Horizon and Range of Motion Study
Rev 1
Summary
This document will detail the limitations of the equatorial mount, and
quantify the duration of time where the sun has become visible above the
tree line, but is not trackable. There are times during the summer months
where the sun is behind the dish, and the dish cannot physically turn that
far. An important assumption made for the calculations in this document is
that the RA-axis (rotation about the axis pointed at Polaris) is limited to
180 degrees. Based on the visit to Ionia and the images of the dish, the
range of motion should be close to 180°. If the dish can move slightly
farther, the extra movement does not remedy the tracking issue, as will be
discussed later in the document.
Motion of the sun
Over the course of the year, the apparent plane of motion for the sun
shifts north and south. In the northern hemisphere, the plane is shifted
north in the summer and south in the winter (see Figure 1). During the
equinox, the sun will rise directly east, and set directly west.
Independent of the earth’s position around the sun, the pole of the
earth is always pointed towards Polaris (the North Star). Polaris is usually
used as a way of aligning the polar axis of an equatorial mount for a
telescope. Once the polar axis is aligned, rotation at 15° per hour will
account for the rotation of the earth and essentially de-rotate the motion
of the sky. For our application, tracking the sun in this exact manner
will not work using the bracket the way it functions now. The reason
being that the dish only has two axis of movement, and pointing the
polar axis towards the North Star limits us to a single plane of
movement.
Figure 1: Sun’s plane of
motion (yellow line) in the
summer (top) and winter
(bottom)
As an example, imagine that the stick figure in Figure 1 is the
location of the dish. Notice that between the seasons, the suns path
has moved, yet the pole (pointed towards Polaris) has not. Figure 2
also demonstrates this. Looking at the degrees of freedom of our
equatorial mount (Figure 3), if the polar axis is always pointed at
Polaris, the dish will always be confined to sweeping the same path.
Figure 2: Another representation
With an estimated 2° beam width, and a declination axis pointed at
of the suns path
Polaris, we will only be able to see the sun when it enters our 2°
sweep zone (right around the equinox). In order to see the sun at all times of the year, declination and
right ascension will have to be changed on-the-fly to track the sun.
Horizon and Range of Motion Study
Rev 1
Polar Axis (Towards Polaris)
Declination Screw location
Declination Axis pivot
Figure 3: Bracket Motions
Figure 4 on the right illustrates the adjustments needed to get to
the suns path during one of the summer months. In the top of the
figure, the plane that the dish can sweep based on a set declination
(polar axis towards Polaris) is shaded in green. The suns path, in orange,
is offset from the dish and out of the 2° beam window. To reach the
suns position, the declination axis must be increased such that the RAaxis can swing down and reach the suns position (Figure 4 bottom).
The motion of the dish can be described in two angles; 𝛾, the
distance above the horizon the dish should aim, and 𝜂, the angle the
RA-axis should move from its nominal due-south orientation. Given an
azimuth and altitude value for the sun, the necessary 𝛾 and 𝜂 can be
calculated. These calculations and a model verifying their accuracy will
be included at the end of this document.
E
S
N
W
E
N
S
W
Figure 4: Dish plane and sun path
Horizon and Range of Motion Study
Rev 1
Limitations:
Figures 1, 2 and 4 illustrate that is possible that the sun can set behind the dish. Since the dish
cannot move beyond 90 degrees from due south, there are times when the sun is still visible but out of
the range of motion. Against an ideal horizon, data loss would begin starting after the vernal equinox
and end on the autumnal equinox (anytime the suns plane is north of pure east-west. However, given
that there is a real horizon with obstructions, the sun may be obscured by trees or other obstacles that
make this a non-issue. To quantify the severity of the potential data loss, the images in Figure 5 were
created.
Figure 5: Sun Path
Figure 5 shows true horizon data overlaid on sun position data. The sun position data was taken for
the days in the spring and fall when the sun has just appeared above the tree line and is in range of
motion of the dish. The point of interest was where the suns path intercepts the east heading (one
extreme of the dishes motion) and also the true horizon (blue line). From the charts, it can be expected
that the sun-rise cannot be tracked in its entirety between the days of April 25th and October 17th. At
most, (during the summer solstice) the sun would appear above the tree line for about 1 hour 30
minutes before entering the trackable region of the sky.
Similar charts can be generated for the sun set times. From Figure 5 it can be seen that the sun is
still above the tree line for about an hour beyond the western-most motion of the dish. During the
solstice, there is a two hour window where the sun has not set but the dish cannot track it.
Our question to the customer is if this is acceptable. Are there other eCallisto sights capable of
detecting solar phenomenon during the times when the current design cannot operate? If data
collection is critical during these times, bracket modifications or possibly a redesign of the bracket
may be required.
Horizon and Range of Motion Study
Rev 1
Dish Position Calculations:
Z
Dish Views
Southward
Define 𝜙 and 𝜃:
𝜙 = 𝐴𝐿𝑇 (𝑎𝑙𝑡𝑖𝑡𝑢𝑑𝑒)
X (W)
𝜃 = 180 − 𝐴𝑍 (𝑎𝑧𝑖𝑚𝑢𝑡ℎ)
Using trigonometry, the following must
be satisfied:
tan(𝜃) =
Y (S)
Altitude
-Y (N)
𝑦
𝑧
𝑎𝑛𝑑
= tan(𝜙)
𝑥
√(𝑥 2 + 𝑦 2 )
-X (E)
Azimuth
Equation of a sphere must be satisfied:
𝑥 2 + 𝑦 2 + 𝑧 2 = 𝑅2
Z
Using the first equations:
𝑥=
Dish Views
Southward
𝑦
𝑦2
∴ 𝑥2 =
tan(𝜃)
tan(𝜃)2
Sun (x,y,z)
R
𝑧 = tan(𝜙) ∗ √𝑥 2 + 𝑦 2 ∴
-Y (N)
θ
90-θ
Insert these into the sphere equation:
𝑅2 =
φ
Y (S)
𝑧 2 = tan(𝜙)2 ∗ (𝑥 2 + 𝑦 2 )
2
X (W)
-X (E)
2
𝑦
𝑦
2
2
+
𝑦
+
tan(𝜙)
∗
+ 𝑦2)
(
tan(𝜃)2
tan(𝜃)2
Assuming a unit sphere (R=1), solve for 𝑦:
𝑦 is now known. Solve for x and z:
𝑦=
√
1
1
tan(𝜙)2
+
+ tan(𝜙)2 + 1
tan(𝜃)2 tan(𝜃)2
1
tan(𝜙)2
1
+
+tan(𝜙)2 +1
tan(𝜃)2 tan(𝜃)2
√
𝑥=
𝑦
tan(𝜃)
=
tan(𝜃)
𝑧 = tan(𝜙) ∗ √𝑥 2 + 𝑦 2
The dish needs to sweep a plane that intersects sun(x,y,z). The necessary elevation angle is given by 𝑦 and 𝑧:
𝛾, 𝐷𝑖𝑠ℎ𝐸𝑙𝑒𝑣𝑎𝑡𝑖𝑜𝑛𝐴𝑛𝑔𝑙𝑒 = tan−1(𝑧/𝑦)
The right ascension pivot angle from the nominal position is found by the relative angle between ‘R’ and due-south in a
titled plane:
Verification:
𝜂, 𝐷𝑖𝑠ℎ𝑅𝐴𝐴𝑛𝑔𝑙𝑒 = tan−1(𝑥/√𝑦 2 + 𝑧 2 )
Horizon and Range of Motion Study
Rev 1
A 3D CAD model was created that included a ground plane, sky, sun, antenna, and beam cone.
Using the equations above to direct the dish, the dish stays centered on the suns position anywhere in
the field of view.