Download Similarities Between Electric and Gravitational Forces • Coulomb’s force: q F

Similarities Between Electric and Gravitational Forces
• Coulomb’s force:
F12 =
•
q1 q2
r2
Felectric
ee/r 2
e2
42
=
=
4.17
×
10
=
Fgravity
Gme me /r 2
Gm2e
1
(1)
(2)
Gravitational Instability
• The equations modeling the hydrodynamic
effects governing the systems we consider:
Hydrodynamic equation, Continuity
equation, and Poisson equation
• A homogeneous gas, without rotation and
magnetic field, an analysis due to J. H. Jeans
(1902, 1928)
• Hydrodynamic equation, or conservation of
momentum equation:
∂v
ρ
+ ρ(v·∇)v = −∇P − ρ∇Φ (3)
∂t
where ρ is the density, v is the macroscopic
velocity of the “gas,” P is the pressure, and
Φ is the gravitational potential
2
• Between ρ and v there exists also the
continuity equation
∂ρ
+ ∇·(ρv) = 0
∂t
(4)
• The Poisson equation: connects Φ and ρ
∆Φ = −4πGρ
(5)
with the boundary condition Φ → 0 for
|r| → ∞
• In our modeling three properties are of
immediate interest: mass (density), velocity,
and temperature (energy). Here velocity is a
three dimensional vector and hence we need
at least five independent equations to
determine the properties of the system
3
• Additional equation of state, or energy
conservation equation:
P =
2
cs ρ
(6)
where cs = const is the sound speed. That is,
we consider an isotermic “gas”
• So, we have five independent equations
(3)–(5) for unknowns ρ, v, and Φ (and
additional Eq. (6))
4
• An equilibrium state: ρ = ρ0 = constant,
P = P0 = constant, v = v0 = constant, and
Φ = Φ0 = constant
• Consider small perturbations ρ = ρ0 + ρ1 ,
P = P0 + P1 , v = v0 + v1 , and
Φ = Φ0 + Φ1
with |ρ1 | ≪ ρ0 , . . . , |Φ1 | ≪ |Φ0 |
• Linearised equations (3)–(6) are (under the
additional assumption v0 = 0)
1
∂v1
= − P1 − ∇Φ1
∂t
ρ0
∂ρ1
= −ρ0 ∇·v1
∂t
P1 = c2s ρ1
∆Φ1 = −4πGρ1
5
(7)
(8)
(9)
(10)
• If the functions ρ1 , P1 , v1 , Φ1 representing
the solution of this system of equations
increase with time the medium is called
unstable; otherwise it is called stable
• We form the divergence of (8), with P1 from
(10), and the time derivative of (9)
∂v1
2 ρ1
= −∆ cs + Φ1
(11)
∇·
∂t
ρ0
∂ 2 ρ1
∂v1
= −ρ0 ∇·
(12)
2
∂t
∂t
• Eliminating of ρ0 ∇·(∂v1 /∂t) from these
equations, and using (10), leads to the basic
wave equation
∂ 2 ρ1
2
=
4πGρ
ρ
+
c
0 1
s ∆ρ1
2
∂t
6
(13)
• Of the well known solutions of (13) we
consider the plane wave propagating in the
x-direction
ρ1 = A exp[i(kx−ωt)]+A exp[−i(kx−ωt)]
(14)
where A = const is the amplitude, k = 2π/λ
= const is the wavenumber, λ is the
wavelength, and ω is the wavefrequency
• Substituting (14) in (13), since
ρ̈1 = −ω 2 ρ1 and ∆ρ1 = −k 2 ρ1 , we obtain
the dispersion equation
ω 2 = k 2 c2s − 4πGρ0
(15)
• For ρ1 to increase (exponentially) with time
at fixed x it is necessary to have a real
coefficient of t in (14), i.e. ω 2 < 0
7
Gravitational (Jeans) Instability
• According to (15), instability can only occur
for k 2 < 4πGρ0 /c2s or for wavelengths
s
4c2s
λ > λjeans =
(16)
Gρ0
where λjeans is called the Jeans length
• Examples of Jeans-unstable systems:
Schematic view
(a)
(a)
(b)
(b)
(c)
8
Trigonometric Parallax
9
Trigonometric Parallax
10
Trigonometric Parallax
• Parsec (pc)
1′′
360◦ ×60′ ×60′′
1.5×1013
= 2πdpc
18
→ 1 pc ≡ dpc = 3.1 × 10 cm
• Light year (ly) = c × 1 yr
= 3 × 1010 × 3.16 × 107 = 9.5 × 1017 cm
• 1 pc = 3.1 × 10−1 ly
11
Trigonometric parallax – Example
• Parallax of α Cen is 0′′ .751
0′′ .751
360◦ ×60′ ×60′′
• Distance to α Cen →
→ dα = 4.1 × 1018 cm =
4.1×1018
1′′
3.1×1018 = 0′′ .751 = 1.3 pc
• Distance to α Cen →
4.1×1018
9.5×1017
12
=
1.5×1013
2πdα
= 4.3 ly
Trigonometric Parallax
13
The Hertzsprung–Russell Diagram
14
The Hertzsprung–Russell Diagram
• The pattern of lines depend on the
temperatures (and pressures)
• The spectral type of a star yields an estimate
of its temperature:
spectral type = function of Te
15
H-R Diagram – Spectra of Stars
• The pattern of line depends of temperature
(and pressures)
16
H-R Diagram – Spectra of Stars
17
The Hertzsprung–Russell Diagram
• The spectral type of a star yields an estimate
of its temperature:
spectral type = function of Te
18
The Hertzsprung–Russell Diagram
• O B A F G K M (RNS)
• I: supergiants II: bright giants III: giants
IV: subgiants V: dwarfs
19
The Hertzsprung–Russell Diagram
20
The Hertzsprung–Russell Diagram
• Stellar luminosity classes
21
The Hertzsprung–Russell Diagram
• Distances calculated durectly
(“trigonometric parallax”)
• Giants, main-sequence and white dwarfs
22
The Hertzsprung–Russell Diagram
• All nearby stars
23
The Hertzsprung–Russell Diagram
• Bright stars
24
The Hertzsprung–Russell Diagram
• Evolution tracs of protostars
25
The Hertzsprung–Russell Diagram
• T Tauri stars’ evolution tracs
26
The Hertzsprung–Russell Diagram
• Main sequence stars
27
The End States of Stars
• Ordinary stars sustain the thermal pressure
• Heat flows from the star to the universe
• Without nuclear-energy sources: the star
will contract and get hotter
• But violent or not, death is as inevitable
• Astronomers belive four are possible:
• 1. Nothing may be left: explosion
• 2. White dwarf: mass ∼ 0.7M⊙ , radius
∼ 109 cm
• 3. Neutron star: mass ∼ 1.4M⊙ , radius
∼ 106 cm
• 4. Black hole: mass > 2M⊙ , radius > 1016
cm (Earth–Sun separation ≈ 1013 cm)
28