The trajectory and the orbital velocity are determined for an object moving in a gravitational system, in terms of fundamental and independent variables. In particular, considering a path on equipotential line, the elliptical orbit is naturally traced, verifying evidently the keplerian laws. The case of the planets of the solar system is presented

Modestino, Giuseppina

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               Laboratori Nazionali di Frascati   ISTITUTO NAZIONALE DI FISICA NUCLEARE                  INFN-16-03/LNF         17th February 2016       Unconventional physical way for the orbital velocity evaluation  Giuseppina Modestino INFN-Laboratori Nazionali di Frascati Via E. Fermi 40, Frascati, Italy    Abstract  The trajectory and the orbital velocity are determined for an object moving in a gravitational system, in terms of fundamental and independent variables. In particular, considering a path on equipotential line, the elliptical orbit is naturally traced, verifying evidently the keplerian laws. The case of the planets of the solar system is presented.           Pubblicato da SIDS–Pubblicazioni      Laboratori Nazionali di Frascati     Unconventional physical way for the orbital velocity evaluationGiuseppina Modestino∗INFN, Laboratori Nazionali di Frascati,via Enrico Fermi 40, I-00044, Frascati (Roma) Italy(Dated: February 16, 2016)The trajectory and the orbital velocity are determined for an object moving in a gravitationalsystem, in terms of fundamental and independent variables. In particular, considering a path onequipotential line, the elliptical orbit is naturally traced, verifying evidently the keplerian laws. Thecase of the planets of the solar system is presented.I. INTRODUCTIONAlthough there are a myriad of treatises dealingthe velocity of an object passing trough a gravita-tional field, we can hardly find procedures or expres-sions in terms of independent and fundamental vari-ables. The formulas are extracted from the keplerianand classical gravitation laws, and generally theycontain a series of orbital parameters most of themachieved from experimental observations. So, it isnot so easy to distinguish the exact role of each phys-ical variable, even in the most rigorous procedures.In the present article, a series of calculations willbe performed to express the orbital speed in termsof fundamental and physically observable variables.Principally, the results of a previous report will beused [1]. In that note, the intensity field evolutionwas predicted at a given point moving at relativeconstant speed respect to an electromagnetic source.Taking into account the source emission mechanism,fundamentally a decreasing potential with the dis-tance growth, and pointing out the geometrical pa-rameters of the physical environment, the dynamicsof the system was analytically described. As shown,the results agree with the solutions obtained by theclassical physical treatises, for instance the retardedpotential evaluation, despite the greater simplicity.Synthetically, in the cited note, the space-time scalarparameter which determines the potential energy,the quantity R does not correspond simply to thereal distance between the source and the observa-tion point, but it is the sum of the real distanceplus the distance between the same objects evalu-ated at the instant when the intensity field was null.In the present note, taking into account the analogyof the electromagnetic and gravitational field, thesame criterion will be employed. In summary, in thenext sections, the basic statements of the previousprocedure will be highlighted, then the analyticalformula will be used to calculate the orbital speed,adopting as orbital parameters the potential energy∗Electronic address: modestino@lnf.infn.itof the system and the initial momentum of the mov-ing object. The specific case of a constant energypotential will reveal a natural and strong connec-tion with the elliptical trajectory. In light of thesestatements, the trajectories and the orbital velocityof the solar system planets will be examined.II. REFERENCE SYSTEM AND ANDFUNDAMENTAL PARAMETERSLet’s consider a body with mass m at the posi-tion (xt, yt) at the instant t, in a spatial referencesystem with origin into O, as shown in fig1. (Note:from now on, only a two-dimensional space will beconsidered). Initially, in absence of field, it moves atvelocity v0. Presuming the point O as the center ofthe mass generating the gravitational field, the aimis to characterize the system at instant t, from thephysical point of view. As previously demonstrated[1], at that space-time coordinates, the intensity of afield propagating at light-speed c, depends on 1/R2,where R is the real path covered during the timeinterval (t− t0) [1], being t0 such thatR = c(t− t0) (1)andβR =√(xt − x0)2 + (yt − y0)2 (2)where β ≡ v0/c. Considering the time parameteras an independent and linear variable, t and t0 mustbelong to the same time reference line (fig.2). Itis fundamental and useful to define an intermediateinstant t′, on the same time line, in such a wayt = t′+√x2t + y2tcand t0 = t′−√x20 + y20c(3)At same time, in terms of space coordinates, we findx′, y′ and the relative length D′ ≡√x′2 + y′2.Being Dt ≡√x2t + y2t and D0 ≡√x20 + y20 thedistances of the object from the center point O re-spectively at time t and t0, it resultsR = Dt +D0. (4)2FIG. 1: Space coordinates and other geometrical pa-rameters relative a body (red bullet) bumping into agravitational field at time t.FIG. 2: The figure shows the time variables of the eqs.1and 3, and the time arrow. That one is pointed towardsthe right side with the red tip corresponding to the in-stant t.FIG. 3: Rotating the reference system of fig1 of anangle α. Thus, the ordinate value at instant t assumesthe same value at time t0, but the metric of the systemis unchanged. The vector v0 becomes [v0, 0].Rotating the reference system of an angle α sothat yt = y0 (fig.3), the relation 2 becomesβR = |xt − x0| (5)and the same length R can be expressed also in thefollowing waysR = 2(Dt − βxt)1− β2, (6)R = 2(D0 + βx0)1− β2. (7)III. TRAJECTORY WITH CONSTANTPOTENTIAL ENERGYMeaning the trajectory as just the set of the cov-ered spatial positions, the four eqs.4-7 are enoughfor defining it. Also a reference system where yt =y′ = y0 is necessary, and it is essential to underlinethat it is always possible to perform such a choice,without invalidating the metric properties of the sys-tem. That is easily understandable looking at theeqs.4-7, and considering the spatial transformationinvariance of the quantities R, Dt and D0. So, re-garding that equations, they describe the physicaland spatial properties of the body of mass m, thatruns into a gravitational field, just at instant t. Therelative potential energy is E ∼ 1/R, and the mo-mentum is q = βmc, so considering invariant thattwo quantities, means keeping constant R and β.In that conditions, for the body m, the locus (xt, yt)of the allowed positions is an ellipse- with one focus centered into O- with 2a = R, and- with eccentricity ε = β, being a the major semiaxis.In a natural way, these statements reveal the geo-metrical properties of such a physical system. Onthe other hand, corresponding with the first keple-rian law, they disclose the intrinsic physical meaningof an orbital path. In that sense, trying to betterunderstanding its physical properties, we can saythat the eccentricity indicates the specific momen-tum of the orbiting body, coinciding to β = v0/c,where v0 ≡ q/m can be interpreted as the velocityof the body respect to the gravity center, before itis trapped into the gravitational field. In fact, itis natural to perceive an initial high speed objectgenerating a strongly elliptical track, and obviouslyvice-versa. Then, being R expression of the poten-tial energy, it can be invariant although it is the sumof the two variables (eq.4), just like the ellipse out-line is the locus which the sum of the distances tothe two focal points is constant.Let’s consider a practical example relative to abody having initial momentum q = 0.6mc that is3(x0$,y0$)$ ($xt$,yt$)$l (AU) l (AU) y$x$Dt$D0$F’$ O$ F’’$.$.$R/2$.$.$a$βR$.$ .$βR$FIG. 4: In the figure, the parameters solving the systemequations 4-7 are indicated for a body with momentumq = 0.6 mc, that is β = 0.6, and R = 14.57. The realorbit is described by the continuos line forming an ellipsemainly developing on the right side, with ε = 0.6 = β,and semi axis length a = 7.285 = R/2. It represents theonly space locus (xt, yt), for the defined reference system.On the left, the set of the points (x0, y0) is crosshatched.Both lines define two identical ellipses surrounding thefocal points F ′, F ′′, and O in common, correspondingthat to the origin of reference system, as well as to thecenter of the gravitational system. A generic point onthe trajectory is drawn with real distance Dt, and therelative length D0.β = 0.6, and R = 14.57 in arbitrary units (AU). Therelative elliptical trace is outlined in fig.4. It repre-sents the set of possible solutions of the equationsystem 4-7. In the same figure, all the variables sat-isfying that equations are shown. In particular, theset of point (x0, y0) corresponds to the crosshatchedellipse whose dimensions are exactly equal to the(xt, yt) ellipse, with one focus anyway into O. It isimportant noticing that also (−x0,±yt) satisfies theeq.6, as though (−xt,±y0) satisfies 7, making so ev-ident the complete overlapping of the two curves forless than a space-time phase quantity, and reaffirm-ing so the stationary configuration of the system. Inthe fig.5, there are ten closed orbits, each one withthe same R but different value for β. The curvesflatten oneself while the β value grows. One of thetwo focal points is anyway steady in the origin ofreference system, for each orbit.IV. ORBITAL VELOCITYThe orbital speed expresses the dynamics ofa body passing through a gravitational field, sol (AU) l (AU) a$FIG. 5: In a reference system in arbitrary unit (AU),some trajectories are shown, that are the couple sets(xt, yt) solving the eqs.4-7, each one with the same lengthR = 14.57, and with different β values included intothe interval [0.33, 0.83]. The reference system is chosenin order to v0 gets null contribute on the y axis, i.e.v0[vx, 0, ]. The elliptical figures are easily recognizable,all of that with 2a = R, but different eccentricities ε,each one corresponding to the fixed β value. The curvesflatten themselves while the β value grows.it is characterized by the physical and spatialparameters, each time. Trying to define vorb, thefollowing hypotheses are considered: the gravity isthe only force, the total mass M of the system isconcentrated into O, so if MO is the mass at center,M = m+MO ≈MO. Then, taking into account thefundamental properties of the system, consecutivereasoning leads to following formvorb(t) =D′Dt√2GMOR(1− β2)=D′Dt√2γ2GMOR(8)where G is the gravitational constant, and γ ≡1/√1− β2.The formula 8 can be rewritten for the specificcase of a stationary orbit, that is R = const = 2a,and choosing a special reference system like in fig.3where x′ = xt − βDt. So, taking into the accountthe relation D′ =√(x′2 + y2t ), the eq.8 becomesvorb(t) =√γ2GMOa[1 + β2 − 2βxtDt]. (9)In fig.6, the orbital velocity is reported in terms ofthe geometrical parameters. It has been calculatedby eq.9, using the the same numerical values of fig.4.4v orb$v orb$ vorb$vorb$yt$xt$Dt$xt$yt$FIG. 6: The orbital velocity is reported as a function ofdifferent parameters, all of them in terms of AU. Thecalculation is related to the same example of fig.4.Although that quantity is obviously depending ontime, the last two expressions do not contain an ex-plicit temporal term. Effectively, tracing the trajec-tory, until now, the relative points have been consid-ered as positions temporally independent each fromthe others, connecting each point to the real time t.In spite of the presence of several time quantities,only t defines a real position of the object, while thevalues t′ and t0 contribute to describe the physicalstate anyway concerning t. Thus, let’s divide a bodytrajectory in n points (xn, yn). Each of them corre-sponds to the time t = t1, t2, ..., tn, entailing so adifferent system configuration with own t′1, t′2, ..., t′n,and t01, t02, ..., t0n and the set related variables thatsatisfies all the previous equations. Such a way, thevorb(tn) value is univocally assigned to each point,by eq.8.With the aim to calculate the orbital time period,let’s consider an elliptical trajectory. Coming af-ter some logical steps, the following relation can beadoptedTorb =Πel√8GMOR(γ1/2∑n|xi − xi−1| − γ2/3Pel)(10)where Πel is the length of the orbit, xi−1 and xi areconsecutive abscissa coordinates related to the in-stants ti−1 and ti, and Pel is the perihelion length.The relation is probably to improve with furtherstudy, since in the present report the approximationlevel is not well defined.FIG. 7: The trajectories of the four planets nearestthe Sun are drawn. They have been calculated takinginto account the eqs.4-7 and consequently the statementscontained in the section II and III. The experimental pa-rameters 2a and ε are extracted from the data tables in[2],[3].V. THE SOLAR SYSTEMFor performing the experimental test on the pre-vious formulas, the solar system is considered, sinceit is well conformed to the physical cases debated inthe previous sections. Summarizing, that cases ex-pect the trajectory that lies in a plane, a fixed centerof mass O, and M0  m, with M0 equal to the so-lar mass, and m corresponding to the single planetmass.A. Planet orbitsReferring to the planets (including Pluto) of thesolar system, let’s trace the nine relative orbits try-ing to find as much as couple possible set (x, y) ofthe object positions. Essentially, two parameters areessential for each orbit, the major axis length 2a andthe eccentricity ε, two geometrical quantities to re-veal from the data tables [2, 3]. Assigning the physi-cal meaning as seen, that is 2a = R and ε = β, thesevariables can be introduced into the eqs.4-7. Theresult is shown in fig.7 and 8. As predicted in theprevious section, the resulting trajectories are ellip-tical and coinciding with the true planet orbits, withthe Sun naturally centered in one focus, accordingto the keplerian law.5FIG. 8: The trajectories of the four planets more dis-tant from Sun are shown, plus that of Pluto, the largestone. As in the previous figure, they derive from the state-ments contained in the sections II and III, and from datatables in [2],[3] regarding the ellipse dimensions.B. Orbital velocity and periodThe orbital speed is evaluated for each of the ninebody belonging to the solar system. The eq.9 isapplied along the whole trajectories, tabulating theaverage with the minimum and maximum values intab.I. That two values correspond to the object po-sitions relatively at aphelion and perihelion. Theagreement with the measurements [2, 3] is good es-pecially for the minimum and maximum values thatregard punctual body positions. The major discrep-ancy about the average value can be due to the dif-ferent algorithms used for the average elaboration.In fig.9, regarding the same system, the specific en-ergy spectrum is reported. The upper graph de-scribes the distribution of the objects in terms ofmajor semi axis ′a′ relative to each own orbit, mean-ing each average distance from Sun. The lower oneconcerns the same data but reporting the averagedistance dE from Earth, that is watching the samemass distribution but from another point of view.The comparison between the two curves (the lowergraph shows both) is useful also from another per-spective, considering that the variable a is not sim-ply a geometrical quantity but it contains an im-portant physical significance being a = R/2, whiledE is just a spatial distance. So, not giving suffi-cient consideration to that difference, the interpre-tation of results might be ambiguous or even erro-neous. The time period is calculated for each planetof the solar system and for Pluto, using the eq.10.Also here, there are some discrepancies with respectexperimental values [2, 3], as reported in the lastcolumn of the tab.I. Anyway, the real errors can beE/m$(J/Kg)$E/m$(J/Kg)$MeVEaMaJ     SaUNPa(m)dE(m)FIG. 9: Spatial distribution for the orbiting objects inthe solar system. In the upper graph, where the orbitalsemi axis length a is reported, the observational point isthe Sun. In the lower one, the average distances dE areevaluated from Earth. The crosshatched line is just theline of the upper graph, also there reported to improvethe comparison. Remembering that a is also a physicalvariable being a = R/2, while dE represents just a spa-tial distance, the discrepancy between the two curves isbetter interpreted.TABLE I: Velocity values and orbital period for the plan-ets of the solar system, including Pluto. The formula 9 isapplied. The orbital period Tcalc is calculated by eq.10,while Tmea is extracted principally from the NASA data[2, 3].vmin v̄ vmax Tcalc Tcalc/Tmea(Km/s) (Km/s) (Km/s) (days)Mercury 38.858 48.215 58.976 87.918 0.9994Venus 34.784 35.020 35.258 224.706 1.00002Earth 29.291 29.786 30.286 365.257 1.000002Mars 21.972 24.164 26.497 686.914 0.99993Jupiter 12.440 13.063 13.705 4334.60 1.0003Saturn 9.138 9.649 10.179 10756.6 1.00003Uranus 6.485 6.802 7.128 30702.1 0.9998Neptune 5.385 5.432 5.478 60227.4 1.00007Pluto 3.676 4.790 6.112 90548.3 1.0004hardly calculated since they can due to different kindof normalization with respect the experimental data.Further, as said before, the eq.10 probably must bebetter adapted, then the formula have been consid-ered for a simplified system with just two bodies withM0  m, without considering other perturbations.6VI. CONCLUSIONSThe orbital parameters have been determinedfor an object traveling in a gravitational field.Two scalar quantities are necessary, R and q. Thefirst one, R defines the gravitational potential,being that depending on the rate 1/R. It doesn’trepresent simply the distance between the orbitingobject and the gravity center, it is the sum of thesame real distance plus the distance at the timewhen R was null. That distance is virtual, in thatsense it locates a precise instant but not an uniquespatial point. The second parameter, q = βmc,is the momentum value relative to the orbitingbody. The applied formulae have been previouslyand more generally determined considering theintensity variation of an electromagnetic field,depending on the dynamic distance between thesource and the observation point. In any case theyhave been derived from geometrical analysis ofthe physical environment, considering the space aseuclidean and the time as linear. In the presentarticle, the simple case of a fixed center of masshas been treated. Testing the agreement with theexperimented physical formulae, the orbits of thesolar system have been analyzed. The agreementswith the experimental data is very satisfactory andencouraging for extending the same criterion toother physical structures.AcknowledgementsIt is a great pleasure to thank Giorgio Fornetti forhis precious support and for the certainly interestingdiscussions.[1] G. Modestino, arXiv:1406.4670 [physics.class-ph](2014)[2] NASA Space Science Data Coordinated Archive,http://nssdc.gsfc.nasa.gov[3] https://commons.wikimedia.org

Unconventional physical way for the orbital velocity evaluation

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Unconventional physical way for the orbital velocity evaluation

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