# This paper is also taken for the relevant Examination for the Associateship. For Second Year Physics Students Wednesday, 4th June 2008: 14:00 to 16:00

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1 Imperial College London BSc/MSci EXAMINATION June 2008 This paper is also taken for the relevant Examination for the Associateship SUN, STARS, PLANETS For Second Year Physics Students Wednesday, 4th June 2008: 14:00 to 16:00 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 3 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 corresponding answer book. Hand in 3 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 /PO/2.1 1 Turn over for questions

2 FORMULA SHEET 2008/PO/2.1 2 Turn over for questions

3 SECTION A 1. (i) In a plane parallel atmosphere, the pressure p(z) and density ρ(z) depend only on the coordinate z (upwards). Take the gravitational acceleration g to be constant. Derive from first principles an inequality that must hold (Schwarzschild s criterion) for the atmosphere to be stable to convection. Explain briefly, giving reasons, where convection occurs in the Sun s interior. (ii) Star α Cen A is observed to have an apparent visual magnitude, m v, of 0.01, and its parallax is measured to be arcsec. Calculate the distance to α Cen A giving your answer in parsecs. What is the absolute visual magnitude M v of α Cen A? If the absolute visual magnitude of the Sun, M v, is +4.8 how does the luminosity of α Cen A compare to that of the Sun? (iii) Explain what is meant by the effective temperature of a star. Main sequence star X exhibits its peak black body radiation at a wavelength λ max = 290nm. For the Sun λ max = 500nm and the effective temperature is 5780K. Calculate the effective temperature of star X. If the ratio of the luminosity of star X, L x, to that of the luminosity of the Sun, L, Lx L = 100, calculate the radius of star X. [Solar radius R = 6.96 x 10 8 m] What type of star is star X? (iv) A binary star system has stars of masses m 1 and m 2 and semimajor axis a. By considering the circular orbits of the two stars derive the following expression for the binary period P: P = 2π ( a 3 G(m 1 + m 2 ) ) 1/2 The ratio of the stars masses, m 1 /m 2, is 0.55, and they lie at a distance of 13.2 parsecs from the Earth. The stars have an observed angular separation of 0.05 arc seconds, and orbital period of 104 days. Calculate the separation, a, of the stars, giving your answer in metres. Calculate the mass of each star. (v) Considering asteroids in our solar system, explain what Kirkwood Gaps are. Calculate the distances, in AU, from the Sun (semi-major axis), that correspond to the 5:2 and 2:1 resonance Kirkwood Gaps. [Jupiter lies at a distance of 5.2 AU from the Sun] [3 marks] (vi) In an H-R diagram for stars of a star cluster, the main sequence turn-off point is at L L = Estimate the age, in years, of this star cluster. [You may take the main sequence lifetime for a 1 solar mass star, 1 M, as years.] [3 marks] [Total 24 marks] 2008/PO/2.1 3 Please turn over

4 SECTION B 2. A star-planet system, has a star of luminosity L, with a planet Y situated at a distance d from the star, with albedo a, planetary radius R p = 6000km,and planet mass M p = kg. (i) What is the Greenhouse Effect? [2 marks] (ii) Derive the following expression for the surface temperature T p of planet Y. For this derivation assume the greenhouse effect is negligible for this planet. T p = ( ) 1/4 L(1 a) 16πd 2 σ Calculate the no atmosphere temperature, T p for planet Y, for L = 5 L, d = 2.21 AU, and a = (iii) What is meant by the habitable zone? Calculate the minimum and maximum distance from the star of the habitable zone. Is planet Y habitable? (iv) Derive an expression for, and calculate, the escape velocity of planet Y in terms of the planet s mass M p, and planet radius R p. Consider now the possible atmosphere for planet Y. Derive also an expression for the thermal velocity of gas particles in the planet s atmosphere, v th, in terms of the atmospheric temperature and particle mass. Hence, giving reasons and assumptions, determine whether planet Y is likely to have an atmosphere. Comment again on the likelihood of planet Y being habitable. [7 marks] 2008/PO/2.1 4 Please turn over

5 3. (i) Using relevant equations and sketch diagrams, show by considering conservation of angular momentum why an accretion disk will form in a gas cloud that is collapsing to form a star. Explain also why bipolar jets and mass outflows are seen in stars that are in the process of formation. (ii) In the context of stars, what is meant by hydrostatic equilibrium? Using the equation of hydrostatic equilibrium [provided with this exam paper], derive an expression for, and evaluate, an estimate of the Sun s central pressure. Also derive an expression for and calculate an estimate of the Sun s central temperature T c. Comment on how and why you would expect stars source of energy, particular thermonuclear reactions, to vary with stellar mass. [6 marks] (iii) Using the equation of hydrostatic equilibrium, and that internal energy per unit mass u is given by 1 P u = (γ 1) ρ derive the virial theorem 2U tot + Ω = 0 where U tot is the total thermal energy, and Ω is the gravitational potential energy. You may use the approximation that the pressure at the stellar surface can be neglected. [ Take γ = 5/3] Assuming a star is a perfect gas, and that there are no other sources of energy, using the virial theorem show and explain what happens to a star as it radiates into space. [7 marks] 2008/PO/2.1 5 Please turn over

6 4. Consider, hypothetically, that a civilization in a star system 5 parsecs from our Solar System is trying to discover whether there are any planets orbiting our Sun. [In answering the following questions you may take: Jupiter radius R J = 70,000km, Earth radius R E = 6371 km, Jupiter-Sun separation D J = 5.2AU, Jupiter Period =11.86 years, Jupiter mass = 1900 x kg] (i) Consider the direct detection of Jupiter by this hypothetical civilization, by observation of Jupiter shining by reflected light from the Sun. Jupiter lies at distance D J from the Sun and has radius R J. Show that: L J R2 J L DJ 2 Evaluate L J / L for the case of Jupiter. Calculate the angular separation in arc seconds of Jupiter and the Sun as observed from a distance of 5 parsecs from our Solar System. Comment on the suitability of direct detection as a method for finding exoplanets. (ii) In the astrometric detection of exoplanets, the angular size β of a star s orbit is measured. Show that β is given by: ( )1/3 ( ) G P 2/3 m p β = 4π 2 M s d where P is the orbital period of the star, d is the distance from which the star is observed, M s is the mass of the star, m p is the mass of the planet, G is the Gravitational constant. Evaluate β in units of arcseconds for the case of our Sun being observed from a distance of 5 parsecs. Consider only the effect of Jupiter on the Sun s motion. Assume the hypothetical civilization has similar technological limits to ourselves, so that their limit for measurement of β is of the order of 1 milli arc second. Would Jupiter be detectable by this technique? [6 marks] (iii) If the distant civilization now monitors our Sun for transits, planets passing directly across the face of the Sun, show that the duration of a transit τ is given by: τ = P R s πa where P is the period of the planetary orbit, R s is the stellar radius, a is the Sun-planet separation, and you may assume an inclination of 90 o. Calculate the duration, in hours, of the transits of Jupiter and the Earth. (iv) During a planetary transit the observed luminosity of the Sun drops. Calculate the fractional drop in luminosity during a transit of the Earth and during a transit of Jupiter. [3 marks] 2008/PO/2.1 6 Please turn over

7 5. Write concisely (1-2 pages each) on TWO of the topics below. In what you write, draw attention to the main features of the phenomena involved, and identify the relevant physics underpinning your understanding of them. Illustrate your answer with sketch diagrams where this is helpful. (i) The origins of planetary satellites. (ii) The comparative histories of Venus and Neptune. (iii) The H-R diagram. (iv) The ideal location for life in the Universe (v) The evolution of a 1 solar mass star from the Main Sequence stage to the end of its life. [18 marks] 2008/PO/2.1 7 End of examination paper

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