On the influence of the proper rotation of a central body on the motion of the planets and the moon, according to Einstein s theory of gravitation.

Transcription

1 Übe die Einfluß de Eigenotation de Zentalköpe auf die Bewegung de Planeten und Monde nach de Einsteinschen Gavitationstheoie Zeit. Phys. 9 (98) On the influence of the pope otation of a cental body on the motion of the planets and the moon accoding to Einstein s theoy of gavitation. By J. Lense and H. Thiing Tanslated by D. H. Delphenich In a pape that appeaed ecently ( ) one us computed the field that is found inside of a otating hollow sphee appoximately in Einstein s theoy of gavitation. That example seemed to be inteesting pimaily in the context of answeing the question of whethe the otation of distant masses actually poduces a gavitational field that is equivalent to a centifugal field in Einstein s theoy of gavitation. In anothe espect it is also inteesting to pefom the same easily-pefomed integation of the field equations fo a otating solid sphee. As long as one stands on the basis of Newton s theoy one can eplace the field in the space that is outside of a sphee of constant mass density (which is at est o otating) with the field of a mateial point of equal mass pecisely. Moeove in Einstein s theoy the field of a sphee at est is equivalent to that of a mass point ( ) as an incompessible fluid but that will no longe be tue fo otating sphees. As we will show in what follows additional tems will then appea that will coespond to the centifugal and Coiolis foces. Now since the planets move in the field of a Sun that otates aound itself and the moon moves in the field of a planet that otates aound itself it does not seem out of the question at the outset to obtain a new astonomical confimation of Einstein s theoy by obseving the petubations that the additional tems yield. The numeical computations that ae pefomed in what follows will poduce petubations of the obital elements of the planets that lie beyond the limits of obsevability. Howeve they will yield elatively lage petubations fo the moons of Jupite that might in fact lie within the limits of measuement. Notations:. The computation of g µν fo the field of a otating solid sphee. l M ω the adius of a sphee its mass its angula velocity ( ) Hans Thiing this Zeit. 0 (98) ; efeed to as loc. cit. in what follows. ( ) K. Schwazschild Bel. Be. (96) 44.

2 Lense and Thiing On the influence of the pope otation of cental bodies x y z the ectangula coodinates of a point in the integation space x y z the coodinates of the oigin k the gavitational constant the natually-measued spatial density of matte ρ 0 The computation poceeds in a manne that is completely analogous to what was done in the pape that was cited in the intoduction: Einstein s appoximate method of integation ( ) is used except that this time in the constuction of the enegy tenso fo matte the velocity of the mass that ceates the field is egaded as small enough in compaison to (viz. the speed of light) that one can neglect the squaes and poducts of the velocity components. (As a esult the diffeence between the pesent teatment and the example that was teated in the pevious pape is that centifugal foce tems which ae popotional to ω will go away and the Coiolis tems will emain.) In hindsight neglecting these tems is justified completely since lω will be vey small fo the Sun and all of the planets when it is measued in any system of measuement fo which the speed of light is. Fo that eason we will conside the field at a geat distance fom the bounday suface of the ball in the case that will be teated hee. If stands fo the distance fom the oigin to the cente of the sphee stands fo the distance fom the cente to the integation element and R stands fo the distance fom the oigin to the integation element then we will develop / R into a seies in / which we will tuncate with the quadatic tems. We shall now go on to the appoximate solution that was given by Einstein exactly as we did in loc. cit. ( ): if µ = ν g µν = δ µν + γ µν δ µν = 0 if µ ν () γ µν δ µν µν αα α k T ( x y z t R) dv π. R µν µν = 0 We then constuct the enegy tenso fo stess-fee matte: () T µν = T µν dxµ dxν = ρ 0 = ds ds dx dx dx ρ0 dx dx µ ν ds with the following expessions fo the velocity components: dx dx 4 = i dx dt = i ω sin ϑ sin ϕ ( ) A. Einstein Bel. Be. (96) 688. ( ) The facto δ µν was obviously omitted fom the coesponding eq. () in loc. cit..

3 Lense and Thiing On the influence of the pope otation of cental bodies dx () dx4 dx dx 4 = i dy dt = 0 = i ω sin ϑ sin ϕ ( ϑ ϕ ae the pola coodinates of a point of the ball; otation takes place aound the Z-axis) and upon neglecting the tems in ω we will get: (4) T µν = ρ i ω sinϑ sinϕ dx i ω sinϑ cosϕ 4. ds i ω sinϑ sinϕ i ω sinϑ cosϕ 0 Accoding to equations (7) and (8) of loc. cit. we will have to set: dx4 (5) dv 0 = i ds d sin ϑ dϑ dϕ. In ode to expess / R in tems of the integation vaiables we choose the coodinate system in such a way that its oigin lies in the ZX-plane. With the intoduction of pola coodinates we will then have: and we will get: x = sin ϑ y = 0 z = cos ϑ R = ( sin ϑ cos ϑ sin ϑ) + ( sin ϑ sin ϑ + ( cos ϑ cos ϑ cos ϑ) = (sinϑ cosϕ sinϑ cosϑ cos ϑ) + +. We develop this into a binomial seies and tuncate it afte the second tem: (6) R = + (sinϑ cosϕ sinϑ + cosϑ cos ϑ) + (sinϑ cosϑ sinϑ + cosϑ cos ϑ). We futhe denote the expession in the culy backets by K and wite: (6a) R = K.

4 Lense and Thiing On the influence of the pope otation of cental bodies 4 If we now intoduce (4) (5) and (6a) into the last of equations () then we will get: 44 = iκ ρ π l π π 0 dx4 d dϕ dϑ sinϑ K ds (7) 4 = 4 = κ ρ π κ ρ π l π π 0 dx4 ω d dϕ dϑ sinϑ sinϕ K ds l π π 0 dx4 ω d dϕ dϑ sinϑ cosϕ K ds 4 = 0. By neglecting the tems in ω and assuming the fist viewpoint the appoximation will yield: dx 4 ds = i [cf. eq. () loc. cit.]. dx If one intoduces this value fo 4 as well as the expessions fo K fom (6) and ds (6a) into (7) then upon evaluating the integals one will get: κ π 44 = M (8) 4 = 0 4 = i κ M ω l sin ϑ π 5 4 = 0. Accoding to () when one once moe intoduces ectangula coodinates and uses the Newtonian gavitational constant k = κ / 8π in place of Einstein s it will then follow fom this that: (9) g = g = g = km g 44 = + km

5 Lense and Thiing On the influence of the pope otation of cental bodies 5 4κ M lκ g 4 = i ω l 5 g = g = g = g 4 = g 4 = 0. If one now makes the special choice of coodinate system in which the oigin falls in the ZX-plane by means of a otation of the system then one will ultimately get the following coefficient matix: (0) g µν = km 4kM ly 0 0 i ωl 5 km 4kM lx 0 0 i ωl 5 km kM ly 4kM lx km i ωl i ωl The equations of motion of a mass-point in the field of a otating solid sphee. In what follows the equations of motion of a mass-point in the field of a otating solid sphee will be pesented in which we will assume that its speed is so small that we can neglect the squaes and poducts of its velocity components in compaison to. Thus it shall be emphasized fom the outset that all we have to do hee is to find the petubational tems to the planetay motion that oiginate in the otation of the cental body. In ode to obtain a sufficiently exact solution to the planetay poblem in the sense of Einstein s theoy one must add the tems that lead to the known motion of the peihelion to the petubing tems that wee computed ( ). Howeve if the tems that oiginate in the pope otation of the cental body aleady come fom the fist appoximation of Einstein s theoy then since the afoementioned petubation of the peihelion was fist obtained fom the second appoximation it would cetainly not be uneasonable to conside the fome and neglect the latte. The eason that one cannot do that comes fom the following consideation: Any additional tems that make the futhedeveloped foce expession diffe fom the Newtonian one will be popotional to ω l v whee v epesents the velocity of the planet (moon esp.) while ω l epesents the velocity of a point on the equato of a cental body. Now fo the Sun-planet system as well as fo the planet-moon systems that come unde consideation we will have the inequality: () v > ω l. Thus when we include the tems in ω l v in ou calculation we must also popely conside any tems in the equations of motion that involve the squaes and poducts of the ( ) A. Einstein Bel. Be. (9) pp. 8.

6 Lense and Thiing On the influence of the pope otation of cental bodies 6 velocity components of the mass-points. Howeve if we do that then we can no longe compute in the fist appoximation alone since any tems that combine with the Newtonian tems in the second appoximation will compae to them like α / : (α = km). The squae of the velocity of a planet likewise has an ode of magnitude of α /. The consideation of the quadatic tems in velocity will thus logically imply that one must conside the tems that aise fom the second appoximation. It will then follow that the calculations that wee employed hee will have no meaning in and of themselves due to the validity of the inequality (). Howeve we can use them in pactice if we ealize that all of the petubations that come unde consideation hee ae small enough that one can egad them as linea with espect to each othe. One will then aive at the desied esult of an obital calculation that includes all elativistic effects when one stats with Einstein s calculations in the equations of motion that he gave fo the pecession of the peihelion of Mecuy as a basis and adds the petubing tems that ae computed in what follows to them. As was shown in loc. cit. by the use of the afoementioned omission of tems and the coodinates x = x x = y x = z x 4 = it the geneal equations of motion: will go ove into: () d x d x τ = ds τ µν dx µ ds dx ds τ dx dx dx = i ds + + dt dt dt τ τ τ τ ν. Fom the initial viewpoint of the stationay field appoximation the 6 quantities τ σ 4 that appea hee will ead like: 4 = 0 g4 g 4 4 = x x g4 g 4 4 = x x g44 44 = x g 4 g 4 4 = x x 4 = 0 g 4 g 4 4 = x x g44 44 = x g4 g 4 4 = x x g4 g 4 4 = x x 4 = 0 g44 44 = x 4 g44 4 = x 4 g44 4 = x 4 g44 4 = x 4 44 = 0. Fo ou field which is given by equation (0) this table will go to: km ω l x + y + z 0 i 5 6kM ωl yz i 5 km x

7 Lense and Thiing On the influence of the pope otation of cental bodies 7 km ω l x + y + z + i 5 (4) 6kM ωl xz 0 + i 5 km y 6kM ωl yz i 5 6kM ωl xz i 5 km z 0 km x km y km z 0 τ If we substitute these values fo the σ 4 in () then we will get the desied equations of motion: km xɺɺ = ω l 4 x + y + z yz y z 5 ɺ + 5 ɺ km x km (5) yɺɺ = ω l 4 x + y + z z x z 5 ɺ + 5 ɺ km y zɺɺ = ɺ ɺ km ωl z xy yx 5 km z. The last tems on the ight-hand side epesent the Newtonian foce; as explained above one must eplace them with the foce components that follow fom Einstein s wok with Mecuy. The fist tem on the ight-hand side is the petubing tem that is of inteest to us since it aises fom the pope otation of the cental body.. The computation of the afoementioned petubations that ae due to the pope otation of the cental body. The petubing tems that appea in equations (5) ae seen to be the components of the petubing foce that oiginates in the pope otation of the cental body. We decompose them into thee othe mutually-othogonal components S T W whee S can be the adial component T the tansvesal and W the othogonal one (i.e. the one that is nomal to the planetay obital plane) and intoduce the following customay astonomical nomenclatue: a e p = a( e ) i = yωπ Ω = XOΩ semi-majo axis eccenticity semi-paamete inclination longitude of the ascending node

8 Lense and Thiing On the influence of the pope otation of cental bodies 8 ϖ = boken XOΠ L 0 v = ΠOP u = WOP = v + ϖ U n = π km = U a longitude of peiapsis mean longitude of the epoch = mean longitude of the planet o satellite at the time t = 0 (likewise a boken angle that is measued fom the X-axis) tue anomaly agument of latitude peiod of the planet o satellite in days mean daily motion C = v = na e twice the aeal velocity Futhemoe in ode to abbeviate we will set the constant K that appeas in equations (5) equal to 4kMω l / 5. z x O Ω Π y P Π and P mean the positions of the peiapsides of the planets and satellites when pojected fom the cente O of the cental body onto the sphee. We now have: x = (cos u cos Ω sin u sin Ω cos i) y = (cos u sin Ω + sin u cos Ω cos i) z = sin u cos i P = + e cos v xyɺ yxɺ = C cos i

9 Lense and Thiing On the influence of the pope otation of cental bodies 9 S = X(cos u cos Ω sin u sin Ω cos i) + Y(cos u sin Ω + sin u cos Ω cos i) + Zsin u sin i T = X(sin u cos Ω + cos u sin Ω cos i) + Y(sin u sin Ω cos u cos Ω cos i) + Zcos u sin i W = X sin Ω sin i Y cos Ω sin i + Z cos i. If one insets the values of X Y Z that ae povided by equations (5) into these fomulas fo S T W then by using the given elations and notations one will obtain afte some lengthy calculations: KC cosi S = 4 Kɺ cosi (6) T = 4 KCecos isin v = p K sin i KC sin i esin v cosu W = 4 ( C sin u + ɺ cos u ) = sin u 4 + P. The change in the obital elements unde the petubing foce is given by the equations: da dt = P Sesin v T + n e de dt = e a S sin v T e + cos v na + + a di dt = W cos u C dω = W sin u dt C sin i dϖ e = i d S cosv T sin v sin v sin dt nae Ω P dt dl 0 dt e dϖ i dω = S + + e sin na + e dt dt which can be epesented in the following fom when one eplaces the values (6): da dt = 0

10 Lense and Thiing On the influence of the pope otation of cental bodies 0 de dt = K cos i sin v vɺ Ca di dt = K sin i cos u [ e sin v cos u + ( + e cos v )sin u ] vɺ Cp dω K = sin u [ e sin v cos u + ( + e cos v )sin u ] vɺ dt Cp dϖ K cos i + e = i dω + cosv vɺ + sin dt Ca e dt dl 0 dt K cosi e dϖ i dω + ecosv vɺ + + e sin. na P + e dt dt = ( ) In the spiit of petubation theoy we conside the obital element that appeas on the ight-hand side with the infinitesimally small facto K to be constant and integate ove just v while obseving that u = v + ϖ Ω. Thus we compute the fist-ode petubations. If we intoduce K = K / na then we will get: a = 0 K cosi e = e K cosi i = e cos v (cos u + e cos v cos u) K Ω = ( e ) / [v sin u + e (sin v sin u cos v)] ϖ = K cos i + e v sin v sin i / + + Ω ( e ) e K cosi e i v + esin v + ϖ + e sin Ω. e + e L 0 = ( ) The inteesting esult follows fom this that the petubation of the semi-majo axis vanishes pecisely. Wheeas only peiodic tems aise in e and i secula tems will also appea in the emaining elements namely since v = nt + peiod:

11 Lense and Thiing On the influence of the pope otation of cental bodies (7) ϖ = L 0 = K Ω = ( e ) / nt K i ( ) sin / e nt. 4. Numeical esults. Numeical analysis shows that these secula petubations will emain below the theshold of obsevability ove the span of a centuy fo the Sun-planets system since they will each a maximum of 0.0 (fo the peihelion of Mecuy). The situation is diffeent fo the planet-moons systems: Lage numbes will appea in that case. Fo the sake of numeical calculations it is bette to tansfom fomulas (7). We shall use the following notations: l Radius of the planet in cm. τ Rotational duation of the planet in days a Semi-majo axis of the satellite obit in cm. a planetay obit in cm. U Peiod of the satellite in days U planet J Numbe of days in a yea ε Velocity of light in cm sec The following fomula which esults fom (7): (8) Ω = ϖ = L 0 = π Jl 9c τu will give us the afoementioned petubation of the satellite elements that is due to the otation of the planets in ac seconds pe centuy. We have set e = i = 0 in it since that would be pemitted to the desied degee of accuacy fo the moons unde consideation. In the spiit of the petubations that wee discussed by Einstein in his eseach on Mecuy will then emain additive as a contibution that oiginates in the diect action of the planet and a contibution that oiginates in petubing foce of the Sun. The fome contibution is given by: 5π J a (9) Ω = 0 ϖ = L 0 = 4 c U ( e ) and the latte ( ) by: ( ) W. de Sitte Planetay motion and the motion of the Moon accoding to Einstein s theoy Amstedam Poc. 6 (96). The obital plane is subsequently used fo the XY-plane in fomula (0). In de Sitte s teatment the fomula (8) fo δϖ on pp. 79 is missing a facto of /4.

12 Lense and Thiing On the influence of the pope otation of cental bodies (0) 4 Ω = ϖ = L 0 = 5π J a c U all of which ae in ac seconds pe centuy. Both the eccenticity and the inclination of the planetay and satellite obital planes wee neglected in the latte which is justified by the infinitesimal magnitude of these tems as is shown by Table I: Table I. Ω ϖ = L 0 Eath moon Both moons of Mas The values ae much smalle fo all of the emaining moons. The petubations that ae due to the pope otations of the planets ae included in Table II. Table II. Jupite Satun V I II 4 5 Ω ϖ = L The numbes ae less than 0.5 fo all of the othe satellites. The lagest tems ae analogous to the ones that elate to Einstein s pecession of the peihelion of Mecuy [fomula (9)] as Table III shows: Table III. ( Ω = 0) ϖ = L 0 ϖ = L 0 Mas Jupite I 4 8 II 4 Satun 5 46 III 6 0 IV 6 47 V Uanus Neptune moon 5 They ae less than 0.5 fo all of the moons that wee not enteed.

13 Lense and Thiing On the influence of the pope otation of cental bodies If we would now wish to add togethe all thee types of tems in ode to obtain the total elativistic influence then we would have to conside the following: The coection to the Newtonian laws that was teated by Einstein s Mecuy eseach was caused by a petubing foce along the adius vecto whose components in the cited efeence wee: n a C vɺ S = T = W = 0; c hence it is independent of the choice of coodinate system. In what follows the coesponding petubations [fomula (9) and Table III] can be efeed to an abitay XY-plane. The vaiations of the elements that ae included in fomulas (0) that aise fom the petubing foce of the Sun and that deviate fom the classical fom as we have aleady mentioned ae efeed to the obital plane of the planets and thus the numbes that computed fo them in Table I as well while eveything in Table II which includes the petubing tems that oiginate in the otation of the plane is efeed to the choice of coodinate system that was made in the pesent teatment egading the equatoial plane of the cental body. The total elativistic influence is then summaized in Table IV: Only the tems (9) and (0) appea fo the Eath moon and both moons of Mas so the efeence plane will then be the obital plane of the planets. On the othe hand the plane of the cental body in question is used fo the satellites of Jupite and Satun since once moe only the tems (9) and (0) will appea. The petubation of the moon of Uanus and the moon of Neptune include only the tem (9) so the efeence planes can be chosen abitaily. Table IV. Ω ϖ = L 0 t Eath moon 8.9 Mas. Phobos Deimos Jupite I II III IV V 5 5 m 5.4 Satun. Mimas Enceladus Thetys Dione Rhea Titan Hypeion Themis 0.9 Uanus. Aiel 0.7. Umbiel 0 0.7

14 Lense and Thiing On the influence of the pope otation of cental bodies 4. Titania Obeon 0.0 Neptune moon 0 5. Let us say this about the column that is labeled t: The secula petubations to the mean longitude poduce a vaiation in the mean daily motion; i.e. in the time that is elapsed between two events (e.g. the eclipses of the moons of Jupite) which adds a cetain coection to the case in which thee ae no elativistic influences. This coection is given in the last column of Table IV fo a span of one hunded yeas and is obtained fom the following fomula: t = U L 0. Summay The petubing tems fo the planetay and moon obits that oiginate in Einstein s theoy of the pope otation of a cental body ae smalle than the ones that come fom the second appoximation and lead to the pecession of the peihelion of Mecuy. We do not encounte these tems fo the planetay obits but they must be intoduced fo the computation of the obits of the moons of Jupite and Satun. The secula petubations that oiginate upon consideing the total elativistic influence wee computed fo the moons of the oute planets. Whethe o not they will individually (e.g. fo the fifth moon of Jupite) attain a magnitude that is sufficient to pemit a poof of the theoy fo the petubations of the moon obits lies beyond the limits of pecision fo existing obsevations. Vienna Febuay 98 Institute fo Theoetical Physics of the Univesity. (Received Febuay 98)