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Celestial navigation series, part nine

Oct 31, 2019

Star sights and precalculation

Editor’s note: We’re revisiting this series on navigating by the sun, moon, planets and stars in the age of GPS because celestial nav is not only a viable backup to satellite navigation, but it is also a skill that ocean voyagers should have in their toolkit. In this series, we’ll cover all the basic knowledge you’ll require to get up to speed on this elegant and rewarding technique for finding your way at sea. Click to read Part 1, Part 2, Part 3, Part 4Part 5, Part 6, Part 7 and Part 8.

In this installment, we’ll cover how to reduce a star sight and how to use HO 249 Vol. 1 to precalcuate what stars will be available.

 

The GHA of a star is made up of two components: the GHA of Aries and the SHA of the star. Aries is a benchmark in the sky that “circles” the Earth at the same speed as the stars. The SHA of a star is fairly constant. It changes only a few minutes each year.

So far, we have concentrated on sights of the sun. We’ve looked at how to reduce a sun sight and then how to advance a sun line of position along our DR track and construct a running fix. Now we move on to another type of celestial body that we can use for celestial navigation: stars (technically, of course, a sun sight is a star sight since the sun is a star, but because the sun is so close to us, it is a special case).

The stars we see in the sky are often called the “fixed stars.” They are actually moving at great speed but are at such a great distance that they appear to us as fixed and unmoving. The nearest stars are Alpha and Proxima Centauri, 4.3 light years away. Light travels at 186,282 miles per second, or 5.88 billion miles in a year, which means that Alpha and Proxima are approximately 25 trillion miles away. At that distance, it’s clear why we can’t see them moving.

The stars’ motion comes from the Earth’s spin. This causes them to rise in the east and set in the west. Some stars are visible all night, circling the pole, and thus are called “circumpolar stars.”

There is a seasonal change in the stars. They rise four minutes earlier every night because the celestial sphere slips about 1 degree west every night. This accounts for the difference between the “winter sky” and the “summer sky.” The stars that are visible during the summer nights are behind the sun during the daylight hours in the winter. As the Earth slowly orbits the sun, the night sky slowly changes as different stars become visible.
 

Because the stars remain relatively stationary in the sky, their relationship to each other is pretty much unchanging. This allows for a shortcut in describing the Greenwich Hour Angle, or GHA, of each star. All of the stars are measured from a benchmark in the sky, a meridian called Aries. Aries is determined by the position of the sun at the time of the vernal (spring) equinox. That point was in the constellation Aries during the time of the ancient Greeks, but because of the wobble of the Earth’s axis — known as precession — the vernal crossing point is in the constellation Pisces and is slowly moving toward the constellation Aquarius. Aries cannot be seen, just the way the projection of the Greenwich meridian cannot be seen in the sky.

Since the relative position of the stars remains constant, the distance west from this Aries meridian to each star also remains constant over the course of a day. This distance is called the star’s sidereal hour angle (SHA). Aries has a GHA, which changes second by second just the way the GHA of the sun does. To find the GHA of a star, it is necessary to first find the GHA of Aries. This is the distance west from the Greenwich meridian to Aries. The SHA measures west from Aries to the meridian of the star. So, the GHA of Aries plus the SHA of the star equals the GHA of the star.

Each star has its own declination, which changes very slowly — only a few minutes each year.

Once a star sight has been taken, there are only a few differences between reducing a star sight and the familiar sun sight reduction.

There is a different apparent altitude correction for stars than for the sun. The table is in the front inside cover of the Nautical Almanac right in the middle of the page between the sun correction and the dip table. This is a critical table, and it works just the same way the sun correction table does. The table is entered with the apparent altitude for the star (that is, the sextant reading corrected for index error and dip). The result is applied to the apparent altitude to give the Ho.

 

Finding the GHA of a star is a little more involved than finding the GHA of the sun. Take a look at the accompanying Aries column from the Nautical Almanac on the opposite page. Take the hours value of sight time in GMT and enter the Aries column. For the minutes and seconds of time, find an increment in the back of the almanac in the increments and corrections boxes. In these boxes there is a special column for Aries; the increment must not be taken from the “SUN” column. This is added to the GHA for Aries for hours to give the total GHA Aries. Next, it is necessary to look up the SHA of the star in the list on the daily page. This SHA is always added to the GHA Aries to find the GHA of the star.

The declination of the star is found in the column next to the SHA. It does not need to be modified.

The rest of the process is just the same as a sun sight reduction.

Sample problem
On May 5 at 10:26:17, you shoot Alpheratz. Find the GHA and declination of Alpheratz. First, find the GHA of Aries at 1000 on May 5 (see table above). That is 13° 31.6’. Next find the increment from the colored pages in the back of the almanac for 26:17. Make sure to use the “ARIES” column. This is 6° 35.3’.

GHA Aries (1000)
increment (26:17)
GHA Aries (10:26:17)
  13° 17.3’
+     6° 35.3’
19° 52.6’

Now back to the daily pages to find the SHA and declination for Alpheratz. The SHA is 358° 00.2’, and the declination is 29° 02.8’ N. The declination does not need any further work. The SHA must be added to the GHA of Aries:

GHA Aries
SHA Alpheratz 
GHA Alpheratz
   
GHA Alpheratz
  19° 52.6’
+     357° 59.4’
377° 52.0’
–     360° 00.0’
17° 52.0’

HO 249 Vol. 1
While HO 249 Volumes 2 and 3 can be used to reduce sights of the sun, moon, planets and many stars, there is also a sight reduction table specifically designed for star sight reduction: Volume 1 of HO 249. This has a number of advantages over other sight reduction volumes.

The primary advantage is that the table is entered with the assumed latitude, the LHA Aries (not of the star) and the name of the star. This streamlines the process for getting to the sight reduction volume. The table directly yields Hc and Zn; there is no need to use interpolation tables or formulas.

The disadvantages are that only seven stars are available at any one time. This limits the stars that can be shot, especially in partly cloudy conditions where the navigator may not have the flexibility to pick stars. The volume is only useable for 10 years, unlike other tables, which are good indefinitely.

For each group of seven stars, three of them have a black diamond. These are the recommendation by the editors of Vol. 1 (or more likely a computer someplace) as to the best three stars to shoot for nice crossing angles. Some of the stars are written in all capital letters. These have a magnitude of 1.5 or better. They are brighter than those with only the first letter capitalized.

Sample problem
On May 4 at 21:26:18 GMT you shoot Arcturus. Your DR latitude is 5° 14’ N, and DR longitude is 23° 46’ W. Find the Hc and Zn of Arcturus.

To enter Vol. 1, we need the LHA Aries, so first find the GHA Aries. GHA Aries at 2100 is 177° 59.6’ (see table above), and at 26:18 it is 6° 35.6’.

GHA Aries (2100) 
GHA Aries (26:18)
GHA Aries (21:26:18) 
  177° 45.3’
+     6° 35.6’
184° 20.9’

Now subtract the west assumed longitude in order to find the LHA Aries:

GHA (21:26:18)
assumed longitude
LHA Aries    
  184° 20.9’
–     23° 20.9’
161° 00.0’

The other entering argument is the assumed latitude. Round off 5° 14’ to the nearest degree to get 5°. Use this to enter Vol. 1 (see table below). After finding the page for 5° N, move down the LHA columns to 161°. Then follow that row across to the right to the column under the headline “Arcturus.” There are two numbers there: 36° 41’, which is Hc, and 070°, which is Zn.

There are no further corrections necessary.

  
 

This excerpt is from the daily pages of the Nautical Almanac for May 5. To use the table, enter with latitude. It is not necessary to interpolate for most precalculation (click to enlarge).

Precalculating stars
In order to determine where to look for which stars, even the experienced navigator should precalculate the position of stars before sight time. One method for doing this is to use HO 249 Vol. 1. This requires the LHA Aries as a beginning argument. There are also mobile nav apps you can use to determine what stars to shoot.

The first job is to determine the time of twilight. Generally speaking, both the horizon and the brighter navigational stars are visible between civil twilight and nautical twilight. These are both listed in the daily pages by latitude. In the evening it makes sense to determine civil twilight (which will occur first), and in the morning find nautical twilight. When using the table, it is usually not necessary to interpolate between latitudes. The time extracted is for twilight at 0° longitude, so it needs to be converted to GMT at our longitude.

Take the expected DR longitude at the rough time of twilight and use it to enter the arc-to-time conversion table (the first colored page in the back of the Nautical Almanac). If the longitude is west, then this should be added to the time of twilight (subtract if it’s east). This is the time of twilight at your location in GMT. Use this time to find the GHA Aries and then convert that to LHA Aries.

The advantage to using HO 249 Vol. 1 is that then the stars that you precalculated may be reduced using this relatively quick method. Its drawback is that only seven stars are available through precalculation.

Sample problem
On May 5, you desire to find the time for twilight this evening. Your DR position should be about 5° 10’ N and 123° 15’ W. Determine the time of civil twilight in GMT at your position.

First find the time for civil twilight at 5° N: Because 5° is conveniently between 0° and 10°, this may be interpolated to 1828. Now we need to find the time that it takes for twilight to travel to our longitude of 123° W. After consulting the arc-to-time conversion table, 123° gives us 0812. Because we are in west longitude, this is added to 1828.

civ. twilight (0°)
123° W
civ. twilight (123° W)
 
civ. twilight (123° W)   
  
  GMT 1828
+     0812
GMT 2640
–     2400
May 6 0240

Now find the LHA Aries. The GHA Aries at 0200 is 253° 11.9’ (see Aries table above). The increment at 40:00 is 10° 01.6’.

GHA Aries (0200)
increment (40:00)
GHA Aries (02:40:00)   
   
  253° 56.7’
+     10° 01.6’
263° 58.3’

Next, find the LHA by subtracting assumed longitude:

GHA Aries (02:40:00)
assumed longitude
LHA Aries     
    
  263° 58.3’
–     123° 58.3’
140° 00.0’

Take 140° LHA Aries and the assumed latitude, which would be 5° 10’ N rounded off to 5° N here, and enter Vol. 1. Find the page for 5° N and go down the LHA column to 140°.

Moving along the row to the right, there are seven stars available. The Hc and Zn for each star are listed in this row. Write down this information in a notebook so that it is accessible while on deck taking the sights. A list in the notebook might start like this:

Dubhe
Spica
30° 03’
27° 02’
014°
105°

Write down all seven stars in case it is partially cloudy and the stars you want are obscured. When taking the sights, use this precalculated data to search for the stars in the sky. The Zn will indicate which direction to look (remember this is true direction, not magnetic), and the Hc will tell how high up in the sky the star should be. Generally these stars are not near any other bright stars, so if you see a star where one is precalculated, it’s probably the right star.

It is also possible to preset your sextant for the Hc of the star and then just search the horizon for it through the sextant.

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