Engineering NotesNear-Circular Orbits in a Central Force

Field with Variable Drag

Mayer Humi∗

Worcester Polytechnic Institute, Worcester,

Massachusetts 01609

DOI: 10.2514/1.57678

I. Introduction

T HE search for an analytical approach to orbit determination andrelative-motion studies has taken place over many years using

various assumptions [1–28]. Some representative studies of two-bodyorbits and their related relative motion are found in [2–4,7–10]. Someearly studies include the J2 effect and atmospheric drag [1,6,11,12].Much of the early work is summarized in the paper by Liu [13]. Somelater studies have concentrated on atmospheric drag [14,16,18,19] ofa satellite in the gravitational field of a spherical body and drag-freeorbits in the gravitational field of an oblate body [20–26].Some of the early work, as summarized by Liu [13], includes

atmospheric drag with rotating atmosphere and orbits of arbitraryinclinations about an oblate planet. Other effects are sometimestreated as higher-order perturbations. Typically, a Keplerian solutionto the two-body problem is adjusted through a method of variation ofconstants with approximations found from dominant terms in a seriesexpansion. One of the most important of these approaches is found inthework of Brouwer andHori [11]. They use Delaunay variables, buttransform the equations of motion into canonical variables for thedrag-free problem. The drag acceleration is expanded in powers ofeccentricity and in multiples of the mean anomaly and then inte-grated. Although the approximating formulas are in analytic form,they are long and cumbersome.In a recent series of papers, Carter and Humi [14–16,18,19,29]

considered the motion of a satellite in a central force field withquadratic drag and a stationary atmosphere. For nearly circular orbits,in which the radial velocity of the satellite is negligible and theatmospheric density gradient is radial, closed-form approximateanalytic solutions for the orbit were found. In all these papers, thedrag coefficientwas considered to be constant. It was demonstrated inthese papers that deviation between the analytic closed-form for-mulas and the numerical solution of the exact equations ofmotion didnot exceed 10 m after 160 revolutions [27,28].However, as a matter of fact, the expression for the drag force FD

acting on a satellite is

FD �SCD2

ρv2

where ρ is the atmospheric density, v is the satellite speed, S is itscross-sectional area normal to the direction of motion, and CD is thedrag coefficient, which, for satellites, is assumed to be equal to two.

However, experimentally, it was found that CD and the “effectivecross-sectional area” of the satellite can vary widely. For this reason,these two parameters are usually lumped together as the “effectivedrag coefficient,”which is taken to be constant. Additional variationsin the drag force are due to the periodic variations in the ther-mospheric density [30]. Motivated by these observations, it is theobjective of this Note to study near-circular satellite orbits under theinfluence of a drag force, which vary with time, namely,

FD � αD�t�ρv2

where D�t� is a general (continuous) function and α is an orderparameter.The outline of the Note is as follows: Sec. II introduces the basic

equations ofmotion and derives the orbit equation in a general centralforce field subject to quadratic drag. Section III discusses approx-imations to the expression of the angularmomentum for near-circularorbits. Section IV presents approximate analytic expressions fornear-circular orbits with time-varying drag force around a sphericalbody where the atmospheric density is modeled by the standard“exponential density model.” Section V does the same for near-circular equatorial orbits around an oblate body. SectionVI comparesthe orbits obtained from these analytic expressions with the numer-ical solutions of the exact equations of motion.

II. Basic Equations

Themotion of a satellite in a central force field with quadratic dragand a stationary atmosphere is modeled by the following equation:

�r � −f�r�r − αD�t�ρ�r��_r · _r�1∕2 _r (1)

Two special cases of this equation are important in many appli-cations. For motion around a spherical body,

f�r� � μ

r3(2)

Similarly, the motion in the equatorial plane of an oblate body(including the J2 effect) is modeled by

f�r� � μ

�1

r3� 3R2J2

2r5

�(3)

In these equations, r is the radius vector and r � jrj, ρ�r� is theatmospheric density, μ is a constant that characterizes the gravi-tational field, andαD�t� stands for the drag coefficient. [Typicallyα isO�10−12�]. R is the equatorial radius of the body and dots representdifferentiation with respect to time.When the motion of a satellite is modeled by Eq. (1), the resulting

orbit is in a plane. In this plane, one can use standard polar coor-dinates (r, θ) to describe the orbit of the satellite (where θ representsthe true anomaly). The equations of motion become

r�θ� 2_r _θ � −αD�t�ρ�r��_r · _r�1∕2r_θ (4)

�r − r_θ2 � −f�r�r − αD�t�ρ�r��_r · _r�1∕2 _r (5)

Multiplying Eq. (4) by r and integrating, it follows that

J � r2 _θ � h exp

�−αZD�t�ρ�r��_r · _r�1∕2 dt

�(6)

Received 19 January 2012; revision received 24 May 2012; accepted forpublication 9 June 2012; published online 11 December 2012. Copyright ©2012 by the American Institute of Aeronautics and Astronautics, Inc. Allrights reserved. Copies of this paper may be made for personal or internal use,on condition that the copier pay the $10.00 per-copy fee to the CopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; includethe code 1533-3884/12 and $10.00 in correspondence with the CCC.

*Professor of Mathematics, Department of Mathematical Sciences, 100Institute Road; mhumi@wpi.edu.

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where h is a constant of integration that represents the initial valueof J. Using the fact that

1

J

dJ

dt� −αD�t�ρ�r��_r · _r�1∕2 (7)

rewriting Eq. (5) as

�r − r_θ2 � −f�r�r� 1

J

dJ

dt_r (8)

and changing the independent variable from t to θ, one obtains (aftersome algebra) the orbit equation

rr 0 0 − 2�r 0�2 � r2 − f�r�r6

J2(9)

where primes denote differentiation with respect to θ. This is anintegrodifferential equation, which can be solved analytically only bymaking proper approximations for J.

III. Approximations for J

From Eq. (6), it is easy to see that one of the hurdles to the explicitevaluation of J is due to the term

�_r · _r�1∕2 ���������������������_r2 � r2 _θ2

q(10)

To overcome this problem, the approximation

��������������������_r2 � r2 _θ2

q� r_θ (11)

was used in [14,18,19]. That is, it was assumed that the radialcomponent of the velocity is very small relative to the transversecomponent of the velocity (i.e., j_rj≪ jr_θj). This assumption is exactlytrue only for circular orbits. It is also a reasonable approximation fororbits that are initially elliptic with very low eccentricity [29].Under this approximation, the expression for J becomes

J � h exp

�−αZD�θ�ρ�r�r dθ

�(12)

IV. Almost-Circular Orbits Around a Spherical Body

Using the approximation given by Eq. (12), one can derive theexpression for the orbit equation using the exponential density modelthat is used routinely in the literature to model the Earth atmosphericdensity as a function of height:

ρ�r� � A exp

�R0 − rH

�

where A, R, andH are constants. (In the following, the constant A islumped with α.)Substituting this form of ρ�r� in Eq. (12), the orbit equation (9)

becomes

rr 0 0 − 2�r 0�2 � r2 − r6f�r�h2

exp

�2α

ZD�θ�r exp

�R0 − rH

�dθ

�

(13)

Introducing the transformation r�θ� � 1∕u�θ� and f�r� � μ∕r3, thisequation becomes

u 0 0 � u � μ

h2exp

�2α

ZD�θ�u

exp

�R0 − 1∕u

H

�dθ

�(14)

Because α is a small parameter, it is appropriate to solve this equationin the form

u�θ� � u0�θ� � αu1�θ� (15)

Substituting this expression in Eq. (14) and using first-order Taylorexpansion in α, one obtains to order zero

u 0 00 � u0 �μ

h2(16)

whose general solution is

u � μ

h2�1� e cos�θ� ϕ�� (17)

where e and ϕ are constants. To first order in α, one obtains

u 0 01 � u1 �μ

h2

Z �2D�θ�u0

exp

�R0 − 1∕u0

H

�dθ

�(18)

When the initial orbit is circular and u0�θ� � μ∕h2, Eq. (18) reducesto a linear inhomogeneous equation

u 0 01 � u1 − 2 exp

�R0μ − h2

μH

�Zθ

θ0

D�s� ds � 0 (19)

For initial orbits that are near circular, where e≪ 1, one can approxi-mate Eq. (18) bymaking a first-order expansion of the integral on theright-hand side in powers of e, which yields

u 0 01 � u1 − 2 exp

�R0μ − h2

μH

�

�e

�h2

μH− 1

�Zθ

θ0

D�s� cos�s� ds�Z

θ

θ0

D�s� ds��O�e2� � 0

(20)

Equations (19) and (20) are linear inhomogeneous equations that canbe solved by standard methods once D�θ� has been specified.We now exhibit some explicit solutions for u1 with variable drag

functions.Example: If D�θ� � 1� cos�ωθ� (where ω is a nondimensional

constant), then the general solution of Eq. (19) with θ0 � 0 is

u1 � C1 cos θ� C2 sin θ� 2 exp

�R0μ − h2

μH

��θ −

sin ωθ

ω�ω2 − 1�

�;

ω ≠ �1 (21)

When ω � �1, the second term in D�θ� is in resonance with one ofthe solutions of the homogeneous part of Eq. (19) and we have

u1 � C1 cos θ� C2 sin θ� exp

�R0μ − h2

μH

�θ�2 − cos θ� (22)

When the initial orbit is not circular (i.e., e ≠ 0), the modifiedexpression for u1 [using Eq. (20)] is

u1 � C1 cos θ� C2 sin θ� 2 exp

�R0μ − h2

μH

��θ −

sin ωθ

ω�ω2 − 1�

�

� e�μH − h2� exp�R0μ − h2∕μH�μHω

�

sin��ω − 1�θ��ω − 1��ω − 2� �

sin��ω� 1�θ��ω� 1��ω� 2� � θω cos θ� ω

2sin θ

�;

ω ≠ �1;�2 (23)

When ω � �1 or ω � �2, the expression for u1 is given,respectively, by

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u1 � C1 cos θ� C2 sin θ� exp

�R0μ − h2

μH

�θ�2 − cos θ�

� e exp

�R0μ − h2

μH

��h2

μH− 1

��θ −

�1

3sin θ� θ

�cos θ

�

(24)

u1 � C1 cos θ� C2 sin θ� exp

�R0μ − h2

μH

��2θ −

1

3sin 2θ

�

� e exp

�R0μ − h2

μH

��h2

μH− 1

��1

6sin3 θ� 5

8sin θ −

3

2θ cos θ

�

(25)

Example:D�θ� � r 0�θ�r�θ� . Substituting this form ofD�θ� in Eq. (13),

evaluating the integral, and making the substitution u�θ� � 1∕r�θ�,one obtains the following equation for u

d2u

dθ2� u − μ

h2exp

�−2Hα

�exp

�R0u − 1

uH

�− 1

��� 0 (26)

Circular orbits u�t� � v0, which are the solutions of this equation,must satisfy

v0 �μ

h2exp

�−2Hα

�exp

�R0v0 − 1

v0H

�− 1

��(27)

To order α, these solutions are given (implicitly) by

v0 �μ

h2

�1 − 2αH

�exp

�R0v0 − 1

Hv0

�− 1

���O�α2� (28)

For near-circular orbits, one can use Eq. (15). The solution for u0 isgiven by Eq. (17) and the equation for u1 to first order in e is

u 0 01 � u1 − 2μH

h2� 2

�μH

h2� e cos θ

�exp

�R0μ − h2

μH

�� 0 (29)

The general solution of this equation is

u1 � C1 cos θ� C2 sin θ� 2μH

h2

−�2μH

h2� e�cos θ� θ sin θ�

�exp

�R0μ − h2

μH

�(30)

V. Almost-Circular Equatorial Orbits Around anOblate Body

Although a spherical body is a special case of an oblate body, thesetwo cases are treated separately because they require differenttransformations and approximations.The orbit equation (9) for the gravitational force in Eq. (3) is

r 0 0

r− 2

�r 0

r

�2

� 1 −μ

J2

�r� k

r

�(31)

where k � 3∕2R2J2 and the expression for J is given by Eq. (12).Introducing the transformation r � 1∕u, this equation becomes

u 0 0 � u � μ

h2�1� ku2� exp

�2α

Z �D�θ�u

exp

�R0 − 1∕u

H

��dθ

�

(32)

To solve this equation, one uses the approximation given by Eq. (15).For u0, one obtains

u 0 00 � u0 �μ

h2�1� ku20� (33)

For circular orbits, this equation reduces to the following algebraicequation:

u0 �h2 −

���������������������h4 − 4μ2k

p2μk

; k ≠ 0 (34)

In this formula, the minus sign in front of the square root is chosento insure that as k → 0, u0 → μ∕h2. We assumed also thath4 − 4μ2k ≥ 0.The equation for u1 becomes

u 0 01 ��1 −

2μku0h2

�u1

� 2μ

h2�1� ku20�

exp��R0u0 − 1�∕�u0H��u0

Zθ

θ0

D�s� ds (35)

This is a linear inhomogeneous equation which can be solved bystandard techniques once D�θ� has been specified.In themore general case (where u0 is not constant), Eq. (33) can be

solved in terms of Weierstrass’ elliptic function P [29,30] or ellipticintegrals [31]. The equation for u1�t� becomes

u 0 01 ��1 −

2μku0�θ�h2

�u1

� 2μ

h2�1� ku20��

Zθ

θ0

D�s� exp��R0u0�s� − 1�∕�u0�s�H��u0�s�

ds (36)

Dividing this equation by 2�μ∕h2��1� ku20�θ�� and differentiatingthe result with respect to θ, one obtains a third-order equation for u1[29], which can be solved numerically.Example: For D�θ� � 1� cos�ωθ� with u0 as a constant [see

Eq. (34)], the solution for u1 is

u1 � C1 cos�βθ� � C2 sin�βθ� − 2μ�1� ku20� exp�R0u0 − 1

u0H

�

�

sin�ωθ�u0ω�h2�ω2 − 1� � 2μku0�

−θ

u0�h2 − 2μku0�

�(37)

where β ������������������������h2 − 2μku0

p∕h, where we assumed that h2 − 2μku0 is

positive. (This is a plausible assumption because, for almost-circularorbits, u0 ≈ h2∕μ).Example: D�θ� � r 0�θ�∕r�θ�. Substituting this function in

Eq. (32) and evaluating the integral yields the following differentialequation for u�θ�

u 0 0 � u � μ

h2�1� ku2� exp

�−2αH

�exp

�R0u − 1

uH

���(38)

Setting u � u0 � αu1, one obtains for u0 the same solution as inEq. (33). For constant u0 [see Eq. (34)], one derives the followingresult for u1:

u1 � C1 cos�βθ� � C2 sin�βθ�

−2μH�1� ku20� sin�ωθ�u0ω�h2 − 2μku0�

exp

�R0u0 − 1

u0H

�(39)

VI. Model Verification

In this section, a test is performed to find the accuracy of theanalytical models presented in the preceding sections and their rangeof applicability. To do so, the orbits obtained from the analyticalformulas are compared with those obtained from the numericalsimulation of the exact equation of motion [namely, Eqs. (4) and (5)]using the same variable drag coefficient function. In all cases, thesimulations started from a circular orbit of 6700 km from the Earthcenter for a duration of 160 revolutions. The drag “order parameter”in all simulations was α � 10−11. This led to a drop of approximately

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900 m after 160 revolutions. For the numerical simulations, animplicit adaptive step Runge–Kutta method of order 4–5 with step-wise accuracy of 10−13was used. Furthermore, all computationswerecarried out in 128 bit mode.The maximum deviation between the analytical and numerical

orbits throughout the duration of these simulations as a function of θfor a spherical body with D�θ� � 1� cos�θ� (first example inSec. IV) is presented in Fig. 1. The corresponding orbits are shown inFig. 2. Similarly, for an oblate bodywith the sameD�θ� (first examplein Sec. V), the same results are presented in Figs. 3 and 4. Figure 5 is azoom on part of the trajectory of Fig. 4, which exhibits the finestructure of the orbit due to the fluctuations in the drag.It follows from these figures that the maximum deviation between

the analytic formulas and the numerical solution of the exact equationof motion is less than 6 m after 160 revolutions.

VII. Conclusions

Several recent papers considered near-circular satellite orbitsaround spherical and oblate bodies under the influence of quadraticdrag with constant drag coefficient. The results obtained in thesepapers show excellent agreement between the analytical formulas forthese orbits and the numerical simulation of the exact equations ofmotion. The present study is an extension of these studies to asituation where the drag coefficient varies with time. These changesin the drag coefficient may be the result of many factors as doc-umented in various studies.The formulas for satellite orbits under the influence of variable

drag coefficient were derived using a perturbation expansions in the

0 100 200 300 400 500 600 700 800 900 1000−7

−6

−5

−4

−3

−2

−1

0

1

2x 10

−3

θ (radians)

Dev

iatio

n (k

m)

Fig. 1 Differences between analytic and exact numerical solutions.

Spherical body withD�θ� � 1� cos θ.

0 100 200 300 400 500 600 700 800 900 10006699.1

6699.2

6699.3

6699.4

6699.5

6699.6

6699.7

6699.8

6699.9

6700

θ (radians)

heig

ht (

km)

Fig. 2 Satellite orbit around a spherical body as a function of θ with

D�θ� � 1� cos θ.

0 100 200 300 400 500 600 700 800 900 10006699.1

6699.2

6699.3

6699.4

6699.5

6699.6

6699.7

6699.8

6699.9

6700

θ (radians)

heig

ht (

km)

Fig. 4 Satellite orbit around an oblate body as a function of θ with

D�θ� � 1� cos θ.

914.8 915 915.2 915.4 915.6 915.8 916 916.26699.1785

6699.179

6699.1795

6699.18

6699.1805

6699.181

θ (radians)

heig

ht (

km)

Fig. 5 Zoom of part of the orbit of a satellite around an oblate body as

depicted in Fig. 4 showing its fine structure.

0 1 2 3 4 5 6 7 8 9x 10

5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2 x 10−3

θ (radians)

Dev

iatio

n (k

m)

Fig. 3 Differences between analytic and exact numerical solutions.

Oblate body withD�θ� � 1� cos θ.

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drag coefficient and the eccentricity. The accuracy of these newformulas was verified and they were found to be as accurate as thosethat were derived for nearly circular orbits with constant dragcoefficients. (The comparison in each case is between the analyticalsolution and a high-accuracy numerical solution of the exactequations of motion with the same drag coefficient.)

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[28] Humi, M., “Low Eccentricity Elliptic Orbits in a Central Force Fieldwith Drag,” Journal of Guidance, Control, and Dynamics, Vol. 33,No. 5, 2010, pp. 1368–1375.doi:10.2514/1.48693

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[30] Xu, J., Wang, W., Lei, J., Sutton, E. K., and Chen, G., “The Effectof Periodic Variations of Thermospheric Density on CHAMP andGRACE Orbits,” Journal of Geophysical Research, Vol. 116, 2011,p. A02315.doi:10.1029/2010JA015995

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