A method is suggested to determine of minimum distance between orbits of two celestial bodies. The method is based on finding of the singular points of one variable function. The orbits of celestial bodies can be elliptic, parabolic or hyperbolic. The orbits of celestial bodies can also be described universal variable. On the basis of the given method the program is developed, using which the accounts of minimal distance between orbits of comets and orbits of planets are executed. The given method can also be used for finding of the minimal distances both between orbits of asteroids and orbits of planets, and between asteroids orbits

Babenko, Yu.

Kinematika i fizika nebesnyh tel (Online)

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

DETERMINATION OF MINIMUM DISTANCEBETWEEN ORBITS OF CELESTIAL BODIESYu. BabenkoAstronomical Observatory, National Taras Shevchenko University of Kyiv3 Observatorna Str., 04053 Kyiv, Ukrainee-mail: babenko@observ.univ.kiev.uaA method is suggested to determine of minimum distance between orbits of two celestial bodies.The method is based on ﬁnding of the singular points of one variable function. The orbits ofcelestial bodies can be elliptic, parabolic or hyperbolic. The orbits of celestial bodies can also bedescribed universal variable. On the basis of the given method the program is developed, usingwhich the accounts of minimal distance between orbits of comets and orbits of planets are executed.The given method can also be used for ﬁnding of the minimal distances both between orbits ofasteroids and orbits of planets, and between asteroids orbits.INTRODUCTIONThe minimal distances between orbits of celestial bodies are heavy weighed. If they are small enough thatprobably close approach celestial bodies. It is especially important in case of comet, when the small value ofthis minimal distance can serve indication that the close approach of the comet and planet could take place, asa result of which there could be an essential change of elements of an orbit of a comet. There is one more case,when it is important to know this minimal distance is a case of orbits asteroids and orbit of the Earth, whensmall distance between their orbits can be indication on an opportunity of collision this asteroid with the Earth.DETERMINATION OF MINIMUM DISTANCE BETWEEN ORBITSIn [2] the case of elliptic orbits of two bodies is considered and is shown that the given task can be reduceto ﬁnding of zero function by one variable. By the author [1] is shown that the ﬁnding of minimal distancebetween an elliptic orbit both hyperbolic and parabolic orbits can also be reduce to a ﬁnding of zero function byone variable. In connection with this it represents the great interest, including practical, whether a ﬁnding ofminimal distance between orbits of any type, including in a case, when for the description of an orbit are useduniversal variable, is possible, to reduce a ﬁnding of zero function by one variable, to that the present work isdevoted.Let us consider a task of a ﬁnding of minimal distance between orbits of two celestial bodies in the mostgeneral case.The system of coordinates, where the XY plane coincides with a celestial body orbital plane, and the X axisis directed in an orbit perihelion may be written as{x = rx(a, e, φ),y = ry(a, e, φ),(1)where rx and ry are also functions dependent on major axis, eccentricity and variable φ, which will be concretizedbelow. It is clear that the radius-vector r =√x2 + y2 will be function of the same variable.As it is known, in ecliptic system of coordinates the radius-vector r can be expressed by rpx, rpy , and also byargument perihelion ω, longitude of the ascending node Ω and inclination of an orbit i:r = Prx +Qry, (2)where P and Q are the orbital vectors of the constants, which are expressed by ω, Ω, and i:Px = cosω cosΩ− sinω sinΩ cos i, Qx = − sinω cosΩ + cosω sinΩ cos i,Py = cosω sinΩ + sinω cosΩ cos i, Qy = − sinω sinΩ− cosω cosΩ cos i,Pz = sinω sin i, Qz = cosω sin i.(3)c© Yu. Babenko, 2004423Marking everything, concerns to the ﬁrst body by 1, and to the second one by 2 it is possible to writethe following formula for a square of distance between bodies ρ :ρ2 = (r2 − r1, r2 − r1) = r22 − 2(r2, r1)− r21, (4)where ρ is a function of two variables φ1 and φ2. It will depend on parameters of orbits of two celestial bodies.As it is known, for a ﬁnding of its minimum it is necessary to equate with zero its derivative of variable andto solve concerning them the turned out equations system. If such decisions will be a little bit that to ﬁnddistances for all values of the decisions and minimal of the found distances will be required minimal distancebetween orbits of these two celestial bodies. From above-stated it is received the following equations system:⎧⎪⎪⎨⎪⎪⎩∂ρ2∂φ1= 2r1dr1dφ1− 2(r2,dr1dφ1)= 0,∂ρ2∂φ2= 2r2dr2dφ2− 2(dr2dφ2, r1)= 0.The obtained system can also be rewrote as follows:⎧⎪⎪⎨⎪⎪⎩r1dr1dφ1−(r2,dr1dφ1)= 0,r2dr2dφ2−(dr2dφ2, r1)= 0.(5)With (1) and (2) the equations system (5) is possible to rewrite in the following kind:⎧⎪⎨⎪⎩r1dr1dφ1− S1 r1xdφ1− S3 r1ydφ1= 0,r2dr2dφ2− S2r1x − S4r1y = 0,(6)where S1 = P1P2r2x + P1Q2r2y , S2 = P1P2dr2xdφ2+ P1Q2dr2ydφ2, S3 = Q1P2r2x + Q1Q2r2y, S4 = Q1P2dr2xdφ2+Q1Q2dr2ydφ2, and where by P1P2, Q1P2, P1Q2, and Q1Q2 deﬁned scalar products (P1,P2), (Q1,P2), (P1,Q2),(Q1,Q2).The further consideration we shall take at once for all types of orbits, including for a case of use forthe description of an orbital universal variable. In Table 1 for all these cases the values of rx, ry, r, drx/dφ,dry/dφ, dr/dφ are given without their expressions in order to save place for our paper. For an elliptic orbitthe eccentric anomaly E is chosen as independent variable, for a parabolic orbit σ = q tanf2is chosen, where qis the perihelion distance, f is the true anomaly, for a hyperbolic orbit H is chosen, which is as follows expressedby true anomaly: tanhH2=√e + 1e− 1 tanf2, for a case of use for the description of an orbital universal variable,i.e., generalized anomaly s, determined by expression μdtds= r, where μ is the gravitational constant, p isan orbital parameter. The functions Cn(z) are deﬁned by the formulas:Cn(z) =k=∞∑k=0(−1)k zk(2k + n)!; n = 1, 2, 3, ... . (7)Let us note that there are 10 variants of the equations system (6): E − E, E − P , E −H , E − U , P − P ,P −H , P −U , H−H , H−U , U−U , where for orbits types the following designations are entered: E is elliptic,P is parabolic, H is hyperbolic, U is used for the description of an orbital universal variable. We shall consideronly four cases, when the ﬁrst orbit is consistently chosen elliptic, parabolic, hyperbolic and written down inuniversal constant while for the second orbit we shall keep recording in a general view.Elliptic orbit. The equations system (6) with the accounting Table 1 accepts the following form:{M sinE1 + N cosE1 = K sinE1 cosE1,A sinE1 + B cosE1 = 0,(8)424Table 1. Expression for rx, ry, r, drx/dφ, dry/dφ, dr/dφ for various types of orbitsExpression Elliptic orbit Parabolic orbit Hyperbolic orbit Universal variablerx a(cos E − e) q(1− σ2) | a | (e− coshH) q − s2C2(αs2)rx a√1− e2 sinE 2q σ | a | √e2 − 1 sinhH) √qsC1(αs2)r a(1− e cosE) q(1 + σ) | a | (e coshH + 1) q C0(α s2)s2C2(αs2)drxdφ−a sinE −2qσ − | a | sinhH sC1(αs2)√μrdrydφa√1− e2 cos E 2q | a | √e2 − 1 coshH C0(αs2)√pμrdrdφae sinE 2q σ | a | e sinhH (1− αq)sC1(αs2)where M = S1 − a1e1, N = −√1− e21S3, K = −a1e1, A = a1√1− e21S4, B = a1S2, C = r2dr2dφ2. It followsfrom (8) that A, B, C, M , N , and K depend only on φ2 and do not depend on E1. Thus, to reduce the decisionof the given equations system to a ﬁnding of zero function by one variable it is necessary to ﬁnd sinE1 andcosE1 from the second equation and to substitute the received expressions in the ﬁrst equation.By solving the second equation from the equations system (8) we obtain the following expressions for sinE1and cosE1:sinE1 =AC ±B√A2 + B2 − C2A2 + B2,cosE1 =BC ∓A√A2 + B2 − C2A2 + B2.(9)As follows from (9) the decision of the equations system (8) only exists, when the condition: A2+B2−C ≥ 0is satisﬁed.Substituting the obtained expressions for sinE1 and cosE1 in the ﬁrst equation of the system (8) afteralgebraic transformations one can get:F = [CE˜(MA + NB) + KAB(G− C2)]−G[KC(B2 −A2)− E(MB −NA)]2 = 0, (10)where E˜ = A2 + B2, G = E˜ − C2.Parabolic orbit. The equations system (6) with the accounting Table 1 accepts the following form:{Mσ31 + Nσ1 = K,Aσ21 + 2Bσ1 = C,(11)where M = q1, N = q1 − S1, K = S3, A = q1S2, B = q1S4, C = r2 dr2dφ2− q1S2. Solving the second equation ofthe system (11) the following equation for σ may be written:σ1,2 =−B ±√B2 −ACA. (12)As follows from (12) the decision of the equations system (11) only exists, when the condition: B2 −AC ≥ 0 issatisﬁed.In this case substituting the obtained expression for σ1,2 in the second equation of the system (11) similarlyprevious one the following function F by one variable is obtained:F = (3MB2 + B2 −AC + A2N)(B2 −AC)− (A3K + MB3 + 3B2 − 3BAC + A2NB) = 0. (13)Thus, and in this case the searching for a minimal distance between a parabolic orbit and other orbit form wasreduced to a ﬁnding of zero function by one variable.Hiperbolic orbit. The equations system (6) with the accounting Table 1 may be written by the equations:{M sinhH1 + N coshH1 = K sinhH1 coshH1,A sinhH1 + B coshH1 = 0,(14)425where M =| a | −S1, N = −√e21 − 1S3, K =| a1 | e1, A =| a1 |√e21 − 1S4, B = − | a1 | S2, C = r2dr2dφ2S2.From the second equation in (14) we ﬁnd:sinhH1 =−AC ±B√A2 + C2 −B2B2 − C2 ,coshH1 =BC ∓A√A2 + C2 −B2B2 − C2 .(15)As follows from (15) the decision of the equations system (14) only exists, when the condition: A2+C2−B2 ≥ 0is satisﬁed. The values sinhH1, coshH1 in the ﬁrst equation of the system (14) were substituted for these valuesfrom (15) and the function F was obtained:F = [CE˜(AM −NB) + KAB(C2 + G)]2 −G[E˜(MB −NA) + KC(A2 + B2)]2 = 0, (16)where E˜ = A2 −B2, G = E˜ + C2.The case of use for the description of an orbital universal variable. We can write the equationssystem (6) with the accounting Table 1 as follows:{Ms1C1(α1s21) + Ns21c2(α1s21)− L = Ks1C1(α1s21)s21C2(α1s21),A(s1C1(α1s21))2 + 2Bs1C1(α1s21) + G = 0,(17)where M =μS1r1− q1(1 − α1e1), N = −α1√μp1r1, L =√μp1S3, K = 1 − α1q1, A = α1 + (α1q1S4)2,B =√q1S4(α1r2dr2xdφ2− (1 − α1q1)S2), C =(α1r2dr2xdφ2− (1− α1q1)S2)2− 1.Let us note that at reception of the second equation of the system (17) the following expression is used:αs2C2(αs2) = 1−√1− α(sC1(αs2))2.A solution of the second equation may be presented as:(sC1(αs2))1,2 =−B ±√B2 −AGA. (18)As follows from (18) the decision of the equations system (17) only exists, when the condition: B2 − AQ ≥ 0is satisﬁed. The following expression for function F is derived after substitution of the obtained decision inthe ﬁrst equation of the system (17) taking into account (18):F = U2 − E2K2WA2, (19)where U = (N − L)2 − 2(N − L) (M−KA )B + (M−KA )B2 + W (M−KA )2 − RA2(V 2 + WK2A)− 4BWVA2 , E =2B(M−KA)2− 2RVA2 − 2BA2(V 2 + K2WA)− 2RA2(KBA), V = KBA +N, W = B2−AG, R = A2+B2−AG−α1B2.Thus, one see in this case the searching for a minimal distance between orbits of two bodies was reduced toa ﬁnding of zero function by one variable as well as in the three above-mentioned cases too.It is necessary to note that our research was carried out without a concrete deﬁnition of the second orbit.However, from stated above clearly that the transition to the concrete kind of the second orbit is reduced toreplacement in the received formulas of general expressions by the concrete expressions according to Table 1.In summary we shall note that from our consideration is followed that the deﬁnition of minimal distancebetween orbits of two celestial bodies can be executed as follows. To ﬁnd a dependence on types of orbits orform of their description it is necessary to ﬁnd zero function by one variable. Then, for these values variablethe second variable is founded, and the distances between celestial bodies orbits are founded for the obtainedpairs of variables. Minimum of these distances will be minimal distance between the given orbits of celestialbodies.[1] Babenko Yu. G. // Astrometry and Astrophys.–1983.–49.–P. 22–26.[2] Vasiljev N. N. // Bull. Inst. Theor. Astron.–1978.–14.–P. 266–268.426

Determination of minimum distance between orbits of celestial bodies

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Determination of minimum distance between orbits of celestial bodies

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)