Aldo Dell'OroINAF- Observatory of Turin

Detailed analysis of the signal from

asteroids by GAIA and their size estimation

BesançonNovember 6-7, 2003

Size determination of main belt asteroids: simulating the GAIA signal.

Best fit approach:

If we can simulate the detailed features of the signal produced by GAIA for a given asteroid model and given observing circumstances, after processing all the single detections of the same object, we can determine the best asteroid model reproducing the full set of single observations.

Tool (simulator) requirements:

In order to obtain this goal we have to reproduce not only the optical propertiesof the object, like magnitude and photometric surface distribution, but also the exact “instrumental processing” of the collected photons, their aquisition, storage and transmission.

In the last meeting in Paris a simplified analytical model of the signal was developed, in order to perform a preliminary assessment of theexpected accuracy in the size estimation of asteroids.

The mean conclusions were: the limit angular size (uncertainty 100 %) that can be estimated is ~ 6 mas at

magnitude G ~ 12, and ~ 40 mas at magnitude G ~ 20 the size of the main belt asteroids with diameter larger than 20 km can be

estimated with an accuracy equal or better than 10 %, at least once during the operative life of GAIA below 20 km, no size estimation can be done

That model did not take into account: the role of finite size of the CCD pixels the exact (and variable) position of the image in the CCD grid the finite number of pixels used in signal acquisition (windowing)

Incoming angulardistribution of photonsfrom the object

The number anddistribution of photonsare determined by:

shape scattering law observing conditions Poisson statistics

How does the instrument work? (1)

Diffraction-spreadimage on focal planeproduced by the instrument optics(convolution with PSF)

How does the instrument work? (2)

CCD grid superposition:distribution of photonsinside CCD pixels

The binned distribution of photons depends on:

pixel size relative position image- -grid TDI motion

How does the instrument work? (3)

Photocenter determination andwindow definition aroundthe image(astrometric sky mapper)

How does the instrument work? (4)

Window selection andread-out of the signalin the window(astrometric field)

How does the instrument work? (5)

Binning and final signal (recorded)

How does the instrument work? (6)

Proposed window sizesE. Høg et al. (2003) GAIA-CUO-117

1 pixel: 10x30 m

vertical binning(across-scan integration)

windowsize(pixels)

6

12 6

along-scan direction

read-out signal(photoelectron distribution)

G=12-16 G=16-20

The signal is a vector of 6 or 12 numbers, corresponding to the numbers of collected photoelectrons in each of 6 or 12 column of (6) pixels in the window. The signal is nothing else than the along-scan photoelectrons distribution.

What do we mean by “signal (measure)”?

PhotocenterThe photocenter is the mean (in pixels) of the photoelectron distribution.

WidthThe width of the signal is the standard deviation (in pixels) of thephotoelectron distribution.

From the signal we can derive:

Photocenter and width

The signal cannot be predicted in a purely deterministic way

The number of detected photoelectrons per bin is subject to random fluctuationsdue to:

“Internal” image sources of fluctuation: photons statistics: difference in number, and in time and spatial distribution of

photons arrivals; differences in relative position between object and CCD grid (i. e., the center of the optical image can be in the center of a pixel or on its edge).

“External” sources of fluctuation: photon statistics of background; cosmic rays; electronic-instrumental noise;

Stochastic nature of the signal

Photons statistics and magnitude

G = 20

50 mas

Stochastic signal fluctuationsFour repeated observations of the same asteroid model (same object) in identical observing circumstances

Dispersion of the measured width

Due to the stochastic nature of signal formation, a single measurementof a given object gives a width belonging in principle to a more or less wide range of possible values.

The dispersion of the width values depends on different parameters: apparent magnitude, number of sampling pixels, etc...

Width measurements of the signals from two different objects with slightlydifferent sizes, can give identical values.

Can we distinguish among different bodies, in such a way as to appreciate small size differences?

Dispersion of the measured width

Single width measurement of the signal of a 200 mas asteroid

width dispersion

Repeated width measurement of the signal of a 200 mas asteroid

Repeated width measurement of all asteroids

Range of sizes compatible with a single measured signal width

single width measure

Dispersion ofcompatible sizes

Predicting the error in estimating the size of a 200 mas object

width dispersion

Dispersion ofcompatible sizes

The theoretical dispersion of compatible sizes can be used to predict the GAIA precision in estimating apparent asteroid sizes.

The relative precision is the dispersion of compatible sizesdivided by the real size of the object.

The relative precision vs. size provides the real limits in sizeestimation.

Precision in estimating the size of an object

Spheres: phase = 0o velocity = 0 mas/sec

Spheres: phase = 0o velocity = 0 mas/sec

Spheres: phase = 0o velocity = 0 mas/sec

Spheres: phase = 0o velocity = 0 mas/sec

Number of pixels and accuracy

What is the best number of pixels in the window for asteroid size measurements?

Increasing the number of pixels in the window, the number of sampling bins increases but so does also the noise due to pixels collecting the tails of the photon distribution.

As a consequence, by increasing the pixel number we improve the accuracy in measuring large sizes, but we worsen the accuracy in measuring smaller sizes.

Summary

The conclusions of the preliminary semi-analytical analysis are substantially confirmed: Main belt asteroids with diameter larger than 20÷30 km can be measured with an accuracy equal or better than 10 %, at least once during the operative life of GAIA; below 20 km, no reliable size estimate can be obtained;

The minimum angular size that can be measured with an accuracy of 10 % is ~ 20 mas at magnitude G ~ 12, and ~ 120 mas at magnitude G ~ 20

The 6-pixel window represents a reasonable trade-off between accuracy and number of asteroids that can be measured