Mapping the V-infinity Globe

AAS 07-277
MAPPING THE V-INFINITY GLOBE
Nathan Strange, Ryan Russell∗, and Brent Buffington
Jet Propulsion Laboratory / California Institute of Technology
4800 Oak Grove Drive, Pasadena, CA, 91109-8099
Abstract
This paper presents a graphical method for the design of transfers between the same gravityassist body (i.e., same-body transfers). This graphical method collapses a large and complex
space of possible trajectories to a map on which a tour designer may use intuition and experience to design a gravity-assist tour. This method was used with great success in the Cassini
extended mission design.
Introduction
The theory of patched conics [1, 2, 3] has long been used to solve for scientifically interesting, low
∆V trajectories in the 3-body problem. In particular, the Galileo mission [4] to Jupiter and the
Cassini mission [5] to Saturn have used very complex sequences of multiple gravity-assist flybys to
meet a myriad of disparate scientific goals. In using gravity-assists, these missions have achieved a
diversity of orbit geometries that otherwise would have been unachievable.
The complexity in choosing a sequence of gravity-assist flybys and associated orbital transfers
between the flybys, (i.e. the pathfinding problem [6]) grows geometrically as the number of flybys
increases. The combinatorics of the problem quickly overwhelms attempts at solving the problem
computationally and forces tour designers to adopt experience based methods that are essentially
trial and error driven by flashes of inspiration.
This pathfinding problem is just another case of the traveling salesman problem [6] that has
confounded mathematicians for decades. And although finding an optimal solution to this problem
is extraordinarily difficult, traveling salesmen themselves seem to have little problem finding near
optimal solutions that allow them to conduct their business. To do this, they use their experience
coupled with a very important aid: a map.
This analogy has inspired us to seek a similar tool that would be applicable to tour design.
Earlier efforts led to a Tisserand criterion based method useful for planar transfers between different
gravity assist bodies (i.e., multi-body transfers). That method is extremely useful for the heliocentric
case [6] as well as for Jovian [7] and Uranian [8] tour design. But it is of limited utility for same-body
transfers used for tour design at Saturn and Neptune.
In this paper, we use the assumption of a same-body transfer to develop a new method based on
the v-infinity globe [9] where all possible orbits around the central body are plotted as points on the
v-infinity globe. Having plotted these points for all possible orbits, we can then plot a sequence of
flybys to move from point to point. To do this, we calculate a bending angle for a flyby of a given
altitude. This bending angle then translates to an angular distance between asymptotes before and
after a flyby. A tour can then be mapped out by navigating across the surface in steps equal to or
smaller than the bending angle.
∗ Currently
Assistant Professor, School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA
30332-0150.
1
A previous work [9] introduced the idea of mapping all the free-return trajectories on a 3dimensional velocity diagram (v-infinity globe) as a mission analysis tool. The resulting diagrams
were successfully applied in the search and optimization of periodic interplanetary and intermoon
(cycler) trajectories [9, 10, 11, 12]. We expand upon this concept, and map a variety of reachable
orbital parameters onto the globe including inclination, node crossings, and periods. We also change
the orientation of the v-infinity globe so that contours on the globe correspond directly to the
asymptotic sub-pointsa when the gravity-assist body is tidally locked. This globe orientation (a
90 degree shift from that proposed earlier [9]) is therefore also convenient for designing flybys over
specific regions on a flyby body.
Figure 1 shows a simple version of the v-infinity globe with blue contours representing orbits
with the same orbit period and green contours with the same inclination. Such a map may have
contours that represent properties of orbits other than resonance and inclination. For example orbits
that impact Saturn or its rings may be shown as well as orbits that cross the orbits of other moons.
Contours describing many useful properties of orbits may be drawn on this globe to assist in the
design of a tour.
0.6
0.4
! c
(v /v ) " p
3
0.2
0
!0.2
!0.4
!0.6
!0.5
0
0.5
0.4
0.6
!0.2
0
0.2
!0.4
!0.6
(v /v ) " p
! c
(v!/vc) " p2
1
Figure 1: The V-Infinity Globe
a The hyperbolic asymptote is normal to the surface of the gravity-assist body and therefore has a unique sub-point
on the surface.
2
Analysis
This section will develop the equations that will be used to draw various contours on the v-infinity
globe (also known as the crank sphere) and an associated coordinate system. We will then relate
regions on this sphere to various qualities of orbits around the central body. This will enable us to
construct a tour using a map of surface the globe.
The Velocity Triangle
By the patched-conic assumption, the spacecraft’s velocity relative to the central body (~vsc ) is given
by the vector sum of the gravity-assist body velocity (~vga ) and the spacecraft’s v-infinity (~v∞ ) with
respect to the gravity-assist body. This sum is shown both in Eqn. 1 and Fig. 2.
~vsc = ~vga + ~v∞
(1)
!vsc
!vga
!v∞
!vsc
!vga
!v∞
!vsc
!vga
!v∞
α
Figure 2: Gravity-Assist Vector Diagram
The time of a flyby yields the position and velocity of the gravity-assist body. The spacecraft
position is assumed to be the same as the gravity-assist body at this time (i.e. assuming zero sphere
of influence patched conics). And by Eqn. 1, once we know ~v∞ , we can get the spacecraft’s velocity
(~vsc ) from the gravity-assist body’s velocity (~vga ).
Pump Angle
The angle drawn in Fig. 2 is the pump angle[3] (α). Applying the law of cosines to Fig. 2, we may
write:
2
2
2
vsc
= v∞
+ vga
+ 2v∞ vga cos(α)
(2)
or, in terms of cos(α):
cos(α) =
2
2
2
vsc
− v∞
− vga
2v∞ vga
(3)
where α is always positive by definition.
Notice, since we know vga from the flyby time, that vsc is a function of v∞ and α. And, by vis
viva, vsc is directly related to asc (or vi for a hyperbolic orbit):
1
2
2
−
(4)
vsc = µcb
renc
asc
2
vsc
=
2µcb
+ vi2
renc
(5)
1
3
1
1
1
where renc is the distance to the central body at the time of the flyby. These equations can be
written more elegantlyb by dividing by the square of the local circular velocity (vc ):
vsc
vc
2
vsc
vc
2
=2−
renc
asc
=2+
vi
vc
(6)
2
(7)
where vc is given by:
r
µcb
vc =
renc
(8)
This then allows us to write asc as:
renc
1 2
2
= 2 − 2 v∞
+ vga
+ 2v∞ vga cos(α)
asc
vc
(9)
For elliptical and circular orbits, the semi-major axis is directly related to the orbit period (for
an ellipse) by Kepler’s third law:
a3sc = µcb
Tsc
2π
2
(10)
Which we may also re-write using the local circular orbit period (Tc ) to:
asc
renc
3
=
Tsc
Tc
2
(11)
where Tc is given by:
s
3
renc
Tc = 2π
µcb
(12)
This allows us to write the spacecraft period in terms of the pump angle:
Tsc
=
Tc
2−
3
−2
1 2
2
v
+
v
+
2v
v
cos(α)
∞ ga
ga
vc2 ∞
(13)
So if we know the v∞ magnitude, the pump angle gives us the orbit period (or vi ). Thus, α is a
measure of the energy relative to the central body. This relationship is independent of inclination
(isc ), i.e. ~vsc can rotate around ~vga without changing the relations Fig. 2 or Eqn. 2.
Since π ≥ α ≥ 0 (i.e. α is a positive angle), we may now bound asc (and therefore energy) for
all possible orbits with a given v∞ :
2−
renc
1 2
1 2
2
2
v∞ + vga
− 2v∞ vga ≤
≤ 2 − 2 v∞
+ vga
+ 2v∞ vga
2
vc
asc
vc
(14)
The pump angle also determines whether the spacecraft orbit is in the same direction as the
gravity-assist bodies orbit or not. We shall use prograde to denote ~vsc in the same direction as the
b This
form suggests a non-dimensionalization given in the appendix that is helpful when using these equations
programatically, because it avoids passing constants like µcb to subroutines.
4
gravity-asist body’s orbital motionc and a retrograde for the opposite direction. Then by inspection
from Fig. 2:
if prograde:
vga > −v∞ cos(α)
(15)
if retrograde:
vga < −v∞ cos(α)
(16)
When vga = v∞ cos(α) the spacecraft orbit is rectilinear and is headed on a straight line either away
from or towards the central body.
Crank Angle
Since the pump angle is independent of inclination, in order to understand how inclinationd is a
function of the direction of the ~v∞ , we need the other angle in our spherical coordinate system,
crank (κ). This angle describes the orientation of the plane comprised of the ~vsc , ~vga , and ~v∞
vectors (i.e. the plane of Fig. 2) with respect to the plane of the gravity-assist body’s orbit.
The pump and crank angles are illustrated in Fig. 3 with the sphere of possible ~v∞ directions.
q̂1 p̂1
q̂3 p̂2
ĉ1 q̂1
ĉ3 q̂2
κ γga
q̂1
q̂3
ĉ1
ĉ3
κ
ga-bo
!v!vscsc
!v!vgaga
!v!v∞∞
!vsc
!vga
!v∞
dy or
κ
α
bit pla
ne
to cb
Figure 3: Crank Angle
To describe this orientation, we construct a reference framee from the gravity-assist body’s orbit
tied to the direction of ~vga , the q̂i -frame:f
q̂2 = v̂ga
r̂enc × v̂ga
q̂3 =
cos(γga )
(17)
q̂1 = q̂2 × q̂3
(19)
(18)
Next we define the ĉi -frame which is tied to the plane of Fig. 2, with ĉ2 in the same direction as q̂2 :
ĉ2 = v̂ga
v̂ga × v̂∞
ĉ3 =
sin(α)
ĉ1 = ĉ2 × ĉ3
(20)
(21)
(22)
1
c For
Neptune’s moon Triton, this means what we will call a prograde orbit is retrograde with respect to Neptune’s
11
pole.
d Traditionally, inclination is strictly positive. But a negative inclination may be used to correspond to inclination
1
measured at the descending node.
1
e All reference frames and their transformations
are defined in the appendix.
1 1
f By the convention in the appendix, v̂
vga .
ga is a unit vector in the direction of ~
1
5
q̂1
q̂3
q̂1 ĉ1
q̂3 ĉ3
ĉ1 κ
ĉ3
κ
q̂1
q̂3
ĉ1
ĉ3 q̂1
κ q̂3
ĉ1
ĉ3 q̂q̂1 1
κ q̂q̂3 3
ĉĉ1 1
ĉĉ3 3
κκ
Figure 4: Definition of Crank Angle
The crank angle (κ) is then defined as the angle between the gravity-assist body’s orbit normal,
i.e. q̂3 , and the plane of the triangle in Fig. 2 (ĉ3 ). In the ĉi -frame, it is straightforward to write the
components of ~v∞ from Fig. 2:
~v∞ = v∞ sin(α)ĉ1 + v∞ cos(α)ĉ2
(23)
Which, in the q̂i -frame is:
~v∞ =
v∞ sin(α) cos(κ)q̂1 + v∞ cos(α)q̂2 − v∞ sin(α) sin(κ)q̂3
(24)
Next, the components of ~vsc can be used from Eqn. 1 to write another equation for ~v∞ . To do
this, we will introduce two new reference frames tied to the position vector at the flyby (~renc ): a
p̂i -frame in the gravity-assist body’s orbit plane,g and a ŝi -frame in the spacecraft’s orbit plane.
p̂1 = r̂enc
r̂enc × v̂ga
p̂3 =
cos(γga )
(25)
(26)
1
p̂2 = p̂3 × p̂1
ŝ1 = r̂enc
r̂enc × v̂sc
ŝ3 =
cos(γsc )
ŝ2 = ŝ3 × ŝ1
1
(27)
1
(28)
1
(29)
(30)
At the time of the flyby, the location both the spacecraft and the gravity assist body with respect
to the central body can now be described as:
11
~renc = renc ŝ1 = renc p̂1
(31)
We may write each velocity vector from Fig. 2 in the p̂i -frame:
g If
the gravity-assist body is in a circular orbit, the p̂i -frame is identical to the q̂i -frame.
6
~vga = vga q̂2
(32)
= vga sin(γga )p̂1 + vga cos(γga )p̂2
~vsc = vsc sin(γsc )ŝ1 + vsc cos(γsc )ŝ2
= vsc sin(γsc )p̂1 + vsc cos(γsc ) cos(isc )p̂2 + vsc cos(γsc ) sin(isc )p̂3
(33)
We may write ~v∞ in the p̂i -frame either from Eqn. 1:
~v∞ = ~vsc − ~vga
= [vsc sin(γsc ) − vga sin(γga )]p̂1 + [vsc cos(γsc ) cos(isc ) − vga cos(γga )]p̂2
(34)
+ vsc cos(γsc ) sin(isc )p̂3
or from Eqn. 24:
~v∞ =
v∞ [sin(α) cos(κ) cos(γga ) + cos(α) sin(γga )]p̂1 + v∞ [cos(α) cos(γga )
− sin(α) cos(κ) sin(γga )]p̂2 − v∞ sin(α) sin(κ)p̂3
(35)
By combining Eqn. 34 and Eqn. 35, we get an equation from each of the p̂i :
vsc sin(γsc ) − vga sin(γga ) = v∞ sin(α) cos(κ) cos(γga ) + v∞ cos(α) sin(γga )
(36)
vsc cos(γsc ) cos(isc ) − vga cos(γga ) = v∞ cos(α) cos(γga ) − v∞ sin(α) cos(κ) sin(γga )
(37)
vsc cos(γsc ) sin(isc ) = −v∞ sin(α) sin(κ)
(38)
We may solveEqn. 36 for cos(κ):
cos(κ) =
1
vsc sin(γsc ) − vga sin(γga ) − v∞ cos(α) sin(γga )
v∞ sin(α) cos(γga )
(39)
In the case of γga = 0 this simplifies to:
cos(κ) =
vsc sin(γsc )
v∞ sin(α)
(40)
To resolve quadrant ambiguities in Eqn. 39, from Eqn. 38:
vsc cos(γsc ) sin(isc )
v∞ sin(α)
(41)
sign(κ) = sign(isc cos(γsc ))
(42)
sin(κ) =
Which tells us:
When the orbit is prograde, cos(γsc ) ≥ 0 and κ will have the same sign as isc .
A further derivation from the above relations can be found in the appendix. Relations are given
that allow conversion between pump & crank and a full set of orbit elements.
The V-Infinity Globe
Figure 3 shows a sphere for all v-infinity vectors with the same magnitude. Each point on this
sphere represents a direction of ~v∞ , the corresponding ~vsc , and therefore an orbit about the central
body. This v-infinty globe represents all orbits with a given v-infinty magnitude with repsect to the
gravity-assist body.
By relating the pump and crank of a v-infinity vector to the orbit of the central body, we can
then draw contours of orbits on the v-infinty globe representing orbits with a given characteristic
such as orbit period or inclination as shown in Fig. 1. To draw contours of constant period we use
Eqn. 13, and to draw contours of constant inclination we use Eqn. 39.
7
Relating Pump & Crank to Latitude and Longitude
Most moons in the solar systemh are tidally locked to their planet. In terms of coordinate systems,
this means that the moon’s prime meridian, on average, points towards the planet, i.e. the −p̂1
direction. In addition, the poles of most moons point in the direction of their orbit normals.
This creates a correspondence in the orientation of the v-infinity globe shown in Fig. 1 and the
latitude and longitude on a body. And we may write the v-infinty unit vector in terms of latitude
(φ) and longitude (λ) for such moons[14] (assuming no libration of the prime-meridian):
v̂∞ = − cos(φ)cos(λ)p̂1 − cos(φ) sin(λ)p̂2 + sin(φ)p̂3
(43)
Equation 43 may be used with Eqn. 35 to convert pump and crank to latitude and longitude, and
Eqn. 34 may be used with inclination and flight path angle.
We have used outbound asymptotes to draw our tour map. The inbound flyby asymptotes may
also be plotted as the negative vector of the previous flybys outbound asymptote. If this is done,
a flyby’s groundtrack is then a great circle connecting these two points. Analytic expressions for
drawing these groundtracks are given in [14].
Resonant Transfers
If a the time of flight of a transfer is an integer multiple of the gravity-assist body’s period that
transfer is resonant. Resonant transfers are labeled m : n, where m is the number of gravity-assist
body revs and n is the number of spacecraft revs.
m
Tsc
=
Tga
n
(44)
In resonant transfers both flybys occur at the same point in the gravity-assist body’s orbit. Therefore,
the spacecraft orbit plane is only constrained to contain the line connecting this point to the central
body and the transfer may achieve a wide range of inclination. Because of this flexibility, period
contours on the v-infinity globe are only plotted for resonant transfers.
Nodal Control of Resonant Transfers
During inclined resonant transfers, one node crossing is at the gravity-assist body (e.g. Titan) and
the other node radius, called the vacant node,[16] is a function of the orbit geometry[6]. Figure 5
illustrates vacant node. By controlling the location of this node crossing the spacecraft may be
targeted to cross the orbit of another moon or to avoid the rings.
x direction
L
des
of no
line
vacant
node
encounter
Figure 5: Vacant Node
h Saturn’s moon Hyperion is not tidally locked. In fact, it is chaotically rotating. But all moons at Saturn larger
than Hyperion and Jupiter’s Galilean satellites are all tidally locked and their poles point towards their orbit normals.
8
Manipulation of Eqn. 103 and Eqn. 104 from the appendix gives the following equation[16] for
the vacant node:
asc
(1 − e2sc )
rvac
= rencasc
renc
2 − renc (1 − e2sc )
=
2
(45)
psc
renc
sc
− rpenc
Equation 45 is can be used with other relations in the appendix to find the pump & crank angles for
orbits that avoid ring debris or radiation fields. It may also be used to place a vacant node crossing
at the orbit of another moon to set up a geometry favorable to non-targeted encounters.
Non-Resonant & Backflip Transfers
A transfer in which the flybys occur at different places in the gravity-assist body’s orbit is called a
non-resonant transfer. In general, the two flybys and the central body do not fall on a line and the
spacecraft inclination is constrained to be in the gravity-assist body’s orbit plane. The special case
of a non-resonant transfer where the angle between flybys is π radians is called a backflip transfer[3]
or a pi-transfer. Because the encounters fall on a line in a backflip transfer, these transfers are
typically inclined.
These transfers can be found analytically via the method given in [15]. An interesting result is
that backflip transfers have an inclination specified by the v-infinty:
renc
v2
1
3−
(46)
− ∞
cos(isc ) =
2
asc
vc2
The pump angle for these transfers can be found from the period given by the method given in
[15]. The crank angle for a non-resonant transfer is either 0 or π depending on whether the flyby
is inbound or outbound respectively, and the crank angle for a backflip transfer may be found from
the inclination given in Eqn. 46.
On the v-infinity globe these transfers appear as a set of two points with the same pump angle,
one inbound and the other outbound. In plotting a tour across the globe they act as wormholes,
teleporting to the opposite hemisphere of the v-infinity globe.
Non-Dimensionalisation of Tour Map
If we assume that the gravity-assist body is in a circular orbit we can non-dimensionalize the tour
map so that it is applicable to any moon. To do this we divide distances by renc , velocities by vc , and
times by Tc . Figure 6 shows this map for v∞ /vc = 0.75 with vacant node contours, non-resonant,
and backflip transfers added. Figure 7 through Fig. 11 shows this for other values of v∞ /vc .
In these plots, contours of constant period are shown in blue and labeled with their resonance
with respect to gravity-assist body. Inclination contours are shown in green for steps of 10 degrees.
Node radius contours non-dimesionlized in terms of rvac /renc are shown in cyan. The red period
contour represents escape energy and the orbits in the region bounded by it are hyperbolic with
respect to the central body. Similarly, the dashed black contour bounds orbits that are retrograde.
Finally the magenta ’x’ and ’o’ marks represent some non-resonant and backflip transfers. The ‘x’
represents the asymptote before a transfer, and the ‘o’ is the asymptote after the transfer.
On these maps, the “northern hemisphere” is comprised of orbits which have the gravity-assist
encounter at the ascending node, and the “southern hemisphere” of encounters at the descending
node. The hemisphere from 90◦ E to 90◦ W (i.e. the moon’s anti-planet hemisphere) is comprised
of asymptotes for orbits which encounter the gravity-assist body outbound from periapsis, and the
opposite hemisphere is comprised of orbits with inbound flybys.
9
Figure 6: Non-Dimensional Tour Map, v∞ /vc = 0.75
Figure 7: Non-Dimensional Tour Map, v∞ /vc = 0.25
10
Figure 8: Non-Dimensional Tour Map, v∞ /vc = 0.5
Figure 9: Non-Dimensional Tour Map, v∞ /vc = 1.0
11
Figure 10: Non-Dimensional Tour Map, v∞ /vc = 1.5
Figure 11: Non-Dimensional Tour Map, v∞ /vc = 2.0
12
Bending Angle
Now that we can draw a map, we need to deduce how flybys can move us across the v-infinity globe
so that we can plan a route between orbits that interest us (i.e. to design a tour). To do this we use
the bending angle from the flyby (derived in the appendix):
sin
δ
µga
=
2
2
µga + rpf b v∞
(47)
This bending angle translates to the v-inifnty globe as simply a distance on the surface. If we
had a 3D globe we could measure this distance with a pair of spanners much as with terrestrial
navigation. In practice we can also do this with a ruler on more convenient 2D projections as long
as we stay away form the high latitudes of the tour map.
Application To Cassini Extended Mission Design
90
80
6:1
5:1
4:1
3:1
5:2
Asymptote North Latitude
70
2:1
5:3
3:2
5:4
6:5
60
1:1
Hy
6:7
5:6
4:5
3:4
5:7
2:3
5:8
3:5
4:7
5:9
6:11
Ti
50
40
Rh
Di
1:2
Te
6:13
5:11
4:9
30
En
T44
T43
T42
Mi
3:7
5:12
T41
2:5
20
60
70
85
80
75
65
50
45
40
10
35
30
15
10
0
5
0
0
20
40
T38
25
20
T40
T39
55
60
80
100
Asymptote East Longitude
120
140
T37
160
180
Figure 12: Tour Map For Cassini Extended Mission
Figure 12 is the tour map that was used for much of the early Cassini extended mission design.
It is a detailed map for the Cassini v-infinty at the end of the prime mission (5.8 km/s) showing
many various Titan resonances. Node crossing distances are shown for the orbits of all of the major
satellites except for Iapetusi , and node crossings that intersect the rings are shown as a grey region.
Orbits with periapses below Saturn’s surface are in the central white region.
The last few orbits of the Cassini prime mission are plotted on Fig. 12. Each orbit is a point, and
the flybys are the lines connecting the points. The last orbit of the prime mission is marked with an
asterisk and a red circle is drawn around it showing the 8◦ of bending from a 1000 km Titan flyby.
The extended mission will start from this point and move in steps of 8◦ or less across the map.
With this map, the Cassini tour designers were able to draw in tours with a pencil and a ruler.
The flight times of the tours could then be estimated by adding up the Titan revs of the various
i Iapetus
orbits in a different plane from the other satellites end encounters with it do not necessarily occur on the
line of nodes.
13
resonant orbits. From this map the tour designers could quickly assess the various methods for
achieving equatorial orbits or of placing the node crossings at the orbits of other moons (particularly
Enceladus).
Conclusion
This method allows the tour designer to plot a course through many possible gravity-assist transfers
to find a tour that meets scientific objectives in a systematic and efficient manner. It is especially
useful with inclined orbits. It’s use in the Cassini extended mission was important in developing
insight and intuition into the complex space of possible tours.
In the future, we hope to extend the application of this method to tours that use ∆V to change
the v-infinity. This would be equivalent to adding different layers to the map which can be shifted
between with maneuvers. Similarly, we would like to look at extensions to the Jovian system where
flybys of other moons could also be used to modulate v-infinity.
We are also interested in possible applications of this method to planetary science. This method
could possibly be used to reconstruct the orbits of objects that have hit a tidally-locked moon. A
crater’s size can give an estimate of the energy of the impact, and the latitude and longitude of a
crater corresponds to a point on the v-infinity globe.
Acknowledgments
We would like to thank Jerry Jones and Robert Mitchell for their support of the development of
these techniques for the Cassini extended mission design. The research described in this paper was
carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with
the National Aeronautics and Space Administration.
14
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Aerospace Sciences Meeting, AIAA Paper 83-0101, Reno, Nevada, Jan. 1983.
[5] Wolf, A.A., “Touring the Saturnian System,” Space Science Reviews, Vol. 104, 2002, pp. 101-128.
[6] Strange, N. J., and Longuski, J. M., “Graphical Method for Gravity-Assist Trajectory Design,” Journal
of Spacecraft and Rockets, Vol. 39, No. 1, Jan. 2001, pp. 9-16.
[7] Heaton, A. F., Strange, N. J., Longuski, J. M., and Bonfiglio, E. P., “Automated design of the Europa
Orbiter tour,” Journal of Spacecraft and Rockets, Vol. 39, No. 1, Jan. 2001, pp. 17-22
[8] Heaton, A. F., and Longuski, J. M., “The feasibility of a Galileo-style tour of the Uranian satellites,”
Journal of Spacecraft and Rockets, Vol. 40, No. 4, July 2003, pp. 591-596
[9] Russell, R. P., and Ocampo, C. A., “Geometric Analysis of Free-Return Trajectories Following a
Gravity-Assisted Flyby,” Journal of Spacecraft and Rockets, Vol. 42, No. 1, 2005, pp. 694-698
[10] Russell, R. P., and Ocampo, C. A., “Systematic Method for Constructing Earth-Mars Cyclers Using
Free-Return Trajectories,” Journal of Guidance, Control, and Dynamics, Vol. 27, No. 3, 2004, pp.
321-335.
[11] Russell, R. P., and Ocampo, C. A., “Global Search for Idealized Free-Return Earth-Mars Cyclers,”
Journal of Guidance, Control, and Dynamics, Vol. 28, No. 2, 2005, pp. 194-208.
[12] Russell, R. P., Strange N.J., “Planetary Moon Cycler Trajectories,” Paper AAS 07-118, AAS/AIAA
Space Flight Mechanics Meeting, Sedona, AZ, Jan 2006.
[13] Battin, R.H., An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition,
AIAA Press, Reston, Va., 1999.
[14] Buffington, B.B., and Strange, N. J., “Science Driven Design of Enceladus Flyby Geometry,” International Astronautical Congress, IAC Paper 06-C1.6.05, Valencia, Spain, Oct. 2006
[15] Strange, N.J. and Sims, J.A., “Methods for the Design of V-Infinity Leveraging Maneuvers,”
AAS/AIAA Astrodynamics Conference, AAS Paper 01-473, Québec, Québec, July/Aug. 2001.
[16] Strange, N.J., “Control of Node Crossings in Saturnian Gravity-Assist Tours,” AAS/AIAA Astrodynamics Conference, AAS Paper 03-545, Big Sky, MT, Aug. 2003.
15
A
Nomenclature Appendix
A.1
α
αe
γ
δ
θ
η
κ
µ
τ
φ
ψ
Ω
ω
a
B
CT iss
E
e
f
f∞
h
I
i
J
L
M
m
A.2
cb
fb
ga
n
IO
OI
sc
0
00
Variables
pump angle
pump angle for ve
flight path angle
flyby bending angle
b-plane angle
angle from asc. node (f + ω)
crank angle
gravitational parameter
time wrt periapsis
declination or latitude
right ascension or longitude
longitude of the ascending node
argument of periapsis
semi-major axis
impact parameter
Tisserand invariant
eccentric anomaly
eccentricity
true anomaly
true anomaly of asymptote
specific angular momentum
inclination wrt cb pole
inclination wrt ga-body orbit
a cost function
longitude of encounter
integer number of ga-body orbits
number of ga-body orbits
N
n
p
q
r
ra
rasc
rdsc
renc
rp
rpf b
rvac
s
T
Tc
t
tof
u
v
v∞
va
vc
ve
vi
vp
vpf b
integer number of sc orbits
number of sc orbits
semi-latus rectum
non-dimensional orbit period
distance from center of body
apoapsis radius wrt central body
radius of asc. node crossing wrt cb
radius of desc. node crossing wrt cb
radius wrt central body at encounter
periapsis radius wrt central body
periapsis radius wrt ga-body
radius of vacant node crossing wrt cb
non-dimensional radius
orbit period
circular orbit period at renc
time
time of flight between encounters
non-dimensional velocity
velocity
hyperbolic excess velocity wrt ga-body
velocity at apoapsis wrt cb
circular velocity wrt cb at renc
local escape velocity wrt cb
hyperbolic excess velocity wrt cb
velocity at periapsis wrt cb
velocity at periapsis wrt ga-body
Subscripts / Superscripts
pertains to central body
pertains to patched-conic spacecraft orbit relative to gravity assist body at encounter
pertains to gravity-assist body orbit relative to central body at encounter
nondimensional patched-conic spacecraft orbit relative to central body at encounter
inbound to outbound transfer
outbound to inbound transfer
pertains to patched-conic spacecraft orbit relative to central body at encounter
quantity before flyby or maneuver
quantity after flyby or maneuver
16
A.3
~x
x̂
x
Vectors
A.5.6
a vector
a unit vector
a vector magnitude
A.4
0
Ŝ = b̂3 = v̂∞
(63)
v̂ 0 × (r̂enc × v̂ga ) ~
T~ = b̂1 = ∞
R
cos(φ0 ) cos(γga )
Nondimensionalization
posiition
time
velocity
b̂i : b-plane wrt ga-body orbit
= b̂2 = b̂3 × b̂1
(64)
divide by renc
divide by Tc
divide by vc
Frame Transformations
p̂1
A.6.1 p̂i ↔ q̂i
p̂2
p̂1
p̂
A.5 Reference Frame Definitions
1
q̂1 p̂1
p̂2
p̂2
p̂q̂12
A.5.1 p̂i : gravity-assist body position
p̂2
q̂1
q̂1
p̂ga
2
γ
q̂1
q̂2
p̂1 = r̂enc
(48)
q̂
q̂
2
1
p̂1γ
q̂2
ga
r̂enc × v̂ga
γ
q̂
ga
2
(49)
p̂3 =
p̂2
γga
cos(γga )
γ
ga
q̂1
p̂2 = p̂3 × p̂1
A.5.2
A.6
(50)
q̂2
p̂1 = cos(γ
γ ga )q̂1 + sin(γga )q̂2
q̂i : gravity-assist body velocity
q̂2 = v̂ga
r̂enc × v̂ga
q̂3 =
cos(γga )
(51)
q̂1 = q̂2 × q̂3
(53)
A.5.3
A.5.4
p̂2 = − sin(γga )q̂1 + cos(γga )q̂2
(66)
p̂3 = q̂3
(67)
q̂1 = cos(γga )p̂1 − sin(γga )p̂2
(68)
q̂2 = sin(γga )p̂1 + cos(γga )p̂2
(69)
q̂3 = p̂3
(70)
A.6.2
p̂2
p̂i p̂
↔2 ŝi p̂
3
p̂
p̂3 ŝ22
ŝ2 ŝp̂33
ŝ2
ŝ3 isc
ŝ3
isc
isc
p̂2
p̂2
p̂3
p̂3
ŝ2
ŝ2
ŝ3
ŝ3
isc p̂2
isc
1p̂3
ŝ2
1
p̂1 = ŝ1
1
ŝ3
p̂2 = cos(isc )ŝ2 − sin(isc )ŝ3
1 isc
(54)
(55)
(56)
r̂i : spacecraft velocity
r̂2 = v̂sc
r̂enc × v̂sc
r̂3 =
cos(γsc )
r̂1 = ŝ2 × ŝ3
A.5.5
(52)
ŝi : spacecraft position
ŝ1 = r̂enc
r̂enc × v̂sc
ŝ3 =
cos(γsc )
ŝ2 = ŝ3 × ŝ1
(57)
(58)
(59)
(71)
1
(72)
p̂3 = sin(isc )ŝ2 + cos(isc )ŝ3
(73)
ŝ1 = p̂1
(74)
ŝ2 =
ĉi : crank plane
cos(i1
sc )p̂2
+ sin(isc )p̂3
ŝ3 = − sin(isc )p̂2 + cos(isc )p̂3
ĉ2 = v̂ga
v̂ga × v̂∞
ĉ3 =
sin(α)
ĉ1 = ĉ2 × ĉ3
(65)
ga
(60)
(61)
(62)
17
1
1
1
1
1
(75)
(76)
A.6.3
ŝ1
ŝŝ1
2
ŝr̂2
1
r̂r̂1
2
r̂γ2
sc
γsc
ŝi ↔ r̂i
ŝ1
ŝ1
ŝ2
ŝ2
r̂1
r̂1
r̂2
ŝ1γsc r̂2
ŝ2 γsc
r̂1
r̂2
ŝ1 = cos(γsc )r̂1 + sin(γsc )r̂2
γsc
ŝ1
ŝ2
r̂1
r̂2
γsc
(77)
ŝ2 = − sin(γsc )r̂1 + cos(γsc )r̂2
(78)
ŝ3 = q̂3
(79)
r̂1 = cos(γsc )ŝ1 − sin(γsc )ŝ2
(80)
r̂2 = sin(γsc )ŝ1 + cos(γsc )ŝ2
(81)
r̂3 = ŝ3
(82)
A.6.4
q̂1
q̂i ↔ ĉiq̂
3
q̂1 ĉ1
q̂3 ĉ3
ĉ1 κ
ĉ3
κ
1
q̂1
q̂3
ĉ1
ĉ3 q̂1
κ q̂3
ĉ1
ĉ3 q̂1q̂111
1
κ q̂q̂3 3
1
ĉĉ1 1
ĉĉ3 3
κκ
1
q̂1 = cos(κ)ĉ1 + sin(κ)ĉ3
(83)
q̂2 = ĉ2
(84)
q̂3 = − sin(κ)ĉ1 + cos(κ)ĉ3
(85)
ĉ1 = cos(κ)q̂1 − sin(κ)q̂3
(86)
ĉ2 = q̂2
(87)
ĉ3 = sin(κ)q̂1 + cos(κ)q̂3
(88)
18
1
1
1
B
Flight Path Angle Relations
In Eqn. 39 to Eqn. 42, orbital angular momentum will give us γsc as a function of asc and esc from
this cross product:
~hsc = ~renc × ~vsc
(89)
= hsc ŝ3
= renc vsc cos(γsc )ŝ3
but also it is also given by this equation[13] from the solution to the two-body problem:
p
hsc = µcb asc (1 − e2sc )
Combining Eqn. 89 with Eqn. 90 gives:
p
µcb asc (1 − e2sc )
cos(γsc ) =
r v
r enc sc
vc
asc
=
(1 − e2sc )
vsc renc
(90)
(91)
Which we may combine with Eqn. 6:
s
cos(γsc ) =
asc
renc
(1 − e2sc )
2−
(92)
renc
asc
and then sin(γsc ) is:
s
sin(γsc ) = sign(fsc ) 1 −
asc
renc
(1 − e2sc )
2−
(93)
renc
asc
The sign of γsc in Eqn. 91 may be determined by noting whether the flyby occurs before periapsis
(i.e. inbound) or after periapsis (i.e. outbound) of the orbit relative the central body. For inbound
flybys γsc < 0, and for outbound flybys γsc > 0. When γsc = 0 the flyby occurs at either periapsis
(if asc < renc ) or apoapsis (if asc > renc ). Or from Eqn. 36:
sign(γsc ) =
sign[v∞ sin(α) cos(κ) cos(γga ) + v∞ cos(α) sin(γga ) + vga sin(γga )]
(94)
which when γga = 0 simplifies to:
sign(γsc ) = sign[cos(κ)]
(95)
But what if we’re given κ and want to solve for γsc ? From Eqn. 37:
cos(γsc ) =
1
[vga cos(γga ) + v∞ cos(α) cos(γga ) − v∞ sin(α) cos(κ) sin(γga )]
vsc cos(isc )
Equation 91 may then be used to find eccentricity:
renc renc
2
2
esc = 1 − 2 −
[cos(γsc )]
asc
asc
(96)
(97)
Equation 38 can be rewritten for isc :
sin(isc ) = −
v∞ sin(α) sin(κ)
vsc cos(γsc )
(98)
19
if we then substitue in Eqn. 96:
tan(isc ) = − v∞ sin(α) sin(κ)/ vga cos(γga ) + v∞ cos(α) cos(γga )
−v∞ sin(α) cos(κ) sin(γga )
(99)
Which when γga = 0 reduces to:
tan(isc ) = −
v∞ sin(α) sin(κ)
vga + v∞ cos(α)
(100)
Since 0 ≤ |isc | ≤ π2 for prograde orbits and π2 ≤ |isc | ≤ π for retrograde orbits, we know the
quadrant of isc based on whether the orbit is prograde or retrograde.
Although inclination is traditionally a strictly positive quantity, we have allowed negative values
for isc in Eqn. 99 and elsewhere. In Eqn. 33, the p̂3 component of ~vsc will have the same sign as
isc . This means that a negative inclination corresponds to a flyby at the spacecraft’s descending
node. By allowing isc to go negative, we do not have to track whether a flyby is at the ascending or
descending node separately.
The maximum value for inclination happens when, in Eqn. 99, κ = ± π2 . Therefore, inclination,
for a given α, is boundedj by:
| tan(isc )| ≤
C
v∞ sin(α)
vga cos(γga ) + v∞ cos(α) cos(γga )
(101)
Orbit Orientation
We start by describing the spacecraft orbit orientation. To do this, we first describe the longitude
of encounter (L) as the angle from some reference direction to the encounter, as shown in Fig. 5.
The longitude of the encounter is the longitude of the ascending node (Ωsc ) if the encounter is
at the ascending node, and π − Ωsc if the encounter is at the descending node. Since isc > 0 when
a flyby is at the ascending node, and isc < 0 for the descending node, we made write:
(
Ωsc =
L
L+π
if isc ≥ 0
if isc < 0
(102)
When isc = 0, the choice of Ωsc is arbitrary, so the relation above sets it equal to L in that case.
The longitude of the ascending node and the inclination gives the orientation of the spacecraft’s
orbit plane. To orient the orbit in this plane we use the argument of periapsis (ωsc ) as shown in
Fig. 13.
The argument of periapsis is defined as the angle from the ascending node to periapsis. Since
the true anomaly of the encounter (fsc ) is the angle from periapsis to the encounter:
fsc
(
−ωsc
=
π − ωsc
if isc ≥ 0
if isc < 0
(103)
If we substitute these values for fsc into the conic equation, setting r = renc :
renc =
j due
asc (1 − e2sc )
1 ± esc cos(ωsc )
(104)
to our definition of prograde, cos(γga ) is always positive
20
lin
ap e o
sid f
es
wsc
line of
nodes
isc
e
lan
ep
c
n
ere
ref
Figure 13: Argument of Periapsis
we may then write ωsc as the following:
if at ascending node (i.e., isc ≥ 0):
1
asc
cos(ωsc ) =
(1 − e2sc ) − 1
esc renc
(105)
if at descending node (i.e., isc < 0):
1
asc
2
cos(ωsc ) =
(1 − esc )
1−
esc
renc
(106)
This gives us cos(ωsc ), but not the sign of ωsc . To resolve this quadrant ambiguity, we note from
Fig. 13 that the sign of ωsc depends of whether the flyby is inbound (i.e. before periapsis) or
outbound (i.e. after periapsis):
(
sign(ωsc ) =
1
−1
if flyby inbound
if flyby outbound
(107)
This means that ωsc will have the opposite sign of γsc , and, for prograde orbits, ωsc will also have
the opposite sign of κ.
D
Summary of Pump & Crank Relations
Pump & Crank to Orbit Elements
We now have derived equations to relate the flyby ~v∞ vector to the orbit around the central central
body. This section summarizes these relations and presents them in the correct order.
The time of the flyby gives us vga , γga , renc , and L from the gravity-assist body’s ephemeris.
From Eqn. 9, we may write the orbit semi-major axis from v∞ , and α:
asc =
2−
1
vc2
2
v∞
+
renc
2
vga + 2v∞ vga
cos(α)
(108)
Which also gives us period from Eqn. 11
Tsc
Tc
2
=
asc
renc
3
(109)
21
Next, Eqn. 99 gives us the flight path angle:
tan(isc ) = − v∞ sin(α) sin(κ)/ vga cos(γga ) + v∞ cos(α) cos(γga )
−v∞ sin(α) cos(κ) sin(γga )
(110)
where the quadrant is given by:
(
1
sign(cos(isc )) =
−1
if orbit prograde
if orbit retrograde
(111)
and from Eqn. 15 and Eqn. 16:
vga > −v∞ cos(α) ⇒ prograde
(112)
vga < −v∞ cos(α) ⇒ retrograde
(113)
The eccentricity is given by Eqn. 97:
e2sc
renc renc
2
=1− 2−
[cos(γsc )]
asc
asc
(114)
where:
cos(γsc ) =
1
[vga cos(γga ) + v∞ cos(α) cos(γga )
vsc cos(isc )
− v∞ sin(α) cos(κ) sin(γga )]
(115)
The longitude of the ascending node is then given by the longitude of the encounter:
(
Ωsc =
L
L+π
if isc ≥ 0
if isc < 0
(116)
and the argument of periapsis by Eqn. 105 and Eqn. 106:
cos(ωsc ) =
sign(isc ) asc
(1 − e2sc ) − 1
esc
renc
(117)
where Eqn. 94 and Eqn. 107 tell us the quadrant of ωsc :
sign(ωsc ) = − sign[v∞ sin(α) cos(κ) cos(γga ) + v∞ cos(α) sin(γga ) + vga sin(γga )]
(118)
And finally, the true anomaly of the flyby is:
fsc
(
−ωsc
=
π − ωsc
if isc ≥ 0
if isc < 0
(119)
Orbit Elements to Pump & Crank
The v∞ magnitude is given by Eqn. ??:
2
2
2
v∞
= vsc
+ vga
− 2vsc vga [sin(γsc ) sin(γga ) + cos(γsc ) cos(γga ) cos(isc )]
22
(120)
where:
s
cos(γsc ) =
asc
renc
(1 − e2sc )
2−
(121)
renc
asc
s
sin(γsc ) = sign(fsc ) 1 −
asc
renc
(1 − e2sc )
2−
(122)
renc
asc
The semi-major axis gives us the pump angle (which is always positive):
cos(α) =
2
2
2
vsc
− v∞
− vga
2v∞ vga
(123)
And crank is given by Eqn. 39 and Eqn. 42:
cos(κ) =
1
vsc sin(γsc ) − vga sin(γga ) − v∞ cos(α) sin(γga )
v∞ sin(α) cos(γga )
sign(κ) = sign(isc cos(γsc ))
E
(124)
(125)
Bending Angle Relations
In the theory of patched conics, we model the flyby as a hyperbola and calculate the angle, the
00
0
vector. The flyby is then
vector and the outgoing ~v∞
bending angle, between the incoming ~v∞
modeled as a rotation of the ~v∞ , which is treated as an instantaneous ∆V applied to the orbit
around the central body.
δ
f∞
f∞
"
"v∞
""
"v∞
δδ
f∞f∞
" "
"v∞
"v∞
"" ""
"v∞"v∞
δ π
+
2 2
Figure 14: Hyperbolic Flyby
Figure 14 shows the hyperbolic flyby in detail. The bending angle (δ) is the angle between the
two asymptotes. To find this angle, we will find the true anomaly of the asymtotes (f∞ ). We begin
with the conic equation:
rf b =
af b (1 − e2f b )
1 + ef b cos(ff b )
(126)
23
which we solve for true anomaly:
1 af b
cos(ff b ) =
(1 − e2f b ) − 1
ef b rf b
(127)
and then take the limit as radius goes to infinity:
#
"
af b (1 − e2f b )
1
−
cos(f∞ ) = lim
rf b →∞
rf b ef b
ef b
(128)
1
=−
ef b
and the eccentricity of the flyby is given by:
ef b = 1 −
rpf b
af b
(129)
µf b
2
v∞
2
rpf b v∞
=1+
µga
af b = −
(130)
ef b
(131)
By the geometric construction in Fig. 14, the f∞ angle is also given by:
f∞ =
δ
π
+
2
2
(132)
which means:
− cos(f∞ ) = − cos
δ
π
+
2
2
= sin
δ
2
(133)
So, the bending angle of the flyby is:
µga
δ
sin
=
2
2
µga + rpf b v∞
(134)
Which we can translate into ∆V :
∆Vf b =
2v∞ µga
2
µga + rpf b v∞
(135)
24