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On the possibility of detecting some terms of the diurnal polar motion by the study of satellite orbits
Adv. Space Res. Vol. 6. No. 9. pp. 3~36.19S6
O27~1177. 86 SO.(X~+ .50 Copynght © COSPAR
Printed in Great Britain. All rights reserved.
ON THE POSSIBILITY OF DETECTING SOME TERMS OF THE DIURNAL POLAR MOTION BY THE STUDY OF SATELLITE ORBITS Daniel Gambis Bureau International de I’Hetire, Observatoire de Paris, 61 Avenue de I’Observatoire, F-75014 Paris, France
ABSTRACT The high precision of satellite laser ranging data obtained by the laser stations and the improvement of the models of forces acting on the satellite permit us to estimate the pole position with a few milliarcsecond precision. In addition to the well—known precession and nutation in a space—fixed system the earth’s rotation axis is affected by a diurnal motion which is superimposed to the polar motion in the terrestrial system. However, the way the pole position is usually estimated by determining a constant value and assuming a linear drift over a few days, the diurnal polar motion is averaged to zero. After some study of the geometry of the system and the relation between the diurnal polar motion and the satellite orbit, we have performed a series of simulations to see if this variation can be separated from variations in the orbit. The main result is that a specific quasi—diurnal term (corresponding to the fortnightly term in space) can be estimated with a milliarcsecond accuracy. Due to the effect of the tides at the same frequency, no significant value has been determined using real data. I. INTRODUCTION The diurnal polar motion or forced diurnal nutation, is (like precession—nutation) a forced effect of the earth’s rotation axis but one that appears in the earth and consequently is superposed on the polar motion. This diurnal polar motion is in fact the addition of several terms which have been theoritically determined (Woolard 1953, Mc Clure 1973) but poorly observed — a strictly diurnal term with a one sidereal day period — various terms with quasi—diurnal terms of nearly one
sidereal
day period
The total amplitude of the variation ranges from a few centimeters to about 60 cm.
ii
D. Gambis
34
Let us see at first how the diurnal polar motion appears in the course of the transformation between the earth—fixed and the space—fixed systems in which the station coordinates and the motion of the satellite are respectively expressed. II. GEOMETRICAL CONFIGURATION Let (X,Y,Z) and (x,y,z) terrestrial systems. The following matrices product
(~) =
be point coordinates in the space and the transformation can be represented as the
(~)
R~R~R0 ~
where R~and R~ precession and nutation matrix with the usual notations. R0 : matrix corresponding to the rotation of the sidereal time R~~ : matrix corresponding to the rotation due to the pole
components Over some days we can consider the pole components to be the sum of a constant value, a linear drift and the expansion of the diurnal polar motion limited here to the main terms of amplitude a, and argument a u = u0 + u1 (t —.t0 ) +~a~ sin (9+a1 ) v v0 + v1 (t — to ) —ia; cos (O+a1 ) for any nearly diurnal term = a,sin (0 +a1 ) ~i2 = — a1 cos ( 0 + a1 ) with a, = 1 + k,2 F + k13 D + k140 linear combination of luni—solar motions arguments. In the transformation matrix we can easily show that R~(A~ L~v • ~e ) R0 R71,~ = R~(A~ ,Au —q11 ,AE ÷q.2
) R0
In conclusion the effect of the diurnal polar motion is not separable from errors in the nutation model. That means that it will not be possible, in the best cases, to determine the diurnal polar motion of any quasi—diurnal term but only the sum of the error on nutation and the tern itself. III.RELATION BETWEEN DIURNAL POLAR MOTION AND THE ORBITAL ELEMENTS In orbit computation, precession and nutation are considered to be known, so the diurnal polar motion, if not taken into account in the solution, will affect the realization of the inertiel system, in other words th~angular orbital elements determining the orientation of the orbital plane U and i.
)
Let (i,, Q~) and (ii, U~ the node respectively in systems.
&i= i~ I, ~, —Q~ —
=
—
=
—
u sin (Q~—~ )
be the
the inclination and the longitude of terrestrial and in the instantaneous
v cos (O~ 0 ) u cos (Q~—~ ) cotg i. v sin (U, —
—
—
—
0) cotg 1,
These differences show periodic terms with a period of one sidereal day for constant value of the pole.
Diurnal Polar Motion Terms
If we introduce a quasi—diurnal term in these expressions we get = ~ cos ( U, + a, )
~a1
sin ( U, + a~) cotg i, Beside a small dynamical effect, the results are: 1 — A constant effect affecting the orbital elements 2 — For the quasi—diurnal terms long periodic variations appearing with the same periods as those of nutation. =
The first effect will not be detected. The analysis of Ai and A~) can theoretically lead to the determination of the transformed terms of the quasi diurnal term in space. Let us see now how these terms can be determined and separated from the polar motion. We have performed here some simulations in order to see which terms could be us.efully observed. IV.SIM1JLATIONS. POSSIBILITY OF DETERMINING THE AMPLITUDE OF SOME SPECIFIC TERMS OF THE DIURNAL POLAR VARIATION As usual, the simulations are performed intwo steps: 1 — Creation of the “pseudo—data” Taking into account a complete model of forces, a lageos orbit is computed over 20 or 30 days. Nutation model is Woolard’s for a rigid earth. The pole components introduced are assumed to be the sum of a constant and a periodic term arbitraly ranging from 12 h to one sidereal day. t U
=
u,~ + asin
—
P t
v
v0
(a is an arbitrary selected amplitude) P The network of stations consists of 15 well—distributed stations; the data generated every 3 mm are corrupted by a white noise with a 5 cm root—mean square magnitude. =
—
a cos
—
2 — Estimation of the amplitude In this step, the stimulated data are processed as though they were real data; beside the six initial orbital elements, we estimated two pole components ~ and ~ and the amplitude ~ for the period P fixed to the successive values input in the first step. The results were as follows: First the values estimated were equal to u0 and v0 to a 1O~ precision. As could be expected the precision with which is estimated the amplitude a is good far from the sidereal day period and degrades towards one sidereal day (fig.1). Among the various quasi—diurnal terms, only those of 22h 18mm corresponding to the fortnightly period in space, can be determined with a 1 mas precision, the others being too small in amplitude or too long in period. This positive result allows us to try to determine this term, using real data. For all these determinations, Wahr’s model has been used. In this model the reference pole is the Celestial Ephemeris Pole (CEP) which is assumed to have no diurnal motion with respect to the terrestrial reference frame (Seidelman 1982). Thus what we expect to determine are the possible errors in the value of the fortnightly term of nutation. However at the period of about 13.66d. the tides
35
36
D. Gambis
effects (in particular M2 and 01) perturbe the orbit. Using real data it doesn’t seem possible to separate the errors on the tides perturbation model and the searched fortnightly term; the correlation is about .8 between the estimated amplitude, the inclination and the longitude of the node of the satellite. The tentative of determination of this amplitude over several one—month arcs failed in the sense that no significant result has been obtained.
AMPLITUDE A
(“)
~aa-a
10
15
20
22 23
HOUR
Fig.1—Results of the simulations: estimated—initial value V.CONCLUSION We have shown that the strictly diurnal term is not detectable by orbit computation. The term of 22h 18mm corresponding to the fortnightly term in space can be estimated with a 1 mas precision. The lageos orbit is perturbed by the tides due to the luni—solar forces into the earth. Its orbit errors will therefore have many of the same periodicities as the nutation errors. Separation between the two sources of errors is difficult and so far no significant result has been obtained using real data. REFERENCES P. McCLURE : 1973, “Diurnal polar motion”, GSFC X592 — 73 —259 P.K. SEIDELMANN : 1982, “IAU theory of Nutation, of the IAU working group on Nutation”, Celes. Mech. 27 , 79—106. E.W. WOOLARD : 1953, “Theory of the Earth around its center of mass”, Astron. Pap. Amer Eph. , vol. 15 , part.1.
O27~1177. 86 SO.(X~+ .50 Copynght © COSPAR
Printed in Great Britain. All rights reserved.
ON THE POSSIBILITY OF DETECTING SOME TERMS OF THE DIURNAL POLAR MOTION BY THE STUDY OF SATELLITE ORBITS Daniel Gambis Bureau International de I’Hetire, Observatoire de Paris, 61 Avenue de I’Observatoire, F-75014 Paris, France
ABSTRACT The high precision of satellite laser ranging data obtained by the laser stations and the improvement of the models of forces acting on the satellite permit us to estimate the pole position with a few milliarcsecond precision. In addition to the well—known precession and nutation in a space—fixed system the earth’s rotation axis is affected by a diurnal motion which is superimposed to the polar motion in the terrestrial system. However, the way the pole position is usually estimated by determining a constant value and assuming a linear drift over a few days, the diurnal polar motion is averaged to zero. After some study of the geometry of the system and the relation between the diurnal polar motion and the satellite orbit, we have performed a series of simulations to see if this variation can be separated from variations in the orbit. The main result is that a specific quasi—diurnal term (corresponding to the fortnightly term in space) can be estimated with a milliarcsecond accuracy. Due to the effect of the tides at the same frequency, no significant value has been determined using real data. I. INTRODUCTION The diurnal polar motion or forced diurnal nutation, is (like precession—nutation) a forced effect of the earth’s rotation axis but one that appears in the earth and consequently is superposed on the polar motion. This diurnal polar motion is in fact the addition of several terms which have been theoritically determined (Woolard 1953, Mc Clure 1973) but poorly observed — a strictly diurnal term with a one sidereal day period — various terms with quasi—diurnal terms of nearly one
sidereal
day period
The total amplitude of the variation ranges from a few centimeters to about 60 cm.
ii
D. Gambis
34
Let us see at first how the diurnal polar motion appears in the course of the transformation between the earth—fixed and the space—fixed systems in which the station coordinates and the motion of the satellite are respectively expressed. II. GEOMETRICAL CONFIGURATION Let (X,Y,Z) and (x,y,z) terrestrial systems. The following matrices product
(~) =
be point coordinates in the space and the transformation can be represented as the
(~)
R~R~R0 ~
where R~and R~ precession and nutation matrix with the usual notations. R0 : matrix corresponding to the rotation of the sidereal time R~~ : matrix corresponding to the rotation due to the pole
components Over some days we can consider the pole components to be the sum of a constant value, a linear drift and the expansion of the diurnal polar motion limited here to the main terms of amplitude a, and argument a u = u0 + u1 (t —.t0 ) +~a~ sin (9+a1 ) v v0 + v1 (t — to ) —ia; cos (O+a1 ) for any nearly diurnal term = a,sin (0 +a1 ) ~i2 = — a1 cos ( 0 + a1 ) with a, = 1 + k,2 F + k13 D + k140 linear combination of luni—solar motions arguments. In the transformation matrix we can easily show that R~(A~ L~v • ~e ) R0 R71,~ = R~(A~ ,Au —q11 ,AE ÷q.2
) R0
In conclusion the effect of the diurnal polar motion is not separable from errors in the nutation model. That means that it will not be possible, in the best cases, to determine the diurnal polar motion of any quasi—diurnal term but only the sum of the error on nutation and the tern itself. III.RELATION BETWEEN DIURNAL POLAR MOTION AND THE ORBITAL ELEMENTS In orbit computation, precession and nutation are considered to be known, so the diurnal polar motion, if not taken into account in the solution, will affect the realization of the inertiel system, in other words th~angular orbital elements determining the orientation of the orbital plane U and i.
)
Let (i,, Q~) and (ii, U~ the node respectively in systems.
&i= i~ I, ~, —Q~ —
=
—
=
—
u sin (Q~—~ )
be the
the inclination and the longitude of terrestrial and in the instantaneous
v cos (O~ 0 ) u cos (Q~—~ ) cotg i. v sin (U, —
—
—
—
0) cotg 1,
These differences show periodic terms with a period of one sidereal day for constant value of the pole.
Diurnal Polar Motion Terms
If we introduce a quasi—diurnal term in these expressions we get = ~ cos ( U, + a, )
~a1
sin ( U, + a~) cotg i, Beside a small dynamical effect, the results are: 1 — A constant effect affecting the orbital elements 2 — For the quasi—diurnal terms long periodic variations appearing with the same periods as those of nutation. =
The first effect will not be detected. The analysis of Ai and A~) can theoretically lead to the determination of the transformed terms of the quasi diurnal term in space. Let us see now how these terms can be determined and separated from the polar motion. We have performed here some simulations in order to see which terms could be us.efully observed. IV.SIM1JLATIONS. POSSIBILITY OF DETERMINING THE AMPLITUDE OF SOME SPECIFIC TERMS OF THE DIURNAL POLAR VARIATION As usual, the simulations are performed intwo steps: 1 — Creation of the “pseudo—data” Taking into account a complete model of forces, a lageos orbit is computed over 20 or 30 days. Nutation model is Woolard’s for a rigid earth. The pole components introduced are assumed to be the sum of a constant and a periodic term arbitraly ranging from 12 h to one sidereal day. t U
=
u,~ + asin
—
P t
v
v0
(a is an arbitrary selected amplitude) P The network of stations consists of 15 well—distributed stations; the data generated every 3 mm are corrupted by a white noise with a 5 cm root—mean square magnitude. =
—
a cos
—
2 — Estimation of the amplitude In this step, the stimulated data are processed as though they were real data; beside the six initial orbital elements, we estimated two pole components ~ and ~ and the amplitude ~ for the period P fixed to the successive values input in the first step. The results were as follows: First the values estimated were equal to u0 and v0 to a 1O~ precision. As could be expected the precision with which is estimated the amplitude a is good far from the sidereal day period and degrades towards one sidereal day (fig.1). Among the various quasi—diurnal terms, only those of 22h 18mm corresponding to the fortnightly period in space, can be determined with a 1 mas precision, the others being too small in amplitude or too long in period. This positive result allows us to try to determine this term, using real data. For all these determinations, Wahr’s model has been used. In this model the reference pole is the Celestial Ephemeris Pole (CEP) which is assumed to have no diurnal motion with respect to the terrestrial reference frame (Seidelman 1982). Thus what we expect to determine are the possible errors in the value of the fortnightly term of nutation. However at the period of about 13.66d. the tides
35
36
D. Gambis
effects (in particular M2 and 01) perturbe the orbit. Using real data it doesn’t seem possible to separate the errors on the tides perturbation model and the searched fortnightly term; the correlation is about .8 between the estimated amplitude, the inclination and the longitude of the node of the satellite. The tentative of determination of this amplitude over several one—month arcs failed in the sense that no significant result has been obtained.
AMPLITUDE A
(“)
~aa-a
10
15
20
22 23
HOUR
Fig.1—Results of the simulations: estimated—initial value V.CONCLUSION We have shown that the strictly diurnal term is not detectable by orbit computation. The term of 22h 18mm corresponding to the fortnightly term in space can be estimated with a 1 mas precision. The lageos orbit is perturbed by the tides due to the luni—solar forces into the earth. Its orbit errors will therefore have many of the same periodicities as the nutation errors. Separation between the two sources of errors is difficult and so far no significant result has been obtained using real data. REFERENCES P. McCLURE : 1973, “Diurnal polar motion”, GSFC X592 — 73 —259 P.K. SEIDELMANN : 1982, “IAU theory of Nutation, of the IAU working group on Nutation”, Celes. Mech. 27 , 79—106. E.W. WOOLARD : 1953, “Theory of the Earth around its center of mass”, Astron. Pap. Amer Eph. , vol. 15 , part.1.