Searching for deviations from
General Relativity near
gravitational Saddle Points with
LISA Pathfinder
Christian Trenkel and Steve Kemble
Astrium, Stevenage, UK
Joao Magueijo and Neil Bevis
Imperial College, London, UK
Fabrizio deMarchi and Giuseppe Congedo
University of Trento, Trento, Italy
Overview
Motivation
LISA Pathfinder as Gravitational Laboratory
Gravitational Saddle Points in the Sun-Earth-Moon System
Potential Trajectories for LISA Pathfinder
MOND/TEVES tests with LISA Pathfinder
Summary and Conclusions
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Marcel Grossman 12 – Paris, 2009
Motivation
LISA Pathfinder is becoming a reality as we speak – not some
distant, proposed, potential mission
LISA Pathfinder nominally intended as “mere” Technology
Demonstrator for LISA – but can we also do some Science with it?
LISA Pathfinder total mission cost O(108€) – there is a “moral
obligation” to exploit it to the maximum
Timescale until another “LISA Pathfinder” comes along O(10years) –
so there is also a strong practical incentive to exploit it
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Marcel Grossman 12 – Paris, 2009
Motivation
The scientific community is therefore strongly encouraged to
propose science to be done with LISA Pathfinder!
One obvious constraint: LISA Pathfinder is being built at the
moment, and it has to demonstrate the technology for LISA
Any proposal has to be based on LISA Pathfinder “as is”, and is also
not allowed to interfere with the nominal mission
One could say LISA Pathfinder is “a solution looking for a problem”
Here we present one interesting “problem” that we think LISA
Pathfinder can solve:
to search for deviations from General Relativity (one in
particular) by flying through gravitational Saddle Points
following the nominal mission
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Marcel Grossman 12 – Paris, 2009
LISA Pathfinder as Gravitational Laboratory
LISA Pathfinder and its Payload will offer the following (see
ESA-SCI(2007)1):
Differential Force Measurement Sensitivity:
1.3x10-14N / Hz around 1mHz
Drag-Free Platform Stability:
Platform Free-Fall Quality of
10-13ms-2/ Hz around 1mHz
10-9ms-2 at DC
Gravity Gradiometer Sensitivity:
1.5x10-14s-2/ Hz around 1mHz
We want to
exploit this!
… and more…
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Marcel Grossman 12 – Paris, 2009
Gravitational SPs in the Sun–Earth–Moon System
Defined by zero total gravitational field
Not to be confused with Lagrangian points L1, L2, etc
SPs are not a stable location for spacecraft
In the Sun-Earth-Moon system, there are two SPs:
Earth
Sun
Moon
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Marcel Grossman 12 – Paris, 2009
Gravitational SPs in the Sun–Earth–Moon System
The Sun-Earth SP is perturbed by the motion of the Moon:
12000km
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Marcel Grossman 12 – Paris, 2009
Gravitational SPs in the Sun–Earth–Moon System
The Earth-Moon SP is hugely perturbed by the Sun (maybe it
should more accurately be called the Sun-Moon SP):
Sun “orbits” the EarthMoon system
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Marcel Grossman 12 – Paris, 2009
Gravitational SPs in the Sun–Earth–Moon System
Should we target the Sun-Earth or the Earth-Moon SP?
Gradients around Earth-Moon SP around 3 times larger than
around Sun-Earth SP – if region with certain maximum
gravitational field is to be targeted, the Sun-Earth SP offers
a larger target
But: If a spacecraft is to be flown through the SP region,
then attitude constraints due to Solar Array illumination
requirements are more stringent for the Sun-Earth SP – the
Moon-Earth SP may offer the chance to measure different
gradients
Here we will focus on targeting the Sun-Earth SP
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Marcel Grossman 12 – Paris, 2009
Potential Trajectories for LISA Pathfinder
Nominal LPF orbit is Lissajous orbit around L1
Only very weak stability – in fact over long time scales orbit propagation
is chaotic (deterministic over the time scales of interest here)
Even small dV manoeuvres (as possible with FEEP micropropulsion
system) allow LPF to reach many different destinations
Back-up LPF orbit is Highly Elliptic Orbit (HEO) around Earth
Less room for manoeuvre, given limited micropropulsion system
authority, and Earth-bound orbit
Synchronising HEO with lunar motion may still offer some possibilities
Here we will focus on the nominal LPF orbit as starting point
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Marcel Grossman 12 – Paris, 2009
Potential Trajectories for LISA Pathfinder
Assumptions and approximations entering orbit propagation
dV manoeuvres up to 1m/s have been considered
compatible with residual FEEP control authority following nominal
mission
reasonable timescales for manoeuvres (10-20days)
Orbit propagation time limited to 2 years (cut-off to some extent
arbitrary, but eventually determined by lifetime concerns)
Nominal Solar Radiation Pressure on LPF assumed – in practice this
will be determined in-flight, and the trajectory adapted accordingly
Propagator includes standard gravitational environment
Note: aim is not to find the “exact” trajectory, but to demonstrate that
solutions exist, and to assess some general properties!
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Marcel Grossman 12 – Paris, 2009
Potential Trajectories for LISA Pathfinder
Illustration of the chaotic nature of the problem:
1.5mio km
Earth
Sun
Single dV manoeuvres between 0.5m/s and 1m/s applied at 0.25 day intervals
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Marcel Grossman 12 – Paris, 2009
Potential Trajectories for LISA Pathfinder
42.2
Miss distance (km)
47.5
0.00
0.06
0.11
0.17
0.22
0.28
0.33
0.39
0.44
0.50
31.7
36.9
21.1
26.4
10.6
tim e (days)
100000
90000
80000
70000
60000
50000
40000
30000
20000
10000
0
15.8
0.0
5.3
SP miss distances from a simple two parameter search
dV between 0.5 and 1.0m/s
time of manoeuvre (arbitrary reference point)
90000-100000
80000-90000
70000-80000
60000-70000
50000-60000
40000-50000
30000-40000
20000-30000
10000-20000
0-10000
DeltaV (m /s)
“Best” solutions found this way achieve miss distances of 2000 –
3000 km after transfer times between 450 and 500 days
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Marcel Grossman 12 – Paris, 2009
Potential Trajectories for LISA Pathfinder
Summary of first Iteration
Chaotic nature of the problem makes it difficult to find the “right”
local minimum to search for
The main challenge to targeting the SP is to remove the out-of
ecliptic component of LISA Pathfinder motion
In general, smallest miss distances are found for longer
timescales – eg 30000km are obtained after 340days, but 20003000km require 450-500day transfers and more than one
apogee
Although this cannot be demonstrated conclusively, this seems
to be the general trend
One family of solutions however has been found that bucks this
trend…
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Marcel Grossman 12 – Paris, 2009
Potential Trajectories for LISA Pathfinder
Trajectories including Lunar fly-bys
Lunar fly-bys can be specifically targeted to amplify the effect of
any active dV manoeuvre
This amplification effect applies to both the magnitude and the
direction of the dV manoeuvre
Out-of ecliptic motion can be “killed” much sooner
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Marcel Grossman 12 – Paris, 2009
Potential Trajectories for LISA Pathfinder
Trajectory with lowest miss distance found:
Miss distance 600km
Transfer time from nominal orbit departure 410days
Lunar flyby (60000km) after 300days
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Marcel Grossman 12 – Paris, 2009
Potential Trajectories for LISA Pathfinder
Lunar fly-bys have the following potential spin-offs:
Fly-by could be used as direct absolute calibration for the
gradiometer, by providing the external gravity gradient due to the
Moon
If we are very lucky, we could fly through both the Earth-Moon
and the Sun-Earth SPs – unfortunately highly unlikely!
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Marcel Grossman 12 – Paris, 2009
Potential Trajectories for LISA Pathfinder
Summary and Conclusions
It is possible to fly LISA Pathfinder through the region around the
Sun-Earth SP following the nominal mission, based on the
residual propulsion system (FEEP) control authority
The most promising trajectories combining low miss distances
and acceptable transfer times include a lunar flyby
Best result found (by chance) has 600km miss distance after
410days
In practice, the real trajectory will be flown iteratively through
continuous navigation and trajectory correction manoeuvres –
actual miss distances of the order of 10-20km can be achieved
Only one SP region crossing event possible – any experiment
will be a “one-off”
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Marcel Grossman 12 – Paris, 2009
MOND/TEVES Tests with LISA Pathfinder
Brief background to MOND / TEVES:
Newtonian dynamics are modified within composite system when
centre-of-mass of system falls below 10-10ms-2 (Milgrom 1983)
Phenomonological formula with no underlying theory
Extremely successful in describing galactic rotation curves without
Dark Matter
Relativistic theory (TEVES) developed with non-relativistic MOND
limit (Bekenstein 2004)
Remains controversial (Saturn rings, Bullett cluster)
Prospects for tests within Solar system poor, eg solar acceleration
at 1AU still 6x10-3ms-2
But: Saddle points may offer opportunities (Bekenstein and
Magueijo 2006)
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Marcel Grossman 12 – Paris, 2009
MOND/TEVES Tests with LISA Pathfinder
TEVES predicts anomalous gravity gradients near the Saddle
Points (Bekenstein and Magueijo 2006)
Size and location of “bubble” in which gradients become
significant (unperturbed, considering Sun and Earth only):
SP miss distance should
be less than a few 100km –
ok!
1532km
766km
259000km
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Marcel Grossman 12 – Paris, 2009
MOND/TEVES Tests with LISA Pathfinder
TEVES gradients have been calculated for the 3 body problem in
3D – brief numerical method description:
In the non-relativistic limit, TeVeS yields:
∇(Φ + φ)
a = -∇
(Φ is the Newtonian potential and φ is an additional scalar field)
This new scalar field obeys the equation:
G
∇⋅[ µ( |∇φ| / a0 ) ∇φ ] = 4
(note the presence of the non-linear function µ and the
parameters
and a0)
Equation for φ solved for a cube volume containing the saddle
point but large enough that the situation is Newtonian (φ =
Φ)
on the boundary
Use relaxation algorithm to solve for φ
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Marcel Grossman 12 – Paris, 2009
MOND/TEVES Tests with LISA Pathfinder
Prediction of anomalous gravity gradients that LISA Pathfinder
will see:
Numerical method yields cube volume with gradients at its grid
points (as a function of Sun, Earth and Moon position)
Representative LPF trajectory is propagated through the volume
and the anomalous gradients are extracted at each point:
Sun
Sun-Earth SP
45
45
Earth
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Marcel Grossman 12 – Paris, 2009
Typical LPF
Trajectory
MOND/TEVES Tests with LISA Pathfinder
Prediction of anomalous gravity gradients that LISA Pathfinder
will see
Only gradients in the sensitive direction of the LPF gradiometer
are relevant
In principle, there is a choice of orientation for LPF around the
Earth - Sun axis (in practice the difference in gradients is small):
Test Mass
Earth
Sun
Test Mass
Test Mass
Earth
or
Sun
Test Mass
Solar Array
Finally, the spacecraft speed (typically around 1-1.5km/s) is used
to predict the temporal gradient variations
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Marcel Grossman 12 – Paris, 2009
MOND/TEVES Tests with LISA Pathfinder
Results
Presented for new Moon case – no significant difference found for
other Moon positions (except SP shift as shown before) – robust
signal prediction
Anomalous TEVES gradients as a function of miss distance:
0km
50km
100km
400km
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Marcel Grossman 12 – Paris, 2009
MOND/TEVES Tests with LISA Pathfinder
Results
Of course LPF will see the (much larger) Newtonian background
gradient as well. For the 50km miss distance, it will see:
Newtonian only
Newtonian + TEVES
Note: TEVES signal
roughly at 1/1000s =
1mHz
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Marcel Grossman 12 – Paris, 2009
MOND/TEVES Tests with LISA Pathfinder
Results
Will the LPF gradiometer have adequate sensitivity to detect the
TEVES gradient? We can compare the amplitude spectral density
of the signal abovePower
with the
predicted
noiseinspectral
in Signal
/ Power
Noise density:
20 around 1mHz!
Newtonian
TEVES
LPF Sensitivity
LPF should have more than adequate sensitivity!
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Marcel Grossman 12 – Paris, 2009
MOND/TEVES Tests with LISA Pathfinder
Results
For completeness: TEVES signal spectral density for various miss
distances
0km
50km
100km
400km
LPF Sensitivity
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Marcel Grossman 12 – Paris, 2009
Summary and Conclusions
With regard to solving the original “problem”, the following
conclusions can be drawn:
LISA Pathfinder can be flown through the Sun-Earth Saddle Point
following the nominal mission, and miss distances of 10-20km will
be achievable for transfer times of order 400days
The predicted TEVES signal is robust against lunar position, and
in conjunction with the spacecraft speed falls precisely in the mHz
region – ideally suited to LISA Pathfinder
If LISA Pathfinder achieves its nominal performance, the
predicted anomalous TEVES gradients could easily be detected
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Marcel Grossman 12 – Paris, 2009
Summary and Conclusions
Having demonstrated the feasibility of the proposal, a few
questions remain (feedback welcome and requested):
Is the scientific motivation for such a MOND/TEVES test in itself
strong enough? The mission would be extended by just over a year,
for an experiment lasting around 1000s – at some financial cost
Are there other meaningful GR tests that would benefit from the
special gravitational environment found around SPs? If so, this
might tip the balance!
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Marcel Grossman 12 – Paris, 2009