Subject: Re: LSST Astrometric Survey

From: Nick Kaiser

Submitted: Wed, 9 Oct 2002 07:59:05 -1000 (HST)

Message number: 11 (previous: 10, next: 12 up: Index)

On Wed, 9 Oct 2002, Michael Strauss wrote:

>   Having said this, I would like to hear more about the rationale
> behind Dave's number of 30-50 mas absolute astrometry on a single
> observation.  In the 8.4m design, the exposure time is 10 seconds,
> which is short enough that atmospheric refraction effects completely
> dominate the astrometric residuals.  With SDSS (55 second exposure
> time), and using the UCAC dense network of standards, it was my
> understanding that the residuals are roughly 60 mas per coordinate,
> rms.  I don't know how this number will increase with a 1/5-as-long
> exposure time, but it will certainly be higher. 

From a more theoretical perspective, if we observe through a
layer of Kolmogorov turbulence of thickness L_0 that creates
seeing disk of width FWHM and is moving at velocity v and we
observe at wavelength lambda for a time t_exp, the
1-sigma limit on astrometric precision is
\begin{equation}
\sigma_x \simeq 0''.16 \left({{\rm FWHM} \over 0''.6}\right)^{5/6}
\left({L_{\rm o} \over 20 {\rm m}}\right)^{1/3} 
\left({v \over 10 {\rm m/s}}\right)^{-1/2}
\left({\lambda \over 5 \times 10^{-7} {\rm m}}\right)^{1/6}
\left({t_{\rm exp} \over 1 {\rm s}}\right)^{-1/2}.
\end{equation}

These deflections have an angular coherence scale
\begin{equation}
\theta \simeq L_{\rm o} / H \simeq 12' {L_{\rm o} \over 20 {\rm m}}
{5 {\rm km} \over H}
\end{equation}
where $H$ is the altitude of the layer and a temporal
coherence scale
\begin{equation}
\tau \simeq {L_{\rm o} \over v} = 2 {\rm s} \left({L_{\rm o} \over 20 {\rm 
m}}\r
ight)
\left({v \over 10 {\rm m/s}}\right)^{-1}.
\end{equation}

Features of this:

a) Quite sensitive to (poorly known) layer thickness.  We can try
to pin this down using Dave's empirical numbers.  This factor
is likely to be quite dependent on site.

b) independent of telescope size (assumes outer scale L_0 > D),
though an array of N telescopes does better, as you can get N
times the integration time t_exp.

c) The deflection field is highly spatially coherent. If one does
an astrometry dedicated pre-survey then the appropriate 
t_exp is the total time spent on each patch of sky during the
pre-survey.  Once a set of standards are established, the
deflections in a later short exposure will be taken out (provided
the model of the detector is sufficiently flexible --- "Jello
detector model").

d) This is all systematic, of course, the random errors are

sigma_x = sqrt(2) FWHM / (2.35 nu) \simeq 70mas (5/nu)

where nu is the significance of the detection.

Nick



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