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Angular Distances between
two celestial objects and Lunar Calculation
Every celestial navigator knows how to take object
altitudes with the sextant but few have ever measured the angular
distances between two celestial objects. Yet these distances
can be useful to check the sextant accuracy and find the the
time. 

The time and the lunar calculation
It is not much of a problem to keep the time today. We can keep
it using a cheap  and yet good  quartz watch. We can get it from
the Internet, from a GPS receiver or from all kinds of radio and
cell phone devices. But keeping the time was a nightmare for many
centuries. Primitive mechanical clocks worked poorly on land and
even more so on sea conditions.
Temperature variation conspired to change the dimensions of clocks
pendulums, making the time pass slower when hot. Lubrication of
mechanisms influenced their pace. The problems seemed so complex
that at one point even clever fellows, like Isaac Newton, though
it was impossible to built a reliable clock to use on the sea.
Why is it so important to know the correct time on a boat ? It
is because the time is necessary to calculate the longitude using
celestial navigation. If the navigator doesn't know the correct
time, the boat is basically lost. Of course latitude can be obtained,
but longitude cannot. This problem caused many accidents in the
past. Captains had to make blind decisions regarding navigation.
Entire fleets of ships run into rocks, killing hundreds of sailors
and sinking fortunes.
At some point it started to damage national economies. Kings tried
to direct scientific efforts to solve the problem of longitude determination
at sea, by supporting astronomy research and offering large prizes
for a workable solution.
One of the early scientists to think about the time problem was
Galileu Galilei. He suggested that the pendulum could be used as
a clock (for the period of a pendulum swing is constant, no matter
the amplitude). He also noticed, using the newly invented telescope,
that the moons of Jupiter had a fast speed and could be used as
clocks in the sky. But observing the jovian moons with the telescope
proved to be difficult, even on land.
Another larger moon was available: our own. So the quest for the
method to find the time using the Moon was started: the search for
the perfect lunar based time calculation.
The
Lunar calculation
To understand the lunar
calculation, consider the movement of celestial objects
in the celestial sphere. It is the movement observed when
the Earth rotation is removed from the picture. Stars are
nearly fixed in the sky. Their so called proper motion is
very slow. The angular distance between two stars is constant
in time and they remain in their exquisite constellation
designs. The Sun goes a complete turn in one year, traveling
along the ecliptic plane in the sky. Planets also move slowly.
In this animated picture, the Moon
is the fastest thing. It goes a complete turn in 28 days.
That's 12 degrees per day, or about 30 minutes of arc per
hour. If we draw an imaginary scale in the sky, between
a given star and the Moon, we can use it as
a clock . The celestial sphere (here represented by a star)
would be the clock face. The Moon would be the clock hand
(see animation below).
Being a one handed clock,
it's difficult to read the time from the Moon. Its movement
is not so fast when confronted to our ability to read the
angle with a sextant. Reading the Lunar time fine detail
requires ultimate accuracy and care. Even with a good instrument,
the time resolution is poor. In fact this method became
obsolete when affordable marine chronometers appeared, around
1780.

How good is your sextant ?
Every celestial navigator dreams with a shinning metal sextant,
solid is a rock, with a clear eyepiece and a perfectly divided scale.
No perceptible construction error along the arc all the way to 120
degrees. Such things exist, but can be expensive. There are cheaper
options: plastic, paper and even doityourself
sextants.
Even the best instrument has to be checked from time to time. This
can be done using the distances between two celestial objects. By
taking many different distances and comparing to the Nautical Almanac
derived values, we can draw a chart of the sextant error as a function
of the angle. This can be used to correct the sextant construction
errors (not the index error) and improve the precision of any instrument.
The calculation and corrections
Most of the work in celestial navigation goes to correct the sextant
angle from a number of optical factors. These corrections are supposed
to clear the sextant reading from apparent effects and bring it
to the same assumptions of the Nautical Almanac.
The Nautical Almanac lists the geocentric positions of celestial
objects at fixed time intervals (hourly for Sun, Moon and planets,
each 3 days for stars). All coordinates refer to object centers.
The Almanac is an animated map of geometric points that represent
the positions of objects in the celestial sphere.
Sextant corrections are familiar to celestial navigators: index
correction, height of the eye, refraction, parallax and semidiameter
(more
details on corrections) . To understand the way
corrections are worked in Lunars and object distance calculations,
it's convenient to draw a distinction between two classes of optical
factors: those that affect the apparent position of the object and
those related to the observer position or instrument peculiarities.
Refraction and parallax work to displace the object
from its theorical Nautical Almanac place, producing the apparent
image, different from the real. This "real' means the theorical
universe model used by the Almanac, where objects are points and
there is no air. These two factors move the apparent image along
the vertical plane: the plane were altitudes are inscribed.
The refraction error is caused by light rays "bending"
while crossing the multiple air layers. It is a tricky correction,
because it depends on atmospheric conditions on the different air
layers. The best thing is to avoid too much refraction, by selecting
objects with altitudes larger then 20 degrees. The refraction correction
calculation uses the following variables: altitude, temperature
and pressure.
Parallax error is illustrated to the right. Since the navigator
is not in the Earth's center, but on the surface, the apparent
object position is below the true geocentric position. Parallax
correction is only necessary for the Moon and some planets.
Other objects are so far, their parallax is very small.
The Almanac lists the Horizontal Parallax. It is the parallax
objects show when they are low, on the horizon.


The correction we must use is the Parallax in Altitude. It is the
parallax observed when the object is at some altitude. This can
be calculated with the formula:
Par. in altitude = Horizontal parallax * Cos(altitude)
Both apparent corrections share their dependency on altitude. This
leads to a circular dependency inside the calculation: the corrections
depend on the altitude, which depends on the corrections. Among
all corrections one must be noticed, because it is so large: the
Moon parallax. Since the Moon is very close to the Earth, the Moon
Parallax can be almost a full degree.
The Height of the eye (Dip) does not apply to object distances,
since it corrects the position of the sea horizon, not used when
distance between two objects is measured. It will be necessary only
when handling the two objects altitudes (also required in this calculation).
The Index Correction applies to any sextant reading.
Semi diameter (SD) are only considered to the Sun and Moon. Other
objects are basically points, with no visible dimension. While for
altitude readings we talk about lower and upper Limb, with object
distance we use the terms near and far limb. If the near
limb is used when observing the Sun or Moon, we must add the semi
diameter. If the far limb is used, the SD must be subtracted.
We must consider yet another correction for the Moon. The Moon
is pretty close to the Earth. The linear distance EarthMoon is
about 360.000 Km, and the earth radius is about 6.000 Km. So, the
observer is 6000 km closer when the Moon is high in the sky. As
a consequence, the Moon disc is a little larger than the Almanac
listed geocentric number for the semi diameter. This augmentation
of the Moon semi diameter can be as much as 0.2 minutes of arc.
It is small, but relevant in this case, since the Lunar calculation
requires maximum precision.
The order of calculations is:

sextant distance
index error
semi diameters of objects + [SD augmentation for the Moon
]

apparent distance
projected refraction & parallax

observed distance

Refraction and parallax projection

Both refraction and parallax displace the
object in the vertical plane. In fig.2 the combined
effect of parallax and refraction is represented by C
(The solid circle is the apparent object and the dotted
circle is the true object).
We can see that C is a vertical displacement,
not in the direction of the other object. DC is the
correction C projected in the arc of D.
We have:
DC = C . cos ß
where ß is the angle between the
vertical and the geodesic line to the other celestial object
(see fig.3). The total distance correction is:
Corr = SDM + SDO+
DC
SDM and SDO
are semidiameters of the Moon and Object (when the Sun
is used)
Since the celestial positions of the objects
(Right Ascension and Declination) can be calculated using
the Nautical Almanac, one solve the spherical triangle ZenithMoonSun
using the law of cosines. This will give all angles necessary
for the projections.
Due to the large parallax of the Moon, C
can be large (as much as a degree) , requiring an extra
second order correction to the calculated distance (see
reference
1 for more details).

The cleared Distance can be compared with calculated distances
obtained using Almanac tables. Since the Almanac lists object positions
for full hours, we must interpolate the time inside the one hour
interval that includes the cleared distance. A linear interpolation
is used.
Measuring the angular distance

Measuring angular distances is different from measuring altitudes.
Instead of the vertical plane, the sextant must be held inclined,
so that the instrument plane includes de Earth's center and
the two celestial objects. This will challenge your sense
of geometry. Try to rock the instrument, to find the inclination
that gives the lowest possible sextant reading.
The Moon normally displays two borders: the sharp, real disc
border and the blurred division between the dark and light
sides. The sharp disc border is the one to be used. While
this seems obvious, it is sometimes difficult to see which
is which when the Moon is full.

