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

1968 ◽  
Vol 303 (1472) ◽  
pp. 17-35 ◽  
Keyword(s):  
Mathematical Theory ◽  
Angular Motion ◽  
Satellite Orbits ◽  
Orbital Eccentricity ◽  
Air Drag ◽  
Rates Of Change ◽  
Gravitational Forces ◽  
Air Density ◽  
Complicated Form ◽  
Time Variations

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. II. With oblate atmosphere

1961 ◽  
Vol 264 (1316) ◽  
pp. 88-121 ◽  
Keyword(s):  
Orbital Period ◽  
Rate Of Change ◽  
Satellite Orbits ◽  
Constant Density ◽  
Circular Orbits ◽  
Air Drag ◽  
Small Eccentricity ◽  
Separate Section ◽  
Air Density ◽  
Independent Parameters

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 IV. With scale height dependent on altitude

1963 ◽  
Vol 275 (1362) ◽  
pp. 357-390 ◽  
Keyword(s):  
Orbital Period ◽  
Scale Height ◽  
Rate Of Change ◽  
Time Interval ◽  
Satellite Orbits ◽  
Air Drag ◽  
Small Eccentricity ◽  
Best Constant ◽  
Air Density ◽  
Linear Manner

The effect of air drag on satellite orbits of small eccentricity e (< 0.2) was studied in part I on the assumption that atmospheric density p varies exponentially with distance r from the earth’s centre, so that the ‘density scale height’ H , defined as — p /(d p /d r ), is constant. In practice H varies with height in an approximately linear manner, and in the present paper the theory is developed for an atmosphere in which H varies linearly with r . 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- H equations of part I. The second (and usually better) method is to obtain the best constant value of H for use with the equations of part I. For example, it is found that the constant- H equations connecting perigee distance (or orbital period) and eccentricity can be used unchanged without loss in accuracy, if H is taken as the value of the variable H at a height y H above the mean perigee height during the time interval being considered, where y — 3/2 for e > 0.02, and y decreases from § towards zero as e decreases from 0.02 towards 0. Similarly the constant- H equations for air density at perigee can still be used if H is evaluated at a height above perigee, where £ = 3/4 for e >0.01, and £ decreases towards zero as e decreases from 0.01 towards 0. For circular orbits the constant- H equations for radius in terms of time can still be used if H is evaluated at one scale height below the initial height. Variation of H with altitude has a small effect on the lifetime—about 3 %— and on the curve of e against time. 1 *

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

1987 ◽  
Vol 411 (1840) ◽  
pp. 1-17 ◽  
Keyword(s):  
Major Part ◽  
Atmospheric Model ◽  
Scale Height ◽  
Satellite Orbits ◽  
High Eccentricity ◽  
Air Drag ◽  
Air Density ◽  
Exponential Variation ◽  
Constant Part ◽  
Density Scale

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.

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

1965 ◽  
Vol 259 (1096) ◽  
pp. 33-67 ◽  
Keyword(s):  
Orbital Period ◽  
Angular Distance ◽  
Maximum Density ◽  
Rate Of Change ◽  
Satellite Orbits ◽  
The Sun ◽  
Air Drag ◽  
Air Density ◽  
Main Geometrical Parameter ◽  
Long Term Effect

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, III. High-eccentricity orbits (0·2≤ e <1)

1962 ◽  
Vol 267 (1331) ◽  
pp. 541-557 ◽  
Keyword(s):  
Orbital Period ◽  
Rate Of Change ◽  
Satellite Orbits ◽  
High Eccentricity ◽  
Air Drag ◽  
Air Density

The effect of air drag on satellite orbits of eccentricity e less than 0·2 was studied in parts I and II. Here the theory for values of e between 0·2 and 1 is presented. Equations are derived which show how perigee distance and orbital period vary with eccentricity during the satellite’s life, and how eccentricity is related to time; and formulae are obtained for the lifetime and the air density at perigee, in terms of the rate of change of period. The results are also presented graphically and their implications and limitations are discussed.

scholarly journals King-Hele orbit theory for periodic orbit and attitude variations

2020 ◽  
Vol 501 (1) ◽  
pp. 1168-1187
Author(s):  
Vishal Ray ◽  
Daniel J Scheeres
Keyword(s):  
Drag Coefficient ◽  
Dynamic Parameter ◽  
Mission Design ◽  
Fourier Series Expansion ◽  
Satellite Orbits ◽  
Physical Interaction ◽  
Satellite Mission ◽  
Simplifying Assumptions ◽  
Time Variations ◽  
Specific Factors

ABSTRACT The analytical theory of satellite orbits in an atmosphere developed by King-Hele remains widely in use for satellite mission design because of its accurate approximation to numerical integration under simplifying assumptions. Over the course of six decades, modifications to the theory have addressed many of its weaknesses. However, in all subsequent modifications of the original theory, the assumption of a constant drag-coefficient has been retained. The drag-coefficient is a dynamic parameter that governs the physical interaction between the atmosphere and the satellite and depends on ambient as well as satellite specific factors. In this work, Fourier series expansion models of the drag-coefficient are incorporated in the original King-Hele theory to capture time-variations of the drag-coefficient in averaging integrals. The modified theory is validated through simulations that demonstrate the attained improvements in approximating numerical results over the original King-Hele formulation.

A discussion on orbital analysis - Variations in air density at 280 km height

1967 ◽  
Vol 262 (1124) ◽  
pp. 195-199
Keyword(s):  
Diurnal Variation ◽  
Local Time ◽  
Atmospheric Density ◽  
Air Drag ◽  
Average Value ◽  
Air Density ◽  
Maximum Value ◽  
Orbital Analysis

The effect of air drag on the orbits of six Cosmos satellites having low perigee heights has been investigated. The diurnal variation of neutral atmospheric density at about 280 km is shown to have an amplitude of about 25% of its average value and to have a maximum value at about 14 h local time.

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

1987 ◽  
Vol 414 (1847) ◽  
pp. 271-295 ◽  
Keyword(s):  
Narrow Band ◽  
Great Majority ◽  
Satellite Orbits ◽  
Orbital Theory ◽  
Air Drag ◽  
Zonal Harmonics ◽  
The Earth ◽  
Upper Limits

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 effect of air drag on satellite orbits: Advances in 1687 and 1987

1987 ◽  
Vol 30 ◽  
pp. 269-289 ◽  
Author(s):  
D.G. King-Hele ◽  
D.M.C. Walker
Keyword(s):  
Satellite Orbits ◽  
Air Drag
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