Non-Keplerian Motion
5.4 Gauss’ Variation of Parameters
Example 5.4 Compute the secular drift rates for the longitude of the ascending node and argument of perigee for the satellite orbit in Example 5.1. Compare the analytical secular change in Ω and ω with the numerical simulation results presented in Figures 5.3d and 5.3e.
Recall that the orbital elements from Example 5.1 are Semimajor axis a = 8,059 km Eccentricity e = 0.15
Inclination i = 20 The mean motion of this orbit is
n= μ
a3= 0 000872664 rad/s
Using this mean motion with J2= 0.0010826267 and RE= 6,378.14 km in Eq. (5.70), the secular change in ascending node is
dΩ
dt = −3nJ2
2 1−e2 2 RE
a
2
cosi =– 8 7297 10−7 rad/s Or, in degrees per day we have dΩ/dt = −4 32 deg/day
Recall that Figure 5.3d (Example 5.1) shows the periodic and secular changes in the lon-gitude of the ascending node resulting from numerical integration (i.e., special perturba-tion methods). The dashed line in Figure 5.3d is the secular drift inΩ. The longitude of the ascending node decreases approximately–1.8 over 10 h (or 0.4167 days) so the approx-imate secular rate is (–1.8 )/(0.4167 days) = –4.32 deg/day. Hence, an approximate linear fit through the numerically simulated responseΩ(t) shows a good match with the analytically determined mean drift rate dΩ/dt.
Equation (5.73) gives us the secular change in argument of perigee dω
dt = 3nJ2
4 1−e2 2 RE
a
2
4−5sin2i = 1 5863 10−6 rad/s Or, in degrees per day dω/dt = 7 85 deg/day
Figure 5.3e shows that argument of perigee increases by about 3.25 over 10 h or 7.8 deg/day. Again, the analytical secular change exhibits a good match with the approx-imate linear fit through the simulation results.
secular changes in the elements. Example 5.1 demonstrated this approach, where the total acceleration is the gradient of a geopotential function that includes a single zonal harmonic term (J2) associated with Earth’s oblateness. Next, we presented a general per-turbation method where the goal is to derive analytical expressions for the non-Keplerian motion. We outlined Lagrange’s variation of parameters which culminated with six first-order differential equations for the orbital elements. Here the time-rates of the elements are in terms of a disturbing function R, and therefore Lagrange’s variation of parameters applies to conservative perturbations such as a non-spherical central body and third-body accelerations. We applied orbital averaging techniques to Lagrange’s equations and developed analytical expressions for the secular (or mean) changes in the orbital ele-ments due to oblateness. The zonal harmonic J2has a zero net effect on semimajor axis, eccentricity, and inclination but produces a secular change in the longitude of ascending nodeΩ and argument of perigee ω.
Gauss developed a form of the variation of parameters where the perturbing accelera-tions are expressed in a satellite-based coordinate frame that moves with the vehicle.
Gauss’ variation of parameters can handle conservative and non-conservative perturba-tions; the only constraint is that the perturbing accelerations must be expressed in terms of a satellite-fixed frame. The general form for Gauss’ variation of parameters is
dα
dt =f α,aP, t (5.75)
Recall thatα is the 6 × 1 vector of the orbital elements and aPis the 3 × 1 vector of per-turbing accelerations in a convenient satellite-based frame. Although it is possible to derive all six variational equations, we will only present derivations for da/dt, de/dt, and di/dt. These three equations will be used in Chapter 9 when we analyze low-thrust orbit transfers, where the low-thrust propulsion force (divided by satellite mass) is trea-ted as the perturbing accelerationaP.
We will derive Gauss’ variation of parameters by using basic concepts from mechanics and by applying calculus to the orbital relationships developed in Chapter 2. Let us begin with the time-rate of semimajor axis, da/dt. Because semimajor axis is directly related to total energy, we start with an expression for power or the time-rate of energy:
dξ dt=FP v
m =aP v (5.76)
Equation (5.76) is a familiar result from basic mechanics: the time-rate of energy is the dot product of the perturbing force vectorFP and the satellite’s velocity v (the reader should note that if the perturbing force is zero, then we have Keplerian motion where energy is constant). Becauseξ is total energy per unit mass, its time-rate is specific power where the perturbing acceleration vector isaP=FP/m. Now relate energy to semima-jor axis
ξ = −μ
2a (5.77)
Using the chain rule and Eq. (5.77), the time-rate of energy is dξ
dt=dξ da
da dt = μ
2a2 da
dt (5.78)
Substituting Eq. (5.76) for the left-hand side, Eq. (5.78) becomes μ
2a2 da
dt =aP v (5.79)
Therefore, we can derive the time-rate da/dt if we find an expression for the dot prod-uct of the perturbing accelerationaPand velocity v. Gauss used the orthogonal RSW coordinate frame where theR unit vector is along the radius vector, S is in the orbital plane and along the local horizon in the direction of motion, andW is along the angular momentum vector h. We will use the normal-tangent coordinate frame NTW.
Figure 5.11 shows theRSW and NTW frames; these frames are fixed to the satellite.
TheT (tangent) unit vector is always tangent to the orbit in the direction of motion, theN axis is in the orbital plane and normal to the T axis (pointing away from the central body) and the commonW axis is along h, or N × T = W. The RSW and NTW frames can be aligned by a rotation through the flight-path angle γ. For circular orbits, the RSW frame is always aligned with theNTW frame. Suppose we express the perturbing accel-eration vectoraPin theNTW frame
aP= anN + atT + awW (5.80)
where an, at, and aware components along the orthogonalNTW axes. It should be clear that the velocity vectorv has a single component when expressed in the NTW frame (i.e., v = vT). Hence, the dot product is
aP v = atv (5.81)
Substituting Eq. (5.81) into Eq. (5.79) and solving for the time-rate of semimajor axis yields
da dt=2a2v
μ at (5.82)
Equation (5.82) is Gauss’ variation of parameter equation for semimajor axis where the perturbing acceleration is expressed in theNTW frame. Component atis the sum of all
R T
N
(horizon)S
r
Waxis is normal to orbital plane
(tangent) (radial)
Satellite γ
γ R × S = W
N × T = W
Figure 5.11 NTW and RSW satellite-based frames.
perturbing force/mass vectors resolved into the local T-axis direction (tangent to the orbital path). Therefore, atmay be due to conservative forces (e.g., gravity) or non-conservative forces (e.g., aerodynamic drag, solar radiation pressure, or thrust).
Next, we seek the time-rate de/dt. We begin with the orbital relationship for para-meter p
p=h2
μ = a 1−e2 (5.83)
Solving Eq. (5.83) for eccentricity yields
e= 1−h2
μa (5.84)
Taking the time derivative of Eq. (5.84), we obtain de
dt= −h
μa 1−h2/ μa dh
dt+ h2
2μa2 1−h2/ μa da
dt (5.85)
We can substitute Eq. (5.84) for the common denominator term in Eq. (5.85) to produce
de dt= h
μae −dh dt + h
2a da
dt (5.86)
We can use Eq. (5.82) for the time-rate da/dt. In addition, we need the time rate of change of the magnitude of angular momentum dh/dt. From a basic dynamics course, we know that the time-rate of the angular momentum vector is the moment or torque.
This is equal to the cross product of positionr and applied force F. Recall that h is the total angular momentum per unit mass, and therefore its time-rate is
h = r × aP (5.87)
whereaP=FP/m is the perturbing acceleration (we already know that two-body gravity does not change angular momentum because the gravity force is aligned with r). For now, let us express the perturbing accelerationaP as components ar, as, and awin the RSW frame (see Figure 5.11 for the RSW directions). It should be clear that radial pertur-bation arwill not change angular momentum because it is aligned with radial position vec-torr. A transverse perturbation aswill increase the magnitude of the angular momentum vector. The orbit-normal perturbation awwill cause the angular momentum vectorh to rotate and change direction. Using these arguments, the time-rate of the magnitude of the angular momentum dh/dt is solely due to the in-plane perturbation along theS axis, or
dh
dt= ras (5.88)
Because we want to develop the variation equations with perturbations expressed in theNTW frame, we can replace asin Eq. (5.88) with the in-plane perturbing accelera-tions anand at
dh
dt = r atcosγ −ansinγ (5.89)
The reader should be able to easily identify the projections of theT and N components onto theS axis by reviewing Figure 5.11. Equation (5.89) shows that we need expressions for the cosine and sine of the flight-path angleγ. Equations (2.69) and (2.71) present the radial and transverse velocity components, r and rθ, in terms of e, h, and true anomaly θ
r=μ
hesinθ (5.90)
rθ =μ
h 1 + e cosθ (5.91)
We know that sinγ = r/v and cosγ = rθ/v. Using Eqs. (5.90) and (5.91), we obtain the expressions for the sine and cosine of the flight-path angle
sinγ =μesinθ
hv (5.92a)
cosγ =μ 1 + ecosθ
hv (5.92b)
Substituting Eqs. (5.92a) and (5.92b) into Eq. (5.89), we obtain dh
dt =rμ
hv 1 + e cosθ at−esinθan (5.93) Finally, substitute Eqs. (5.93) and (5.82) into Eq. (5.86) to yield an expression for the time-rate de/dt in terms ofNTW perturbations anand at
de dt= h
μae −rμ
hv 1 + e cosθ at−esinθan +hav
μ at (5.94)
The final steps involve substitutions of orbital relationships (such as the trajectory equation) and simplifications. These algebraic steps are omitted here. Equation (5.94) can be simplified to yield
de dt=1
v 2 e + cosθ at+rsinθ
a an (5.95)
Equation (5.95) is Gauss’ variational equation for eccentricity in terms of perturbing accelerations expressed in the satellite-based NTW frame. Only perturbations in the orbital plane cause eccentricity to change over time.
The variational equation for inclination can be obtained using calculus and geomet-rical methods. We start with the expression for the cosine of inclination, Eq. (3.8)
cosi =K h
h (5.96)
whereK is the unit vector along the Z axis of the ECI frame. Taking a time derivative yields
−sinidi
dt=h K h −h K h
h2 (5.97)
We may use Eq. (5.87) to compute the time-rate of vectorh in terms of perturbation accelerations in theRSW frame
h = r × aP=
R S W
r 0 0
ar as aw
=−rawS + rasW (5.98)
We may use Eq. (5.96) to substituteK h = hcosi in Eq. (5.97). The dot product K h will involve the following dot products betweenK and unit vectors S and W:
K S = sinicos ω + θ (5.99)
K W = cosi (5.100)
The angleω + θ (the argument of latitude; see Section 3.3) is measured in the orbital plane from the ascending node to the satellite. Making these substitutions (along with h= ras), Eq. (5.97) becomes
−sinidi
dt=h −rawsin i cos ω + θ + rascosi−rashcosi
h2 (5.101)
Canceling the two terms hrascos i in Eq. (5.101), we obtain di
dt=rcos ω + θ
h aw (5.102)
Equation (5.102) is Gauss’ variational equation for inclination. Only perturbations that are normal to the orbital plane (aw) will change inclination.
We can follow the same basic procedures and derive the remaining Gauss variational equations. These steps will not be presented here. The interested reader may consult Val-lado [1; pp. 633–636] or Bate et al. [3; pp. 402–406]. Gauss’ variational equations in NTW coordinates are
da dt =2a2v
μ at (5.103)
de dt=1
v 2 e + cosθ at+rsinθ
a an (5.104)
di
dt=rcos ω + θ
h aw (5.105)
dΩ
dt =rsin ω + θ
hsini aw (5.106)
dω dt = 1
ev 2sinθat− 2e +rcosθ
a an −rsin ω + θ cosi
hsin i aw (5.107) dθ
dt = h r2−1
ev 2 sinθat− 2e +rcosθ
a an (5.108)
Note that, when all perturbing acceleration components vanish (an= at= aw= 0), Gauss’ variational equations show that the five elements (a, e, i, Ω, ω) remain constant while the time-rate of true anomaly is governed by conservation of angular momentum, or h = r rθ = r2θ. It is also interesting to note that the orbit-normal perturbation awonly affects the orientation of the orbital plane in three-dimensional space (i.e., orbital ele-ments i,Ω, and ω).
Gauss’ variation of parameters (5.103)–(5.108) may be used in special (numerical) or general (analytical) perturbation methods. Gauss’ variational equations provide two dis-tinct advantages for a special perturbation method: (1) numerical integration provides the time histories of the orbital elements without a coordinate transformation step;
and (2) a relatively large time step may be used in the numerical integration process.
Referring back to Example 5.1, we see that the special perturbation method was applied to a perturbed system in Cartesian coordinates [see Eqs. (5.18)–(5.20)], and therefore a coordinate transformation was required to obtain time histories of the orbital elements.
In order to use Gauss’ equations, we must provide the perturbing accelerations (third-body gravity, drag, thrust, etc.) as components in the satellite-based NTW frame. In Chapter 9 we will apply a general perturbation method to develop analytical solutions for low-thrust transfers where the small perturbing accelerationaP is produced by an onboard propulsion system.
Finally, we should note that Gauss’ variational equations possess singularities for equa-torial orbits (Ω is not defined) and circular orbits (ω and θ are not defined). As inclination approaches zero, the time-rate dΩ/dt becomes infinite even if the perturbations are small. The same problem occurs for the time-rates dω/dt and dθ/dt as eccentricity approaches zero. One solution is to use a non-singular set of elements that are nonlinear functions of the classical orbital elements. The non-singular element for angular position is measured from the inertialI axis to the satellite. Gauss’ variational equations may be written in terms of non-singular elements (the so-called equinoctial elements). The inter-ested reader may consult Battin [2; pp. 490–494] for the definition and use of non-singular orbital elements.