This ``forward'' error calculator of the Astrometrical
Operations computes variations
in the baseline length and optical path difference (OPD) given
variations in the telescope coordinates and star positions.
This is correlated with but not the same as answering the question
how errors in the baseline and OPD propagate backwards to errors
in the star coordinates.
Graphical User Interface, Applet
Fields above the "Update" button are editable
after the telescope stations have been selected. Fields below the "Update"
button are not editable with the exception of the MIDI subsection. Image
snapshots can be generated with xwd(1) and printed after using convert(1),
or running the appletviewer(1) and using its print option.
The Paranal Origin of (U,V,W) coordinates for generic
sky chart calculators
are 70° 24' 17.953'' West, 24° 37' 38.782'' South.
Optionally edit the set of station coordinates with their geodetic parameters
(longitude in degrees, latitude in degrees, including their signs, altitude
above the tangential plane of the geoid) and error estimates.
Nonzero error estimates must follow the values separated by "+-".
The name "ZE" at the end of the list refers to the nominal origin of the
array; for the VLTI this is the U=V=W point. Names "Ke1" and "Ke2" have
been intialized with the Keck coordinates.
Optionally set the model of the Earth flattening and radius to the values
selectable from the WGS84 or two IAU conventions. The 1/f entry for the
inverse of the flattening factor and equatorial radius are only editable
if the selector "others" has been chosen.
(For comparison: see D Egger's Location Observer applet.)
Select stations for telescope 1 (T1) and 2 (T2),
specify two star coordinates for the primary (PS) and secondary (SS) star with
associated error bars.
These are apparent coordinates, proper motion already included.
Type in a local sidereal time between 0 and 86400 seconds with an error
bar separated from its reference value by a +- sign. (The default represents
the time the GUI is started at the location of the telescope array.)
Click on "Update below!" to compute
The square plot at the bottom shows a green circle symbolizing a diameter of
2 arcsec (radius of 1 arcsec) on the sky in the x-y FITS coordinates of the
North (N), East (E) and projected baseline (b) directions to T2
are also marked with green lines (and these may occur twice if hybrid telescope
pairs are selected) as explained in the reference given above. Red
symbols show the primary star (PS) and the secondary star (SS)
at a location compatible with the star orientation and field orientation angles.
(This plot does not yet take into account optical trains that use the
star separator or the differential delay lines which may be installed
at some time in the future.)
Blue lines of equal differential delay run perpendicular to this
direction of the projected baseline at a stride [micron per line]
that can be selected to the right of the graph by editing the blue number.
Optionally a brown rectangular grid of the detector pixel mesh can be added
for the two different MIDI cameras. Changing the zoom factor helps
to locate the SS if the star separation is larger than 1 arcsec.
- A star separation with an error [as]
- OIFITS geocentric Cartesian coordinates [m], as in eq (15)
- A baseline length between the telescope stations with an error [m]
- projected baseline and star position angles [deg] referring to the
direction from the PS to the North Celestial Pole as zero degrees.
The angles represent rotation angles within the tangent plane pinned
at the PS to the celestial sphere around the pointing direction of
the first telescope, see the section on "How does the baseline project onto
the sky?" in the reports
MIDI Optical Path Differences and Phases
and Serb. Astron J. 179 (2009).
- Four azimuth angles (A), zenith angles (z) and parallactic angles (p)
[deg] for the two stars as seen
from the local coordinate systems of the telescopes,
- OPD's for both stars with their error [m], and the differential OPD
DOPD as their difference [mm]. The sign convention of OPD's is that
they are greater than zero if the star is closer to T2 than to T1.
Conversion to the VLTI sign convention is explained in the aforementioned
- time derivatives of these three OPD's with their error [mm/s and
Mathematics of Error Model
Calculations are based on linear error propagation of maximum, correlated
The model represents telescopes and delay lines in vacuum.
Note that OPD's are signed variables: the definition here is OPD=OPL(T1)-OPL(T2)
and the value is positive if the star is closer to T2 than to T1.
The Azimuths are 0 in the North direction
and becoming positive when turning to the West. Subtract this value
from 180 degrees (modulo 360)
to obtain the values reported by the ISS in ISS AZ FITS header
The color of the output field of the star separation
changes if the separation becomes larger than 2 arcmin. The color
of the altitude output field z changes if the zenith distance
becomes larger than 60 deg.
by the skyline of neighbouring domes (see the issshadow files in
my VLTI web page)
are not considered.
If the external OPD becomes too large for compensation by the main delay line
ranges, the field of the OPD and DOPD is colorized to indicate that restriction.
This is an estimate based on a simple geometric model, and the issgui files
VLTI web page
should be consulted to obtain more accurate pointing limits.
"Maximum" and "linear" indicate that the error of a dependent
variable f is calculated from errors in the independent source errors
Err(i) as Err(f)=sum over i |grad f_i *Err(i)|; ie, the maximum norm and not
the square root norm is used to the first order in the Taylor expansion.
A typical effect of the linear approximation is that the error in the w-coordinate
does not effect the baseline length between pairs of AT's or pairs of UT's,
but only for mixed-type telescopes.
"Correlated" indicates that the error in the calculation of any variable
via intermediate variables, g(f1(i),f2(i),..)) does not uncorrelate the errors
in the intermediate variables as with Err(g)= sum over j |grad g_j *Err(fj)|,
but tracks them back to the gradient in the independent source errors.
Example in simple case of the error in the differential dopd=opd(PS)-opd(SS):
Let the error in the OPD of the primary star be formed by errors assumed
on the u-coordinate of T1 and the right ascension of the primary star,
and the equivalent hold for the secondary
Then the error in the difference becomes Err(dopd) =
This correlation propagation often results in smaller errors in the dependent
variables than guessed from using the error bars in the intermediate variables,
which is obvious from using the triangular inequality in the example.
Richard J Mathar,