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Satellite orbit perturbations in a dusty Martian atmosphere
Acta Astronautica 72 (2012) 27–37
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Satellite orbit perturbations in a dusty Martian atmosphere Ioannis Haranas a,n, Spiros Pagiatakis b a b
Department of Physics and Astronomy, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3 Department of Earth & Space Science & Engineering, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3
a r t i c l e i n f o
abstract
Article history: Received 25 January 2011 Received in revised form 2 September 2011 Accepted 12 September 2011 Available online 4 October 2011
In this paper we calculate the effect of atmospheric dust on the orbital elements of a satellite. Dust storms that originate in the Martian surface may evolve into global storms in the atmosphere that can last for months can affect low orbiter and lander missions. We model the dust as a velocity-square depended drag force acting on a satellite and we derive an appropriate disturbing function that accounts for the effect of dust on the orbit, using a Lagrangean formulation. A first-order perturbation solution of Lagrange’s planetary equations of motion indicates that for a local dust storm cloud that has a possible density of 8.323 10 10 kg m 3 at an altitude of 100 km affects the orbital semimajor axis of a 1000 kg satellite up 0.142 m day 1. Regional dust storms of the same density may affect the semimajor axis up to of 0.418 m day 1. Other orbital elements are also affected but to a lesser extent. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Lagrange’s equations Gravity field Mars Mars atmosphere Dust storms
1. Introduction There is strong evidence that dust and variable amounts of aerosols are fundamental components of the Martian atmosphere [13,43]. In addition, heating of the dust by absorption of visible radiation strongly affects the dynamical and thermal state of its atmosphere. Martin and Zurek [20] demonstrate such variabilities that are of inter-annual nature. In certain years, this dust cycle manifests itself into dust storms in form of small size dust devil vortices that can spread all over the Martian space. During Martian spring, local or regional dust storms take place in the northern and southern hemisphere. Planet encircling dust storms occur about perihelion time just before the southern summer solstice during which solar heating is maximum, although such storms do not occur every year [20]. Global dust storms can probably be classified among the most important meteorological phenomena; in approximately two weeks, a small localized storm can expand to cover the whole planet
n
Corresponding author. Tel.: þ1 416 932 3155. E-mail address: [email protected] (I. Haranas).
0094-5765/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2011.09.005
[42]. Depending on the particle size, Mars atmospheric dust can remain suspended in the atmosphere for months [21]. After the storm sets, the atmospheric dust can be at high levels for months [21]. For example, it takes dust particles of diameter d E1 mm (in this paper we study particles with diameter dE1.25 mm) 1–2 months to settle when falling from h¼10 km. In November 1971, during its arrival in Mars, Mariner 9 experienced the effects of a major global dust storm that lasted several weeks and the predictions of Martian eolian processes were verified. After the dust cleared, Mariner 9 cameras revealed abundant features attributed to eolian activity including dunes, various pits, etc. [8]. The dust clouds of this particular dust storm that Mariner 9 experienced upon its arrival, reached an altitude of 70 km [38], something that was determined by analyzing Martian limb pictures. In 1993, Mars Global Surveyor mapped the temperature and opacity of the atmosphere using a thermal emission spectrometer. As a result of that dust storm, the density of the atmosphere and therefore the orbit of the spacecraft were significantly affected. The November 1971 dust storm resulted in an atmospheric mass density change at 110 km altitude by at least a factor of 10 [2]. Finally, dust storm heating constitutes
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I. Haranas, S. Pagiatakis / Acta Astronautica 72 (2012) 27–37
a low frequency unpredictable perturbation to the entire Martian atmosphere [2]. New studies of the Martian atmosphere have been possible with the use of large data sets acquired from spacecrafts such as, data from the thermal emission spectrometer (TES) aboard Mars Global Surveyor (MGS) [39]. TES instruments measure dust opacity, also called absorption coefficient an, which expresses the loss of intensity of a light beam as it travels through the dust, and by convention, it is positive when energy is taken away from the beam [36]. The motivation for this paper is simple. Our goal is not to engage in any precise atmospheric computations in order to accurately predict the Martian thermospheric densities, nor to predict the effect of the dust on the Martian thermosphere. Instead we follow the work of Martin [22] in order derive an upper estimate for the dust atmospheric density at the orbital altitude of 100 km We also calculate the dust drag coefficient, in order to study the effect of Mars atmospheric dust drag on the satellite. Using a formula for the atmospheric density as given in [22], we calculate that certain dust storms can produce dust atmospheric densities at 100 km that are two orders less than the magnitude with those of Martian atmosphere. For example, we use the peak of the 1977b storm, which produced an estimated suspended mass of 4.3 1011 kg actually produced an equivalent of 4.3 10–3 kg m 2 or a layer of 1.4 mm thickness [22]. Next we distribute the dust in a band located between the latitudes of f ¼ 7601 in order to obtain an upper bound for the dust density, which at the altitude of 100 km gives a dust atmospheric density rd ¼8.323 10 10 kg m 3. This density is 84 times smaller than the Martian atmospheric density at the altitude of 100 km given in Magalha~ es et al. [19] namely ra ¼ 7.03 10 8 kg m 3. Even though storms like that one are not very common on Mars we reckon that dust after all might be an important perturbation that should be taken into account in orbital calculations. 2. Satellite motion in Martian atmospheric dust As discussed in the introduction, dust particles from large-scale sandstorms can reach altitudes of 50 km or higher [35]. Jaquin et al. [13] also report that in the 1977a storm a continuous haze extended to 50 km where a detached haze appeared from 50–90 km. Recent models also predict, that meteoroid impacts on the surface of Mars can produce and rise dust particles up to different altitudes depending on the size of the impactor [17]. To derive the dust disturbing potential Rd to be included in the equations of motion of a satellite, we consider a Mars orbiter, with its position being a function of time t in an areocentric, body-fixed coordinate system [r(t), y(t), l(t)]. Further, we assume that dust particles of certain size and known density rd(r) that in general a function of the distance r from the Martian center exist along the satellite orbit. Next, let us assume that the dust fluid exerts a force opposite to the satellite’s orbital path. The result of this force is acceleration onto the satellite that can be described in a similar way to that of the atmospheric drag as a velocity-square depended force according to the formula [30]. The justification of our assumption
emanates from recent Phoenix lander data. With reference to atmospheric optics the backscatter signal amplitude from dust in the atmosphere on Mars is quite similar to the lidar-signal from molecules in the atmosphere of Earth, when comparing, figure 4 for the Earth in [41] with figure 1 in [6] for Mars. On the basis of the two graphs we propose a velocity square force. Also with reference to Rubincam [32], the author uses a velocity square force to describe the semimajor axis time rate of change due to the interplanetary dust drag force exerted on the LAGEOS satellite. ad ¼
1 As C d rd v2s ; 2ms
ð1Þ
and Bds ¼ms/AsCd represents the satellite dust ballistic coefficient that also describes the susceptibility of the satellite to dust drag forces, a higher B will imply less dust drag effect onto the satellite, and As is the frontal area of the satellite, Cd is the satellite dust drag coefficient, rd is the dust atmospheric density at the orbital altitude and vs is the magnitude of the relative velocity of the satellite with respect to the atmosphere. Since, force Fd acting by the dust ‘‘fluid’’ onto the satellite is velocity-square depended, it is of the general form F d ¼ K i vni [40,33] where Ki is a constant of proportionality, vi is the magnitude of the velocity vector, and n is an index taking different values for different types of forces such as, n ¼0 for dry friction or n ¼1 for viscous drag. If the non-conservative force is of the above general form then the dissipation function (Rayleigh’s dissipation function) due to Mars atmospheric dust can be written as follows [40]: Rd ¼
s X Ki vn þ 1 : n þ1 i i¼1
ð2Þ
Assuming a velocity-square depended dust drag force, then n ¼2, and Rd ¼ ðK 1 =3Þv31 (cf. Eq. (2); see also [14]). Since the dust drag has been assumed to be mathematically equivalent to that of the atmospheric drag, we can equate the force per unit mass due to atmospheric drag (F a ¼ ðAS C r=2ms Þv2s ) [30] to that of dust drag per unit mass, and by analogy obtain the corresponding dust drag constant that multiplies the velocity square term to be (ASCd/2ms)rd. Here the dust density rd is assumed to have a constant value at the orbital altitude of 100 km. Next, with the help of Eq. (2) above we can write the dust disturbing (dissipation) function in the areocentric, bodyfixed coordinate system as follows: As C d rd ! ! Rd ðr, y, l, r_ , y_ , l_ Þ ¼ ðvs 9 o M r 9Þ3 ; 6ms
ð3Þ
which represents the satellite dissipated energy density per ! unit time due to its interaction with the dust. o M is the ! angular velocity vector of Mars and r is the position vector of the satellite. Here we consider that the dust co-rotates ! ! ! with the Martian atmosphere with velocity v d ¼ o M r [28]. If the satellite velocity relative to the dust is described ! ! ! ! according to v r ¼ v s o M r , maximum observed deviations from this assumption are of the order of 40% leading to uncertainties in the drag force of less than 5% [28].
I. Haranas, S. Pagiatakis / Acta Astronautica 72 (2012) 27–37
In the last few decades different authors have investigated the problem of atmospheric drag (e.g., [29,16,27,18,37]) with a force that was quadratic in the magnitude of the velocity, but analytical solutions have not been obtained except for very special cases (e.g., [10,11] personal communication). The work on quadratic atmospheric drag found in the literature has been presented in closed form solutions that deal with the two-body case and can be classified in two categories: (a) use of simplified orbital equations, where the motion is mostly tangential, and (b) drag is treated using the method of the variation of the orbital elements and the solutions are given in terms of the six orbital elements using Gauss’s planetary equations, which describe the rates of change of the orbital elements as functions of the components of the disturbing forces. More details on these two approaches can be found in the papers cited above. However, our approach in the study of the dust drag is that of the cube velocity dissipation function given in Eq. (3) above. In this contribution we follow the Lagrangean approach; we use the disturbing function given by Eq. (3) that expresses energy density dissipation per unit time. This allows us to write the planetary equations as functions of disturbing potentials; these equations are the Lagrangean form of Gauss’s planetary equations. The application of Lagrange’s equations is widespread and the calculations required to obtain the time rates of change of the orbital elements are generally straightforward, and simple to interpret [9]. Lagrange’s equations can be preferably used in orbiting bodies of low eccentricities [3,34] (a satellite with an eccentricity e¼0.01 is considered in this paper—see below). It should also be noted that Lagrange’s equations are rigorous, originally derived for a perturbation exerted by one planet to another, and they also hold true when the disturbing function is due to many other causes thus, widening the spectrum of applications and for a wide variety of phenomena. The analytical form of the disturbing function depends upon the acting force(s). This provides the advantage that we only really need to define the appropriate disturbing function that corresponds to the phenomenon under study (here the dust drag force). Furthermore, Lagrange’s equations are the simplest means of obtaining the first approximation effect of the disturbing function on the orbital elements [14], which is also our main goal in this paper. However, Lagrange’s equations are less suitable for numerical treatment [34], which also justifies the first order perturbation method used in this paper (Section 4). 3. The dust disturbing function
29
Following Kaula [15], we can now simply define the dust disturbing function Rd to be equal to Rayleigh’s dissipation function (cf. Eq. (3)). The dust disturbing function Rd, and more precisely its derivatives with respect to the orbital elements, can then be substituted in Eqs. (7)–(12) (see below). If vr is the satellite’s orbital velocity relative to the dust and if it takes the satellite a time t¼Ld/vr ¼ xd/vr to travel through a large dust cloud of angular size x (radians) that is situated at distance d from the center of Mars (d ¼RM þh) (Fig. 1) then, using Eq. (3) we derive the disturbing function Rd that represents the total energy density dissipated during the passage of the satellite through the dust cloud Rd ¼
C d AS rd dx ! ! ðvs 9 o M r 9Þ2 : 6ms
Further, we assume that all equatorial areocentric ! ! components of v s and also r in Eq. (4) are functions of time t but are not indicated for simplicity. Given Eq. (4), we can write the disturbing function Rd as a function of the orbital elements. Following Kaula’s [15] known standard transformations and rewriting sin f ¼ sin i sin (o þf), or cos y ¼sin(o þf) sin i and also using that sin2 y ¼ 1 sin2 i sin2(o þf) we substitute in Eq. (4). All these ! substitutions take place after we express v s in equatorial areocentric components, perform the cross product and ! ! obtain a final expression for the ðvs 9 o M r 9Þ2 term. We also substitute r, y, l, and r_ , y_ , l_ with their transformed equivalents that contain the orbital elements and their time derivatives. Finally, we also use the relation l ¼ O cos i þ o þ f [1], where o is the argument of the periareon, f is the true anomaly of the satellite, and i is the inclination of the orbit. The final form of Rd (as a function of the orbital elements) becomes Rd ¼
As C d rd dx Fd ; 6ms
ð5Þ
where Fd ¼ Fs Frot 2" #2 _ ð1e2 Þa2ae e_ að1e2 Þðe_ cos ueu_ sin uÞ 1 þe cos u ð1 þe cos uÞ2
¼4
þ
a2 ð1e2 Þ2 ði_ cos i sinuu_ sin i cos uÞ2 ð1 þe cos uÞ2 ð1sin2 i sin2 uÞ
Satellite Ldust RS
The Lagrange’s planetary equations give the time rates of change of the six orbital elements in terms of the partial derivatives of the disturbing function Rd with respect to the orbital elements. In general, the disturbing function Rd comprises all potential field terms except the central one [15] and expresses energy density per unit time. It is only the disturbing function Rd that contributes to the orbit perturbations and its partial derivatives with respect to the (osculating) orbital elements that appear in the right hand side of the Lagrange’s planetary equations.
ð4Þ
ξdust
Satellite's orbit
Rd Dust
MARS
Fig. 1. This figure demonstrates the idealized scenario of a cylindrical type satellite entering an episodic dust cloud.
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I. Haranas, S. Pagiatakis / Acta Astronautica 72 (2012) 27–37
þ
_ cos i_i O siniÞ2 a2 ð1e2 Þ2 ð1sin2 i sin2 uÞðu_ þ O
#
where qk and q_ k are the generalized coordinates and velocities, respectively.
2
ð1 þe cos uÞ
" 2oM a2 ð1e2 Þ2 ðu_ cos u sini þ _i cos i sin uÞcos i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ð1 þe cos uÞ2 1sin i sin u
o
2 2 2 2 2 2 M a ð1e Þ cos i 2
ð1 þ ecos uÞ
4. Lagrange’s equations of motion and first order perturbation solution
2 2 2 2 2 2 M a ð1e Þ sin i cos u 2
o
ð1þ e cos uÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# _ cos ii_ sin iOÞ 1sin2 i sin2 u 2oM a2 ð1e2 Þ2 sin i cos uðu_ þ O ; ð1þ e cos uÞ2
ð6Þ and u ¼ o þf. Eq. (5) describes the total energy loss of the satellite per unit mass due to dust and all the orbital elements and their derivatives in Eq. (6) have been assumed to be functions of time but not indicated for simplicity. At this point, we must clarify that the presence of the derivatives of the orbital elements in Eq. (6) is due to the fact that the components of the satellite velocity containing r_ , y_ , l_ and their relations to the orbital elements. However, in a first order perturbation treatment after taking the derivatives with respect to the orbital elements in the RHS of these equations, we substitute for the initial values of all the orbital elements taking into account that a_ q ð0Þ ¼ a€ q ð0Þ ¼ 0, q¼ 1,2, y, 6. Given Rd by Eq. (5), we can write Lagrange’s planetary equations of the satellite in the following form [15]: da 2 @Rd ¼ , dt na @M
ð7Þ
pffiffiffiffiffiffiffiffiffiffiffiffi de ð1e2 Þ @Rd 1e2 @Rd ¼ , dt ne a2 @M ne a2 @o
ð8Þ
pffiffiffiffiffiffiffiffiffiffiffiffi do cos i @Rd 1e2 @Rd pffiffiffiffiffiffiffiffiffiffiffiffi ¼ , dt na2 e @e na2 1e2 sin i @i
ð9Þ
In general, Lagrange’s equations of planetary motion (Eqs. (7)–(12)), cannot be solved analytically, but only approximately by successive approximations or by numerical integration. The method of successive approximations to first order is identical to that of perturbation treatment and makes possible to write the osculating orbital elements of the satellite in form of a power series of a small parameter [4] that includes the orbital elements. For instance, we write as follows only the equations for a and e; other equations are similar: a ¼ a0 þ D1 a0 þ D2 a0 þ þ Dn a0 þ e ¼ e0 þ D1 e0 þ D2 e0 þ þ Dn e0 þ :
ð14Þ
Quantities Dna, Dne, etc., are called nth order perturbations. As a first approximation in Lagrange’s Eqs. (7)–(12), we set the disturbing function Rd ¼ 0, and as consequence, the equations imply that a ¼a0, e¼e0, i¼i0, y, M¼M0 þ _ 0 ¼o € 0 ¼ 0, n0t, and a_ 0 ¼ a€ 0 ¼ 0, e_ 0 ¼ e€ 0 ¼ 0, i_0 ¼ €i 0 ¼ 0, o _0 ¼O € 0 ¼ 0, where subscript ‘‘0’’ indicates initial (undisO turbed) values. Next, we use Rd in the right hand side of Lagrange’s planetary equations, we take the total derivatives with respect to the osculating orbital elements, and then we substitute for the initial orbital element values and also for their corresponding values of the first and second derivatives as indicated above, in order to obtain the first-order perturbation solution for the time rates of the orbital elements of the satellite (Appendix A). We start with Eq. (7), which becomes da 2 @Rd 2K 0 @Fd ¼ ¼ , dt na @M na @M
di cos i @Rd 1 @Rd pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi ¼ , dt na2 1e2 sin i @o na2 1e2 sin i @O
ð10Þ
dO 1 @Rd pffiffiffiffiffiffiffiffiffiffiffiffi ¼ , dt na2 1e2 sin i @i
ð11Þ
where K0 ¼AsCdrddx/6ms. In order to differentiate Eq. (7) with respect to the mean anomaly M, we express the true anomaly f and its time derivative f_ as a function of M using the following series expansion [4] up to order e2
dM ð1e2 Þ @Rd 2 @Rd ¼ n : dt ne a2 @e na @a
ð12Þ
f ffi M þ 2e sin M þ
These are the equations of motion of the satellite also given by [15], in which the left-hand-sides describe the time variability of the Keplerian orbital elements (osculating elements) a, e, o, i, O and M, in terms of Rd, which in our case is the dust dissipation function, and n is the mean motion of the satellite in s 1 (e.g., [15]). It is also possible to study the motion of a body (i.e., satellite) in the presence of dissipative forces, using a standard Lagrangean treatment, via Lagrange’s second order differential equations. In the case where velocity dependent dissipative forces are involved we can still derive Lagrange’s equations of motion if the two scalar functions i.e. the Lagrangean L, and also Rayleigh’s dissipation function Rd are known. In that case Lagrange’s equations can be written in the following form [7] d @L @L @R þ ¼ 0, ð13Þ _ dt @q k @qk @q_ k
5 2 e sin 2M þ , 4
ð15Þ
ð16Þ
5 _ _ _ þ 2e_ sinðMÞþ 5 ee_ sinð2MÞ þ 2MecosðMÞ þ e2 M cosð2MÞ, f_ ffi M 2 2
ð17Þ _ ¼ n. At this point we should where M¼n(t t0) [2] and M mention that we do not write explicitly @Fd/@M because it is long and cumbersome. Carrying out similar differentiations for the other orbital elements and substituting their initial values indicated by the subscript ‘‘0’’ we obtain da As C a ra dx ffi ðP þ P 2d oM þ P 3d o2M Þ, dt 3ms n0 a0 1d
ð18Þ
where all quantities given in the derived equations are given in Appendix B and represent the first order perturbation coefficients as functions of the initial values of the orbital elements. Substituting initial numerical values for a polar orbit and for the orbital elements o0 ¼1801, f0 ¼2251, i0 ¼901, O0 ¼2701 we obtain the time rate of
I. Haranas, S. Pagiatakis / Acta Astronautica 72 (2012) 27–37
change of the semi-major axis in m day 1, when the satellite surface to mass ratio, the spatial density of the dust, and the dust drag coefficient are known.q The initial ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi value of mean motion was calculated to be n0 ¼ GM=a30 ¼ 9:891 1024 s1 . For a local dust storm cloud with length Ld ffi1230 km we obtain the following numerical form of the equations: da As C d rd ffi 4:471 103 dt ms 1
: ð19Þ
Similarly, the time rate of change of the eccentricity is qffiffiffiffiffiffiffiffiffiffiffiffi As C d rd dx 1e20 de ffi ðP 1d þ P 2d oM þ P 3d o2M Þ dt 6ms n0 e0 a20 As C d ra dxð1e20 Þ ðE1d þE2d oM þ E3d o2M Þ: 6ms n0 e0 a20
ð20Þ
dO As C d rd dx qffiffiffiffiffiffiffiffiffiffiffiffi csc i0 ðF 1d þ F 2d oM þ F 3d o2M Þ, ffi dt 6m n a2 1e2 s 0 0
ð25Þ
0
which numerically becomes h i dO As C d rd 1 : ¼ 3:304 104 doM deg: day dt ms
ð26Þ
de As C d rd ffi64:117 dð11:907 103 oM dt ms þ 9:360 105 o2M Þ½day
1
:
dM C As rd dx ffi n0 d ðC þ C 2d oM þC 3d o2M Þ dt 3ms n0 a0 1d ð1e20 ÞAs C d rd dx ðC 4d þ C 5d oM þC 6d o2M Þ, 6ms n0 e0 a20
ð27Þ
and the last of Lagrange’s planetary equations numerically becomes dM As C d rd d ¼ 0:75055:866 dt ms ð1661:21oM þ 1:642 105 o2M Þ½rev day
1
Substituting again the initial values of the orbital elements as above, we obtain the following equation:
ð21Þ
Next, we obtain
:
ð28Þ
It’s easy to see that Eqs. (18)–(28) are scaling linear with the ballistic coefficient Bds defined shortly after Eq. (1). The introduction of the ballistic coefficient as a single parameter is used in satellite and spacecraft design that can help scaleout final numerical results. 5. Numerical results
do As C d rd dx cot i0 qffiffiffiffiffiffiffiffiffiffiffiffi ðD1d þ D2d oM þ D3d o2M Þ ffi dt 6m n a2 1e2 s 0 0
þ
For the argument of the ascending node we obtain:
Finally
dð11:907 103 oM þ 9:360 105 o2M Þ½m day
þ
31
0
pAs C d rd dxð1e20 Þ 6ms n0 e0 a20
ðD4d þ D5d oM þ D6d o2M Þ,
ð22Þ
and similarly, do As C d rd ffi 2:0 104 dð16:60 102 oM þ 1:621 dt ms 105 o2M Þ½deg: day
1
:
ð23Þ
Next h i di As C d rd dx qffiffiffiffiffiffiffiffiffiffiffiffi cot i0 ðR1d þ R2d oM þ R3d o2M Þ deg: day1 , ffi dt 2 2 6m0 n0 a0 1e0
ð24Þ and similarly di ¼ 0: dt
ð25Þ
In Table 1 we present first order results for the time variability of the orbital elements. To derive the dust drag coefficient we consider stream lines of dust particles impinging on the frontal surface area of the satellite, where the individual particles within these stream lines are separated by a certain distance s0 and with the assumption that the dust particles do not accumulate on the satellite surface. Considering elastic collisions, between the satellite frontal surface area and the dust and conserving momentum and kinetic energy we obtain the following expression of the dust drag coefficient: 4md Cd ¼ ð29Þ N S ¼ 4:0: As rd s0 This is only the dust drag coefficient when the satellite orbits at a certain orbital altitude through a Martian atmosphere, contaminated by dust a contamination resulting from the development of Martian surface dust storms. This value corresponds to a higher value drag
Table 1 First order perturbation effects for the orbital elements time ratesa of the satellite resulting from a dust cloud at an altitude of 100 km, with density rd ¼8.323 10 10 kg m 3 and Cd ¼4.0 have been considered in our calculation of the orbital element changes. Dust cloud size (km)
da/dt (m day 1)
800
615 1230 1844
0.044 0.177 0.400
1000
615 1230 1844
0.035 0.142 0.320
Satellite mass ms (kg)
a
de/dt (day 1)
do/dt (deg. day 1)
dO/dt (deg. day 1)
dM/dt (rev day 1)
3.800 10 8 1.523 10 7 3.427 10 7
0.218 0.872 0.962
2.674 10 5 1.070 10 4 2.410 10 4
6.100 10 4 2.435 10 3 5.500 10 3
3.046 10 8 1.218 10 7 2.742 10 7
0.174 0.700 1.570
2.140 10 5 8.560 10 5 1.925 10 4
4.900 10 4 1.950 10 3 4.500 10 3
The rate of change of the inclination for the polar satellite used is di/dt¼ 0 and therefore it has been omitted from Table 1.
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I. Haranas, S. Pagiatakis / Acta Astronautica 72 (2012) 27–37
coefficient when compared to the coefficient of atmospheric drag of a cylinder that is given to be Ca ¼3.0 [5]. Our derived dust drag coefficient depends proportionally to: the total mass of the dust particles impinging on the frontal surface NSmd, which is also equal to As rds0 where NS, is the number of dust particle stream lines, and inversely proportional to: the frontal surface area of the satellite As ¼3.14 m2 the atmospheric dust density rd, the and the dust particle separation s0. The radius of the dust particles is taken to be rd ¼1.25 mm, a value given in [25], and material density rm ¼3000 kg m 3 [25]. Finally we consider dust clouds of different sizes namely Ld ¼dx ¼ 615, 1230, 1844 km deg. (cf. Fig. 2) situated at the same orbital altitude of 100 km. Dust in Martian atmosphere constitutes an important orbital perturbation that depending on the type of the storm and the size of the cloud can in time seriously affect the orbit of the satellite. Dust can also interact, interfere, and also affect the electronics of the satellite. Rovers and landers can also be seriously affected, not to mention man missions in the distant future. The avoidance of possible hazards for the safety of future missions will depend on more detailed studies of the dust properties and size of its particles, abundance and distribution, or even possible electrification under different atmospheric conditions thus, increasing the hazards of future missions. From Viking lander data [26] it is estimated that 800 kg km 2 of dust are lifted daily from the Martian surface. This would imply a total dust mass of Md ¼1.225 1011 kg over the Martian surface under a homogeneous distribution assumption for the dust. Our calculations using first order perturbations in the equations of motion are, summarized in Table 1. The table contains results that correspond to the dust drag coefficient Cd ¼4.0. An upper bound for the atmospheric dust density has been calculated using the 1997b storm [22] and using a dust scaling height hd ¼ 13 km [12]. In this calculation we choose to distribute the dust in a latitude band situated between f ¼ 7601 and considering the possibility of a local dust storm the density could be by a factor of 4 higher [22] we obtain an upper bound for the dust density to be rd(h)¼ 4r0e 100/13 ¼8.323 10 10 kg m 3. Furthermore, the table
Fig. 2. Semimajor axis change for satellites (The second line of each pair represents the ms ¼ 800 kg satellite higher density effect) with masses ms ¼ 800 and 1000 kg orbiting in a 615, 1230 and 1845 km dust clouds of density rd ¼ 8.323.1 10 10 kg m 3.
shows three different calculations of the orbital element perturbations corresponding to the same dust density as mentioned above and for two different satellites with masses ms ¼800 and 1000 kg. The most significant effect is on the semi-major axis of the orbital ellipse as it is intuitively expected. A 1000 kg satellite may lose up to 0.320 m per day in altitude (see last raw of Table 1, for instance). The loss of the satellite altitude also depends linearly on the angular size of the dust cloud and the time that the satellite travels in it. For example in the extreme case scenario of a global Mars dust storm where the dust cloud totally surrounds the planet we calculate that the same 1000 kg satellite can lose up to 46 m d 1. When speaking about lander missions going through violent storms, these 46 m of loss in altitude may become critical for a successful landing. Since, both atmospheric drag and dust drag can be described as velocity square depended forces, the same mathematical modeling that describes dust can be also applied to the atmospheric drag. As a test of our model, we apply the same mathematical formulation to the GRACE-A Earth satellite system. Using the appropriate parameters for GRACE-A, (http://www.csr. utexas.edu/grace/newsletter/archive/august2002.html) i.e. a¼6876.4816 km, h¼498.3416 km, i¼89.02544661, O ¼ 354.447141, e¼0.00040989, o ¼302.41424441, AGRACE ¼ 0.9486 m2, mGRACE ¼487 kg, 1.9rCa,r2.2, the mean motion of the satellite n¼9.90 10–4 rad s 1, and finally the atmospheric density at the orbital altitude of GRACE-A, rA ¼7.0 10 13 kg/m3 [23] we obtain that the time rate of change of the semimajor axis a lies in the range 25.28 m/drda/dtr 29.28 m/d. In particular for CdGRACE ¼2.2 we obtain that da/dtr 29.28 m/d, which is within 98% of the total value predicted by [31] who predict da/dtr 29.60 m/d. Martian atmospheric drag is the perturbation that most strongly influences the motion of a satellite. In fact, during the final revolutions of the orbital life of a satellite, drag effects can be more dominant than those of Martian oblateness. Comparing the two drags, we say that atmospheric drag with no dust content constitutes itself a very important perturbation. In particular, for a polar orbit satellite, which is the satellite considered in this study at the orbital altitude in the range 100–140 km, atmospheric drag is a very significant perturbation and becomes critical over long periods. A Martian polar satellite orbiting at 100 km at the end of 24 h period looses 0.7% of its initial altitude, but there is no change in its inclination. The rotation of the Martian atmosphere introduces an additional 14% increase in the time rates change of the orbital elements of the satellite. The time rate of change of the rest of the orbital elements for a 1000 kg satellite due to atmospheric drag with no 1 _ ¼ 0:048 deg: d1 , dust included is e_ ¼ 0:00073 d , o _ ¼ 0:517 deg: d1 , M _ ¼ 2:973 revd1 . In the case where O the atmosphere contains dust particles of certain size in our case rd ¼1.25 mm, the effect of atmospheric drag will be larger due to the fact that the dust drag effect is now added in the total effect, increasing by 0.005% per day the semimajor axis loss, or 0.15% in a month. Furthermore, dust drag will further increase those orbital elements that atmospheric drag increases the most, namely the semimajor axis, the argument of the periareon, and the mean anomaly,
I. Haranas, S. Pagiatakis / Acta Astronautica 72 (2012) 27–37
by the dust drag amount calculated in Table 1. Therefore, it accelerates the change of an elliptical orbit to circular, and therefore reduces further the satellite’s lifetime, each time the satellite passes through the dust clouds. 6. Conclusions In this paper the effects of dust on low orbiters around Mars were investigated. Because Mars dust might be able to reach up to heights of 100 km or even higher, we anticipate that the study of this particular perturbation will be important for satellite orbits at an altitude of 100 km, not to mention that such an orbital altitude has already been achieved as a periareon altitude by missions such as Mars Odyssey [24]. To model the effects of dust, we considered a velocity-square dependent force that was similar to that of the aerodynamic drag. Subsequently, a disturbing function was developed so that Lagrange’s planetary equations for the time variability of the orbital elements could be studied in a first order perturbation approach. For one estimated dust density and one calculated dust drag coefficient we calculated the effects on the orbital elements of the satellite orbit as a function of time (see Table 1). Our calculations showed that in the first order perturbation, all orbital elements change significantly except the inclination i, with the semimajor axis a, followed by the argument of the periareon o, and the eccentricity e and the mean anomaly M being affected the most and finally the argument of the ascending node O. The orbital element time rates of change
are directly related to the product of the dust density and the dust aerodynamic coefficient and can create a very critical cumulative effect for the satellite if it orbits in dust for long periods of time. Dust on Mars and its atmosphere in particular constitutes an important orbital perturbation that can seriously affect the orbit and therefore have an impact on the satellite’s lifetime. An increased drag effect can result in higher fuel consumption that is also related to the orbital maintenance of the satellite. An increased drag can also accelerate the spacecraft’s degradation affecting its solar cells, external spacecraft materials, and also the spacecraft physical surface morphology, and it is predominant in the velocity vector exposure (ram direction). It is also particularly important for example for low orbit gravimetric satellites for precise Mars gravity models. Dust can also interact, interfere, and also negatively affect its instruments. Rovers and Landers on the surface can also be seriously affected, not to mention man mission in the nearer future.
Acknowledgments This research was financially supported by the GEOIDE National Centre of Excellence (Canada), Phase III funding. The authors want to thank an unknown reviewer who with his/her valuable comments help improved this manuscript considerably.
Appendix A. Summary of all symbols ad ms As CD
rd vs Fd Rd Fa ! oM ! r ! vd ! vr Ld
x d RM a h
f y l
o f
O e
33
dust drag acceleration exerted on the satellite [m/s2] satellite’s mass¼800 and 1000 [kg] satellite’s frontal surface area ¼3.14 [m2] dust drag aerodynamic coefficient¼4.0 dust density at the orbital altitude of the satellite¼8.323 10 10 [kg/m3] satellite’s orbital velocity¼3.508 [km/s] force acting by the dust ‘‘fluid’’ onto the satellite [N] Rayleigh’s dissipation function [kg m2/s3] drag force acting on to the satellite [N] Mars’ angular velocity¼7.0854 10 5 [rad/s] satellite’s position vector [m] dust atmospheric velocity¼5 [m/s] satellite’s velocity relative to the dust [m/s] length of dust cloud 615, 1230 and 1844 [km] dust cloud angular size ¼10, 20 and 30 [1] dust cloud distance from the center of Mars ¼100 [km] Mars’s radius ¼3397.2 [km] satellite’s semimajor axis¼3522.30 [km] orbital altitude of the satellite¼100 [km] satellite’s areocentric latitude¼90 [1] satellite’s areocentric colatitude¼0 [1] satellite’s Areocentric longitude¼0 [1] satellite’s argument of the periareon¼180 [1] satellite’s true anomaly¼225 [1] satellite’s argument of the node ¼270 [1] satellite’s eccentricity¼0.01
34
i M
Fd Fs Fs n L qk q_ k
I. Haranas, S. Pagiatakis / Acta Astronautica 72 (2012) 27–37
satellite’s inclination¼90 [1] satellite’s mean anomaly ¼0.057 [1] satellite’s total disturbing potential [m2/s2] as a function of orbital elements satellite’s non rotational disturbing [m2/s2] potential as a function of orbital elements satellite’s rotational disturbing potential [m2/s2] as a function of orbital elements satellite’s mean motion¼9.9 10–4 [rad/s] Lagrangean function [kg m2/s2] satellite’s generalized coordinate [m] satellite’s generalized velocity [m/s]
Appendix B. First order perturbation coefficients of Lagrange’s equations Using the disturbing function Eq. (5) in Eqs. (7) and (12), and taking the indicated derivative qF/qM in the first Lagrange orbital equation, we then substitute for the initial values of the orbital element in the derived expression, which gives 5e2 X0 ¼ o0 þ n0 t þ2e0 sinðn0 tÞ þ 0 sinð2n0 tÞ , ð1:1Þ 4
D0 ¼ 1 þ 2e0 cosðn0 tÞ þ
5e20 cosð2n0 tÞ , 2
ð1:2Þ
5e2 K 0 ¼ n0 1þ 2e0 cosðn0 tÞ þ 0 cosð2n0 tÞ , 2
ð1:3Þ
T 0 ¼ ð1sin2 i0 sin2 X0 Þ,
ð1:4Þ
5e0 sinð2n0 tÞ , A0 ¼ 2sinðn0 tÞ þ 2
ð1:5Þ
B0 ¼ ða20 ð1e20 Þ2 sin2 i0 Þ,
ð1:6Þ
C 0 ¼ ðe0 a20 ð1e20 ÞÞ,
ð1:7Þ
D0 ¼ ðcos X0 e0 A0 sin X0 Þ,
ð1:8Þ
E0 ¼ n0 ð2cosðn0 tÞ þ 5e0 cosð2n0 tÞÞ,
ð1:9Þ
F 0 ¼ sin i0 ðE0 cos X0 þ A0 K 0 sin X0 Þ,
ð1:10Þ
S0 ¼ ð1 þ e0 cos X0 Þ,
ð1:11Þ
G0 ¼ sin i0 ðE0 cos X0 þ A0 K 0 sin X0 Þ,
ð1:12Þ
H0 ¼ ð1e20 ÞðK 0 sin X0 e0 A0 K 0 cos X0 e0 E0 sin X0 Þ,
ð1:13Þ
L0 ¼ sin i0 ðE0 cos X0 A0 K 0 sin X0 Þ,
ð1:14Þ
a0 H0 2a0 e20 K sin X0 2a0 e0 ð1e20 ÞD0 K 0 N0 ¼ , 2 2 S S
ð1:15Þ
M 0 ¼ ða0 ð1e20 Þsin i0 Þ,
ð1:16Þ
M 0 sin i0 ¼ ða0 ð1e20 Þ2 sin2 i0 Þ,
ð1:17Þ
P 0 ¼ ða0 e20 ð1e20 Þ2 Þ,
ð1:18Þ
R0 ¼ ða20 ð1e20 Þ2 Þ,
ð1:19Þ
X 0 ¼ n0 ð2e0 sinðn0 tÞ þ 5e20 sinð2n0 tÞÞ,
ð1:20Þ
G0 ¼ sin i0 ðX 0 cos X0 þ K 0 D0 sin X0 Þ,
ð1:21Þ
J 0 ¼ e0 a0 ð1e20 Þ,
ð1:22Þ
I. Haranas, S. Pagiatakis / Acta Astronautica 72 (2012) 27–37
U0 ¼
P1d ¼
e2 ð1e20 ÞK 0 D0 sin2 X0 2a 0 0 S30
S20
2K 0 R0 T 0 X 0 þ 2J0 K 0 U 0 sin X0 2B0 D0 K 20 sin X0 cos X0 S20 þ
P2d ¼
! a0 e0 ð1e20 ÞðK 0 D0 cos X0 X 0 sin X0 Þ
2e0 K 20 R0 D0 T 0 sin X0 S30
2R0 X 0
þ
2e0 B0 D0 K 20 sin X0 cos2 X0
pffiffiffiffiffiffi pffiffiffiffiffiffi T 0 sin i0 cos X0 2K 0 R0 D0 T 0 sin i0 sin X0
2K 0 R0 D0 cos i0 sin i0 sin X0 cos2 X0 3=2
þ
P3d ¼
S20 T 0 pffiffiffiffiffiffi 4e0 K 0 R0 D0 T 0 sin i0 sin X0 cos X0 S30
B0 D0 sin 2X0 S20
þ
ð1:23Þ
þ
S20 T 0
2K 20 R0 D0 sin4 i0 sin X0 cos3 X S20 T 20
,
S20
,
2K 0 R0 G0 cos i0 cos X0
!
S30 T 0
3
35
ð1:24Þ 2K 0 R0 D0 sin3 i0 sin X0 cos2 X0 pffiffiffiffiffiffi S20 T 0
2R0 cos i0 sin i0 ðX 0 cos X0 K 0 D0 sin X0 Þ pffiffiffiffiffiffi S20 T 0 ! 4e0 K 0 R0 D0 sin i0 cos i0 cos X0 sin X0 p ffiffiffiffiffi ffi , S30 T 0
e0 B0 D0 sin 2X0 cos X0 þ 2e0 R0 D0 cos i0 sin X0
ð1:25Þ
!
S30
:
ð1:26Þ
Using the disturbing function (cf. Eq. (5)) in Eqs. (7) and (12), and taking the indicated derivatives @F/@M and @F/@o in the second Lagrange orbital equation, and substituting for the initial values of the orbital element in the derived expressions, the following expressions indicated by the Greek letters in the first order derived equation, are given below where u0 ¼ o0 þf0: E1d ¼
2K 0 R0 T 0 X 0 þ 2J 0 K 0 U 0 sin X0 K 0 B0 D0 sin 2X0 S20 þ
2e0 R0 T 0 D0 K 20 sin X0 S30
þ
2K 0 R0 G0 sin i0 cos X0 S20 T 0
!
2e0 B0 D0 K 20 sin X0 cos2 X S30 T 0
þ
2K 20 R0 D0 sin4 i0 sin X0 cos3 X0 S20 T 20
,
ð1:27Þ
W 0 ¼ sin i0 ðX 0 cos X0 K 0 D0 sin X0 Þ, E2d ¼
E3d ¼
ð1:28Þ
pffiffiffiffiffiffi pffiffiffiffiffiffi 2R0 X 0 T 0 sin i0 cos X0 2R0 K 0 D0 T 0 sin i0 sin X0
2R0 W 0 cos i 2R0 K 0 D0 sin3 i0 sin X0 cos2 X0 pffiffiffiffiffiffi pffiffiffiffiffiffi S20 T 0 S20 T 0 ! pffiffiffiffiffiffi 2R0 K 0 D0 cos i0 sin3 i0 sin X0 cos2 X0 2e0 R0 K 0 D0 T 0 sin i0 sin 2X0 2e0 R0 K 0 D0 cos 2i0 sin X0 cos X0 þ p ffiffiffiffiffi ffi , 3=2 S30 S30 T 0 S20 T 0 S20
B0 D0 sin 2X0 S20
þ
2e0 B0 D0 sin X0 cos2 X0 þ 2e0 R0 D0 cos2 i sin X0 S30
ð1:29Þ
! :
ð1:30Þ
Using the disturbing function (Eq. (13)) in Eqs. (7) and (12), and taking the indicated derivatives @F/@e and @F/@i in the third Lagrange orbital equation, and substituting for the initial values of the orbital element in the derived expressions, the following expressions indicated by the Greek letters in the first order derived equation, are given below where u0 ¼ o0 þf0 where: ! R0 K 20 sin2i0 cos2 X0 K 20 R0 sin 2i0 sin2 X0 R0 K 20 sin 2i0 sin2 i0 sin2 X0 D1d ¼ þ , ð1:31Þ S20 T S20 S20 T 20 2
D2d ¼
D3d ¼
D4d ¼
!
pffiffiffiffiffiffi
2B0 K 0 cos X0 2K 0 R0 cos2 i0 cos X0 2B0 K 0 cos i0 cos X0 sin X0 2K 0 R0 T 0 cos i0 cos X0 2B0 K 0 cos2 i cos X0 sin2 X0 pffiffiffiffiffiffi , 3=2 S20 S20 T 0 S20 T 0
R0 sin 2i0 þ R0 sin 2i0 cos2 X0 S20
2D0 R0 T 0 K 20 S30
ð1:32Þ
! ,
ð1:33Þ
4C 0 T 0 K 20 þ 2E0 K 0 R0 T 0 þ 2J0 K 0 N 0 sin X0 A0 B0 K 20 sin 2X0
2F 0 K 0 R0 sin i0 cos X0 4C 0 K 20 sin2 i0 cos2 X0 S20 T 0
S20 þ
2A0 R0 K 20 sin4 i0 sin X0 cos3 X0 S20 T 20
2B0 D0 K 20 cos2 X0 ! ,
S30 T 0 ð1:34Þ
36
I. Haranas, S. Pagiatakis / Acta Astronautica 72 (2012) 27–37
D5d ¼
D6d ¼
2R0 L0 cos i0 2A0 K 0 R0 sin3 i0 sin X0 cos2 X0 þ4C 0 K 0 sin 2i0 cos X0 pffiffiffiffiffiffi S20 T 0 pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 8C 0 K 0 T 0 sin i0 cos X0 2A0 K 0 R0 T 0 sin i0 sin X0 þ 2E0 R0 T 0 sin i0 cos X0 pffiffiffiffi 4D0 K 0 R0 T sin i0 cos X0 S30
S20
þ
! 2D0 K 0 R0 sin 2i0 cos X0 A0 K 0 R0 sin 2i0 sin2 i0 cos2 X0 p ffiffiffiffiffi ffi , 3=2 S30 T 0 S20 T 0
4C 0 cos2 iA0 B0 sin 2X0 4C 0 sin2 i0 cos2 X0 S20
2B0 D0 cos2 X0 þ2D0 R0 cos2 i0 S30
ð1:35Þ
!! :
ð1:36Þ
Using the disturbing function (Eq. (13)) in Eqs. (7) and (12) and taking the indicated derivatives @F/@o and @F/@O in the forth Lagrange orbital equation and substituting for the initial values of the orbital element in the derived expressions, the following expressions indicated by the Greek letters in the first order derived equation, are given below where u0 ¼ o0 þf0: ! 2J0 K 0 V 0 sin X0 B0 K 20 sin 2X0 B0 K 20 sin 2X0 2e0 K 20 R0 T 0 sin X0 e0 B0 K 20 sin 2X0 cos X0 þ þ R1d ¼ , ð1:37Þ S20 S20 T 0 S30 S30 T 0 R2d ¼
R3d ¼
pffiffiffiffiffiffi T 0 sin i0 sin X0
K 0 R0 sin 2i0 sin X0 K 0 R0 sin3 i0 sin 2X0 cos X0 pffiffiffiffiffiffi S20 S20 T 0 ! pffiffiffiffiffiffi K 0 R0 sin 2i0 sin2 i0 sin X0 cos2 X0 2e0 K 0 R0 T 0 sin i0 sin 2X0 e0 K 0 R0 sin 2i0 sin 2X0 þ pffiffiffiffiffiffi , 3=2 S30 S30 T 0 S20 T 0 2K 0 R0
B0 sin 2X0 S20
þ
þ
e0 B0 sin 2X0 cos X0 þ 2R0 cos2 i0 sin X0 S30
ð1:38Þ
! :
ð1:39Þ
Using the disturbing function (Eq. (13)) in Eqs. (7) and (12) and taking the indicated derivatives @F/@i and in the fifth Lagrange orbital equation, and substituting for the initial values of the orbital element in the derived expressions, the following expressions indicated by the Greek letters in the first order derived equation, are given below where u0 ¼ o0 þf0: ! R0 K 20 sin 2i0 sin2 X0 R0 K 20 sin 2i0 cos2 X0 R0 K 20 sin 2i0 sin2 i0 sin2 X0 cos2 X0 þ þ F 1d ¼ , ð1:40Þ S20 S20 T 0 S20 T 20 F 2d ¼
2K 0 R0 cos X0 2K 0 R0 cos2 i0 cos X0 B0 K 0 cos i0 sin X0 sin 2X0 2K 0 R0 pffiffiffiffiffiffi þ S20 T 0
pffiffiffiffiffiffi T 0 cos i0 cos X0 S20
B0 K 0 cos2 i0 sin X0 sin 2X0 3=2
S20 T 0
! ,
ð1:41Þ F 3d ¼
R0 sin 2i0 cos2 X0 R0 sin 2i0 S20
! :
ð1:42Þ
Using the disturbing function (Eq. (13)) in Eqs. (7) and (12) and taking the indicated derivatives @F/@e and in @F/@a the sixth Lagrange orbital equation, and substituting for the initial values of the orbital element in the derived expressions, the following expressions indicated by the Greek letters in the first order derived equation, are given below where u0 ¼ o0 þf0: C 1d ¼
C 2d ¼
S20
2F 0 K 0 R0 sin i0 cos X0 4C 0 K 0 sin2 i0 cos2 X0 S20 T 0
2J 0 N 0 K 0 sin X0 A0 B0 K 20 sin 2X0 þ 2E0 K 0 R0 T 0 4C 0 T 0 K 20
8C 0 K 0
þ
2D0 R0 T 0 K 20 S30
2B0 D0 K 20 cos2 X0
A0 K 20 R0 sin4 i0 sin 2X0 cos2 X0
!
S20 T 20
S30 T 0
,
ð1:43Þ
pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi T 0 sin i0 cos X0 þ2E0 R0 T 0 sin i0 cos X0 2A0 K 0 R0 T 0 sin i0 sin X0 S20
2R0 L0 cosi0 þ 4C 0 K 0 sin2i0 cosX0 A0 K 0 R0 sin3 i0 sin2X0 cosX0 pffiffiffiffiffiffi S20 T 0
2D0 K 0 R0 sin 2i0 cos X0 A0 K 0 R0 sin 2i0 sin2 i0 sin X0 sin2 X0 4K 0 D0 R0 pffiffiffiffiffiffi þ 3=2 S30 T 0 S20 T 0
! pffiffiffiffiffiffi T 0 sin i0 cos X0 , S30
ð1:44Þ
I. Haranas, S. Pagiatakis / Acta Astronautica 72 (2012) 27–37
C 3d ¼
2B0 D0 cos2 X0 2D0 R0 cos2 i0 S30
4C 0 cos2 i0 þ A0 B0 sin 2X0 þ 4C 0 sin2 i0 cos2 X0
S20
Q 0 ¼ a0 ð1e20 Þ2 , C 4d ¼
C 5d ¼
C 6d ¼
2Q 0 T 0 K 20 S20 4K 0 M 0
37
!! ,
ð1:45Þ
ð1:46Þ
þ
2M 0 K 20 sin i0 cos2 X0
þ
S20 T 0
pffiffiffiffiffiffi T 0 cos X0 S20
S40
! 4K 0 M 0 cos i0 cos X0 p ffiffiffiffiffi ffi , S20 T 0
2Q 0 cos2 i0 þ 2M 0 sin i0 cos2 X0 S20
2P0 K 20 sin2 X0
! ,
ð1:47Þ
ð1:48Þ
! :
References [1] R.M.L. Baker, M.W. Makemson, An Introduction to Astrodynamics, Academic Press, 1960, pp. 189. [2] S.W. Bougher, J. Murphy, R.M. Haberle, Dust storm impact on the mars upper atmosphere, Adv. Space Res. 19 (8) (1997) 1255–1260. [3] D. Brouwer, G.M. Clemence, Celestial Mechanics, Academic Press, 1961, pp. 299. [4] G.A. Chebotarev, Analytical and Numerical Methods of Celestial Mechanics, American Elsevier 6. Publishing Company, 1967, pp. 148. [5] V. Chobotov, Orbital Mechanics, American Institute of Aeronautics and Astronautics, 2002. [6] C. Dickinson, L. Komguema, J.A. Whiteway, M. Illnicki, V. Popovici, W. Junkermann, P. Connolly, J. Hacker, Lidar atmospheric measurements on mars and earth, Planet. Space Sci. doi:10.1016/ j.pss.2010.03.004. [7] H. Goldstein, Classical Mechanics, Addison-Wesley, 1981, pp. 24. [8] R. Greely, Aeolian modification of planetary surfaces. The comparative study of the planets, in: Proceedings of the NATO Advanced Study Institute, Italy, September 14–25, 1981, pp. 423. [9] F.T. Greyling, H.R. Westerman, Introduction to Orbital Mechanics, Addison-Wesley, 1971, pp. 175. [10] M. Humi, T. Carter, Models of motion in a central force field with quadratic drag, Celest. Mech. Dyn. Astron. 84 (2002) 245–262. [11] M. Humi, T. Carter, Functional Relations Between Time and True Anomaly for Near-circular Orbits with Quadratic Drag, August 2007. [12] N.M. Hoekzema, W.J. Markievic, K. Gwinner, J.A. Meima, H.U. Keller, A. Inada, H. Hoffmann, H. Neukum, HRSC Team, American Astronomical Society, DPS Meeting #37, #33, #36, Bulletin of the American Astronomical Society, vol. 37, 2005, pp. 1567. [13] F. Jaquin, P. Gierasch, R. Khan, The vertical structure of limb hazes in the Martian atmosphere, Icarus 68 (1986) 442–461. [14] W.M. Kaula, Celestial Geodesy, NASA Technical Note D-1155, Washington, 1962, pp. 7. [15] W.M. Kaula, Satellite Geodesy, Dover Publications Inc., 2000, pp. 29. [16] D.G. King-Hele, D.W. Scott, The effects of atmospheric rotation on a satellite orbit, when scale height varies with height, Planet. Space Sci. 17 (1969) 217. [17] B.I. Kosarev, T.V. Losseva, I.V. Nemtchinov, V.V. Shuvalov, R. Greely, Atmospheric disturbances and radiation impulses caused by largemeteoroid impacts on the surface of mars. Formation and evolution of dust cloud, Solar Syst. Res. 36 (3) (2002) 175–192. [18] P.G.L. Leach, The first integral and orbit equation for the Kepler equation with drag, J. Phys. A 20 (1987) 1997–2002. [19] J.A. Magalha~ es, J.T. Schofield, A. Seiff, Results of the mars pathfinder atmospheric structure investigation, J. Geophys. Res. 104 (1999) 8943–8955. [20] L.J. Martin, R.W. Zurek, An analysis of the history of dust activity on mars, J. Geophys. Res. 98 (1993) 3221–3246. [21] T.Z. Martin, M.I. Richardson, New dust opacity mapping from viking infrared thermal mapper data, J. Geophys. Res. 98 (10) (1993) (941-10, 949).
ð1:49Þ
[22] T.M. Martin, Mass of dust in the Martian atmosphere, J. Geophys. Res. 100 (E4) (1995) 7509–7512. [23] D.D. Mazanek, R.R. Kumar, M. Qu, H. Seywald, Aerothermal Analysis and Design of the Gravity Recovery and Climate Experiment (GRACE) Spacecraft, NASA/TM-2000-210095, 2001. [24] E. Mazarico, M. Zuber, Atmospheric density during the aerobraking of mars odyssey from radio tracking data, J. Spacecraft Rockets 44 (6) (2007) 1165–1171. [25] S.M. Metzger, J.R. Carr, T.J. Johnson, M.T. Lemmon, Dust devil vortices seen by the mars pathfinder camera, Geophys. Res. Lett. 26 (1999) 2781. [26] S.M. Metzger, Recent advances in understanding dust devil processes and sediment flux on earth and mars, Lunar Planet. Sci. XXXII (2001). [27] D. Mittleman, D. Jelewski, An analytic solution to the class of 2-body motion with drag, Cell Mech. 28 (1982) 401–413. [28] O. Montenbruck, E. Gill, Satellite Orbits, Models, Methods, Applications, Springer, 2001, pp. 85. [29] D.G. Parkyn, Elliptic orbits in a frictional atmosphere, Am. J. Phys. 26 (1985) 436. [30] A.E. Roy, Orbital Motion, Adam Hilger, 1988, pp. 324. [31] M. Rutkowska, J.B. Zielinski, Preliminary results of orbit estimation for GRACE-A and B, in: Proceedings of the Thirteenth International Workshop on Laser Ranging, Washington DC, 7–11 October 2002. [32] D.P. Rubincam, On the secular decrease in the semimajor axis of Lageos’s orbit, Cell Mech. 26 (1982) 361–382. [33] R., M. Santilli, Foundations of Theoretical Mechanics I, The Inverse Problem in Newtonian Mechanics, Springer-Verlag, 1978, pp. 231. [34] G. Seeber, Satellite Geodesy, Springer, 1989, pp. 421. [35] T.F. Spohn, Sohl, D. Breuer, Mars, Astron. Astrophys. Rev. 8 (1998) 181–235. [36] G.B. Rybicki, A.P. Lightman, Radiative Processes in Astrophysics, John Wiley and Sons, 1979, pp. 9. ˇ [37] S. Segan, Analytical computation of drag effects, Celest. Mech. 41 (1987) 381–388. [38] A.E. Smith, Mars: The Next Step, Adam Hilger, 1989, pp. 22. [39] M.D. Smith, J.C. Pearl, B.J. Conrath, P.R. Christiensen, Mars global surveyor thermal emission spectrometer (TES) observations of dust opacity during aerobraking and science phasing, J. Geophys. Res. 105 (2000) 9539–9552. [40] D.A. Wells, Lagrangian Dynamics, Schaum’s Outlines, McGraw Hill, 1967, pp. 105. [41] James Whiteway, Michael Daly, Allan Carswell, Thomas Duck, Cameron Dickinson, Leonce Komguem, Clive Cook, Lidar on the phoenix mission to mars, J. Geophys. Res. 113 (2008) E00A08, doi:10.1029/2007JE003002. [42] R.W. Zurek, Martian great dust storms: an update, Icarus 50 (1982) 288–310. [43] L. Montabone, S.R. Lewis, P.L. Read, Interannual variability of Martian dust storms in assimilation of several years of Mars global surveyor observations, Adv. Space Res. 36 (11) (2005) 2146–2155.
Contents lists available at SciVerse ScienceDirect
Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Satellite orbit perturbations in a dusty Martian atmosphere Ioannis Haranas a,n, Spiros Pagiatakis b a b
Department of Physics and Astronomy, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3 Department of Earth & Space Science & Engineering, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3
a r t i c l e i n f o
abstract
Article history: Received 25 January 2011 Received in revised form 2 September 2011 Accepted 12 September 2011 Available online 4 October 2011
In this paper we calculate the effect of atmospheric dust on the orbital elements of a satellite. Dust storms that originate in the Martian surface may evolve into global storms in the atmosphere that can last for months can affect low orbiter and lander missions. We model the dust as a velocity-square depended drag force acting on a satellite and we derive an appropriate disturbing function that accounts for the effect of dust on the orbit, using a Lagrangean formulation. A first-order perturbation solution of Lagrange’s planetary equations of motion indicates that for a local dust storm cloud that has a possible density of 8.323 10 10 kg m 3 at an altitude of 100 km affects the orbital semimajor axis of a 1000 kg satellite up 0.142 m day 1. Regional dust storms of the same density may affect the semimajor axis up to of 0.418 m day 1. Other orbital elements are also affected but to a lesser extent. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Lagrange’s equations Gravity field Mars Mars atmosphere Dust storms
1. Introduction There is strong evidence that dust and variable amounts of aerosols are fundamental components of the Martian atmosphere [13,43]. In addition, heating of the dust by absorption of visible radiation strongly affects the dynamical and thermal state of its atmosphere. Martin and Zurek [20] demonstrate such variabilities that are of inter-annual nature. In certain years, this dust cycle manifests itself into dust storms in form of small size dust devil vortices that can spread all over the Martian space. During Martian spring, local or regional dust storms take place in the northern and southern hemisphere. Planet encircling dust storms occur about perihelion time just before the southern summer solstice during which solar heating is maximum, although such storms do not occur every year [20]. Global dust storms can probably be classified among the most important meteorological phenomena; in approximately two weeks, a small localized storm can expand to cover the whole planet
n
Corresponding author. Tel.: þ1 416 932 3155. E-mail address: [email protected] (I. Haranas).
0094-5765/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2011.09.005
[42]. Depending on the particle size, Mars atmospheric dust can remain suspended in the atmosphere for months [21]. After the storm sets, the atmospheric dust can be at high levels for months [21]. For example, it takes dust particles of diameter d E1 mm (in this paper we study particles with diameter dE1.25 mm) 1–2 months to settle when falling from h¼10 km. In November 1971, during its arrival in Mars, Mariner 9 experienced the effects of a major global dust storm that lasted several weeks and the predictions of Martian eolian processes were verified. After the dust cleared, Mariner 9 cameras revealed abundant features attributed to eolian activity including dunes, various pits, etc. [8]. The dust clouds of this particular dust storm that Mariner 9 experienced upon its arrival, reached an altitude of 70 km [38], something that was determined by analyzing Martian limb pictures. In 1993, Mars Global Surveyor mapped the temperature and opacity of the atmosphere using a thermal emission spectrometer. As a result of that dust storm, the density of the atmosphere and therefore the orbit of the spacecraft were significantly affected. The November 1971 dust storm resulted in an atmospheric mass density change at 110 km altitude by at least a factor of 10 [2]. Finally, dust storm heating constitutes
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a low frequency unpredictable perturbation to the entire Martian atmosphere [2]. New studies of the Martian atmosphere have been possible with the use of large data sets acquired from spacecrafts such as, data from the thermal emission spectrometer (TES) aboard Mars Global Surveyor (MGS) [39]. TES instruments measure dust opacity, also called absorption coefficient an, which expresses the loss of intensity of a light beam as it travels through the dust, and by convention, it is positive when energy is taken away from the beam [36]. The motivation for this paper is simple. Our goal is not to engage in any precise atmospheric computations in order to accurately predict the Martian thermospheric densities, nor to predict the effect of the dust on the Martian thermosphere. Instead we follow the work of Martin [22] in order derive an upper estimate for the dust atmospheric density at the orbital altitude of 100 km We also calculate the dust drag coefficient, in order to study the effect of Mars atmospheric dust drag on the satellite. Using a formula for the atmospheric density as given in [22], we calculate that certain dust storms can produce dust atmospheric densities at 100 km that are two orders less than the magnitude with those of Martian atmosphere. For example, we use the peak of the 1977b storm, which produced an estimated suspended mass of 4.3 1011 kg actually produced an equivalent of 4.3 10–3 kg m 2 or a layer of 1.4 mm thickness [22]. Next we distribute the dust in a band located between the latitudes of f ¼ 7601 in order to obtain an upper bound for the dust density, which at the altitude of 100 km gives a dust atmospheric density rd ¼8.323 10 10 kg m 3. This density is 84 times smaller than the Martian atmospheric density at the altitude of 100 km given in Magalha~ es et al. [19] namely ra ¼ 7.03 10 8 kg m 3. Even though storms like that one are not very common on Mars we reckon that dust after all might be an important perturbation that should be taken into account in orbital calculations. 2. Satellite motion in Martian atmospheric dust As discussed in the introduction, dust particles from large-scale sandstorms can reach altitudes of 50 km or higher [35]. Jaquin et al. [13] also report that in the 1977a storm a continuous haze extended to 50 km where a detached haze appeared from 50–90 km. Recent models also predict, that meteoroid impacts on the surface of Mars can produce and rise dust particles up to different altitudes depending on the size of the impactor [17]. To derive the dust disturbing potential Rd to be included in the equations of motion of a satellite, we consider a Mars orbiter, with its position being a function of time t in an areocentric, body-fixed coordinate system [r(t), y(t), l(t)]. Further, we assume that dust particles of certain size and known density rd(r) that in general a function of the distance r from the Martian center exist along the satellite orbit. Next, let us assume that the dust fluid exerts a force opposite to the satellite’s orbital path. The result of this force is acceleration onto the satellite that can be described in a similar way to that of the atmospheric drag as a velocity-square depended force according to the formula [30]. The justification of our assumption
emanates from recent Phoenix lander data. With reference to atmospheric optics the backscatter signal amplitude from dust in the atmosphere on Mars is quite similar to the lidar-signal from molecules in the atmosphere of Earth, when comparing, figure 4 for the Earth in [41] with figure 1 in [6] for Mars. On the basis of the two graphs we propose a velocity square force. Also with reference to Rubincam [32], the author uses a velocity square force to describe the semimajor axis time rate of change due to the interplanetary dust drag force exerted on the LAGEOS satellite. ad ¼
1 As C d rd v2s ; 2ms
ð1Þ
and Bds ¼ms/AsCd represents the satellite dust ballistic coefficient that also describes the susceptibility of the satellite to dust drag forces, a higher B will imply less dust drag effect onto the satellite, and As is the frontal area of the satellite, Cd is the satellite dust drag coefficient, rd is the dust atmospheric density at the orbital altitude and vs is the magnitude of the relative velocity of the satellite with respect to the atmosphere. Since, force Fd acting by the dust ‘‘fluid’’ onto the satellite is velocity-square depended, it is of the general form F d ¼ K i vni [40,33] where Ki is a constant of proportionality, vi is the magnitude of the velocity vector, and n is an index taking different values for different types of forces such as, n ¼0 for dry friction or n ¼1 for viscous drag. If the non-conservative force is of the above general form then the dissipation function (Rayleigh’s dissipation function) due to Mars atmospheric dust can be written as follows [40]: Rd ¼
s X Ki vn þ 1 : n þ1 i i¼1
ð2Þ
Assuming a velocity-square depended dust drag force, then n ¼2, and Rd ¼ ðK 1 =3Þv31 (cf. Eq. (2); see also [14]). Since the dust drag has been assumed to be mathematically equivalent to that of the atmospheric drag, we can equate the force per unit mass due to atmospheric drag (F a ¼ ðAS C r=2ms Þv2s ) [30] to that of dust drag per unit mass, and by analogy obtain the corresponding dust drag constant that multiplies the velocity square term to be (ASCd/2ms)rd. Here the dust density rd is assumed to have a constant value at the orbital altitude of 100 km. Next, with the help of Eq. (2) above we can write the dust disturbing (dissipation) function in the areocentric, bodyfixed coordinate system as follows: As C d rd ! ! Rd ðr, y, l, r_ , y_ , l_ Þ ¼ ðvs 9 o M r 9Þ3 ; 6ms
ð3Þ
which represents the satellite dissipated energy density per ! unit time due to its interaction with the dust. o M is the ! angular velocity vector of Mars and r is the position vector of the satellite. Here we consider that the dust co-rotates ! ! ! with the Martian atmosphere with velocity v d ¼ o M r [28]. If the satellite velocity relative to the dust is described ! ! ! ! according to v r ¼ v s o M r , maximum observed deviations from this assumption are of the order of 40% leading to uncertainties in the drag force of less than 5% [28].
I. Haranas, S. Pagiatakis / Acta Astronautica 72 (2012) 27–37
In the last few decades different authors have investigated the problem of atmospheric drag (e.g., [29,16,27,18,37]) with a force that was quadratic in the magnitude of the velocity, but analytical solutions have not been obtained except for very special cases (e.g., [10,11] personal communication). The work on quadratic atmospheric drag found in the literature has been presented in closed form solutions that deal with the two-body case and can be classified in two categories: (a) use of simplified orbital equations, where the motion is mostly tangential, and (b) drag is treated using the method of the variation of the orbital elements and the solutions are given in terms of the six orbital elements using Gauss’s planetary equations, which describe the rates of change of the orbital elements as functions of the components of the disturbing forces. More details on these two approaches can be found in the papers cited above. However, our approach in the study of the dust drag is that of the cube velocity dissipation function given in Eq. (3) above. In this contribution we follow the Lagrangean approach; we use the disturbing function given by Eq. (3) that expresses energy density dissipation per unit time. This allows us to write the planetary equations as functions of disturbing potentials; these equations are the Lagrangean form of Gauss’s planetary equations. The application of Lagrange’s equations is widespread and the calculations required to obtain the time rates of change of the orbital elements are generally straightforward, and simple to interpret [9]. Lagrange’s equations can be preferably used in orbiting bodies of low eccentricities [3,34] (a satellite with an eccentricity e¼0.01 is considered in this paper—see below). It should also be noted that Lagrange’s equations are rigorous, originally derived for a perturbation exerted by one planet to another, and they also hold true when the disturbing function is due to many other causes thus, widening the spectrum of applications and for a wide variety of phenomena. The analytical form of the disturbing function depends upon the acting force(s). This provides the advantage that we only really need to define the appropriate disturbing function that corresponds to the phenomenon under study (here the dust drag force). Furthermore, Lagrange’s equations are the simplest means of obtaining the first approximation effect of the disturbing function on the orbital elements [14], which is also our main goal in this paper. However, Lagrange’s equations are less suitable for numerical treatment [34], which also justifies the first order perturbation method used in this paper (Section 4). 3. The dust disturbing function
29
Following Kaula [15], we can now simply define the dust disturbing function Rd to be equal to Rayleigh’s dissipation function (cf. Eq. (3)). The dust disturbing function Rd, and more precisely its derivatives with respect to the orbital elements, can then be substituted in Eqs. (7)–(12) (see below). If vr is the satellite’s orbital velocity relative to the dust and if it takes the satellite a time t¼Ld/vr ¼ xd/vr to travel through a large dust cloud of angular size x (radians) that is situated at distance d from the center of Mars (d ¼RM þh) (Fig. 1) then, using Eq. (3) we derive the disturbing function Rd that represents the total energy density dissipated during the passage of the satellite through the dust cloud Rd ¼
C d AS rd dx ! ! ðvs 9 o M r 9Þ2 : 6ms
Further, we assume that all equatorial areocentric ! ! components of v s and also r in Eq. (4) are functions of time t but are not indicated for simplicity. Given Eq. (4), we can write the disturbing function Rd as a function of the orbital elements. Following Kaula’s [15] known standard transformations and rewriting sin f ¼ sin i sin (o þf), or cos y ¼sin(o þf) sin i and also using that sin2 y ¼ 1 sin2 i sin2(o þf) we substitute in Eq. (4). All these ! substitutions take place after we express v s in equatorial areocentric components, perform the cross product and ! ! obtain a final expression for the ðvs 9 o M r 9Þ2 term. We also substitute r, y, l, and r_ , y_ , l_ with their transformed equivalents that contain the orbital elements and their time derivatives. Finally, we also use the relation l ¼ O cos i þ o þ f [1], where o is the argument of the periareon, f is the true anomaly of the satellite, and i is the inclination of the orbit. The final form of Rd (as a function of the orbital elements) becomes Rd ¼
As C d rd dx Fd ; 6ms
ð5Þ
where Fd ¼ Fs Frot 2" #2 _ ð1e2 Þa2ae e_ að1e2 Þðe_ cos ueu_ sin uÞ 1 þe cos u ð1 þe cos uÞ2
¼4
þ
a2 ð1e2 Þ2 ði_ cos i sinuu_ sin i cos uÞ2 ð1 þe cos uÞ2 ð1sin2 i sin2 uÞ
Satellite Ldust RS
The Lagrange’s planetary equations give the time rates of change of the six orbital elements in terms of the partial derivatives of the disturbing function Rd with respect to the orbital elements. In general, the disturbing function Rd comprises all potential field terms except the central one [15] and expresses energy density per unit time. It is only the disturbing function Rd that contributes to the orbit perturbations and its partial derivatives with respect to the (osculating) orbital elements that appear in the right hand side of the Lagrange’s planetary equations.
ð4Þ
ξdust
Satellite's orbit
Rd Dust
MARS
Fig. 1. This figure demonstrates the idealized scenario of a cylindrical type satellite entering an episodic dust cloud.
30
I. Haranas, S. Pagiatakis / Acta Astronautica 72 (2012) 27–37
þ
_ cos i_i O siniÞ2 a2 ð1e2 Þ2 ð1sin2 i sin2 uÞðu_ þ O
#
where qk and q_ k are the generalized coordinates and velocities, respectively.
2
ð1 þe cos uÞ
" 2oM a2 ð1e2 Þ2 ðu_ cos u sini þ _i cos i sin uÞcos i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ð1 þe cos uÞ2 1sin i sin u
o
2 2 2 2 2 2 M a ð1e Þ cos i 2
ð1 þ ecos uÞ
4. Lagrange’s equations of motion and first order perturbation solution
2 2 2 2 2 2 M a ð1e Þ sin i cos u 2
o
ð1þ e cos uÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# _ cos ii_ sin iOÞ 1sin2 i sin2 u 2oM a2 ð1e2 Þ2 sin i cos uðu_ þ O ; ð1þ e cos uÞ2
ð6Þ and u ¼ o þf. Eq. (5) describes the total energy loss of the satellite per unit mass due to dust and all the orbital elements and their derivatives in Eq. (6) have been assumed to be functions of time but not indicated for simplicity. At this point, we must clarify that the presence of the derivatives of the orbital elements in Eq. (6) is due to the fact that the components of the satellite velocity containing r_ , y_ , l_ and their relations to the orbital elements. However, in a first order perturbation treatment after taking the derivatives with respect to the orbital elements in the RHS of these equations, we substitute for the initial values of all the orbital elements taking into account that a_ q ð0Þ ¼ a€ q ð0Þ ¼ 0, q¼ 1,2, y, 6. Given Rd by Eq. (5), we can write Lagrange’s planetary equations of the satellite in the following form [15]: da 2 @Rd ¼ , dt na @M
ð7Þ
pffiffiffiffiffiffiffiffiffiffiffiffi de ð1e2 Þ @Rd 1e2 @Rd ¼ , dt ne a2 @M ne a2 @o
ð8Þ
pffiffiffiffiffiffiffiffiffiffiffiffi do cos i @Rd 1e2 @Rd pffiffiffiffiffiffiffiffiffiffiffiffi ¼ , dt na2 e @e na2 1e2 sin i @i
ð9Þ
In general, Lagrange’s equations of planetary motion (Eqs. (7)–(12)), cannot be solved analytically, but only approximately by successive approximations or by numerical integration. The method of successive approximations to first order is identical to that of perturbation treatment and makes possible to write the osculating orbital elements of the satellite in form of a power series of a small parameter [4] that includes the orbital elements. For instance, we write as follows only the equations for a and e; other equations are similar: a ¼ a0 þ D1 a0 þ D2 a0 þ þ Dn a0 þ e ¼ e0 þ D1 e0 þ D2 e0 þ þ Dn e0 þ :
ð14Þ
Quantities Dna, Dne, etc., are called nth order perturbations. As a first approximation in Lagrange’s Eqs. (7)–(12), we set the disturbing function Rd ¼ 0, and as consequence, the equations imply that a ¼a0, e¼e0, i¼i0, y, M¼M0 þ _ 0 ¼o € 0 ¼ 0, n0t, and a_ 0 ¼ a€ 0 ¼ 0, e_ 0 ¼ e€ 0 ¼ 0, i_0 ¼ €i 0 ¼ 0, o _0 ¼O € 0 ¼ 0, where subscript ‘‘0’’ indicates initial (undisO turbed) values. Next, we use Rd in the right hand side of Lagrange’s planetary equations, we take the total derivatives with respect to the osculating orbital elements, and then we substitute for the initial orbital element values and also for their corresponding values of the first and second derivatives as indicated above, in order to obtain the first-order perturbation solution for the time rates of the orbital elements of the satellite (Appendix A). We start with Eq. (7), which becomes da 2 @Rd 2K 0 @Fd ¼ ¼ , dt na @M na @M
di cos i @Rd 1 @Rd pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi ¼ , dt na2 1e2 sin i @o na2 1e2 sin i @O
ð10Þ
dO 1 @Rd pffiffiffiffiffiffiffiffiffiffiffiffi ¼ , dt na2 1e2 sin i @i
ð11Þ
where K0 ¼AsCdrddx/6ms. In order to differentiate Eq. (7) with respect to the mean anomaly M, we express the true anomaly f and its time derivative f_ as a function of M using the following series expansion [4] up to order e2
dM ð1e2 Þ @Rd 2 @Rd ¼ n : dt ne a2 @e na @a
ð12Þ
f ffi M þ 2e sin M þ
These are the equations of motion of the satellite also given by [15], in which the left-hand-sides describe the time variability of the Keplerian orbital elements (osculating elements) a, e, o, i, O and M, in terms of Rd, which in our case is the dust dissipation function, and n is the mean motion of the satellite in s 1 (e.g., [15]). It is also possible to study the motion of a body (i.e., satellite) in the presence of dissipative forces, using a standard Lagrangean treatment, via Lagrange’s second order differential equations. In the case where velocity dependent dissipative forces are involved we can still derive Lagrange’s equations of motion if the two scalar functions i.e. the Lagrangean L, and also Rayleigh’s dissipation function Rd are known. In that case Lagrange’s equations can be written in the following form [7] d @L @L @R þ ¼ 0, ð13Þ _ dt @q k @qk @q_ k
5 2 e sin 2M þ , 4
ð15Þ
ð16Þ
5 _ _ _ þ 2e_ sinðMÞþ 5 ee_ sinð2MÞ þ 2MecosðMÞ þ e2 M cosð2MÞ, f_ ffi M 2 2
ð17Þ _ ¼ n. At this point we should where M¼n(t t0) [2] and M mention that we do not write explicitly @Fd/@M because it is long and cumbersome. Carrying out similar differentiations for the other orbital elements and substituting their initial values indicated by the subscript ‘‘0’’ we obtain da As C a ra dx ffi ðP þ P 2d oM þ P 3d o2M Þ, dt 3ms n0 a0 1d
ð18Þ
where all quantities given in the derived equations are given in Appendix B and represent the first order perturbation coefficients as functions of the initial values of the orbital elements. Substituting initial numerical values for a polar orbit and for the orbital elements o0 ¼1801, f0 ¼2251, i0 ¼901, O0 ¼2701 we obtain the time rate of
I. Haranas, S. Pagiatakis / Acta Astronautica 72 (2012) 27–37
change of the semi-major axis in m day 1, when the satellite surface to mass ratio, the spatial density of the dust, and the dust drag coefficient are known.q The initial ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi value of mean motion was calculated to be n0 ¼ GM=a30 ¼ 9:891 1024 s1 . For a local dust storm cloud with length Ld ffi1230 km we obtain the following numerical form of the equations: da As C d rd ffi 4:471 103 dt ms 1
: ð19Þ
Similarly, the time rate of change of the eccentricity is qffiffiffiffiffiffiffiffiffiffiffiffi As C d rd dx 1e20 de ffi ðP 1d þ P 2d oM þ P 3d o2M Þ dt 6ms n0 e0 a20 As C d ra dxð1e20 Þ ðE1d þE2d oM þ E3d o2M Þ: 6ms n0 e0 a20
ð20Þ
dO As C d rd dx qffiffiffiffiffiffiffiffiffiffiffiffi csc i0 ðF 1d þ F 2d oM þ F 3d o2M Þ, ffi dt 6m n a2 1e2 s 0 0
ð25Þ
0
which numerically becomes h i dO As C d rd 1 : ¼ 3:304 104 doM deg: day dt ms
ð26Þ
de As C d rd ffi64:117 dð11:907 103 oM dt ms þ 9:360 105 o2M Þ½day
1
:
dM C As rd dx ffi n0 d ðC þ C 2d oM þC 3d o2M Þ dt 3ms n0 a0 1d ð1e20 ÞAs C d rd dx ðC 4d þ C 5d oM þC 6d o2M Þ, 6ms n0 e0 a20
ð27Þ
and the last of Lagrange’s planetary equations numerically becomes dM As C d rd d ¼ 0:75055:866 dt ms ð1661:21oM þ 1:642 105 o2M Þ½rev day
1
Substituting again the initial values of the orbital elements as above, we obtain the following equation:
ð21Þ
Next, we obtain
:
ð28Þ
It’s easy to see that Eqs. (18)–(28) are scaling linear with the ballistic coefficient Bds defined shortly after Eq. (1). The introduction of the ballistic coefficient as a single parameter is used in satellite and spacecraft design that can help scaleout final numerical results. 5. Numerical results
do As C d rd dx cot i0 qffiffiffiffiffiffiffiffiffiffiffiffi ðD1d þ D2d oM þ D3d o2M Þ ffi dt 6m n a2 1e2 s 0 0
þ
For the argument of the ascending node we obtain:
Finally
dð11:907 103 oM þ 9:360 105 o2M Þ½m day
þ
31
0
pAs C d rd dxð1e20 Þ 6ms n0 e0 a20
ðD4d þ D5d oM þ D6d o2M Þ,
ð22Þ
and similarly, do As C d rd ffi 2:0 104 dð16:60 102 oM þ 1:621 dt ms 105 o2M Þ½deg: day
1
:
ð23Þ
Next h i di As C d rd dx qffiffiffiffiffiffiffiffiffiffiffiffi cot i0 ðR1d þ R2d oM þ R3d o2M Þ deg: day1 , ffi dt 2 2 6m0 n0 a0 1e0
ð24Þ and similarly di ¼ 0: dt
ð25Þ
In Table 1 we present first order results for the time variability of the orbital elements. To derive the dust drag coefficient we consider stream lines of dust particles impinging on the frontal surface area of the satellite, where the individual particles within these stream lines are separated by a certain distance s0 and with the assumption that the dust particles do not accumulate on the satellite surface. Considering elastic collisions, between the satellite frontal surface area and the dust and conserving momentum and kinetic energy we obtain the following expression of the dust drag coefficient: 4md Cd ¼ ð29Þ N S ¼ 4:0: As rd s0 This is only the dust drag coefficient when the satellite orbits at a certain orbital altitude through a Martian atmosphere, contaminated by dust a contamination resulting from the development of Martian surface dust storms. This value corresponds to a higher value drag
Table 1 First order perturbation effects for the orbital elements time ratesa of the satellite resulting from a dust cloud at an altitude of 100 km, with density rd ¼8.323 10 10 kg m 3 and Cd ¼4.0 have been considered in our calculation of the orbital element changes. Dust cloud size (km)
da/dt (m day 1)
800
615 1230 1844
0.044 0.177 0.400
1000
615 1230 1844
0.035 0.142 0.320
Satellite mass ms (kg)
a
de/dt (day 1)
do/dt (deg. day 1)
dO/dt (deg. day 1)
dM/dt (rev day 1)
3.800 10 8 1.523 10 7 3.427 10 7
0.218 0.872 0.962
2.674 10 5 1.070 10 4 2.410 10 4
6.100 10 4 2.435 10 3 5.500 10 3
3.046 10 8 1.218 10 7 2.742 10 7
0.174 0.700 1.570
2.140 10 5 8.560 10 5 1.925 10 4
4.900 10 4 1.950 10 3 4.500 10 3
The rate of change of the inclination for the polar satellite used is di/dt¼ 0 and therefore it has been omitted from Table 1.
32
I. Haranas, S. Pagiatakis / Acta Astronautica 72 (2012) 27–37
coefficient when compared to the coefficient of atmospheric drag of a cylinder that is given to be Ca ¼3.0 [5]. Our derived dust drag coefficient depends proportionally to: the total mass of the dust particles impinging on the frontal surface NSmd, which is also equal to As rds0 where NS, is the number of dust particle stream lines, and inversely proportional to: the frontal surface area of the satellite As ¼3.14 m2 the atmospheric dust density rd, the and the dust particle separation s0. The radius of the dust particles is taken to be rd ¼1.25 mm, a value given in [25], and material density rm ¼3000 kg m 3 [25]. Finally we consider dust clouds of different sizes namely Ld ¼dx ¼ 615, 1230, 1844 km deg. (cf. Fig. 2) situated at the same orbital altitude of 100 km. Dust in Martian atmosphere constitutes an important orbital perturbation that depending on the type of the storm and the size of the cloud can in time seriously affect the orbit of the satellite. Dust can also interact, interfere, and also affect the electronics of the satellite. Rovers and landers can also be seriously affected, not to mention man missions in the distant future. The avoidance of possible hazards for the safety of future missions will depend on more detailed studies of the dust properties and size of its particles, abundance and distribution, or even possible electrification under different atmospheric conditions thus, increasing the hazards of future missions. From Viking lander data [26] it is estimated that 800 kg km 2 of dust are lifted daily from the Martian surface. This would imply a total dust mass of Md ¼1.225 1011 kg over the Martian surface under a homogeneous distribution assumption for the dust. Our calculations using first order perturbations in the equations of motion are, summarized in Table 1. The table contains results that correspond to the dust drag coefficient Cd ¼4.0. An upper bound for the atmospheric dust density has been calculated using the 1997b storm [22] and using a dust scaling height hd ¼ 13 km [12]. In this calculation we choose to distribute the dust in a latitude band situated between f ¼ 7601 and considering the possibility of a local dust storm the density could be by a factor of 4 higher [22] we obtain an upper bound for the dust density to be rd(h)¼ 4r0e 100/13 ¼8.323 10 10 kg m 3. Furthermore, the table
Fig. 2. Semimajor axis change for satellites (The second line of each pair represents the ms ¼ 800 kg satellite higher density effect) with masses ms ¼ 800 and 1000 kg orbiting in a 615, 1230 and 1845 km dust clouds of density rd ¼ 8.323.1 10 10 kg m 3.
shows three different calculations of the orbital element perturbations corresponding to the same dust density as mentioned above and for two different satellites with masses ms ¼800 and 1000 kg. The most significant effect is on the semi-major axis of the orbital ellipse as it is intuitively expected. A 1000 kg satellite may lose up to 0.320 m per day in altitude (see last raw of Table 1, for instance). The loss of the satellite altitude also depends linearly on the angular size of the dust cloud and the time that the satellite travels in it. For example in the extreme case scenario of a global Mars dust storm where the dust cloud totally surrounds the planet we calculate that the same 1000 kg satellite can lose up to 46 m d 1. When speaking about lander missions going through violent storms, these 46 m of loss in altitude may become critical for a successful landing. Since, both atmospheric drag and dust drag can be described as velocity square depended forces, the same mathematical modeling that describes dust can be also applied to the atmospheric drag. As a test of our model, we apply the same mathematical formulation to the GRACE-A Earth satellite system. Using the appropriate parameters for GRACE-A, (http://www.csr. utexas.edu/grace/newsletter/archive/august2002.html) i.e. a¼6876.4816 km, h¼498.3416 km, i¼89.02544661, O ¼ 354.447141, e¼0.00040989, o ¼302.41424441, AGRACE ¼ 0.9486 m2, mGRACE ¼487 kg, 1.9rCa,r2.2, the mean motion of the satellite n¼9.90 10–4 rad s 1, and finally the atmospheric density at the orbital altitude of GRACE-A, rA ¼7.0 10 13 kg/m3 [23] we obtain that the time rate of change of the semimajor axis a lies in the range 25.28 m/drda/dtr 29.28 m/d. In particular for CdGRACE ¼2.2 we obtain that da/dtr 29.28 m/d, which is within 98% of the total value predicted by [31] who predict da/dtr 29.60 m/d. Martian atmospheric drag is the perturbation that most strongly influences the motion of a satellite. In fact, during the final revolutions of the orbital life of a satellite, drag effects can be more dominant than those of Martian oblateness. Comparing the two drags, we say that atmospheric drag with no dust content constitutes itself a very important perturbation. In particular, for a polar orbit satellite, which is the satellite considered in this study at the orbital altitude in the range 100–140 km, atmospheric drag is a very significant perturbation and becomes critical over long periods. A Martian polar satellite orbiting at 100 km at the end of 24 h period looses 0.7% of its initial altitude, but there is no change in its inclination. The rotation of the Martian atmosphere introduces an additional 14% increase in the time rates change of the orbital elements of the satellite. The time rate of change of the rest of the orbital elements for a 1000 kg satellite due to atmospheric drag with no 1 _ ¼ 0:048 deg: d1 , dust included is e_ ¼ 0:00073 d , o _ ¼ 0:517 deg: d1 , M _ ¼ 2:973 revd1 . In the case where O the atmosphere contains dust particles of certain size in our case rd ¼1.25 mm, the effect of atmospheric drag will be larger due to the fact that the dust drag effect is now added in the total effect, increasing by 0.005% per day the semimajor axis loss, or 0.15% in a month. Furthermore, dust drag will further increase those orbital elements that atmospheric drag increases the most, namely the semimajor axis, the argument of the periareon, and the mean anomaly,
I. Haranas, S. Pagiatakis / Acta Astronautica 72 (2012) 27–37
by the dust drag amount calculated in Table 1. Therefore, it accelerates the change of an elliptical orbit to circular, and therefore reduces further the satellite’s lifetime, each time the satellite passes through the dust clouds. 6. Conclusions In this paper the effects of dust on low orbiters around Mars were investigated. Because Mars dust might be able to reach up to heights of 100 km or even higher, we anticipate that the study of this particular perturbation will be important for satellite orbits at an altitude of 100 km, not to mention that such an orbital altitude has already been achieved as a periareon altitude by missions such as Mars Odyssey [24]. To model the effects of dust, we considered a velocity-square dependent force that was similar to that of the aerodynamic drag. Subsequently, a disturbing function was developed so that Lagrange’s planetary equations for the time variability of the orbital elements could be studied in a first order perturbation approach. For one estimated dust density and one calculated dust drag coefficient we calculated the effects on the orbital elements of the satellite orbit as a function of time (see Table 1). Our calculations showed that in the first order perturbation, all orbital elements change significantly except the inclination i, with the semimajor axis a, followed by the argument of the periareon o, and the eccentricity e and the mean anomaly M being affected the most and finally the argument of the ascending node O. The orbital element time rates of change
are directly related to the product of the dust density and the dust aerodynamic coefficient and can create a very critical cumulative effect for the satellite if it orbits in dust for long periods of time. Dust on Mars and its atmosphere in particular constitutes an important orbital perturbation that can seriously affect the orbit and therefore have an impact on the satellite’s lifetime. An increased drag effect can result in higher fuel consumption that is also related to the orbital maintenance of the satellite. An increased drag can also accelerate the spacecraft’s degradation affecting its solar cells, external spacecraft materials, and also the spacecraft physical surface morphology, and it is predominant in the velocity vector exposure (ram direction). It is also particularly important for example for low orbit gravimetric satellites for precise Mars gravity models. Dust can also interact, interfere, and also negatively affect its instruments. Rovers and Landers on the surface can also be seriously affected, not to mention man mission in the nearer future.
Acknowledgments This research was financially supported by the GEOIDE National Centre of Excellence (Canada), Phase III funding. The authors want to thank an unknown reviewer who with his/her valuable comments help improved this manuscript considerably.
Appendix A. Summary of all symbols ad ms As CD
rd vs Fd Rd Fa ! oM ! r ! vd ! vr Ld
x d RM a h
f y l
o f
O e
33
dust drag acceleration exerted on the satellite [m/s2] satellite’s mass¼800 and 1000 [kg] satellite’s frontal surface area ¼3.14 [m2] dust drag aerodynamic coefficient¼4.0 dust density at the orbital altitude of the satellite¼8.323 10 10 [kg/m3] satellite’s orbital velocity¼3.508 [km/s] force acting by the dust ‘‘fluid’’ onto the satellite [N] Rayleigh’s dissipation function [kg m2/s3] drag force acting on to the satellite [N] Mars’ angular velocity¼7.0854 10 5 [rad/s] satellite’s position vector [m] dust atmospheric velocity¼5 [m/s] satellite’s velocity relative to the dust [m/s] length of dust cloud 615, 1230 and 1844 [km] dust cloud angular size ¼10, 20 and 30 [1] dust cloud distance from the center of Mars ¼100 [km] Mars’s radius ¼3397.2 [km] satellite’s semimajor axis¼3522.30 [km] orbital altitude of the satellite¼100 [km] satellite’s areocentric latitude¼90 [1] satellite’s areocentric colatitude¼0 [1] satellite’s Areocentric longitude¼0 [1] satellite’s argument of the periareon¼180 [1] satellite’s true anomaly¼225 [1] satellite’s argument of the node ¼270 [1] satellite’s eccentricity¼0.01
34
i M
Fd Fs Fs n L qk q_ k
I. Haranas, S. Pagiatakis / Acta Astronautica 72 (2012) 27–37
satellite’s inclination¼90 [1] satellite’s mean anomaly ¼0.057 [1] satellite’s total disturbing potential [m2/s2] as a function of orbital elements satellite’s non rotational disturbing [m2/s2] potential as a function of orbital elements satellite’s rotational disturbing potential [m2/s2] as a function of orbital elements satellite’s mean motion¼9.9 10–4 [rad/s] Lagrangean function [kg m2/s2] satellite’s generalized coordinate [m] satellite’s generalized velocity [m/s]
Appendix B. First order perturbation coefficients of Lagrange’s equations Using the disturbing function Eq. (5) in Eqs. (7) and (12), and taking the indicated derivative qF/qM in the first Lagrange orbital equation, we then substitute for the initial values of the orbital element in the derived expression, which gives 5e2 X0 ¼ o0 þ n0 t þ2e0 sinðn0 tÞ þ 0 sinð2n0 tÞ , ð1:1Þ 4
D0 ¼ 1 þ 2e0 cosðn0 tÞ þ
5e20 cosð2n0 tÞ , 2
ð1:2Þ
5e2 K 0 ¼ n0 1þ 2e0 cosðn0 tÞ þ 0 cosð2n0 tÞ , 2
ð1:3Þ
T 0 ¼ ð1sin2 i0 sin2 X0 Þ,
ð1:4Þ
5e0 sinð2n0 tÞ , A0 ¼ 2sinðn0 tÞ þ 2
ð1:5Þ
B0 ¼ ða20 ð1e20 Þ2 sin2 i0 Þ,
ð1:6Þ
C 0 ¼ ðe0 a20 ð1e20 ÞÞ,
ð1:7Þ
D0 ¼ ðcos X0 e0 A0 sin X0 Þ,
ð1:8Þ
E0 ¼ n0 ð2cosðn0 tÞ þ 5e0 cosð2n0 tÞÞ,
ð1:9Þ
F 0 ¼ sin i0 ðE0 cos X0 þ A0 K 0 sin X0 Þ,
ð1:10Þ
S0 ¼ ð1 þ e0 cos X0 Þ,
ð1:11Þ
G0 ¼ sin i0 ðE0 cos X0 þ A0 K 0 sin X0 Þ,
ð1:12Þ
H0 ¼ ð1e20 ÞðK 0 sin X0 e0 A0 K 0 cos X0 e0 E0 sin X0 Þ,
ð1:13Þ
L0 ¼ sin i0 ðE0 cos X0 A0 K 0 sin X0 Þ,
ð1:14Þ
a0 H0 2a0 e20 K sin X0 2a0 e0 ð1e20 ÞD0 K 0 N0 ¼ , 2 2 S S
ð1:15Þ
M 0 ¼ ða0 ð1e20 Þsin i0 Þ,
ð1:16Þ
M 0 sin i0 ¼ ða0 ð1e20 Þ2 sin2 i0 Þ,
ð1:17Þ
P 0 ¼ ða0 e20 ð1e20 Þ2 Þ,
ð1:18Þ
R0 ¼ ða20 ð1e20 Þ2 Þ,
ð1:19Þ
X 0 ¼ n0 ð2e0 sinðn0 tÞ þ 5e20 sinð2n0 tÞÞ,
ð1:20Þ
G0 ¼ sin i0 ðX 0 cos X0 þ K 0 D0 sin X0 Þ,
ð1:21Þ
J 0 ¼ e0 a0 ð1e20 Þ,
ð1:22Þ
I. Haranas, S. Pagiatakis / Acta Astronautica 72 (2012) 27–37
U0 ¼
P1d ¼
e2 ð1e20 ÞK 0 D0 sin2 X0 2a 0 0 S30
S20
2K 0 R0 T 0 X 0 þ 2J0 K 0 U 0 sin X0 2B0 D0 K 20 sin X0 cos X0 S20 þ
P2d ¼
! a0 e0 ð1e20 ÞðK 0 D0 cos X0 X 0 sin X0 Þ
2e0 K 20 R0 D0 T 0 sin X0 S30
2R0 X 0
þ
2e0 B0 D0 K 20 sin X0 cos2 X0
pffiffiffiffiffiffi pffiffiffiffiffiffi T 0 sin i0 cos X0 2K 0 R0 D0 T 0 sin i0 sin X0
2K 0 R0 D0 cos i0 sin i0 sin X0 cos2 X0 3=2
þ
P3d ¼
S20 T 0 pffiffiffiffiffiffi 4e0 K 0 R0 D0 T 0 sin i0 sin X0 cos X0 S30
B0 D0 sin 2X0 S20
þ
ð1:23Þ
þ
S20 T 0
2K 20 R0 D0 sin4 i0 sin X0 cos3 X S20 T 20
,
S20
,
2K 0 R0 G0 cos i0 cos X0
!
S30 T 0
3
35
ð1:24Þ 2K 0 R0 D0 sin3 i0 sin X0 cos2 X0 pffiffiffiffiffiffi S20 T 0
2R0 cos i0 sin i0 ðX 0 cos X0 K 0 D0 sin X0 Þ pffiffiffiffiffiffi S20 T 0 ! 4e0 K 0 R0 D0 sin i0 cos i0 cos X0 sin X0 p ffiffiffiffiffi ffi , S30 T 0
e0 B0 D0 sin 2X0 cos X0 þ 2e0 R0 D0 cos i0 sin X0
ð1:25Þ
!
S30
:
ð1:26Þ
Using the disturbing function (cf. Eq. (5)) in Eqs. (7) and (12), and taking the indicated derivatives @F/@M and @F/@o in the second Lagrange orbital equation, and substituting for the initial values of the orbital element in the derived expressions, the following expressions indicated by the Greek letters in the first order derived equation, are given below where u0 ¼ o0 þf0: E1d ¼
2K 0 R0 T 0 X 0 þ 2J 0 K 0 U 0 sin X0 K 0 B0 D0 sin 2X0 S20 þ
2e0 R0 T 0 D0 K 20 sin X0 S30
þ
2K 0 R0 G0 sin i0 cos X0 S20 T 0
!
2e0 B0 D0 K 20 sin X0 cos2 X S30 T 0
þ
2K 20 R0 D0 sin4 i0 sin X0 cos3 X0 S20 T 20
,
ð1:27Þ
W 0 ¼ sin i0 ðX 0 cos X0 K 0 D0 sin X0 Þ, E2d ¼
E3d ¼
ð1:28Þ
pffiffiffiffiffiffi pffiffiffiffiffiffi 2R0 X 0 T 0 sin i0 cos X0 2R0 K 0 D0 T 0 sin i0 sin X0
2R0 W 0 cos i 2R0 K 0 D0 sin3 i0 sin X0 cos2 X0 pffiffiffiffiffiffi pffiffiffiffiffiffi S20 T 0 S20 T 0 ! pffiffiffiffiffiffi 2R0 K 0 D0 cos i0 sin3 i0 sin X0 cos2 X0 2e0 R0 K 0 D0 T 0 sin i0 sin 2X0 2e0 R0 K 0 D0 cos 2i0 sin X0 cos X0 þ p ffiffiffiffiffi ffi , 3=2 S30 S30 T 0 S20 T 0 S20
B0 D0 sin 2X0 S20
þ
2e0 B0 D0 sin X0 cos2 X0 þ 2e0 R0 D0 cos2 i sin X0 S30
ð1:29Þ
! :
ð1:30Þ
Using the disturbing function (Eq. (13)) in Eqs. (7) and (12), and taking the indicated derivatives @F/@e and @F/@i in the third Lagrange orbital equation, and substituting for the initial values of the orbital element in the derived expressions, the following expressions indicated by the Greek letters in the first order derived equation, are given below where u0 ¼ o0 þf0 where: ! R0 K 20 sin2i0 cos2 X0 K 20 R0 sin 2i0 sin2 X0 R0 K 20 sin 2i0 sin2 i0 sin2 X0 D1d ¼ þ , ð1:31Þ S20 T S20 S20 T 20 2
D2d ¼
D3d ¼
D4d ¼
!
pffiffiffiffiffiffi
2B0 K 0 cos X0 2K 0 R0 cos2 i0 cos X0 2B0 K 0 cos i0 cos X0 sin X0 2K 0 R0 T 0 cos i0 cos X0 2B0 K 0 cos2 i cos X0 sin2 X0 pffiffiffiffiffiffi , 3=2 S20 S20 T 0 S20 T 0
R0 sin 2i0 þ R0 sin 2i0 cos2 X0 S20
2D0 R0 T 0 K 20 S30
ð1:32Þ
! ,
ð1:33Þ
4C 0 T 0 K 20 þ 2E0 K 0 R0 T 0 þ 2J0 K 0 N 0 sin X0 A0 B0 K 20 sin 2X0
2F 0 K 0 R0 sin i0 cos X0 4C 0 K 20 sin2 i0 cos2 X0 S20 T 0
S20 þ
2A0 R0 K 20 sin4 i0 sin X0 cos3 X0 S20 T 20
2B0 D0 K 20 cos2 X0 ! ,
S30 T 0 ð1:34Þ
36
I. Haranas, S. Pagiatakis / Acta Astronautica 72 (2012) 27–37
D5d ¼
D6d ¼
2R0 L0 cos i0 2A0 K 0 R0 sin3 i0 sin X0 cos2 X0 þ4C 0 K 0 sin 2i0 cos X0 pffiffiffiffiffiffi S20 T 0 pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 8C 0 K 0 T 0 sin i0 cos X0 2A0 K 0 R0 T 0 sin i0 sin X0 þ 2E0 R0 T 0 sin i0 cos X0 pffiffiffiffi 4D0 K 0 R0 T sin i0 cos X0 S30
S20
þ
! 2D0 K 0 R0 sin 2i0 cos X0 A0 K 0 R0 sin 2i0 sin2 i0 cos2 X0 p ffiffiffiffiffi ffi , 3=2 S30 T 0 S20 T 0
4C 0 cos2 iA0 B0 sin 2X0 4C 0 sin2 i0 cos2 X0 S20
2B0 D0 cos2 X0 þ2D0 R0 cos2 i0 S30
ð1:35Þ
!! :
ð1:36Þ
Using the disturbing function (Eq. (13)) in Eqs. (7) and (12) and taking the indicated derivatives @F/@o and @F/@O in the forth Lagrange orbital equation and substituting for the initial values of the orbital element in the derived expressions, the following expressions indicated by the Greek letters in the first order derived equation, are given below where u0 ¼ o0 þf0: ! 2J0 K 0 V 0 sin X0 B0 K 20 sin 2X0 B0 K 20 sin 2X0 2e0 K 20 R0 T 0 sin X0 e0 B0 K 20 sin 2X0 cos X0 þ þ R1d ¼ , ð1:37Þ S20 S20 T 0 S30 S30 T 0 R2d ¼
R3d ¼
pffiffiffiffiffiffi T 0 sin i0 sin X0
K 0 R0 sin 2i0 sin X0 K 0 R0 sin3 i0 sin 2X0 cos X0 pffiffiffiffiffiffi S20 S20 T 0 ! pffiffiffiffiffiffi K 0 R0 sin 2i0 sin2 i0 sin X0 cos2 X0 2e0 K 0 R0 T 0 sin i0 sin 2X0 e0 K 0 R0 sin 2i0 sin 2X0 þ pffiffiffiffiffiffi , 3=2 S30 S30 T 0 S20 T 0 2K 0 R0
B0 sin 2X0 S20
þ
þ
e0 B0 sin 2X0 cos X0 þ 2R0 cos2 i0 sin X0 S30
ð1:38Þ
! :
ð1:39Þ
Using the disturbing function (Eq. (13)) in Eqs. (7) and (12) and taking the indicated derivatives @F/@i and in the fifth Lagrange orbital equation, and substituting for the initial values of the orbital element in the derived expressions, the following expressions indicated by the Greek letters in the first order derived equation, are given below where u0 ¼ o0 þf0: ! R0 K 20 sin 2i0 sin2 X0 R0 K 20 sin 2i0 cos2 X0 R0 K 20 sin 2i0 sin2 i0 sin2 X0 cos2 X0 þ þ F 1d ¼ , ð1:40Þ S20 S20 T 0 S20 T 20 F 2d ¼
2K 0 R0 cos X0 2K 0 R0 cos2 i0 cos X0 B0 K 0 cos i0 sin X0 sin 2X0 2K 0 R0 pffiffiffiffiffiffi þ S20 T 0
pffiffiffiffiffiffi T 0 cos i0 cos X0 S20
B0 K 0 cos2 i0 sin X0 sin 2X0 3=2
S20 T 0
! ,
ð1:41Þ F 3d ¼
R0 sin 2i0 cos2 X0 R0 sin 2i0 S20
! :
ð1:42Þ
Using the disturbing function (Eq. (13)) in Eqs. (7) and (12) and taking the indicated derivatives @F/@e and in @F/@a the sixth Lagrange orbital equation, and substituting for the initial values of the orbital element in the derived expressions, the following expressions indicated by the Greek letters in the first order derived equation, are given below where u0 ¼ o0 þf0: C 1d ¼
C 2d ¼
S20
2F 0 K 0 R0 sin i0 cos X0 4C 0 K 0 sin2 i0 cos2 X0 S20 T 0
2J 0 N 0 K 0 sin X0 A0 B0 K 20 sin 2X0 þ 2E0 K 0 R0 T 0 4C 0 T 0 K 20
8C 0 K 0
þ
2D0 R0 T 0 K 20 S30
2B0 D0 K 20 cos2 X0
A0 K 20 R0 sin4 i0 sin 2X0 cos2 X0
!
S20 T 20
S30 T 0
,
ð1:43Þ
pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi T 0 sin i0 cos X0 þ2E0 R0 T 0 sin i0 cos X0 2A0 K 0 R0 T 0 sin i0 sin X0 S20
2R0 L0 cosi0 þ 4C 0 K 0 sin2i0 cosX0 A0 K 0 R0 sin3 i0 sin2X0 cosX0 pffiffiffiffiffiffi S20 T 0
2D0 K 0 R0 sin 2i0 cos X0 A0 K 0 R0 sin 2i0 sin2 i0 sin X0 sin2 X0 4K 0 D0 R0 pffiffiffiffiffiffi þ 3=2 S30 T 0 S20 T 0
! pffiffiffiffiffiffi T 0 sin i0 cos X0 , S30
ð1:44Þ
I. Haranas, S. Pagiatakis / Acta Astronautica 72 (2012) 27–37
C 3d ¼
2B0 D0 cos2 X0 2D0 R0 cos2 i0 S30
4C 0 cos2 i0 þ A0 B0 sin 2X0 þ 4C 0 sin2 i0 cos2 X0
S20
Q 0 ¼ a0 ð1e20 Þ2 , C 4d ¼
C 5d ¼
C 6d ¼
2Q 0 T 0 K 20 S20 4K 0 M 0
37
!! ,
ð1:45Þ
ð1:46Þ
þ
2M 0 K 20 sin i0 cos2 X0
þ
S20 T 0
pffiffiffiffiffiffi T 0 cos X0 S20
S40
! 4K 0 M 0 cos i0 cos X0 p ffiffiffiffiffi ffi , S20 T 0
2Q 0 cos2 i0 þ 2M 0 sin i0 cos2 X0 S20
2P0 K 20 sin2 X0
! ,
ð1:47Þ
ð1:48Þ
! :
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