ResearchArticle...
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Research Article Balanced Low Earth Satellite Orbits A. Mostafa 1,2 and M. H. El Dewaik 3 1 University of Jeddah, College of Science and Arts at Khulis, Department of Mathematics, Jeddah, Saudi Arabia 2 Ain Shams University, Faculty of Science, Department of Mathematics, Cairo, Egypt 3 Department of Basic Science, Faculty of Engineering, e British University in Egypt, Cairo, Egypt Correspondence should be addressed to A. Mostafa; [email protected] Received 28 May 2020; Accepted 10 July 2020; Published 19 August 2020 Guest Editor: Euaggelos E. Zotos Copyright © 2020 A. Mostafa and M. H. El Dewaik. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e present work aims at constructing an atlas of the balanced Earth satellite orbits with respect to the secular and long periodic effects of Earth oblateness with the harmonics of the geopotential retained up to the 4th zonal harmonic. e variations of the elements are averaged over the fast and medium angles, thus retaining only the secular and long periodic terms. e models obtained cover the values of the semi-major axis from 1.1 to 2 Earth’s radii, although this is applicable only for 1.1 to 1.3 Earth’s radii due to the radiation belts. e atlas obtained is useful for different purposes, with those having the semi-major axis in this range particularly for remote sensing and meteorology. 1. Introduction e problem of the motion of an artificial satellite of the Earth was not given serious attention until 1957. At this time, little was known about the magnitudes of the coef- ficients of the tesseral and sectorial harmonics in the Earth’s gravitational potential. It was pretty well known at this time (1957–1960) that the contributions of the 3rd, 4th, and 5th zonal harmonics were of order higher than the contribution of the 2nd zonal harmonic, but the values of the coefficients C 30 , C 40 , and C 50 were not very well established. No reliable information was available for the tesseral or the sectorial coefficients except that the obser- vations of orbiting satellites indicated that these coefficients must be small, certainly no more than the first order with respect to C 20 . For low Earth orbits within an altitude less than 480 km, if the satellite attitude is stabilized, or at least a mean projected area could be estimated, the perturbative effects of atmospheric drag should be included. Unfortunately, the literature is still void of even a mention of this topic of balancing this kind of very low Earth orbits. e reason may be the present increased interest in space communications and broadcasting, which still make use of the geostationary orbits that lie beyond the effects of atmospheric drag, though they still suffer the effects of drift solar radiation pressure. With the advance of the space age, it became clear that most space applications require fixing, as strictly as pos- sible, the areas covered by the satellite or the constellation of satellites. In turn, fixing the coverage regions requires fixed nodes and fixed apsidal lines. is in turn leads to the search for orbits satisfying these requirements. e families of orbits satisfying such conditions are called “frozen or- bits” [1–5]. Clearly, the design of such orbits includes the effects of the perturbing influences that affect the motion of the satellite. As the present work is interested in low Earth orbits, only the effect of Earth oblateness is taken into concern. ese have been extensively treated in the liter- ature [6–10]. is paper is aiming at constructing an atlas of the balanced low Earth satellite orbits, which fall in the range from 600 km to 2000 km above sea surface, in the sense that the variations of the elements are averaged over the fast angle to keep only long periodic and secular variations that affect the orbit accumulatively with time. In this paper, a model is given for the averaged effects (over the mean anomaly) of Hindawi Advances in Astronomy Volume 2020, Article ID 7421396, 12 pages https://doi.org/10.1155/2020/7421396
Transcript of ResearchArticle...
Research ArticleBalanced Low Earth Satellite Orbits
A Mostafa 12 and M H El Dewaik3
1University of Jeddah College of Science and Arts at Khulis Department of Mathematics Jeddah Saudi Arabia2Ain Shams University Faculty of Science Department of Mathematics Cairo Egypt3Department of Basic Science Faculty of Engineering e British University in Egypt Cairo Egypt
Correspondence should be addressed to A Mostafa ahmedmostafasciasuedueg
Received 28 May 2020 Accepted 10 July 2020 Published 19 August 2020
Guest Editor Euaggelos E Zotos
Copyright copy 2020 A Mostafa and M H El Dewaik is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited
e present work aims at constructing an atlas of the balanced Earth satellite orbits with respect to the secular and long periodiceffects of Earth oblateness with the harmonics of the geopotential retained up to the 4th zonal harmonic e variations of theelements are averaged over the fast and medium angles thus retaining only the secular and long periodic terms e modelsobtained cover the values of the semi-major axis from 11 to 2 Earthrsquos radii although this is applicable only for 11 to 13 Earthrsquosradii due to the radiation belts e atlas obtained is useful for different purposes with those having the semi-major axis in thisrange particularly for remote sensing and meteorology
1 Introduction
e problem of the motion of an artificial satellite of theEarth was not given serious attention until 1957 At thistime little was known about the magnitudes of the coef-ficients of the tesseral and sectorial harmonics in theEarthrsquos gravitational potential It was pretty well known atthis time (1957ndash1960) that the contributions of the 3rd 4thand 5th zonal harmonics were of order higher than thecontribution of the 2nd zonal harmonic but the values ofthe coefficients C30 C40 and C50 were not very wellestablished No reliable information was available for thetesseral or the sectorial coefficients except that the obser-vations of orbiting satellites indicated that these coefficientsmust be small certainly no more than the first order withrespect to C20
For low Earth orbits within an altitude less than 480 kmif the satellite attitude is stabilized or at least a meanprojected area could be estimated the perturbative effects ofatmospheric drag should be included Unfortunately theliterature is still void of even a mention of this topic ofbalancing this kind of very low Earth orbits e reason maybe the present increased interest in space communications
and broadcasting which still make use of the geostationaryorbits that lie beyond the effects of atmospheric drag thoughthey still suffer the effects of drift solar radiation pressure
With the advance of the space age it became clear thatmost space applications require fixing as strictly as pos-sible the areas covered by the satellite or the constellationof satellites In turn fixing the coverage regions requiresfixed nodes and fixed apsidal lines is in turn leads to thesearch for orbits satisfying these requirements e familiesof orbits satisfying such conditions are called ldquofrozen or-bitsrdquo [1ndash5] Clearly the design of such orbits includes theeffects of the perturbing influences that affect the motion ofthe satellite As the present work is interested in low Earthorbits only the effect of Earth oblateness is taken intoconcern ese have been extensively treated in the liter-ature [6ndash10]
is paper is aiming at constructing an atlas of thebalanced low Earth satellite orbits which fall in the rangefrom 600 km to 2000 km above sea surface in the sense thatthe variations of the elements are averaged over the fast angleto keep only long periodic and secular variations that affectthe orbit accumulatively with time In this paper a model isgiven for the averaged effects (over the mean anomaly) of
HindawiAdvances in AstronomyVolume 2020 Article ID 7421396 12 pageshttpsdoiorg10115520207421396
Earth oblatenessen the Lagrange planetary equations forperturbations of the elements are investigated to get sets oforbital values at which the variations of the elements can becancelled simultaneously
2 Earth Potential
e actual shape of the Earth is that of an eggplante centerof mass does not lie on the spin axis and neither the meridiannor the latitudinal contours are circles e net result of thisirregular shape is to produce a variation in the gravitationalacceleration to that predicted using a point mass distributionis variation reaches its maximum value at latitude 45 degand approaches zero at latitudes 0 and 90 deg
e motion of a particle around the Earth can be visu-alized best by resolving it into individual motions along themeridian and the latitudinal contoursemotion around themeridian can be thought of as consisting of different periodicmotions called ldquozonal harmonicsrdquo Similarly the motionalong a latitudinal contour can be visualized as consisting ofdifferent periodic motions called ldquotesseral harmonicsrdquo ezonal harmonics describe the deviations of a meridian from agreat circle while the tesseral harmonics describe the devi-ations of a latitudinal contour from a circle
At points exterior to the Earth the mass density is zerothus at external points the gravitational potential satisfies
nabla2 V 0 (1)
where V is a scalar function representing the potential Alsothe gravitational potential of the Earth must vanish as werecede to infinitely great distances With these conditions onthe above equation the potential V at external points can berepresented in the following form [11]
V minus μr
1113944nge0
1113944
n
m0
R
r1113874 1113875
n
Pmn (sin δ) Cnm cosmα + Snm sinmα( 1113857
(2)
is expression of the potential is called ldquoVenti poten-tialrdquo and it was adopted by the IAU (International Astro-nomical Union) in 1961 e terms arising in the aboveequation are Cnm and Snm are harmonic coefficients (they arebounded as is always the case in physical problems) R is theequatorial radius of the Earth μ GM is the Earthrsquosgravitational constant G is the universal constant of gravityM is the mass of the Earth and (r α δ) are the geocentriccoordinates of the satellite (Figure 1 [12]) with α measuredeast of Greenwich and Pm
n (sin δ) represents the associatedLegendre polynomials
e terms with m 0 are called ldquozonal harmonicsrdquoe terms with 0ltmlt n are called ldquotesseralharmonicsrdquoe terms with m n are called ldquosectorial harmonicsrdquo
e case of axial symmetry is expressed by taking m 0while if equatorial symmetry is assumed we consider onlyeven harmonics since P2n+1 (minus x) minus P2n+1 (x) Also thecoefficients C21 and S21 are vanishingly small Further if the
origin is taken at the center of mass the coefficients C10 C11and S11 will be equal to zero
Considering axial symmetry with origin at the center ofmass we can write
V minus μr
1113944nge0
R
r1113874 1113875
n
Pn(sin δ)Cn0
minus μr
+ 1113944nge2
Jn
Rn
rn+1Pn(sin δ)
(3)
where Jn minus Cn0Taking terms up to j4 we can write V in the following
form
V 11139444
i1Vi (4)
where
V0 minusμr
V1 0
V2 J2R2
r3P2(sin δ)
12J2
R2
r33 sin2 δ minus 11113872 1113873
V3 J3R3
r4P3(sin δ)
12J3
R3
r45 sin3 δ minus 3 sin δ1113872 1113873
V4 J4R4
r5P4(sin δ)
18J4
R4
r53 minus 30 sin2 δ + 35 sin4 δ1113872 1113873
(5)
It is a purely geometrical transformation to express thepotential function V(r δ) given by the above equations as afunction of the Keplerian orbital elements a e i Ω ω and Iin their usual meanings (Figure 2 [12]) where a and e are thesemi-major axis and the eccentricity of the orbit respec-tively i is the inclination of the orbit to the Earthrsquos equatorialplane Ω and ω describes the position of the orbit in spacewhere Ω is the longitude of the ascending node and ω is theargument of perigee and finally l is the mean anomaly todescribe the position of the satellite with respect to the orbit
en V (a e i Ω ω I) is in a form suitable to use inLagrangersquos planetary equations and in canonical
Right ascension α
Celestial equator
SatelliteCelestial north pole
Celestial sphere
Earthrsquos equatorial plane
Intersection of equatorialand ecliptic planes
r
X
v
Y
K
I
J
Z
Vernal equinox γ
Declination δ
Figure 1 Geocentric coordinates of the satellite
2 Advances in Astronomy
perturbations methods through the relation From thespherical trigonometry of the celestial sphere we have
sin δ Si sin(f + ω) (6)
where f is the true anomalyω is the argument of perigee andSi sin i Ci cos i
Substituting for δ in Pn (sin δ) we get
P2(sin δ) 12
3Si2 sin2(f + g) minus 11113966 1113967
14
1 minus 3Ci2
1113872 1113873 minus 3Si2 cos(2g + 2f)1113966 1113967
P3(sin δ) 12
5Si3 sin3(g + f) minus 3Si sin(f + g)1113966 1113967
18
15Si3
minus 12Si1113872 1113873sin(f + g) minus 5Si3 sin(3f + 3g)1113966 1113967
P4(sin δ) 18
3 minus 30Si2 sin2(g + f) + 35Si
4 sin4(g + f)1113966 1113967
164
9 minus 90Ci2
+ 105Ci4
1113872 11138731113966
+ minus 20 + 160Ci2
minus 140Ci4
1113872 1113873cos(2f + 2g)
+ 35Si4 cos(4f + 4g)1113967
(7)
us we get V2 V3 and V4 as functions of the orbitalelements
We now proceed to evaluate the effects of Earth ob-lateness considering the geopotential up to the zonal har-monic J4
In the present solution we consider only the secular andlong periodic terms averaging over the mean anomaly l
21 e Disturbing Function e disturbing function R isdefined as follows
R minus V2 + V3 + V4( 1113857 (8)
where V2 V3 and V4 are the 2nd 3rd and 4th terms of thegeopotential namely
V2 μj2R
2
4r31 minus 3Ci
21113872 1113873 minus 3Si
2 cos(2f + 2ω)1113960 1113961
V3 μj3R
3
8r415Si
3minus 12Si1113872 1113873sin(f + ω) minus 5Si
3 sin(3f + 3ω)1113960 1113961
V4 μj4R
4
64r59 minus 90Ci
2+ 105Ci
41113872 11138731113960
+ minus 20 + 160Ci2
minus 140Ci4
1113872 1113873cos(2f + 2ω)
+ 35Si4 cos(4f + 4ω)
(9)
where Ci cos(i) Si sin(i) and f is the true anomalySince terms depending only on the fast variable l will not
affect the orbit in an accumulating way with time we averagethe perturbing function R over the mean anomaly with itsperiod 2π e average function is defined by
langF(l)rang 12π
11139462π
0F(l)dl (10)
As the perturbing function R is a function of the trueanomaly f not the mean anomaly l we use the relation
dl 1
1 minus e2
radicr
a1113874 1113875
2df (11)
where both angles have the same period and the same end points0 and 2π and therefore the average function will be given by
langF(f)rang 1
2π1 minus e2
radic 11139462π
0
r
a1113874 1113875
2F(f)df (12)
Applying the required integrals we get the averageddisturbing function langRrang
langRrang minus langV2rangl +langV3rangl +langV4rangl1113858 1113859 (13)
where
langV2rangl μj2R
2
4a3 1 minus e2
1113872 1113873minus (32)
1 minus 3Ci2
1113872 1113873
langV3rangl μj3R
3
8a4 e 1 minus e2
1113872 1113873minus (52)
15Si3
minus 12Si1113872 1113873sin(ω)
langV4rangl μj4R
4
64a5 1 minus e2
1113872 1113873minus (72)
1 +3e2
21113888 1113889 9 minus 90Ci
2+ 105Ci
41113872 1113873
+3μj4R
4
256a5 e2 1 minus e
21113872 1113873
minus (72)minus 20 + 160Ci
2minus 140Ci
41113872 1113873cos(2ω)
(14)
22 Lagrange Equations for the Averaged Variations of theElements e Lagrange planetary equations for the varia-tions of the elements for a disturbing potential R are asfollows [13]
Ω
ω
i
i
r
fv
Y
Z
Node line N
Ascending node
SatellitePerigee
Earthrsquos north polar axis
Earthrsquos equatorial plane
J
I
X
h
e
γ
Figure 2 Keplerian orbital elements of the satellite
Advances in Astronomy 3
_a 2
na
zR
zl
_e minus
1 minus e2
radic
na2e
zR
zω+1 minus e2
na2e
zR
zl
_i minus 1
na2Si1 minus e2
radiczR
zh+
cot(i)
na21 minus e2
radiczR
zω
_Ω 1
na2Si1 minus e2
radiczR
zi
_ω
1 minus e2
radic
na2e
zR
zeminus Ci _Ω
_l n minus2
na
zR
za1113888 1113889
n
minus1 minus e2
na2e
zR
ze
(15)
where n is the mean motion given by n μa3
1113968
Substituting for the averaged disturbing function langRrang
due to Earth oblateness the Lagrange equations become
_a 0 (16)
_e minus
1 minus e2
radic
na2e
zlangRrang
zω (17)
_i minus 1
na2Si1 minus e2
radiczlangRrang
zΩ+
cot(i)
na21 minus e2
radiczlangRrang
zω (18)
_Ω 1
na2Si1 minus e2
radiczlangRrang
zi (19)
_ω
1 minus e2
radic
na2e
zlangRrang
zeminus Ci _Ω (20)
where the equation for l is neglected because we concentrateon balancing the orbit position not the satellite motion in theorbit
23 Variations of the Elements due to Earth OblatenessSubstituting for the required derivatives in equations (16) to(20) yields
_e 3 μradic
j3R3
2a92 1 minus e2
1113872 1113873minus 2 5
4Si
2minus 11113874 1113875Si cos(ω)
minus15 μradic
j4R4
32a112 e 1 minus e2
1113872 1113873minus 3
minus 1 + 8Ci2
minus 7Ci4
1113872 1113873sin(2ω)
(21)
Defining
η 1 minus e2
1113872 1113873
c1 3j3
2
F1i 54
Si2
minus 11113874 1113875Si
c2 15j4
32
F2i minus 1 + 8Ci2
minus 7Ci4
1113872 1113873
(22)
We get
_e
μradicR4
η3a112 c1ηa
RF1iCos(ω) minus c2eF2i sin(2ω)1113874 1113875
_i cot(i)
n
3μj3R3
2a6 e 1 minus e2
1113872 1113873minus 3 5
4Si
2minus 11113874 1113875Si cos(ω)1113896
minus15μj4R
4
16a7 e2 1 minus e
21113872 1113873
minus 4minus 1 + 8Ci
2minus 7Ci
41113872 1113873sin(2ω)1113897
(23)
or
_i minuse cot(i)
η_e
_Ω minus Ci
n
3μj2R2
2a5 1 minus e2
1113872 1113873minus 2
1113896
+μj3R
3
8a6 e 1 minus e2
1113872 1113873minus 3
45Si minus12Si
1113874 1113875sin(ω)
+15μj4R
4
16a7 1 minus e2
1113872 1113873minus 4
1 +3e2
21113888 1113889 3 minus 7Ci
21113872 11138731113890
+ e2
minus 4 + 7Ci2
1113872 1113873cos(2ω)11138911113897
(24)
Defining
c3 3j22
c4 3j3
8
c5 15j4
16
F3i 15Si minus4Si
F4i 3 minus 7Ci2
F5i minus 4 + 7Ci2
(25)
We get
4 Advances in Astronomy
_Ω minus Ci
μradicR4
η4a112 c3a
R1113874 1113875
2η2 + c4
a
ReηF3i sin(ω)1113896
+ c5 1 +3e2
21113888 1113889F4i + e
2F5i cos(2ω)1113890 11138911113897
(26)
_ω minus Ci _Ω minus3μj2R
2
4na5 1 minus e2
1113872 1113873minus 2
1 minus 3Ci2
1113872 1113873
minus3μj3R
3
2na6e1 + 4e
21113872 1113873 1 minus e
21113872 1113873
minus 3 54
Si2
minus 11113874 1113875Si sin(ω)
minus15μj4R
4
32na7 1 minus e2
1113872 1113873minus 4
1 +3e2
41113888 1113889 3 minus 30Ci
2+ 35Ci
41113872 11138731113890
+ 1 +52e2
1113874 1113875 minus 1 + 8Ci2
minus 7Ci4
1113872 1113873cos(2ω)1113891
(27)
Defining as in equation (28) we get equation (29)
c6 3j2
4
F6i 1 minus 3Ci2
1113872 1113873
F7i 3 minus 30Ci2
+ 35Ci4
(28)
_ω minus
μradicR4
η4a112 c6a
R1113874 1113875
2η2F6i + c1
a
R1 + 4e
21113872 1113873
ηeF1i sin(ω)1113890
+ c2 1 +3e2
41113888 1113889F7i + 1 +
52e2
1113874 1113875F2i cos(2ω)1113888 11138891113891 minus Ci _Ω
(29)
Equations (22)ndash(29) give the average effects of Earthoblateness including the zonal harmonics of the geopotentialup to J4 on the Keplerian elements of the satellite orbit
3 Balanced Low Earth Satellite Orbits
In what follows we try to find orbits that are balanced in thesense that the averaged (over the fast variable l) variations ofthe orbit elements are set equal to zero In equation (23) weput it equal to zero and get a relation between the argumentof perigee ω and the inclination i while treating the ec-centricity e and the semi-major axis a as parameters iswill give a range of values for ω and i at different values of eand a which are all give balanced orbits with respect to boththe eccentricity e and the inclination ie same is done withequations (26) and (29) while putting _Ω 0 and _ω 0
e applicable ranges for this model of the semi-majoraxis a are 11Rle ale 13R where the range 14Rndash2R isavoided due to the predominance of the radiation belts atthese levels to avoid the damages of the equipment that itmay produce besides its fatal effects on human life (forinhabited spacecrafts) e values for the eccentricity e aretaken between 001 (almost circular orbit) and 05
31 Orbits with Fixed Eccentricity and Inclination Byequating the variation of e by zero we get
_e
μradicR4
η3a112 c1a
RηF1i cos(ω) + c2eF2i sin(2ω)1113874 1113875 0 (30)
is implies
c1a
RηF1i cos(ω) + c2eF2i sin(2ω) 0 (31)
So either cos(ω) 0 or
sin(ω) a 1 minus e2( 1113857
eRC1F(i) (32)
where
C1 8J3
5J4
F(i) 1 minus (54)sin2(i)( 1113857sin(i)
1 minus 8cos2(i) + 7cos4(i)
(33)
Equations (32) and (33) give the family of low orbits thathave both the eccentricity and the inclination fixed
e condition for the existence of such orbits is clearlythat
minus 1le sin(ω)le 1 (34)
which is guaranteed by
minus 1lea 1 minus e2( 1113857
eRC1F(i)le 1 (35)
We put it as a condition on the eccentricity i only when
minus eR
C1a 1 minus e2( )leF(i)le
eR
C1a 1 minus e2( ) (36)
32 Orbits with Fixed Node e families of orbits for whichdΩdt 0 (ie with fixed nodes) are obtained from
minus Ciμradic
R4
η4a112 c3a
R1113874 1113875
2η2 + c4
a
ReηF3i sin(ω)1113896
+ c5 1 +3e2
21113888 1113889F4i + e
2F5i cos(2ω)1113890 11138911113897 0
(37)
is implies
c3a
R1113874 1113875
2η2 + c4
a
ReηF3i sin(ω)
+ c5 1 +3e2
21113888 1113889F4i + e
2F5i 1 minus 2sin2(ω)1113872 11138731113890 1113891 0
(38)
Advances in Astronomy 5
Table 1 Real values of i (in degree) corresponding to the given values of F(i)
F(i) i1 180 minus i1 i2 minus 180 minus i2001 6363 11637 minus 6322 minus 11678005 6427 11573 minus 6211 minus 11789010 6484 11516 minus 5976 minus 12024015 6525 11475 minus 5514 minus 12486020 6556 11444 minus 4716 minus 13284
01 02 03 04 05
e
F (i) = 001
ω
π
3π4
π2
π4
0
Figure 3 e curve ω(e) at which _e _i 0 for F(i) 001
secndash1
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
ndash10 times 10ndash6
01 02 03 04 05
e
Ωmiddot
ωmiddot
(a)
secndash1
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
ndash10 times 10ndash6
01 02 03 04 05
e
Ωmiddot
ωmiddot
(b)
secndash1
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
01 02 03 04 05e
Ωmiddot
ωmiddot
(c)
secndash1
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
01 02 03 04 05e
Ωmiddot
ωmiddot
(d)
Figure 4 Curves of _Ω and _ω with e for the 4 cases corresponding to F(i) 001 (a) i 6363 (b) i minus 6322 (c) i 11637 and (d) i 11678
6 Advances in Astronomy
is can be arranged as a second order equationfor sin(ω) giving a family of orbits with fixed argumentof perigee for different values of ω as a function ofthe inclination i the semi-major axis a and the eccen-tricity e
minus 2c5e2F5isin
2(ω) + c4
a
ReηF3i sin(ω)
+ c3a
R1113874 1113875
2η2 + c5 1 +
3e2
21113888 1113889F4i + e
2F5i1113890 1113891 0
(39)
e
F (i) = 005
ω
π
3π4
π2
π4
001 02 03 04 05
Figure 5 e curve ω(e) at which _e _i 0 for F(i) 005
secndash1
01 02 03 04 05
Ωmiddot
ωmiddot
e
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
ndash10 times 10ndash6
(a)
secndash1
01 02 03 04 05
Ωmiddot
ωmiddot
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
ndash10 times 10ndash6
ndash12 times 10ndash6
e
(b)
secndash1
Ωmiddot
ωmiddot
ndash10 times 10ndash6
ndash80 times 10ndash7
ndash60 times 10ndash7
ndash40 times 10ndash7
ndash20 times 10ndash7
01 02 03 04 05e
(c)
secndash1
01 02 03 04 05
Ωmiddot
ωmiddot
12 times 10ndash6
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
e
(d)
Figure 6 Curves of _Ω and _ωwith e for the 4 cases corresponding to F(i) 005 (a) i 6427 (b) i minus 6211 (c) i 11573 and (d) i minus 11789
Advances in Astronomy 7
When solving for sin(ω) we get the family of orbits forwhich the longitude of the node is balanced
sin(ω) 2c4zF3i plusmn
4c24z
2F23i + 16c5F5i 2c3z
2 + c5F4i 2 + 3e2( ) + 2c5e2F5i(
1113969
8c5eF5i
(40)
F (i) = 01
ω
π
3π4
π2
π4
0
e
01 02 03 04 05
Figure 7 e curve ω(e) at which _e _i 0 for F(i) 001
secndash1
01 02 03 04 05
e
ndash50 times 10ndash7
ndash10 times 10ndash6
Ωmiddot
ωmiddot
(a)
secndash1
01 02 03 04 05e
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
Ωmiddot
ωmiddot
(b)
secndash1
01 02 03 04 05e
12 times 10ndash6
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
Ωmiddot
ωmiddot
(c)
secndash1
01 02 03 04 05e
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
Ωmiddot
ωmiddot
(d)
Figure 8 Curves of _Ω and _ω with e for the 4 cases corresponding to F(i) 01 (a) i minus 5976 (b) i 6484 (c) i minus 12024 and (d) i 11516
8 Advances in Astronomy
where
z a
Rη (41)
e condition for having the orbit if real solution exists isagain that minus 1le sin(ω)le 1 which gives
minus 1le2c3z
2 + 2 + 3e2( 1113857c5F4i + 2e2 c5 minus 1( 1113857F5i
c4ezF3i
le 1 F5i F3i ne 0
(42)
33Orbitswith FixedPerigee For the argument of perigee tobalance we solve _ω 0We substitute from equation (26)into equation (29) then expand cos(2ω) and collect termswith respect to sin(ω) We get
_ω A sin2(ω) + B sin(ω) + C (43)
us for _ω 0 we get
sin(ω) minus B plusmn
B2 minus 4AC
radic
2A (44)
F (i) = 015
ω
π
3π4
π2
π4
0
e
01 02 03 04 05
Figure 9 e curve ω(e) at which _e _i 0 for F(i) 015
01 02 03 04 05
50 times 10ndash7
ndash50 times 10ndash7
ndash10 times 10ndash6
ndash15 times 10ndash6
e
ωmiddotΩmiddot
secndash
1
(a)
01 02 03 04 05
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
e
ωmiddotΩmiddot
secndash
1
(b)
01 02 03 04 05
14 times 10ndash6
10 times 10ndash6
12 times 10ndash6
80 times 10ndash7
60 times 10ndash7
e
ωmiddotΩmiddot
secndash
1
(c)
01 02 03 04 05
10 times 10ndash6
60 times 10ndash7
80 times 10ndash7
40 times 10ndash7
20 times 10ndash7
ndash20 times 10ndash7
e
ωmiddotΩmiddot
secndash
1
(d)
Figure 10 Curves of _Ω and _ω with e for the 4 cases corresponding to F(i) 015 (a) i minus 5514 (b) i 6525 (c) i minus 12486 and (d) i 11475
Advances in Astronomy 9
where
A minus 2c5e2F5cos
2(i) + 2c2 1 +
52e2
1113874 1113875F2 (45)
B c4ezF3cos2(i) minus c1z 4e +
1e
1113874 1113875F1 (46)
C cos2(i) c3z2
+ c5 e2F5 + 1 +
3e2
21113888 1113889F41113888 11138891113888 1113889
minus c6z2F6 minus c2 1 +
3e2
41113888 1113889F7 + 1 +
5e2
21113888 11138891113888 1113889F21113888 1113889
(47)
F (i) = 02
ω
π
3π4
π2
π4
0
e
01 02 03 04 05
Figure 11 e curve ω(e) at which _e _i 0 for F(i) 02
ωmiddotΩmiddot
01 02 03 04 05
e
15 times 10ndash6
10 times 10ndash6
50 times 10ndash7
ndash50 times 10ndash7
ndash10 times 10ndash6
ndash15 times 10ndash6
secndash
1
(a)
ωmiddotΩmiddot
01 02 03 04 05e
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
secndash
1
(b)
ωmiddotΩmiddot
01 02 03 04 05e
16 times 10ndash6
14 times 10ndash6
12 times 10ndash6
10 times 10ndash6
secndash
1
(c)
ωmiddotΩmiddot
01 02 03 04 05
e
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
ndash20 times 10ndash7
secndash
1
(d)
Figure 12 Curves of _Ω and _ωwith e for the 4 cases corresponding to F(i) 02 (a) i minus 4716 (b) i 6556 (c) i minus 13284 and (d) i 11444
10 Advances in Astronomy
Equations (44)ndash(47) give the relation between sin(ω) andthe inclination i the semi-major axis a and the eccentricitye which gives the family of orbits that balance the argumentof perigee ω subject of course to the restriction that sin(ω) isa real value between minus 1 and 1
4 Numerical Results
In this section numerical results and graphs are obtained forthe case of seasat a 7100 km by putting ω as a function of eand i from equations (32) and (33) e curves are plottedwithin the possible range given by condition (36) to give curvesof balanced e and ie curves are against the eccentricity e inthe range [001 05] e numerical values involved areJ2 0001082645 J3 minus 0000002546 J4 minus 0000001649 R
6378165km α 7100m and μ 3986005 km3secminus 2e condition (36) gives the upper and lower bounds of
the function F(i) as a function of e and since the functioneR(C1a(1 minus e2)) is increasing with e and has no criticalpoints in the interval [001 05] then the minimum valueoccurs at e 001 and the maximum at e 05 which givesminus 024leF(i)le 024 e graphs are plotted for differentvalues of F(i) which corresponds to specific values of i foundby solving the equation resulting from setting F(i) equal tothe required values After that we plot _Ω 0 and _ω 0 si-multaneously for the same values of F(i) at each specific i-value to find the orbit values at which we have nearest valuesfor _Ω 0 and _ω 0 In the graphs of _Ω and _ω the relation(32) was kept to ensure that e and i are already balanced
Five selected values of F(i) are chosen F(i) 001 05010 015 and 020 Negative values will give the same resultswith negative sign since F(i) is odd with respect to i Also foreach value the solution of F(i) x with x equals one of theabove values we get four real values of i on the form i1180 minus i1 i2 and minus 180 minus i2 where i1 and i2 are near the criticalinclination one of them is positive and the other is negativeTable 1 gives the values of i corresponding to the selectedvalues of F(i)
We note that as F(i) increases i gets away from thecritical inclination and the curve gets shorter indicating lessstability of ω as expected
e graphs are plotted for each value of F(i) first for thebalanced values of e and i then for the corresponding fourvalues of i four graphs are plotted for _Ω and _ω to find thenearest values of zero variation for both elements
Figures 3ndash12 show the possibility of balancing ω with eand i while Ω will have a variation of order 10minus 6sec or itmust be balanced alone at i 90 deg (as shown in equation(26)) according to the orbit kind Figures 3 5 7 9 and 11show the possible values of ω(e) that balance e and i whileFigures 4 6 8 10 and 12 show the curves of _Ω and _ω at thevalues of i at which _e _i 0
5 Conclusion
Let balanced orbits be defined as those for which the orbitalelements are set equal to zero under the effect of secular andlong periodic perturbations In this work the effect of Earthoblateness is the considered perturbing force because of
dealing with low Earth orbits e above analysis then showsthat such an orbit will be balanced within a reliable toleranceonly for few weeks since we are forced to accept the motionof either the node or the perigee by about 10minus 6 degsec ereason is that under the influence of the Earth oblateness_Ω 0 (exactly) for i (π2) while _ω 0 only near thecritical inclination ic 634 deg Hence the best procedure isto design a satellite constellation for which the nodal shiftsdue to the perturbative effects and Earth rotation aremodeled to yield continuous coverage e perigees areeither fixed or arranged to realize that the perigee (or theapogee) be overhead the coverage region (regions) in duetimes is may require near commensurability with theadmitted nodal periods
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was funded by the University of Jeddah SaudiArabia under grant no UJ-26-18-DR us the authortherefore acknowledges with thanks the university technicaland financial support
References
[1] N Delsate P Robutel A Lemaıtre and T Carletti ldquoFrozenorbits at high eccentricity and inclination application tomercury orbiterrdquo Celestial Mechanics and Dynamical As-tronomy vol 108 no 3 pp 275ndash300 2010
[2] E Condoleo M Cinelli E Ortore and C Circi ldquoStable orbitsfor lunar landing assistancerdquo Advances in Space Researchvol 60 no 7 pp 1404ndash1412 2017
[3] A Elipe andM Lara ldquoFrozen orbits about the moonrdquo Journalof Guidance Control and Dynamics vol 26 no 2 pp 238ndash243 2003
[4] C Circi E Condoleo and E Ortore ldquoA vectorial approach todetermine frozen orbital conditionsrdquo Celestial Mechanics andDynamical Astronomy vol 128 no 2-3 pp 361ndash382 2017
[5] E Condoleo M Cinelli E Ortore and C Circi ldquoFrozenorbits with equatorial perturbing bodies the case of Gany-mede Callisto and Titanrdquo Journal of Guidance Control andDynamics vol 39 no 10 pp 2264ndash2272 2016
[6] V Kudielka ldquoBalanced earth satellite orbitsrdquo Celestial Me-chanics amp Dynamical Astronomy vol 60 no 4 pp 455ndash4701994
[7] S L Coffey A Deprit and E Deprit ldquoFrozen orbits forsatellites close to an earth-like planetrdquo Celestial Mechanics ampDynamical Astronomy vol 59 no 1 pp 37ndash72 1994
[8] M Lara A Deprit and A Elipe ldquoNumerical continuation offamilies of frozen orbits in the zonal problem of artificialsatellite theoryrdquo Celestial Mechanics amp Dynamical Astronomyvol 62 no 2 pp 167ndash181 1995
[9] D Brouwer ldquoSolution of the problem of artificial satellitetheory without dragrdquo e Astronomical Journal vol 64p 378 1959
Advances in Astronomy 11
[10] Y Kozai ldquoe motion of a close earth satelliterdquo e Astro-nomical Journal vol 64 p 367 1959
[11] P M Fitzpatrick Principles of Celestial Mechanics AcademicPress Cambridge MA USA 1970
[12] H D Curtis Orbital Mechanics for Engineering StudentsElsevier Amsterdam Netherlands 2005
[13] D Brower and G M Clemence Methods of Celestial Me-chanics Academic Press Cambridge MA USA 1961
12 Advances in Astronomy
Earth oblatenessen the Lagrange planetary equations forperturbations of the elements are investigated to get sets oforbital values at which the variations of the elements can becancelled simultaneously
2 Earth Potential
e actual shape of the Earth is that of an eggplante centerof mass does not lie on the spin axis and neither the meridiannor the latitudinal contours are circles e net result of thisirregular shape is to produce a variation in the gravitationalacceleration to that predicted using a point mass distributionis variation reaches its maximum value at latitude 45 degand approaches zero at latitudes 0 and 90 deg
e motion of a particle around the Earth can be visu-alized best by resolving it into individual motions along themeridian and the latitudinal contoursemotion around themeridian can be thought of as consisting of different periodicmotions called ldquozonal harmonicsrdquo Similarly the motionalong a latitudinal contour can be visualized as consisting ofdifferent periodic motions called ldquotesseral harmonicsrdquo ezonal harmonics describe the deviations of a meridian from agreat circle while the tesseral harmonics describe the devi-ations of a latitudinal contour from a circle
At points exterior to the Earth the mass density is zerothus at external points the gravitational potential satisfies
nabla2 V 0 (1)
where V is a scalar function representing the potential Alsothe gravitational potential of the Earth must vanish as werecede to infinitely great distances With these conditions onthe above equation the potential V at external points can berepresented in the following form [11]
V minus μr
1113944nge0
1113944
n
m0
R
r1113874 1113875
n
Pmn (sin δ) Cnm cosmα + Snm sinmα( 1113857
(2)
is expression of the potential is called ldquoVenti poten-tialrdquo and it was adopted by the IAU (International Astro-nomical Union) in 1961 e terms arising in the aboveequation are Cnm and Snm are harmonic coefficients (they arebounded as is always the case in physical problems) R is theequatorial radius of the Earth μ GM is the Earthrsquosgravitational constant G is the universal constant of gravityM is the mass of the Earth and (r α δ) are the geocentriccoordinates of the satellite (Figure 1 [12]) with α measuredeast of Greenwich and Pm
n (sin δ) represents the associatedLegendre polynomials
e terms with m 0 are called ldquozonal harmonicsrdquoe terms with 0ltmlt n are called ldquotesseralharmonicsrdquoe terms with m n are called ldquosectorial harmonicsrdquo
e case of axial symmetry is expressed by taking m 0while if equatorial symmetry is assumed we consider onlyeven harmonics since P2n+1 (minus x) minus P2n+1 (x) Also thecoefficients C21 and S21 are vanishingly small Further if the
origin is taken at the center of mass the coefficients C10 C11and S11 will be equal to zero
Considering axial symmetry with origin at the center ofmass we can write
V minus μr
1113944nge0
R
r1113874 1113875
n
Pn(sin δ)Cn0
minus μr
+ 1113944nge2
Jn
Rn
rn+1Pn(sin δ)
(3)
where Jn minus Cn0Taking terms up to j4 we can write V in the following
form
V 11139444
i1Vi (4)
where
V0 minusμr
V1 0
V2 J2R2
r3P2(sin δ)
12J2
R2
r33 sin2 δ minus 11113872 1113873
V3 J3R3
r4P3(sin δ)
12J3
R3
r45 sin3 δ minus 3 sin δ1113872 1113873
V4 J4R4
r5P4(sin δ)
18J4
R4
r53 minus 30 sin2 δ + 35 sin4 δ1113872 1113873
(5)
It is a purely geometrical transformation to express thepotential function V(r δ) given by the above equations as afunction of the Keplerian orbital elements a e i Ω ω and Iin their usual meanings (Figure 2 [12]) where a and e are thesemi-major axis and the eccentricity of the orbit respec-tively i is the inclination of the orbit to the Earthrsquos equatorialplane Ω and ω describes the position of the orbit in spacewhere Ω is the longitude of the ascending node and ω is theargument of perigee and finally l is the mean anomaly todescribe the position of the satellite with respect to the orbit
en V (a e i Ω ω I) is in a form suitable to use inLagrangersquos planetary equations and in canonical
Right ascension α
Celestial equator
SatelliteCelestial north pole
Celestial sphere
Earthrsquos equatorial plane
Intersection of equatorialand ecliptic planes
r
X
v
Y
K
I
J
Z
Vernal equinox γ
Declination δ
Figure 1 Geocentric coordinates of the satellite
2 Advances in Astronomy
perturbations methods through the relation From thespherical trigonometry of the celestial sphere we have
sin δ Si sin(f + ω) (6)
where f is the true anomalyω is the argument of perigee andSi sin i Ci cos i
Substituting for δ in Pn (sin δ) we get
P2(sin δ) 12
3Si2 sin2(f + g) minus 11113966 1113967
14
1 minus 3Ci2
1113872 1113873 minus 3Si2 cos(2g + 2f)1113966 1113967
P3(sin δ) 12
5Si3 sin3(g + f) minus 3Si sin(f + g)1113966 1113967
18
15Si3
minus 12Si1113872 1113873sin(f + g) minus 5Si3 sin(3f + 3g)1113966 1113967
P4(sin δ) 18
3 minus 30Si2 sin2(g + f) + 35Si
4 sin4(g + f)1113966 1113967
164
9 minus 90Ci2
+ 105Ci4
1113872 11138731113966
+ minus 20 + 160Ci2
minus 140Ci4
1113872 1113873cos(2f + 2g)
+ 35Si4 cos(4f + 4g)1113967
(7)
us we get V2 V3 and V4 as functions of the orbitalelements
We now proceed to evaluate the effects of Earth ob-lateness considering the geopotential up to the zonal har-monic J4
In the present solution we consider only the secular andlong periodic terms averaging over the mean anomaly l
21 e Disturbing Function e disturbing function R isdefined as follows
R minus V2 + V3 + V4( 1113857 (8)
where V2 V3 and V4 are the 2nd 3rd and 4th terms of thegeopotential namely
V2 μj2R
2
4r31 minus 3Ci
21113872 1113873 minus 3Si
2 cos(2f + 2ω)1113960 1113961
V3 μj3R
3
8r415Si
3minus 12Si1113872 1113873sin(f + ω) minus 5Si
3 sin(3f + 3ω)1113960 1113961
V4 μj4R
4
64r59 minus 90Ci
2+ 105Ci
41113872 11138731113960
+ minus 20 + 160Ci2
minus 140Ci4
1113872 1113873cos(2f + 2ω)
+ 35Si4 cos(4f + 4ω)
(9)
where Ci cos(i) Si sin(i) and f is the true anomalySince terms depending only on the fast variable l will not
affect the orbit in an accumulating way with time we averagethe perturbing function R over the mean anomaly with itsperiod 2π e average function is defined by
langF(l)rang 12π
11139462π
0F(l)dl (10)
As the perturbing function R is a function of the trueanomaly f not the mean anomaly l we use the relation
dl 1
1 minus e2
radicr
a1113874 1113875
2df (11)
where both angles have the same period and the same end points0 and 2π and therefore the average function will be given by
langF(f)rang 1
2π1 minus e2
radic 11139462π
0
r
a1113874 1113875
2F(f)df (12)
Applying the required integrals we get the averageddisturbing function langRrang
langRrang minus langV2rangl +langV3rangl +langV4rangl1113858 1113859 (13)
where
langV2rangl μj2R
2
4a3 1 minus e2
1113872 1113873minus (32)
1 minus 3Ci2
1113872 1113873
langV3rangl μj3R
3
8a4 e 1 minus e2
1113872 1113873minus (52)
15Si3
minus 12Si1113872 1113873sin(ω)
langV4rangl μj4R
4
64a5 1 minus e2
1113872 1113873minus (72)
1 +3e2
21113888 1113889 9 minus 90Ci
2+ 105Ci
41113872 1113873
+3μj4R
4
256a5 e2 1 minus e
21113872 1113873
minus (72)minus 20 + 160Ci
2minus 140Ci
41113872 1113873cos(2ω)
(14)
22 Lagrange Equations for the Averaged Variations of theElements e Lagrange planetary equations for the varia-tions of the elements for a disturbing potential R are asfollows [13]
Ω
ω
i
i
r
fv
Y
Z
Node line N
Ascending node
SatellitePerigee
Earthrsquos north polar axis
Earthrsquos equatorial plane
J
I
X
h
e
γ
Figure 2 Keplerian orbital elements of the satellite
Advances in Astronomy 3
_a 2
na
zR
zl
_e minus
1 minus e2
radic
na2e
zR
zω+1 minus e2
na2e
zR
zl
_i minus 1
na2Si1 minus e2
radiczR
zh+
cot(i)
na21 minus e2
radiczR
zω
_Ω 1
na2Si1 minus e2
radiczR
zi
_ω
1 minus e2
radic
na2e
zR
zeminus Ci _Ω
_l n minus2
na
zR
za1113888 1113889
n
minus1 minus e2
na2e
zR
ze
(15)
where n is the mean motion given by n μa3
1113968
Substituting for the averaged disturbing function langRrang
due to Earth oblateness the Lagrange equations become
_a 0 (16)
_e minus
1 minus e2
radic
na2e
zlangRrang
zω (17)
_i minus 1
na2Si1 minus e2
radiczlangRrang
zΩ+
cot(i)
na21 minus e2
radiczlangRrang
zω (18)
_Ω 1
na2Si1 minus e2
radiczlangRrang
zi (19)
_ω
1 minus e2
radic
na2e
zlangRrang
zeminus Ci _Ω (20)
where the equation for l is neglected because we concentrateon balancing the orbit position not the satellite motion in theorbit
23 Variations of the Elements due to Earth OblatenessSubstituting for the required derivatives in equations (16) to(20) yields
_e 3 μradic
j3R3
2a92 1 minus e2
1113872 1113873minus 2 5
4Si
2minus 11113874 1113875Si cos(ω)
minus15 μradic
j4R4
32a112 e 1 minus e2
1113872 1113873minus 3
minus 1 + 8Ci2
minus 7Ci4
1113872 1113873sin(2ω)
(21)
Defining
η 1 minus e2
1113872 1113873
c1 3j3
2
F1i 54
Si2
minus 11113874 1113875Si
c2 15j4
32
F2i minus 1 + 8Ci2
minus 7Ci4
1113872 1113873
(22)
We get
_e
μradicR4
η3a112 c1ηa
RF1iCos(ω) minus c2eF2i sin(2ω)1113874 1113875
_i cot(i)
n
3μj3R3
2a6 e 1 minus e2
1113872 1113873minus 3 5
4Si
2minus 11113874 1113875Si cos(ω)1113896
minus15μj4R
4
16a7 e2 1 minus e
21113872 1113873
minus 4minus 1 + 8Ci
2minus 7Ci
41113872 1113873sin(2ω)1113897
(23)
or
_i minuse cot(i)
η_e
_Ω minus Ci
n
3μj2R2
2a5 1 minus e2
1113872 1113873minus 2
1113896
+μj3R
3
8a6 e 1 minus e2
1113872 1113873minus 3
45Si minus12Si
1113874 1113875sin(ω)
+15μj4R
4
16a7 1 minus e2
1113872 1113873minus 4
1 +3e2
21113888 1113889 3 minus 7Ci
21113872 11138731113890
+ e2
minus 4 + 7Ci2
1113872 1113873cos(2ω)11138911113897
(24)
Defining
c3 3j22
c4 3j3
8
c5 15j4
16
F3i 15Si minus4Si
F4i 3 minus 7Ci2
F5i minus 4 + 7Ci2
(25)
We get
4 Advances in Astronomy
_Ω minus Ci
μradicR4
η4a112 c3a
R1113874 1113875
2η2 + c4
a
ReηF3i sin(ω)1113896
+ c5 1 +3e2
21113888 1113889F4i + e
2F5i cos(2ω)1113890 11138911113897
(26)
_ω minus Ci _Ω minus3μj2R
2
4na5 1 minus e2
1113872 1113873minus 2
1 minus 3Ci2
1113872 1113873
minus3μj3R
3
2na6e1 + 4e
21113872 1113873 1 minus e
21113872 1113873
minus 3 54
Si2
minus 11113874 1113875Si sin(ω)
minus15μj4R
4
32na7 1 minus e2
1113872 1113873minus 4
1 +3e2
41113888 1113889 3 minus 30Ci
2+ 35Ci
41113872 11138731113890
+ 1 +52e2
1113874 1113875 minus 1 + 8Ci2
minus 7Ci4
1113872 1113873cos(2ω)1113891
(27)
Defining as in equation (28) we get equation (29)
c6 3j2
4
F6i 1 minus 3Ci2
1113872 1113873
F7i 3 minus 30Ci2
+ 35Ci4
(28)
_ω minus
μradicR4
η4a112 c6a
R1113874 1113875
2η2F6i + c1
a
R1 + 4e
21113872 1113873
ηeF1i sin(ω)1113890
+ c2 1 +3e2
41113888 1113889F7i + 1 +
52e2
1113874 1113875F2i cos(2ω)1113888 11138891113891 minus Ci _Ω
(29)
Equations (22)ndash(29) give the average effects of Earthoblateness including the zonal harmonics of the geopotentialup to J4 on the Keplerian elements of the satellite orbit
3 Balanced Low Earth Satellite Orbits
In what follows we try to find orbits that are balanced in thesense that the averaged (over the fast variable l) variations ofthe orbit elements are set equal to zero In equation (23) weput it equal to zero and get a relation between the argumentof perigee ω and the inclination i while treating the ec-centricity e and the semi-major axis a as parameters iswill give a range of values for ω and i at different values of eand a which are all give balanced orbits with respect to boththe eccentricity e and the inclination ie same is done withequations (26) and (29) while putting _Ω 0 and _ω 0
e applicable ranges for this model of the semi-majoraxis a are 11Rle ale 13R where the range 14Rndash2R isavoided due to the predominance of the radiation belts atthese levels to avoid the damages of the equipment that itmay produce besides its fatal effects on human life (forinhabited spacecrafts) e values for the eccentricity e aretaken between 001 (almost circular orbit) and 05
31 Orbits with Fixed Eccentricity and Inclination Byequating the variation of e by zero we get
_e
μradicR4
η3a112 c1a
RηF1i cos(ω) + c2eF2i sin(2ω)1113874 1113875 0 (30)
is implies
c1a
RηF1i cos(ω) + c2eF2i sin(2ω) 0 (31)
So either cos(ω) 0 or
sin(ω) a 1 minus e2( 1113857
eRC1F(i) (32)
where
C1 8J3
5J4
F(i) 1 minus (54)sin2(i)( 1113857sin(i)
1 minus 8cos2(i) + 7cos4(i)
(33)
Equations (32) and (33) give the family of low orbits thathave both the eccentricity and the inclination fixed
e condition for the existence of such orbits is clearlythat
minus 1le sin(ω)le 1 (34)
which is guaranteed by
minus 1lea 1 minus e2( 1113857
eRC1F(i)le 1 (35)
We put it as a condition on the eccentricity i only when
minus eR
C1a 1 minus e2( )leF(i)le
eR
C1a 1 minus e2( ) (36)
32 Orbits with Fixed Node e families of orbits for whichdΩdt 0 (ie with fixed nodes) are obtained from
minus Ciμradic
R4
η4a112 c3a
R1113874 1113875
2η2 + c4
a
ReηF3i sin(ω)1113896
+ c5 1 +3e2
21113888 1113889F4i + e
2F5i cos(2ω)1113890 11138911113897 0
(37)
is implies
c3a
R1113874 1113875
2η2 + c4
a
ReηF3i sin(ω)
+ c5 1 +3e2
21113888 1113889F4i + e
2F5i 1 minus 2sin2(ω)1113872 11138731113890 1113891 0
(38)
Advances in Astronomy 5
Table 1 Real values of i (in degree) corresponding to the given values of F(i)
F(i) i1 180 minus i1 i2 minus 180 minus i2001 6363 11637 minus 6322 minus 11678005 6427 11573 minus 6211 minus 11789010 6484 11516 minus 5976 minus 12024015 6525 11475 minus 5514 minus 12486020 6556 11444 minus 4716 minus 13284
01 02 03 04 05
e
F (i) = 001
ω
π
3π4
π2
π4
0
Figure 3 e curve ω(e) at which _e _i 0 for F(i) 001
secndash1
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
ndash10 times 10ndash6
01 02 03 04 05
e
Ωmiddot
ωmiddot
(a)
secndash1
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
ndash10 times 10ndash6
01 02 03 04 05
e
Ωmiddot
ωmiddot
(b)
secndash1
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
01 02 03 04 05e
Ωmiddot
ωmiddot
(c)
secndash1
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
01 02 03 04 05e
Ωmiddot
ωmiddot
(d)
Figure 4 Curves of _Ω and _ω with e for the 4 cases corresponding to F(i) 001 (a) i 6363 (b) i minus 6322 (c) i 11637 and (d) i 11678
6 Advances in Astronomy
is can be arranged as a second order equationfor sin(ω) giving a family of orbits with fixed argumentof perigee for different values of ω as a function ofthe inclination i the semi-major axis a and the eccen-tricity e
minus 2c5e2F5isin
2(ω) + c4
a
ReηF3i sin(ω)
+ c3a
R1113874 1113875
2η2 + c5 1 +
3e2
21113888 1113889F4i + e
2F5i1113890 1113891 0
(39)
e
F (i) = 005
ω
π
3π4
π2
π4
001 02 03 04 05
Figure 5 e curve ω(e) at which _e _i 0 for F(i) 005
secndash1
01 02 03 04 05
Ωmiddot
ωmiddot
e
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
ndash10 times 10ndash6
(a)
secndash1
01 02 03 04 05
Ωmiddot
ωmiddot
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
ndash10 times 10ndash6
ndash12 times 10ndash6
e
(b)
secndash1
Ωmiddot
ωmiddot
ndash10 times 10ndash6
ndash80 times 10ndash7
ndash60 times 10ndash7
ndash40 times 10ndash7
ndash20 times 10ndash7
01 02 03 04 05e
(c)
secndash1
01 02 03 04 05
Ωmiddot
ωmiddot
12 times 10ndash6
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
e
(d)
Figure 6 Curves of _Ω and _ωwith e for the 4 cases corresponding to F(i) 005 (a) i 6427 (b) i minus 6211 (c) i 11573 and (d) i minus 11789
Advances in Astronomy 7
When solving for sin(ω) we get the family of orbits forwhich the longitude of the node is balanced
sin(ω) 2c4zF3i plusmn
4c24z
2F23i + 16c5F5i 2c3z
2 + c5F4i 2 + 3e2( ) + 2c5e2F5i(
1113969
8c5eF5i
(40)
F (i) = 01
ω
π
3π4
π2
π4
0
e
01 02 03 04 05
Figure 7 e curve ω(e) at which _e _i 0 for F(i) 001
secndash1
01 02 03 04 05
e
ndash50 times 10ndash7
ndash10 times 10ndash6
Ωmiddot
ωmiddot
(a)
secndash1
01 02 03 04 05e
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
Ωmiddot
ωmiddot
(b)
secndash1
01 02 03 04 05e
12 times 10ndash6
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
Ωmiddot
ωmiddot
(c)
secndash1
01 02 03 04 05e
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
Ωmiddot
ωmiddot
(d)
Figure 8 Curves of _Ω and _ω with e for the 4 cases corresponding to F(i) 01 (a) i minus 5976 (b) i 6484 (c) i minus 12024 and (d) i 11516
8 Advances in Astronomy
where
z a
Rη (41)
e condition for having the orbit if real solution exists isagain that minus 1le sin(ω)le 1 which gives
minus 1le2c3z
2 + 2 + 3e2( 1113857c5F4i + 2e2 c5 minus 1( 1113857F5i
c4ezF3i
le 1 F5i F3i ne 0
(42)
33Orbitswith FixedPerigee For the argument of perigee tobalance we solve _ω 0We substitute from equation (26)into equation (29) then expand cos(2ω) and collect termswith respect to sin(ω) We get
_ω A sin2(ω) + B sin(ω) + C (43)
us for _ω 0 we get
sin(ω) minus B plusmn
B2 minus 4AC
radic
2A (44)
F (i) = 015
ω
π
3π4
π2
π4
0
e
01 02 03 04 05
Figure 9 e curve ω(e) at which _e _i 0 for F(i) 015
01 02 03 04 05
50 times 10ndash7
ndash50 times 10ndash7
ndash10 times 10ndash6
ndash15 times 10ndash6
e
ωmiddotΩmiddot
secndash
1
(a)
01 02 03 04 05
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
e
ωmiddotΩmiddot
secndash
1
(b)
01 02 03 04 05
14 times 10ndash6
10 times 10ndash6
12 times 10ndash6
80 times 10ndash7
60 times 10ndash7
e
ωmiddotΩmiddot
secndash
1
(c)
01 02 03 04 05
10 times 10ndash6
60 times 10ndash7
80 times 10ndash7
40 times 10ndash7
20 times 10ndash7
ndash20 times 10ndash7
e
ωmiddotΩmiddot
secndash
1
(d)
Figure 10 Curves of _Ω and _ω with e for the 4 cases corresponding to F(i) 015 (a) i minus 5514 (b) i 6525 (c) i minus 12486 and (d) i 11475
Advances in Astronomy 9
where
A minus 2c5e2F5cos
2(i) + 2c2 1 +
52e2
1113874 1113875F2 (45)
B c4ezF3cos2(i) minus c1z 4e +
1e
1113874 1113875F1 (46)
C cos2(i) c3z2
+ c5 e2F5 + 1 +
3e2
21113888 1113889F41113888 11138891113888 1113889
minus c6z2F6 minus c2 1 +
3e2
41113888 1113889F7 + 1 +
5e2
21113888 11138891113888 1113889F21113888 1113889
(47)
F (i) = 02
ω
π
3π4
π2
π4
0
e
01 02 03 04 05
Figure 11 e curve ω(e) at which _e _i 0 for F(i) 02
ωmiddotΩmiddot
01 02 03 04 05
e
15 times 10ndash6
10 times 10ndash6
50 times 10ndash7
ndash50 times 10ndash7
ndash10 times 10ndash6
ndash15 times 10ndash6
secndash
1
(a)
ωmiddotΩmiddot
01 02 03 04 05e
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
secndash
1
(b)
ωmiddotΩmiddot
01 02 03 04 05e
16 times 10ndash6
14 times 10ndash6
12 times 10ndash6
10 times 10ndash6
secndash
1
(c)
ωmiddotΩmiddot
01 02 03 04 05
e
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
ndash20 times 10ndash7
secndash
1
(d)
Figure 12 Curves of _Ω and _ωwith e for the 4 cases corresponding to F(i) 02 (a) i minus 4716 (b) i 6556 (c) i minus 13284 and (d) i 11444
10 Advances in Astronomy
Equations (44)ndash(47) give the relation between sin(ω) andthe inclination i the semi-major axis a and the eccentricitye which gives the family of orbits that balance the argumentof perigee ω subject of course to the restriction that sin(ω) isa real value between minus 1 and 1
4 Numerical Results
In this section numerical results and graphs are obtained forthe case of seasat a 7100 km by putting ω as a function of eand i from equations (32) and (33) e curves are plottedwithin the possible range given by condition (36) to give curvesof balanced e and ie curves are against the eccentricity e inthe range [001 05] e numerical values involved areJ2 0001082645 J3 minus 0000002546 J4 minus 0000001649 R
6378165km α 7100m and μ 3986005 km3secminus 2e condition (36) gives the upper and lower bounds of
the function F(i) as a function of e and since the functioneR(C1a(1 minus e2)) is increasing with e and has no criticalpoints in the interval [001 05] then the minimum valueoccurs at e 001 and the maximum at e 05 which givesminus 024leF(i)le 024 e graphs are plotted for differentvalues of F(i) which corresponds to specific values of i foundby solving the equation resulting from setting F(i) equal tothe required values After that we plot _Ω 0 and _ω 0 si-multaneously for the same values of F(i) at each specific i-value to find the orbit values at which we have nearest valuesfor _Ω 0 and _ω 0 In the graphs of _Ω and _ω the relation(32) was kept to ensure that e and i are already balanced
Five selected values of F(i) are chosen F(i) 001 05010 015 and 020 Negative values will give the same resultswith negative sign since F(i) is odd with respect to i Also foreach value the solution of F(i) x with x equals one of theabove values we get four real values of i on the form i1180 minus i1 i2 and minus 180 minus i2 where i1 and i2 are near the criticalinclination one of them is positive and the other is negativeTable 1 gives the values of i corresponding to the selectedvalues of F(i)
We note that as F(i) increases i gets away from thecritical inclination and the curve gets shorter indicating lessstability of ω as expected
e graphs are plotted for each value of F(i) first for thebalanced values of e and i then for the corresponding fourvalues of i four graphs are plotted for _Ω and _ω to find thenearest values of zero variation for both elements
Figures 3ndash12 show the possibility of balancing ω with eand i while Ω will have a variation of order 10minus 6sec or itmust be balanced alone at i 90 deg (as shown in equation(26)) according to the orbit kind Figures 3 5 7 9 and 11show the possible values of ω(e) that balance e and i whileFigures 4 6 8 10 and 12 show the curves of _Ω and _ω at thevalues of i at which _e _i 0
5 Conclusion
Let balanced orbits be defined as those for which the orbitalelements are set equal to zero under the effect of secular andlong periodic perturbations In this work the effect of Earthoblateness is the considered perturbing force because of
dealing with low Earth orbits e above analysis then showsthat such an orbit will be balanced within a reliable toleranceonly for few weeks since we are forced to accept the motionof either the node or the perigee by about 10minus 6 degsec ereason is that under the influence of the Earth oblateness_Ω 0 (exactly) for i (π2) while _ω 0 only near thecritical inclination ic 634 deg Hence the best procedure isto design a satellite constellation for which the nodal shiftsdue to the perturbative effects and Earth rotation aremodeled to yield continuous coverage e perigees areeither fixed or arranged to realize that the perigee (or theapogee) be overhead the coverage region (regions) in duetimes is may require near commensurability with theadmitted nodal periods
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was funded by the University of Jeddah SaudiArabia under grant no UJ-26-18-DR us the authortherefore acknowledges with thanks the university technicaland financial support
References
[1] N Delsate P Robutel A Lemaıtre and T Carletti ldquoFrozenorbits at high eccentricity and inclination application tomercury orbiterrdquo Celestial Mechanics and Dynamical As-tronomy vol 108 no 3 pp 275ndash300 2010
[2] E Condoleo M Cinelli E Ortore and C Circi ldquoStable orbitsfor lunar landing assistancerdquo Advances in Space Researchvol 60 no 7 pp 1404ndash1412 2017
[3] A Elipe andM Lara ldquoFrozen orbits about the moonrdquo Journalof Guidance Control and Dynamics vol 26 no 2 pp 238ndash243 2003
[4] C Circi E Condoleo and E Ortore ldquoA vectorial approach todetermine frozen orbital conditionsrdquo Celestial Mechanics andDynamical Astronomy vol 128 no 2-3 pp 361ndash382 2017
[5] E Condoleo M Cinelli E Ortore and C Circi ldquoFrozenorbits with equatorial perturbing bodies the case of Gany-mede Callisto and Titanrdquo Journal of Guidance Control andDynamics vol 39 no 10 pp 2264ndash2272 2016
[6] V Kudielka ldquoBalanced earth satellite orbitsrdquo Celestial Me-chanics amp Dynamical Astronomy vol 60 no 4 pp 455ndash4701994
[7] S L Coffey A Deprit and E Deprit ldquoFrozen orbits forsatellites close to an earth-like planetrdquo Celestial Mechanics ampDynamical Astronomy vol 59 no 1 pp 37ndash72 1994
[8] M Lara A Deprit and A Elipe ldquoNumerical continuation offamilies of frozen orbits in the zonal problem of artificialsatellite theoryrdquo Celestial Mechanics amp Dynamical Astronomyvol 62 no 2 pp 167ndash181 1995
[9] D Brouwer ldquoSolution of the problem of artificial satellitetheory without dragrdquo e Astronomical Journal vol 64p 378 1959
Advances in Astronomy 11
[10] Y Kozai ldquoe motion of a close earth satelliterdquo e Astro-nomical Journal vol 64 p 367 1959
[11] P M Fitzpatrick Principles of Celestial Mechanics AcademicPress Cambridge MA USA 1970
[12] H D Curtis Orbital Mechanics for Engineering StudentsElsevier Amsterdam Netherlands 2005
[13] D Brower and G M Clemence Methods of Celestial Me-chanics Academic Press Cambridge MA USA 1961
12 Advances in Astronomy
perturbations methods through the relation From thespherical trigonometry of the celestial sphere we have
sin δ Si sin(f + ω) (6)
where f is the true anomalyω is the argument of perigee andSi sin i Ci cos i
Substituting for δ in Pn (sin δ) we get
P2(sin δ) 12
3Si2 sin2(f + g) minus 11113966 1113967
14
1 minus 3Ci2
1113872 1113873 minus 3Si2 cos(2g + 2f)1113966 1113967
P3(sin δ) 12
5Si3 sin3(g + f) minus 3Si sin(f + g)1113966 1113967
18
15Si3
minus 12Si1113872 1113873sin(f + g) minus 5Si3 sin(3f + 3g)1113966 1113967
P4(sin δ) 18
3 minus 30Si2 sin2(g + f) + 35Si
4 sin4(g + f)1113966 1113967
164
9 minus 90Ci2
+ 105Ci4
1113872 11138731113966
+ minus 20 + 160Ci2
minus 140Ci4
1113872 1113873cos(2f + 2g)
+ 35Si4 cos(4f + 4g)1113967
(7)
us we get V2 V3 and V4 as functions of the orbitalelements
We now proceed to evaluate the effects of Earth ob-lateness considering the geopotential up to the zonal har-monic J4
In the present solution we consider only the secular andlong periodic terms averaging over the mean anomaly l
21 e Disturbing Function e disturbing function R isdefined as follows
R minus V2 + V3 + V4( 1113857 (8)
where V2 V3 and V4 are the 2nd 3rd and 4th terms of thegeopotential namely
V2 μj2R
2
4r31 minus 3Ci
21113872 1113873 minus 3Si
2 cos(2f + 2ω)1113960 1113961
V3 μj3R
3
8r415Si
3minus 12Si1113872 1113873sin(f + ω) minus 5Si
3 sin(3f + 3ω)1113960 1113961
V4 μj4R
4
64r59 minus 90Ci
2+ 105Ci
41113872 11138731113960
+ minus 20 + 160Ci2
minus 140Ci4
1113872 1113873cos(2f + 2ω)
+ 35Si4 cos(4f + 4ω)
(9)
where Ci cos(i) Si sin(i) and f is the true anomalySince terms depending only on the fast variable l will not
affect the orbit in an accumulating way with time we averagethe perturbing function R over the mean anomaly with itsperiod 2π e average function is defined by
langF(l)rang 12π
11139462π
0F(l)dl (10)
As the perturbing function R is a function of the trueanomaly f not the mean anomaly l we use the relation
dl 1
1 minus e2
radicr
a1113874 1113875
2df (11)
where both angles have the same period and the same end points0 and 2π and therefore the average function will be given by
langF(f)rang 1
2π1 minus e2
radic 11139462π
0
r
a1113874 1113875
2F(f)df (12)
Applying the required integrals we get the averageddisturbing function langRrang
langRrang minus langV2rangl +langV3rangl +langV4rangl1113858 1113859 (13)
where
langV2rangl μj2R
2
4a3 1 minus e2
1113872 1113873minus (32)
1 minus 3Ci2
1113872 1113873
langV3rangl μj3R
3
8a4 e 1 minus e2
1113872 1113873minus (52)
15Si3
minus 12Si1113872 1113873sin(ω)
langV4rangl μj4R
4
64a5 1 minus e2
1113872 1113873minus (72)
1 +3e2
21113888 1113889 9 minus 90Ci
2+ 105Ci
41113872 1113873
+3μj4R
4
256a5 e2 1 minus e
21113872 1113873
minus (72)minus 20 + 160Ci
2minus 140Ci
41113872 1113873cos(2ω)
(14)
22 Lagrange Equations for the Averaged Variations of theElements e Lagrange planetary equations for the varia-tions of the elements for a disturbing potential R are asfollows [13]
Ω
ω
i
i
r
fv
Y
Z
Node line N
Ascending node
SatellitePerigee
Earthrsquos north polar axis
Earthrsquos equatorial plane
J
I
X
h
e
γ
Figure 2 Keplerian orbital elements of the satellite
Advances in Astronomy 3
_a 2
na
zR
zl
_e minus
1 minus e2
radic
na2e
zR
zω+1 minus e2
na2e
zR
zl
_i minus 1
na2Si1 minus e2
radiczR
zh+
cot(i)
na21 minus e2
radiczR
zω
_Ω 1
na2Si1 minus e2
radiczR
zi
_ω
1 minus e2
radic
na2e
zR
zeminus Ci _Ω
_l n minus2
na
zR
za1113888 1113889
n
minus1 minus e2
na2e
zR
ze
(15)
where n is the mean motion given by n μa3
1113968
Substituting for the averaged disturbing function langRrang
due to Earth oblateness the Lagrange equations become
_a 0 (16)
_e minus
1 minus e2
radic
na2e
zlangRrang
zω (17)
_i minus 1
na2Si1 minus e2
radiczlangRrang
zΩ+
cot(i)
na21 minus e2
radiczlangRrang
zω (18)
_Ω 1
na2Si1 minus e2
radiczlangRrang
zi (19)
_ω
1 minus e2
radic
na2e
zlangRrang
zeminus Ci _Ω (20)
where the equation for l is neglected because we concentrateon balancing the orbit position not the satellite motion in theorbit
23 Variations of the Elements due to Earth OblatenessSubstituting for the required derivatives in equations (16) to(20) yields
_e 3 μradic
j3R3
2a92 1 minus e2
1113872 1113873minus 2 5
4Si
2minus 11113874 1113875Si cos(ω)
minus15 μradic
j4R4
32a112 e 1 minus e2
1113872 1113873minus 3
minus 1 + 8Ci2
minus 7Ci4
1113872 1113873sin(2ω)
(21)
Defining
η 1 minus e2
1113872 1113873
c1 3j3
2
F1i 54
Si2
minus 11113874 1113875Si
c2 15j4
32
F2i minus 1 + 8Ci2
minus 7Ci4
1113872 1113873
(22)
We get
_e
μradicR4
η3a112 c1ηa
RF1iCos(ω) minus c2eF2i sin(2ω)1113874 1113875
_i cot(i)
n
3μj3R3
2a6 e 1 minus e2
1113872 1113873minus 3 5
4Si
2minus 11113874 1113875Si cos(ω)1113896
minus15μj4R
4
16a7 e2 1 minus e
21113872 1113873
minus 4minus 1 + 8Ci
2minus 7Ci
41113872 1113873sin(2ω)1113897
(23)
or
_i minuse cot(i)
η_e
_Ω minus Ci
n
3μj2R2
2a5 1 minus e2
1113872 1113873minus 2
1113896
+μj3R
3
8a6 e 1 minus e2
1113872 1113873minus 3
45Si minus12Si
1113874 1113875sin(ω)
+15μj4R
4
16a7 1 minus e2
1113872 1113873minus 4
1 +3e2
21113888 1113889 3 minus 7Ci
21113872 11138731113890
+ e2
minus 4 + 7Ci2
1113872 1113873cos(2ω)11138911113897
(24)
Defining
c3 3j22
c4 3j3
8
c5 15j4
16
F3i 15Si minus4Si
F4i 3 minus 7Ci2
F5i minus 4 + 7Ci2
(25)
We get
4 Advances in Astronomy
_Ω minus Ci
μradicR4
η4a112 c3a
R1113874 1113875
2η2 + c4
a
ReηF3i sin(ω)1113896
+ c5 1 +3e2
21113888 1113889F4i + e
2F5i cos(2ω)1113890 11138911113897
(26)
_ω minus Ci _Ω minus3μj2R
2
4na5 1 minus e2
1113872 1113873minus 2
1 minus 3Ci2
1113872 1113873
minus3μj3R
3
2na6e1 + 4e
21113872 1113873 1 minus e
21113872 1113873
minus 3 54
Si2
minus 11113874 1113875Si sin(ω)
minus15μj4R
4
32na7 1 minus e2
1113872 1113873minus 4
1 +3e2
41113888 1113889 3 minus 30Ci
2+ 35Ci
41113872 11138731113890
+ 1 +52e2
1113874 1113875 minus 1 + 8Ci2
minus 7Ci4
1113872 1113873cos(2ω)1113891
(27)
Defining as in equation (28) we get equation (29)
c6 3j2
4
F6i 1 minus 3Ci2
1113872 1113873
F7i 3 minus 30Ci2
+ 35Ci4
(28)
_ω minus
μradicR4
η4a112 c6a
R1113874 1113875
2η2F6i + c1
a
R1 + 4e
21113872 1113873
ηeF1i sin(ω)1113890
+ c2 1 +3e2
41113888 1113889F7i + 1 +
52e2
1113874 1113875F2i cos(2ω)1113888 11138891113891 minus Ci _Ω
(29)
Equations (22)ndash(29) give the average effects of Earthoblateness including the zonal harmonics of the geopotentialup to J4 on the Keplerian elements of the satellite orbit
3 Balanced Low Earth Satellite Orbits
In what follows we try to find orbits that are balanced in thesense that the averaged (over the fast variable l) variations ofthe orbit elements are set equal to zero In equation (23) weput it equal to zero and get a relation between the argumentof perigee ω and the inclination i while treating the ec-centricity e and the semi-major axis a as parameters iswill give a range of values for ω and i at different values of eand a which are all give balanced orbits with respect to boththe eccentricity e and the inclination ie same is done withequations (26) and (29) while putting _Ω 0 and _ω 0
e applicable ranges for this model of the semi-majoraxis a are 11Rle ale 13R where the range 14Rndash2R isavoided due to the predominance of the radiation belts atthese levels to avoid the damages of the equipment that itmay produce besides its fatal effects on human life (forinhabited spacecrafts) e values for the eccentricity e aretaken between 001 (almost circular orbit) and 05
31 Orbits with Fixed Eccentricity and Inclination Byequating the variation of e by zero we get
_e
μradicR4
η3a112 c1a
RηF1i cos(ω) + c2eF2i sin(2ω)1113874 1113875 0 (30)
is implies
c1a
RηF1i cos(ω) + c2eF2i sin(2ω) 0 (31)
So either cos(ω) 0 or
sin(ω) a 1 minus e2( 1113857
eRC1F(i) (32)
where
C1 8J3
5J4
F(i) 1 minus (54)sin2(i)( 1113857sin(i)
1 minus 8cos2(i) + 7cos4(i)
(33)
Equations (32) and (33) give the family of low orbits thathave both the eccentricity and the inclination fixed
e condition for the existence of such orbits is clearlythat
minus 1le sin(ω)le 1 (34)
which is guaranteed by
minus 1lea 1 minus e2( 1113857
eRC1F(i)le 1 (35)
We put it as a condition on the eccentricity i only when
minus eR
C1a 1 minus e2( )leF(i)le
eR
C1a 1 minus e2( ) (36)
32 Orbits with Fixed Node e families of orbits for whichdΩdt 0 (ie with fixed nodes) are obtained from
minus Ciμradic
R4
η4a112 c3a
R1113874 1113875
2η2 + c4
a
ReηF3i sin(ω)1113896
+ c5 1 +3e2
21113888 1113889F4i + e
2F5i cos(2ω)1113890 11138911113897 0
(37)
is implies
c3a
R1113874 1113875
2η2 + c4
a
ReηF3i sin(ω)
+ c5 1 +3e2
21113888 1113889F4i + e
2F5i 1 minus 2sin2(ω)1113872 11138731113890 1113891 0
(38)
Advances in Astronomy 5
Table 1 Real values of i (in degree) corresponding to the given values of F(i)
F(i) i1 180 minus i1 i2 minus 180 minus i2001 6363 11637 minus 6322 minus 11678005 6427 11573 minus 6211 minus 11789010 6484 11516 minus 5976 minus 12024015 6525 11475 minus 5514 minus 12486020 6556 11444 minus 4716 minus 13284
01 02 03 04 05
e
F (i) = 001
ω
π
3π4
π2
π4
0
Figure 3 e curve ω(e) at which _e _i 0 for F(i) 001
secndash1
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
ndash10 times 10ndash6
01 02 03 04 05
e
Ωmiddot
ωmiddot
(a)
secndash1
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
ndash10 times 10ndash6
01 02 03 04 05
e
Ωmiddot
ωmiddot
(b)
secndash1
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
01 02 03 04 05e
Ωmiddot
ωmiddot
(c)
secndash1
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
01 02 03 04 05e
Ωmiddot
ωmiddot
(d)
Figure 4 Curves of _Ω and _ω with e for the 4 cases corresponding to F(i) 001 (a) i 6363 (b) i minus 6322 (c) i 11637 and (d) i 11678
6 Advances in Astronomy
is can be arranged as a second order equationfor sin(ω) giving a family of orbits with fixed argumentof perigee for different values of ω as a function ofthe inclination i the semi-major axis a and the eccen-tricity e
minus 2c5e2F5isin
2(ω) + c4
a
ReηF3i sin(ω)
+ c3a
R1113874 1113875
2η2 + c5 1 +
3e2
21113888 1113889F4i + e
2F5i1113890 1113891 0
(39)
e
F (i) = 005
ω
π
3π4
π2
π4
001 02 03 04 05
Figure 5 e curve ω(e) at which _e _i 0 for F(i) 005
secndash1
01 02 03 04 05
Ωmiddot
ωmiddot
e
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
ndash10 times 10ndash6
(a)
secndash1
01 02 03 04 05
Ωmiddot
ωmiddot
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
ndash10 times 10ndash6
ndash12 times 10ndash6
e
(b)
secndash1
Ωmiddot
ωmiddot
ndash10 times 10ndash6
ndash80 times 10ndash7
ndash60 times 10ndash7
ndash40 times 10ndash7
ndash20 times 10ndash7
01 02 03 04 05e
(c)
secndash1
01 02 03 04 05
Ωmiddot
ωmiddot
12 times 10ndash6
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
e
(d)
Figure 6 Curves of _Ω and _ωwith e for the 4 cases corresponding to F(i) 005 (a) i 6427 (b) i minus 6211 (c) i 11573 and (d) i minus 11789
Advances in Astronomy 7
When solving for sin(ω) we get the family of orbits forwhich the longitude of the node is balanced
sin(ω) 2c4zF3i plusmn
4c24z
2F23i + 16c5F5i 2c3z
2 + c5F4i 2 + 3e2( ) + 2c5e2F5i(
1113969
8c5eF5i
(40)
F (i) = 01
ω
π
3π4
π2
π4
0
e
01 02 03 04 05
Figure 7 e curve ω(e) at which _e _i 0 for F(i) 001
secndash1
01 02 03 04 05
e
ndash50 times 10ndash7
ndash10 times 10ndash6
Ωmiddot
ωmiddot
(a)
secndash1
01 02 03 04 05e
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
Ωmiddot
ωmiddot
(b)
secndash1
01 02 03 04 05e
12 times 10ndash6
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
Ωmiddot
ωmiddot
(c)
secndash1
01 02 03 04 05e
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
Ωmiddot
ωmiddot
(d)
Figure 8 Curves of _Ω and _ω with e for the 4 cases corresponding to F(i) 01 (a) i minus 5976 (b) i 6484 (c) i minus 12024 and (d) i 11516
8 Advances in Astronomy
where
z a
Rη (41)
e condition for having the orbit if real solution exists isagain that minus 1le sin(ω)le 1 which gives
minus 1le2c3z
2 + 2 + 3e2( 1113857c5F4i + 2e2 c5 minus 1( 1113857F5i
c4ezF3i
le 1 F5i F3i ne 0
(42)
33Orbitswith FixedPerigee For the argument of perigee tobalance we solve _ω 0We substitute from equation (26)into equation (29) then expand cos(2ω) and collect termswith respect to sin(ω) We get
_ω A sin2(ω) + B sin(ω) + C (43)
us for _ω 0 we get
sin(ω) minus B plusmn
B2 minus 4AC
radic
2A (44)
F (i) = 015
ω
π
3π4
π2
π4
0
e
01 02 03 04 05
Figure 9 e curve ω(e) at which _e _i 0 for F(i) 015
01 02 03 04 05
50 times 10ndash7
ndash50 times 10ndash7
ndash10 times 10ndash6
ndash15 times 10ndash6
e
ωmiddotΩmiddot
secndash
1
(a)
01 02 03 04 05
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
e
ωmiddotΩmiddot
secndash
1
(b)
01 02 03 04 05
14 times 10ndash6
10 times 10ndash6
12 times 10ndash6
80 times 10ndash7
60 times 10ndash7
e
ωmiddotΩmiddot
secndash
1
(c)
01 02 03 04 05
10 times 10ndash6
60 times 10ndash7
80 times 10ndash7
40 times 10ndash7
20 times 10ndash7
ndash20 times 10ndash7
e
ωmiddotΩmiddot
secndash
1
(d)
Figure 10 Curves of _Ω and _ω with e for the 4 cases corresponding to F(i) 015 (a) i minus 5514 (b) i 6525 (c) i minus 12486 and (d) i 11475
Advances in Astronomy 9
where
A minus 2c5e2F5cos
2(i) + 2c2 1 +
52e2
1113874 1113875F2 (45)
B c4ezF3cos2(i) minus c1z 4e +
1e
1113874 1113875F1 (46)
C cos2(i) c3z2
+ c5 e2F5 + 1 +
3e2
21113888 1113889F41113888 11138891113888 1113889
minus c6z2F6 minus c2 1 +
3e2
41113888 1113889F7 + 1 +
5e2
21113888 11138891113888 1113889F21113888 1113889
(47)
F (i) = 02
ω
π
3π4
π2
π4
0
e
01 02 03 04 05
Figure 11 e curve ω(e) at which _e _i 0 for F(i) 02
ωmiddotΩmiddot
01 02 03 04 05
e
15 times 10ndash6
10 times 10ndash6
50 times 10ndash7
ndash50 times 10ndash7
ndash10 times 10ndash6
ndash15 times 10ndash6
secndash
1
(a)
ωmiddotΩmiddot
01 02 03 04 05e
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
secndash
1
(b)
ωmiddotΩmiddot
01 02 03 04 05e
16 times 10ndash6
14 times 10ndash6
12 times 10ndash6
10 times 10ndash6
secndash
1
(c)
ωmiddotΩmiddot
01 02 03 04 05
e
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
ndash20 times 10ndash7
secndash
1
(d)
Figure 12 Curves of _Ω and _ωwith e for the 4 cases corresponding to F(i) 02 (a) i minus 4716 (b) i 6556 (c) i minus 13284 and (d) i 11444
10 Advances in Astronomy
Equations (44)ndash(47) give the relation between sin(ω) andthe inclination i the semi-major axis a and the eccentricitye which gives the family of orbits that balance the argumentof perigee ω subject of course to the restriction that sin(ω) isa real value between minus 1 and 1
4 Numerical Results
In this section numerical results and graphs are obtained forthe case of seasat a 7100 km by putting ω as a function of eand i from equations (32) and (33) e curves are plottedwithin the possible range given by condition (36) to give curvesof balanced e and ie curves are against the eccentricity e inthe range [001 05] e numerical values involved areJ2 0001082645 J3 minus 0000002546 J4 minus 0000001649 R
6378165km α 7100m and μ 3986005 km3secminus 2e condition (36) gives the upper and lower bounds of
the function F(i) as a function of e and since the functioneR(C1a(1 minus e2)) is increasing with e and has no criticalpoints in the interval [001 05] then the minimum valueoccurs at e 001 and the maximum at e 05 which givesminus 024leF(i)le 024 e graphs are plotted for differentvalues of F(i) which corresponds to specific values of i foundby solving the equation resulting from setting F(i) equal tothe required values After that we plot _Ω 0 and _ω 0 si-multaneously for the same values of F(i) at each specific i-value to find the orbit values at which we have nearest valuesfor _Ω 0 and _ω 0 In the graphs of _Ω and _ω the relation(32) was kept to ensure that e and i are already balanced
Five selected values of F(i) are chosen F(i) 001 05010 015 and 020 Negative values will give the same resultswith negative sign since F(i) is odd with respect to i Also foreach value the solution of F(i) x with x equals one of theabove values we get four real values of i on the form i1180 minus i1 i2 and minus 180 minus i2 where i1 and i2 are near the criticalinclination one of them is positive and the other is negativeTable 1 gives the values of i corresponding to the selectedvalues of F(i)
We note that as F(i) increases i gets away from thecritical inclination and the curve gets shorter indicating lessstability of ω as expected
e graphs are plotted for each value of F(i) first for thebalanced values of e and i then for the corresponding fourvalues of i four graphs are plotted for _Ω and _ω to find thenearest values of zero variation for both elements
Figures 3ndash12 show the possibility of balancing ω with eand i while Ω will have a variation of order 10minus 6sec or itmust be balanced alone at i 90 deg (as shown in equation(26)) according to the orbit kind Figures 3 5 7 9 and 11show the possible values of ω(e) that balance e and i whileFigures 4 6 8 10 and 12 show the curves of _Ω and _ω at thevalues of i at which _e _i 0
5 Conclusion
Let balanced orbits be defined as those for which the orbitalelements are set equal to zero under the effect of secular andlong periodic perturbations In this work the effect of Earthoblateness is the considered perturbing force because of
dealing with low Earth orbits e above analysis then showsthat such an orbit will be balanced within a reliable toleranceonly for few weeks since we are forced to accept the motionof either the node or the perigee by about 10minus 6 degsec ereason is that under the influence of the Earth oblateness_Ω 0 (exactly) for i (π2) while _ω 0 only near thecritical inclination ic 634 deg Hence the best procedure isto design a satellite constellation for which the nodal shiftsdue to the perturbative effects and Earth rotation aremodeled to yield continuous coverage e perigees areeither fixed or arranged to realize that the perigee (or theapogee) be overhead the coverage region (regions) in duetimes is may require near commensurability with theadmitted nodal periods
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was funded by the University of Jeddah SaudiArabia under grant no UJ-26-18-DR us the authortherefore acknowledges with thanks the university technicaland financial support
References
[1] N Delsate P Robutel A Lemaıtre and T Carletti ldquoFrozenorbits at high eccentricity and inclination application tomercury orbiterrdquo Celestial Mechanics and Dynamical As-tronomy vol 108 no 3 pp 275ndash300 2010
[2] E Condoleo M Cinelli E Ortore and C Circi ldquoStable orbitsfor lunar landing assistancerdquo Advances in Space Researchvol 60 no 7 pp 1404ndash1412 2017
[3] A Elipe andM Lara ldquoFrozen orbits about the moonrdquo Journalof Guidance Control and Dynamics vol 26 no 2 pp 238ndash243 2003
[4] C Circi E Condoleo and E Ortore ldquoA vectorial approach todetermine frozen orbital conditionsrdquo Celestial Mechanics andDynamical Astronomy vol 128 no 2-3 pp 361ndash382 2017
[5] E Condoleo M Cinelli E Ortore and C Circi ldquoFrozenorbits with equatorial perturbing bodies the case of Gany-mede Callisto and Titanrdquo Journal of Guidance Control andDynamics vol 39 no 10 pp 2264ndash2272 2016
[6] V Kudielka ldquoBalanced earth satellite orbitsrdquo Celestial Me-chanics amp Dynamical Astronomy vol 60 no 4 pp 455ndash4701994
[7] S L Coffey A Deprit and E Deprit ldquoFrozen orbits forsatellites close to an earth-like planetrdquo Celestial Mechanics ampDynamical Astronomy vol 59 no 1 pp 37ndash72 1994
[8] M Lara A Deprit and A Elipe ldquoNumerical continuation offamilies of frozen orbits in the zonal problem of artificialsatellite theoryrdquo Celestial Mechanics amp Dynamical Astronomyvol 62 no 2 pp 167ndash181 1995
[9] D Brouwer ldquoSolution of the problem of artificial satellitetheory without dragrdquo e Astronomical Journal vol 64p 378 1959
Advances in Astronomy 11
[10] Y Kozai ldquoe motion of a close earth satelliterdquo e Astro-nomical Journal vol 64 p 367 1959
[11] P M Fitzpatrick Principles of Celestial Mechanics AcademicPress Cambridge MA USA 1970
[12] H D Curtis Orbital Mechanics for Engineering StudentsElsevier Amsterdam Netherlands 2005
[13] D Brower and G M Clemence Methods of Celestial Me-chanics Academic Press Cambridge MA USA 1961
12 Advances in Astronomy
_a 2
na
zR
zl
_e minus
1 minus e2
radic
na2e
zR
zω+1 minus e2
na2e
zR
zl
_i minus 1
na2Si1 minus e2
radiczR
zh+
cot(i)
na21 minus e2
radiczR
zω
_Ω 1
na2Si1 minus e2
radiczR
zi
_ω
1 minus e2
radic
na2e
zR
zeminus Ci _Ω
_l n minus2
na
zR
za1113888 1113889
n
minus1 minus e2
na2e
zR
ze
(15)
where n is the mean motion given by n μa3
1113968
Substituting for the averaged disturbing function langRrang
due to Earth oblateness the Lagrange equations become
_a 0 (16)
_e minus
1 minus e2
radic
na2e
zlangRrang
zω (17)
_i minus 1
na2Si1 minus e2
radiczlangRrang
zΩ+
cot(i)
na21 minus e2
radiczlangRrang
zω (18)
_Ω 1
na2Si1 minus e2
radiczlangRrang
zi (19)
_ω
1 minus e2
radic
na2e
zlangRrang
zeminus Ci _Ω (20)
where the equation for l is neglected because we concentrateon balancing the orbit position not the satellite motion in theorbit
23 Variations of the Elements due to Earth OblatenessSubstituting for the required derivatives in equations (16) to(20) yields
_e 3 μradic
j3R3
2a92 1 minus e2
1113872 1113873minus 2 5
4Si
2minus 11113874 1113875Si cos(ω)
minus15 μradic
j4R4
32a112 e 1 minus e2
1113872 1113873minus 3
minus 1 + 8Ci2
minus 7Ci4
1113872 1113873sin(2ω)
(21)
Defining
η 1 minus e2
1113872 1113873
c1 3j3
2
F1i 54
Si2
minus 11113874 1113875Si
c2 15j4
32
F2i minus 1 + 8Ci2
minus 7Ci4
1113872 1113873
(22)
We get
_e
μradicR4
η3a112 c1ηa
RF1iCos(ω) minus c2eF2i sin(2ω)1113874 1113875
_i cot(i)
n
3μj3R3
2a6 e 1 minus e2
1113872 1113873minus 3 5
4Si
2minus 11113874 1113875Si cos(ω)1113896
minus15μj4R
4
16a7 e2 1 minus e
21113872 1113873
minus 4minus 1 + 8Ci
2minus 7Ci
41113872 1113873sin(2ω)1113897
(23)
or
_i minuse cot(i)
η_e
_Ω minus Ci
n
3μj2R2
2a5 1 minus e2
1113872 1113873minus 2
1113896
+μj3R
3
8a6 e 1 minus e2
1113872 1113873minus 3
45Si minus12Si
1113874 1113875sin(ω)
+15μj4R
4
16a7 1 minus e2
1113872 1113873minus 4
1 +3e2
21113888 1113889 3 minus 7Ci
21113872 11138731113890
+ e2
minus 4 + 7Ci2
1113872 1113873cos(2ω)11138911113897
(24)
Defining
c3 3j22
c4 3j3
8
c5 15j4
16
F3i 15Si minus4Si
F4i 3 minus 7Ci2
F5i minus 4 + 7Ci2
(25)
We get
4 Advances in Astronomy
_Ω minus Ci
μradicR4
η4a112 c3a
R1113874 1113875
2η2 + c4
a
ReηF3i sin(ω)1113896
+ c5 1 +3e2
21113888 1113889F4i + e
2F5i cos(2ω)1113890 11138911113897
(26)
_ω minus Ci _Ω minus3μj2R
2
4na5 1 minus e2
1113872 1113873minus 2
1 minus 3Ci2
1113872 1113873
minus3μj3R
3
2na6e1 + 4e
21113872 1113873 1 minus e
21113872 1113873
minus 3 54
Si2
minus 11113874 1113875Si sin(ω)
minus15μj4R
4
32na7 1 minus e2
1113872 1113873minus 4
1 +3e2
41113888 1113889 3 minus 30Ci
2+ 35Ci
41113872 11138731113890
+ 1 +52e2
1113874 1113875 minus 1 + 8Ci2
minus 7Ci4
1113872 1113873cos(2ω)1113891
(27)
Defining as in equation (28) we get equation (29)
c6 3j2
4
F6i 1 minus 3Ci2
1113872 1113873
F7i 3 minus 30Ci2
+ 35Ci4
(28)
_ω minus
μradicR4
η4a112 c6a
R1113874 1113875
2η2F6i + c1
a
R1 + 4e
21113872 1113873
ηeF1i sin(ω)1113890
+ c2 1 +3e2
41113888 1113889F7i + 1 +
52e2
1113874 1113875F2i cos(2ω)1113888 11138891113891 minus Ci _Ω
(29)
Equations (22)ndash(29) give the average effects of Earthoblateness including the zonal harmonics of the geopotentialup to J4 on the Keplerian elements of the satellite orbit
3 Balanced Low Earth Satellite Orbits
In what follows we try to find orbits that are balanced in thesense that the averaged (over the fast variable l) variations ofthe orbit elements are set equal to zero In equation (23) weput it equal to zero and get a relation between the argumentof perigee ω and the inclination i while treating the ec-centricity e and the semi-major axis a as parameters iswill give a range of values for ω and i at different values of eand a which are all give balanced orbits with respect to boththe eccentricity e and the inclination ie same is done withequations (26) and (29) while putting _Ω 0 and _ω 0
e applicable ranges for this model of the semi-majoraxis a are 11Rle ale 13R where the range 14Rndash2R isavoided due to the predominance of the radiation belts atthese levels to avoid the damages of the equipment that itmay produce besides its fatal effects on human life (forinhabited spacecrafts) e values for the eccentricity e aretaken between 001 (almost circular orbit) and 05
31 Orbits with Fixed Eccentricity and Inclination Byequating the variation of e by zero we get
_e
μradicR4
η3a112 c1a
RηF1i cos(ω) + c2eF2i sin(2ω)1113874 1113875 0 (30)
is implies
c1a
RηF1i cos(ω) + c2eF2i sin(2ω) 0 (31)
So either cos(ω) 0 or
sin(ω) a 1 minus e2( 1113857
eRC1F(i) (32)
where
C1 8J3
5J4
F(i) 1 minus (54)sin2(i)( 1113857sin(i)
1 minus 8cos2(i) + 7cos4(i)
(33)
Equations (32) and (33) give the family of low orbits thathave both the eccentricity and the inclination fixed
e condition for the existence of such orbits is clearlythat
minus 1le sin(ω)le 1 (34)
which is guaranteed by
minus 1lea 1 minus e2( 1113857
eRC1F(i)le 1 (35)
We put it as a condition on the eccentricity i only when
minus eR
C1a 1 minus e2( )leF(i)le
eR
C1a 1 minus e2( ) (36)
32 Orbits with Fixed Node e families of orbits for whichdΩdt 0 (ie with fixed nodes) are obtained from
minus Ciμradic
R4
η4a112 c3a
R1113874 1113875
2η2 + c4
a
ReηF3i sin(ω)1113896
+ c5 1 +3e2
21113888 1113889F4i + e
2F5i cos(2ω)1113890 11138911113897 0
(37)
is implies
c3a
R1113874 1113875
2η2 + c4
a
ReηF3i sin(ω)
+ c5 1 +3e2
21113888 1113889F4i + e
2F5i 1 minus 2sin2(ω)1113872 11138731113890 1113891 0
(38)
Advances in Astronomy 5
Table 1 Real values of i (in degree) corresponding to the given values of F(i)
F(i) i1 180 minus i1 i2 minus 180 minus i2001 6363 11637 minus 6322 minus 11678005 6427 11573 minus 6211 minus 11789010 6484 11516 minus 5976 minus 12024015 6525 11475 minus 5514 minus 12486020 6556 11444 minus 4716 minus 13284
01 02 03 04 05
e
F (i) = 001
ω
π
3π4
π2
π4
0
Figure 3 e curve ω(e) at which _e _i 0 for F(i) 001
secndash1
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
ndash10 times 10ndash6
01 02 03 04 05
e
Ωmiddot
ωmiddot
(a)
secndash1
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
ndash10 times 10ndash6
01 02 03 04 05
e
Ωmiddot
ωmiddot
(b)
secndash1
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
01 02 03 04 05e
Ωmiddot
ωmiddot
(c)
secndash1
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
01 02 03 04 05e
Ωmiddot
ωmiddot
(d)
Figure 4 Curves of _Ω and _ω with e for the 4 cases corresponding to F(i) 001 (a) i 6363 (b) i minus 6322 (c) i 11637 and (d) i 11678
6 Advances in Astronomy
is can be arranged as a second order equationfor sin(ω) giving a family of orbits with fixed argumentof perigee for different values of ω as a function ofthe inclination i the semi-major axis a and the eccen-tricity e
minus 2c5e2F5isin
2(ω) + c4
a
ReηF3i sin(ω)
+ c3a
R1113874 1113875
2η2 + c5 1 +
3e2
21113888 1113889F4i + e
2F5i1113890 1113891 0
(39)
e
F (i) = 005
ω
π
3π4
π2
π4
001 02 03 04 05
Figure 5 e curve ω(e) at which _e _i 0 for F(i) 005
secndash1
01 02 03 04 05
Ωmiddot
ωmiddot
e
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
ndash10 times 10ndash6
(a)
secndash1
01 02 03 04 05
Ωmiddot
ωmiddot
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
ndash10 times 10ndash6
ndash12 times 10ndash6
e
(b)
secndash1
Ωmiddot
ωmiddot
ndash10 times 10ndash6
ndash80 times 10ndash7
ndash60 times 10ndash7
ndash40 times 10ndash7
ndash20 times 10ndash7
01 02 03 04 05e
(c)
secndash1
01 02 03 04 05
Ωmiddot
ωmiddot
12 times 10ndash6
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
e
(d)
Figure 6 Curves of _Ω and _ωwith e for the 4 cases corresponding to F(i) 005 (a) i 6427 (b) i minus 6211 (c) i 11573 and (d) i minus 11789
Advances in Astronomy 7
When solving for sin(ω) we get the family of orbits forwhich the longitude of the node is balanced
sin(ω) 2c4zF3i plusmn
4c24z
2F23i + 16c5F5i 2c3z
2 + c5F4i 2 + 3e2( ) + 2c5e2F5i(
1113969
8c5eF5i
(40)
F (i) = 01
ω
π
3π4
π2
π4
0
e
01 02 03 04 05
Figure 7 e curve ω(e) at which _e _i 0 for F(i) 001
secndash1
01 02 03 04 05
e
ndash50 times 10ndash7
ndash10 times 10ndash6
Ωmiddot
ωmiddot
(a)
secndash1
01 02 03 04 05e
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
Ωmiddot
ωmiddot
(b)
secndash1
01 02 03 04 05e
12 times 10ndash6
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
Ωmiddot
ωmiddot
(c)
secndash1
01 02 03 04 05e
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
Ωmiddot
ωmiddot
(d)
Figure 8 Curves of _Ω and _ω with e for the 4 cases corresponding to F(i) 01 (a) i minus 5976 (b) i 6484 (c) i minus 12024 and (d) i 11516
8 Advances in Astronomy
where
z a
Rη (41)
e condition for having the orbit if real solution exists isagain that minus 1le sin(ω)le 1 which gives
minus 1le2c3z
2 + 2 + 3e2( 1113857c5F4i + 2e2 c5 minus 1( 1113857F5i
c4ezF3i
le 1 F5i F3i ne 0
(42)
33Orbitswith FixedPerigee For the argument of perigee tobalance we solve _ω 0We substitute from equation (26)into equation (29) then expand cos(2ω) and collect termswith respect to sin(ω) We get
_ω A sin2(ω) + B sin(ω) + C (43)
us for _ω 0 we get
sin(ω) minus B plusmn
B2 minus 4AC
radic
2A (44)
F (i) = 015
ω
π
3π4
π2
π4
0
e
01 02 03 04 05
Figure 9 e curve ω(e) at which _e _i 0 for F(i) 015
01 02 03 04 05
50 times 10ndash7
ndash50 times 10ndash7
ndash10 times 10ndash6
ndash15 times 10ndash6
e
ωmiddotΩmiddot
secndash
1
(a)
01 02 03 04 05
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
e
ωmiddotΩmiddot
secndash
1
(b)
01 02 03 04 05
14 times 10ndash6
10 times 10ndash6
12 times 10ndash6
80 times 10ndash7
60 times 10ndash7
e
ωmiddotΩmiddot
secndash
1
(c)
01 02 03 04 05
10 times 10ndash6
60 times 10ndash7
80 times 10ndash7
40 times 10ndash7
20 times 10ndash7
ndash20 times 10ndash7
e
ωmiddotΩmiddot
secndash
1
(d)
Figure 10 Curves of _Ω and _ω with e for the 4 cases corresponding to F(i) 015 (a) i minus 5514 (b) i 6525 (c) i minus 12486 and (d) i 11475
Advances in Astronomy 9
where
A minus 2c5e2F5cos
2(i) + 2c2 1 +
52e2
1113874 1113875F2 (45)
B c4ezF3cos2(i) minus c1z 4e +
1e
1113874 1113875F1 (46)
C cos2(i) c3z2
+ c5 e2F5 + 1 +
3e2
21113888 1113889F41113888 11138891113888 1113889
minus c6z2F6 minus c2 1 +
3e2
41113888 1113889F7 + 1 +
5e2
21113888 11138891113888 1113889F21113888 1113889
(47)
F (i) = 02
ω
π
3π4
π2
π4
0
e
01 02 03 04 05
Figure 11 e curve ω(e) at which _e _i 0 for F(i) 02
ωmiddotΩmiddot
01 02 03 04 05
e
15 times 10ndash6
10 times 10ndash6
50 times 10ndash7
ndash50 times 10ndash7
ndash10 times 10ndash6
ndash15 times 10ndash6
secndash
1
(a)
ωmiddotΩmiddot
01 02 03 04 05e
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
secndash
1
(b)
ωmiddotΩmiddot
01 02 03 04 05e
16 times 10ndash6
14 times 10ndash6
12 times 10ndash6
10 times 10ndash6
secndash
1
(c)
ωmiddotΩmiddot
01 02 03 04 05
e
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
ndash20 times 10ndash7
secndash
1
(d)
Figure 12 Curves of _Ω and _ωwith e for the 4 cases corresponding to F(i) 02 (a) i minus 4716 (b) i 6556 (c) i minus 13284 and (d) i 11444
10 Advances in Astronomy
Equations (44)ndash(47) give the relation between sin(ω) andthe inclination i the semi-major axis a and the eccentricitye which gives the family of orbits that balance the argumentof perigee ω subject of course to the restriction that sin(ω) isa real value between minus 1 and 1
4 Numerical Results
In this section numerical results and graphs are obtained forthe case of seasat a 7100 km by putting ω as a function of eand i from equations (32) and (33) e curves are plottedwithin the possible range given by condition (36) to give curvesof balanced e and ie curves are against the eccentricity e inthe range [001 05] e numerical values involved areJ2 0001082645 J3 minus 0000002546 J4 minus 0000001649 R
6378165km α 7100m and μ 3986005 km3secminus 2e condition (36) gives the upper and lower bounds of
the function F(i) as a function of e and since the functioneR(C1a(1 minus e2)) is increasing with e and has no criticalpoints in the interval [001 05] then the minimum valueoccurs at e 001 and the maximum at e 05 which givesminus 024leF(i)le 024 e graphs are plotted for differentvalues of F(i) which corresponds to specific values of i foundby solving the equation resulting from setting F(i) equal tothe required values After that we plot _Ω 0 and _ω 0 si-multaneously for the same values of F(i) at each specific i-value to find the orbit values at which we have nearest valuesfor _Ω 0 and _ω 0 In the graphs of _Ω and _ω the relation(32) was kept to ensure that e and i are already balanced
Five selected values of F(i) are chosen F(i) 001 05010 015 and 020 Negative values will give the same resultswith negative sign since F(i) is odd with respect to i Also foreach value the solution of F(i) x with x equals one of theabove values we get four real values of i on the form i1180 minus i1 i2 and minus 180 minus i2 where i1 and i2 are near the criticalinclination one of them is positive and the other is negativeTable 1 gives the values of i corresponding to the selectedvalues of F(i)
We note that as F(i) increases i gets away from thecritical inclination and the curve gets shorter indicating lessstability of ω as expected
e graphs are plotted for each value of F(i) first for thebalanced values of e and i then for the corresponding fourvalues of i four graphs are plotted for _Ω and _ω to find thenearest values of zero variation for both elements
Figures 3ndash12 show the possibility of balancing ω with eand i while Ω will have a variation of order 10minus 6sec or itmust be balanced alone at i 90 deg (as shown in equation(26)) according to the orbit kind Figures 3 5 7 9 and 11show the possible values of ω(e) that balance e and i whileFigures 4 6 8 10 and 12 show the curves of _Ω and _ω at thevalues of i at which _e _i 0
5 Conclusion
Let balanced orbits be defined as those for which the orbitalelements are set equal to zero under the effect of secular andlong periodic perturbations In this work the effect of Earthoblateness is the considered perturbing force because of
dealing with low Earth orbits e above analysis then showsthat such an orbit will be balanced within a reliable toleranceonly for few weeks since we are forced to accept the motionof either the node or the perigee by about 10minus 6 degsec ereason is that under the influence of the Earth oblateness_Ω 0 (exactly) for i (π2) while _ω 0 only near thecritical inclination ic 634 deg Hence the best procedure isto design a satellite constellation for which the nodal shiftsdue to the perturbative effects and Earth rotation aremodeled to yield continuous coverage e perigees areeither fixed or arranged to realize that the perigee (or theapogee) be overhead the coverage region (regions) in duetimes is may require near commensurability with theadmitted nodal periods
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was funded by the University of Jeddah SaudiArabia under grant no UJ-26-18-DR us the authortherefore acknowledges with thanks the university technicaland financial support
References
[1] N Delsate P Robutel A Lemaıtre and T Carletti ldquoFrozenorbits at high eccentricity and inclination application tomercury orbiterrdquo Celestial Mechanics and Dynamical As-tronomy vol 108 no 3 pp 275ndash300 2010
[2] E Condoleo M Cinelli E Ortore and C Circi ldquoStable orbitsfor lunar landing assistancerdquo Advances in Space Researchvol 60 no 7 pp 1404ndash1412 2017
[3] A Elipe andM Lara ldquoFrozen orbits about the moonrdquo Journalof Guidance Control and Dynamics vol 26 no 2 pp 238ndash243 2003
[4] C Circi E Condoleo and E Ortore ldquoA vectorial approach todetermine frozen orbital conditionsrdquo Celestial Mechanics andDynamical Astronomy vol 128 no 2-3 pp 361ndash382 2017
[5] E Condoleo M Cinelli E Ortore and C Circi ldquoFrozenorbits with equatorial perturbing bodies the case of Gany-mede Callisto and Titanrdquo Journal of Guidance Control andDynamics vol 39 no 10 pp 2264ndash2272 2016
[6] V Kudielka ldquoBalanced earth satellite orbitsrdquo Celestial Me-chanics amp Dynamical Astronomy vol 60 no 4 pp 455ndash4701994
[7] S L Coffey A Deprit and E Deprit ldquoFrozen orbits forsatellites close to an earth-like planetrdquo Celestial Mechanics ampDynamical Astronomy vol 59 no 1 pp 37ndash72 1994
[8] M Lara A Deprit and A Elipe ldquoNumerical continuation offamilies of frozen orbits in the zonal problem of artificialsatellite theoryrdquo Celestial Mechanics amp Dynamical Astronomyvol 62 no 2 pp 167ndash181 1995
[9] D Brouwer ldquoSolution of the problem of artificial satellitetheory without dragrdquo e Astronomical Journal vol 64p 378 1959
Advances in Astronomy 11
[10] Y Kozai ldquoe motion of a close earth satelliterdquo e Astro-nomical Journal vol 64 p 367 1959
[11] P M Fitzpatrick Principles of Celestial Mechanics AcademicPress Cambridge MA USA 1970
[12] H D Curtis Orbital Mechanics for Engineering StudentsElsevier Amsterdam Netherlands 2005
[13] D Brower and G M Clemence Methods of Celestial Me-chanics Academic Press Cambridge MA USA 1961
12 Advances in Astronomy
_Ω minus Ci
μradicR4
η4a112 c3a
R1113874 1113875
2η2 + c4
a
ReηF3i sin(ω)1113896
+ c5 1 +3e2
21113888 1113889F4i + e
2F5i cos(2ω)1113890 11138911113897
(26)
_ω minus Ci _Ω minus3μj2R
2
4na5 1 minus e2
1113872 1113873minus 2
1 minus 3Ci2
1113872 1113873
minus3μj3R
3
2na6e1 + 4e
21113872 1113873 1 minus e
21113872 1113873
minus 3 54
Si2
minus 11113874 1113875Si sin(ω)
minus15μj4R
4
32na7 1 minus e2
1113872 1113873minus 4
1 +3e2
41113888 1113889 3 minus 30Ci
2+ 35Ci
41113872 11138731113890
+ 1 +52e2
1113874 1113875 minus 1 + 8Ci2
minus 7Ci4
1113872 1113873cos(2ω)1113891
(27)
Defining as in equation (28) we get equation (29)
c6 3j2
4
F6i 1 minus 3Ci2
1113872 1113873
F7i 3 minus 30Ci2
+ 35Ci4
(28)
_ω minus
μradicR4
η4a112 c6a
R1113874 1113875
2η2F6i + c1
a
R1 + 4e
21113872 1113873
ηeF1i sin(ω)1113890
+ c2 1 +3e2
41113888 1113889F7i + 1 +
52e2
1113874 1113875F2i cos(2ω)1113888 11138891113891 minus Ci _Ω
(29)
Equations (22)ndash(29) give the average effects of Earthoblateness including the zonal harmonics of the geopotentialup to J4 on the Keplerian elements of the satellite orbit
3 Balanced Low Earth Satellite Orbits
In what follows we try to find orbits that are balanced in thesense that the averaged (over the fast variable l) variations ofthe orbit elements are set equal to zero In equation (23) weput it equal to zero and get a relation between the argumentof perigee ω and the inclination i while treating the ec-centricity e and the semi-major axis a as parameters iswill give a range of values for ω and i at different values of eand a which are all give balanced orbits with respect to boththe eccentricity e and the inclination ie same is done withequations (26) and (29) while putting _Ω 0 and _ω 0
e applicable ranges for this model of the semi-majoraxis a are 11Rle ale 13R where the range 14Rndash2R isavoided due to the predominance of the radiation belts atthese levels to avoid the damages of the equipment that itmay produce besides its fatal effects on human life (forinhabited spacecrafts) e values for the eccentricity e aretaken between 001 (almost circular orbit) and 05
31 Orbits with Fixed Eccentricity and Inclination Byequating the variation of e by zero we get
_e
μradicR4
η3a112 c1a
RηF1i cos(ω) + c2eF2i sin(2ω)1113874 1113875 0 (30)
is implies
c1a
RηF1i cos(ω) + c2eF2i sin(2ω) 0 (31)
So either cos(ω) 0 or
sin(ω) a 1 minus e2( 1113857
eRC1F(i) (32)
where
C1 8J3
5J4
F(i) 1 minus (54)sin2(i)( 1113857sin(i)
1 minus 8cos2(i) + 7cos4(i)
(33)
Equations (32) and (33) give the family of low orbits thathave both the eccentricity and the inclination fixed
e condition for the existence of such orbits is clearlythat
minus 1le sin(ω)le 1 (34)
which is guaranteed by
minus 1lea 1 minus e2( 1113857
eRC1F(i)le 1 (35)
We put it as a condition on the eccentricity i only when
minus eR
C1a 1 minus e2( )leF(i)le
eR
C1a 1 minus e2( ) (36)
32 Orbits with Fixed Node e families of orbits for whichdΩdt 0 (ie with fixed nodes) are obtained from
minus Ciμradic
R4
η4a112 c3a
R1113874 1113875
2η2 + c4
a
ReηF3i sin(ω)1113896
+ c5 1 +3e2
21113888 1113889F4i + e
2F5i cos(2ω)1113890 11138911113897 0
(37)
is implies
c3a
R1113874 1113875
2η2 + c4
a
ReηF3i sin(ω)
+ c5 1 +3e2
21113888 1113889F4i + e
2F5i 1 minus 2sin2(ω)1113872 11138731113890 1113891 0
(38)
Advances in Astronomy 5
Table 1 Real values of i (in degree) corresponding to the given values of F(i)
F(i) i1 180 minus i1 i2 minus 180 minus i2001 6363 11637 minus 6322 minus 11678005 6427 11573 minus 6211 minus 11789010 6484 11516 minus 5976 minus 12024015 6525 11475 minus 5514 minus 12486020 6556 11444 minus 4716 minus 13284
01 02 03 04 05
e
F (i) = 001
ω
π
3π4
π2
π4
0
Figure 3 e curve ω(e) at which _e _i 0 for F(i) 001
secndash1
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
ndash10 times 10ndash6
01 02 03 04 05
e
Ωmiddot
ωmiddot
(a)
secndash1
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
ndash10 times 10ndash6
01 02 03 04 05
e
Ωmiddot
ωmiddot
(b)
secndash1
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
01 02 03 04 05e
Ωmiddot
ωmiddot
(c)
secndash1
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
01 02 03 04 05e
Ωmiddot
ωmiddot
(d)
Figure 4 Curves of _Ω and _ω with e for the 4 cases corresponding to F(i) 001 (a) i 6363 (b) i minus 6322 (c) i 11637 and (d) i 11678
6 Advances in Astronomy
is can be arranged as a second order equationfor sin(ω) giving a family of orbits with fixed argumentof perigee for different values of ω as a function ofthe inclination i the semi-major axis a and the eccen-tricity e
minus 2c5e2F5isin
2(ω) + c4
a
ReηF3i sin(ω)
+ c3a
R1113874 1113875
2η2 + c5 1 +
3e2
21113888 1113889F4i + e
2F5i1113890 1113891 0
(39)
e
F (i) = 005
ω
π
3π4
π2
π4
001 02 03 04 05
Figure 5 e curve ω(e) at which _e _i 0 for F(i) 005
secndash1
01 02 03 04 05
Ωmiddot
ωmiddot
e
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
ndash10 times 10ndash6
(a)
secndash1
01 02 03 04 05
Ωmiddot
ωmiddot
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
ndash10 times 10ndash6
ndash12 times 10ndash6
e
(b)
secndash1
Ωmiddot
ωmiddot
ndash10 times 10ndash6
ndash80 times 10ndash7
ndash60 times 10ndash7
ndash40 times 10ndash7
ndash20 times 10ndash7
01 02 03 04 05e
(c)
secndash1
01 02 03 04 05
Ωmiddot
ωmiddot
12 times 10ndash6
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
e
(d)
Figure 6 Curves of _Ω and _ωwith e for the 4 cases corresponding to F(i) 005 (a) i 6427 (b) i minus 6211 (c) i 11573 and (d) i minus 11789
Advances in Astronomy 7
When solving for sin(ω) we get the family of orbits forwhich the longitude of the node is balanced
sin(ω) 2c4zF3i plusmn
4c24z
2F23i + 16c5F5i 2c3z
2 + c5F4i 2 + 3e2( ) + 2c5e2F5i(
1113969
8c5eF5i
(40)
F (i) = 01
ω
π
3π4
π2
π4
0
e
01 02 03 04 05
Figure 7 e curve ω(e) at which _e _i 0 for F(i) 001
secndash1
01 02 03 04 05
e
ndash50 times 10ndash7
ndash10 times 10ndash6
Ωmiddot
ωmiddot
(a)
secndash1
01 02 03 04 05e
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
Ωmiddot
ωmiddot
(b)
secndash1
01 02 03 04 05e
12 times 10ndash6
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
Ωmiddot
ωmiddot
(c)
secndash1
01 02 03 04 05e
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
Ωmiddot
ωmiddot
(d)
Figure 8 Curves of _Ω and _ω with e for the 4 cases corresponding to F(i) 01 (a) i minus 5976 (b) i 6484 (c) i minus 12024 and (d) i 11516
8 Advances in Astronomy
where
z a
Rη (41)
e condition for having the orbit if real solution exists isagain that minus 1le sin(ω)le 1 which gives
minus 1le2c3z
2 + 2 + 3e2( 1113857c5F4i + 2e2 c5 minus 1( 1113857F5i
c4ezF3i
le 1 F5i F3i ne 0
(42)
33Orbitswith FixedPerigee For the argument of perigee tobalance we solve _ω 0We substitute from equation (26)into equation (29) then expand cos(2ω) and collect termswith respect to sin(ω) We get
_ω A sin2(ω) + B sin(ω) + C (43)
us for _ω 0 we get
sin(ω) minus B plusmn
B2 minus 4AC
radic
2A (44)
F (i) = 015
ω
π
3π4
π2
π4
0
e
01 02 03 04 05
Figure 9 e curve ω(e) at which _e _i 0 for F(i) 015
01 02 03 04 05
50 times 10ndash7
ndash50 times 10ndash7
ndash10 times 10ndash6
ndash15 times 10ndash6
e
ωmiddotΩmiddot
secndash
1
(a)
01 02 03 04 05
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
e
ωmiddotΩmiddot
secndash
1
(b)
01 02 03 04 05
14 times 10ndash6
10 times 10ndash6
12 times 10ndash6
80 times 10ndash7
60 times 10ndash7
e
ωmiddotΩmiddot
secndash
1
(c)
01 02 03 04 05
10 times 10ndash6
60 times 10ndash7
80 times 10ndash7
40 times 10ndash7
20 times 10ndash7
ndash20 times 10ndash7
e
ωmiddotΩmiddot
secndash
1
(d)
Figure 10 Curves of _Ω and _ω with e for the 4 cases corresponding to F(i) 015 (a) i minus 5514 (b) i 6525 (c) i minus 12486 and (d) i 11475
Advances in Astronomy 9
where
A minus 2c5e2F5cos
2(i) + 2c2 1 +
52e2
1113874 1113875F2 (45)
B c4ezF3cos2(i) minus c1z 4e +
1e
1113874 1113875F1 (46)
C cos2(i) c3z2
+ c5 e2F5 + 1 +
3e2
21113888 1113889F41113888 11138891113888 1113889
minus c6z2F6 minus c2 1 +
3e2
41113888 1113889F7 + 1 +
5e2
21113888 11138891113888 1113889F21113888 1113889
(47)
F (i) = 02
ω
π
3π4
π2
π4
0
e
01 02 03 04 05
Figure 11 e curve ω(e) at which _e _i 0 for F(i) 02
ωmiddotΩmiddot
01 02 03 04 05
e
15 times 10ndash6
10 times 10ndash6
50 times 10ndash7
ndash50 times 10ndash7
ndash10 times 10ndash6
ndash15 times 10ndash6
secndash
1
(a)
ωmiddotΩmiddot
01 02 03 04 05e
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
secndash
1
(b)
ωmiddotΩmiddot
01 02 03 04 05e
16 times 10ndash6
14 times 10ndash6
12 times 10ndash6
10 times 10ndash6
secndash
1
(c)
ωmiddotΩmiddot
01 02 03 04 05
e
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
ndash20 times 10ndash7
secndash
1
(d)
Figure 12 Curves of _Ω and _ωwith e for the 4 cases corresponding to F(i) 02 (a) i minus 4716 (b) i 6556 (c) i minus 13284 and (d) i 11444
10 Advances in Astronomy
Equations (44)ndash(47) give the relation between sin(ω) andthe inclination i the semi-major axis a and the eccentricitye which gives the family of orbits that balance the argumentof perigee ω subject of course to the restriction that sin(ω) isa real value between minus 1 and 1
4 Numerical Results
In this section numerical results and graphs are obtained forthe case of seasat a 7100 km by putting ω as a function of eand i from equations (32) and (33) e curves are plottedwithin the possible range given by condition (36) to give curvesof balanced e and ie curves are against the eccentricity e inthe range [001 05] e numerical values involved areJ2 0001082645 J3 minus 0000002546 J4 minus 0000001649 R
6378165km α 7100m and μ 3986005 km3secminus 2e condition (36) gives the upper and lower bounds of
the function F(i) as a function of e and since the functioneR(C1a(1 minus e2)) is increasing with e and has no criticalpoints in the interval [001 05] then the minimum valueoccurs at e 001 and the maximum at e 05 which givesminus 024leF(i)le 024 e graphs are plotted for differentvalues of F(i) which corresponds to specific values of i foundby solving the equation resulting from setting F(i) equal tothe required values After that we plot _Ω 0 and _ω 0 si-multaneously for the same values of F(i) at each specific i-value to find the orbit values at which we have nearest valuesfor _Ω 0 and _ω 0 In the graphs of _Ω and _ω the relation(32) was kept to ensure that e and i are already balanced
Five selected values of F(i) are chosen F(i) 001 05010 015 and 020 Negative values will give the same resultswith negative sign since F(i) is odd with respect to i Also foreach value the solution of F(i) x with x equals one of theabove values we get four real values of i on the form i1180 minus i1 i2 and minus 180 minus i2 where i1 and i2 are near the criticalinclination one of them is positive and the other is negativeTable 1 gives the values of i corresponding to the selectedvalues of F(i)
We note that as F(i) increases i gets away from thecritical inclination and the curve gets shorter indicating lessstability of ω as expected
e graphs are plotted for each value of F(i) first for thebalanced values of e and i then for the corresponding fourvalues of i four graphs are plotted for _Ω and _ω to find thenearest values of zero variation for both elements
Figures 3ndash12 show the possibility of balancing ω with eand i while Ω will have a variation of order 10minus 6sec or itmust be balanced alone at i 90 deg (as shown in equation(26)) according to the orbit kind Figures 3 5 7 9 and 11show the possible values of ω(e) that balance e and i whileFigures 4 6 8 10 and 12 show the curves of _Ω and _ω at thevalues of i at which _e _i 0
5 Conclusion
Let balanced orbits be defined as those for which the orbitalelements are set equal to zero under the effect of secular andlong periodic perturbations In this work the effect of Earthoblateness is the considered perturbing force because of
dealing with low Earth orbits e above analysis then showsthat such an orbit will be balanced within a reliable toleranceonly for few weeks since we are forced to accept the motionof either the node or the perigee by about 10minus 6 degsec ereason is that under the influence of the Earth oblateness_Ω 0 (exactly) for i (π2) while _ω 0 only near thecritical inclination ic 634 deg Hence the best procedure isto design a satellite constellation for which the nodal shiftsdue to the perturbative effects and Earth rotation aremodeled to yield continuous coverage e perigees areeither fixed or arranged to realize that the perigee (or theapogee) be overhead the coverage region (regions) in duetimes is may require near commensurability with theadmitted nodal periods
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was funded by the University of Jeddah SaudiArabia under grant no UJ-26-18-DR us the authortherefore acknowledges with thanks the university technicaland financial support
References
[1] N Delsate P Robutel A Lemaıtre and T Carletti ldquoFrozenorbits at high eccentricity and inclination application tomercury orbiterrdquo Celestial Mechanics and Dynamical As-tronomy vol 108 no 3 pp 275ndash300 2010
[2] E Condoleo M Cinelli E Ortore and C Circi ldquoStable orbitsfor lunar landing assistancerdquo Advances in Space Researchvol 60 no 7 pp 1404ndash1412 2017
[3] A Elipe andM Lara ldquoFrozen orbits about the moonrdquo Journalof Guidance Control and Dynamics vol 26 no 2 pp 238ndash243 2003
[4] C Circi E Condoleo and E Ortore ldquoA vectorial approach todetermine frozen orbital conditionsrdquo Celestial Mechanics andDynamical Astronomy vol 128 no 2-3 pp 361ndash382 2017
[5] E Condoleo M Cinelli E Ortore and C Circi ldquoFrozenorbits with equatorial perturbing bodies the case of Gany-mede Callisto and Titanrdquo Journal of Guidance Control andDynamics vol 39 no 10 pp 2264ndash2272 2016
[6] V Kudielka ldquoBalanced earth satellite orbitsrdquo Celestial Me-chanics amp Dynamical Astronomy vol 60 no 4 pp 455ndash4701994
[7] S L Coffey A Deprit and E Deprit ldquoFrozen orbits forsatellites close to an earth-like planetrdquo Celestial Mechanics ampDynamical Astronomy vol 59 no 1 pp 37ndash72 1994
[8] M Lara A Deprit and A Elipe ldquoNumerical continuation offamilies of frozen orbits in the zonal problem of artificialsatellite theoryrdquo Celestial Mechanics amp Dynamical Astronomyvol 62 no 2 pp 167ndash181 1995
[9] D Brouwer ldquoSolution of the problem of artificial satellitetheory without dragrdquo e Astronomical Journal vol 64p 378 1959
Advances in Astronomy 11
[10] Y Kozai ldquoe motion of a close earth satelliterdquo e Astro-nomical Journal vol 64 p 367 1959
[11] P M Fitzpatrick Principles of Celestial Mechanics AcademicPress Cambridge MA USA 1970
[12] H D Curtis Orbital Mechanics for Engineering StudentsElsevier Amsterdam Netherlands 2005
[13] D Brower and G M Clemence Methods of Celestial Me-chanics Academic Press Cambridge MA USA 1961
12 Advances in Astronomy
Table 1 Real values of i (in degree) corresponding to the given values of F(i)
F(i) i1 180 minus i1 i2 minus 180 minus i2001 6363 11637 minus 6322 minus 11678005 6427 11573 minus 6211 minus 11789010 6484 11516 minus 5976 minus 12024015 6525 11475 minus 5514 minus 12486020 6556 11444 minus 4716 minus 13284
01 02 03 04 05
e
F (i) = 001
ω
π
3π4
π2
π4
0
Figure 3 e curve ω(e) at which _e _i 0 for F(i) 001
secndash1
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
ndash10 times 10ndash6
01 02 03 04 05
e
Ωmiddot
ωmiddot
(a)
secndash1
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
ndash10 times 10ndash6
01 02 03 04 05
e
Ωmiddot
ωmiddot
(b)
secndash1
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
01 02 03 04 05e
Ωmiddot
ωmiddot
(c)
secndash1
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
01 02 03 04 05e
Ωmiddot
ωmiddot
(d)
Figure 4 Curves of _Ω and _ω with e for the 4 cases corresponding to F(i) 001 (a) i 6363 (b) i minus 6322 (c) i 11637 and (d) i 11678
6 Advances in Astronomy
is can be arranged as a second order equationfor sin(ω) giving a family of orbits with fixed argumentof perigee for different values of ω as a function ofthe inclination i the semi-major axis a and the eccen-tricity e
minus 2c5e2F5isin
2(ω) + c4
a
ReηF3i sin(ω)
+ c3a
R1113874 1113875
2η2 + c5 1 +
3e2
21113888 1113889F4i + e
2F5i1113890 1113891 0
(39)
e
F (i) = 005
ω
π
3π4
π2
π4
001 02 03 04 05
Figure 5 e curve ω(e) at which _e _i 0 for F(i) 005
secndash1
01 02 03 04 05
Ωmiddot
ωmiddot
e
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
ndash10 times 10ndash6
(a)
secndash1
01 02 03 04 05
Ωmiddot
ωmiddot
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
ndash10 times 10ndash6
ndash12 times 10ndash6
e
(b)
secndash1
Ωmiddot
ωmiddot
ndash10 times 10ndash6
ndash80 times 10ndash7
ndash60 times 10ndash7
ndash40 times 10ndash7
ndash20 times 10ndash7
01 02 03 04 05e
(c)
secndash1
01 02 03 04 05
Ωmiddot
ωmiddot
12 times 10ndash6
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
e
(d)
Figure 6 Curves of _Ω and _ωwith e for the 4 cases corresponding to F(i) 005 (a) i 6427 (b) i minus 6211 (c) i 11573 and (d) i minus 11789
Advances in Astronomy 7
When solving for sin(ω) we get the family of orbits forwhich the longitude of the node is balanced
sin(ω) 2c4zF3i plusmn
4c24z
2F23i + 16c5F5i 2c3z
2 + c5F4i 2 + 3e2( ) + 2c5e2F5i(
1113969
8c5eF5i
(40)
F (i) = 01
ω
π
3π4
π2
π4
0
e
01 02 03 04 05
Figure 7 e curve ω(e) at which _e _i 0 for F(i) 001
secndash1
01 02 03 04 05
e
ndash50 times 10ndash7
ndash10 times 10ndash6
Ωmiddot
ωmiddot
(a)
secndash1
01 02 03 04 05e
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
Ωmiddot
ωmiddot
(b)
secndash1
01 02 03 04 05e
12 times 10ndash6
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
Ωmiddot
ωmiddot
(c)
secndash1
01 02 03 04 05e
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
Ωmiddot
ωmiddot
(d)
Figure 8 Curves of _Ω and _ω with e for the 4 cases corresponding to F(i) 01 (a) i minus 5976 (b) i 6484 (c) i minus 12024 and (d) i 11516
8 Advances in Astronomy
where
z a
Rη (41)
e condition for having the orbit if real solution exists isagain that minus 1le sin(ω)le 1 which gives
minus 1le2c3z
2 + 2 + 3e2( 1113857c5F4i + 2e2 c5 minus 1( 1113857F5i
c4ezF3i
le 1 F5i F3i ne 0
(42)
33Orbitswith FixedPerigee For the argument of perigee tobalance we solve _ω 0We substitute from equation (26)into equation (29) then expand cos(2ω) and collect termswith respect to sin(ω) We get
_ω A sin2(ω) + B sin(ω) + C (43)
us for _ω 0 we get
sin(ω) minus B plusmn
B2 minus 4AC
radic
2A (44)
F (i) = 015
ω
π
3π4
π2
π4
0
e
01 02 03 04 05
Figure 9 e curve ω(e) at which _e _i 0 for F(i) 015
01 02 03 04 05
50 times 10ndash7
ndash50 times 10ndash7
ndash10 times 10ndash6
ndash15 times 10ndash6
e
ωmiddotΩmiddot
secndash
1
(a)
01 02 03 04 05
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
e
ωmiddotΩmiddot
secndash
1
(b)
01 02 03 04 05
14 times 10ndash6
10 times 10ndash6
12 times 10ndash6
80 times 10ndash7
60 times 10ndash7
e
ωmiddotΩmiddot
secndash
1
(c)
01 02 03 04 05
10 times 10ndash6
60 times 10ndash7
80 times 10ndash7
40 times 10ndash7
20 times 10ndash7
ndash20 times 10ndash7
e
ωmiddotΩmiddot
secndash
1
(d)
Figure 10 Curves of _Ω and _ω with e for the 4 cases corresponding to F(i) 015 (a) i minus 5514 (b) i 6525 (c) i minus 12486 and (d) i 11475
Advances in Astronomy 9
where
A minus 2c5e2F5cos
2(i) + 2c2 1 +
52e2
1113874 1113875F2 (45)
B c4ezF3cos2(i) minus c1z 4e +
1e
1113874 1113875F1 (46)
C cos2(i) c3z2
+ c5 e2F5 + 1 +
3e2
21113888 1113889F41113888 11138891113888 1113889
minus c6z2F6 minus c2 1 +
3e2
41113888 1113889F7 + 1 +
5e2
21113888 11138891113888 1113889F21113888 1113889
(47)
F (i) = 02
ω
π
3π4
π2
π4
0
e
01 02 03 04 05
Figure 11 e curve ω(e) at which _e _i 0 for F(i) 02
ωmiddotΩmiddot
01 02 03 04 05
e
15 times 10ndash6
10 times 10ndash6
50 times 10ndash7
ndash50 times 10ndash7
ndash10 times 10ndash6
ndash15 times 10ndash6
secndash
1
(a)
ωmiddotΩmiddot
01 02 03 04 05e
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
secndash
1
(b)
ωmiddotΩmiddot
01 02 03 04 05e
16 times 10ndash6
14 times 10ndash6
12 times 10ndash6
10 times 10ndash6
secndash
1
(c)
ωmiddotΩmiddot
01 02 03 04 05
e
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
ndash20 times 10ndash7
secndash
1
(d)
Figure 12 Curves of _Ω and _ωwith e for the 4 cases corresponding to F(i) 02 (a) i minus 4716 (b) i 6556 (c) i minus 13284 and (d) i 11444
10 Advances in Astronomy
Equations (44)ndash(47) give the relation between sin(ω) andthe inclination i the semi-major axis a and the eccentricitye which gives the family of orbits that balance the argumentof perigee ω subject of course to the restriction that sin(ω) isa real value between minus 1 and 1
4 Numerical Results
In this section numerical results and graphs are obtained forthe case of seasat a 7100 km by putting ω as a function of eand i from equations (32) and (33) e curves are plottedwithin the possible range given by condition (36) to give curvesof balanced e and ie curves are against the eccentricity e inthe range [001 05] e numerical values involved areJ2 0001082645 J3 minus 0000002546 J4 minus 0000001649 R
6378165km α 7100m and μ 3986005 km3secminus 2e condition (36) gives the upper and lower bounds of
the function F(i) as a function of e and since the functioneR(C1a(1 minus e2)) is increasing with e and has no criticalpoints in the interval [001 05] then the minimum valueoccurs at e 001 and the maximum at e 05 which givesminus 024leF(i)le 024 e graphs are plotted for differentvalues of F(i) which corresponds to specific values of i foundby solving the equation resulting from setting F(i) equal tothe required values After that we plot _Ω 0 and _ω 0 si-multaneously for the same values of F(i) at each specific i-value to find the orbit values at which we have nearest valuesfor _Ω 0 and _ω 0 In the graphs of _Ω and _ω the relation(32) was kept to ensure that e and i are already balanced
Five selected values of F(i) are chosen F(i) 001 05010 015 and 020 Negative values will give the same resultswith negative sign since F(i) is odd with respect to i Also foreach value the solution of F(i) x with x equals one of theabove values we get four real values of i on the form i1180 minus i1 i2 and minus 180 minus i2 where i1 and i2 are near the criticalinclination one of them is positive and the other is negativeTable 1 gives the values of i corresponding to the selectedvalues of F(i)
We note that as F(i) increases i gets away from thecritical inclination and the curve gets shorter indicating lessstability of ω as expected
e graphs are plotted for each value of F(i) first for thebalanced values of e and i then for the corresponding fourvalues of i four graphs are plotted for _Ω and _ω to find thenearest values of zero variation for both elements
Figures 3ndash12 show the possibility of balancing ω with eand i while Ω will have a variation of order 10minus 6sec or itmust be balanced alone at i 90 deg (as shown in equation(26)) according to the orbit kind Figures 3 5 7 9 and 11show the possible values of ω(e) that balance e and i whileFigures 4 6 8 10 and 12 show the curves of _Ω and _ω at thevalues of i at which _e _i 0
5 Conclusion
Let balanced orbits be defined as those for which the orbitalelements are set equal to zero under the effect of secular andlong periodic perturbations In this work the effect of Earthoblateness is the considered perturbing force because of
dealing with low Earth orbits e above analysis then showsthat such an orbit will be balanced within a reliable toleranceonly for few weeks since we are forced to accept the motionof either the node or the perigee by about 10minus 6 degsec ereason is that under the influence of the Earth oblateness_Ω 0 (exactly) for i (π2) while _ω 0 only near thecritical inclination ic 634 deg Hence the best procedure isto design a satellite constellation for which the nodal shiftsdue to the perturbative effects and Earth rotation aremodeled to yield continuous coverage e perigees areeither fixed or arranged to realize that the perigee (or theapogee) be overhead the coverage region (regions) in duetimes is may require near commensurability with theadmitted nodal periods
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was funded by the University of Jeddah SaudiArabia under grant no UJ-26-18-DR us the authortherefore acknowledges with thanks the university technicaland financial support
References
[1] N Delsate P Robutel A Lemaıtre and T Carletti ldquoFrozenorbits at high eccentricity and inclination application tomercury orbiterrdquo Celestial Mechanics and Dynamical As-tronomy vol 108 no 3 pp 275ndash300 2010
[2] E Condoleo M Cinelli E Ortore and C Circi ldquoStable orbitsfor lunar landing assistancerdquo Advances in Space Researchvol 60 no 7 pp 1404ndash1412 2017
[3] A Elipe andM Lara ldquoFrozen orbits about the moonrdquo Journalof Guidance Control and Dynamics vol 26 no 2 pp 238ndash243 2003
[4] C Circi E Condoleo and E Ortore ldquoA vectorial approach todetermine frozen orbital conditionsrdquo Celestial Mechanics andDynamical Astronomy vol 128 no 2-3 pp 361ndash382 2017
[5] E Condoleo M Cinelli E Ortore and C Circi ldquoFrozenorbits with equatorial perturbing bodies the case of Gany-mede Callisto and Titanrdquo Journal of Guidance Control andDynamics vol 39 no 10 pp 2264ndash2272 2016
[6] V Kudielka ldquoBalanced earth satellite orbitsrdquo Celestial Me-chanics amp Dynamical Astronomy vol 60 no 4 pp 455ndash4701994
[7] S L Coffey A Deprit and E Deprit ldquoFrozen orbits forsatellites close to an earth-like planetrdquo Celestial Mechanics ampDynamical Astronomy vol 59 no 1 pp 37ndash72 1994
[8] M Lara A Deprit and A Elipe ldquoNumerical continuation offamilies of frozen orbits in the zonal problem of artificialsatellite theoryrdquo Celestial Mechanics amp Dynamical Astronomyvol 62 no 2 pp 167ndash181 1995
[9] D Brouwer ldquoSolution of the problem of artificial satellitetheory without dragrdquo e Astronomical Journal vol 64p 378 1959
Advances in Astronomy 11
[10] Y Kozai ldquoe motion of a close earth satelliterdquo e Astro-nomical Journal vol 64 p 367 1959
[11] P M Fitzpatrick Principles of Celestial Mechanics AcademicPress Cambridge MA USA 1970
[12] H D Curtis Orbital Mechanics for Engineering StudentsElsevier Amsterdam Netherlands 2005
[13] D Brower and G M Clemence Methods of Celestial Me-chanics Academic Press Cambridge MA USA 1961
12 Advances in Astronomy
is can be arranged as a second order equationfor sin(ω) giving a family of orbits with fixed argumentof perigee for different values of ω as a function ofthe inclination i the semi-major axis a and the eccen-tricity e
minus 2c5e2F5isin
2(ω) + c4
a
ReηF3i sin(ω)
+ c3a
R1113874 1113875
2η2 + c5 1 +
3e2
21113888 1113889F4i + e
2F5i1113890 1113891 0
(39)
e
F (i) = 005
ω
π
3π4
π2
π4
001 02 03 04 05
Figure 5 e curve ω(e) at which _e _i 0 for F(i) 005
secndash1
01 02 03 04 05
Ωmiddot
ωmiddot
e
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
ndash10 times 10ndash6
(a)
secndash1
01 02 03 04 05
Ωmiddot
ωmiddot
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
ndash10 times 10ndash6
ndash12 times 10ndash6
e
(b)
secndash1
Ωmiddot
ωmiddot
ndash10 times 10ndash6
ndash80 times 10ndash7
ndash60 times 10ndash7
ndash40 times 10ndash7
ndash20 times 10ndash7
01 02 03 04 05e
(c)
secndash1
01 02 03 04 05
Ωmiddot
ωmiddot
12 times 10ndash6
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
e
(d)
Figure 6 Curves of _Ω and _ωwith e for the 4 cases corresponding to F(i) 005 (a) i 6427 (b) i minus 6211 (c) i 11573 and (d) i minus 11789
Advances in Astronomy 7
When solving for sin(ω) we get the family of orbits forwhich the longitude of the node is balanced
sin(ω) 2c4zF3i plusmn
4c24z
2F23i + 16c5F5i 2c3z
2 + c5F4i 2 + 3e2( ) + 2c5e2F5i(
1113969
8c5eF5i
(40)
F (i) = 01
ω
π
3π4
π2
π4
0
e
01 02 03 04 05
Figure 7 e curve ω(e) at which _e _i 0 for F(i) 001
secndash1
01 02 03 04 05
e
ndash50 times 10ndash7
ndash10 times 10ndash6
Ωmiddot
ωmiddot
(a)
secndash1
01 02 03 04 05e
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
Ωmiddot
ωmiddot
(b)
secndash1
01 02 03 04 05e
12 times 10ndash6
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
Ωmiddot
ωmiddot
(c)
secndash1
01 02 03 04 05e
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
Ωmiddot
ωmiddot
(d)
Figure 8 Curves of _Ω and _ω with e for the 4 cases corresponding to F(i) 01 (a) i minus 5976 (b) i 6484 (c) i minus 12024 and (d) i 11516
8 Advances in Astronomy
where
z a
Rη (41)
e condition for having the orbit if real solution exists isagain that minus 1le sin(ω)le 1 which gives
minus 1le2c3z
2 + 2 + 3e2( 1113857c5F4i + 2e2 c5 minus 1( 1113857F5i
c4ezF3i
le 1 F5i F3i ne 0
(42)
33Orbitswith FixedPerigee For the argument of perigee tobalance we solve _ω 0We substitute from equation (26)into equation (29) then expand cos(2ω) and collect termswith respect to sin(ω) We get
_ω A sin2(ω) + B sin(ω) + C (43)
us for _ω 0 we get
sin(ω) minus B plusmn
B2 minus 4AC
radic
2A (44)
F (i) = 015
ω
π
3π4
π2
π4
0
e
01 02 03 04 05
Figure 9 e curve ω(e) at which _e _i 0 for F(i) 015
01 02 03 04 05
50 times 10ndash7
ndash50 times 10ndash7
ndash10 times 10ndash6
ndash15 times 10ndash6
e
ωmiddotΩmiddot
secndash
1
(a)
01 02 03 04 05
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
e
ωmiddotΩmiddot
secndash
1
(b)
01 02 03 04 05
14 times 10ndash6
10 times 10ndash6
12 times 10ndash6
80 times 10ndash7
60 times 10ndash7
e
ωmiddotΩmiddot
secndash
1
(c)
01 02 03 04 05
10 times 10ndash6
60 times 10ndash7
80 times 10ndash7
40 times 10ndash7
20 times 10ndash7
ndash20 times 10ndash7
e
ωmiddotΩmiddot
secndash
1
(d)
Figure 10 Curves of _Ω and _ω with e for the 4 cases corresponding to F(i) 015 (a) i minus 5514 (b) i 6525 (c) i minus 12486 and (d) i 11475
Advances in Astronomy 9
where
A minus 2c5e2F5cos
2(i) + 2c2 1 +
52e2
1113874 1113875F2 (45)
B c4ezF3cos2(i) minus c1z 4e +
1e
1113874 1113875F1 (46)
C cos2(i) c3z2
+ c5 e2F5 + 1 +
3e2
21113888 1113889F41113888 11138891113888 1113889
minus c6z2F6 minus c2 1 +
3e2
41113888 1113889F7 + 1 +
5e2
21113888 11138891113888 1113889F21113888 1113889
(47)
F (i) = 02
ω
π
3π4
π2
π4
0
e
01 02 03 04 05
Figure 11 e curve ω(e) at which _e _i 0 for F(i) 02
ωmiddotΩmiddot
01 02 03 04 05
e
15 times 10ndash6
10 times 10ndash6
50 times 10ndash7
ndash50 times 10ndash7
ndash10 times 10ndash6
ndash15 times 10ndash6
secndash
1
(a)
ωmiddotΩmiddot
01 02 03 04 05e
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
secndash
1
(b)
ωmiddotΩmiddot
01 02 03 04 05e
16 times 10ndash6
14 times 10ndash6
12 times 10ndash6
10 times 10ndash6
secndash
1
(c)
ωmiddotΩmiddot
01 02 03 04 05
e
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
ndash20 times 10ndash7
secndash
1
(d)
Figure 12 Curves of _Ω and _ωwith e for the 4 cases corresponding to F(i) 02 (a) i minus 4716 (b) i 6556 (c) i minus 13284 and (d) i 11444
10 Advances in Astronomy
Equations (44)ndash(47) give the relation between sin(ω) andthe inclination i the semi-major axis a and the eccentricitye which gives the family of orbits that balance the argumentof perigee ω subject of course to the restriction that sin(ω) isa real value between minus 1 and 1
4 Numerical Results
In this section numerical results and graphs are obtained forthe case of seasat a 7100 km by putting ω as a function of eand i from equations (32) and (33) e curves are plottedwithin the possible range given by condition (36) to give curvesof balanced e and ie curves are against the eccentricity e inthe range [001 05] e numerical values involved areJ2 0001082645 J3 minus 0000002546 J4 minus 0000001649 R
6378165km α 7100m and μ 3986005 km3secminus 2e condition (36) gives the upper and lower bounds of
the function F(i) as a function of e and since the functioneR(C1a(1 minus e2)) is increasing with e and has no criticalpoints in the interval [001 05] then the minimum valueoccurs at e 001 and the maximum at e 05 which givesminus 024leF(i)le 024 e graphs are plotted for differentvalues of F(i) which corresponds to specific values of i foundby solving the equation resulting from setting F(i) equal tothe required values After that we plot _Ω 0 and _ω 0 si-multaneously for the same values of F(i) at each specific i-value to find the orbit values at which we have nearest valuesfor _Ω 0 and _ω 0 In the graphs of _Ω and _ω the relation(32) was kept to ensure that e and i are already balanced
Five selected values of F(i) are chosen F(i) 001 05010 015 and 020 Negative values will give the same resultswith negative sign since F(i) is odd with respect to i Also foreach value the solution of F(i) x with x equals one of theabove values we get four real values of i on the form i1180 minus i1 i2 and minus 180 minus i2 where i1 and i2 are near the criticalinclination one of them is positive and the other is negativeTable 1 gives the values of i corresponding to the selectedvalues of F(i)
We note that as F(i) increases i gets away from thecritical inclination and the curve gets shorter indicating lessstability of ω as expected
e graphs are plotted for each value of F(i) first for thebalanced values of e and i then for the corresponding fourvalues of i four graphs are plotted for _Ω and _ω to find thenearest values of zero variation for both elements
Figures 3ndash12 show the possibility of balancing ω with eand i while Ω will have a variation of order 10minus 6sec or itmust be balanced alone at i 90 deg (as shown in equation(26)) according to the orbit kind Figures 3 5 7 9 and 11show the possible values of ω(e) that balance e and i whileFigures 4 6 8 10 and 12 show the curves of _Ω and _ω at thevalues of i at which _e _i 0
5 Conclusion
Let balanced orbits be defined as those for which the orbitalelements are set equal to zero under the effect of secular andlong periodic perturbations In this work the effect of Earthoblateness is the considered perturbing force because of
dealing with low Earth orbits e above analysis then showsthat such an orbit will be balanced within a reliable toleranceonly for few weeks since we are forced to accept the motionof either the node or the perigee by about 10minus 6 degsec ereason is that under the influence of the Earth oblateness_Ω 0 (exactly) for i (π2) while _ω 0 only near thecritical inclination ic 634 deg Hence the best procedure isto design a satellite constellation for which the nodal shiftsdue to the perturbative effects and Earth rotation aremodeled to yield continuous coverage e perigees areeither fixed or arranged to realize that the perigee (or theapogee) be overhead the coverage region (regions) in duetimes is may require near commensurability with theadmitted nodal periods
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was funded by the University of Jeddah SaudiArabia under grant no UJ-26-18-DR us the authortherefore acknowledges with thanks the university technicaland financial support
References
[1] N Delsate P Robutel A Lemaıtre and T Carletti ldquoFrozenorbits at high eccentricity and inclination application tomercury orbiterrdquo Celestial Mechanics and Dynamical As-tronomy vol 108 no 3 pp 275ndash300 2010
[2] E Condoleo M Cinelli E Ortore and C Circi ldquoStable orbitsfor lunar landing assistancerdquo Advances in Space Researchvol 60 no 7 pp 1404ndash1412 2017
[3] A Elipe andM Lara ldquoFrozen orbits about the moonrdquo Journalof Guidance Control and Dynamics vol 26 no 2 pp 238ndash243 2003
[4] C Circi E Condoleo and E Ortore ldquoA vectorial approach todetermine frozen orbital conditionsrdquo Celestial Mechanics andDynamical Astronomy vol 128 no 2-3 pp 361ndash382 2017
[5] E Condoleo M Cinelli E Ortore and C Circi ldquoFrozenorbits with equatorial perturbing bodies the case of Gany-mede Callisto and Titanrdquo Journal of Guidance Control andDynamics vol 39 no 10 pp 2264ndash2272 2016
[6] V Kudielka ldquoBalanced earth satellite orbitsrdquo Celestial Me-chanics amp Dynamical Astronomy vol 60 no 4 pp 455ndash4701994
[7] S L Coffey A Deprit and E Deprit ldquoFrozen orbits forsatellites close to an earth-like planetrdquo Celestial Mechanics ampDynamical Astronomy vol 59 no 1 pp 37ndash72 1994
[8] M Lara A Deprit and A Elipe ldquoNumerical continuation offamilies of frozen orbits in the zonal problem of artificialsatellite theoryrdquo Celestial Mechanics amp Dynamical Astronomyvol 62 no 2 pp 167ndash181 1995
[9] D Brouwer ldquoSolution of the problem of artificial satellitetheory without dragrdquo e Astronomical Journal vol 64p 378 1959
Advances in Astronomy 11
[10] Y Kozai ldquoe motion of a close earth satelliterdquo e Astro-nomical Journal vol 64 p 367 1959
[11] P M Fitzpatrick Principles of Celestial Mechanics AcademicPress Cambridge MA USA 1970
[12] H D Curtis Orbital Mechanics for Engineering StudentsElsevier Amsterdam Netherlands 2005
[13] D Brower and G M Clemence Methods of Celestial Me-chanics Academic Press Cambridge MA USA 1961
12 Advances in Astronomy
When solving for sin(ω) we get the family of orbits forwhich the longitude of the node is balanced
sin(ω) 2c4zF3i plusmn
4c24z
2F23i + 16c5F5i 2c3z
2 + c5F4i 2 + 3e2( ) + 2c5e2F5i(
1113969
8c5eF5i
(40)
F (i) = 01
ω
π
3π4
π2
π4
0
e
01 02 03 04 05
Figure 7 e curve ω(e) at which _e _i 0 for F(i) 001
secndash1
01 02 03 04 05
e
ndash50 times 10ndash7
ndash10 times 10ndash6
Ωmiddot
ωmiddot
(a)
secndash1
01 02 03 04 05e
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
Ωmiddot
ωmiddot
(b)
secndash1
01 02 03 04 05e
12 times 10ndash6
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
Ωmiddot
ωmiddot
(c)
secndash1
01 02 03 04 05e
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
Ωmiddot
ωmiddot
(d)
Figure 8 Curves of _Ω and _ω with e for the 4 cases corresponding to F(i) 01 (a) i minus 5976 (b) i 6484 (c) i minus 12024 and (d) i 11516
8 Advances in Astronomy
where
z a
Rη (41)
e condition for having the orbit if real solution exists isagain that minus 1le sin(ω)le 1 which gives
minus 1le2c3z
2 + 2 + 3e2( 1113857c5F4i + 2e2 c5 minus 1( 1113857F5i
c4ezF3i
le 1 F5i F3i ne 0
(42)
33Orbitswith FixedPerigee For the argument of perigee tobalance we solve _ω 0We substitute from equation (26)into equation (29) then expand cos(2ω) and collect termswith respect to sin(ω) We get
_ω A sin2(ω) + B sin(ω) + C (43)
us for _ω 0 we get
sin(ω) minus B plusmn
B2 minus 4AC
radic
2A (44)
F (i) = 015
ω
π
3π4
π2
π4
0
e
01 02 03 04 05
Figure 9 e curve ω(e) at which _e _i 0 for F(i) 015
01 02 03 04 05
50 times 10ndash7
ndash50 times 10ndash7
ndash10 times 10ndash6
ndash15 times 10ndash6
e
ωmiddotΩmiddot
secndash
1
(a)
01 02 03 04 05
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
e
ωmiddotΩmiddot
secndash
1
(b)
01 02 03 04 05
14 times 10ndash6
10 times 10ndash6
12 times 10ndash6
80 times 10ndash7
60 times 10ndash7
e
ωmiddotΩmiddot
secndash
1
(c)
01 02 03 04 05
10 times 10ndash6
60 times 10ndash7
80 times 10ndash7
40 times 10ndash7
20 times 10ndash7
ndash20 times 10ndash7
e
ωmiddotΩmiddot
secndash
1
(d)
Figure 10 Curves of _Ω and _ω with e for the 4 cases corresponding to F(i) 015 (a) i minus 5514 (b) i 6525 (c) i minus 12486 and (d) i 11475
Advances in Astronomy 9
where
A minus 2c5e2F5cos
2(i) + 2c2 1 +
52e2
1113874 1113875F2 (45)
B c4ezF3cos2(i) minus c1z 4e +
1e
1113874 1113875F1 (46)
C cos2(i) c3z2
+ c5 e2F5 + 1 +
3e2
21113888 1113889F41113888 11138891113888 1113889
minus c6z2F6 minus c2 1 +
3e2
41113888 1113889F7 + 1 +
5e2
21113888 11138891113888 1113889F21113888 1113889
(47)
F (i) = 02
ω
π
3π4
π2
π4
0
e
01 02 03 04 05
Figure 11 e curve ω(e) at which _e _i 0 for F(i) 02
ωmiddotΩmiddot
01 02 03 04 05
e
15 times 10ndash6
10 times 10ndash6
50 times 10ndash7
ndash50 times 10ndash7
ndash10 times 10ndash6
ndash15 times 10ndash6
secndash
1
(a)
ωmiddotΩmiddot
01 02 03 04 05e
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
secndash
1
(b)
ωmiddotΩmiddot
01 02 03 04 05e
16 times 10ndash6
14 times 10ndash6
12 times 10ndash6
10 times 10ndash6
secndash
1
(c)
ωmiddotΩmiddot
01 02 03 04 05
e
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
ndash20 times 10ndash7
secndash
1
(d)
Figure 12 Curves of _Ω and _ωwith e for the 4 cases corresponding to F(i) 02 (a) i minus 4716 (b) i 6556 (c) i minus 13284 and (d) i 11444
10 Advances in Astronomy
Equations (44)ndash(47) give the relation between sin(ω) andthe inclination i the semi-major axis a and the eccentricitye which gives the family of orbits that balance the argumentof perigee ω subject of course to the restriction that sin(ω) isa real value between minus 1 and 1
4 Numerical Results
In this section numerical results and graphs are obtained forthe case of seasat a 7100 km by putting ω as a function of eand i from equations (32) and (33) e curves are plottedwithin the possible range given by condition (36) to give curvesof balanced e and ie curves are against the eccentricity e inthe range [001 05] e numerical values involved areJ2 0001082645 J3 minus 0000002546 J4 minus 0000001649 R
6378165km α 7100m and μ 3986005 km3secminus 2e condition (36) gives the upper and lower bounds of
the function F(i) as a function of e and since the functioneR(C1a(1 minus e2)) is increasing with e and has no criticalpoints in the interval [001 05] then the minimum valueoccurs at e 001 and the maximum at e 05 which givesminus 024leF(i)le 024 e graphs are plotted for differentvalues of F(i) which corresponds to specific values of i foundby solving the equation resulting from setting F(i) equal tothe required values After that we plot _Ω 0 and _ω 0 si-multaneously for the same values of F(i) at each specific i-value to find the orbit values at which we have nearest valuesfor _Ω 0 and _ω 0 In the graphs of _Ω and _ω the relation(32) was kept to ensure that e and i are already balanced
Five selected values of F(i) are chosen F(i) 001 05010 015 and 020 Negative values will give the same resultswith negative sign since F(i) is odd with respect to i Also foreach value the solution of F(i) x with x equals one of theabove values we get four real values of i on the form i1180 minus i1 i2 and minus 180 minus i2 where i1 and i2 are near the criticalinclination one of them is positive and the other is negativeTable 1 gives the values of i corresponding to the selectedvalues of F(i)
We note that as F(i) increases i gets away from thecritical inclination and the curve gets shorter indicating lessstability of ω as expected
e graphs are plotted for each value of F(i) first for thebalanced values of e and i then for the corresponding fourvalues of i four graphs are plotted for _Ω and _ω to find thenearest values of zero variation for both elements
Figures 3ndash12 show the possibility of balancing ω with eand i while Ω will have a variation of order 10minus 6sec or itmust be balanced alone at i 90 deg (as shown in equation(26)) according to the orbit kind Figures 3 5 7 9 and 11show the possible values of ω(e) that balance e and i whileFigures 4 6 8 10 and 12 show the curves of _Ω and _ω at thevalues of i at which _e _i 0
5 Conclusion
Let balanced orbits be defined as those for which the orbitalelements are set equal to zero under the effect of secular andlong periodic perturbations In this work the effect of Earthoblateness is the considered perturbing force because of
dealing with low Earth orbits e above analysis then showsthat such an orbit will be balanced within a reliable toleranceonly for few weeks since we are forced to accept the motionof either the node or the perigee by about 10minus 6 degsec ereason is that under the influence of the Earth oblateness_Ω 0 (exactly) for i (π2) while _ω 0 only near thecritical inclination ic 634 deg Hence the best procedure isto design a satellite constellation for which the nodal shiftsdue to the perturbative effects and Earth rotation aremodeled to yield continuous coverage e perigees areeither fixed or arranged to realize that the perigee (or theapogee) be overhead the coverage region (regions) in duetimes is may require near commensurability with theadmitted nodal periods
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was funded by the University of Jeddah SaudiArabia under grant no UJ-26-18-DR us the authortherefore acknowledges with thanks the university technicaland financial support
References
[1] N Delsate P Robutel A Lemaıtre and T Carletti ldquoFrozenorbits at high eccentricity and inclination application tomercury orbiterrdquo Celestial Mechanics and Dynamical As-tronomy vol 108 no 3 pp 275ndash300 2010
[2] E Condoleo M Cinelli E Ortore and C Circi ldquoStable orbitsfor lunar landing assistancerdquo Advances in Space Researchvol 60 no 7 pp 1404ndash1412 2017
[3] A Elipe andM Lara ldquoFrozen orbits about the moonrdquo Journalof Guidance Control and Dynamics vol 26 no 2 pp 238ndash243 2003
[4] C Circi E Condoleo and E Ortore ldquoA vectorial approach todetermine frozen orbital conditionsrdquo Celestial Mechanics andDynamical Astronomy vol 128 no 2-3 pp 361ndash382 2017
[5] E Condoleo M Cinelli E Ortore and C Circi ldquoFrozenorbits with equatorial perturbing bodies the case of Gany-mede Callisto and Titanrdquo Journal of Guidance Control andDynamics vol 39 no 10 pp 2264ndash2272 2016
[6] V Kudielka ldquoBalanced earth satellite orbitsrdquo Celestial Me-chanics amp Dynamical Astronomy vol 60 no 4 pp 455ndash4701994
[7] S L Coffey A Deprit and E Deprit ldquoFrozen orbits forsatellites close to an earth-like planetrdquo Celestial Mechanics ampDynamical Astronomy vol 59 no 1 pp 37ndash72 1994
[8] M Lara A Deprit and A Elipe ldquoNumerical continuation offamilies of frozen orbits in the zonal problem of artificialsatellite theoryrdquo Celestial Mechanics amp Dynamical Astronomyvol 62 no 2 pp 167ndash181 1995
[9] D Brouwer ldquoSolution of the problem of artificial satellitetheory without dragrdquo e Astronomical Journal vol 64p 378 1959
Advances in Astronomy 11
[10] Y Kozai ldquoe motion of a close earth satelliterdquo e Astro-nomical Journal vol 64 p 367 1959
[11] P M Fitzpatrick Principles of Celestial Mechanics AcademicPress Cambridge MA USA 1970
[12] H D Curtis Orbital Mechanics for Engineering StudentsElsevier Amsterdam Netherlands 2005
[13] D Brower and G M Clemence Methods of Celestial Me-chanics Academic Press Cambridge MA USA 1961
12 Advances in Astronomy
where
z a
Rη (41)
e condition for having the orbit if real solution exists isagain that minus 1le sin(ω)le 1 which gives
minus 1le2c3z
2 + 2 + 3e2( 1113857c5F4i + 2e2 c5 minus 1( 1113857F5i
c4ezF3i
le 1 F5i F3i ne 0
(42)
33Orbitswith FixedPerigee For the argument of perigee tobalance we solve _ω 0We substitute from equation (26)into equation (29) then expand cos(2ω) and collect termswith respect to sin(ω) We get
_ω A sin2(ω) + B sin(ω) + C (43)
us for _ω 0 we get
sin(ω) minus B plusmn
B2 minus 4AC
radic
2A (44)
F (i) = 015
ω
π
3π4
π2
π4
0
e
01 02 03 04 05
Figure 9 e curve ω(e) at which _e _i 0 for F(i) 015
01 02 03 04 05
50 times 10ndash7
ndash50 times 10ndash7
ndash10 times 10ndash6
ndash15 times 10ndash6
e
ωmiddotΩmiddot
secndash
1
(a)
01 02 03 04 05
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
e
ωmiddotΩmiddot
secndash
1
(b)
01 02 03 04 05
14 times 10ndash6
10 times 10ndash6
12 times 10ndash6
80 times 10ndash7
60 times 10ndash7
e
ωmiddotΩmiddot
secndash
1
(c)
01 02 03 04 05
10 times 10ndash6
60 times 10ndash7
80 times 10ndash7
40 times 10ndash7
20 times 10ndash7
ndash20 times 10ndash7
e
ωmiddotΩmiddot
secndash
1
(d)
Figure 10 Curves of _Ω and _ω with e for the 4 cases corresponding to F(i) 015 (a) i minus 5514 (b) i 6525 (c) i minus 12486 and (d) i 11475
Advances in Astronomy 9
where
A minus 2c5e2F5cos
2(i) + 2c2 1 +
52e2
1113874 1113875F2 (45)
B c4ezF3cos2(i) minus c1z 4e +
1e
1113874 1113875F1 (46)
C cos2(i) c3z2
+ c5 e2F5 + 1 +
3e2
21113888 1113889F41113888 11138891113888 1113889
minus c6z2F6 minus c2 1 +
3e2
41113888 1113889F7 + 1 +
5e2
21113888 11138891113888 1113889F21113888 1113889
(47)
F (i) = 02
ω
π
3π4
π2
π4
0
e
01 02 03 04 05
Figure 11 e curve ω(e) at which _e _i 0 for F(i) 02
ωmiddotΩmiddot
01 02 03 04 05
e
15 times 10ndash6
10 times 10ndash6
50 times 10ndash7
ndash50 times 10ndash7
ndash10 times 10ndash6
ndash15 times 10ndash6
secndash
1
(a)
ωmiddotΩmiddot
01 02 03 04 05e
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
secndash
1
(b)
ωmiddotΩmiddot
01 02 03 04 05e
16 times 10ndash6
14 times 10ndash6
12 times 10ndash6
10 times 10ndash6
secndash
1
(c)
ωmiddotΩmiddot
01 02 03 04 05
e
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
ndash20 times 10ndash7
secndash
1
(d)
Figure 12 Curves of _Ω and _ωwith e for the 4 cases corresponding to F(i) 02 (a) i minus 4716 (b) i 6556 (c) i minus 13284 and (d) i 11444
10 Advances in Astronomy
Equations (44)ndash(47) give the relation between sin(ω) andthe inclination i the semi-major axis a and the eccentricitye which gives the family of orbits that balance the argumentof perigee ω subject of course to the restriction that sin(ω) isa real value between minus 1 and 1
4 Numerical Results
In this section numerical results and graphs are obtained forthe case of seasat a 7100 km by putting ω as a function of eand i from equations (32) and (33) e curves are plottedwithin the possible range given by condition (36) to give curvesof balanced e and ie curves are against the eccentricity e inthe range [001 05] e numerical values involved areJ2 0001082645 J3 minus 0000002546 J4 minus 0000001649 R
6378165km α 7100m and μ 3986005 km3secminus 2e condition (36) gives the upper and lower bounds of
the function F(i) as a function of e and since the functioneR(C1a(1 minus e2)) is increasing with e and has no criticalpoints in the interval [001 05] then the minimum valueoccurs at e 001 and the maximum at e 05 which givesminus 024leF(i)le 024 e graphs are plotted for differentvalues of F(i) which corresponds to specific values of i foundby solving the equation resulting from setting F(i) equal tothe required values After that we plot _Ω 0 and _ω 0 si-multaneously for the same values of F(i) at each specific i-value to find the orbit values at which we have nearest valuesfor _Ω 0 and _ω 0 In the graphs of _Ω and _ω the relation(32) was kept to ensure that e and i are already balanced
Five selected values of F(i) are chosen F(i) 001 05010 015 and 020 Negative values will give the same resultswith negative sign since F(i) is odd with respect to i Also foreach value the solution of F(i) x with x equals one of theabove values we get four real values of i on the form i1180 minus i1 i2 and minus 180 minus i2 where i1 and i2 are near the criticalinclination one of them is positive and the other is negativeTable 1 gives the values of i corresponding to the selectedvalues of F(i)
We note that as F(i) increases i gets away from thecritical inclination and the curve gets shorter indicating lessstability of ω as expected
e graphs are plotted for each value of F(i) first for thebalanced values of e and i then for the corresponding fourvalues of i four graphs are plotted for _Ω and _ω to find thenearest values of zero variation for both elements
Figures 3ndash12 show the possibility of balancing ω with eand i while Ω will have a variation of order 10minus 6sec or itmust be balanced alone at i 90 deg (as shown in equation(26)) according to the orbit kind Figures 3 5 7 9 and 11show the possible values of ω(e) that balance e and i whileFigures 4 6 8 10 and 12 show the curves of _Ω and _ω at thevalues of i at which _e _i 0
5 Conclusion
Let balanced orbits be defined as those for which the orbitalelements are set equal to zero under the effect of secular andlong periodic perturbations In this work the effect of Earthoblateness is the considered perturbing force because of
dealing with low Earth orbits e above analysis then showsthat such an orbit will be balanced within a reliable toleranceonly for few weeks since we are forced to accept the motionof either the node or the perigee by about 10minus 6 degsec ereason is that under the influence of the Earth oblateness_Ω 0 (exactly) for i (π2) while _ω 0 only near thecritical inclination ic 634 deg Hence the best procedure isto design a satellite constellation for which the nodal shiftsdue to the perturbative effects and Earth rotation aremodeled to yield continuous coverage e perigees areeither fixed or arranged to realize that the perigee (or theapogee) be overhead the coverage region (regions) in duetimes is may require near commensurability with theadmitted nodal periods
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was funded by the University of Jeddah SaudiArabia under grant no UJ-26-18-DR us the authortherefore acknowledges with thanks the university technicaland financial support
References
[1] N Delsate P Robutel A Lemaıtre and T Carletti ldquoFrozenorbits at high eccentricity and inclination application tomercury orbiterrdquo Celestial Mechanics and Dynamical As-tronomy vol 108 no 3 pp 275ndash300 2010
[2] E Condoleo M Cinelli E Ortore and C Circi ldquoStable orbitsfor lunar landing assistancerdquo Advances in Space Researchvol 60 no 7 pp 1404ndash1412 2017
[3] A Elipe andM Lara ldquoFrozen orbits about the moonrdquo Journalof Guidance Control and Dynamics vol 26 no 2 pp 238ndash243 2003
[4] C Circi E Condoleo and E Ortore ldquoA vectorial approach todetermine frozen orbital conditionsrdquo Celestial Mechanics andDynamical Astronomy vol 128 no 2-3 pp 361ndash382 2017
[5] E Condoleo M Cinelli E Ortore and C Circi ldquoFrozenorbits with equatorial perturbing bodies the case of Gany-mede Callisto and Titanrdquo Journal of Guidance Control andDynamics vol 39 no 10 pp 2264ndash2272 2016
[6] V Kudielka ldquoBalanced earth satellite orbitsrdquo Celestial Me-chanics amp Dynamical Astronomy vol 60 no 4 pp 455ndash4701994
[7] S L Coffey A Deprit and E Deprit ldquoFrozen orbits forsatellites close to an earth-like planetrdquo Celestial Mechanics ampDynamical Astronomy vol 59 no 1 pp 37ndash72 1994
[8] M Lara A Deprit and A Elipe ldquoNumerical continuation offamilies of frozen orbits in the zonal problem of artificialsatellite theoryrdquo Celestial Mechanics amp Dynamical Astronomyvol 62 no 2 pp 167ndash181 1995
[9] D Brouwer ldquoSolution of the problem of artificial satellitetheory without dragrdquo e Astronomical Journal vol 64p 378 1959
Advances in Astronomy 11
[10] Y Kozai ldquoe motion of a close earth satelliterdquo e Astro-nomical Journal vol 64 p 367 1959
[11] P M Fitzpatrick Principles of Celestial Mechanics AcademicPress Cambridge MA USA 1970
[12] H D Curtis Orbital Mechanics for Engineering StudentsElsevier Amsterdam Netherlands 2005
[13] D Brower and G M Clemence Methods of Celestial Me-chanics Academic Press Cambridge MA USA 1961
12 Advances in Astronomy
where
A minus 2c5e2F5cos
2(i) + 2c2 1 +
52e2
1113874 1113875F2 (45)
B c4ezF3cos2(i) minus c1z 4e +
1e
1113874 1113875F1 (46)
C cos2(i) c3z2
+ c5 e2F5 + 1 +
3e2
21113888 1113889F41113888 11138891113888 1113889
minus c6z2F6 minus c2 1 +
3e2
41113888 1113889F7 + 1 +
5e2
21113888 11138891113888 1113889F21113888 1113889
(47)
F (i) = 02
ω
π
3π4
π2
π4
0
e
01 02 03 04 05
Figure 11 e curve ω(e) at which _e _i 0 for F(i) 02
ωmiddotΩmiddot
01 02 03 04 05
e
15 times 10ndash6
10 times 10ndash6
50 times 10ndash7
ndash50 times 10ndash7
ndash10 times 10ndash6
ndash15 times 10ndash6
secndash
1
(a)
ωmiddotΩmiddot
01 02 03 04 05e
ndash20 times 10ndash7
ndash40 times 10ndash7
ndash60 times 10ndash7
ndash80 times 10ndash7
secndash
1
(b)
ωmiddotΩmiddot
01 02 03 04 05e
16 times 10ndash6
14 times 10ndash6
12 times 10ndash6
10 times 10ndash6
secndash
1
(c)
ωmiddotΩmiddot
01 02 03 04 05
e
10 times 10ndash6
80 times 10ndash7
60 times 10ndash7
40 times 10ndash7
20 times 10ndash7
ndash20 times 10ndash7
secndash
1
(d)
Figure 12 Curves of _Ω and _ωwith e for the 4 cases corresponding to F(i) 02 (a) i minus 4716 (b) i 6556 (c) i minus 13284 and (d) i 11444
10 Advances in Astronomy
Equations (44)ndash(47) give the relation between sin(ω) andthe inclination i the semi-major axis a and the eccentricitye which gives the family of orbits that balance the argumentof perigee ω subject of course to the restriction that sin(ω) isa real value between minus 1 and 1
4 Numerical Results
In this section numerical results and graphs are obtained forthe case of seasat a 7100 km by putting ω as a function of eand i from equations (32) and (33) e curves are plottedwithin the possible range given by condition (36) to give curvesof balanced e and ie curves are against the eccentricity e inthe range [001 05] e numerical values involved areJ2 0001082645 J3 minus 0000002546 J4 minus 0000001649 R
6378165km α 7100m and μ 3986005 km3secminus 2e condition (36) gives the upper and lower bounds of
the function F(i) as a function of e and since the functioneR(C1a(1 minus e2)) is increasing with e and has no criticalpoints in the interval [001 05] then the minimum valueoccurs at e 001 and the maximum at e 05 which givesminus 024leF(i)le 024 e graphs are plotted for differentvalues of F(i) which corresponds to specific values of i foundby solving the equation resulting from setting F(i) equal tothe required values After that we plot _Ω 0 and _ω 0 si-multaneously for the same values of F(i) at each specific i-value to find the orbit values at which we have nearest valuesfor _Ω 0 and _ω 0 In the graphs of _Ω and _ω the relation(32) was kept to ensure that e and i are already balanced
Five selected values of F(i) are chosen F(i) 001 05010 015 and 020 Negative values will give the same resultswith negative sign since F(i) is odd with respect to i Also foreach value the solution of F(i) x with x equals one of theabove values we get four real values of i on the form i1180 minus i1 i2 and minus 180 minus i2 where i1 and i2 are near the criticalinclination one of them is positive and the other is negativeTable 1 gives the values of i corresponding to the selectedvalues of F(i)
We note that as F(i) increases i gets away from thecritical inclination and the curve gets shorter indicating lessstability of ω as expected
e graphs are plotted for each value of F(i) first for thebalanced values of e and i then for the corresponding fourvalues of i four graphs are plotted for _Ω and _ω to find thenearest values of zero variation for both elements
Figures 3ndash12 show the possibility of balancing ω with eand i while Ω will have a variation of order 10minus 6sec or itmust be balanced alone at i 90 deg (as shown in equation(26)) according to the orbit kind Figures 3 5 7 9 and 11show the possible values of ω(e) that balance e and i whileFigures 4 6 8 10 and 12 show the curves of _Ω and _ω at thevalues of i at which _e _i 0
5 Conclusion
Let balanced orbits be defined as those for which the orbitalelements are set equal to zero under the effect of secular andlong periodic perturbations In this work the effect of Earthoblateness is the considered perturbing force because of
dealing with low Earth orbits e above analysis then showsthat such an orbit will be balanced within a reliable toleranceonly for few weeks since we are forced to accept the motionof either the node or the perigee by about 10minus 6 degsec ereason is that under the influence of the Earth oblateness_Ω 0 (exactly) for i (π2) while _ω 0 only near thecritical inclination ic 634 deg Hence the best procedure isto design a satellite constellation for which the nodal shiftsdue to the perturbative effects and Earth rotation aremodeled to yield continuous coverage e perigees areeither fixed or arranged to realize that the perigee (or theapogee) be overhead the coverage region (regions) in duetimes is may require near commensurability with theadmitted nodal periods
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was funded by the University of Jeddah SaudiArabia under grant no UJ-26-18-DR us the authortherefore acknowledges with thanks the university technicaland financial support
References
[1] N Delsate P Robutel A Lemaıtre and T Carletti ldquoFrozenorbits at high eccentricity and inclination application tomercury orbiterrdquo Celestial Mechanics and Dynamical As-tronomy vol 108 no 3 pp 275ndash300 2010
[2] E Condoleo M Cinelli E Ortore and C Circi ldquoStable orbitsfor lunar landing assistancerdquo Advances in Space Researchvol 60 no 7 pp 1404ndash1412 2017
[3] A Elipe andM Lara ldquoFrozen orbits about the moonrdquo Journalof Guidance Control and Dynamics vol 26 no 2 pp 238ndash243 2003
[4] C Circi E Condoleo and E Ortore ldquoA vectorial approach todetermine frozen orbital conditionsrdquo Celestial Mechanics andDynamical Astronomy vol 128 no 2-3 pp 361ndash382 2017
[5] E Condoleo M Cinelli E Ortore and C Circi ldquoFrozenorbits with equatorial perturbing bodies the case of Gany-mede Callisto and Titanrdquo Journal of Guidance Control andDynamics vol 39 no 10 pp 2264ndash2272 2016
[6] V Kudielka ldquoBalanced earth satellite orbitsrdquo Celestial Me-chanics amp Dynamical Astronomy vol 60 no 4 pp 455ndash4701994
[7] S L Coffey A Deprit and E Deprit ldquoFrozen orbits forsatellites close to an earth-like planetrdquo Celestial Mechanics ampDynamical Astronomy vol 59 no 1 pp 37ndash72 1994
[8] M Lara A Deprit and A Elipe ldquoNumerical continuation offamilies of frozen orbits in the zonal problem of artificialsatellite theoryrdquo Celestial Mechanics amp Dynamical Astronomyvol 62 no 2 pp 167ndash181 1995
[9] D Brouwer ldquoSolution of the problem of artificial satellitetheory without dragrdquo e Astronomical Journal vol 64p 378 1959
Advances in Astronomy 11
[10] Y Kozai ldquoe motion of a close earth satelliterdquo e Astro-nomical Journal vol 64 p 367 1959
[11] P M Fitzpatrick Principles of Celestial Mechanics AcademicPress Cambridge MA USA 1970
[12] H D Curtis Orbital Mechanics for Engineering StudentsElsevier Amsterdam Netherlands 2005
[13] D Brower and G M Clemence Methods of Celestial Me-chanics Academic Press Cambridge MA USA 1961
12 Advances in Astronomy
Equations (44)ndash(47) give the relation between sin(ω) andthe inclination i the semi-major axis a and the eccentricitye which gives the family of orbits that balance the argumentof perigee ω subject of course to the restriction that sin(ω) isa real value between minus 1 and 1
4 Numerical Results
In this section numerical results and graphs are obtained forthe case of seasat a 7100 km by putting ω as a function of eand i from equations (32) and (33) e curves are plottedwithin the possible range given by condition (36) to give curvesof balanced e and ie curves are against the eccentricity e inthe range [001 05] e numerical values involved areJ2 0001082645 J3 minus 0000002546 J4 minus 0000001649 R
6378165km α 7100m and μ 3986005 km3secminus 2e condition (36) gives the upper and lower bounds of
the function F(i) as a function of e and since the functioneR(C1a(1 minus e2)) is increasing with e and has no criticalpoints in the interval [001 05] then the minimum valueoccurs at e 001 and the maximum at e 05 which givesminus 024leF(i)le 024 e graphs are plotted for differentvalues of F(i) which corresponds to specific values of i foundby solving the equation resulting from setting F(i) equal tothe required values After that we plot _Ω 0 and _ω 0 si-multaneously for the same values of F(i) at each specific i-value to find the orbit values at which we have nearest valuesfor _Ω 0 and _ω 0 In the graphs of _Ω and _ω the relation(32) was kept to ensure that e and i are already balanced
Five selected values of F(i) are chosen F(i) 001 05010 015 and 020 Negative values will give the same resultswith negative sign since F(i) is odd with respect to i Also foreach value the solution of F(i) x with x equals one of theabove values we get four real values of i on the form i1180 minus i1 i2 and minus 180 minus i2 where i1 and i2 are near the criticalinclination one of them is positive and the other is negativeTable 1 gives the values of i corresponding to the selectedvalues of F(i)
We note that as F(i) increases i gets away from thecritical inclination and the curve gets shorter indicating lessstability of ω as expected
e graphs are plotted for each value of F(i) first for thebalanced values of e and i then for the corresponding fourvalues of i four graphs are plotted for _Ω and _ω to find thenearest values of zero variation for both elements
Figures 3ndash12 show the possibility of balancing ω with eand i while Ω will have a variation of order 10minus 6sec or itmust be balanced alone at i 90 deg (as shown in equation(26)) according to the orbit kind Figures 3 5 7 9 and 11show the possible values of ω(e) that balance e and i whileFigures 4 6 8 10 and 12 show the curves of _Ω and _ω at thevalues of i at which _e _i 0
5 Conclusion
Let balanced orbits be defined as those for which the orbitalelements are set equal to zero under the effect of secular andlong periodic perturbations In this work the effect of Earthoblateness is the considered perturbing force because of
dealing with low Earth orbits e above analysis then showsthat such an orbit will be balanced within a reliable toleranceonly for few weeks since we are forced to accept the motionof either the node or the perigee by about 10minus 6 degsec ereason is that under the influence of the Earth oblateness_Ω 0 (exactly) for i (π2) while _ω 0 only near thecritical inclination ic 634 deg Hence the best procedure isto design a satellite constellation for which the nodal shiftsdue to the perturbative effects and Earth rotation aremodeled to yield continuous coverage e perigees areeither fixed or arranged to realize that the perigee (or theapogee) be overhead the coverage region (regions) in duetimes is may require near commensurability with theadmitted nodal periods
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was funded by the University of Jeddah SaudiArabia under grant no UJ-26-18-DR us the authortherefore acknowledges with thanks the university technicaland financial support
References
[1] N Delsate P Robutel A Lemaıtre and T Carletti ldquoFrozenorbits at high eccentricity and inclination application tomercury orbiterrdquo Celestial Mechanics and Dynamical As-tronomy vol 108 no 3 pp 275ndash300 2010
[2] E Condoleo M Cinelli E Ortore and C Circi ldquoStable orbitsfor lunar landing assistancerdquo Advances in Space Researchvol 60 no 7 pp 1404ndash1412 2017
[3] A Elipe andM Lara ldquoFrozen orbits about the moonrdquo Journalof Guidance Control and Dynamics vol 26 no 2 pp 238ndash243 2003
[4] C Circi E Condoleo and E Ortore ldquoA vectorial approach todetermine frozen orbital conditionsrdquo Celestial Mechanics andDynamical Astronomy vol 128 no 2-3 pp 361ndash382 2017
[5] E Condoleo M Cinelli E Ortore and C Circi ldquoFrozenorbits with equatorial perturbing bodies the case of Gany-mede Callisto and Titanrdquo Journal of Guidance Control andDynamics vol 39 no 10 pp 2264ndash2272 2016
[6] V Kudielka ldquoBalanced earth satellite orbitsrdquo Celestial Me-chanics amp Dynamical Astronomy vol 60 no 4 pp 455ndash4701994
[7] S L Coffey A Deprit and E Deprit ldquoFrozen orbits forsatellites close to an earth-like planetrdquo Celestial Mechanics ampDynamical Astronomy vol 59 no 1 pp 37ndash72 1994
[8] M Lara A Deprit and A Elipe ldquoNumerical continuation offamilies of frozen orbits in the zonal problem of artificialsatellite theoryrdquo Celestial Mechanics amp Dynamical Astronomyvol 62 no 2 pp 167ndash181 1995
[9] D Brouwer ldquoSolution of the problem of artificial satellitetheory without dragrdquo e Astronomical Journal vol 64p 378 1959
Advances in Astronomy 11
[10] Y Kozai ldquoe motion of a close earth satelliterdquo e Astro-nomical Journal vol 64 p 367 1959
[11] P M Fitzpatrick Principles of Celestial Mechanics AcademicPress Cambridge MA USA 1970
[12] H D Curtis Orbital Mechanics for Engineering StudentsElsevier Amsterdam Netherlands 2005
[13] D Brower and G M Clemence Methods of Celestial Me-chanics Academic Press Cambridge MA USA 1961
12 Advances in Astronomy
[10] Y Kozai ldquoe motion of a close earth satelliterdquo e Astro-nomical Journal vol 64 p 367 1959
[11] P M Fitzpatrick Principles of Celestial Mechanics AcademicPress Cambridge MA USA 1970
[12] H D Curtis Orbital Mechanics for Engineering StudentsElsevier Amsterdam Netherlands 2005
[13] D Brower and G M Clemence Methods of Celestial Me-chanics Academic Press Cambridge MA USA 1961
12 Advances in Astronomy
