Orbital Motion & Gravity


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1 Astonomy: Planetay Motion 1 Obital Motion D. Bill Pezzaglia A. Galileo & Fee Fall Obital Motion & Gavity B. Obits C. Newton s Laws Updated: 013Ma05 D. Einstein A. Galileo & Fee Fall 3 1. Pojectile Motion 4 1. Pojectile Motion Path of a pojectile. Centipetal Acceleation Old view was that pojectile tavelled in an ac until it an out of impetus and then it fell staight down. 3. Galileo & Obits Galileo shows the natual path is a paabola (which is a combination of constant speed in the hoizontal motion with constant acceleation in the vetical) 1b. Simultaneous Fall Simultaneous Fall: Galileo shows a bullet fied hoizontal will hit gound at same time as bullet dopped. 5 1c Simultaneous Fall Both balls fall in the vetical diection at the same acceleation. Thei paths only diffe because of the constant hoizontal velocity 6 (why?) 1
2 Galileo poposed that thowing a ball at diffeent speeds causes it to tavel fathe befoe it falls to Eath. Thow it fast enough, and as it falls the eath s cuve falls undeneath it, and it falls foeve ( fee fall ) The citical speed is called Obital Velocity. Fo Eath, obital velocity is 17,500 miles/h, o 8 km/sec Obital velocity 7 Fig 11, p.54. Centipetal Acceleation Unifom cicula motion: the tangential speed is constant, but the diection of the velocity changes, so thee is acceleation towads the cente. v ac R v R a c R 8 3. Obital Speed Galileo deduces that if the cause of the centipetal acceleation is gavity (centipetal foce) then we can calculate the obital speed 9 1. Histoy B. Obits 10 a c v v R gr g. Keple s Laws 3. Newton s Laws 1a. Claudius Ptolemy Claudius Ptolemaeu ( A.D.). Geocentic Model the eath is at the cente of the univese Planets move on epicycles 11 1b. Nicolaus Copenicus ( AD) Common belief was that the eath was the cente of the univese, and eveything evolved aound us. 1 Copenicus developed the Suncenteed (heliocentic) view of the Univese, which impoved the pedictions of planetay positions.
3 1b. Copenican System Instead of having 5 defeents with 5 epicycles, you only need 5 cicles fo the planets. The only thing that obits the eath is the moon. 13 One of the most impotant books eve Nicolaus Copenicus On the Revolution of Heavenly Sphees (1543) 14 The 1000 Zlotych bill featues Copenicus. Due to inflation, it was woth about 10 cents USD when I was last in Poland 15 a. Tycho 16 Tycho Bahe measuing sta positions (without a telescope) Measuements of position of Mas showed deviations fom Copenican model! He built a big obsevatoy with gigantic potactos (no telescopes yet!) b Tycho Bahe s Uanibog Obsevatoy 17 b.3 Tycho Bahe ( ) 18 He suggested a weid hybid model whee planets go aound sun, but sun goes aound eath 3
4 c. Johannes Keple ( ) Tycho at fist invited Keple to help in analysis of his data, but then jealously wouldn t let him have the infomation. On his deathbed he gave Keple the data. Keple used it (paticula data on Mas), to develop thee laws of planetay motion. 19 c.1 Keple s 1 st Law: Obits ae Ellipses 1605: Keple ealized that the motion of Mas could not be explained with a cicula obit, o the multiple cicles poposed by Ptolemy. He accepted Copenicus view that Mas was in obit aound the Sun, athe than aound the Eath. He expeimented (mathematically) with obits of vaious shapes, and found that Mas obit best fits an ellipse. 0 c.1 Keple s 1 st Law (1605) Law No. 1. Each planet moves aound the Sun in an obit that is an ellipse, with the Sun at one focus. This is contay to the ealie belief that the obits wee pefect cicles o combinations of cicles. 1 Ellipses, cicles (paabolas and hypebolas) ae conic sections, studied fist by the geeks. But it would NEVER occu to the geeks that an obit is an ellipse. (why?) Fig 3, p.45 Dawing an ellipse 3 The Ellipse Do you emembe any of this fom high school geomety? 4 Focus Focus Fig 4, p.45 4
5 5 6 Planet obits tend to have low eccenticity (nealy cicula). Highly eccentic Comet obits tend to be highly eccentic. Focus Focus Not vey eccentic Fig 10, p.53 7 c. Keple s nd Law (1609) 8 Keple also noticed that when Mas is closest to the Sun in its elliptical obit, it moves faste than when it is fathe away. This led him to fomulate his Second Law of Planetay Motion. c. Keple s nd Law (Equal aeas in Equal Times) 9 c. Keple s nd Law Accoding to his second law, a planet moves fastest when closest to the Sun (at peihelion) and slowest when fathest fom the Sun (at aphelion). As the planet moves, an imaginay line joining the planet and the Sun sweeps out equal amounts of aea (shown as coloed wedges in the animation) in equal intevals of time. Keple s nd law is actually a fom of consevation of angula momentum A t v mv m 1 1 L m 5
6 c.3 Keple s 3 d Law: Hamonic Law 31 c.3 Keple s 3 d Law (1618) 3 Planets close to the sun move faste. This is consistent with his nd law, that showed a planet will move faste at peihelion. He seached fo a elationship between obital peiod and distance to the sun. The squae of the obital peiod (P) is diectly popotional to the cube of the semimajo axis of the obit (a). P = a 3 This law explains the popotions of the sizes of the obits of the planets and the time that it takes them to make one complete cicuit aound the Sun. [Note: in physics, the symbol a is also used to epesent acceleation. Confused?] Why is it called the hamonic law? Keple thought the spacing between planets was elated to musical intevals. 33 An example of Keple s thid law: The obit of Mas (Recall: P = a 3 ) Mas obit peiod (P) is 1.88 yeas. P = Keple s law says that P = a 3, so 3.53 = a 3. So then a = (3.53) 1/3 (the cube oot of 3.53), o 1.5. Thus, the semimajo axis (aveage distance of Mas fom the Sun) is 1.5 Astonomical Units. But how big is an Astonomical Unit? Keple didn t know. c. The distances of the planets fom the Sun In the Copenican wold view, the planets ae in obit aound the Sun. Astonomes knew the elative distances of the planets, but not the absolute distances. Known: Jupite is 5 times fathe fom the Sun than the Eath is. It takes Jupite 1 times longe to go aound the Sun than it does fo the Eath. Not known: How many kilometes (o miles) ae the Eath and Jupite fom the Sun? Fundamental Question: What is the absolute scale of the Sola System? 35 How lage is the Astonomical Unit? Astonomical Unit (AU) The aveage distance fom the Eath to the Sun 150,000,000 kilometes, o 93,000,000 miles But how was this measued? 36 6
7 The NewtonKeple Law In the Pincipia Newton also deduced Keple's thid law, but in an impotant new fom 37 In SI units With Newton s law of gavity, his nd law and Galileo s centipetal acceleation, we can deive: 38 Mass of cental body: M = a 3 /P Obital Radius a (in astonomical units) Peiod P (in yeas) Mass M in units of sola masses To measue mass of Eath, use moon s obit Jupite, use Galilean moons Sun, use obits of planets Galaxy, use obits of stas aound galaxy v mm m G R R R v P v R 4 M G G R P 3 C. Law of Gavity 39 1a. Invese Squae Law Invese Squae Law. Newton s 4 th law 3. Acceleation of Gavity 1b. Invese Squae Law 41 Appaent Luminosity dops off invesely popotional to squaed distance. Sun at planet Satun (10 futhe away than eath) would appea 1/100 as bight. Sound behaves the same way So do electic and magnetic foces 1c Gavity obeys invese squae law 4 Acceleation of gavity is invesely popotional to distance (fom cente of eath) Example: At the suface of the eath (one eath adii distance) the acceleation of gavity is nealy g=10 m/s The moon is 60 futhe away Acceleation of moon towads eath is hence 60 smalle (about a=0.003 m/s ). 1 g g g a
8 . Gavity: Newton s 4 th Law (a) The apple tee stoy "Afte dinne, the weathe being wam, we went into the gaden and dank tea, unde the shade of some apple tees," wote Stukeley, in the papes published in 175 and peviously available only to academics. "He told me, he was just in the same situation, as when fomely, the notion of gavitation came into his mind. It was occasion'd by the fall of an apple, as he sat in contemplative mood. Why should that apple always descend pependiculaly to the gound, thought he to himself." 43 (b) The Law of Gavitation The mutual foce between two bodies is popotional to thei masses, and invesely popotional to squae of distance. Newton could not detemine the Gavitation Constant G 44 (c) Cavendish Expeiment: The Acceleation of Gavity (a) Galileo s Law of Falling Bodies 46 Ove 100 yeas late Cavendish measues the constant: G= Nm /kg Vey Small! To have 1 N of foce would need 10 kg masses 1 cm apat! Combining Newton s nd and 4 th laws, we see that the mass of the test body cancels out! Hence we deive Galileo s law that all test bodies fall at the same acceleation g, independent of thei mass m GmM ma F GM a g 3b. Measue Mass of Eath 47 3c. Escape Speed 48 Hence if we measue g, and know the adius of the eath (measued by ancient geeks), we can detemine the mass of the eath! GM g The gavitational potential enegy is the amount of wok we would have to do to lift a mass m fom suface of eath to infinity. Equivalently, it s the amount of Kinetic Enegy an meteooid would have if it fell to the eath. U GmM g M G m m s m kgs kg Note mass m cancels out (all bodies fall at same ate!). Hence, thee is a minimum escape speed such that a body will not fall back to eath! [about 11 km/sec o 5,000 miles pe hou] 1 mv GmM GM v g 8
9 D. Gavity Field Action at a Distance 50 Action at a Distance (no touching) 1. Action at a Distance. Gavitational Field 3. Black Holes etc. Huygens citicized: How can one believe that two distant masses attact one anothe when thee is nothing between them? Nothing in Newton's theoy explains how one mass can possible even know the othe mass is thee. actio in distans (action at a distance), no mechanism poposed to tansmit gavity Newton himself wites: "...that one body may act upon anothe at a distance though a vacuum without the mediation of anything else, by and though which thei action and foce may be conveyed fom one to anothe, is to me so geat an absudity that, I believe no man, who has in philosophic mattes a competent faculty of thinking, could eve fall into it.". The field concept 51 b. Definition of Mass Faaday poposes ideas of Lines of Foce Example: ion filings ove a magnetic show field lines Thee ae 3 ways to think about mass 1. Inetial Mass F=ma. Passive Gavitational Mass F=mg Gavitational Analogy: Eath s mass M ceates a gavity field g Foce of field on mass m is: F=mg (i.e. weight ) This eliminates action at a distance Michael Faaday Active Gavitational Mass GM g The Weak Equivalence pinciple says that inetial mass equals passive gavitational mass 3a. The Equivalence Pinciple Refeence at est with Gavity is indistinguishable to a efeence fame which is acceleating upwad in gavity fee envionment. 53 3b. Bending of Stalight Newton: Light is NOT affected by gavity Einstein: Elevato example shows light must be affected by gavity. Pedicts stalight will be bent aound sun! 1919 Measued by Eddington! 54 The apple acceleating downwad due to gavity looks the same as an apple at est in space, with the floo acceleating upwad towads it. 9
10 3c. Black Hole If the mass of a sta is vey big and its size shinks vey small the escape speed becomes bigge than the speed of light, and not even light can escape! Any mass is compessed into a size smalle than the Schwazschild Radius R s, it will become a black hole This can happen duing a supenova explosion, o late by additional mass falling on a neuton sta. Anything that comes close than the Schwazschild Radius, will fall in and neve escape. 55 3d. Obseving a Black Hole 56 If black how do we see them? Mateial shed fom anothe sta falls towads black hole. Not all the mateial falls into the hole. Some is ejected at vey high enegies out jets along the axis of the black hole. 3e. Radio Lobes fom galaxy Centauus A 57 Black holes at the cente of galaxies have a mass of ove a billion stas combined! 10
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