Expected Precision of Europa Clipper Gravity Measurements
Ashok K. Verma Jean-Luc Margot
astro-ph.EP physics.data-an physics.geo-ph physics.space-ph































epss]Ashok K. Verma
epss,aa]Jean-Luc Margot
[epss]Department of Earth, Planetary, and Space Sciences, University of California, Los Angeles, CA 90095, USA
[aa]Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA



The primary gravity science objective of NASA's Clipper mission to
Europa is to confirm the presence or absence of a global subsurface
ocean beneath Europa's Icy crust.  Gravity field measurements obtained
with a radio science investigation can reveal much about Europa's
interior structure. Here, we conduct extensive simulations of the
radio science measurements with the anticipated spacecraft trajectory
and attitude (17F12v2) and assets on the spacecraft and the ground,
including antenna orientations and beam patterns, transmitter
characteristics, and receiver noise figures.  In addition to two-way
Doppler measurements, we also include radar altimeter crossover range
measurements.  We concentrate on ±2 hour intervals centered on the
closest approach of each of the 46 flybys.  Our covariance analyses
reveal the precision with which the tidal Love number k_2,
second-degree gravity coefficients C̅_20 and C̅_22,
and higher-order gravity coefficients can be determined.  The results
depend on the Deep Space Network (DSN) assets that are deployed to
track the spacecraft. We find that some DSN allocations are sufficient
to conclusively confirm the presence or absence of a global
ocean. Given adequate crossover range performance, it is also possible
to evaluate whether the ice shell is hydrostatic.




  
  Europa;
  Geophysics;
Tides, solid body;
Satellites, shapes;
Orbit determination  










§ INTRODUCTION

 
The spacecraft mission design process at NASA relies on design
requirements that flow from measurement requirements, which themselves
flow from science objectives.  The Europa Clipper mission has a set of
compelling science objectives <cit.> that
emerged out of strategic planning
documents <cit.> and other studies.  Here we
investigate some of the measurement requirements that may be needed to
enable a gravity science investigation.  Gravity science experiments
provide powerful data for investigating the physical state of
planetary bodies.  Examples include mapping the gravity field,
estimating the rotational state, and probing the internal structure of
Mercury <cit.>, 
Venus <cit.>, 
Mars <cit.>, 
and Titan <cit.>.



In 2015, NASA appointed a Gravity Science Working Group (GSWG) to help
refine science objectives for the Europa Clipper mission (then known
as the Europa Multiple Flyby Mission).  NASA's charge to the GSWG
included the following statement: “The GSWG will define and recommend
to the science team science investigations related to understanding
the response of the satellite to gravity, specifically, but not
limited to, understanding the tidal distortion of Europa, its internal
structure, precession, and moments of inertia.”  The GSWG produced a
report <cit.> that specifies the precision with which certain
quantities must be measured in order to meet specific science
objectives (Table <ref>).  The GSWG focused primarily on
measurements that pertain to the ice shell and the presence of an
ocean.



One of the primary objectives of a mission to Europa is to confirm the
presence of a global ocean.  A gravity science investigation can
address this objective in a number of ways <cit.>.  Here, we
focus on  measurements of the tidal Love number k_2. 
An alternate approach consists of measuring the tidal Love number
h_2, as examined by <cit.>.  Calculations by
<cit.> indicate that k_2 is expected to range from 0.14 to
0.26, depending on the thickness and strength of the ice shell, if a
global ocean is present underneath the ice shell.  In contrast, k_2
is expected to be less than 0.015 if there is no global ocean.
Therefore, a measurement of k_2 is sufficient to test the global
ocean hypothesis <cit.>, provided that the
uncertainties do not exceed the 0.06 level recommended by the GSWG.

Another important objective of a gravity science investigation is to
confirm whether the ice shell is in hydrostatic equilibrium.
Galileo-based estimates of second-degree gravity coefficients rely on
the assumption of hydrostatic equilibrium <cit.>, but it
is unclear whether hydrostatic equilibrium applies.  It is possible to
test the hydrostatic equilibrium hypothesis by measuring the
second-degree gravitational harmonic coefficients, C̅_20 and
C̅_22, to the level prescribed by the GSWG
(Table <ref>).  Trajectories being designed for the Clipper
mission offer promising prospects for measuring these quantities.

In section <ref>, we provide an overview of the anticipated
Clipper trajectory.  In section <ref>, we review measurements,
uncertainties, and model assumptions.  Our dynamical model, solution
strategy, and estimated parameters are discussed in section
<ref>. In Section <ref>, we discuss our covariance analysis
results.  Our conclusions are provided in section <ref>.





§ SPACECRAFT TRAJECTORY AND ATTITUDE



Europa Clipper will orbit Jupiter and execute repeated close flybys of
Europa, Ganymede, and Callisto with science observations at Europa and
gravitational assists at Ganymede and Callisto <cit.>.  To
achieve the science goals of the mission, Clipper trajectories are
designed to obtain globally distributed regional coverage of Europa
with multiple low-altitude flybys <cit.>.  The current
trajectory, named 17F12v2, includes 46 flybys with altitudes as low as
25 km (Figure <ref>) and 126 crossovers below 1000 km
altitude.  Crossovers are locations where two ground tracks intersect
and where altimetric measurements are particularly valuable.

We examined the suitability of trajectories 15F10, 16F11, and 17F12v2
for gravity science investigations, with a particular emphasis on
17F12v2.  All of these trajectories were designed to obtain
  globally distributed regional coverage of Europa with 42, 43, and 46
  flybys, respectively (Table <ref>).



An important consideration for a gravity science investigation is the
distribution of sub-spacecraft latitudes when the spacecraft is at
closest approach.  Trajectory 17F12v2 provides an adequate
distribution for gravity science purposes (Table
<ref>). Details about the spacecraft's anticipated
trajectory and orientation in space (attitude) are available at
  ftp://naif.jpl.nasa.gov/pub/naif/EUROPACLIPPER in the form of SPICE
kernels <cit.>.









§ MEASUREMENTS



The gravity science investigation will utilize a radio link between
Earth-based stations and the spacecraft's radio frequency
telecommunications subsystem to provide range and Doppler measurements 
(see Section <ref>) and solve for Clipper's trajectory.  These data will yield
measurements of Europa's gravity field and tidal response.  The
investigation will also rely on spacecraft-to-Europa ranging data from the 
Radar for Europa Assessment and Sounding: Ocean to Near-surface (REASON)
instrument <cit.>. 
Analysis of REASON data may be enhanced with digital elevation models
obtained by stereo imaging from the Europa Imaging System
(EIS) <cit.>.  The primary observables will be two-way,
coherent Doppler measurements (section <ref>) and radar range
measurements (section <ref>).  Our assumptions about the radio
link and the Doppler and crossover measurements are summarized in
Table <ref> and explained in detail below.




 §.§ Radio link

In order to meet mission requirements, it is anticipated that Clipper
will carry at least three fan-beam (medium-gain) antennas and two
wide-beam (low-gain) antennas.  Nominal antenna gain patterns for
these antennas were provided by Peter Ilott (pers. comm.) and Avinash
Sharma (pers. comm.).  We used these gain patterns in conjunction with 
the spacecraft attitude to compute the signal to noise ratio of the radio link.
Clipper will carry a 20 W transmitter operating in the X band.  Transmitter parameters 
were provided byDipak Srinivasan (pers. comm.). Ground stations are expected to be
primarily 34 m or 70 m antennas of the Deep Space Network (DSN)
equipped with low-noise receivers.  Nominal DSN system temperatures were 
provided by Ryan Park (pers. comm.).
Typical spacecraft radio links are established in a closed-loop mode
with a signal to noise ratio of 7 dB-Hz or above.  
A typical spacecraft radio link operates 
with a signal to noise ratio of 7 dB-Hz or above.
However, it may be possible to use an open-loop receiver and obtain radio science data with a signal to noise ratio of 4 dB-Hz or less <cit.>.However, it may be possible to  establish a radio link with 4 dB-Hz.
The DSN has the capability to array two or three
34 m antennas to improve the radio link budget.



 §.§ Doppler measurements



Gravity science investigations rely primarily on two-way Doppler shift
measurements between spacecraft and Earth-based antennas.  These
measurements yield the velocity of the spacecraft along the observer
line-of-sight (LOS). Because the Doppler shift measures a component
of the spacecraft velocity that is affected by the gravitational
field, the radio tracking of spacecraft can produce detailed
information about the distribution of mass in planetary bodies. To
first order,

    f_D ≃ 2f_TV_r/c,

where f_D is the Doppler shift, f_T is the transmitted frequency,
V_r is the LOS component of the relative velocity between spacecraft
and observer, and c is the speed of light.


In this study, we assumed that the spacecraft telecom subsystem
receives an X-band signal with a carrier frequency of ∼7.2 GHz
from a DSN ground station and coherently transmits this signal back to
the DSN with a carrier frequency of ∼8.4 GHz.  The uncertainties
of the Doppler measurements depend on a number of factors that include
fluctuations in the ionospheric and solar wind plasmas, variations in
the water content in the troposphere, as well as instrumental noise
<cit.>.  However, at small Sun-Earth-Probe (SEP) angles, the
interplanetary plasma noise dominates.
We modeled the Doppler uncertainties in a 60 s integration time as:

    σ_D = √(σ_plasma^2 + σ_other^2) + 0.01  mm/s,

where σ_plasma represents the noise due to interplanetary
plasma according to the model of <cit.> and σ_other is
a Europa Project estimate of the noise contribution due to other sources, 
including thermal noise (0.053 mm/s), spacecraft jitter (0.020 mm/s), and ionosphere (0.015 mm/s).  
The last term (0.01 mm/s) represents a margin added to the noise model (Figure <ref>).  
See <cit.> for a detailed review of noise sources in radio science experiments. 
where σ_plasma represents the noise due to interplanetary
plasma according to the model of Iess et al. (2012), σ_other is
a Europa Project estimate (0.0596 mm/s) of the noise contribution due
to other sources, including media, DSN clocks, and spacecraft and DSN
instruments, and 0.01 mm/s is a small margin added to the noise 
model (Figure <ref>).





 §.§ Crossover measurements
 


Improvements to the quality of a spacecraft's orbit determination can
be obtained when ranging measurements to a body of known shape and
surface of known topography are available.  Even when the shape varies
with time and when the topography is unknown, as is the case for
Europa, it is still possible to benefit from altimetry at so-called
crossover points.  Crossovers points are the geographic locations on
the surface where two ground tracks intersect (Figure <ref>).
Ranges obtained at the same location on two separate tracks can be
subtracted to yield a crossover height difference:

    Δ h(t_1, t_2) = h(t_1)-h(t_2),

where h(t) is the altimeter height measurement at time t.  These
crossover measurements eliminate the uncertainty introduced by the
unknown local topography and can provide powerful constraints in the
orbit determination process.  The analysis of multiple crossover
measurements obtained throughout Europa's orbital cycle will also be
important in determining the radial amplitude of the tidal signal,
i.e., in estimating the tidal Love number h_2
<cit.>.

To simulate crossover measurements, we assumed a nadir-pointed
altimeter and incorporated spacecraft attitude data in our
calculations.  It is anticipated that the REASON instrument will range
to the surface when the spacecraft is below altitudes of 1000 km with
respect to the surface of Europa.  We identified 126 crossovers with
spacecraft altitudes < 1000 km in trajectory 17F12v2.  However, 14
of these crossovers were discarded because they occur when the terrain
under the spacecraft is not illuminated by the Sun during one or both
of the encounters.  Without proper illumination, it may not be
possible to produce a Digital Terrain Model (DTM) from stereo analysis
of EIS images.
And without a DTM, the enhanced cross-over analysis approach that
combines REASON and EIS data, as described by <cit.>,
would not be possible.


We assigned uncertainties to the crossover measurements given by

    σ_t_1,t_2 = √((σ_t_1^2 + σ_t_2^2)),

where σ_t is the altimeter height measurement
uncertainty at time t.  These uncertainties were provided by the
REASON team (Gregor Steinbruegge, pers. comm., Sept. 30, 2017) with
heritage from the procedure described by <cit.>.
The median and standard deviation of
crossover uncertainties are 4.5 m and 6.5 m, respectively, with
minimum and maximum values of 2.7 and 24.9 m, respectively.

 





§ METHODOLOGY


Our approach consisted in simulating the precision with which the
Love number k_2 and degree-two gravitational harmonics can be
determined with realistic mission scenarios and assumptions about
measurement uncertainties (section <ref>).  


We used version 124 of the Mission Operations and Navigation Toolkit
Environment (MONTE) <cit.>, an astrodynamics computing
platform that is developed at NASA's Jet Propulsion Laboratory (JPL).
In addition to its uses in trajectory design and spacecraft
navigation, MONTE has been used for a variety of scientific purposes,
including gravity analysis <cit.>, orbit determination
<cit.>, and sensitivity analysis for tests of general
relativity <cit.>.


MONTE's observation model uses <cit.>'s formulation to compute
Doppler measurements (section <ref>) and a ray-intersect method
to compute altitude measurements (section <ref>).  MONTE can
specify arbitrary gravity fields (section <ref>) and spin
states (section <ref>).  We used MONTE's observation model to
compute simulated observables and their partial derivatives with
respect to solve-for parameters (section <ref>).
Finally, we used MONTE's tools to perform covariance analyses and quantify the
precision with which geophysical parameters can be determined (section <ref>).
In the sections below, we provide details about
the gravity and spin state models, solve-for parameters, and
covariance analyses.




 §.§ Representation of Europa's gravity field


MONTE represents gravity fields as spherical harmonic expansions <cit.>:

    U =     GM/r + GM/r∑_l=2^∞∑_m=0^l(R/r)^l P̅_l,m (sinϕ)  
       (C̅_l,mcos(mλ) + S̅_l,msin(mλ) ),


where G is the gravitational constant, M is the mass of
Europa, P̅_l,m are the normalized associated Legendre
polynomials of degree l and order m, R is the reference radius
of Europa (1562.6 km, <cit.>), and λ, ϕ, and r
are the longitude, latitude, and distance of Clipper from the origin
of the reference frame, which is chosen to coincide with Europa's
center of mass.  C̅_l,m and S̅_l,m are the
4π-normalized dimensionless spherical harmonic coefficients.  In
this work, we limited gravity field expansions to degree and order 20.


Jupiter's gravity field induces tides in Europa.  Because of the small
eccentricity of Europa's orbit, the tidal amplitude varies as a
function of time.  MONTE represents the tidal signal by applying
time-varying corrections to the spherical harmonics coefficients
<cit.>:

    ΔC̅_2,m - iΔS̅_2,m =    (k_2,m/5)    (M_J/M)  
    (R/r_EJ)^3
    P̅_2,m (sinϕ_J) e^-imλ_J,

where ΔC̅_2,m and ΔS̅_2,m are the
time-varying corrections to C̅_2,m and S̅_2,m,
respectively, k_2,m is the degree-2 Love number for degree 2 and order m, M_J is the mass
of Jupiter, r_EJ is the distance between Jupiter and Europa, and
λ_J and ϕ_J are the East longitude and latitude of the
sub-Jupiter point in Europa's body-fixed frame.  Here, we assume k_2,0 = k_2,1 = k_2,2 = k_2.



 §.§ Representation of Europa's spin state



The orientation of planetary bodies in inertial space can reveal
important insights about interior properties.  Europa's spin state is
not know well.  Its spin period is thought to be closely synchronized
to its orbital period, and its obliquity is thought to be
small <cit.>.  Analysis of existing Earth-based radar
measurements is expected to provide a measurement of Europa's spin
axis orientation to arcminute precision <cit.>.

The orientation of Europa can be modeled as:

    α = α_0 + α̇Δ t + ∑_i A_i sin M_i,


    δ = δ_0 + δ̇Δ t + ∑_i B_i sin M_i,


    W = W_0 + ẆΔ t + ∑_i C_i sin M_i,

where α and δ are the right ascension and declination of
the spin pole, respectively, W is the orientation of the prime
meridian, α_0, δ_0, and W_0 are the values at the
J2000 reference epoch, α̇, δ̇, and Ẇ are
the corresponding rates of change, Δ t is the time since the
reference epoch, and the
A_i, B_i, C_i, and M_i describe Fourier expansions of the
nutation-precession and libration signatures.  In this work, we use
the rotation model that is recommended by the International
Astronomical Union's Working Group on Cartographic Coordinates and
Rotational Elements <cit.>, which has its origin in
<cit.>'s model.
The values of the coefficients can be found in the current version
(pck00010.tpc) of the planetary constants kernel published by NASA's
Navigation and Ancillary Information Facility (NAIF) <cit.>.



 §.§ Solve-for parameters



In our simulations, we solved for the spacecraft's initial state
vectors, unmodeled accelerations, and geophysical parameters of
interest.  We divided our solve-for parameters into two categories:
local and global. The local parameters are applicable to a
single flyby only, whereas the global parameters are common to all
flybys.  Each parameter was assigned an a priori uncertainty for the
purpose of our covariance analyses (Table <ref>).



We placed a priori constraints on the uncertainties of coefficients of
degree l>2 in the expansion of the gravity field.  The constraints
follow a Kaula rule and we adopted the formulation given by
<cit.>:

    K = (28×10^-5)/l^2(R_m/R)^l,

where K is the a priori constraint for coefficients of
degree l and R_m is the assumed mantle radius of Europa (1465 km).


All local parameters are estimated for each flyby. The constant
acceleration is necessary in order to account for unmodeled
non-gravitational forces (e.g., solar radiation pressure, Jupiter
radiation pressure, etc.).  Its components are expressed in the
Radial-Transverse-Normal (RTN) frame associated with the spacecraft
trajectory.



 §.§ Covariance analysis



To evaluate the precision with which geophysical parameters of Europa
can be determined, we performed covariance analyses <cit.>.

Given z observables and p solve-for parameters, the normal
equations can be written as:

    η = H^TWH + C_0^-1,

where H is the matrix of partial derivatives of z with respect to
p, W is the matrix of weights appropriate for z, and C_0 is
the a priori covariance matrix of p.  We computed and stored the
elements of the normal equations for all flybys, using 11,040 Doppler
measurements, 112 crossover measurements, and 856 solve-for
parameters.  We computed the covariance matrix as:

    C = η^-1.

The formal uncertainties in the estimated parameters are obtained by
taking the square root of the diagonal elements of the covariance
matrix:

    σ_i = √(C_ii).



The GSWG emphasized that formal uncertainties from covariance analyses
are typically too optimistic, i.e., too small.  The GSWG recommended
multiplying formal uncertainties by a factor of 2 to arrive at more
realistic one-standard-deviation uncertainties.  In this work, we
consistently multiplied formal uncertainties by a factor of 2 per
GSWG recommendations.  All uncertainty values listed or displayed have
the factor of 2 applied.


The covariance analysis technique quickly enables the examination of a
variety of scenarios.  The normal equations are computed and stored
once and for all. If one chooses to restrict the analysis to certain
observables or certain parameters, one simply selects the relevant
subset of lines and columns in η (equation <ref>) and
performs a new matrix inversion.





§ RESULTS



We assumed that the spacecraft is tracked within ±2 hours of each
closest approach, when the altitude of the spacecraft with respect to
Europa's surface is ≤ 28,000 km.  The radio link budget depends on
the DSN assets that are deployed to track the spacecraft and on the
spacecraft telecommunication assets, including antenna gain patterns.
The Clipper spacecraft design currently includes two low-gain antennas
and three medium-gain antennas.  We assumed that Doppler measurements
are available only when the radio link budget exceeds a nominal value
(4 dB-Hz) that enables the establishment of a coherent, two-way link.
Our calculations included the spacecraft position and attitude
relevant to the 17F12v2 trajectory, variations due to spacecraft
antenna gain patterns, occultations by Europa and Jupiter, and other
assumptions listed in Table <ref>.

We examined two scenarios. In the first scenario, we assumed that one
of the three most sensitive DSN antennas, the 70 m diameter antennas
at Goldstone, Madrid, and Canberra, was used to track the spacecraft.
This scenario reveals the best performance that one can hope to
achieve with typical ground assets.  In the second scenario, we
assumed that 34 m diameter antennas were deployed either as single
antennas or as arrays of antennas, and we examined the minimum set of
assets that are necessary to meet the gravity science objectives.

We conducted three separate case studies in each of the two scenarios.
In the first case study, we examined the precision that is achieved as
the data from each consecutive flyby is processed.  In the second case
study, we asked how many flybys are necessary to meet the required
measurement precision on k_2, C̅_20, and C̅_22 if
the flybys tracked with DSN antennas are selected randomly from the
set of all available flybys.  In the third case study, we quantified
the minimum number of flybys that are necessary to meet the
measurement objectives when the flybys are grouped according to their
sub-spacecraft latitude at closest approach.  Results from these case
studies allowed us to gain a progressively deeper understanding of the
measurement precision that can be achieved in a variety of
circumstances.



 §.§ Scenario 1: 70 m DSN antennas



In Scenario 1, we considered that 70 m DSN antennas were available for
tracking, and we analyzed all 46 flybys of trajectory 17F12v2.  After
applying a 4 dB-Hz cutoff to the radio link budget, we were left with
a total of 10,048 Doppler measurements (Figure <ref>).   We also
considered up to 112 crossover measurements (section <ref>).
The exact number of Doppler and crossover measurements included in our
analysis depends on the specific flyby selections in the various case
studies.

We note that all flybys in 17F12v2 except E5 can be tracked for at
least 1 h within ±2 h of closest approach with a 70 m antenna.
Flybys with less than 1 h of tracking time are problematic: they
generally contribute little to the realization of measurement
objectives and they have a high ratio of DSN overhead time to useful
tracking time.  Therefore, in this work, we focus on flybys that can
be tracked for a total duration of at least 1 h (not necessarily
continuous).






  §.§.§ Case Study 1: Consecutive flybys



In this case study, we examined the precision of the estimates as
data from consecutive flybys becomes available.  At step n, the
available observables consist of observables acquired during flybys 1
through n, and the solve-for parameters consist of global parameters
and local parameters relevant to flybys 1 through n.

We conducted simulations for both Doppler-only and Doppler+crossover
situations (Figure <ref>).  We found that measurement
requirements for k_2 and C̅_20 can be met (Table
<ref>).  The precision of the C̅_22 gravity
coefficient estimates in Doppler-only simulations does not meet the
measurement objective.  We found that C̅_30 and
C̅_40 are never determined with sufficient precision to
estimate the ice shell thickness at the ±20% level.  The results
indicate that k_2 and C̅_20 measurement objectives can be
met with fewer than 46 flybys, provided that tracking of the
spacecraft is accomplished consistently with 70 m antennas.  The
results also demonstrate that measurement objectives can be achieved
with fewer flybys when crossover measurements are included in the
analysis.









  §.§.§ Case Study 2: Random sets of flybys



In this case study, we quantified the minimum number of flybys that
are necessary to meet measurement objectives when flybys tracked with
DSN antennas are selected randomly from the set of available flybys.
In actuality, DSN scheduling would likely take into consideration the
flybys that provide the best possible science return.  For this
reason, we eliminated 4 flybys (E1, E25, E26, E46) that do not
approach Europa's surface within

100 km and 3 flybys (E5, E6, E7) with SEP angles < 20^∘.
These seven flybys are expected to be less valuable than others from a
gravity science perspective. Thus, we considered a maximum of 39
flybys in this case study.  Because the previous case study revealed
the value of combining Doppler and crossover measurements, we included
both types of measurements in this case study.


Our simulations used a Monte Carlo scheme in which we considered n_c
randomly selected flybys out of n_a available flybys (here,
n_a=39).  The number of possible combinations is (1<= n_c <= 39):

    N = n_a!/n_c!(n_a-n_c)!.

If the number of combinations N was smaller than 10,000, we examined
all possible combinations.  Otherwise, we randomly selected 10,000
cases from the pool of available combinations.  We gradually increased
the number of considered flybys from 1 to 39.
We found that it is possible to meet measurement objectives for k_2,
C̅_20, and C̅_22 100% of the time with 34, 33, and
38 flybys, respectively (Figure <ref>).




 
An important goal of this case study was to gain information about the
distribution of sub-spacecraft latitudes at closest approach that
provides the best prospect of meeting measurement objectives.  Based
on our extensive set of simulations, we were able to quantify the
number of flybys that are required according to specific latitude
regions when considering sets of randomly selected flybys (Table
<ref>).




 


  §.§.§ Case Study 3: Preferred sets of flybys



In this third and final case study of Scenario 1, we used knowledge
gained in previous case studies to inject some intelligence in the
selection of flybys that can meet the primary (k_2) measurement
objective.  As in the previous case study, we discarded 7 flybys that
either have low (< 20^∘) SEP angles or high (> 100 km)
closest-approach altitudes.  The remaining 39 flybys were classified
into latitude regions.  Because of the arrangement of the fan-beam
(medium gain) antennas on the spacecraft, the low-latitude flybys are
easiest to track with DSN assets.  Thus, our selection started with
all 13 low-latitude flybys.  In the first step, we evaluated the
performance with all 13 low-latitude flybys plus a randomly selected
flyby from each mid-latitude band, for a total of 15 flybys.  In the
next step, we considered all 13 low-latitude flybys, a randomly
selected flyby from each mid-latitude band, and a randomly selected
flyby from each high-latitude band, for a total of 17 flybys.  We
gradually increased the number of mid- and high-latitude flybys in
this fashion.  At each step, we examined all possible combinations of
flybys.  We found that, with 70 m DSN assets, it is possible to meet
the k_2 measurement objective with 23 flybys (Table
<ref>).
Referring back to Figure <ref>, about 10% of
the simulations with 23 randomly selected flybys were successful with
respect to k_2.






We evaluated k_2, C̅_20, and C̅_22 parameter
uncertainties with the minimum number of flybys specified in Table
<ref>.  We performed covariance analyses for all
possible combinations and found that it is possible to meet the k_2
measurement objective 100% of the time, whereas the requirements for
C̅_20 and C̅_22 are met only 92% and 0% of the
time, respectively.  It is possible to meet the C̅_20 and
C̅_22 measurement objectives by increasing the number of
flybys that are tracked (section <ref>).




 §.§ Scenario 2: Minimum DSN assets



Scenario 1 provided estimates of what can be achieved with 70 m
antennas.  However, 70 m antenna time is difficult to schedule.  In
Scenario 2, we considered situations that place fewer demands on the
ground telecommunication assets.  We identified the minimal set of
ground assets that achieve the k_2 measurement objective.

Similar to Scenario 1, we assumed that Doppler measurements were
available only when the radio link budget exceeded 4 dB-Hz within
±2 hours of each closest approach. One can deploy a variety of
ground assets to maintain such a radio link.  We considered four
configurations: a 34 m antenna, an array composed of two 34 m
antennas, an array composed of three 34 m antennas, and a 70 m
antenna.  For each of these configurations, we computed the time
intervals during which a 4 dB-Hz radio link can be maintained (Figure
<ref>).




We examined the distribution of flybys according to tracking duration
with various DSN assets (Table <ref>).  A 34 m
antenna can track 9 flybys for ±2 hours of each closest approach.
With a two-antenna or three-antenna array, the number of flybys that
can be tracked increases to 12 and 16, respectively.  A 70 m antenna
can track 19 flybys for ±2 hours of each closest approach.




As in Scenario 1 (section <ref>), we only considered flybys
that provide the best possible science return. We therefore discarded
7 flybys either with high closest approach altitudes (> 100 km) or
low SEP angles (< 20^∘). For the remaining 39 flybys, we
identified the number of flybys for which Clipper can be tracked for
at least 1 h (not necessarily continuous) within ±2 h of closest
approach with a link budget above 4 dB-Hz.  We discarded flybys with
less than 1 h of total DSN tracking because these flybys generally
contribute little to the realization of measurement objectives and
because of their high ratio of DSN overhead time to useful tracking
time.  A 34 m antenna is sufficient to track 26 out of the 39
considered flybys for more than 1 h.  We label this configuration
DSN_34m.  If we combine two 34 m antennas into a
two-antenna array, 5 additional flybys can be tracked for more than 1
h.  The union of DSN_34m and these 5 additional flybys
yields a total of 31 flybys, a configuration that we label
DSN_2×34m.  If we combine three 34 m antennas into a
three-antenna array, 5 additional flybys can be tracked for more than
1 h.  The union of DSN_2×34m and these 5 additional
flybys yields a total of 36 flybys, a configuration that we label
DSN_3×34m.  If a 70 m antenna is used, 3 additional
flybys can be tracked for more than 1 h.  The union of
DSN_3×34m and these 3 additional flybys yields a
total of 39 tracked flybys, a configuration that we label
DSN_70m.  The number of available flybys with each DSN
configuration are summarized in Table <ref>, and the
corresponding tracking coverage is illustrated in Figure
<ref>. In order to minimize the use of ground assets, we
select, for each flyby, the least sensitive antenna configuration that
can provide at least 1 h of tracking.  Specifically, we do not use
additional assets to extend the duration of flybys that are already
tracked for at least 1 h (compare Figure <ref>(d) to Figure
<ref>).





The available number of Doppler and crossover measurements varies
according to the chosen DSN configuration (Table <ref>).
However, the exact number of Doppler and crossover measurements
included in our analysis depends on the specific flyby selections in
the various case studies.







  §.§.§ Case Study 1: Consecutive flybys



This Scenario 2 case study is based on the same principles as its
analog in Scenario 1 (section <ref>). We analyzed the Doppler and
crossover data from consecutive flybys as it becomes available.




We found that measurement objectives cannot be met if tracking is
restricted to a single 34 m antenna (Figure <ref>).
However, it is possible to meet the k_2 measurement objective with
the DSN_2×34m configuration by tracking 26 flybys
with a single 34 m antenna and 5 additional flybys with a two-antenna
array (2x34 m), for a total of 31 tracked flybys.

The estimated uncertainties in tidal Love number k_2 and low-order
gravity field coefficients using a variety of DSN assets are shown in
Table <ref>. The measurement objective pertaining to
verifying whether the ice shell is hydrostatic (Table <ref>)
is barely met with the most sensitive antenna configuration.  The
gravity field coefficients C̅_30 and C̅_40 are never
measured at the level required to measure the ice shell thickness with
±20% uncertainties (section <ref>).








  §.§.§ Case Study 2: Random sets of flybys



This Scenario 2 case study is based on the same principles as its
analog in Scenario 1 (section <ref>).  We quantified the minimum
number of flybys that are necessary to meet measurement objectives
when tracked (> 1 h) flybys are selected randomly from the set of available flybys.  

Because it is not possible to meet science objectives with a single 34
m antenna (DSN_34m), we selected the
DSN_2×34m antenna configuration, where a 2x34 m
antenna array is used on up to 5 occasions to supplement the up to 26
flybys tracked with a single 34 m antenna.  Thus, a total of up to 31
tracked flybys are available in this case study (Table
<ref>).




We considered n_c randomly selected flybys out of n_a available
flybys (here, n_a=31).  The number of possible combinations is given
by equation <ref>.  If the number of combinations N was
smaller than 10,000, we examined all possible combinations.  Otherwise,
we randomly selected 10,000 cases from the pool of available
combinations.  We gradually increased the number of considered flybys
from 1 to 31.

We found that it is possible to meet the k_2 and C̅_20
measurement objectives 100% of the time with 31 and 30 randomly
chosen flybys, respectively.  The measurement objective for
C̅_22 was never achieved (Figure <ref>).  The
counter-intuitive result of meeting the k_2 (C̅_20)
measurement objectives with 31 (30) flybys in this scenario versus 34
(33) flybys in the Case Study 2 of Scenario 1 (section <ref>) is
due to the fact that, in Scenario 2, a larger proportion of equatorial
flybys is represented in the pool of 31 flybys than in the pool of 39
flybys in Scenario 1.  Equatorial flybys provide a better
determination of k_2 than high-latitude flybys.

As in Scenario 1, we identified the latitudinal distribution of flybys
in successful simulations.  (Table <ref>).








  §.§.§ Case Study 3: Preferred sets of flybys




In this third and final case study for Scenario 2, we examined the
number of tracked (> 1 h duration) flybys required to meet the k_2
measurement objective with careful selection of the flybys according
to latitude region. As shown in section <ref>, a single 34 m
antenna is not sufficient to meet this objective. Therefore we
selected the DSN_2×34m antenna configuration (Table
<ref>) to perform this case study with 31 tracked flybys.

As in section <ref>, we started with all low-latitude flybys and gradually
increased the number of randomly selected mid- and high-latitude
flybys until the measurement objective was achieved.


We found that it is possible to meet the k_2 measurement objective
with 25 flybys (Table <ref>, Figure
<ref>), as long as they include all 13 low-latitude
flybys, at least 8 mid-latitude flybys, and at least 4 high-latitude
flybys.  This result applies to the 17F12v2 trajectory and
DSN_2×34m antenna configuration.  The measurement
objective for C̅_20 is also met, but that of C̅_22
is never met for any combination of 25 flybys.  Failure to track a
single low-latitude flyby from the 13 available in 17F12v2 would
result in a failure to meet Clipper's primary gravity science
objective.  This fact highlights an element of risk associated with
relying on the DSN_2×34m antenna configuration.
This risk is reduced when using 70 m antennas.

 








§ OTHER TRAJECTORIES



In an attempt to generalize our results, we examined the
suitability of the other trajectories available to us, 15F10 and
16F11, for meeting the k_2 measurement objective.




The 15F10 and 16F11 trajectories consist of 42 and 43 flybys,
respectively.  They yield 88 and 106 illuminated crossover points
below 1000 km altitude, respectively.  Similar to Scenario 2 of
trajectory 17F12v2 (section <ref>), Doppler observations were
simulated with the least sensitive DSN configuration that maintains
the 4dB-Hz radio link budget for track durations of at least 1
h. Figure <ref> shows the time intervals for which such a
link can be maintained with a variety of DSN assets within ±2 h
of closest approach. After further discarding flybys with SEP angle
< 20^∘ and closest approach altitude > 100 km,

a total of 34 and 37 tracked flybys remain with the 15F10 and 16F11
trajectories, respectively.  The numbers of Doppler and crossover
measurements that can be obtained with the available flybys are
shown in Table <ref>.








Similar to the first case study in Scenario 2 (section
<ref>), we examined the precision of the Love number k_2
as data from consecutive flybys becomes available. We found that the
k_2 measurement objective is not achievable with a single 34 m
antenna (Figure <ref>). However, as in the situation
with the 17F12v2 trajectory, the k_2 measurement objective can be
met (Table <ref>) with both 15F10 and 16F11
trajectories and DSN_2×34m antenna configurations,
where the majority of flybys are tracked with a single 34 m antenna
and a few additional flybys are tracked with a 2x34 m antenna array
(Table <ref>).




We also examined the minimum number and latitudinal distribution
of tracked flybys that are required to meet the k_2 measurement
objective.  We selected the DSN_2×34m antenna
configuration (Table <ref>) to perform this case study
with 28 and 29 tracked flybys for trajectories 15F10 and 16F11,
respectively.  We selected the flybys as in Case Study 3 of Scenario 2
(section <ref>).



We found that a minimum of 24 flybys with a specific latitudinal
distribution (Table <ref>) are sufficient to meet
the k_2 measurement objective for both trajectories.
As with the 17F12v2 trajectory, the measurement objective for
C̅_20 is also met with the same distribution of flybys,
whereas the measurement objective for C̅_22 is never met for
any combination of 24 flybys.

In summary, it takes at least 24 carefully selected flybys in 15F10
and 16F11 (Table <ref>) and at least 25 carefully
selected flybys in 17F12v2 (Table <ref>) to meet
the k_2 objective with the DSN_2×34m antenna
configuration.  The similarity in the required number of flybys
suggests that it may be possible to generalize the results to other
trajectories that are similar in character (section <ref>).






§ GENERALIZED TRACKING REQUIREMENTS




Here, we examine whether results that apply to trajectories 15F10,
16F11, and 17F12v2 are sufficiently similar that they can be
extrapolated to other trajectories that are similar in character.  Our
results are summarized in Table <ref>.  Taking the maximum
values of these results as a basis for extrapolation, we find that a
minimum of ∼13 low-latitude, ∼8 mid-latitude, and ∼5
high-latitude flybys are necessary to meet the k_2 objective.
Likewise, we find that a total tracking duration that is at least 50%
of the total potential tracking time (±2 h of each closest
approach) is necessary. Expressed as a fraction of total potential
tracking time for selected flybys only (tracking duration > 1 h,
altitude < 100 km, SEP angle > 20^∘ deg), the required
percentage is 88%.  The availability of at least 52 crossover points
in illuminated regions completes the requirements.








§ CONCLUSIONS


A Europa Clipper gravity science investigation can address important
mission objectives, such as confirming the presence of an ocean,
determining Europa's gravity field, quantifying the time-varying tidal
potential, verifying whether the ice shell is hydrostatic, and
providing high-precision reconstructed trajectories that other
instrument teams will greatly benefit from.

We performed covariance analyses to quantify the precision with which
geophysical parameters can be determined with a radio science
investigation and a nominal mission profile with trajectory 17F12v2.
We found that the availability of crossover measurements allows
measurement objectives to be achieved with substantially fewer tracked
flybys than in a Doppler-only scenario.  Even with 70 m antennas, the
measurement objective for the second degree and order gravitational
harmonic cannot be achieved without crossover measurements.

By simulating hundreds of thousands of combinations of tracked flybys,
we were able to quantify the distribution of flybys with
sub-spacecraft latitudes at closest approach within certain latitude
regions that provides the best prospects for meeting measurement
objectives.  We found that tracking a dozen low-latitude flybys and a
dozen mid- to high-latitude flybys are both essential.

We found that it is not possible to maintain a 4 dB-Hz radio link
budget during a ±2 hour interval centered on each flyby's closest
approach epoch, even with the most sensitive ground-based assets of
the DSN.  However, we found that 45 out of 46 flybys can be tracked
for a total duration of at least one hour with a 70 m antenna.  With a
34 m antenna, 26 out of 46 flybys can be tracked for a total duration
of at least one hour, and 5 additional flybys can be tracked for a
total duration of at least one hour with a two-antenna array of 34 m
diameter antennas.  

If 70 m antennas are used, the tidal Love number k_2 measurement
objective can be met by tracking at least 23 methodically selected
flybys, with good resilience in case certain flybys are unexpectedly
missed.  If 34 m antennas are used without arraying, the k_2
measurement objective is not achievable.  However, it is achievable by
tracking 26 flybys with 34 m antennas and 5 additional flybys with
arrays of two 34 m antennas.  If flybys are carefully selected with
respect to the latitudinal distribution of closest approaches, the
k_2 objective can be met by tracking at least 25 flybys with 34 m
antennas and two-antenna arrays, provided that all low-latitude flybys
are tracked.  Such a configuration provides little margin for error.

By comparing our 17F12v2 results to 15F10 and 16F11 results, we showed
that our conclusions are roughly generalizable to trajectories that
are similar in character.










































§ ACKNOWLEDGMENTS

AKV and JLM were supported in part by NASA Europa
Mission Project Subcontract 1569162.  This work was enabled in part by
the Mission Operations and Navigation Toolkit Environment (MONTE) and
the Spacecraft, Planet, Instrument, Camera-matrix, Events (SPICE)
toolkit.  MONTE and SPICE are developed at the Jet Propulsion
Laboratory, which is operated by Caltech under contract with NASA.  
The Europa Clipper trajectory data used in this work is available from the NAIF server at
<ftp://naif.jpl.nasa.gov/pub/naif/EUROPACLIPPER>. We
thank James Roberts and Robert Pappalardo with assistance in
specifying gravity science measurement requirements.  We thank Dipak
Srinivasan, Peter Ilott, Avinash Sharma, and Ryan Park with assistance
in providing values of system parameters.  We thank Gregor
Steinbrügge for providing estimates of range uncertainties at
crossover locations. We thank Francis Nimmo for providing helpful
comments on the manuscript and GSWG members for informative
discussions.






























§ REFERENCES


