Download Introduction to Astronomy and Astrophysics - 1

Introduction to Astronomy and Astrophysics - 1
IUCAA-NCRA Graduate School 2013
Instructor: Dipankar Bhattacharya
IUCAA
August - September 2013
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Coordinate Systems, Units
and the Solar System
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Locating Objects
• Angular position can be measured accurately; distance difficult
• Spherical Polar Coordinate System:
latitude = Declination (δ)
longitude = Right Ascension (α)
Time
Solar Day = 24 h
(time between successive
solar transits)
Equatorial
plane of
Earth’s
rotation
Pole
Earth’s
Orbital
plane
Origin of α
Earth’s Spin Period: 23h56m
(time between successive
stellar transits)
24h “Sidereal Time”
= 23h56m Solar Time
α is expressed in units of time
Transit time of a given α = Local Sidereal Time
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Different coordinate systems
Convention: Latitude: (π/2 − θ) ; Longitude: Φ
• Equatorial: Poles: extension of the earth’s spin axis
• Ecliptic: Poles: Normal to the earth’s orbit around the sun
• Galactic: Poles: Normal to the plane of the Galaxy
For Equatorial and Ecliptic: same longitude
reference (ascending node - vernal equinox)
θ
Φ
For Galactic coordinates: longitude reference is
the direction to the Galactic Centre
Equatorial coordinates: larger Φ, later rise: RA (α)
latitude = Declination (δ)
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Rise and Set
Horizon: tangent plane to the earth’s surface at observer’s location
geographic latitude λ
zenith
OP = sin δ ; QP = cos δ = PB
AP = OP tan λ = sin δ tan λ
∠APB = cos-1 (AP/PB) = cos-1 {tan δ tan λ}
Total angle spent by the source above the horizon
= 360o − 2 cos-1 {tan δ tan λ}
Time spent by the source above the horizon
= ([360o − 2 cos-1 {tan δ tan λ}]/15) sidereal hours
= (1436/1440) ([360o − 2 cos-1 {tan δ tan λ}]/15)
hours by solar clock
Q
90 o
-δ
horizon
O
P
C
λ
A
B
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Earth-Moon system
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Tidally locked. Moon’s spin Period = Period of revolution around the Earth
Mearth = 5.97722 x 1024 kg ; Mmoon = 7.3477 x 1022 kg
Orbital eccentricity = 0.0549
Semi-major axis = 384,399 km, increasing by 38 mm/y (1ppb/y)
Earth’s spin angular momentum being pumped into the orbit
Origin of the moon possibly in a giant impact on earth by a mars-sized body;
moon has been receding since formation.
At present the interval between two new moons = 29.53 days
Moon’s orbital plane inclined at 5.14 deg w.r.t. the ecliptic
Earth around the sun, Moon around the earth: same sense of revolution
Moon is responsible for total solar eclipse as angular size of the sun and the
moon are roughly similar as seen from the earth.
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Bodies in The Solar System
A. Feild (STSCI)
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Measuring Distance
• Inside solar system: Radio and Laser Ranging
Earth-Sun distance: 1 Astronomical Unit = 149597871km
• Nearby stars: Parallax
Parsec = distance at which 1 AU subtends 1 sec arc
= 3.086 x 1013 km = 3.26 light yr
• Distant Objects: Standard Candles
‣ Cepheid and RR Lyrae stars
‣ Type Ia Supernovae
• Cosmological distance: Redshift
Linear size = angular size x distance
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Measure of Intensity
1 Jansky = 10-26 W/m2/Hz
Optical Magnitude Scale
Logarithmic Scale of Intensity: m = −2.5 log (I/I0) apparent magnitude
Absolute Magnitude (measure of luminosity):
M = −2.5 log (I10pc /I0)
The scale factor I0 depends on the waveband, for example:
Band
Johnson U
B
V
R
I
I0
1920 Jy
4130
3690
3170
2550
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
References
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The Physical Universe : Frank H. Shu
An Introduction to Modern Astrophysics : B.W. Carroll & D.A. Ostlie
Astrophysical Quantities: C.W. Allen
Astrophysical Formulae: K.R. Lang
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Orbits
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Kepler Orbit
1. Elliptical orbit with Sun at one focus
2. Areal velocity = constant
3. T2 ∝R3
1
⌧⇡p
G⇢
Dynamical time scale in gravity:
Two body problem:
l
r=
1 + e cos
;
J
l=
;
2
GM µ
M
CM
M1
2
Conserved Energy E
and Angular Momentum J
r

e= 1+
r
2
2EJ
G2 M 2 µ 3
1/2
E < 0 =) e < 1 ; bound elliptical orbit:
GM µ
a=
2|E|
;
J =µ
μ
M2
p
GM a(1
e2 )
M = M1 + M2
M1 M2
µ=
M1 + M2
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Motion in a Central Force Field
1 2
J2
+ U (r) ;
Energy E = µṙ +
2
2
2µr
Ue↵ (r)
Z
(J/r2 )dr
=
+ const.
1/2
[2µ {E Ue↵ (r)}]
30
Setting Ueff = E gives turning points:
"
1
GM µ2
=
1±
2
rmin
J
max
and for E < 0
= 2 [ (rmax )
s
2J 2 E
1+
GM 2 µ3
#
10
(rmax does not exist for E > 0)
(rmin )] = 2⇡
Ue↵ (r)]
Intersections of Ueff (r) with E
give turning points in the orbit
Effective potential for U(r) ∝-r -1
Newtonian Gravity
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Ueff (r)
Newtonian Gravity: U (r) =
GM µ
r
2
ṙ = [E
µ
2
E>0
0
E<0
-10
-20
-30
0.1
1
10
r
i.e. orbit is closed. Departure from 1/r or r2 potential give
6= 2⇡
precession of
periastron
100
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Departure from 1/r2 gravity
Common causes of departure from
1/r form of gravitational potential:
• Distributed mass
• Tidal forces
• Relativistic effects
In relativity, effective potential
near a point mass
Ē =
(including rest energy)
Newtonian approximation r̄
U=
↵
+ 2
r
r
U=
↵
+ 3
r
r
✓
1
per orbit
◆✓
◆
2
1
ā
1+ 2
r̄
r̄
1/2
2⇡ µ/J 2
=
6⇡↵ µ2 /J 4
Ē = E/mc2
ā = J/mcrg
rg = 2GM/c2
1
Next order correction, upon expanding the square root:
6⇡GM
=
Gives
a(1 e2 )c2
per orbit
per orbit
=
ā2
2r̄3
Precession of perihelion of Mercury
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Schwarzschild Gravity: Equation of Motion &
Effective Potential
✓
1
1 1/r̄
◆✓
1
1 1/r̄
◆✓
dr̄
d⌧
◆2
d
d⌧
◆2

1
= 2 Ē 2
Ē
ā2
= 2 4
Ē r̄
A binary orbit decays due to
Gravitational Wave radiation
dE
32 G4
2
2
=
M
M
dt
5 c 5 a5 1 2
⇥(M1 + M2 )f (e)
da
2a2 dE
=
dt
GM1 M2 dt
1 da
e)
a dt
Effective potential by
setting LHS = 0
1.2
Schwarzschild
Newtonian
1.15
6.0
6.0
1.05
1
0.95
4.0
0.9
3.0
0.0
0.85
de
= (1
dt
ā2
ā2
+ 3
2
r̄
r̄
1
1+
r̄
1.1
E (in units of rest energy)
✓
0.8
numbers on curves indicate the square
of the normalised angular momentum
1
10
r (in units of gravitational radius)
100
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Photon Orbit in Schwarzschild Gravity
hence
✓
dr
d
◆2
4
r
= 2
b
✓
2
1
2
b
b
+ rg 3
2
r
r
integrate to get orbit.
3rd term on RHS causes
curvature of light path
Gravitational Lensing
◆
y/r_g
ā
b
⌘ b̄
!
setting m = 0, Ē ! 1, ā ! 1,
rg
Ē
✓
◆ ✓ ◆2
1
dr̄
b̄2
b̄2
10
=1
+
1 1/r̄
d⌧
r̄2
r̄3
✓
◆ ✓ ◆2
1
d
b̄2
5
= 4
1 1/r̄
d⌧
r̄
b = impact parameter at 1
photon orbits in Schwarzschild spacetime
0
-5
impact parameter l/r_g =
2.0, 2.5981, 3.0, 4.0, 6.0, 8.0
from innermost to
outermost trajectory
-10
-10
-5
0
x/r_g
5
10
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
References
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Classical Mechanics : H. Goldstein
Relativistic Astrophysics : Ya B. Zeldovich & I.D. Novikov
Gravitation : C. Misner, K.S. Thorne & J.A. Wheeler
Classical Theory of Fields : L.D. Landau & E.M. Lifshitz
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Tidal forces and Roche Potential
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Tidal effect
Gradient of external gravitational force across an extended body tends to
deform the object - responsible for tides on Earth
R
A
M
m
d
B
l
2GM m
l
Tidal Force: FT =
3
d
Self gravity
Gm2
of object B: Fg = 2
l
Object B would not remain intact if FT > Fg
⇣
⌘
m
1/3
l
q
m
q⌘
<
∴ condition for stability: l3 <
or
,
d3 ,
M
d
2
2M
✓
◆✓
◆
3
M
m
l
3
3
Disruption would occur if d < 2 M = 2R
3 /3
3 /3
m
4⇡R
8
⇥
4⇡(l/2)
✓
◆1/3
⇢M
4/3
i.e. for d < 2 R
: Roche limit
⇢m
1
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Roche Potential in a binary system
The surface of a fluid body is an equipotential
⊥r to orbital plane
In
corotating
frame
axis of revolution
P
In orbital plane
r1
r3
r3
M1
M1
C.M.
r2
r1
r2
M2
P
C.M.
M2
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Roche Potential in the equatorial plane
M2
M1
M2
q⌘
= 0.5
M1
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Lagrangian points and the Roche Lobe
L4
In orbital plane
L3
L1
M1
L5
M2
q⌘
= 0.5
M1
M2
L2
A star
overfilling
its Roche
Lobe
would
transfer
matter to
its
companion
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
References
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Theoretical Astrophysics, vol. 1 sec. 2.3 : T. Padmanabhan
Close Binary Systems : Z. Kopal
Hydrodynamic and Hydromagnetic Stability: S. Chandrasekhar
Astrophysical Journal vol. 55, p. 551 (1984) ; S.W. Mochnacki
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Hydrostatic Equilibrium
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Equations of Fluid Mechanics
@⇢ ~
Continuity Equation:
+ r · (⇢~v ) = 0
@t

@~v
~ v =
Euler Equation:
⇢
+ (~v · r)~
@t
Equation of State:
The fluid is described
by the quantities
~ + ⇢~g
rP
P = P (⇢)
⇢, P, ~v
that are functions of
space and time.
Viscosity is ignored
@
Stationarity follows by setting time derivatives
to zero
@t
@
Hydrostatic equilibrium follows by setting both
and ~v to zero:
@t
~ = ⇢~g
rP
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Hydrostatic Equilibrium
~ = ⇢~g
rP
In spherical symmetry:
In relativity:
dP
=
dr
GM (r)⇢(r)
r2
dP
=
dr
dM (r)
= 4⇡r2 ⇢(r)
dr
h
(Tolman, Oppenheimer, Volkoff)
ih
G M (r) + 4⇡r3 Pc(r)
⇢(r) +
2
⇣
⌘
2GM (r)
2
r 1
c2 r
P (r)
c2
Supplement with appropriate equation of state and solve for the structure of
self-gravitating configurations such as stars, planets etc
i
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Virial Theorem
dP
=
dr
GM (r)⇢(r)
r2
(hydrostatic equilibrium)
3
4⇡r
Multiply both sides by
and integrate over the full configuration: r = 0 ! R

Z R
Z R
GM (r)⇢(r)
3
2
2
4⇡R P (R)
4⇡r · 3P dr =
4⇡r dr
r
0
0
RHS = Total gravitational energy of the configuration Eg
and since P = (
1)uth , where uth is the Thermal (kinetic) energy density,
the 2nd term in LHS = 3(
Hence
(<0)
Eg + 3(
1)Eth , Eth being the total thermal (kinetic) energy
1)Eth = 4⇡R3 P (R) :
Virial Theorem
must be obeyed by all systems in hydrostatic equilibrium
For
= 5/3 and P (R) = 0 :
Note:
Eg + 2Eth = 0
Etot = Eg + Eth = Eg /2 =
Eth
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
A Rough Guide to Stellar Structure
dP
=
dr
stellar radius R
central pressure Pc
GM (r)⇢(r)
r2
total mass M
Using a linear approximation:
0
Pc
=
R
2
3 GM
Pc =
=
4
4⇡ R
✓
4⇡
3
GM
M
· 4⇡ 3
2
R
3 R
◆1/3
GM 2/3 ⇢4/3
(“gravitational pressure”Pgrav)
For equilibrium, this pressure needs to be matched by the Equation of State
kT
e.g. Thermal Pressure: P =
⇢
µmp
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Thermal pressure support
2/3
M
∝
P
M1 < M2 < M3
log Pc
Tc
∝
P
ρ
M3
T3
T2
M2
T1
M1
log ρ
4/3
ρ
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
References
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Astrophysics I : Stars : R.L. Bowers
The Physical Universe : F.H. Shu
Stellar Structure and Evolution : R. Kippenhahn and A. Weigert
Physics of Fluids and Plasmas : A. Rai Choudhuri
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Stars
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Main Sequence
Hydrogen burning star
Pc ⇡ GM
2/3 4/3
⇢
Tc ⇡ TH
kTH
=
⇢
µmp
::
⇢/M
2
M
; R / M and Tc /
R
Luminosity L = Radiative Energy Content / Radiation Escape Time
✓
◆2
R
l
1
1
·
=
Radiation Escape Time =
where l = mean free path =
l
c
n
⇢
opacity
4 3
aT R
4
L
⇡
⇡
aclT
R
Hence
In Main Sequence: L / lR / lM
2
(R /lc)
R3
:: LMS / M 3
High mass stars: opacity: Thomson scattering  = constant; l /
M
T 3.5
Low mass stars: l /
:: LMS / M 5
⇢2
On average
LMS / M 4
tMS / M 3
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Effective Temperature
A typical stellar spectrum is nearly a blackbody
Color Temperature
= Temp. of best-fit
Planck Function
Total Luminosity
Solar Spectrum
4
L = 4⇡ Te↵
R2
Te↵ is defined as
the Effective
Temperature
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Hertzsprung Russell Diagram for Stars
A plot of Luminosity vs. Temperature
(Absolute Magnitude vs. Color)
4
On Main Sequence L / M ; R / M
Near Blackbody spectrum:
Te↵ ⇡ Tcol
4
Te↵
-5
Horizontal
8
8
LMS / Te↵
⇠ Tcol
Nearby stars
measured
by the
Hipparcos
satellite
n
ai
5
Sun
nc
ue
q
Se
MV (mag)
Gi
an
ts
0
M
log Luminosity
Branch
L
/ 2 / M2
R
e
log Color Temperature
10
15
-0.5
0
0.5
1
B-V (mag)
1.5
2
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Theoretical
H-R diagram
Stellar evolutionary tracks
by Schaller et al 1992
O
B
A
F G K
M
Stellar
spectral
classification
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Degeneracy Pressure
Occupation No.
Momentum space occupation in cold Fermi gas
pF
momentum
⇣ g ⌘ 4⇡
3
No. of particles per unit volume n =
p
F
3
h
3
✓
◆1/3
3
hn1/3
hence pF =
4⇡g
Pressure P ⇠ n · v · pF / v · n4/3
Electron degeneracy: v = pF /me (non-relativistic) and v = c (relativistic)
ne = ⇢/(µe mp ) in both regimes
∴ Electron Degeneracy Pressure
Pdeg / ⇢5/3
/ ⇢4/3
(non-relativistic)
(relativistic)
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Stellar Equilibrium
log Pc
P
av
gr
Pth
T3
TH
Excluded Zone
T1
M
3
M
cr
M2
M1
log ρ
P
g
de
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Stellar Mass Function
The mass distribution
is typically a powerlaw.
In the Milky Way
the index
dN/dM
M
↵ ⇡ 2.35
Salpeter (1966)
M
↵
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
References
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Astrophysics I : Stars : R.L. Bowers
The Physical Universe : F.H. Shu
Stellar Structure and Evolution : R. Kippenhahn and A. Weigert
Structure and Evolution of the Stars: M. Schwarzcshild
http://www.rssd.esa.int/index.php?project=HIPPARCOS&page=HR_dia
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Compact Stars
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
White Dwarfs
Configurations supported by Electron Degeneracy Pressure
Pdeg = K1 me 1 (⇢/µe mp )5/3 (non-relativistic)
Pdeg = K2 (⇢/µe mp )4/3
>m c
(relativistic) :: when pF ⇠
e
> 106
⇢
( ⇠
g cm-3 )
Equilibrium condition: Pdeg ⇡ GM 2/3 ⇢4/3
Non-relativistic: R / me 1 µe
Relativistic: M ⇠
✓
K2
G
5/3
◆3/2
M
1/3
(µe mp )
( R ~ 104 km for M ~ 1 Msun )
2
: Limiting Mass
(Chandrasekhar Mass)
MCh = 5.76µe 2 M
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Limiting Mass of White Dwarf
log Pc
P
M
Ch
log ρ
g
de
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Chandrasekhar 1931, 1935
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Neutron Stars
Supported by Neutron degeneracy pressure and repulsive strong interaction
TOV equation + nuclear EOS required for description
Beta equilibrium : > 90% neutrons, < 10% protons and electrons
Uncertainty in the knowledge of nuclear EOS leads to uncertainty in the
prediction of Mass-radius relation and limiting mass of neutron stars
(upon exceeding the max. NS mass a Black Hole would result)
Inter-nucleon distance ~ 1 fm
n ~ 1039 , ρ ~ 1015 g cm-3
R ~ 10 km for M ~ 1 Msun
Neutron stars spin fast: P ~ ms - mins
and have strong magnetic field: Bsurface ~ 108 - 1015 G
Exotic phenomena: Pulsar, Magnetar activity
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Demorest et al 2010
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
References
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The Physical Universe : F.H. Shu
Stellar Structure and Evolution : R. Kippenhahn and A. Weigert
An Introduction to the Study of Stellar Structure: S. Chandrasekhar
White Dwarfs, Neutron Stars and Black Holes : S.A. Shapiro and S.L. Teukolsky
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Stellar Evolution
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Schönberg-Chandrasekhar limit
Core-envelope configuration: Inert core surrounded by burning shell
R
Pe ,Te
Rc
Mc T c
Pc
M
Mc2
c2 4
Rc
2Eth + Eg
Mc T c
= c1
Core surface pressure Pc =
3
4⇡Rc
Rc3
Te4
Envelope base pressure Pe = c3 2
M
For mechanical and thermal balance Te = Tc and Pe = Pc
But Pc has a maximum as a function of Rc
Pc,max
Tc4
= c4 2
Mc
So balance is possible only if Pe  Pc,max
Mc

i.e. q0 ⌘
M
r
c4
⌘ qsc ⇡ 0.37
c3
✓
µenv
µcore
◆2
if core mass grows beyond this, then
core collapse would occur.
contraction until degeneracy support
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Post Main Sequence Evolution
Low mass star (< 1.4 M⦿):
gradual shrinkage of the core to
degenerate configuration
He ign at Mc = 0.45 M⦿ : L=100 L⦿
varied mass loss; horizontal branch
later AGB WD+planetary nebula
High mass star:
sudden collapse of the core from
thermal to degenerate branch
quick progress to giant
Multiple burning stages
If final degen. support at Mc < Mch ,
WD+PN will result
Else burning all the way to Fe core.
Once Mch exceeded : collapse,
neutronization, supernova
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Nuclear burning stages
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Stellar
Evolutionary
Tracks
Star Cluster Studies
• Spectroscopic
Parallax: distance
• Turnoff mass: age
Schaller et al 1992
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
A star’s journey to a supernova
Woosley et al
Core-collapse Supernovae:
Etot ~ 1053 erg; Ekin ~ 1051 erg; Erad ~ 1049 erg
Massive, fast spinning stars
r-process nucleosynthesis
jets
GRB
heavy elements
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Supernovae of Type Ia
Occur due to mass transfer in double WD binary followed by complete explosion.
No core collapse
WD composition: C+O
Accretion increases WD mass
Mch approached
degenerate C-ignition
rapid contraction
thermal runaway
heating
explosion
Etot ~ 1051 erg ; Generates radioactive Ni which powers light curve
Standard conditions, standard appearance
Distance Indicators
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
References
•
•
•
•
Stellar Structure and Evolution : R. Kippenhahn and A. Weigert
Stars: Their Birth, Life and Death : I.S. Shklovskii
An Introduction to the Theory of Stellar Structure and Evolution : D. Prialnik
An Invitation to Astrophysics : T. Padmanabhan
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Radiation
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Radiative Transfer
I⌫ : Specific Intensity
Radiation is modified while propagating through matter
dI⇥
=
ds
⇥ I⇥
absorption coef
(energy/area/s/Hz/sr)
dI
= I +S
d⇥
+ j⇥
I⌫ = S⌫ + (I⌫0
emission coef
⌧⌫ : Optical Depth
S⌫ )e
⌧⌫
S⌫ : Source Function
1
2h⌫3
For a thermal source S⌫ = Planck function B⌫ = 2
c exp(h⌫/kT)
(blackbody)
⌧⌫
1
Blackbody radiation
Blackbody function B⌫ (T)
A source can be optically thick at some
frequencies and optically thin at others
Emitted bolometric flux:
⌧⌫ ⌧ 1
(optically thin)
I⌫ =
I⌫0
+ (S⌫
I⌫0 )⌧⌫
Tsrc > Tbg : emission
log intensity
(optically thick)
F = T4
= 1034
kT
1 keV
1
h⌫max = 2.82 kT
T1
!4
T2
erg/s/km2
T3
Tsrc < Tbg : absorption
For non-thermal distribution of particle
energies, S⌫ , B⌫
= 2⌫2 kT/c2
(h⌫ ⌧ kT)
T4
T1 > T2 > T3 > T4
log frequency
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Bremsstrahlung
(free-free process)
Radiation
Emission coefficient :
✏↵⌫ = 6.8 ⇥ 10
Bolometric:
38
Z2 ne ni T
1/2
✏↵ = 1.4 ⇥ 10
e
27
h⌫/kT
ḡ↵ erg s-1 Hz-1 cm-3
Z2 ne ni T1/2 ḡ↵ erg s-1 cm-3
Free-free absorption coefficient:
⇣
↵↵⌫ = 3.7 ⇥ 108 Z2 ne ni T 1/2 ⌫ 3 1
⇡ 4.5 ⇥ 10
25
e
h⌫/kT
⌘
ḡ↵ cm-1
3/2
2
(
)
Z ne ni kT keV ⌫MHz ḡ↵ cm-1
2
all n values in
cm-3
Compare Thomson:
↵T = ne T
= 6.65 ⇥ 10
25
ne cm-1
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Synchrotron
(relativistically moving charged particle in a magnetic field)
Single Particle Spectrum:
n
qB
Gyration frequency: !H =
mc
io
at
di
Ze
Ra
lorentz factor
x=
Opt. thin synchrotron
spectrum from powerlaw particle energy
distribution
pitch angle
c
⇤c = ⇥3 ⇤H sin
✓
The radiation spectrum generated by
such a distribution is also a power-law:
◆
!c
B Zme
2
⌫peak = 0.29
⇡ 0.81
MHz
2⇡
1G m
!2
2
4
Z me
2 2
UB
Power =
Tc
3
m
!2
✓
◆2
2
Z me erg / s
B
14 2
= 2.0 ⇥ 10
per particle
m
1G
log intensity
B
Particle acceleration process often
generates non-thermal, power-law
energy distribution of the relativistic
charged particles: N( ) / p
I⌫ /
⌫
(p
1)/
2
log frequency
j⌫ / ⌫
↵⌫ / ⌫
(p 1)/2
(p+4)/2
S⌫ / ⌫5/2
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Compton scattering
high-energy
photon
in
=c
/⌫
in
c
✓
sc
me c
=
s
c /⌫
2
4
Inverse compton power emitted per electron:
3
Non-thermal comptonization
Thermal comptonization
In the frame where the electron is initially
at rest,
h
(1 cos ✓)
sc
in =
me c
and cross section: Klein-Nishina
KN
⇡ T
/ ⌫in1
(h⌫in ⌧ me c2 )
(h⌫in
me c 2 )
In the observer’s frame, where the electron
is moving with a lorentz factor ,
⌫sc ⇡ 2 ⌫in
for h⌫in ⌧ me c2 /
(inverse compton scattering)
Tc
2 2
Uph ; Uph = photon energy density
spectral shape akin to synchrotron process
number conserving photon diffusion in energy space
power-law, modified blackbody or Wien spectrum
(for different limiting cases of opt. depth and y-parameter)
Compton y parameter = (av. no. of scatterings) x (mean fractional energy change per scattering)
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
References
•
•
•
•
Radiative Processes in Astrophysics : G.B. Rybicki and A.P. Lightman
High Energy Astrophysics : M. Longair
The Physics of Astrophysics vol. I: Radiation : F.H. Shu
Theoretical Astrophysics vol. 1 : T. Padmanabhan
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Diffuse Matter
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Diffuse matter between stars: the ISM
Interstellar matter exists in a number of phases, of different temperatures and
densities. Average density of Interstellar medium is ~ 1 atom / cm3
At such low densities heat transfer between different phases is very slow.
So phases at multiple temperatures coexist at pressure equilibrium.
Cooling: free-free/free-bound continuum, atomic and molecular lines, dust radn.
Heating: Cosmic Rays, Supernova Explosions, Stellar Winds, Photoionization
!3/2
2
2gi 2⇡me kT
x
/kT
Ionization fraction x :
=
e
(Saha equation)
1 x
g0 n
h2
Hydrogen is fully ionized at T > 104 K
Ionization states are denoted by roman numeral: I: neutral, II: singly ionized.......
Gas around hot stars are photoionized by the stellar UV photons.
Ionization balance: Recombination rate = Rate of supply of ionizing photons
!1/3
3ṄUV
4⇡ 3
Strömgren sphere:
(singly ionized: HII region)
R=
R ne ni ↵ = ṄUV
2
3
4⇡ne ↵
recombination coef
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
ISM phases and constituents
H217: Aug-Dec 2003
4
•
•
•
•
•
at gas and Coronal
stars in the galactic
disk
rotate
about
6 K the galactic centre, with
Gas:
T
~
10
e angular speed Ω a function of the galactocentric radius R. It turns out
Warm
Ionized
T ~at8000
Kkm/s,
at the circular
velocity
(vc = RΩ) Medium:
is nearly constant,
about 220
om R ∼ 2 kpc outwards, much beyond the solar circle (orbit of the Sun,
Warm Neutral Medium: T ~ 6000 K
∼ 8 kpc). The inferred mass included in an orbit of galactocentric radius
2
given by M(R)
= Rv
to increase
well
Cold
Neutral
Medium:
T ~ linearly
80 Kwith
(HIR clouds)
c /G, then continues
yond most of the “visible” matter. This is a property shared by all disk
Molecular
Clouds:
Ta<large
20amount
K of matter that
laxies. These
galaxies presumably
contain
ves out almost no light. This is called the “missing mass” or the “dark
atter” problem, and this non-luminous mass is assumed to lie in a dark
HI gas
produces
1420
MHz
lo. In Distributed
our galaxy the luminous
component
adds up
to about
1011 M(21-cm)
!,
hile the dark halo contributes several times this amount.
Cosmic rays, diffuse starlight,
magnetic field: ~ 1eV/cm3 each
Dust: solid particles - graphite,
silicates, PAHs etc.
hyperfine transition line.
vital probe of density, temperature, kinematics (e.g. rotation curve)
Rotation Curve of the Milky Way
Molecular regions can be studied through
mm-wave rotational transitions, e.g. of CO
300
circular speed (km/s)
250
200
Rvc2 (R)
M (R) =
G
150
100
evidence for dark matter
50
0
0
0.5
1
1.5
R/R0
2
2.5
3
Dust causes extinction and reddening,
re-radiates energy in Infrared, polarizes
starlight, provides catalysis for molecule
formation, shields molecular clouds from
radiation damage, depletes the diffuse gas
of some elements.
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Centres of
main seq
stars
IGM
ionised
90%
Interplanetary
medium
at 1 AU
50%
ISM
DLAs
neutral
HI clouds
Giant
Molecular
Clouds
Air
on
Earth
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Star Formation
Stars form by gravitational collapse and fragmentation of dense molecular clouds
Gravitational Instability occurs at masses larger than the Jeans’ scale MJ ⇠ ⇢0 L3J
"
#1/2
"
#3/2
GMJ
kT0
kT0
1
LJ =
where
MJ =
µmp = kT0
Gµmp ⇢0
Gµmp
LJ
⇢1/2
0
Collapse can proceed only in presence of cooling. Hence star formation rate is
strongly dependent on cooling. Cooling is provided by atomic and molecular
transitions. More molecules faster cooling.
Dust aids the formation and survival of molecules.
Formation of dust needs heavy elements
In early epochs, star formation was slow; fewer, very massive stars formed
With enrichment, star formation rate (SFR) increased, many small stars produced
Infrared emission from hot dust is tracer of star formation activity
Ultra-Luminous Infra Red Galaxies (ULIRGs): example of high SFR
High SFR → High SN rate → More CR → stronger Sychrotron emission (radio)
Radio - FIR correlation
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Free Electrons
Ionization provides free electrons in the ISM: hne i ⇠ 0.03 cm
3
Propagation through this plasma causes Dispersion of e.m. radiation,
measurable at radio wavelengths. Can be used to infer distances of, e.g. pulsars.
Plasma frequency of the ISM !p ⇠ 6 kHz
Magnetic Field permeates the Interstellar Medium.
Polarized radio waves undergo Faraday Rotation while propagating through
the magnetized plasma. key probe of distributed magnetic field
Typical interstellar field strength: ⇠ 1 µG
Cyclotron resonance: !c ⇠ 20 Hz
Dispersion relation of circularly polarized eigenmodes:
2
3
2 ✓
◆
!p
! 666
!c 777
!
!p , !c
for
kr,l = 641
1
⌥
7
5
c
!
2!2
Dispersion
Z L Measure:
ne ds
DM =
0
Rotation
Z Measure:
L
RM =
ne Bk ds
0
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Intergalactic Medium
ISM of line-of-sight galaxies, gas clouds and the diffuse intergalactic medium
can show up in absorption against the radiation of distant galaxies and QSOs
Lyman Alpha provides a strong absorption at 1216 Å in the rest frame of the
absorbing
gas. Due to cosmological redshift, 24
absorption by different gas clouds
PH217:
Aug-Dec 2003
in the line of sight occur at different wavelengths Lyman Alpha Forest
Spectrum of QSO0913+072 (z=2.785)
1200
1000
Clouds with large Hydrogen
content (e.g. galaxies) produce
deep absorption with damping
wings: Damped Lyman Alpha
systems (DLAs)
QSO spectrum showing
Lyman Alpha Forest
Intensity (arbitrary units)
800
600
DLA
400
200
0
-200
3700
Bechtold 1994
3800
3900
4000
4100
4200
4300
4400
4500
4600
4700
Wavelength (Angstrom)
Figure 5: Lyman alpha absorption in the spectrum of the Quasar Q0913+072 (Bechtold
Diffuse Intercloud Medium is
almost fully ionized. If not, then
radiation shortward of emitted
Ly-α would have been completely
absorbed: Gunn-Peterson effect
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
References
• Physical Processes in the Interstellar Medium : L. Spitzer Jr.
• The Physical Universe: F.H. Shu
• An Invitation to Astrophysics : T. Padmanabhan
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Galaxies
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Hubble Classification
Galaxies are the basic
building blocks of the
universe
Various shapes and
sizes:
- flattened spirals with
high net ang. mom.
- ellipticals with lower
net ang. mom.
- early galaxies mainly
irregular
Baryon content
~106 M⦿ (dwarfs) to
~1012 M⦿ (giant ellipt.)
skyserver.sdss.org
Dark Matter
~10-100 x Baryons
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Properties of Galaxies
- Disk galaxies and irregulars are gas-rich, Ellipticals gas poor
- Star formation more prevalent in spirals/irregulars, more old stars in Ellipticals
- More Ellipticals found in galaxy clusters
- Ellipticals grow by merger: giants (cDs) found at centres of rich clusters
- Every galaxy appears to contain a central supermassive black hole
- Correlations:
R : size
1.4±0.15 0.9±0.1
Ellipticals: Fundamental Plane: R /
I
: velocity dispersion
I : surface brightness
↵
Spirals: Tully-Fisher relation: L / W
L : luminosity
↵ ⇠ 3 4 depending on wavelength
W : rotation velocity
Central Black Hole Mass: MBH /
5
,
= velocity dispersion of elliptical galaxy or of
central bulge in a spiral galaxy
- If central BH is fed by copious accretion, Active Galactic Nucleus (AGN)
results: High nuclear luminosity, relativistic jets, non-thermal emission
> Broad Line Region, Narrow Line Region, Disk, Torus
> Diversity of appearance depending on viewing angle & jet strength:
Seyferts, Radio Galaxies, QSOs, Blazars, LINERs......
- Emission is variable
- Reverberation mapping of BLR allows measurement of BH mass:
light echo → R ; spectrum → v ; MBH = Rv2/G
Urry & Padovani
heasarc.gsfc.nasa.gov
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Superluminal Motion
Proof of relativistic bulk motion in AGNs
A
v (tB
tA
v (tB
(t2
vapp
t1 ) = (tB
tA )
sin ✓
=c
1
sin ✓
!
dB )
c
= (tB
t1
tA ) sin
tA ) cos
(dA
t2
observer
tB
~
~
B
~
~
dA

tA ) 1
dB
max. vapp = c
faster than light
for β ≥ 0.71
v cos ✓
c
at cos ✓ =
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Galaxy Populations
Luminosity Function
Smith 2012
Analytical fit: Schechter Function
(L)dL = ⇤ (L/L⇤ )↵ e (L/L⇤ ) (dL/L⇤ ) ,
↵ ⇠ 1.25
L⇤ depends on galaxy type and redshift
1993ApJ...405..538B
Luminosity and spectrum of a galaxy would
change with time as stellar population evolves
Multiple starburst
episodes may be
required to
characterize a
galaxy
Large fraction of galaxies are found in groups and clusters.
Interaction with other galaxies and with the cluster gas can
strip a galaxy of gas and terminate star formation
Bruzual & Charlot 1993
t (Gy)
=
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Galaxy Clusters
- A rich cluster can contain thousands of galaxies bound to a large dark matter halo.
- Diffuse gas in clusters fall into the deep potential well, get heated and emit X-rays.
- Hot gas Compton scatters the cosmic microwave background: Sunyaev-Zeldovich.
- Gravitational lensing can be used to measure cluster mass, revealing dark matter.
- Groups and Clusters grow via collision and mergers.
i
h
2
Density profile of a DM halo: ⇢(r) = ⇢0 / x(1 + x) , x ⌘ r/Rs (NFW: from simulations)
¯ vir ) = 200⇢c (z). If hot gas at cluster core has time to cool then it will
Virial radius: ⇢(r
condense and flow inwards: Cooling Flow. Found to be rare: energization by AGN?
Abell 2218
Bullet Cluster
Markevich et al
Clowe et al
Fruchter et al HST
arcs caused by weak lensing help estimate mass
blue: dark matter; pink: x-ray gas.
colliding clusters
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
References
• Galactic Astronomy : J. Binney & M. Merrifield
• An Invitation to Astrophysics : T. Padmanabhan
• www.astr.ua.edu/keel/galaxies/
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Cosmology
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Hubble Expansion
Over large scales, the Universe is homogeneous and isotropic, and it is expanding.
Scale factor a(t) multiplies every coordinate grid.
Let object A receive radiation from object B. The coordinate (comoving) distance
between them is dc , which remains constant, and the proper distance is d = a(t)dc . Due
to cosmological expansion, the proper distance between these two points increases
at a rate v = ȧ(t)dc . This is an apparent relative velocity which causes a Doppler shift:
!
aem
ȧ(t)dc
⌫
ȧ d
ȧ
a
em
⌫(t)a(t)
=
constant
,
giving
=
=
=
=
t=
aobs
⌫
c
a c
a
a
obs
aobs
a0
aobs
subscript 0 denotes a
1
+
z
=
=
,
or
=
1
Thus redshift z =
quantity at present epoch
aem
a(z)
aem
em
✓ ◆
✓ ◆
ȧ
ȧ
d = Hd ; H ⌘
and v =
= Hubble parameter. Present value: H0 ' 67 km/s/Mpc
a
a
obs
em
a
1
Hubble Time tH = =
~ age of the universe. Present value: tH,0 ⇡ 14.5 Gy
ȧ H(t)
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Dynamics of the Universe
Dynamical equations for cosmology follow from a fully relativistic framework.
However a Newtonian analogy may be drawn to mimic the basic equations:
✓
◆
1
1 2
4⇡ 3 1
k
per unit mass K.E.+P.E. = const. : ȧ
= const. =
G
⇢a
2
2
3
a
✓ ◆2
ȧ
k
8⇡G
k
8⇡G
⌦(t) ⌘ ⇢(t)/⇢c (t)
2
hence
+ 2 =
⇢(t) or H + 2 =
⇢(t)
a
3
3
a
a
⌦(t) = 1 + k/a2= 1 for k = 0
3H2
29
In our Universe k ⇡ 0, i.e. ⇢ = ⇢c ⌘
. At present ⇢c,0 ⇡ 0.84 ⇥ 10 g/cm3
8⇡G
(flat geometry)
Solution of the dynamical equation requires knowledge of ⇢(a) : Equation of State
Adiabatic evolution: P / V
/a
3
while P = (
1)u = (
Using this the dynamical equation yields the solution a / t
For non-relativistic matter P / ukin ⌧ ⇢c2 , so
For relativistic matter or radiation
For “Dark Energy” ⇢c2 ⇡ const., i.e.
= 4/3
⇡0
⇡ 1.
2
3
1)⇢c2 . Thus ⇢ / a
( , 0)
⇢/a
⇢/a
⇢ / a0
3
4
In cosmology the EoS is usually labelled by the parameter w ⌘ (
3
flat universe
a/t
a/t
2
3
matter dominated
1
2
radiation dominated
a / et
acceleration, inflation
1)
Present day universe: ⌦tot ⇡ 1, ⌦m ⇡ 0.3, ⌦DE ⇡ 0.7, ⌦b ⇡ 0.048, ⌦rad ⇡ 5 ⇥ 10
5
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Distance measures in Cosmology
Coordinate system: origin at the observer. Comoving (coordinate) distance to a source: rc
Proper distance now: d = a0 rc , Redshift: z
Light leaves source at tc , received at t0
Z 0
Z t0
dt
a
dr
=
c
dt
Propagation from r = rc to r = 0 :
i.e.
dr = c
rc
tc a(t)
Z z
Z 0
Z z
dz
a
1 dt da
c
= c⌧; ⌧ = lookback
Hence rc = c
time
dz and d = a0 rc = c
dz =
Z
a0 0 ȧ
z
0 H(z)
z a da dz
dz
=
0 H
Luminosity
Luminosity distance dL :
✓
◆✓
◆
1
L
L
1
Received Flux F ⌘
=
2
4⇡d2 1 + z 1 + z
4⇡dL
Reduction of
photon energy
Z
∴ dL = d(1 + z) = c(1 + z)
z
0
dz
H
Reduction of
photon arrival rate
Angular Diameter distance dA :
l
;
Angular size ✓ =
rc a(tc )
Transverse
linear size
dA ⌘
l
a0 rc
c
= a(tc )rc =
=
✓
1+z 1+z
Z
z
0
dz
H
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Supernova Ia Hubble Diagram
46
44
Distance Modulus
42
40
Flat Universe Models:
with Dark Energy
without Dark Energy
38
36
Data from Supernova Cosmology Project
Kowalski et al (2008) Ap J: Union Compilation
34
32
0
0.2
0.4
0.6
0.8
Redshift z
1
1.2
1.4
1.6
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Cosmological Expansion History
1e+15
15
z
000
6
⇠
0. 3
20
1000005
⇢m
⇢rad
10
z⇠
log(⇢/⇢c,0 )
1e+10
10
z=0
1e+20
⇢DE
1e-05
-5
log a
-10
1e-10
1e-05
-4
0.0001
Radiation
dominated
era
-3
0.001
-2
0.01
Matter dominated era
-1
0.1
01
10
log(a/a0 )
Acceleration
era
e
t
t2/3
t1/2
log t
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Thermal History of the Universe
Radiation and matter in thermal equilibrium in early universe, at a common temperature
4
⇢
/
T
rad
Radiation is Planckian, with blackbody spectrum and energy density
4
In cosmological evolution ⇢rad / a , thus T / a 1 ; or T = 2.73(1 + z) K
Seen today at microwave bands: Cosmic Microwave Background Radiation
If kT > 2mc2 for any particle species, relativistic pairs of the species can be freely
created. All relativistic particle species behave similar to radiation in the evolution of
2
4
their energy density. Total ⇢rel c = ḡaR T where ḡ = total stat wt of all rel. particle species
In the Early, radiation-dominated universe a / t1/2 . So t = 1 s (kT/1MeV)
[ ḡ ⇠ 100 at kT > 1 GeV, ~10 at 1-100 MeV, ~ 3 at < 0.1 MeV ]
2
ḡ
1/2
Expansion → cooling → pair annihilation of relevant species → energy added to radiation
At kT ⇠< 1 GeV, nucleon-antinucleon annihilation → small no. of baryons survived
n
⌘=
⇠ 109 : “entropy per baryon”, photon-to-baryon ratio. “Baryon asymmetry” ~10-9
nb
kT ⇠< 1 MeV: e± annihilation → ~10-9 of the pop. left as electrons → charge neutrality
Contents at this stage: neutrons, protons, electrons, neutrinos, dark matter and photons
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Primordial Nucleosynthesis
At kT > 0.5 MeV neutrons and protons are in beta equilibrium: nn /np = exp( 1.3 MeV/kT)
kT ≈ 0.5 MeV: beta reactions become inefficient, n-p freeze at nn /(nn + np ) ⇡ 15%
( no n-decay yet as tH << tdecay )
Neutrons will then undergo decay with lifetime of 881 s until locked up in nuclei
Nucleosynthesis begins in earnest only after kT drops to ~0.07 MeV
(decided by the rate of first stage synthesis: that of Deuterium. Most of this 2D then converts to 4He)
Neutron fraction drops to ~11% at this
point, giving primordial mass fraction:
4He
: ~22%; 1H : ~78%
Synthesis does not progress beyond
this stage due to expansion and cooling
The entire nucleosynthesis takes place
within approx. the first three minutes
after Big Bang: z ~ 3x108
http://www.astro.ucla.edu/~wright/BBNS.html
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Recombination, CMB, Reionization
After nucleosynthesis the universe has ionized H and He, and electrons. Material is
very optically thick due to electron scattering. Cooling continues as T / a 1. There
are also neutrinos, and leptonic Dark matter which do not interact with photons.
Once Dark Matter becomes non-relativistic, gravitational instability develops and selfgravitating collapsed halos start to form. Baryonic matter cannot collapse yet because
of strong coupling with radiation.
As temperature drops to ~3x103 K, electrons and ions recombine to form atoms.
Universe becomes transparent to the Cosmic Background radiation. The CMB we see
today comes from this “surface of last scattering” at z~1100. Diffuse gas is now neutral.
Decoupled from radiation, baryons now fall into the potential wells already created by
Dark Matter, Luminous structures begin to form by z~20.
UV radiation from the luminous structures starts ionizing diffuse gas again. These
“Stromgren sphere”s grow and overlap, completely reionizing the diffuse intergalactic
medium by z~10.
Cosmic Background radiation temperature continues to fall as 1/a, reaching 2.73 K at
present epoch.
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
CMB spectrum
120
As measured by COBE satellite
Blackbody T=2.726 K
Intensity (10-6 erg cm-2 s-1 sr-1 cm)
100
Data from Mather et al 1994
80
60
40
Error bars enlarged 100x
20
0
0
5
10
Frequency (cm-1)
15
20
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
CMB sky distribution
COBE Satellite Maps
Bennett et al 1996
Monopole
Dipole
caused by our peculiar velocity
~ 370 km/s w.r.t. the Hubble flow
Higher order anisotropies
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Planck Collaboration: The Planck mission
CMB sky distribution
Ade et al 2013
Fig. 14. The SMICA CMB map (with 3 % of the sky replaced by a constrained Gaussian realization).
CMB high-order anisotropy map from Planck Satellite
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
CMB Anisotropies
Epoch of Decoupling:
It follows from Saha equation that the universe recombines when the radiation
temperature drops to ~3000 K. This corresponds to a redshift zdec ' 1100
h
Defines the last scattering surface (LSS)
i1/2
In a flat universe H(z) = H0 ⌦rad,0 (1 + z)4 + ⌦m,0 (1 + z)3 + ⌦DE,0
!
2
1
⇡ 4 ⇥ 105 y
Age of the universe at decoupling tdec ⇡
3 H(zdec )
H(zdec ) ⇡ 22000 H0
CMB anisotropies developed until tdec : Primary Anisotropy. Later: Secondary Anisotropy
Sources of Primary Anisotropy:
• Intrinsic, from primordial perturbations
• Gravitational Redshift from fluctuating potential at LSS (Sachs-Wolfe effect)
• Doppler shifts due to scattering from moving gas
• Acoustic oscillations of the photon-baryon fluid
Modified by:
• Finite width of LSS
• Photon Diffusion (Silk Damping)
Secondary Anisotropies from: Integrated Sachs-Wolfe effect, Sunyaev-Zeldovich effect,
Gravitational Lensing etc
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Acoustic Horizon Anisotropy Scale
l = cs tdec
sc
att
eri
ctlook back (zdec )
dA =
1 + zdec
ng
su
l=
cs t
de
c
last
rfa
ct0
⇡
1 + zdec
ce
ec =
: zd
Δθ
c
= p tdec
3
1100
✓ = l/dA
✓
1 + zdec tdec
) ✓= p
t0
3
⇡1
◆
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Anisotropy spectrum of the CMB
Planck Collaboration: The Planck mission
Angular scale
90
18
1
0.2
0.1
6000
0.07
Planck 2013 results
Ade et al
5000
D [µK2]
4000
3000
2000
1000
0
2
10
50
500
1000
1500
2000
2500
Multipole moment,
Fig. 19. The temperature angular power spectrum of the primary CMB from Planck, showing a precise measurement of seven acoustic peaks, that
are well fit by a simple six-parameter ⇤CDM theoretical model (the model plotted is the one labelled [Planck+WP+highL] in Planck Collaboration
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
Formation of Structures
Gravitational Instability leads to growth of density perturbations of Dark Matter.
- overdense [⇢ = (1 + )⇢bg ] regions expand less slowly than Hubble flow, eventually
stop expanding, turn around and collapse to form halos.
- “critical overdensity” c ⇡ 1.68
- Halo mass distribution at a redshift z : fraction of bound objects with mass > M :
2
3
66 c (1 + z) 77
77
f (> M, z) = erfc 664 p
(Press-Schechter Formula)
5
2 0 (M)
where 0 (M) is the linearly extrapolated rms at mass scale M at present epoch
Typical density contrast today at scale 8 (100/H0 ) Mpc is 8 ⇡ (0.5 0.8)
Initial density perturbations result from quantum fluctuations.
- However perturbations seen today would require scales much larger than horizon
size in the early universe.
- Made possible by accelerated (exponential) growth of a(t) for a brief period: inflation
- Energy provided by a decaying quantum field, which also generates fluctuations
Introduction to Astronomy and Astrophysics - 1 IUCAA-NCRA Graduate School 2013 Instructor: Dipankar Bhattacharya
References
•
•
•
•
•
•
•
An Introduction to Cosmology : J.V. Narlikar
An Invitation to Astrophysics : T. Padmanabhan
Theoretical Astrophysics vol. 3 : T. Padmanabhan
Cosmological Physics : J.A. Peacock
The First Three Minutes : S. Weinberg
C.L. Bennett et al 1996 ApJ 464, L1
P.A.R. Ade et al arXiv 1303.5602 (2013)
IUCAAAAIntro12013DB