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Course, academic year 2019/2020
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Astrophysics II - NAST014
Title in English: Astrofyzika II
Guaranteed by: Astronomical Institute of Charles University (32-AUUK)
Faculty: Faculty of Mathematics and Physics
Actual: from 2012 to 2019
Semester: summer
E-Credits: 6
Hours per week, examination: summer s.:4/0 Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech
Teaching methods: full-time
Additional information:
Guarantor: prof. RNDr. Petr Harmanec, DrSc.
doc. Mgr. Miroslav Brož, Ph.D.
Classification: Physics > Astronomy and Astrophysics
Annotation -
Last update: prof. RNDr. David Vokrouhlický, DrSc. (10.01.2019)
Fundamentals of thermodynamics in stellar interiors: Mean molecular weight, Avogadro law, equations of state of the stellar matter. Basic equation of stellar structure and their mathematical form. Boundary conditions, current methods of solution of the equations of stellar structure and evolution. Evolution of single stars. Models and evolution of rotating stars. Stellar wind and mass loss from stars. Structure and evolution of binaries. Tests of stellar evolution theory: Clusters, apsidal motion of binaries. Simple (polytropic) stellar models and their practical application.
Course completion requirements -
Last update: prof. RNDr. David Vokrouhlický, DrSc. (11.06.2019)

Oral examination.

Literature - Czech
Last update: doc. Mgr. Miroslav Brož, Ph.D. (09.04.2019)

Harmanec P., Brož M.: Stavba a vývoj hvězd, Matfyzpress, Praha, 2011

Carrol B.W., Ostlie D.A: An Introduction to Modern Astrophysics, Pearson, Addison Wesley, San Francisco, 2007

Hansen C.J., Kawaler S.D., Trimble, V.: Stellar Interiors, Springer, New York 2004

Kippenhahn R., Weigert A.: Stellar structure and evolution, Springer, Heidelberg 1994

Schatzman E.L., Praderie F.: The Stars, Springer, Heidelberg 1993

Stix, M.: The Sun. An Introduction, Astronomy and Astrophysics Library, Springer-Verlag, Berlin, 2002

Rose W.K.: Advanced Stellar Astrophysics, Cambridge University Press, Cambridge, 1998

Bohm-Vitense E.: Introduction to Stellar Astrophysics, Vol. 1-3, CUP 1992

Harwitt M.: Astrophysical Concepts, J. Willey and Sons, New York 1973

Mihalas D.: Stellar Atmospheres, W.H. Freeman and Co., San Francisco 1980

Tayler R.J.: The Stars: their structure and evolution, CUP 1994

Vanýsek V.: Základy astronomie a astrofyziky, Academia, Praha 1980

Teaching methods - Czech
Last update: T_AUUK (31.03.2008)


Requirements to the exam - Czech
Last update: doc. Mgr. Miroslav Brož, Ph.D. (06.10.2017)

Zkouška je ústní, sestávající ze 3 obsáhlejších otázek.

Požadavky odpovídají syllabu, resp. základní učebnici Harmanec a Brož (2011), v tom rozsahu, který byl prezentován na přednášce. Známka se stanovuje dle správnosti nebo chybnosti odpovědí, včetně doplňujících otázek.

Syllabus -
Last update: prof. RNDr. David Vokrouhlický, DrSc. (10.01.2019)
1 Introduction

1.1 Origin of the theory

1.2 Model of our Sun

Lithium problem

Neutrino problem

3 State equation

3.1 Mean molecular weight

Atomic mass

Amount of substance, gram-atom, gram-molecule

Molar weight, molecular weight

Mean molecular weight

3.2 Ideal gas

3.3 Radiation pressure

3.4 Electron degeneracy

Full degeneracy

3.5 Partial ionisation in subsurface layers

Iterative solution

More complicated state equations

Compact objects

4 Basic equations of stellar structure

4.1 Equation of mass conservation

4.2 Equation of motion and equation of hydrostatic equilibrium

4.3 Equation of thermal equilibrium

4.3.1 Proton-proton chain

4.3.2 CNO cycle

4.3.3 Transformation of helium to carbon and other reactions

4.3.4 Thermal equilibrium and changes of entropy

4.4 Equation of energy transfer

4.4.1 Equation for radiation energy transfer

Equation of radiation transfer in spherical symmetry

Integral quantities

1st integral of the transfer equation

2nd integral of the transfer equation

Development of almost-isotropic intensity

Kirchhoff law

Rosseland mean opacity

An estimate of mean free path and flux

A note on diffusion formalism

4.4.2 Equation for convective energy transfer

A condition for convection

Derivation of adiabatic gradient of temperature

A common formalism of radiative and convective equilibrium

Subsurface layers

5 Mathematic structure of equation of stellar interior

5.1 Stationary models

5.2 Evolutionary model

5.3 Dynamic model

6 Initial and boundary conditions

6.1 Initial conditions

6.2 Boundary conditions in the centre

6.3 Boundary conditions at the surface

6.3.1 Photosphere

6.3.2 Subphotospheric layers

7 Henyey method for integration of interior parts of a star

7.1 Method of complete linearisation


Boundary conditions in the centre

Outer boundary conditions



Time step

7.2 Limits of discretization

8 Evolution of a solitary star

8.1 Illustrative example: evolution of a star with 4 M_Sun

8.2 Differences of stellar evolution dependent on stellar mass

The role of initial content of helium and more massive elements

9 Comparison of theoretical predictions of stellar evolution and observations

9.1 How to acquire observational data?

Luminosities of stars

Effective temperature of stars

Masses and radii of stars

V versus (B-V) diagram for clusters

9.2 Explanation of major features of Hertzsprung-Russell diagram

9.3 Stellar evolution in star clusters

9.4 Stellar evolution in double stars

9.5 Changes of chemical composition observed in spectra

9.6 Test of internal structure with help of apsidal motion

9.6.1 Apsidal motion in classical mechanics

9.6.2 Relativistic apsidal motion

9.6.3 Total apsidal motion

9.7 Stellar evolution in course of human history

10 Simple analytical models and estimates

10.1 Polytropic process

A concrete example of state equation of stellar matter

A mode general derivation from the 1st law of thermodynamics

10.2 Lane-Emden differential equation

10.3 Polytropic models of stars




Mass contained in a sphere

Comparison of polytropic models with the standard solar model

Chandrasekhar limit

11 Stellar wind and mass loss from stars

11.1 Observational facts

Observational confirmation of wind around cool stars

Confirmations for hot stars

Escape velocity

11.2 Parker theory for cold stars

Instability of isothermal atmosphere

Hydrodynamic equations

11.3 CAK theory of stellar wind driven by radiation

Acceleration caused by radiation

Influence of metallicity on wind

Temporal modulation of stellar wind

11.4 Influence of stellar wind on evolution of stars

Parametric description of wind

Influence of wind

12 Influence of rotation

12.1 Roche model and simple estimates

Estimates of radii of stars

Minimum rotation period

Maximum rotation period

12.2 Models of stellar evolution with rotation

Vectorial form of stellar equations

Various models of rotating stars

12.3 Selected results for evolution of rotating stars

Evolution of rotational velocity

Influence on evolutionary paths in the HR diagram

Influence on surface chemical composition

Comparison with observations

Influence of metallicity on rotational instability

13 Evolution of double stars

13.1 Roche model and simple estimates

Physical classification of double stars

13.2 Calculation of stellar evolution in the phase of mass exchange

Distance of components

Non-conservative mass transfer

Model of stellar interior

13.3 Selected results of double stars modelling

An example of a double star 4 M_Sun and 3.2 M_Sun

13.4 Models of double stars evolution versus observations

Evolutionary paradox

Be stars

Eccentric orbits

Magnetic polars

14 Pulsations of stars

14.1 Radial pulsations of spherical stars

14.1.1 Condition for onset of pulsations

14.1.2 Opacity mechanism of pulsations

14.1.3 A crude estimate of period of radial pulsations

14.1.4 Relations period - luminosity - colour

14.2 Kinematics of non-radial pulsations

14.2.1 Sectoral pulsations of rotating stars

14.3 Hydrodynamics for simple waves

Basic equations of hydrodynamics

Equilibrium state


14.3.1 Acoustic waves in homogeneous medium (p-modes)

14.3.2 Internal gravitation waves (g-modes)

14.3.3 Surface gravitation waves (f-modes)

Exact spherical solutions

15 Gravitational collapse of protostars

15.1 Cooling processes

15.2 Evolution before main sequence, Hayashi line

Initial collapse of protostars (towards the Hayashi line)

Collapse towards main sequence (from the Hayashi line)

Position of Hayashi line

15.3 Minimum Jeans mass

15.4 Eddington limit

16 Explosive phase of stellar evolution

16.1 Core-collapse supernovae

Energetic balance

Observations of neutrinos from SN 1987 A

16.1.1 Mechanism of neutrino bomb

16.1.2 Gamma-ray bursts (GRB)

16.1.3 Nucleosynthesis by r-process

16.1.4 Afterglow and supernova remnants

16.2 Supernovae originating in an explosion of a white dwarf

16.2.1 Laminar velocity of deflagration

16.2.2 Chapman-Jouguet velocity of detonation

16.2.3 Rayleigh-Taylor instability

17 Types of observed stars and their evolutionary stages *

17.1 Hot stars of spectral type O and Wolf-Rayet stars

O stars

Wolf-Rayet stars

O subdwarfs

17.2 Stars of spectral type B

17.2.1 Chemically peculiar Bp stars

17.2.2 Pulsating beta Cep stars

17.2.3 Slowly-pulsating B stars (SPB)

17.2.4 Be stars

17.2.5 Luminous blue variables (LBV)

17.3 Stars of spectral types A to F

17.3.1 Chemically peculiar Am stars

17.3.2 Magnetic Ap stars

17.3.3 Pulsating delta Scuti stars

17.3.4 SX Phe stars

17.3.5 gamma Dor stars

17.3.6 Lithium anf beryllium in F and G stars

17.4 Cold G, K and M stars

17.4.1 Chromospherically active stars: UV Cet, BY Dra, etc.

Stars of UV Cet type

Stars of BY Dra type

Spotted stars of RS CVn type

Close binaries of W UMa type

Stars of FK Com type

17.4.2 Pulsating stars: Cepheids, Miras, R CrB and AGB stars


Stars of W Vir type

Stars of RR Lyr type


Stars of R CrB type

Asymptotic Giant Branch stars (AGB)

Stars of RV Tau type

17.5 Stars in early evolutionary stages

17.5.1 T Tauri stars

17.5.2 FU Ori stars

17.6 Stars in late evolutionary stages

17.6.1 White dwarfs and ZZ Cet stars

White dwarfs

ZZ Cet stars

17.6.2 Novae, cataclysmic variables and polars

Recurrent novae

Dwarf novae

Polars of AM Her type

Intermediate polars DQ Her

Stars of AM CVn type

17.6.3 Supernovae

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