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 This paper presents the expected accuracy of the tides and gravity of Europa that can be measured by tracking a spacecraft during close flybys of Europa. A reference trajectory was designed for flyby science observations and consists of a total of 36 flybys of Europa at 100 km altitude. Earth-based Doppler measurements were created during ±2 hours of each periapsis passage and were simulated with realistic dynamical and measurement assumptions. The result shows that the degree 2 tidal Love number,k2, can be estimated to σk2 = 0.045 and σk2= 0.009 (1-sigma formal uncertainty) assuming X-band and Ka-band tracking capabilities, respectively, which is sufficient to confirm the existence of a global subsurface ocean.
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 Since the flyby of Europa by NASA's Galileo spacecraft, there has been a growing interest in exploring Europa mainly due to the possible existence of a global subsurface ocean, which would be an extremely important astrobiological discovery. From analysis of Galileo's Doppler tracking data, Anderson et al. have shown that Europa is likely to be a differentiated body with metallic core, rocky mantle, and water ice-liquid outer shell. Other evidence of Europa's subsurface ocean comes from Galileo spacecraft's magnetometer data during flybys [Kivelson et al., 2000; Zimmer et al., 2000]; however, the measurement was not accurate enough to provide a definitive answer. The high-resolution images taken during the flybys also support the theory of a global ocean [Greenberg, 2005]. All the measurements taken from the Galileo spacecraft indicate that Europa is geologically active with an ocean beneath its icy crust. If there is a subsurface ocean, the driving source for the heat necessary to maintain the internal temperature is likely from the energy due to tidal flexing.
 Recently, a mission called Jupiter Europa Orbiter was proposed to NASA in conjunction with ESA's Jupiter Ganymede Orbiter mission. However, due to its high cost, the mission concept has been divided into an orbiter or multiple flybys. The orbiter mission would provide a high-resolution gravity field (radiometric tracking), a detailed shape model (altimetry and imagery), and highly accurate determinations of the tidal Love numbersk2 and h2. Previous studies have shown the expected accuracy of tides and gravity field from a dedicated orbiter mission [Wu et al., 2001; Wahr et al., 2006]. Such an orbiter mission can answer not only the existence of a global ocean, but also the variations in crustal thickness, existence of regional oceans, and librations [Wahr et al., 2006]. The flyby mission would use remote-sensing measurements, such as ice-penetrating radar, imagery, etc.
 The purpose of this paper is to show the feasibility of detecting tides at Europa by tracking a spacecraft during close flybys of Europa, which has not been explored previously. It is important to note that the tracking data during flybys provide less global and temporal coverage than that of an orbiter. However, this study shows that the tracking data from multiple flybys were strong enough to detect if there is a global ocean at Europa. The expected degree 2 tidal Love number, k2, of Europa is on the order of 0.2 assuming a tidal equilibrium state with 0-100 km ice thickness [Wu et al., 2001; Wahr et al., 2006], and about 25% accuracy on k2(i.e., 0.05) should be a convincing evidence in answering if a global ocean exists. In case Europa is in a non-equilibrium ocean tidal state, as discussed by Tyler as a possible condition for Europa [Tyler, 2008, 2011], the tidal signature would be generally larger; thus easier to measure k2 and to detect if a global ocean exists.
 The tidal force is a temporal effect, but it is highly correlated with the static low-degree gravity coefficients since thek2signature appears as a variation in the degree 2 gravity coefficients. Therefore, the reference trajectory must be designed to optimally sample the tidal signature, and at the same time, consider geometric requirements for other science instruments. A nominal, realistic two-year-long trajectory has been designed which consists of a total of 36 flybys at 100 km altitude. This trajectory was optimized with geometric constraints (rather than for tides detection) and was designed to target three right ascensions of Europa (with respect to Jupiter) for different illuminating conditions. Based on this reference trajectory, a detailed covariance analysis has been performed with realistic orbit dynamics and measurement assumptions. Although the trajectory was not optimized for tides detection, this study shows thatk2 can be measured to the level that is of scientifically important, which suggests that an improved estimates can be achieved with a dedicated flyby mission for tidal detection.
2. Trajectory Design
 The trajectory used in this study stems from an ongoing study to assess the feasibility of a Europa flyby mission that exhibits a high level of scientific return and possible advantages (cost, simplicity, radiation dosage, etc.) when compared with an orbiter mission. The specific trajectory used for this analysis consists of two major phases: 1) energy dissipation via Ganymede gravity assists and 2) multiple Europa flybys for scientific observations for optimal imaging.
 The first mission phase, consisting of four Ganymede flybys, was necessary to reduce the spacecraft's Jovian orbital period and attain the correct lighting conditions and V∞ at Europa compatible with the type of observations envisioned at Europa. The second mission phase, the science phase, consists of 36 close Europa flybys (with the closest approach altitude of 100 km) at a number of different geometries over the course of ∼1.4 years (see Figure 1).
 This observation strategy was to conduct imaging on the inbound asymptote over the illuminated trailing hemisphere of Europa and to observe Europa using ice-penetrating radar near the close approach (altitude < 400 km). In addition, the majority of these flybys were designed such that a continuous spacecraft-to-Earth communication link could be maintained (i.e., no occultations), thus allowing for gravity science measurements. The spacecraft trajectory was optimized with numerically integrated Jovian satellite ephemerides [Jacobson et al., 2000]. Figure 2shows the right ascension, mean anomaly, and radius of Europa with respect to Jupiter at closest flyby distances. The nominal trajectory targets three geometric constraints (i.e., right ascensions), but varies temporally (i.e., mean anomaly and radius) due to the high perijove precession-rate of Europa.
3. Covariance Analysis
 Covariance analysis is a powerful technique for analyzing the expected estimation performance and statistics [Bierman, 1977]. However, it must be noted that covariance analysis is meaningful only if the problem is parameterized properly as it can easily become an under- or over-determined system depending on the choice of estimated parameters. All simulations were carried out using JPL's Multiple Interferometric Ranging And GPS Ensemble (MIRAGE) software set, which is high-precision orbit determination software and was used to process the data from a number of space missions, including Magellan, Lunar Prospector, GRACE, Mars Global Surveyor, Mars Reconnaissance Orbiter, and many others. The spacecraft trajectory integration included all planets and all major Jovian satellites as perturbing bodies. Note that a Doppler measurement provides the line-of-sight velocity of the spacecraft relative to the tracking station. The candidate trajectory simulated in this study gives favorable geometries for Doppler tracking.
 The non-spherical gravitational potential of Europa can be represented using the fully normalized spherical harmonics coefficients, i.e., nm and nm [Kaula, 2000], which is related to the conventional un-normalized spherical harmonics as , where is usually referred to as the Kaula normalization. In this study, the nominal values of Europa's gravity spherical harmonics (up to degree 3) were taken from the solution of the JPL's Jovian satellite ephemeris file JUP230, which is based on the tracking data from the Galileo mission and is an improved version of the ephemerides presented by Jacobson et al. . The gravity coefficients with degree 4 and higher were assumed to be zero since a nominal field is not significant for a covariance study.
 The tidal contribution at Europa due to the variations in distance to Jupiter can be written as [Kaula, 2000]:
where knm represent degree n and order m Love numbers, GMj represents the gravitational constant of Jupiter, GMe represents the gravitational constant of Europa, Re represents the mean radius of Europa (1565 km), rej represents the distance of Jupiter from Europa, λj and ϕjrepresent Europa-fixed longitude and latitude of Jupiter, respectively. Note that the time dependence of tides comes from the variablesrej, λj, and ϕj, which depend on the orbital frequency of Europa. This study only considers the real part of the second-degree tides (i.e.,k2m) with the assumed nominal value of 0.2 [Wahr et al., 2006; Wu et al., 2001]. The third-degree tidal effect was ignored since it's smaller by a factor ofrej/Re ≈ 400. To first order, decoupling the tidal effect from the static gravitational effect comes from tracking the spacecraft at different Europa orbit distances and observing the same surface location more than once. This is because the tidal force varies temporally and Europa is tidally locked (i.e., 1:1 spin/orbit resonance).
 For each flyby, Earth-based Doppler observables were created during the ±2 hours of periapsis passage and were simulated assuming two different tracking capabilities, i.e., X-band and Ka-band. For the X-band tracking (8.4 GHz), the assumed measurement accuracy was 0.1 mm/s at 60-second count time, which is typical for interplanetary missions [Asmar et al., 2005]. The Ka-band tracking (32 GHz) gives about an order of magnitude improvement on the Doppler accuracy, i.e., 0.01 mm/s accuracy at 60-second count time, assuming the media effect is calibrated using the advanced water-vapor radiometer (AWVR) data [Armstrong, 2006; Bertotti et al., 2003].
 The estimated parameters were three position parameters (σa = 100 km), three velocity parameters (σa = 1 m/s), three constant acceleration parameters (σa = 5 × 10−11 km/s2), Europa's GM (σa = 320 km3/s2), second-degree Love numbers (σa = 0.2), and 20 × 20 normalized spherical harmonics coefficients, where σa represents the a prioriuncertainty of an estimated parameter. Note that the position, velocity, and acceleration parameters were estimated for each flyby. The three constant acceleration parameters represent the accelerations in the radial, transverse, and normal directions and their nominal values were set to zero. These accelerations were used to model the errors in the non-gravitational forces, such as solar and planetary radiation pressures, spacecraft thermal emission, etc. Since flybys occur over a relatively short time frame, three constant accelerations should be a good representation of the errors in the non-gravitational models.
 For the gravity field, a 20 × 20 field should be sufficient for this study since beyond degree 20 is insignificant to the tidal contribution. The a priori uncertainty for the gravity field was constrained by the Kaula rule (28 × 10−5)(Rmantle/Re)n/n2, where Rmantle represents the mantle radius (1465 km). This Kaula constraint was derived from half the power of Earth's Kaula spectrum and by scaling with the assumed mantle radius. Since Europa is a differentiate body, probably isostatically compensated, and has relatively smoother surface and lower bulk density compared to Earth, this Kaula rule should be a conservative constraint for the a priori uncertainty in the gravity field of Europa. The effect of the uncertainties in other important parameters, such as Europa ephemeris, media calibration, Jupiter's gravity field, etc., were considered in parallel studies, but shown to be insignificant (less than, at most, 10% change in the formal uncertainties) and were discarded in this study.
4. Results and Discussion
Table 1 shows the expected accuracy of k2m from the flyby simulations. The cases X-Band and Ka-Bandgive the 1-sigma formal uncertainty of the estimated Love numbers based on X-band and Ka-band tracking capabilities, respectively. The result shows that the Love numberk22 can be estimated with the best accuracy, i.e., σk2= 0.045 for X-band tracking andσk2= 0.009 for Ka-band tracking. The correlations among the estimated Love numbers were generally small except for the pairs (k20, 2 ) and (k22, ). Note that both uncertainties and correlations strongly depend on the flyby geometry and should not be considered as a general behavior.
Table 1. Error in the Estimated k2m(1-Sigma Formal Uncertainty)a
The k22 parameter gives the best estimate, which strongly depends on the flyby geometry and should not be considered as a general behavior.
 The overall result indicates that the existence of a subsurface global ocean can be answered with either X-band or Ka-band tracking cases. However, based on practical experience, the formal uncertainty is usually an optimistic statistical realization of the estimates and is usually scaled up based on different estimation schemes and techniques. Therefore, Ka-band tracking capability is probably necessary in order to definitively detect the presence of a global ocean.
Figure 3shows the error root-mean-square (RMS) of the estimated gravity spherical harmonics, which is defined as:
where is called the error degree variance and and represent the standard deviation of the estimated nm and nm, respectively. Note that RMSn represents the error spectrum of the estimated degree nspherical harmonics and is typically used to represent the uncertainties in the estimated gravity field. As expected the Ka-band tracking yields much better accuracy compared to the X-band tracking for the low-degree harmonics. The difference is not exactly a factor of 10, as in the measurement accuracy difference, because of the assumeda prioriuncertainty model. The high-degree harmonics asymptotically approach thea priori uncertainty model, i.e., (28 × 10−5)(Rmantle/Re)n/n2. This gravity result, when combined with the estimated tides, can provide a constraint on the ice thickness through a forward modeling (i.e., assumed interior structure) using a similar approach shown by Wahr et al. .
 When measuring the overall static gravity field, it would be ideal to design flybys so that the surface is uniformly sampled, which, however, is not favorable for tides detection since tidal effect is temporally varied. Understanding how to design a mission that optimally weights tides and gravity is beyond the scope of this study. The purpose of this paper is to present a stress case showing that both tides and gravity can be measured to the level that is scientifically important from a Europa flyby mission.
 This paper presented the expected accuracy of tides and gravity that can be measured at Europa from a candidate flyby mission scenario. A detailed covariance analysis was performed with realistic dynamics and measurement assumptions. The result showed that the second-degree tidal Love number can be estimated toσk2 = 0.045 and σk2= 0.009 accuracy (1-sigma formal uncertainty) assuming X-band and Ka-band tracking capabilities, respectively. Although both cases were sufficient in answering the existence of a subsurface global ocean at Europa, the Ka-band tracking capability is probably necessary considering an estimation aliasing that often occurs in practice.
 The research described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.
 The Editor thanks Robert H. Tyler and an anonymous reviewer for their assistance in evaluating this paper.