Celestial Navigation in Your Head

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How to do Celestial Navigation in your Head

Introduction

These are the notes of a learning student and are meant to be taken as such. It is very possible that I am way off in the methods that I will use below. Please consider this a theory.

It's my dream that someday I can look at the night sky and get a Roller Coaster Intuitive feel for where I'm at in the Cosmos. The hardest part about understanding the heavens is that we are literaly inside the experiment, looking out, trying to figure out our angle as part of the whole. In essance we are riding on the outside of a sphere that is revolving around its own axis, and that is revolving around another solar body, the Sun. The fact that we can look out at other solar bodies, and stars, and use them to determine where we are physically located on the ball has got to excite us.

What kind of man would we be if we never pondered our current physical location on the Earth in relationship to other celestial bodies such as the Sun, Moon, Planets and Stars.

The ancient people would use the word 'Wanderers' in reference to the Sun, Moon, and Planets but to them they knew that the positions of the stars never changed, that the stars were isotropic.

Knowing from our example below that we can estimate the longitude point on the Earth where Jupiter intersects the Earth with a tangent line drawn from Jupiter to the Center of the Earth, we can go outside and during some times of the right years, view Jupiter and imagine it passing through that point.

These are my notes on doing a Celestial Navigation 'Fix'. A Fix is a calculation that places you at a Latitude and Longitude on the Earth that is calculated by viewing three celestial bodies above the Horizon. Once three bodies are fixed, three circles are drawn on a globe of the earth and where these circles converge is where you are at on the Earth. In these notes I will attempt to show how you can do this fix very roughly in your head without an ephemeris, or nautical almanac. In this manner, we can better understand how it works when we use a nautical almanac. We will only calculate one celestial body Jupiter since once we know how to do one, we can do others.

Project Requirements

You can read this and build some tools or you can just read this and imagine that you are using the tools. The requirements for this project is as follows:

World globe
Time piece
School protractor
String
Soda straw
Paper and pencil
A bolt or nut to be used at the end of the string for a weight.
Web access is optional

Pictures and tables

How to get the general idea of how to measure a celestial bodies distance from Zenith (directly over head)

Using a school protractor with a straw and string with a weight attached. We call this our own quadrant.

Remember that Jupiter is at a geocentric longitude of 191 degree's on our work date of April 26, 2005 Noon Just for fun, if you where born on this day, the Astrologers would say that Jupiter was in Libra or opposite Aries and note that Aries is 0 to 29 degree's longitude and Libra is 180 to 209 degree's longitude.

We will be using April 26, 2005 - 21:00 hours for our calculations so notice that in the Nautical Almanac below that 9:00 PM (21:00 Military time) in Salt Lake City Utah, USA is 3:00 Am the next day in Greenwich England where the Main time Meridian starts. We will use this table to see if the figures that we calculate match the ones shown here. See the red circled '21' and that we added 6 hours to that to get to the Greenwich time which is also circled. This information came from this online nautical almanac site
Notice left column day 26 hour 21 the data: | 169 57.3 and that we are adding 6 hours to this to arive at last line day 27 hour 3 | 260 12.1

View from a planetarium Program to find the 'First Point of Aries' or right ascension point 0

A picture of the sky from LAT 0, LON 84, (local) ST 0, facing North. As my friend recently showed me the picture he says: "You can see the North Star just on the horizon, with the big dipper hanging down under the horizon (the green line), as if you could see through the Earth. The celestial prime meridian can be drawn by extending a particular star in the big-dipper to Polaris (North star), and then a line from Polaris past Cassiopeia and then continuing to your Zenith over-head and beyond to the Southern polar region on the horizon". The big dipper is shown directly under the North star and is labeled 'Uma' for Ursa Major. Cassiopeia is aligned directly above the North Star and is labeled 'Cas'. We will use this Celestial Prime Meridian in our calculation so just remember that it runs through a star in Cassiopeia, the North Star, and then a star in the Big Dipper. My friend calls this line the 24 hour clock hand. Note that the Cassiopeia side is the Prime Celestial Meridian (0-180 degrees) and the Big Dipper side is the 180th meridian (180-360 degrees).

(!! Warning !! Note that according to this more scientific site, the last star of the 'M' of Cassiopeia and the last star of the handle of the big dipper are used to determine what they say : These three stars,Cassiopeiae, Polaris, and Alkaid, form a line through the N. Celestial Pole (approximately). Remember, I'm a learning student to say the most. See this site where the image below was taken from: See page 382)

Since Cassiopeia may look like an 'M' or a 'W' (W when on the bottom), you must remember a little rule to determine which side of the M or W is the 0 index also known as the 'first point of aries. Just remember:
'a broken Mast' which means 'M' it's the last star (M-last)
M comes before W in the alphabet so the leading left side of the 'M' is the 0 point. If it's a 'W' then the trailing side of the 'W' is the first point of Aries.

Note that now we can find the first point of Aries in the heavens, it is possible for instance to roughly estimate that Jupiter is at 191 degrees (not on this star map, see planet map above) opposite from it (on the Big Dipper side) of the imaginary line we can draw through Cassiopia, the North Star, and then the Big Dipper. Since we are doing celestial navigation in our head, you can see that it could be difficult to calculate a celestial bodies position this way however perhaps it could be done easier if we used a transparent piece of plexiglass with the center point being the north star.

Just in case you live to be thousands of years old, remember that With the 'Precession of the Equinox', measured in March each year, this map would rotate 360 degrees clockwise each 25,765 years (the cycle known as the 'Great Year'

Also notice that Cassiopeia spans the degrees from 0 to 30 and knowing that, you should be able to locate something if I told you it was at 191 degrees. Finding a point at 191 degrees is what this is all about, like reading longitude on a map of the Earth, simple once you break through and get it you are now beginning to see things in a beautiful new way, especially knowing that these degrees relate to time because in a two hour period, the Earth will rotate 30 degrees or put another way, the movement of the Earth will make the stars in the sky appear to be moving accross the sky 30 degress each and every two hours (double hour).

Real quick point is that if a pilot was flying due north at night, and say his destination was 3 degrees of latitude north, and it was a 12 hour flight, then the north star would slowly climb 3 degrees as he flew, and the celestial meridian line (24 hour clock hand) that we've been talking about would turn counter clockwise 180 degress, thereby guiding him on his flight.

Remember that star maps are two diminsional images of a three diminsional space, therefore imagine yourself inside a canalope that has been put back together but was sliced up into 12 even peices and halved perpendicular to the stem. The halved line is the celestial equator or the part of each piece where they are the widest. If you're viewing from a latitude of 45 North, then the halved line will be 45 degrees south of your zennith over-head. You could saw a hola-hoop in half, place one sawn end at due east and the other at due west, then tilt it to the south the same number of degrees that your latitude is (45), and that defines the celestial equator where each piece of the canalope is the widest. The view from inside this canalope is now very similar to what you could imagine the lat and long lines of the Earth would look like if you were in the center of the Earth. So now you know that the Celestial Spear has lat and long lines and that the Celestial Equator is the projection of the Earths equator into this celestial sphere. Also note that the Earths longitude lines go from 0 to 180 East and 0 to 180 West however the Celestial Sphere is 360 degrees. If you are looking South, the first point of aries would raise, then 2 hours later 30 degrees from the first point of aries, then 2 hours later 60 degress would raise, etc. So when looking North, the celestial bodies rotate counter clockwise around the north star, 30 degrees every 2 hours.

In Dutton's Nautical Navigation article, right ascention RA, is measured eastward from the first point of aries. When you think eastward in terms of the Celestial Sphere, imagine walking on top of it in an eastward direction similar to walking east on the sphere of the Earth. Dutton's PDF Article
He goes on to say that most of the time, astronomers use Right Ascention (RA) which is measured eastward from the first point of aries but that (We) navigators use hour angle which is measured westward and expressed in units of arc rather than time as with RA. It gets confusing here but again navigators use hour angle measured in arcs even though they call those measurements hour angles. He further states that GHA refers to Greenwich hour angel, LHA means local hour angel, and SHA means sidereal hour angel.

So now using the solar system model above, move around counter clockwise from the first point of aries found at the top, to 191 degrees and you will see that Jupiter is there on our test date of April 26, 2005.

First do the calculations to get the latitude of the fix pin stick by the book, not in our head

According to the nautical almanac, Jupiters GHA on this date was 69. That means that our pin we use to do the fix will be stuck into the Earth globe somewhere on latidute 69.

So by looking at the nautical almanac above we can say that on April 26, 2005, the GHA of Aries was 260 and for Jupiter was 69. These are the numbers that we will now try to calculate in our head. Write these down or remember them now and lets do some math with them to get a feel of how this stuff works.

Now we will do the calculations to get the latitude using our head rather than using the almanac or ephemerus

First we need to calculating the GHA(Aries) in our head

The vernal equinox was on march 21, 2005 ( noon was 359.06.2 ) On April 26, at 21:00 hours: Calculate GHA of aries as follows: 1 day for each day past vernal equinox = 36 days multiplied by ((360/365.25) gives an answer of int(35) days. Remember this 35.
21:00 hours here in Salt Lake City is 03:00(Am) in Greenwich and 03:00 converted to degree's is 225 because noon is the 0 point when calculating GHA(Aries).
So 12:00 Midnight is 180 degree's plus 45 degree's for the 03:00 makes the 225.

225 + 35 = 260 and bingo, the nautical almanac above says GHA(Aries) for this date and time is in fact 260 (See Almanac above). Note that this is a measurement of how far the Earth has gone around the Sun since the Vernal Equinox, and how the time of day tells us how much the Earth has spun around on this date.

Now we can estimate where Jupiter is so that we can make the calculation to find the fix longitude point

GHA=Greenwich Hour Angle
SHA=Sidereal Hour Angle, usually used to locate fixed stars on the celestial sphere. It uses the hour circle of the first point or Aries for it point of origin.
GP=geographical point where a celestial sphere intersects the Earth surface on a tangent to the center of the Earth
Sidereal=Relative to the stars

Looking at Jupiter in the night sky on this night and knowing where the first point of aries is, one could estimate that Jupiter is 191 degrees counter clockwise (westward) from the first point of aries. Note that an ephemerus would also provide this same number.

Again, I will use April 26, 2005 at 21:00 hours SLC Utah (latitude 40 North Longitude 111 West) for this example.

Again we know from the above almanac that on the test date, Jupiter's GHA is 69, and this is the number we want to arrive in our computation.

Since an our observation or an Ephemurus shows that Jupiter is at 191 degree's, we convert to RA by taking the Complement of that to 360 and getting an SHA of 169.

GHA = SHA + GHA(aries)
GHA = SHA(Jupiter) + GHA(first point of aries) note: see nautical almanac above, or see how we will compute it shown below to get the GHA(aries))

Now we can calculate the GHA of Jupiter

So Jupiter GHA = (Jupiter's SHA) + (GHA (Aries))
    x = 169 + 260
    430 = 169 + 260
    -360 (subtract 360 if answer is greater than 360)
    -------
    70
And notice that the Almanac (see above) says Jupiter's GHA is 69
Note: not sure how to add in extra day since 21:00 plus 6 hours equals :03:00 next day.

So now we have the longitude and it's the same as Venezuela in South America. But we really need to use the celestial spheres declination to gain the latitude before we can say that it's Venezuela.

The fact that we calculated the 69 is very significant because it is the same number that the nautical almanac shows. It is the longitude point where we will stick the pin in the Earth's globe and will be somewhere around the same longitude as Venezuela South America, however, we now need to calculate the latitude for the pin to see if if will be in Venezuala or Brazil. Note that the middle of Venezuela is about latitude 7 north and longitude 67 East and it can be found at the northern most part of South America.

Now to get the latitude for the fix pin stick from the celestial sphere's declination (in this case Jupiter)

If we are using the nautical almanac to make the fix, we can simply do a lookup for Jupiter's Declination for that given day and hour. The declanation shown in the nautical chart above shows 2 hours 58 minutes south which we can round off to 3 degrees of latitude south of the equator. This will mean that the pin would be pegged at 3 degrees south latitude and 69 degrees west longitude, a place in Brazil south of Venezuela.

According to my studies, Jupiter's declination will head north for 6 years, then south for 6 years as it makes it's 12 year cycle around the Sun. In the year 2002 it made it's farthest journey north to about 23 degrees and then in 2008 it traveled 23 degrees south. So as I write this, Jupiter will be seen lower in the sky than I've seen it for 12 years.

If however, we are doing the fix in our head then we must continue by saying that declination is measured positive if a body is north of the celestial equatore and it is measured negitive if it is south of the celestial equator.

Since I live at latitude 40 North, the celestial equator is 40 degree's down from my zenith toward the south. You could take a hula-hoop and saw it in half and hold one end at the east and the other at the west (not magnetic). Now tilt it to the south so it aligns with your celestial equator. Now look through the angle of the hula-hoop to Jupiter. See how many degree's it is North or South of the hula-hoop line (Celestial equator) and this is the declination. You can also use the Quadrant that you made to make this measurement. This declination of Jupiter is used to plot your latitude of the point where Jupiter passes through the earth at this time. Remember that at the equator, the latitude is 0, and then moves north or south 90 degree's. This same way, Jupiter may be north or south of the celestial equator, so simply convert Jupiter's declination to latitude.

Now that the longitude and latitude are known, we can press a pin into the globe and it should be in northern Brazil just south of Venezuela. If you don't have a globe, you won't be able to use the string but you can see the latitude and longitude lines on the Earth at this Site

How to calculate the length of the string

Next we will calculate the length of a string that we will attached to the pin we stuck in Venezuela and use it to draw a "circle of equal latitude" on the globe. Using the protractor (quadrant) we made above, we measure the angle down from our zenith to Jupiter to get the zenith distance. I think that my reading on April 26th was roughly 61 degree's. I used my self made quadrant. Now multiply this 61 by the number 60 to get 3660. This is now the number of nautical miles that the length of the string must be cut to. Knowing that a nautical mile is 1 minute, of a degree will also help. Cutting the string to the length and attaching it to a pin, you should be able to draw one of the three arcs that are needed to calculate where these three arcs converge thus giving you the fix or position where you are located on Earth.

If the the number of nautical miles is 3660 then divide that by 60 to get 61 and this is a rough approximation of the number of degrees latitude (preferred) or longitude that the string length must represent.

Now we know where to place the pin in the globe and mark the geographical point or place where Jupiter is intersecting the surface of the Earth as it transits to the center of the Earth. I have often wondered when looking at the night sky how to calculate where the center of the Earth would be and then knowing this in relation to where Jupiter passes for instance through Venezuela would be remarkable. Could somehow knowing another Celestial body that passed through your location on its way to the center of the Earth provide this viewing?

To Calculate where we are located

You need three celestial bodies to make this calculation and there are some real tricky aspects about using the Sun and the Moon that you had better learn before using them. If you are lucky, you can use three planets. Since you placed a pin in the globe where Jupiter's intersected the Earth, and cut the string, (or used dividers) at the calculated length, you know that you can draw an arc on the globe using that string. You can do these same calculations with stars or other planets and by drawing 3 arcs, the place that they intersect is where you are on the earth. You can use 2 celestial bodies instead of three if you have a rough idea of where you are at. You can also use the Pythagorean theorem for other calculations.

References and Links:

An excellent site on celestial navigation (pdf)
Nautical Almanac
Dutton's Nautical Navigation book
View Dividers tools used in place of string
A better more scientific approach can be found here and Note that this site uses the star Alkaid (last star in handle of big dipper) for the first point of aries alignment

Written by SDH