1 1 (gravitation) 1.1 m F a ma = F (1.1) F a m F 1.1 m F a (1.1) m a F m F a m a F F a m 0 0 1.2 (universal gravitation) (potential) M m gravitational force F r F = ; GMm r 2 (1.2) G = 6:67 10 ;8 dyn cm 2 g ;2 gravitational acceleration g g = F m = ;GM r 2 (1.3) 1.1 M M =5:9810 27 g r R = 6378km
2 g g = GM R 2 = 980 cm s ;1 M m potential energy E r E = ; GMm r (1.4) potential = E m = ;GM r (1.5) r F E F = ; de dr (1.6) g g = ; d dr (1.7) g g g = ;r (1.8) r (x y z) r = @ @x @ @y @! (1.9) @z (r ' z) r = @ @r 1 @ r @' @! @z (1.10) 1.2 M m 0-1 LA T EX -2-3 0 1 2 3 1.3 M g
3 2 z = ; 1 2 gt2 + v 0 t + z 0 (2.3) g g = 980 cm s ;2 t = 0 z =0 v =0 dz = v = ;gt (2.4) dt z = ; 1 2 gt2 (2.5) 2.1 2.1 x y z m d2 z dt 2 = ;mg d2 z = ;g (2.1) dt2 m g 2.2 z 0 v 0 dz dt = v = ;gt + v 0 (2.2) 2.3 v z 2.2 z m dv dt = ;mg ; kv (2.6) m g k v z
4 2.4 t =0 z =0 v =0 2.6 dv v + mg=k = ; k dt (2.7) m ln v + mg k = ;kt m + C 1 (2.8) v = ; mg k + C 2e ;kt=m (2.9) t =0 v =0 C 2 = mg=k z v = mg k (e;kt=m ; 1) (2.10) t =0 z =0 z = ; mg k t + m k (e;kt=m ; 1) (2.11) t!1 2.10 2.11 0 v! v1 = ; mg k (2.12) z!; mg k t (2.13) 2.12 v1 terminal speed 2.6 ;mg ;kv 0 1mm 7m/s 2.1 2.2 500 km 1km km 10 km
5 3 r t v 2 2 ; GM r = E = v2 0 2 ; GM r 0 (3.3) E 3.3 v 2 =2 ;GM=r E s dr dt = v = ; 2E + 2GM (3.4) r 3.1 3.1 M r g = ;GM=r 2 m m d2 r dt 2 = ;GMm r 2 d 2 r dt 2 = ;GM r 2 (3.1) t = t 0 r = r 0 v = v 0 v 0 0 E = ;GM=r 0 s dr dt = v = ; 2GM r ; 2GM r 0 (3.5) 0 3.3 3.2 v = dr=dt dr d 2 r dt dt + dr GM =0 (3.2) 2 dt r 2 3.1 3.5 r = r 0 r =0 t r r 3 0 t = (3.6) 8GM 3.5 r = r 0 cos 2 3.2
6 3.2 r = R r = 0 T v T = u t R2 2 GM (3.12) T =21 (3.13) 42 3.3 3.4 r M r M r = 4 3 r2 (3.7) 3.4 1pc 10 5 M 1M =1:9910 33 g r g = ; GM r 2 = ; 4G r (3.8) 3 r d 2 r dt 2 = ;4G r (3.9) 3 t =0 r = R v =0 v 2 2 + 2G 3 r 2 = E = 2G R 2 3 (3.10) M =(4=3)R 3 v s v GM u v = t r 2 1 ; (3.11) R R 2
7 4 Kepler's laws 1 v v = r 4.2 s GM = (4.2) r 3 v s GM v = r = r (4.3) 4.1 4.1 M m r M m M m r M m m GMm r 2 1 = mr 2 = m v2 r (4.1) 1. 2. 3. T 2 a 3 T 2 =a 3 P s P = 2r v = 2 =2 r 3 GM (4.4) 4.3 v 4.1 84.3 4.2 1 42297km 4.3
8 4.2 V R g V < p gr V = p gr p gr <V < p 2gR V = p 2gR V > p 2gR 4.4 4.4 1 4.5 2 V 2 = p 2g R = 11:2kms ;1 escape velocity
9 5 binary binary system 5.1 5.1 M 1 M 2 separation center of mass orbital plane inclination angle orbital period mass ratio primary star companion star 5.2 M 1 M 2 a 1 a 2 a distant binary close binary visual binary spectroscopic binary eclipse occultation eclipsing binary i
10 5.2 1 M 1 2 M 2 1 2 a 1 a 2 a a 1 +a 2 = a a 1 : a 2 : a = M 2 : M 1 :(M 1 + M 2 ) (5.1) 1 2 3 T 2 =a 3 = 5.2 A -1.46 A B 8.3 8.6 P 50.0 a 20.0 3 A B a 1 : a 2 M 1 : M 2 A B GM 1 M 2 a 2 = M 1 a 1 2 (5.2) a 1 a 1 G P (= 2=) 2 2 GM 2 = a 1 a 2 (5.3) P 2 5.3 2 2 GM 1 = a 2 a 2 (5.4) P 5.3 5.4a = a 1 + a 2 M 1 + M 2 = a3 G 2 2 (5.5) 3 P 5.1 3 1 M 2 P T
11 6 star cluster 6.1 open cluster galactic cluster association OB OB OB association 6.1 M50 NASA globular cluster 150 500 6.2 M80 NASA/STScI M b 10 5 M < M < 10 6 M (6.1) 50 < b < 500 (6.2) 6.2 ;GM=r 1 1
12 Plummer Plummer M r = ; p GM (6.3) r2 + b 2 b 0 r b 6.5 d 2 r dt 2 = ; GMr (r 2 + b 2 ) 3=2 = ; GMr b 3 (1 + r2 b 2 ) 3=2 ; GM b 3 r (6.7)! s GM! = (6.8) b 3 A B r = A sin!t + B cos!t (6.9) -GM/b 6.3 r ; d dr = ; GMr (6.4) (r 2 + b 2 ) 3=2! P =2=! 6.1 M 10 5 M b 50 M 10 6 M b 500 N d 2 r dt 2 = ;d dr = ; GMr (6.5) (r 2 + b 2 ) 3=2 1 2 v2 ; p GM = E() (6.6) r2 + z2 6.5 6.6
13 7 galaxy The Galaxy (dark matter) 10 Hubble classication 10 7.2 7.1 M51 7.1 elliptical galaxy disk galaxy irregular galaxy peculiar galaxy spiral galaxy barred spiral galaxy 220km/s 2 rotation curve V [/] 300 200 100 7.3 0 0 10 20 30 r []
14 7.2 r m V mv 2 =r r M(r) GM(r)m=r 2 GM(r)m=r 2 GM(r)m=r 2 GM(r)m r 2 = mv 2 r 2 (7.1) M(r) M(r) = rv 2 (7.2) G r V M(r) 7.1 7.2 300 V [/] 200 100 0 0 10 20 30 r [] 7.4 M31
15 8 group of galaxies cluster of galaxies 8.1 NAO 8.1 5900 M87 50 3 100 1933 1000km 500 dark matter 8.2 8.2
16 virial theorem T 2K +=0 (8.1) M R V T 8.3 0024 NASA/STScI K 1 2 MV 2 (8.2) ; GM 2 (8.3) R 8.1 M RV 2 G (8.4) 8.1 14.5 184 1000 z =0:0232 1 900 km s ;1 2.9 H =75km s ;1 500 500
17 9 3K 150 137 reball big bang Big Bang Universe H z =(H=c)r (9.1) v v = cz v = Hr (9.2) 9.2 9.1 9.1 1929 Hubble's law z r c H Hubble constant 1Mpc [km/s] H =71 4km=s=Mpc (9.3) 9.2 1917
18 lambda term cosmological constant 1922 1927 expanding universe model 3 closed universe at universe open universe 9.3 =0 9.3 M a M a t d 2 a dt 2 = ;GM a 2 + c2 3 a (9.4) 9.4 1 2 c dark energy 9.4 9.4 1 2! 2 da ; GM dt a ; c2 6 a2 = E = ; 1 2 kc2 (9.5)
19 9.5 1 2 3 E 0 E = ;kc 2 =2 k k =1 E k =0 E 0 k = ;1 E 9.5 6= 0 k da=dt 9.4 6= 0 9.1 9.4 9.2 9.5 9.3 0 E = ;kc 2 =2 9.4 =0 k da=dt