K2 Star Tracker


K2 is a motor-driven star tracker that I designed and built to allow photographs of the night sky to be taken without the photographs showing star trails due to the rotation of the Earth. I wanted to be able to take the star tracker on holiday to Kenya (on the equator) and photograph the skies there that are unaffected by light pollution.
So the star tracker had to be:
  • Compact
  • Light
  • Strong
  • Accurate
  • Battery operated
  • Low power
  • Cheap
  • Easy to construct
In practice, this means:
  • Footprint no larger than A4
  • Less than 1 kg
  • Support a digital SLR camera
  • Exposures of up to 15 minutes
  • No mains electricity in the bush
  • Operate for up to 6 hours
  • Less than about 50 for all components
  • Manual tools + electric screwdriver

Construction

As this was my second star tracker (I built my first one when I was a student at college) I called it K2. I decided to construct K2 from lengths of L-section and U-section aluminium bolted together as this is very strong and very light. The sections make two 'T' shapes, the static base T and the moving top T. The pivot is provided by two brass hinges that connect the two T's together, positioned at the ends of the top bars of the T's. When operating the two T's are pushed apart by a bolt driven at 1 rpm by a small motor and gearbox. The design is optimised for low latitudes (within 20 of the equator), but will also work in the UK (latitude 52) provided that the camera and lens are not so heavy that they put the whole system out of balance.


K2 in operating configuration

K2 opened to show components

Hinges give a smooth movement
The M6 bolt has a 1 mm pitch and is positioned 230 mm from the hinge axis. A nut sits on the bolt and is driven upwards as the bolt rotates (the nut cannot rotate as it is constrained within the U-section arm that forms the length of the top T). Driving the bolt at 1 rpm forces the top T to rotate at approximately 1/230 of a radian per minute, or one revolution per day, to counteract the rotation of the Earth. The design is much more accurate than a conventional tangent drive and much simpler than a double-arm drive. The reason for the high accuracy is explained, with technobabble, here.
The long arm of the base T is made from two lengths of L-section aluminium bolted together to form a 'U' section of width 40 mm (the width of two AA batteries side-by-side). If my local B&Q store had had some U-section aluminium in stock that was 40 mm wide I would have used that instead. The arm forms a tray in which all the components fit - batteries, switches, potentiometer, voltmeter and circuit board (see close-up image on the right). The motor/gearbox is bolted underneath the end of this arm with the M6 bolt poking through a hole in the arm and in the circuit board. The head of the bolt is filed down to fit snugly into a slot in the gearbox output shaft.

The camera is mounted onto an adjustable mount bolted to the top T, close to the hinge axis. If the hinge axis of K2 is aligned with the Earth's axis, the the camera should follow the stars as they appear to rotate around the sky and hence the stars will not trail when a long-exposure photograph is taken. A plastic tube is fitted to the underside of the top T, parallel to the hinge axis, visible in the image above in which K2 is opened up. This can be used as a sighting tube to align on Polaris when operating in the UK. It does not work in Kenya as Polaris is on the northern horizon and is usually hidden by trees, hills, lions or elephants.

Close-up of components
In practise, the tracking accuracy of K2 is determined by the degree of alignment between the hinge axis and the Earth's axis. A polar scope, used in some commercial star trackers, is useless if Polaris is on the horizon. An alternative method of alignment is to use an inclinometer and a compass to set the altitude and azimuth, respectively. A digital inclinometer is accurate to 0.1, but even a digital compass is only accurate to about 1 and you have to know the offset between magnetic North and true North for your location. Inaccurate polar alignment is the biggest factor that affects the overall tracking accuracy of K2 and I am thinking about ways to improve it.


Movement after 15 minutes driving
The image on the left shows K2 at the start (top) and end (bottom) of a 15-minute run. The motor has turned the bolt 15 revolutions, pushing the nut 15 mm upwards along the bolt and rotating the top T by about 4 from its starting position. At this point the top T can be lifted off the nut and the nut spun back down the bolt by hand, ready to start another 15-minute run.

Note that the direction of the bolt moves slightly as the top T moves relative to the base T. The nut moving on the bolt is cylindrical (see close-up image below right) and so the contact point on the underside of the top T 'rolls' over the cylinder. Small strips of teflon on the underside of the top T ensure that the contact between the nut and top T is smooth.
The layout of the components inside the base T is shown in the close-up image above right. The resistors are arranged in two sets. One set is a potential divider to drop the 3 V supplied by the two AA batteries down to the 2.3 V that is needed to drive the motor at 1 rpm (3 V drives it at 1.3 rpm). The second set is used to drop either (i) the voltage supplied by the batteries or (ii) the voltage across the motor down to the appropriate value for display on the LCD voltmeter. As the voltmeter is set to read a maximum of 1.999 V (for maximum resolution) the battery voltage is divided by 2 and so displays 1.5 V when the batteries are fresh. The voltage across the motor is divided by 2.3 so that the voltmeter reads 1 V when the motor has 2.3 V across it (and is rotating at 1 rpm). Thus, the voltmeter effectively reads rpm.

Cylindrical nut on drive bolt

Layout of circuit components
The layout of the components inside the base T is shown in the circuit diagram on the left. The potentiometer is in parallel with one of the resistors in the potential divider that drops the battery supply voltage down from 3 V to 2.3 V. This allows the voltage across the motor, and hence the speed of the drive, to be adjusted to compensate for the slow drop in voltage of the batteries as they gradually run down. When starting a photography session, the potentiometer is adjusted until the voltmeter, set to read the voltage across the motor (divided by 2.3), reads 1.000 rpm. Over a 15-minute period, the battery voltage will hardly change at all, but it can be checked at the end of each 15-minute session when the top T is lifted to spin the nut back to its starting position. The voltage supplied by a battery changes with temperature, so it's worth keeping an eye on it over the course of a night as the temperature drops.

As K2 was built with hand tools, I could not guarantee that the dimensions were
exactly as per theoretical design. This is not a problem. For instance, if when constructed it turns out that the distance from the drive bolt to the hinge axis is 231 mm, rather than 230 mm, then the motor can be set to drive at 231/230 = 1.004 rpm.




Performance

So, does it work? I have tried it with a Nikon D200 digital SLR and a Sigma 100-300 mm f/4 zoom lens. The combined mass of camera plus lens was about 2 kg, so this was a good test of the rigidity of the system. I took 30 sec exposures rather than anything longer as I was not sure about the alignment of the hinge axis with respect to the Earth's axis.




30 sec exposures with K2 off (left) and K2 on (right)


The images are 600 x 800 pixel areas cropped from the original 10 Mpixel images. They were taken with the zoom lens set to a focal length of 100 mm. They are single exposures (no stacking of multiple frames) and no dark frames were subtracted to reduce the noise levels. Note the red dot in the bottom left corner of each image, a pixel that is not recording the correct light intensity. Such 'hot' pixels would be removed by subtracting a dark frame, which is common practice with astrophotography.

The image taken with K2 switched off shows the extent of trailing that would be expected due to the rotation of the Earth. The image taken with K2 switched on shows essentially no trailing. So, yes, it works.

This image of the Milky Way shows what you can do with K2 and a 35 mm lens from a dark sky site, in this case the Teide Observatory in Tenerife.




Components

The components used to construct K2 are listed below:

2 aluminium sections 200 mm long
2 aluminium sections 250 mm long
1 aluminium sections 210 mm long

2 plastic corner braces
1 metal 'T' brace
2 brass hinges

Motor and gearbox (25)
M6 bolt with barrel nut

2 AA batteries in holder
2 mini switches
Mini LCD voltmeter (20)
1 kohm potentiometer
Resistors
Circuit board

L-section 40 x 20 mm for hinged sides of both Ts
L-section 40 x 20 mm for U-section arm of base T
U-section 20 x 10 mm for arm of top T

To brace junction of base T
To brace junction of top T
To provide the pivot axis

http://www.precisionmicrodrives.com - part #256-101*
60 mm, sold as 'furniture bolt'

Standard
Single pole double throw
3.5 digit reading 0-1.999 volts - RS stock num 223-199
Standard
Values will depend on motor - see circuit diagram
100 x 40 mm



* This motor is an integrated motor/gearbox/microswitch. Although the motor and gearbox are ideally suited for the job of driving K2, the integrated microswitch was an unnecessary addition. A cam on the output shaft of the gearbox operated the microswitch every revolution, which caused a small but noticeable periodic error in the motor's speed. If you use a motor like this, I suggest that you remove the microswitch.



Steve Barrett    July 2013

S.D.Barrett@liv.ac.uk

All images Steve Barrett 2010


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