Astonom 10 Section 1 MWF 1500-1550 134 Astonom Building This Class (Lectue 7): Gavitation Net Class: Theo of Planeta Motion HW # Due Fida! Missed nd planetaium date. (onl 5 left), including tonight Stadial Obseving epot is available Music: Eathbound Dain Dda What is F? Angula Momentum L = mv What about gavit guaanteed that dl/dt =0? Conseved!!!! Outline How things move in a gavit field. Keple's 1 st Law b Newton Newton also found moe options that satisfied his univesal law of gavit Fo an attactive invese squae law foce, can show that the obits ae conic sections with the othe bod at one focus.
Cicula Obit: Special Case mv c F = ma = Eample: Cicula speed at 1 AU fom Sun is 3 10 4 m/s Keple's nd Law b Newton 1 da = ω dt v θ Keple's nd Law b Newton Do obits in Newton s fomalism sweep out equal aea in equal time? d θ = ωdt v θ Keple's nd Law b Newton 1 da = ω dt v = ω da = 1 v dt v θ
Keple's nd Law b Newton v θ Keple's nd Law b Newton The poduct of mass, obital velocit, and obital adius is constant while in obit. Keple's nd Law b Newton Then θ v da dt 1 L m = Constant!! Keple's 3 d Law b Newton Remembe v c = G fo a M cicula motion so GM 4 π 3 a = P
Keple's 3 d Law b Newton Remembe Keple's 3 d Law Newton s found a geneal vesion with his Law of Gavit P = a 3 GM 4 π k P = (P in eas, a in AUs, M in sola masses) k =1 Spaceflight Rockets opeate b Newton's 3d Law Action: Gas epelled out the end of the ocket Reaction: Rocket is pushed upwad 3 a The Powe of Newton s Laws An astonome named Edmund Halle was intigued that a comet appeaed about eve 76 eas Was it the same comet eve time? Using Newton s Laws, Halle detemined the comet s obit aound the Sun Pedicted its etun in 1758 It will be back in 061! Spaceflight Rockets opeate b Newton's 3d Law Action: Gas epelled out the end of the ocket Reaction: Rocket is pushed upwad
Obiting the Eath In obit, spacecaft is falling aound the Eath A spacecaft must have a velocit of 8,000 km/h (17,000 mph) to sta in obit If its velocit eceeds 11 km/sec (5,000 mph), it can escape the Eath s gavit Result Now we know that the obiting planets ae just pepetuall falling bodies. This includes the shuttle, satellites, etc. Weightlessness is just like falling. Thee is gavit on the shuttle, but as one is in feefall it is not noticeable. Keple had thought biefl about this, but he decided he needed foces along the diection of the velocit, not pependicula to it. So Newton ealized that like an apple falling fom a tee o a eall big tee, the moon must have a foce towad the Eath. Newton did not discove gavit, but he ealized that it was univesal. Geosnchonous Obit At a distance of 4,00 km fom the Eath's cente, a satellite takes 4 hous to obit the Eath Equal to the Eath's otation If a satellite is above the Eath's equato, it will appea stationa fom the gound This obit is called geosnchonous Useful fo communication satellites Receives can point at one spot in the sk Applications We now have two tools in ou bo: Newton's laws of motion: how things move with, without foces applied Newton's gavit: the net foce on sola sstem bodies Now, we need to out the two togethe: how do bodies move unde the invese squae foce of gavit.
Kinetic Eneg http://www.nap.edu/html/oneunivese/images/eneg_1_small.jpg http://www.am-technolog.com/pojects/thaad/thaad3.html Potential Eneg of Gavit U = G Mm Potential Eneg of Gavit f U = F i d Potential Eneg of Gavit U = G Mm Negative potential eneg? What s up with that? It eall implies that the object is bound b the gavitational field. In ode to escape the pull of gavit, it has to be given that much eneg. The onl phsicall meaningful quantit is elative changes in potential eneg.
Total Eneg in Gavit TE = KE + PE Conseved!!!! Obits Conside a cicle with e=0 Obits Can classif obits with total eneg. Can wite a cicle o ellipse in pola coodinates as: Obits Conside a cicle with e=0 TE < 0!!
What? Negative total eneg? Obits Is that due to spook antimatte? No it means that ou have to suppl eneg to sstem to beak it apat. When paticles at est, fa awa, KE = PE = 0. To get to this situation, have to input eneg to an obiting sstem. Since eneg is conseved, can't spontaneousl get this eneg, so sstem is bound! Note: KE = -1/PE geneall tue fo gavitating sstems in equilibium: viial theoem" Geneal Obits ellipse: 0 < e < 1 It tuns out that TE depends onl on a, not on e Fom consevation of eneg can see: Geneal Obits ellipse: 0 < e < 1 It tuns out that TE depends onl on a, not on e Fom consevation of eneg can see: < 0!! Bound! Geneal Obits ellipse: 0 < e < 1 It tuns out that TE depends onl on a, not on e Fom consevation of eneg can see: v = GM 1 a This is called the Vis Viva equation. The most hand obital equation. http://goups.msn.com/andsusingsciencefictionfoeducationpage/visvivaequation.msnw
Geneal Obits paabola: e = 1 Can wite as: Whee p is distance at closest appoach. Just unbound sstem (can bael escape to = ) E = 0 o KE = -PE Fo Eath. V escape = 11km/s = 5000 mph. v escape = G M Geneal Obits hpebola: e > 1 Can wite as: ( θ ) = (1 1 + e e ) a cos θ Completel unbound sstem (escapes to = ) E > 0 That implies that v > 0 at =