5.3-4

Question 1

Evolution of Stars

which two diagrams can be used?

Answer 1

Two ways to describe the evolution:

• HR-diagram changes
• Changes in central parameters (log ρ - log T-diagram)

Question 2

Equation of States of gas (4)

Answer 2

• A: Ideal Gas (common in normal stars), P = K₀ ρ T
• B: Degenerate Gas (high ρ, low T), P = K₁ ρ^5/3
• C: Relativistic Degenerate Gas, P = K₂ ρ^4/3
• D: Radiation Pressure Dominated Gas, P = a T⁴ / 3

Question 3

Sun in log ρ-log T-diagram

Answer 3

• In the ideal gas region
• ρ and T decrease from center to surface (convection)

Question 4

Nuclear Burning Zones

properties of gas

Answer 4

• Mainly in ideal gas region, extending into degenerate gas region
• Higher ρ and T for more advanced burning states

Question 5

Evolution of Stars in log ρ-log T-diagram

Answer 5

• Towards higher ρ and T (Virial theorem)
• Pause on contraction at burning zones
• Evolution mass dependent

Question 6

Gas Planets and Brown Dwarfs

Answer 6

Degenerate before reaching H-burning

Question 7

Low Mass Stars

(less than solar mass)

Answer 7

• Reach H-burning
• Takes longer than Hubble time to evolve further

Question 8

Burning / Evolution of Solar Mass Stars

Answer 8

• H-burning
• He-burning
• Then evolve into degenerate white dwarf

Question 9

High Mass Stars

Answer 9

• All burning stages
• Explosion during Si-burning (Fe-core > 1.4 M_⊙)

Question 10

Neutron Stars and Pulsars (5)

also state densities

Answer 10

• Free neutrons decay spontaneously (n → p + e^- + 𝛎_bar, half-life 13 min)
• Inverse neutron decay requires energy (ΔE=1.3 MeV), occurs at critical density ρ_c ≈ 10¹⁰ kg/m³
• Collapse of degenerate iron core leads to neutron star formation
• Density for collapse: ρ_c ≈ 10¹⁴ kg/m³ for Fe-nuclei
• Neutron stars form a degenerate gas

Question 11

Pulsars

Answer 11

• Emit radio pulses with stable periodicity (~1 s)
• First pulsar (Crab) found in nebulosity
• Associated with past supernovae (potentially)
• Pointed emission from magnetic poles causes light-house effect

Question 12

Rotation Picture for Pulsars

which conditions have to be met and what is the density?

Answer 12

• Gravitational force > centrifugal force
• Ω < (G ρ)^1/2
• For P = 1 s, ρ > 6 · 10¹¹ kg/m³
• Higher density than white dwarfs (600x)

Question 13

Energy of Pulsars

and their lifetime

Answer 13

• Rotation energy E_rot = 1/5 m r² Ω²
• Yields E_rot = 2 · 10³⁹ J for typical pulsar parameters
• Pulsars slow down due to radiation and nebula powering
• Typical life-time 10⁷ yr

Question 14

Millisecond Pulsars (5)

Answer 14

• Old
• Low magnetic field
• Fast rotation (P ≈ 10 ms)
• Often in binary systems, spun-up by mass transfer
• Extremely accurate clocks (P_dot ≈ 10^-17)

Question 15

Binary Pulsar and Gravitational Waves

Answer 15

• Rotational period 0.059 s
• Orbital period 8 hours
• Measured orbital period change matches prediction for gravitational wave emission

Question 16

Statistics of Pulsars

not sure if statistics is the best way of putting it

Answer 16

• Born fast rotating in supernovae
• Slow down and magnetic field decays with age
• In binaries, can undergo spin-up and become millisecond pulsars
• Visible as X-ray sources during mass transfer