Equations of 2-body motion

Transcription

1 Equation of -body motion The fundamental eqn. of claical atodynamic i Newton Law of Univeal Gavitation: F g = Gm i i i ˆ i (1) We ae inteeted in atellite in obit about ingle planet, o (1) educe to the -body fom: F = G m ˆ () whee F i the gavitational foce on the atellite and i it poition vecto elative to the cente of the planet. We can ue Newton nd Law to get id of m : m F = m & = G & + G = 0 3 ˆ (3) Thi eqn can be olved, leading to the eqn of the atellite obit:

2 = a(1 - e ) 1 + e coθ and to Keple nd Law: ( & θ ) 0 d = dt (4) (5) Eqn (4) ay the atellite obit i a conic ection with the planet at one focu; we ll ee that the type of conic ection depend on the eccenticity, e, of the obit. An impotant conequence of thi i that the obit, planet and atellite-cente ae confined to a plane (the obital plane ) fo -body motion. The quantity ( θ & ) i the angula momentum, o (5) i a tatement of the conevation of angula momentum. Enegy- & θ = contant h (6) Gavity i a conevative field, theefoe the total enegy of the atellite:

3 KE o: E + PE = 1 v G m v G m 1 = contant (7) Notice that the PE pat of the enegy, a defined hee, i zeo fo at infinity, and negative fo anything le. Alo, E i eally the enegy pe unit ma. Thu: - only object that each with ome exce velocity (ie. that ecape the planet) have poitive value fo E. - object that can baely each (ie. whoe v 0 a ) have E=0. - object in bound (ie. cloed) obit have E < 0. Now we e eady to look at obit hape. Obit and enegy- The the negative-enegy, cloed obit obviouly have to be ellipe. Thee coepond to value of

4 e between 0 and 1. The cae of e=0 educe eqn (4) to = a (a being the emi-majo axi); the pecial cae of a cicula obit. The othe pecial cae, e=1 coepond to an ellipe with an infinite emi-majo axi (ty it and ee) a.k.a., a paabola. The paabola epeent a citical cae; e>0 obit coepond to hypebolic obit, which in pacecaft tem ae ecape obit. Cicula obit- Notice that, accoding to (5), both the adiu (thu altitude) and the angula velocity of a atellite in an elliptical obit vay with time; atellite move lowe the moe ditant they ae fom thei pimay and (5) goven how. Thi i vey bad fo imaging!

5 The ed dot ae poition of a atellite obitting the oigin, at fixed time inteval. The fact that the yellow tiangle all have equal aea i a conequence of (5). BUT, (5) alo ay that if i contant, o i the angula velocity. Thi i a big eaon why cicula obit ae pefeed fo almot all planetay emote ening application. Phyic 101 fact: The acceleation of an object moving in a cicle at contant angula velocity i: & v = cic ˆ But accoding to (3): & = G ˆ

6 thu: v cic = G (8) Thi equation give the peed of any object in cicula obit at adiu about a pimay. Hee ae ome example: (1) EOS-1 Tea i in low cicula obit (LEO) at an altitude of ~700 km o ~ 7080 km. It peed elative to Eath cente at thi altitude i 7.5 km/. The cicumfeence of thi obit i 44,500 km, o Tea peiod (T) i 44,500/7.5 = 5933 (1h 39min). () Geoynchonou atellite obit Eath in a cicle once each day. Solving: = geo geo π v T and v geo = G imultaneouly, and plugging in geo T = 4h one find that Eath GEO i at an altitude of 36,000 km ( geo = 4,400 km) and v geo = 3.1 km/ (notice how much malle thi i than the velocity of Tea, in LEO). When placed ove the equato, geoynch ae tationay ove one point on Eath uface, o they ae geotationay. Thee ae ton of emote ening (e.g. weathe) and comm at in thee obit. Obital element- In addition to an oigin, 6 numbe ae needed to pecify an obit. In the Cateian cae, we would need 3 poition component and 3 velocity component. An altenative, and much moe intuitive et ae called the obital element. We aleady have of thee: Semi-majo axi and eccenticity decibe the hape of the elliptical obit. But we till need to pecify it' oientation in pace:

7 "N" = node, "P" = peiapi, "S" = pimay body. Thee angle take cae of thi: the "longitude of the acending node" (Ω), the "inclination" (i) and the "agument of peiapi" (ω). Note that: - pefectly cicula obit would have no apide, theefoe ω would be undefined; howeve eal obit ae neve pefect...

8 - fom example () above that planeto-tationay atellite mut have inclination of zeo (theefoe Ω and ω ae undefined). - the highet latitude a atellite eache i equal to it' inclination; fo example, pola obitting atellite... (3) The vat majoity of emote ening atellite ae put in nea-pola LEO, fo a numbe of eaon: a. Highly inclined atellite can cove the actic/antactic. b. They can obeve mid- and low-latitude eveal time a day, with coveage of mot aea evey -10 day, depending on wathwidth, latitude, obit detail... One day' woth of a pola LEO atellite' coveage. c. They can be obitted o that they pa ove the day ide of the planet at the ame time evey day... A high- i atellite obitting a pheical planet.... and one obitting an oblate planet. a LEO i looely defined a an obit with apogee le than ~000 km in altitude.

9 Thi i becaue eal planet ae oblate. Thi hape caue the obit plane of inclined atellite (including planet!) to pece: J R E Ω = 3π co( i) adian pe obit. a ( 1 e ) Fo Eath, J = The obit chaacteitic a and i can be "tuned" (e i eentially zeo) o that the peceion ate i 1/365th of a degee pe Eathday. Thi why ton of Eath atellite (e.g. Tea, Aqua) have altitude ~700 km and i = ~98 o. (4) a' oblatene i ~ i geate than Eath' and it' adiu ~3390 km. To put a Global Suveyo in a "pm", un-ynchonou LO, GS' aeocentic obit i inclined at 9.9 o to a' equato, at an altitude of ~380 km. (5) The Hubble Space Telecope obit at alt. = 560 km, i = 8.5 o ; it' obit pecee about Eath at a highe ate, and in the oppoite diection than thoe of Tea and Aqua. The lat obital element i a time index; afte all, we alo need to know whee in it' obit the atellite i. A common one i the "time of peiapi paage". Launching atellite and pace pobe- -Launch ite nea the equato ae pefeed, becaue you can launch into mot pogade inclination of obit fom thee. -You alo get a boot of ~05. km/ of " v" fom the Eath' pin -Retogade obit (i > 90 o )alway equie an outof-plane maneuve which i vey expenive, vwie.

10 Inteplanetay tanfe- Pobe to othe planet obviouly have to get fom Eath to thei detination. Thi poce can be boken down into 4 tep: Eath ecape, tanfe (a.k.a. cuie phae ), planetay endezvou, and obit inetion o landing. Each of thee tep equie the expenditue of v. To leave Eath: E = 0 = 1 v ec G eath v = ec G eath = ~11. km/. Thi tep only get u clea of Eath and into a helio-centic obit eentially the ame a Eath, with a heliocentic velocity: v = o G R un eath-un = 9.8 km/. The next tep i getting into a 'tanfe' obit. Obviouly, the tanfe obit ha to inteect both the pacecaft' oiginal obit and the obit of the planetay taget. It alo ha to be planned o that when the pacecaft aive at it' taget' obit, the planet i, in fact, thee.

11 The mot efficient tanfe obit i called the Hohmann Tanfe: Hohmann tanfe (dahed) to upeio planet (left) and infeio planet (ight). (6) A lage numbe of pobe have been ent to a; afte Eath-ecape mot ae put on baically Hohmann tajectoie to a. To poceed we need thi Peliminay Fact: Combining eqn (4), (6), (7), it can be hown that E G a = (9) which elate the enegy of an obit to it ize. Applying eqn (9) and (7) to the peihelion of an Eath-a Hohmann obit: E = G eath-un un + ma-un = 1 v 1 G un eath-un Thi can be olved fo v 1 which tun out to be about 3.7 km/. Thu the pacecaft mut be given a v of ~3 km/. Exactly the ame method can be ued to find the peed of a in it obit (4.1 km/), and the peed of the pacecaft when it eache a (1.5 km/). Thi mean that, when the pacecaft eache a' obit, a i actually 'catching up to' it. Thu to endezvou with a it need about.6 km/ moe peed, fo a total of ~5.6 km/ jut fo the tanfe (not including the >11. km/ equied to ecape Eath). (7) To ente matian obit about a pobe mut loe moe peed till, othewie it will imply decibe a hypebolic path aound a and "e-

12 ecape". To actually land on a it ha to loe the equivalent of a' ecape velocity of ~5 km/. Theefoe, it take a total of at leat 1.8 km/ of v to land a pobe on a. Not all of thi ha to be via ocket buning, howeve. Note that thee ARE othe planet in the ola ytem... Alo, notice that obit inetion o landing actually equie de-celeation, elative to a... Gavity ait- -Caini, eenge Gavity aited tajectoie of Caini (left) and eenge (ight). Baking into obit o landing- -Reto-ocket -Aeobaking: agellan, a Global Suveyo -Paachute: Viking, a Rove, Huygen -Impact: Pathfinde, a Exploation Rove