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 2015 Introduction to Astrophysics 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 of the pper is ment s guide to the mrks tht cn be obtined from ech prt. 2 hours 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
SECTION A 1. A str hs n bsolute mgnitude of 7.0. Wht is the pprent mgnitude of n unresolved str cluster, if it consists of 300 of these strs, nd is locted t distnce of 5.0 kpc. [5] 2. A source is observed to hve continuous spectrum (the Plnck curve) with series of emission lines superimposed. For ech of the sources below, stte whether the source could produce such spectrum, nd why: () high density ccretion disc with hot, low density tmosphere; (b) the Sun; (c) neon dvertising sign. [5] 3. A str hs photospheric temperture of 22,000K, rdius of 6.0 10 6 m, nd mesured prllx of 0.010 rcseconds. Estimte: () the bolometric luminosity of the str in W; (b) the bolometric flux of the str mesured t Erth in W m 2 ; (c) the wvelength t which the str s spectrum peks in nm. Stte ny ssumptions mde. [3] [1] In wht prt of the electromgnetic spectrum does the str s spectrum pek? [1] PHY104 CONTINUED 2
4. The figure below shows concept mp, which depicts the reltionships between properties of strs which cn be deduced from observtions. There re seven blnk boxes. In your nswer book, mtch the following terms with the numbers in the blnk boxes: () prllx; (b) blck bodies; (c) luminosity; (d) the inverse squre lw; (e) distnce; (f) bolometric flux; (g) temperture. [5] PHY104 TURN OVER 3
SECTION B 5. The wvelengths of lines corresponding to trnsitions in the energy levels of hydrogen re given by λ = 1 1 1 1, R n 2 1 n 2 2 where R is the Rydberg constnt nd n 1,2 re the quntum numbers of the energy levels involved. Without writing down ny equtions, explin in words the steps tken to derive the eqution bove. [6] The infrred spectrum of str shows series of bsorption lines due to tomic hydrogen. The lines become more closely spced s wvelength decreses until the series converges t limit of 91.13 nm. () Clculte the principl quntum number (n) of the lowest energy level involved in producing the lines. (b) Drw n energy level digrm for hydrogen t n pproprite scle, nd use it to explin how the series of lines described bove rises. Indicte on your digrm the ionistion energy nd ground stte. [4] (c) Explin how nd why the strengths of the bsorption lines due to the Blmer series of hydrogen vry with photospheric temperture long the Hrvrd spectrl clssifiction sequence. [2] [3] 6. Write detiled ccounts of ny THREE of the following: () the history of stronomicl spectroscopy; [5] (b) evidence for drk mtter in the Universe; [5] (c) the serch for extr-solr plnets; [5] (d) blck body rdition nd its pplictions in strophysics. [5] PHY104 CONTINUED 4
7. Show tht, for two strs (A nd B) orbiting on circulr orbits round common centre of mss, the period (P) is relted to the msses of the two strs (m A nd m B ) nd the distnce between the centres of the two strs (r) by [5] GP 2 (m A + m B ) 4π 2 = r 3. Two strs in visul binry system re observed to trvel on circulr orbits, with rdii of 0.34 ± 0.01 nd 1.37 ± 0.04 rcseconds for the primry str nd secondry str respectively. The orbitl period is 100 yers, nd the prllx of the system is 0.100 ± 0.003 rcseconds. Find the msses (with uncertinties) of the two strs, in Solr msses. [5] Briefly describe why visul binries cn only be used to mesure the msses of nerby strs. [2] Explin why spectroscopic eclipsing binries re so useful in strophysics. [2] Why re eclipsing visul binries so rre? [1] 8. 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 15 W m 2. Its spectrum shows pek flux t 800 ± 20 nm, nd it is locted in cluster with known distnce of 2.0 ± 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. [3] Stte ny ssumptions mde. [1] A second str hs the sme rdius nd temperture s the str bove, but is 8.0 mgnitudes brighter. Wht is the distnce to this second str, nd could you mesure the prllx from ground bsed telescope? [2] END OF EXAMINATION PAPER 5
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