1. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Orbital Dynamics About Small Bodies Stardust Opening Training School University of Strathclyde, 21 st November 2013 Juan L. Cano, ELECNOR DEIMOS, Spain 1

2. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Relevant Items 1. Small bodies and NEAs 2. Past and current missions to small bodies 3. The dynamical environment 4. The effect of the solar radiation pressure 5. Application to space missions 6. Conclusions 2

3. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Connection to other Talks “Manipulation of asteroids and space debris” by Prof. H. Yamakawa “Methods and techniques for asteroid deflection”, Prof. M. Vasile “On the accessibility of NEAs”, E. Perozzi “From regular to chaotic motion in Dynamical Systems with application to asteroids and debris dynamics”, Prof. A. Celleti “Physical properties of NEOs from space missions and relevant properties for mitigation”, Dr. Patrick Michel 3

4. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Small Bodies and NEAs 4 1

5. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Small Bodies Asteroids and comets Lecture centred on NEAs Perihelion < 1.3 AU …and particularly on very small NEAs Size < few km 5

6. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. What are the NEAs NEAs are asteroids that have migrated from the Main Belt into the inner Solar System Most are relatively small (< few kms) As other asteroids, they are remnants from the origins of the Solar System They also inform us on the dynamical evolution of the rest of bodies in the Solar System They have shaped life on Earth … and they are more reachable than main-belt asteroids 6

7. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Advances in recent years Studies on their population, properties, evolution, dynamics, etc have boomed in recent years Such advances have been reached after: Increasing the detection and observation programs (mainly in USA) Improving the knowledge on the Solar System dynamics and evolution Performing a number of deep space missions targeted to small bodies (NEAR, Hayabusa, Rosetta) Increasing the level of awareness of the threat that NEAs can pose to life on Earth 7 Image of the Chelyabinsk event

8. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Increase in discovery of NEAs 8 Start of the SpaceGuard Survey in USA

9. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Current knowledge on NEA population 9 Source: A.W. Harris 2011

10. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Why is it important to fly to NEAs? Science! This is currently the primary interest, targeted to better understand the Solar System origin, the original materials and their properties, its dynamics and evolution, etc. In many cases, we would like the S/C orbiting the asteroid And in some others have very close operations and even landing 10

11. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. What relevant information on NEAs can we obtain from a close mission? Proximity missions to asteroids allow determining: Type and albedo Size and shape Rotation state Existence and characterisation of secondary objects orbiting the primary Central gravity field (and maybe first terms of the harmonic expansion) Density Surface material distribution and properties Thermal properties Constraints on internal structure (cohesion, density changes, etc) Accurate measurement of the asteroid orbit 11

12. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Why is it important to fly to NEAs? Mitigation / Prevention! Relevant field gaining importance in the last decade in order to understand how to deviate an asteroid and actually test deflection strategies Many of those rely on actual asteroid rendezvous and close in orbit operations: Gravity tractor Ion beam shepherd Laser beaming Explosive techniques Pre-impact surveying 12

13. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Why is it important to fly to NEAs? Exploration and exploitation! This is today a “trending topic” boosted by NASA from 2013 and aimed at favouring manned missions to asteroids and the future exploitation of NEA resources Currently targeting very small NEOs (few metres) with the intention of graping one object and actually bringing it down to an orbit within the Earth-Moon system 13 Image Credit: NASA/Advanced Concepts Lab

14. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Past and Current Missions to Small Bodies 14 2

15. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Past and On-going Missions Initially, a number of missions only flew by small bodies: Giotto (Halley), Galileo (Gaspra & Ida), Deep Space 1 (Braille and comet Borrelly) But in more recent cases missions have done much more than just passing by: NEAR Hayabusa Dawn 15 Deep Impact Rosetta Images Credit: NASA

16. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Flown missions: NEAR NEAR (NASA) was the first mission to orbit a small body Launched in Feb. 1996, it orbited and landed on EROS (Feb. 2001) EROS features: 34.4 km x 11.2 km x 11.2 km, 2.67 g/cm 3, 6.69E+15 kg S type, rotation period of 5.27 h 16 Images Credit: NASA

17. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Flown missions: Hayabusa Hayabusa (JAXA) was the first mission to reach a very small body and bring back to Earth asteroid samples Launched in 2003, reached Itokawa in 2005 and returned to Earth in 2010 Itokawa’s features: 535 m × 294 m × 209 m, 1.95 g/cm 3, 3.58E+10 kg S-type, rotation period of 12.13 h 17 Image Credit: JAXA Image Credit: J.R.C. Garry

18. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Flown missions: Rosetta Rosetta (ESA) is a comet rendezvous mission launched in 2004 It will reach its target 67P/Churyumov-Gerasimenko in mid 2014 It will orbit the comet and deliver a lander to the surface Comet’s features: 4 km, rotation period of 12.76 h 18 Images Credit: ESA

19. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. The Dynamical Environment about Small NEAs 19 3

20. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. What is the environment about NEAs? Complex gravity field derived from irregular shapes and mass distributions Solar radiation pressure acting on the S/C Solar gravity tide 20

21. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. NEA Shape and Gravity Field Asteroids come in a wide diversity of sizes, shapes, composition, rotation states, etc This means that the shape of the gravity field can be very complex… … as well as the rotation state (fast rotators, slow rotators, nutation rates, etc.) Shape and rotation have a prominent role in cases were the asteroid is large or when operating very close to the surface in small asteroids 21

22. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Solar Radiation Pressure The solar radiation pressure mainly depends on the exposed S/C surface to the Sun Also on the optical properties of the exposed surfaces Simple models assume a constant exposed surface and a constant reflectivity parameter The case of the electric propulsion satellites is particularly important, as this is a common solution to fly to asteroids (Hayabusa, Deep Space 1, Dawn,Don Quijote, Proba-IP, etc.) In such cases the area of the solar panels can be large, which increases the surface to mass ratio of the S/C and thus the effects of SRP forces 22

23. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Solar Gravity Tide This effect can be considered as a minor perturbation Except in cases where the S/C orbits at some large distances from the asteroid In those cases, the perturbation can compete with the SRP In many analyses, as the required operational distances to the asteroids are small, this interaction is neglected 23

24. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. The result of all that is… Forget about Keplerian motion Orbits can be quite distorted, chaotic, unstable… … and in some particular cases stable enough for a S/C to operate close to the asteroid 24 Images Credit: D. Scheeres

25. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Orbital Stability about NEAs 25 4

26. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Typical questions to be answered at mission design level Can we safely orbit an asteroid? Can a S/C remain uncontrolled for long periods around an asteroid? Is it possible to hover wrt the asteroid or wrt a fixed point on the surface? Is it possible to land on them? 26

27. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Approach to the Assessment Typically and for simplicity the uncontrolled motion about an asteroid has been analysed separating the perturbation effects: SRP dominated orbits Gravity dominated orbits Combined effect orbits We will review in detail the SRP dominated orbits, which are applicable to small NEOs Furthermore, we will consider single asteroid systems 27

28. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. SRP Dominated Motion These motions are typically analysed in a reference frame rotating as the asteroid moves about the Sun 28 Origin at the centre of the asteroid X axis in the direction of sunlight (Sun in the negative side of the axis) Z axis in the direction of orbit angular momentum Y axis forming a right-handed reference system

29. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. SRP Dominated Motion In such reference system: Although the motion is not inertial, the reference frame is quasi- inertial (negligible inertial accelerations derived from rotation) SRP pulsates as the asteroid moves in its orbit, peaking at perihelion 29

30. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. SRP Dominated Motion Methods of analysis of such motion involve: Introduction of additional simplifications Averaging methods Full propagation of the equations of motion Examples are: Point mass, non-rotating with constant acceleration (SRP) Averaged method over a circular asteroid orbit Full averaged problem The theroretical aspects presented in the following are taken from several articles published by D. Scheeres 30

31. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. SRP Dominated Motion Point masses + Constant acceleration problem We shall start analysing the motion of an object close to a point mass and affected by a constant acceleration This is also called the Two-body Photo-gravitational Problem This problem was initially analysed by Dankowicz (1994- 1997) and then by Scheeres (1999-2001) The problem can be more easily formulated in a cylindrical reference system in the direction of the constant acceleration 31

32. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. SRP Dominated Motion SRP formulation Let the SRP acceleration be expressed as: With  being the reflectivity of the S/C (0 full absorption / 1 full reflection) And B the ratio of mass to exposed surface 32

33. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. SRP Dominated Motion Point masses + Constant acceleration problem Formulation: Which has a Jacobi integral: 33

34. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. SRP Dominated Motion Point masses + Constant acceleration problem There are very interesting properties of these equations It is demonstrated that the total angular momentum in the direction of the SRP is conserved Mostly interesting the existence of an equilibrium solution which is a circular orbit This solution is offset from the centre of attraction and is perpendicular to the uniform acceleration 34

35. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. SRP Dominated Motion Point masses + Constant acceleration problem Equilibrium conditions: As then 35

36. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. SRP Dominated Motion Point masses + Constant acceleration problem Analysing the stability of the solutions, one obtains this condition: or: Which represents a ~43% of the maximum equilibrium distance 36 Example for:  = 10 -9 km 3 /s 2 g = 10 -10 km/s 2 Instable branch Stable branch Locus of equilibrium circular orbits Maximum equilibrium offset Asteroid point mass

37. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. SRP Dominated Motion Point masses + Cte acceleration + Solar tide In case adding the tidal effects from the Sun, the zero velocity curves have the following shape: The sun-ward equilibrium point can be used as a monitoring site for a comet when passing through perihelion The anti-sun point provides a sufficient condition for escape 37 Image Credit: D. Scheeres

38. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. SRP Dominated Motion General SRP problem with averaging The problem is now analysed assuming the actual motion of the small body about the Sun Formulation is now posed with the SRP as a perturbation and averaging on the Lagrange Planetary equations After averaging, it is obtained that the averaged semi- major axis is constant (the orbit energy is preserved in average) Mignard and Hénon (1984) demonstrated that the equations can be integrated in closed form

39. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. SRP Dominated Motion General SRP problem with averaging Richter and Keller (1995) arrived at a compact formulation based on the use of the angular momentum vector h and the eccentricity vector e further generalised by Scheeres (2009): Being the averaged direction of the SRP acceleration This is a linear differential equation with non time-invariant terms, as and g depend on 1/d 2

40. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. SRP Dominated Motion General SRP problem with averaging However, is time invariant, which leads to: Where A is the SMA of the asteroid and E its eccentricity The following constant is then defined for a given asteroid, spacecraft and S/C orbit: SRP is strong for and weak for

41. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. SRP Dominated Motion General SRP problem with averaging By introducing a change of variables a time invariant formulation can be derived: Which solution can be obtained in the form of elementary functions (introducing ):

42. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. SRP Dominated Motion General SRP problem with averaging The solutions are periodic in : For large SRP perturbation the solution will repeat many times in a solar period of the asteroid For small SRP perturbation the solution will repeat only once per heliocentric orbit Looking for frozen orbits, two kinds of solutions appear: One in which is parallel to and is parallel to Another with parallel to and parallel to

43. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. SRP Dominated Motion General SRP problem with averaging In the first case the conditions that are needed for solution are: These are the so called Ecliptic frozen orbits and are contained in the orbital plane of the asteroid If the orbit normal is in the same direction as the asteroid orbit normal the periapsis must be directed to the Sun and opposite for the contrary case For large SRP the orbits are quite elliptic, which is not desireable Furthermore they suffer eclipses

44. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. SRP Dominated Motion General SRP problem with averaging In the second case the conditions are: These are the Solar Plane of the Sky orbits which are the continuation of the solution in the non-rotating case If the orbit normal points to the Sun the periapsis must be in the direction of the asteroid orbit normal and opposite for the contrary case For large SRP the orbits are more circular, which then tends to stabilise the orbits Furthermore they do not suffer eclipses, however, asteroid visibility conditions are not optimal (solar aspect angle > 90 deg)

45. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. SRP Dominated Motion Stability of the terminator plane orbit First considerations are derived from the variability of the SRP between aphelion and perihelion Larger SRP at perihelion decreases the value of a max possibly leading to escape View from the Sun Side view

46. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. SRP Dominated Motion Stability of the terminator plane orbit To analyse the stability of the TP orbits, this is done by linearising the Lagrange Planetary equations around the TP solution: And including the effect of asteroid oblateness: Two uncoupled harmonic oscillators:

47. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. SRP Dominated Motion Stability of the terminator plane orbit For long term stability we search to bound eccentricity variations, which complies with the following: Introducing : Which has the smallest perturbation effects at aphelion In the case of the ellipticity of the asteroid Equator, S/C can be safe of its interaction when:

48. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. SRP Dominated Motion Stability of the terminator plane orbit The destabilisation mechanisms of the TPOs are the following: The asteroid oblateness alone that might induce large oscillations in the frozen orbit elements which can excite the longer-term oscillations and thus make the eccentricity grow. However this is a not very fast interaction Combined action of oblateness and ellipticity can lead in non- favourable cases to resonant effects that introduce large variations in semi-major axis, eccentricity and inclination. This is a faster mechanism that needs to be avoided by the mentioned criteria:

49. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Gravity Dominated Motion Asteroid gravity dominates the motion of objects already for asteroids of several km in size Or in case motion about a small asteroid is brought to very close distances Such motions and their combined effect with other perturbations will not be analysed in this lecture 49

50. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Application to Space Missions 50 5

51. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Application to Missions Points of equal SRP and central gravity acceleration for current and future missions 51

52. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Application to Missions For SRP dominated missions, the actual stable TO would be at 43% of the reported distances Rosetta, although having large solar panels, is expected to operate at large distance from perihelion Clearly, as NEAR operated at close distances to Eros and actually landed on it, the mission was "gravity dominated" Hayabusa and Osiris REX represent a challenge, as SRP dominates the mission design in many mission phases Rosetta falls in between both extremes 52

53. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. The Hayabusa case Being a low-thrust mission to Itokawa, solar panels were comparatively large The obtained value of a max is 1.6 km which is rather small The value of  is about 87 deg 53 Due to the uncertainty in the knowledge of the asteroid mass it was decided to take a safe approach and design a hovering strategy Image Credit: JAXA

54. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. The Hayabusa case An a posteriori analysis was done with the available information and it was determined that TO would have been feasible with SMA between 1.0 and 1,.5 km 54 Image Credit: D. Scheeres

55. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Application to Missions Don Quijote mission study During the Don Quijote phase A study for ESA a number of stability assessments were done for 1989 ML and 2002 AT4 Mission design called for an impacting mission to an asteroid accompanied by an orbiter arriving first to the asteroid TOs were required in order to perform a radio-tracking experiment Stable solutions were found for both asteroids 55

56. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Application to Missions Don Quijote mission study 56

57. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Application to Missions Proba-IP mission study During the Proba-IP phase 0 for ESA we did also performed a number of stability assessments for the target asteroids, which were smaller than the ones considered for Don Quijote:1989 UQ 2001 CC21 and Apophis TOs were again required to perform a radio-tracking experiment Solutions were found for the two first asteroids However, Apophis presented a large problem because of its small size and its large rotation period (30 h) which was commensurate with the orbital period of the TOs  resonant perturbation which leads to orbit instability 57

58. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Application to Missions Proba-IP mission study 58 Apophis – No altitude range guarantees safety for every rotational state 1989 UQ – Safe orbits between 1.1 km and 3 km 2001 CC21 – Safe orbits between 3.5 km and 16+ km

59. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Conclusions 59 6

60. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Conclusions Orbiting a minor body is affected by a set of perturbations that make the motion of an object in its vicinity quite complex In many cases the trajectories will be unstable due particularly to the combination of a large SRP with other perturbations In some cases, stable solutions can be preliminary found, whose stability needs to be double-checked with full perturbation simulations 60

61. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. Conclusions One of those examples are the so called terminator orbits which allow circling about the asteroid in an off-set orbit behind the asteroid Stability of these orbits is mainly affected by the eccentricity of the asteroid orbit and the gravity / rotation state of the asteroid Lack of a priori knowledge of the asteroid properties is a major source of mission complexity and cost 61

62. DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U. 62 Thank you!