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JUPITER ICY MOONS ORBITER INTERPLANETARY INJECTION

PERIOD ANALYSIS

Theresa D. Kowalkowski,

Julie A. Kangas,

†

and Daniel W. Parcher

‡

This paper investigates the sensitivity of the planned Jupiter Icy Moons Orbiter
mission  to  variations  in  interplanetary  injection  date,  magnitude,  and  direction,
starting in a low-Earth assembly orbit.  These results are used to determine the
frequency  and  number  of  injection  opportunities  from  a  precessing  assembly
obit.  It is shown that the use of a low-thrust propulsion system with a nuclear-
electric  power  source  would  allow  the  interplanetary  trajectory  performance  to
be relatively insensitive to variations in injection conditions.  This result yields
many  injection  opportunities  due  to  the  long  injection  period  and  consecutive
orbits with favorable geometry.

INTRODUCTION

The proposed Jupiter Icy Moons Orbiter (JIMO) mission would employ a nuclear-electric propul-

sion (NEP) system to both rendezvous with Jupiter and explore three of its “icy” satellites: Callisto, Gany-
mede, and Europa. It is well-documented that highly-efficient electric-propulsion systems allow inter-
planetary missions to deliver more mass and/or reduce the flight times to their targets.

1-12

The NEP system

is an enabling technology for the planned JIMO mission because of the mission’s high ∆V requirements.

The JIMO architecture considered here calls for multiple launches into a low-Earth assembly orbit.

After assembly completion, the spacecraft would inject to a positive Earth-departure energy (C

). In this

paper, we investigate the mass-performance impact of varying the interplanetary injection date on four dif-
ferent trajectories: 6-year direct, 7.5-year direct, 6-year Earth gravity assist, and the JIMO Reference Tra-
jectory representation (a 5.5-yr direct case). Next, we explore the effects of variations in the injection burn
direction and magnitude. Finally, these results are used to ascertain the frequency and number of injection
opportunities from the precessing assembly orbit.

DEFINITIONS AND MODELS

The term

injection

refers to the departure from the assembly orbit onto an interplanetary trajectory.

The

injection period

is defined as the number of consecutive days during which the spacecraft can inject

Member of Engineering Staff; Guidance, Navigation, and Control Section; Jet Propulsion Laboratory, California Institute of

Technology, Mail Stop 301-150, 4800 Oak Grove Drive, Pasadena, California, 91109-8099.

E-mail: Theresa.D.Kowalkowski@jpl.nasa.gov. Phone: (818) 354-4699. Fax: (818) 393-7116.

†

Senior Member of Engineering Staff; Guidance, Navigation, and Control Section; Jet Propulsion Laboratory, California Institute of

Technology, Mail Stop 301-150, 4800 Oak Grove Drive, Pasadena, California, 91109-8099.
E-mail: Julie.A.Kangas@jpl.nasa.gov. Phone: (818) 354-8285. Fax: (818) 393-7413.

‡

Member of Engineering Staff; Guidance, Navigation, and Control Section; Jet Propulsion Laboratory, California Institute of

Technology, Mail Stop 230-205, 4800 Oak Grove Drive, Pasadena, California, 91109-8099.

E-mail: Daniel.W.Parcher@jpl.nasa.gov. Phone: (818) 393-0457. Fax: (818) 393-4215.

and still deliver sufficient mass to Jupiter. This is analogous to a conventional launch period, but the pro-
posed JIMO architecture requires the distinction between launch and injection because the spacecraft could
spend a substantial amount of time in the assembly orbit (weeks or more).

We define the

injection vector sensitivity

to be the effects of deviations in the magnitude and di-

rection of the injection burn from their optimized values. Although we do not model the finite burn, we
introduce dispersions in the optimal values of the Earth departure V

∞

(hyperbolic excess velocity) vector.

Finally, an

injection opportunity

is defined as each time the spacecraft and the assembly orbit are

aligned in the proper injection direction, within a specified tolerance. Because the node of the assembly
orbit precesses around the Earth, not every orbit will have an injection opportunity.

The spacecraft is assumed to have a 180 kW nuclear power supply with a 7000 second specific

impulse (Isp) and an efficiency of 70% (corresponding to 3.67 N of thrust and a 5.35x10

-5

kg/s mass flow

rate). The spacecraft is constrained to coast for 60 days after injection for reactor commissioning. When
the initial mass and departure energy are optimized, the injection vehicle curve shown in Figure 1 is used.

27500

30000

32500

35000

37500

40000

42500

Departure C

(km

)

(

)

Figure 1 – Injection vehicle performance curve

The majority of the trajectory optimization studies in this paper were performed with the Mission

Analysis Low-Thrust Optimization program (MALTO). MALTO is a patched-conic propagator that mod-
els low-thrust arcs as a series of small, impulsive maneuvers and maximizes the final mass of the space-
craft.

The only forces modeled are the sun’s gravity and the spacecraft thrust. No other bodies have

mass, and solar radiation pressure is not included. For rendezvous with Jupiter, the spacecraft matches
Jupiter’s position and velocity, but the planet itself has no gravitational pull. Earth departure is from a
massless Earth, and gravity-assist maneuvers are modeled as instantaneous rotations of the V

∞

vector.

A few select analyses were carried out using Mystic, a low-thrust trajectory optimization tool that

also maximizes the final mass of the spacecraft.

Mystic fully integrates trajectories with low-thrust arcs

modeled as a series of continuous-thrust segments.  The thrust has a constant magnitude and direction over
a given segment.  In these studies, the forces modeled are the Sun’s gravity, Mars’s gravity, Jupiter’s grav-
ity, (with no oblateness for any of the planets) and the spacecraft thrust.  Earth’s gravity was not modeled;
the  trajectory  starts  at  the  center  of  the  massless  Earth  (as  in  the  MALTO  analyses).    No  solar  radiation

pressure was modeled, either. Mystic targeted a final distance from Jupiter of less than 2,000,000 km at
less than 0 km

two-body energy with respect to Jupiter.

Although it is a lower-fidelity tool than Mystic, MALTO’s speed and ease of use makes it particu-

larly well-suited to performing parametric studies. We will show that the results from MALTO are some-
what conservative relative to Mystic but the MALTO findings are an excellent indicator of trends and over-
all behavior.

For the investigation of injection opportunities, a representative assembly orbit of 400 km altitude

and inclination of 28.5º was propagated with a 50x50 Earth gravity field. The initial right ascension of the
ascending node (RAAN) was chosen such that the first injection opportunity would be available soon after
the open of the injection period.

TRAJECTORY CASES

Although the planned JIMO mission assumed a baseline trajectory, called the Reference Trajec-

tory,

we present results from three additional trajectory types. Many of the analyses on these other trajec-

tories preceded and influenced the selection of the Reference Trajectory type. In addition, the results pre-
sented here are documented for the sake of potential future mission design efforts.

Three of the four representative trajectories in this paper use the injection vehicle performance

curve given in Figure 1. The fourth trajectory assumes a bounded departure C

magnitude and mass, dis-

cussed below.  All four trajectories were optimized assuming a fixed flight time, but the software optimized
the injection and arrival dates, the injection vector, and the corresponding initial mass.  Each trajectory is
assigned a case number for the sake of discussion, but no priority is implied by the numbering scheme.

The trajectory designated Case 1 is a 6-year direct case to Jupiter with a little more than 1.5 helio-

centric revolutions (see Table 1).  Case 1 has only one optimal coast arc and the rest of the time is spent
thrusting (Figure 2).  Case 2 is also a direct trajectory but with a 7.5-year flight time and just under 2.5 so-
lar revolutions.  This case has two optimal coasting arcs: the first is after about 1.5 revolutions, and the sec-
ond is prior to the Jupiter-rendezvous thrust period.

Table 1 – Summary of trajectory characteristics

Case Software

Traj.
Type

Revs.

Flight

Time

(yrs)

Optimized

Injection

Date

Optimized

Arrival

Date

Injection

(km

)

Injection

Mass

(kg)

Arrival

Mass

(kg)

MALTO

Direct

1.5

10/20/2015 12/20/2021

5.06

37,925

30,451

MALTO

Direct

2.5

7.5

12/21/2014 06/21/2022

0.33

40,932

32,471

MALTO

EEJ

2.5

02/19/2015 02/19/2021

2.73

39,385

32,945

Mystic

Direct

1.5

5.5

12/09/2015 06/10/2021

10.0

36,000

30,886

MALTO

Direct

1.5

5.7

12/26/2015 09/19/2021

10.0

36,000

29,957

The Case 3 trajectory is the final trajectory that uses the injection vehicle performance curve.

This trajectory has a 6-year time of flight, like Case 1, but it includes an Earth gravity assist, making it an
Earth-Earth-Jupiter (EEJ) trajectory. The trajectory is referred to as a 3:2i trajectory

because there are

approximately  three  Earth  revolutions  during  the  nearly  two  spacecraft  revolutions  on  the  Earth-Earth
transfer,  and  the  Earth  flyby  occurs  when  the  spacecraft  is  moving  inbound  toward  the  Sun,  as  shown  in
Figure  2.    The  spacecraft  was  constrained  to  an  Earth  flyby  radius  of  at  least  7000  km.    After  the  Earth
gravity assist, there is roughly half a revolution on the Earth-Jupiter leg.

Figure 2 – Trajectory Cases 1-3. Case 1 is a 6-year direct trajectory, Case 2 is a 7.5-year

direct trajectory, and Case 3 is a 6-year Earth-Earth-Jupiter (EEJ) trajectory.

The last trajectory is Case 4, and it is a representation of the interplanetary portion of the proposed

JIMO Reference Trajectory.

This trajectory does not use the injection vehicle curve given in Figure 1 but

instead has a constrained injection energy and mass. The JIMO architecture assumed for Case 4 has upper
stages capable of injecting 36,000 kg to C

=10 km

. Since the planned launch vehicle would only be

able to loft a 36,000 kg spacecraft into the assembly orbit, injecting to a lower C

could not result in a

higher initial mass. The optimization programs described above were free to use a lower energy, but lower
injection energies did not yield a greater initial mass.

Case 4 is a direct trajectory and is subdivided into Cases 4a and 4b (see Figure 3). Case 4a was

generated  using  Mystic,  the  high-fidelity  low-thrust  optimization  tool,  and  was  constrained  to  a  5.5-year
flight time.  Case 4b was produced with MALTO and was crafted to resemble the Mystic representation in
terms of optimal coast arcs and thrusting periods.

Figure 3 – Mystic (4a) and MALTO (4b) representations of the Reference Trajectory

Case 4b is allowed a longer flight time (5.7 years) and an arrival V

∞

magnitude of 0.5 km/s to ac-

commodate the two major differences between MALTO and Mystic in the Jupiter arrival models and con-
straints. First, MALTO requires an exact match in position with Jupiter, but a non-zero V

∞

magnitude can

be permitted (positive energy). Mystic, on the other hand, allows the spacecraft to be 2,000,000 km from
the center of Jupiter at the trajectory’s end state, but the two-body energy with respect to Jupiter was re-

quired to be less than or equal to 0 km

. Second, Mystic models Jupiter’s gravity while MALTO does

not.

Hence, Case 4b uses different constraints than Case 4a, but both trajectories exhibit very similar

thrust profiles, which was desired to yield comparable sensitivities to injection conditions. While Case 4b’s
final mass is lower than Mystic’s (Table 1), it is shown in a later section that the sensitivity to the injection
vector is quite similar for both.

INJECTION PERIOD

Approach

The objective behind a launch period analysis is to determine the sensitivity of the delivered mass

to sub-optimal launch dates near the optimal value.  This is normally done to account for any issues with
equipment, software, weather, or any other circumstances that might occur just prior to launch preventing
launch from occurring at the optimal time.  The analysis performed here differs from that somewhat since
we  are  examining  the  performance  sensitivity  to  interplanetary  injection  from  the  assembly  orbit,  rather
than launch from the ground.  As a result, the injection period must account for a different suite of prob-
lems  such  as  delays  in  on-orbit  assembly,  delays  in  component  launches,  or  spacecraft  hardware  or  soft-
ware problems.

To determine the injection period sensitivity, we examine each of the trajectories listed in the pre-

vious  section,  in  addition  to  examining  different  mission  scenarios  and  alternate  trajectory  families.    We
begin  with  a  discussion  of  the  injection  periods  for  trajectory  Cases  1  through  4a.    Next,  we  expand  this
analysis to include alternative arrival dates and trajectory types in the hopes of increasing the injection win-
dow.  Since delays to injection  may occur after the spacecraft has been launched into the assembly orbit,
the spacecraft mass and injection C

must remain fixed at the optimized values for the nominal trajectory.

While different trajectory types are explored to extend the injection period, the fixed injection mass and C

may be sub-optimal for these additional trajectory families.

The injection period performance profile for the different trajectory scenarios was generated by

first optimizing the injection and arrival dates of each case for the fixed flight time given in Table 1. In
Cases 1-3, the injection C

and mass were also optimized. Then, the optimal arrival date, injection C

, and

initial spacecraft mass were held fixed while the injection date was varied parametrically. The optimized
arrival date was retained to preserve the tour of the Jovian system, which eliminated the need for a compli-
cated and time-consuming redesign of the moon tour. Finally, the thrust profile was re-optimized at each
value of the injection date to deliver the greatest possible mass to Jupiter.

Results

The injection period for trajectory Cases 1-4a is shown in Figure 4 in percent change of delivered

mass with respect to the optimal value. Normalizing the delivered mass by the optimal value for each case
and plotting the results together provides some insight into the relative sensitivities of the different trajec-
tory types.

Case 1

Trajectory Case 1 is the most sensitive to suboptimal injection dates in terms of delivered mass.

However,  the  performance  penalty  for  injecting  prior  to  the  optimal  date  is  less  than  the  penalty  for  late
injection.  A 100-day injection period requires only a 1.5% loss in performance with respect to the optimal
delivered mass.  The fact that Case 1 is the most sensitive to suboptimal injection dates is partly due to the

short flight time. Only Case 4a is shorter, and it benefits from Jupiter’s gravitational pull on approach to
rendezvous, whereas this case does not. Case 3 has a comparable flight time, but the Earth gravity assist
appears to offer reduced sensitivity to the injection date.

As injection is moved past the optimal injection date, Case 1, as with all of the cases here, reaches

a point where the delivered mass begins to drop off quite quickly.  This “cliff” effect in the performance is
caused by the shortening, and eventually the closing, of optimal coast arcs in the trajectory.  The result is a
fairly inefficient thrust profile that sacrifices a significant amount of propellant to achieve the required arri-
val  date.    The  performance  “cliff”  in  Case  1  occurs  almost  immediately  as  the  injection  date  is  delayed
from the optimal value.  Case 1 has the fewest optimal coast arcs of all the trajectories considered (Figure
2), so it is expected that it would be the most sensitive to sub-optimal injection conditions.

Figure 4 – Interplanetary trajectory performance sensitivity to injection date

Case 2

The performance curve for trajectory Case 2 is also shown in Figure 4. The optimal C

at the

starting point for this trajectory is the lowest of all of the cases (0.33 km

). The low initial C

is a result

of the longer flight time and additional heliocentric revolution for this Case. Both of these characteristics
contribute to a substantially reduced sensitivity to the injection period.

Early injection in this case results in an increase in mass due to increased flight time. Since the C

is low, the trajectory benefits more from the increased flight time than it loses from the change in injection
geometry. As a result, an injection period for this is indefinite. Consider as an example the case where the
injection C

is zero. In this case, “injection” simply refers to waiting for the most opportune time to begin

thrusting for Earth departure. This means that the spacecraft can be “injected” to a C

of zero at any time

prior to the optimal date, and as long as thrusting starts at the optimal point, no performance is lost.

similar effect is occurring in Case 2.

The Earth is modeled as a massless body in these trajectory scenarios.

Note that the “cliff” in performance exists for this case much as it did in Case 1. In this case, the

sudden drop in performance occurs roughly 70 days after the optimal injection point. This is significantly
later than the results for the other trajectories. Again, the low C

results in a lack of sensitivity to subopti-

mal injection geometries, while having additional flight time (1.5 years over trajectory Case 1, for example)
means that optimal coast arcs are longer, and don’t close until the injection date delay is more significant.

Case 3

Also shown in Figure 4 is the injection period performance for the 3:2i Earth gravity-assist case

(trajectory Case 3).   For the  gravity  assist case, injecting  less than 30 days early results in an increase in
delivered  mass  due  to  the  corresponding  increase  in  flight  time.    However,  the  delivered  mass  begins  to
decrease  again  when  injection  occurs  more  than  30  days  early.    This  is  due  to  the  change  in  phase  angle
between Earth and Jupiter.  Once again,  we see that injecting after the optimal point causes a sudden, al-
though less severe, drop off in performance.  However, this does not occur until injection has been delayed
by two months.

When compared to the other 6-year trajectory, Case 1, the Earth gravity-assist case appears to

have less sensitivity to variations in injection date. An injection period of 100 days would yield slightly
more than 0.5% loss in delivered mass as compared to the 1.5% loss in Case 1.

Case 4a

Finally, the performance vs. injection curve for the JIMO baseline case, trajectory Case 4a, is also

shown in Figure 4.  The trajectory exhibits low sensitivity to variations in injection date until over a month
past the optimal injection date.  Injecting early, however, has little performance impact for all of the injec-
tion dates investigated here.  Allowing the mass at Jupiter capture to reduce by only ~0.35% of the optimal
delivered  mass  results  in  a  generous  injection  period  of  over  80  days.    As  previously  stated,  the  reduced
sensitivity of this case versus Case 1 is likely due to the presence of Jupiter’s gravity in the trajectory simu-
lation  and  shortening  the  final  approach  thrust  arc,  which  reduces  the  sensitivity  of  the  trajectory  to  the
changes in flight time and phasing introduced by varying the injection date.

Delays in Arrival Date

Our first attempt to mitigate the performance drop off seen in each of the cases in Figure 4 is to

simply delay the Jupiter arrival date in integer numbers of Callisto periods. By restricting the delay to inte-
ger numbers of Callisto periods, the impact on the capture at Jupiter is minimized since a Callisto flyby on
Jupiter approach has been baselined.

The arrival date was allowed to “slip” 1 and then 5 Callisto periods,

and the parametric variation of injection date was performed as before, though the C

and mass were not

reoptimized. This simulates what affect a post-launch, but pre-injection delay to the arrival date might
have on performance. The Case 1 injection period performance for both the nominal and delayed arrival
dates is shown in Figure 5.

The results of all three scenarios shown in Figure 5 indicate that, as before, significantly less per-

formance is lost by injecting early than is lost by injecting late. At injections 20 days past the optimal in-
jection date all three cases lose roughly 5% deliverable mass with respect to each case’s optimal value.
Delaying the arrival by 5 Callisto periods (83.45 days) delays the “cliff” in performance by only a few days
and does little to extend the injection period.

The delivered masses in the delayed arrival cases are higher than the nominal case due to the in-

creased flight time.  By increasing the flight time, coast arcs can be extended slightly in the interplanetary
trajectory, thereby increasing the efficiency of the thrust and reducing the propellant consumption.  Had the
case  been  completely  re-optimized  (including  injection  C

) the mass delivered to Jupiter for the delayed

arrival case would be even larger. However, the changes to the upper stage propellant loading to achieve
these C

’s would be difficult to achieve after launch, so this scenario was not investigated further.

Figure 5 – Direct 6-year (Case 1) trajectory injection period and effect of delayed arrival

Since the delays to the arrival date were unable to extend the injection period past the optimal in-

jection date significantly, we began investigating alternative ways to take advantage of a delay in arrival
date.

Alternate Trajectory Types

Figure 6 shows how considering alternative trajectory families can offer extended injection oppor-

tunities. Case A in the figure indicates the performance capability of trajectory Case 1 with its nominal
arrival date, also shown in Figure 5. Case descriptions for the other scenarios in Figure 6 are listed in Table
2.

Case B is the same trajectory type as Case A but with the arrival date delayed 5 Callisto periods.

Case C in Figure 6 illustrates the performance obtained for a trajectory that arrives roughly 7 months after
Case A and performs 2.5 revolutions about the sun, one revolution more than Case A. For this case, the
injection C

and mass are still constrained to the optimal value for Case A. This is done to simulate the

need to switch families after the spacecraft has been launched to the assembly orbit but before it has been
injected onto its interplanetary path. This constraint on C

and injected mass is the reason that Case C un-

derperforms Cases A and B for injection dates less than +25 days from the nominal.

Finally, we verified that it is indeed possible to delay the injection one Earth-Jupiter synodic pe-

riod by delaying the arrival date ~13 months in integer numbers of Callisto periods (Case D). Note the
sinusoidal behavior of the delivered mass as the injection oscillates between favorable and unfavorable
geometries.

Table 2 – Descriptions of the four trajectory scenarios illustrated in Figure 6

Description

Case A

Base Case

: Optimized for a 6-year flight time with free injection

parameters and Jupiter arrival date. The injection C

(and

corresponding initial mass) and Jupiter arrival date were fixed while
the injection date was varied.

Case B

Base Case with Modified Arrival Date:

Injection parameters fixed at

the values for Case A, but the arrival date is delayed 5 Callisto Periods.

Case C

Suboptimal 2.5 Rev Case:

Parameters are fixed at the values for Case

A, but the arrival date is delayed 12 Callisto Periods (~7 mo.) and the
trajectory family includes an additional half-revolution about the Sun.

Case D

Base Case One Synodic Period Later

: Parameters are fixed at the

values for Case A, but the trajectory (including arrival date) is shifted
one synodic period later.

Figure 6 – Direct 6-year (Case 1) trajectory injection period results; multiple families considered

As with trajectory Case 1, Case 2 shows little performance improvement from a delayed arrival

date within a particular trajectory family.  This is illustrated by Case B in Figure 7.  Each case in Figure 7 is
described in detail in Table 3.  Case B is the result of moving the arrival date for Case A approximately 6
months into the future, thereby increasing the flight time.  Due to the geometry of the trajectory, however,
there is very little benefit in this delay.  However, when an alternate trajectory family is considered (in this
case a 2-rev trajectory indicated by Case C in Figure 7) with an optimal injection date that occurs one year
later, it is apparent that one could inject far in advance of the optimal injection date for that trajectory and
be able to delay injection further than the previous family would allow with the same delay in arrival date.
In the figure, this second trajectory family was constrained to have the same injection C

and corresponding

initial mass as Case A to simulate an inability to change the spacecraft configuration after launching into
the assembly orbit. Note again the sinusoidal behavior in the delivered mass as the injection geometry os-
cillates between favorable and unfavorable geometries.

Table 3 – Descriptions of the three trajectory scenarios illustrated in Figure 7

Description

Case A

Base Case

: 2.5 revolution trajectory optimized for a 7.5-year flight time

with free injection parameters and Jupiter arrival date. Then the injection
C

(and corresponding initial mass) and Jupiter arrival date were fixed

while the injection date was varied.

Case B

Base Case with Modified Arrival Date:

Parameters are fixed at the

values for Case A, but the arrival date is delayed 12 Callisto Periods (~6
months).

Case C

Suboptimal 2 Rev Case:

Parameters are fixed at the values for Case A,

but the arrival date is delayed 11 Callisto Periods.

Figure 7 – Injection period for trajectory Case 2 and an alternate solution

The drop off in performance for Case C in Figure 7 occurs roughly one year after Cases A and B

even though the arrival date was only delayed by half a year relative to the nominal (Case A).  This is due
to the change in trajectory family.  Case C is a trajectory that performs roughly 2 heliocentric revolutions
rather than 2.5 revolutions as in Cases  A and B.  This difference allows Case  C to be a generally shorter
trajectory at the cost of some delivered mass.  The rest of the difference in delivered mass can be accounted
for by the  suboptimal C

and initial mass used for Case C to maintain consistency with the nominal case

(Case A). An advanced mission study could be performed to determine an intermediate injection mass and
C

that would provide nearly equivalent performance for both the 2-rev and 2.5-rev solutions.

In both the 6-year and 7.5-year trajectory examples (Figure 6 and Figure 7), allowing for different

families of trajectory and arrival times offers the ability to extend the injection period indefinitely with mi-
nor impacts to delivered mass. By examining the alternative solutions that exist, one can trade changes in
arrival date with injection period size and performance.

INJECTION VECTOR SENSITIVITY

Approach

The purpose of the injection-vector sensitivity study is two-fold. First of all, the study aims to

characterize the trajectories’ sensitivities to errors in the injection execution to allow sufficient margins in
the mission design. Second, the injection direction perturbations feed directly into the injection opportuni-
ties analysis by defining the range of acceptable departure directions about the optimal. This is explained
in greater detail in the next section.

In each of the injection vector sensitivity experiments, the departure and arrival dates and the in-

jection mass were held fixed at the optimized values for each trajectory case.  The arrival date is maintained
to preserve the Jovian tour (as described in the Injection Period section).  The injection date and mass are
held fixed because it is not reasonable to be able to either predict the injection errors, and therefore opti-
mize the injection date and mass around the errors, or to plan the assembly orbit’s orientation in anticipa-
tion of exactly when during the injection period the spacecraft will inject.  The latter would negate the need
for an injection period.  Hence, these assumptions are consistent with the study objectives stated above.

The first injection-vector sensitivity experiment explores the effects of introducing deviations in

the injection C

magnitude. In order to isolate the impact of magnitude versus direction variations, the in-

jection direction is held constant while the magnitude is incrementally increased or decreased from the op-
timal value. The thrust profile is re-optimized at each step to yield the greatest mass at Jupiter arrival. It is
reasonable to assume that the thrust profile could be re-designed in such an event because each trajectory
requires coasting for the first 60 days after injection.

The second experiment in this study is the reverse of the first: the injection C

is held constant

while the direction is modified.  Both the right ascension of the injection asymptote (RLA) and the declina-
tion of the injection asymptote (DLA) are parametrically varied.  (Although we distinguish between launch
and injection, we use the conventional terms of RLA and DLA to describe the Earth departure asymptote.)
This experiment serves to characterize the sensitivity to small errors in the injection execution as well as to
define the acceptable RLA and DLA ranges used in calculating the injection opportunities.

Injection C

Variation Results

The mass-performance impacts of varying the injection C

by ±10% for trajectory Cases 1 and 3

and ±20% for Cases 2 and 4 are shown in Figure 8. Each case has its own reference value for the C

(see

Table 1), however, so a 20% change in Case 4 (2 km

), for example, is very different from the same per-

cent change in Case 2 (0.07 km

). The horizontal axis shows the percent change from the optimal C

and the vertical axis gives the percent change in the final mass (mass at Jupiter arrival) relative to the opti-
mal mass.

In Case 1, the performance impact of an injection under-burn is more significant than with any

other trajectory. Case 1 suffers from having only one optimal coast period (Figure 2), so there are limited
opportunities to apply additional thrusting to compensate for an underperforming injection burn. The worst
instance shown, however, is less than a 1.7% mass decrease for a rather substantial under-burn of 10% of
the C

. An over-burn, on the other hand, means less ∆V must be applied by the electric propulsion system

to achieve the Jupiter arrival date. An increase in the C

therefore yields a higher final mass, as shown in

Figure 8.

-1.75%

-1.50%

-1.25%

-1.00%

-0.75%

-0.50%

-0.25%

0.00%

0.25%

0.50%

0.75%

-20%

-15%

-10%

-5%

10%

15%

20%

Variation from Optimal Injection C

Magnitude

)

Case 1) Direct, 6-yr

Case 2) Direct, 7.5-yr

Case 3) EEJ, 6-yr

Case 4a) Ref. Traj. Mystic

Case 4b) Ref. Traj. MALTO

Injection C

direction

held constant

Case Nominal C

Direct, 6-yr 5.06 km

Direct, 7.5-yr 0.33 km

EEJ 3:2 2.73 km

Ref. Traj. 10.0 km

Case Nominal C

1) Direct, 6-yr 5.06 km

2) Direct, 7.5-yr 0.33 km

3) EEJ, 6-yr 2.73 km

4) Ref. Traj. 10.0 km

Figure 8 – Mass Impact of Varying the Injection C

Magnitude

Case 2, on the other hand, is relatively insensitive to proportionally sizeable changes in the injec-

tion C

. This is due in a large part to the optimal injection C

of only 0.33 km

, which is an order of

magnitude lower than for the other cases. A 20% change in the C

is a delta of only 0.07 km

, so a 20%

under-burn would not burden the NEP system with much additional ∆V. Although not shown in Figure 8,
C

variations of up to ±50% were computed, but the relative mass changes only ranged from -0.32% to

+0.23%. This trajectory also benefits from two optimal coasting periods during its 7.5-year flight time
(Figure 2), so there are ample opportunities to make up for any energy lost from the reduced injection C

Because it is the only trajectory with a gravity assist, Case 3 has a unique set of constraints com-

pared to the other cases. Not only does the spacecraft have to rendezvous with Jupiter on a fixed date, it
must also target an Earth flyby to get to Jupiter. Despite this added complexity, Case 3 handles ±10% C

magnitude variations better than Case 1. Even if one considers the actual value of injection energy change,
a loss of 0.25 km

results in a shortfall of 0.47% mass in Case 1 but only 0.39% in Case 3. Figure 2 also

shows that Case 3 has several optimal coast arcs where additional thrusting can be applied, which tends to
reduce the sensitivity to injection magnitude errors.

The plots of both Cases 4a and 4b in Figure 8 are especially interesting because of how the results

from these cases compare to one another. From +20% down to nearly -15% of the optimal C

, the results

from Mystic and MALTO are nearly indistinguishable. In fact, if the C

is increased by as much as 50%

(not shown in Figure 8), the Case 4b results are still in extremely close agreement with the Case 4a values.
This  discovery  is  significant  because  it  validates  the  use  of  MALTO,  a  lower-fidelity  preliminary-design
tool,  for  analyzing  the  effects  of  injection  perturbations.    At  C

magnitude deviations beyond -15%,

MALTO provides a more conservative estimate than Mystic, but there is only a 0.3% mass difference be-
tween them at a 20% decrease in the injection C

Overall, the NEP system could produce trajectories that are fairly insensitive to errors in the C

magnitude. In all cases studied, an over-burn means that less low-thrust ∆V is required, so less propellant
is expended on the interplanetary trajectory. As for instances of an under-burn, even the most sensitive

case, Case 1, can tolerate a 5% reduction in the C

and still only have the mass at Jupiter diminished by

0.5%. The same mass reduction limit of 0.5% can be maintained on Case 4, the Reference Trajectory, with
an injection underperformance of 15%!

Injection Asymptote Variation Results

Having shown that using nuclear-electric propulsion can result in a low sensitivity to injection

magnitude errors, we now turn to the question of how a NEP trajectory is impacted by deviations in the
injection asymptote. The optimal injection directions, RLA and DLA,

are given in Table 4. The differ-

ences in the optimal RLA and DLA for Cases 4a and 4b are attributed to their different injection dates. The
geometry of the injection is tied to the Earth’s location at the injection epoch, and Case 4b injects about 15
days later than Case 4a (see Table 1). Since the Earth moves roughly 1º/day in its orbit, the 15º RLA dif-
ference between Cases 4a and 4b is expected.

Table 4 – Optimal Injection Direction for Each Trajectory Case

Case

Optimal

RLA (deg)

Optimal

DLA (deg)

122.8

20.5

182.8

1.6

234.7

-19.4

170.2

5.7

185.6

-0.9

-5.0

-2.5

0.0

2.5

5.0

-5.0

-2.5

0.0

2.5

5.0

-0.009%

-0.008%

-0.007%

-0.006%

-0.005%

-0.004%

-0.003%

-0.002%

-0.001%

0.000%

)

Delta RLA (deg.)

Delta DLA

(deg.)

Figure 9 – Mass Impact of Varying the in Injection Direction on Trajectory Case 2

Figure 9 shows the mass impact of varying the RLA and DLA by ±5º on Case 2. The two hori-

zontal axes give the amount of change from the optimal injection direction. The vertical axis represents the
percent change in mass at Jupiter arrival relative to the mass when injecting in the optimal direction. Note
that injecting 5º from the optimized values of both RLA and DLA results in a tiny mass change of less than

Earth Equator and Equinox of J2000 inertial reference frame (EME2000)

3 kg, or 0.009%! As with the C

magnitude variations, Case 2 is quite impervious to changes in the injec-

tion V

∞

direction. This insensitivity is due to the very low C

(only 0.33 km

), which means this trajec-

tory does not depend heavily on the precise injection conditions. In addition, the relatively long flight time
of 7.5 years means there is sufficient time to use the NEP engines to impart the required ∆V and therefore
plenty of time to correct any changes to the planned injection conditions.

In Cases 1 and 3, we broadened the RLA range to ±20º to explore the possibility of intentionally

injecting in an off-nominal direction. It is possible that the node of the assembly orbit will not align with
the optimal injection asymptote at the instant injection is desired. Rather than relying on the injection vehi-
cle to perform a costly plane change to reach the optimal departure asymptote, we investigate the perform-
ance impact of using the NEP system to compensate for the sub-optimal injection direction.

Figure 10 shows the results of injection direction variations of ±20º and ±5º in RLA and DLA, re-

spectively,  for  Case  1.    Case  1  exhibits  a  much  greater  sensitivity  to  injection  asymptote  deviations  than
Case 2.  Even if the RLA range is constrained to ±5º, the mass loss is as high as 0.13% (versus 0.009% for
Case 2).  Also note the “cliff” in performance when the RLA deviation exceeds +15º.  While injecting with
an  RLA  value  less  than  the  optimal  (negative  delta-RLA)  causes  the  NEP  system  to  compensate  for  the
sub-optimal  departure  asymptote,  injecting  at  higher  RLA  values  leads  to  issues  with  orbital  energy,  as
well.  The RLA is measured in the counterclockwise direction, so increasing the departure RLA means that
the injection asymptote is directed closer to the Sun.  If the RLA is increased far enough, the injection burn
will leave the spacecraft with notably less orbital energy with respect to the Sun than the optimal injection
would provide.  At that point, the NEP system must compensate for this energy deficiency, and the cost is
nearly 3 km/s in low-thrust ∆V.

-20

-15

-10

-5

-5.0%

-4.5%

-4.0%

-3.5%

-3.0%

-2.5%

-2.0%

-1.5%

-1.0%

-0.5%

0.0%

)

Delta RLA (deg.)

Delta DLA

(deg.)

Figure 10 – Mass Impact of Varying the in Injection Direction on Trajectory Case 1

Despite these obstacles, Case 1 can abide a wide RLA range without unreasonable mass losses. A

range of at least -20º to +13º in delta RLA (RLA deviations beyond -20º were not evaluated) and ±5º in
delta DLA results in a mere 1% mass decrease. If only a 0.5% mass loss can be tolerated, the full extent of
the DLA values and RLA values of -16º to +10º are available.

Trajectory Case 3, however, does not have a steep drop-off in performance when the same DLA

and RLA ranges are computed (see Figure 11). It was expected that this trajectory would be more sensitive

to variations in the injection conditions than Case 1, which has the same total flight time, because it must
target the intermediate flyby of Earth in addition to arriving at Jupiter on a fixed date. This is obviously not
the case, however, and the full RLA and DLA ranges of ±20º and ±5º, respectively, result in mass losses of
less than 0.5%. (Note that the “indentations” in the Figure 11 surface plot are due to the convergence toler-
ances used in performing this analysis.)

-20

-15

-10

-5

-0.50%

-0.45%

-0.40%

-0.35%

-0.30%

-0.25%

-0.20%

-0.15%

-0.10%

-0.05%

0.00%

)

Delta RLA (deg.)

Delta DLA

(deg.)

Figure 11 – Mass Impact of Varying the in Injection Direction on Trajectory Case 3

One reason Case 3 is less affected by injection asymptote changes is the greater number of optimal

coast arcs in this trajectory compared with Case 1.  Case 1 only has one optimal coast arc during which it
can compensate for sub-optimal injection conditions.  This coast arc also occurs a long time after injection
(see Figure 2).  If there were an optimal coast arc in Case 1 closer to injection, less ∆V would be required
to maintain the Jupiter arrival date.  Case 3, though, has several optimal coast arcs, many of which are early
in the trajectory, as shown in Figure 2.  Also, Case 3 has a lower optimal C

than Case 1 (see Table 1), so it

can be thought of as a smaller vector going in the “wrong” direction, which suggests it should be easier to
correct post-injection. This trajectory has more flexibility to cope with sub-optimal injections and therefore
does not exhibit the “cliff” in performance that Case 1 has, shown in Figure 10.

The response to injection asymptote perturbations of Case 4b, the MALTO representation of the

Reference Trajectory, is given in Figure 12.  This figure shows a greater RLA range of ±25º, versus ±20º
for Cases 1 and 3.  Like Case 1 (Figure 10), there is a steep drop-off in performance at large increases in
the RLA.  However, the “cliff” for Case 4b does not manifest itself until the RLA is increased by approxi-
mately 22º,  which is in a range  not covered by the analyses of Cases 1 through 3.  In the  RLA  variation
range  of  ±20º,  the  largest  mass  loss  for  Case  4b  is  0.5%,  which  puts  this  trajectory  on  par  with  Case  3
(shown in Figure 11).

A comparison of the results of varying the injection direction for Cases 4a and 4b is given in

Figure 13 with both cases plotted against the same vertical scale.  (Case 4a was computed at a finer mesh,
so the grid lines in Figure 13 are more closely-spaced in the plot on the left.)  The correspondence between
the two plots is quite remarkable, especially the negative delta RLA values.  It is only at the largest RLA
increases that the Mystic and MALTO representations begin to differ significantly.  At an RLA increase of
20º, the greatest mass loss in both cases occurs when the DLA is increased by 5º, and the disparity between
the Case 4a and 4b mass changes at that point is only 0.05%.

-25

-20

-15

-10

-5

-1.50%

-1.25%

-1.00%

-0.75%

-0.50%

-0.25%

0.00%

)

Delta RLA (deg.)

Delta DLA

(deg.)

Figure 12 – Mass Impact of Varying the Injection Direction on Trajectory Case 4b

-25

-20

-15

-10

-5

-0.7%

-0.6%

-0.5%

-0.4%

-0.3%

-0.2%

-0.1%

0.0%

)

Delta RLA (deg.)

Delta DLA

(deg.)

Case 4a

-25

-20

-15

-10

-5

-0.7%

-0.6%

-0.5%

-0.4%

-0.3%

-0.2%

-0.1%

0.0%

)

Delta RLA (deg.)

Delta DLA

(deg.)

Case 4b

Figure 13 – Mass Impact of Varying the Injection Direction on Trajectory Cases 4a and 4b

When the differences in how Mystic and MALTO model trajectories are considered (i.e. level of

fidelity, force models, etc.), the good agreement between their results, evident in Figure 13 as well as
Figure 8, is an additional validation of the use of MALTO for performing injection V

∞

variation analyses.

Furthermore, Figure 13 illustrates the low sensitivity of Case 4a to variations in the injection asymptote.
For an extensive RLA range of ±25º and a DLA range of ±5º, the mass hit is less than 0.7%. Using the
NEP system for this mission can enable a wide range of injection directions without substantial mass per-
formance impacts.

INJECTION OPPORTUNITIES

Approach

For a launch from the surface of the Earth to an interplanetary departure trajectory, the primary

problem is to match the required V

∞

vector to the specified site location.

In the mission architecture ana-

lyzed in this paper, however, the V

∞

vector must lie within the plane of a regressing assembly orbit for an

injection to be possible. Furthermore, the spacecraft must be phased within the assembly orbit such that the
final burn of the final stage takes place at the perigee of the interplanetary hyperbolic trajectory.

Unlike the case of launching from the surface of the Earth, for which in general two injection solu-

tions exist per day, an assembly orbit that is properly located for injection on one day may be incorrectly
phased on the following day.  The RAAN of an assembly orbit at 400 km altitude and at an inclination of
28.5º regresses at a rate of 7.1º/day.  For typical values and rates of change of DLA and RLA of interplane-
tary trajectories, such an assembly orbit will quickly regress away from the desired targets and another op-
portunity  will  not  be  available  until  the  orbit  has  regressed  180º.    As  a  consequence,  there  may  be  many
days within the injection period in which injection from the assembly orbit is not possible.

To increase the number of injection opportunities during the injection period, a targeting tolerance

is placed on the nominal RLA and DLA. By allowing the spacecraft to inject into a range of departure as-
ymptotes that are close to the optimal, the number of days when injection is possible is greatly increased.
The number of injection opportunities per day and the duration of the injection window are also increased.

Results

A representative assembly orbit, with an altitude of 400 km, an inclination of 28.5º, and a RAAN

chosen to place the first injection opportunity near the beginning of the injection period, was propagated for
the duration of the 84-day injection period given in Figure 14.  Each orbit rev during that time was checked
to determine if the optimal departure asymptote or any of the nearby asymptotes lay within the orbit plane.
If the orbit  was aligned correctly and the spacecraft properly phased to allow  for an injection burn at the
correct position and time, an injection opportunity was defined to exist.  The injection period in Figure 14
is  taken  from  the  injection  period  analysis  of  trajectory  Case  4a  (also  shown  in  Figure  4).    The  optimal
RLA and DLA targets for that case are given in Figure 15.

Figure 14 – Reference injection period (Case 4a)

Targeting tolerances of ±25º in RLA and ±5º in DLA, which increase the number of possible in-

jection days to 40, can be sustained without a substantial impact on the delivered mass, as shown in Figure
13, but this assumes a perfect injection by the final stage of the injection vehicle. Injection dispersions
must be considered when determining tolerances on the RLA and DLA, but even modest tolerances greatly
increase the number of days and opportunities for injection and thus the probability of injection.

CONCLUSIONS

Nuclear-electric propulsion trajectories to Jupiter can allow for a substantial injection period,

broad RLA ranges, and multiple Earth-departure opportunities over the injection period without great sacri-
fices in mass performance.  Using the NEP system and allowing for different trajectory families and arrival
times can offer the ability to extend the injection period indefinitely with minor impacts to delivered mass.
By examining alternative solutions, one can trade changes in arrival date with injection period length and
performance.   In  addition,  employing  a  NEP  system  could  yield  trajectories  that  are  rather  insensitive  to
errors in the injection V

∞

vector as well as planned variations in the injection direction.

Trajectories that had lower optimal injection C

values, longer flight times, and/or longer and more

numerous coast arcs were less sensitive to injection vector variations and injection date. Lower C

solu-

tions reduced the sensitivity of the injection period and the injection vector due to the diminished impact of
unfavorable geometry. The longer flight times and increased coasting durations allowed higher C

cases to

compensate for sub-optimal injection conditions. The large RLA range provided by the NEP system, cou-
pled with long injection periods, increases the probability of achieving Earth departure and fulfilling the
mission requirements.

ACKNOWLEDGEMENTS

The authors would like to thank Jon Sims and Louis D’Amario for their guidance and leadership

in directing the work presented in this paper.  We are also grateful Paul Finlayson, Edward Rinderle, Greg-
ory  Whiffen,  and  Benjamin  Engebreth  for  their  work  developing  and  enhancing  the  low-thrust  trajectory
optimization software used in these analyses.  This work was performed at the Jet Propulsion Laboratory,
California Institute of Technology, under a contract with the National Aeronautics and Space Administra-
tion.

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