This little document is intended to clarify (and perhaps document) some of the issues related to the third axis (the image derotator) on the MMT telescope. Note that this can be called the rotator or the derotator depending on your mood at the time. Calling it the derotator is perhaps more accurate, although in this case you should go the full route and call it the field or image derotator. More commonly it is just called the rotator.

Some background

There are equatorial telescopes and then there are alt-az telescopes. The MMT is an alt-az telescope. An equatorial telescope is built so that one of its axes is parallel to the earths rotational axis. If this is done accurately enough, all that is necessary to keep the telescope pointing at a star is to move this one axis at a rate equal and opposite to the rate the earth rotates. For an alt-az telescope like the MMT, things are more complicated. Both the altitude and azimuth axes must both be moved at varying rates to track objects in the sky. The rate at which the earth rotates is commonly known as the "sidereal rate" (you would think it might be called the geodereal rate or some such, but the name "sidereal" caves in to the illusion that the stars are moving and the earth is standing still. The sidereal rate is pretty much 360 degrees per 24 hours, or 15 degrees per hour, or .004167 degrees per second. For small angles it is handier to use units of arc-minutes or arc-seconds, where 1 degree is 60 arc-minutes is 3600 arc-seconds. The prefix "arc-" avoids confusing these angular measurements with time units. The sidereal rate is 15 arc-seconds per second.

RA and Dec

Astronomers use a coordinate system in the sky that is a lot like latitude and longitude. Any point in the sky (or object in the sky) can be specified by giving a pair of coordinates. RA (Right Ascension) is the analog of longitude (it measures things east and west), and declination is the analog of latitude (it measures north and south). This analogy conveys the basic concept, but many of the details are different. RA is measured in time units (hours, minutes, and seconds) rather than degrees, so when making a full circuit from west to east, 24 hours of RA are covered. Objects in the east have a bigger RA value than objects in the west (ignoring the fact that RA jumps from 24 to 0 at a certain place). The place where RA is zero is the vernal equinox (the place where the sun is in the sky when spring begins). Declination is measured in degrees from the celestial equator (the projection of the earths equator on the sky) and is positive in the northern hemisphere.

Sidereal Time

As most people have noticed, the sun tends to rise in the east and set in the west. Of course it is the earth rotating (the earth is simply a gigantic clock) not the sun roaming around. Over the course of a day the sun is (crudely speaking, it actually moves about 4 minutes in 24 hours) fixed in RA and dec, and we can use it to meter "solar time" by its position relative to the horizon or zenith or some chosen point of reference. The sidereal time is the RA that is directly overhead at a given location on earth. In general the sidereal time is different for different places on earth (it is the same for all locations that have the same longitude). Since the RA is bigger for objects to the east, as the earths rotation causes these objects to seemingly move towards the west, sidereal time increases.

The image derotator - parallactic angle

As an alt-az telescope tracks an object, the image of that object will rotate. For projects that just want to track a single star (like spectroscopy) it may be possible to not care about this. For any use of an alt-az telescope that involves imaging or tracking more than one object in a field, it is necessary to do something about this image rotation, hence the need for a derotator.

When an alt-az telescope is pointed exactly north or south, the elevation axis moves along a line of constant RA (given by the current sidereal time) and a change in telescope elevation produces only a change in declination. For any other azimuth orientation there is some angular difference between a line of constant RA and the line accessed by moving the telescope solely in elevation. For any point in the sky, this angle can be calculated and is known at the parallactic angle. The parallactic angle is the angle between a line of constant azimuth and a line of constant RA. Lines of constant azimuth converge at the point directly overhead (the zenith). Lines of constant RA converge at the projection of the earths north pole on the sky.

Without a derotator, the top of the image produced by the telescope is always "toward the zenith". What we might prefer would be the top to always be aligned to point "toward the north pole". We can accomplish the later by moving the derotator by the value of the parallactic angle. Notice that as we track an object, we generate a continually changing motion in altitude and azimuth which cause the telescope to keep pointing to a constant value of RA and dec. To avoid field rotation, we must impose a third constraint, namely that the rotator points to the constantly changing value of the parallactic angle.

Position Angle

It leads to no end of confusion that Parallactic Angle and Position Angle have such similar names. In particular it is uniquely unfortunate that they both are commonly abbreviated "PA" and this leads to endless confusion. Position Angle is a way of specifying an angle centered on some object in the imaginary RA and Declination grid on the sky. The zero line is the line of constant RA through the object going towards the north celestial pole (the projection of the earths north pole). The angle is measured from this north line positive towards the east.

When a position angle is specified for an object (typically this is an "extended" object such as a galaxy that has some prefered angular alignment), this value is treated as a constant offset and added to the parallactic angle to produce the rotator command. This causes the field to be rotated relative to the usual "north at the top" orientation.