Lecture 12 - Stellar Structure II:
Energy Transport (2/23/99)
Stellar Structure I --- | ---
ISM
Reading:
Chapter 16-1, 16-2 (ZG4)
Notes:
pages 46 - 51
The
Herbig-Haro Object HH32, as seen by HST WFPC2. This is a young star
that still has its accretion disk and outflow jet! (Courtesy
Chris Burrows, WFPC2 Science Team, NASA)
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Key Question: |
How is Energy Transported in a Star?
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Key Principle: |
Radiation Pressure and Convection
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Key Problem: |
Find the Main-Sequence
Mass-Luminosity Relation
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Investigations:
- Equations of State
- What is an equation of state?
- What is the equation of state of an ideal gas?
- What the relation between dP/dr and dP/drho?
- Why do we parameterize P=P0*rho^gamma?
- Energy Generation and Radiative Transport
- What is the luminosity dL generated in a shell of mass dM?
- What is dL/dr in a spherical star?
- For radiative transport of energy, what is dP_rad/dr in terms of
dT/dr?
- What is dP_rad/dr in terms of L(r)?
- What is the radiative temperature gradient dT/dr in a star?
- How does this depend on the opacity, density, T and L(r) at the shell?
- Convective Energy Transport
- What is the expected scale of convection (ie. the pressure scale
height in the interior) in the Sun?
- How does convection transport energy?
- For an ideal gas P=nkT, what is dP/dr in terms of dT/dr and dn/dr
(for number density n)?
- What is a polytropic equation of state?
- What is the convective temperature gradient dT/dr for a pressure
gradient dP/dr?
- When is a gas stable against convection?
- Where might you expect energy to be carried by convection in a star?
- Main Sequence Scaling Relations
- How does the mean density scale with mass and radius?
- How does the central pressure scale with mass and radius?
- How does the central temperature scale with mass and radius?
- How does the luminosity scale with mass and opacity?
- Why does this relation depend on the opacity but not the energy
generation?
- What is the Kramer's Opacity Law for bound-free and free-free
opacity?
- What is electron scattering cross section and opacity?
More Stellar Structure in Outline:
- Stellar Structure continued...
- The mass-luminosity relation (L/Lsun) = (M/Msun)^3.5 is an
approximation to the solution of the equations of stellar structure
for a star of mass close to that of the Sun. The actual exponent
ranges from 2.5 to over 4 over the main sequence, and of course
does not obey anything like this off the main sequence.
- It turns out that the stellar structure that you find depends only
upon the mass M of the star! If the energy is produced by fusion
as given by E(r)=E(T(r),rho(r)), then the star falls on the main
sequence and the surface temperature T, and luminosity
L depend on the mass M such that they increase as you increase the
mass.
- Because higher mass stars are much more luminous, they burn their
fuel faster, and have shorter lifetimes on the main sequence.
Note that the lifetime t is proportional to M/L (fuel/rate of burning)
and for stars of mass near the Sun's:
t/tsun = (M/Msun)/(L/Lsun) = M/M^3.5 = 1/M^2.5
Thus, a star of twice the mass of the Sun would live only 18% of the
life span of the Sun. A star of 0.5 Msun would live 5.6 times longer.
- The most massive stars live only a few millions of years! (Short
by cosmic standards.)
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Steven T. Myers