helpful. Using a distant object in daylight or the moon, the
finder scope can be adjusted to match the view through
the hollow hinge pin. Viewing Polaris through the hinge
pin can be difficult — especially with a bright sky in an
urban environment. The step size can be changed to
400 steps per turn, but this improvement in accuracy is
overshadowed by the first three items in the list. The
dominant factor in accuracy is pointing the hinge pin at
the north celestial pole.
■ FIGURE 10.
Figure 10 is a portion of a four minute exposure at
F/8 taken using the barn door tracker with a Nikon D810
digital camera, Nikon 50 mm F/1.4 lens, and a Lumicon
high contrast filter to reduce the bright sky in my
neighborhood. These shots are some of the first ones
taken with the barn door tracker as a proof of concept,
and are not the best quality. The stars shown are from the
bend in the neck of the constellation Draco at approx.
right ascension 19 hours, declination 70 degrees.
■ FIGURE 11.
Figure 11 shows the same area with the barn door
tracker turned off. Further details on taking photos with a
barn door tracker can be found on Internet sites. NV
6. Using the Law of Sines which relates the lengths of the sides
of a triangle to the sines of its opposite angles:
psi = (PI - theta) / 2.0
d / sin(theta) = r / sin(psi)
d = r sin(theta) / sin(psi)
Referring back to Figure 1, look at the isosceles triangle created
by the two wooden plates and threaded rod. The length of the base
of the triangle created by the threaded rod is labeled d, while the
length of the sides are labeled r. The angle created by the two
wooden plates is labeled theta and the two equal base angles of the
isosceles triangle are labeled psi. Here is the sequence of the
algorithm for calculating how high to raise the top plate at a distance
d for a given elapsed time in solar seconds, beginning with the two
plates closed creating an angle theta of zero:
7. The number of steps the stepper motor needs to have taken for
this time is:
steps = (d 28.0) 200.0
given 28 threads per inch and 200 steps needed for each rotation of
the threaded rod.
1. Set solar_time = 0, total_steps = 0, r = 11. 5.
2. Read elapsed solar_time in seconds.
3. Convert solar_time to sidereal time by multiplying solar seconds by
8. As the elapsed time progresses, keep track of the number of steps
the stepper motor has completed, adding steps as necessary to
maintain each newly calculated distance d:
sidereal = solar_time 1.0027379
4. Calculate angle theta in radians. A circle contains 2 PI radians,
or about 6. 24 radians. Angles are calculated in radians because
this is the unit necessary when using trigonometry functions in the
Arduino program. Assume the barn door will run for at most three
hours, or 10,800 seconds. Therefore:
while (total_steps < steps)
have stepper motor do one step
total_steps = total_steps + 1
9. Loop back to number 2.
theta = (sidereal time / 10800.0) (PI / 4.0)
where (sidereal time / 10800.0) is the fraction of three hours using
seconds for both values, and (PI / 4.0) is the angle in radians theta
that would subtend in three solar hours.
The algorithm here works fine for the first few rotations of the
stepper motor. However, as the threaded rod moves the top plate up,
the rod itself starts to angle to the left on its swivel mounts. This
changes the geometry as the distance r is reduced. The center of the
swivel mount is located 1.25 inches below the top of the bottom
plate. The angle of the triangle created when the threaded rod
moves over is the complement of psi, or 90 degrees psi, which is
5. Next, calculate the angle psi. The sum of a triangle's angles is
180 degrees, or PI radians. Because the two base angles are equal in
an isosceles triangle:
PI / 2 psi in radians. These formulas are:
comp_psi = (PI / 2.0) - psi
correction = 1.25 tan(comp_psi)
This correction amount must be subtracted from r before r is used in
number 6 above.
January 2015 31