Scope

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.

Usage

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

A star separation with an error [as]
OIFITS geocentric Cartesian coordinates [m], as in eq (15) in arXiv:math.MG/0711.0642.
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 report.
time derivatives of these three OPD's with their error [mm/s and μm/s].

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 MIDI detector. 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.

Mathematics of Error Model

Calculations are based on linear error propagation of maximum, correlated errors.

"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, Err(opd(PS))= |opdPut1*Err(uT1)|+...+|opdPra*Err(Pra)|+..., and the equivalent hold for the secondary Err(opd(SS))= |opdSut1*Err(uT1)|+...+|opdSra*Err(Sra)|+.... Then the error in the difference becomes Err(dopd) = |(opdPut1-opdSut1)*Err(uT1)+..+|opdPra*Err(Pra)|+...+|opdSra*Err(Sra)|, not Err(opd(PS))+Err(opd(SS)). 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.

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 keywords. 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. Limitations 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 in the VLTI web page should be consulted to obtain more accurate pointing limits.

Richard J Mathar, www.mpia.de/~mathar