Mechanics Physics 151

Transcription

1 Mechanics Physics 151 Lectue 6 Kepe Pobem (Chapte 3)

2 What We Did Last Time Discussed enegy consevation Defined enegy function h Conseved if Conditions fo h = E Stated discussing Centa Foce Pobems Reduced -body pobem into centa foce pobem Pobem is educed to one equation Used angua momentum consevation Enegy consevation gives m 1 E = + + V() = const m Now we must sove this L t = m = + f () 3 m

3 Goas fo Today Anayze quaitative behavio of centa-foce pobem Soutions: bounded o unbounded Detemined by the shape of the potentia Sove the Kepe pobem Get the shape of the obit As if we don t know yet Deive Kepe s 3 d Law Peiod of otation is popotiona to the 3/ powe of the majo axis

4 Quaitative Behavio Integating the adia motion isn t aways easy Moe often impossibe You can sti te genea behavio by ooking at E V () V() + m Enegy E is conseved, and E V must be positive m = + V () m Pot V () and see how it intesects with E = E V() m m Quasi potentia incuding the centifuga foce = E V () > E > V ()

5 Invese-Squae Foce Conside an attactive 1/ foce k k f() = V() = Gavity o eectostatic foce k V () = + m 1/ foce dominates at age Centifuga foce dominates at sma A dip foms in the midde V () m k

6 Unbounded Motion Take V simia to 1/ case Ony genea featues ae eevant E = E 1 > min E = V ( ) 1 min Patice can go infinitey fa E 1 V () 1 m Aive fom = E Tuning point E = V = Go towad = E 3 A 1/ foce woud make a hypeboa

7 Bounded Motion E = E min < < max Patice is confined between two cices E 1 V () Goes back and foth between two adii E 1 m E 3 Obit may o may not be cosed. (This one isn t) A 1/ foce woud make an eipse

8 Cicua Motion E = E 3 = (fixed) Ony one adius is aowed Stays on a cice Cassification into unbounded, bounded and cicua motion depends on the genea shape of V Not on the detais (1/ o othewise) E = V ( ) = = const = E 1 E E 3 V ()

9 Anothe Exampe V a = f = 3 4 Attactive -4 foce V has a bump 3a Patice with enegy E may be eithe bounded o unbounded, depending on the initia a V = + m 3 E V m V

10 Stabe Cicua Obit Cicua obit occus at the bottom of a dip of V m = E V = Top of a bump woks in theoy, but it is unstabe Initia condition must be exacty = and = = const Stabe cicua obit equies dv m = = d dv d > E E stabe unstabe

11 Obit Equation We have been tying to sove = (t) and θ = θ(t) We ae now inteested in the shape of the obit = Switch fom dt to dθ = m θ const dv m + = m d 3 Switch fom to u 1 d d d d d = dt m dθ du d 1 1 d = = θ θ θ ( θ ) d d dv 3 + = dθ m dθ m d d d = u d du

12 Obit Equation Switching vaiabes 1 du mdv ( u ) u dθ + + du = Soving this equation gives the shape of the obit Not that it s easy (How coud it be?) Wi do this fo invese-squae foce ate One moe usefu knowedge can be extacted without soving the equation

13 Symmety of Obit 1 du mdv ( u ) u dθ + + du = Equation is even, o symmetic, in θ Repacing θ with θ does not change the equation Soution u(θ) must be symmetic if the initia condition is Choosing θ = at t =, θ θ makes du du du u() u() OK () () OK if () = dθ dθ dθ Obit is symmetic at anges whee du/dθ =

14 Symmety of Obit Obit is symmetic about evey tuning point = apsidas Obit is invaiant unde efection about apsida vectos That s why I didn t cae too much about the sign of Sove the obit between a pai of apsida points Entie obit is known Now it s time to sove the equation du dθ =

15 Soving Obit Equation 1 du mdv ( u ) u dθ + + du = Integating the diff eqn wi give enegy consevation One can use enegy consevation to save effot E m = + + V() m du Switch vaiabes = mdθ = E V() m m 1 mv ( u ) du me u dθ = Integate this

16 Invese Squae Foce f k = V Look it up in a math text book and find dx β + γx = accos α + βx+ γx γ β 4αγ Just substitute α, β and γ O k du me mku = u dθ = + du = dθ me mku + u

17 Woking It Out Yousef du dθ = = = mku + u me m k mk + 4 me m k + 4 mk u me 1 1 du me 4 sinω = d ω = ω du sinω cosω = cos( θ θ ) = mk me u m k m k du = + ( u) Define as cosω me m k 4 sin Sove this fo u = 1/ ωdω

18 Soution 1 mk E u = = cos( θ θ ) mk This matches the genea equation of a conic 1 ( 1 ecos( θ θ )) = C + e is eccenticity One focus is at the oigin e > 1 e = 1 e < 1 e = E > E = E < E = mk hypeboa paaboa eipse cice Matches the quaitative cassification of the obits

19 Enegy and Eccenticity E = sepaates unbounded and bounded obits Bodeine = Paaboa Cicua obit equies k V ( ) = + = m dv k = = d m 3 mk E = E Hypeboa Paaboa Eipse Cice V () m k

20 Unbound Obits = C( 1+ ecos( θ θ )) 1 θ θ θ e > 1 hypeboa θ is the tuning point (peiheion) cos(θ θ ) > 1/e imits θ e = 1 paaboa θ

21 Bound Obits Ends of the majo axis ae 1 ( 1 ecos( θ θ )) = C + ( ) 1/ = C 1± e Length of the majo axis k e a = + = C(1 + e) C(1 e) E Majo axis is given by the tota enegy E mk C = E = 1+ mk b θ Mino axis is b a 1 e = = me a

22 Rotation Peiod Aea of the obit a = k E We know that the aea veocity is constant da 1 = θ = dt m Expess τ in tems of a b = me Peiod of otation k A= πab= π 8mE da mk τ = A = π dt E 3 3 τ = π m a k 3/ Peiod of otation is popotiona to the 3/ powe of the majo axis Kepe s Thid Law of Panetay Motion

23 Kepe s Thid Law Kepe s thid aw is not exact The eason: educed mass k is given by the gavity Mm k f = G = Peiod of otation becomes µ 1 τ = π a = π a k G( M + m) 3/ 3/ τ = µ = M + m panet Coefficient is same fo a panets ony if M >> m k = GMm π Sun m a k 3/

24 Time Dependence So fa we deat with the shape of the obit: = (θ) We don t have the fu soutions = (t) and θ = θ(t) Why aen t we doing it? It s awfuy compicated Not that bad to get t = t(θ) See Godstein Section 3.8 Inveting to θ = θ(t) impossibe Physicists spent centuies cacuating appoximate soutions Aeady got physicay inteesting featues of the soution Leave it to the computes

25 Summay Studied quaitative behavio of the obits Bounded o unbounded Shape of Deived obit equation fom the eqn of adia motion (o u = 1/) as a function of θ Anayzed the Kepe Pobem Soved the obit Conic depending on E Fo eiptic obit, majo axis depends ony on E Kepe s thid aw of panetay motion V () V() + m 1 mk E = cos( θ θ ) mk a = k E