3.1

Question 1

Physical Setup/ Assumptions for stellar structure equations

Answer 1

• spherically symmetric
• self-gravitating cloud
• ideal gas
• mass M
• radius R.

Question 2

Approach to finding basic equations of Stellar Structure

Answer 2

• Ignoring nuclear reactions
• Treat stars as spherical gas clouds
• Consider a spherical shell at radius r with thickness dr.

Question 3

Mass continuity

stellar structure

Answer 3

dM_r / dr = 4 π r² ρ

First stellar structure equation

Question 4

Hydrostatic Equilibrium

Formula

Answer 4

Balance of gravity and pressure in each gas layer.

dP / dr = - G M_r ρ / r²

Second stellar structure equation

Question 5

Estimates for the central pressure and temperature of sun

Answer 5

• linear interpolation: dP/dr ≈ - P_c/R_⊙
• appropriate averages: M_r ≈ M_⊙/2 and r ≈ R_⊙/2
• plug in values for M_⊙ & R_⊙:

P_c ≈ 6 · 10¹⁴ Nm^-2

• ideal gas law and assuming gas only hydrogen:
• P = 2k_B/m_H ρ T

T_c ≈ 10⁷ K

Question 6

Virial Theorem for Stars

Answer 6

• thermal energy E_T & gravitational energy E_G

2E_T + E_G = 0

Total energy:

E = E_G + E_T = 1/2 E_G = -1/2 |E_G|

• which is always negative!

E_T = -1/2 E_G = 1/2 |E_G|

Question 7

Why is the total energy of the system always negative?

(in stars)

Answer 7

• star is formed by slow gravitational contraction of material which was initially spread over much larger volume
• as star contracts: becomes hotter and must radiate away some energy
• energy of star becomes negative

Question 8

Radiative Energy Transport

inside stars

(luminosity)

Answer 8

Energy flux L_r through a layer at r.

dL_r / dr = 4 π r²𝛒 ε

Radiation driven by temperature gradient.

• ε is rate of energy generation per unit mass per unit time

Third stellar structure equation

Question 9

Energy transfer via radiation

outwards of star

Answer 9

dT/dr = -3/(4a_Bc) · χρ/T³ · L_r /(4πr²)

∝ χρ/T³ · L_r /r²

if heat flux is carried outward by radiative transfer

Fourth stellar structure equation

Question 10

Changes in gas blobs due to convection

formula

Answer 10

• convection involves motions of gas
• blob of gas displaced adiabatically:
• ρ density, P pressure initially
• ρ’ density, P’ pressureof new surrounding
• ρ^* density of blob at new surrounding (P^* = P’)

ρ^* = ρ (P’ / P)^1/γ

γ is adiabatic index

Question 11

Stability of system in relation to convection

Answer 11

• ρ’ density of new surrounding
• ρ^* density of blob at new surrounding
• ρ^* < ρ’: displaced blob buyant and will continue to move further away
• system becomes unstable giving rise to convection
• ρ^* > ρ’: displaced blob will try to return to original position
• system is stable and there is no convection

Question 12

Convective energy transport inside stars

formula

Answer 12

dT/dr = (1 - 1/γ) · T/P · dP/dr

• Schwarzschild stability condition

|dT/dr| < (1 - 1/γ) · T/P · |dP/dr|

• if temperature gradient of atmosphere is steeper than critical value then atmosphere is unstable to convection
• ciritcal value (1 - 1/γ) · T/P · |dP/dr|

Also fourth stellar structure equation

Question 13

Boundary Conditions

Answer 13

• At center: M_r(r = 0) = 0, L_r(r = 0) = 0
• At edge: ρ(r = R) = 0, T(r = R) = 0
• Used for solving structure equations

Question 14

Numerical Challenges

Answer 14

dT/dr = - 3/ (4a_Bc) · χρ/T³ · L_r / (4πr²)
- T^-3 near surface
dP/dr = - (GM_r)/r² · ρ
- r^-2 near center

Question 15

Solution Uniqueness

Answer 15

Stars with specific mass M and radius R have unique structures, barring degenerate conditions.