Guy Artzner (1), Frédéric Auchère (2), Jean-Pierre Delaboudinière (1) and Jean-François Hochedez (3)

(1)Institut d'Astrophysique Spatiale bâtiment 121 Université Paris Sud F 91405 Orsay France
(2)USRA -GSFC NASA Mail code 682,3 Greenbelt Maryland 20771
(3)Hochedez Consultants 43, rue de Constantinople. Paris France

• Abstract.
• Introduction.
• Requirement for and implementation of local length mesurements.
• Calibration results.
• Discussion.
• Conclusion.
• Acknowledgment.
• References.

ABSTRACT

Converting linear coordinates in the plane of the detector of an astronomical instrument to celestial coordinates involves in principle the equivalent focal length of the instrument. However, most methods in astrometry manage to reduce observations in a global manner whitout actually measuring a focal length. We point out a case for solar space observations where the long term stability of angular distance measurements is better than the ground calibration of the angular value of a pixel. We report and discuss this ground calibration.

Key words: astrometry, equivalent focal length, telescopes, calibration, ultraviolet, aberrations, geometry, detectors, mirrors.

1. INTRODUCTION

We indicate how methods in the field of astrometry allow a determination of the equivalent focal length of astronomical instruments.

1.1 absolute astrometry.

The first fundamental visual star catalogs were obtained by individual measurements of each star. The absolute measurement of each coordinate, right ascension and declination, is made according to two different references: with a meridian telescope the declination is measured along graduated circles whereas the right ascension is determined from a clock. The Danjon astrolabe similarly uses two references: the angle of a prism, and a clock. The angle of the prism is not calibrated before observing, but is obtained as a by product of a global reduction of the measurements. The angular accuracy obtained is a small fraction of the angular diameter of the Airy disk for the pointing instrument.

Secondary star catalogs are obtained from sky photographs as the "Carte du Ciel". Let be the angular distance between two stars and d the linear distance measured between their images. When using a radian as angular unit the linear term in the polynomial expression of d as a function of a is measured in units of length. This coefficient depends upon the location in the field of view and upon the direction of the stars. When neglecting the distorsions, due to the optical scheme or due to misalignment and manufacturing defects, this coefficient amounts to the focal length of the telescope. The polynomial coefficients are obtained by fitting linear measurements to celestial coordinates of a few reference stars obtained from a fundamental catalog. In this case the telescope's focal length is a by-product of the data reduction.

Catalogs of the positions of radio-sources are produced by very long baseline interferometry (VLBI), using precisely coordinated atomic clocks as a reference. In the field of interferometric astrometry in the visible range a laser source is used as a length reference. In both cases the Earth's rotation is the other reference used to construct a celestial sphere.

A new stellar catalog has been recently obtained from space observations of the HIPPARCOS mission, using as references the angle of a prism and the angular velocity of a spinning spacecraft. The global accuracy obtained, 0.001", is a very small fraction of the angular size of the diffraction pattern of the observing telescope.

We reviewed five astrometric methods. In each case the construction of a celestial sphere relies upon at least two references: the angular velocity of a solid body, Earth or spacecraft, spinning in vacuum, and a clock. The angular distances between stars are then obtained as the product of an angular velocity and a time interval. The focal length of the imaging telescopes appears as a by-product of the global data reduction. We consider in the next subsection local solar astrometry from space. In this case the reviewed methods do not apply and we have to consider a ground calibration of the instrument.

1.2 relative astrometry.

Observing the Universe from balloons, rockets or spacecraft is motivated by the need for access to infrared and ultraviolet light blocked by the Earth's atmosphere. The HIPPARCOS program demonstrated that astrometry in the visible range also greatly benefits from the elimination of the turbulence and refraction. As the diurnal turbulence is stronger than the night turbulence, moving from ground to space should be even more beneficial for solar astrometry. We indicate below some recent advances in this field.

Since january 1996 twelve instruments on board the SOlar an Heliospheric Observatory (SOHO) have been quasi continuously observing the Sun and its vicinity. Solar images are obtained on visible and UV CCD detectors. According to a 2"62 pixel size the 1024x1024 field of view of the Extreme Ultraviolet Imaging Telescopemaps the base of the solar corona in addition to the solar disk. The solar limb observed at four different wavelengths is sharply defined by some inflexion point of radial intensity. We constructed a data base of several thousands of fits of a circle to the solar limb on individual images. The measured long term pointing stability measured in this way is better than one pixel per year. For a single image the rms value of residuals amounts to 0.1 pixel (0"25) or less. At each wavelength the residuals are not a random function of position angle. The solar limb is located higherin areas of the solar atmosphere known as coronal holes. In addition to obtaining this result from relative astrometry we investigated the absolute size of the solar sphere. The distance between SOHO and the Sun is known with high precision from Doppler radio measurements. We observed that the measured value of the average angular radius of the solar disk, corrected for the satellite to Sun distance, remains stable to within 0.1 pixel, i.e. the focal length of the telescope remains stable to within one part in 8000.

2. REQUIREMENT FOR AND IMPLEMENTATION OF LOCAL LENGTH MESUREMENTS

2.1 necessity of focal length measurements.

We look toward a determination of the linear diameter of the solar sphere as viewed at four different wavelengths with the EIT instrument. As the distance between SOHO and the Sun is known to high precision from Doppler radio measurements this is equivalent to measuring an angular diameter. Observing simultaneously, in the field of view of the telescope, two stars from a catalog obtained as described in section 1 would allow a calibration of the angular size of a pixel. This is not feasible. It therefore remains to calibrate the focal length from laboratory measurements.

2.2 implementation of focal length measurements.

Due to practical reasons we could not calibrate the angular value of a pixel by using a complete instrument, including telescope and detector. This calibration has to be made in two steps involving separate measurements of the pitch of the detector and of the focal length of the telescope. We report here only on measurements from August to Novembre 1998 of a spare telescope after a photometric rocket calibration flight on October,18, 1997.

Taking into account cleanliness requirements for handling EUV reflecting mutilayer coatingsa calibration from ground based star field measurements is not feasible. We therefore proceeded to make measurements in a room at a controlled temperature close to the expected, and actually obtained, temperature of the SOHO instrument. A large granite main table supports an horizontal star simulator made of a red laser point source and a long focal length newtonian collimator. The telescope is located on an "azimuth plate" rotating around the vertical axis. The calibration is performed by measuring simultaneously the linear position of the star's image and the angular position of the azimuth table as described below.

The azimuth plate is kinematically supported by three steel balls. The forward ball, located on the left of the figure, rests in a hollow cone rigidly attached to the main table. The two end balls are supported by two steel discs rolling on the main table via a ball-cage. A compression spring pushes the azimuth plate againt a micrometer screw A attached to the main table. As the distance between the forward ball and the ball pushed by the micrometer screw is accurately known the linear readings on the micrometer screw A are easily converted into angular displacements of the azimuth plate.

The position of the star's image at the focus of the telescope is located by using a Point Diffraction Interferometer, PDI. This is a three axis device for positioning a vertical, thin glass window covered uniformly with a low transmission coating. A small disk at the center of the window is left transparent as a hole. The relative centering of the hole and of the simulated stellar image is obtained when the amount of light diffracted by this hole outside of the geometrical pupil is at a maximum. The precision of centering is a small fraction of the size of the stellar image. As the friction of the azimuth plate is low the repeatability of centering is as good as the reading of the micrometer screw A.

A measurement of focal length proceeds as follows. Firstly, the point source is located by autocollimation from a flat reference mirror at the focus of the collimator. Secondly, the diffracting window is located longitudinally at the nominal position of the CCD detector. The lateral position is set at the boundary of the telescope’s field of view. The angular position of the azimuth plate is then adjusted in order to match the positions of the artificial star and of the diffracting hole. The readings on the micrometer screws A and B are noted. The parallelism between the translation axis of the PDI with respect to the azimuth plate and the plane of the table upon wich the plate rolls is as good enough to allow to repeated readings for several positions, from one side of the field of view to the other side, whitout centering the diffracting window in the vertical direction.

figure : view from above the azimuth plate

A azimuth micrometer screw; B micrometer screw for adjusting the lateral position of a diffracting hole

We describe the results of measurements in section 3. before discussion in section 4.

3. CALIBRATION RESULTS

3.1 internal accuracy of micrometer measurements.

The micrometer screws A and B are read to within 1µm by visual interpolation on 0.01mm graduated thimbles. The lateral linear and angular displacements are performed, respectively, along 27mm by 1mm steps at the focus of the telescope with screw B, and along 16mm by smaller steps with screw A. The length of the lever arm between the center of the fixed steel ball and the center of the moving ball pushed by screw A amounts to 1032mm. The adjusted values read from A are linearly adjusted to the values set on B. The rms value of the residuals varies from 0.9µm to 2.0µm for the 16 sets of measurements described in the next subsection 3.2.

3.2 influence of the longitudinal position of the measurement plane upon the measured focal length.

The theoretical depth of field at =633nm for our 1650mm focal length 124mm diameter telescope is 0.24mm. We made 16 sets of measurements, 0.05mm longitudinally apart on each side of the best focus in order to experimentaaly determine how the focal length varies with respect to the longitudinal position of the detector. We found a linear relationship with a 2.55 ±0.15 slope and a 0.977 correlation coefficient: when moving the detector by a small amount dx away from the telescope, the measured focal length increases by an amount approximately 2.6 times larger.

These numbers are obtained by fitting a straight line to each of 16 sets of 28 A and B readings, i.e. by fitting 32 parameters to 488 data points. The global rms value of residuals is 1.30µm. We independently adjusted the data points by a priori setting the focal length for each set to a common linear function of its longitudinal position. In this way when adjusting 488 data points to two parameters, a focal length at central longitudinal position and a correcting factor according to this longitudinal position, we obtain a global value of rms residuals only slightly larger at 1.35µm. We then easily compute that the variation of the first parameter, focal length at the central position, corresponding to a 50% (25%) increase of the rms value of the residuals is equal to 0.50mm (0.17mm).

The equivalent focal length adjusted for the nominal location of the focus is equal to 1649.1mm ±0.5mm, close to the design value of 1650mm.

This complete procedure has been repeated for a second ensemble of 26 sets of 28 couples of readings, yielding a 1648.5mm ±0.5mm focal length and a 2.46±0.14 correlation factor.

3.3 influence of the adjustment of the collimator upon the measured focal length.

The focus of the 30cm diameter 3m focal length collimator is experimentally located by autocollimation within 0.02mm. We investigated the influence of a longitudinal displacement of the light source, or collimation defect, by repeating a focal length measurement for 22 positions of the point source spaced by 0.25mm from one another. When adjusting, independantly, a focal length for each of these positions we find a linear relationship between the telescope’s focal length and the longitudinal position of the source. The correlation coefficient is 0.76. The focal length of the telescope decreases by 0.14mm ±0.02mm when the source moves 1mm towards the collimator. This coefficient amounts to 0.12mm / 1mm by globally adjusting the data points as described in section 3.2. The measured equivalent focal length is located between 1645.0 and 1646.2mm.

4. DISCUSSION

We discusss the numerical values we obtained before considering how to implement better measurements.

4.1 accuracy.

The difference found for the measurements in sections 3.2 and 3.3 is eight times larger than the estimated internal accuracy of measurements. This displeasing inconsistency is attributed to some manipulation of the telescope away from and back to its supporting bracket. One may therefore a fortiori expects that any ground based calibration could not be used for any space observations. We however note that during our experiments the handling fixture of the secondary mirror was a much lighter system than the flight system. We clearly need to repeat our measurements on a flight ready instrument before and after some tests simulating the vibrations during launch.

The 0.6mm difference between the results of two sets of measurements in section 3.2 is only slightly larger than the variation of adjusted focal length corresponding to a 50% increase of the rms value of residuals. Our estimated relative accuracy for a 1650mm focal length is slightly better than one part in three thousand whereas the in flight observed stability is better than one part in eight thousand. Even if we neglect the discrepancy between the values found in sections 3.2 and 3.3 we note that the estimated accuracy of our laboratory focal length determination is short by a factor of nearly three in order to fully use the in flight observed astrometric stability.

We made our experiments using the full aperture of a four quadrants telescopes. Each quadrant has a different multilayer coating made for reflecting a precise EUV bandpas. The four lobe diffraction figure is more complex than an Airy disk. One could expect a better accuracy when calibrating an instrument with a simpler pupil. Operating at a wavelength shorter than 633nm, i.e. at the mercury 254nm line, using a UV sensitive CCD detector and some centroiding algorithm should also help obtain a better accuracy

4.2 Equivalent focal length variations according to the detector’s longitudinal position.

The same result is obtained from the two sets of measurements in section 3.2: in a compound telescope with a magnifying ratio of three, a small variation dx of the longitudinal position of the detector induces a variation of 2,5dx in the equivalent focal length. This effect arises from the fact that the rays converging to the focal caustic come from the virtual image of the primary mirror given by the secondary mirror. This effect has to be taken into account when considering the astrometric use of two mirrors telescopes.

4.3 Equivalent focal length variations according to the adjustment of the reference collimator.

Let dc be the longitudinal displacement of a source away from the focus of a 3m collimator illuminating a 1.65m equivalent focal length telescope. Our measurements indicate that the adjusted focal length of the telescope varies as 0.13 dc. As the position of the focal plane of the collimator is easily determined to within 0.05mm, we see that the uncertainty induced in the adjusted focal length of the telescope is negligible.

4.4 Field of view, gravity effects.

We explored in section 3 a single diameter of the field of view of the telescope. We could repeat our measurements by rotating the telescope around its axis in order to simultaneously map the complete field of view and estimate the influence of gravity. In order to practically implement this point and possibly to increase the accuracy we consider using as references an engraved precision slide located at the telecope's focus, viewed by a modern electronic theodolite.

5. CONCLUSION

We estimated as 3[-4] the relative accuracy of the ground calibration of the equivalent focal length of a telescope, whereas the observed 1.2 [-4] inflight stability is better.

6. ACKNOWLEDGMENT

SOHO is a joint program of ESA and NASA. The Institut d'Astrophysique Spatiale (IAS) is supported by CNRS and by Université Paris XI. References to the EIT consortium are found in

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