Appendix: Tracking Slit Position Angle at Nasmyth Foci

A fixed spectrograph slit at a Nasmyth focus of an Alt-Az telescope changes its position angle (PA) on the sky as the telescope tracks an object. Conversely, a spectrograph slit must rotate if it is to maintain a constant PA on the sky as the telescope tracks. This document demonstrates the mathematics necessary for smooth tracking of the PA as the earth rotates.

Assumptions

Finding the PA for a given Dec and HA

First, solve the problem under the assumption that the slit is fixed to the Nasmyth platform; i.e., it cannot rotate to track objects. Finding the PA at any given time is a straightforward application of spherical trigonometry.

Finding the Vertical Angle (VA)

The declination of an object is delta, with north being positive. The latitude of an observer is phi, with north being positive. The Hour Angle of an object is HA; zero on the meridian, negative to the east, and positive to the west. In the equatorial (RA,Dec) coordinate system the position angle of a directed vector on the sky is measured from equatorial North toward equatorial East; this is EquPA. In the Horizon (Alt-Az) coordinate system the position angle of a directed vector on the sky is measured from Up toward an upright observer's Left; this is HorPA. The elevation of an object is h, and cos(h) is always positive. The Vertical Angle (VA) is defined as the PA of the local vertical as expressed in the equatorial system.

VA = EquPA - HorPA

(Note that in many applications this angle is mis-named the ``Parallactic Angle''. Lately this has been promoted by terminology found in the extremely useful SLALIB software by Pat Wallace, but the misnomer can be traced back at least as far as Spherical Astronomy by Smart. The correct usage of Parallactic Angle is in the context of the Equatorial and Ecliptic coordinate systems. In that context it describes the orientation of the ellipse that stellar coordinates traverse due to annual parallax. This was a necessary calculation for correcting the catalog position of objects to their apparent position, and it uses exactly the same spherical trigonometry, but it was a mistake to use the term ``Parallactic Angle'' in the context of the Horizon system.)

Spherical trigonometry gives

sin(h) = sin(phi) * sin(delta) + cos(phi) * cos(delta) * cos(HA)
cos(VA) = (sin(phi)*cos(delta) - cos(phi)*sin(delta)*cos(HA)) / sqrt(1 - sin(h)**2)
sin(VA) = sin(HA) * cos(phi) / sqrt(1 -sin(h)**2)

Finding the EquPA of a Nasmyth slit

The slit has a certain position angle as measured in the coordinate system of a fixed platform at the Nasmyth focus. This angle is NaPA. NaPA = 0 denotes a slit pointing toward the nadir. NaPA increases in the sense of N->E on the focal plane. This means that, all else the same, rotating the slit so as to increase NaPA also increases the EquPA.

For a typical 3 mirror Nasmyth configuration this means that NaPA increases in a counter-clockwise direction as seen by an observer who is blocking the light as it attempts to enter the spectrograph.

There are 2 Nasmyth platforms. The telescope elevation is not permitted to pass through the zenith. Using a metaphor wherein the telescope ``faces'' toward some (non-zenith) location in the sky, the ``Right'' Nasmyth platform is on the telescope's right. Similarly for the ``Left'' Nasmyth platform. These definitions hold for any Alt-Az telescope at any latitude. Define NaSign such that NaSign = +1 for the Right platform, and NaSign = -1 for the Left platform.

Given these definitions, the position angle of the slit on the sky is

EquPA = NaPA + (NaSign * h) + VA

Choosing the correct branch for EquPA

As the telescope tracks an object across the sky the EquPA changes. If it is desired to rotate the slit to follow the object then it is also important to prevent jumps from one branch of this result to another. The above formulae produce an angular result which is ambiguous by 2*PI radians. It is possible to choose families of EquPA which have values in the vicinity of NaPA.

The families of EquPA solutions for objects at various declinations have the property that they tend to cluster together at certain hour angles and to spread apart at others. There are two families of solutions for values in the vicinity of NaPA:

East
The values of EquPA for objects at various declinations cluster together when rising in the eastern part of the sky. BrSign = +1.
West
The values of EquPA for objects at various declinations cluster together when setting in the western part of the sky. BrSign = -1.
The choice of correct EquPA is demonstrated by the following pseudo-code, with angles in radians. This is the essential result of this article. 
NearBranch(b, x)
# b is an approximately correct value to which the result must be near
# x is the exact PA which is uncertain by 2*n*PI
{
    return x + nint((b-x) / (2*PI)) * (2*PI)
}

CorrectEquPA(x)
# x is the exact EquPA which is uncertain by 2*n*PI
{
    if ( delta > abs(phi) ) {
	# the object is too far north to encircle the local vertical
	return NearBranch(NaPA - HA - BrSign * PI, x)
    } else if ( abs(delta) <= abs(phi) ) {
	# the diurnal motion of the object encircles the zenith/nadir
	if (phi >= 0) {
	    # by using >= we demonstrate a N hemisphere bias
	    return NearBranch(NaPA, x)
	} else {
	    return NearBranch(NaPA - BrSign * PI, x)
	}
    } else if ( delta < -abs(phi) ) {
	# the object is too far south to encircle the local vertical
	return NearBranch(NaPA + HA, x)
    }
}

Tracking the sky with a movable slit

In the case of DEIMOS, the spectrograph can rotate to maintain a constant position angle on the sky, EquPA. It is straightforward to invert the above algorithm. From above we have

NaPA = EquPA - (NaSign * h) - VA

The pseudo-code which chooses the branch nearest to EquPA is then 

CorrectNaPA(x)
# x is the exact NaPA which is uncertain by 2*n*PI
{
    if ( delta > abs(phi) ) {
	# the object is too far north to encircle the local vertical
	return NearBranch(EquPA + HA + BrSign * PI, x)
    } else if ( abs(delta) <= abs(phi) ) {
	# the diurnal motion of the object encircles the zenith/nadir
	if (phi >= 0) {
	    # by using >= we demonstrate a N hemisphere bias
	    return NearBranch(EquPA, x)
	} else {
	    return NearBranch(EquPA + BrSign * PI, x)
	}
    } else if ( delta < -abs(phi) ) {
	# the object is too far south to encircle the local vertical
	return NearBranch(Sky - HA, x)
    }
}

Rates

In either of the above cases the rate of change of the position angle depends on d(VA)/d(HA) and dh/d(HA).

Warning: Next two equations not verified. 

 dh     -cos(phi) * cos(delta) * sin(HA)
----- = --------------------------------
d(HA)          sqrt(1 - sin(h)**2)
and 
d(VA)    cos(phi) * ( cos(HA) * sin(phi) * cos(delta) - cos(phi) * sin(delta) )
------ =  ----------------------------------------------------------------------
d(HA)                            1 - sin(h)**2

Implementations

Example Fortran 77 codes which implement these algorithms are available.
Steve Allen <sla@ucolick.org>
$Date: $