20.5 Sources of Position Uncertainty

We have given evidence that extended baselines are best for accurate position measurements. In addition, as long as sensitivity is not an issue and that observed sources are not resolved by the array, the outermost stations should always be preferred (Section 20.4). The great asset of the IRAM array is clearly sensitivity coupled with resolving power, although atmospheric fluctuations and instrumental limitations may limit the accuracy of position measurements.

We further discuss below the boundary conditions or requirements in astrometric observations. Table 2 at the end of this section summarizes the limitations with respect to the IRAM array.

20.5.1 Known Limitations

Among several practical limitations it is worth mentioning: wind effects, thermal effects in the antenna structure (including de-icing instabilities), the influence of refraction effects, imperfections in subreflector displacements, imperfections in the azimuth and elevation bearings of the antennas and, not least, uncertainties in the ``crossing point" of the azimuth and elevation axes. These imperfections, and in fact the resulting differential effects of each antenna pair in the array, have adverse effects on the visibility phase measurements; however, many of them can be removed to a large extent by phase calibration (a posteriori), and thus will not be discussed further.

In order to make the reader more aware of these questions, we just mention that the large-scale uneveness of the azimuth bearings gives rise, in places, to optical path deviations of about $40 \mu$m which translate into position offsets of $0.04''$ with 200m baselines. Likewise, position uncertainties result from imperfections in the ``crossing point" of the azimuth and elevation (nodal point) of each antenna in the array (see Chapter 6). Slow drifts in the focal position are also corrected to first order by the calibration procedure. Only large and rapid focal drifts are problematic if not recognized as such in the phase of a reference calibrator.

20.5.2 Practical Details

We elaborate here on some properties of the IRAM array related to inaccuracies in the determination of baseline lengths, and we briefly discuss how atmospheric phase noise and source strength can limit the accuracy of position measurements.

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Baselines: are easily measured with the IRAM interferometer on Plateau de Bure with a precision of a few degrees or a small fraction of one millimetre. As a reference, good winter conditions allow us to measure baselines at 86GHz, using a number of quasars well-distributed in hour-angle and declination, with uncertainties of $5^\circ-8^\circ$ in the D configuration (the most compact one at IRAM) and $10^\circ-20^\circ$ in the A configuration. But even the most accurate baseline measurement will be limited in precision. Residual uncertainties in the baselines will finally produce phase errors that scale with $\Delta\vec{k} = \vec{k}_{\rm quasar}-\vec{k}_{\rm source}$, the distance between a calibration quasar and the source. Combining the different forms of the phase equation defined in Section 20.2, we can then derive a rough estimate of the mean uncertainty in the absolute position of a source from

$$\Delta \theta \simeq (\delta \vec{\cal B} \cdot \Delta \vec{k})/{... ...(\delta\phi / 2\pi) (\lambda/{\cal B}) \simeq (\delta\phi/2\pi) \theta_B$$ (20.22)

where $\delta\phi$ is the phase error due to inaccuracies in the baselines. This formula is convenient, as it associates uncertainties in the knowledge of the baseline length at a given frequency with $\theta_B$, the synthesized beam. For instance, observations at 86GHz in the D configuration with baseline phase residuals $\delta\phi$ between $2^\circ$ and $5^\circ$ (i.e. assuming baseline errors, $\delta \vec{\cal B} = 0.2$mm, and typical phase calibrator distances, $\Delta \vec{k}=5^\circ-15^\circ$) appear to have position uncertainties smaller than $0.20''$. See Subsection 20.4.2 for suggestions to improve these uncertainties.

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Atmospheric phase fluctuations are among the most important limitations that affect the accuracy of position measurements. Poor seeing conditions imply phase decorrelation which in turn implies reduced flux density sensitivity and larger apparent source sizes (see Chapters 9 and 10). When the atmospheric phase noise dominates, phase decorrelation can be estimated by least-square fitting in time the phase profile of a reference calibrator. Under the assumptions made at the end of Section 20.3 or assuming here that the atmospheric phase fluctuations remain unchanged, namely $\sigma_\phi$ is similar for each phase sample, we can estimate the mean angular uncertainty from

$$\Delta \theta \simeq \sigma_\phi / (2\pi \sqrt{n_p} ({\cal B}/\lambda)) = \sigma_\phi \theta_B / (2\pi \sqrt{n_p})$$ (20.23)

where ${n_p}$ is the number of phase samples. The size of the associated ``seeing disk" is defined as $(\sigma_\phi / 2\pi) \sqrt{8\ln{2}} \theta_B$. For instance, measuring mean atmospheric phase fluctuations $\sigma_\phi \simeq 10^\circ$ at 86GHz on a 60m baseline is equivalent to observe in $\simeq 0.78''$ seeing conditions (which is small since $\theta_B \sim \lambda/{\cal B} \sim 12''$ and corresponds to a small fraction of the synthesized beam). Observations at the same frequency, on the same baseline and with similar atmospheric conditions will then provide a position accuracy of order $\Delta \theta \simeq 0.33''/\sqrt{n_p}$ (or $3 \%$ of $\theta_B \sim 12''$).

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Source strength and finite signal to noise ratio is another important limitation to astrometric accuracy. While reference calibrators have to provide enough sensitivity for rapid detection, detection of program sources may require hours of integration. In addition, observed sources are sometimes resolved out with extended configurations. Therefore, the interesting astrometric case described in Section 20.4 where spectral sources are rather easily detected may not be common in mm-wave astronomy. As mentioned in Section 20.4.2 we can use the result of [Reid et al. 1988] to estimate the one sigma position uncertainty

$$\Delta\theta = \sigma_\theta \simeq (\theta_B/2)(\sigma_S/S) = \theta_B/(2\cdot\rm SNR)$$ (20.24)

where $\sigma_S$ is the noise in the map and $S$ the source flux density. With the IRAM array in the D configuration, a source at mean declination (e.g. $30^\circ-40^\circ$) detected with a signal-to-noise SNR$\simeq 5$, cannot be located with a precision better than $10\%$ of $\theta_B$ (e.g. $0.25''$ at 230 GHz). Uncertainties in declination measurements will obviously be larger for southern sources owing to the elongation of the synthesized beam.

On the other hand, astrometric observations of bright sources such as the SiO line sources presented in Section 20.4 are not limited by SNR issues in general, but by the accuracy of the bandpass calibration. While delay calibration (see Chapter 5) already removes the bulk of the phase gradient across the band selected for observations, residual variations can only be removed by observing strong calibrators. Using the classical radiometric equation, bandpass calibration requires the following:

$$\Delta t^C = (S\cdot\sigma_S)^2/(C\cdot\sigma_C)^2 \Delta t^S$$ (20.25)

In this expression, $S$ and $C$ are the flux densities of the source and calibrator, $\sigma_S$ and $\sigma_C$ the respective r.m.s. noise levels, and $\Delta t^S$ and $\Delta t^C$ the integration times on the source and calibrator. For instance, a 1sec integration on a 15Jy calibrator like 3C273 (at the time of writing, the strongest calibrator at 86GHz available in the northern sky) is sufficient for bandpass calibration in the case of a $5\sigma-$detection of a 2mJy source. (In practice, however, several seconds integration would be better.) On the other hand, a 10min integration on the same calibrator would just be sufficient to meet the minimum requirement ( $\sigma_S = \sigma_C$) to calibrate 1min observations of a 50Jy strong source.

There are a few other issues which we list below. They are worth mentioning although there is little implication for observations with the IRAM array. (For other effects such as bandwidth smearing and visibility averaging, we recommend reading the book of [Thompson et al. 1986]; see Chapter 6.)

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Pointing offsets is a potential source of position errors. Ideally, the phase of the incoming wavefront does not depend on pointing offsets across the Airy (or diffraction) pattern. However, imperfections in the optical system may result in differences in the Airy pattern from an antenna to another in the array (although all antennas are of comparable quality). Experience at 86GHz shows, that rather strong phase differences (up to $10^\circ$) may appear when antennas are individually offset from the target position by a distance equal to half the primary beamwidth (HPBW).

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Primary beam attenuation produces a radial displacement for off-axis targets. It needs to be corrected for targets at large angular distances ($\simeq$ HPBW/2) from the phase tracking center of the interferometer.

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Gravitational Lensing by the Sun introduces positional offsets $\sim M_\odot/D_E (\sqrt{1+\cos\theta}/\sqrt{1-\cos\theta})$ which are negligible for targets outside the Sun avoidance region of the IRAM antennas ( $\theta\ge 45^\circ$). For instance $\Delta\theta\ \simeq 0.1''$ at $\sim 5^\circ$ from the Sun limb.

A summary of the main practical position uncertainties for the IRAM array is given in Table 2 in arc second or in terms of the synthesized beam $\theta_B$; $\cal B$ is the baseline length in meter. Only instrumental errors are removed to first order by calibration.

Table 2. Plateau de Bure Interferometer - Main Sources of Position Uncertainty

TELESCOPE$^\star$	$\Delta\theta$	Calibration
Focus Offset	$\leq 0.20''\cdot(100/{\cal B})$	Yes
Axes Non-Intersection	$\leq 0.10''\cdot(100/{\cal B})$	Yes
AzEl Bearings	$\leq 0.08''\cdot(100/{\cal B})$	Yes
OBSERVATION
Atmospheric Seeing$^\dagger$	$\leq 0.06\cdot\theta_B$	No
Calibrator Distance$^\dagger$	$\leq 0.02\cdot\theta_B$	No
Pointing Offset	$\leq 0.02\cdot\theta_B$	Partially
Source Intensity	$\leq (0.5/{\rm SNR})\cdot\theta_B$	No

$^\dagger$ Upper limits are illustrative for astrometric observations in limiting conditions. See text for more details.
$^\star$ Instrumental values are all calibrated out to first order.