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Lawrlwytho'r Atodiad Gwreiddiol Imperial College London BSc/MSci EXAMINATION June 2014 This paper is also taken for the relevant Examination for the Associateship SUN, STARS AND PLANETS For 2nd, 3rd and 4th -Year Physics Students Monday, 2nd June 2014: 10:00 to 12:00 Answer ALL parts of Section A and TWO questions from Section B. Marks shown on this paper are indicative of those the Examiners anticipate assigning. General Instructions Complete the front cover of each of the FOUR answer books provided. If an electronic calculator is used, write its serial number at the top of the front cover of each answer book. USE ONE ANSWER BOOK FOR EACH QUESTION. Enter the number of each question attempted in the box on the front cover of its corre- sponding answer book. Hand in FOUR answer books even if they have not all been used. You are reminded that Examiners attach great importance to legibility, accuracy and clarity of expression. c Imperial College London 2014 2013/PO2.1 1 Go to the next page for questions Fundamental physical constants a radiation density constant 7.6 × 10−16 J m−1 K−4 c speed of light 3.0 × 108 m s−1 G Gravitational constant 6.7 × 10−11 N m2 kg−2 h Planck’s constant 6.6 × 10−34 J s k Boltzmann’s constant 1.4 × 10−23 JK−1 e electron charge 1.6 × 10−19 C −31 me mass of electron 9.1 × 10 kg −27 mH mass of hydrogen atom 1.7 × 10 kg 23 −1 NA Avogadro’s number 6.0 × 10 mol σ Stefan-Boltzmann constant 5.7 × 10−8 W m−2 K−4 −12 −1 0 permittivity of free space 8.9 × 10 F m −7 −1 µ0 permeability of free space 4π × 10 H m R Gas constant 8.3 × 103 JK−1 kg−1 Astrophysical quantities 26 L solar luminosity 3.8 × 10 W 30 M solar mass 2.0 × 10 kg 8 R solar radius 7.0 × 10 m Teff effective temperature of Sun 5780 K AU astronomical unit 1.5 × 1011 m pc parsec 3.1 × 1016 m Equations of Stellar Structure dm = 4π r 2 ρ dr dP Gmρ = − dr r 2 dT 3κ ρ L = − if heat transport is radiative dr 16π a c r 2 T 3 ! dT 1 T dP = 1 − if heat transport is convective dr γ P dr dL = 4π r 2 ρ dr 2013/PO2.1 2 Go to the next page for questions SECTION A 1. (i) Explain the parallax method for measuring distances to stars. Define parsec. The star known as Bellatrix in Orion has a measured parallax of 13.42 milliarc- secs. Calculate the distance to Bellatrix in parsecs and light years. [6 marks] (ii) Estimate the mean density and central pressure of brown dwarf Teide 1, given that its radius is 0.1R and its mass is 0.052 M . Estimate also the central temperature of Teide 1. Comment on the accuracy of your estimates. [7 marks] (iii) Draw and label the axes of an HR diagram, and sketch carefully the evolutionary track of a 1M star. List the main evolution phases of the star’s life and clearly label these on the diagram. [7 marks] [Total 20 marks] 2013/PO2.1 3 Please go to the next page 2. (i) The Roche limit is the distance from a large massive body, such as a planet, within which a smaller gravitationally bound object will be ripped apart by tidal forces. A simple formula for the Roche Limit can be obtained by comparing the gravitational force between two small bodies of mass m a distance r apart along their orbital radius, that are orbiting a larger body of mass M with an orbital radius of d, where r << d. Show that the value of the Roche Limit obtained in this way is: !1=3 M d = 2 × r R m [10 marks] (ii) Using the result in part (i), derive an expression for the Roche limit for a small body of density ρs orbiting a planet of density ρp in terms of the densities and the radius R of the planet. [5 marks] (iii) Give an example in the Solar System where the Roche Limit plays an important role in the formation of a specific class of structures. [5 marks] [Total 20 marks] 2013/PO2.1 4 Please go to the next page SECTION B 3. (i) Star X has peak blackbody radiation observed at a wavelength λmax = 960nm. For the Sun λmax = 500nm and the Sun’s effective temperature is 5800K. Calcu- late the effective temperature of star X. What is the likely spectral class of star X ? Give a likely characteristic feature of the spectrum of star X. [7 marks] (ii) For a stellar atmosphere to be stable to convection Schwarzschild’s criterion 1 dP 1 dρ γP dr > ρ dr must hold. Assuming the plasma of a star behaves as an ideal gas, show that in regions of convective heat transport: ! dT 1 T dP = 1 − dr γ P dr Explain briefly, giving reasons, where convection occurs in the Sun’s interior. [5 marks] (iii) A star cluster has stars of differing masses M and radii R of the same chemical composition. The stars are supported by gas pressure. Giving full working, show by homologous transform that Tc / M=R. 3 2 4 You may take ρc / M=R and Pc / M =R . [5 marks] (iv) Explain what is meant by opacity. Give two examples of sources of opacity, and explain their nature. For the stars in the cluster referred to in part (iii), the opacity is given by κ = −7=2 κoρT . If it is assumed that heat transport in the cluster stars is radiative, by homologous transform, derive a relation between luminosity and mass for the cluster stars. Using this mass-luminosity relation show that the main-sequence lifetime t of the cluster stars varies with mass M as: 1 t / M4.5 [7 marks] (v) What is the significance of the main-sequence turn-off in an HR diagram of a star cluster? The main sequence turn-off in an HR diagram of this star cluster is at log L = −0.1. Explaining your assumptions, estimate the age of this star 10 L cluster. Comment briefly on how and why the HR diagram of a young star cluster would differ to that of a very old star cluster. [6 marks] [Total 30 marks] 2013/PO2.1 5 Please go to the next page 4. (i) Explain what is meant by the thermal timescale τth of the Sun. Derive an ex- pression for τth for the Sun, and calculate its value in years. There was considerable controversy during the 19th century regarding the source of the Sun’s energy. Comment briefly on the possible sources of the Sun’s energy that were considered before the discovery of thermonuclear fu- sion, and why they were thought to be inadequate to explain the luminosity of the Sun. [6 marks] (ii) Write down the overall PPI reaction occuring in the Sun, and, with explanations, calculate the energy released per reaction. −27 You may take the proton mass mH = 1.673 × 10 kg and the He nucleus mass = 3.97mH. Comment on where this reaction occurs in the solar interior, and briefly explain why we observe absorption lines of heavier elements (for example oxygen, iron, gold etc) given the nature of the nuclear fusion reactions in the Sun. [5 marks] (iii) Star Procyon, at a distance of 3.51 pcs has apparent visual magnitude mv = 0.34. Calculate the absolute visual magnitude of Procyon. Also estimate the luminosity of Procyon in solar units L . You may take the absolute visual magnitude of the Sun Mv = 4.83. Assuming Procyon has the same type of nuclear fusion reactions in its core as the Sun, estimate the flux of neutrinos at Earth from Procyon. [7 marks] (iv) In fact Procyon has a fainter companion star, and the two stars Procyon A and B form a binary system. Given the angular separation of Procyon A and B is 4.26 arcsecs, their binary period is 40.82 years and that they have a mass ratio mA =mB = 2.4, calculate the individual masses of the two stars in units of solar mass M . [6 marks] (v) Given the absolute visual magnitude of Procyon B is 13.04, calculate its lumi- nosity. The effective temperature of Procyon B is 7740K, calculate the radius of Pro- cyon B and give your answer in units of solar radius R . What type of star might Procyon B be? what will be its eventual fate? Why might your estimates of luminosity be incorrect? [6 marks] [Total 30 marks] 2013/PO2.1 6 Please go to the next page 5. (i) The transit method for finding exoplanets relies on detecting the dip in a star’s brightness that occurs when a planet passes through our line of sight to the star. For a star of measured, uneclipsed, flux Fs, and radius Rs, show that the fractional change in received flux from this star due to the transit of a planet of radius Rp is: !2 ∆F Rp s = Fs Rs [5 marks] 8 6 (ii) For our own Solar System, RSun = 6.96 ×10 m, rEarth = 6.4 ×10 m and RJupiter is 7.1×107m. Calculate the fractional change in flux received from the Sun by a distant observer as a result of transits of both the Earth and Jupiter across the Sun.
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