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PHY216
PHY472
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physical constants
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DEPARTMENT
OF PHYSICS
DEPARTMENT
OF PHYSICS
& Spring&Semester 2015-2016
ASTRONOMY
ASTRONOMY
Autumn Semester 2009-2010
GALAXIES
2 hours
2 hours
ADVANCED QUANTUM MECHANICS
Answer
question
ONE (Compulsory)
and TWO
questions,
Answer ALL
questions
from Section
A, and TWO questions
fromother
Section
B. Please one each from
clearly indicate
theA question
numbers
section
and section
B. on which you would like to be examined on
the front cover of your answer book. Cross through any work that you do not
want to All
be examined.
questions are marked out of ten. The breakdown on the right-hand side of the
paper is meant as a guide to the marks that can be obtained from each part.
Questions from Section A are marked out of ve, whereas those from Section B
are marked out of fteen. The breakdown on the right-hand side of the paper is
meant as a guide to the marks that can be obtained from each part.
PHY216
TURN OVER
1
PHY216
SECTION A – answer ALL questions
1. A Cepheid variable in a nearby galaxy has a mean apparent magnitude in the V band
of hmV i = 24.45 magnitudes, a period of P = 8 days, and suffers extinction of AV =
0.35 magnitudes due to foreground dust in the Milky Way. If Cepheid variables
follow a V-band period-luminosity relation of form
hMV i = −2.76 log10 P(days) − 4.16,
estimate the distance to the galaxy. What are the main uncertainties involved when
using Cepheid variables as distance indicators?
[5]
2. Briefly explain each of the following.
- The blue colours of reflection nebulae.
- The detection of stars moving at very high velocities (>1000 km s−1 ) close to
the Galactic Centre.
- The low metallicities and high velocities relative to the Sun of stars in the Galactic Halo.
- The red colours of elliptical galaxies.
- The fact that the colours of spiral galaxies become bluer along the Hubble sequence from type Sa to Sc.
[5]
3. If the stellar bulge of a nearby spiral galaxy has a mass of 109 M and an effective
radius of 2 kpc, estimate its velocity dispersion (σ ).
[5]
4. HI 21cm observations show that a spiral galaxy has a flat rotation curve at radial
distances between 5 and 30 kpc from its nucleus, with an inclination corrected rotation velocity of 250 km s−1 . Estimate the mass density in the dark matter halo at a
distance of 15 kpc from the nucleus.
[5]
PHY216
CONTINUED
2
PHY216
SECTION B — answer TWO questions
5. Write an essay on the determination of the Hubble Constant (H0 ). Your answer should
include reference to the following:
- the cosmic distance ladder and the steps required to produce an accurate value
of H0 ;
- the various primary and secondary distance indicators required (give a brief
description of two of each);
- the problems caused by peculiar galaxy velocities relative to the Hubble Flow;
- how estimates of the Hubble Constant have changed since the first determinations in the 1920s;
- the reasons for the early inaccuracy in the Hubble Constant;
[15]
- the impact of the Hubble Space Telescope.
6.
(a) Oort’s constants (A and B) are defined as follows:
"
"
#
#
1
1
dvc
dvc
A=
, B = − ω0 +
,
ω0 −
2
dr R0
2
dr R0
where ω0 is the angular velocity of the local standard of rest (LSR) around the
Galactic centre in km s−1 kpc−1 , vc is the circular velocity in km s−1 , and r is
the radial distance in kpc. Currently the best estimates of Oort’s constants and
the distance to the centre of the Milky Way are A = 14.8 ± 0.8 km s−1 kpc−1 ,
B = −12.4 ± 0.6 km s−1 kpc−1 , and R0 = 8.0 ± 0.5 kpc respectively.
Estimate, with associated uncertainties, (i) the circular rotation speed of the
LSR around the centre of the Galaxy in km s−1 , and (ii) the velocity gradient of
the disk of the Galaxy in the vicinity of the Sun in km s−1 kpc−1 .
[6]
(b) Describe in detail the tangent point method for determining the inner rotation
curve of the Milky Way.
[4]
(c) Why is it much harder to measure the outer rotation rotation curve of the Milky
Way than its inner rotation curve?
[2]
(d) The rotation curves of the Milky Way and other spiral galaxies are observed to
remain flat out to large radii — well beyond the main concentrations of starlight.
Explain why this might imply that they contain large amounts of dark matter.
[3]
PHY216
TURN OVER
3
PHY216
7.
(a) Describe the general features that the various types of active galactic nuclei
(AGN) have in common.
[4]
Based on optical spectroscopic observations, how would you distinguish an
AGN from the nucleus of a normal, non-active galaxy?
[2]
(b) Compare and contrast, in terms of observational features, the following pairs of
objects:
- Type 1 and Type 2 Seyfert galaxies;
- quasars and Type 1 Seyfert galaxies;
- FRI and FRII radio galaxies.
(c) If a quasar in the distant universe has a bolometric luminosity of 1039 W, estimate its mass accretion rate. State any assumptions made.
[2]
[2]
[2]
[3]
8. Write detailed accounts of any three of the following.
(a) The general differences between spiral and elliptical galaxies.
[5]
(b) Determining the masses of elliptical galaxies.
[5]
(c) The observational differences between boxy and disky elliptical galaxies.
[5]
(d) The light profiles of elliptical galaxies.
[5]
(e) Theories for the formation of elliptical galaxies.
[5]
END OF QUESTION PAPER
PHY216
4
PHYSICAL CONSTANTS & MATHEMATICAL FORMULAE
Physical Constants
electron charge
electron mass
proton mass
neutron mass
Planck’s constant
Dirac’s constant (~ = h/2π)
Boltzmann’s constant
speed of light in free space
permittivity of free space
permeability of free space
Avogadro’s constant
gas constant
ideal gas volume (STP)
gravitational constant
Rydberg constant
Rydberg energy of hydrogen
Bohr radius
Bohr magneton
fine structure constant
Wien displacement law constant
Stefan’s constant
radiation density constant
mass of the Sun
radius of the Sun
luminosity of the Sun
mass of the Earth
radius of the Earth
e = 1.60×10−19 C
me = 9.11×10−31 kg = 0.511 MeV c−2
mp = 1.673×10−27 kg = 938.3 MeV c−2
mn = 1.675×10−27 kg = 939.6 MeV c−2
h = 6.63×10−34 J s
~ = 1.05×10−34 J s
kB = 1.38×10−23 J K−1 = 8.62×10−5 eV K−1
c = 299 792 458 m s−1 ≈ 3.00×108 m s−1
ε0 = 8.85×10−12 F m−1
µ0 = 4π×10−7 H m−1
NA = 6.02×1023 mol−1
R = 8.314 J mol−1 K−1
V0 = 22.4 l mol−1
G = 6.67×10−11 N m2 kg−2
R∞ = 1.10×107 m−1
RH = 13.6 eV
a0 = 0.529×10−10 m
µB = 9.27×10−24 J T−1
α ≈ 1/137
b = 2.898×10−3 m K
σ = 5.67×10−8 W m−2 K−4
a = 7.55×10−16 J m−3 K−4
M = 1.99×1030 kg
R = 6.96×108 m
L = 3.85×1026 W
M⊕ = 6.0×1024 kg
R⊕ = 6.4×106 m
Conversion Factors
1 u (atomic mass unit) = 1.66×10−27 kg = 931.5 MeV c−2
1 astronomical unit = 1.50×1011 m
1 eV = 1.60×10−19 J
1 atmosphere = 1.01×105 Pa
1 Å (angstrom) = 10−10 m
1 g (gravity) = 9.81 m s−2
1 parsec = 3.08×1016 m
1 year = 3.16×107 s
Polar Coordinates
x = r cos θ
y = r sin θ
∂
1 ∂2
1 ∂
2
r
+ 2 2
∇ =
r ∂r
∂r
r ∂θ
dA = r dr dθ
Spherical Coordinates
x = r sin θ cos φ
y = r sin θ sin φ
z = r cos θ
dV = r2 sin θ dr dθ dφ
1
∂
1
∂2
1 ∂
∂
2
2 ∂
∇ = 2
r
+ 2
sin θ
+ 2 2
r ∂r
∂r
r sin θ ∂θ
∂θ
r sin θ ∂φ2
Calculus
f (x)
f 0 (x)
f (x)
f 0 (x)
xn
ex
nxn−1
ex
tan x
sin−1
ln x = loge x
1
x
sin x
cos x
cos x
− sin x
cosh x
sinh x
sinh x
cosh x
a
cos−1 xa
tan−1 xa
sinh−1 xa
cosh−1 xa
tanh−1 xa
cosec x
−cosec x cot x
uv
sec x
sec x tan x
u/v
sec2 x
x
√ 1
a2 −x2
− √a21−x2
a
a2 +x2
√ 1
x2 +a2
√ 1
x2 −a2
a
a2 −x2
0
0
u v + uv
u0 v−uv 0
v2
Definite Integrals
Z
∞
xn e−ax dx =
0
Z
+∞
n!
an+1
r
(n ≥ 0 and a > 0)
π
a
−∞
r
Z +∞
1 π
2 −ax2
xe
dx =
2 a3
−∞
Z b
b Z b du(x)
dv(x)
Integration by Parts:
u(x)
dx = u(x)v(x) −
v(x) dx
dx
dx
a
a
a
−ax2
e
dx =
Series Expansions
(x − a) 0
(x − a)2 00
(x − a)3 000
f (a) +
f (a) +
f (a) + · · ·
1!
2!
3!
n X
n n−k k
n
n!
n
Binomial expansion: (x + y) =
x y
and
=
(n − k)!k!
k
k
k=0
Taylor series: f (x) = f (a) +
(1 + x)n = 1 + nx +
ex = 1 + x +
n(n − 1) 2
x + ···
2!
x 2 x3
+ +· · · ,
2! 3!
sin x = x −
ln(1 + x) = loge (1 + x) = x −
Geometric series:
n
X
rk =
k=0
Stirling’s formula:
(|x| < 1)
x3 x5
+ −· · ·
3! 5!
x2 x3
+
− ···
2
3
and
cos x = 1 −
x2 x4
+ −· · ·
2! 4!
(|x| < 1)
1 − rn+1
1−r
loge N ! = N loge N − N
or
ln N ! = N ln N − N
Trigonometry
sin(a ± b) = sin a cos b ± cos a sin b
cos(a ± b) = cos a cos b ∓ sin a sin b
tan a ± tan b
1 ∓ tan a tan b
sin 2a = 2 sin a cos a
tan(a ± b) =
cos 2a = cos2 a − sin2 a = 2 cos2 a − 1 = 1 − 2 sin2 a
sin a + sin b = 2 sin 21 (a + b) cos 12 (a − b)
sin a − sin b = 2 cos 12 (a + b) sin 12 (a − b)
cos a + cos b = 2 cos 12 (a + b) cos 12 (a − b)
cos a − cos b = −2 sin 12 (a + b) sin 21 (a − b)
eiθ = cos θ + i sin θ
1 iθ
1 iθ
cos θ =
e + e−iθ
and
sin θ =
e − e−iθ
2
2i
1 θ
1 θ
cosh θ =
e + e−θ
and
sinh θ =
e − e−θ
2
2
sin a
sin b
sin c
Spherical geometry:
=
=
and cos a = cos b cos c+sin b sin c cos A
sin A
sin B
sin C
Vector Calculus
A · B = Ax Bx + Ay By + Az Bz = Aj Bj
A×B = (Ay Bz − Az By ) î + (Az Bx − Ax Bz ) ĵ + (Ax By − Ay Bx ) k̂ = ijk Aj Bk
A×(B×C) = (A · C)B − (A · B)C
A · (B×C) = B · (C×A) = C · (A×B)
grad φ = ∇φ = ∂ j φ =
∂φ
∂φ
∂φ
î +
ĵ +
k̂
∂x
∂y
∂z
∂Ax ∂Ay ∂Az
+
+
∂x
∂y
∂z
∂Ax ∂Az
∂Ay ∂Ax
∂Az ∂Ay
−
î +
−
ĵ +
−
k̂
curl A = ∇×A = ijk ∂ j Ak =
∂y
∂z
∂z
∂x
∂x
∂y
div A = ∇ · A = ∂ j Aj =
∇ · ∇φ = ∇2 φ =
∂ 2φ ∂ 2φ ∂ 2φ
+
+ 2
∂x2 ∂y 2
∂z
∇×(∇φ) = 0
and
∇ · (∇×A) = 0
∇×(∇×A) = ∇(∇ · A) − ∇2 A