Orbits and Kepler s Laws


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1 Obits nd Keple s Lws This web pge intoduces some of the bsic ides of obitl dynmics. It stts by descibing the bsic foce due to gvity, then consides the ntue nd shpe of obits. The next section consides how velocity chnge ( V, ponounced deltvee ) is used to initite mnoeuves, nd looks t souces of velocity chnge. Hving estblished these bsic ides it then looks t inteplnety mnoeuves, stting with simple EthMoon tnsfe nd moving on to moe complex inteplnety mnoeuves. By the end of this section edes should be in we of the guys t JPL who do this fo living! Gvittionl Foce Thee is much discussion mong physicists bout the ntue of gvity. Fo the pupose of this website we ll use Newtonin mechnics nd ignoe notions of gvity wves o ny othe cuent theoies. In the Newtonin wold ny two lumps of mtte will exet gvittionl foce on ech othe. Imgine we hve two msses m1 nd m, nd they e distnce pt (msses in kg, distnce in metes). F F m 1 m The ttctive foce between the two msses cn be clculted fom: Gm1m F = whee G is the univesl gvittionl constnt, G = m 3 /kg sec. If one of the msses is fixed nd vey much gete thn the othe, fo exmple plnet nd spcecft, then it is sometimes witten: GMm F = whee M is the mss of the plne n m is the mss of the spcecft. The poduct GM is clled the gvittionl pmete, witten s the Geek lette. The vlue of is diffeent fo ech plnet, so emembe to chnge vlue when woking on inteplnety mnoeuves. Some useful gvittionl pmetes e: Body (m 3 /s ) (km 3 /s ) Sun Eth
2 Moon Ms If spcecft ws sttiony bove the plnet it would simply fll out the sky due to gvity. In ode to sty in obit it needs to move in n obit. Bsic obits Let s debunk one myth stight wy spcecft is not kept in obit by centifugl foce. Let s conside Newton s fist lw: A body will emin t est o unifom motion in stight line unless cted on by foce If spcecft is tvelling in empty spce with its motos off. Thee e no foces on the spcecft so it will tvel in stight line, s pedicted by Newton s fist lw. Imgine tht plnet suddenly ppes beneth the spcecft. The only foce intoduced is gvity, nd the effect of this is to deflect the pth of the spcecft towds the plnet. If the spcecft is tvelling vey fst its pth will bend, but it will escpe the gvittionl foce of the plnet. If it is tvelling vey slowly it will spil into the plnet nd csh. Ove now nge of speeds it will fll towds the plnet, but it will neve lnd. It is in obit. Spcecft No plnet Low velocity (spils into plnet) High velocity (flyby) Optimum velocity (entes obit) One wy of thinking bout this is tht the cuvtue of the plnet is such tht it flls wy t the sme te s gvity is pulling the spcecft towds it. The spcecft is in fee fll ound the plnet. If the spcecft is tvelling with velocity v m/s t distnce of metes fom the cente of the plnet, then cicul obit will occu only when the following eqution is tue:
3 If the spcecft is tvelling slightly fste thn this it will just fil to escpe the gvittionl pull nd emin in n obit which is not cicul but ellipticl. It cn be shown tht, if the velocity is equl to: then it will just escpe fom the plnet s gvittionl pull. The shpe of the obit will be pbolic. If we exceed this velocity the shpe of the obit becomes hypebolic nd the spcecft escpes fom the plnet s gvittionl pull t highe velocity. We cn summise the shpe of obits in the following tble: Spcecft Velocity Obitl shpe Comments v< < v v > Unsustinble Cicul Ellipticl Pbolic Hypebolic Spcecft spils into plnet Closed obit Closed obit Open obit, minimum escpe velocity Open obit, escpe velocity The shpes of obits e eithe cicles, ellipses, pbole o hypebole. These shpes my ppe to be vey diffeent but mthemticlly they ll hve something in common: they e deived fom the cuts of cone. The obitl shpes e thus efeed to s conic sections.
4 Hypebol Pbol Cicle Ellipse Cicle Ellipse Hypebol Pbol Lte on we ll see tht these conic section obit shpes hve mthemticl eltionship to ech othe, but fo now we ll just ccept this s fct. Keples lws Between 1609 nd 1619 Johnnes Keple published his fmous 3 lws of plnety motion, bsed on obsevtions mde by the stonome Tycho Bhe. Keple s lws descibed the obits of bodies to emkble ccucy, nd stted tht ll obits would be ellipticl o cicul. In 1687 Si Isc Newton supplied the theoeticl explntion fo why the obits wee this shpe nd llowed clcultion of the velocities which stellite would need to ech if it ws to sustin n obit. Keple s lws stte: 1. A body obiting ound plnet will descibe n obit tht is n ellipse with the plnet t one of the foci.. If we dw line fom the plnet to the body in obit ound it, the line will sweeps out equl es in equl intevls of time. 3. The sque of the time tken fo body to complete one obit is popotionl to the cube of the mjo xis of the obit. Wht do these men in pctice? Keple s Fist Lw Keple s fist lw descibes the shpe of n obit, bsed on obsevtions of the motions of plnets.
5 Stellite (t some point on obit) b θ (t focus) b (1+e) (1e) The geomety of n ellipse tells us tht it is n eccentic cicle, whee the degee of eccenticity is denoted by the lette e. The vlue of e is defined fom the dimensions of the ellipse nd b, by the eqution: b = (1e ) Fom the geomety of n obit it cn be shown tht, fo stellite t ny point on the obit t distnce fom the plnet, the following eltionship is lwys tue: (1 e ) = 1+ ecos( θ ) Let s think bout these equtions bit. If the eccenticity is zeo, then wht do we get? Substituting e=0 into the fist eqution we find tht =b. Fom the second eqution we find tht =. Both of these cses indicte tht we hve cicle when e=0, so cicul obit is just specil cse of n ellipticl obit whee the eccenticity is zeo. Keple s Second Lw Imgine ou stellite is t point ❶ in its obit ound plnet. A time t seconds lte it hs moved on to point❷. The line between the plnet nd the stellite will sweep out n e A in tht time. Lte on in its obit it moves fom point ❸ to point ❹ in time t, sweeping out n e B.
6 Time = t seconds 4 3 B A Time =t seconds 1 Keple s second lw tells us tht e A will lwys equl e B, egdless of whee we stt to mesue the time t. Keple s Thid Lw The digm below shows n ellipticl obit with mjo xis. It tkes time, which we ll cll T seconds, to complete one obit of the plnet. Instinct sys tht if we mke the obit bigge, in othe wods incese, the stellite will tke longe to complete one obit. Time = T seconds metes 1 Stellite Keple s thid lw tells us how T vies s we chnge : T = π 3
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