PH607/14/test3
UNIVERSITY OF KENT
PH607: GALAXIES
Test 3, Wednesday Week 24, 09:00
. FORMULA AND DATA ARE PROVIDED OVERLEAF.
This is a 50 minute test. Please solve ALL questions.
Students with an ILP that gives them more time in exams, please
indicate on the answer sheet and do NOT solve Question 5.
1. An E3 galaxy at a redshift of 0.04 is measured to have a radius of 2 arcseconds.
Apply Hubble’s Law to estimate its radius in kiloparsecs (hint: convert redshift into
distance and arcseconds into radians).
The redshift 0.04 correspo9nds to a recession speed of v = 0.04c.
According to Hubble’s law, its distance is
D = v / Ho
= (0.04 x 3 x 105 ) / 70
Mpc
= 1200/7 = 170 Mpc
Angle subtended by galaxy = 2 arcseconds = 2 (1 AU/ 1 pc) radians
= 2 x (1.5 ×1011 m / 3 ×1016 m )
= 10-5 radians
Galaxy size = angle subtende x distance = 10-5 x 170 Mpc
= 170 x 10-5 Mpc = 1.7 kpc – a smallish galaxy
2. State the two major observational differences between a Seyfert 1 and a Seyfert 2
galaxy. State briefly the major reason for both the differences.
2
o
o
Seyfert 1 galaxies are bright in the optical, and have both broad and
narrow lines in their spectra.
Seyfert 2 galaxies are fainter in the optical (but the same as Seyfert 1s
in the infrared), and they only have narrow lines in their spectra.
Seyfert 2s are obscured Seyfert 1s; a dust torus hides the inner broad line region and
nucleus but is too small to hide the narrow line region. Hence it is a an orientation
effect.
3. Calculate the Schwarzschild radius in units of the Sun’s radius for a black hole with
the Sun’s mass and for a super-massive black hole of mass 108 M
.
Using
(
ds = cdt 1-2GM/rc
2
2
)
2
æ
ö
dr
-ç
- r 2 ( dq 2 + sin 2 q dj 2 )
÷
è 1-2GM/rc2 ø
2
The equation indicates a singularity occurring at the Schwarzschild radius which is
2GM/c2
= 2 x 6.67 × 10-11 x 2.0 × 1030 / (3 × 108 ) 2
= 3.000 m
= 3,000 / 7.0 ×108
= 4.3 x 10 -6 R
The radius is proportional to the mass, hence for the SMBH, the Schwarzschikd
radius is 430 R
4 The rate of expansion of the Universe can be described by
3
 1 dR  2 8
1 2 2
2
  G  c  R  kc ,

3
3
 R dt 

where R is the scale factor,
is the density,  is the cosmological constant and k is
the curvature. Define the Hubble parameter, H(t), in both words and an equation, and
the Hubble constant, Ho, in words.
H(t) is the Hubble parameter, the rate of expansion of the universe at any time:
H=
1 dR
R dt
The Hubble constant is the present value of the Hubble parameter.
5. Taking a Hubble constant of Ho = 70 km s-1 Mpc-1, and  = 0 and k = 0, determine
the critical density of the Universe in units of solar masses per cubic megaparsec.
Taking  and k = 0, and re-arranging, setting H = Ho , yields the critical density
c = 3Ho2/(8G).
Ho = 70 x 1000 / (106 x 3.086 x 1016 ) s-1
= 2.26831e-18
Substitute for Ho and G yields:
c = 9.208 x 10-27
kg m-3
M
= 1.989 x 1030 kg
Mpc = 3.086 x 1022 m
c = 1.36 x 10-7 M
/Mpc3
turn over
4
In spherical coordinates, the Schwarzschild metric is
(
ds = cdt 1-2GM/rc
2
2
)
2
æ
ö
dr
-ç
- r 2 ( dq 2 + sin 2 q dj 2 )
÷
è 1-2GM/rc2 ø
.
Ho
= 70 (km/s)/Mpc
G
= 6.67 × 10-11 N m2 kg-2,
1 M = 2.0 × 1030 kg ,
1 R = 7.0 ×108 m,
c
= 3 × 108 m s-1,
1 pc
= 3 ×1016 m,
1 AU = 1.5 ×1011 m
1 radian = (1 pc / 1 AU) arcseconds
2