Determination of the Ppn Parameter with the Hipparcos Data

Home , Eddington (spacecraft)

DETERMINATION OF THE PPN PARAMETER WITH THE HIPPARCOS DATA

1 1 2

M. Frschl e , F. Mignard , F. Arenou

Observatoire de la C^ote d'Azur, CERGA, UMR CNRS 6527, Avenue Cop ernic, F06130 Grasse, France

Observatoire de Paris-Meudon, DASGAL, URA CNRS 335, F92195 Meudon Principal Cedex, France

ABSTRACT Assuming that the de ection for all stars is of the

form:

1 +

mas 4

In the pro cessing of the Hipparcos data, relativistic

e ects in the propagation of light, like the ab erra-

tion and the b ending of light rays, were intro duced

could be known to b etter than 0.001. Unfortu-

at an early level in the mo delling. Thanks to the

nately, the correlation between the global parallax

accumulation of very accurate measurements of star

zero p oint and is very large 0:92 and increases

p ositions at various elongations from the Sun, it was

the exp ected formal error of by an estimated factor

p ossible to assess byhowmuch the PPN parameter

of ab out 2.5, so that it b ecomes 0.003. However, such

deviates from its General Relativityvalue. A series

a result would rank among the b est determinations

of tests has b een implemented to determine this pa-

obtained byvarious means.

rameter and evaluate the magnitude of the p ossible

sources of errors. From the results we can conclude

A fairly large number of determinations of have

that the co ecient is within 0.3 p er cent of unity,

app eared in the past twentyyears Will 1981, 1984,

with =0:997 0:003. This value is not signi cantly

1987 involving b oth the b ending of light, the Shapiro

di erent from unity, as predicted by General Relativ-

time delay in the solar system or the motion of the

ity.

Mo on.

Concerning the measurement of the b ending in the

visible from the Hipparcos data, it is simply a rep-

Keywords: Astrometry, General Relativity.

etition at large angles of the celebrated 1919 ex-

p editions to Brazil and to Princip e Island Dyson et

al. 1920, which to ok advantage of a solar eclipse to

observe stars with resp ect to each other in the solar

1. INTRODUCTION

neighb ourho o d. This was followed by a subsequent

comparison to the same stellar eld without the Sun.

Since then, a total of nine solar eclipses have b een

The accuracy of the Hipparcos measurements implied

used for light de ection measurements. In this kind

that the relativistic e ect in the light propagation

of exp eriment, the error is very large and generally

had to b e included in the mo delling of the data reduc-

exceeds 10 p er cent. In addition to the intrinsic di-

tion. This concerned the second order light ab erra-

culty in observing at small distances from the Sun, in

tion which is of the order of 1 mas 1 mas=0.001 arc-

the adverse conditions of an eclipse, the main source

sec and the light de ection whose magnitude is

of uncertainty is the astrometry over two indep en-

4.07 mas at right angles to the solar direction for

dent exp osures of the same eld with two di erent

an observer at one astronomical unit from the Sun.

techniques, involving a delicate calibration of scaling

After a careful calibration of the instrument parame-

factors.

ters, each abscissa on a reference great-circle RGC

had a typical precision of 3 mas for a star of 8{9

The adventofvery{long{base interferometry at radio

mag. Each consortium has generated for the full

wavelengths has pro duced greatly improved determi-

mission ab out 3.5 million abscissae. Thus, ideally,

nations of , thanks to the capability of measuring

if the reduction is without systematic errors and by

the distance b etween radio{sources as they pass very

neglecting all kind of correlations, an unknown an-

close to the Sun. Somewhat akin to Hipparcos in

gular parameter in the abscissa mo delling common

its principle, a recent global pro cessing of the VLBI

to all stars could b e determined with a precision as

data used to monitor the Earth's rotation includes

go o d as:

the determination of from observations carried out

over almost the entire celestial sphere. This yields

=0:0016 mas

Rob ertson & Carter 1984, Rob ertson et al. 1991:

3:5 10

1 +

Based on observations made with the ESA Hipparcos

=1:000 0:001

satellite. 2

In all these observations, the refraction of the radio 133 degrees. At rst glance, this is a drawback,

waves in the solar corona, which b ends the ray more but it allows a b etter randomisation of the sys-

strongly than in the visible, limits the accuracy as tematic errors thanks to the large coverage in

one must mo del this additional b ending to correct elongation;

the data. The Hipparcos data retains the b est of the

This is the rst optical measurement of p er-

twoabove metho ds: the optical wavelength where no

formed without need for solar eclipses;

corona e ect is to b e feared and the accuracy and full

sky coverage of the VLBI technique.

All observations treated have b een made with

the same instrument, well calibrated all over the

In addition to its b ending, a light signal takes a longer

sky and over a p erio d of 37 months.

time to travel a given distance in a gravitational eld

than in the vacuum and the magnitude of this `time

delay' Shapiro 1964 dep ends also on . The analysis

by Reasenb erg et al. 1979 of the radar time-delay

2. THE LIGHT DEFLECTION MODEL

using Mariner 9 and the Viking landers and orbiters

has resulted in:

The general derivation of the equation of light-rays

1 +

in the PPN formalism is given in So el 1989 and

=1:000 0:001

yields the measurable angle b etween two sources in

a gravitational eld as a function of the co ordinates.

Despite the use of a dual-frequency at 2.3 and 8.4

Let us de ne the co ordinate de ection as the an-

GHz, the propagation in the solar corona remains

gle between the unp erturb ed and p erturb ed paths,

the ma jor factor limiting the accuracy.

by:

A global evaluation of the PPN parameters by 1 + sin 2GM

1 =

Hellings 1984 has allowed the use of a large data

rc 2 1 cos

sample of solar system observations: meridian tran-

sits of planets and the Mo on, lunar laser ranging mea-

where G is the gravitational constant, M the solar

surements, radar range measurements on Mercury,

mass, r the distance between the satellite and the

Venus and Mars. The dynamics of all these ob jects

center of the Sun and c the sp eed of light. The angle

and the observations were mo delled in the PPN for-

is the angular distance between the star and the

malism and the b est t yielded:

Sun.

1 +

The e ect of the light de ection on the RGC abscissa

=0:9994 0:0008

is given by:

' = cos

1 1

However the great heterogeneity of the observational

and

material makes the assessment of the true uncertainty

dicult.

2GM 1 + cos " sin

' =

rc 2 1 cos " cos

From the measurementof the rate of change of the

fringe delays of radiosources during a solar o cculta-

tion, Lebach et al. 1995 have obtained:

where " is the latitude of the Sun with resp ect to the

RGC and the abscissa di erence on the RGC b e-

1 +

tween the star and the Sun as shown in Figure 1. In

=0:9998 0:0008

the Hipparcos data pro cessing a value of = 1 has

b een used consistently, so that the true unknown to

where the quoted error includes the standard error

b e used in the analysis of the residuals is 1. The

of the t and accounts for systematic errors brought

geometry of the light de ection and that of the par-

ab out by the source structure and the propagation

allactic e ect are very similar, di ering in direction

e ects.

and in their dep endence in , with = sin for

the parallactic e ect and the expression of Equation 1

Very recently Williams et al. 1996 have made sev-

for the de ection. This di erence proves essential in

eral determinations of PPN parameters from lunar

separating the two e ects in the observation equa-

laser ranging. The test of geo detic precession based

tions. However the separation is not p erfect, despite

on the rotation of the lunar orbit can b e taken as a

the go o d coverage in elongation, and the two e ects

1 p er cent test of . In addition, a careful discussion

remain largely correlated.

of the di erent terms in the lunar motion yields even-

tually =1:000 0:005 as the present LLR result.

3. OBSERVATION EQUATIONS

With the Hipparcos data, the information related to

the is unique in several resp ects:

We have used all the star abscissa residuals result-

ing from the FAST and NDAC solutions, expressed

The measurements are carried out in the visible

in the Hipparcos catalogue system. For each mea-

and are then free from the large b ending of ra-

surement the calibrated standard errors and the cor-

dio waves by the solar corona, which limits the

relation between the two nearly simultaneous mea-

accuracy of the astrometry by VLBI;

surements are available in the Hipparcos and Tycho

Catalogues ESA 1997, Volume 1, Section 2.8. In All the observations are obtained at very large

this data set, wehave selected only the single star's angular distance from the Sun, b etween 47 and 51

% 7

RGC 6

δθ1 δθ2 5 8 χ ψ star 4

Sun 3 @ 2 ε Ecliptic . φ 1 0 (deg) 35 45 55 65 75 85 95

105 115 125 135 145

Figure2. Distribution of the angular distancebetween the

stars and the Sun.

Figure 1. Geometry of the light de ection and stel lar

paral lax. stands for the light de ection and for

1 2

the paral lactic e ect. They di er in direction and in their

analytical dependencein .

4. DETERMINATION OF THE PARAMETER

measurements, which are on the average of b etter

Wehave designed several exp eriments with the data

quality than those of the double and multiple stars.

to assess the reliability of the solution through its

The total number of abscissae used in this analysis

stability according to the mo del tted to the obser-

is 5 429 065. The observation equations express the

vations.

di erence between the observed and computed ab-

scissae in terms of the di erent unmo delled factors.

Presently wehave considered three kinds of terms:

An error in the reference value of ;

4.1. The Exp eriments

A p ossible zero p oint of the parallaxes, common

Exp eriment1: Half of the single stars were used

to all stars and unmo delled in the astrometric

and the parameters tted were restricted to 1

reduction;

and the zero p oint parallax. The distribution of the

An error in the chromaticityvalue.

stars is fairly uniform on the sky and that of the

elongation with resp ect to the Sun is regular b etween

the two extreme p ossible values of 47 and 133 degrees

The corrected abscissa for each star i observed on the

as shown in Figure 2. After the least squares we

j th RGC is:

recover the exp ected correlation b etween and the

zero p oint of the parallaxes.

' = d

ij k

Exp eriment 2: In this exp erimentwehave used

k =1

the same data set, but three general parameters were

considered and solved together with 1. The solu-

with the partial derivatives for the unknown =

tion gives in particular the correlations b etween the

chromatic parameters and . These co ecients are

indeed very small 0.003, 0.012, 0.000.

@' cos " sin

ij j ij

=2:036 mas 3

@ 1 cos " cos

1 j ij

Exp eriment 3: All the stars known to be non-

problem stars in the data reduction were selected,

= cos " sin 4

j ij

which amounts to a sample of 87 382 out of the

117 955 in the catalogue. Only observations on even

RGCs were retained in the least squares solution.

for the global zero p oint error of the parallaxes:

Exp eriment 4: Same data set of stars as in ex-

= V I 0:5 5

p eriment 3, but only observations on o dd RGCs were

used.

=[V I 0:5] 6

Exp eriment 5: Same data set of stars as in ex-

p eriment 3 with all the observations. The unknowns

and

are 1 and the zero p oint of the parallaxes

=[V I 0:5] T T

j 0

Exp eriment6: Same data set of stars and obser-

vations as in exp eriment5. General parameters, zero where T = 1991:25 in the last of the three chromatic

p oint parallax and 1were tted. terms.

Table 1. Summary of the results. The eight columns give

respectively: the number of the experiment, the number of Hipparcos

3 1997

stars in each experiment in 10 , the number of abscis-

d in 10 , the number of independent parameters

sae use 1996 Lunar Laser Ranging

tted to the data, the estimates of 1 and and the

1984 Planetary data set

values of the zeropoint of the paral laxes with their in as.

1991 VLBI

2 3 4 5 6 7 8 1 1979 Time-delay Eclipse

1973

1 44 2.7 2 -0.0049 0.0043 -11 11

2 44 2.7 5 -0.0052 0.0043 -11 11

87 2.7 5 -0.0017 0.0044 -12 11

3 0.94 0.96 0.98γ 1 1.02 1.04 1.06

4 87 2.7 5 -0.0053 0.0043 - 4 11

5 87 5.4 2 -0.0033 0.0031 - 7 8

6 87 5.4 5 -0.0035 0.0031 - 8 8

Figure3. The best determinations of in each technique.

= 0:997 0:003. This is the b est determination

based on the light de ection in the visible and is com-

parable in accuracy to the measurement rep orted by

4.2. Results

Rob ertson & Carter 1984 from the VLBI observa-

tions of the de ection in radio wavelengths. It is still

The main parameters of each exp eriment together

short by a factor three of the b est estimates of , but

with 1 and the formal errors are listed in ta-

should help spur future space astrometric missions

ble 1. It app ears that in no case di ers signi -

like GAIA with an exp ected improvementby at least

cantly from its General Relativityvalue. The formal

two orders of magnitude.

errors dep end primarily on the number of observa-

tions and on the correlations b etween the parallaxes

and . As long as one can consider the formal er-

ACKNOWLEDGMENTS

ror as representative of a true uncertainty, the Hip-

parcos based determination of is the b est ever de-

rived from the light de ection in the visible and the

We are grateful to Lennart Lindegren for his com-

rst to use observations made at wide angles from

ments regarding the imp ortance of the correlation of

the Sun. The comparison of exp eriments di ering

the light de ection with the parallactic e ect.

only by the numb er of adjusted parameters Exp eri-

ments 1{2 and 5{6 shows that the results are rather

insensitive to the intro duction or neglect of general

REFERENCES

parameters which are weakly correlated with the par-

allaxes and , thanks to a wide coverage of the con-

gurations. The formal error is signi cantly b etter

Dyson, F., Eddington, A., Davidson,C., 1920, Mem.

than the results obtained separately by FAST and

Trans. Roy. So c. 220A, 291

NDAC ESA 1997, Volume 3 with their own sphere

ESA, 1997, The Hipparcos and Tycho Catalogues,

solution b efore the astrometric catalogue merging.

ESA SP{1200

Hellings, R., 1984, in General Relativity and Gravita-

The Hipparcos solution is plotted in Figure 3 with

tion, eds B. Bertotti, F. de Felice and A. Pascolini,

the other determinations cited in the Intro duction.

365-85, Reidel.

Although the Hipparcos determination is not as ac-

curate as the b est determinations of based on the

Lebach, D.E., Corey, B.E., Shapiro, I.I., et al. 1995,

Shapiro e ect, it constitutes a rst full pro of of the

Phys. Rev. Letters, 75, 1439

p ossibility of determining the space curvature from

Reasenb erg, R. D., et al., 1979, ApJ, 234, L219

global astrometric measurements that could b e p er-

Rob ertson, D.S., Carter, W.E., 1984, Nature, 310,

formed in the future by a mission like GAIA. More-

572

over, the fact that we reach with Hipparcos a re-

Rob ertson, D.S., Carter, W.E., Dillinger, W.H.,

sult which do es not depart signi cantly from the GR

1991, Nature, 349, 768

value, may be taken as an additional indication of

the absence of systematic bias in the Hipparcos data.

Shapiro, I.I., 1964, Phys. Rev. Lett., 13, 789

So el, M.H. 1989, Relativity in Astrometry, Celestial

Mechanics and Geo desy, 59-65, Springer-Verlag

Will, C.M., 1981, Theory and Exp erimentinGravi-

5. CONCLUSION

tational Physics, Cambridge U.P

Will, C.M., 1984, Phys. Rev., 113, 345

Will, C.M., 1987, Exp erimental gravitation from

A determination of the parameter of the PPN for-

Newton's Principia to Einstein's general relativ-

mulation of the gravitation theory has b een made

ity in 300 Years of Gravitation, eds S. Hawking

by using all the astrometric observations carried out

and W. Israel, Cambridge U.P.

by Hipparcos. Various numerical exp eriments were

designed in order to control the e ect of hidden cor-

Williams, J.G., Newhall, X.X., Dickey, J.O., 1996,

relations on the results and the stability of the so-

Phys. Rev. D, 53, 6730-6739

lution with varying data sets. The nal result gives