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[email from M. Shao, July 8, 2010] :

It turns out the 1 yr period is not a serious limitation for an
astrometry-only mission. In our double blind test for SIM, there were ~8
earthlike (in hz) planets and we found all 8. We were only slightly
lucky.

1) with a 5 yr mission, the "avoidance" zone is +/- 10% of a 1 yr period.

2) parallax has a well defined motion (depending on the ecliptic
long/lat)

2a) planet  orbits are more likley edge on than face on. For an edge on
planet if the motion is in the dec direction while the parallax is in RA
direction, we will see the planet.

2b) the orbital phase of the planet is random with respect to the
orbital phase of the parallax motion. If the orbital phase is 90deg from
the parallax phase we'll see the whole thing.

From the two above there is a 75% chance that the orbital motion is
orthogonal to the parallax motion even at 1.00 yr orbit.

In the double blind test, there was one such example, 1.01yr orbit but
orthogonal to the parallax motion and we found the earth.

3) The HZ is from 0.7 to 1.5 AU, from 0.59 yr to 1.83yr orbit. While the
parallax confusion is from 0.9~1.1yr. So in total there is only a 5%
chance that astrometry won't find an Earth in the HZ.



We were only slightly lucky finding 8 for 8 in the double blind test.
If we had found only 7 of the 8 (12% missed) we'd be slightly unlucky
since the expected value is 5%.

Of course imaging + astrometry is closer to 100%. But imaging+astrometry
and getting the mass, is still 95% not 100%.

[email from M. Shao, Oct 13, 2009] :

In uas astrometry we have to solve separately, we have to separately solve for the parallax of
the ref star(s) and the target star. The traditional route of assuming all the ref stars are at 1kpc
or at a distance estimated by photometry is not viable when the accuracy is < 10uas.  Also the
0.3 deg field is large enough that the parallax signature will vary across the field at levels >> 10uas.

A solution for the parallax of all the objects will absorb on average 50% of the signal of a planet with
a 1 year period, meaning there will be large errors in the orbital parameters derived. But no planet other
than our own Earth has a 1.00000 yr period and the orbit of the spacecraft will not be 1.00000 yrs.
The "zone of avoidance" is proportional to 1/mission length, a 5 year mission has a +/-10% window
around a 1 yr period where a significant fraction of the planet's signature will be absorbed into the
parallax solution.

[email from M. Shao, Oct 4, 2009] :

Technical issues with Coronagraphic Astrometry

M. Shao

I  Background.

It's useful to put 0.12 vs 30uas into perspective. 0.1uas astrometry 
is measuring the l/D image to 1 part per million. The implication is
that the wavefront is in some way accurate to 1 part per million, that
optical paths are matched to ~ 1 picometer. (1 million'th of a wavelength)

In most cases optics do not have to be accurate to a million'th of a
wave, but the wavefronts from the target star and reference star
must be the same to 1e-6 lambda, over some angle on the sky.

The third piece of background information, is that the stability we
need in astrometry is over 5 years, not 30 seconds or 10 minutes. 
The wavefront stability of a coronagraphic telescope is only ~50 picometers
over ~10 minutes.

II  Basic Concept

Shaklan and Pravdo have studied astrometry with a normal telescope and
large CCD focal plane and arrived at a systematic error limit of ~50uas.
This type of astrometry measures the position of the target star with
respect to all the reference stars over a several arcmin field. One
major source of error comes from using a cassagrain or other telescope
with more than 1 surface. The wavefront of a star's light is the sum
of the wavefront errors in all the optics traversed, added together.
Since different stars in different parts of the field of view, follow
different paths through a cass telescope, the wavefront from each
star (target or reference) will be slightly different, typically
by lambda/100 or many orders of magnitude larger than the 1 picometer
requirement for stability. The difference from different paths is
sometimes called the non-common path errors.

The concept of doing astrometry not of the target star, but of the
diffraction spikes generated by the target star gets around many of
the non-common path errors mentioned above. 

The basic concept behind doing diffractive astrometry, is ultimate
based on the assumption that the non-common path errors do not 
have a solution. But since the diffracted light is diffracted in
the same direction as the reference star light, there is no non-common
path error.

III  Isoplanatic angle and its effect on photon noise.

The diffracted light is spread over the whole 0.3*0.3 deg field
of view. If we do very narrow angle astrometry, that is measure
the position of a reference star relative only to the diffraction
features 1 arcsec away, then the photon noise from the diffracted
target star is a serious limitation. If we perform relative 
astrometry only over 1 arcsec out of a 0.3 deg field we only
use 1e-6 of the diffracted light photons. If the total diffracted
light is 1% of the aperture, the photons from the target star
useable for astrometry is lower by 1e-8. A 5 mag target star will
have photon limited performance of a 25 mag star, much fainter
than the ~13 mag ref stars.

What is the size of the isoplanatic patch? The area over which
one can make 0.1uas precision measurments? A very quick back of
the envelop calculations based on SIM characterization of beam
walk metrology errors give roughly a ~4 arcsec radius where
differential astrometry would be accurate to ~10 uas, and with
~10,000 ref stars, the ensemble precision would be ~0.1uas.

The size of the patch is a strong function of the desired
accuracy, the patch size of 1uas astrometry is 10 times the
size of the patch for 0.1uas astrometry.  

A more detailed description for a rough estimate of the isoplanatic
patch will be made later.

IV   Focal Plane Errors.

Instrumental or systematic errors in the focal plane are also
serious limitations, at levels well above 0.1 uas.

The diff limit of a 1.4m telescope is ~70mas, and if we have
2 pixels/(l/D) we are attempting to centroid to roughly 2 million'th
of a pixel.  This is about 1000 times better than what has been
done with space imaging systems like HST.

For a mission like GAIA, stars brighter than ~9 mag are not photon
limited but instrument limited. A single epoch measurement by GAIA
is roughly ~70 uas. (The mission accuracy is ~7uas and GAIA makes
~100 visits to an "average" star.) With a 70mas PSF GAIA astrometry
is similar to HST 1/1000 PSF only slightly better (2X) than HST.

GAIA moves the stellar image across multiple CCDS, averaging the
pixelation errors across many thousands of pixels.  In the end,
this averaging by scanning the image across many CCDs is only slightly 
better than the modeling done with "stareing" ccds in HST.

Still needed is a procedure for image-centroiding that is a few
hundred times better than GAIA or HST.