orbital dynamics about small bodies

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

Orbital DynamicsAbout Small Bodies

Stardust Opening Training SchoolUniversity of Strathclyde, 21st November 2013

Juan L. Cano, ELECNOR DEIMOS, Spain

1


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


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


Small Bodies and NEAs

4

1


Small Bodies

• Asteroids and comets

• Lecture centred on

NEAs

• Perihelion < 1.3 AU

• …and particularly on

very small NEAs

• Size < few km

5


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


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


Increase in discovery of NEAs

8

Start of the SpaceGuard

Survey in USA


Current knowledge on NEA population

9

Source: A.W. Harris 2011

100

102

104

106

108

1010

910111213141516171819202122232425262728293031

10-1

102

105

108

100

102

104

106

108

0.01 0.1 1 10

Brown et al. 2002Constant power lawDiscovered to 7/21/1020072010

K-T

Im

pact

or

Tun

gu

ska

Absolute Magnitude, H

Diameter, Km

N(<

H)

Impa

ct I

nte

rva

l, ye

ars

Impact Energy, MT


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


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


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


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


Past and Current Missions to Small Bodies

14

2


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


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, rotation period of 5.27 h

16

Images Credit: NASA


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

17

Image Credit: JAXAImage Credit: J.R.C. Garry


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


The Dynamical Environment about Small NEAs

19

3


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


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


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


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


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


Orbital Stability about NEAs

25

4


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


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


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



• 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

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0

1

2

3

4

5

6

7

8

9

10

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

SRP

ratio

bet

wee

n pe

rihel

ion

and

aphe

lion

Asteroid orbit eccentricity



• 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


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 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

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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

32



• Formulation:

• Which has a Jacobi integral:

33



• 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



• Equilibrium conditions:

• As then

• As then

35

Orbital plane

Asteroid

To sun

acc (gravity) acc (SRP)

Orbiter



• Analysing the stability of the solutions, one obtains this

condition: or:

• Which represents a ~43% of the maximum equilibrium

distance

36

0.0

0.5

1.0

1.5

2.0

2.5

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50

Circ

ular

orb

it ra

dius

(km

)

Orbital offset (km)

Example for:

= 10-9 km3/s2

g = 10-10 km/s2

Instable branchStable branch

Locus of equilibrium circular orbits

Maximum equilibrium offsetAsteroid

point mass


SRP Dominated MotionPoint 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


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



• 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

d~ˆ



• 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

g/

constant

2

0



• 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 ):



• 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

e d h z

e z h d



• 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



• 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)


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



• 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:



• 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:



• 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:


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


Application to Space Missions

50

5


Application to Missions

• Points of equal SRP and central gravity acceleration for current

and future missions

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1.E-12

1.E-11

1.E-10

1.E-09

1.E-08

1.E-07

1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04

Acc

eler

ation

(km

/s2 )

Distance to the small body (km)

Gravity (10 m)Gravity (100 m)Gravity (1 km)Gravity (10 km)SRP (low)SRP (medium)SRP (high)

EROS - NEAR

Itokawa - Hayabusa

Rosetta - CGBennu - Osiris-REX


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


The Hayabusa case

• Being a low-thrust mission to Itokawa, solar panels were

comparatively large

• The obtained value of amax 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


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


Application to MissionsDon 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


Application to MissionsDon Quijote mission study

56

X vs. Y coord. (inertial trajectory) X vs. Y coord. (rotating trajectory)

X vs. Z coord. (inertial trajectory) X vs. Z coord. (rotating trajectory)

Y vs. Z coord. (inertial trajectory) Y vs. Z coord. (rotating trajectory)

Semi major axis evolution for minimal and maximal initial boundary altitude

Eccentricity evolution for minimal and maximal boundary altitudes

Inclination evolution for minimal and maximal boundary altitudes

Asteroid distance (rotating trajectory)


Application to MissionsProba-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


Application to MissionsProba-IP mission study

58

• Apophis– No altitude range

guarantees safety for every rotationalstate

• 1989 UQ– Safe orbits between

1.1 km and 3 km

• 2001 CC21– Safe orbits between

3.5 km and 16+ km


Conclusions

59

6


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


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

orbital dynamics about small bodies

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