PHY104 PHY472 Dt Provided: Formul sheet nd physicl constnts Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. DEPARTMENT OF PHYSICS & Autumn Semester 2009-2010 ASTRONOMY DEPARTMENT OF PHYSICS AND ASTRONOMY ADVANCED QUANTUM MECHANICS 2 hours Spring 2016 Introduction to Astrophysics 2 hours Answer question ONE (Compulsory) nd TWO other questions, one ech from section A nd section B. Instructions: All questions Answer ll re FOUR mrked questions out from of ten. Section The Abrekdown nd TWO questions on the from right-hnd Section B. side Plese of the clerly indicte is ment the question s guide numbers to on the which mrks you tht wouldcn like be to be obtined exminedfrom on theech front cover prt. of your nswer pper book. Cross through ny work tht you do not wnt to be exmined. Section A is worth 20 mrks in totl nd ll questions in section B re worth 15 mrks ech. The brekdown on the right-hnd side of the pper is ment s guide to the mrks tht cn be obtined from ech prt. Plese clerly indicte the question numbers on which you would like to be exmined on the front cover of your nswer book. Cross through ny work tht you do not wish to be exmined. PHY104 TURN OVER 1
PHY104 SECTION A 1. Define the following terms: () stronomicl prllx; (b) colour index; (c) bolometric luminosity; (d) redshift; (e) distnce modulus. [5] 2. Write down the three lws of plnetry motion introduced by Johnnes Kepler. The orbit of dwrf plnet Mkemke hs semi-mjor xis of 45.4 AU nd n eccentricity of 0.16. Clculte: () its orbitl period (in yers), (b) its perihelion nd phelion distnces (in AU), (c) the rtio of its mximum to minimum orbitl velocities. [5] 3. Fomlhut hs mesured ngulr dimeter of 2.1 millircsec, nd bolometric flux of 8.6 10 9 W m 2. Wht is its effective temperture? Stte ny ssumptions mde. If Fomlhut lies t distnce of 7.7 prsec, wht is its bsolute bolometric mgnitude? Note: The bsolute bolometric mgnitude of the Sun is +4.74 mg. [5] 4. The binry system Spic hs n pprent V-bnd mgnitude of 0.97 mg. If the system hs prllx of 0.013 rcsec, nd the primry is 2.0 mgnitudes brighter thn the secondry in the V-bnd, wht re the bsolute V-bnd mgnitudes of the two components? Estimte the spectrl types of the components if B V = 0.25 for both strs. [5] PHY104 CONTINUED 2
PHY104 SECTION B 5. () Show tht for single-lined spectroscopic binry system, the period, P, nd the mesured rdil velocity of the visible component, v, re relted to the msses of 1 the two strs in the system by m 3 2 (m 1 + m 2 ) 2 sin3 i = P 2πG v 3 1, where the term on the right-hnd side is the mss function, m 1 is the mss of the visible component, m 2 is the mss of the invisible component nd i is the inclintion of the orbit to the plne of the sky. Assume the orbit is circulr. [6] (b) Explin why the mss function represents lower limit to the mss of the invisible component, m 2. [2] (c) How my the inclintion of spectroscopic binry be found? [1] (d) Use the mss function defined bove to show tht for n exoplnet system, whose plnet is much less mssive thn the str, the observed Doppler wobble of the str obeys (m p sin i) 3 P 2πG v 3. s m 2 s Explin why the rdil velocity technique for detecting plnets is bised towrds mssive plnets with orbits close their host str. [5] 6. Write detiled ccounts of ny THREE of the following: () the history of stronomicl spectroscopy; [5] (b) mesuring distnce in strophysics; [5] (c) evidence for drk mtter in the Universe; [5] (d) mesuring the msses of strs. [5] PHY104 TURN OVER 3
PHY104 7. Describe the conditions under which blck body rdition is produced. Give two exmples from stronomy of sources of blck body rdition. [3] Briefly describe one other continuum emission mechnism, nd the physicl process tht gives rise to it. [1] A str hs mesured bolometric flux of (1.0 ± 0.1) 10 16 W m 2. Its spectrum shows pek flux t 500 ± 20 nm, nd it is locted in cluster t distnce of 1.0 ± 0.2 kpc. Estimte the following quntities, with errors: () the temperture of the photosphere of the str; [2] (b) the bolometric luminosity of the str; [3] (c) the rdius of the str. Stte ny ssumptions mde. A second str hs the sme rdius nd temperture s the str bove, but is 6.0 mgnitudes brighter. Wht is the distnce to this second str, nd could you mesure the prllx from ground-bsed telescope? [2] [3] [1] 8. () Briefly describe the Kirchhoff-Bunsen lws, nd their predictions for the type of stronomicl spectr seen from different types of object. [3] (b) The Blmer bsorption line series rises from trnsitions out of the n = 2 energy level in n tom of neutrl hydrogen. If the degenercy of the n th energy level of hydrogen is 2n 2, clculte N 2 /N 1, the number of electrons in the n = 2 level, reltive to the number of electrons in the ground stte, t temperture of i. 8 000 K, nd ii. 15 000 K. (The energy difference between the n = 1 nd n = 2 levels of hydrogen is 10.2 ev.) [3] (c) Sketch the vrition in strength of the Blmer bsorption lines with spectrl clss, ensuring tht you lbel spectrl clsses ppropritely. With reference to your nswer bove, explin why the Blmer bsorption line strength peks in A strs, including reference to pproprite equtions. [5] (d) Estimte the photospheric temperture for str whose spectrum peks t wvelength of 1.1 µm. Wht is the men energy of gs prticles in the photosphere? If the men gs pressure in its tmosphere is 1000 N m 2, clculte the men number density of prticles in the stellr tmosphere. [4] END OF EXAMINATION PAPER 4
PHYSICAL CONSTANTS & MATHEMATICAL FORMULAE Physicl Constnts electron chrge e = 1.60 10 19 C electron mss m e = 9.11 10 31 kg = 0.511 MeV c 2 proton mss m p = 1.673 10 27 kg = 938.3 MeV c 2 neutron mss m n = 1.675 10 27 kg = 939.6 MeV c 2 Plnck s constnt h = 6.63 10 34 J s Dirc s constnt ( = h/2π) = 1.05 10 34 J s Boltzmnn s constnt k B = 1.38 10 23 J K 1 = 8.62 10 5 ev K 1 speed of light in free spce c = 299 792 458 m s 1 3.00 10 8 m s 1 permittivity of free spce ε 0 = 8.85 10 12 F m 1 permebility of free spce µ 0 = 4π 10 7 H m 1 Avogdro s constnt N A = 6.02 10 23 mol 1 gs constnt R = 8.314 J mol 1 K 1 idel gs volume (STP) V 0 = 22.4 l mol 1 grvittionl constnt G = 6.67 10 11 N m 2 kg 2 Rydberg constnt R = 1.10 10 7 m 1 Rydberg energy of hydrogen R H = 13.6 ev Bohr rdius 0 = 0.529 10 10 m Bohr mgneton µ B = 9.27 10 24 J T 1 fine structure constnt α 1/137 Wien displcement lw constnt b = 2.898 10 3 m K Stefn s constnt σ = 5.67 10 8 W m 2 K 4 rdition density constnt = 7.55 10 16 J m 3 K 4 mss of the Sun M = 1.99 10 30 kg rdius of the Sun R = 6.96 10 8 m luminosity of the Sun L = 3.85 10 26 W mss of the Erth M = 6.0 10 24 kg rdius of the Erth R = 6.4 10 6 m Conversion Fctors 1 u (tomic mss unit) = 1.66 10 27 kg = 931.5 MeV c 2 1 Å (ngstrom) = 10 10 m 1 stronomicl unit = 1.50 10 11 m 1 g (grvity) = 9.81 m s 2 1 ev = 1.60 10 19 J 1 prsec = 3.08 10 16 m 1 tmosphere = 1.01 10 5 P 1 yer = 3.16 10 7 s
Polr Coordintes x = r cos θ y = r sin θ da = r dr dθ 2 = 1 ( r ) + 1r 2 r r r 2 θ 2 Sphericl Coordintes Clculus x = r sin θ cos φ y = r sin θ sin φ z = r cos θ dv = r 2 sin θ dr dθ dφ 2 = 1 ( r 2 ) + 1 r 2 r r r 2 sin θ ( sin θ ) + θ θ 1 r 2 sin 2 θ 2 φ 2 f(x) f (x) f(x) f (x) x n nx n 1 tn x sec 2 x e x e x sin ( ) 1 x ln x = log e x 1 x cos 1 ( x sin x cos x tn ( 1 x cos x sin x sinh ( ) 1 x cosh x sinh x cosh ( ) 1 x sinh x cosh x tnh ( ) 1 x ) ) 1 2 x 2 1 2 x 2 2 +x 2 1 x 2 + 2 1 x 2 2 2 x 2 cosec x cosec x cot x uv u v + uv sec x sec x tn x u/v u v uv v 2 Definite Integrls 0 + + x n e x dx = n! (n 0 nd > 0) n+1 π e x2 dx = π x 2 e x2 dx = 1 2 Integrtion by Prts: 3 b u(x) dv(x) dx dx = u(x)v(x) b b du(x) v(x) dx dx
Series Expnsions (x ) Tylor series: f(x) = f() + f () + 1! n Binomil expnsion: (x + y) n = (1 + x) n = 1 + nx + k=0 ( ) n x n k y k k n(n 1) x 2 + ( x < 1) 2! (x )2 f () + 2! nd (x )3 f () + 3! ( ) n n! = k (n k)!k! e x = 1+x+ x2 2! + x3 x3 +, sin x = x 3! 3! + x5 x2 nd cos x = 1 5! 2! + x4 4! ln(1 + x) = log e (1 + x) = x x2 2 + x3 3 n Geometric series: r k = 1 rn+1 1 r k=0 ( x < 1) Stirling s formul: log e N! = N log e N N or ln N! = N ln N N Trigonometry sin( ± b) = sin cos b ± cos sin b cos( ± b) = cos cos b sin sin b tn ± tn b tn( ± b) = 1 tn tn b sin 2 = 2 sin cos cos 2 = cos 2 sin 2 = 2 cos 2 1 = 1 2 sin 2 sin + sin b = 2 sin 1( + b) cos 1 ( b) 2 2 sin sin b = 2 cos 1( + b) sin 1 ( b) 2 2 cos + cos b = 2 cos 1( + b) cos 1 ( b) 2 2 cos cos b = 2 sin 1( + b) sin 1 ( b) 2 2 e iθ = cos θ + i sin θ cos θ = 1 ( e iθ + e iθ) 2 nd sin θ = 1 ( e iθ e iθ) 2i cosh θ = 1 ( e θ + e θ) 2 nd sinh θ = 1 ( e θ e θ) 2 Sphericl geometry: sin sin A = sin b sin B = sin c sin C nd cos = cos b cos c+sin b sin c cos A
Vector Clculus A B = A x B x + A y B y + A z B z = A j B j A B = (A y B z A z B y ) î + (A zb x A x B z ) ĵ + (A xb y A y B x ) ˆk = ɛ ijk A j B k A (B C) = (A C)B (A B)C A (B C) = B (C A) = C (A B) grd φ = φ = j φ = φ x î + φ y ĵ + φ z ˆk div A = A = j A j = A x x + A y y + A z z ) curl A = A = ɛ ijk j A k = ( Az y A y z φ = 2 φ = 2 φ x + 2 φ 2 y + 2 φ 2 z 2 ( φ) = 0 nd ( A) = 0 ( A) = ( A) 2 A ( Ax î + z A ) ( z Ay ĵ + x x A ) x y ˆk