Controlling a Three Meter Mirror Array Reflector to Track Stars

Controlling a Three Meter Mirror Array Reflector to Track Stars Benjamin Adams University of Utah Department of Physics and Astronomy August, 2009

Abstract The University of Utah Gamma Ray Group and the Department of Physics and Astronomy have assembled two 3 meter optical reflector telescopes in the desert outside of Grantsville, Utah. This thesis project was to provide software programming and associated hardware to control the positioning and star tracking operations of the telescopes. The hardware included the use of servo motors and position encoders, a communications link between the telescopes and computer, and mechanical safety switches for position limits. The software was completely written in National Instrument's LabVIEW. The end goal was to provide a system that would, at a greater precision than the telescope's resolution, move the telescope to any fixed point, as well as track a star's position over time.

Table of Contents Page 1.0 Introduction 5 2.0 The System Design 2.1 Hardware 2.1.1 Mechanical Limit Safety Switches 7 2.1.2 Communication Link 11 2.1.3 Servo and Encoder Wiring 13 2.2 Software 2.2.1 Communication Protocol 16 2.2.2 Tracking Algorithm with Servos and Encoders 17 2.2.3 Coordinate System Transformations 19 2.2.4 Precession and Corrections 21 2.2.5 Software Limit Safety Switches 22 3.0 Operations Guide 3.1 Power-up 24 3.2 Starting the program 25 3.3 Finding the Reference Zeros for Each Axis 26 3.4 Positioning 3.4.1 Track a Star 27 3.4.2 Go to a Position 28 3.4.3 Manual Entry 30 3.5 Ending Communication 30 3.6 Power Down 31 4.0 Conclusions and System Validation 32 5.0 References 38

Table of Figures Page Figure 1, Horizon Coordinate System 5 Figure 2, Equatorial Coordinate System 6 Figure 3, Safety Switch Wiring Diagram 8 Figure 4, Power Relays 9 Figure 5, Elevation Safety Switch 10 Figure 6, Azimuth Safety Plug 10 Figure 7, Communications Wiring Diagram 11 Figure 8, Servo and Encoder Wiring Diagram 13 Figure 9, Encoders, Encoder Unit 14 Figure 10, Servo Amplifier/Controller, Motor 15 Figure 11, Software Code Example Servo Commands 16 Figure 12, Servo Movement Velocity vs. Position 17 Figure 13, Servo Movement Position vs. Time 18 Figure 14, Software Code Calculating Sidereal Time 19 Figure 15, Software Code Transf. from Equatorial to Horizonal Coord. 20 Figure 16, Software Code Precession Function 21 Figure 17, Software Code Corrections Function 22 Figure 18, Software Main Tab 25 Figure 19, Software Manual Control Tab 26 Figure 20, Software Positioning Tab 27 Figure 21, Software Track Selection Sub-Program 28 Figure 22, Software Position Selection Sub-Program 29 Figure 23: Tracking Data Azimuth Axis 32 Figure 24: Tracking Data Elevation Axis 33 Figure 25: Tracking Data Elevation Axis Closer 34 Figure 26: Tracking Data Sample from Data File 34 Figure 27: Star Images, Alp Lyr 36

1.0 Introduction The University of Utah Gamma Ray Group and the Department of Physics and Astronomy acquired a number of mirror array reflecting telescopes that will be used for a wide range of astronomical research. Two of these 3 meter antennas have been assembled in the Grantsville desert to begin this work. The telescopes are an altazimuth design which provides simple pointing in its native horizon coordinate system. The horizon coordinate system consists of two axes, the azimuth and the altitude or elevation. An object s natural horizon is the primary plane and the zero degree elevation point. The azimuth maps out 360 degrees around this primary plane. In the Duffet- Smith convention, 0 degrees azimuth is North, and increases clockwise as you go East. Though more simple to build and operate, the altazimuth design does not easily lend itself to tracking stars. Figure 1: Horizon Coordinate System The equatorial coordinate system is what the Grantsville telescopes use to chart and track stars. This system bases the star s position on its location on the celestial sphere rather than on the observer s location on Earth. The celestial sphere is a sphere that places the Earth at the center, and all celestial bodies at its surface. The Earth s poles and equator are projected on to this sphere to help with charting. The two axes for locating positions in this system is the declination and right ascension. The vernal equinox, where the sun passes through the celestial equator in March, is located at declination zero, and right ascension zero. Right ascension is measured in hours, minutes, and seconds. It is much like Earth s longitude. A star sits on a line that goes from one celestial pole to the other and passes through it, and the

celestial equator. Right ascension is the angle between the vernal equinox and the point that this line passes through the celestial equator. Declination is much like Earth s latitude. It is the angle between the stars position and the celestial equator. Figure 2: Equatorial Coordinate System To use an altazimuth telescope to point at stars that are charted in the equatorial coordinate system, a transformation must be done. Because the Earth constantly rotates, these two coordinate systems are always changing positions in relationship to each other. To do the transformation, it is necessary to use time in the calculation. To go from equatorial to horizon coordinates: Where A is for azimuth, a is for altitude or elevation. δ is for declination, H for hour angle. φ is the observer s geographic latitude. From hour angle we can find the right ascension (α), using the local sidereal time (LST): H = LST α

Sidereal time is a time system that s day is approximately 23 hours, 56 minutes, and 4.1 seconds relative to our standard time. As opposed to basing a day on the time it takes for the Earth to make a revolution relative to the sun, it bases it on the time it takes for the Earth to make a revolution relative to the distant stars on the celestial sphere. Since sidereal time is required to make the necessary transformations, it is also necessary to calculate this from standard time. In the software section, the actual code containing the transformation is included. When tracking stars using charts that are based on the equatorial coordinate system, depending on the level of accuracy that is required, it may be necessary to correct for precession. Precession of the Earth is the slow shift in the orientation of Earth s axis of rotation over time. Its motion can be described as a wobbling top that traces out a cone once every 26,000 years. This slow change results in a stars location in the sky shifted slightly from the charts. So the process to account for this shift looks at the position in the sky when the chart was made, and calculates where it will be currently. The calculation for this is shown in the actual program code later on in the thesis. These are the basic elements that need to be understood when pointing at a star with an altiazimuth telescope.

2.0 The System Design 2.1 Hardware 2.1.1 Mechanical Limit Safety Switches The telescopes have mechanical limits to their movement. Due to data and power cables, the telescope can't move more than 360 degrees in either azimuth direction. The mirror array's size prohibits movement much below 0 degrees in elevation. Moving too far in either of these two axes with cause tremendous damage to the structure and electronics. As a result, limit safety switches are a necessary part of a control system. Because software can not be fully trusted with the protection of the telescope and users, mechanical safety switches have been installed to ensure that movement stops when limits are reached. To be certain that movement stops when necessary, the mechanical switches directly control the power source that powers the servo motors through relays. Figure 3: Safety Switch Wiring Diagram

24 volts DC continually holds the relays in the on position, allowing power to flow to the servo motors. This 24 volt loop includes a plug at the azimuth axis and a toggle switch at the elevation axis. The 24 Volts is provided by a din-rail mounted power supply that also powers the serial server, and the fiber optic extender. Figure 4: Power Relays The elevation axis has two metal brackets that move as the elevation moves, one for the upper limit, and one for the lower limit. As a limit is reached, the bracket makes contact with the switch, trips the relay, and stops movement.

Figure 5: Elevation Safety Switch Since the azimuth limit is based on the cables being pulled too tight as the telescope spins, the safety mechanism is connected directly to the cables. If a limit is reached and the cables are pulled to tightly, the plug with pull apart, the relay trips, and movement stops. Figure 6: Azimuth Safety Plug

2.1.2 Communication Link Figure 7: Communications Wiring Diagram Communication with the encoders and the servos are done with special serial cables. Wiring diagrams for each type of cable is located in their operations manuals which will be made

available to the group. Those serial cables are plugged in an Ethernet Serial Server. This device allows the serial data to be communicated over the network. On the computer side, this serial data can be transmitted and received by either creating a virtual serial port (a driver provided by the manufacturer of the serial server), or by TCP/IP. TCP/IP was selected as the method of communication because of its stability and because it required no additional drivers or operating system settings. The network consists of a four port router that is connected to the workstation computer, the two serial servers, and a GPS clock. It is not connected to a gateway. Between the serial servers and the network router there are fiber optic extenders. The communication link takes up considerable network bandwidth, so this primary network should not be added upon to provide internet access, etc.

2.1.3 Servo and Encoder Wiring Figure 8: Servo and Encoder Wiring Diagram

Figure 9: Encoder Unit The Encoder system consists of a main display unit, and two position encoders. The main display unit, located in the telescope rack, has a front panel display of the current raw degree position of each encoder. It is the main unit that communicated the positions via serial as well. There is a single cable that provides both low voltage power and data communication to each position encoder located on the telescope. This is a pre-made cable with mil-style connectors.

Figure 10: Servo Amplifier/Controller, Motor Each servo has an amplifier/controller that is located in the telescope rack. It requires 220V power that is provided by a 3-phase generator that is installed behind the control shed. A breaker is located in the rack next to the amplifier/controllers for safety. Two cables connect the servo motor to the amplifier/controller. One provides power and uses a mil-style connector on the motor end, and screw down terminals on the amplifier/controller side. The other provides control data and uses a D-sub style connector on the amplifier/controller side, and a mil-style connector on the motor side.

2.2 Software 2.2.1 Communication Protocol Communicating with the servos and encoders is done through a TCP/IP connection to each serial port on the ethernet serial server connected to the devices. Entering the serial server's local IP address and the serial's port number is all that is required for addressing. All of the specific port settings are saved in the non-volatile memory of the serial server. Both the servos and encoders communicate using hexadecimal strings. The encoder is very simple to communicate with. Sending a simple request signal, /02, you are than able to read back a string of data that has position information for each axis. One iteration of sending the command and reading the information takes 140 milliseconds. The servo is a bit more difficult to communicate with. Though it has a serial port, it was designed only to use with its own proprietary software. After signing a non-disclosure form, and obtaining the communication protocol, I was able to determine what commands would make it work. Every routine for the servo, from setting up communication, to changing velocity takes 3 steps. The first step is a communications check. The second is a routine request. The third confirms the request. Each step is answered by the servo acknowledging the requests. Once initial communication is established, these 3 steps continue to repeat without break during the entire session. This requires special timing and network resources. The following is an example of the code for one routine. Figure 11: Software Code Example Servo Commands

2.2.2 Tracking Algorithm with Servos and Encoders When the group acquired the telescopes, they came with some control hardware that had already been designed to use. This was the case with the servo motors and position encoders. The encoders are made by Heidenhain and have a resolution of 0.001 degrees. One is mounted on each axis to provide continual position feedback. These are absolute encoders that have a built in reference zero, so that repeatable positioning could be accomplished. The servo motors are made by Sanyo-Denki. They are operated essentially by varying their speed incrementally over a range slower that we would want to go, to much faster than the telescope would want to move. Using the combination of a motor and an encoder, positioning can be accomplished by writing a programming algorithm commonly known as a servo loop. In this program, the speed of the motor is a function of how far it is from where it needs to be as well as from where it started. For example, when it is very close to its start position, its speed is slow. As well, as it gets further away from its start position, the speed increases to a limit. Equally so with the end position, the closer it gets, the slower it goes. This function is evaluated 5 times every second in a programmed loop. This is depicted below. Figure 12: Servo Movement Velocity vs. Position

Figure 13: Servo Movement Position vs. Time This algorithm is the basis for all of the positioning that the telescopes perform. For the tracking routine, the end position is not a fixed point, but changes over time based on position of a star. The telescope follows this end position, and thus tracks the star. In repeated observations, positioning can be accomplished with 0.001 degree precision, and tracking can be accomplished with greater than 0.010 degree precision. This accomplishes the goal of positioning with a better precision than the approximate 0.100 degree resolution of the telescopes.

2.2.3 Coordinate System Transformations In the introduction, it was explained that in order to point at stars in the equatorial coordinate system, it requires a transformation from the altazimuth telescope's horizontal coordinate system. This transformation requires that both the current time and location be known. Below is the actual code for this transformation. Figure 14: Software Code Calculating Sidereal Time

Figure 15: Software Code Transformation from Equatorial to Horizonal Coordinates

2.2.4 Precession and Corrections There are some corrections that need to be made so that after the transformation, the telescope is pointing at the desired location. Precession is the gradual shift of the Earth's axis of rotation. As a result of this shift, the stars location is slightly shifted over time as well. Once the telescope is in the equatorial coordinate system, a correction for precession must be made. The actual code for this correction is shown below. Figure 16: Software Code Precession Function Before the raw azimuth and elevation data from the encoders can be transformed to the equatorial coordinate system, it needs to be corrected to the true azimuth and elevation. There are a number of factors that cause the raw data to be incorrect. The first order is that the reference zeros of the encoders are not at true elevation and azimuth zeros. This is relatively easy to correct. The more difficult problems come from the telescope itself. Though I don't know all of the sources of misalignment, there are a few common ones. The telescope is not level. If the base in not completely level, this will cause a large misalignment in the elevation axis as well as a change in the azimuth axis.

The two axes are not orthogonal. If the elevation axis is not exactly orthogonal to the azimuth axis, this will cause incorrect reading in both the elevation and the azimuth axis. This thesis makes an attempt to understand the main causes for misalignment, but does not attempt to correct for it alone. Below is the actual code for some initial corrections that have been applied to account for misalignment. This section of code will be adjustable as further data is processed. 2.2.5 Software Limit Safety Switches Figure 17: Software Code Corrections Function Programmed in to the code of the system are limit switches. These prevent the user from using the software to move the telescope too far in any direction. The limits are as follows: Azimuth: Lower Limit = 0 degrees Upper Limit= 450 degrees Elevation Lower Limit = -2 degrees Upper Limit = 90 degrees

Just past these software limits, mechanical limit switches are installed as a backup. The limits will manifest themselves to the user in a couple of ways. During the selection of a position or star, the software will determine if it is within its moveable limits. If it is not, the user will be notified, and movement will be denied. If the user is manually controlling the speed of the movement with no programmed end point, the software will immediately cease communication with motors if a limit is reached, causing all movement to stop. Software switches are the first line of protection and work fine, assuming no software malfunction.

3.0 Operations Guide 3.1 Power-up When the site is not in use, the power is shut off to all but the GPS clock unit. Inside the control shed. Plug in the two 110V extension cords inside the control shed. This will provide power for some of the electronics in the telescope racks. Power on the battery backup box in the control shed. Power on the workstation computer. On the back side of the shed Pull the 220V power switch to the off position. Press the green start button on the 3-phase generator, and wait for 5 seconds for warm up. Pull the 220V power switch to the on position. At the telescope rack location Power on the encoder main unit by flipping the main switch in the back. Press 'Ent' on the encoder main unit keypad to prepare to find reference zeros. Flip the breaker to the on position. The servo amplifier/controllers should indicate a ready signal by displaying three horizontal bars.

3.2 Starting the program On the workstation computer On the desktop of the computer, double click on the main telescope program. Click the white run arrow in the upper left hand corner to start the program. Select the telescope you would like to control. Click the 'Set Up Communication' button on the Main tab. Follow the onscreen prompts. Figure 18: Software Main Tab At the telescope rack location Ensure that you see a figure 8 pattern on each of the amplifier/controllers signifying successful communication.

3.3 Finding the Reference Zeros for Each Axis When the telescope is in the home or park position, it is located just negative of the raw reference zeros for each axis. When the antennas are powered up, it is necessary to manually move the telescope with the software up and to the right to find each of the zeros. Finding reference zeros is only necessary at the beginning of the session or if the encoder main unit has power cycled. On the workstation computer Go to the Manual Control tab. Click 'Start Manual Control' Click the 'UP' button, and the 'RIGHT' button. Set each axis speed to 10. Figure 19: Software Manual Control Tab At the telescope rack location Note: the telescope will be moving slowly now in both axis. Watch the encoder main unit until the positions begin to register on each axis. On the workstation computer Set each axis speed to 0. Un-click the 'UP' button and the 'RIGHT' button. Un-click 'Start Manual Control'.

3.4 Positioning There are 3 different positioning activities: Track a Star, Go to a Position, or Manual Entry. 3.4.1 Track a Star Figure 20: Software Positioning Tab On the workstation computer Go to the Positioning tab. Click on the 'Track a Star' button. Note: a sub-program will launch. Select a data file containing your star position data Add any position filters to narrow your search for a star. Select and confirm a star by double clicking on its line and clicking yes. Note: the subprogram will load the star data into the tracker. When you are ready to track, click 'Start' Note: if your selected position fall outside of the telescopes limits, it will notify you and deny movement.

Figure 21: Software Track Selection Sub-Program At the telescope Confirm that the telescope is moving to your selected location. Caution: the telescope may move quickly at times, and it is much stronger than you. On the workstation computer Click stop when you have completed your task. 3.4.2 Go to a Position On the workstation computer Go to the Positioning tab. Click on the 'Go to a Position' button. Note: a sub-program will launch. Select a data file containing your position data Add any position filters to narrow your search for a position.

Select and confirm a position by double clicking on its line and clicking yes. Note: the subprogram will load the position data into the tracker. When you are ready to move, click 'Start' Note: if your selected position fall outside of the telescopes limits, it will notify you and deny movement. Figure 22: Software Position Selection Sub-Program At the telescope Confirm that the telescope is moving to your selected location. Caution: the telescope may move quickly at times, and it is much stronger than you. On the workstation computer Click stop when you have completed your task.

3.4.3 Manual Entry This mode allows you to manually enter a fixed azimuth/elevation position, or a moving right ascension/declination position. On the workstation computer Go to the Positioning tab. Click on the 'Manual Entry' button. Toggle to either the azimuth/elevation side or the right ascension/declination side. Enter your position information When you are ready to move, click 'Start'. Note: if your selected position fall outside of the telescopes limits, it will notify you and deny movement. At the telescope Confirm that the telescope is moving to your selected location. Caution: the telescope may move quickly at times, and it is much stronger than you. On the workstation computer Click stop when you have completed your task. 3.5 Ending Communication On the workstation computer Follow the instructions in the 'Go to a Position' section to move the telescope to the park or home position, using the main position data file. At the telescope Confirm that the telescope is moving to the park or home position. Caution: the telescope may move quickly at times, and it is much stronger than you. On the workstation computer Go to the Main tab. Click the 'End Communication' button. Follow the same procedure for the other telescope if that one is also running. Click the 'End Program' button.

3.6 Power Down At the telescope rack location Power off the encoder main unit by flipping the main switch in the back. Flip the breaker to the off position. On the back side of the shed Pull the 220V power switch to the off position. Press the red stop button on the 3-phase generator. Inside the control shed Unplug the two 110V extension cords. Power down the workstation computer. Power off the battery backup unit.

1.0 Conclusions and System Validation To demonstrate that the system is stable and tracks stars at the desired precision, I performed a one hour test in which I moved to a star from a stopped position and tracked its location. I collected time, elevation and azimuth target and actual positions, and photos of the stars image at the focal plane of the telescope. This test included only the first order corrections described in the corrections section. As a result, the image of the star does not stay perfectly centered over the hour of tracking. Below are the graphs of the entire tracking session. 210 190 Azimuth Angle (degrees) 170 150 130 110 90 70 50 Az Position Az Target 63832 64029 64226 64422 64619 64815 65011 65207 65403 65599 65796 65991 66188 66383 66580 66776 66973 67171 Time in Seconds (LST) Figure 23: Tracking Data Azimuth Axis

95 90 Elevation Angle (degrees) 85 80 75 70 65 60 55 50 El Position El Target 63832 64018 64203 64388 64573 64758 64943 65128 65314 65498 65684 65868 66053 66238 66423 66608 66793 66979 67165 Time in Seconds (LST) Figure 24: Tracking Data Elevation Axis In the graphs, you will see that at the beginning, the telescope moves towards the star and begins tracking it. From the point that it reaches the star, it continues to track without interruption. To better see the two traces, we can zoom in on the graph to show how well it tracks.

80.45 Elevation Angle (degrees) 80.4 80.35 80.3 80.25 El Position El Target 80.2 63953 63955 63958 63960 63963 63965 63967 63970 63972 63975 63977 63979 63982 63984 63987 63989 63991 Time in Seconds (LST) Figure 25: Tracking Data Elevation Axis - Closer The telescope does not have the ability to exactly match the speed of a star s motion because it has a finite number of different speeds. However, you can see that it still has the ability to track with excellent precision and the variations are imperceptible since they are much smaller than the resolution of the telescope. To quantify how well it tracks, I calculated the angle between the target and actual positions over the entire session. A sample of that data is below. Time in Seconds (LST) Az Position (degrees) Az Target (degrees) El Position (degrees) El Target (degrees) Angle (degrees) 65488.9824 106.519 106.536489 85.021 85.028753 0.005586 65489.98512 106.521 106.550043 85.022 85.03176 0.007127 65489.98512 106.524 106.550043 85.023 85.03176 0.006397 65490.98784 106.532 106.550043 85.025 85.03176 0.004906 65490.98784 106.541 106.56361 85.027 85.034767 0.005664 65490.98784 106.551 106.56361 85.03 85.034767 0.003458 65491.99056 106.558 106.577191 85.032 85.037774 0.004248 65491.99056 106.561 106.577191 85.033 85.037774 0.003518 65492.99328 106.562 106.577191 85.034 85.037774 0.002826 65492.99328 106.564 106.590787 85.034 85.040781 0.005067 65492.99328 106.566 106.590787 85.035 85.040781 0.00436

65493.996 106.575 106.604396 85.036 85.043787 0.005792 65493.996 106.584 106.604396 85.039 85.043787 0.003607 65494.99872 106.602 106.604396 85.041 85.043787 0.001976 65494.99872 106.618 106.618019 85.042 85.046793 0.003389 65494.99872 106.625 106.618019 85.042 85.046793 0.003416 65496.00144 106.625 106.631656 85.043 85.049799 0.004825 65496.00144 106.625 106.631656 85.044 85.049799 0.004121 65497.00416 106.626 106.631656 85.045 85.049799 0.003411 65497.00416 106.628 106.645308 85.046 85.052804 0.004926 65497.00416 106.63 106.645308 85.047 85.052804 0.004209 65498.00688 106.632 106.658973 85.048 85.05581 0.005762 65498.00688 106.636 106.658973 85.049 85.05581 0.005015 65499.0096 106.645 106.658973 85.05 85.05581 0.004196 65499.0096 106.654 106.672652 85.052 85.058815 0.004951 65499.0096 106.663 106.672652 85.055 85.058815 0.002761 65500.01232 106.668 106.686345 85.056 85.06182 0.004264 65500.01232 106.67 106.686345 85.057 85.06182 0.003551 65501.01504 106.673 106.686345 85.058 85.06182 0.002821 65501.01504 106.675 106.700053 85.058 85.064824 0.005061 65501.01504 106.679 106.700053 85.059 85.064824 0.004313 65502.018 106.689 106.713775 85.06 85.067829 0.005737 65502.018 106.711 106.713775 85.062 85.067829 0.004125 65503.02072 106.72 106.713775 85.065 85.067829 0.002036 65503.02072 106.721 106.727511 85.067 85.070833 0.002739 65503.02072 106.721 106.727511 85.068 85.070833 0.002042 Figure 26: Tracking Data Sample from Data File It can be seen that in this sample, no angle is greater than.01 degrees. This is consistent with the entire data set. In the figure below, you will see the image of the star. The diameter of its image on the screen is ~0.5 cm. The focal length of the telescope is 300 cm. This is corresponds to a roughly 0.095 degree resolution. Since the telescopes resolution is.095 degrees, the tracking algorithm is very successful in tracking with roughly 10 times the resolution. This session was performed with only first order corrections applied. As a result, the actual position that is depicted above is not exact. As further corrections are added, the actual position will better match the stars actual position. Below are some pictures of the star tracked during this session. This is the telescope s behavior with first order corrections.

63900 64522 64535 64726 64872 65090 65291 65582 65776 66165 66428 66591 66701 67075 Figure 27: Star Images, Alp Lyr

During a portion of the session, clouds covered the star. However, this gives you a good idea at the telescope s current alignment, and ability to track. The software has been an evolving process over the past couple of years. Many features and have been added that improve usability and functionality. Some of these features are outside of the scope of this thesis, but were added as a contribution to the group. The original scope included the setup of only one telescope, but I have put in extra time and effort to assist in getting a second telescope up and running. I do this happily and have enjoyed my cooperation in the project. At this stage, the software is fully functional. A number of group members have been trained first hand, and with the inclusion of the guide, they should be well supported. I have expressed to the group that I will be available for support on this system after this thesis process is over. I would like to see these telescopes become a valuable research asset for all of those involved.

5.0 References 1. Meeus, Jean (2005). Astronomical Algorithms, Second Edition. Richmond, Va: Willmann-Bell, Inc. 2. Wikipedia, retrieved July 2009. en.wikipedia.org/wiki/precession_(astronomy) en.wikipedia.org/wiki/sidereal_time 3. University of Cincinnati Physics Department Website, retrieved July 2009. www.physics.uc.edu/~sitko/fall2002/1-sky/sky.html images only