Galaxies II – Dr Martin Hendry 10 lectures to A3/A4, beginning January 2008.

Galaxies II – Dr Martin Hendry10 lectures to A3/A4, beginning January 2008

10 lectures to A3/A4, beginning January 2006

Course Topics

1. Galaxy Kinematicso Spectroscopy and the LOSVDo Measuring mean velocities and velocity dispersionso Rotation curves of disk systemso Evidence for dark matter haloso The Tully-Fisher and Fundamental Plane relations

Galaxies II – Dr Martin Hendry


Course Topics

1. Galaxy Kinematicso Spectroscopy and the LOSVDo Measuring mean velocities and velocity dispersionso Rotation curves of disk systemso Evidence for dark matter haloso The Tully-Fisher and Fundamental Plane relations

2. Abnormal and Active Galaxies

o Starburst galaxieso Galaxies with AGNo The unified model of AGNo Radio lobes and jetso Evidence for supermassive black holes



Course Topics

3. Galaxy Formation and Evolutiono Galaxy mergers and interactionso Polar rings, dust lanes and tidal tailso Star formation in ellipticals and spiralso Chemical evolution models



Course Topics

3. Galaxy Formation and Evolutiono Galaxy mergers and interactionso Polar rings, dust lanes and tidal tailso Star formation in ellipticals and spiralso Chemical evolution models

4. Galaxies and Cosmology

o Hierarchical clustering theorieso Galaxy clusters as cosmological probeso Proto-galaxies and the Lyman-alpha foresto 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)

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




c

v

0

z

Radial velocity(can be +ve or –ve)

Speed of lightWavelength of light as measured in the laboratory

Change in wavelength(can be +ve or –ve)

(Formula OK if v << c)

(1.1)




c

v

0

z

Radial velocity(can be +ve or –ve)

Speed of lightWavelength of light as measured in the laboratory

Change in wavelength(can be +ve or –ve)

(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)

(1.1)

A B

A

A

AB

B

B

0 0 0 0

AB A+B A+BBA

Spectroscopic Binaries

Orbits, from above

Spectral lines

To Earth

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 Hline: SDSS)





We define the Line of Sight Velocity Distribution (LOSVD) via:

LOSLOS v)v( dF Fraction of stars contributing to spectrum with radial velocities between and LOSv LOSLOS vv d

(1.2)

It is useful to define the observed spectrum not in terms of wavelength or frequency, but spectral velocity, , viau

lncu (1.3)

It is useful to define the observed spectrum not in terms of wavelength or frequency, but spectral velocity, , via

Hence, a Doppler shift of corresponds to

u

lncu

LOSv

c

u

(1.3)

(1.4)

It is useful to define the observed spectrum not in terms of wavelength or frequency, but spectral velocity, , via

Hence, a Doppler shift of corresponds to

Light observed at spectral velocity was emitted at spectral velocity

u

lncu

LOSv

c

u

uLOSvu

(1.3)

(1.4)

measures the (relative)

intensity of radiation at spectral

velocity

Intensity received from a star with

line of sight velocity is

Suppose that all stars have intrinsically identical spectra, )(uS

)(uSu

Rel

ativ

e in

tens

ity (

arbi

trar

y un

its)

Wavelength (Angstroms)

)v( LOSuSLOSv



velocity




)(uSu

Rel

ativ

e in

tens

ity (

arbi

trar

y un

its)


Observed composite spectrum:

LOSLOSLOS vvv)( duSFuG

(1.5)

LOSv )v( LOSuS



velocity




)(uSu


(1.5)

LOSv )v( LOSuS




velocity




)(uSu


(1.5)

Galaxy spectrum is smoothed version of stellar spectrum –

‘smeared out’ by LOSVD

LOSv )v( LOSuS


replaced by (local) average spectrum

which depends on :

Of course, stars don’t all have identical spectra,

)(uS )v,( LOSav uS

o ageo metallicityo galaxy environment

Spectral SynthesisSpectral Synthesis

(See Section 3)


which depends on :


)(uS



(See Section 3)

LOSLOSLOSavLOS vv,vv)( duSFuG

(1.6)

)v,( LOSav uS


which depends on :


)(uS )v,( LOSav uS



(See Section 3)

We consider here only the simpler case where is the same throughout the galaxy

)(uS LOSLOSLOSavLOS vv,vv)( duSFuG

Generally the slowly varying continuum component of the spectrum is removed first – i.e. we write:

)()()( linecont uSuSuS Emission:

Absorption:(1.7)

0)(line uS

0)(line uS


so that

)()()( linecont uSuSuS

LOSLOSlineLOSline vvv)( duSFuG

Emission:

Absorption:

0)(line uS(1.7)

(1.8)

0)(line uS


so that

)()()( linecont uSuSuS

Rel

ativ

e in

tens

ity (

arbi

trar

y un

its)


Emission:

Absorption:(1.7)

(1.8) LOSLOSlineLOSline vvv)( duSFuG

0)(line uS

0)(line uS

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).

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).

It is a particular type of integral equation: a convolution

Observed galaxy spectrum LOSVD ‘Template’ stellar spectra

dxxysxfyg

)(

‘Data’ function ‘Source’ function ‘Kernel’ function

(1.9)


We want to estimate the source function, , given the observed galaxy spectrum, , and using a kernel function, , computed from e.g. a stellar spectral synthesis model.

How can we extract from inside the integral?…

)(uS)(uG

LOSvF

LOSvF

We want to estimate the source function, , given the observed galaxy spectrum, , and using a kernel function, , computed from e.g. a stellar spectral synthesis model.

How can we extract from inside the integral?…

Fourier Convolution Theorem

Consider a convolution equation of the form

The Fourier transforms of the functions , and satisfy the

)(uS)(uG

dxxysxfyg

)(

gf s

)(~)(~

)(~ kskfkg

relation

Here

dxexfkf ikx)()(~

(1.10)

For proof, see Examples 1

LOSvF

LOSvF

In the context of our problem:

And

Hence, we can in principle invert the integral equation and reconstruct the LOSVD, LOSvF

)(~

)(~

~v 1

LOSkS

kGFF

)(~

)(~

)(~

kSkFkG (1.11)

(1.12)

Inverse Fourier transform

In the context of our problem:

And

Hence, we can in principle invert the integral equation and reconstruct the LOSVD,

In practice, this method is vulnerable to noise on the observed galaxy spectrum, , and uncertainties in the kernel .

Need to filter out high frequency (k) noise

)(~

)(~

)(~

kSkFkG (1.11)

(1.12)

)(uG )(uS

Inverse Fourier transform

)(~

)(~

~v 1

LOSkS

kGFF

LOSvF

Ratio of two small quantities: very noisy

Filter, denoting range of wavenumbers which give reliable inversion

If we cannot easily reconstruct the complete LOSVD , we can at least constrain some of the simplest properties of this function

LOSF

LOSLOSLOSLOS vvvv dF

Mean value (1.13)

Variance LOSLOS2

LOSLOS2LOS vvvv dF

(1.14)

Velocity dispersion2LOSLOS (1.15)

The Cross-Correlation Function MethodThe Cross-Correlation Function Method

This is a common method for estimating and .

Pioneered by e.g. Tonry & Davis (1979)

We define:

(We use continuum-subtracted galaxy and template spectra)

LOSv LOS

duuSuGCCF

LOSLOS v)()v( (1.16)


For a random value of the product

fluctuates between +ve and –ve values

duuSuGCCF

LOSLOS v)()v(

LOSv LOSv)( uSuG

)v( LOSCCF is small

LOSvuS

)(uG+ve-ve

+ve-ve


For a random value of the product

fluctuates between +ve and –ve values

duuSuGCCF

LOSLOS v)()v(

LOSv LOSv)( uSuG

)v( LOSCCF is small

LOSvuS

)(uG+ve

+ve

-ve

-ve

When emission and absorption features

line up,

and the product is large everywhere


duuSuGCCF

LOSLOS v)()v(

LOSv)( uSuG

)v( LOSCCF is large and positive

LOSvuS

)(uG

LOSLOS vv

+ve

+ve

-ve

-ve

We estimate by

finding the maximum of

the cross-correlation

function.

LOSv


duuSuGCCF

LOSLOS v)()v(LOSv

We estimate by

finding the maximum of

the cross-correlation

function.

Width of CCF peak allows

estimation of

Advantages:

LOSv

LOS


duuSuGCCF

LOSLOS v)()v(LOSv

Fast, objective, automatic

What do we learn from the LOSVD?…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)




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




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


We can construct a rotation curve : a graph of rotation

speed versus distance from the centre of the galaxy.

Milky Way Rotation CurveMilky Way Rotation Curve

Inside 1 kpc ‘rigid-body’ rotation

This is consistent with a spherical matter distribution, of

constant matter density

Consider a mass, , at distance from the centre of the

Galaxy.

Equating circular acceleration and gravitational force:

rr v

m r

2

2v

r

mMG

r

m r

Mass interior to radius r

(1.17)


This is consistent with for constant

32v

rG

rM r (1.18)

334 rM r


This is consistent with for constant

At large radii (well beyond the limit of the optical disk)

the Milky Way’s rotation curve is flat

32v

rG

rM r (1.18)

334 rM r

Evidence for a halo of dark matter around the Galaxy

0

10

20

30

40

50

60

0 10 20 30 40 50

In Our Solar System:

Distance from the Sun (AU)

Orb

ital v

eloc

ity (

km/s

)

0

10

20

30

40

50

60

0 10 20 30 40 50

Distance from the Sun (AU)

Orb

ital v

eloc

ity (

km/s

)

2/1v r

rM

constantv2 r

constant for all

SunRr

2/1v r (1.19)

Observed rotation curve

Rotation curve predicted from luminous matter

2/1v rSame argument gives

in outer

regions of the

Galaxy, if only a

roughly spherical

distribution of

luminous matter

contributes to the

rotation curve.

Instead rotation

curve is flat.

Same behaviour seen

for external galaxies

From the Mathewson et al ‘Mark III’ Spirals survey

Outer regions:

This is consistent with a roughly spherical distribution of

dark matter , with density

Consider a mass, , at distance from the centre of the

Galaxy.


const.v r

m r

2

2v

r

mMG

r

m r

Mass interior to radius r

(1.17)

2 r

Outer regions:

This is consistent with a roughly spherical distribution of

dark matter , with density

const.v r

rM r

2 r

(1.18)

const.dr

dM r but…

)(4 2 rrdr

dM r

for a spherical distribution

2)( rr as required (1.19)

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…Points to note…



least 100 kpc.

We cannot have to arbitrary radii, however, if

the halo mass is to remain finite.

2)( rr




least 100 kpc.

We cannot have to arbitrary radii, however, if

the halo mass is to remain finite.

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

2)( rr


Link between galaxy formation and cosmology – see later!

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

has the correct properties (but see later)

For the Milky Way:


220)(ra

Cr

(1.20)

-1Sun

80 kpc106.4 MC

kpc8.2a

So what is the Dark Matter?…So what is the Dark Matter?…

(Revision of A1Y Cosmology) (Revision of A1Y Cosmology)

Simplest candidates: Baryonic Dark Matter:Brown dwarfs

White dwarfs

Can constrain mass and distribution of MACHOs via gravitational microlensing

Large Magellanic Cloud

A MACHO

MACHO’s gravity focuses the light of the background star on the Earth

So the background star briefly appears brighter

Detecting MACHOs withGravitational Microlensing

Lightcurve of a microlensing event

Time

The shape of the curve tells about the mass and position of the dark matter which does the lensing

Lightcurve of a microlensing event

Time

The shape of the curve tells about the mass and position of the dark matter which does the lensing

Results indicate not nearly enough MACHOs to explain rotation curves




White dwarfs


Can also measure X-ray emission from galaxy clusters: baryonic cold gas

Cluster baryons from X-ray maps

2bEM

Optical X-ray




White dwarfs


Can also measure X-ray emission from galaxy clusters: baryonic cold gas

Again, not enough baryons to explain motion of galaxies in clusters!

Isotopes of hydrogen

+ + +

Deuterium(1 proton + 1 neutron)

Tritium(1 proton + 2 neutrons)

Hydrogen(1 proton)

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 SpiralsThe 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




To Earth

-1skmvelocity

HI

flux

den

sity

(Jy

)15001000




To Earth

-1skmvelocity

HI

flux

den

sity

(Jy

)15001000

79.4sin

Vlog7.68- max

10

iI




To Earth

-1skmvelocity

HI

flux

den

sity

(Jy

)15001000

79.4sin

Vlog7.68- max

10

iI

If disk is inclined to the line of sight, we see only a component of maxV

Absolute magnitude

Origin of the Tully-Fisher Relation Origin of the Tully-Fisher Relation

The disk surface brightness distribution of spirals can be well described by an exponential law:

DRRIRI /exp)0()( (1.21)

Central surface brightnessDisk scale length

M51



DRRIRI /exp)0()( (1.21)


NGC 7331



DRRIRI /exp)0()( (1.21)


Luminosity of disk:

22

0 0Disk

)0(2)()( DD RIRdRdRIdARIL

(1.22)

I-band SB profile of NGC 7331

DR


Formally the exponential disk extends to , but the

luminosity converges after a few disk scale lengths, at

(say).

(e.g. for ; see example sheet 1)

DRR R

DLL 96.0 5


Formally the exponential disk extends to , but the

luminosity converges after a few disk scale lengths, at

(say).

(e.g. for ; see example sheet 1)

By this radius, rotation velocity

Hence, from eq. (1.17)

DRR

D

R

R

MGD

2

maxV

R

DLL 96.0 5

maxVV

(1.23)

Mass inside radius DRR


Squaring eq. (1.23) and substituting from eq. (1.22)

D

R

D

R

L

IMG

R

MGDD

)0(2V

2

22

22

224

max

(1.24)



Defining as the disk mass-to-light ratio :

Hence

D

R

D

R

L

IMG

R

MGDD

)0(2V

2

22

22

224

max

(1.24)

D

R

D

DL

ML

MD

D

D

L

LGI2

2224

max

)0(2V

(1.25)



Defining as the disk mass-to-light ratio :

Hence

Assume and the same for all galaxies

D

R

D

R

L

IMG

R

MGDD

)0(2V

2

22

22

224

max

(1.24)

D

R

D

DL

ML

MD

D

D

L

LGI2

2224

max

)0(2V

(1.25)

)0(I 4

maxVDL



Easy to show (see Examples 1) that this implies:

Compare this with the empirical result:

Why the different slope?…

)0(I 4maxVDL

max10 Vlog10M k

Absolute magnitude

(1.26)

79.4sin

Vlog7.68- max

10

iI






)0(I 4maxVDL

max10 Vlog10M k

Absolute magnitude

(1.26)

79.4sin

Vlog7.68- max

10

iI

Spirals don’t all have same and )0(I






)0(I 4maxVDL

max10 Vlog10M k

Absolute magnitude

(1.26)

79.4sin

Vlog7.68- max

10

iI

Spirals don’t all have same and

Agreement with

prediction better at longer

wavelengths

)0(I4

maxVDL


2.8maxVDL

B band: 440nm H band: 1.65m3.8

maxVDL

4

1-max

Sun,'10

'

kms205

V

103

K

K

L

L

K’ band: 2.2m

(1.27)


Why the different slope?… Spirals don’t all have same and

Agreement with

prediction better at longer

wavelengths.

)0(I4

maxVDL

2.8maxVDL

B band: 440nm H band: 1.65m3.8

maxVDL

Bluer wavelengths dominated by hot, young stars – luminosity

sensitive to current star formation rate; greater scatter

The Fundamental Plane Relation for Ellipticals The Fundamental Plane Relation for Ellipticals

In A1Y cosmology we introduced another relationship,

analogous to the Tully-Fisher relation, but applicable to

ellipticals – the 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.

nD


In A1Y cosmology we introduced another relationship,

analogous to the Tully-Fisher relation, but applicable to

ellipticals – the 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

nD

0

R

MG

5virial2

0

(1.28)

(See A1Y cosmology, and Example Sheet 2)


Exact result depends on the ellipticity (triaxiality) of the

elliptical, but in any case we get

What is ?…

Depends on surface brightness profile of the elliptical.

e.g. the de Vaucouleurs law, special case of Sersic’s

formula :

with

R

MG virial20 (1.29)

Radius of galaxy

R

1

1

)()(n

eRRb

e eRIRI 327.02

4

nb

n(1.30)

e.g. NGC3379 (M105) in Leo.Very good fit to de Vaucouleurs law

As for exponential disk, strictly the SB profile extends to

but we can again treat the luminosity as converged within

some finite value of (which we can express as a multiple of

).


1

1

)()(n

eRRb

e eRIRIeR = effective radius; contains half of

the galaxy luminosity (also

sometimes known as ‘half light’

radius

(See Example Sheet 2)R

R eR

As for exponential disk, strictly the SB profile extends to

but we can again treat the luminosity as converged within

some finite value of (which we can express as a multiple of

).

We can write

Squaring eq. (1.29)

and substituting

from eq. (1.31)


1

1

)()(n

eRRb

e eRIRIeR = effective radius; contains half of

the galaxy luminosity (also

sometimes known as ‘half light’

radius

(See Example Sheet 2)R

R eR

(1.31)2RIL

Mean SB inside radiusR

L

ILG

R

MG 222

2

2240

(1.32)

Assume and the same for all ellipticals

This is known as the Faber-Jackson relation

More luminous ellipticals are also more massive

Stars in their central regions are moving faster.


I 40L (1.33)





(Also applicable to dwarf

spheroidals and spiral bulges)


I 40L (1.33)





(Also applicable to dwarf

spheroidals and spiral bulges)

But the relation shows

considerable scatter:

and are not the

same for all ellipticals


I 40L (1.33)

I


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 eR

(Core radius = radius at which SB drops to half its central value)


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

relation

nD

Isophotal diameter


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

relation

2. (better!) Include effective radius, , as an

extra parameter in the Faber-Jackson relation

nD

Isophotal diameter

eR

Fundamental Plane Fundamental Plane 65.02.650 eRL (1.34)

Taking logarithms of eq. (1.34), the FP relation can be

written in the linear form:

Or, re-writing eq. (1.31) and taking

logarithms:


CRBA e 10010 loglogM (1.35)

2eeRIL

cIbaR ee 1001010 logloglog (1.36)

Mean surface brightness inside effective radius, eR


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


z = 2.2


z = 2.3


z = 2.4


z = 2.5


z = 2.6


z = 2.7


z = 2.8


z = 2.9


z = 3.0


z = 3.1


z = 3.2


z = 3.3


z = 3.4


z = 3.6


z = 3.7


z = 3.8


z = 4.0


z = 4.1


z = 4.3


z = 4.4


z = 4.5


z = 4.6


z = 5.0


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Galaxies II – Dr Martin Hendry 10 lectures to A3/A4, beginning January 2008.

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