Expected Precision of Europa Clipper Gravity Measurements
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 , second-degree gravity coefficients and , 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.
keywords:Europa; Geophysics; Tides, solid body; Satellites, shapes; Orbit determination
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 (e.g., Pappalardo et al., 2017) that emerged out of strategic planning documents (e.g., National Research Council, 1999, 2011) 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 (e.g., Smith et al., 2012; Mazarico et al., 2014; Verma and Margot, 2016), Venus (e.g., Sjogren et al., 1997; Konopliv et al., 1999), Mars (e.g., Smith et al., 1999; Konopliv et al., 2011), and Titan (e.g., Iess et al., 2010).
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 (Gravity Science Working Group, 2016) that specifies the precision with which certain quantities must be measured in order to meet specific science objectives (Table LABEL:tab-gswg). The GSWG focused primarily on measurements that pertain to the ice shell and the presence of an ocean.
|Confirm the presence of an ocean||Tidal Love number||0.06|
|Verify whether ice shell is hydrostatic||Gravitational harmonics||8e-6 and 9e-6|
|Measure shell thickness (to 20)||Gravitational harmonics||4e-7 and 4e-7|
|Confirm the presence of an ocean||Tidal Love number||0.3|
|Confirm the presence of an ocean||Obliquity||0.01|
|Measure elastic shell thickness (to 10 km)||Tidal Love numbers||0.015 and 0.015|
|Confirm ice shell is decoupled from interior||Amplitude of longitude libration||50 m at tidal period|
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 (Gravity Science Working Group, 2016). Here, we focus on measurements of the tidal Love number . An alternate approach consists of measuring the tidal Love number , as examined by Steinbrügge et al. (2018). Calculations by Moore and Schubert (2000) indicate that 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, is expected to be less than 0.015 if there is no global ocean. Therefore, a measurement of is sufficient to test the global ocean hypothesis (Park et al., 2011, 2015; Mazarico et al., 2015), 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 (Anderson et al., 1998), 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, and , to the level prescribed by the GSWG (Table LABEL:tab-gswg). Trajectories being designed for the Clipper mission offer promising prospects for measuring these quantities.
In section 2, we provide an overview of the anticipated Clipper trajectory. In section 3, we review measurements, uncertainties, and model assumptions. Our dynamical model, solution strategy, and estimated parameters are discussed in section 4. In Section 5, we discuss our covariance analysis results. Our conclusions are provided in section 8.
2 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 (Lam et al., 2015). 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 (Pappalardo et al., 2017). The current trajectory, named 17F12v2, includes 46 flybys with altitudes as low as 25 km (Figure 1) 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. \addedAll of these trajectories were designed to obtain globally distributed regional coverage of Europa with 42, 43, and 46 flybys, respectively (Table 2).
|<50 km||50–100 km||100–1000 km||>1000 km|
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 LABEL:nomTrajTable). Details about the spacecraft’s anticipated trajectory and orientation in space (attitude) are available \addedat ftp://naif.jpl.nasa.gov/pub/naif/EUROPACLIPPER in the form of SPICE kernels (Acton et al., 2017).
|Europa region||Latitude range||Altitude range||Number of flybys|
|High latitude north||90 45||25 km 1442 km||9|
|Mid latitude north||45 15||50 km 100 km||4|
|Low latitude||15 -15||25 km 100 km||13|
|Mid latitude south||-15 -45||25 km 2554 km||12|
|High latitude south||-45 -90||25 km 100 km||8|
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 \added(see Section 3.2) 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 \addedspacecraft-to-Europa ranging data from the Radar for Europa Assessment and Sounding: Ocean to Near-surface (REASON) instrument (Blankenship et al., 2014). Analysis of REASON data may be enhanced with digital elevation models obtained by stereo imaging from the Europa Imaging System (EIS) (Turtle et al., 2016). The primary observables will be two-way, coherent Doppler measurements (section 3.2) and radar range measurements (section 3.3). Our assumptions about the radio link and the Doppler and crossover measurements are summarized in Table LABEL:tab-assumptions and explained in detail below.
3.1 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.). \addedWe 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.). \replacedTypical 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. \replacedHowever, 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 (Deep Space Network, 2016).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.
3.2 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,
where is the Doppler shift, is the transmitted frequency, V is the LOS component of the relative velocity between spacecraft and observer, and 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 (Asmar et al., 2005). However, at small Sun-Earth-Probe (SEP) angles, the \addedinterplanetary plasma noise dominates. We modeled the Doppler uncertainties in a 60 s integration time as:
where represents the noise due to interplanetary plasma according to the model of Iess et al. (2012) and 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 2). See Asmar et al. (2005) for a detailed review of noise sources in radio science experiments. where represents the noise due to interplanetary plasma according to the model of Iess et al. (2012), 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 2).
3.3 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 1). Ranges obtained at the same location on two separate tracks can be subtracted to yield a crossover height difference:
where is the altimeter height measurement at time . 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 (e.g., Wahr et al., 2006).
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 Steinbrügge et al. (2018), would not be possible.
We assigned uncertainties to the crossover measurements given by
where is the altimeter height measurement uncertainty at time . These uncertainties were provided by the REASON team (Gregor Steinbruegge, pers. comm., Sept. 30, 2017) with heritage from the procedure described by Steinbrügge et al. (2018). 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.
|X-band uplink||reference frequency 7.2 GHz|
|X-band downlink||reference frequency 8.4 GHz|
|Doppler||60 s count time||integration time for Doppler measurements|
|measurements||up to 2 h of tracking per flyby||centered on closest approach epochs|
|0.07 mm/s 0.096 mm/s||at 20 SEP 180|
|0.096 mm/s 0.21 mm/s||at 10 SEP 20|
|surface illumination by the Sun||excludes crossovers over non-illuminated terrain|
|1000 km altitude||spacecraft altitude at epochs of crossovers|
|Crossover||range uncertainties:||from REASON team|
|measurements||median = 4.5 m||from REASON team|
|standard deviation = 6.5 m||from REASON team|
|[min, max] = [2.7, 24.9] m||from REASON team|
|+40 dBm transmitter power||includes all losses due to waveguides and switches|
|Spacecraft||3 fan-beam antennas (FBAs)||orientation per 2c spacecraft model (Version 0.1)|
|transmitter||2 low-gain antennas (LGAs)||orientation per 2c spacecraft model (Version 0.1)|
|and||LGAs: Gaussian beam pattern||peak at 7.5 dB and width of 42|
|antennas||FBAs: fan-beam pattern||from table lookup and bi-linear interpolation|
|No obscuration by spacecraft structures||assumes free path, no scatter losses included|
|68.4 dBi antenna gain||for 34 m antenna (NASA DSN, 2017)|
|31.29 K system temperature||for 34 m antenna (from Ryan Park)|
|74.3 dBi antenna gain||for 70 m antenna (NASA DSN, 2017)|
|26.39 K system temperature||for 70 m antenna (from Ryan Park)|
|DSN antennas||1 Hz downlink loop bandwidth||used in noise power calculation|
|2.71 dB two-antenna array gain||assumes typical 0.3 dB combining loss (NASA DSN, 2017)|
|4.47 dB three-antenna array gain||assumes typical 0.3 dB combining loss (NASA DSN, 2017)|
|Jupiter radio noise contribution||model from NASA DSN (2017); assumes 152 K black body|
|No elevation or horizon mask||assumes continuous view of Jupiter by DSN assets|
|Earth-Jupiter distance:||trajectory 17F12v2|
|mean = 5.36 au||trajectory 17F12v2|
|Radio link||median = 5.39 au||trajectory 17F12v2|
|[min, max] = [4.36, 6.28] au||trajectory 17F12v2|
|4 dB-Hz link budget||best-effort basis|
Our approach consisted in simulating the precision with which the Love number and degree-two gravitational harmonics can be determined with realistic mission scenarios and assumptions about measurement uncertainties (section 3).
We used version 124 of the Mission Operations and Navigation Toolkit Environment (MONTE) (Evans et al., 2016), 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 (Verma and Margot, 2016), orbit determination (Greenberg et al., 2017), and sensitivity analysis for tests of general relativity (Verma et al., 2017).
MONTE’s observation model uses Moyer (2003)’s formulation to compute Doppler measurements (section 3.2) and a ray-intersect method to compute altitude measurements (section 3.3). MONTE can specify arbitrary gravity fields (section 4.1) and spin states (section 4.2). We used MONTE’s observation model to compute simulated observables and their partial derivatives with respect to solve-for parameters (section 4.3). Finally, we used MONTE’s tools to perform covariance analyses and quantify the precision with which geophysical parameters can be determined (section 4.4). In the sections below, we provide details about the gravity and spin state models, solve-for parameters, and covariance analyses.
4.1 Representation of Europa’s gravity field
MONTE represents gravity fields as spherical harmonic expansions (Kaula, 2000):
where is the gravitational constant, is the mass of Europa, are the normalized associated Legendre polynomials of degree and order , is the reference radius of Europa (1562.6 km, Archinal et al. (2011)), and , , and 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. and are the -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 (McCarthy and Petit, 2004, p.59):
where and are the time-varying corrections to and , respectively, is the \deleteddegree-2 Love number \addedfor degree 2 and order , is the mass of Jupiter, is the distance between Jupiter and Europa, and and are the East longitude and latitude of the sub-Jupiter point in Europa’s body-fixed frame. \addedHere, we assume .
4.2 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 (Bills et al., 2009). Analysis of existing Earth-based radar measurements is expected to provide a measurement of Europa’s spin axis orientation to arcminute precision (Margot et al., 2013).
The orientation of Europa can be modeled as:
where and are the right ascension and declination of the spin pole, respectively, is the orientation of the prime meridian, , , and are the values at the J2000 reference epoch, , , and are the corresponding rates of change, is the time since the reference epoch, and the A, B, C, and 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 (Archinal et al., 2011), which has its origin in Lieske (1998)’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) (Acton et al., 2017).
4.3 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: and . The parameters are applicable to a single flyby only, whereas the parameters are common to all flybys. Each parameter was assigned an a priori uncertainty for the purpose of our covariance analyses (Table LABEL:aprioriPara).
|Type||Parameter||A priori uncertainty||Number of parameters|
|position||100 km||3 per flyby|
|Local||velocity||1 m/s||3 per flyby|
|constant acceleration||510 km/s||3 per flyby|
|Global||2020 gravity field||Kaula rule (see text)||437|
|spin pole||1 degree||2|
|rotation rate||110 degree/day||1|
We placed a priori constraints on the uncertainties of coefficients of degree in the expansion of the gravity field. The constraints follow a Kaula rule and we adopted the formulation given by Park et al. (2015):
where is the a priori constraint for coefficients of degree and is the assumed mantle radius of Europa (1465 km).
All 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.
4.4 Covariance analysis
To evaluate the precision with which geophysical parameters of Europa can be determined, we performed covariance analyses (Bierman, 1977).
Given observables and solve-for parameters, the normal equations can be written as:
where is the matrix of partial derivatives of with respect to , is the matrix of weights appropriate for , and is the a priori covariance matrix of . 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:
The formal uncertainties in the estimated parameters are obtained by taking the square root of the diagonal elements of the covariance matrix:
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 (equation 11) and performs a new matrix inversion.
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 LABEL:tab-assumptions.
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 , , and 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.
5.1 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 3). We also considered up to 112 crossover measurements (section 3.3). 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).
5.1.1 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 , the available observables consist of observables acquired during flybys 1 through , and the solve-for parameters consist of global parameters and local parameters relevant to flybys 1 through .
We conducted simulations for both Doppler-only and Doppler+crossover situations (Figure 4). We found that measurement requirements for and can be met (Table LABEL:seqRecTable). The precision of the gravity coefficient estimates in Doppler-only simulations does not meet the measurement objective. We found that and are never determined with sufficient precision to estimate the ice shell thickness at the 20% level. The results indicate that and 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.
|Radial distances||2157 m||1953 m||–|
5.1.2 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 randomly selected flybys out of available flybys (here, ). The number of possible combinations is ():
If the number of combinations 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 , , and 100% of the time with 34, 33, and 38 flybys, respectively (Figure 5).
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 LABEL:tab-random).
|Europa region||Latitude range||Available flybys|
|High latitude north||90 45||8||71||71||70|
|Mid latitude north||45 15||4||41||31||40|
|Low latitude||15 -15||13||111||111||131|
|Mid latitude south||-15 -45||8||71||71||80|
|High latitude south||-45 -90||6||51||51||60|
5.1.3 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 () 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 measurement objective with 23 flybys (Table LABEL:tab-preferred). Referring back to Figure 5, about 10% of the simulations with 23 randomly selected flybys were successful with respect to .
|Europa region||Latitude range||Avail. flybys||Req. flybys|
|High latitude north||90 45||8||2|
|Mid latitude north||45 15||4||3|
|Low latitude||15 -15||13||13|
|Mid latitude south||-15 -45||8||3|
|High latitude south||-45 -90||6||2|
We evaluated , , and parameter uncertainties with the minimum number of flybys specified in Table LABEL:tab-preferred. We performed covariance analyses for all possible combinations and found that it is possible to meet the measurement objective 100% of the time, whereas the requirements for and are met only 92% and 0% of the time, respectively. It is possible to meet the and measurement objectives by increasing the number of flybys that are tracked (section 5.1.2).
5.2 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 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 7).
We examined the distribution of flybys according to tracking duration with various DSN assets (Table LABEL:dsnAssetsNotused). 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.
|DSN assets||Number of flybys|
|4 h||1–4 h||1 h|
As in Scenario 1 (section 5.1), 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. 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 and these 5 additional flybys yields a total of 31 flybys, a configuration that we label DSN. 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 and these 5 additional flybys yields a total of 36 flybys, a configuration that we label DSN. If a 70 m antenna is used, 3 additional flybys can be tracked for more than 1 h. The union of DSN and these 3 additional flybys yields a total of 39 tracked flybys, a configuration that we label DSN. The number of available flybys with each DSN configuration are summarized in Table LABEL:dsnAssets, and the corresponding tracking coverage is illustrated in Figure 8. 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 7(d) to Figure 8).
|Configuration||Number of available flybys||Total|
|34 m||2x34 m||3x34 m||70 m|
The available number of Doppler and crossover measurements varies according to the chosen DSN configuration (Table LABEL:msrPoints). However, the exact number of Doppler and crossover measurements included in our analysis depends on the specific flyby selections in the various case studies.
|Tracking of 39 flybys with 70 m antennas||8756||95||79%|
5.2.1 Case Study 1: Consecutive flybys
This Scenario 2 case study is based on the same principles as its analog in Scenario 1 (section 5.1.1). 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 9). However, it is possible to meet the measurement objective with the DSN 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 and low-order gravity field coefficients using a variety of DSN assets are shown in Table LABEL:seqRecTable2x34m. The measurement objective pertaining to verifying whether the ice shell is hydrostatic (Table LABEL:tab-gswg) is barely met with the most sensitive antenna configuration. The gravity field coefficients and are never measured at the level required to measure the ice shell thickness with 20% uncertainties (section 1).
|(26 flybys)||(26+5=31 flybys)||(26+5+5=36 flybys)||(26+5+5+3=39 flybys)|
5.2.2 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 5.1.2). 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), we selected the DSN 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 LABEL:dsnAssets).
We considered randomly selected flybys out of available flybys (here, ). The number of possible combinations is given by equation 14. If the number of combinations 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 and measurement objectives 100% of the time with 31 and 30 randomly chosen flybys, respectively. The measurement objective for was never achieved (Figure 10). The counter-intuitive result of meeting the () measurement objectives with 31 (30) flybys in this scenario versus 34 (33) flybys in the Case Study 2 of Scenario 1 (section 5.1.2) 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 than high-latitude flybys.
As in Scenario 1, we identified the latitudinal distribution of flybys in successful simulations. (Table LABEL:tab-random2x34m).
|Europa region||Latitude range||Available flybys|
|High latitude north||90 45||7||70||61|
|Mid latitude north||45 15||4||40||40|
|Low latitude||15 -15||13||130||130|
|Mid latitude south||-15 -45||5||50||50|
|High latitude south||-45 -90||2||20||20|
5.2.3 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 measurement objective with careful selection of the flybys according to latitude region. As shown in section 5.2.1, a single 34 m antenna is not sufficient to meet this objective. Therefore we selected the DSN antenna configuration (Table LABEL:dsnAssets) to perform this case study with 31 tracked flybys.
As in section 5.1.3, 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 measurement objective with 25 flybys (Table LABEL:tab-preferred2x34m, Figure 11), 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 antenna configuration. The measurement objective for is also met, but that of 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 antenna configuration. This risk is reduced when using 70 m antennas.
|Europa region||Latitude range||Avail. flybys||Req. flybys|
|High latitude north||90 45||7||2|
|Mid latitude north||45 15||4||4|
|Low latitude||15 -15||13||13|
|Mid latitude south||-15 -45||5||4|
|High latitude south||-45 -90||2||2|
6 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 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 5.2), 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 12 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 LABEL:tab-15N16Msr.
Similar to the first case study in Scenario 2 (section 5.2.1), we examined the precision of the Love number as data from consecutive flybys becomes available. We found that the measurement objective is not achievable with a single 34 m antenna (Figure 13). However, as in the situation with the 17F12v2 trajectory, the measurement objective can be met (Table 16) with both 15F10 and 16F11 trajectories and DSN 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 LABEL:tab-15N16Msr).
We also examined the minimum number and latitudinal distribution of tracked flybys that are required to meet the measurement objective. We selected the DSN antenna configuration (Table LABEL:tab-15N16Msr) 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 5.2.3).
|Europa region||Latitude range||15F10||16F11|
|Avail. flybys||Req. flybys||Avail. flybys||Req. flybys|
|High latitude north||90 45||7||3||4||2|
|Mid latitude north||45 15||5||5||7||7|
|Low latitude||15 -15||11||11||12||12|
|Mid latitude south||-15 -45||3||3||4||1|
|High latitude south||-45 -90||2||2||2||2|
We found that a minimum of 24 flybys with a specific latitudinal distribution (Table LABEL:tab-preferred16F11) are sufficient to meet the measurement objective for both trajectories. As with the 17F12v2 trajectory, the measurement objective for is also met with the same distribution of flybys, whereas the measurement objective for is never met for any combination of 24 flybys.
In summary, it takes at least 24 carefully selected flybys in 15F10 and 16F11 (Table LABEL:tab-preferred16F11) and at least 25 carefully selected flybys in 17F12v2 (Table LABEL:tab-preferred2x34m) to meet the objective with the DSN 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 7).
7 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 LABEL:tab-4.5m. 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 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.
|No. of||Tracking||No. of||No. of||Tracking||No. of||No. of||Tracking||No. of|
|High latitude: 90 45||5||4||4|
|Mid latitude: 15 15||8||50%||38||8||49%||50||8||47%||52|
|Low latitude: 15 0||11||12||13|
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 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 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 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.
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.
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