Chapter 5: Evolution of main sequence star Stellar Physics PHYS3040 . Jumpei Takata takata@hku.hk. 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 star Stellar Physics PHYS3040

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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

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.

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

(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

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

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

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