# «Received ; accepted PASP revised manuscript, 12 Aug 96, proofs to NZ 1 with Universities Space Research Association (USRA), Division of Astronomy and ...»

**Measuring the Atmospheric Inﬂuence on Diﬀerential Astrometry:**

a Simple Method Applied to Wide Field CCD Frames

N. Zacharias1

U.S. Naval Observatory, 3450 Mass. Ave. N.W., Washington D.C. 20392,

Electronic mail: nz@pyxis.usno.navy.mil

Received ; accepted

PASP revised manuscript, 12 Aug 96, proofs to NZ

1

with Universities Space Research Association (USRA), Division of Astronomy and Space

Physics, Washington D.C., based on observations made at KPNO and CTIO –2–

## ABSTRACT

Subject headings: astrometry: limits by the atmosphere, guided CCD frames –3–

1. INTRODUCTION Turbulence in the Earth’s atmosphere adds noise to ground–based astrometric observations. Semi–empirical and empirical results have been published previously, e.g.

(Lindegren 1980; Kleine 1983; Han 1989; Monet and Monet 1992, Han and Gatewood 1995). This eﬀect ultimately limits the accuracy of ground–based astrometric observations, and it is important to ﬁnd these limits. The eﬀect is largest, about 100 mas, for absolute astrometry. For diﬀerential astrometry, previous investigations have dealt with angular separation measures. The eﬀect is found to be at the 1–2 mas level for arcminute separations and several minutes integration time (Han and Gatewood 1995), e.g. applicable to double star and parallax observations.

Here we will go one step further and deﬁne σatm as the added noise introduced by the atmosphere to astrometric observations, after an orthogonal or linear mapping model has been applied to the x, y data of guided exposures. This is more appropriate for astrometric imaging observations, because such a mapping model is used for the calibration of the x, y data to the reference star positions anyway, thus absorbing terms like scale factor and ﬁeld rotation. The proposed technique in principle can be used with photographic plates as well as with CCD imaging, although the use of CCDs is more likely to show any atmospheric eﬀect due to usually shorter exposure times and higher internal precision.

The dependence of σatm on integration time is well known to be σatm ∼ t−1/2 and we assume this relationship here in our deﬁnition of σatm. The dependence of σatm on the ﬁeld of view (FOV) is approximately known to be σatm ∼ (F OV )−1/3 (e.g. Han 1989), at least for ﬁelds smaller than about half a degree, and will not be investigated here. Our goal is to determine the range of σatm for diﬀerent nights and atmospheric conditions and look for a dependence on seeing, as determined from the full width at half maximum (FWHM) of the image proﬁles.

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** 2. METHOD**

A simple method is introduced here to measure σatm based on direct CCD imaging without the need for further instrumentation. CCD frames have been taken of ﬁelds with a high star density and reduced by standard procedures including bias removal and ﬂat–ﬁelding. Circular symmetric 2–dimensional Gaussian image proﬁles have been ﬁtted by least–squares methods to the ﬂat–ﬁelded CCD pixel data. Fig. 1 gives an example for the standard error in position plotted vs. instrumental magnitude for individual images. Stars within a dynamic range from the saturation limit (here set to 10th magnitude) to about 5 magnitudes fainter, display an almost constant level of precision for the x, y position as obtained by the image proﬁle ﬁt. The positional error increases for fainter stars because of the smaller S/N ratio and for brighter stars because of the model insuﬃciencies (saturated pixels, diﬀraction spikes). Images well above the average position error for their magnitudes are either from double stars or galaxies and have been excluded from this investigation.

This diagram does not change with exposure time, except for a shift along the magnitude scale and the number of images available in a given range of instrumental magnitudes.

Assume 2 CCD frames of equal exposure times have been taken within a short period of time under the same conditions (atmosphere and telescope). The ﬁeld center of the second exposure has been shifted by a few pixels with respect to the ﬁrst one. Thus, independent observations have been obtained with images of the same star located on diﬀerent pixels of the CCD for both frames. Only the repeatability of the observations is investigated here, so no attempt has been made to convert the x, y measures into right ascensions and declinations. All error contributions related to ﬁeld distortions are avoided because the same approximate ﬁeld center has been used for both exposures.

model. Only those stars within the magnitude range of almost constant ﬁt precision as described above, have been used for this transformation. The variance of the transformation 2 between the 2 exposures, σtrans, is

with σatm being the contribution from the atmosphere and σb the remaining error contribution – the base–level – as inherent in our procedure and instrument (model insuﬃciencies, digitization errors, etc.), for each individual CCD frame. Deﬁning σa from σatm = σa t−1/2 with exposure time t in seconds we arrive at

which is a linear relationship between the observable quantity σ 2 and the nearly error free parameter t−1. Assuming constant observational conditions for the time to take more sets of CCD frame pairs for other exposure times, we can solve for σa and σb.

Observing runs for the Radio–Optical Reference Frame (RORF) project (Johnston et al. 1991) have been conducted from 1994 to 1996 at the 0.9–meter telescopes on Kitt Peak and Cerro Tololo (Zacharias et al. 1995). The KPNO 0.9–meter telescope has a scale of 0.68”/pixel and a FOV of 23.2’, while those numbers are 0.40”/pixel and 13.6’ for the CTIO telescope.

A few CCD frames were speciﬁcally taken to investigate the limits set by the atmosphere on astrometric accuracy. Fields with a high star density (close to the Milky Way) and close to the zenith, if possible, were observed close to the meridian. For most of –6– the selected ﬁelds, 2 frames of 40, 20 and 10 seconds exposure time each were taken with the same Gunn r ﬁlter in addition to the long exposures for the RORF project. An oﬀ–axis autoguider was used with guide stars selected close to the edge of the main FOV. The integration and correction cycle time was set to about 1 second and the system was guiding for at least 10 seconds before the start of a new exposure.

Figure 2 shows an example of σ 2 plotted vs. t−1 for the 4 exposure times. A linear model has been used for the x, y transformations. The ﬁlled circles and open squares are the results for the x and y coordinates respectively (along δ, α for the KPNO telescope).

Because there are more faint than bright stars, the longer exposure frames show more stars near the saturation limit than the short exposure frames. Also, for a given constant number of stars used for diﬀerent transformations, the value of σtrans is better determined for longer exposure times because of the smaller scatter in the x, y transformation data due to better image deﬁnition from the longer integration time. Thus, weights have been 2 assigned to each measured σtrans value. Let E(y) be an estimate of the standard error of the quantity y and y = x2 with x = σtrans, then we have from the error propagation law

The weighting is not critical for the determination of the slope itself, i.e. for the atmospheric contribution. The determination of the base–level and the error estimates on the results depend more strongly on the choice of the weighting algorithm. This conclusion was obtained from tests made with diﬀerent weighting algorithms, including unweighted reductions.

A weighted least–squares ﬁt was performed with the σ 2 vs. t−1 data points, in order to obtain the slope and constant of the straight line predicted by the theory. Fit results for each axis separately (dotted, full line) as well as for the combined data (dashed line), are shown in Fig. 2.

** 4. RESULTS**

From the straight line ﬁt of the σ 2 vs. t−1 plots for the combined data of both axes, σa and σb, and their errors were calculated. Table 1 lists all observations and results.

The ﬁrst line for each observation shows the result from the linear transformation model, while the second line shows the result as obtained with the orthogonal model. The last column displays σan, which is σa normalized to 100 seconds exposure time and a FOV of 20 arcminutes for the zenith, obtained from

with fov being the ﬁeld of view in arcminutes as used for the x, y transformations and z being the mean zenith distance while observing the ﬁeld. Fig. 3 shows results for σan obtained with the linear x, y transformation model plotted vs. FWHM, scaled to the zenith with a cos z factor, adopted from (Lindegren 1980).

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** 5. DISCUSSION**

There is a large nightly variation in the atmospheric inﬂuence on the observed star positions, which is correlated with seeing (FWHM), but ”the seeing value” alone does not determine σan. On the average we obtain σan ≈ 3 mas and 6 mas for 1 and 2 arcsecond seeing respectively.

The results obtained with the orthogonal x, y transformation model give on the average larger numbers for σan by about 10%. This is expected, because fewer parameters will model less of the real atmospheric eﬀects.

All our results hold only for this type of guided imaging observing procedure. For diﬀerential transit circle or scanning mode observations, the modelling of the atmospheric eﬀects is diﬀerent, as will be the residual eﬀects caused by the atmosphere on the astrometric results (Stone et al. 1996).

Lindegren (1980) obtained standard errors for observing the separation of stars, i.e. a diﬀerent kind of diﬀerential astrometry from that investigated here. His result, scaled to 100 seconds exposure time and a mean separation of 10’ (comparable to our 20’ FOV), is about 19 mas. Results by Han (1989) would lead to 14 mas for this case. Both Lindegren’s and Han’s results are obtained in medium seeing conditions (≈ 2”), thus they have to be compared to our 6 mas, which is a factor of 2 to 3 smaller. Scaled to a 100 second exposure time and a star separation of 10’, Han and Gatewood (1995) found σan = 5.4 mas from star trail observations obtained from Mauna Kea. Our result is 3 mas for good seeing, which is smaller by almost a factor of 2.

Separation measurements made at the 61–inch Flagstaﬀ telescope result in an atmospheric contribution of 9 to 28 mas for this case, depending on the night (Monet and Monet 1992, Monet 1996). Again our result is a factor of 2 to 3 smaller. Similar –9– to our results, Monet and Monet ﬁnd a lose correlation with seeing, which ranged from FWHM=1.2” to 2.3”. According to a hypothesis (Monet 1996), local eﬀects near the dome cause some of the ”astrometric seeing”, not correlated with the general FWHM.

This diﬀerence between our results and those obtained by other investigations can be explained by the diﬀerent types of observations. The simultaneous observation of all stars in the FOV seems to be important. Moreover, our guided exposures follow correlated image motions of a star ﬁeld, caused by the atmosphere, and thus reduce the noise contribution compared to other diﬀerential observing procedures. Also, a linear reduction model will give smaller residuals as compared to angular separation measurements with fewer free parameters.

As a by–product of this method, the base–level of accuracy has been determined as well. The mean of all σb with a standard error smaller than 1.0 mas is found to be 8.5 mas =

0.012 pixel for all KPNO observations. The corresponding result for the CTIO instrument is 6.0 mas = 0.015 pixel. These numbers are for a single observation per coordinate. The slightly smaller value for σb (in pixel units) for the KPNO instrument can be explained by the better optical quality of that telescope, which includes a ﬁeld ﬂattener corrector lens.

Imperfections in the CCD, the optics of the telescope and model deﬁciencies contribute to this base–level of astrometric accuracy, which is under further investigation (Zacharias and Raﬀerty 1995; Winter 1996; Zacharias 1997).

A large nightly variation (factor of 2) of the noise added by the atmosphere to diﬀerential astrometry has been found. This added noise is correlated with the seeing. In good seeing conditions (≈ 1”) the contribution of the atmosphere to diﬀerential astrometry – 10 – can be as small as 3 mas for guided 100–second exposures and a FOV of 20 arcminutes for 0.9–meter aperture telescopes.

Guided exposures with simultaneous observation of all stars in the FOV give a considerable (about a factor of 2) advantage over angular separation measurements made with other diﬀerential astrometric observing techniques.