Zeroth Assignment - 9th January, 2004

1. Observation :
   The star Betelgeuse is a supergiant with distance 150 pc and a radius equal to the radius of the earth's orbit about the sun. How large a telescope is needed to just "resolve" the disk of this star at a wavelength of 550nm? At a wavelength of 5 microns? (Consider diffraction limited observations only, and ignore seeing.)
2. Time-Scales :
   a. Define the thermo-nuclear time-scale of a star. What is the value of $\tau_{nuc}$ in case of Sun ?
   b. Define the thermal time-scale, $\tau_{\rm thermal}$. This is also known as the Kelvin-Helmholtz time-scale. Verify that this is $\sim 10^7$ years for Sun.
3. Generation of Stars :
   Assume that the uranium isotopes $^{235} U$ and $^{238} U$ are produced in supernovae in net ratio of $^{235} U$ to $^{238} U$ of 1.5. If the present ratio of $^{235} U$ to $^{238} U$ is 0.007 and formation occurred in a single event, calculate the elapsed time since the explosion of the supernova that created uranium. The half-lives of $^{235} U$ and $^{238} U$ are $7.1 \times 10^8$ yrs and $4.6 \times 10^9$ yrs, respectively.
4. Radiation :
   How much more energy would be emitted by a star if its temperature were increased from 5,000K to 20,000K? Would the star have to shrink or expand in order to emit the same amount of energy? By how much?
5. Compact Objects in the Universe :
   A. Obtain the expression for the escape velocity, $V_E$, from the surface of a spherical, self-gravitating object of mass $M$ and radius $R$. Assume perfect Newtonian gravity. In the $M-R$ plane, draw the $V_E = C$ line, where $C$ is the velocity of light. Indicate the position of the following objects in the $M-R$ plane - a. Moon, Earth (terrestrial planets), b. Jupiter (giant gaseous planet), c. Sun (low-mass main sequence star), d. Sirius B (a typical white dwarf), e. Neutron Star of mass $1.4 \mbox{{\rm M}$_{\odot}$}$, f. Stellar Mass Black Hole, g. Globular Cluster, h. Dwarf/Giant Elliptical Galaxy, i. Milky Way, Andromeda (Spiral Galaxy), j. Local Group Notice that, the region right of the $V_E = C$ line is where all the observable objects reside.
   B. Assume all the above-mentioned self-gravitating objects to have uniform density. Re-express the escape velocity in terms of density and indicate the escape velocity line and the position of the above objects in the $\rho-R$ plane.
6. Energy Scales :
   A. Compare the magnitude of electrostatic and gravitational forces between a) two electrons and b) two protons. Notice that, due to the inverse square nature of both the force laws, the distance dependence drops out of the ratio. Therefore, the relative importance of the two forces remains the same at all distances.
   B. The normal solids that we encounter in everyday life are bound together by electrostatic forces. The average energy of such cohesion is $\sim 1$ eV per atom for crystalline solids. Calculate the gravitational binding energy and the total electrostatic/cohesive energy of the following objects - $\bullet$ a hydrogen atom, $\bullet$ an asteroid ($R \sim 10$ km, $\rho \sim 5 {\rm g} {\rm cm}^{-3}$), $\bullet$ Earth, $\bullet$ Jupiter. (Assume the material of asteroid/Earth to have the same average cohesive energy as ionic crystals.) C. Estimate the minimum mass of a self-gravitating object (using the average value of the cohesive energy given above). Assume an average density of $\sim 5 {\rm g} {\rm cm}^{-3}$. What is the gravitational binding energy of this object ? Compare this with the binding energies of - $\bullet$ Sun, $\bullet$ A White Dwarf of solar mass, $\bullet$ A Neutron Star of solar mass.
7. Maximum Height of the Mountains :
   a. This could be an interesting measure of the compactness of a self-gravitating object. Assuming the density of terrestrial rock to be $\sim 5 {\rm g} {\rm cm}^{-3}$ find the maximum height of mountain on the surface of earth. Compare this with the height of Mt. Everest.
   b. If similar material is used to build mountain on the surface of a white dwarf what would be the maximum height ? What about a mountain on the surface of a Neutron Star (use Iron at a density of $\sim 10^6 {\rm g} {\rm cm}^{-3}$ instead of rock) ?
8. Gas Pressure :
   a. Obtain an expression for the central pressure of a self-gravitating object in terms of its mass and radius. Use order-of-magnitude arguments.
   b. Prove that in the centre of the Sun, ideal gas law holds. Estimate the central pressure by using ideal gas law and compare with the result obtained by using the expression obtained in the previous problem.
   c. Check the validity of ideal gas laws in the centres of a White Dwarf, a Neutron Star.
9. Neutron Stars :
   a. The half-life of free neutrons is approximately 12 minutes. Why do they remain stable in Neutron stars ? Find a density above which neutrons will be stable.
   b. Find the minimum density above which neutrons will interact, assuming that the maximum range of strong interactions is $\sim \hbar/{m_{\pi}c}$ where $m_{\pi}$ is the mass of the pion.
10. Accretion : Consider a typical White Dwarf of mass 1.4  $\mbox{{\rm M}$_{\odot}$}$ and radius 10,000 km.
    a. Calculate the energy release when a proton/electron falls on the surface of the White Dwarf.
    b. Estimate the rise in temperature of one gram of completely ionized hydrogen falling on the surface of the White Dwarf from infinity, if - i. the energy release is completely taken up by the proton gas, ii. the energy is shared by the electron and the proton gas (calculate the temperature for the components separately).
    c. Estimate the energy release when one gram of hydrogen is completely converted to helium. Compare this with the energy release if one gram of hydrogen falls on the surface of a) a neutron star, b) a white dwarf. From this argue why the Nova outbursts are seen only in white dwarfs not in neutron stars.