Transcript National Institute of Standards and Technology, Boulder

Tests of gravitational physics by
ranging to Mercury
Neil Ashby*, John Wahr
Dept. of Physics, University of Colorado at Boulder
Peter Bender
Joint Institute for Laboratory Astrophysics, Boulder
*Affiliate, National Institute of Standards and Technology, Boulder
email: [email protected]
1
Outline
1. History of the present calculation
2. Characterizing the approach
a. Analytical vs. numerical
b. Worst-case systematics
3. The range observable
4. Choice of parameters
a. orbital parameters
b. solar system parameters
c. cosmological parameters
d. relativity parameters
5. Assumptions
6. Results
2
History and purpose
•
•
•
•
Began 1974
1980-1982: NASA funding
1989-1995: various publications, conference talks/proceedings
Most recent results published in Phys. Rev. D 75, 022001-022020 (2007)
applied to BepiColombo mission to Mercury.
•
The purpose is to develop theory and associated computer code to:
support experiments to test alternative gravitational theories;
determine important solar system parameters (e.g. GM , J 2 ).
in ranging experiments between the Earth and
- Mercury
- Mars or Mars & Mercury
- A close solar probe.
3
Characterizing the approach-theoretical
• Orbital perturbations of the planets due to various relativity and/or
cosmological effects are treated analytically to first order.
• The theoretical perturbation expressions are implemented in code for
simulation of ranging missions of varying duration.
4
Characterizing the approach-statistical
The approach is is a “modified worst-case” approach.
This means that errors are presumed to be highly correlated.
Specifically, systematic ranging errors have time signatures that
have the worst possible effect on determination
of final uncertainties of a parameter of interest.
However, systematic errors cannot maximize the uncertainties in
all parameters simultaneously, so we adopt a more
reasonable “modified” approach:
the worst-case uncertainties are divided by 3.
The true worst-case uncertainties can be recovered by multiplying all
quoted errors by 3.
For random uncorrelated errors, the estimated uncertainties can
be recovered by multiplying the quoted errors by 3 N
where N is the number of observations.
5
Parameters--Keplerian Orbital Elements
Range is constructed from Keplerian orbital elements of Earth and Mercury:
a1 , a2  semimajor axes;
e1 , e2  eccentricities;
1 , 2  longitude of perihelion;
0,i 2  inclination;
--,  2  longitude of ascending node;
L10 , L20  initial longitude;
6
Unperturbed range observable
 2  r12  r22  2r1r2 cos 12 ;
cos 12  cos  f1  (1  L10 )  ( 2  L20 )  ( L20  L10 )  
cos[ f 2  ( 2  L20 )  ( 2  L20 )] 
cos I 2 sin  f1  (1  L10 )  ( 2  L20 )  ( L20  L10 )  
sin[ f 2  ( 2  L20 )  ( 2  L20 )] 
a(1  e2 )
r
1  e cos f
9 Orbital Parameters are selected:
a1 , a2 , e1 , e2 ;
1  L10 ,  2  L20 ,  2  L20 , I 2 ;
L20  L10
7
Additional parameters
GM
J2
G
G
product of Newtonian gravitational constant and solar mass
quadrupole moment of the sun
cosmological change in gravitational constant
Number of parameters so far: 9+3 = 12
8
Relativity parameters


1
2
 3


measures nonlinear contribution of gravitational potential to
g 00
measures spatial curvature produced by mass; of interest since
some scalar-tensor theories predict values of order 10-7
preferred frame parameter
preferred frame parameter

speed 377 m/s relative to CMWBR
preferred frame parameter, (deleted from consideration
because it is now very well determined)
Whitehead parameter: solar system -- milky way interaction
Nordvedt parameter: effect of a third massive body on
gravitational interaction of two bodies (violation of
strong equivalence principle)
Total number of parameters: 9+3+6=18
9
Quadrupole Moment of the Sun--J2
Objective: to develop better models of the solar interior, explain
-- energy generation, solar evolution
-- 11-year sunspot cycle
-- neutrino flux
-- …
Some information comes from observations of the surface:
Flattening
Rotation
Helioseismometry
10
Orbital perturbations--solar J2
The effect of the solar quadrupole
moment on orbital elements was
taken from the literature on
Lagrangian planetary perturbation
theory, after checking by numerical
integration.
This sample is from
“Principles of Celestial Mechanics,”
by P. M. Fitzpatrick (Academic Press,
New York (1971).
11
Sample perturbations-
These perturbations are expressed with the help of the integrals
Sij   df  sin f   cos f 
f
i
j
f0


 a  2me 2 S10  2eS11 / 
4
m
2
 e    a / 2ea
GM
c2
F  F  F0
  1  e2
   I  0
 

2
 ~  m  S00  2 S01 / e   (sin f cos f / a

 M   3m 1  e cos f 0

 n (t ) / a 4 
2

 m 2 S01 / e   (sin f cos f / (a ) .
12
Strong equivalence principle violation
The nonlinear effect of the sun’s gravitational self-energy on two falling
bodies (such as Jupiter and Mercury) is described by differential
equations for the radial and tangential perturbations:
G( M  M i )
d 2q i
  GM j


q

i
dt 2
qi3
M c 2 qij 3
qij  qi  q j ,

6 (ratio of sun’s self energy


3.52

10
to rest energy)
M c2
qi is heliocentric position of planet i. The driving term can be expanded in
power series in ratios such as
aearth amercury
,
.
a jupiter a jupiter
13
Strong equivalence principle violation-cont’d
If the planets are coplanar, the equations for radial and tangential
(to the orbit) perturbations can be expanded and expressed in the
form

2
 rr  2i rt  3i  rr   An cos(nSi );
n 1

 rt  2i rr   Bn sin(nSi ),
n 1
where for planet i,
i2 
G( M  M i )
,
ai3
Si  (i   j )t    i t  .
Particular solutions are:
ni An  2i Bn
 rr  
,
2
2
2
n 1 ni ( n i  i )

2ni i An  (n 2i 2  3i 2 ) Bn
 rt  
.
2
2
2
ni (n i  i )
n 1

14
Strong equivalence principle violation-cont’d
Range perturbation (earth-moon)
in meters
The second-order differential equations have solutions that are
superpositions of:
(a) particular solutions
(g) general solutions of homogeneous equations
--i.e., without driving terms.
Numerical solutions pick up contributions including the general solutions
unless the boundary conditions are chosen properly. Example:
for the earth-moon-sun system, the solutions to the differential equations
typically look like this:
It is known that
the lunar range
perturbation is
about 8 m in
amplitude if   1.
Time in days
15
Covariance Analysis &
Worst Case Systematic Error
Ckj 
N 18

i 1
 (ti )  (ti )
d k d j
dl    C
18
j 1
1
(correlation matrix)
 (ti )
lj  d  (ti )
i 1
j
N
(correction to parameter)
where  (ti ) is the range residual, the difference between theoretically
predicted range with nominal values for the parameters, and the measured range.
If errors in the range residuals were random and uncorrelated, such that
 (ti ) (t j )   ij 2 ,
Then it follows that the parameter error would be
 dl   (C 1 )ll .
16
Correlations between  and J2
However, the time signatures of various perturbations are
instead highly correlated. Here is an example.
17
Worst-case analysis
The error in a parameter di could be bigger if the error in the residual
i / d n ,
is correlated with the partial derivative
for some n.
Suppose that over the entire data set we were confident that the rms error in
the residuals could be limited or constrained by:
1
N
  (ti )   2 .
2
i
Then we look for the maximum error in di subject to the above constraint.
 dl   N (C 1 )ll .
Note there is a factor of N. Generally this error decreases but approaches
a limit as the number of observations continues to increase.
18
Error Correlations
If the residuals are such that the mth parameter is most poorly determined,
Then the error in the nth parameter is:
N
1
 dn  
(
C
)nl
1
(C )ll
So the inverse of the covariance matrix contains a huge amount of information,
For the BepiColombo Mission, simulations have been carried out with
19, then 18 parameters.
19
Assumptions
Launch January 1, 2012--Julian Date 2455928.0;
Unperturbed Keplerian elements taken from American Ephemeris;
Known Newtonian perturbations assumed to be removed from data;
Worst-Case uncertainties divided by 3 are presented;
Mission duration is extended to 8 years in the calculation;
Nordtvedt parameter can be treated as independent, or can be
viewed as dependent on other parameters, e.g.,
2
  4     3  1   2
3
(Simulations have been done in this case but are not presented here.)
20
Further assumptions
• One normal range point per day is obtained;
• No a priori knowledge of uncertainties of parameters is assumed;
o
• Data is excluded if the line-of-sight passes within 5 of the sun’s center;
• Systematic range errors are subject to the constraint
1
N
N
2
2
2

(
t
)



(4.5cm)
 i
i 1
21
Perturbation Theory--outline of simulation
1. Use Relativistic equations of motion to obtain perturbing accelerations;
2. Resolve perturbing accelerations into cartesian components: radial (R),
normal to radius in orbit plane (S), normal to orbit plane (T);
3. Integrate Lagrange Planetary Perturbation Equations to find the
perturbed orbital elements (analytical, not numerical);
4. Calculate partial derivatives of the range with respect to each of the
parameters of interest;
5. Construct the covariance matrix each day (for up to 8 years)
6. Invert and calculate the worst-case uncertainties
7. Divide by 3 to get “modified worst-case uncertainties”
22
Systematic error (m)

Time in days
23
Systematic error (m)
G /G
Time in days
24
Systematic error (m)
Comparisons of worst-case systematic errors
G /G

Time in days
Since the “worst-case” systematic error cannot simultaneously be
worst for any two parameters, the worst-case errors are divided by 3.
25
Results for 
5 105
26
Uncertainty in solar quadrupole moment
Calculation results-uncertainty in solar quadrupole moment
Present uncertainty level
If J2 and  are included in the parameter list
--isotropic case
With all 18 parameters
If General Relativity is correct
27
Present uncertainties
in nonorbital parameters
28
Results--assuming GR is correct
with dG/dt
Significant improvements
are obtained after 1 year
in all these parameters.
29
Nonmetric theory results--15 &18 parameters
*
*
*
*
*
*
*
*
*
*
*
*
*
*
* Significant improvement over present uncertainties
30