23 July 2010

Calculating Latitude

Having taken our sights, graphed them to estimate Hs, and applied corrections to determine Ho, we are now ready to calculate the latitude of my position when I took the sights.

Since Ho is the observed height of the sun above the horizon, you might be wondering if Ho is the latitude value. No, it's not that simple, but almost.

Here is a diagram that explains the simple relationship between Ho and Latitude.

The geometrical relationship between Ho and Latitude
drawing jalmberg

Let's take this diagram step by step.

First, the inside circle represents the Earth. The outside circle represents the Celestial Sphere. Forget Copernicus and all that 'Earth rotating around the Sun' nonsense. Copernicus was clearly wrong. The Earth is at the center of the universe, and the Sun, Moon, stars, and planets all rotate around the earth, on the surface of the Celestial Sphere.

At the top of the Earth, you can see me with my sextant, taking my sights.

There are three lines drawn through the center of the earth, and projected out onto the celestial sphere:

  1. The Zenith line starts at the center of the earth, goes through me, and straight up to the Celestial Sphere. My Zenith is the point on the celestial sphere directly above my head
  2. The Horizon is at a right angle to the Zenith line
  3. The Equator is, well, the Equator
Study these three lines until they make sense to you, before continuing. They're pretty simple, but if you gloss over these three lines without really taking them in, the rest of this explanation will make no sense to you.

Got it? Okay, my Latitude is the angle between my Zenith line, and the equator. This is the angle we are trying to find.

Note that the angle of Latitude is equal to the sum two other angles, labled 'Dec.' and 'Z.D.'

Latitude = Dec + Z.D.

Z.D. is the Zenith Distance of the Sun -- the angle between the Sun and my Zenith. Since the Zenith and Horizon lines are at a right angle, it's pretty obvious that you can calculate the zenith distance by subtracting Ho from 90°.

Z.D. = 90° - Ho

Since we know that Ho is 71° 14.6', then Z.D. must be 18° 45.4'

Get it? Again, stop and study the diagram until this makes sense. If you just skip over this part, you won't really understand what's going on. It's easy, but you must stop a second to understand it.

Dec. is the Declination of the Sun -- the height of the Sun above the Equator at solar noon on 12 Jul 2010.

Note the difference between Ho and Declination: Ho is the height of the Sun above the horizon, and Dec is the height of the Sun above the equator.

How do we know what the Sun's declination was at noon on 12 Jul 2010? We look it up in the Nautical Almanac.

 Look up Sun's Declination in the Nautical Almanac
click for larger image

Actually, I circled the wrong entry. It's an easy mistake to make, so it's helpful that I made it (again!).

We don't want the declination of the Sun when it was solar noon in Greenwich (12:00 GMT), we want it when it was solar noon at my location, roughly 17:00 GMT.

You can see that the Sun's declination at 17:00 GMT on 12 July 2010 was 21° 54.7'. The 'N' indicates that the Sun was North of the Equator. (In January, for example, it would be South of the Equator.)

So, we are almost there.

We used Ho to find the Sun's Zenith Distance. We looked up the Sun's declination in the Nautical Almanac. If we just add those two angles together, we should get an angle that is equal to my Latitude:

ZD + Dec = Latitude

ZD:   18° 45.4'
Dec:  21° 54.7'
Lat:  40° 40.1'

So that's was my Latitude, according to my sextant sights. If  you've followed along, and studied the diagram above just a bit, it should make sense to you. And note that the only math we've done is some simple addition and subtraction. No rocket science, and hopefully, no mystery.


Next time we tackle the slightly more difficult problem of Longitude.

>>> Next Episode: Time of Meridian Passage

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  1. DeeP: I guess you understood it? Cool! It's pretty simple, right?

  2. Ok John, I figure your just near where Ocean Ave turns into Hartshorne Drive. But where did you park the car? is that a restaurant carpark? What is that just up the road? Maybe Tollbooths?
    This is assuming you walked to the edge of the water to take your reading? I am also assuming this is the only place at 40.40 that you can see the horizon.
    Cheers john, keep em coming


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