ASTR 400/700:
Stellar Astrophysics
Stephen Kane
Models of Stars
The parameters used for studying and modeling stellar interiors
include:
r
= radial distance from the center of the star
M(r)
= mass interior to r
T(r)
= temperature at r
P(r)
= pressure at r
L(r)
= luminosity at r
ε(r)
= energy generation at r
κ(r)
= opacity at r
ρ(r) = density at r
In modern models mass M is used as the dependent variable
rather than radial distance r, but it is more informative to initiate
the study of stellar interiors using the geometrical variable r.
Stellar Models:
The complete set of differential equations describing the interiors
of stars is therefore:
Equation of Continuity:
Hydrostatic Equilibrium:
Energy Generation:
Temperature Gradient:
dM ( r )
= 4π r 2 ρ
dr
dP − G M ( r ) ρ
=
dr
r2
dL
= 4π r 2 ρ ε
dr
− 3 κ ρ Lr
 dT 

 =
3
2
dr
4
ac
T
4
π
r

 rad
− 1 GM ( r )
 dT 

 =
2
dr
C
r

 ad
P
Solar Interior
Chapter 11.1
The Sun
General Properties
• Spectral type G2V
• Age: ~ 4.52 Gyr
• Diameter = 696,342 km (109 x Earth)
• Mass = 1.989 x 1030 kg (333,000 x Earth)
• Absolute visual magnitude MV = 4.83
• Absolute bolometric magnitude Mbol = 4.76
• Initial abundances:
X = 0.73, Y = 0.25, Z = 0.02
• Central temperature = 15 million 0K
• Effective (surface) temperature = 5770 0K
The Moon’s orbit around the
Earth would easily fit within
the Sun!
The Standard Solar Model
Sun does not rotate as a rigid sphere. The equator of the
Sun rotates faster than the poles of the Sun. This is called
the differential rotation. Sunspots and many other solar
activities are due to this differential rotation.
The Solar Interior - “The Standard
Model”
o Core
o Energy generated by
nuclear fusion (the
proton-proton chain).
o Radiative Zone
o Energy transport
radiation.
by
o Convective Zone
o Energy transport
convection.
by
The Solar Interior
The Solar Core
o R: 0.0 - 0.25 Rsun
o T(r): 15 - 8 MK
ο
ρ(r): 150 - 10 g cm-3
o Temperatures and densities
sufficiently high to drive
hydrogen burning (H->He).
o Ultimate source of energy in
the Sun and Sun-like stars.
IN
4 protons
OUT
4
He nucleus
2 gamma rays
2 positrons
2 neutrinos
Total mass is
0.7% lower.
The Solar Interior
The Solar Interior
Energy Production
The Solar Interior
The Radiative Zone
o
R: 0.25 - 0.8 Rsun
o
T(r): 8 - 0.5 MK
ο
ρ(r): 10 - 0.01 g cm-3
o
Hydrogen burning cuts off abruptly
at r ~ 0.25 Rsun.
o
Interior becomes optically thin or
transparent as density decreases.
o
Energy transported radiatively.
o
Photons cannot be absorbed in the
radiative zone as the temperature are
too high to allow atoms to form.
Therefore no mechanism for the
absorption of photons.
The Radiative Zone
o For T = 15MK Wien’s displacement law
implies λmax = 0.19 nm i.e., the center of the
Sun is full of X-rays.
o Photons do 3D random walk out of Sun.
o Assume photon moves l between interactions
(mean free path) and takes a total number of
steps N.
o On average it will have moved a distance d = l N
o As tdiffusion = N l / c and R = l N => t diffusion = R 2 / lc
=> tdiffusion >104 yrs!