Back to Pytheia 2.1.1

Pytheia v2.1.0 — pointing terms

These 2.1.0 term definitions and control-system conventions are retained in Pytheia 2.1.1.

This document describes every built-in pointing term of Pytheia: what each one corrects, its exact formula, and the conventions needed to apply a fitted model to a telescope. The formulas are those of standard_terms.py, which is what Pytheia computes.

Conventions

Coordinates

• Azimuth (Az) is measured from north through east: 0° is north, 90° is east.
• Elevation (El) is measured from the horizon; z = 90° − El is the zenith distance.
• Hour angle (h) is positive west of the meridian; declination (δ) is positive north.
• φ is the latitude of the site, positive north.

The direction of a correction

Every term describes how the position of a star differs from where the encoders point:

star position = encoder position + Σ coefficient × correction

The encoder position is given by columns 1–2 of the data file, the star position by columns 3–4. Coefficients are in arcseconds; the corrections in the tables below are for a coefficient of 1.

For example, IE has the correction −1 in elevation: with IE = +10″, every star is found 10″ below the elevation the encoder reads.

The star position is the star's observed place: its catalogue position corrected for proper motion, precession, nutation, aberration, parallax and atmospheric refraction — where the star actually appears in the sky from the site. Pytheia's data must use it, and a control system computes it before applying the pointing model.

Corrections in azimuth and hour angle are changes of that coordinate. On the sky, such a change shrinks with cos El, or cos δ; Pytheia applies that factor itself, so it does not appear in the tables. A term describing a fixed shift on the sky therefore contains sec El, or sec δ.

Kinds

• rotation — applied exactly, as a rotation of the mount geometry; see How the rigorous model works in the README. The tables give their first-order form.
• encoder — evaluated at the encoder position.
• flexure — caused by gravity, and evaluated where the telescope actually points.

Applying a model in a telescope control system

The modules in the tcs/ folder apply a fitted model exactly as Pytheia does, in Python and in C, which C++ can use as it is; tcs/README.md shows how. Use them if you can. This section describes the model completely, for a control system in another language or for anyone who wants to know exactly what the modules compute. A control system must reproduce it in full: the same terms, the same signs, and the same steps.

From the encoders to the sky

Given the encoder position and the coefficients, in radians, for an alt-azimuth mount:

1. Roll and pitch from the encoders: a = 180° − Az, b = El.
2. The four rotation angles: ca = −CA, np = −NPAE, an = −AN, aw = −AW.
3. Pitch: x₁ = cos b · cos ca, y₁ = sin ca, z₁ = sin b · cos ca.
4. Axis non-perpendicularity: y₂ = cos np · y₁ + sin np · z₁, and z₂ = −sin np · y₁ + cos np · z₁.
5. Roll: Aₓ = cos a · x₁ − sin a · y₂, A_y = sin a · x₁ + cos a · y₂, A_z = z₂.
6. Tilt of the roll axis: V = Mᵀ · A, where M = R₁(aw) · R₂(−an), with R₁(t) = [[1, 0, 0], [0, cos t, sin t], [0, −sin t, cos t]] and R₂(t) = [[cos t, 0, −sin t], [0, 1, 0], [sin t, 0, cos t]].
7. Back to the sky: Az₀ = 180° − atan2(V_y, Vₓ), El₀ = asin(V_z).
8. Encoder terms: add, for every other term of kind encoder, coefficient × correction, evaluated at the encoder position. The correction in azimuth is added to the azimuth itself, as listed in the tables.
9. Flexure terms: add, for every term of kind flexure, coefficient × correction, evaluated at the result, then evaluate them again at the new result, and repeat until it changes by less than 10⁻⁸″ — usually two to five times. Flexure follows where the telescope actually points, which is what is being computed.

For an equatorial mount, the same steps with a = −h, b = δ; ca = +CH, np = +NP, an = −ME, aw = +MA; and in step 7, h₀ = −atan2(V_y, Vₓ), δ₀ = asin(V_z).

Pointing at a star: from the sky to the encoders

To point at a star at position S, start with the encoder position P = S, and repeat: compute where the telescope would point from P, with the steps above, and move P by the difference between S and that result. Stop when the move is below 10⁻⁸″ — usually after three to five repetitions. P is where the encoders must go; P − S is the correction.

Reference values

These are the values each module checks itself against, computed by Pytheia. An implementation of the steps above must reproduce them to 10⁻⁵″, in both directions: from each encoder position to the star position, and back.

Az/El mode, latitude -29.259°, auxiliary reading aux1 = 0.5. Coefficients, in arcsec: IA -1238.04, IE +598.41, CA -129.11, NPAE +20, AN -64.37, AW -751.69, TF +71.11, TX -14.2, PZZ3 -20.84, ACEC -7.11, ECES +229.69, HZSZ2 +7.44, HZSA2 +3.1, NRX +4.5, NRY -2.2, AUX1E +3.

HA/Dec mode, latitude 33.3563°, auxiliary reading aux1 = 0.5. Coefficients, in arcsec: IH +94.91, ID +123.66, CH -46.42, NP +1.14, ME -59.75, MA +2.92, TF -4.53, TX +2.1, FO +3.3, HCES +1.7, HXSH2 -9.59, DCEC +2.2, HDSH2 -1.1, AUX1D +1.5.

The rotations are part of the model

CA, NPAE, AN and AW, or CH, NP, ME and MA, must be applied through the rotation steps. Replacing them with their first-order formulas from the tables changes where the telescope points. In our tests the difference was below 0.3″ for coefficients of a few tens of arcseconds, but reached 6″ below 70° elevation, and 24″ below 85°, for an azimuth-axis tilt of 750″.

Coefficients belong to their definitions

A fitted model is valid only with the term definitions it was fitted with. Version 2.1.0 changed the definitions of some terms (see CHANGELOG.md): coefficients obtained with an earlier version must not be used with these definitions, nor with these modules.

Az/El mode

Geometry of the mount

Flexure

Encoder centring

Harmonics

Nasmyth rotator

Auxiliary readings

HA/Dec mode

In the formulas for TX, sin a = sin δ sin φ + cos δ cos h cos φ is the sine of the altitude.

Geometry of the mount

Flexure

Encoder centring

Harmonics

Auxiliary readings

How the names are built

Most harmonic terms follow one pattern: H, then what is corrected, then S or C for sine or cosine, then the coordinate it varies with, then the multiple, which is left out when it is 1.

For example, HZSZ2 corrects the zenith distance (Z), varying as the sine (S) of twice (2) the zenith distance (Z). The polynomial terms follow the same idea: PZZ3 corrects the zenith distance (Z) in proportion to the zenith distance (Z) to the third power.

Terms that are identical

Some terms have the same formula, up to its sign, under different names:

The data cannot tell two such terms apart: fitting both only splits one effect between them arbitrarily, and the report says so. Fit the one whose physical meaning matches the effect you expect.

Your own terms

Terms of your own go in additional_terms.py, in the same form and with the same convention as the built-in ones: each correction is the change from the encoder position to the star position, for a coefficient of 1. The instructions at the top of that file and the Terms section of the README explain the rest.

The control-system modules need them too: the Python module reads additional_terms.py directly, and the C module has a marked function, user_term(), where each term is added in one line. tcs/README.md explains both.