Appendix L: The Pointing Terms

This appendix defines the TPoint pointing terms. 

The pointing terms obey the following nomenclature and sign conventions:

·       Hour angle: h is positive west of the meridian.

·       Declination: d is positive north of the equator.

·       Parallactic angle: q is positive west of the meridian.

·       Azimuth: TPoint’s internal convention such that A is zero for due south and 90 degrees for due east. 

·       Zenith distance: z the distance from 90 degrees altitude.

·       Elevation: E = 90 - zenith distance.

·       Hour angle and east-west corrections: Dh and DX (= Dh cos d) are positive when corrected telescope position is west of the uncorrected position.

·       Declination corrections: Dd is positive when corrected telescope position is north of the uncorrected position.

·       Azimuth and left right corrections: DA and DS (= DA cos E) are positive when the corrected telescope position is to the left of the uncorrected position as seen by someone standing at the telescope looking at the sky.

·       Elevation corrections: DE is positive when the corrected telescope position is above the uncorrected position.

TheSky provides the sequence of transformations and corrections needed to bridge the gap between:

(i.)   the cataloged object’s position and

(ii.)   where the telescope needs to be pointed to see the object, that is, transforming them to the “observed place.”

The cosine component of the once-per-revolution cycle errors produced by a mis-centered azimuth setting-circle.

Notes: The sine component is ACES.

The sine component of the once-per-revolution cycle errors produced by a mis-centered azimuth setting-circle.

Notes: The cosine component is ACEC.

In an alt-azimuth mount, misalignment of the azimuth axis north south: rotation about a horizontal east-west axis equal to coefficient AN.

Notes: AN is one of the six purely geometrical terms that affect all alt-azimuth mounts, the others being IA, IE, CA, NPAE, AW.

If AN is positive, the pole of the mounting is north of the zenith.

The other (that is east-west) component of the azimuth axis misalignment is the term AW.

In an alt-azimuth mount, misalignment of the azimuth axis east west: rotation about a horizontal north-south axis equal to coefficient AW. 

Notes: AW is one of the six purely geometrical terms that affect all alt-azimuth mounts, the others being IA, IE, CA, NPAE, AN.

If AN is positive, the pole of the mounting is west of the zenith.

The other (that is north-south) component of the azimuth axis misalignment is the term AN.

In an alt-azimuth mount, the collimation error is the non-perpendicularity between the nominal pointing direction and the elevation axis.  It produces a left-right shift on the sky that is constant for all elevations. 

Notes: CA is one of the six purely geometrical terms that affect all alt-azimuth mounts, the others being IA, IE, AW, NPAE, AN.

A non-zero CA on its own means there is an area round the zenith inside which the telescope cannot point (irrespective of velocity and acceleration limits close to the zenith). 

Collimation error has many causes–misalignment of the optics, an inaccurately centered eyepiece reticle and so on.

In an equatorial mount, the collimation error is the non-perpendicularity between the nominated pointing direction and the declination axis.  It produces an east-west shift on the sky that is constant for all declinations.

CH is one of the six purely geometrical terms that affect all equatorial mounts, the others being IH, ID, NP, ME and MA.

A non-zero CH on its own means there is an area around the pole inside which the telescope cannot point.

Collimation error has many causes -- misalignment of the optics, an inaccurately centered eyepiece reticle and so on.

Downward sag of cantilevered declination axis.  English cross-axis and German equatorial mountings), proportional to the sine of the zenith distance of the declination axis.

Note: The sign convention corresponds to a declination axis that emerges from the polar axis towards the west when the telescope points at the meridian.

The cosine component of the once-per-revolution cyclic errors produced by a mis-centered declination setting-circle.

The sine component is the term DCES.

The sine component of the once-per-revolution cyclic errors produced by a mis-centered declination setting-circle.

The cosine component is the term DCEC.

For an equatorial mount, a change in the non-perpendicularity proportional to sin h.

Notes: The static form is the term NP.

The cosine component of the once-per-revolution cyclic errors produced by a mis-centered elevation setting-circle.

The sine component of the once-per-revolution cyclic errors produced by a mis-centered elevation setting-circle.

Flexure in an equatorial fork mounting: change in declination proportional to cos h.

Notes: This model of fork flexure is based in the idea that the flexure is at a maximum on the meridian, that it affects only declination, and that at h = 6^h the flexural displacement leaves the pointing direction unaffected.  These assumptions are usually borne out well in practice.

The cosine component of the once-per-revolution cyclic errors produced by a mis-centered hour angle setting-circle.

The sine component of the once-per-revolution cyclic errors produced by a mis-centered hour angle setting-circle.

Azimuth index error in an alt-azimuth mount: the zero-point error in A.

Take care with sign conventions: inside TPoint, azimuth increases counterclockwise.

Notes: On many alt-azimuth telescopes, IA is arbitrarily set at the start of each night by pointing at a calibration star, and on telescopes without absolute encoders or readouts there is no choice; under these circumstances the IA value returned by TPoint has no lasting significance.  However, note that IA is not the same thing as a horizontal error on the sky, such as might be caused by a badly adjusted primary mirror; the latter, addressed by the CA term, would produce the same image displacement at all elevations, whereas IA has a lessening effect as the zenith is approached.

IA is one of the six purely geometrical terms that affect all equatorial mounts, the others being IE, CA, NPAE, AN and AW.

Declination index error in an equatorial mount: the zero-point error in declination.

Notes: ID produces a fixed offset of the image north south.  On many telescopes, it is arbitrarily set at the start of each night by pointing at a calibration star, and on telescopes without absolute encoders or readouts, there is no choice; under these circumstances the ID value returned by TPoint has no lasting significance.  It is indistinguishable from other north-south shifts, such as those caused by a badly adjusted primary mirror.

ID is one of the six purely geometrical terms that affect all equatorial mounts, the others being IH, CH, NP, ME and MA.

Elevation index error in an alt-azimuth mount: the zero-point error in E.

IE produces a fixed vertical offset of the image.  On many alt-azimuth telescopes IE is arbitrarily set at the start of each night by pointing at a calibration star, and on telescopes without absolute encoders or readouts there is no choice; under these circumstances the IE value returned by TPoint has no lasting significance.  It is indistinguishable from various other vertical shifts, such as those caused by a badly adjusted primary mirror.

IE is one of the six purely geometrical terms that affect all alt-azimuth mounts, the others being IA, CA, NPAE, AN and AW.

Hour angle index error in an equatorial mount.

Notes: IH has the same effect on pointing as an error in the clock or using an inaccurate value for the site longitude.  On many telescopes, it is arbitrarily set at the start of each night by pointing at a calibration star, and on telescopes without absolute encoders or readouts, there is no choice; under these circumstances the IH value returned by TPoint has no lasting significance.  However, note that it is not the same thing as an east-west error on the sky, such as might be caused by a badly adjusted primary mirror; the latter, addressed by the CH term, would produce the same image displacement at all declinations, whereas IH has a lessening effect as the pole is approached. 

IH is one of the six purely geometrical terms that affect all equatorial mounts, the others being ID, CH, NP, ME and MA.

Misalignment of the polar axis of an equatorial mount to the left or right of the true pole: a rotation about an axis through h = d = 0  equal to coefficient MA.

Notes: In the Northern Hemisphere, positive MA means that the pole of the mounting is to the right of due north.

In the Southern Hemisphere, positive MA means that the pole of the mounting is to the right of due south.

To avoid unwanted field rotation effects it is best to eliminate MA mechanically by an appropriate azimuth adjustment.  To eliminate MA arcseconds of polar axis azimuth error, rotate the mounting MA sec f arcseconds counterclockwise as seen from above.

The other (that is, up and down) component of the polar axis misalignment is the term ME.

Vertical misalignment of the polar axis of an equatorial mount: a rotation about an east-west axis equal to coefficient ME.

Notes: In the Northern Hemisphere, positive ME means that the pole of the mounting is below the true (unrefracted) pole.  A mounting aligned the refracted pole (for most telescopes probably the simplest and best thing to aim for in order to avoid unwanted field rotation effects) will have negative ME.

In the southern hemisphere, positive ME means that the pole of the mounting is above the true (unrefracted) pole, and a mounting aligned the refracted pole will have positive ME.

The other (that is left-right) component of the polar axis misalignment is the term MA.

In an equatorial mount, if the polar axis and declination axis are not exactly at right angles, east-west shifts of the image occur that are proportional to sin d. 

Notes: NP is one of the six purely geometrical terms that affect all equatorial mounts, the others being IH, ID, CH, ME and MA.

The image displacement produced by NP is zero on the celestial equator and reaches a maximum at the pole.

Most telescope sites lie well away from the equator of the Earth and hence the correction for NP tends to be the same sign most of the time.  Furthermore, because there is more sky at lower declinations than near the pole there are likely to be fewer pointing observations in a given declination-band near the pole than in a similar band nearer the equator.  These two effects conspire so that the NP effect tends to be poorly sampled when pointing tests are made, and hence the NP coefficient is apt to be poorly determined.  In some cases, it can be better to omit NP from the model and rely on other terms mopping up any NP-like effects that are preset.

A non-zero NP on its own means there is an area around the pole inside which the telescope cannot point.

NPAE is one of the six purely geometrical terms that affect all alt-azimuth mounts, the others being IA, IE, CA, AN and AW.

The image displacement produced by NPAE is zero on the horizon and reaches a maximum at the zenith.

Just as E is always positive, the correction for NPAE never changes sign.  Furthermore, because there is more sky at lower elevations than near the zenith there are likely to be fewer pointing observations in a given elevation-band

near the zenith than in a similar band nearer the horizon.  These two effects conspire so that the NPAE effect tends to be very poorly sampled when pointing tests are made, and hence the NPAE coefficient is apt to be poorly determined.  In some cases, it can be better to omit NPAE from the model and rely on other terms mopping up any NPAE-like effects that are preset.

A non-zero NPAE on its own means there is an area around the zenith inside which the telescope cannot point (irrespective of velocity and acceleration limits close to the zenith).

In a Nasmyth alt-azimuth mount, a horizontal displacement between the elevation axis of the mount and the rotation axis of the Nasmyth instrument-rotator produces an image shift on the sky with a horizontal component proportional to cosine elevation and a vertical component proportional to minus sine elevation.

(Take care with sign conventions: inside TPoint, azimuth increases counterclockwise.)

Notes: The same corrections apply even if there is no Nasmyth rotator, except that the displacement is between the elevation axis and the nominated pointing-origin in the Nasmyth focal-plane.

NRX is positive when the elevation axis emerges from the Nasmyth focal plane to the right of the rotator axis.

In a Nasmyth alt-azimuth mount, a vertical displacement between the elevation axis of the mount and the rotation axis of the Nasmyth instrument-rotator produces an image shift on the sky with a horizontal component proportional to sine elevation and a vertical component proportional to cosine elevation.  (Take care with sign conventions: inside TPoint, azimuth increases counterclockwise.)

Notes: The same corrections apply even if there is no Nasmyth rotator, except that the displacement is between the elevation axis and the nominated pointing-origin in the Nasmyth focal-plane.  NRY is positive when the elevation axis emerges from the Nasmyth focal plane above the rotator axis.

The NRY correction has an identical form to the two terms NPAE and TF acting in combination.  This unfortunately prevents simultaneous fitting of pointing observations to a model containing all three terms.  The recommended procedure for Nasmyth telescopes is to determine a canonical value of TF from observations at a non-Nasmyth focus (prime, Cassegrain, Newtonian etc., and then to fit Nasmyth pointing observations to a model which includes TF fixed at this value.

Classical tube flexure: change in zenith distance proportional to sin z.

Notes: This is the classic tube flexure model, which assumes that the telescope obeys Hooke's Law.  In practice there is often a rapid increase in the vertical displacement towards the horizon, and it is sometimes found that a tangent law is a better approximation than a sine law: see the TX term.

Empirical tube flexure: change in zenith distance proportional to tan z.

Notes: The classical tube flexure model, TF, which assumes that the telescope obeys Hooke's Law, is a sine rather than tangent law.  In practice there is often a rapid increase in the vertical displacement towards the horizon, and it is sometimes found that the present term, TX, is a better approximation than TF.

Note that TX will compensate for any scaling error in the refraction corrections.