Introduction to Astrophysics

Transcription

1 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 ASTRONOMY DEPARTMENT OF PHYSICS AND ASTRONOMY ADVANCED QUANTUM MECHANICS 2 hours Spring 2017 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 of from ten. Section The brekdown A nd TWO questions on the right-hnd from Section side B. of Plese the clerly pper indicte is ment the s question guide numbers to the mrks on which tht you cn would be obtined like to be from exmined ech on prt. the front cover of your nswer 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

2 SECTION A 1. Two strs (A nd B) hve the sme rdius nd temperture, but dierent pprent mgnitudes. Str A hs V-bnd mgnitude of 14.0, while str B hs V-bnd mgnitude of If str A hs prllx of 0.15, clculte the distnce to str B. Could the prllx of str B be observed by the strometric stellite Gi? [5] 2. The monochromtic ux of rdio source hs frequency dependence of F ν ν 2.5 when mesured in frequency units. Wht is the slope of grph of log F λ ginst log λ? At which end of the electromgnetic spectrum would you expect to see the most ux? [5] 3. Estimte the photospheric temperture of str whose spectrum peks t wvelength of 450 nm. Wht is the men energy of gs prticles in the photosphere? If the men number density of prticles in the photosphere is m 3, clculte the men gs pressure in the photosphere. Stte ny ssumptions you mke. [5] 4. Below re descriptions of the bsorption line spectr of ve strs. Use these descriptions to plce the strs in order of incresing temperture. Str A: medium strength lines of hydrogen nd singly ionised clcium. Str B: strong hydrogen lines, wek lines of neutrl helium. Str C: strong moleculr lines nd neutrl clcium lines. Str D: strong singly ionised clcium lines, moderte neutrl metl lines, wek hydrogen lines. Str E: strong ionised helium lines. [5] PHY104 CONTINUED 2

3 SECTION B 5. () Give detiled ccount of mesuring distnces in strophysics, from the nerest strs to the most distnt glxies. [5] (b) Aldebrn (α Turi) hs mesured prllx of 50.0 milli-rcseconds. Spectroscopic observtions of Aldebrn show tht the nm bsorption line of hydrogen is redshifted by 0.12 nm. Clculte its distnce nd its rdil velocity with respect to the Erth. (c) An unresolved str cluster is composed of individul, identicl strs, ech with n pprent V-bnd mgnitude of If the V-bnd mgnitude of the cluster s whole is 17.5, clculte: i. the number of strs in the cluster, nd [3] ii. the monochromtic V-bnd ux of single str. [2] (d) If the cluster discussed bove is t distnce of 300 prsecs, wht is its bsolute mgnitude? Wht is the bsolute mgnitude of ech component str? [2] Note: the str Veg hs monochromtic ux of W m 2 nm 1 in the V-bnd. [3] 6. The infrred spectrum of str shows series of bsorption lines due to tomic hydrogen. The lines become more closely spced s wvelength increses until the series converges t limit of 1458 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 nd use it to explin how the series of lines described bove rises. Indicte on your digrm the ionistion energy nd ground stte. (c) Explin, with the id of sketch, how nd why the strengths of the Blmer (n = 2) bsorption lines due to hydrogen vry with photospheric temperture long the Hrvrd spectrl clssiction sequence. [4] (d) The opticl 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 K, nd ii K. Note: The ionistion energy of hydrogen is 13.6 ev. [3] [4] [4] PHY104 TURN OVER 3

4 7. () Describe the conditions under which blck body rdition is produced. Give two exmples from stronomy of sources of blck body rdition. [3] (b) Briey describe one other type of continuum emission, nd the physicl process tht gives rise to it. (c) Wht type of spectrum (emission line, bsorption line, or continuum) would be expected to be observed in ech of the following sources: i. molten steel in smelting furnce; ii. neon dvertising sign; iii. str like the Sun; iv. the low density outer tmosphere of the Sun (the coron) observed during totl solr eclipse? [1] [4] (d) A str hs mesured bolometric ux of W m 2. Its spectrum shows pek ux t 430 nm, nd it is locted in cluster t distnce of 1.1 kpc. Estimte the following quntities: i. the temperture of the photosphere of the str; [1] ii. the bolometric luminosity of the str; [2] iii. the rdius of the str. [2] Stte ny ssumptions mde. [1] (e) The str bove (cll this Str A) hs colour index (B V) of 0.3. Another str in the cluster (cll this Str B) hs colour index (B V) of 0.1. Which str is bluer nd why? [1] PHY104 CONTINUED 4

5 8. () 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. (b) A visul binry str is observed with the Hubble Spce Telescope. Its orbitl period is seen to be 12 yers, nd the ngulr size of the orbits of the two components (A nd B) re α A = 0.20 nd α B = 0.50 respectively. If the binry system is 15 prsecs wy, clculte: i. the totl mss of the system; [2] ii. the msses of the two components, in Solr msses. [3] Note: Assume tht the orbits of the two strs re circulr, nd in the plne of the sky. (c) How could you tell if the binry's orbit ws not in the plne of the sky? [1] (d) Visul binries re rre. Wht other type of strophysicl binry system cn be used to mesure exct msses? [1] (e) For glxy cluster in sphericl equilibrium, the viril theorem cn be used to derive the following sttisticl estimte for the totl mss of the cluster: M c = 5R c v(r) 2, G where M c is the mss of the cluster, R c is the rdius of the cluster, nd v(r) 2 is the men squred rdil velocity of glxies within the cluster. Estimte the mss of the Com cluster, which sits t n verge distnce of 103 Mpc, hs n ngulr rdius of 6.0, nd whose glxies hve men rdil velocity reltive to ech other of 215 km s 1. [3] END OF EXAMINATION PAPER 5

6 PHYSICAL CONSTANTS & MATHEMATICAL FORMULAE Physicl Constnts electron chrge e = C electron mss m e = kg = MeV c 2 proton mss m p = kg = MeV c 2 neutron mss m n = kg = MeV c 2 Plnck s constnt h = J s Dirc s constnt ( = h/2π) = J s Boltzmnn s constnt k B = J K 1 = ev K 1 speed of light in free spce c = m s m s 1 permittivity of free spce ε 0 = F m 1 permebility of free spce µ 0 = 4π 10 7 H m 1 Avogdro s constnt N A = mol 1 gs constnt R = J mol 1 K 1 idel gs volume (STP) V 0 = 22.4 l mol 1 grvittionl constnt G = N m 2 kg 2 Rydberg constnt R = m 1 Rydberg energy of hydrogen R H = 13.6 ev Bohr rdius 0 = m Bohr mgneton µ B = J T 1 fine structure constnt α 1/137 Wien displcement lw constnt b = m K Stefn s constnt σ = W m 2 K 4 rdition density constnt = J m 3 K 4 mss of the Sun M = kg rdius of the Sun R = m luminosity of the Sun L = W mss of the Erth M = kg rdius of the Erth R = m Conversion Fctors 1 u (tomic mss unit) = kg = MeV c 2 1 Å (ngstrom) = m 1 stronomicl unit = m 1 g (grvity) = 9.81 m s 2 1 ev = J 1 prsec = m 1 tmosphere = P 1 yer = s

7 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 x 2 2 +x 2 1 x x 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 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

8 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

9 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