Pencil-Beam Searches for faint Kuiper Belt objects.

Motivated by a desire to understand the size distribution of objects in the Kuiper belt, an observing program has been conducted at the CFHT and Palomar 5-m telescopes. We have conducted pencil-beam searches for outer solar system objects to a limiting magnitude of $R\sim26$ .Fields were searched using software recombinations of many short exposures shifted at different angular rates in order to detect objects at differing heliocentric distances. Five new trans-neptunian objects were detected in these surveys. Our combined data set provides an estimate of $\sim90$ trans-neptunian objects per square degree brighter than $\simeq 25.9$ .This estimate is a factor of 3 above the expected number of objects based on an extrapolation of previous surveys with brighter limits, and appears consistent with the hypothesis of a single power-law luminosity function for the entire trans-neptunian region. Maximum likelihood fits to all self-consistent published surveys with published efficiency functions predicts a cumulative sky density $\Sigma(<R)$ obeying $\log \Sigma$ = 0.76(R-23.4) objects per square degree brighter than a given magnitude R.

INTRODUCTION

The size distribution of TNOs is of great interest. Although originally it had been hoped that the population might be collisionless and thus might retain the signature of the planetesimal formation process, recent work has shown that collisional effects cannot be neglected over the age of the solar system. However, knowledge of the size distribution is still important for understanding the link between the Kuiper Belt and both the short-period comets (Levison and Duncan 1997) and Pluto. The HST results of Cochran et al. (1995) provided another strong observational motivation by statistically detecting a very large population of faint trans-neptunian objects near R=28; our research was partially motivated by attempting to work at intermediate magnitudes $R\sim26$ .

The goal of our program, begun in 1994, has been to find small TNOs rather than more objects with diameters larger than $\sim$ 100 km. Instead of searching large areas of sky to limiting magnitudes of $R\simeq 22$ - 24, we chose to concentrate on one or two fields for each observing run, and integrate for 4-6 hours to reach a limiting magnitude of $R \simeq 25-26$ for the target field. In essence, the idea is that a power law increase in the sky density of objects at fainter magnitudes will produce objects in the field if the search is faint enough. Thus, going deep enough will allow us to extend the range over which the luminosity function is determined.

The Luminosity Function

The cumulative number of objects (per square degree) brighter than magnitude R, is denoted the luminosity function $\Sigma$ of the Kuiper Belt. Models often hypothesize a power-law behaviour of the form $\log \Sigma = \alpha (R - R_o)$ , where $\alpha$ is the slope on a $\log \Sigma$ vs. R plot, and R_o is a normalization (the magnitude at which the luminosity function reaches one object per square degree). Although there is general agreement in the literature that $R_o\sim23$ ,estimates of the slope have varied from $\alpha$ =0.4-0.8. The slope is crucial because it is related to the size distribution of the belt: if there is no upper or lower size cutoff then $\alpha$ =(q-1)/5, where q is the exponent of the differential diameter distribution of the Kuiper Belt objects ( $N(D)\propto D^{-q}$ ). If the slope is steep then there will be only a few more (or none) Pluto-sized objects, and there will be vast numbers of small ( $D\sim 1$ km) comets in the Kuiper Belt to supply the short-period comets. If the slope is shallow, there should be dozens of Pluto/Triton sized bodies in the main part of the belt, and there may not be enough comets to supply the observed short-period comet flux directly from the dynamically eroding Kuiper Belt. The latest surveys of Luu and Jewitt (1998) performed at the UH 2.2m and Keck seem to imply a relatively shallow slope of $\alpha$ =0.54 $\pm$ 0.03, while the surveys of our group, when combined with all other published surveys with known detection efficiency functions gives $\alpha=0.76\pm 0.10$ .

Technique

In order to work to the maximum possible depth, our group spends entire nights integrating on target fields chosen in order to have low densities of background objects (to reduce confusion while looking for moving Kuiper Belt objects). Since moving objects produce trailing losses, exposures of roughly 10 minutes duration limit motion (of order 3''/hour) to less than a pixel. Exposures are then recombined by `shifting and adding' for the suspected rates of motion and directions of Kuiper Belt objects. Although the parameter space would seem rather large, this is limited by searching near opposition, where object motions are largely dominated by the Earth-induced retrograde; this eliminates the need to search directions other than anti-parallel to the ecliptic, leaving only the rate to vary (with each rate corresponding to first order to a specific heliocentric distance). By using a median recombination instead of a sum, all stationary objects have their signal progressively eliminated (the longer the time-base of the observations the better).

Figure 1 shows the result of such a search using UH8K data obtained April 1997. This particular field was chosen because the TNO 1994 GV9 (with R=23) was known to be there; this TNO was effectively invisible (S/N $\sim$ 2) in individual frames, but the recombination of 37 images easily brings it out of the noise. This recombination at the retograde rate of GV9 revealed a second TNO about 3 arcmin northeast, and thus moving at similar rate (and at a similar heliocentric distance). The new TNO ( $R\sim23.7$ ) is designated 1997 GA45, and was the only new TNO discovered in the CFHT data set, which had a limit of about R=24.6 from the single night. A observing run using the same technique at Palomar in September 1998 (in much better weather conditions than CFHT), revealed 4 TNOs in a single 10' CCD field, with a limit near R=25.6. Results are discussed in Gladman et al. (1998).

**Figure 1:** Median-combined image from UH8K data taken on April 2 1997, showing the Kuiper Belt objects 1994 GV9 (lower left) and 1997 GA45 (upper right). This image was constructed by recombining 37 8-minute exposures at the known angular rate of 1994 GV9. This rate was near enough to the apparent rate of 1997 GA45 that we discovered this new object in this recombination; normally other angular rates are searched in order to discover objects at varying heliocentric distances. The FOV shown is 2 $\times$ 4'.
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The future

We will use the new CFH12K camera to continue this project. The chief uncertainty in the determination of the luminosity function remains small number statistics; tens of TNOs need to be discovered in a single survey under constant observing conditions to alleviate this. The CFH12K has the necessary combination of a large-field of view and depth (Figure 2) to generate tens of detections in a single night of observation. This will allow us not only to determine the luminosity function, but also to explore the gross morphology of the belt by observing the variation of surface density with ecliptic longitude and latitude.

**Figure 2:** Figure of merit showing the relative number of Kuiper Belt comets one would find using different instruments, for the Gladman (1998) luminosity function (filled squares) and the Luu and Jewitt (1998) luminosity function (open). Identical instrumental seeing is assumed in this comparison. The relative number of objects found is determined by the field of view of the instrument and its zero point; since the luminosity function rises steeply, depth is very important (explaining the jump between the CFH12K and UH8K). Depth is unable to compensate for the tiny relative field of view of LRIS on Keck however; the 1/3rd of a degree field of view of CFH12K with high throughput on the CFHT makes it the best platform for this science.
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