Model of non-gravitational perturbations for CESAR experiment with MACEK accelerometer

Pergamon 0273-I 177(95)00409-2

Adv. Space Res. Vol. 16, No. 12. pp. (12)3-(12)13, 1995 Capyright0 1995 COSPAR Printed in Great Britain. All rights resewed. 0273-I 177I95 $9.50 + 0.00

MODEL OF NON-GRAVITATIONAL PERTURBATIONS FOR CESAR EXPERIMENT WITH MACEK ACCELEROMETER L. Sehnal* and D. Vokrouhlicky** *Astronomical Institute of the Czech Academy of Sciences, 251 65 Ondrejov, Czech Republic ** Institute of Astronomy, Charles University Prague, &dskd 8, IS0 00 Prague 5, Czech Republic

ABSTRACT Non-gravitational perturbations have recently achieved great interest among the theoreticians in satellite dynamics. Both short- and long-term residual phenomena in many space missions are connected with the non-gravitational effects. However, our theoretical understanding of such phenomena is rather limited. In many cases, only rough phenomenological models are available. In situ (instantaneous) measurements of the non-gravitational effects have thus great importance for checking our theoretical concepts. Microaccelerometer MACEK (embodied in the CESAR mission) is devoted to such measurements with expected threshold up to 5 x lo-” m/s2. Model of the non-gravitational phenomena developed for the data analysis of the MACEK device include two principal parts: (i) atmospheric drag and lift model, (ii) radiation pressure model. With the aid of developed theory we assume to be able to fulfil the principal goal of the mission: the determination of the distribution and variations of the thermospheric density. Radiative effect package includes: (i) direct solar radiation pressure, with emphasis of the Earth shadow penumbra transitions; (ii) pressure of the Earth reflected/scattered sunlight (‘the albedo effect’), with special attention to the atmospheric phenomena and finite cone pattern of the specular reflection on the ocean surface; (iii) pressure of the Earth thermal (IR) radiation. 1 INTRODUCTION Project CESAR is a common endeavour of the S.C. Central European Initiative (former “Hexagonale”) countries in the region of the space research /l/. It was initiated at the beginning of 1992 under the leading role of the Italian “Alenia Spazio” enterprise. After considering possible areas of a cooperative effort a decision was made to launch into an elliptical orbit a satellite for atmospheric and ionospheric studies, supplemented by some supporting instruments for solar research. In the mass-center of the satellite body, a high-sensitivity accelerometer will be placed to trace the surface forces of non-gravitational origin /2/_ The satellite has been named “CESAR” as an acronym for “Central European Satellite for Advanced Research”and is now its “Phase A” stage. The satellite should look approximately as shown on Figure 1, with its longitudinal axis oriented towards the Sun, with the accuracy of altitude determination better then 1 deg. As for the radiative effects, we shall in this communication present our ideas concerning special phenomena which are particularly important for the analysis of the long-term effects observed in the LAGEOS-type satellite orbit residuals. Besides of this task, we also plan to achieve ‘a global solution’ for the Earth albedo distribution (and if possible, also for the Earth thermal radiation) from a complementary set of arcs (to those discussed before), which will be free of intricate dynamical effects. Comments on this task will be given elsewhere.

L. Sehnal and D. Vokmhlicky

WM

Fig. 1. Satellite

CESAR

Fig. 2. Drag and lift effective areas

2 SATELLITE

CESAR

The surface of the satellite CESAR can be approximated by a rectangular parallelepiped’with two solar panels and two antennas. The respective surfaces giving rise to drag and lift effects are summarized in Table 1 together with their respective areas and atomic masses. Moreover, the fourth column of Table 1 contains the expressions characterizing the changing effective surfaces during one revolution (the satellite is oriented towards the Sun). TABLE No. 1

2 3 4 5 6

Surface BU BS SF SR

Area

1

AM

time-dep 1cos fq

ss

2t4 3.45 3.45 0.2

ti::; 1cos(8+n/2)1 28.06 (1 - Se)1 cos81 26.98 $91cosel 26.98 1cos(0+ n/2)1

NT

0.15

26.98

I cos el

In the above table, we introduced 8 as the angle between the direction to the Sun and the velocity vector. In the fifth column, we introduce the coefficient Se to distinguish between the front and rear side of the solar panels (having different surface properties), as 5’0 = (1 - sgn8)/2. It is possible to determine 8 by formulas of spherical astronomy as a function kJ = B(cra,60, El,v), where cya, 60 are the coordinates of the Sun, El are the orbital elements and v denotes the orbital true anomaly. The explicit form of the above equations enables us to determine the drag effective cross-section of the satellite at any moment. Figure 2 gives an example of these conditions at a configuration of the mutual position of the Sun and the satellite orbit with R = 280 deg, w = 0, at the date of 1998.0.

3 LIFETIME The choice of the orbit, the schedule of the time-intervals of the instruments’ measurements, the choice of launch as well as the fuel-consumption and many other problems connected with the exploitation of the proposed “CESAR” satellite may be affected by its lifetime. Therefore, we tried to predict its lifetime in detail to enable the choice of the orbit with the highest possible accuracy.

for MACEK Micrmcceleratneler

Per(urb&xw

(IV5

The prediction method is in principle that of analytical determination of the orbital perturbations for a longer time interval (10 days) with the adjustment of a new set of initial orbital elements after the interval in question /4/. The longer length of the time step is enabled by a relatively accurate analytical determination of the drag effects on such a time interval. Within the principal time step, we use a second-order theory of the atmospheric drag effects (in a celestial mechanics sense) and a first-order theory of the Earth oblateness effects. The orbital elements integrated simultaneously are the semi-major axis and the eccentricity; the computation is stopped when the eccentricity drops below zero. The second-order drag theory enables to choose much longer integration steps than usual (several days) without a loss of accuracy. The theory is based on the usage of a special thermospheric total density model TD88 /5/. Our solar flux forecast uses the prediction of the solar activity as revealed by several solar astrophysicists /6/ and suggests a minimum of Flo.7 = 60 at 1.1.1995 and a maximum of F~o.7 = 220 at 30.6.2000. The method and the whole program has been already used and checked on an observed example of a satellite decay /7/. We believe that the accuracy of the predictions presented in this report is better than 5% provided the solar activity prediction is estimated with the same accuracy. The effective area A was taken as one revolution average of the changing instantaneous area as indicated in Figure 2; the initial orbital elements are summarized in Table 2: TABLE Area m2 4.58

mass kg 245

e I;“, 7078

0.0424

effective

2. R deg 0

i deg 70

w deg 0

h(per) km 400

h(apo) km 1000

The time interval over which we computed analyticaHy the simultaneous changes of the eIements a and e was 10 days. The computation lead to the result of 722 days. The changes of the perigee and apogee heights show a rapid decrease of the apogee height after approximately 650 days, from the apogee height of 600 km. The individual changes of the perigee and apogee heights are plotted on Figure 3. This means, the satellite will provide useful data for at least two years.

-

---

pwl@e.

opog..

Fig. 3. Lifetime

0 h

IO

0 - - -

of the CESAR

satellite.

drog

lift

Fig. 4. Drag and Lift

”

(d-d

L. Sehnal and D. Vokmhticky

(W

4 ACCELERATIONS

CAUSED

BY DRAG AND LIFT

The final shape, dimensions and mass are not yet definitely determined, nevertheless, a preliminary determination of lifetime as well as of the disturbing accelerations can be done. The basic formula for the determination of drag and lift accelerations ?D and CL read as follows :

iD,L

=

-

2:,[cCt),LA;TL(eff)] pv’uD,L

-

I

where UD and UL are the unit vectors in the direction of drag and lift force, resp.; the vector UL is given by UL = Uj-J x (ug x ni) m is the satellite’s mass, Ak are the effective areas, V is the velocity with respect to the rotating thermosphere. CD and CL are the drag and lift coefficients, resp. Vector ni is the surface normal to the area ALceff) and the summation in the equation (1) is done over all the areas in question. A difficult problem of the treatment of the dynamical effects of the atmosphere lies in the determination of the drag and lift coefficients, usually considered to be constant. In our theory, we determine these coefficients separately for each flat surface of the satellite body, with respect to the respective thermal accommodation coefficient. Also, the theory considers the angles of incidence of the incoming molecules and their individual atomic masses. The formula

used for the determination

cgi = 2 [l

-

of the drag and lift coefficients

dn

the products

cD

L -_

molecules & depends on the thermal accommoand the angle of reflection given by cos 0, =

CE,)L belonging

values of the coefficients,

areas Afff), we can determine cient according to

:

47 {COS(D)}{sin(L)}(@i t %)I,

where the angle of the dispersion of the reflected dation coefficient by & = cos(O.535185 arcsin cos” O;, where v = l/(1 - 6)2. Using the individual

reads as follows /lo/

k

CD,,

Ck

A@ff)

k

and

C~,LA:'~(~~

to the respective

,

consequently,

effective incident

the mean CDJ

coeffi-

f,

AW(eff).

From the above analysis, it follows that it is necessary to know as precisely as possible the composition of the thermosphere and the number densities of the individual constituents. From that point of view, the most reliable thermospheric model is most probably the CIRA86 /9/ which covers a larger altitudinal region and considers more of the thermospheric constituents than the other models now available. The equations (I)-(3) all ow us to determine the drag and lift effects at any position of the satellite. However, lift consists of two components, one within the orbital plane (2) and the other perpendicular to it (Y). Since the accelerometer placed on board of the CESAR satellite is a tri-axial one, we have to predict all three components and to try to determine them from the observations. Figure 4 shows us drag and both components of lift for the same orbital position as used in Figure 3, whereas

Petturbationsfor MACEK Microaccelerometer

Figure

5 depicts

the Y- and Z- components

(1217

of lift..

From the figures, we see a big difference in the disturbing accelerations during the orbital motion, m.s- ‘. This is due, of course, by the relatively large orbital within the range of 10d5 till lo-” eccentricity; the satellite revolves within the altitudes of 400 km till 1000 km. We hope we shall be able to provide measurements in such a large region of magnitudes with a unique measurement system. However, we shall concentrate to the fainter values of accelerations; we would like to use the measurements performed at the apogee for the calibration of the device. At that point, the principal acceleration should come out from the direct solar radiation pressure which is constant in magnitude as well as in the direction.

0 I-I-I

0

v (ded

1;:; ;

Fig. 5. Y- and Z- lift components

Fig. 6. Drag and lift coefficients

At certain moments, we can observe the values of lift to exceed those of drag. This can be explained by the important role of large solar panels inclined to the velocity vector in a favorable way. In such cases, even the lift coefficient may surpass that of drag, as shown in Figure 6. 5 RADIATIVE

FORCES

A set of radiative forces (i.e. those involving momentum interchange between a radiative field and the satellite) attracted large interest in recent past, because they often cause tiny but still observable long-periodic or even secular effects /ll/. Despite of their importance, the state of art of their modeling is quite often unsatisfactory, based only on some phenomenological suggestions. Thus, testing the physical assumptions and models for the radiative forces (exp. by accelerometers) is of great interest, having large implications for current state of the orbit determination generally. Moreover, investigation of the direct solar radiation pressure (hereafter DSRP) has autonomous importance for the MACEK microaccelerometer experiment as it will serve for the device calibration in the early stages of the mission. It appears, that only the group of the external radiative forces is relevant for our treatment /12/. The major representatives of them are: (i) DSRP, (ii) light pressure of the radiation reflected on the Earth surface or redifused in the Earth atmosphere (hereafter called the albedo effect), and (iii) the Earth thermal (IR) radiation pressure. According to simple estimates only these phenomena are reliably measurable regarding to the device precision threshold. Nevertheless, it would be argued that previously mentioned (external) effects give possibility to investigate different complex phenomena. We devote

current

study

to the discussion

of the physical

phenomena

modeling,

skipping

at the

L. Sehnal and D. Vokmuhlicky

WY3

moment geometrical effects connected with the CESAR satellite complex shape. Mutual shadowing of the different satellite parts or multiple photon reflections will naturally affect the finally measured signal, however from the tutorial point of view we do not want to mix up them with the physically important effects. Moreover the final shape of our approach to the various geometrical effects (inspired with the Monte Carlo ray tracing method /13/) is under consideration. The synthetic data presented in the following graphs thus apply to the case of a spherical satellite with the ‘A/m’ parameter effectively corresponding to the CESAR satellite. However, it should be kept in mind that in the CESAR case this parameter may vary by a factor of about 4 during one revolution around the Earth as a function of its attitude (see Table 3). TABLE 3. Comparison of the estimated magnitudes (in m/s’) of the external radiative effects perturbations of the CESAR satellite. Expected factor accounting for variation of the perturbations due to the geometrical effects induced by changing satellite attitude with respect to the radiation source in the last column (equal to 1 for DSRP as CESAR is supposed to be Sun oriented).

DSRP albedo

IR

effect out of shadow penumbra diffusive clouds specular

magnitude 5 x 1o-s (5 - 10) x 10-s 1 x 10-s 1 x 10-s 1 x 10-e 7 x 1o-g

geometrical 1 1 4 4 4 4

factor

5.1 DSRP Very rough, but still viable extensively used approach to the DSRP modeling considers the solar radiation as a homogeneous force field in the Earth vicinity directed towards the instantaneous solar position. Indeed, sophistication of the model shows only tiny differences except one case: satellite passage to the Earth shadow. First important attempt in solving this problem has been suggested by Ferraz-Mello who introduced so called shadow function 1141. It shows very good properties for semianalytical theories of the orbit long-term evolution. However, significant drawback of this approach arises from the fact that it is based uniquely on the mathematically suitable truncated development of simple physical ideas. Recently, necessity of the physically more advanced models has been recognized, e.g. 1151, 1161. The crucial idea commented by those authors concerns the fact that the dominant phenomena ruling the penumbra transition are due to the atmospheric effects while the solar disk eclipsing by the Earth body is of the minor importance (this property has been in fact originally known from large number of studies of the lunar eclipses dating back to the end of the last century; in context of the artificial satellites clearly spelled out by Link 1171). Interestingly, this fact has some implications for the long-term effects acting on geodetical satellites of the LAGEOS-type 1161. Without entering into somewhat intricate algebra we shall remind the essential features of our penumbra model (for details see 1151). W e carefully account for all geometrical effects connected with the individual light rays in the Earth atmosphere (‘refraction’). Very efficient algorithm can be derived in case of the spherical Earth and spherically stratified Earth atmosphere. Extension to the case of ‘weakly flattened’ Earth and its atmosphere is under progress. Reliable approximation has been developed for computing the atmospheric extinction, notably influence of the Rayleigh model is an type scattering and high atmosphere absorption layers. A good Earth atmosphere important part of our penumbra theory. At the moment, we implemented Garfinkel’s theory of the Earth atmosphere refraction 1181, 1191, together with classical treatment of the Rayleigh extinction 1201. Basic set of data supplied to the penumbra theory concerns the Earth surface state of the atmosphere (i.e. temperature, pressure, humidity etc.).

WYJ

Perturbationsfor MACEK Micmaccelerometer

0 -10

‘I’

0

10

20

time

30

40

50

0

1

2

3

4

5

M

(s)

Penumbra phase acceleration due to the DSRP for the suggested CESAR satellite orbit. Fig. 7. Solid curve 1 includes the refractive features of the Earth atmosphere only, curve 2 accounts for the Rayleigh type extinction in more. Short dashed curve 4 stands for the simplified penumbra model based on the geometric portion of the solar disk visible above the Earth horizon disregarding the value estimated for the CESAR totally the atmospheric effects. Ordinate units in 10-sm/s2, satellite. Amplitude of the albedo perturbations vs. mean anomaly during one CESAR revoluFig. 8. tion. Equatorial homogeneous cloud belt bounded by latitudes (-loo, loo) assumed. Solid lines correspond to the model with anisotropic local reflection, while short dashed curves to the locally isotropic reflection. Several values of the planar cloud albedo considered: (i) ii = 1 for curves 1, (ii) ti = 0.8 for curves 2, (iii) a = 0.6 for curves 3. Curves labeled t describe results based on the hypothetical total cover of the Earth surface with correspond cloud type. Satellite crosses the equator at M = x. Ordinate units in 10-sm/s2 as in Figure 7. Figure 7 shows a simulated penumbra transition for the CESAR expected orbital elements (the Sun has been fixed to the satellite orbit so that this configuration is the least favourable as concerns the duration of the effect; maximal duration of the penumbra transition has been estimated to 400 s). Perturbing acceleration on the ordinate, time since the beginning of the penumbra phase at abscissa. The important lesson driven from this figure concerns difference of the curves 2, corresponding to our penumbra theory including the atmospheric refraction and extinction, and 4, following simple geometrical model of the solar disk eclipse. We observe large difference between a simply motivated model disregarding the atmospheric effects and the full theory accounting for the atmospheric effects. It has been demonstrated that such difference is maximal for the low orbit satellites /15/. We hope to analyze a set of the penumbra transition with the aim of testing out penumbra theory. The shadow crossing configuration occurs on more than 70 % of the CESAR lifetime due to the low orbit 1211. 5.2 Albedo

Effect and the Earth

IR Radiation

The albedo effect clearly involves several additional difficulties if compared to the DSRP: (i) modeling of the physical phenomena resulting from the sunlight reflection on the Earth surface, (ii) it unavoidably involves the atmospheric phenomena, either of the global character - Rayleigh extinction, absorption in the high atmosphere layers - or of the local character - huge cloud complexes. 3ne can trace two streams in the development of the albedo effect analytical techniques for rapid evaluation of the diffusive part of Earth surface 1221, 1231, (ii) purely numerical techniques involved complex phenomena. The aim of our work determines the choice

theory in recent past: (i) large the sunlight reflection on the for modeling of the physically of approach close to the item

(12)lO

L. Sehnal and D. Vokrcuhlicky

(ii), however still considering the diffusive reflection on the continents. Two special effects which desire interest: (i) importance of the optically thick clouds, (ii) specular reflection on the ocean surface. Both effects have consequences on the long-term residuals for the LAGEOS-type satellites as recently demonstrated in a series of papers 1241, 1251, contrary to the purely diffusive albedo part as was proven in the quoted references. The effect of the Earth thermal (IR) radiation pressure is often discussed together with the diffusive part of the albedo effect as the nature of both phenomena is common 1231. Usually, the Earth emissivity is expanded in the series of the spherical functions and only pole and quadrupole zonal terms are considered. Though, the dipole term might have some applications, it appears that the IR radiation cannot possess long-term effects and is thus on the edge of our primary interest. 5.2.1 Cloud patterns in the albedo effect context. Existence of huge cloud fields seemingly influence the Earth surface reflected radiation. A qualitative estimate of this importance can be easily performed realizing that the effective planar cloud albedo can reach values close to 1, while the average Earth surface albedo is roughly 0.3. One can thus expect enhancing of the albedo signal by a factor of about 3 when the Earth surface visibility cap for a given satellite is largely filled with optically thick clouds. Nevertheless, more careful treatment of the cloud influence is currently desirable mainly in view of theoretical results 1241, 1251, pointing out that the non-diffusive patterns of the reflection modes only can rise observable long-term effects in the orbit determination. Photometric data collected from the Nimbus mission presented do show such asymmetries of the local reflection isophotes mainly for larger zenith angles 1261. We have adopted a theoretical scheme developed by Chandrasekhar for light scattering in planar planetary atmospheres with the infinite optical thickness /25/,/28/. Despite of the model simplicity, it was argued that it is endowed with necessary local reflection asymmetry patterns desirable for the long-term effects modeling /28/. I n order to get idea of the cloud influence, we present in Figures 8 and 9 resulting from two hypothetical experiments with two different cloud distribution on the Earth surface: (i) uniform cloud belt around the Earth equator (Figure 8), and (ii) random cloud distribution on the Earth surface with the 40 % filling factor (Figure 9). Different values of the effective (plane) cloud albedo ii = 0.6, resp. 0.8 and 1, accepted for particular solid lines, while the corresponding short-dashed lines relate to the same plane albedo but isotropic local reflection. We observe that situations with large cloud filling of the visible cap by the satellite does result in considerable increase of the total albedo signal, however the difference between the anisotropic and isotropic reflection signals is small. The desirable anisotropy patterns enhance for larger zenith angles both of the solar incident radiation and the local direction to the satellite (this situation has not been met in Figure 8). In some selected configuration the cloud anisotropy influence can be largely amplified. In the MACEK program we would search for such situations and try to interpret then, even keeping in mind that not often the information about the anisotropy factor can be estimated from the meteorological data 1291, w h ic h must be carefully interpreted. 5.2.2 Specular reflection on the oceans. Despite the fact that the specular mode of the sunlight reflection can reach roughly 10 of the total albedo effect some interest has been focused on this after Barlier recognized that this part of the albedo effect can contribute to the long-term residuals of the LAGEOS-type satellites 1241. However, his scheme has been criticized as it considered ideal (mirror-like) reflection on the ocean. In view of the ocean surface roughness, one expects forming the finite cone pattern of the reflected sunlight having as a consequence a decrease of the perturbing signal. First steps towards understanding these phenomena have been done by Rubincam et al. 1301, who adopted a simple model inspired by the photometric data. Vokrouhlicky and Farinella followed a simple theoretical model based on the statistical description of the ocean surface roughness 1311, 1321. They showed that the effects due to reflected field dilution connected with the finite cone phenomenon results in expected decrease of the long-term effects only by about 20 30 %, so that the specularly reflected radiation still remains an interesting viable effect in discussion of the long-term satellite orbit phenomena.

Pettttrb;ltions for MACEK Micmaccelerotneter

(12)ll

M

Influence of randomly distributed clouds with 40 percent filling of the Earth surface. Fig. 9. Radial component of the albedo acceleration on the ordinate (lo-’ m/s2 units) vs. mean anomaly. The three solid curves for different values of the plane albedo parameter of the clouds: (i) si = 1 for curve 3; (ii) zi = 0.8 for curve 2; (iii) 7i = 0.6 for curve 1. Bottom dashed curve corresponds to the cloudless signal (as in previous figure). Comparison of a step-like pattern for the simple treatment of the sunlight specular Fig. 10. reflection (dashed curves) with smooth behaviour of the same effect modeling based on the precise theory by Vokrouhlicky and Farinella /31/. Different effective values of the finite cone aperture accepted: (i) 5” for curve 1, (ii) 10” for curve 2, (iii) 20” for curve 3. Short dashed lines represent mixture of the ideal mirror approach with more precise theory involving generalized Fresnel reflection functions introduced by Vokrouhlicky and Farinela 1311. Figure 10 shows a typical example of the difference between the exact solution for the specularly reflected sunlight pressure accounting for the finite cone phenomenon - solid lines - (notice, that from the standpoint of the satellite we can also speak in terms of finite spot of the reflected radiation on the Earth surface), and the simple approach disregarding the ocean surface roughness - dashed lines. Various curves correspond to different parameters of the effective aperture of the reflected sunlight cone. 6 CONCLUSION We predict the accelerations of the satellite CESAR caused by the non-gravitational forces to be variable in a relatively large range of 10s5 till 10-l’ m.sm2, and to act in all directions, defined in the orbital frame. This is due to a non-negligible lift force, produce by solar panels. This poses, in turn, extensive technical requirements to the construction of the accelerometer. On the other side, the acceleration at apogee caused by the direct solar radiation pressure should be of at least one order larger than that produced by the atmospheric effects. This means that it will be able to use the acceleration measured at apogee to calibrate the whole measuring device. In the part the analysis arcs would this task is of the Earth phenomena

devoted to the radiative effects, we discussed here phenomena potentially important for of the long-term residuals of the geodetic satellites. A carefully chosen set of the orbital hopefully allow testing of the theoretical models of those effects. Detailed strategy for under consideration. Aside of this special goal, we also hope to achieve global solution albedo distribution considering data from larger set of the orbital arcs, free of peculiar mentioned previously.

Acknowledgement. This work has been supported by the Czech Republic research grant No. 205/93/0893 and the research grant of the Academy of Sciences of the Czech Republic No. 94/303105.

L. Sehanl and D. Vokmhlicky

(12)12

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