Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. Picture of glxy cluster Abell 2218 required for question 4(c) is ttched. DEPARTMENT OF PHYSICS AND ASTRONOMY Autumn Semester (2015-2016) DARK MATTER AND THE UNIVERSE 2 HOURS Answer question ONE (Compulsory) nd TWO other questions. All questions re mrked out of twenty. 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. 1 TURN OVER
COMPULSORY 1 () Briefly explin three techniques tht cn be used to serch for Brown Dwrfs nd stte why it is unlikely tht popultion of Brown Dwrfs in the Universe cn explin the drk mtter problem. (b) With the id of sketch describe how observtions of the Cosmic Microwve Bckground Rdition cn be used to determine the density of bryons in the Universe. (c) Outline why we might expect sufficiently lrge nd sensitive nucler recoil detector to observe n nnul modultion in the mesured flux of WIMP drk mtter prticles. Stte briefly why the positive detection of such modultion might resonbly be viewed with suspicion. (d) Using suitble plot, briefly explin how dt from observtions of type I supernove suggest the existence of drk energy. (e) Briefly outline two resons why ntimtter cnnot solve the drk mtter problem. Include n outline of how the presence of ntimtter in the Universe might be observed. 2 CONTINUED
2 The rottion curves of glxies provide some of the clerest evidence for the existence of drk mtter. () Describe briefly the strophysicl process in glxies tht llows such rottion curves to be mesured out to lrge distnces from glctic centres. (b) With the id of suitble grph explin how results from glxy rottion curves led to the interprettion tht glxies must contin more mtter thn cn be ccounted for by the luminous mteril. (c) Consider glxy tht contins centrl group of strs with combined mss M within rdius r = r M. In ddition it hs sphericl distribution of drk mtter with density distribution given by ρ(r) = ρ ο r 2 o r 2 where ρ o nd r o re constnts. Show tht the mss enclosed by sphere of rdius r, where r > r M is given by M(r) = M + 4πρ ο r 2 0 r. (d) A smll gs cloud is observed in the glxy to be moving in the grvittionl potentil creted by the combintion of the two components bove, with circulr orbit t rdius r > r M. Show using Newton s second lw tht the velocity v of the cloud is given by v 2 = GMr 1 + 4πGρ ο r o 2. [2] (e) At rdius r = 1 kpc where r M r r o the velocity is found to be v = 1100 km s -1. At rdius r b = 100 kpc where r b r o r M the velocity is found to be v = 180 km s -1. Use this informtion to estimte vlues for M (in solr mss 2 units), the constnt ρ o r o (in M pc -1 ) nd hence ρ(r) t 10 kpc (in M pc -3 ), distnce similr to tht of our Sun in the Milky Wy Glxy. [6] 3 TURN OVER
3 Although WIMPs re fvoured cndidte to explin the drk mtter, there re other exotic possibilities including xions nd the concept of Modified Newtonin Dynmics (MOND). () Briefly stte the prticle physics motivtion for the existence of xions nd describe one technique tht my llow them to be detected in the lbortory. (b) Sterile neutrinos re nother exotic possible drk mtter cndidte. Stte briefly wht distinguishes these prticles from neutrinos in the Stndrd Model of prticle physics. Roughly wht frction of the mss-energy content of the Universe do stndrd neutrinos contribute? (c) The bsis of MOND is chnge to Newton's second lw such tht F = m (), where = 1 except in the cse of very smll ccelertions when = / 0, where 0 is constnt. By considering the grvittionl force cting on test mss t distnce r from the centre of glxy of mss 4M, show tht = 2(GM 0 ) 1/2 /r. (d) Now consider the mss of the glxy in (c) split into four equl prts, ech seprted by distnce Δr from the centre of mss s in the digrm below (where r >> Δr). Reclculte s the sum of the ccelertion due to ech of the msses nd show tht this produces totl twice s big s the result for the single mss 4M. (Hint: use binomil expnsion for ech of the four terms using the substitution x = Δr/r nd ssume x << 1.) (e) Wht does the result in prt (d) tell us bout MOND? [2] [6] 4 CONTINUED
4 Grvittionl lensing nd the viril theorem provide powerful techniques for the study of drk mtter in glxy clusters. () Briefly describe the min differences between strong grvittionl lensing nd wek grvittionl lensing. Outline the process by which observtion of wek lensing cn be used to mp the distribution of drk mtter in glxy cluster. (b) Describe briefly one other technique tht cn be used to mp drk mtter in glxy clusters. [2] (c) Consider the imge in the ttched sheet of the glxy cluster Abell 2218. The cluster is t redshift of 0.17. The imge is 1.3 rcminutes wide nd 1.6 rcminutes high. Estimte the dimeter of the outermost Einstein ring in units of Mpc. (d) Write n eqution for the opening ngle of the Einstein ring in terms of the lensing mss, the grvittionl constnt nd the distnce from Erth to the cluster, ssuming this distnce is the sme s the distnce from the cluster to the bckground source. Hence estimte the lensing mss for Abell 2218 in solr mss units. [6] (e) Show tht the grvittionl potentil energy of sphere of uniform density, mss M nd rdius R is V = 3GM2 5R nd hence estimte the totl kinetic energy in the cluster. 5 TURN OVER
5 Wekly Intercting Mssive Prticles (WIMPs) re fvoured cndidte for drk mtter, thought likely to occur in glxies with distribution in the form of n isotherml sphere. Such prticles re expected to be observble on Erth vi detection of nucler recoil events tht my be induced in suitble detector mterils tht hve sufficiently low nturl bckground contmintion. () Briefly describe the three min chrcteristics of n isotherml sphere in the context of drk mtter in glxies. Explin why it is tht the luminous bryonic mtter in glxies does not hve this form of distribution. (b) The result of dt nlysis from WIMP detector, in the sitution where lrge popultion of WIMP events is ctully seen, would be nucler recoil energy spectrum in the form of plot of dr/de obs vs. E obs, where E obs is the observed nucler recoil energy in kev. Briefly explin the four min fctors tht determine the shpe of such plot. (c) The results of WIMP experiments in which no events re seen re usully drwn in the form of n exclusion plot comprising line on grph of WIMP-nucleon cross section vs. WIMP mss. Explin the term exclusion plot nd with the id of sketch of such plot explin wht determines the shpe of the curve t low nd high WIMP msses. (d) Consider the cse of popultion of WIMPs with mss 830 GeV hving velocity 270 km s -1 reltive to the trget nuclei in liquid rgon detector. Clculte the mximum recoil energy tht cn be imprted to single rgon tom, tomic mss 40, in such detector. (e) An 800 kg liquid rgon detector contins 1 nucleus in 10 11 of 39 Ar, which hs hlf life of 256 yers. Estimte the bckground rte in the rgon detector from these decys, ssuming they re detected with 100% efficiency. END OF EXAMINATION PAPER 6 CONTINUED
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