Non-Keplerian Motion
5.2 Special Perturbation Methods
5.2.1 Non-Spherical Central Body
The first perturbation we will consider is the non-spherical shape of the central gravi-tational body. Recall that in Chapter 3, we used an ellipsoidal model for the Earth: that is, a“flattened” sphere where the equatorial radius is about 21 km greater than the polar radius. In addition, the Earth has uneven mass distribution that varies with latitude and longitude. Spherical harmonics are used to model a planet’s surface and subsequent grav-itational field. We will provide a brief introduction to this topic by focusing on the spher-ical harmonics that represent the“oblate ellipsoid” shape of the Earth.
We may model the Earth’s (or any planet’s) gravitation field using a scalar potential function U(r,λ, ϕ ) that depends on radius r, longitude λ, and geocentric latitude ϕ (recall that geocentric latitude is measured from the equatorial plane to a line from the center of the gravitational body to the surface; see Figure 3.17). For the Earth, we will call U(r,λ, ϕ ) the geopotential function. The gravitational acceleration of an Earth-orbiting satellite is the gradient of the geopotential function. To show this, let us consider the two-bodygeopotential function
U2b r =μ
r (5.4)
Note that we have used the subscript“2b” to indicate a “two-body” potential function.
The two-body geopotential function U2b is the negative potential energy and only depends on radius r (there is no dependence on latitude or longitude because the two-body problem assumes that the Earth is a homogeneous sphere). The absolute accel-eration due to gravity is the gradient of the geopotential function
r = ∇U2b r (5.5)
where the“del” or vector differential operator for the ECI Cartesian frame is
∇ = ∂
∂xI + ∂
∂yJ + ∂
∂zK (5.6)
Recall thatIJK are unit vectors associated with the ECI coordinate system. The satellite’s position vectorr has ECI components (x, y, z), or
r = xI + yJ + zK (5.7)
and the magnitude of the position vector is r = x2+ y2+ z2. Therefore, the two-body geopotential (5.4) is rewritten as
U2b= μ
x2+ y2+ z2 (5.8)
The gradient of the two-body geopotential is computed by applying the“del” operator defined by Eq. (5.6):
∇U2b= −μx
x2+ y2+ z2 3/2I + −μy
x2+ y2+ z2 3/2J + −μz
x2+ y2+ z2 3/2K (5.9) Note that the common denominator term in Eq. (5.9) is r3. Furthermore, we can substi-tuter = xI + yJ + zK so that Eq. (5.9) becomes
∇U2b=−μ
r3r (5.10)
Using this result as the right-hand side of Eq. (5.5), we obtain r = −μ
r3r (5.11)
Equation (5.11) is the governing equation of motion for the two-body problem. This simple exercise shows that the governing equation for two-body (Keplerian) motion can be derived from the two-body geopotential function U2b=μ/r.
We desire a more accurate representation of the Earth’s gravitational field (in partic-ular, the gravitational field of an oblate, flattened sphere). Let us express the total geopotential function U(r,λ, ϕ ) as the sum of a two-body potential function and a disturbingpotential function
U r,λ,ϕ =μ
r+ R r,λ,ϕ (5.12)
It should be clear thatμ/r is the two-body potential in Eq. (5.12) and R(r, λ, ϕ ) is the dis-turbing potential function. The disdis-turbing potential function R(r,λ, ϕ ) represents per-turbations due to a non-spherical Earth with an uneven mass distribution. Hence, it depends on radius, longitude, and geocentric latitude. It is possible to express the dis-turbing potential function R in terms of spherical harmonics or periodic functions on the surface of a unit sphere. These spherical harmonics consist of zonal harmonics (bands of latitude), sectoral harmonics (sections of longitude), and tesseral harmonics (“checkerboard tiles” that depend on latitude and longitude). Characterizing the disturb-ing potential R usdisturb-ing a complete set of spherical harmonic functions is beyond the scope of this textbook (the interested reader may consult Vallado [1; pp. 538–550] for details).
Instead, let us focus on a total geopotential function that is axially symmetric about theK axis and therefore only depends on radius and latitude:
U r,ϕ =μ r 1− ∞
k= 2
Jk RE
r
k
Pk sinϕ (5.13)
where Jkare the zonal harmonic coefficients, REis the equatorial radius of the Earth, and Pk is a Legendre polynomial of order k. The input to the Legendre polynomial is sinϕ = z/r. The Legendre polynomials represent the “harmonic fluctuations” of the Earth’s surface relative to a spherical shape as latitude varies. The dimensionless zonal coefficients Jkrepresent the“dips” and “bulges” of the Earth’s surface (relative to a sphere) and they are empirically determined from satellite observations. Zonal coefficient J2 models the Earth’s “bulge” at its equator and it is nearly 1000 times larger than all other Jkcoefficients. Therefore, if we only consider the J2coefficient, Eq. (5.13) becomes
U r,ϕ =μ r 1−J2
RE r
2
P2 sinϕ (5.14)
Equation (5.14) is the total geopotential function for an oblate Earth or a“flattened” sphere.
This geopotential function has axial symmetry about the Earth’s polar axis (i.e., the Earth’s mass is equally distributed with longitude), and the second-order Legendre polynomial P2(sinϕ ) models the Earth’s mass bulge at its equator. The reader should note that Eq. (5.14) represents the simplest possible non-spherical model of the Earth because we have neglected zonal harmonics greater than order 2 as well as all sectoral and tesseral harmonics.
Finally, the reader should note that if we set J2= 0 (no oblateness) in Eq. (5.14), then we have U r =μ/r and we are back to a spherical Earth and two-body (Keplerian) motion.
The reader may have some difficulty comprehending or visualizing the geopotential function Eq. (5.14) that models Earth’s equatorial bulge. In order to enhance our under-standing of the geopotential function, let us plot U(r,ϕ ) for various radial distances from the Earth’s center. First, we must present the second-order Legendre polynomial
P2 u =1
2 3u2−1 (5.15)
where the input is u = sinϕ . In order to evaluate Eq. (5.14), we also need numerical values for Earth’s equatorial radius and Earth’s second zonal harmonic coefficient J2. Let us use RE= 6,378.14 km, J2= 0.0010826267, andμ = 3.986(105) km3/s2(Earth’s gravitational parameter). Using these values in Eq. (5.14), we can evaluate the geopotential function U(r,ϕ ) for a fixed radius r and geocentric latitude ϕ ranging from –90 (South Pole) to +90 (North Pole) [of course we must use Eq. (5.15) to evaluate the second-order Legendre polynomial as latitude varies]. Figure 5.1 shows how the oblate geopotential
–90 –60 –30 0 30 60 90
Geocentric latitude, deg 59.4
59.45 59.5 59.55 59.6 59.65 59.7 59.75 59.8
Two-body geopotential
r = RE+ 300 km Oblate geopotential r = RE+ 300 km
Oblate geopotential r = RE+ 310 km
Oblate geopotential r = RE+ 320 km Geopotential function, U, km2/s2
Figure 5.1 Oblate geopotential function [Eq. (5.14)] vs. latitude: polar low-Earth orbit.
function varies with latitude for a polar low-Earth orbit (LEO). It is clear that U(r,ϕ ) exhibits a“bulge” at the equator (ϕ = 0) and a minimum value at the poles. Figure 5.1 also shows that the“strength” of the oblate geopotential function diminishes as radial distance increases. The two-body geopotential function evaluated at rLEO = RE + 300 km [i.e., U rLEO =μ/rLEO] is shown in Figure 5.1 as the dashed line. Because the two-body geopotential corresponds to a spherical, homogeneous Earth, it does not exhibit any variation with latitude. Figure 5.2 shows the oblate geopotential function for circular polar orbits with radii approximately equal to the radius of geostationary orbit, that is, 42,164 km. Note that the geopotential function’s “equatorial bulge” is sig-nificantly reduced for near-geostationary orbits (i.e., the geopotential function appears to behave more like the two-body potential). Therefore, we can expect that an oblate-Earth gravity model will exhibit a more pronounced effect on satellites in low-altitude orbits as compared with high-altitude orbits.
The next step is to take the gradient of Eq. (5.14) in order to determine the satellite’s absolute acceleration in an oblate-Earth gravity field. After substituting Eq. (5.15) for the Legendre polynomial, Eq. (5.14) becomes
U r,ϕ =μ r 1−J2
2 RE
r
2 3z2
r2 −1 (5.16)
Note that we have used u = sinϕ = z/r in the Legendre polynomial (5.15). We must also substitute r = x2+ y2+ z2 and r2= x2+ y2+ z2 into Eq. (5.16) so that the geopotential function U is in terms of Cartesian coordinates (x,y,z). The satellite’s absolute accelera-tion due to the oblate-Earth gravity field is the gradient of Eq. (5.16):
–90 –60 –30 0 30 60 90
Geocentric latitude, deg 9.43
9.44
9.435 9.445 9.45 9.455 9.46
r = 42,250 km r = 42,200 km r = 42,164 km (geostationary)
Geopotential function, U, km2/s2
Figure 5.2 Oblate geopotential function [Eq. (5.14)] vs. latitude: geostationary orbital radius.
r = ∇U r,ϕ =∂U
∂xI +∂U
∂yJ +∂U
∂zK (5.17)
After taking the partial derivatives (and performing some algebra), we can write the three acceleration components in the ECI frame:
x=∂U
∂x =−μx r3 1−J2
3 2
RE
r
2 5z2
r2 −1 (5.18)
y=∂U
∂y =−μy r3 1−J2
3 2
RE
r
2 5z2
r2 −1 (5.19)
z=∂U
∂z =−μz r3 1−J2
3 2
RE
r
2 5z2
r2 −3 (5.20)
Equations (5.18)–(5.20) are the absolute acceleration components of a satellite orbiting an oblate spheroid Earth. These equations fit the form of Eq. (5.3), the perturbed two-body equations of motion. Note that the terms outside the brackets in Eqs. (5.18)–(5.20) are the two-body gravitational components of −μr/r3. The terms involving J2are the components of the perturbing accelerationaP.
Now we can apply the special perturbation method to a satellite orbiting a non-spherical (oblate) Earth. Numerically integrating Eqs. (5.18)–(5.20) will yield the velocity components,v = x y z T, and numerically integratingv will yield the position vector, r = x y z T. The following example demonstrates the special perturbation technique for determining the non-Keplerian motion of an Earth satellite.
Example 5.1 Use the special perturbation method to obtain the non-Keplerian motion of a LEO that is perturbed by Earth-oblateness (J2) effects. The initial orbital elements at time t = 0 are
Semimajor axis a0= 8,059 km Eccentricity e0= 0.15
Inclination i0= 20
Longitude of the ascending nodeΩ0= 60 Argument of perigeeω0= 30
True anomalyθ0= 50
The special perturbation method requires that we numerically integrate the satellite’s perturbed equations of motion. Equations (5.18)–(5.20) are the satellite’s absolute accel-eration components due to central-body gravity perturbed by the Earth-oblateness (or J2) effect. We can use a numerical integration algorithm (such as a Runge–Kutta scheme) to integrate Eqs. (5.18)–(5.20) from time t = 0 to an arbitrary final time t = t1. Of course, we must integrate these three acceleration equations to obtain the ECI velocity vectorv, and integrate the three velocity components to obtain the ECI position vectorr. The numer-ical integration must begin at the initial ECI state vector (r0,v0) associated with the initial LEO orbital elements. Using the coordinate transformation algorithm presented in Section 3.5, the initial Cartesian coordinates are
r0=
−5,134 41 4, 405 01 2, 420 05
km , v0=
−5 5265
−5 5142 0 7385
km/s
Here we use MATLAB’s M-file ode45.m to numerically integrate Eqs. (5.18)–(5.20) starting from the initial state vector (r0,v0). The final end time is set at 10 h (note that because semimajor axis is 8,059 km, the orbital period is 120 min = 2 h). The numerical values of the constants used here are J2 = 0.0010826267, RE = 6,378.14 km, and μ = 3.986004(105) km3/s2. Numerical integration produces ECI vectors r(t) and v(t).
Semimajor axis, a, km
0 1 2 3 4 5 6 7 8 9 10
Time, hr 8052
8054 8056 8058 8060 8062 8064 8066
(a) (b)
(c) (d)
(e)
a0 = 8059 km
1 orbital period
Inclination, i, deg
0 1 2 3 4 5 6 7 8 9 10
Time, hr 19.99
19.995 20 20.005 20.01 20.015 20.02 20.025 20.03
i0 = 20 deg 1 orbital period
Longitude of the ascending node, Ω, deg
0 1 2 3 4 5 6 7 8 9 10
Time, hr 58
58.5 59 59.5 60 60.5
Secular change in Ω
0 1 2 3 4 5 6 7 8 9 10
Time, hr 29
29.5 30 30.5 31 31.5 32 32.5 33 33.5 34
Argument of perigee, ω, deg
Secular change in ω
Eccentricity, e
0 1 2 3 4 5 6 7 8 9 10
Time, hr 0.148
0.1485 0.149 0.1495 0.15 0.1505 0.151
e0 = 0.15
1 orbital period
Figure 5.3 LEO with J2perturbation: (a) semimajor axis; (b) eccentricity; (c) inclination; (d) longitude of the ascending node; and (e) argument of perigee (Example 5.1).
Because the histories of these Cartesian coordinates provide very little insight into how the orbit is affected by the J2perturbation, the state vector (r,v) simulation data are trans-formed to orbital elements using the methods presented in Section 3.4.
Figure 5.3 presents the time histories of the five classical orbital elements after numerical integration of the perturbed equations of motion. Figure 5.3a shows that semimajor axis a is perturbed by the Earth’s oblateness and varies during an orbital revolution (recall that the orbital period is 2 h). However, semimajor axis returns to its initial value (a0= 8,059 km) at the end of every 2 h orbit. This behavior shows that the J2perturbation is conservative because its net effect on energy is zero over an orbital revolution. Figure 5.3b shows that eccentricity also exhibits oscillations during each 2-h orbit and Figure 5.3c shows that incli-nation has two periodic cycles each revolution. Figures 5.3a–c show that Earth’s oblateness causes very small periodic variations in semimajor axis, eccentricity, and inclination and that the net change is zero for these three elements over each orbital revolution. The obl-ateness perturbation, however, causes the longitude of the ascending nodeΩ and argu-ment of perigee ω to drift over time as illustrated in Figures 5.3d and 5.3e. Periodic fluctuations inΩ and ω are evident, but these oscillations are superimposed on to a linear function with time. The secular changes inΩ and ω are the linear variations with time and are illustrated by the dashed lines in Figures 5.3d and 5.3e. Figure 5.3d shows that Ω diminishes by about 1.8 over 10 h and hence the ascending node vector n is drifting westward at an average rate of 0.18 deg/h or 4.32 deg/day. Figure 5.3e shows that the argument of perigeeω increases by about 3.25 over 10 h. Therefore, the perigee direction e is drifting away from the ascending node n at an average rate of 0.325 deg/h or 7.8 deg/day.
Example 5.1 is a demonstration of the special perturbation method where the only per-turbation is due to a non-spherical (oblate) Earth. Numerical integration of the perturbed equations of motion illustrates that Earth oblateness causes periodic variations in the orbital elements a, e, and i but with a net zero change after a full revolution. The longi-tude of ascending nodeΩ and argument of perigee ω show periodic and secular changes.
The dashed lines in Figures 5.3d and 5.3e show the secular changes (or linear drift) inΩ andω. Because Ω shows a steady drift rate, the orbital plane is rotating about the Earth’s pole. The linear drift inω indicates that the apse line is rotating in the orbital plane about the angular momentum vectorh.
We should reiterate that Example 5.1 has demonstrated non-Keplerian motion where the only perturbation is due to an oblate Earth. Furthermore, this example illustrates the
“special” part of the special perturbation method; that is, we gleaned the oblateness effect only after performing numerical integration and plotting the results. The secular changes inΩ and ω pertain to the “specific” or “special” initial orbit presented in Example 5.1. At this point, we cannot make general statements regarding the secular drift rates forΩ and ω. General perturbation methods use analytical techniques to develop “general” expres-sions that convey the effects of perturbations. We will revisit and characterize the Earth-oblateness effect with general perturbation methods in the next section.