A numerical method for computing optimum radii of host stars and orbits of planets, with application to Kepler-11, Kepler-90, Kepler-215, HD 10180, HD 34445, and TRAPPIST-1
aa r X i v : . [ a s t r o - ph . E P ] S e p A numerical method for computing optimum radii of host starsand orbits of planets, with application to Kepler-11, Kepler-90,Kepler-215, HD 10180, HD 34445, and TRAPPIST-1Vassilis S. GeroyannisDepartment of Physics, University of Patras, [email protected] 10, 2020
Abstract
In the so-called “global polytropic model”, we assume planetary sys-tems in hydrostatic equilibrium and solve the Lane–Emden equation in thecomplex plane. We thus find polytropic spherical shells providing hostingorbits to planets. On the basis of this model, we develop a numericalmethod which has three versions. In its three-dimensional version, themethod is effective for systems with substantial uncertainties in the ob-served host star radius, and in the orbit of a particular planet (comparedto the uncertainties in the orbits of the other planets); the method usesas fixed entry values the observed orbits of the remaining planets. In itstwo-dimensional version, the method is effective for systems with substan-tial uncertainty in the host star radius; in this case, the method uses asfixed entry values the observed orbits of the planets. The one-dimensionalversion was previously developed and applied to several systems; in thisversion, the observed values of the host star radius and of the planetaryorbits are taken as fixed entry values. Our method can compute optimumvalues for the polytropic index of the global polytropic model which simu-lates the exoplanetary system, for the orbits of the planets, and (excludingthe one-dimensional version) for the host star radius.
Keywords: exoplanets: orbits; global polytropic model; hydrostatic equi-librium: Lane–Emden equation; stars: individual (Kepler-11, Kepler-90,Kepler-215, HD 10180, HD 34445, TRAPPIST-1)
Planetary orbits in the solar system and in exoplanetary systems have beenstudied by several authors within the framework of classical mechanics (see e.g.[1]-[6]). Alternatively, other investigators use the frameworks of scale relativity,relativity theory regarding the finite propagation speed of gravitational interac-tion, and quantum mechanics (see e.g. [7]-[11]).1e develop here a numerical method based on the equations of hydrostaticequilibrium of classical mechanics. These equations lead to the Lane–Emdendifferential equation, solved in the complex plane by the the so-called “com-plex plane strategy” (see e.g. [12], Section 3), which is effective for numericallystudying several astrophysical problems (see e.g. [13], [14]). The solution is thecomplex Lane–Emden function. According to the so-called “global polytropicmodel” (for preliminary concepts regarding this model, see [15], Sections 2-3),polytropic spherical shells defined by succesive roots of the real part of theLane–Emden function are appropriate places for accomodating planets.Our method has three versions. The “one-dimensional version”, developedearlier, has been applied to several exoplanetary systems ([16]-[20]). This ver-sion uses fixed entry values for the host star radius and for the planetary orbits.Entry values for the polytropic index of the global polytropic model which sim-ulates the system are taken from a properly defined interval of values. Thisversion can compute optimum values for the polytropic index and for the plan-etary orbits.The “two-dimensional version” is developed and used here for the first time.It takes fixed entry values for the planetary orbits. Entry values for the poly-tropic index of the global polytropic model and for the host star radius aretaken from two properly defined intervals of values. The method can computeoptimum values for the polytropic index, for the radius of the host star, and forthe planetary orbits.The “three-dimensional version” is also developed and used here for the firsttime. It works with fixed entry values for the orbits of the planets, exceptfor a particular planet with substantial uncertainty in its orbit in comparisonwith the uncertainties in the orbits of the other planets. Entry values for thepolytropic index of the global polytropic model, for the host star radius, and forthe orbit of the particular planet are taken from three properly defined intervalsof values. The method can compute optimum values for the polytropic index,for the radius of the host star, and for the planetary orbits.
For convenience, we will use hereafter the definitions and symbols adopted in[15].The real part ¯ θ ( ξ ) of the complex function θ ( ξ ) has a first root at ξ =¯ ξ + i ˘ ξ , a second root at ξ = ¯ ξ + i ˘ ξ with ¯ ξ > ¯ ξ , a third root at ξ =¯ ξ + i ˘ ξ with ¯ ξ > ¯ ξ , etc. The polytropic sphere of polytropic index n andradius ¯ ξ is the central component of a resultant polytropic configuration withfurther components the polytropic spherical shells S , S , . . . , defined by thepairs of radii ( ¯ ξ , ¯ ξ ), ( ¯ ξ , ¯ ξ ), . . . , respectively. Each polytropic shell can beconsidered as an ideal hosting place for a planet. The most appropriate orbitradius ¯Ξ j ∈ [ ¯ ξ j − , ¯ ξ j ] is that at which | ¯ θ | takes its maximum value inside S j ,max | ¯ θ [ S j ] | = | ¯ θ (¯Ξ j + i ˘ ξ ) | . (1)2here are two further proper orbits with radii ¯Ξ Lj and ¯Ξ Rj , such that¯Ξ Lj < ¯Ξ j < ¯Ξ Rj , (2)at which | ¯ θ | becomes equal to its average value inside S j ,avg | ¯ θ [ S j ] | = | ¯ θ (¯Ξ Lj + i ˘ ξ ) | = | ¯ θ (¯Ξ Rj + i ˘ ξ ) | . (3)Accordingly, up to three planets can be hosted in S j on orbits with radii ¯Ξ Lj ,¯Ξ j , and ¯Ξ Rj .Our method can be applied to a system with N P planets, { P m } = { P m , m = 1 , . . . , N P } , (4)and with corresponding observed distances from the host star { A m } = { A m , m = 1 , . . . , N P } , such that A < A < · · · < A N P . (5)The method is based on an algorithm which takes action over a three-dimensional parametric space S = ( α [p] i , R j , n k ) , (6)where α [p] i are entry values for the orbit radius of a particular planet p withsubstantially large uncertainty in its observed value in comparison with theuncertaities in the orbit radii of the other planets; R j are entry values for thehost star radius R ; and n k are entry values for the polytropic index n of theglobal polytropic model which simulates the system.It is expected that appropriate values of the polytropic index n for modelingthe planetary systems under consideration are about n ∼ n are provided by an array { n k } = { n k , k = 1 , . . . , N n + 1 } (7)with elements n k = 2 .
400 + 0 . × ( k − , k = 1 , , . . . , N n + 1 , (8)and N n = 900 . (9)The 901 complex “initial value problems” (IVP, IVPs) counted in Eq. (8)are solved by the Fortran package dcrkf54.f95 [14] which is a Runge–Kutta–Fehlberg code of fourth and fifth order modified for solving complex IVPs es-tablished on ordinary differential equations of various complex functions in onecomplex variable, along contours prescribed as continuous chains of straight-linesegments; details on the usage of dcrkf54.f95 are also given in [15] (Section 4).Integrations proceed along the contour C = { ξ = (10 − , − ) → ξ end = (10 , − ) } . (10)3his contour belongs to the special form (8) of [15]; various contours and theircharacteristics are defined in [14] (Section 5).Since physical interest focuses on real parts of complex orbit radii, we willhereafter quote only such values and, for simplicity, we will drop overbars de-noting real parts.For each n k , the algorithm computes the array { ξ l , l = 1 , . . . , N r } n k , wherethe integer N r is chosen adequately large. Thus, next to the first root { ( ξ ) } n k = { ( ξ ) k , k = 1 , . . . , N n } , (11)there are computed N r − N r − N H = 3 ( N r −
1) (12)hosting orbits; namely, { Ψ l , l = 1 , . . . , N H } n k = { Ξ L2 , Ξ , Ξ R2 , . . . , Ξ LN H , Ξ N H , Ξ RN H } n k . (13)Accordingly, computations over all entry values n k give the two-dimensionalarray { Ψ kl } = { Ψ kl , k = 1 , . . . , N n + 1 , l = 1 , . . . , N H } . (14)Entry values for the host star radius are taken from the array { R j } = { R j , j = 1 , . . . , N R + 1 } . (15)In particular, if the quoted host radius (observed or estimated by an appropriatemodel) is R q with uncertainty ± ( R q ) u , then the array elements R j are given by R j = R q − ( R q ) u + ( j −
1) 2 ( R q ) u N R , j = 1 , . . . , N R + 1 . (16)Entry values for the orbit radius of a particular planet p with substantiallylarge uncertainty in its orbit radius, compared to those of the other planets, areprovided by the array { α [p] i } = { α [p] i , i = 1 , . . . , N α + 1 } . (17)In detail, if the quoted observed distance of the planet p is A [p]q with uncertainty ± ( A [p]q ) u , then the array elements α [p] i are given by α [p] i = A [p]q − ( A [p]q ) u + ( i −
1) 2 ( A [p]q ) u N α , i = 1 , . . . , N α + 1 . (18)For the model with current indices i , j , k , the method assigns to the planet P m the orbit radius α m for which the ratio α ml = Ψ kl / ( ξ ) k has the minimumabsolute percent difference with respect to the ratio A m /R j among all the in-dices l . Next, the method computes the sum of the minimum absolute percentdifferences of the assigned orbits over all the planets of the system.4mong the ( N α +1) × ( N R +1) × ( N n +1) resolved models, “optimum model”,with indices I , J , K , is the model for which: (a) the sum of the minimumabsolute percent differences becomes minimum among all the models (denotedby ∆ opt ), and (b) the indices I and J are not coinciding with their startingvalues, i = 1 and/or j = 1, or with their terminating values, i = ( N α +1) and/or j = ( N R +1). The meaning of the condition (b) is that the observed uncertaintiesinvolved in the definitions of the arrays (16) and (18) of the entry values shouldindeed bound the host star radius and the orbit radius of the particular planetp. Failure in obeying this condition points out that the observed uncertaintieshave been probably underestimated. In such a case, the array intervals definedby the relations (16) and (18) are properly extended, either to the left or to theright, and the computations are repeated for these new intervals.The polytropic index n K is the “optimum polytropic index” for the globalpolytropic model simulating the system; the host star radius R J is the “optimumhost star radius” predicted by the method; and the orbit radius α [p] I is the“optimum orbit radius” for the particular planet p predicted by the method. Our method is implemented by a Fortran code consisting of two packages. Thefirst package treats a particular system up to Eq. (14) by solving the IVPs in-volved in the problem. Basic constituent of the first package is the Fortrancode dcrkf54.f95 [14]. The use of this code has been adequately describedin previous investigations ([14], [15]). In fact, the first package performs thebookkeeping of the numerical results computed by dcrkf54.f95 . The secondpackage controls all necessary iterations (DO Loops) over the entry values re-lated to Eqs. (7)-(8), (15)-(16), and (17)-(18).
To proceed with a macro-description of the second package by using Fortranconventions, we need to assign Fortran names to the following involved vari-ables:
N P = N P (Eq. (4)), A P(1:N P )= { A m , m = 1 , . . . , N P } (Eq. (5)), N n = N n + 1 (Eq. (7)), PLI(1:N n )= { n k , k = 1 , . . . , N n + 1 } (Eq. (7)), x1(1:N n )= { ( ξ ) k , k = 1 , . . . , N n + 1 } (Eq. (11)), N H = N H (Eq. (12)), PSI(1:N n,1:N H )= { Ψ kl , k = 1 , . . . , N n + 1 , l = 1 , . . . , N H } (Eq. (14)), N R = N R + 1 (Eq. (15)), R(1:N R )= { R j , j = 1 , . . . , N R + 1 } (Eq. (15)), Position of App = N [p] (Eqs. (17)-(18)), N a = N α + 1 (Eq. (17)), app(1:N a )= { α [p] i , i = 1 , . . . , N α + 1 } (Eq. (17)). SUM min opt =∆ opt (Section 2) 5hen a macro-description of the second package has as follows: ! Start of Loop AppOrbitRadiiAppOrbitRadii: DO I=1,N_a! Start of Loop HostStarRadiiHostStarRadii: DO J=1,N_R! Start of Loop PolytropicIndicesPolytropicIndices: DO K=1,N_n! Start of Loop HostingOrbitsHostingOrbits: DO L=1,N_Halpha(L)=PSI(K,L)/x1(K)END DO HostingOrbits ! End of Loop HostingOrbits! Start of Loop HostedPlanetsHostedPlanets: DO M=1,N_PIF (M==Position_of_App) THENA_P(M)=app(I)END IFAUX_A_P(1:N_H)=A_P(M)/R(J)difmin_P(M)=MINVAL(ABS(AUX_A_P-alpha))MINPOS_M =MINLOC(ABS(AUX_A_P-alpha))orbits_P(M)=MINPOS_MEND DO HostedPlanets ! End of Loop HostedPlanetsSUM_min_n(K) =SUM(difmin_P(1:N_P))orbits_n(K,1:N_P)=orbits_P(1:N_P)END DO PolytropicIndices ! End of Loop PolytropicIndicesSUM_min_R(J) =MINVAL(SUM_min_n(1:N_n))MINPOS_J =MINLOC(SUM_min_n(1:N_n))PLI_opt_R(J) =PLI(MINPOS_J)ORBITS_opt_R(J,1:N_P)=orbits_n(MINPOS_J,1:N_P)END DO HostStarRadii ! End of Loop HostStarRadiiSUM_min_a(I) =MINVAL(SUM_min_R(1:N_R))MINPOS_I =MINLOC(SUM_min_R(1:N_R))R_opt_a(I) =R(MINPOS_I)PLI_opt_a(I) =PLI_opt_R(MINPOS_I)ORBITS_opt_a(I,1:N_P)=ORBITS_opt_R(MINPOS_I,1:N_P)END DO AppOrbitRadii ! End of Loop AppOrbitRadii! Final Session: Overall EstimatesSUM_min_opt =MINVAL(SUM_min_a(1:N_a))MINPOS =MINLOC(SUM_min_a(1:N_a))app_opt =app(MINPOS)RADIUS_opt =R_opt_a(MINPOS)POLIND_opt =PLI_opt_a(MINPOS)ORBITS_opt(1:N_P)=ORBITS_opt_a(MINPOS,1:N_P) Loop
HostingOrbits
For the current K-th polytropic index
PLI(K) , the array alpha(1:N H) isset equal to the array subobject
PSI(K,1:N H)/x1(K) having elements the orbitradii measured with unit the first root x1(K) .Loop
HostedPlanets
The elements of the auxiliary array
AUX A P(1:N H) are set equal to theobserved distance
A P(M) of the current M-th planet measured with unit the6urrent J-th host’s radius
R(J) . If
M=Position of App , then
A P(M) is setequal to the current I-th entry app(I) . MINPOS M is the position in the array alpha(1:N H) occupied by the orbit radius having the minimum absolute per-cent diference difmin P(M) relative to the distance
A P(M) of the current M-thplanet, and the polytropic orbit orbits P(M) hosting this planet is set equal to
MINPOS M .Loop
PolytropicIndices
For the current K-th polytropic index
PLI(K) , the element
SUM min n(K) isset equal to the sum of the elements of the array difmin P(1:N P) . Next, the ar-ray subobject orbits n(K,1:N P) is set equal to the array orbits P(1:N P) ;so, the rank-2 array orbits n(1:N n,1:N P) is the extension of the rank-1array orbits P(1:N P) over the dimension .Loop
HostStarRadii
For the current J-th host’s radius
R(J) , the element
SUM min R(J) is setequal to the minimum value of the array
SUM min n(1:N n) , occupying the po-sition
MINPOS J . The element
PLI opt R(J) having the minimum sum
SUM minR(J) is set equal to the element
PLI(MINPOS J) of the array
PLI(1:N n) . Thearray subobject
ORBITS opt R(J,1:N P) is set equal to the array subobject orbits n(MINPOS J,1:N P) ; so, the rank-2 array
ORBITS opt R(1:N R,1:N P) is the extension of
ORBITS opt R(J,1:N P) over the dimension .Loop
AppOrbitRadii
For the current I-th entry app(I) , the element
SUM min a(I) is set equalto the minimum value of the array
SUM min R(1:N R) , occupying the position
MINPOS I . The element
R opt a(I) having the minimum sum
SUM min a(I) is set equal to the element
R(MINPOS I) of the array
R(1:N R) . The element
PLI opt a(I) having the minimum sum
SUM min a(I) is set equal to the ele-ment
PLI opt R(MINPOS I) of the array
PLI opt R(1:N R) . The array subob-ject
ORBITS opt a(I,1:N P) is set equal to the array subobject
ORBITS opt R(MINPOS I, 1:N P) ; so, the rank-2 array
ORBITS opt a(1:N a,1:N P) is theextension of
ORBITS opt a(I,1:N P) over the dimension .Final Session: Overall EstimatesThe “optimum minimum sum”,
SUM min opt , of the absolute percent dif-ferences of the computed planetary orbit radii relative to their observed valuesis the minimum value of the array
SUM min a(1:N a) , occupying the position
MINPOS . The “optimum orbit radius”, app opt , of a particular planet p with asubstantially high uncertainty (compared to the uncertainties in the distancesof the other planets) is the element app(MINPOS) of the array app(1:N a) . The“optimum host star radius”,
RADIUS opt , is the element
R opt a(MINPOS) ofthe array
R opt a(1:N a) . The “optimum polytropic index”
POLIND opt isthe element
PLI opt a(MINPOS) of the array
PLI opt a(1:N a) . The “opti-mum planetary orbits”,
ORBITS opt(1:N P) , are the respective elements of thearray
ORBITS opt a(MINPOS,1:N P) . 7
The Two- and One-Dimensional Versions
There are numerous exoplanetary systems listed in NExA, as well as in otherexoplanet archives, for which the uncertainties in the orbits of their planetsare comparable. In addition, for several exoplanetary systems appearing in thearchives, observational data regarding uncertaities in the planetary orbits aremissing. In both cases, we cannot distinguish a planet with a substantially largeruncertainty in its orbit. Hence, the two-dimensional version of our method is theeffective one for such systems. It is easy to implement this version, instead ofits three-dimensional counterpart, by simply setting N α = 0 in the relation (18)and α [p]1 = A for the unique entry value.In this paper, the one-dimensional version of our method is not implemented.Typically, this version is appropriate for studying systems with small or missingobservational uncertainties in the radii of the host stars and, as said above, withcomparable or missing uncertainties in the planetary orbits. Implementation ofthis version can be achieved by additionally setting N R = 0 in the relation (16)and R = R q for the unique entry value. Numerical results of several exoplan-etary systems computed by the one-dimensional method are given in [16]-[20];some of these results are discussed below.In terms of the quantities involved in the Fortran code (Section 3.2), use ofthe two-dimensional version is achieved by setting N a=1 and app(1)=A P(1) .To apply the one-dimensional version, we additionally set
N R=1 and
R(1)=R q . We select the exoplanetary systems Kepler-11, Kepler-90, Kepler-215, HD 10180,HD 34445, and TRAPPIST-1 as paradigms for applying our method. Relevantobservational data are included in the “NASA Exoplanet Archive” (https://exo-planetarchive.ipac.caltech.edu/ — hereafter abbreviated “NExA”) unless explic-itly stated otherwise. The systems TRAPPIST-1 and Kepler-90 are the onlyones in NExA with number of planets N P ≥
7. Next, among the six sys-tems listed in NExA with N P = 6, we have selected three of them: Kepler-11,HD 10180, and HD 34445. On the other hand, the system Kepler-215 has fourplanets, the uncertainties in their orbits are missing from NExA, and the un-certainty in the host star radius is substantially large when compared to theuncertainties of other cases. The numerical results for the selected paradigmsreveal some interesting aspects of the method, verifying in turn its flexibilityand reliabity (to be discussed below).The symbols involved in Tables 1-6 have the following meaning: n opt is theoptimum polytropic index for the global polytropic model which simulates theexoplanetary system. ξ is the optimum radius of the host star given inclassical polytropic units, in which the length unit is equal to the polytropic8arameter α ([15], Eq. (3b)). R opt is the optimum radius of the host starexpressed in solar radii ( R ⊙ ). R q is the quoted radius of the host star, eitherobserved or computed by a model, and ( R q ) u is its uncertainty, both given insolar radii. Shell radii and orbit radii are given in astronomical units (AU). Forsuccessive shells S j and S j +1 , inner radius of S j +1 is the outer radius of S j .Percent differences % D j in the computed orbit radii α j are given with respectto the corresponding distances A j , % D j = 100 × | ( A j − α j ) | /A j . Parenthesizedsigned integers denote powers of 10.Hereafter, radii of the host stars and their uncertainties will be expressed insolar radii ( R ⊙ ); planetary orbit radii and their uncertainties will be expressedin astronomical units (AU). Furthermore, the meaning assigned to the term“difference” will be that of “absolute percent difference”. Regarding the 6-planet system Kepler-11 (see e.g. [22]-[25]), the computed op-timum minimum sum ∆ opt is found to be∆ opt ≃ . . (19)The average difference in the computed distances of the six planets is ≃ . ≃ . ≃ . α b = α ; and the latter ishosted on the “average-density orbit” (Eq. (3)) with radius α c = α R5 . Likewise,the shell No 6 is occupied by the planets d and e. The former is resident ofthe average-density orbit with radius α d = α L6 ; the latter is resident of theaverage-density orbit with radius α e = α L6 .The computed optimum radius R opt for the star Kepler-11 lies to the left ofthe interval [1 . , . R q ) u = +0 . − . in thequoted radius R q = 1 . R opt = 1 . < . , (20)and its absolute percent difference relative to R q is% D ( R opt ) ≃ . . (21)On the other hand, however, in [38] (Sect. 3) the revised value 1 .
021 is assignedto the radius of the star Kepler-11, with an uncertainty ± . . , . R opt = 1 . ∈ [1 . ± . , (22)with a difference ≃ .
93% relative to the revised radius.It is worth mentioning here that in [16] (Eq. (8) and Table 5) we appliedthe one-dimensional method to the system Kepler-11, with fixed radius 1 . n opt = 2 .
779 and ∆ opt ≃ . .2 The System Kepler-90 (KOI-351) For the 8-planet system Kepler-90 (see e.g. [26]-[28]), there is no information inNExA for the orbit radius A i of the planet i; concerning this distance, we adoptfrom [28] (Table 5) the value A i = 0 . opt ≃ . . (23)The average difference in the computed distances of the eight planets is ≃ . ≃ . ≃ . R opt for the star Kepler-90 lies in the interval deter-mined by the uncertainty ( R q ) u of the quoted radius R q , R opt = 1 . ∈ [1 . ± . , (24)and its difference relative to R q is equal to% D ( R opt ) ≃ . . (25)Note that in [16] (Eq. (10) and Table 9) we studied the system Kepler-90with the one-dimensional method by taking fixed radius 1 . n opt = 2 .
819 and ∆ opt ≃ . For the 4-planet system Kepler-215 (see e.g. [29]), the computed optimum modelgives ∆ opt ≃ . . (26)The average difference in the computed distances of the four planets is ≃ . ≃ . ≃ . R opt for the star Kepler-215 lies in the in-terval determined by the uncertainty ( R q ) u of the quoted radius R q , R opt = 1 . ∈ [1 . ± . , (27)and its difference relative to R q is equal to% D ( R opt ) ≃ . . (28)10 .4 The System HD 10180 For the 6-planet system HD 10180 (see e.g. [30]-[32]), the computed optimummodel gives ∆ opt = 14 . . (29)The average difference in the computed distances of the six planets is ≃ . ≃ . ≃ . R opt for the star HD 10180 lies in the interval deter-mined by the uncertainty ( R q ) u of the quoted radius R q , R opt = 1 . ∈ [1 . ± . , (30)and its difference relative to R q is equal to% D ( R opt ) ≃ . . (31)It is interesting to mention here that in [19] (Eq. (1) and remarks followingthis equation) we studied the system HD 10180 by applying the one-dimensionalmethod, taking fixed radius 1 . n opt = 3 .
060 and ∆ opt ≃ . Concerning the 6-planet system HD 34445 (see e.g. [33]-[34]), it is apparentfrom the available data that the planet g has a substantially larger uncertainty,( A [g]q ) u = ± .
02, in its observed distance, A [g]q = 6 .
36, in comparison with theuncertainties in the distances of the other five planets of the system. Thus, thethree-dimensional version of our method is the effective one for this system.For the optimum model, we find∆ opt ≃ . . (32)So, the average difference in the computed distances of the six planets is ≃ . ≃ . ≃ . R opt for the star HD 34445 lies to the leftof the interval [1 . , .
40] determined by the uncertainty ( R q ) u = ± .
02 in thequoted radius R q = 1 .
38, that is R opt = 1 . < . , (33)and its difference relative to R q is equal to% D ( R opt ) ≃ . . (34)However, according to the data given in [33] (Table 1; see also [34], Table 1),the uncertainty in the radius of HD 34445 is assigned the value ± .
08; so, theinterval of values becomes [1 . , .
46] and the computed optimum radius lies inthis interval, R opt = 1 . ∈ [1 . ± . . (35)Regarding the optimum distance a [g]opt of the planet g, we find that it liesin the interval defined by the uncertainty ( A [g]q ) u of the quoted distance A [g]q , a [g]opt = 6 . ∈ [6 . ± .
02] (36)with a difference relative to A [g]q equal to% D ( a [g]opt ) ≃ . . (37) For the 7-planet system TRAPPIST-1 (see e.g. [5], [35]-[39]), there is no infor-mation in NExA for the orbit radius A h of the planet h; for this orbit radius,we adopt from [36] (Table 1) the value A h = 0 . A [h]q ) u = +0 . − . , in its observed distance, A [h]q = 0 . opt is∆ opt ≃ . , (38)and the average difference in the computed distances of the seven planets is ≃ . ≃ .
06% (the zero differencefor h’s distance is excluded from the comparison, since this distance plays aparametric role in our method). Larger difference is that for g’s distance, ≃ . R opt for the star TRAPPIST-1 lies beyondthe right bound 0 . . , . R q ) u = ± . R q = 0 . R opt = 0 . > . , (39)and its absolute percent difference relative to R q is equal to% D ( R opt ) ≃ . . (40)It is worth remarking, however, that in Table 1 of [38] the updated value for theradius of the star TRAPPIST-1 is 0 .
121 with an uncertainty ± . . , . R opt = 0 . ∈ [0 . ± . , (41)with a difference ≃ .
5% relative to the updated radius.Regarding the optimum distance of the planet h, we find that it lies in theinterval defined by the uncertainty ( A [h]q ) u of the quoted distance A [h]q , a [h]opt = 0 . ∈ [0 . +0 . − . ] , (42)with a difference relative to A [h]q equal to% D ( a [h]opt ) ≃ . . (43)It would be useful to quote here a previous investigation ([20]; Eq. (1) andTable 1), in which we treated numerically the system TRAPPIST-1 by applyingthe one-dimensional method, with fixed radius 0 .
117 for the host star and fixedorbit radius 0 .
063 for the planet h. Our computations resulted in the values n opt = 2 .
525 and ∆ opt ≃ . The predictions given by our method can be eventually verified by future ob-servations and/or new numerical models. An interesting case pointing to pre-diction(s) arises when the method fails to satisfy the condition (b) (Section 2)in its first run. As discussed in Section 2, failure in fulfilling the condition (b)shows that the corresponding observed uncertainties have been probably under-estimated. If so, the method extends properly the intervals (16) and/or (18) ofentry values for the host star radius and/or for the orbit radius of a particularplanet p with substantial uncertainty in its orbit, and then proceeds with asecond run.Such a case has emerged during the study of the system Kepler-11. As saidin Section 5.1, the optimum host star radius given in Table 1 lies to left of theinterval determined by the observed uncertainties. This optimum value has been13omputed by a second run, with the interval of entry values properly extendedto the left. By taking into account the revised value 1 .
021 and its uncertainty ± .
025 quoted in [25] (Section 3), we verify that the computed optimum radiuslies in the revised interval of values.Likewise, as said in Section 5.5, in the case of the system HD 34445 theoptimum radius of the host star given in Table 5 lies to the left of the intervaldetermined by the observed uncertainties. This optimum radius has been com-puted by a second run of the method with the interval of entry values properlyextended to the left. By taking into account the revised uncertainty ± .
08 inthe radius of HD 34445 given in [33] (Table 1; see also [34], Table 1), we deducethat the computed optimum radius lies in the revised interval of values.A third similar case has emerged in the treatment of the system TRAPPIST-1. As said in Section 5.6, the optimum host star radius given in Table 6 lies toright of the interval determined by the observed uncertainties. This value hasbeen computed by a second run, with interval of values properly extended tothe right. In accordance with the updated value 0 .
121 for the radius of the starTRAPPIST-1 and its uncertainty ± .
003 given in [38] (Table 1), the computedoptimum radius lies in this updated interval of values.Furthermore, our method can show flexibility when studying systems withsmall or missing uncertainties in the radii of the host stars. In such systems,the method can ‘pretend’ that there are certain appreciable uncertainties inthe observed radii and, accordingly, resolve the systems by the two-dimensionalversion instead of its one-dimensional counterpart (which could typically used).A relevant case is that of the system Kepler-11. The quoted uncertainties in theradius of the host star are small (in fact, the smallest ones among the selectedparadigms), ( R q ) u = +0 . − . , i.e. ∼ ( ± R q = 1 . . opt is relatively large when comparedto the corresponding values found by the two- and three-dimensional methods.For example, for the system Kepler-11 (Section 5.1), this minimum is ∼ ≃ .
7% estimated by the two-dimensional method; forthe system HD 10180 (Section 5.4), this minimum is ∼ ≃ .
2% estimated by the two-dimensional method; and for the systemTRAPPIST-1 (Section 5.6), the minimum is ∼ ≃ .
3% estimated by the three-dimensional method.In addition, there is a relevant question arising here: Can we reduce ‘de-liberately’ the dimensions of our method in a particular application withoutdecreasing the accuracy of the respective results? As discussed above, the as-trophysical data available for a given exoplanetary system show themselves theneed for applying to this system the three-dimensional method. Namely, if thereis a planet p in this system having substantially larger uncertainty in its orbitwith respect to the uncertainties in the orbits of the other planets, then thesystem must be treated numerically by the three-dimensional method. A ‘delib-14rate’ application of the two-dimensional method in the given system, instead,reduces the accuracy of the simulation. For instance, when applying the two-dimensional method to the system HD 34445, with fixed orbit radius 6.36 forthe planet g, we find n opt = 2 . opt ≃ . R opt = 1 . opt is ∼ ≃ .
96 foundby the three-dimensional method. Likewise, when applying the two-dimensionalmethod to the system TRAPPIST-1, with fixed orbit radius 0.063 for the planeth, we find n opt = 2 . opt ≃ . R opt = 0 . opt is ∼ . ≃ . N P < . . . . . . . . S , i.e. the host star Kepler-11,and polytropic spherical shells of the planets b, c, d, e, f, and g.Host star Kepler-11 – Shell No 1 R q ( R q ) u n opt .
833 (+00) ξ . R opt . . +1 . − − . − Planets of the system A % D b – Shell No 5Inner radius, ξ . − ξ . − α b = α . −
02) 9 . −
02) 1 . − α c = α R5 . −
01) 1 . −
01) 1 . − ξ . − α d = α L6 . −
01) 1 . −
01) 4 . α e = α R6 . −
01) 1 . −
01) 1 . ξ . − α f = α L7 . −
01) 2 . −
01) 1 . ξ . − α g = α R8 . −
01) 4 . −
01) 9 . − S , i.e. the host star Kepler-90,and polytropic spherical shells of the planets b, c, i, d, e, f, g, and h.Host star Kepler-90 – Shell No 1 R q ( R q ) u n opt .
784 (+00) ξ . R opt . . ± . − A % D b – Shell No 4Inner radius, ξ . − ξ . − α b = α R4 . −
02) 7 . −
02) 8 . − ξ . − α c = α L5 . −
02) 8 . −
02) 2 . ξ . − α i = α . −
01) 2 . −
01) 4 . ξ . − α d = α . −
01) 3 . −
01) 3 . − ξ . − α e = α R8 . −
01) 4 . −
01) 1 . − ξ . − α f = α L9 . −
01) 4 . −
01) 2 . ξ . − α g = α . −
01) 7 . −
01) 1 . − ξ . α h = α R11 . . . − S , i.e. the host star Kepler-215,and polytropic spherical shells of the planets b, c, d, and e.Host star Kepler-215 – Shell No 1 R q ( R q ) u n opt .
884 (+00) ξ . R opt . . ± . − A % D b – Shell No 4Inner radius, ξ . − ξ . − α b = α L4 . −
02) 8 . −
02) 7 . − ξ . − α c = α . −
01) 1 . −
01) 2 . − ξ . − α d = α . −
01) 1 . −
01) 3 . − ξ . − α e = α . −
01) 3 . −
01) 9 . − S , i.e. the host star HD 10180,and polytropic spherical shells of the planets c, d, e, f, g, and h.Host star HD 10180 – Shell No 1 R q ( R q ) u n opt .
096 (+00) ξ . R opt . . ± . − A % D c – Shell No 3Inner radius, ξ . − ξ . − α b = α R3 . −
02) 6 . −
02) 7 . ξ . − α c = α . −
01) 1 . −
01) 2 . − ξ . − α d = α . −
01) 2 . −
01) 3 . − ξ . − α e = α . −
01) 4 . −
01) 1 . ξ . ξ . α f = α R9 . . . ξ . ξ . α g = α . . . S , i.e. the host star HD 34445,and polytropic spherical shells of the planets e, d, c, f, b, and g.Host star HD 34445 – Shell No 1 R q ( R q ) u n opt .
421 (+00) ξ . R opt . . ± . − A % D e – Shell No 9Inner radius, ξ . − ξ . − α e = α . −
01) 2 . −
01) 3 . − ξ . − ξ . − α d = α L12 . −
01) 4 . −
01) 7 . − ξ . − ξ . − α c = α R14 . −
01) 7 . −
01) 3 . − ξ . ξ . α f = α R20 . . . − ξ . ξ . α b = α L24 . . . − A [g]q ( A [g]q ) u g – Shell No 41Inner radius, ξ . ξ . α [g]opt = α L . . ± . S , i.e. the host starTRAPPIST-1, and polytropic spherical shells of the planets b, c, d, e, f, g,and h.Host star TRAPPIST-1 – Shell No 1 R q ( R q ) u n opt .
466 (+00) ξ . R opt . −
01) 1 . − ± . − A % D b – Shell No 6Inner radius, ξ . − ξ . − α b = α R6 . −
02) 1 . −
02) 3 . − ξ . − α c = α . −
02) 1 . −
02) 2 . ξ . − α d = α L8 . −
02) 2 . −
02) 5 . − ξ . − α e = α R9 . −
02) 2 . −
02) 1 . − ξ . − α f = α R10 . −
02) 3 . −
02) 1 . ξ . − α g = α R11 . −
02) 4 . −
02) 1 . A [h]q ( A [h]q ) u h – Shell No 13Inner radius, ξ . − ξ . − α [h]opt = α . −
02) 6 . − +2 . − − . − eferences [1] D. M. Christodoulou and D. Kazanas, arXiv:0706.3205v2 [astro-ph], 2008.[2] D. M. Christodoulou and D. Kazanas, arXiv:0811.0868v1 [astro-ph], 2008.[3] D. M. Christodoulou and D. Kazanas, arXiv:1901.10642v2 [astro-ph.EP],2019.[4] D. M. Christodoulou and D. Kazanas, arXiv:1903.01019v1 [astro-ph.EP],2019.[5] D. M. Christodoulou and D. Kazanas, Research Notes of the AmericanAstronomical Society, Vol. 3, No 3, 2019.[6] P. Pintr, V. Peˇrinov´a and A. Lukˇs, Chaos, Solitons and Fractals , Vol. 36,2008, pp. 1273-1282.[7] R. Hermann, G. Schumacher and R. Guyard,
Astronomy and Astrophysics ,Vol. 335, 1998, pp. 281-286.[8] J. Gin´e,
Chaos, Solitons and Fractals , Vol. 32, 2007, pp. 363-369.[9] A. G. Agnese and R. Festa,
Physics Letters A , Vol. 227, 1997, pp. 165-171.[10] A. Rubˇci´c and J. Rubˇci´c,
FIZIKA B , Vol. 7, 1998, pp. 1-13.[11] M. de Oliveira Neto, L. A. Maia and S. Carneiro,
Chaos, Solitons andFractals , Vol. 21, 2004, pp. 21-28.[12] V. S. Geroyannis and V. G. Karageorgopoulos,
New Astronomy , Vol. 28,2014, pp. 9-16.[13] V. S. Geroyannis,
The Astrophysical Journal , Vol. 327, 1988, pp. 273-283.[14] V. S. Geroyannis and F. N. Valvi,
International Journal of Modern PhysicsC , Vol. 23, 2012, Article Code: 1250038, 15 pages.[15] V. Geroyannis, F. Valvi and T. Dallas,
International Journal of Astronomyand Astrophysics , Vol. 4, 2014, pp. 464-473.[16] V. S. Geroyannis, arXiv:1410.5844v2 [astro-ph.EP], 2014.[17] V. S. Geroyannis, arXiv:1411.5390v1 [astro-ph.EP], 2014.[18] V. S. Geroyannis, arXiv:1501.04189v1 [astro-ph.EP], 2015.[19] V. S. Geroyannis, arXiv:1506.06344v1 [astro-ph.EP], 2015.[20] V. S. Geroyannis, Gravitational quantization of exoplanet orbits: The sys-tem TRAPPIST-1, 13th Hellenic Astronomical Society Conference, Herak-lion, Crete, 2-6 July 2017, helas.gr/conf/2017/posters/S 4/vgeroyannis.pdf2221] G. P. Horedt,
Polytropes: Applications in Astrophysics and Related Fields (Kluwer Academic Publishers, New York, 2004).[22] J. J. Lissauer, D. Jontof-Hutter, J. F. Rowe, et al.,
Astrophysical Journal ,Vol. 770, 2013, pp. 131-145.[23] N. Mahajan and Y. Wu, arXiv:1409.0011v1 [astro-ph.EP], 2014.[24] L. Borsato, F. Marzari, V. Nascimbeni, et al.,
Astronomy and Astrophysics ,Vol. 571, 2014, A38.[25] M. Bedell, J. L. Bean, J. Melendez, et al., arXiv:1611.06239v2 [astro-ph.EP], 2017.[26] J. Cabrera, Sz. Csizmadia, H. Lehmann, et al., arXiv:1310.6248v2 [astro-ph.EP], 2013.[27] J. R. Schmitt, J. Wang, D. A. Fischer, et al., arXiv:1310.5912v3 [astro-ph.EP], 2014.[28] C. J. Shallue and A. Vanderburg, arXiv:1712.05044v1 [astro-ph.EP], 2017.[29] J. F. Rowe, S. T. Bryson, G. W. Marcy, et al., arXiv:1402.6534v1 [astro-ph.EP] (2014)[30] C. Lovis, D. S´egransan, M. Mayor, et al., arXiv:1011.4994v1 [astro-ph.EP],2010.[31] M. Tuomi, arXiv:1204.1254v1 [astro-ph.EP], 2012.[32] S. R. Kane and D. M. Gelino, arXiv:1408.4150v1 [astro-ph.EP], 2014.[33] A. W. Howard, J. A. Johnson, G. W. Marcy, et al.,