Hour Angle Differences
Using the Chart:
For each of the five naked-eye planets and the moon, the
curves plot, for the current calendar year, the angular distance between the
object and the sun as measured along the celestial equator. One then thinks of
the central vertical line as representing the sun's position. Each horizontal
line on the grid corresponds, on the vertical time axis, Sat/Sun midnight on the
labeled Sunday date.
The angular distance is plotted in 'hours' instead of degrees
or radians--an hour being one twenty-fourth of a complete rotation and thus
corresponding to 360/24 or 15 degrees. Since the earth rotates one hour of arc
in one hour of time, an 'angle' read from the graph translates directly into an
hour of time.
From northern latitudes we tend to find solar system objects
in our southern sky and the curves are plotted this way so 'left and right' on
the graph corresponds to 'left and right' of the sun in the sky. Thus, left is
east. If its curve shows a planet to be, say, 3 hours east of the sun it would
set approximately 3 hours after the sun sets. And, prior to sunset, be visible
in the southwestern sky.
Or, if 'reading' Venus's curve shows it two hours to the
right (west) of the sun it would come up in the east some two hours ahead of the
sun and probably be referred to as the 'morning star.'
One can use the graph to estimate when and where a planet or
the moon might be in the sky or to check the possibility that a bright 'star' is
really a planet. One can deduce from the curves when two or more planets are
close together in the sky or when a planet and the moon are in the same
vicinity. The charts are no help, however, in predicting the north and south
positions of objects. Nor does it contain any information about an object's
brightness.
More from the charts:
A planet's change of position from day-to-day (as
opposed to during a day) is a function of its position in the solar system and
its orbital motion about the sun as well as of the earth's own orbital motion.
Venus and Mercury are 'inner' planets, closer to the sun than the earth.
Mercury, for example, lies some .4 of the earth's distance from the sun. Hence
it never gets more than two 'hours' from the sun in our sky and its period (synodic)
is under four months. Notice how these facts are shown by Mercury's curve on the
Hour Angle Differences graph. Venus's curve also moves back and forth across the
sun line but more slowly and with greater maximum angles. While the inner
planets move back and forth across the sun line, the outer planets--Mars,
Jupiter and Saturn--never do but move from the west side to the east side by
drifting off the edge of the chart at the maximum hour angle.
And, you might want to think how the moon's 'curve' relates
to its physical position and motion in the solar system.
Creating the Charts:
The first steps in crafting the charts uses the wonderful software created at the U.S. Naval Observatory, Multiyear Interactive Computer Almanac, 1800 - 2050. Seven tables are generated providing. for each day of the calendar year. positional data for the Sun, Moon and naked-eye planets. The AutoCAD drawing consisting of the grid is opened. Horizontal drawing units correspond to hours and vertical to days. An AutoLISP program is loaded and run. This program I wrote a number of years ago and have amended several times to take advantage of later versions of the Almanac. This program reads in sequence from the Almanac-generated tables the right ascensions of the object and of the sun, subtracts them, computes a plotting point for each day and draws a line segment from the previous day's point. Each planet curve is on a separate layer so it can be drawn in a different color. The Sunday dates are different for each year so these need to be amended individually, the AutoLISP program doesn't do it. AutoCAD can print copies of the chart but for displaying here the completed graph is exported in PDF format.