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Chapter 5: Evolution of main sequence star Stellar Physics PHYS3040 . Jumpei Takata email@example.com. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A. Introduction. Main sequence stars ; -Main components in Hertzsprung –Russell diagram (HR-diagram).
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Chapter 5: Evolution of main sequence starStellar Physics PHYS3040 JumpeiTakata firstname.lastname@example.org TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA
Introduction • Main sequence stars; -Main components in Hertzsprung–Russell diagram (HR-diagram). -Hydrogen burning in the core -Sun - -
5.1 Overview 1, Molecular cloud -molecular hydrogen 2, Collapsing under the self-gravity 3, Increasing temperature at the core, and slow down the contraction Formation of the protostar. Horsehead nebular
4, If the temperature at the core reaches at the nuclear reaction (hydrogen burning) is turned on • Stop of the contraction • Main-sequence star
5.2 Pre-main sequence collapse • Molecular cloud • -Star formation region Optical W5 region Infrared Carina Nebula
How does the cloud collapse to from a star? • Molecular cloud; basic parameters • -Molecular hydrogen • - • -size • -10-100K
Jeans Mass • Equation Motion Gravity (inward) Pressure (outward) • If Gravity > Pressure, the cloud is collapsed. • If Pressure>Gravity, the cloud expand. There is a critical mass, above which the cloud is collapsed under the self gravity.
Virial Theorem • Hydrostatical equilibrium is established when Gravitational potential +2 ×Kinetic energy=0 • Gravitational potential • Kinetic energy ; Boltzmann constant ;mean molecular weight Kinetic energy per particle
Virial theorem • The condition of the collapse The molecular cloud larger than Jeans mass can collapse to form a star.
Initially, large scale structure is collapsed. • But, as the collapsing continue, the density inside the cloud increases. with ↑ Local structure is formed fragmentation
As the collapsing proceeds, temperature inside the cloud increases, which tends to increase Jean’s mass. • The fragmentation will cease when the temperature is sufficiently high. What is the smallest mass of the collapsing cloud produced by the fragmentation?
Consider the constant temperature during the collapse and the fragmentation. • The release rate of the gravitational potential is • The rate of radiation can not exceed the blackbody radiation; The fragmentation of the collapse of a giant molecular cloud prodces the collapsing cloud with a masse of the order of the solar mass or above
5.3 The track of pre-main sequence star in H-R diagram • Let us consider the track of pre-main sequence star with the solar mass in H-R diagram (). • Initial stage • Typical temperature • Initial radius can be estimate from
Initial stage of the contraction, the cloud is not opaque. • Typical opacity • Mean density • The released gravitational potential energy is efficiently radiation away without heating up the cloud T ～ constant • Luminosity decreases as
(2) Formation of the protostar • As the collapsing proceeds, the central region becomes opaque (). The released energy is efficiently converted into the energy of the particles. • Increasing the temperature at the core • At of the core, the radiation energy can be used to dissociation of the hydrogen molecules, ionization of the hydrogen and helium atoms. • The collapsing is now slowing down. Formation of protostar
Initial radius of the proto star; Gravitational energy ≈ Energy required to dissociate the hydrogen molecules and ionize the hydrogen and helium atoms. X: Mass fraction of hydrogen Y=1-X: Mass fraction of helium
Now, the core of the temperature is very high, but the surface of the cloud is still Formation of protostar
(3)Formation of the convection zone • The core temperature is K • The surface temperature is K • If the gradient of the temperature becomes bigger than the case of the adiabatic case, the convection occurs (Chapter 4). • The released energy is too large to be carried by the radiation. • The macroscopic motion of material carries the energy (convection) More efficient energy transportation. • The convection increases the temperature of the surface of the cloud to K • Increaser of the luminosity by factor
Formation of convection zone L/
5.4 Hayashi track and contraction onto the Main sequence star • If inside of protostar becomes almost fully convective, the increase of the temperature at the protostar surface will stop. • As the contraction proceeds, the luminosity decreases as • This later stage of the protostar is called Hayashi state.
Let us quantitative derive the relation between L and • We consider the fully convective and collapsing protostar. • is defined by the boundary between the convection zone photo sphere. The effective temperature is defined by the value at R, • Assuming the hydrostatic equilibrium, pressure at R is estimated;
2. The equation of ideal gas (1) (2) 3. The convection tends to form the temperature gradient close to that of the adiabatic case. (3) is the adiabatic index
(1) (2) (3) • In Hayashi stage, as the contraction proceeds, the luminosity decreases as • For , the luminosity will decrease to
Contraction onto main sequence • In the late state of the Hayashi stage, the radioactive zone, where the energy is carried by the radiation, is formed at the center. • Both hydrostatic equilibrium and thermal equilibrium are nearly satisfied inside the star. hydrostatic equilibrium thermal equilibrium
(1) (2) (3)
Eventually, the temperature at the core becomes sufficiently high so that the hydrogen burning initiate. • Contraction of the star stop. • Zero age main-sequence star.
Summary • Molecular cloud with a mass larger than can collapse under the self-gravity. • As collapsing proceeds, the Jeans mass becomes smaller, and small structure is formed (fragmentation).
Fully convective star Initial state for Hayashi phase L/ Zero age main sequence star Protostar
Time scale of evolution of the protostar to main sequence star • It is difficult to detect the protostar. • However they are found with a variable star with a circumstellar disk and a feature of the jet (T Tauri star)
Equations for the main sequence stars (Chapter 2) Energy release rate • Dimensional analysis provides the predicted relation between the luminosity and mass or effective temperature.
Dimensionless mass is a constant with the dimension of the radius, which will be determined in later Dimensionless radius
Hydrostatic equilibrium ;dimensionless pressure Physical dimension Dimensionless equation for hydrostatic equilibrium
Equation of mass • Equation of state • Radiative transfer • Thermal equilibrium
Solution does not Depend on any physical properties. Coefficients depends only mass
The profiles of the pressure, density, etc. from the center to the surface of the star does not depend on the mass. Coefficients depends only mass Homology
Radius Increase with the mass • Density Decrease with increases of the mass • Luminosity Observation • Life time
Luminosity vs. effective temperature (Relatively low mass) (Relatively high mass)
Scaling law can be applied to estimate the minimum mass of the main sequence star. • Sun: • p-p chain =4) reaction for hydrogen burning;
5.6 Evolution of the main sequence stars • After staring the hydrogen burning at the core, the contraction is sopped. • The evolution of the main-sequence star is characterized by evolution of the temperature and the density at the core due to the nuclear processes. • The equation of the state at the core depends on the temperature and the density.
Ideal gas • Equation of states • Non-relativistic degeneracy • Relativistic degeneracy • Radiation pressure
Ideal gas Pi= No-relativistic degeneracy Pn,d • Ideal gas Pi= Relativistic degeneracy Pr,d • No-relativistic degeneracy = Relativistic degeneracy • Ideal gas Pi = Radiation pressure
How does the central density evolve with the temperature? • For the Polytropic star, • For Ideal gas, the core density varies as a cube of the temperature is the factor which is insensitive to the polytropic index n
or • For degenerate star, the core density is independent of the core temeprature