmax
Origin of the Tully-Fisher Relation
M51
The disk surface brightness
distribution of spirals can
be well described by an
exponential law:
I ( R)  I (0) exp R / RD 
Central surface brightness
(1.21)
Disk scale length
Origin of the Tully-Fisher Relation
The disk surface brightness
distribution of spirals can
be well described by an
exponential law:
I ( R)  I (0) exp R / RD 
Central surface brightness
(1.21)
Disk scale length
NGC 7331
Origin of the Tully-Fisher Relation
The disk surface brightness
distribution of spirals can
be well described by an
exponential law:
I ( R)  I (0) exp R / RD 
Central surface brightness
(1.21)
Disk scale length
NGC 7331
Origin of the Tully-Fisher Relation
The disk surface brightness
distribution of spirals can
be well described by an
exponential law:
I ( R)  I (0) exp R / RD 
Central surface brightness
I-band SB profile of NGC 7331
RD
(1.21)
Disk scale length
Luminosity of disk:
LD 
 I ( R) dA 
Disk
2 

 I ( R) RdRd 
0 0
 2 I (0) RD
2
(1.22)
Origin of the Tully-Fisher Relation
Formally the exponential disk extends to R   , but the luminosity
converges after a few disk scale lengths, at R  a RD (say).
(e.g. L  0.96LD for a  5 ; see example sheet 1)
Origin of the Tully-Fisher Relation
Formally the exponential disk extends to R   , but the luminosity
converges after a few disk scale lengths, at R  a RD (say).
(e.g. L  0.96LD for a  5 ; see example sheet 1)
By this radius, rotation velocity V  Vmax
Hence, from eq. (1.17)
Vmax
2

Mass inside radius
G M a RD
a RD
R  a RD
(1.23)
Origin of the Tully-Fisher Relation
Squaring eq. (1.23) and substituting from eq. (1.22)
2
Vmax
4

G M a RD
a RD
2
2
2
G M a RD 2 I (0)

a2
LD
2
2
(1.24)
Origin of the Tully-Fisher Relation
Squaring eq. (1.23) and substituting from eq. (1.22)
2
Vmax
4

G M a RD
a RD
2
2
2
G M a RD 2 I (0)

a2
LD
2
2
Defining  as the disk mass-to-light ratio :
(1.24)
  MD L 
D
Hence
Vmax
4

2 I (0)G  LD
a 2 LD
2
2
2
(1.25)
M a RD
LD
Origin of the Tully-Fisher Relation
Squaring eq. (1.23) and substituting from eq. (1.22)
2
Vmax

4
G M a RD
a RD
2
2
2
G M a RD 2 I (0)

a2
LD
2
2
Defining  as the disk mass-to-light ratio :
(1.24)
  MD L 
D
Hence
Vmax
Assume

and
4

I ( 0)
2 I (0)G  LD
a 2 LD
2
2
M a RD
LD
2
the same for all galaxies
(1.25)

LD  Vmax
4
Origin of the Tully-Fisher Relation
Assume

and
I ( 0)
the same for all galaxies

LD  Vmax
4
Easy to show (see Examples 1) that this implies:
M  k  10 log 10 Vmax
(1.26)
Absolute magnitude
Compare this with the empirical result:
Why the different slope?…
 Vmax 
 I  - 7.68 log 10 
  4.79
sin
i


Origin of the Tully-Fisher Relation
Assume

and
I ( 0)
the same for all galaxies

LD  Vmax
4
Easy to show (see Examples 1) that this implies:
M  k  10 log 10 Vmax
(1.26)
Absolute magnitude
Compare this with the empirical result:
Why the different slope?…
 Vmax 
 I  - 7.68 log 10 
  4.79
sin
i


Spirals don’t all have same

and
I ( 0)
Origin of the Tully-Fisher Relation
Assume

and
I ( 0)
the same for all galaxies

LD  Vmax
4
Easy to show (see Examples 1) that this implies:
M  k  10 log 10 Vmax
(1.26)
Absolute magnitude
Compare this with the empirical result:
Why the different slope?…
 Vmax 
 I  - 7.68 log 10 
  4.79
sin
i


Spirals don’t all have same

and
I ( 0)
Agreement with LD  Vmax prediction
better at longer wavelengths
4
Origin of the Tully-Fisher Relation
B band: 440nm
LD  Vmax
H band: 1.65m
LD  Vmax
2.8
K’ band: 2.2m
LK '
3 1010 LK ',Sun
 Vmax 

 
-1 
 205 kms 
4
(1.27)
3.8
Origin of the Tully-Fisher Relation
Why the different slope?…
Spirals don’t all have same

and
I ( 0)
Agreement with LD  Vmax prediction
better at longer wavelengths.
4
Bluer wavelengths dominated by hot, young stars – luminosity
sensitive to current star formation rate; greater scatter
B band: 440nm
2.8
LD  Vmax
H band: 1.65m
3.8
LD  Vmax
The Fundamental Plane Relation for Ellipticals
In A1Y cosmology we introduced another relationship, analogous to
the Tully-Fisher relation, but applicable to ellipticals – the Dn  
relation. This is a special case of a more general relationship for
ellipticals: the Fundamental Plane.
Ellipticals do not exhibit large systemic rotation velocities. However,
their stars are moving rapidly on a variety of (often quite complex)
orbits, determined by the galaxy’s gravitational potential.
The Fundamental Plane Relation for Ellipticals
In A1Y cosmology we introduced another relationship, analogous to
the Tully-Fisher relation, but applicable to ellipticals – the Dn  
relation. This is a special case of a more general relationship for
ellipticals: the Fundamental Plane.
Ellipticals do not exhibit large systemic rotation velocities. However,
their stars are moving rapidly on a variety of (often quite complex)
orbits, determined by the galaxy’s gravitational potential.
If we observe the spectrum along the line of sight through the
centre of the elliptical, we will see a central velocity dispersion ,
We can use the virial theorem to show that
(See A1Y cosmology, and Example Sheet 2)

2
0

0
G M virial
5R
(1.28)
The Fundamental Plane Relation for Ellipticals
Exact result depends on the ellipticity (triaxiality) of the elliptical,
but in any case we get

2
0
G M virial

R
(1.29)
Radius of galaxy
What is
R
?…
Depends on surface brightness profile of the elliptical.
e.g. the de Vaucouleurs law, special case of Sersic’s formula :
I ( R)  I ( Re ) e

b 

  1
1
R n
Re
with
n4
b  2n  0.327
(1.30)
e.g. NGC3379 (M105) in Leo.
Very good fit to
de Vaucouleurs law
The Fundamental Plane Relation for Ellipticals
I ( R)  I ( Re ) e

b 

 
1
R n
Re

1 

Re
= effective radius; contains half of
the galaxy luminosity (also sometimes
known as ‘half light’ radius
(See Example Sheet 2)
As for exponential disk, strictly the SB profile extends to R  
but we can again treat the luminosity as converged within some finite
value of R (which we can express as a multiple of Re ).
The Fundamental Plane Relation for Ellipticals
I ( R)  I ( Re ) e

b 

 
1
R n
Re

1 

Re
= effective radius; contains half of
the galaxy luminosity (also sometimes
known as ‘half light’ radius
(See Example Sheet 2)
As for exponential disk, strictly the SB profile extends to R  
but we can again treat the luminosity as converged within some finite
value of R (which we can express as a multiple of Re ).
We can write
L 
I R2
(1.31)
Mean SB inside radius R
Squaring eq. (1.29)
and substituting
from eq. (1.31)

2
4
0
2
G M


2
R
G 2  2 L2 I
L
(1.32)
The Fundamental Plane Relation for Ellipticals
Assume

and
I
the same for all ellipticals
 L0
This is known as the Faber-Jackson relation
More luminous ellipticals are also more massive

Stars in their central regions are moving faster.
4
(1.33)
The Fundamental Plane Relation for Ellipticals
Assume

and
I
the same for all ellipticals
 L0
This is known as the Faber-Jackson relation
More luminous ellipticals are also more massive

Stars in their central regions are moving faster.
(Also applicable to dwarf
spheroidals and spiral bulges)
4
(1.33)
The Fundamental Plane Relation for Ellipticals
Assume

and
I
the same for all ellipticals
 L0
This is known as the Faber-Jackson relation
More luminous ellipticals are also more massive

Stars in their central regions are moving faster.
(Also applicable to dwarf
spheroidals and spiral bulges)
But the relation shows
considerable scatter:
I
and

are not the
same for all ellipticals
4
(1.33)
The Fundamental Plane Relation for Ellipticals
The SB of some ellipticals is more centrally concentrated than for
others.
Effect correlates with luminosity :
more luminous ellipticals have fainter central SB, and larger core
radii

larger effective radii Re
(Core radius = radius at which SB drops to half its central value)
The Fundamental Plane Relation for Ellipticals
Can improve the Faber-Jackson relation in two ways:
1.
Define radius of galaxy to a fixed isophotal value – i.e.
to a given SB level – analogous to ‘sea level’: defines a
standard galaxy size which reduces effect of variation in
SB profile between galaxies

Dn  
Isophotal diameter
relation
The Fundamental Plane Relation for Ellipticals
Can improve the Faber-Jackson relation in two ways:
1.
Define radius of galaxy to a fixed isophotal value – i.e.
to a given SB level – analogous to ‘sea level’: defines a
standard galaxy size which reduces effect of variation in
SB profile between galaxies

Dn  
relation
Isophotal diameter
2. (better!)
Include effective radius, Re , as an extra
parameter in the Faber-Jackson relation

Fundamental Plane
L0
2.65
Re0.65
(1.34)
The Fundamental Plane Relation for Ellipticals
Taking logarithms of eq. (1.34), the FP relation can be
written in the linear form:
M  A log 10  0  B log 10 Re  C
Or, re-writing eq. (1.31)
logarithms:
L 
(1.35)
I e Re
and taking
2
Mean surface brightness
inside effective radius, Re
log 10 Re  a log 10  0  b log 10 I e  c
(1.36)
The Fundamental Plane Relation for Ellipticals
Some recent real data,
from the EFAR galaxy
survey (Colless et al 2001)
z = 2.0
Light travel time =
10.3 billion years
z = 2.1
Light travel time =
10.5 billion years
z = 2.2
Light travel time =
10.6 billion years
z = 2.3
Light travel time =
10.8 billion years
z = 2.4
Light travel time =
10.9 billion years
z = 2.5
Light travel time =
11.0 billion years
z = 2.6
Light travel time =
11.1 billion years
z = 2.7
Light travel time =
11.2 billion years
z = 2.8
Light travel time =
11.3 billion years
z = 2.9
Light travel time =
11.4 billion years
z = 3.0
Light travel time =
11.5 billion years
z = 3.1
Light travel time =
11.6 billion years
z = 3.2
Light travel time =
11.6 billion years
z = 3.3
Light travel time =
11.7 billion years
z = 3.4
Light travel time =
11.8 billion years
z = 3.6
Light travel time =
11.9 billion years
z = 3.7
Light travel time =
11.9 billion years
z = 3.8
Light travel time =
12.0 billion years
z = 4.0
Light travel time =
12.1 billion years
z = 4.1
Light travel time =
12.1 billion years
z = 4.3
Light travel time =
12.2 billion years
z = 4.4
Light travel time =
12.2 billion years
z = 4.5
Light travel time =
12.3 billion years
z = 4.6
Light travel time =
12.3 billion years
z = 5.0
Light travel time =
12.5 billion years
Re-run
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