恒星结构与演化

Star Formation

1. The detailed process of star formation:

   • giant molecular clouds
   • dense core: fragmentation and collapse, rotating disk (accretion disk)
   • protostar
   • ignition: the Main Sequence

2. Brown dwarfs: if the initial mass of the protostar is too low, the process fails.

The Observed Properties of Stars

1. Distance:

   • parallax
   • Cepheid variables (Period-luminosity relation)
   • Hubble's law

2. Mass:

   • astrometric binaries
   • visual binaries
   • spectroscopic binaries
   • eclipsing binaries （食双星）

3. Radii:

   • angular diameter
   • interferometry and eclipsing binaries

4. Magnitude:

   • colour-magnitude:

     $$\text{B-V}=-2.5\log \left(\frac{\text{flux in B}}{\text{flux in V}}\right)$$

   • absolute magnitude, apparent magnitude:

     $$m-M=5\log d-5$$

   • bolometric magnitude （热星等）:

     $$M_{\text{bol}}-M_{\text{bol}\odot }=-2.5\log \frac{L}{L_{\odot }}$$

5. Stellar spectral types, effective temperature: OBAFGKM

6. The Hertzsprung-Russell diagram

7. Mass-luminosity relation

8. Age: star clusters

star clusters

• stars all at same distance
• dynamically bound
• same age
• same initial chemical composition

Open cluster & Globular cluster

MS turn-off points: Globular cluster all old

open cluster has the Hertzsprung gap

9. Metallicity: spectrum

The Equations of Stellar Structure

4 equations: momentum transport (hydrostatic equilibrium), mass conservation, energy conservation, energy transport.

• Stars: gravity v.s. internal thermal pressure

• Assumptions: isolated, static, spherically symmetric, and inital homogeneous

  conditions

  • sphere (rotating, magnetic field)

• 3 supplements:

  Equation of state

  Opacity

  Core nuclear energy generation rate

1. Equation of hydrostatic equilibrium:

   $$\frac{\mathrm{d}P(r)}{\mathrm{d}r}=-\frac{GM(r)\rho (r)}{r^2}$$

   derivation

   consider a mass of element

   $$\delta m=\rho (r)\delta s\delta r$$

   outward force: pressure exerted by stellar material on the lower face

   $$P(r)\delta s$$

   inward force: pressure on the upper face, and gravitational attraction of all stellar material lying within $r$:

   $$P(r+\delta r)\delta s+\frac{GM(r)}{r^2}\delta m=P(r+\delta r)\delta s+\frac{GM(r)}{r^2}\rho (r)\delta s\delta r$$

   In hydrostatic equilibrium

   $$\frac{\mathrm{d}P(r)}{\mathrm{d}r}=-\frac{GM(r)\rho (r)}{r^2}$$

2. Equation of mass conservation:

   $$\frac{\mathrm{d}M(r)}{\mathrm{d}r}=4\pi r^2\rho (r)$$

applications

1. Minimun values for central pressure of a star:

   $$\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}P}{\mathrm{d}r}}=-{\displaystyle \frac{GM\rho }{r^2}}\\ {\displaystyle \frac{\mathrm{d}M}{\mathrm{d}r}}=4\pi r^2\rho \end{array}$$$$\Rightarrow \frac{\mathrm{d}P}{\mathrm{d}M}=-\frac{GM}{4\pi r^4}$$$$P_c-P_s={\int }_0^{M_s}\frac{GM}{4\pi r^4}\mathrm{d}M>{\int }_0^{M_s}\frac{GM}{4\pi r_s^4}\mathrm{d}M=\frac{G{M}_s^2}{8\pi r_s^4}$$

2. The Virial theorem:

   $$\Rightarrow 4\pi r^3\mathrm{d}P=-\frac{GM}{r}\mathrm{d}M$$$$\Rightarrow 3{\int }_{P_c}^{P_s}V\mathrm{d}P=-{\int }_0^{M_s}\frac{GM}{r}\mathrm{d}M$$$$\Rightarrow 3[PV]{|}_c^s-3{\int }_{V_c}^{V_s}P\mathrm{d}V=-{\int }_0^{M_s}\frac{GM}{r}\mathrm{d}M$$$$\Rightarrow 3{\int }_0^{V_s}P\mathrm{d}V+\mathrm{\Omega }=0$$

   where $\mathrm{\Omega }$ is the total gravitational potential energy of the star.

3. Minimun mean temperature of a star:

   $$-\mathrm{\Omega }=3{\int }_0^{V_s}P\mathrm{d}V=3{\int }_0^{M_s}\frac{P}{\rho }\mathrm{d}M$$

   for ideal gas $P=nkT=k\rho T/m$,

   $$3{\int }_0^{M_s}\frac{kT}{m}\mathrm{d}M=-\mathrm{\Omega }={\int }_0^{M_s}\frac{GM}{r}\mathrm{d}M>{\int }_0^{M_s}\frac{GM}{r_s}\mathrm{d}M=\frac{G{M}_s^2}{2r_s}$$$$\bar{T}=\frac{{\int }_0^{M_s}T\mathrm{d}M}{M_s}>\frac{G{M}_{s}m}{6k{r}_s}$$

4. Physical state of stellar material: plasma (highly ionized gas)

5. the radiation

   $$P_{rad}=\frac{a{T}^4}{3}$$

   radiation v.s. gas pressure

   $$\frac{P_r}{P_g}=\frac{a{T}^4/3}{kT\rho /m}\approx \frac{am{T}^3}{3k\left({\displaystyle \frac{3M_s}{4\pi r_s^3}}\right)}=\frac{4\pi am{r}_s^3{T}^3}{9k{M}_s}\propto M_s^2$$

6. Gravitational instability:

   using the Virial theorem $3{\int }_0^{V_s}P\mathrm{d}V+\mathrm{\Omega }=0$

   $$3{\int }_0^{V_s}nkT\mathrm{d}V=3{\int }_0^{V_s}\frac{{\rho }_0}{\mu m_H}kT\mathrm{d}V=3\frac{{\rho }_0}{\mu m_H}kT\frac{4\pi }{3}{r}_s^3=\frac{3M_{s}kT}{\mu m_H}$$$${\int }_0^{M_s}\frac{GM}{r}\mathrm{d}M={\int }_0^{r_s}\frac{G{\displaystyle \frac{4\pi }{3}}{\rho }_0{r}^3}{r}4\pi r^2{\rho }_0\mathrm{d}r=\frac{3}{5}\frac{G{M}_s^2}{r_s}$$

   Jeans criterions:

   $$M_s>{\left(\frac{5kT}{G\mu m_H}\right)}^{3/2}{\left(\frac{3}{4\pi {\rho }_0}\right)}^{1/2}\equiv M_J$$$$r_J>{\left(\frac{15kT}{4\pi G\mu m_H{\rho }_0}\right)}^{1/2}\equiv r_J$$

• Source of energy generation:

  • Cooling or Contraction
  • Chemical Reactions
  • Nuclear Reactions (Fission / Fusion)

3. Equation of energy production:

   $$\frac{\mathrm{d}L(r)}{\mathrm{d}r}=4\pi r^2\rho (r)\epsilon$$

   derivation

   $$L(r+\delta r)=L(r)+4\pi r^2\rho (r)\delta r\epsilon$$

   where $\epsilon$ is the energy release per unit mass per unit time.

• Method of energy transport:

  • Conduction: by collisions of gas particles

  • Radiation: by photons

  • Convection: by mass motions of the gas

    condition for occurrence of convection (Schwarzschild criterion):

    $$\frac{\mathrm{d}T}{\mathrm{d}r}>\frac{\gamma -1}{\gamma }\frac{T}{P}\frac{\mathrm{d}P}{\mathrm{d}r}={\left(\frac{\mathrm{d}T}{\mathrm{d}r}\right)}_{\text{adia}}$$

    derivation

    The density of the convective element should be lower than the surroundings

    $$\rho -\delta \rho <\rho -\mathrm{\Delta }\rho$$

    Assuming adiabatical

    $$P{V}^{\gamma }=\text{const}\Rightarrow \frac{P}{{\rho }^{\gamma }}=\text{const}$$$$\Rightarrow \delta \rho \approx \frac{\rho }{\gamma P}\delta P$$$$\frac{\rho }{\gamma P}\frac{\mathrm{d}P}{\mathrm{d}r}>\frac{\mathrm{d}\rho }{\mathrm{d}r}$$$$P=\frac{\rho kT}{m}\Rightarrow 1=\frac{P}{\rho }\frac{\mathrm{d}\rho }{\mathrm{d}P}+\frac{P}{T}\frac{\mathrm{d}T}{\mathrm{d}P}$$

• The characteristic timescales

  • The dynamical timescale (Newtonian timescale)$$t_d=\sqrt{\frac{2r^3}{GM}}$$
  • The thermal timescale (Kelvin-Helmholtz timescale)$$t_{th}=\frac{G{M}^2}{Lr}$$
  • The nuclear timescale (Einstein timescale)$$t_{nuc}\sim \frac{\eta M{c}^2}{L}$$

4. Equation of radiative transport:$$\frac{\mathrm{d}T(r)}{\mathrm{d}r}=-\frac{3\rho \kappa }{16\pi ac{r}^2{T}^3}L(r)$$

   derivation

   Only considering the radiation

   $$E=a{T}^4,p_R=\frac{1}{3}a{T}^4$$$$-\mathrm{d}{p}_R=F\kappa \rho \mathrm{d}x/c$$$$F=-\frac{4ac{T}^3}{3\kappa \rho }\frac{\mathrm{d}T}{\mathrm{d}r}$$$$L=4\pi r^{2}F$$

summary

Four equation:

$$\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}P(r)}{\mathrm{d}r}=-\frac{GM(r)\rho (r)}{r^2}}\\ {\displaystyle \frac{\mathrm{d}M(r)}{\mathrm{d}r}=4\pi r^2\rho (r)}\\ {\displaystyle \frac{\mathrm{d}L(r)}{\mathrm{d}r}=4\pi r^2\rho (r)\epsilon }\\ {\displaystyle \frac{\mathrm{d}T(r)}{\mathrm{d}r}=-\frac{3\rho \kappa }{16\pi ac{r}^2{T}^3}L(r)}\end{array}$$

with 7 variablies $M(r),\rho (r),P(r),L(r),\epsilon (r),T(r),\kappa (r)$.

Solvable with 3 supplements:

• Equation of state: $P$
• Equation of Opacity: $\kappa$
• Equation of nuclear reactions: $\epsilon$