## Finding the Apparent Lunar Distance from Hamal, 3 October 1805

### Step 1

Plot the observed data. The data sets have a "fair" scatter, but the scatter seems to be equally distributed on either side of an average line; the average angular distance is 78°00'10".

### Step 2

Take the average time of the observations and correct it for the chronometer's error and rate of loss. Because the chronometer sometimes ran erratically, it is best to average this time with the time projected for the observation from the chronometer's error and the calculated time of the PM Equal Altitudes observation. The Local Apparent Time of the Lunar observation averaged from the two methods is 7:55:28 PM. From the estimated or "dead-reckoned" longitude of the place of observation, determine the Greenwich Apparent Time of the observation = 3:40:47 AM October 7.

### Step 3

Because the captains did not take measurementrs of the altitude of the moon and the star at the time of the observation, these altitudes must be computed by formula. This is a lengthy process because it involves making several sets of calculations using data from the Nautical Almanac and adjusting them to the calculated time of the observation. Needed are the Right Ascension^{1} and the sun for noon on October 6 and 7, the Right Ascension and Declination of the moon for midnight October 6 -7 and at noon on October 7, and the Right Ascension and Declination of the star Hamal.

### Step 4

For the time of the observation, determine the Right Ascension of the sun (12h 38m 25.2s) and the Right Ascension (21h 15m 18.5s) and Declination (–12°53'59.3") of moon. The Right Ascension and Declination of the star (01h 56m 17.3d, and 22° 32' 26") remain nearly the same for many years. Subtract the Right Ascension of sun from that of the moon and Hamal to find their angular separation from the sun with respect to the Celestial Equator.^{2}

### Step 5

The Local Apparent Time of the observation is the sun's hour angle. The difference in degrees from the sun's hour angle gives the hour angle for the moon and Hamal (– 10°21'18" and –80°36'00", respectively). From the hour angle, declination and known latitude, the true altitude of the moon and Hamal at the time of observation are calculated. The parallax and refraction for those altitudes then are calculated. The parallax is subtracted from the true altitude and the refraction is added to it to produce the "observed" altitude of the moon and Hamal, 29°05'48" and 22°29'19", respectively. This process is repeated for the star Hamal: 22°29'19".

### Step 6

From the observed angular distance 78° 00' 10" subtract the sextant's index error (8' 45"). Find the moon's semidiameter in the 1805 Nautical Almanac (15' 03") and subtract that, because the star was "brought" to the side of the moon opposite the star. Next subtract the moon's augmentation (8" from the Tables Requisite for the moon's altitude) and the result is the apparent distance: 77° 36' 14".

Now, let the calculations begin.

1.

Right Ascension:The Right Ascension of a celestial body is the angle expressed in hours-minutes-seconds between that body and a point in the sky where the sun will be or was at the mooment it crossed the celestial equator during the spring equinox. this point is called both the first point of Aries and the vernal equinox; it has a Right Ascension of 0h 00m 00.0s. the Right Ascension of the sun, moon, planets and brighter stars are calculated years in advance and published in a book called a Nautical Almanac, a copy of which was in the expedition's portable library.The difference in Right Ascension between two bodies is the difference in time that they cross the upper meridian ( line due south from a person in the northern hemisphere). This difference, multiplied by 15, gives the angular weparation o f the two bodies referenced to the celestial equator. The celestial equator is essentially the plane of the earth's equator projected to the celestial sphere, which, in turn, is the imaginary globe upon whose surface all celestial bodies are imagined to be affixed.

2.

Celestial Equator:The celestial equator is essentially the plane of the earth's equator projected to the celestial sphere, which, in turn, is the imaginary globe upon whose surface all celestial bodies are imagined to be affixed.

Funded in part by the Idaho Governor's Lewis and Clark Trail Committee