# Tether orientation and mag

Robert H. McNaught, Anglo-Australian Observatory (RMN@aaocbn2.aao.gov.au)
Thu, 14 Mar 1996 15:41:43 +1100 (EST)

```TETHER ORIENTATION
The problem of deriving the 3-D orientation is relatively easy to describe,
but will need a bit of effort to code it.

The simplest thing to recognise is that if it is vertical, then it will be
seen as perpendicular to the horizon no matter where you see it from, at
what angle or distance.  If you were lucky enough to have it pass high in
the sky and the tether diminished to a point, then that defines the
orientation.  Some simple geometry would then define the in-orbit orientation.

In general terms, however, you could define a rectangular coord system centred
on the satellite with Z up (extension of the radius vector) and X forward in
the orbital plane, and Y normal (perpendicular) to the plane.  Two estimates
of the tether orientation (angle relative to the observer's zenith) are needed
to make a determination and these should be from as wide an arc as possible
during one pass, or from well separated observers during the same pass.  (It
is assumed that tether orientation is stable over a period of several minutes).

The observer's spatial coordinates and the observed orientation for each
observation have to be transformed into the satellite coordinate system as
defined above.  These define two planes that contain the tether.  It is then a
simple procedure of finding the intersection of the two planes.

Other than the transformation of coordinates, this is formally equivalent to
determining the radiant of a meteor.  Of course, more than two observations
can be combined, in which case a least squares solution is used for the
intersection of all the planes.

Regarding Paul Maley's request for the magnitude of the tether, it is not
possible to give a number that is comparable with stellar magnitudes.  The
brightness to the naked eye is greater than in binoculars, as the brightness
decreases linearly with magnification until the seeing disk is resolved.
It may be possible to define mags/arcmin or some such, but this is not trivial.
Naked-eye visibility relative to the sky background may be the easiest way
of defining the visibility, but you will require estimates of the naked-eye
limiting magnitude and how obvious the tether is.  The tether is a strange
intermediate between a point source (star) and an extended source (sky
background) as it is only resolved in one dimension.

Rob McNaught
```