Equtions fo the illenniu heoy of Ineti nd vity Copyight 004 Joseph A. Rybzyk ollowing is oplete list of ll of the equtions used o deived in the illenniu heoy of Ineti nd vity. o ese of efeene the equtions e gouped into tegoies ost elevnt to thei use. Note: Eh eqution is shown only one, in the ode of its fist use. Definitions Unless othewise shown, ltentely given s is the piy, o gete of two sses. ltentely given s is the seondy, o lesse of two sses. is the distne fo the ente of objet to the ente of ss, o byente. is the distne fo the ente of objet to the ente of ss, o byente. is the su of distnes, nd, o the totl ente-to-ente distne between nd. is the eletion te of ss eltive to the ente of ss. is the eletion te of ss eltive to the ente of ss. is the eltive foe, inetil, o gvittionl, ting between the two sses, nd. is the su of eletion tes nd, o the totl te of one objet eltive to the othe. is the equivlent foe, inetil o gvittionl, equied fo to elete t te,. is the equivlent foe, inetil o gvittionl, equied fo to elete t te,. is the popotionlity onstnt, univesl gvittionl onstnt, whose vlue depends on the units of ss, distne, nd foe used. he pesent vlue fo is, 6.67 x 0 - /kg s, whih is in geeent with the Intentionl Syste of Units SI used in the ssoited wok.. Newton s Lws of ehnis Cente of ss n n 4 Newton s Seond Lw of ehnis * n 5 Newton s Seond Lw Restted to ive Rte of Aeletion * n * Subsipts dded to void onfusion with vibles used in expnded lws of ehnis.. Newton s Lw of vittion Cente of ss Newton s Univesl Lw of vittion vittionl oe Between n Objet nd the Eth e Clssil oul fo vittionl Aeletion * e
6 Lw of vittion Renged to ive Aeletion of Eth s vity e e 7 Lw of vittion Renged to ive Aeletion of ss s vity * 5 Clssil oul fo the Aeletion of n Objet Reltive to Eth * e 6 Newton s Univesl Lw of vittion nd 7 Aeletions Reltive to the Cente of ss Newton s Lw of vittion, Restted 4 fo Newton s Lw of vittion * Appopite nueil subsipt dded to vible,, to void onfusion with expnded lw vible.. illenniu heoy Expnded Lws of ehnis - Reltive Ref. Eq. 8 outh Lw of ehnis Restted - Ref. Eq. 9 o Ref. Eq. 6 o Ref. Eq. 9 nd Ref. Eq. 8
nd Ref. Eq. 4. illenniu heoy Expnded Lw of vittion - Reltive Whee s is the ss of the sun, o piy body, nd p is the ss of the plnet, o seondy body, p s p oul fo vittionl Aeletion Between wo Bodies * vittionl Aeletion Between wo Bodies * illenniu Univesl vittion Lw * 4 Altente o of illenniu vittion Lw * * Appopite nueil subsipt dded to vible,, to void onfusion with Newton s ente of ss vible,. 8 Su of Rtes 9 he Cobined Rte of Aeletion Between sses 0 illenniu Univesl Lw of vittion fo ss illenniu Univesl Lw of vittion fo ss 5 fo the oth Lw of ehnis
5. Convesion Between Newtonin nd illenniu vittionl Lws 4 Reltionship of to,,,, nd giving 5 Reltionship of to,, nd 6 tken fo Eq. 0 7 nd 8 Reltionship of to 9 tken fo Eq. 0 nd Reltionship of to 6 7 oe/ss to Aeletion/ss, Reltionship 8 ss/oe to ss/aeletion, Reltionship 9 oe of vity Between sses nd 40 Reltive Aeletion Between sses nd 4
4 he to, nd 4 to Reipol 4 Reltionship of to,, nd 44 45 46 Reltionship of to,, nd 47 Reltionship of, nd, to, nd 48 Reltionship of, to,, nd 49 Reltionship of, to,, nd 50 nd 5 5 nd 5 Distne,, Reltionships Unless othewise shown, is the peiod of the obit. C o is the seijo xis of n elliptil obit, o dius of iul obit. v is the obitl veloity of piy body of ss,, bout the ente of ss, o byente. v is the obitl veloity of seondy body of ss,, bout the ente of ss, o byente. K is nueil onstnt tht edues to if stonoil units e used fo nd C. t is n intevl of tie duing whih onstnt eletion tkes ple. d is distne tveled duing onstnt eletion. 5
6. Keple s hid Lw of Plnety otion KC 8 Keple s hid Lw of Plnety otion C 9 Keple s hid Lw Siplified C 7. Cente of ss ouls 0 Newton s Deivtion of Keple s hid Lw 0 v IURE Cente of ss Between wo Bodies v v 54 oe to Aelete ss into Ciul Obit v 55 oe to Aelete ss into Ciul Obit π nd v π v 56 Obitl Speeds in es of Peiod nd Distne 57 Distne-to-ss Rtios nd 58 utul oe Between sses nd t d 59 Distne fo Constnt Aeletion d 60 t 6
t nd t nd 6 Aeletion Rtes,, nd t 6 utul oe Between sses nd t 6 64 Su of Cente of ss Distnes 65 66 Aeletion Distne, 67 Altente oul fo Aeletion Distne, Using 68 Diffeene of Distnes,, nd 69 Altente oul fo Aeletion Distne, Using 8. Point of vittionl Neutlity ouls v n n 0 N v IURE 4 Point of vittionl Neutlity Between wo Bodies Unless othewise shown, n is the distne fo the ente of objet to the point of vittionl Neutlity N. n is the distne fo the ente of objet to the point of vittionl Neutlity N. 70 oe of vity Between sses,, nd n 7
7 oe of vity Between sses,, nd n n 7 Squed-Distnes-to-ss Rtios n n 7 Distnes-to-Sque-Root-of-sses Rtios n n n 74 vittionl Neutlity Distne, n n n 75 vittionl Neutlity Distne, n n 76 Su of vittionl Neutlity Distnes n 77 Diffeene of Distnes,, nd n n n n 78 vittionl Neutlity Distne, n 9. Cente of ss Reltionship to Point of vittionl Neutlity n 79 vittionl Neutlity Distne, n 0. Ciul Obits 8
v v ente of iles & ente of ss v v A IURE 5 Ciul Obit of Seondy Body bout Piy Body B 4π 80 oe to Aelete ss into Ciul Obit 4π 8 oe to Aelete ss into Ciul Obit 8 Newton s Univesl Lw of vittion Restted 4π 8 Newtonin Vesion of Keple s hid Lw fo Ciul Obits 4π 84 Popotionlity Constnt I II Units of Sun s ss eth s ss Eth s ss oon s ss Units of Sideel ye Sideel onth Units of Astonoil unit en distne of oon fo eth. Elliptil Obits ble Systes of Units fo Whih 4π / 9
foi v v ente of ellipse ente of ss v v A IURE 6 Septions of oi fo Cente of Ellipse B v v A v v IURE 7 Elliptil Obit of Seondy Body bout Piy Body B jo xis d seiino xis seijo xis ino xis IURE 8 Axes of Elliptil Obits 4π 4π nd 85 & 86 Newtonin oe & Ael. 0
4 π nd 4 π 87 & 88 Newtonin oe & Ael. 4 π 89 Definition of the Popotionlity to, K 4 π 90 Newtonin Vesion of Keple s hid Lw fo Elliptil Obits 4 K π 9 Newton s Popotionlity Constnt, K 4 K π 9 illenniu heoy Popotionlity to, K Sheti Copison of Oiginl nd Expnded Lws of ehnis nd vittion he illenniu heoy s Six Lws of ehnis Cente of ss Newton s Univesl Lw of vittion illenniu heoy Univesl Lw of vittion B A IURE Copison of Univesl Lws of vittion
Newton s hee Oiginl Lws of ehnis. If n objet hs zeo eletion, the veto su of the foes on the objet is zeo. he lw n be expessed thetilly s, Σ 0 whee is foe nd the sybol Σ ens tht the foe oponents, whih e el nubes, e to be sued.. he eletion of ptile is given by the eltion, n Σ o Σ n n n whee n is lled the inetil ss of the ptile nd whee Σ is the veto su of the foes on the ptile.. If one objet exets foe on seond objet, then the seond objet lso exets foe on the fist. hese foes hve equl gnitude nd opposite dietions. Note: In the theoy of eltivity, the eletion of n objet is not the se eltive to diffeent inetil fes. illenniu heoy s hee Expnded Lws of ehnis 4. he eltive eletion between two ptiles, objets, o bodies, is equl to the su of the individul eletions. his lw y be expessed thetilly s, Ref. Eq. 8 whee is the equivlent eltive te of eletion, nd nd e the individul eletion tes fo sses, nd espetively, s given by the seond lw of ehnis. 5. he equivlent foe equied to elete eh of the two ptiles, objets, o bodies to the eltive te of eletion,, is popotionl to the ss of tht ptile, objet, o body nd hs the eltion, o nd o Ref. Eq s. 6 & 9 whee is the foe equied fo ss to elete t te, nd is the foe equied fo ss to elete t te.
6. he eltionship between the eltive foe,, expeiened by the ptile, objet, o body of ss, nd the ente of ss foe,, is given by the fouls, nd Ref. Eq. 8 nd, the eltionship between the eltive foe,, expeiened by the ptile, objet, o body of ss, nd the ente of ss foe,, is given by the fouls, nd Ref. Eq. Note: If you enteed this pge dietly duing seh, you n visit the illenniu Reltivity site by liking on the Hoe link below: Hoe