In the celestial navigation world, if you have a zenith angle of a star and the time of it, you can calculate the ground position of said star, (from the time), and the distance to you, (in nautical miles), by multiplying the zenith angle by 60.
This is obviously a simple "spherical earth" formula (1'=1nm),good enough for navigation but doesn't account for the ellipsoid, and can be several nm off.
My question: is there a better "geodetic" formula that can use the just the zenith angle and an approximate bearing, to improve the accuracy?
I know the forward and inverse great circle formulas require either lat/lon's of both ends of the line, or the lat/lon of one and the distance and azimuth to the other - but what if you only have the zenith angle and you want to know the distance. Is there a way to improve on zenith angle x 60? I'm thinking if you at least have a compass bearing or angle from polaris to establish a basic direction then maybe you can account for the ellipsoid somewhat.
I'm talking about lines in the 1000's of nautical miles range - and this is just for hobby interest that I'm asking. Just curious.
Thanks!
Not sure why you wouldn't just do a proper astro observation, but check out this:
http://www.rollanet.org/~eksi/Handbook.htm
There is a procedure here for using the sun via the "Altitude" method, that might work with stars. There are also all the usual Polaris observations used to determine latitude.
For a more comprehensive treatment of celestial navigation, I recommend?ÿ https://msi.nga.mil/NGAPortal/MSI.portal?_nfpb=true&_pageLabel=msi_portal_page_62&pubCode=0002
The American Practical Navigator is a free PDF at the site above. Start Chapter 15 page 255 print/233 PDF numbering
I do not understand from the original post how his observations are made nor the equipment used. Sextant angles are made with respect to the horizon (or reduced to it) and not zenith angles. There are a number of corrections described in the referenced text. Time accuracy and corrections to it are also important. I have not looked around for a Nautical Almanac in years. I thought the printed versions were discontinued be see a 2019 version that is on backorder at the US GPO site. <post edited to remove out-of-date reference to a free digital almanac>
If you know your geographic position you can easily compute the various radii of curvature at that point i.e. Prime Vertical, Meridian, Gaussian.
I have not done any celestial navigation problems in years. Now that I live inland, the issue for me is unlikely to spur further exploration.
Cheers,
DMM?ÿ
I vaguely recall that the navigation calcs include a correction for height of eye above the sea. With angle above horizon, refraction correction, and height of eye correction you should have 90 - zenith angle so you can measure either way.?ÿ?ÿ?ÿ
If you get to an accuracy well under a nautical mile, there is a geodetic correction that gets from astro latitude and longitude to geodetic lat-lon.?ÿ Look up deflection of the vertical due to gravity variations tilting your apparent zenith or horizon. That is usually negligible at navigation accuracy but significant when astro measurements are included in surveys on the ground.
I used a total station. I observed Mars crossing my meridian by the equal altitudes method. I recorded a zenith angle and time shortly before the actual passage, then kept tracking the planet to determine the absolute high point, then reset the first angle and?ÿ timed the moment Mars hit that angle on the way down. Thus with GHA and declination from an ephemeris, and my observations, I could determine my lat/lon. Was within a few km...ok 3..but it was just for fun, I already know where I am within a cm thanks to GPS ?????ÿ
Solving the distance by the geodetic inverse routine in "Adjust" I got 5839km from the ground position of Mars, to me.
My interest in cel nav did start with aquiring a sextant - and it does measure altitude from horizon, but after correcting for refraction, index error, etc you subtract it from 90 degrees, which gives you the zenith angle, then multiply by 60 to get the radius of a circle of equal altitude, which you are therefore located on.
So, multiplying my observed zenith angle by 60, I got 5860km, a difference of 21km.
So if instead of doing a meridian passage, I wanted to shoot 3 stars at any time and find the intersection of the 3 circles of equal altitude, it would, I assume, improve the result if you could "improve" the way the distance is calculated by scaling it to account for the ellipsoid flattening. The only info you would have would be the zenith angle and a bearing.
I'll shut up now.. ?????ÿ
Bill93 has captured the essence of sextant positioning accuracy.?ÿ A sextant is handheld so the shakiness factor negates any consideration of ellipsoid vs. a round globe, it's immaterial.?ÿ I own a sextant and have used it on the West Coast of the North America from SanFran to PV on sailboats over the years.?ÿ There's lots of inaccuracies, determination of the horizon, internal sextant errors, improper tilting of the sextant, etc.
That being said, after practice ashore and developing the skill to hold really still during the perfect shot when the boat is stable I could get to 10-15 miles typical accuracy, and skilled observers could get down to 5 miles or less.?ÿ Multiple shots to various stars would tighten the circle but basic celestial is Sun, Polaris and Moon shots.?ÿ Sextant observations are inherently not a part of land surveying.
Terrestrial celestial observations are still a part of a well rounded surveyor's bag, to determine the deflection of the vertical, for example.?ÿ The difference, an accurate theodolite on a stable tripod with sub second time accuracy and an absolute bubble determined level reference.?ÿ On more mundane surveys?ÿ I've observed Polaris maybe 50 times professionally and when working for BLM on a crew running solar traverses?ÿ a dozen solar OBS/day,?ÿ both which tightened up the closure immensely.?ÿ That was back in the T2?ÿ invar tape traverse days.
But of course in the modern GPS era orientation and absolute position are available with the push of a button in clear skies. So time passes on and skills using terrestrial equipment has become obsolete except in the less than 2,000 feet arena & obscured sky view situations.?ÿ ?ÿSad.?ÿ Celestial objects are still worthy backsights/checkshots in certain situations.
