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    Re: Lunar Calculation Presentation
    From: Frank Reed
    Date: 2022 Sep 5, 10:09 -0700

    Cade Crockett, you wrote:
    "I am taking a world history class this year, and we just covered some of the history of navigation.  One of the topics we covered was about lunars and how helpful they were to sailors of the past.  For an assignment, I have to give a presentation about a topic found in the book.  I would love to learn how to complete a lunar calculation and share the method with my class.  Is this possible?  Is there a lunar method that highschoolers who have no prior experience with navigation could understand?"

    For other NavList readers, I think it's clear that this is a case where the class in question has already learned some rudiments of navigation, maybe even the calculation for latitude by Noon Sun. 

    Cade, to your question, absolutely it's possible. You should probably limit yourself to a special case, and emphasize in your presentation that you're doing a special case for the sake of a short presentation. The process of "clearing" lunar distances, in general cases, was once an important problem in practical mathematics, and some of the greatest mathematicians in the world developed clever, efficient techniques and analyzed the techniques already published. This was primarily in the second half of the 18th century though there were some important streamlined methods developed late in the game, even as late as the 1830s (see PS). These general mathematical methods are probably too much trouble for a short presentation (do you have 25 minutes before the class? More? Less?). But we can skip most of the heavy lifting by analyzing a "vertical lunar". Just a few weeks ago I started a discussion of a special "vertical lunar" case here on the NavList message boards (see Liechtenstein Lunar). There's still more to do there, but the idea is simple. When the Moon and the other body (Sun, planet, star) are aligned vertically, the corrections to "clear" or correct an observed lunar distance for the effects of refraction and parallax are just simple additions and subtractions. Nothing to it! It's easy. While the circumstances for vertical lunars are not rare, they are not a daily event, and a navigator would have to plan to shoot that specific geometry. This can be done easily today. Historically the benefits of vertical lunars were not well known, and most lunarian navigators learned a general method anyway. 

    The general process:

    • Observe the angle between the Moon and star. This is the critical "lunar distance" arc, and it must be observed to exacting precision using the best available sextant. Also observe, nearly simultaneous, the altitudes of the Moon and star. These "altitudes" are the angles between the celestial body (Moon or star, planet, Sun, etc.) above the sea horizon. These altitudes can be measured with relatively lower precision using a more basic octant or sextant.
    • Acquire some almanac date corresponding to your sights, specifically you'll need tabulated, predicted "geocentric" lunar distances for known GMT times on either side of your sights (before/after) and you will need the so-called "horizontal parallax" or HP of the Moon at that time.
    • Correct all observations for any sextant "index error".
    • Pre-clear the altitudes of the two bodies: +20' (minutes of arc) for a Moon or Sun Lower Limb sight, -12' for an Upper Limb, -4' for a star or planet (these numbers assumes rather "average" values for the dip of the horizon and negligible refraction.
    • Pre-clear the measured lunar distance by carefully adding on the exact SD (semi-diameter) of the Moon, including augmentation. This computation depends on the Moon's HP and its altitude, but it's not difficult. If the other body is the Sun, add that SD, too. In so-called "Far Limb" lunar distances, you would subtract the Moon SD instead of adding, but that's a detail not worth attention in a short presentation. After this step, you have the observed center-to-center angle between the Moon and the other body.
    • Using the pre-cleared altitudes of the Moon and the other body, look up or calculate the exact altitude corrections which include refraction and parallax in altitude.
    • Depending on the relative positions of the bodies, you will add or subtract the altitude corrections from this last step to/from the pre-cleared lunar distance. This is the "clearing" step, and it's very short for vertical lunars --30 seconds of thought and 10 seconds of easy math. For general cases, this step might take 5 or 10 minutes of computation (and it's not very interesting for a captive audience which is why I suggest focusing on a "vertical lunar").
    • Now compare your cleared lunar with tabulated lunars. Historically these were provided for every three hours of Greenwich time. This is a "relatively simple" linear interpolation process. If the predicted tabulated lunar distance at 1800 GMT is 85°30' and at 2100 GMT it is 87°01', and if your "cleared" lunar distance (derived from your observation) is 86°00', then you can see that your distance is just about a third of the way from the 1800 GMT value to the 2100 GMT value, and that means the time must be nearly 19:00:00 GMT (a little less in fact, but nearly so).

    That's not too bad, right? To accomplish this, you would need observational details on an actual vertical lunar example. We can get that for you, or you can simulate something in an astronomy simulation app, like Stellarium. Next you'll need to learn about correcting altitudes (Lower Limb versus Upper Limb sights, what is dip of the horizon? etc.). And you'll need to learn the basicsof looking up (or calculating, if you prefer) the values for the refraction and parallax in altitude. You'll also need to find proper values for the before and after geocentric lunar distances and the HP of the Moon. 

    Frank Reed
    Clockwork Mapping / ReedNavigation.com
    Conanicut Island USA

    PS: Though some methods for clearing lunars were being published in the 1830s, at sea, lunars were rapidly fading into obsolescence by c.1835, and after about 1850-60 they were almost dead in maritime use. Lunars were rarely used at sea after this though their very rarity gained attention and even notoriety in later decades. 

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