PERIHELION ADVANCE OF MERCURY

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Aubrey Perry
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1 PERIHELION ADVANCE OF MERCURY The peihelion of Mecuy, that is, the point in its obit closest to the sun, is obseed to adance 5600 seconds of ac pe centuy. The Eath s pecession, that is, the changing oientation of the Eath s axis, and the Newtonian gaitational inteaction with othe planets accounts fo all but 43 seconds of this adance (Stom 987). Classical Newtonian mechanics pedicts all but 43 seconds of the total obseed adance of 5600 seconds of ac pe centuy. The missing 43 seconds was a mystey befoe Einstein. Fo a planet of mass m taeling at speed in a cicula obit of adius, Einstein theoy gies 6 = as a small coection to the gaitational foce F. F = m + 6 = m (.) This coection accounts fo the unexplained obseed adance of the peihelion of Mecuy of 43 seconds of ac pe centuy (Stom 987). This is an execise poided by Fench (97). We late show that the coection facto 6 = may be obtained by a method othe than Einstein s equations.

2 8. PERIHELION ADVANCE OF MERCURY. Calculation of the Peiod of Reolution of Mecuy when the E ectie Value of G is G + 6 = m m Fig.. The peiod of the unpetubed (Newtonian) obit is obtained fom m = m = ; = ; = (.) Time fo one obit is the peiod T 0 T 0 = = () 3 T 0 = p 3= (.3) (.4) To calculate the peiod, T; gien by the foce law, Eq. (.), eplace G in Eq. (.3) by G + 6 Fom Eq. (.) and Eq. (.4) 3= p p + 6 = = T 0 3 = (.5) c = = 4 3 T0 = 4 ; = () 3 T0 T0 (.6)

3 . Calculation of the Peiod of Reolution of Mecuy 9 Put Eq. (.6) fo = in Eq. (.5) to obtain T 0 T 0 (.7) Calculation of Mecuy adance fom Eq. (.7). Without petubation the angula elocity is:! 0 = T 0 With petubation the angula elocity is:! = 4 = = T 0 Angula distance taeled in time T 0 is: T 0 ( 4) = ( + 4) T 0 =!T 0 = ( + 4) T 0 T 0 = ( + 4) Theefoe, fo each eolution of the planet, it will tael an angle geate by about 4 = = 43 T 0 = 43 4 = 6 3 adians pe eolution fathe than unde the pue Newtonian foce. In one centuy 6 N = numbe of eolutions pe centuy. Time fo one eolution = 88 days. N N = centuy = 6 adian = 360 centuy days in centuy 88 days = 43 seconds = adius centuy = 6 3: degees = seconds of ac (36) 0 3 sec of ac centuy

4 0. PERIHELION ADVANCE OF MERCURY. Relatiistic E ect on the Peiod Gien the following non-elatiistic elations, calculate the peiod of a paticle in a gaitational eld and then coect it fo elatiistic elocity. m = m = = (.8) The non-elatiistic time fo one obit is the peiod gien by T 0 = = p 3= (.9) Now ealuate the peiod including the elatiistic coection to, namely! = p =. Then the elatiistic peiod is p = = T0 (.0) Conside a peiod gien by 3= p p + =c = 3= p MG ( + = ) = T 0 (.) that is, change G to G ( + = ). Eq.(.) agees with Eq.(.0), which shows that changing G by + = gies the elatiistically calculated peiod. Theefoe the elatiistic coection to poides one = of the equied 6 = coection to G to gie the coect peihelion adance. Theefoe, 5 = additional coections to G ae needed to agee with the expeimental alue fo the peihelion adance. To adapt the peceding esults to gaity, eplace qt by [i (ce 0 + )] and take the scala pat. No contaction is inoled. [i (ce 0 + )] = i + t = c t = t = Fo futhe details, see Chaptes 6 and 8.

5 .3 Obital Peiod in a Schwazschild metic! m = m = T 0 =.3 Obital Peiod in a Schwazschild metic p =c! q = ' q = ' = T0 = ' c ' = ( a) c a = ' = ( a) = 4a = ( + 4a) c T 0 = T 0 = T 0 ( + 4a) ( 4a) c + 4 = T 0 c + 34 c 4 The thid tem is a negligible coection to the elatiistic coection. Theefoe we can ignoe the Schwazschild metic elatiistic coection to the elatiistic coection in at space.

6 . PERIHELION ADVANCE OF MERCURY.4 Summay of the Souces of = to Obtain a Total Value of 6 = Seconds of Ac pe Centuy fo the Adance of the Peihelion of Mecuy These additional coections ae as follows:. A special elatiity coection contibutes to G: c the equiement of 6 to G to a equiement of 5. In othe wods, it educes. A non-commutatie coection is. It includes a toque density contibution. c Eqs. (8.5), (8.85), (8.86). This is a new foce, item below. 3. A classical coection following Page and Adams (945) is 3. Fo the linea c momentum coection potion of this tem see Eq. (0.6). Fo the associated toque coection, see Eq. (0.38). 4. A Schwazschild metic coection is. Eq. (7.4). c 5. The 0 (E t B t ) coection is : Section. c The total of all coections to G is: Non-commutatie Page & Adams (945) Schwazschild Contibution of 0 (E t B t ), (classical) (cued space) Total coection : 5 The gaitational eld descibed in the following wok follows fom electomagnetism and has a simila stuctue. Theefoe the esulting gaitational eld may be quantized as it is fo the electomagnetic eld.

Determining solar characteristics using planetary data

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