Contraction Of Satellite Orbits Research Articles

Contraction of satellite orbits using KS uniformly regular canonical elements in an oblate diurnally varying atmosphere

A new non-singular analytical theory for the motion of near Earth satellite orbits with the air drag effect is developed in terms of the Kustaanheimo and Stiefel (KS) uniformly regular canonical elements, by assuming the atmosphere to be oblate diurnally varying with constant density scale height. The series expansions include up to third-order terms in eccentricity and c (a small parameter dependent on the flattening of the atmosphere). Only two of the nine equations are solved analytically to compute the state vector and change in energy at the end of each revolution, due to symmetry in the equations of motion. Numerical comparisons of the important orbital parameters semimajor axis and eccentricity up to 1000 revolutions, obtained with the present solution, with the third-order analytical theories of Swinerd and Boulton and in terms of the KS elements, with respect to the numerically integrated values, show the superiority of the present solution over the other two theories over a wide range of eccentricity, perigee height and inclination.

Contraction of satellite orbits using KS elements in an oblate diurnally varying atmosphere

A non‐singular analytical theory for the motion of near‐Earth satellite orbits with air drag is developed in terms of KS elements, utilizing a model of the atmosphere that allows for oblateness and in which the density behaviour approximates to the observed diurnal variation. The series expansions include up to cubic terms in e (eccentricity) and c (a small parameter dependent on the flattening of the atmosphere). Only two of the nine equations are solved analytically to compute the state vector at the end of each revolution due to symmetry in the KS element equations. Numerical comparisons between the analytical and numerically integrated values of the drag perturbed orbital parameters (semi‐major axis, eccentricity and argument of perigee) are made up to 100 revolutions for the test cases with different inclinations, over a range of perigee and apogee heights, which experience significant diurnal bulge effects. The comparisons are found to be good. The accuracies of the numerical computations are checked with the bilinear relation in KS elements, and are found to be very satisfactory.

The contraction of satellite orbits under the influence of air drag. VIII. Orbital lifetime in an oblate atmosphere, when perigee distance is perturbed by odd zonal harmonics in the geopotential

This paper is devoted to developing the necessary orbital theory for predicting the lifetimes of satellites moving in an oblate atmosphere and subjected to the perturbations due to odd zonal harmonics in the geopotential. The effects of odd zonal harmonics and atmospheric oblateness are expressed as multiplying factors, F (oz) and F (ao), to be applied to the lifetime predictions calculated in the absence of the perturbations. The results are valid for the great majority of orbits about the Earth, and in particular for all orbital eccentricities between 0 and 1; but the limits set for the controlling parameters exclude ( a ) near-polar orbits with perigee heights lower than about 180 km, and ( b ) orbits having inclinations within a narrow band centred on 63.4°. The results show that, when the controlling parameters are at their upper limits, either F (oz) or F (ao) can change the lifetime by up to about 35%, and taken together they can produce changes of up to 60%, if the initial and final positions of perigee are at specified points on the orbit and the eccentricity exceeds 0.2. Such combinations of values rarely arise, however, and the effects are more often of order 10-20%. Even at these moderate levels, the effects need to be taken into account in order to make realistic estimates of the decay dates of satellites in the last few months of their lives.

The contraction of satellite orbits under the influence of air drag. VII. Orbits of high eccentricity, with scale height dependent on altitude

Part III of this series of papers developed the theory of high-eccentricity orbits ( e > 0.2) in an atmosphere having an exponential variation of air density with height, that is, with the density scale height H taken as constant. Part IV derived the appropriate theory for low-eccentricity orbits ( e < 0.2) in a more realistic atmosphere where H varies linearly with height y (and μ = d H /d y < 0.2). The present paper treats the orbits of part III when they meet the air drag specified by the atmospheric model of part IV. Equations are derived showing how the perigee height varies with eccentricity, and the eccentricity varies with time, over the major part of the satellite’s life. It is shown that the theory of part III remains valid, to order μ 2 , if H is evaluated at a specific height above perigee.

Contraction of satellite orbits in an oblate atmosphere with a diurnal density variation

The effect of air drag on satellite orbits of small eccentricity, e < 0.2, is considered. A model of the atmosphere that allows for oblateness is adopted, in which the density behaviour approximates to the observed diurnal variation. The equations governing the changes due to drag in the semi-major axis a , and in x = ae , during one revolution of the satellite are integrated, the density scale-height H being assumed constant. The resulting expressions for ∆ a and ∆ x are presented to third order in e . Compact expressions for the gradient d a /d x , and for the mean air density at perigee altitude ρ 1 are obtained, when H is allowed to vary with altitude. An equivalence between the variable- H and the constant- H equations is demonstrated, provided that the value of H used in the latter is chosen appropriately.

The contraction of satellite orbits under the influence of air drag - VI. Near-circular orbits with day-to-night variation in air density

The theory for the effect of air drag on satellite orbits was developed in Parts I to V of this series of papers on the assumption that the angular motion of perigee is controlled by gravitational forces and is not affected by air drag. If the orbital eccentricity is less than about 0·01, however, and the atmosphere exhibits a substantial day-to-night variation in density, the air drag itself significantly affects the angular motion of perigee. In these circumstances the mathematical theory takes a different and more complicated form, which is developed in the present paper. General equations are derived for the rates of change of parameters specifying the eccentricity and argument of perigee. Complete analytical solutions for the time variations of these parameters are obtained when e is of order 0·001, in two different forms, for (1) high drag, i. e. short lifetime, and (2) low drag. The results are illustrated by numerical examples.

The contraction of satellite orbits under the influence of air drag V. With day-to-night variation in air density

The effect of air drag on satellite orbits of small eccentricity (< 0-2) was studied in part I on the assumption that the atmosphere was spherically symmetrical. In reality the density of the upper atmosphere depends on the elevation of the Sun above the horizon and has a maximum when the Sun is almost overhead. In the present paper the theory is extended to an atmosphere in which the air density at a given height varies sinusoidally with the geocentric angular distance from the maximum-density direction. Equations are derived which show how perigee distance and orbital period vary with eccentricity throughout the satellite’s life, and how eccentricity varies with time. Expressions are also obtained for lifetime and air density at perigee in terms of the rate of change of orbital period. The main geometrical parameter determining the long-term effect of this day-to-night variation is the angular distance <f>p of perigee from the maximum-density direction. Results are obtained for <})pconstant and <j)pvarying linearly with time.

The contraction of satellite orbits under the influence of air drag IV. With scale height dependent on altitude

The effect of air drag on satellite orbits of small eccentricitye(< 0.2) was studied in part I on the assumption that atmospheric densitypvaries exponentially with distancerfrom the earth’s centre, so that the ‘density scale height’H, defined as —p/(dp/dr), is constant. In practiceHvaries with height in an approximately linear manner, and in the present paper the theory is developed for an atmosphere in whichHvaries linearly withr. Equations are derived which show how perigee distance and orbital period vary with eccentricity, and how eccentricity varies with time. Expressions are also obtained for the lifetime and air density at perigee in terms of the rate of change of orbital period. The main results are presented graphically. The results are formulated in two ways. The first is to specify the extra terms to be added to the constant-Hequations of part I. The second (and usually better) method is to obtain the best constant value ofHfor use with the equations of part I. For example, it is found that the constant-Hequations connecting perigee distance (or orbital period) and eccentricity can be used unchanged without loss in accuracy, ifHis taken as the value of the variableHat a height yHabove the mean perigee height during the time interval being considered, where y — 3/2 fore> 0.02, and y decreases from § towards zero as e decreases from 0.02 towards 0. Similarly the constant-Hequations for air density at perigee can still be used ifHis evaluated at a height above perigee, where £ = 3/4 fore>0.01, and £ decreases towards zero as e decreases from 0.01 towards 0. For circular orbits the constant-Hequations for radius in terms of time can still be used ifHis evaluated at one scale height below the initial height. Variation ofHwith altitude has a small effect on the lifetime—about 3 %— and on the curve of e against time. 1 *

The estimation of atmospheric scale heights from the contraction of satellite orbits

A numerical investigation is carried out into the behaviour of the ratio Δr p / Δe, where Δr p and Δe are the changes in the perigree distance and eccentricity per revolution due to air drag. It is shown that Δr p / Δe can be related to both the density scale height H, and the pressure scale height h, of the upper atmosphere. The results are presented in a graphical form and are used to obtain H and h values from the observed changes in satellite orbits for which any reasonable orbital data are available. Estimates are also made for the temperature/molecular weight ratio and the temperature in the 180–240 km height region.

The contraction of satellite orbits under the influence of air drag. II. With oblate atmosphere

The effect of air drag on satellite orbits of small eccentricity e (< 0.2) was studied in part I on the assumption that the atmosphere was spherically symmetrical. Here the theory is extended to an atmosphere in which the surfaces of constant density are spheroids of arbitrary small ellipticity. Equations are derived which show how perigee distance and orbital period vary with eccentricity, and how eccentricity is related to time. Expressions are also obtained which give lifetime and air density at perigee in terms of the rate of change of period. In most of the equations, terms of order e 4 and higher are neglected. The results take different forms according as the eccentricity is greater or less than about 0.025, while circular orbits are dealt with in a separate section. The influence of atmospheric oblateness is difficult to summarize fairly, because it depends on four independent parameters. If these simultaneously assume their ‘worst’ values, some of the spherical-atmosphere results can be altered by up to 30% as a result of oblateness. But usually the influence of atmospheric oblateness is much smaller, and 5 to 10% would be a more representative figure.

The contraction of satellite orbits under the influence of air drag. I. With spherically symmetrical atmosphere

The effect of air drag on satellite orbits of small eccentricity e (< 0.2) is studied analytically by a perturbation method, on the assumption that the atmosphere is spherically symmetrical. Equations are derived which show (1) how perigee distance and orbital period vary with eccentricity as the orbit contracts, and (2) how each of these quantities varies with time. The equations of type (1) are nearly independent of the oblateness of the atmosphere. In all the equations, terms of order e 4 and higher are usually neglected. The results are also presented graphically, in a manner designed for practical use. The theory will be extended to an oblate atmosphere in part II, and will later be compared with observation.