1 - Planetary Moon Science Orbits

1. Figure 10:relative error as a function of the altimeter s. Data points obtained with a two-day data set from an 84° inclination orbit. Figure 3: Enceladus, effects of body tides on the orbit of the spacecraft. Sensitivity of different altimetric orbiters for the detection of subsurface oceans on Europa and Enceladus S. Casotto,* S. Padovan,§ R. Russell,# M. Lara¶ *University of Padua, Department of Astronomy and Center for Space Studies (CISAS), Padua, Italy (stefano.casotto@unipd.it), § University of Padua, Department of Astronomy, Padua, Italy, #Georgia Institute of Technology, Guggenheim School of Aerospace Engineering, Atlanta, Georgia, ¶ Real Observatorio de la Armada, San Fernando, Spain The unambiguous detections of the putative subsurface oceans of Europa and Enceladus are among the main objectives of the future missions to the outer Solar System: EJSM (Europa Jupiter System Mission) and TSSM (Titan and Saturn System Mission). In the absence of landing elements, one of the most promising ways to obtain strong, althoug indirect, indications on the presence of an undercrust global ocean is the analysis of the effects of the tidal mass deformations on the measurements carried out on or by the spacecraft. This analysis can be performed through the combined use of ground-based radio tracking to the spacecraft and the measurements performed by an onboard altimeter to the surface of the satellite. Through the inversion of synthetic altimeter and Earth-based range measurements, this study simulates the detection of the Love numbers h2 and k2, in the case of a Europa orbiter and of an Enceladus orbiter. In both cases reference orbital geometries are obtained from families of repeat ground-track orbits.This analysis provides insight into the sensitivity of the solve-for model parameters to the orbital geometry. In addition, result will show how the precision and time-rate of the simulated measurements relates to the errors associated with the estimated parameters. This information can be of value for the preliminary planning phases of EJSM and TSSM. 1 - Planetary Moon Science Orbits 4 - Sensitivity vs Orbit Inclination and Altitude The topic of science orbit design around planetary moons is broad and has been the subject of many important studies in recent years. In order to ensure global coverage, adequate surface mapping, and tidal bulge detection, science orbits are required to have high inclination, low altitude, and low eccentricity. Unfortunately, it is well known that most orbits about planetary satellites with these properties are dynamically unstable. For example at Enceladus, instability leads to impact of typical uncontrolled mapping orbits in 2-3 days. Maximizing orbit lifetimes of planetary orbiters is a challenging task. Multiple solution methods are possible including averaging methods and a variety of brute force numerical searches. The technique used here is to seek PERIODIC ORBITS in the unaveraged dynamical system. Periodic orbits represent equilibria (high horder “frozen orbits”) in the dynamical system. Despite being unstable, the “frozen orbit” impact times for near polar solutions can increase to 1~2 weeks at Enceladus for ballistic orbits. Initial guesses are simple to derive and are based on two-body repeat ground track solutions. Different orbital configurations have different sensitivities to Love’s numbers h2 and k2. With the aim to understand which configurations can better detect these tidal parameters, plots of the relative errors on h2 and k2 versus orbital inclination are shown, both for Enceladus and Europa. The data sets considered are longer than the revolution period of the satellite around the planet, which is 3.55d for Europa and 1.37dfor Enceladus. The main tidal signals have the same period. Sensitivity to k2 Enceladus, and the pair of supplementary orbits for Europa). Furthermore, on Enceladus the comparison of a 250 km altitude orbit with a 50 km altitude orbit with approximately the same inclination (not shown), displays a relative error decrease on k2 of about 0.06. As a consequence, inclination seems to play a more important role than altitude in determining the sensitivity. Near equatorial orbits—both direct and retrograde—give larger errors compared to mid-latitude to polar orbits. The subset of the high-inclination orbits displays a quite flat behavior in both data sets. In order to obtain a good result in the determination of k2 a high-inclination orbit is required. Then a high-altitude, high-inclination orbit should be chosen, at least in a first stage of the mission. Such a configuration minimizes the impact risk and gives a good response. In Figure 4 the results from the Enceladus orbiter are less smooth when compared to the regular shape of the case of Europa. This may be due to the more hostile dynamical environment experienced of Enceladus. Sensitivity to h2 k2 is a dynamical parameter and the perturbations it brings about are altitude-dependent—the higher the altitude, the lower the sensitivity—but remarkable differences among the orbits plotted (Figure 4) are not evident at least for the altitude range considered here. (compare errors of the 87°/57km orbit with the 97°/16 km orbit for Natural families of solutions are then achievable via predictor corrector procedures. Families are categorized as M:N resonances where M is approximately the number of sidereal satellite periods, and N is exactly the number of spacecraft periods. Figure 1 gives an example of a highly inclined, unstable trajectory that will impact balistically in a perturbed ephemeris model in approximately 8 days. Figure 4: relative errors on k2 for the orbiter around Europa (left) and Enceladus (right). Numbers labeling each dot indicates the mean altitude of the orbit. 2 - Ocean Detection Simulation Model As already noted both for Europa and Enceladus, an ocean would enhance the tides raised on the satellite by the parent planet and in turn would increase the dynamical perturbations on an orbiter. Appropriate measurements of ocean related parameters could give strong indication on the presence of the ocean. Tides are here modeled through the classical Love formulation: for the parameters k2 (related to tidal variation of the gravitational field) and h2 (related to the surface radial displacement) there is no reliable information. The following values have been adopted: k2= 0.261 and h2 = 1.26. These are compatible with ocean models for Europa, and it is likely that if there is a global ocean also on Enceladus these parameters have the right order of magnitude. Dynamical model: The approach followed in this study makes use of numerical integration to generate the orbits of the spacecraft (starting with the initial conditions obtained via the approach described in the previous section) in the dynamical system defined by: 1) the natural satellite (Europa or Enceladus) modeled as an extended body; 2) the parent planet (Jupiter or Saturn), which raises tides on the satellite; 3) point-mass perturbers (Saturn, Titan, Jupiter and Sun in the case of the Enceladus orbiter; Jupiter, Io, Ganymede, Callisto, Saturn and Sun in the case of the Europa orbiter). The parent planet is modeled as an extended body with gravitational field U. The Stokes coefficients account both for the static field of the satellite and its time-variable part due to the tidally redistributed mass: where Measurements: For the detection of k2 and h2 two kinds of measurements have been simulated: Earth-based range to an orbiter and altimetry from the orbiter to the surface of the satellite. The adopted observational system parameters are: s, the single measurement error and Dt, the time interval between two subsequent observations. Value listed below have been used: Range: s = 3 cm, Dt = 300 s; Altimetry: s = 9 cm, Dt = 150 s. In simulating range data, occultation events by Enceladus and Saturn are duly accounted for. In the model, h2 is a geometrical parameter and its detection rests on the precision of the instrument used. The orbital altitude theoretically makes no difference, as long as the spacecraft is able to measure its distance from the surface. It is apparent from Figure 5 that low inclination orbits—both direct and retro-grade—represent the best configuration for the precise Figure 5: relative errors on k2 for the orbiter around Europa (left) and Enceladus (right). Numbers labeling each dot indicates the mean altitude of the orbit. detection of h2. The double-peaked appearance of the data sets indicates that mid-inclination orbits should be avoided in order to obtain a good signal for the determination of h2. The double peak appearance is compatible both with the tidal signal of Figure 2 and with the pictorial explanation of Figure 6. Figure 6 5 - Consideration on the role of the dataset length Clearly, the longer the measurement arc, the better the determination of the parameters. For an Enceladus orbiter, determining k2 with a 175° inclination orbit is the worst case among the data in Figure 4 (right panel); however, with a long batch of data the error on the determination tends to level off at a value comparable with the error of a 75° inclination orbit (Figure 7). Thus, a long enough data set can, at some extent, compensate for unfavorable, but more stable, configurations. Nevertheless orbital configuration plays key role in the sensitivity to tidal parameters, as apparent in Figures 8 and 9. Despite the relative error decreases with increasing orbital arc length, there is a gap between the 3 - Tidal Signals This section shows how tides deform the surface of the satellite and affect the orbit of the spacecraft through the modification of the gravitational field. Surface radial deformations of Enceladus (h2 Love number) location is plotted. Amplitude varies from 17 m to 35 m and deformation about the mean can be very small at mid-latitudes. Therefore the altimetric measurements will be more sensitive to surface radial displacement (i.e., to h2) at lower inclinations (see Figure 6). A qualitatively similar scenario holds for Europa. Tidal variations of the gravitational field (k2 Love number) In the lower graphs the time variation of the deformation with respect to the mean radius of Enceladus (252.1 km) is plotted, both on the equator and on the prime meridian. Results are compatible with a prolate ellipsoid aligned along the satellite-planet direction, according to the figure of a synchronous satellite deformed by the tides raised by the parent planet. In the upper graphs the variation about the mean for each Figure 7: influence of the dataset length on the determination of k2 for a Enceladus orbiter curves plotted in the figures. Even with a 30-days data set the gap does not decrease. A 5% error is usually considered an accetable accuracy for geophysical modeling. Some configurations, no matter how long the orbital arc length, do not attain this result. Figure 2: Enceladus: surface radial deformations and variations about the mean deformation for selected points along the prime meridian and along the equator. Figure 8:influence of the dataset length on the determination of h2 for a Europa orbiter Figure 8:influence of the dataset length on the determination of k2 for a Europa orbiter 6 - A critical parameter: altimeter measurement error The error associated to the single altimeter measurement plays a crucial role in the determination of h2, the parameter related to the surface radial displacement. As discussed in the “Tidal signals” box, mid-inclination latitudes undergo small radial displacements of the surface, as small as a few meters. The s of the altimeter has thus to be carefully chosen, since the signal it must detect could be quite small, depending on the orbital inclination of the spacecraft. As illustrated in Figure 10 in the case of an Enceladus orbiter, a 1-meter s leads to a 100% error in the estimated value of h2. As already noted in Figure 5, mid-inclination orbit are the worst configurations for the determination of h2. This can be explained by considering the time variation of the intersection between the orbital plane and the figure of Enceladus, which is smaller for mid-inclination orbits, see Figure 6. The effects of the tidal variations of the gravitational field can be appreciated by comparing two orbits propagated from the same initial conditions, but with different tide models. One orbit is propagated with the static field of the satellite only (zero-tide orbit), the other accounting also for the tidal variations of the Stokes coefficients induced by the parent planet (the direct gravitational effect of the planet on the spacecraft is not taken here into account). Figure 3 refers to the Enceladus orbiter. The radius vector difference in the RTN reference frame of the zero-tide orbit is shown. Although the parent planet does not directly affect the orbit of the spacecrafts, it raises such large tides on the satellite that in just one day the distance increases to some tens of kilometers. In the case of Enceladus, in only one day the position difference grows to approximately 35 km, or 15% of the radius of Enceladus. This clearly point to the necessity of Conclusions Results show that altitude plays a less important role than inclination in k2 determination (a measure of the satellite’s dynamical response to tides). Future mission to Europa or Enceladus (or to other “likely oceanic” moons) should then first use a high-altitude, more stable orbit which represents a good compromise between safety and scientific usefulness. The spacecraft could then move to lower altitude orbits for finer analysis. In designing appropriate configurations it should be also taken into account that h2 determination is easier for low-inclination orbits, and that mid-inclination orbit should be avoided. determining the tidal response properties of Enceladus—which essentially amounts to detecting the presence of its putative subsurface ocean—well before lowering the spacecraft to more risky altitudes. In the case of Europa the quantitative effect of the tides on the orbiter is the same—the distance grows to about 30 km in one day—which correspond to 2% of the radius of Europa. The situation is then more benign. See also Figure 4 and 5.