Download Stellar Astrophysics: Introduction Q. Daniel Wang Astronomy Department University of Massachusetts

Stellar Astrophysics: Introduction
Q. Daniel Wang
Astronomy Department
University of Massachusetts
Why should we care about stellar astrophysics?
Why should we care about stellar astrophysics?
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stars are a major constituent of the visible universe
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understanding how stars work is probably the earliest
major triumph of astrophysics
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stars are responsible for the chemical composition of the
universe
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mass and energy feedback (supernovae)
stellar astrophysics is the foundation for the astronomy and
astrophysics in general
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cosmic yardsticks to estimate the distances (e.g., Cepheid
variables)
kinematic status of large scale structure (redshifts)
time scales and mechanisms for the formation of stellar
populations and galaxies (stellar ages)
Outline
Review of the course syllabus
Basic equations
Mass conservation
Momentum Conservation and Hydrostatic Equilibrium
Application example: constant-density model
Molecular Weights
Energy Generation
Energy Transportation
The HR diagram
Homology and Scaling Law
Virial theorem
Dimensional Analysis
Summary and Conclusions
Approaches
A lot to cover! Emphasize on the physical principles,
methodologies, and connections to what you might be
interested in doing in your research.
I
lectures are mainly to provide a coherent thread of the
course and to cover essential and/or difficult parts.
I
reading materials will be discussed and quizzed; your
questions can be part of the class discussion
I
homework is to consolidate and expand your
understanding
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exams to review systematically what has been learned and
to check the mastering of the materials
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Research and writing assignments to further expand and
integrate your knowledge.
Scopes of this chapter
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introduce some basic concepts and physical processes.
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estimate various time and physical scales of stars.
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paint a preliminary picture of the stellar interior.
To be expanded later to the modern development of stellar
structure and evolution.
See next chapter for a narrative description of the development.
Assumptions: single isolated, spherical symmetric, and
mechanically steady.
Outline
Review of the course syllabus
Basic equations
Mass conservation
Momentum Conservation and Hydrostatic Equilibrium
Application example: constant-density model
Molecular Weights
Energy Generation
Energy Transportation
The HR diagram
Homology and Scaling Law
Virial theorem
Dimensional Analysis
Summary and Conclusions
Mass conservation
What is a star?
Mass conservation
What is a star?
A steady shining body, self-sustained, getting most energy from
nuclear burning.
Mass conservation
What is a star?
A steady shining body, self-sustained, getting most energy from
nuclear burning.
How is the balance achieved in a star?
Mass conservation
What is a star?
A steady shining body, self-sustained, getting most energy from
nuclear burning.
How is the balance achieved in a star?
Consider a shell with a thickness of dr , the mass included is
dMr = ρ4πr 2 dr .
(1)
The integration of this equation gives the mass Mr within the
radius r .
Since the radius of a star can change greatly during its lifetime,
sometimes quickly, while the change in mass is relatively small
and slow, it is convenient to use the Mr as the coordinate; i.e.,
expressing various equations in the Lagrangian form.
Momentum Conservation and Hydrostatic Equilibrium
Stars are mostly steady, spending most of their lifetime
converting H into He. Mostly forces are balanced between
gravity - only the mass within r matters and acts as if it is at the
center of a star (the gravitational potential is different).
I pressure - most importantly its differences between the inner
and outer boundaries of a shell considered here.
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The equation of motion:
GMr ρ
dP
−
(2)
dr
r2
For a steady star, the balance of forces leads to the hydrostatic
equilibrium:
GMr ρ
dP
=− 2 ,
(3)
dr
r
which in the Lagrangian form is
ρr̈ = −
dP
GMr
=−
.
(4)
dMr
4πr 4
This form is especially useful when the stellar evolution (hence
the structure change) is discussed.
Application example: constant-density model
Working out a simple model of a star with a constant density,
which satisfies the above hydrostatic equilibrium equation and
the boundary condition: P = 0 at the star surface.
Application example: constant-density model
Working out a simple model of a star with a constant density,
which satisfies the above hydrostatic equilibrium equation and
the boundary condition: P = 0 at the star surface. First, we
need to eliminate r . The constant density implies
Mr =
r3
M
R3
(5)
Now, we have
GM 4/3 −1/3
dP
M
=−
dMr
4πR 4 r
Integrating this equation from the center to Mr gives
P − Pc = −
3 GM 4/3 2/3
M
2 4πR 4 r
(6)
(7)
Application example: constant-density model
Working out a simple model of a star with a constant density,
which satisfies the above hydrostatic equilibrium equation and
the boundary condition: P = 0 at the star surface. First, we
need to eliminate r . The constant density implies
Mr =
r3
M
R3
(5)
Now, we have
GM 4/3 −1/3
dP
M
=−
dMr
4πR 4 r
Integrating this equation from the center to Mr gives
3 GM 4/3 2/3
M
2 4πR 4 r
Using the boundary condition: P = 0 at Mr = M, we have
P − Pc = −
Pc =
3 GM 2
2 4πR 4
(6)
(7)
(8)
We then have
Mr 2/3 P = Pc 1 −
(9)
M
This central pressure is a lower limit if density always
decreases outward.
By using the equation of state (EoS) for a monatomic ideal gas,
P = nkT , one can get the temperature structure.
But what is an ideal gas?
We then have
Mr 2/3 P = Pc 1 −
(9)
M
This central pressure is a lower limit if density always
decreases outward.
By using the equation of state (EoS) for a monatomic ideal gas,
P = nkT , one can get the temperature structure.
But what is an ideal gas?
It is composed of a set of randomly-moving, non-interacting
point particles except when they collide elastically.
We also need to know how to calculate n.
Molecular Weights
The thermodynamic relations between P, ρ, and T , as well as
the calculation of stellar opacity requires knowledge of the
mean molecular weight µ (the mean mass of a particle in
Atomic Mass Units).
ρ
= ni + ne
(10)
n≡
µmA
1
1
1
≡
+
µ
µi
µe
If the mass fraction of species i is xi , then we have
1
x
= Σi i
µi
Ai
(11)
(12)
1
x
= Σi i Zi fi
(13)
µe
Ai
where Ai is the atomic weight of the species, Zi be the atomic
number of species i, and fi be the species ionization fraction,
i.e., the fraction of electrons of i that are free.
Note that in the case of total ionization (fi = 1), this equation
simplifies greatly. Since Zi /Ai = 1 for hydrogen, and ∼ 1/2 for
everything else,
µe =
−1
2
1
=
X + (Y + Z )
2
1+X
(14)
1
.
(X + Y /4)
(15)
µi ≈
For a “zero-age main sequence” star (ZAMS), X ≈ 0.7,
Y = 0.3, and Z ≈ 0.03:
µi = 1.3, µe = 1.2 and µ = 0.6.
Using this µ value in the EoS, we can now get the temperature
structure. In particular, the central temperature for the Sun can
be estimated as
Tc =
1 GM µmA
≈ 6 × 106 K
2 R k
(16)
To balance the gravity, a star must have a high pressure, which
is realized with both high temperature and density. How to
maintain such a high temperature?
Using this µ value in the EoS, we can now get the temperature
structure. In particular, the central temperature for the Sun can
be estimated as
Tc =
1 GM µmA
≈ 6 × 106 K
2 R k
(16)
To balance the gravity, a star must have a high pressure, which
is realized with both high temperature and density. How to
maintain such a high temperature?
We need to consider both the energy generation and
transportation.
Energy Generation
Neglecting terms due to the energy input or loss to gravitational
expansion or contraction and due to neutrinos, we can express
the energy equation as
dLr
= .
dMr
(17)
Over some sufficiently restricted range of T , ρ, and
composition, one may approximate in a power law form
= 0 ρλ T µ ,
(18)
where λ = 1, 1, and 2 and µ ≈ 4, 15, and 40 for pp-chain,
CNO-cycles, and triple-α modes, respectively.
For the conversion of hydrogen to Helium, about 0.7% of the
rest mass energy is released, which is 6 × 1018 ergs for energy
per gram of hydrogen consumed.
Energy Generation
Neglecting terms due to the energy input or loss to gravitational
expansion or contraction and due to neutrinos, we can express
the energy equation as
dLr
= .
dMr
(17)
Over some sufficiently restricted range of T , ρ, and
composition, one may approximate in a power law form
= 0 ρλ T µ ,
(18)
where λ = 1, 1, and 2 and µ ≈ 4, 15, and 40 for pp-chain,
CNO-cycles, and triple-α modes, respectively.
For the conversion of hydrogen to Helium, about 0.7% of the
rest mass energy is released, which is 6 × 1018 ergs for energy
per gram of hydrogen consumed. With this energy efficiency,
how long will sun last? (assuming that about 10% of the Sun’s
mass will be fused and 70% of it is hydrogen).
Energy Generation
Neglecting terms due to the energy input or loss to gravitational
expansion or contraction and due to neutrinos, we can express
the energy equation as
dLr
= .
dMr
(17)
Over some sufficiently restricted range of T , ρ, and
composition, one may approximate in a power law form
= 0 ρλ T µ ,
(18)
where λ = 1, 1, and 2 and µ ≈ 4, 15, and 40 for pp-chain,
CNO-cycles, and triple-α modes, respectively.
For the conversion of hydrogen to Helium, about 0.7% of the
rest mass energy is released, which is 6 × 1018 ergs for energy
per gram of hydrogen consumed. With this energy efficiency,
how long will sun last? (assuming that about 10% of the Sun’s
mass will be fused and 70% of it is hydrogen). t ∼ 1010 yrs
Energy Transportation modes
Now, what determines the luminosity of a star?
Energy Transportation modes
Now, what determines the luminosity of a star?
The energy leak makes a star shine.
But this leakage must be slow to maintain a star steady. A
feedback mechanism is needed, like a thermometer.
Energy Transportation modes
Now, what determines the luminosity of a star?
The energy leak makes a star shine.
But this leakage must be slow to maintain a star steady. A
feedback mechanism is needed, like a thermometer.
The energy generation also needs to be balanced by energy
removal, but not too fast. Then we call the material is in
“thermal balance”.
Energy Transportation modes
Now, what determines the luminosity of a star?
The energy leak makes a star shine.
But this leakage must be slow to maintain a star steady. A
feedback mechanism is needed, like a thermometer.
The energy generation also needs to be balanced by energy
removal, but not too fast. Then we call the material is in
“thermal balance”.
Three major modes of energy transportation:
1 radiation (photon) transfer
2 convection of hotter and cooler materials
3 heat conduction (only important under degenerate condition,
i.e., in white dwarfs).
Energy Transportation modes
Now, what determines the luminosity of a star?
The energy leak makes a star shine.
But this leakage must be slow to maintain a star steady. A
feedback mechanism is needed, like a thermometer.
The energy generation also needs to be balanced by energy
removal, but not too fast. Then we call the material is in
“thermal balance”.
Three major modes of energy transportation:
1 radiation (photon) transfer
2 convection of hotter and cooler materials
3 heat conduction (only important under degenerate condition,
i.e., in white dwarfs).
For the time being we consider the radiation transfer or
diffusion.
Radiation transfer
Consider a system of particles diffusing across a boundary in
the z direction. The net number flux of particles diffuse across
the boundary is
F ≈
1
1 dn
v̄ [nz−l − nz+l ] = − v̄ l ,
6
3 dz
(19)
where l ≡ 1/nσ is the particle mean free
path, while σ is the cross section of the
collision. We have the Fick’s law of
diffusion:
dn
F = −D ,
(20)
a
dz
a
v̄
Diffuse of particles
where D =
is the diffusion
across
a boundary in z
3nσ
direction.
coefficient.
Radiation transfer
Similarly, we can compute the energy flux of radiative energy
across a boundary. Treat the photons as particles and recall the
energy density U = aT 4 and lph = 1/κρ, where κ is the
absorption coefficient. Thus, the energy transportation
equation is
clph dU
4caT 3 dT
=−
,
(21)
F =−
3 dz
3κρ dz
or
(4πr 2 )2 4caT 3 dT
,
(22)
Lr = −
3κ
dMr
where a = 4σs /c and σs = 5.7 × 10−5 cgs is the
Stefan-Boltzmann’s constant.
Opacity
The calculation of the opacity is a whole industry. Here we use
a generic opacity form
κ = κ0 ρn T −s cm2 g−1 .
(23)
For the Thompson scattering in an ionized medium, n = 0 and
s = 0, where n = 1 and s = 3.5 for Kramers’ opacity,
characteristic of radiative processes involving atoms.
Summary
In summary, under the steady and spherical assumptions, we
have described the basic equations (mass, momentum, energy,
and heat transfer):
dr
1
=
,
(24)
dMr
4πr 2 ρ
GMr
dP
=−
,
dMr
4πr 4
dLr
= 0 ρλ T µ ,
dMr
(25)
(26)
(4πr 2 )2 4caT 3 dT
(27)
3κ
dMr
These equations, together with an E.O.S., assumed to be
P ∝ ρχρ T χT , and appropriate boundary conditions, allow one to
solve for r , ρ, Lr , and T as functions of Mr .
Lr = −
Summary
In summary, under the steady and spherical assumptions, we
have described the basic equations (mass, momentum, energy,
and heat transfer):
dr
1
=
,
(24)
dMr
4πr 2 ρ
GMr
dP
=−
,
dMr
4πr 4
dLr
= 0 ρλ T µ ,
dMr
(25)
(26)
(4πr 2 )2 4caT 3 dT
(27)
3κ
dMr
These equations, together with an E.O.S., assumed to be
P ∝ ρχρ T χT , and appropriate boundary conditions, allow one to
solve for r , ρ, Lr , and T as functions of Mr .
Based on these equations, we may also get some ideas about
various scales and dependencies among various stellar
properties.
Lr = −
Outline
Review of the course syllabus
Basic equations
Mass conservation
Momentum Conservation and Hydrostatic Equilibrium
Application example: constant-density model
Molecular Weights
Energy Generation
Energy Transportation
The HR diagram
Homology and Scaling Law
Virial theorem
Dimensional Analysis
Summary and Conclusions
The HR diagram
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Please draw a Hertzsprung-Russell (HR) diagram, both
theorist’s and observer’s versions, and explain the axes, as
well as directions of stellar radius and mass).
I
Show an example of the diagram and point out the main
sequence, giants, white dwarf etc.
I
How to get R, M, and age of a star? In the HR diagram?
The HR diagram
a
a
a
Theorist’s version of HR
diagram.
a
HR (or CMD) diagram of ”nearby”
stars measured by Hipparcos. Colors
indicate multiple stars at that position.
An important way to characterize the properties of stars: power
output vs. temperature or equivalent. The exact units of the
axes depend on the context and who present them. For historic
reason, the temperature axis has the highest value on the left.
The HR diagram
As the radiation from a star
photosphere is close to a
black-body, one can define Teff
as
1/4
L
Teff ≡
, (28)
4πσs R 2
a
Density of stars depends on the
lifetime of individual processes
that govern the evolution at
different stages.
a
Theorist’s version of HR diagram.
where R is the radius of the
photosphere.
The HR diagram
As the radiation from a star
photosphere is close to a
black-body, one can define Teff
as
1/4
L
Teff ≡
, (28)
4πσs R 2
a
Density of stars depends on the
lifetime of individual processes
that govern the evolution at
different stages.
a
Theorist’s version of HR diagram.
where R is the radius of the
photosphere.
The L − Teff diagram itself gives
no further information than
L, Teff , and R and says nothing
about mass, composition, or
state of evolution. But these
may be inferred from from the
distribution of stars in the
diagram, plus the modeling.
Outline
Review of the course syllabus
Basic equations
Mass conservation
Momentum Conservation and Hydrostatic Equilibrium
Application example: constant-density model
Molecular Weights
Energy Generation
Energy Transportation
The HR diagram
Homology and Scaling Law
Virial theorem
Dimensional Analysis
Summary and Conclusions
Homology and Scaling Law
A very useful technique, not only for the study of stars, but for
other astrophysical problems.
Without getting all the solutions, such analysis can provide
insights into how physical quantities depend on each other and
scaled.
Homology and Scaling Law
A very useful technique, not only for the study of stars, but for
other astrophysical problems.
Without getting all the solutions, such analysis can provide
insights into how physical quantities depend on each other and
scaled. Consider the four basic equations and remove all the
constants:
dr
1
∝ 2
dMr
r ρ
dP
Mr
∝− 4
dMr
r
dLr
∝ ρλ T µ
dMr
r 4 T 3 dT
Lr ∝ − n −s
,
ρ T dMr
together with the E.O.S.,
P ∝ ρ χρ T χT .
a
a
assuming the radiation transfer
and implicitly a uniform chemical
When all the constants and
exponents are assumed to be
the same, solutions from one
star to another with different
masses are scalable. This
family of stars is said to be in a
homologous sequence.
Homology and Scaling Law
To show this, we do the following
variable transformation in the
above equations:
1
dr
∝ 2
dMr
r ρ
dP
Mr
∝− 4
dMr
r
dLr
∝ ρλ T µ
dMr
r 4 T 3 dT
Lr ∝ − n −s
ρ T dMr
r = r 0 M αr , ρ = ρ 0 M αρ ,
Lr = Lr ,0 M αL , T = T0 M αT ,
Mr = Mr ,0 M,
where M is any constant. We then
obtain, for example,
M αr −1
1
dr0
∝ M −2αr −αρ 2 .
dMr ,0
r0 ρ0
If we have αr − 1 = −2αr − αρ or
3αr + αρ = 1, the above two mass
equations are the exactly the
same.
Homology and Scaling Law
1
dr
∝ 2
dMr
r ρ
Mr
dP
∝− 4
dMr
r
dLr
∝ ρλ T µ
dMr
r 4 T 3 dT
Lr ∝ − n −s
ρ T dMr
All together, we have
3αr + αρ = 1
4αr + χρ αρ + χT αT = 2
λαρ − αL + µαT = −1
4αr − nαρ − αL + (4 + s)αT = 1
This set of linear equations can be
solved (if the determinant is not
zero) to get the α∗ values.
Homology and Scaling Law
For example, consider upper MS stars as a homologous
sequence. For these stars, the opacity is mostly due to electron
scattering (i.e., n = s = 0) and the nuclear reaction is due to
CNO cycles (i.e., λ = 1 and µ = 15. The pressure is still
dominated by the ideal gas law (i.e., χρ = χT = 1).
Homology and Scaling Law
For example, consider upper MS stars as a homologous
sequence. For these stars, the opacity is mostly due to electron
scattering (i.e., n = s = 0) and the nuclear reaction is due to
CNO cycles (i.e., λ = 1 and µ = 15. The pressure is still
dominated by the ideal gas law (i.e., χρ = χT = 1).
The inferred αr = 0.78 and αL = 3.0 may be compared with the
empirical fitted values of 0.75 and 3.5, respectively.
In addition, we get αT = 0.22 and αρ = −1.3 so that T should
increase with the mass whereas ρ decreases. This is indeed
what happens!
Homology and Scaling Law
For example, consider upper MS stars as a homologous
sequence. For these stars, the opacity is mostly due to electron
scattering (i.e., n = s = 0) and the nuclear reaction is due to
CNO cycles (i.e., λ = 1 and µ = 15. The pressure is still
dominated by the ideal gas law (i.e., χρ = χT = 1).
The inferred αr = 0.78 and αL = 3.0 may be compared with the
empirical fitted values of 0.75 and 3.5, respectively.
In addition, we get αT = 0.22 and αρ = −1.3 so that T should
increase with the mass whereas ρ decreases. This is indeed
what happens!
For lower MS stars, the physics are different (different λ, µ, χρ ,
and χT ). The convection is also more important. The homology
does not work as well.
A more general discussion of the scalability of the
hydrodynamic solution or simulation and application examples
can be found in Tang & Wang (2009, MNRAS, 397, 2106).
Outline
Review of the course syllabus
Basic equations
Mass conservation
Momentum Conservation and Hydrostatic Equilibrium
Application example: constant-density model
Molecular Weights
Energy Generation
Energy Transportation
The HR diagram
Homology and Scaling Law
Virial theorem
Dimensional Analysis
Summary and Conclusions
Virial theorem
1 d 2I
= 2K + Ω,
2 dt 2
(29)
where the momentum of inertia I = Σi mi ri2 , the kinetic energy
Gmi mj
1
.
K = Σi mi vi2 , and the potential energy Ω = −Σ
2
ri,j
Virial theorem
1 d 2I
= 2K + Ω,
2 dt 2
(29)
where the momentum of inertia I = Σi mi ri2 , the kinetic energy
Gmi mj
1
.
K = Σi mi vi2 , and the potential energy Ω = −Σ
2
ri,j
For a gaseous system (e.g., a star),
Z
3P
dMr
(30)
K =
M 2ρ
and
Z
Ω=−
M
GMr
dMr .
r
(31)
The system is called to be virialized (in kind of dynamic
equilibrium), if the r.h.s of Eq. 29, averaged over time, is equal
to zero (i.e., statistically does not change with time).
dI i
1 h dI
−
/(t2 − t1 ) = 2K + Ω = 0,
2 dt t1 dt t2
(32)
where t2 − t1 is sufficiently large. Note that the above
expression of the theorem applies to the entire system (e.g.,
the entire star). For a portion of a star, one needs to account for
external force (e.g., the pressure at the boundary).
Consider a so-called “γ−law” equation of state (EoS):
P = (γ − 1)ρE,
(33)
where E is the mass specific internal energy density. Eq. 30
R
3
then can be written as K = (γ − 1)U, where U = EdMr .
2
Thus K = U only if γ = 5/3; that is, the total kinetic energy is
the same as the total internal energy only under certain
circumstances!
Application of the Virial theorem
From Eq. 32, we have
1
(34)
K =− Ω
2
Think about the formation of a star and a galaxy from a cloud of
gas. When it is virialized, only half of the potential energy loss
is turned into the kinetic energy (thermal or orbit motion).
Where is the rest of the energy?
Application of the Virial theorem
From Eq. 32, we have
1
(34)
K =− Ω
2
Think about the formation of a star and a galaxy from a cloud of
gas. When it is virialized, only half of the potential energy loss
is turned into the kinetic energy (thermal or orbit motion).
Where is the rest of the energy?
Radiated away!
Application of the Virial theorem
From Eq. 32, we have
1
(34)
K =− Ω
2
Think about the formation of a star and a galaxy from a cloud of
gas. When it is virialized, only half of the potential energy loss
is turned into the kinetic energy (thermal or orbit motion).
Where is the rest of the energy?
Radiated away!
Can you think any example for the application?
Outline
Review of the course syllabus
Basic equations
Mass conservation
Momentum Conservation and Hydrostatic Equilibrium
Application example: constant-density model
Molecular Weights
Energy Generation
Energy Transportation
The HR diagram
Homology and Scaling Law
Virial theorem
Dimensional Analysis
Summary and Conclusions
Dimensional Analysis
Contraction and nuclear and their time scales.
What is the dynamic scale? We can use simple dimensional
analysis and central temperature estimate.
Outline
Review of the course syllabus
Basic equations
Mass conservation
Momentum Conservation and Hydrostatic Equilibrium
Application example: constant-density model
Molecular Weights
Energy Generation
Energy Transportation
The HR diagram
Homology and Scaling Law
Virial theorem
Dimensional Analysis
Summary and Conclusions
Summary and Conclusion
In summary, under the steady and spherical assumptions, we
have described the basic equations (mass, momentum, energy,
and heat transfer). These equations, together with an equation
of the state as well as the energy generation and opacity forms,
allow for the construction of a complete stellar model (e.g.,
solving for dependent variables: r , ρ, T , and L as function of
Mr ).
Summary and Conclusion
In summary, under the steady and spherical assumptions, we
have described the basic equations (mass, momentum, energy,
and heat transfer). These equations, together with an equation
of the state as well as the energy generation and opacity forms,
allow for the construction of a complete stellar model (e.g.,
solving for dependent variables: r , ρ, T , and L as function of
Mr ).
We may also get some ideas about various scales and
dependencies among various stellar properties, based on some
scaling or dimension analyses.
Summary and Conclusion
In summary, under the steady and spherical assumptions, we
have described the basic equations (mass, momentum, energy,
and heat transfer). These equations, together with an equation
of the state as well as the energy generation and opacity forms,
allow for the construction of a complete stellar model (e.g.,
solving for dependent variables: r , ρ, T , and L as function of
Mr ).
We may also get some ideas about various scales and
dependencies among various stellar properties, based on some
scaling or dimension analyses.
While the above outlines the general approach to construct
stellar models, we will have in-depth look of the physics of the
EoS, the heat transfer, and energy generation, before showing
how the above equations can actually be solved.
Review Questions
1. How is the hydrostatic balance achieved in a star?
2. What is the molecular weight (which you should be able to derive, depending on
the chemical composition of the gas)?
3. Now what is the lifetime of the Sun with the nuclear power?
4. What is the virial theorem?
5. Can you have a quick estimate of the central temperature of the Sun, based on a
simple dimensional analysis? How may the temperature depend on the average
density and mass of a star (assuming ideal gas)?
6. Can you estimate the lifetime of the Sun if it were powered by the gravitational
energy alone?
7. What is the dynamic (or free-fall) time scale of the Sun?
8. Can you derive an expression of the gravitational potential for a star (assuming
spherical symmetry)?
9. What is the H-R diagram?
Point out the directions of increasing temperature, luminosity, mass, and radius.
Mark the locations of various types of stars you know.
Trace evolutionary tracks of stars.
10. How do the central density and temperature of a main-sequence star depend on
its mass, qualitatively?