Star Tracker

In order to fix the mount to the tripod and the camera, the easiest solution I could find was to add some piece of wood fastened with screws, taking advantage of the slots in the steel profiles.

Once finished the drills , was time to bend the threaded rod. It must have a radius of curvature equals to the distance measured between the axis of the hinge and the center of its through hole. In my model this was 170mm, designed in the model and measured in the real part. Otherwise, all subsequent calculations must be repeated with the new measurement.

Threaded rods are usually sold in 1m lengths. This helps to bend it with less force since it is very difficult to shape a short rod without using machines. To get the curvature closer to the required one, we must make a pattern that guides us when we bend. The pattern consists of drawing a circle of radius equal to the radius r of the mount (170mm).This will be the center of the diameter rod D (6mm). Then we draw two more circumferences to guide the outer and inner edge of the rod, one with radius of r+(D ÷ 2) (173mm) and another with radius of r-(D ÷ 2) (167mm). Using this pattern we only have to bend little by little and place the rod on the pattern centered between the circles to see its accuracy.

Once finished bending the threaded rod I cut the piece of rod with better accuracy. It is not necessary a long piece, with 15 to 20 cm we will have several hours of tracking. Later I will explain the formulas to find the needed length given a required time or the tracking time given a fixed length. You must also check that the length of the rod does not collide with the tripod.After fixing it with nuts to the upper plate it's necesary to force it inwards matching the rod and the hole in the lower plate, checking that the mount opens and closes completely without friction. Finally I inserted the female nut into the gear and checked the total adjustment of the mount.

We move

Some motorized barn door mounts are using either an analog motor and circuits for speed control (by varying the voltage) or fixed speed motor (typically 1 RPM). In such projects the distance between hinge and threaded rod must be calculated basedon choosen motor. On Internet there are a myriad of such designs.

I had already decided the components that would form the mechanical part of the mount, so I had some restrictions on the mechanism maximum dimensions . Because of this I ruled out the use of an analog motor (DC) and opted for the use of stepper motors that allow greater accuracy in movement and speed control.

Based on the premise of reusing as much as possible junk, before starting the 3D model, I decided to find the right stepper motor among all those available. The engines was removed from old devices so I didn't know the specifications of many of them. I was luckier identifying others, thanks to the information printed on its body (see 1, 2). Some already had a pinion or gear on its axle. I'll need a secondary gear for controlling the ascension of the curved threaded rod.

Finally I found a compatible set of motor, pinion and gear with the correct dimensions to fit the rest of the components of mount. But before going on, I had to do some calculations to find out whether I could achieve the correct angular velocity with all this componets. It's time for mathematics!

We already know that the mount must rotate exactly at speed of earth's rotation, but in the opposite direction, so that the stars appear stationary. The assembly must also be aligned at the correct angle (according to latitude) aligning the mount and Earth rotation axes. My barn door design requires a constant lifting speed, thanks to curved threaded rod, and also we don't have to worry about the tangent error.

But what is that speed? The sky has a continuous movement around a fixed point, the North / South Celestial Pole. As we know, the Earth rotates once a day, which means full 360 ​​° in ~ 24 hours. This translates to approximately 15 ° per hour (360/24 = 15) or 1/4 ° per minute (360/24/60 = 0.25). If we raise the top plate at constant speed of 15 ° / hour, we can follow the sky quite accurately.

But to be more exact, according to Wikipedia, "... solar time is measured by the apparent diurnal movement of the Sun and local noon is defined as the moment when the Sun is at its zenith (the projected shadow points exactly towards the north in the northern hemisphere and to the south in the southern hemisphere.) By definition, the time it takes the Sun to return to its highest point is on average 24 hours.

However, the stars have a slightly different apparent movement. During the course of a day, the Earth will have moved a little along its orbit around the Sun, so it must rotate a small extra angular distance before the Sun reaches its highest point. Instead, the stars are so far apart that the movement of the Earth along its orbit generates a barely noticeable difference with respect to its apparent direction (see, in any case, parallax), so that they return to their highest point in something less than 24 or solar day. An average sidereal day takes about 23 h and 56 min (it is almost 4 minutes shorter than the solar day). Due to variations in the Earth's rotation index, the index of an ideal sidereal clock deviates from any simple multiple of a civil clock. In practice, it is taken into account through the UTC-UT1 difference, which is measured using radio telescopes, and stored and offered to the public through the IERS and the United States Naval Observatory.

As shown in the figure, the time elapsed between successive culminations is not the same for the Sun as for the distant stars. When the Earth moves from B to C the star culminates again but the Sun does not, and it is said that it delays the angle DCA that is what it needs to repeat its culmination. The time corresponding to the arc BC is a sidereal time. "

When the mount is aligned and resting, it describes a circumference identical to the one followed by the stars in its relative motion, but with a smaller radius and opposite direction. This is because the mount is on the Earth's surface and moves along the planet. The earth rotates clockwise in the northern hemisphere and in the opposite direction in the south. So the stars seem to move counter-clockwise in the northern hemisphere and clockwise in the southern hemisphere.

To stop the stars we must counteract the terrestrial movement, this means to stop the Earth. But, because that's really hard to get, what we're going to do is make our mount move at the same speed as the ground but in the opposite direction, either in the northern or southern hemisphere.

We need to find out how fast the planet moves! If we remember the years of the school, is known that a full rotation of the planet needs 360° sexagesimal or 2π radians and it costs 24h to do it, according to solar time. But as we are interested in following the stars we must use the sidereal clock, as we know 23 hours, 56 minutes, 4.0916 seconds and thus avoid accumulating errors in long photo sessions due to those 4 minutes difference.

With this data we can calculate the Earth's rotation speed, we know that in 23 hours, 56 minutes, 4.0916 seconds it turns 360°, or what is the same, 360° each (23)+(56÷60)+(4.0916÷3600)=23.934469 hours. With a simple rule of three we find out that it runs 360÷23.93446989=15.041068°/h or what is the same 2π÷23.93446989=0.262516 rad/h. So 15°/h or 0.26 rad/h corresponds to the Earth's angular velocity.

The problem is that our stepper motor doesn't directly control the swivel angle of the mount, nor would it have sufficient resolution to achieve such a slow continuous movement. They typically revolve between 7 ° and 30 ° per step and no intermediate angles can be achieved without a good gear train.

Our assembly shift a curved screw up, with the same radius of curvature as the mount. And it have to shift it at a constant velocity that opens the mount those 15 ° per hour. We can get this thanks to the gear reducer, the motor torque and the "Law of the lever" because we require less force of thrust if the threaded rod is located at the end farthest from the hinge.

But, what is the speed with which to rotate the engine to get this? This velocity is tangential or linear velocity. Would be the distance that moves, tracing a curve, the upper end of the screw per unit of time, remember that v=d/T. We must therefore find out the relationship between angular and tangential velocity.

Let's draw it on the blackboard and continue...

Our camera, aligned to the Earth's rotation axis, describes a circumference with radius r, corresponding to the distance between the hinge axis and the center of the threaded rod. In the hypothetical case that we let the mount working for a full day, the end of the mount would travel a distance d corresponding the length of the circumference with radius r, this is d = 2π r .

On the other hand we have seen that v=d ÷ t, therefore clearing d and combined with the above we get [4] v=2π r ÷ z, being z the time it takes to turn 2π radians (one whole day). This is the tangential speed of the mount corresponding to the angular velocity of the Earth's rotation.

But what happens with the gear system, the nut and the screw... how is this? Well, we have to keep calculating. Now we are going to model the linear speed at which the screw is raised through our mechanical system. We'll call M to engine revolutions per unit of time, frequency, which is what we're trying to figure out.

On the other hand we have to find the transmission ratio between two circular gears with a certain number of teeth, mathematically expressed as τ = S÷L, being S the number of teeth of the input gear (motor) and L the number of teeth of the output gear (screw). This ratio multiplied by M shows us the motor frequency (rotations/time) around the screw, ie its angular velocity. Once again we have to find the relationship with its linear velocity.

To do this we have to realize that, for every turn of the output gear, the nut pressed into it makes the threaded rod advance a fixed distance. This will come determined by the distance from the crest of one thread to the next (pitch), in this case equals to lead. If, as in my case, you aren't sure about the metric of the screw and its specifications, it is best to measure that distance as accurately as possible and figure out how many turns it gives per unit of distance, A.

Now, if we divide the rotations per unit of time of the threaded rod's gear (rotations/t) between the number of threads of the rod per unit of distance (rotations/d) we will have, simplifying, the linear speed (vertical movement) of the threaded rod. This is, using the above formulas v=(M(S÷L))÷A=MS÷LA.

We already have the two formulas that we were looking for, the linear velocity of the mount turning at the same angular velocity as the Earth (v=2π r ÷ z) and the raising speed of the threaded rod (v=MS÷LA). But if we think about it, they represent the same speed. Both describe the speed at which a point (located at center of threaded rod over the top plate) moves around the circumference of mount radius r. So we can solve the problem now, because as 2π r ÷ z =MS÷LA we can calculate M, which is the unknown data in our project, as M=2π r LA ÷ z S.