Expert: Courtney Seligman - 9/16/2008

Question
QUESTION: Hello Courtney,
It's a long story...but I'm working on a historical novel, and have gone off on a tangent about celestial navigation.  I think I'm finally making some headway with Bowditch's American Practical Navigator, but I'm wondering: given the declination and right ascension of the moon at midnight, is it possible to calculate its coordinates at a different time?
Thank you,
Carissa

ANSWER: If you only know the Moon's right ascension and declination at one date and time, you can't calculate its exact position at a later date and time (some estimates can be made, but they would be relatively crude). You need either its position at some other time, so you can do an interpolation, or the current elements of its orbit. Even then, you need an approximate idea of your position on the Earth (and your local time) to get truly accurate results, because the Earth's radius, as seen from the Moon, is about one degree, and that can affect the apparent position of the Moon by the same amount.

This doesn't mean that the situation is hopeless. If you can let me know what information you have, and what you're trying to do with it, I can probably give you a more useful answer.

Courtney Seligman

(P.S. I'm sorry I couldn't get back to you earlier, but your message came in after I left for work, and I don't get back till nearly midnight, on Mondays.)

---------- FOLLOW-UP ----------

QUESTION: Okay!  Sounds easier than going through the logarithmic sines, cosines, secants, etc. backwards!
The Nautical Almanac and Astronomical Ephemeris has the declination and right ascension of the moon at noon and midnight every day, which I'm assuming is its apparent position from the observatory at Greenwich?
I thought I knew where my characters were, but I now wonder if they should be a few degrees southwest of where I thought they were.  Oops. I can easily come up with new coordinates, and with those and the local time I can find the time in Greenwich as well, if that helps.  
Local time, now...I may have painted myself into a corner here, because I don't know when the best time is to be taking this kind of lunar observation.  The book only says that it's easiest if the moon is not at its zenith, but higher than the star from which its distance is being measured (either Fomalhaut or Altair.)  Do you know?
If there's a formula I can just plug these things into as soon as I know them, that would be great!
Carissa

Answer
The coordinates in the Ephemeris should be geocentric -- that is, as seen from the center of the Earth, or from the place where the Moon is at the zenith (directly overhead).

It sounds like you're trying (or having your characters try) to determine a position from observations of the stars, in the absence of an accurate clock; for if your characters have an accurate clock, they don't need to use lunar distances.

The easiest way to determine your position is to observe the altitude of a star at the time it crosses the local meridian (that is, when it is due north, due south, or directly overhead). Presuming the star is observed at upper culmination (when it is at the top of its diurnal path), your latitude is the star's declination, plus or minus its zenith distance (plus, if it passes to the South of overhead, and minus, if it passes to the North of overhead). Longitude is determined by the solar clock time (e.g., PST, EDT, UT), plus a correction for the time zone (8 hrs for PST, 4 hrs for EDT, 0 for UT), plus a correction for the difference between solar and sidereal time (tables giving this difference are in the Ephemeris, and many other publications), less the right ascension of the star. (There is a summary of this on my website, at http://cseligman.com/laboratory/navcalc.htm)

Any navigator who had tables of lunar distances would also have tables of the sort mentioned in the previous paragraph, so the only question is, do your characters have an accurate clock? If they don't, they would use lunar distances to calculate their longitude. Time was (in the 1800's), those distances would be in a nautical almanac. Nowadays, they aren't always given, as they are "simple" to calculate, with a computer, or a scientific calculator. As an example of such distances, you might look at http://www.historicalatlas.com/lunars/, which has a lengthy discussion of the matter, and handy (and accurate) calculators. (If you use one of the calculators, you'll find that it gives the geocentric lunar distances, rather than the local ones; corrections of up to a degree are required, based on the Moon's altitude and azimuth)

If the discussion above is not quite what you are looking for, and you would like to calculate the declination and right ascension of the Moon directly, you can estimate the geocentric values by interpolation from the nearest noon and midnight positions. Because the Moon's motion is not absolutely uniform, there would be a small error here, but it would be only a fraction of a degree, and if the tables contain second-order correction values, they could be used to eliminate that error.

To convert to the local apparent position of the Moon, you would need an additional correction, based on the Moon's observed altitude and azimuth. For non-meridian observations, the correction requires spherical trigonometry; but for meridian observations, it's fairly simple. Namely, the right ascension would be unchanged, and the declination would change by a fraction of a degree, being further south/north than the geocentric value by one degree (the radius of the Earth as seen from the Moon), multiplied by the cosine of its south/north altitude (that is, the correction would be zero, it if were overhead, one degree, if on the southern/northern horizon, and an intermediate value, anywhere in-between). Unfortunately, this correction for the declination does not apply to lunar distances, because those depend on the altitude and azimuth of both the Moon, and the star it is being compared to.

Hopefully, this gives you a better idea of what you need; but if you would like further clarification, just let me know.

Best wishes,

Courtney Seligman