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. 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 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.

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

1960 ◽  
Vol 257 (1289) ◽  
pp. 224-249 ◽  
Keyword(s):  
Perturbation Method ◽  
Orbital Period ◽  
Satellite Orbits ◽  
Air Drag ◽  
Small Eccentricity

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.

The advance of the perigee of a satellite in a rotating oblate atmosphere with a diurnal variation in density

1983 ◽  
Vol 389 (1797) ◽  
pp. 433-444 ◽  
Keyword(s):  
Diurnal Variation ◽  
Scale Height ◽  
Satellite Orbits ◽  
Third Order ◽  
Air Drag ◽  
Small Eccentricity ◽  
Density Scale

The effect of air drag on satellite orbits of small eccentricity, 0.01 ≾ е ≾ 0.2, is considered. A model of the atmosphere that allows for oblateness, and in which the density behaviour approximates to the observed diurnal variation, is adopted. The equation governing the changes due to drag in the argument of perigee ω , during one revolution of the satellite, is integrated with the assumption that the density scale height H is constant. The resulting expression for e ∆ ω is presented to third order in e . Compact expressions for e ∆ ω and e ∆ ω / ∆ T D , where ∆ T D is the corresponding change in the period, are obtained when H is allowed to vary with altitude. It is shown that there is an equivalence between the variable- H and the constant- H equations, provided that the value of H used in the latter is chosen appropriately.

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

1982 ◽  
Vol 383 (1784) ◽  
pp. 127-145 ◽  
Keyword(s):  
Major Axis ◽  
Density Variation ◽  
Scale Height ◽  
Satellite Orbits ◽  
Semi Major Axis ◽  
Third Order ◽  
Small Eccentricity ◽  
Air Density ◽  
The Mean ◽  
Density Scale

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.

Scale height in the upper atmosphere, derived from changes in satellite orbits

1962 ◽  
Vol 270 (1343) ◽  
pp. 562-587 ◽  
Keyword(s):  
Molecular Weight ◽  
Solar Activity ◽  
Air Temperature ◽  
Scale Height ◽  
Upper Atmosphere ◽  
Satellite Orbits ◽  
Small Eccentricity ◽  
Air Density ◽  
Density Scale ◽  
Eccentricity Orbit

The 'density scale height’ H in the upper atmosphere is a measure of the rate at which air density ρ varies with height y , being given by H = — ρ/(dρ/d y ). The value of H , although important because (with the molecular weight of the air) it determines the air temperature, has not as yet been well determined at heights above 200 km. This paper develops methods for finding H from the decrease in a satellite’s perigee height and from the decrease in the orbital period of a satellite in a small-eccentricity orbit. These methods are then applied to all the 14 satellites found suitable for the purpose. The 44 values of H obtained, for heights of 200 to 450 km, represent an average over day and night and probably have errors (s.d.) of 5 to 10 %. It is found that, as solar activity declined between 1957 and 1961, H decreased greatly: e.g. at height 275 km, H decreased from 60 km in early 1958 to 40 km in 1960-61. The results also show that the increase of H with height becomes much less rapid above 350 km, and are consistent with the supposition that H had low values, near 35 km, at heights near 250 km, for 1959-61, The results could be greatly extended in scope and improved in accuracy if more accurate orbits were available for short-lifetime satellites.

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.

Determination of upper-atmosphere air density and scale height from satellite observations

1959 ◽  
Vol 252 (1268) ◽  
pp. 16-27 ◽  
Keyword(s):  
Satellite Orbit ◽  
Major Axis ◽  
Scale Height ◽  
Rate Of Change ◽  
Satellite Observations ◽  
Upper Atmosphere ◽  
Semi Major Axis ◽  
Air Density ◽  
Standard Deviations

A solution is obtained for the rate of change of semi-major axis and perigee distance of a satellite orbit with time due to the resistance of the atmosphere. The logarithm of air density is assumed to vary quadratically with height, and the oblateness of the atmosphere is taken into account. The calculation of perigee air density in terms of the rate of change of satellite period is dealt with; and the method is applied to data at present available on six different satellites. The variation of air density with height is obtained as ln ρ = -28·59(±0·15) - ( h - 200 )/46(±5) + 0·028(±0·013) ( h - 200) 2 /(46) 2 for h in the range of approximately 170 to 700 km, where ρ is in grams/cm 3 , h is in kilometres and standard deviations are given in brackets.

The change in satellite mean anomaly due to a rotating oblate atmosphere with a diurnal variation in density

1984 ◽  
Vol 391 (1800) ◽  
pp. 201-214
Keyword(s):  
Atmospheric Model ◽  
Scale Height ◽  
Satellite Orbits ◽  
Atmospheric Drag ◽  
Linear Variation ◽  
Small Eccentricity ◽  
Anomalistic Period ◽  
Compact Expression ◽  
The Mean ◽  
Density Scale

The effect of atmospheric drag on satellite orbits of small eccentricity e ≲ 0.2, is considered. The atmospheric model allows for oblateness, and has a density profile that approximates to the observed day-to-night variation. The equation governing the changes due to drag in the mean anomaly M , during one revolution of the satellite is integrated, assuming that H , the density scale height is constant. Two particular cases are detailed. In the first, the change ∆ M in M over one anomalistic period is given for eccentricity e in the range 0.01 ≲ e ≲ 0.2, and in the second the change in ( M + ω) over one draconic period, ∆( M + ω) is given for eccentricity e in the range 0 ≼ e ≲ 0.01, where ω is the argument of perigee. Finally a compact expression for e ∆ M is derived for an atmosphere with a linear variation of H with altitude, if 0.01 ≲ e ≲ 0.2. This is the final paper in a series describing the secular and long-periodic effects of drag on the six Keplerian elements that determine the orbit of a satellite.

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