18. Small Planetary Satellites: Deimos We finish with perhaps one of the most challenging dynamical environments that can be encountered. That of a planetary satellite which has a strongly non-spherical shape. The prototypical examples of such satellites are the Martian moons Phobos and Deimos. These bodies are also of significant scientific interest, if for no other reason than their formation and subsequent evolution persists as a significant scientific mystery. These bodies exist in a dynamical environment where both the strong tidal perturbations from the planet and their own non-spheroidal gravity fields combine to place stringent limits on feasible orbital operations at these bodies. Orbital mechanics about the Martian planetary satellite Phobos has been of considerable interest to researchers for some time, with some of the earliest papers discussing dynamics about strongly non-spheroidal bodies occurring for this body. The earliest study of dynamics about Phobos was given by Dobrovolskis [32], where he pointed out many interesting features of this system relevant to understanding its current state and presumed natural evolution. Wiesel published the first detailed study of spacecraft orbital dynamics in the vicinity of Phobos [196]. More recently, the Russian Phobos-Grunt mission [98], which had a failed orbit insertion after its launch in November 2011 planned to visit the Martian moon Phobos and place instruments on its surface for its detailed scientific study. Associated with these plans were studies of the orbital mechanics about this specific system [176, 2]. The satellite Deimos has also been of interest to space scientists, and has even been proposed as a potential way-station for astronauts leading up to a visit to the surface of Mars. In this chapter we provide a discussion of the peculiar dynamics encountered at the Martian moon Deimos. While the environment of this moon is not as strongly perturbed as that of Phobos, the same essential dynamics are encountered at both bodies. Given that orbital dynamics about Deimos has not been explicitly studied in the published literature, it is fitting to focus the final analysis of this book on this particular body. The following work was largely performed in support of earlier NASA Discovery mission proposals to this specific body. While none of these missions have been supported for further development
344 18. Small Planetary Satellites: Deimos to date, missions to Deimos remain a perennial contender for space science mission proposals. 18.1 Model of Deimos The main parameters used for the current analysis are the shape model of Deimos, its gravitational parameter and its spin period. Deimos is locked in synchronous rotation with its orbit about Mars, and thus its rotation period and orbit period are equal. Due to this we assume it has a zero obliquity angle with respect to its orbit plane and that it rotates about its maximum moment of inertia. In Table 18.1 we present the basic information needed for this study. The shape model is available at the PDS-SBN [117]. The gravity field is modeled using the Deimos shape model with a uniform density assumption to provide the total gravitational parameter listed in Table 18.1. When sufficiently far from the surface of the asteroid, nominally outside of 10 km from its center of mass, a 16 16 gravity field is used. Table 18.1 Deimos parameters. Parameter Symbol Value Units Gravitational Parameter μ 1.354 10 4 km 3 /s 2 Rotation Period T 32.2986 hours Rotation Rate ϖ 5.404 10 5 rad/s Mean Radius R o 6.234 km Oblateness Gravity Coefficient Ro 2C 20 4.208 km 2 Ellipticity Gravity Coefficient Ro 2C 22 1.251 km 2 18.2 Equations of Motion The equations of motion of a particle about Deimos are modeled by the Hill model developed in Chapter 16 with the addition of a general gravity field instead of a point potential. The dimensional version of the equations are exclusively used for the current analysis. In the following the x-axis is aligned with the minimum moment of inertia of Deimos and is pointed towards Mars, the z-axis is aligned with the maximum moment of inertia and is normal to the orbit plane, and the y-axis completes the triad. The equations of motion in scalar form are then ẍ 2ϖẏ =3ϖ 2 x + U x (18.1) ÿ +2ϖẋ = U y (18.2) z = ϖ 2 z + U z (18.3) where ϖ is the rotation rate and U(x, y, z) is the gravitational potential.
18.3 Dynamics about Deimos 345 We consider one approximation to this model when discussing retrograde periodic orbits, related to the neglect of the attraction of Deimos. Specifically, when far from Deimos the gravitational attraction quickly becomes negligible relative to the tidal accelerations. For a specific example, consider displacing a particle along the x-axis away from Deimos. The tidal and gravitational accelerations will balance at a distance ( μ/3ϖ 2) 1/3 25 km. As the distance of the particle from Deimos is doubled to 50 km, the tidal acceleration doubles and the gravitational acceleration decreases by a factor of 4, yielding a relative change between the two accelerations of a factor of 8. A further doubling to 100 km makes the tidal acceleration a factor of 64 larger than the gravitational acceleration, etc. Thus, as one considers orbits far from Deimos, on the order of hundreds of kilometers, the gravitational attraction of Deimos becomes negligible and can be ignored. Applying the approximation (U x 0, etc.) yields a simplified set of equations ẍ 2ϖẏ =3ϖ 2 x (18.4) ÿ +2ϖẋ = 0 (18.5) z = ϖ 2 z (18.6) which are sometimes called the Hill equations (not to be confused with the Hill problem derived previously in this book) or the Clohessy Wiltshire equations, especially when applied to satellites. These equations, as given above, represent the linearized motion of a particle relative to a circular orbit, written in the frame rotating with the nominal circular orbit about the planet. As these are time invariant, linear equations they can be solved in closed form. 18.3 Dynamics about Deimos In the following several different aspects of orbital motion in the Deimos system are considered. All of the following is based on numerical evaluation of algebraic equations and numerical integration of the dynamical equations of motion, except as noted. 18.3.1 Zero-Velocity Curves and Equilibrium Points As the equations of motion are time invariant a Jacobi integral exists for this system. A traditional way to derive this integral directly from the equations of motion is to multiply them by ẋ, ẏ and ż, respectively, and add them. The Coriolis acceleration terms cancel and the resulting quantity can be reduced to an exact differential. Integrating this then yields the Jacobi integral J = 1 (ẋ2 +ẏ 2 +ż 2) 1 ( 2 2 ϖ2 3x 2 z 2) U(x, y, z) (18.7)
346 18. Small Planetary Satellites: Deimos As has been discussed earlier, this can be used to define zero-velocity surfaces that separate regions of allowable motion. For a specified value of J = C the constraint is C + 1 2 ϖ2 ( 3x 2 z 2) + U(x, y, z) 0 (18.8) In Fig. 18.1 the zero-velocity curves for Deimos are presented along the z = 0 plane. Note that they are similar to the zero-velocity surfaces in the Hill problem (Fig. 16.2), although they have some asymmetry due to the Deimos mass distribution. Fig. 18.1 Zero-velocity curves for Deimos along the z = 0 plane. Evident in the figure are the equilibrium points along the x-axis, analogous to the equilibrium points in the Hill problem. It is relevant to point out that Deimos has no equilibrium points along the y-axis, unlike its uniformly rotating counterpart Eros. The tidal potential of Mars provides a sufficiently large perturbation so that it effectively destroys these equilibrium points. While this is clear in the Hill problem derivation, it is still a bit surprising when applied to a uniformly rotating general shape. The two Deimos equilibrium points share the same stability properties as those in the Hill problem, and given our complete discussion in Chapter 16 we do not consider them in detail here.
18.3 Dynamics about Deimos 347 18.3.2 General Trajectories To start our more detailed discussion of orbits about Deimos, we first point out the strongly unstable nature of motion at this body. The location of the equilibrium points, and hence the Hill radius for Deimos, is approximately 25 km. From numerical and analytical studies carried out in the restricted three-body problem it has been shown that direct orbits outside of half the Hill radius have a strong tendency to escape [61]. Thus, in the following initially circular orbits of radius 10 15 km were considered. Starting these orbits with an inclination ranging between 0 and 90 degrees, they generally will escape or impact with Deimos in less than a single orbit period. A detailed analysis of these dynamics would require a combination of techniques, including the discrete maps developed for the analysis of Eros in Chapter 7 and the conditions for escape found for the Hill problem in Chapter 12. As there is seemingly little hope for developing long-term direct stable orbits about this body, these topics are not pursued further. As an example, Fig. 18.2 shows two different initially circular, polar orbits about Deimos. The only difference between the two is that one has its line of nodes along the y-axis and escapes from the body after a few orbits, and the other has its line of nodes along the x-axis and impacts. As inclination is further increased, regions of orbital stability are found. In general, if the orbital inclination maintains a sustained value above 140 degrees, then an orbit can persist for long periods of time. If the inclination is lower than this, however, the eccentricity tends to grow and impact usually ensues. It is significant to note that this inclination limit is consistent with the averaged analysis in Chapter 17 for when the eccentricity becomes unstable (i >140, specifically). Retrograde orbiters take advantage of having their orbital motion go against the rotation of the mass distribution, and thus can maintain stability if not destabilized by the tide. Figure 18.3 plots two orbits with an initial inclination of 135 degrees. One has its node situated so that the inclination increases to a higher value, while the other s inclination decreases to a lower value and is seen to impact rapidly. As orbital inclinations are moved to higher values, including retrograde, these initially circular orbits are stable in general, and are candidates for sustained spacecraft orbits. 18.3.3 Periodic Orbits Moving from general initial conditions for orbits, it is instructive to consider periodic orbits about Deimos. In the following we present some example direct orbits, all unstable, that exhibit some interesting geometry. Following these examples, a discussion and analysis of retrograde periodic orbits is provided, as these are of most interest for space science missions to such bodies. The following periodic orbits about Deimos were computed using the computational algorithm for periodic orbits presented in Chapter 6. The y = 0 plane is taken to be the surface of section and the ẏ velocity is eliminated using the Jacobi integral.
348 18. Small Planetary Satellites: Deimos Fig. 18.2 Initially polar, circular orbits about Deimos with semi-major axis of 15 km. Top: View looking down the Deimos z-axis. Bottom: Eccentricity of the orbits as a function of time.
18.3 Dynamics about Deimos 349 Fig. 18.3 Initially 135 inclination, circular orbits about Deimos with semi-major axis of 15 km. Top: Eccentricity as a function of time. Bottom: Inclination as a function of time.
350 18. Small Planetary Satellites: Deimos 18.3.3.1 Direct Orbits First consider a family of direct periodic orbits close to Deimos, with members shown in Fig. 18.4. This particular family is interesting as on either end of the family, the members of these orbits impact onto Deimos. From their morphology, they can be identified with the g family of periodic orbits described by Hénon in [67]. Despite their similarity, the current family has significantly different stability characteristics from the family described by Hénon. In the Hill problem this family becomes stable at lower values of radius, scaled to our problem at a distance of 10 km, and at this point the family has a bifurcation into the g family, which is elliptic with a their line of apses aligned with the x-axis. About Deimos, however, this g-like family is unstable for all of its members and does not intersect with any other family. Furthermore, scaling the g family to Deimos indicates that most of its members would have their periapsis beneath the body, and thus do not form a viable periodic orbit family in this situation. This serves as an excellent reminder of how the scaled size of the planetary satellite in the Hill problem can significantly modify conclusions and shape viable orbital strategies for planetary satellites. Fig. 18.4 Members of the direct family of periodic orbits about Deimos.
18.3 Dynamics about Deimos 351 18.3.3.2 Retrograde Periodic Orbits To end this chapter we focus on retrograde periodic orbits. Per the previous discussion on general trajectories, we expect that retrograde orbits may be stable and could serve as safe and viable candidate orbits for missions to planetary satellites. This is definitely the case. We find an entire family of stable periodic orbits, nominally called the f family by Hénon, that exist about Deimos. When far from Deimos, these orbits degenerate to periodic solutions of the Clohessy Wiltshire equations given above in Eqs. 18.4 to 18.6, and are identified as the larger orbits in Fig. 18.5 which describe a 2:1 ellipse centered on Deimos. These all travel retrograde in the rotating frame, and retrograde relative to Deimos, but they are still direct orbits about Mars. As these orbits come closer to Deimos, the mass of the body begins to affect the dynamics and they become more circular. The entire family is stable, down to the lowest orbit radii shown in Fig. 18.5. At these close orbits, however, the mass distribution of Deimos starts to significantly affect the dynamics. Note that they become tilted out of the plane and actually orbit at a radius closer than the maximum radius of Deimos (Fig. 18.6). The stability of this family across all orbital distances is contrasted with the stability of retrograde orbits about Eros, studied in Chapter 7. There it was seen that destabilizing resonances occurred at some close orbit distances. For the current problem the strength of the Mars tidal accelerations apparently prevents these resonant instabilities from occurring. Fig. 18.5 Members of the retrograde family of periodic orbits about Deimos viewed from the z-axis. As a final point of discussion, we also show plots of the orbit period and x-axis crossing speed of the retrograde periodic orbit family, parameterized by its x-axis crossing value. Figure 18.7 shows the period of the family members, and we note that as the orbit becomes large the period approaches that of Deimos about Mars. Figure 18.8 shows the speeds of the family members at their closest approach to
352 18. Small Planetary Satellites: Deimos Fig. 18.6 Members of the retrograde family of periodic orbits about Deimos viewed from the y-axis (left) and the x-axis (right). Deimos. We note that the speed is non-monotonic as the crossing comes closer to Deimos. Now consider the periodic orbits when far from Deimos. As mentioned previously, as the distance from the body increases, the relative strength of its gravitation attraction becomes insignificant as compared to the tidal effects. This enables the Fig. 18.7 Retrograde periodic orbit periods compared to Deimos s orbit period.
18.3 Dynamics about Deimos 353 Fig. 18.8 Retrograde periodic orbit speeds at closest approach to Deimos. Top: Shows the entire family. Bottom: Shows details for the closest orbits. Clohessy Wiltshire equations to be used to analyze the motion. We set z = 0 and only consider planar motion in the following.
354 18. Small Planetary Satellites: Deimos Equation 18.5 corresponds to an integral of motion for this system, and can be immediately integrated once to find ẏ +2ϖx = D (18.9) Substituting this into Eqn. 18.4 to eliminate ẏ yields the second-order differential equation ẍ + ϖ 2 x =2ϖD (18.10) This is just a simple harmonic oscillator with a constant forcing term. Evaluating its solution yields x(t) =A cos(ϖt)+b sin(ϖt)+ 2 ϖ D (18.11) We note that x(t) is completely periodic with a period of 2π/ϖ, equal to the orbit period of Deimos about Mars. Substituting this solution back into Eqn. 18.9 yields a first-order differential equation ẏ = 2Aϖ cos(ϖt) 2Bϖ sin(ϖt) 3D (18.12) For the motion of the particle to be periodic, it is clear that the constant D must equal zero. Otherwise, if it is non-zero there will be a secular drift associated with the orbit and the y component will change secularly, taking the particle away from Deimos. Enforcing this condition then provides the necessary condition for the orbit to be periodic ẏ o = 2ϖx o (18.13) where the condition can be evaluated at any point of the motion, but is usually computed at the x-axis crossing. Figure 18.8 shows the correct dependence predicted by this formula as x o grows large. With this proscription, the corresponding orbit is then periodic with a period equal to the Deimos orbit period, which in turn is consistent with Fig. 18.7.