orbital dynamics about small bodies
DESCRIPTION
Orbital Dynamics About Small Bodies. Stardust Opening Training School. University of Strathclyde, 21 st November 2013 Juan L. Cano, ELECNOR DEIMOS, Spain. Relevant Items. Small bodies and NEAs Past and current missions to small bodies The dynamical environment - PowerPoint PPT PresentationTRANSCRIPT
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
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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
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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
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Small Bodies
• Asteroids and comets
• Lecture centred on
NEAs
• Perihelion < 1.3 AU
• …and particularly on
very small NEAs
• Size < few km
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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
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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
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Image of the Chelyabinsk event
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Increase in discovery of NEAs
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Start of the SpaceGuard
Survey in USA
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Current knowledge on NEA population
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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
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
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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
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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
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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
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Image Credit: NASA/Advanced Concepts Lab
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
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• Deep Impact
• Rosetta
Images Credit: NASA
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/cm3, 6.69E+15 kg
S type, rotation period of 5.27 h
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Images Credit: NASA
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/cm3, 3.58E+10 kg
S-type, rotation period of 12.13 h
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Image Credit: JAXAImage Credit: J.R.C. Garry
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
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Images Credit: ESA
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
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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
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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
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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
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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
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Images Credit: D. Scheeres
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?
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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
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SRP Dominated Motion
• These motions are typically analysed in a reference frame
rotating as the asteroid moves about the Sun
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• 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
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
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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
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
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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
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SRP Dominated MotionPoint masses + Constant acceleration problem
• Formulation:
• Which has a Jacobi integral:
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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
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SRP Dominated MotionPoint masses + Constant acceleration problem
• Equilibrium conditions:
• As then
• As then
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Orbital plane
Asteroid
To sun
acc (gravity) acc (SRP)
Orbiter
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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
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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
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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
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Image Credit: D. Scheeres
DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U.
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
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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
d~ˆ
DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U.
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
g/
constant
2
0
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SRP Dominated MotionGeneral 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 ):
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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
e d h z
e z h d
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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
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SRP Dominated MotionGeneral 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)
DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U.
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
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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:
DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U.
SRP Dominated MotionStability 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:
DMS-DQS-SUPSC03-PRE-10-E © DEIMOS Space S.L.U.
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:
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
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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
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
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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
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• 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
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
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Image Credit: D. Scheeres
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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
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Application to MissionsDon Quijote mission study
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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)
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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
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Application to MissionsProba-IP mission study
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• 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
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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
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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
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