ADVANCED QUANTUM MECHANICS

PHY216 PHY472 Dt Provided: Dt Provided: Formul Formul sheet ndsheet physicl nd constnts physicl constnts DEPARTMENT DEPARTMENT OF PHYSICS OF PHYSICS & Spring& Semester 2015-2016 Autumn Semester 2009-2010 ASTRONOMY ASTRONOMY GALAXIES 2 hours ADVANCED QUANTUM MECHANICS 2 hours Answer ALL Answer questions question from Section ONE (Compulsory) A, nd TWO questions nd TWO fromother Section questions, B. Plese one ech from clerly indicte section the A question nd section numbers B. on which you would like to be exmined on the front cover of your nswer book. Cross through ny work tht you do not wnt to All be exmined. questions re mrked out of ten. The brekdown on the right-hnd side of the pper is ment s guide to the mrks tht cn be obtined from ech prt. Questions from Section A re mrked out of ve, wheres those from Section B re mrked out of fteen. The brekdown on the right-hnd side of the pper is ment s guide to the mrks tht cn be obtined from ech prt. PHY216 TURN OVER 1

PHY216 SECTION A nswer ALL questions 1. A Cepheid vrible in nerby glxy hs men pprent mgnitude in the V bnd of m V = 24.45 mgnitudes, period of P = 8 dys, nd suffers extinction of A V = 0.35 mgnitudes due to foreground dust in the Milky Wy. If Cepheid vribles follow V-bnd period-luminosity reltion of form M V = 2.76log 10 P(dys) 4.16, estimte the distnce to the glxy. Wht re the min uncertinties involved when using Cepheid vribles s distnce indictors? [5] 2. Briefly explin ech of the following. - The blue colours of reflection nebule. - The detection of strs moving t very high velocities (>1000 km s 1 ) close to the Glctic Centre. - The low metllicities nd high velocities reltive to the Sun of strs in the Glctic Hlo. - The red colours of ellipticl glxies. - The fct tht the colours of spirl glxies become bluer long the Hubble sequence from type S to Sc. [5] 3. If the stellr bulge of nerby spirl glxy hs mss of 10 9 M nd n effective rdius of 2 kpc, estimte its velocity dispersion (σ). [5] 4. HI 21cm observtions show tht spirl glxy hs flt rottion curve t rdil distnces between 5 nd 30 kpc from its nucleus, with n inclintion corrected rottion velocity of 250 km s 1. Estimte the mss density in the drk mtter hlo t distnce of 15 kpc from the nucleus. [5] PHY216 CONTINUED 2

PHY216 SECTION B nswer TWO questions 5. Write n essy on the determintion of the Hubble Constnt (H 0 ). Your nswer should include reference to the following: - the cosmic distnce ldder nd the steps required to produce n ccurte vlue of H 0 ; - the vrious primry nd secondry distnce indictors required (give brief description of two of ech); - the problems cused by peculir glxy velocities reltive to the Hubble Flow; - how estimtes of the Hubble Constnt hve chnged since the first determintions in the 1920s; - the resons for the erly inccurcy in the Hubble Constnt; - the impct of the Hubble Spce Telescope. [15] 6. () Oort s constnts (A nd B) re defined s follows: [ A = 1 ( ) ] [ dvc ω 0,B = 1 ( ) ] dvc ω 0 +, 2 dr R 0 2 dr R 0 where ω 0 is the ngulr velocity of the locl stndrd of rest (LSR) round the Glctic centre in km s 1 kpc 1, v c is the circulr velocity in km s 1, nd r is the rdil distnce in kpc. Currently the best estimtes of Oort s constnts nd the distnce to the centre of the Milky Wy re A = 14.8 ± 0.8 km s 1 kpc 1, B = 12.4 ± 0.6 km s 1 kpc 1, nd R 0 = 8.0 ± 0.5 kpc respectively. Estimte, with ssocited uncertinties, (i) the circulr rottion speed of the LSR round the centre of the Glxy in km s 1, nd (ii) the velocity grdient of the disk of the Glxy in the vicinity of the Sun in km s 1 kpc 1. [6] (b) Describe in detil the tngent point method for determining the inner rottion curve of the Milky Wy. [4] (c) Why is it much hrder to mesure the outer rottion rottion curve of the Milky Wy thn its inner rottion curve? [2] (d) The rottion curves of the Milky Wy nd other spirl glxies re observed to remin flt out to lrge rdii well beyond the min concentrtions of strlight. Explin why this might imply tht they contin lrge mounts of drk mtter. [3] PHY216 TURN OVER 3

PHY216 7. () Describe the generl fetures tht the vrious types of ctive glctic nuclei (AGN) hve in common. [4] Bsed on opticl spectroscopic observtions, how would you distinguish n AGN from the nucleus of norml, non-ctive glxy? [2] (b) Compre nd contrst, in terms of observtionl fetures, the following pirs of objects: - Type 1 nd Type 2 Seyfert glxies; [2] - qusrs nd Type 1 Seyfert glxies; [2] - FRI nd FRII rdio glxies. [2] (c) If qusr in the distnt universe hs bolometric luminosity of 10 39 W, estimte its mss ccretion rte. Stte ny ssumptions mde. [3] 8. Write detiled ccounts of ny three of the following. () The generl differences between spirl nd ellipticl glxies. [5] (b) Determining the msses of ellipticl glxies. [5] (c) The observtionl differences between boxy nd disky ellipticl glxies. [5] (d) The light profiles of ellipticl glxies. [5] (e) Theories for the formtion of ellipticl glxies. [5] END OF QUESTION PAPER PHY216 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