PHYS08050 Introductory Astrophysics Full Mock Exam 2021b
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Introductory Astrophysics Full Mock Exam
PHYS08050 (SCQF Level 8)
2021
Section A: Stars
This section is worth 40% of the total mark. Answer all parts of the question.
A.1 (a) The Heisenberg Uncertainty Principle states that it is impossible to define the position
x and momentum p of a particle to an accuracy which is better than ∆x∆p ≥ /2, where = h/2π and h is Planck’s constant. Describe how this quantum-mechanical effect can give rise to a source of pressure, known as ‘Degeneracy Pressure’, when matter is squeezed to very high densities.
(b) For a gas of particles with number density n, comprised of particles with mass m moving with momentum p, the gas pressure is given by
np2
By considering the mean separation between two neighbouring particles in terms of the number density n, and calculating the minimum momentum for those particles given by the Heisenberg Uncertainty Principle, show that the degeneracy pressure is given by
2n
(c) A white dwarf star is in hydrostatic equilibrium where the gravitational forces are balanced by the degeneracy pressure force. Consider a very small cylinder at radius r from the centre of the white dwarf, with mass ∆m, height ∆r and surface area of the top and bottom of ∆A, as shown in the Figure below. The outwards net pressure force on the cylinder is given by Fpressure = ∆P∆A where ∆P is the difference in pressure between the top and the bottom of the cylinder. The inwards gravitational force depends on M (< r), the mass enclosed within a radius of r .
By assuming that the density of the white dwarf is constant, and that it is in ‘hydrostatic equilibrium’, show that
∆P 3GM2r
∆r 4πR6 .
where M and R are the mass and radius of the white dwarf.
Use calculus to show that the central pressure Pc of the Sun is then given by
Pc = 3GM2
(d) By equating the central pressure derived in question (c) to the degeneracy pressure, derived in question (b), derive a proportionality relationship between the mass of the White Dwarf M, and its radius R.
(e) White dwarfs can often be found in a binary pair with a Red Giant Star accreting matter onto the surface of the White Dwarf. With reference to the proportionality rela- tionship between the mass of the White Dwarf M, and its radius R that you have derived, discuss how this binary system will evolve over time.
Section B: Galaxies
This section is worth 40% of the total mark. Answer all parts of the question.
B.1 (a) Briefly summarise the kinematics of stars in spiral and elliptical galaxies respectively,
and how the kinematics relates to their differing morphologies. What morphological features do we observe in some galaxies that lead us to believe they are actually pairs of
galaxies interacting?
(b) A star of radius R is moving at velocity v through a region where the number density
of other stars is n stars per unit volume. Using the “swept out volume” argument, and assuming the other stars have the same radius, show that the typical time between
collisions is tcoll = |
1 |
4πR2nv |
If the typical distance between stars is 0.5pc, and we take v to be the speed at which stars near the Sun rotate around the Milky Way, i.e. v = 225km s-1 , estimate the typical time between stellar collisions for Sun-like stars. How does this compare to the age of the Milky Way? What do we conclude about the importance of stellar collisions?
(c) The nearest star to the Sun, Alpha Centauri, has roughly the same mass as the Sun, and is at a distance of 1.34pc. The Sun is at a distance of 8kpc from the Galactic Centre, and within this radius there are roughly 200 billion stars. From the Sun’s perspective, what is the ratio of the summed force caused by all the distant stars, to the force caused by the nearest star, Alpha Centauri? What do we conclude about what dominates the stellar dynamics within a galaxy?
(d) A radio telescope is used to measure the observed frequency of the HI emission line at two locations at a distance of R = 10 kpc either side of the centre of a galaxy. After correction for the mean velocity of the galaxy as a whole, one of the two spots shows HI emission at ν = 1421.45 MHz and the other at ν = 1419.37 MHz. What is the estimated rotation velocity at this radius? Calculate the mass, in solar masses, implied within the measured radius of R = 10 kpc
(e) For the calculation in part (d), explain briefly why it is reasonable to ignore the mass outside R. Now suppose a second galaxy approaches the first galaxy. Why is the mass of this second galaxy also not ignorable? Briefly explain the relevance to the structures we see in interacting galaxies.
Section C: Cosmology
This section is worth 20% of the total mark. Answer all parts of the question.
C.1 (a) The energy density of matter ρm and the energy density of radiation ρr in the Universe
are related to the cosmic scale factor R(t) as
ρm x R(t)-3 ,
ρr x R(t)-4 .
The ratio between the energy density of matter ρm and the energy density of radiation ρr today is measured to be
ρr (t = today)
What redshift was the epoch of matter-radiation equality, when the energy density of matter and radiation were equal?
(b) What is meant by the term ‘cosmic microwave background’ (CMB) and why does this radiation exist?
(c) At temperatures of T < 3000K, electrons and nuclei form neutral atoms. Use this information to determine the redshift of the ‘epoch of recombination’ when the Universe becomes transparent. Use T = 2.728K as a measurement of the temperature of the CMB today.
2022-08-13