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Contents Figure (1): The effect of atmospheric drag on elliptical orbit [Oubiad, 2002]Figure (2):The effect of atmospheric drag on circular orbit [ Kennewell,1999] 22 Drag force 
Effect of Atmospheric Drag on Artificial Satellite Orbit Rawaa Mezher Oubiad Department of Physics College of Science Babylon University Abstract The perturbation of satellite from low earth orbit and its physical lifetime are studied by their interaction with the atmosphere. The prediction of lifetimes of satellites or their reentry date depends upon a knowledge of the initial satellite orbital parameters, the satellite mass to cross sectional area (in the direction of travel) and a knowledge of the upper atmospheric density and how this responds to space environmental parameters. To establish this task a simple model for atmosphere is used which describes the density as a function of space environmental parameters, and this model is applied to calculate the perturbation rates and orbital lifetimes of satellites in essentially circular orbits below 500 km altitude. A computer program in QBasic is builed to study this perturbation and measure the lifetime of satellite. It has been found that the lifetime of satellite depends on the satellite area to mass ratio, where the shorter orbital lifetime can be obtained when the area to mass ratio increases. الخلاصة تمت دراسة اضطراب القمر الصناعي من المدار الأرضي الواطئ والعمر الفيزياوي له بواسطة تفاعله مع الغلاف الجوي. فالتنبؤ عن أعمار ألأقمار الصناعية أو رجوعها إلى جو الأرض يعتمد على معرفة متغيرات مدار القمر الابتدائية ونسبة الكتلة إلى مساحة المقطع العرضي للقمر( التي هي دائماً باتجاه الحركة) وكذلك على كثافة الغلاف الجوي وكيفية استجابتها لمتغيرات البيئة الفضائية. لإنجاز هذا العمل استعمل نموذج بسيط للغلاف الجوي الذي يصف الكثافة كدالة لمتغيرات البيئة الفضائية، وكذلك فان النموذج يطبق لحساب معدلات الاضطراب والأعمار المدارية للأقمار الصناعية بالخصوص ذات المدارات الدائرية تحت ارتفاع 500km. وتم بناء برنامج حاسوبي لدراسة هذا الاضطراب وقياس عمر القمر الصناعي. وقد وجد بان عمر القمر ذو المدار الدائري يعتمد على نسبة مساحة القمر إلى كتلته، حيث يمكن الحصول على العمر المداري القصير عندما تزداد نسبة المساحة إلى الكتلة. 1. Introduction During recent year, a significant progress has been achieved in modeling of forces acting on an artificial satellite. Partical atmospheric drag force causes significant perturbation of earth satellite orbits. The study of this perturbation is of interest to many researchers .The drag force that such a satellite experiences is due to its interaction with the few air molecules that are present at these altitudes. The density of the atmosphere at low earth orbits heights is controlled by solar flux and particle precipitation from the magnetosphere and so varies with the current space weather conditions. Atmospheric drag models commonly in use for calculation of the perturbation include the Drag Temperature Model DTM [Berger et. Al. 1998] , the Total Density Model TD [Sehnal, 1990] and some other models. These models are based on either in situ atmospheric spectrometer measurement or satellite orbit dynamics. Even with a complete atmospheric model describing variations with time, season, latitude and altitude, a complete specification of orbital perturbation is not possible because of uncertainties in the prediction of satellite attitude (which affects the relevant crosssectional area), and solar and geomagnetic indices (which substantially modify the atmospheric model). Even when most of the quantities are known there appears to be an irreducible level below which it is not possible to predict. This level appears to be around 10% of the remaining satellite lifetime, irrespective of what that lifetime is. In other words, the error in predicting the reentry of a satellite expected to remain aloft for about 10 years is one year, whereas the demise of a satellite expected to reenter in 24 hours time is only accurate to about 2 hours. An appreciation of the uncertainty is shown by a NORAD prediction in April 1979 for the expected reentry of the SKYLAB space station between 11 June and 1 July of that year. The actual reentry occurred on 11 July, outside the stated interval, a prediction error from midinterval of around 15%. 2.Theory In general the orbital perturbation may be defined as the deviation of elements from their mean values due to external forces. These forces may be classified into two types: gravitational and nongravitational forces. [Milani, 1987; Upham, 1991] The gravitational forces arise from mutual gravitational attraction between various bodies such as earth satellite or earthmoon. The non gravitational forces arise from space environment such as radiation pressure, atmospheric drag, and geomagnetic field. etc. The dominant for low earth orbit is the atmospheric drag that is due to continual collision of atmospheric constituents with satellite surfaces. These types of collision take place when the mean free path of molecules is much larger than the satellite dimension, the drag would first circularize the orbit by gradually lowering its apogee and then the radius of circular orbit continues to decrease until deorbiting in the earth atmosphere , as shown in Fig.(1). [Oubiad,2002] Initial orbit Intermediate orbit Final orbit ^ For the circular orbit, these collisions with particles, slowly act to circularize the orbit and slow down the satellite causing it to drop to lower altitudes as shown in Fig.(2). [ Kennewell,1999] ^ 21 The Atmospheric Model The atmospheric model has been confined to satellites with orbits totally below about 500 km altitude. Such orbits can be regarded as essentially circular, with the use of the semi major axis in place of the orbital radius. The atmospheric density ρ is specified by a simple exponential with variable scale height H. For a fixed exospheric temperature T, H is made to vary with altitude h through the use of an effective atmospheric molecular mass m, in which it includes both the actual variation in molecular mass with height and a compensation term for the variation in temperature over the considered range from 180 to 500 km. The variation in density due to the space environment is introduced through T which is specified as a function of the solar radio flux F10.7 and the geomagnetic index Ap. [KingHele 1987] The two terms which are used in the model describe the effects of different agencies, both of which originate from the Sun. The first agency is the solar Xray output incident upon the Earth is generally absorbed at the base of the thermosphere (around 120 km) and this gives rise to a direct heating effect which propagates itself upward from this level. The solar 10 cm radio flux is used as a surrogate for the total solar Xray flux which produces this effect. This flux can vary from a low of about 65 to over 300 Solar Flux Units (1 SFU = 1022 W.m^{2}.s^{1}). The other agency is the precipitation of particles (mainly electrons and protons) from the magnetosphere down into the lower thermosphere. The energy dumped by this precipitation again acts to heat the atmosphere, which subsequently changes the atmospheric density. Most of these particles originate from the Sun. They are expelled in Coronal Mass Ejections, travel through the interplanetary medium, and eventually arrive at the Earth. Precipitation of such particles is well correlated with large variations in the geomagnetic field as measured at ground level and quantified by a number of geomagnetic indices, including the planetary Ap index used here. This index, computed every 24 hours, hovers just above zero in quiet times, but may rise to above 9 (no units) at times of major geomagnetic storms. The set of defining equations for the model are given by: [KingHele 1987] T = 900 + 2.5 (F10.7_{ } 70) + 1.5 Ap (Kelvin) (1) m = 27  0.012 (h  200) 180 < h (km) < 500 (2) H = T / m (km) (3) ρ = 6x10^{10} exp ( (h  175) / H) (kg m^{3}) (4) All constants were empirically derived to give an appropriate fit to the standard models. It should be noted that the only really valid output of this model is the density. The intermediate variables used in deriving this density in general do not correspond to true atmospheric values at any height within the considered range. The temperature may be regarded as the mean asymptotic value for the exosphere at large altitudes. The molecular mass might be regarded as an integrated mean value from the base of the thermosphere up to the specified height. The solar 10 cm radio flux is generally used in an averaged form, and the average preferred is that of the last 90 days prior to the specified date. Sometimes a small correction is made to weight the current flux more strongly. ^ When a spacecraft travels through an atmosphere it experiences a drag force in a direction opposite to the direction of its motion. This drag force is given by the expression:[Baker & Makemson,1960; Sehnal &Vokrouhlicky,1995] D = 1/2ρv^{2}A C_{d } (5) where D is the drag force, is the atmospheric density, v is the speed of the satellite, A is its crosssectional area perpendicular to the direction of motion, and C_{d }is the drag coefficient. At the altitudes at which satellites orbit, C_{d} is a number depends on the interaction of satellite surface with the incoming molecules, which means that its value changes with altitude in according with the composition of the atmosphere layers, in general it is assumed to be equal to two, although experiments have shown that this can vary widely [Sehnal,2003]. Because it is usually difficult to separate out independent variations in the crosssectional area from the variations in the drag coefficient, we shall henceforth use an effective crosssectional area A_{e} =AC_{d} for the rest of the model. We can use this drag force in Newton's second law together with energy considerations of a circular orbit to derive an expression for the change in the orbital radius and period of the satellite with time. For a circular orbit we have the following relation between the period P and semimajor axis a:[ Kennewell,1999] P^{2} G M_{e} = 4π^{2}a^{3} (6) where G is the Universal Gravitational Constant and M_{e} is the mass of the Earth The reduction in the period due to atmospheric drag is given by: [Kennewell ,1999]. dP/dt = 3πaρ (Ae/M) (7) The satellite is flown around its orbit using appropriate past or forecast values for the space environment variables (F10.7 and Ap). Reentry is assumed to occur when the satellite has descended to an altitude of 180 km. In all but the heaviest satellites (those with a mass to area ratio well in excess of 100 kilogram per square metre), the actual lifetime from an altitude of less than 180 km is only a few hours. 3.Results A Simple program in QuickBasic for the above model is used to illustrate the steps involved. The satellite orbit perturbation under the influence of atmospheric drag was studied. Equations (6) and (7), together with the equations modeling the atmospheric density, can be iterated from the starting satellite altitude and time. The input data or the initial boundary conditions ,are given in a form of the Keplerian orbital elements(characterizing the initial position of the orbit in space) are the semi major axis in place of the orbital radius and the physical characteristics of the satellite are given by its mass m and its average effective area A. In the applications for the lifetime determination we are using the area to mass ratio A/m, We are working within the limits 100200kg(mass) and 12 m^{2} (effective area).The orbit shape is not yet fixed, we considered a height to be somewhere 300 km .The environment variables (F10.7 and Ap) can be taken in an averaged form F10.7=70 and Ap=0 . This implementation does not allow for variations in the space environment, and as a result is only suitable for short time periods or during longer times when solar and geomagnetic activity do not show significant variation. This generally only occurs around the years of solar minimum. The lifetime determination in one specific case can be well clarified on the variation of the height .On figures(3) and (4) we see some example of such changes where A/M=0.01 m^{2}/kg in figure (3) and A/M=0.02 m^{2}/kg in figure(4). From the calculations, one can see that, the satellite decays slowly at higher altitudes, and then behaves as a very rapid decay toward the end of its lifetime for two both ratios A/M. Figure (3): variation of semi major axis with time, A/M=0.01m^{2}/kg.Notice how the satellite decays slowly at higher altitudes, then very rapidly toward the end of its life. Figure (4): variation of semi major axis with time, A/M=0.02 m^{2}/kg.Notice how the satellite decays slowly at higher altitudes, then very rapidly toward the end of its life. 4.Discussion and Conclusion According to the results obtained from the model used in this work, we have shown that, there are several remarkable notes: Figure (3) and (4) show a monotonic decrease in semi major axis (height of satellite) with time under the effect of atmospheric drag, until the satellite is unable to make another revolution and then deorbits in dense atmosphere. It can be estimated from these figures that the calculated lifetime is 45 day for the satellite of A/M=0.01 m^{2}/kg and 28 day for the satellite of A/M=0.02 m^{2}/kg. On the same figures, we can see very distinctly the shorter lifetime of the satellite of A/M=0.02 m^{2}/kg in contrast to longer lifetime of the satellite of A/M=0.01 m^{2}/kg. There is a strong dependence of lifetime on the area to mass ratio. The graphs presented may serve for calculation of lifetimes of satellites with a precision of 15%, which we believe is sufficient for most research purposes. The biggest errors may originate in an inaccurate solar activity prediction. Reference Baker, R.M. Makemson, (1960). An Introduction to Astodynamic, Academic, New York and London. Berger, C. Biancale, R. Ill, M Barlier, F. (1998). Improvement of the Empirical Thermospheric Model DTMComparative Review of Various Temporal Variations and Prospects in Space Geodesy Applications, Journal of Geodecy, 72,161178. Kennewell, J. 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