Data Provided: A formula sheet and table of physical constants is attached to this paper.

Transcription

1 PHY106 PHY47 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 hours Spring 016 The Solr System Answer question ONE (Compulsory) nd TWO other questions, one ech from section A nd section B. Instructions: All questions Answer ny re THREE mrked questions 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. hours All questions re mrked out of thirty. 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. PHY106 TURN OVER 1

2 PHY Discuss in detil ny THREE of the following: () Sturn's rings; (b) the steroid belt; [10] [10] (c) cryovolcnism in the solr system; [10] (d) the structure nd composition of the terrestril plnets; [10] (e) the formtion of the Moon. [10]. () Wht ws the Lte Hevy Bombrdment? Wht is the evidence for it nd wht do we think cused it? (b) Discuss the formtion of crter. How re crters used to determine the ge of plnetry surfces? 3. Compre nd contrst the following pirs: () Mrs nd Venus; (b) Pluto nd Triton. 4. () Stte Kepler's lws of plnetry motion [7] (b) In erly 016, stronomers Btygin nd Brown nnounced their predictions for ninth plnet in the solr system, including clculted period of yers, n eccentricity of 0.6, nd mss of 10M Erth. If this plnet exists, where in its orbit is it sttisticlly most likely to be locted t the present time? Wht implictions does this hve for its detection? [3] (c) If this plnet hs n phelion distnce of 100 AU, how close will it come to the Sun t perihelion? (d) Discuss, giving exmples, orbitl resonnces nd their importnce in the outer solr system. [5] 5. Describe in detil the core ccretion model of plnet formtion nd discuss the fetures of the solr system it explins. [30] END OF EXAMINATION PAPER

3 PHYSICAL CONSTANTS & MATHEMATICAL FORMULAE Physicl Constnts electron chrge e = C electron mss m e = kg = MeV c proton mss m p = kg = MeV c neutron mss m n = kg = MeV c Plnck s constnt h = J s Dirc s constnt ( = h/π) = 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 =.4 l mol 1 grvittionl constnt G = N m kg 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 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 1 Å (ngstrom) = m 1 stronomicl unit = m 1 g (grvity) = 9.81 m s 1 ev = J 1 prsec = m 1 tmosphere = P 1 yer = s

4 Polr Coordintes x = r cos θ y = r sin θ da = r dr dθ = 1 ( r ) + 1r r r r θ Sphericl Coordintes Clculus x = r sin θ cos φ y = r sin θ sin φ z = r cos θ dv = r sin θ dr dθ dφ = 1 ( r ) + 1 r r r r sin θ ( sin θ ) + θ θ 1 r sin θ φ f(x) f (x) f(x) f (x) x n nx n 1 tn x sec x e x e x sin ( ) ln x = log e x 1 x cos 1 ( x sin x cos x tn ( cos x sin x sinh ( ) cosh x sinh x cosh ( ) sinh x cosh x tnh ( ) ) ) 1 x 1 x +x 1 x + 1 x x cosec x cosec x cot x uv u v + uv sec x sec x tn x u/v u v uv v Definite Integrls x n e x dx = n! (n 0 nd > 0) n+1 π e x dx = π x e x dx = 1 Integrtion by Prts: 3 b u(x) dv(x) dx dx = u(x)v(x) b b du(x) v(x) dx dx

5 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 + ( x < 1)! (x ) f () +! nd (x )3 f () + 3! ( ) n n! = k (n k)!k! e x = 1+x+ x! + x3 x3 +, sin x = x 3! 3! + x5 x nd cos x = 1 5!! + x4 4! ln(1 + x) = log e (1 + x) = x x + 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 = sin cos cos = cos sin = cos 1 = 1 sin sin + sin b = sin 1( + b) cos 1 ( b) sin sin b = cos 1( + b) sin 1 ( b) cos + cos b = cos 1( + b) cos 1 ( b) cos cos b = sin 1( + b) sin 1 ( b) e iθ = cos θ + i sin θ cos θ = 1 ( e iθ + e iθ) nd sin θ = 1 ( e iθ e iθ) i cosh θ = 1 ( e θ + e θ) nd sinh θ = 1 ( e θ e θ) 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

6 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 = φ = φ x + φ y + φ z ( φ) = 0 nd ( A) = 0 ( A) = ( A) A ( Ax î + z A ) ( z Ay ĵ + x x A ) x y ˆk