Contribution of telescope ROLL to pointing errors
Contribution of telescope ROLL to pointing errors
The diagram below shows the perspective of an observer at the origin, O,
of the telescope coordinate system. Great circle arcs centred on O
will appear as straight lines :
The telescope points towards P;
Z is the zenith and H is the point on the horizon below P defining
the azimuth of P. The elevation of P is E, or arc HP.
Imagine that the telescope rolls an amount r to the right as a
result of track bumps, for instance, and points at A.
The arc HA is then also of length E, while
the elevation of A is E' : so ZA = 90-E'.
The change in azimuth (as read around the horizon)
is the angle a, or HZA.
B is a point with the same elevation as A: i.e. ZA = ZB = 90-E'.
In the spherical triangle HZA
sin (ZA) / sin (ZHA) = sin (AH) / sin (HZA)
i.e.
sin (90-E') / sin (r) = sin (E) / sin (a) -- (1)
and
cos(ZA) = cos(HA)cos(HZ) + sin(HA)sin(HZ)cos(ZHA)
i.e.
cos(90-E') = cos(E)cos(90) + sin(E)sin(90)cos(r)
sin(E') = sin(E)cos(r) -- (2)
It is tempting to jump to the conclusion that, with r small, (2) gives
E=E'.
It is illustrative, however, to set E=89degs and r=3600" and evaluate
E', which turns out to be 88degs35'09"; distinctly not the same as E !
The full solution of (2) for E' is shown below for the case of r=100":
In the limit as E approaches 90 degs, E'-E approaches -r arcseconds
(-100" in this case).
In the case of JCMT r is <30 arcseconds and E never exceeds 85degs
(the 'zone of avoidance'), so the
contribution of roll to the elevation error is <0.1" everywhere.
With a zero approximation for E-E' (1) then gives
cos (E) / sin (r) = sin (E) / sin (a)
i.e.
sin(a) = r*tan(E)
for small r, and 'a' measured around the horizon, recall.
The change in azimuth on the sky is then r*sin(E) arcseconds.
Thus I believe that the formulae at
predict_dazdel.html
are correct (give or take sign conventions), and that despite my explicit
concern of a component of roll in del, expressed at
http://www.jach.hawaii.edu/JACdocs/JCMT/MT/PPN/003.1/node6.html
that there is no need for such and I have edited that document
accordingly.
The current telescope control code uses the correct formulae. (Phew)
I thank Richard Prestage for asking for clarification.
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