Introduction to Stellar Astrophysics

THE UNIVERSITY
of LIVERPOOL

JANUARY 2001 EXAMINATIONS

Degree of Bachelor of Science : Year 3
Degree of Master of Physics : Year 3

FURTHER STELLAR ASTROPHYSICS

TIME ALLOWED : Three Hours

$\begin{displaymath} \frac{dP}{dr} = -\frac{Gm\rho}{r^2}\;\;\;,\;\;\; \frac{dT}{dr} = -\frac{3\kappa \rho L}{16\pi ac r^2 T^3}, \end{displaymath}$

$\begin{displaymath} \frac{dm}{dr} = 4 \pi r^2 \rho\;\;\;,\;\;\; \frac{dL}{dr} = 4 \pi r^2 \rho \epsilon \end{displaymath}$

THE UNIVERSITY
of LIVERPOOL
Section A

1. (a) The top of the earth's atmosphere receives an energy flux of S W/ ( $\rm m^{2}$ ) from a star. Derive the relationship between S, the angular radius of the star ( $\alpha <<$ 1 rad), and its effective temperature ( $\rm T_{eff}$ ). [5]

(b) What is the energy source of Horizontal Branch stars in the H-R diagram of globular clusters? Describe how its luminosity changes with changes in the initial stellar metallicity and Helium content. [5]

(c) After the explosion of a type II supernova, what is left of the original star (remnant) is usually a very compact object. What is the minimum mass of the remnant which will give birth to a Black Hole? How does the Schwarzschild radius scale with the Black Hole mass? Is it allowed by general relativity to observe from our planet a particle crossing the Schwarzschild radius around a very distant Black Hole? Explain your answer. [8]

(d) Derive the value of the mean molecular weight for

i) an ideal He gas completely ionized;

ii) an ideal gas made of 50% He and 50% C both completely ionized.

For what mixture the molecular weight is the highest? [8]

(e) Explain why, during the phases of core nuclear burning, the star central temperature raises. Assume the stellar gas mixture being a perfect gas. [5]

(f) Free neutrons in laboratory conditions decay after $\sim$ 15 minutes. Why are they stable in the core of a neutron star? [5]

(g) Two Red Giant Branch stars with metallicities Z₁ and Z₂ (Z₁ > Z₂) have the same mass (M) and the same luminosity (L). Assuming a mass loss law along the Red Giant Branch of the form

dM/dt=const (LR)/M

(where R is the stellar radius), determine which star is losing more mass [4]

THE UNIVERSITY
of LIVERPOOL
Section B

(i) Discuss if it is possible to find White Dwarfs in a stellar cluster 10⁷ years old. Assume that the pre-WD evolutionary time is t $\sim 10^{10}$ /( $\rm M^{2}$ ) years, where M is the value of the progenitor mass expressed in solar masses. [4]

(ii) What is the source of the energy radiated by White Dwarfs? Describe the phase transitions experienced by the Carbon-Oxygen core during the White Dwarfs evolution, and how they affect the cooling timescale. [9]

(iii) A photon emitted by a White Dwarf loses part of its energy due to the work necessary to overcome the gravitational field of the star. Derive a relationship between the observed frequency $\nu$ of a photon emitted with frequency $\nu_{0}$ , the White Dwarf mass (M) and its radius (R). [12]

(iv) The maximum possible value for the mass of a white dwarf (M_WD^max) can be derived from the condition than the sum of the thermal plus gravitational energy (U+ $\Omega$ ) is equal to zero. Demonstrate that

$\begin{displaymath}M_{\rm WD}^{\rm max} \sim \left(\frac {\rm 5hc}{2G}\right)^{3... ...frac{3}{16 \pi}\right) \left(\frac{\rm 1}{\mu \rm H}\right)^{2}\end{displaymath}$

assuming

$\begin{displaymath}U\sim \left(\frac{3}{4}\right) N_{e}hc \left(\frac{3n_{e}}{8 \pi}\right)^{1/3}\end{displaymath}$

with $\mu$ being the electron mean molecular weight, n_e the number of electrons per unit volume, N_e the total number of electrons in the star and H the proton mass. Assume that the density $\rho$ is constant.

[5]

THE UNIVERSITY
of LIVERPOOL
Section B

(i) When does the Ledoux criterion for convection apply, and when does the Schwarzschild one? Can a non electron degenerate region unstable according to the Ledoux criterion be stable according to the Schwarzschild criterion? Explain your answer [7]

(ii) What is the evolutionary phase affected by the first dredge-up? What is the physical cause of the first dredge-up? What is its effect on the stellar surface abundances? How can it be observed in the luminosity function of globular clusters, and why? [11]

(iii) An isochrone of a given metallicity can be formally described by a curvilinear coordinate s=s(t,M), s being a function of the isochrone age (t) and evolving mass (M). Derive the relationship between the number of stars populating the isochrone between a given s and s+ds, the initial mass function $\Psi$ and the local evolutionary speed $\rm dt\over{\rm ds}$ . What happens if

$\begin{displaymath}\frac {\rm dt}{\rm ds} -> 0 ?\end{displaymath}$

[7]

(iv) Describe two different possible methods to derive the distance to Galactic globular clusters, and their respective ingredients. [5]

THE UNIVERSITY
of LIVERPOOL
Section B

(i) For the outermost layers of a star, recast the equation of hydrostatic equilibrium in terms of the independent variable $\tau$ (the optical depth), where $\rm d\tau=-k \rho \rm dr$ , $\rm k$ being the opacity of the stellar matter. Hence, derive the expression of the radiative gradient $\rm (dlnP/\rm dlnT)_{rad}$ as a function of the effective temperature of the star using the relation

$\begin{displaymath}\rm T= \rm T_{\rm eff} (\frac{1}{2}+\frac{3}{4}\tau)^{1/4}\end{displaymath}$

[6]

(ii) Using the results obtained at point (i) demonstrate that, given an opacity law of this kind

k=A $\rm P^{b}$ (A and b being constants)

the value of the radiative gradient in the external layers of a star depends only on $\tau$ and b. Assume that at the surface $\tau$ =0 and P=0. [7]

iii) The light we receive from a given star is emitted by its photosphere, where $\tau$ =2/3. What is the relationship between b and the adiabatic gradient $\rm (dlnT/\rm dlnP)_{\rm ad}$ in a radiative photosphere? [2]

(iv) The light we receive from stars passes through interstellar clouds in the Galactic disk. What kind of processes affect the starlight during the crossing of an interstellar cloud? What are the parameters affecting the efficiency of these processes? [4]

(v) Sketch the typical (U-B) vs (B-V) colour-colour diagram of Main Sequence stars, and use it to demonstrate that stellar radiation does not have exactly a black-body spectrum. Describe the procedure to derive the colour excess E(B-V) (or E(U-B)) of a Main Sequence star using colour-colour diagrams. [11]

THE UNIVERSITY
of LIVERPOOL
Section B

i) Sketch the typical Colour-Magnitude diagram of globular clusters (except for the White Dwarf sequence) and describe the physical status of the stars populating the main branches of the diagram. [10]

(ii) It is possible to have two globular clusters with the same metallicity but different colour extensions of their Sub Giant Branch (SGB). Explain why and the factors affecting the SGB extension. [5]

(iii) What are RR Lyrae stars? Where are they found in the HR diagram ? Assuming that the velocity of the surface of a RR Lyrae star is constant at v= 15 Km/s, derive the value of the star's radius at two times t₁ and t₂ (where $\rm t_{2} - \rm t_{1} = 10^{6}$ s) along its pulsation cycle, when the star has the same effective temperature. Assume $\rm m_{V}(t_{2})\sim \rm m_{bol}(t_{2})$ , $\rm m_{V}(t_{1})\sim \rm m_{bol}(t_{1})$ and $\rm m_{V}(t_{2})-\rm m_{V}(t_{1})=-$ 1. [10]

iv) The pulsation period of a RR Lyrae star is $\Pi_{p}=\frac{\rm 2R}{c_{s}}$ , where R is the total radius in the hydrostatic equilibrium configuration and c_s an average value of the sound speed in the stellar interior. Derive the relationship between $\Pi_{p}$ and the density (assumed constant) $\rho$ in the assumption that c_s is equal to the value of the sound speed at R/2. Use the boundary condition P=0 at the surface. [5]

PAPER CODE PHYS383page 6 of 6 End