Orbital Dynamics About Small Bodies

1. Orbital DynamicsAbout Small Bodies Stardust Opening Training School University of Strathclyde, 21st November 2013 Juan L. Cano, ELECNOR DEIMOS, Spain

2. Relevant Items • Small bodies and NEAs • Past and current missions to small bodies • The dynamical environment • The effect of the solar radiation pressure • Application to space missions • Conclusions

3. Connection to other Talks • “Manipulation of asteroids and space debris” by Prof. H. Yamakawa • “Methodsandtechniquesforasteroiddeflection”, Prof. M. Vasile • “On the accessibility of NEAs”, E. Perozzi • “From regular tochaoticmotion in DynamicalSystemswithapplicationtoasteroidsanddebrisdynamics”, Prof. A. Celleti • “PhysicalpropertiesofNEOsfromspacemissionsandrelevantpropertiesformitigation”, Dr. PatrickMichel

4. 1 Small Bodies and NEAs

5. Small Bodies • Asteroids and comets • Lecture centred on NEAs • Perihelion < 1.3 AU • …and particularly on very small NEAs • Size < few km

6. 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

7. 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 Image of the Chelyabinsk event

8. Increase in discovery of NEAs Start of the SpaceGuard Survey in USA

9. Current knowledge on NEA population Source: A.W. Harris 2011

10. 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

11. 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

12. 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

13. 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 Image Credit: NASA/Advanced Concepts Lab

14. 2 Past and Current Missions to Small Bodies

15. 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 • Deep Impact • Rosetta Images Credit: NASA

16. 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/cm3, 6.69E+15 kg S type, rotationperiodof 5.27 h Images Credit: NASA

17. 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/cm3, 3.58E+10 kg S-type, rotation period of 12.13 h Image Credit: JAXA Image Credit: J.R.C. Garry

18. 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 Images Credit: ESA

19. 3 The Dynamical Environment about Small NEAs

20. 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

21. 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

22. 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,DonQuijote, 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

23. 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

24. 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 Images Credit: D. Scheeres

25. 4 Orbital Stability about NEAs

26. 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?

27. 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, wewillconsider single asteroidsystems

28. SRP Dominated Motion • These motions are typically analysed in a reference frame rotating as the asteroid moves about the Sun • 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. 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

30. SRP Dominated Motion • Methods of analysis of such motion involve: • Introduction of additional simplifications • Averaging methods • Full propagationoftheequationsofmotion • 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

31. SRP Dominated MotionPoint 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 analysedby 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

32. SRP Dominated MotionSRP 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

33. SRP Dominated MotionPoint masses + Constant acceleration problem • Formulation: • Which has a Jacobi integral:

34. SRP Dominated MotionPoint 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

35. SRP Dominated MotionPoint masses + Constant acceleration problem • Equilibrium conditions: • As then • As then

36. SRP Dominated MotionPoint masses + Constant acceleration problem • Analysing the stability of the solutions, one obtains this condition: or: • Which represents a ~43% of the maximum equilibrium distance Locus of equilibrium circular orbits Instable branch Stable branch Example for:  = 10-9 km3/s2 g = 10-10 km/s2 Maximum equilibrium offset Asteroid point mass

37. SRP Dominated MotionPoint masses + Cte acceleration + Solar tide • In case addingthetidaleffectsfromtheSun, thezero 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 Image Credit: D. Scheeres

38. SRP Dominated MotionGeneral 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. SRP Dominated MotionGeneral 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/d2

40. SRP Dominated MotionGeneral 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. SRP Dominated MotionGeneral SRP problem with averaging • Byintroducing a changeof variables a time invariantformulation can be derived: • Whichsolution can be obtained in theformofelementaryfunctions (introducing ):

42. SRP Dominated MotionGeneral 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. SRP Dominated MotionGeneral 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. SRP Dominated MotionGeneral SRP problem with averaging • In the second case the conditions are: • These are the Solar Plane of the Skyorbits 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. SRP Dominated MotionStability 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 amax possibly leading to escape View from the Sun Side view

46. SRP Dominated MotionStability 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. SRP Dominated MotionStability of the terminator plane orbit • For long termstabilitywesearchtoboundeccentricityvariations, whichcomplieswiththefollowing: • Introducing : • Which has thesmallestperturbationeffectsataphelion • In the case oftheellipticityoftheasteroidEquator, S/C can be safeofitsinteractionwhen:

48. SRP Dominated MotionStability 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. 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

50. 5 Application to Space Missions