Understanding the distribution of the propagated angles-only position vector

Paper number

IAC-19,A6,9,4,x50342

Author

Mr. Shambo Bhattacharjee, United Kingdom, University of Leeds

Coauthor

Prof. John T Kent, United Kingdom, University of Leeds

Coauthor

Mr. Weston Faber, United States, Applied Defense Solutions, Inc.

Coauthor

Dr. Islam Hussein, United States, Applied Defense Solutions, Inc.

Year

2019

Abstract

Consider a space object in an orbit about the earth with an initial
Gaussian uncertainty for position and velocity in ECI coordinates.  In
this paper we explore the propagated uncertainty under Keplerian
dynamics for the angles-only position vector.  If the initial state
defines an orbit in the equatorial plane, then the angles-only
position vector can be described in terms of ``latitude'' and
``longitude'', and various aspects of the distribution under propagation
can be summarized as follows.

The distribution of longitude is approximately Gaussian for small
propagation time.  As the propagation time increases, the distribution
becomes more spread out, eventually wrapping around the circle.  It is
then better described by the wrapped normal distribution, or (nearly
equivalently) the von Mises distribution.

The conditional distribution of the latitude given longitude is more
complicated to describe.  It is conditionally Gaussian with mean 0,
but the conditional variance depends on the longitude.  We will give a
complete description of the joint distribution of latitude and
longitude using first order expansions in terms of the initial
uncertainty, and we call this new distribution the ``distorted Gaussian
distribution''.

In many circumstances, the distorted Gaussian distribution simplifies to the
bivariate Gaussian distribution (small propagation times), or to the
previously developed Fisher-Bingham-Kent (FBK) distribution on the unit sphere (large
propagation times).

However, in the case of a breakup event,
where the initial position is known nearly exactly, but the initial
velocity shows high uncertainty, the distorted Gaussian distribution
shows a pronounced ``pinching'' or ``bow-tie'' effect in a scatterplot of
latitude vs. longitude whenever the propagation time is an integer
multiple of the half-period for the initial state.  This behavior is
well-known in breakup events where the cloud of debris tends to
coalesce every half-integer period.

Implications of this new distorted Gaussian distribution for
association problems will also be explored.

Abstract document

IAC-19,A6,9,4,x50342.brief.pdf

Manuscript document

IAC-19,A6,9,4,x50342.pdf (🔒 authorized access only).

To get the manuscript, please contact IAF Secretariat.