# Implementing Improved Algorithms for Asteroid Shape Reconstruction using Radar Images

Abstract

I present the progress that has been made towards implementing a Square Root Information Filter (SRIF) into the asteroid-modelling software shape. I compare SRIF’s performance with the current fitting algorithm, with the conclusion that SRIF has the potential to perform substantially better than its predecessor. I show the results of SRIF operating on previously collected delay-doppler data for the asteroid 2000 ET70. I also discuss potential future changes to improve shape’s fitting speed and accuracy.

# Introduction

Radar is a powerful tool for gathering information about bodies in the Solar System. Radar allows for determination of orbital elements of both planets and minor bodies (such as asteroids and comets) to high precision. A relatively new use for radar has been computing asteroid shapes based on their radar signature. This problem is not trivial – without information on the spin axis of the asteroid, the underlying shape of the body cannot be uniquely determined from its radar image. Furthermore, even if spin information were known, certain peculiarities of the radar images such as the North South ambiguity make recovery of the physical shape extremely difficult. The best method to accomplish this is by fitting a model for the physical shape to the observed radar image. Unfortnuately, this is a computationally intensive problem, potentially involving hundreds to thousands of free parameters, and millions of data points.

Given the computational intensity, one might raise the question as to the worth of computing shape information from radar data. The most compelling reason to do so is the fact that radar is not diffraction limited, since it does not require spatial resolution of the object. This means that radar can be used to resolve objects substantially smaller than the beamwidth of the detector used to obtain the images. For example, Arecibo, the primary instrument used for the data presented in this paper, has a beamwidth of more than an arcminute. Yet Arecibo can easily gather shape information to an accuracy of decameters for objects which subtend 20 milliarcseconds.

Radar has other advantages as well. Unlike most observational techniques inside the Solar System, radar does not rely on reflected sunlight. This human-controlled illumination allows for greater flexibility with respect to the observations. Radar also has the ability to probe an object’s sub-surface properties, which can give important information about the object such as porosity, surface-ice content, and surface conductivity.

There are also various reasons why asteroid shape data are so important. An asteroid’s orbital future cannot be accurately determined without information on its shape. The Yarkovsky effect, for example, can drastically change an asteroid’s orbit over several periods. This effect occurs because the rotating body absorbs sunlight and then re-emits that light in a non-radial direction. This results in an impulse which perturbs the asteroid’s orbit. The Yarkovsky effect is greatly dependent on the shape of the object, since re-emmission of absorbed sunlight is a surface phenomenon. Orbital determination of asteroids must take the Yarkovksy effect into account, and this is of special importance for determining impact probabilities for near Earth asteroids (http://adsabs.harvard.edu/abs/2013DPS....4510608C). In distinguishing between rubble piles and contact binaries, asteroid shapes also help to test hypotheses about orbital histories, as well as assess statistical properties of asteroid populations. Shapes also effect how two-body interactions between asteroids will develop – a binary with a non-spherically symmetric primary will evolve differently than its spherical counterpart.

# Current

Asteroid shapes are currently modeled using the shape software package. shape takes a model for the asteroid, which is based on both shape and spin parameters, and projects that model into radar space. This projection is compared to observed data for the asteroid, and changes are made to the model parameters in an attempt to minimize the residuals. shape uses increasing model complexity to build up a representation for the asteroid, from a basic ellipsoid model to capture gross features, to a spherical harmonic model which can represent finer surface elements (See section ? (results)).

shape currently uses a Sequential Parameter Fit (SPF) mechanism to adjust the model following a comparison between the projected radar image and the asteroid data. The SPF iterates through each parameter of the model and minimizes the $$\chi^2$$ for that individual parameter using a brent and bracket method [reference: numerical recipes]. This process is not only slow, but it also does not guarantee convergence on a local maximum. My work has been to replace the SPF currently implemented in shape with a modified Square Root Information Filter, as outlined in section ?.