1. The Use of GAIA Photometric Data for the Determination of Physical Parameters of Asteroids
A. Cellino, M. Delbò, A. Dell’Oro, and V. Zappalà
INAF – Astronomical Observatory of Torino

2. Photometry is a classical observing technique which has been widely applied to asteroid studies, with the aim of:
Deriving the spin periods
Deriving the overall shapes (from lightcurve morphology)
Deriving the orientation of the spin axes
The distributions of the above physical properties are important constraints for theories of the collisional evolution of the asteroid population.

3. Historically, two main techniques of asteroid pole determination have been used. The so-called Amplitude-Magnitude method is based on an analysis of the variation of the lightcurve amplitude and magnitude at maximum.
The so-called Epoch method (also called Photometric Astrometry) involves measuring time intervals between a given lightcurve feature observed at different viewing geometries.

4. (p, p) b/a, c/a (p, p) Assumptions:
Asteroids are triaxial ellipsoids with axes a>b>c, spinning around the c axis with period P
AM method
(p, p)
Sense of rotation
E method

5. One major advantage of space missions is that they make it possible to monitor the objects more efficiently and over a wider range of observing conditions.
In this sense, GAIA is an excellent example of a mission which will produce a large set of asteroid photometric data.
The photometric accuracy of GAIA will also be very good compared to usual ground-based photometry.

6. Mignard’s simulations
id/name nobs prec foll spec mbp
1 Ceres
2 Pallas
3 Juno
4 Vesta
5 Astraea
6 Hebe
7 Iris
8 Flora
9 Metis
10 Hygiea
11 Parthenope
12 Victoria
Mignard’s simulations

7. The asteroid photometric data collected by GAIA will be very useful and will be used to derive spin orientations, overall shapes and rotational periods.
With respect to conventional ground-based photometry, the main difference is that the latter has been mostly based on the collection of lightcurves covering single spin periods. In the case of GAIA, we have instead a large number of photometric measurements much more spread in time.

8. We have used the results of the GAIA minor planet survey simulations carried out by F. Mignard.
We have taken as a reference case the simulated GAIA detections of the asteroid 39 Laetitia (typical main-belt orbit).
We have considered only the detections by the GAIA medium band photometer (201 observations).
We have computed the apparent magnitudes, when observed by GAIA, of a triaxial ellipsoid object having the Laetitia’s orbit, as a function of its axial ratios and pole coordinates.

9. Zero-order approximation: no phase, no scattering.

10. Orbit of 39 Laetitia b/a = 0.7 c/a = 0.5
(mag) with respect to first observation

11. Orbit of 39 Laetitia b/a = 0.7 c/a = 0.5
(mag) with respect to first observation

12. Orbit of 39 Laetitia b/a = 0.7 c/a = 0.5
(mag) with respect to first observation

13. Orbit of 39 Laetitia p = 30 p = 60
(mag) with respect to first observation

14. Orbit of 39 Laetitia p = 30 p = 60
(mag) with respect to first observation

15. Orbit of 39 Laetitia p = 30 p = 60
(mag) with respect to first observation

16. Orbit of 39 Laetitia b/a = 0.7 c/a = 0.5
(mag) with respect to first observation

17. Orbit of 39 Laetitia p = 30 p = 60 b/a = 0.7 c/a = 0.5 P = 7h.527
0 = 0.4
GAIA observations
(mag) with respect to first observation

18. Orbit of 39 Laetitia p = 30 p = 60 b/a = 0.7 c/a = 0.5 P = 7h.527
0 = 0.4
GAIA observations
(mag) with respect to first observation

19. Comparison with ground-based observations
Data-base used by Zappalà and Knežević (1984) to derive the pole of 22 Kalliope

20. Magnitude-longitude data are per se very useful to infer preliminary indications about the pole coordinates and the axial ratios.
Maximum amplitude
b/a
Minimum amplitude p
Minimum ampl. ~ 0
p ~ 0
Flat maximum
b/c ~ 1

21. In the case of GAIA observations, we expect to obtain many single photometric measurements of each object.
The measured magnitude at each observing epoch will be determined, for each object by:
its pole coordinates
its axial ratios (assuming triaxial ellipsoid shapes)
its rotational period
an initial rotational phase at the epoch of first detection
The number of unknown parameters (6) is << number of detections (> 100)

22. A numerical approach has been chosen.
We have developed an algorithm aimed at finding the overall solution of the problem (set of p, p, P, b/a, c/a, and 0 which produce the observed magnitudes), using a naïf technique of “genetic evolution” of the solutions.
After a preliminary generation of many random solutions, the best ones are led to “evolve” through mutual coupling of the parameters and “random mutations” of single parameters. An  parameter (function of the O-C) is used to assess the goodness of the solutions.
A maximum of 1,500,000 “evolutionary steps” are performed.

23. The algorithm makes five consecutive attempts to find a global solution after 1,500,000 steps. Due to the kind of approach adopted, not all of the resulting solutions are good, but usually at least one is found which is clearly better than the others ( smaller by a factor of 10 or more, and close to mag). In all the cases explored so far, this corresponds also to the correct solution of the problem.

24. The goodness of the solution is estimated by means of the parameter:
(other parameters could be used as well)

25. Example of a bad solution
 > 0.05
Black boxes, “true” magnitudes

26. Example of a good solution
  mag
Black boxes, “true” magnitudes

27. When the orbit lies on the ecliptic, a spurious pole solution is usually found by methods based on lightcurve analysis P 
Ecliptic P’
Some ambiguity is still found even in many cases in which the inclination is not zero

28. So far, we have analyzed both ideal cases in which the magnitude measurements have zero-error, and more realistic cases in which the photometric accuracy of the single measurements is 0.01 mag.
In both cases, so far, we have been able to find the correct solutions.
Typical errors are 1 deg in p and p, h in P, in b/a and c/a, and 0.01 in 0. In many cases (but not always) we are also able to decide the right sense of rotation.
So far, however, we have not yet introduced the effect of the phase angle. This is expected to make the things significantly worse.

29. Problems The objects are not perfect triaxial ellipsoids.
The observations are performed at fairly large phase angles, and magnitudes are affected by this.
Light-scattering.

30. The assumption of an overall triaxial ellipsoid shape is not expected to lead to tremendous mistakes, since it has found to be a useful first-order approximation in a large variety of cases, dealing with main belt asteroids.
Moreover, there are cases in which a triaxial ellipsoid shape is the expected equilibrium shape for reaccumulated objects (rubble piles), as shown in classical investigations by Farinella et al. (1981, 1982). These cases correspond mostly to objects sufficiently large to be directly measured by GAIA.
In any case, more complicated situations can be simulated (Gaussian shapes, “Cellinoids”, etc.) in the near future.

31. The effect of non-zero phase can be taken into account in different ways.
The easiest way is to use the H,G system and to reduce all observations of the same object to the same average value of phase.
A more accurate method is to include a more analytical computation of the illuminated fraction of the object’s disk (ellipse). Such kind of computations have been published, e.g., by Pospieszlska-Surdey and Surdey (1985) and by Drummond et al., 1985.

32. In any case, we tend to be optimistic, in the sense that non-zero phase will certainly be a problem, but past experience dealing with ground-based lightcurves suggests that probably it will not prevent the possibility of deriving accurate estimates of the pole coordinates (within a few degrees). This should be true also taking account uncertainties on the light scattering law.
Some more uncertainty will probably affect the derived axial ratios.
Work in Progress!

33. Relevance of GAIA asteroid photometry
GAIA will probably produce the biggest data-set of asteroid poles.
There are good reasons to hope that also the sidereal periods will be derived in most cases.
GAIA photometry will be used to make predictions about the shape of the apparent (roughly elliptical) disk exhibited by many objects at the epochs of GAIA measurements. These predictions will be checked by direct measurements in a fraction of the cases, and can be a useful input for techniques of assessment of the likely photocenter displacement.