Heliotropic Orbits at Oblate Asteroids

AAS 14-277
HELIOTROPIC ORBITS AT OBLATE ASTEROIDS: BALANCING
SOLAR PRESSURE AND J2 PERTURBATIONS
Demyan Lantukh∗, Ryan P. Russell†, and Stephen Broschart‡
The combined effect of significant solar radiation pressure and body oblateness
on spacecraft orbits is investigated using both singly and doubly averaged disturbing potentials with the Lagrange Planetary Equations. This combination of
perturbations has applications for potential spacecraft missions to a select class
of primitive bodies. A stable heliotropic equatorial family of orbits is applied in
the current study to the environment near oblate asteroids. This heliotropic family along with new orbit families are identified, analyzed, and extended out of the
equatorial plane. Dynamic bounds for the inclined heliotropic orbits are determined. The resulting orbits provide useful options for low-altitude science orbits
around some small bodies like Bennu, the target for the OSIRIS-REx mission.
INTRODUCTION
Solar radiation pressure (SRP) and irregular central body gravity distribution are often two of the
most significant perturbations to spacecraft orbital dynamics in close proximity to small primitive
bodies. Several solutions for stable spacecraft orbits have been developed for orbit altitudes where
one or the other is the dominant perturbation, including terminator, equatorial Sun-frozen, and quasiterminator orbits (due to SRP); Sun-synchronous and precessing orbits (due to oblate bodies); and
body-fixed orbits (due to irregular gravity). Fewer studies have identified stable or long-lifetime
orbits in situations where both SRP and the irregular gravity perturbation have roughly equal magnitudes and are primary drivers in the orbital motion – a dynamic regime prevalent at low altitudes
at small bodies. The result is a lack of known stable spacecraft orbit options in the range of orbit
altitudes where these perturbations are comparable. This range can be assessed for a particular set
of body and spacecraft parameters; for smaller near-Earth objects, this range often spans from the
surface up to several body radii, necessitating the need for station-keeping maneuvers to maintain
low orbits.1, 2
The combined effect of SRP and J2 are used in the current study to investigate the application of
heliotropic orbits for the close exploration of small primitive bodies, a new application for this class
of orbits. These heliotropic orbits are also extended to the inclined case using a constrained doubly
averaged potential. The resulting design space is explored and bounded in the mean elements.
Heliotropic∗ orbits are an orbit class that was discovered in the context of studying the dynamics
of orbiting planetary dust, with the term coined for orbits in the study of Saturnian ring dynamics.4, 5
∗
Graduate Student, Aerospace Engineering and Engineering Mechanics, University of Texas at Austin, Austin, TX.
Assistant Professor, Aerospace Engineering and Engineering Mechanics, University of Texas at Austin, Austin, TX.
‡
Mission Design and Navigation Engineer, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA.
∗
In botany, ”heliotropic” was coined in 1832 and describes the tendency of plant stems, leaves, and flowers to bend
toward the Sun.3
†
1
The ratio of SRP and planetary oblateness perturbations for dust at orbital distances of several planetary radii are, in fact, often similar to that for a spacecraft operating close to a small primitive body.
The heliotropic orbits are near-planar and eccentric, with the periapsis on the anti-Sun side of the
body.∗ The eccentricity of these orbits is chosen such that the precession caused by the two perturbing potentials keeps the orbit elements frozen with respect to the Sun direction. In the current paper,
the application of heliotropic orbits to small body exploration is proposed, allowing long-lifetime
low-altitude mapping orbits that naturally account for (at least part of) the irregular gravity perturbation. Further, the near-planar character of these orbits provides a promising inclination range for
long-lifetime orbits, as the well-known terminator and quasi-terminators have higher inclinations.
Heliotropic orbits were identified as frozen orbits in the equatorial dynamics of circumplanetary
dust, with some study of the behavior of inclined orbits as well.7, 8, 9 The resulting orbits were
then proposed for high area-to-mass ratio (HAMR) spacecraft orbiting Earth.10, 6 Orbits at Earth
necessitated methods to investigate out-of-plane and non-zero obliquity characteristics.6 The methods introduced in the current study provide a fast, global approach to map zero-obliquity† inclined
heliotropic orbits.
The key and limiting assumptions in the derivation of the heliotropic orbits are: 1) the central
body gravity is oblate, consisting of just the point-mass and second-order zonal (J2 ) gravity terms
and 2) the central body obliquity is near zero or 180 degrees (i.e., the body spin pole is nearly perpendicular to the body orbit plane). Primitive bodies span the entire range of shapes and obliquities,
so heliotropic orbits are not directly applicable to most bodies. According to a recent study, 49 of
100 asteroids with known shapes had a ratio of less than 1.2 between the two equatorial axes of a
best-fit ellipsoid (i.e. were mostly oblate), but only 8 of those 100 had obliquities within twenty
degrees of perpendicular to the orbit plane.11 With these admittedly sparse statistics, it can be estimated that about 4 percent of asteroids would be candidates for exploration with heliotropic orbits
(assuming the above ranges are acceptable deviations from the assumptions). On the bright side,
there is an abundance of asteroids, many of which have yet to be discovered or characterized. Of
those asteroids that are known, Bennu, the target of the upcoming OSIRIS-REx mission, is oblate
and has nearly 180 deg obliquity:12 These characteristics make Bennu suitable for application of
heliotropic orbits. Also, it may be possible to extend the theory to allow for a broader range of
obliquities and shapes, as has been done for applications at the Earth.6
The goal of the present study is to characterize heliotropic orbits as low-altitude science orbits
for asteroid exploration; the fundamental characteristics of a heliotropic orbit are visualized in Figure 1. To find candidate science orbits, heliotropic orbits in the presence of significant SRP and J2
are computed by first applying a singly-averaged potential in the LPEs. This process yields families of Sun-frozen orbits that maintain a frozen average eccentricity vector with respect to the Sun
line. The equatorial heliotropic orbits are a subset of the Sun-frozen orbits. Next, a constrained
doubly-averaged potential is derived that enforces the heliotropic constraint (but not the Sun-frozen
constraint). This doubly-averaged potential is used to develop an analytical formulation for the average orbital elements of inclined heliotropic orbits. The limits of these inclined heliotropic orbits
are investigated and some example orbits are presented.
∗
†
There are also antiheliotropic orbits with periapsis in the direction of the Sun.6
For axisymmetric bodies like those assumed in this study, zero obliquity is equivalent to 180 deg obliquity
2
Heliotropic:
Π=𝑓
Π ≝ 𝜔 + Ω (prograde)
Figure 1: Heliotropic orbit definition.
Figure 2: Reference frame definition and or-
bit geometry.
Table 1: List of parameters for the representative case used in the current study.
Parameter
Body semi-major axis
Body impact / Brouillon sphere radius
Body gravitational parameter
Body oblateness: Second order gravity zonal term
Body oblateness (normalized)
Sun gravitational parameter
SRP acceleration
Solar flux constant
Spacecraft mass / area ratio
Spacecraft reflectivity
Symbol
d
R0
GMbody
J2
GMSun
γ
P0
BSC
ρSC
Value
1.684 × 108
0.2887
−9
4.057 ×
√10
0.1 5
0.1
1.327 × 1011
1.2762 × 10−10
1.0 × 108
35
0.2
Units
km
km
km3 /s2
km3 /s2
km/s2
km3 kg/m2 s2
kg/m2
MODELS
For the current study, the central body is an asteroid around which the significant forces on a
spacecraft are the gravity from the small body and SRP. The small body is considered to be an
oblate spheroid modeled by a point-mass and the J2 gravity term. Solar gravity is neglected because
the orbits of interest – where J2 has an effect comparable to SRP – have sufficiently low altitudes.
However, the motion of the small body about the Sun is included because the Sun line direction
is important for the SRP. The body obliquity is zero: the equatorial plane is the body orbit plane.
Unless otherwise specified, the parameter values used in the current study are given in Table 1.
These parameters approximate a spacecraft around a body similar to Bennu but with higher J2 ;
Initial investigations are all done with the high value of J2 given in Table 1 in order to account for
a broader range of target asteroids. The current study focuses on the process of finding heliotropic
orbits for a given set of body and spacecraft parameters, but changes in those parameters affect the
existence and behavior of the presented orbits.
Orbit conditions are developed in terms of a constant SRP acceleration (γ) directed away from
the Sun, a simplification consistent with the typical preliminary design process.10, 2, 6 The approximation for γ used in the current study is given in Equation (1) and assumes a spherical spacecraft.2
Additionally, for a circular body orbit and where the ratio of spacecraft semi-major axis to body
3
semi-major axis is small, the Sun-spacecraft distance (d) is approximately the body semi-major
axis. Although it is possible to average over one orbit of the body around the Sun to handle body
eccentricity,2 for simplicity the body is currently modeled in a circular orbit about the Sun with
semi-major axis d. With these assumptions, the Sun line moves counterclockwise in the plane of
the small body orbit with the constant rate given by Equation (2).
P0
4
(1 + ρSC )
2
B d
3
rSC
GMSun
f˙ =
d3
γ=
(1)
(2)
Spacecraft orbits are defined by orbital elements in a reference frame centered at the small body
and fixed to the Sun line at a particular epoch, as shown in Figure 2. Orbit propagations are begun
at this epoch: the Sun line is parallel to x̂ with the Sun in the negative x-direction at t = 0. In
addition, all geometric derivations are done assuming this relative geometry. Orbits are visualized
in a body-centered Sun-synodic reference frame where the Sun remains in the negative x-direction
even though the propagation itself is in an inertial reference frame. The Sun-synodic reference frame
is chosen for visualization because heliotropic orbits rotate with the Sun line, as shown in Figure 1.
The orbital elements used in this investigation are the classic set {a, e, i, ω, Ω, ν}, with the angles
defined in Figure 2 (except ν, which is the angle from ê to spacecraft position). LPEs for applying
13 These equations of
a disturbing potential R to this element set have been derived previously.
p
motion and other equations throughout use the intermediates: n = GMbody /a3 ; p = a(1 − e2 );
M ∗ = nt, where M ∗ is mean anomaly.
SINGLE AVERAGING AND RESULTING SUN-FROZEN ORBITS
Sun-frozen orbit solutions in the presence of J2 and SRP – of which equatorial heliotropic orbits
are a subset – are investigated by first applying the singly-averaged disturbing potentials to the
LPEs. In the current study, the disturbing potential applied to the LPEs is the sum of the averaged
potential from J2 and the averaged potential from the SRP, with the singly-averaged potentials
given in Equations (3) and (4) respectively.2 These disturbing potentials are averaged over a single
spacecraft orbit and are valid assuming the change in the slow orbital elements over a single orbit
is sufficiently small. Equation (4) gives the orbital element form of the SRP disturbing potential
assuming that f = 0 and the Sun-line direction is fixed over one period of averaging. With these
assumptions, the orbital element form of Equation (4) takes advantage of the fact that d̂ = x̂ at the
epoch of interest and ê = [cos ω cos Ω − sin ω sin Ω cos i] x̂ + [cos ω sin Ω + sin ω cos Ω cos i] ŷ +
[sin ω sin i] ẑ.
R̄J2
R̄SRP
µR02 J2
3
2
= 3
1 − sin i
2
2a (1 − e2 )3/2
3aeγ
3aeγ
d̂ · ê = −
[cos ω cos Ω − sin ω sin Ω cos i]
=−
2
2
(3)
(4)
Applying these potentials in the five LPEs of the slow-moving variables results in the equations
of motion for the singly-averaged system, given in Equation (5).
4
ȧ = 0
3γ p
1 − e2 [sin ω cos Ω + cos ω sin Ω cos i]
2na
3γ
e
√
i̇ = −
cos ω sin Ω sin i
2na 1 − e2
3nR02 J2
3γ
1
5
2
2
√
ω̇ = −
(1 − e ) cos ω cos Ω − sin ω sin Ω cos i +
2 − sin i
2na e 1 − e2
2p2
2
2
e
3nR0 J2
3γ
√
cos i
sin ω sin Ω −
Ω̇ = −
2
2na 1 − e
2p2
ė = −
(5)
Sun-frozen orbit conditions
In general, a frozen-eccentricity orbit requires that the average eccentricity vector be constant in
time.14 By analogy, a Sun-frozen orbit requires that the average eccentricity vector to has a fixed
magnitude and orientation with respect to the Sun line. The Sun-frozen condition translates to the
orbital element rates given in Equation (6).
ė = i̇ = ω̇ = 0
Ω̇ = f˙
(6)
For equatorial orbits, these conditions are modified to account for the fact that Ω and ω cannot be
defined individually. Following the example of Vallado, a retrograde factor is defined: δdir = 1 for
prograde orbits and δdir = −1 for retrograde orbits.14 Using the retrograde factor, the undefined
elements are replaced by Π = ω + δdir Ω. With this substitution, the frozen orbit conditions for
equatorial orbits are given by Equation (7).
ė = i̇ = 0
Π̇ = δdir f˙
(7)
Although requiring ȧ = 0 is not necessary for an orbit to be frozen with respect to the eccentricity
vector, the LPEs dictate that all orbits averaged over the fast orbital element will indeed have a
constant semi-major axis.
Equatorial orbits For equatorial orbits, sin i = 0 and so i̇ = 0, from Equation (5). Next, the
condition ė = 0 is simplified by using cos i = δdir to write ė with respect to Π, with the result
expressed in Equation (8). Similarly, an expression for Π̇ = ω̇ + δdir Ω̇ is given by Equation (9).
3γ p
1 − e2 sin Π
2na
!
√
3 nR02 J2
γ 1 − e2
Π̇ =
−
cos Π
2
p2
na
e
ė = −
(8)
(9)
Using Equation (8), the corresponding Sun-frozen condition in Equation (7) is satisfied by either
Π = 0 of Π = π. These two possible values of Π correspond to the ”heliotropic” and ”antiheliotropic” orbits, respectively, as described in past studies with different applications.6 Looking at
5
Equation (9), Π̇ can be considered a sum of a J2 term and an SRP term; The J2 term has the sign of
J2 (J2 > 0 for most bodies) and the SRP term has the sign opposite of cos Π. As a result, if J2 > 0
and Π = π, then Π̇ > 0. According to Equation (7), a positive rate in Π means that the Sun-frozen
solution can only exist when the body orbit direction is the same as the spacecraft orbit direction:
for an oblate body in prograde motion, the Π = π solution must be prograde and for an oblate body
in retrograde motion, the Π = π solution must be retrograde.
100
100
−10
0.9
0.8
100
impact
line
1
00
Sun−frozen
contour
0
0.5
−1
e
0.6
1000
0.7
0.4
0.3
0.2
0.1
0
0
−10
2
−1000
4
6
a, body radii
8
10
Figure 3: Sample contours of the Π̇ − f˙ rate equation (deg/day) solved for equatorial prograde
orbits with Π = 0, using parameters from Table 1.
The resulting frozen orbit families are enumerated for each of the four possible cases {δdir =
±1, cos Π = ±1} by choosing one of the two unknowns (a or e) as the independent variable and
solving for the other with Π̇ = f˙ within a specified range of the independent variable. Using e as
the independent variable is convenient because it is bounded. Evaluating the frozen orbit families
for the representative case in the current study results in one prograde family of equatorial orbits
and one retrograde family of equatorial orbits for Π = 0. Figure 3 gives the contour plot of Π̇ − f˙
for the prograde, Π = 0 case. No solutions with Π = π exist for this case, though such solutions
can exist when the perturbations are smaller relative to the body gravitational parameter.6
The Sun-frozen families for the presented case are given in Figure 4. The prograde equatorial
orbit family F1 corresponds to the planar heliotropic types of orbits presented and analyzed in
literature.6, 9 The current analysis verifies that the retrograde equatorial family F6 exists and can be
determined by the process described above.
Nonequatorial orbits For nonequatorial orbits, finding frozen orbit families also begins with
ė = 0 and i̇ = 0; these two conditions together lead to Equation (10), the primary frozen orbit
requirement for nonequatorial orbits.
(cos ω = 0 AN D cos Ω = 0) OR (sin ω = 0 AN D sin Ω = 0)
(10)
The trigonometric functions of ω and Ω always exist in a product of two such terms in the LPEs.
6
F1: Π=0, stable equatorial prograde
F2: ω=−π/2, Ω=π/2, unstable nonequatorial prograde
200
i, deg
F3: ω=−π/2, Ω=π/2, unstable polar terminator
F4: ω=π, Ω=0, unstable polar Sun−line
100
F5: ω=π/2, Ω=π/2, unstable nonequatorial retrograde
1
F6: Π=0, stable equatorial retrograde
0
0
0.5
(a=a*,e=e*,i=0)
5
a, body radii
10 0
e
1
150
0.8
0.6
e
i, deg
(a=a*,e=e*,i=0)
100
0.4
50
0.2
(a=a*,e=e*,i=0)
0
0
0
2
4
6
a, body radii
8
10
0
2
4
6
a, body radii
8
10
Figure 4: Sun-frozen orbit families from the singly-averaged analysis.
The frozen orbit condition then leads to sin ω cos Ω = 0 and sin Ω cos ω = 0. Further, the other
products of trigonometric functions of ω and Ω can be restricted to a small set: cos ω cos Ω =
{−1, 0, 1} and sin ω sin Ω = {−1, 0, 1}. The resulting feasible combinations allow four possible
cases of frozen orbit conditions, enumerated in Table 2.
Of the four cases in Table 2, C1 is considered first: In this case the line of nodes begins normal
to the Sun line and periapsis begins on the night side of the body. Applying the Sun-frozen orbit
conditions maintains this geometry relative to the Sun: Using the rates from Equation (5) in Equation (6), and applying all the simplifications that C1 allows, leads to the Sun-frozen orbit conditions
given in Equation (11).
7
Table 2: Possible frozen orbit cases for nonequatorial orbits.
Case
C1
C2
C3
C4
cos ω cos Ω
0
0
-1
1
sin ω sin Ω
-1
1
0
0
Possible Pairs of {ω, Ω}
{ −π/2, π/2}, { π/2, −π/2}
{ π/2, π/2}, { −π/2, −π/2}
{ 0, π}, { π, 0}
{ 0, 0}, { π, π}
1
3γ
3nR02 J2 2
√
4
−
5
sin
i
cos i +
2na e 1 − e2
4p2
3γ
e
3nR02 J2
√
= f˙ =
−
cos i
2na 1 − e2
2p2
ω̇C1 = 0 = −
Ω̇C1
(11)
Equation (11) provides two equations with three unknowns for a given body: {a, e, i}. Solving
Ω̇C1 = f˙ for cos i and using sin2 i = 1 − cos2 i reduces the Sun-frozen orbit conditions to one
equation in two unknowns: {a, e}. Choosing one of these as an independent variable, the other
can be solved as the dependent variable to generate families of solutions, taking into account that
the dependent variable may be undefined or may not be single-valued for a given value of the
independent variable.
The solution process is repeated for the other three cases, resulting in a pair of non-equatorial
frozen orbit equations that can be reduced to one equation in two parameters for each case. Each
set of equations is mapped globally to bound solution regions, then solved locally using a numerical
root-finding technique to find the frozen orbit families. The resulting Sun-frozen families are shown
in Figure 4. The two near-polar families correspond to J2 -perturbed frozen orbit families that exist
with SRP alone. F2 and F5 span a wide range of inclinations and, at first glance, appear to make
good candidate science orbits. However, these families have large impacting regions and the regions
that do not impact are unstable, as described in the next section. Figure 5a shows the average
expected periapse radius along the different Sun-frozen families in Figure 4. Comparing these
average periapse distances to the impact radius bounds the usable regions of the Sun-frozen families.
The semi-major axis for F1 that marks the border between impacting and nonimpacting orbits is
defined as amax . Further analysis in the following sections shows that inclined Sun-frozen orbits do
not exceed amax .
Stability of Sun-frozen orbit families
The existence of a Sun-frozen mean element solution does not guarantee a valid solution in the
full dynamics: the averaging assumptions may not always hold, or from a practical viewpoint, the
true trajectory may escape or impact the body. To aid in determining practically useful orbits, the
Sun-frozen families can be categorized by evaluating their stability. The LPEs can be written in
vector form as Ẋ = f (X) where X is the vector of the slow orbital elements (or three elements for
the equatorial case: {a, e, Π}). Then, if the state for a frozen orbit is X∗ , the frozen orbit conditions
are Ẋ∗ = f (X∗ ) = 0. Considering a small perturbation around X∗ where X = X∗ + δX, the
dynamics of δX to first order are:
8
F1: Π=0, stable equatorial prograde
F2: ω=−π/2, Ω=π/2, unstable nonequatorial prograde
F4: ω=π, Ω=0, unstable polar Sun−line
F5: ω=π/2, Ω=π/2, unstable nonequatorial retrograde
F6: Π=0, stable equatorial retrograde
0
10
0
10
max[real(λ)]
rp, body radii (log scale)
F3: ω=−π/2, Ω=π/2, unstable polar terminator
0
2
4
6
a, body radii
8
10
−10
10
−20
10
0
(a) Extents of families based on impact with the body.
2
4
6
a, body radii
8
10
(b) Stability criteria
Figure 5: Characteristics of the Sun-frozen families from the singly-averaged equations of motion.
δ Ẋ = AδX
(12)
∂f (X) A=
∂X (13)
X∗
The linear stability of a particular frozen orbit X∗ is determined using the eigenvalues of A. If
any eigenvalue has a positive real part then the orbit is unstable. If all eigenvalues have nonpositive
real parts then the orbit is linearly stable. For the singly-averaged equations of motion given in
Equation (5) applied at the Sun-frozen conditions, the partial derivatives in X are simplified because
sin ω cos Ω = 0 and cos ω sin Ω = 0. For the equatorial case, derivative calculation is simplified by
the condition sin Π = 0.
Evaluating the derivatives shows that one of the eigenvalues of A is always zero. For unstable
orbits, the size of the real component of the eigenvalues gives some indication of the characteristic
time for an orbit to leave the neighborhood of its initial condition. Therefore, the stability metric
for orbits in the current study is the supremum of the real parts of the eigenvalues of A. Figure 5b
shows the maximum real part of the eigenvalues (λ) for each of the families of frozen orbits. The
result is that most of the nonequatorial orbits are unstable and all the equatorial orbits are linearly
stable. The low-eccentricity terminator family becomes stable for very small J2 , which is expected
since terminator orbits are stable with only SRP. The point where the unstable family F2 approaches
the equatorial F1 is designated {a = a∗ , e = e∗ , i = 0} for use as a reference point in studying
nonequatorial orbits from a doubly-averaged potential.
9
CONSTRAINED DOUBLE AVERAGING FOR INCLINED HELIOTROPIC ORBITS
The stable heliotropic orbits from past analytical studies (verified with the singly-averaged Sunfrozen orbit analysis) are equatorial.10, 8 Past work6 has looked at numerically following families of
orbits to find inclined heliotropic orbits, but a more systematic and global means to find inclined heliotropic orbits is desirable. In pursuit of this goal, consider Figure 6 which shows an orbit perturbed
by twenty degrees in inclination from the equatorial family of heliotropic orbits. Although simply
changing the initial inclination from the equatorial family can produce long-lifetime orbits like the
one presented in Figure 6, this method is based on trial and error and provides little understanding
about the design space. However, observing the resulting inclined heliotropic orbit provides at least
one path forward: The secular rates of both Ω and ω appear to be constant. In addition, these angles
are approximately related to each other by Equation (14), which maintains the heliotropic geometry
of the orbit (see Figure 1). Here f gives the angle through which the Sun line has rotated, the angle
from x̂ to d̂ as defined in Figure 2. Observing these characteristics of inclined heliotropic orbits
motivates the determination of the secular rates of Ω and ω. Since the rates in both Ω and ω have
small oscillations about a mean, a second average can isolate the secular components (assuming,
rather liberally, that the typical averaging assumptions hold).
3
2.5
e
0.5
0.4
0
5
10
15
20
z, body radii
a, body radii
1
0.5
0
−0.5
−1
5
10
15
20
5
10
15
20
5
10
15
20
−3
ν, deg
5
10
15
20
10
15
20
100
0
−100
0
5
t, days
−2
−1
0
x, body radii
1
0.4
0.2
e sin ω
Ω, deg
ω, deg
i, deg
0
21
20
19
0
100
0
−100
0
100
0
−100
0
0
−0.2
−0.4
−0.5
0
0.5
e cos ω
Figure 6: Example A: Heliotropic orbit found by perturbing the equatorial family of solutions.
Initial conditions for the orbit are given in Table 3.
Π=f
(14)
Observing that, on average, the orbit elements of the perturbed solution in Figure 6 satisfy Equation (14), leads to a means for double averaging of the disturbing potential. This angular condition
keeps the eccentricity vector in the anti-Sun hemisphere (i.e. makes the orbits heliotropic), so this
one condition is imposed alone to find families of heliotropic inclined orbits.
10
Constrained doubly-averaged LPEs
Double-averaging the potential is achieved by averaging the singly-averaged potential over one
period of Ω (or equivalently ω) as detailed below. For the example in Figure 6, the second averaging
period is approximately 11 days. This process assumes that the change in a, e, and i over one period
of Ω is small, an assumption that may not hold with the large perturbations under consideration. In
addition, since the singly-averaged potential assumes a fixed SRP direction, the doubly-averaged
potential remains subject to this assumption. Even with this limitation, the results in the following
section demonstrate cases of practical use for the doubly-averaged results, validated by simulations
in the full dynamics. Furthermore, it is noted that doubly-averaged potentials have been successfully
applied to the other dynamical systems, such as the restricted three body problem.15 In the problem
investigated here, the doubly-averaged potentials are used to provide the mean values for Ω̇ and ω̇
as well as estimates of initial conditions for inclined heliotropic orbits in the full dynamics.
Performing a simple average of R̄SRP from Equation (4) over either Ω or ω would give zero as a
result. However, by applying a constraint based on the heliotropic condition given by Equation (14)
the resulting doubly-averaged potential is nonzero, as derived below. An approximate heliotropic
condition for averaging is developed by differentiating Equation (14) with respect to time and observing that Ω̇ f˙ and ω̇ f˙ for the example small body case. As a result, over one period of
Ω or ω the contribution of f is relatively small (about 3% for the cases investigated) and so can be
neglected. The resulting approximate heliotropic constraint, Π = 0, leads to a simple, conservative
form for the potential. (If f is explicitly left in the constraint, then it would appear explicitly in R̄
through f = f0 + f˙t.) However, this f˙ ≈ 0 assumption is only necessary for the averaging: the long
term motion of the Sun is accounted for to first order by applying the doubly-averaged equations to
the full heliotropic constraint given in Equation (14).
Equivalently, other constraints could be applied to achieve a different constrained orbit geometry.
One such possibility for future work is to allow for different resonances between full periods of Ω
and ω, leading to the constraint M Ω + N ω = f to produce prograde orbits with an M :N resonance
in Ω and ω, respectively.. Unless M = N = 1 the resulting orbits will not necessarily be heliotropic
but they could still be stable and may be useful as low-altitude science orbits. An example of such
an orbit is given in the section on periodic orbits.
Using the approximate heliotropic constraint (Π = 0) leads to the substitution Ω = −ω (for
prograde orbits) in R̄SRP . The resulting constrained disturbing potential is shown in Equation (15),
where the * superscript indicates the constraint is active.
Averaging the singly-averaged disturbing potential over one period of Ω is only non-zero for
∗
R̄SRP
. The one-period averages of sin2 x and cos2 x are both 1/2, leading to Equation (16). The
resulting non-zero doubly-averaged LPEs for the constraint Π = 0 are presented in Equation (17).
∗
R̄∗ = R̄SRP
+ R̄J∗2
3aeγ 2
∗
R̄SRP
=−
cos Ω + sin2 Ω cos i
2
∗
R̄J2 = R̄J2
(15)
µR02 J2
3
3aeγ
2
∗
¯
R̄ = −
[1 + cos i] + 3
1 − sin i
4
2
2a (1 − e2 )3/2
(16)
11
3nR02 J2
1
1 5
3γ
2
√
1 − e2 + cos i +
−
+
cos
i
4na e 1 − e2
2p2
2 2
2
e
3nR0 J2
3γ
√
cos i
−
Ω̇∗ =
4na 1 − e2
2p2
ω̇ ∗ = −
(17)
Heliotropic orbit doubly-averaged equation of motion
A surface of orbit conditions for heliotropic orbits can be developed from the constrained doublyaveraged LPEs. The long-term change in the SRP direction can be accounted, to first order, through
the reintroduction of f˙ in the heliotropic constraint. Therefore, in the doubly-averaged problem,
there remain three unknowns and one constraint; solving Equations (17) and (18) leads to a quadratic
equation in cos i given by Equation (19).
One of the solutions to this quadratic, Equation (20), is the equation of motion for this constrained
doubly-averaged system (if such a solution exists). Equation (20) gives the solution for inclination
at a given pair {a, e} where such inclination exists. The other solution to the quadratic equation
violates the prograde assumption made in the derivation. An inclination exists that enables a doublyaveraged heliotropic orbit when B4 ≥ 0 and −1 ≤ cos i ≤ 1.
Π̇ = ω̇ + δdir Ω̇ = f˙
(18)
3 f˙ = −
KJ2 + (1 − 2e2 )KSRP + [2KJ2 + KSRP ] cos i − 5KJ2 cos2 i
4
nR02 J2
KJ2 =
p2
γ
√
KSRP =
nae 1 − e2
(19)
√
1 B2 + B4
cos i = +
5
30 B1 B3
B1 = n2 R02 J2
(20)
B2 = 3 γ p 2
p
B3 = ae 1 − e2
B4 = 120 3/5 − e2 B1 B2 B3 + 216 B12 B32 + 240 np2 B1 B32 f˙ + B22
Equation (20) leads to a surface of mean orbital element conditions that produce heliotropic orbits. The resulting surface of orbit conditions, shown in Figure 7, is bounded on its low-eccentricity
end by the equatorial family of frozen heliotropic orbits. On the high-eccentricity end of the surface,
the orbits impact the body.
ORBIT DESIGN FOR HELIOTROPIC ORBITS AT SMALL BODIES
Because the doubly-averaged solutions do not account for the possibility of impact or escape,
defining the useful portion of the surface of heliotropic conditions is desirable. The heliotropic
surface and its bound are shown in Figure 7. In addition, the doubly-averaged solutions assume
12
Table 3: Initial conditions for the integrated orbit examples.
Example
A
B
BP a
C
D
Eb
a (body radii)
2.8987341
2.7454004
3.1417561
4.2500558
4.9121042
1.9048110
e
0.46481127
0.52314381
0.51510321
0.73173913
0.77478261
0.25724790
i (deg)
20
28.715568
30.626542
20.429662
11.734502
18.635145
ω (deg)
-90
-90
-68.447416
-90
-90
-146.26777
Ω (deg)
90
90
77.553083
90
90
-157.51696
ν (deg)
180
180
68.447416
180
0
146.26777
a
Repeat period: 12.995380 days, 18 spacecraft revolutions. Largest eigenvalue magnitude: 1.0000027. Initial
conditions are given at the x-y plane crossing.
b
Repeat period: 4.1014777 days, 11 spacecraft revolutions. Largest eigenvalue magnitude: 1.0255871. Initial
conditions are given at the x-y plane crossing.
that the orbital elements do not change significantly over the course of one orbit or one period of
Ω circulation, so there may be cases where the surface is not always a good predictor of the full
dynamics. Examples B and C, shown in Figures 8 and 9, respectively, illustrate this possibility:
Both examples use initial conditions from the heliotropic surface, but B stays well bounded for 60
days in a heliotropic orbit while C departs from the heliotropic condition and impacts the body.
The low-eccentricity boundary of the heliotropic surface is the Sun-frozen orbit equatorial case
(F1 ) from the singly-averaged analysis. Orbits along this boundary are stable in the singly-averaged
elements. However, there exists a bifurcating unstable family of frozen orbits (F2 ), and orbits
near this unstable family are likely unstable. This unstable family provides a useful boundary,
as discussed below. For a given value of a there is a range of possible e for heliotropic orbits,
as defined by the heliotropic surface. Each of these possible {a, e} pairs also corresponds to a
particular inclination.
The high-eccentricity end of the heliotropic surface consists of body-impacting orbits. This impact boundary can be approximated as the point where the periapse distance is the Brouillon sphere
radius: a(1 − e) = R0 . (In practice a higher impact radius may be desirable.) Substituting this
maximum eccentricity into Equation (20) allows the calculation of the maximum inclination for a
given value of a, a function defined as iimpact since this high eccentricity boundary is also the highinclination boundary for heliotropic orbits. The boundary iimpact begins along the low-e boundary
at {a = R0 , e = 0, i = 0}, then rises steeply in inclination to a maximum value (imax ), and then
returns to the equatorial family (the low-e boundary) again at {a = amax , e = 1−R0 /amax , i = 0}.
For a particular case, the value of imax can be found numerically by solving diimpact /da for aimax
and then calculating iimpact (aimax ). Bounding cases for maximum possible inclination for all heliotropic orbits are discussed below.
Although amax provides an upper bound on a based on impact, the unstable Sun-frozen orbit
family (F2 ) (found in the singly-averaged analysis) may provide a better upper bound on a for
stable of near-stable heliotropic orbits (when F2 is within the bounds [R0 , amax ]). Although F2
only crosses the heliotropic surface at a∗ , it’s proximity suggests that the true dynamics of orbits in
the vicinity are unstable; example C in Figure 9 illustrates such a case. Conveniently, the steepness
of (F2 ) at low inclinations allows the use of its end at i = 0 (a∗ ) as a single upper bound value
for a. For values of a approaching and exceeding a∗ , the singly averaged equations are no longer
stable when integrated. Even so, long lifetime orbits may still exist in the full dynamics with a∗ ≤
a ≤ amax : Example D, shown in Figure 10, is a long-lifetime orbit where a > a∗ . This orbit is
13
50
F2
0.9
F2
46.378 deg
0.8
0.7
imax
40
iimpact
i, deg
e
0.6
0.5
0.4
30
iimpact
20
F1
0.3
0.2
a∗
10
amax
0.1
2
4
a, body radii
0
6
F1
2
a∗
4
a, body radii
amax
6
Figure 7: The heliotropic orbit surface in three dimensions and from two projections. The shaded
region of interest is bounded by F1 and iimpact , the impact boundary. This region is divided by F2 .
notable for its large periodic variation in e. One practical suggestion is to use initial conditions near
periapsis for a ≥ a∗ . For a < a∗ , the initial true anomaly does affect the orbit, but there are often
long-lifetime orbits for any choice of initial true anomaly (contrary to the case of a ≥ a∗ ).
In summary, practical prograde heliotropic orbits are expected to fall within the bounds:
R0 ≤a ≤ min(a∗ , amax )
0 ≤e ≤ emax
0 ≤i ≤ imax . 46.4 deg
The quantities amax , emax , and a∗ come from the singly averaged Sun-frozen orbit solutions.
The value of imax can be found for a particular case with a numerical root-finding procedure as
outlined above. The limit of imax . 46.4 deg, derived below, is independent of SRP and J2 .
14
i, deg
e
0.6
0.4
2
0
0
ω, deg
Ω, deg
5
10
10
15
15
20
25
20
25
30
30
28
26
24
0
ν, deg
5
y, body radii
a, body radii
5
4
3
5
10
15
20
25
30
0
5
10
15
20
25
30
0
5
10
15
20
25
30
0
−1
100
0
−100
−2
100
0
−100
−4
−3
−2
−1
x, body radii
0
1
//
100
0
−100
0
5
10
15
20
25
2
30
t, days
z, body radii
//
1
0.5
e sin ω
1
0
−0.5
−1
−1
−0.5
0
0.5
1
e cos ω
1
0
−1
−2
−4
−3
−2
−1
0
x, body radii
1
Figure 8: Example B: Long-lifetime heliotropic orbit found using the surface of doubly-averaged
heliotropic conditions. Initial conditions for the orbit are given in Table 3.
Limiting cases for heliotropic orbits
The limiting cases for heliotropic orbits can be estimated by manipulating Equation (20). Here
the oblateness effect is contained entirely in B1 and the effect of solar radiation pressure entirely
in B2 . Also, B1 , B2 , B3 , and B4 are all real and nonnegative for captured orbits about an oblate
spheroid. Noting that increasing the strength of SRP (increasing B2 ) decreases i if all else is held
constant, the maximum possible inclination is expected to come about by reducing B2 . Setting
B2 = 0 and ignoring the effect of the small body motion (f˙ = 0), Equation (20) simplifies to
Equation (21), yielding an inclination of ≈ 46.4 deg. Note that B2 also approaches zero when e
approaches one, even with the effect of SRP, so the result still holds for high eccentricity orbits.
Although higher inclination is possible on the heliotropic surface, for the case under consideration,
the highest-inclination areas (visible in the three-dimensional view in Figure 7) occur well into the
15
a, body radii
6
15
10
5
0
10
20
30
40
10
20
30
40
4
ν, deg
Ω, deg
2
10
10
20
20
30
30
z, body radii
ω, deg
i, deg
e
0
2
1
0
0
150
100
50
0
100
0
−100
0
100
0
−100
0
40
0
−2
40
−4
10
20
30
40
−6
100
0
−100
0
10
20
30
40
−8
−8
t, days
−6
−4
−2
x, body radii
0
2
Figure 9: Example C: Unstable orbit from the surface of doubly-averaged heliotropic conditions
near the unstable singly-averaged Sun-frozen family F2 . Initial conditions for the orbit are given in
Table 3.
zone of impacting orbits at relatively high values of a (using the parameter values in Table 1).
cos iB2 =0
√
1+ 6
=
5
(21)
This maximum inclination limit is approached as J2 increases, with everything else held constant. Moving in the opposite direction, trying to eliminate the effect of J2 , leads to B1 = 0 and
causes a singularity in Equation (20). Considering instead a small, positive value for B1 allows the
approximation B4 → B2 , which leads to Equation (22). In this equation, maximizing cos i allows
the minimum possible B1 , so assuming cos i = 1, then the approximate minimum value of B1 is
given by Equation (23). Two intuitive results are verified by Equation (23): first, as the effect of
SRP (B2 ) increases, a larger oblateness is required to maintain the heliotropic orbit and, second,
neglecting the effect of SRP entirely does not drive the requirement on J2 (B1 ) to zero.
3 + B2
15 B1 B3
3 + B2
≈
15 B3
cos iB1 → =
(22)
B1,min
(23)
16
10
20
30
0.8
0.6
0.4
0.2
2
10
0
100
0
−100
0
10
20
30
10
20
30
i, deg
0
14
12
Ω, deg
100
0
−100
100
0
−100
20
30
y, body radii
e a, body radii
3
0
ν, deg
ω, deg
5
4
3
1
0
−1
−2
0
10
20
30
−3
−4
0
10
20
30
−2
0
x, body radii
2
−2
0
x, body radii
2
t, days
z, body radii
e sin ω
0.5
0
−0.5
0.5
0
−0.5
−4
−0.5
0
0.5
e cos ω
Figure 10: Example D: Long-lifetime heliotropic orbit with a > a∗ found using the heliotropic
surface. Initial conditions for the orbit are given in Table 3
Periodic heliotropic orbits
Periodic heliotropic orbits can be found using initial guesses from the averaged dynamics and a
suitable root finding technique applied in the full dynamics.16 Periodic heliotropic orbits are periodic in all the orbital elements except Ω, which experiences a shift equivalent to the change in f to
keep the orbit apoapse towards the Sun. (The orbits are exactly periodic in Cartesian coordinates
with respect to the frame that rotates with the Sun.) Equation (17) is useful for determining the nearest periodic orbit by resonance (number of spacecraft revolutions per circulation in ω). Performing
the described process on example B yields the periodic orbit example BP shown in Figure 11. This
periodic orbit repeats (relative to the Sun line) every 18 spacecraft revolutions, and the eigenvalues
of its monodromy matrix reveal linear stability, at least to the sixth digit. (The largest eigenvalue
magnitude given in Table 3.) Future work includes following and mapping these and other similar
periodic orbit families.
As mentioned previously, it is also possible to find orbits which are not heliotropic but are still
stable; an example of such an orbit is given in Figure 12 (notice the near 2:1 resonance in ω and
Ω, respectively). The orbit shown in Figure 12 is marginally linearly unstable (according to the
eigenvalues of its monodromy matrix) but appears to be nonlinearly stable (when propagated for
17
ω, deg
2
0
2
4
6
8
10
12
0
30
28
26
0
2
4
6
8
10
12
2
4
6
8
10
12
2
4
6
8
10
12
0.65
0.6
0.55
0.5
ŷ, body radii
4
3.5
3
i, deg
e
a, body radii
11,000 orbits). These kinds of orbits can be investigated by modifying the constraint used to develop
the doubly-averaged disturbing potential, resulting in a surface of mean element conditions similar
to the one developed for inclined heliotropic orbits. Investigation of these types of orbits is another
avenue for future work.
100
0
−100
Ω, deg
ν, deg
100
0
−100
−1
−4
−3
−2
−1
0
1
x̂, body radii
0
2
4
0
2
4
6
8
10
12
6
8
10
12
ẑ, body radii
t, days
0.5
e sin ω
0
−2
0
100
0
−100
1
0
1
0
−1
−4
−0.5
−0.5
0
0.5
−2
0
x̂, body radii
e cos ω
Figure 11: Example BP : Long-lifetime periodic heliotropic orbit corrected from example B. Initial
conditions for the orbit are given in Table 3.
CONCLUSION
It is well known that solar radiation pressure in the dynamical environment near small bodies
can be a significant destabilizing perturbation. In the current study, the additional effect of a large
oblateness is considered and several new families are identified as long lifetime orbits suitable for
consideration as low altitude science orbits. An analytical approach based on Lagrange Planetary Equations using a singly-averaged potential is implemented to find Sun-frozen orbit families.
Six families are found this way for the example problem: two perturbations of known near-polar
families, a previously-found equatorial heliotropic family, a new unstable heliotropic family, and
retrograde variants of the two heliotropic families. The previously-discovered equatorial heliotropic
family of orbits is investigated in depth: Specifically a new means for extending this family to
nonequatorial orbits is presented based on a constrained doubly-averaged disturbing potential.
18
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
1
ω, deg
19
18
17
100
0
−100
100
0
−100
0.5
0
−0.5
−1
0
ν, deg
1.5
0.5
0.4
0.3
0.2
0.1
y, body radii
a, body radii
e
2
2
Ω, deg
i, deg
2.5
1
2
3
4
−1.5
−2
0
1
0
1
2
3
4
2
3
4
−2
−1
0
x, body radii
1
−2
−1
0
x, body radii
1
100
0
−100
z, body radii
t, days
e sin ω
0.4
0.2
0
−0.2
−0.5
0
e cosω
0.5
0
−0.5
0.5
Figure 12: Example E: Long-lifetime periodic orbit with similar characteristics to a heliotropic orbit
except with a different resonance in Ω̇ and ω̇. Initial conditions for the orbit are given in Table 3.
This doubly-averaged potential is developed from the heliotropic constraint, but other constraints
can also be applied to potentially find different types of non-heliotropic orbit families. The possible
extents of heliotropic orbits are computed both for the specific case under investigation and for some
limiting cases. In addition, a maximum inclination bound is developed that is independent of body or
spacecraft parameters – that is, heliotropic orbits cannot exceed an average inclination of ≈ 46 deg.
In addition to this general upper bound, for a given set of body and spacecraft parameters, inclined
heliotropic orbits are shown to have mean elements bounded approximately by the equatorial family
and body-impacting orbits. In the case investigated, the surface of inclined heliotropic orbits is
divided by an unstable Sun-frozen orbit found with the singly-averaged analysis. Several specific
orbits are presented to illustrate orbital behavior in the region of heliotropic orbits. In addition, a
periodic, inclined, heliotropic orbit is demonstrated in the unaveraged dynamics: This periodic orbit
is found by numerically correcting initial conditions estimated using the heliotropic surface from
doubly-averaged analysis. The resulting orbital region, bounded and investigated in the current
study, could enable long-lifetime low-altitude orbits around oblate small bodies with low obliquity
angles.
19
ACKNOWLEDGMENT
This work was supported by the NASA Office of the Chief Technologist via a NASA Space
Technology Research Fellowship grant (#NNX12AI77H). In particular, the authors thank Claudia
Meyer for continued interest and support of the project. Part of the work described here was carried
out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the
National Aeronautics and Space Administration.
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20