New general relativistic contributions to Mercury's orbital elements and their measurability
aa r X i v : . [ g r- q c ] A p r New general relativistic contributions to Mercury’s orbital elements andtheir measurability
Lorenzo Iorio Ministero dell’Istruzione, dell’Universit`a e della Ricerca (M.I.U.R.)-IstruzionePermanent address for correspondence: Viale Unit`a di Italia 68, 70125, Bari (BA), Italy [email protected]
Received ; accepted 2 –
Abstract
We numerically and analytically work out the first-order post-Newtonian (1pN)orbital e ff ects induced on the semimajor axis a , the eccentricity e , the inclination I ,the longitude of the ascending node Ω , the longitude of perihelion ̟ , and the meanlongitude at epoch ǫ of a test particle orbiting its primary, assumed static and spheri-cally symmetric, by a distant massive third body X. For Mercury, the rates of changeof the linear trends found are ˙ I X1pN = − . (cid:16) µ as cty − (cid:17) ,˙ Ω X1pN = . µ as cty − , ˙ ̟ X1pN = . µ as cty − , ˙ ǫ X1pN = . µ as cty − , respectively.Such values, which are due to the added actions of the other planets from Venus to Sat-urn, are essentially at the same level of, or larger by one order of magnitude than, thelatest formal errors in the Hermean orbital precessions calculated with the EPM2017ephemerides. The perihelion precession ˙ ̟ X1pN turns out to be smaller than some valuesrecently appeared in the literature in view of a possible measurement with the ongo-ing BepiColombo mission. Linear combinations of the supplementary advances of theKeplerian orbital elements for several planets, if determined experimentally by the as-tronomers, could be set up in order to disentangle the 1pN N -body e ff ects of interestfrom the competing larger precessions like those due to the Sun’s quadrupole moment J and angular momentum S .keywords gravitation − celestial mechanics − ephemerides − methods: miscellaneous
1. Introduction
In its weak-field and slow-motion approximation, general relativity predicts that, in additionto the time-honored first-order post-Newtonian (1pN) gravitoelectric and gravitomagneticprecessions induced by the mass monopole M (Schwarzschild) and the spin dipole S (Lense-Thirring) moments of the central body acting as source of the gravitational field, further 1pNorbital e ff ects due to the presence of other interacting masses arise as well (Will 2018). Let usconsider a nonrotating primary of mass M , assumed as origin of a locally inertial coordinatesystem, orbited by a test particle located at r and moving with velocity v . If a distant, pointlikebody X of mass M X is present at r X and moves with velocity v X with respect to M , the test particleexperiences certain 1pN accelerations which, from Eq. (4) of Will (2018), are A G = G M M X c r h ˆ r − ˆ r · ˆ r X ) ˆ r X + ˆ r · ˆ r X ) ˆ r i , (1) See, e.g., Debono & Smoot (2016) and references therein for a recent overview on its statusand challenges. 3 – A G = GM X rc r { v [( v · ˆ r ) − ˆ r · ˆ r X ) ( v · ˆ r X )] −− v [ ˆ r − ˆ r · ˆ r X ) ˆ r X ] o , (2) A v X = − GM X c r [4 v × ( ˆ r X × v X ) − ˆ r X · v X ) v ] . (3)In Eqs. (1) to (3), which are a particular case of the full 1pN equations of motion for a systemof N pontlike, massive bodies mutually interacting through gravitation (Poisson & Will 2014,Eq. (9.127)), G is the Newton’s gravitational constant, and c is the speed of light in vacuum.Will (2018) looked at the longitude of perihelion ̟ of Mercury finding an additionalcontribution to its 1pN secular precession of about˙ ̟ X1pN = .
22 milliarcseconds per century (cid:16) mas cty − (cid:17) ==
220 microarcseconds per century (cid:16) µ as cty − (cid:17) . (4)Eq. (4) was obtained by making some simplifying assumptions about the orbital geometries ofboth the perturbed and the perturbing bodies, and includes the combined actions of Venus, Earth,Mars, Jupiter and Saturn. It should be a direct e ff ect of the accelerations of Eqs. (1) to (3), andan indirect consequence of the interplay between the usual Newtonian N − body pull by the otherplanets and the Sun-only 1pN gravitoelectric acceleration. Eqs. (1) to (3) and all the standardNewtonian and 1pN N -body dynamics is routinely modeled in the data reduction softwares of theteams of astronomers producing the planetary ephemerides like the Development Ephemeris (DE)by the NASA Jet Propulsion Laboratory (JPL) in Pasadena (Folkner et al. 2014), the Int´egrateurNum´erique Plan´etaire de l’Observatoire de Paris (INPOP) by the Institut de M´ecanique C´eleste etde Calcul des ´Eph´em´erides (IMCCE) at the Paris Observatory (Viswanathan et al. 2018), and theEphemeris of Planets and the Moon (EPM) by the Institute of Applied Astronomy (IAA) of theRussian Academy of Sciences (RAS) in Saint Petersburg (Pitjeva 2015b). Will (2018) claimedthat Equation (4) would likely be detectable with the ongoing BepiColombo mission to Mercury.According to Will (2018), it would be so because the expected ≃ − accuracy with which theparameterized Post-Newtonian (PPN) parameters β, γ should be measured by such a spacecraftwould correspond to an uncertainty in the main contribution to the Mercury’s 1pN perihelionprecession ˙ ̟ = .
98 arcseconds per century (cid:16) ′′ cty − (cid:17) as little as δ ˙ ̟ ≃ .
03 mas cty − = µ as cty − . (5) See also Brumberg & Kopeikin (1989, Eq. (7.11), Eq. (7.12), Eq. (8.18)) with the replacementsEarth → Sun, Sun → Jupiter, and satellite → Mercury. 4 –Iorio (2018), after having pointed out that the indirect, mixed e ff ects should likely be notmeasurable in practical planetary data reductions, analytically worked out the direct perihelionprecessions due to Eqs. (1) to (3) for arbitrary orbital configurations of both the test particle andthe perturbing body X. The total 1pN rate of change induced on the perihelion of Mercury by allthe other planets of the solar system from Venus to Saturn would amount to (Iorio 2018, Table 2)˙ ̟ X1pN = .
15 mas cty − = µ as cty − . (6)Iorio (2018) showed also that Equation (6) would likely be overwhelmed by the larger systematicerrors due to the mismodeling in the competing secular precessions due to the Sun’s oblateness J and angular momentum S (1pN Lense-Thirring e ff ect).In this paper, we will show that the value reported in Equation (6) is, in fact, wrong becauseof an error by Iorio (2018) in the calculation of the precession due to Equation (2). The correctsize of the overall 1pN N − body perihelion precession of Mercury will turn out to be even smallerthan Equation (6), thus enforcing the pessimistic conclusions of Iorio (2018) about its possiblemeasurability. As such, we will further explore the consequences of Eqs. (1) to (3) by numericallyworking out the secular shifts induced by them on all the other orbital elements, i.e. the semimajoraxis a , the eccentricity e , the inclination I , the longitude of the ascending node Ω , and the meanlongitude at epoch ǫ , and will compare them with the uncertainties in the planetary orbital motionsinferred by Iorio (2019) from the most recent version of the EPM ephemerides (Pitjeva & Pitjev2018). Indeed, if and when the astronomers will observationally produce the supplementary ratesof change ∆ ˙ a obs , ∆ ˙ e obs , ∆ ˙ I obs , ∆ ˙ Ω obs , ∆ ˙ ̟ obs , and ∆ ˙ ǫ obs of as many planets as possible, it will bepossible to generalize the approach proposed by Shapiro (1990) by suitably combining them inorder to disentangle the e ff ects of Eqs. (1) to (3) in from the other competing precessions due to,e.g., the Sun’s J and S .
2. The 1pN N − body secular changes of the orbital elements2.1. Numerical integration of the equations of motion We simultaneously integrate the equations of motion of Mercury in Cartesian rectangularcoordinates and the Gauss equations for each orbital element with and without the fifteen terms of To avoid possible misunderstanding, we clarify that Eqs. (1) to (3) are dubbed as “cross-terms”by Will (2018), while here such a definition designates the interplay among the standard Newtonian N -body and 1pN Sun’s monopole accelerations. At that time, the aliasing Newtonian e ff ect which should have been disentangled from theSun-only 1pN gravitoelectric perihelion precession by looking at other planets or highly eccentricasteroids was due to the solar quadrupole mass moment J . 5 –the sum of Eqs. (1) to (3) calculated for Venus, Earth, Mars, Jupiter and Saturn over a time spanas long as 1 cty in order to clearly single out the sought features of motion: both runs share thesame initial conditions retrieved on the Internet from the WEB interface HORIZONS maintainedby the JPL. For consistency reasons with the planetary data reductions available in the literature,we use the equatorial coordinates of the International Celestial Reference System (ICRS). Then,for each orbital element, we plot in Fig. 1 the time series (blue curve) resulting from the di ff erencebetween the runs with and without the 1pN N − body accelerations. Finally, we fit a linear model(yellow line) to its numerically produced signal, and estimate its slope: the outcome is collectedin the caption of Fig. 1. From Fig. 1, the secular trends of I , Ω , ̟, ǫ are apparent, while a and e seem to experience long-term harmonic variations. The size of the slopes of the precessionsof the angular rates of change vary in the range ≃ − µ as cty − = . − . − . Inparticular, it turns out that the secular precession of the perihelion is about five times smaller thanEquation (6) (Iorio 2018, Table 2), being as little as˙ ̟ X1pN = µ as cty − = .
03 mas cty − . (7)Numerical tests conducted by switching o ff from time to time each of Eqs. (1) to (3) for everysingle perturbing planet X showed that the issue resides in the analytical calculation of Eq. (B5) inIorio (2018) and in the consequent numerical results of the third column from the left of Table 2in Iorio (2018). It is also possible to analytically work out the long-term rates of change of the Keplerianorbital elements of the test particle with the Gauss perturbative equations applied to Eqs. (1) to (3)by doubly averaging their right-hand-sides over the orbital periods P b and P X of the perturbedbody and the perturber X, respectively. The resulting expressions, especially those due toEqs. (1) to (2), are very cumbersome. Thus, we display just approximate formulas for them totheir leading order in e . The shifts due to Equation (3), which are relatively less involved, aredisplayed in full. In the next Sects., we use the shorthand ∆Ω (cid:17) Ω − Ω X .It turns out that there is an excellent agreement among the numerical results of Sect. 2.1 andthe analytical results shown below. A G Here, we analytically calculate the doubly averaged rates of change of the Keplerianorbital elements of the test particle, to their leading order in e , due to Equation (1). No furtherapproximations in the orbital configurations of both the perturbed body and X are made. They areas follows. 6 –The semimajor axis a stays constant since˙ a A G = . (8)The rate of change of the eccentricity e turns out to be˙ e A G = − e µ X √ µ a c a (cid:16) − e (cid:17) / E A G ( I , I X , Ω , Ω X ) + O (cid:16) e (cid:17) , (9)with E A G = ω (cid:16) cos I sin I X sin 2 ∆Ω − sin I sin 2 I X sin ∆Ω (cid:17) −− sin 2 ω {− + cos 2 I X [ − + cos 2 I (3 + cos 2 ∆Ω )] −− I X cos 2 ∆Ω + I sin 2 I X cos ∆Ω++ I sin ∆Ω o . (10)As far as the rate of change of the inclination I is concerned, we have˙ I A G = − µ X √ µ ac a (cid:16) − e (cid:17) / I A G ( I , I X , Ω , Ω X ) + O (cid:16) e (cid:17) , (11)with I A G (cid:17) sin I X (cos I cos I X + sin I sin I X cos ∆Ω ) sin ∆Ω . (12)The precession of the node Ω is˙ Ω A G = µ X √ µ a c a (cid:16) − e (cid:17) / N A G ( I , I X , Ω , Ω X ) + O (cid:16) e (cid:17) , (13)with N A G − I csc I sin 2 I X cos ∆Ω++ cos I h cos 2 I X (3 + cos 2 ∆Ω ) + ∆Ω i . (14) 7 –The precession of ̟ due to Equation (1) was correctly worked out, to the zero order in e , inEq. (B2) of Iorio (2018); thus, we do not display it here.The rate of change of the mean longitude at epoch ǫ is˙ ǫ A G = µ X √ µ a c a (cid:16) − e (cid:17) / L A G ( I , I X , Ω , Ω X ) + O (cid:16) e (cid:17) , (15)where L A G = − + I − I X + I cos 2 I X ++
12 sin (cid:18) I (cid:19) sin I X cos 2 ∆Ω++ + I ) tan (cid:18) I (cid:19) sin 2 I X cos ∆Ω . (16) A G Here, we analytically work out the doubly averaged rates of change of the Keplerian orbitalelements of the test particle, to their leading order in e , induced by Equation (2). No furtherapproximations in the orbital configurations of both the perturbed body and X are made. We listthem below.For the semimajor axis a , we have˙ a A G = µ X a / √ µ c a (cid:16) − e (cid:17) / A A G ( I , I X , Ω , Ω X ) + O (cid:16) e (cid:17) , (17)with A A G = sin I X ( − sin I cos I X + cos I sin I X cos ∆Ω ) sin ∆Ω . (18)The rate of change of the eccentricity e is˙ e A G = − e µ X √ µ a c a (cid:16) − e (cid:17) / E A G ( I , I X , Ω , Ω X ) + O (cid:16) e (cid:17) , (19)with E A G = sin I X ( − sin I cos I X + cos I sin I X cos ∆Ω ) sin ∆Ω . (20) 8 –The rate of change of the inclination I turns out to be˙ I A G = µ X √ µ a c a (cid:16) − e (cid:17) / I A G ( I , I X , Ω , Ω X ) + O (cid:16) e (cid:17) , (21)with I A G (cid:17) sin I X (cos I cos I X + sin I sin I X cos ∆Ω ) sin ∆Ω . (22)The precession of the node Ω is˙ Ω A G = − µ X √ µ a c a (cid:16) − e (cid:17) / N A G ( I , I X , Ω , Ω X ) + O (cid:16) e (cid:17) , (23)with N A G (cid:17) − I csc I sin 2 I X cos ∆Ω++ cos I h cos 2 I X (3 + cos 2 ∆Ω ) + ∆Ω i . (24)For the precession of the longitude of perihelion ̟ , we have˙ ̟ A G = − µ X √ µ a csc I c a (cid:16) − e (cid:17) / W ( I , I X , Ω , Ω X ) + O (cid:16) e (cid:17) , (25)with W (cid:17)
92 sin I h − + sin I X (3 + cos 2 ∆Ω ) i ++ sin I { + I + I X −− I X [3 cos I + ( − + cos I ) cos 2 ∆Ω ] o −− (cid:18) I (cid:19) sin 2 I X cos ∆Ω++ + I ) sin I sin 2 I X cos ∆Ω . (26)Eq. (25)-eq. (26), which correct Eq. (B5) of Iorio (2018), allow to calculate the same values forMercury which are obtained with our numerical integrations of Sect. 2.1, limited to Equation (2)only, for each of the perturbing planets at a time. 9 –The rate of change of the mean longitude at epoch ǫ is given by˙ ǫ A G = − µ X √ µ a c a (cid:16) − e (cid:17) / L A G ( I , I X , Ω , Ω X ) + O (cid:16) e (cid:17) , (27)with L A G = ( − + I + I ) (1 + I X ) ++
24 (2 + cos I ) sin (cid:18) I (cid:19) sin I X cos 2 Ω cos 2 Ω X ++ (cid:18) I (cid:19) (cid:20) (cid:18) I (cid:19) + sin (cid:18) I (cid:19)(cid:21) sin 2 I X cos Ω cos Ω X ++ (cid:18) I (cid:19) (cid:20) (cid:18) I (cid:19) + sin (cid:18) I (cid:19)(cid:21) sin 2 I X sin Ω sin Ω X ++
24 (2 + cos I ) sin (cid:18) I (cid:19) sin I X sin 2 Ω sin 2 Ω X . (28) A v X Here, we analytically calculate the doubly averaged rates of change of the Keplerianorbital elements of the test particle caused by Equation (3). No approximations in the orbitalconfigurations of both the perturbed body and X are made; the following expressions are exact.The semimajor axis a and the eccentricity e are constant since˙ a A v X = , (29)˙ e A v X = . (30)The rate of change of the inclination I is˙ I A v X = − µ X √ µ sin I X sin ∆Ω c a / (cid:16) − e (cid:17) . (31)For the precession of the node Ω we have˙ Ω A v X = µ X √ µ (cos I X − cot I sin I X cos ∆Ω ) c a / (cid:16) − e (cid:17) . (32) 10 –The precession of ̟ due to Equation (3) was correctly calculated in Eq. (B8) of Iorio (2018);as such, it is not shown here.The rate of change of the mean longitude at epoch ǫ does depend on e . It turns out to be˙ ǫ A v X = µ X √ µ c a / (cid:16) − e (cid:17) L v X ( I , Ω , I X , Ω X ) , (33)where L v X = (cid:16) + √ − e cos I (cid:17) cos I X ++ (cid:16) + √ − e + √ − e cos I (cid:17) tan (cid:18) I (cid:19) sin I X cos ∆Ω . (34)
3. Confrontation with the observations
Iorio (2019) attempted to calculate the formal uncertainties in the secular rates of changeof a , e , I , Ω , and ̟ of the planets of the solar system from the recently released formal errorsin a and the nonsingular orbital elements e sin ̟, e cos ̟, sin I sin Ω , and sin I cos Ω estimatedfor the same bodies with the EPM2017 ephemerides by Pitjeva & Pitjev (2018). Since, amongother things, the 1pN N -body equations of motion are routinely included in the EPM softwaredynamics, such errors should be overall regarded as representative of the current level ofmodeling the solar system dynamics along with measurement errors. As such, they may beviewed as the uncertainties that would a ff ect a putative measurement of the e ff ects workedout in Sect. 2 if they were explicitly measured in some dedicated data analysis. From thecolumn dedicated to Mercury in Table 1 of Iorio (2019), it can be noted that the 1 − σ errorin ˙ a amounts to δ ˙ a obs = .
003 m cty − , while for the other Keplerian orbital elements we have δ ˙ e obs = . µ as cty − , δ ˙ I obs = µ as cty − , δ ˙ Ω obs = µ as cty − , and δ ˙ ̟ obs = µ as cty − . From acomparison with the expected 1pN rates of change of Fig. 1, it turns out that, with the possibleexception of the perihelion, they are about of the same order of magnitude of the aforementioneduncertainties. Moreover, as discussed in Pitjeva & Pitjev (2018) and Iorio (2019), the latterones may be optimistic. Thus, it is di ffi cult to deem the predicted 1pN N -body precession˙ ̟ X1pN = µ as cty − as realistically measurable compared to a merely formal uncertainty δ ˙ ̟ obs = µ as cty − . It is worth noticing that such a tiny error would correspond to current boundsin the PPN parameters β, γ as little as ≃ − , which are better than the expected accuracy fromthe ongoing BepiColombo mission quoted by Will (2018); see the discussion in Iorio (2019) aboutthe reliability of such an evaluation. The mean longitude at epoch ǫ seem, at first sight, moreinteresting since its 1pN N -body rate is as large as ˙ ǫ X1pN = µ as cty − = .
27 mas cty − . Iorio(2019) did not calculate the uncertainty in ˙ ǫ . In their Table 3, Pitjeva & Pitjev (2018) releasedthe formal uncertainty in the planetary mean longitudes, dubbed there as λ ; for Mercury, it is as 11 –little as δλ obs = . µ as. This implies that, in order to retrieve the uncertainty in ˙ ǫ , the errors inthe mean motion n b due to the mismodeling of the Sun’s gravitational parameter µ and of theplanet’s semimajor axis are required as well. Since δµ obs = × m s − (Pitjeva 2015a), theresulting error in the Hermean mean motion is as large as δ n obsb =
20 mas cty − . It vanishes thepossibility of measuring the 1pN N -body e ff ect on ǫ . As such, only a dramatic improvementin the determination of the Hermean orbit, which might be obtained when all the data fromBepiColombo will be collected and processed, may bring the 1pN N -body precessions due to thedirect e ff ect of Eqs. (1) to (3) in the measurability domain.On the other hand, even should this finally be the case, the concerns raised by Iorio (2018)about the systematic errors caused by the competing Sun’s quadrupole and Lense-Thirring ratesof change are even reinforced by the present analysis since the actual size of the 1pN N -bodyperihelion precession of Mercury turned out to be smaller than the incorrect value of Equation (6).Thus, it is hopeful that the astronomers will finally provide the community with the supplementaryadvances of all the other Keplerian orbital elements in addition to the perihelion. Indeed, if andwhen it will happen, it would, then, be possible to set up linear combinations of them suitablydesigned to cancel out, by construction, the other unwanted precessions. An analogous approach,originally limited just to the perihelia of other planets and asteroids in order to separate thedisturbing Sun’s J action from the Schwarzschild-type rates of changes was proposed by Shapiro(1990). It is also widely used in ongoing relativistic tests with geodetic satellites in the Earth’sfield; see, e.g., Renzetti (2013), and references therein for an overview.
4. Summary and conclusions
Recently, Will (2018) calculated a new general relativistic contribution to the Mercury’sperihelion advance as large as ˙ ̟ X1pN = µ as cty − arising from an approximated form of the1pN N -body equations of motion restricted to a hierarchical three body system. He claimed thatit may be measured in the next future by the ongoing BepiColombo mission to Mercury if it willreach a ≃ − accuracy level in constraining the PPN parameters β, γ . Later, the present authorfirst remarked in Iorio (2018) that the indirect precession due to the interplay of the Newtonian N -body and the 1pN Sun’s Schwarzschild-like accelerations in the equations of motion is likelyundetectable in actual data reductions since it cannot be expressed in terms of a dedicated,solve-for parameter scaling an acceleration di ff erent from the aforementioned ones which areroutinely modeled. Then, he calculated analytically the individual contributions to the perihelionadvance induced directly by each of the approximated 1pN N -body accelerations put forth byWill (2018) by finding an overall precession of ˙ ̟ X1pN = µ as cty − . Iorio (2018) discussed alsothe impact of the systematic aliasing due to the competing perihelion rates induced by the Sun’squadrupole mass moment J and angular momentum A via the Lense-Thirring e ff ect by notingthat their mismodeling would likely compromise a clean recovery of the 1pN e ff ect of interest.Here, the secular rates of change of all the other Keplerian orbital elements a , e , I , Ω , ̟ , and 12 – ǫ caused by the same approximated 1pN N -body accelerations by Will (2018) were analyticallyworked out. A numerical integration of the equations of motion confirmed such findings inthe case of Mercury acted upon by the other planets from Venus to Saturn. The resulting ratesof change amount to ˙ I X1pN = − . (cid:16) µ as cty − (cid:17) , ˙ Ω X1pN = . µ as cty − , ˙ ̟ X1pN = . µ as cty − ,˙ ǫ X1pN = . µ as cty − . As a result, the Hermean 1pN N -body perihelion precession turned out tobe smaller than the previously reported values because of an error explicitly disclosed, at least inthe calculation by Iorio (2018). This makes even more di ffi cult than before its possible present andfuture measurement. A comparison with the merely formal uncertainties in some of the orbitalsecular rates of Mercury, recently obtained by Iorio (2019) from the EPM2017 ephemerides,showed that the sizes of the predicted 1pN N -body precessions are just at the same level or evenbelow them if, more realistically, they are rescaled by a factor of ≃ −
50 (Iorio 2019). If ourfuture knowledge of the orbit of the closest planet to the Sun will be adequately improved, thesystematic bias caused by other competing precessions could be removed by suitably designinglinear combinations of the other Keplerian orbital elements of Mercury, provided that theastronomers will determine also their supplementary advances in addition to the perihelion’s one. 13 –Fig. 1.— Numerically integrated time series, in blue, of the shifts of the semimajor axis a , eccen-tricity e , inclination I , longitude of the ascending node Ω , longitude of perihelion ̟ , and meanlongitude at epoch ǫ of Mercury induced by the sum of all the fifteen 1pN perturbing accelerationsof Eqs. (1) to (3) for X ranging from Venus to Saturn over a time span 1 cty long. The units arem for a and microarcseconds ( µ as) for all the other orbital elements. They were obtained for eachorbital element as di ff erences between two time series calculated by numerically integrating thebarycentric equations of motion of all the planets from Mercury to Saturn in Cartesian rectangularcoordinates with and without the aforementioned 1pN N -body accelerations. The initial condi-tions, referred to the Celestial Equator at the reference epoch J2000, were retrieved from the WEBinterface HORIZONS by NASA JPL; they were the same for both the integrations. The slopesof the secular trends, in yellow, fitted to the blue time series of ∆ I ( t ) , ∆Ω ( t ) , ∆ ̟ ( t ), and ∆ ǫ ( t )are ˙ I X1pN = − . µ as cty − , ˙ Ω X1pN = . µ as cty − , ˙ ̟ X1pN = . µ as cty − , ˙ ǫ X1pN = . µ as cty − ,respectively. 14 – REFERENCES