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**Astrometric reductions: Analysis of the catalogues used for the astrometric reduction**

**Introduction**

To re-reduct or correct old data for our purpose, it is essential to understand what was done for the reductions and what were the reference frames of the data. This will help us to define the process useful to improve old data. In this chapter, our work consists in finding out the reference system, coordinates, the time scales and catalogs used in old and recent observations, reductions and ephemeris of Phoebe. We will recall the general concept of the astrometric reductions and compare the old catalogs and new catalogs.

**Reference Systems and Frames, Fundamental Astron-omy Coordinates**

The motion of a natural satellite can be expressed as position, velocity and acceleration with time. We can define a position relative to a reference system. It is necessary to give some definitions to secure the frame within which we will work by using each of the observations that will be described.

Old data used for the making of Phoebe’s ephemeris were used until today as published by their authors. Doing so, di↵erent reference frames were mixed making biases in the ephemeris. Our purpose is to make a new reduction of old observations or to make correc-tions to old data. So that we must understand what was done in the past by the observers and how to make now useful corrections. In some cases, we will be able to make a com-plete new reduction and otherwise, only partial corrections. In this section we will prepare tools for the making of a new procedure reduction.

**ICRS, ICRF**

The IAU Working Group on Nomenclature for Fundamental Astronomy has recommended the following definitions for the ICRS and ICRF:

International Celestial Reference System (ICRS): The idealized barycentric coordinate system to which celestial positions are referred. It is kinematically non-rotating with re-spect to the ensemble of distant extragalactic objects. It has no intrinsic orientation but was aligned close to the mean equator and dynamical equinox of J2000.0 for continuity with previous fundamental reference systems. Its orientation is independent of epoch, ecliptic or equator and is realized by a list of adopted coordinates of extragalactic sources.

International Celestial Reference Frame (ICRF): A set of extragalactic objects whose adopted positions and uncertainties realize the ICRS axes and give the uncertainties of the axes. It is also the name of the radio catalogue whose 212 defining sources are currently the most accurate realization of the ICRS. Note that the orientation of the ICRF catalogue was carried over from earlier IERS radio catalogs and was within the errors of the standard stellar and dynamic frames at the time of adoption. Successive revisions of the ICRF are intended to minimize rotation from its original orientation.

The ICRS is a fundamental celestial reference system for high-precision positional as-tronomy. It is meant to represent the most appropriate coordinate reference system for expressing reference data on the positions and motions of celestial objects.

A reference frame is the physical realization of a reference system, i.e., the reference frame is the reported coordinates of datum points. The ICRS is an idealization with defined origin and axis. The ICRF consists of a set of identifiable fiducially points on the sky along with their coordinates, which serves as the practical realization of the ICRF. The ICRF is now the standard reference frame used to define the positions of the planets (including the Earth) and other astronomical objects.

Although the directions of the ICRS coordinate axes are not defined by the kinematics of the Earth, the ICRS axes (as implemented by the ICRF) closely approximate the axes that would be defined by the mean Earth equator and equinox of J2000.0 (to within about 0.02 arcsecond), if the latter is considered to be a barycentric system. Because the ICRS axes are non-rotating, there is no date associated with the ICRS. Furthermore, since the defining radio sources are assumed to be so distant that their angular motions, seen from Earth, are negligible, there is no epoch associated with the ICRF. It is technically incorrect, then, to say that the ICRS is a « J2000.0 system », even though for many current data sources, the directions in space defined by the equator and equinox of J2000.0 and the ICRS axes are the same within the errors of the data.

For other reference frame, Geocentric means that the reference system centre is in the Earth. The Topocentric means the centre is the observer.

The numerical integration of the dynamical model allows the computation of the posi-tions of Phoebe in the same reference system as that of the planetary ephemeris. The plan-etary ephemeris develops and the reference system changes and recently we use ICRS. we have to transform all the observations to the same reference system ICRS in order to com-pare computed and observed positions. The classical transformations have been done with the Software Routines from the IAU SOFA Collection [2010]. Copyright at International Astronomical Union Standards of Fundamental Astronomy (http://www.iausofa.org).

**Type of the coordinates**

The coordinates for observation:

1) The « absolute » equatorial coordinates:

Right Ascension and Declination, the notation is (RA, REC) ou (↵, ). The coordinates are usually defined in terrestrial equatorial system. Most of the observations of Phoebe are in absolute coordinates. The absolute coordinates are determined by calculation in using reference stars’ coordinates in star catalogs. The center of the reference system may be topocentric or geocentric.

2) Separation (arcsec) and Position angle (degree):

Separation is the apparent angular distance between the selected satellite and the refer-ence object. The satellite position angle refers to the reference object counted from North to East. Many satellites near the primary are within this coordinates.

3) (X, Y) Di↵erential Coordinates:

The di↵erence of the equatorial coordinates between satellite and its reference body onto the celestial parallel and celestial meridian mutually intersecting in the reference body which can be an object or the geometrical center for all of them. The notation is (Δ↵, Δ ) or (Δ↵ cos , Δ ).

4)(X, Y) tangential coordinates:

The coordinates measured on the tangent plane of the celestial sphere at the point of reference body. Usually X is measured to the east, Y is measured to the north. Sometimes we call them standard coordinates.

If we know the equatorial coordinate (↵0, 0) of the optical center of the instrument and the equatorial position (↵, ) of a celestial body, we can deduce its tangential coordinates (X,Y) relative to the field center, as follows gnomonic projection:

5) (x, y) measured coordinates :

We note (x, y) for the coordinates measured in the photographic plates or CCD images. Theoretically the measured coordinates should be the same as the tangential coordinates if the measurement is from the center of the image and the reference axis are from west to east for x and from south to north for y. But in reality the optical center is not determined exactly and therefore the position (↵0, 0) is an approximation; the inaccuracy of the focal length of the instrument and the orientation of the plate causes a rotation and uneven scale e↵ect of the reference axis; the distortion of the optic and the e↵ect of the atmosphere. These errors will be corrected by least squares method in the astrometric reduction in using the positions of the reference stars.

The transformation of (x, y) to (X, Y) defined by:

The terms T1(x, y), T2(x, y) are the terms with the order higher than 3. The constants a1, b1, c1, d1, e1, f1, a2, b2, c2, d2, e2, f2, are the characteristic of the plate for each observation called plate constants. If we know these constants we can get objet’s tangential coordinates from the measured coordinates of the objet’s, then with the equatorial coordinate of the optical center, the equatorial coordinate of the objet can be solved out.

We can get equatorial coordinates of the reference stars from star catalog, and with the equatorial coordinate of the optical center, the tangential coordinates (X, Y) are provided. The measurements give the (x ,y) of the reference stars, if we have enough reference stars, with the equations above in using the least squares method through several iterations we can determine the plate constants. We need at least 3 stars to define 6 constants for the first order of the equations, which means we ignore the distortion. At least 6 stars are needed to define 12 constants for the second order and 10 stars are needed to define 20 constants for the third order.

The coordinates for ephemerides:

6) x, y, z, Vx, Vy, Vz vectors:

The rectangular vector coordinates and the velocity of satellite. These coordinates are solved out from the equations of motion and used to calculate the orbital elements. The vectors are converted to absolute equatorial coordinates in order to compare computed and observed positions. The vector can be defined in the planetocentric or barycentric and terrestrial ecliptic or equatorial system. The center of the reference system may be another satellite. The vectors from the ephemeris TASS (Vienne & Duriez [1991], [1992]), an ephemeris of main satellites of Saturn, are in the planetocentric and terrestrial ecliptic reference system. The epoch epoch of ecliptic and equinox is J2000.0. The ephemeris of Phoebe provides the positions vector in Saturn-centric ICRS reference system.

**Diferent reference systems where we get observed coordinates and general change to ICRS**

When we want to compare the observed positions of the satellite with the computed posi-tions which at first are the vectors solved from the dynamical equations, we should trans-form these vectors from planetocentric equator or ecliptic system to the reference systems of the observations. We need to find out the reference system for every observation and for simplifying the process of the vectors transformation, we try to reduce the observations in the same reference system. The following reference systems were used in observations of Phoebe and the number that indicate the reference system will be present in the figure 4-4. For each reference system we introduce the general method to change the coordinates in this reference system to the coordinates in the ICRS reference system. We will introduce a new reduction in Chapter 4 with which we do not need to change the reference system as usual for some old observations. The main idea of this new reduction was published in Desmars et al. [2013] at the beginning of our thesis work and only some of earliest old observations had been reduced in that article.

1) ICRS:

The reference frame of most of the modern catalogs are considered to be very close to ICRF and the coordinates of the stars are in ICRS reference system, such as Hipparcos Catalogue, TYCHO catalogs, UCAC catalogs and USNO catalogs except USNO A1.0. The observations reduced with these catalogs are in ICRS reference system.

The coordinates relating to the reference system of the planetary ephemeris DE4XX / LE4XX or INPOP are also considered to be very close to the ICRS.

2) J2000:

The reference system center is in the barycentre of the Solar System, The coordinates reported at the Earth’s mean equator and equinox in J2000.0. There is a very slight di↵er-ence between the ICRS frame and the J2000 frame as mentioned before, we can use the transformation between J2000 and ICRS of the IERS (http://www.iers.org). The accura-cies of the observation and the ephemeris of Phoebe do not reach to the di↵erence between ICRS and J2000, so we can treat the transformation as an identity in our work. But we still distinguish the notation in the observation table for future work.

3) B1950:

The axes of the reference system are defined by equinox and mean equator of B1950.0 in FK4 catalog. The coordinates are transformed to ICRS with Newcomb precession, in-cluding the elliptical aberration. The parameter from Aoki [1983], Kinoshita [1975] and Smith [1989]

4) Apparent:

Case when the coordinates are in true equator and equinox of date. The coordinates are transformed with the precession model including frame bias, and the notation model adopted by IAU in 2006, the SOFA routines are PB06 and MUT06A.

5) B1900:

The axes of the reference system are defined by equinox and mean Equator B1900.0.

The coordinates are transformed to ICRS with Newcomb precession.

6) Year: The coordinates reported at the mean equator and equinox at 1 January of the year of observation. The coordinates are transformed to ICRS by using the precession model adopted by IAU in 2006. The SOFA routine is PB06. The correction of the elliptical aberration must be done at first for the observations using the catalogues before 1984.

7) B1875:

Equinox and mean Equator B1875.0. The coordinates are first corrected from elliptical aberration, then transformed to mean equator and equinox of the date using the precession value of Newcomb (Kinoshita [1975] ). Finally, the coordinates are transformed to ICRS by using current values of precession provided by IAU 2006.

**Time scales**

The time scale used for observations must very carefully verified and converted if neces-sary. Phoebe is moving about 1.8 km/second so that an error of 10 seconds in the timing of an observation leads to an error of 18 km in space.

The time used for dynamical model should be continuous and uniform. But most of the data to record the old observation time are relative to the local time.

1) Greenwich Mean Time (GMT)

It was originally reckoned from noon to noon. In 1925, some countries shifted GMT by 12 hours so that it would begin at Greenwich midnight. This new definition is used for world time and in the navigational publications of English-speaking countries. The designation Greenwich Mean Astronomical Time (GMAT) is reserved for the reckoning of time from noon (and previously called GMT). Before 1805 the Royal Navy Day started 12 hours before local mean solar time, thus the Royal Navy Day was then 24 hours ahead of GMT.

2) Coordinated Universal Time (UTC)

It was introduced in 1972. Now it is the basis of all civilian time throughout the world. Because most daily life is still organized around the solar day, UTC was defined to closely parallel Universal Time, and UTC is uniform between two leaps while UT1 is based on Earth’s rotation, which is gradually slowing. In order to keep the two times within 0.9 seconds of each other, a leap second is added to UTC about once every 12 to 18 months. The GMT presented in the publications before 1925 is 12h after UTC.

GMT and UTC are not a time continuing, and cannot be used as a dynamical time even they are used to date the observations.

3) UT1 – Universal Time

Universal Time (UT1) is a measure of the actual rotation of the earth, independent of observing location. It is the observed rotation of the earth with respect to the mean sun corrected for the observer’s longitude with respect to the Greenwich Meridian and for the observer’s small shift in longitude due to polar motion.

UT1 is not a uniform time since the rate of the earth’s rotation is not constant, and its diference with atomic time is no predictable. As of December 1995, UT1 was drifting about 0.8 seconds per year with respect to atomic time (TAI or UTC). The di↵erence be-tween UT1 and UTC is never greater than 0.9 since the leap seconds defined for UTC to keep this di↵erence.

DUT1 is published weekly in IERS (International Earth Rotation Service) Bulletin A along with predictions for a number of months into the future.

UT1 is continuous but not uniform, so can’t be used as a dynamical time.

4) GMST – Greenwich Mean Sidereal Time

Sidereal time is the measure of the earth’s rotation with respect to distant celestial ob-jects. Compare this to UT1, which is the rotation of the earth with respect to the mean position of the sun. One sidereal second is approximately 365.25/366.25 of a UT1 second. In other words, there is one more day in a sidereal year than in a solar year.

By convention, the reference points for Greenwich Sidereal Time are the Greenwich Meridian and the vernal equinox (the intersection of the planes of the earth’s equator and the earth’s orbit, the ecliptic). The Greenwich sidereal day begins when the vernal equinox is on the Greenwich Meridian. Greenwich Mean Sidereal Time (GMST) is the hour angle of the average position of the vernal equinox, neglecting short term motions of the equinox due to nutation.

In conformance with IAU conventions for the motion of the earth’s equator and equinox GMST is linked directly to UT1 through the equation.

5) International Atomic Time (Temps Atomique International = TAI)

It is defined as the weighted average of the time kept by about 200 atomic clocks in over 50 national laboratories worldwide. UTC is di↵erent from TAI by changing an integral number of seconds.

6) Terrestrial Dynamical Time (TDT, TD)

It was introduced by the IAU in 1979 as the coordinate time scale for an observer on the surface of Earth. It takes into account relativistic e↵ects and is based on TAI. The time TDT is the atomic time used in the theories of motion for bodies in the Solar System. In 1991, the IAU refined the definition of TDT to make it more precise.

**Table of contents :**

**1 Introduction **

1.1 History of Phoebe

1.2 Chapter introduction

2 Astrometric reductions: Analysis of the catalogues used for the astrometric reduction

**2.1 Introduction **

2.2 Reference Systems and Frames, Fundamental Astronomy Coordinates

2.2.1 ICRS, ICRF

2.2.2 Type of the coordinates

2.2.3 Di↵erent reference systems where we get observed coordinates and general change to ICRS

2.3 Time scales

2.4 Catalogs used for reductions

2.4.1 Catalogs used for old reductions

2.4.2 Modern catalogs

2.4.3 Catalog comparison and the statistics on the used catalogues

2.5 Other corrections for astrometric reduction

3 Search and selection of published observations

**3.1 Introduction **

3.2 Old photographic observations

3.2.1 Observatory and telescopes

3.2.2 Di↵erent sets of observations

3.3 New observations

**4 The re-reduction of old observations**

4.1 Introduction

4.2 Introduction to the used method

4.3 Identifying the old reference stars in new catalogs

4.4 Choosing new catalogs and the case of no information on reference stars

4.5 Catalog bias

4.6 Result

**5 A new ephemeris of Phoebe**

5.1 Introduction

5.2 The numerical model

5.2.1 Perturbations

5.2.2 Equations of motion

5.2.3 Observations used to fit the dynamical model

5.2.4 Numerical integration

5.3 The frequency analysis

5.3.1 Definition of the elements

5.3.2 Developpement of quasi-periodic series

5.3.3 The principle of the fine analysis

5.3.4 Data windowing

5.3.5 The procedure of the frequencies analysis

5.3.6 Identification and synthetic representation

5.3.7 Conclusion

**6 Comparisons and validation of the new ephemeris of Phoebe **

6.1 Observations Comparison

6.1.1 CCD observations calibrated with di↵erent catalogs

6.1.2 Old published photographic observations and the reduced observations

6.2 Ephemeris Comparison

**7 Conclusion and future work**