PHYS 20491
GALAXIES
Answers to Examples 2
(1) Surface brightness of Galaxy 1 follows
I r   I o e

r
ro
Hence total luminosity

Ltot   I o e

r
ro
0
ro=4000pc
2rdr  2I r
Io= 150 Lo pc-2
 Ltot =1.5x 1010 Lo
(2)
Inclination i =cos-1 0.3/0.7=60o
Doppler effect

f v
1421
-1
 
f
c 1420400 v=298 km s
2
o o
  xe x dx  1
 0

Circular velocity is inclined to line of sight
Vr  Vcirc sin i cos 
Velocity is measured on major axis cos

298 / 2
Hence
V 
 172 km s 1
circ
cos 60
Radius of galaxy r
r ~ 3000 tan(0.25o) =13.1 kpc
Assume keplerian

rv 2 13.1 3 109  172 103
M

G
6.67 1011

2
 1.78 1041 kg
M=9x 1010 Mo
(3)
Resolving components of velocity along LS
Vradial  V (r ). cos   o Ro sin l 
 Vr   r .r. cos   o Ro sin l 
From triangles
Ro sin l   r sin 90     r cos 
Radial
Vr   r   o Ro sin l 
Velocity
Consider H1 cloud at l=30o with Vr=80km/s
V
vr 
Vr
80
220

 o 

r
Ro sin l Ro 8.2 sin 30  8.2
 46.3 km/sec/kpc
Hence r = 220/46.3= 4.7 kpc
Hence using cosine law
r 2  Ro2  d 2  2 Ro d cos l
d
2 Ro cos l 
4R
 d= 4.8 & 9.4 kpc
(4) Mean Free Path
Stars interact if
Gm2 1
PE  KE
 2 mv 2
d
2Gm
ie d  2
v
2
o
cos 2 l  4 Ro2  4r 2
2

If star density is n, interaction will occur in cylinder radius d if star
travels distance  mean free path
1
v4

  2 
2 2
d n 4G m n
n= 1 star pc-3 = (3x1016)-3 = 3.4x10-50 stars m-3
 
10 10004

4  6.67  10


 = 44 Mpc
[Alternative
  2 10 
11 2
30 2
 3.4  10 50
 1.3 10 24 m
work out relaxation time
v3
100003
tc 

4nG 2 m 2 4  3.4  10 50  6.67  10 11 2  2  1030

 

2
 1.3  10 20 sec
Hence mean free path ~ velocity x relaxation time = 1.3 x 1024 m]
H1 atoms
Atoms interact if d=2r
 
1
1

 3.2  109 m
2
2
d n   10 8  106


=10-7 pc
(5)
for a flat rotation curve
 dv 
  0
 dr  Ro
V
V
 A  12 o & B   12 o
Ro
Ro
Hence if Vo= 180 km s-1 and Ro = 3kpc
A= 30 km s-1 kpc-1 and B = -30 km s-1 kpc-1


 2 r   2 2 1 
Epicyclic frequency

Now
v
r
dv d r  d


r 
dr
dr
dr
&
v  r v 
 r   2 2 1 

r  v r 
2
2
v
  r   2 2
r
r  

2 r 
 dv 
but    0
 dr  Ro
2
2
Hence
 3kpc  2
180
 84.8 km s -1 kpc 1
3
At Inner Lindblad Resonance for two armed spiral
 P   r  
P 
2
180 84.8

 17.6 km s -1 kpc -1
3
2
Corotation is when
ie

17.6 
P  r 
180
r
Hence corotation radius is at 10.2 kpc
(6) for collapse Gravitational PE > KE
GM 2 1
 M v2
R
2
Now
1
3
m p v 2  kT
2
2
4 3
& M  R 
3
Where mp is mass of proton
R2 
Hence substituting
9kT
4m p G
1
2
1
2


 kT 
1.38 10 100

  2.7pc


R

 27 2
9
11 
 m G 

10  6.67 10 
 p

 1.67 10
 23


(7) Unlike spirals, elliptical galaxies have no Neutral Hydrogen which is
the main tracer of spiral galaxy dynamics, particularly at large radii. Also
ellipticals have no O & B stars which means there are no HII regions
and hence H etc emission.
Hence the elliptical dynamics have to be determined from stellar
absorption spectra, but as there are no young stars these lines are weak.
Also as ellipticals are 3-D objects (unlike spirals which are ~2-D) the line
profiles are blended by a range of lines at different depths at differing
distances from the centre of the galaxy.
The main observational result of these studies is that Massive
ellipticals show low rotation velocities (V)~ 50 km/s, but high velocity
dispersion  ~2-300 km/s. By comparing their V/ ratio it appears that
flattened Ellipticals (eg E5-E7) are not supported by rotation.
The other major result from these studies is that the luminosity and stellar
dispersions are linked via the Faber-Jackson relation
L  4
(8) Elliptical galaxies can be described by the 3 parameters Re, e and o ,
where Re is the half light radius, e is the mean surface brightness within
Re, and o is the central velocity dispersion. It is found empirically that
they are related by
Re   o1.4  e0.85
Hence if these parameters are plotted along 3 perpendicular axes, samples
of elliptical galaxies will lie on a plane defined by the above equatiom.
This is known as the ‘fundamental plane’ for elliptical galaxies.
The luminosity –dispersion relation L   shows a scatter of 12 magnitudes and hence while it can give an estimate of distance, there
are large errors.
4
However by plotting Re vs  1.4 e-0.85 a plot with typically a few tenths
of a
magnitude scatter can be
obtained,
and hence significantly
more
accurate distance
estimates.
Origin of relationship
The average surface brightness e within Re is given by
e 
L/2
Re2
Also the potential energy must be comparable to the kinetic energy if the
galaxy is not to expand or collapse.
GM
 k 02
Re
Where k is a ‘structure parameter’ containing information on the
elliptical galaxy
Hence combining both equations
1
Re 
k  M  2 1
   o e
2G  L 
1
 M  2 1
 Re  k    o  e
 L
Show this is consistent with the ‘Fundamental Plane’
M 
k    L0.25   0e .25 Re0.5
 L
1

as L   e Re2

1
M 
substituti ng into RHS of Re  k    o2  e1
 L
hence
Re1.5   e1.25  o2

 Re   e
1.25
1.5

2
1.5
o
  e0.83  o1.33
ie approximately consistent with the ‘fundamental plane’
Re   e0.85 o1.4
(9)
Distance of LMC from Galactic Centre = 50 kpc, rotational velocity =200
km/s
Hence mass of galaxy within 50kpc M


2
V 2 R 200  10 3  50  3  1019
M

 9  10 41 kg  4.5  1011 M o
11
G
6.67  10

Assume it is a point mass, hence from notes
1
3
1
3
 2M 
 m 
R
 r r 
 R
m
2
M





1
3


1010
 m 
  50  11.1 kpc
r 
 R  
11 
2
M
2

4
.
5

10




Hence
So stars & gas will be strongly tidally by Milky Way if they are > 11kpc
from the centre of the LMC.
(10) Using the Virial Theorem
2T+=0
T
1
1
2
m
v

M v2

i i
2 i
2
  
Gmi m j
rij
j i
i
M
Hence
GM 2

R
R v2
G
GM 6.67  1011  1014  2  1030
11
 v 


1
.
5

10
R
3  3  1022
2
hence v
1
2 2
 1.5 1011  3.8 105 m s 1  380 km s 1