KECK INSTRUMENT TECHNICAL NOTE
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Nasmyth Instrument Tracking Velocity & Acceleration
Author
Gregory D. Wirth
Date
2010-May-06
Comment
Original
Synopsis
This document describes the predicted velocity and acceleration required for a Nasmythmounted instrument (e.g., DEIMOS) to maintain constant sky position angle while
tracking a source that passes close to the zenith. These calculations were requested by
Bruce Bigelow of UCO/Lick to aid in the design specifications for the MOBIE wide-field
optical spectrograph for the Thirty-Meter Telescope (TMT) project.
Method
As described in detail in KSD 40 (Lupton 2002), the fundamental equation governing the
physical position of a rotator mounted at the Nasmyth focus of an alt-az telescope such as
Keck is:
R = PA – P – EL
(1)
where
R = instrument physical rotator position (DCS keyword ROTPPOSN)
PA = projected sky position angle
P = parallactic angle
EL = elevation angle
This equation applies equally to proposed Nasmyth instruments on TMT due to its
similar alt-az design. To compute the velocity and acceleration for targets passing close
to zenith, we first compute the value of R as a function of hour angle for various
minimum zenith distances, then obtain the corresponding velocity and acceleration by
taking the first and second derivatives. We compute the position, velocity, and
acceleration for a set of 21 minimum zenith distances which vary logarithmically
between 0.1 deg and 10 deg, thus spanning the range of interest.
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Physical Rotator Position
We employ equation 1 to compute R, the physical rotator position for a Keck Nasmyth
instrument rotator as a function of hour angle; see Figure 1 below. The value of R varies
most slowly for a minimum zenith distance of 10 deg and more rapidly when the
minimum zenith distance is 0.1 deg. The particular case depicted here corresponds to
PA=0; for other values the shape of the curves remains the same but the ordinate shifts
correspondingly.
Figure 1: Physical rotator position for a Keck Nasmyth instrument rotator vs. hour angle as a
function of minimum zenith distance.
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Velocity
We derive the corresponding velocity of the rotator by simply computing dR/dT. Figure
2 shows the resulting velocity as a function of hour angle, and Figure 3 shows the peak
velocity as a function of zenith distance; of course, this will occur at the point HA=0.
Figure 2: Rotator velocity as a function of hour angle for various minimum zenith distance values.
Figure 3: Peak Nasmyth rotator velocity as a function of zenith distance.
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Acceleration
We derive the acceleration of the rotator by computing A = dV/dT. Figure 4 shows the
resulting acceleration as a function of hour angle, and Figure 5 shows the peak
acceleration as a function of zenith distance. Since the velocity peaks at HA=0, this is
actually a point of zero acceleration; the peak acceleration occurs elsewhere.
Figure 4: Rotator acceleration as as function of hour angle.
Figure 5: Peak acceleration as a function of minimum zenith distance.
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References
Lupton, W. F. 2002, Keck Software Document 40: DCS Coordinate Systems
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Appendices
Appendix 1: Tabulated Peak Velocity
Minimum
Zenith
Distance
[deg]
0.10
0.13
0.16
0.20
0.25
0.32
0.40
0.50
0.63
0.79
1.00
1.26
1.58
2.00
2.51
3.16
3.98
5.01
6.31
7.94
10.00
Maximum
Rotator
Velocity
[deg/sec]
2.189
1.755
1.403
1.118
0.891
0.708
0.563
0.448
0.356
0.283
0.225
0.178
0.142
0.113
0.089
0.071
0.057
0.045
0.036
0.028
0.023
Table 1: Peak rotator velocity as a function of minimum zenith distance.
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Appendix 2: Tabulated Peak Acceleration
Minimum
Zenith
Distance
[deg]
0.10
0.13
0.16
0.20
0.25
0.32
0.40
0.50
0.63
0.79
1.00
1.26
1.58
2.00
2.51
3.16
3.98
5.01
6.31
7.94
10.00
Maximum
Rotator
Acceleration
[deg/sec^2]
5.54E-02
3.50E-02
2.24E-02
1.43E-02
9.04E-03
5.70E-03
3.62E-03
2.29E-03
1.45E-03
9.14E-04
5.78E-04
3.66E-04
2.32E-04
1.47E-04
9.31E-05
5.91E-05
3.76E-05
2.39E-05
1.53E-05
9.77E-06
6.25E-06
Table 2: Peak acceleration as a function of minimum zenith distance