Summer Triangle, Vega, Deneb, and Altair

The Summer Triangle is an an imaginary triangle drawn on the northern hemisphere's celestial sphere, with its defining vertices at Altair, Deneb, and Vega. This triangle connects the constellations of Aquila, Cygnus, and Lyra.

Near midnight the Summer Triangle lies virtually overhead at mid-northern latitudes during the summer months, but can also be seen during spring in the early morning. In the autumn the summer triangle is visible in the evening well until November. From the southern hemisphere it appears upside down and low in the sky during the winter months.

Name / Constellation

Vega / Lyra

Deneb / Cygnus

Altair / Aquila

Altair is the brightest star in the constellation Aquila and the twelfth brightest star in the night sky with an apparent visual magnitude of 0.77 .

Deneb is the brightest star in the constellation Cygnus and one of the vertices of the Summer Triangle. It is the 19th brightest star in the night sky, with an apparent magnitude of 1.25. A white supergiant, Deneb is also one of the most luminous stars known.

Vega is the brightest star in the constellation Lyra, the fifth brightest star in the night sky and the second brightest star in the northern celestial hemisphere, after Arcturus. It is a relatively nearby star at only 25 light-years from Earth, and, together with Arcturus and Sirius, one of the most luminous stars in the Sun's neighborhood.

Declination

In astronomy, declination is one of the two coordinates of the equatorial coordinate system, the other being either right ascension or hour angle. Dec is comparable to latitude, projected onto the celestial sphere, and is measured in degrees north and south of the celestial equator. Points north of the celestial equator have positive declinations, while those to the south have negative declinations.

An object on the celestial equator has a dec of 0°.
An object at the celestial north pole has a dec of + 90°.
An object at the celestial south pole has a dec of - 90°.

The sign is customarily included even if it is positive. Any unit of angle can be used for declination, but it is often expressed in degrees, minutes, and seconds of arc. If instead of measuring from and along the equator the angles are measured from and along the horizon, the angles are called azimuth and altitude.

Stars
Because a star lies in a nearly constant direction as viewed from earth, its declination is approximately constant from year to year. However, both the right ascension and declination do change gradually due to the effects of precession of the equinoxes and proper motion.

Sun
The declination of the Sun (δ) is the angle between the rays of the sun and the plane of the earth's equator. Since the angle between the earth axis and the plane of the earth orbit is nearly constant, δ varies with the seasons and its period is one year, that is the time needed by the earth to complete its revolution around the sun.

When the projection of the earth axis on the plane of the earth orbit is on the same line linking the earth and the sun, the angle between the rays of the sun and the plane of the earth equator is maximum and its value is 23°27'. This happens at the solstices. Therefore δ = +23°27' at the northern hemisphere summer solstice and δ = -23°27' at the northern hemisphere winter solstice. Due to the changes in the tilt of the Earth's axis, the angle between the rays of the sun and the plane of the earth equator is slightly decreasing.

Moon
Declination of the Moon is computed by adding Sun's declination (which is called Declination of Place while computing declination of other planets and Moon) to Moon's latitude. Sun's declination (± 23.44°) is much larger in magnitude than Moon's latitude (± 5.14°). The Moon's declination can be said to have an annual cycle synchronous with that of the Sun starting with the vernal equinox.

Moon's latitude is a function of the difference between True Moon and its ascending node. Since lunar nodes make one revolution in nearly 19 years, lunar latitude has an approximately 19 year long cycle. Lunar latitude is equal to inverse sine of the product of sine of maximum lunar latitude and sine of difference between Moon and its node.

For greater accuracy, reduced latitude is used instead of Moon's true latitude, which is obtained by multiplying lunar latitude with a multiplier having a maximum value of 1 for tropical Moon at 180° and 0.91745 for tropical Moon at 0°. This is caused by a third cycle in lunar declination which has a period of one lunar month and a maximum range of ± 0.425°. Summing all three components gives a range of maximum declination from +28°35' to +18°18' and the minimum from -18°18' to -28°35' for lunar declination.

Zenith

In general terms, the zenith is the direction pointing directly above a particular location. The concept of above is more specifically defined in astronomy, geophysics and related sciences as the vertical direction opposite to the force of gravity at a given location. The opposite direction, the direction of the gravitational force is called the nadir. The term zenith is also used to represent the highest point reached by a celestial body during its apparent orbit around a given point of observation. This sense of the word is used to describe the location of the Sun, but it is only technically accurate for one latitude at a time and impossible for latitudes outside the tropics.

Strictly speaking, the zenith is only approximately contained in the local meridian plane because the latter is defined in terms of the rotational characteristics of the celestial body, not in terms of its gravitational field. The two coincide only for a perfectly rotationally symmetric body. On Earth, the axis of rotation is not fixed with respect to the planet so that the local vertical direction, as defined by the gravity field, is itself changing direction in time (for instance due to lunar and solar tides).

Nadir
The nadir is the direction pointing directly below a particular location (perpendicular, orthogonal). Since the concept of being below is itself somewhat vague, scientists define the nadir in more rigorous terms. Specifically, in astronomy, geophysics and related sciences (e.g., meteorology), the nadir at a given point is the local vertical direction pointing in the direction of the force of gravity at that location. The direction opposite of the nadir is the zenith.

Nadir also refers to a downward-facing viewing angle of an orbiting satellite, such as is employed during remote sensing of the atmosphere, as well as when an astronaut faces the Earth while performing an EVA.
Extra-vehicular activity (EVA) is work done by an astronaut away from the Earth, and outside of a spacecraft.

Circumpolar Stars

A circumpolar star is a star that, as viewed from a given latitude on Earth, never sets (it never disappears below the horizon), due to its proximity to one of the celestial poles. Circumpolar stars are visible for the entire night on every night of the year, and would be continuously visible throughout the day were they not overwhelmed by the Sun's glare.

As Earth spins daily on its axis, the stars appear to rotate in circular paths around one of the celestial poles (the north celestial pole for observers in the northern hemisphere, or the south celestial pole for observers in the southern hemisphere). Stars far from a celestial pole appear to rotate in large circles, stars located very close to a celestial pole rotate in small circles and hardly seem to engage in any diurnal motion at all. Depending on the observer's latitude on Earth, some stars, the circumpolar ones, are close enough to the celestial pole to remain continuously above the horizon, while other stars dip below the horizon for some portion of their daily circular path, and others remain permanently below the horizon.

The circumpolar stars appear to lie within a circle that is centered at the celestial pole and tangential to the horizon. At the Earth's North Pole, the north celestial pole is directly overhead, and all stars that are visible at all (that is, all stars in the northern celestial hemisphere) are circumpolar. As one travels south, the north celestial pole moves towards the northern horizon. More and more stars that are at a distance from it begin to disappear below the horizon for some portion of their daily orbit, and the circle containing the remaining circumpolar stars becomes increasingly small. At the Earth's equator this circle vanishes to a single point, the celestial pole itself, which lies on the horizon, and there are no circumpolar stars at all.

As one travels south of the equator the opposite happens. The south celestial pole appears increasingly high in the sky, and all the stars lying within an increasingly large circle centred on that pole become circumpolar about it. This continues until one reaches the Earth's South Pole where, once again, all visible stars are circumpolar.

The north celestial pole is located very close to the North Star (Polaris), so, from the northern hemisphere all circumpolar stars appear to rotate around Polaris. Polaris itself remains almost stationary, always at the north (the azimuth is 0°), and always at the same altitude (angle from the horizon), equal to the latitude of the point of observation on Earth.

The circumstances making a star circumpolar is solely dependant on the observer's hemisphere and their latitude. As the altitude of either the north celestial pole or south celestial pole is the same as the observer's latitude, any star whose position from the pole is less than the latitude, will be circumpolar and will never set below the horizon. If the observer latitude is 45°N and is facing north, then any star will become circumpolar if it lies less than 45° from the north celestial pole. If the observer's latitude is 35°S and is facing south, then these stars are circumpolar within 35° of the south celestial pole. Stars on the celestial equator will not be circumpolar when seen from any latitude in either hemisphere of the Earth.

This is easily calculate if some star will be circumpolar (or not) at the observer's latitude by just knowing the star's declination. Some stars within the far northern constellations, such as Cassiopeia, Cepheus, Ursa Major, and Ursa Minor, roughly north of the Tropic of Cancer (+23½°N), will be circumpolar stars that never rise or set.

Some Stars within the far southern constellations, such as Crux, Musca, and Hydrus, roughly south of the Tropic of Capricorn (-23½°S), will also be circumpolar stars.
Stars (and constellations) that are circumpolar in one hemisphere are always invisible in the high latitudes of the opposite hemisphere, and these never rise above the horizon. For example, the southern circumpolar star Acrux is invisible from most of the Continental United States, likewise, the seven stars of the northern circumpolar Big Dipper asterism are invisible from most of the Patagonia region of South America.

Celestial Poles

The north and south celestial poles are the two imaginary points in the sky where the Earth's axis of rotation, intersects the imaginary rotating sphere of stars called the celestial sphere. The north and south celestial poles appear directly overhead to an observer at the Earth's North Pole and South Pole.

At night the stars appear to drift overhead from east to west, completing a full circuit around the sky in 24 (sidereal) hours. Of course, exactly the same motion occurs during the day, except that the stars are not visible due to the sun's glare. This apparent motion is due to the spinning of the Earth on its axis. As the Earth spins, the celestial poles remain fixed in the sky, and all other points seem to rotate around them.

The celestial poles are also the poles of the celestial equatorial coordinate system, meaning they have declinations of + 90 degrees and - 90 degrees for the north and south celestial poles. The celestial poles do not remain permanently fixed against the background of the stars. Because of a phenomenon known as the precession of the equinoxes, the poles trace out circles on the celestial sphere. The Earth's axis is also subject to other complex motions which cause the celestial poles to shift slightly over cycles of varying lengths. Over very long periods the positions of the stars themselves change, due to the stars proper motions.

A planet's celestial poles are the points in the sky where the projection of the planet's axis of rotation intersects the celestial sphere. These points vary because different planets axes are oriented differently and the apparent positions of the stars also change slightly due to parallax effects.

Polaris

The north celestial pole has nearly the same coordinates as the star Polaris (also called the "pole star"). This makes Polaris useful for navigation in the northern hemisphere. Not only is it always above the north point of the horizon, but its altitude angle is nearly equal to the observer's geographic latitude. Polaris can only be seen from in the northern hemisphere.

To find Polaris, face north and locate the Big Dipper (Plough) and Little Dipper constellations. Looking at the "cup" part of the Big Dipper, imagine that the two stars at the outside edge of the cup form a line pointing upward out of the cup. This line points directly at the star at the tip of the Little Dipper's handle. That star is Polaris, the North Star.

Southern Cross

The south celestial pole is visible only from the southern hemisphere. It lies in the dim constellation Octans, the Octant. Sigma Octantis is identified as the south pole star, over a degree away from the pole, but with a magnitude of 5.5 it is barely visible on a clear night. The south celestial pole can be located from the Southern Cross (Crux) and its two "pointer" stars Centauri and Centauri. Very few bright stars of importance lie between Crux and the pole itself, although the constellation Musca is fairly easily recognised. Canopus (the second brightest star in the sky) and Achernar. Make a large equilateral triangle using these stars for two of the corners. The third imaginary corner will be the south celestial pole.

For a moonless and cloudless night you can use two faint clouds in the southern sky. These are marked in astronomy books as Large and Small Magellanic Clouds. These clouds are actually galaxies close to our own Milky Way. They make an equilateral triangle, the third point of which is the south celestial pole.

What kind of Sextant should I get

The first choice to make is between plastic or metal construction. Today's low cost metal sextants offer better accuracy and are easier to use. This will help you when first starting, and satisfy the professional's demands. Plastic models are perfect if your budget is restricted. They are also acceptable to some professionals who don't mind making frequent adjustments.

New or Used

Older sextants tend to have smaller mirrors and scopes which make them harder to use. Spare parts and maintenance are also more uncertain. Avoid discontinued models, and those out of date. Purchase only from someone you know and trust, or a reputable dealer. You will find that today's low cost metal sextants are very competitive with expensive used ones.

Accuracy

For all practical purposes, metal sextants are error free when compared to the many uncontrollable errors which may exist from such things as refraction, oblateness of the earth, and data tabulation. Generally, a minute of arc (one mile) is about the best anyone can hope to achieve. For these reasons, undue emphasis should not be placed on extreme accuracy guarantees. Plastic sextants can have errors in excess of 5 minutes, even when care is exercised. Although this is sufficient to make landfalls, precision navigation is difficult.

Mirror Size

The size of the mirrors on sextants generally vary directly with the quality of the instrument. Large index and horizon mirrors are desirable because larger mirrors allow more movement of the sextant while taking a sight, and lessen the possibility of losing the image as the body is brought down to the horizon.

Weight

Sextants are available with their major metal parts made of either aluminum, bronze or brass. The alloys of these metals are well suitable for use at sea. Some people feel that the heavier weight of a bronze sextant provides greater steadiness and more accurate readings, especially if it is windy. Others find that the lightweight models are less tiring to their wrist and arm and that the reduced fatigue gives better results. As the observer develops proficiency and speed in sight taking, fatigue becomes less of a factor. Lightweight plastic models can be difficult to use facing into a stiff wind because they tend to "flutter".

Scopes

A 3.5 x 40 scope is a good choice for stars. The large objective 40mm lens admits a great deal of light. The 3.5 power magnification helps you find and maintain stars in view in both calm or pitching seaways. A 7x35 monocular having greater magnification is well suited for sun sights, or the greater heights of eye associated with large ships.The increased magnification allows the sun's diameter to appear larger, and better defines a more distant horizon. This helps the navigator determine the point of tangency of the sun's limb and the horizon. The increased magnification makes finding and holding sights more difficult on a moving deck. A Sight Tube of zero magnification affords a wider field of view for rough weather, horizontal angles, and finding stars. If your sextant is to have only one scope, a 3.5x would be the your best choice for yacht sized vessels.

Traditional or Whole Horizon Mirror

Many sextants have an option of either the traditional (half-silvered) horizon mirror or what is called a "whole horizon mirror". With the traditional mirror, the horizon glass is divided vertically into two halves producing a "split image." The half nearest the frame is a silvered mirror and the other half is clear glass. In some cases this clear glass is eliminated. A later development in sextant technology is the whole horizon mirror. Using specially coated optics, the whole horizon mirror superimposes both the horizon and the celestial body on the entire mirror with no split image. This greatly simplifies "bringing down" the celestial body and makes it easier to hold the body in view. A draw back to this system is a very slight reduction in light transmission and reflection which may affect marginally lighted observations. Some feel these two aspects are a "trade off" that is, one can more quickly take observations with the whole horizon mirror, and be finished before marginal conditions occur. In general, people on stable platforms such as large ships tend to favor the traditional horizon mirror while those on yachts tend to favor the whole horizon mirror.

Illumination

Sextant lighting is the least needed feature on a sextant, since a flashlight should normally be available in any event for recording observations.

What's In It For Me?
It's tempting to think those solemn warnings only applied in the days before we could carry a GPS receiver in every pocket, but even the least paranoid among us knows that's not true: the GPS system itself can be intentionally switched off or degraded very quickly. It can be physically damaged; it can possibly be sabotaged by hackers. None of those things is likely to happen. More likely are on-the-water problems: GPS receivers can be fried by lightning, dropped overboard, crushed and damaged. Batteries do run out or get soaked. Well, as any offshore sailor knows, you just carry a back-up handheld. Or two, or three. Plus batteries and waterproof bags.

With all that, the chances of being far offshore without the ability to find a way home have diminished to a point where the demand for sextants has decreased markedly over the past 10 years. It's not what it was, but the demand for sextants has steadied. It's sailors rather than professional navigators who are interested in sextants now. There's a reasonable and steady demand. It's not so much that people need to have them, it's that they want to use them to navigate or just learn something new. Alot of today's interest in sextants stems from tradition.

There's more to it than that. Reading the sky not only tells us where we are, it connects us with all those who for centuries made their way over the world's waters with nothing more than simple instruments and tables, and the wisdom handed down to them on how to use them. Those skills, in turn, connect us in a practical way to the relationship of the earth to the sun, moon, and stars, a relationship that fewer and fewer sailors understand, to their detriment. The challenge of reading data on a GPS receiver pales next to the challenge of navigating by sextant, and for many people the resultant levels of satisfaction are proportional.

Selecting a Sextant

The clearer you are about your intended uses for the sextant, the better your chance of finding a place along the price range where your standards of value can be met. You can spend anywhere from $19.95 up to $3,000 for a serviceable sextant. The cheapest are made of cardboard (a German-made kit and plastic. While the cardboard kit is something of a novelty item with limited navigational potential (and no waterproofing) some plastic sextants are viable instruments for celestial navigation.

Still, most salts say plastic sextants aren't reliable enough for "real" navigation and that they should be limited to practice and/or lifeboat duty. Plastic sextants can yield results that are very close to those received from metal sextants: While most metal sextants can be shown to yield accuracy within a nautical mile the limitations built into plastic sextants give them a margin of error of five miles at best.

Instrument accuracy is really the least of your worries. Virtually all new sextants have negligible instrument or uncorrectable error. They also come with instructions for removing index (correctable) error. At that point you can bank on your new sextant as being virtually error free. Data reduction, refraction, and the oblateness of the earth are all more likely to be sources of inaccuracy than the sextant itself.

One of the biggest development in sextants over the past 20 years has been an across-the-board improvement of the optics involved. Mirrors have gotten bigger, coatings have been hardened, scopes made more versatile. Whether the sextant you buy comes from Germany, Japan, or China, you're likely to find that the optics are first-class.

Most navigators like a sextant with some "heft." Mass helps to steady the instrument. If, however, extending the sextant strains your arms and you're rushing your sights to be free of its weight, then you've got too much heft. Modern sextants range from 2 to just over 4 pounds. Finding one that's got substance enough to be steady but is light enough not to be taxing is a definite quest. Comfort in the grip of the handle is also important. Beware an overlycocked wrist and a sextant whose weight is not virtually centered around the handle.

Some top of the line sextants are made with brass armatures, but it's more common today to see framesmade of aluminum. This raises the topic of maintenance and the spectre of dissimilar metals. Even though you will obviously do your best to keep your sextant dry, smallboat use makes it likely that sooner or later it will get wet. Rinsing with fresh water and patting the sextant dry with a clean cloth is generally all that's required, but pay particular attention to those spots (like the stainless steel screws holding the mirror frame to the aluminum arm) where dissimilar metals are in contact.

Why should I bother with Celestial Navigation

GPS is great, and with world-wide coverage that helps get accurate positioning of vessels. But do you need a backup. The answer is yes, and that would be celestial navigation, its great to learn and it can be a interesting hobby.

Instances of electronic failure, total electric failure, and flooding are often documented. Even battery powered handhelds can be rendered inoperable in these ways. Batteries can run down, spares can be lost. The GPS system itself is not guaranteed.

GPS will track your boat and steer your boat. Some say it will even take your boat across the ocean for you. Without establishing a discipline, one's navigational skills will be lost. The key to navigation is the time-proven DR track. It should be maintained and updated with fixes, whether electronic or celestial. This gives information about the current and leeway, steering and compass errors.

Who can contemplate an 18th century brass and ebony sextant and not wonder what it was like to peer through it at the heavens, and bring an evening star down to a twilight horizon from the deck of a tall ship? To sense the approval of those who witness this magic-like prowess. To triumph at a land-fall well predicted? To know that they can navigate any ocean with no help from anyone?

Fun is doing something that is both easy and difficult. Easy to get started with, but having enough challenage that mastery does not come easy.

What could be easier than reckoning the longitude by simply observing the time of sunrise or sunset, or steering by a star? Almost as easy is the finding of latitude at noon. But how about identifying the navigational stars? Recognizing the planets? Accounting for the parallax of the moon? More experienced celestial navigators can use an unknown star shot through a hole in the overcast, shoot planets in broad daylight, predict sunrise underway, and calculate great circle distances.

Being familiar with the night sky is like having a giant roadmap overhead. One star may lead to another and before you know it you can identify 12 - 20 navigational stars. It is really cool to beable to take a sight of Sun, Moon, Stars, or Planets and get your location. People enjoy using sextants, even from a backyard with an artificial horizon. It's one of the few nautical activities that can be done without a boat, or even being on the water.

Right Ascension (RA)

Right ascension (RA) is the astronomical term for one of the two coordinates of a point on the celestial sphere when using the equatorial coordinate system. The other coordinate is the declination.

RA is the celestial equivalent of terrestrial longitude. Both RA and longitude measure an east-west angle along the equator, and both measure from a zero point on the equator. For longitude, the zero point is the Prime Meridian, for RA, the zero point is known as the First Point of Aries, which is the place in the sky where the Sun crosses the celestial equator at the March equinox.

RA is measured eastward from the March equinox. Any units of angular measure can be used for RA, but it is most of the time it is measured in hours, minutes, and seconds, with 24 hours being a full circle. The reason for this is that the earth rotates at an approximately constant rate. Since a complete circle has 360 degrees, an hour of right ascension is equal to 1/24 of this, or 15 degrees of arc, a single minute of right ascension is equal to 15 minutes of arc, and a second of right ascension equal to 15 seconds of arc. Sidereal Hour Angle, used in celestial navigation, is similar to RA, but increases westward rather than eastward. Don't confuse SHA with the concept of hour angle as it is usually used in astronomy, which is how far west an object is from your local meridian.

RA can be used to determine a star's location and to determine how long it will take for a star to reach a certain point in the sky. For example, if a star with RA = 01:30:00 is at a location's meridian, then a star with RA = 20:00:00 will be in the meridian 18.5 sidereal hours later.

The concept of right ascension has been known at least as far back as Hipparchos who measured stars in equatorial coordinates in the 2nd century BCE. But Hipparchos and his successors made their star catalogs in ecliptical coordinates, and the use of RA was limited to special cases.

With the invention of the telescope, it became possible for astronomers to observe celestial objects in greater detail, provided that the telescope could be kept pointed at the object for a period of time. The easiest way to do that is to use an equatorial mount for the telescope, which allows the telescope to rotate at the same rate as the earth. As the equatorial mount became adopted for observation, the equatorial coordinate system, which includes right ascension, was adopted at the same time for simplicity. Equatorial mounts could then be accurately pointed at objects with known right ascension and declination by the use of setting circles.

Diurnal Motion

Diurnal motion is an astronomical term referring to the apparent daily motion of stars around the Earth, or more precisely around the two celestial poles. It is caused by the Earth's rotation on its axis, so every star apparently moves on a circle, that is called the diurnal circle. The time for one complete rotation is 23 hours, 56 minutes and 4 seconds (1 sidereal day).

Direction of the motion in the Northern hemisphere:
looking to the north, below the North Star: left-right, West-East
looking to the north, above the North Star: right-left, East-West
looking to the south: left-right, East-West

Northern circumpolar stars move counterclockwise around the North Star.

At the North Pole, North, East and West are not applicable, the motion is simply left-right, or looking vertically upward, counterclockwise around the zenith.

For the southern hemisphere, interchange North / South and left / right, and replace North Star by southern celestial pole. The circumpolar stars move clockwise around it. East / West are not interchanged.

At the equator both celestial poles are at the horizon and motion is counterclockwise (to the left) around the North Star and clockwise (to the right) around the southern celestial pole. All motion is from East to West, except for the two stationary points.

The daily path of an object on the celestial sphere, including the possible part below the horizon, has a length proportional to the cosine of the declination. The speed of the diurnal motion of a celestial object is this cosine times 15° / hr = 15' / min = 15" / s.

Sideral Time

Sidereal time is a measure of the position of the Earth in its rotation around its axis, or time measured by the apparent diurnal motion of the vernal equinox, which is very close to, but not identical to, the motion of stars. They differ by the precession of the vernal equinox in right ascension relative to the stars.

Earth's sidereal day also differs from its rotation period relative to the background stars by the amount of precession in right ascension during one day. Sideral time means to measure time relative to the position of the stars.

Sidereal time is defined as the hour angle of the vernal equinox. When the meridian of the vernal equinox is directly overhead, local sidereal time is 00:00. Greenwich Sidereal Time is the hour angle of the vernal equinox at the prime meridian at Greenwich, England, local values differ according to longitude. When one moves eastward 15° in longitude, sidereal time is larger by one hour (note that it wraps around at 24 hours). Unlike computing local solar time, differences are counted to the accuracy of measurement, not just in whole hours.

Sidereal time is used at astronomical observatories because sidereal time makes it very easy to work out which astronomical objects will be observed at a given time. Objects are located in the night sky using right ascension and declination relative to the celestial equator, and when sidereal time is equal to an object's right ascension, the object will be at its highest point in the sky, or at which time it is best placed for observation.

Solar time is measured by the apparent diurnal motion of the sun, and local noon in solar time is defined as the moment when the sun is at its highest point in the sky (exactly due south or north depending on the observer's latitude and the season). The average time taken for the sun to return to its highest point is 24 hours.

During the time needed by the Earth to complete a rotation around its axis (a sidereal day), the Earth moves a short distance (approximately 1°) along its orbit around the sun. After a sidereal day, the Earth still needs to rotate a small extra angular distance before the sun reaches its highest point. A solar day is, nearly 4 minutes longer than a sidereal day.

The stars, are so far away that the Earth's movement along its orbit makes a generally negligible difference to their apparent direction and so they return to their highest point in a sidereal day. A sidereal day is almost 4 minutes shorter than a mean solar day. Another way to see this difference is to notice that, relative to the stars, the Sun appears to move around the Earth once per year. Which means, there is one less solar day per year than there are sidereal days.

Apparent Time

Solar times are measures of the apparent position of the Sun on the celestial sphere. They are not actually the physical time, but they are hour angles, angles expressed in time units. They are also local times in the sense that they depend on the longitude of the observer.

Apparent solar time or true solar time is the hour angle of the Sun. It is based on the apparent solar day, which is the interval between two successive returns of the Sun to the local meridian. Note that the solar day starts at noon, so apparent solar time 00:00 means noon and 12:00 means midnight.

The length of a solar day varies throughout the year for two reasons. First, Earth's orbit is an ellipse, not a circle, so the Earth moves faster when it is nearest the Sun (perihelion) and slower when it is farthest from the Sun (aphelion) Second, due to Earth's axial tilt, the Sun moves along a great circle (the ecliptic) that is tilted to Earth's celestial equator. When the Sun crosses the equator at both equinoxes, it is moving at an angle to it, so the projection of this tilted motion onto the equator is slower than its mean motion, when the Sun is farthest from the equator at both solstices, it moves parallel to it and closer to the polar axis than the equator, so the projection of this parallel motion onto the equator is faster than its mean motion Consequently, apparent solar days are shorter in March (26–27) and September (12–13) than they are in June (18–19) or December (20–21). These dates are shifted from those of the equinoxes and solstices by the fast / slow Sun at Earth's perihelion / aphelion.

Mean solar time
Mean solar time is the hour angle of the mean Sun. As the mean Sun is a mathematical construction only and cannot be physically observed, the mean solar time is computed from an artificial clock time adjusted via observations of the diurnal rotation of the fixed stars to agree with average apparent solar time. Though the amount of daylight varies significantly, the length of a mean solar day does not change on a seasonal basis. The length of a mean solar day increases at a rate of approximately 1.4 milliseconds each century. An apparent solar day can differ from a mean solar day by as much as 22 seconds shorter to 29 seconds longer. Because many of these long or short days occur in succession, the difference builds up to as much as nearly 17 minutes early or a little over 14 minutes late. Since these periods are cyclical, they do not accumulate from year to year. The difference between apparent solar time and mean solar time is called the equation of time. The mean solar day also starts at noon, with 00:00 meaning noon and 12:00 meaning midnight. The civil time is defined as mean solar time minus 12 hours.

The length of the mean solar day is increasing due to the tidal acceleration of the Moon by Earth, and the corresponding deceleration of the Earth by the Moon.

The mean Sun is defined as follows. First, consider a fictitious Sun that moves along the ecliptic at a constant speed and occupies the same position as the real Sun when Earth passes through the perihelion and also when it passes through the aphelion. Then, the mean sun is a second fictive Sun that moves along the celestial equator at constant speed and passing through the vernal point simultaneously with the first fictive sun.

In astronomy and navigation, the celestial sphere is an imaginary rotating sphere of "gigantic radius", concentric and coaxial with the Earth. All objects in the sky can be thought of as lying upon the sphere. Projected from their corresponding geographic equivalents are the celestial equator and the celestial poles. The celestial sphere projection is a very practical tool for positional astronomy.

In astronomy, the hour angle is one of the coordinates used in the equatorial coordinate system for describing the position of a point on the celestial sphere. The hour angle of a point is the angle between the half plane determined by the Earth axis and the zenith (half of the meridian plane) and the half plane determined by the Earth axis and the given point. The angle is taken with minus sign if the point is eastward of the meridian plane and with the plus sign if the point is westward of the meridian plane.

The hour angle is usually expressed in time units, with 24 hours corresponding to 360 degrees. The hour angle must be paired with the declination in order to fully specify the position of a point on the celestial sphere as seen by the observer at a given time.

Equation of Time

The equation of time is the difference at any moment deduced from the current position of the Sun and time as read from a regulated clock set to the local mean time. The equation of time varies over the course of a year, in way that is almost exactly reproduced from one year to the next. It can be ahead (fast) by as much as 16 minutes 33 seconds (around November 3) or fall behind by as much as 14 minutes 6 seconds (around February 12). It is caused by irregularity in the path of the Sun across the sky, due to a combination of the obliquity of the Earth's rotation axis and the eccentricity of its orbit. The equation of time is the east or west component of the analemma, a curve representing the angular offset of the Sun from its mean position on the celestial sphere as viewed from Earth.

The equation of time was used historically to set clocks. One of two common land based ways to set clocks was by observing the passage of the sun across the local meridian at noon. The moment the sun passed overhead, the clock was set to noon, offset by the number of minutes given by the equation of time for that date. The second method did not use the equation of time, it used stellar observations to give sidereal time, in combination with the relation between sidereal time and solar time.The equation of time values for each day of the year, compiled by astronomical observatories, were listed in almanacs and ephemerides.

Other planets will have an equation of time too. On Mars the difference between sundial time and clock time can be as much as 50 minutes, due to the considerably greater eccentricity of its orbit.

The Earth revolves around the Sun. As such it appears that the Sun makes one rotation around the Earth in one year. If the Earth orbited the Sun with a constant speed, in a circular orbit in a plane perpendicular to the Earth's axis, then the Sun would culminate every day at exactly the same time, and be a perfect time keeper, except for the very small effect of its slowing rotation. But the orbit of the Earth is an ellipse, and its speed varies between 30.287 and 29.291 km/s, according to Kepler's laws of planetary motion, and its angular speed also varies, and the Sun appears to move faster at perihelion (currently around 3 January) and slower at aphelion a half year later. At these extreme points, this effect increases (respectively, decreases) the real solar day by 7.9 seconds from its mean. This daily difference accumulates over a period. As a result, the eccentricity of the Earth's orbit contributes a sine wave variation with an amplitude of 7.66 minutes and a period of one year to the equation of time. The zero points are reached at perihelion (at the beginning of January) and aphelion (beginning of July) while the maximum values are in early April (negative) and early October (positive).

Even if the Earth's orbit were circular, the motion of the Sun along the celestial equator would still not be uniform. This is a because of the tilt of the Earth's rotation with respect to its orbit, or equivalently, the tilt of the ecliptic (the path of the sun against the celestial sphere) with respect to the celestial equator. The projection of this motion onto the celestial equator, along which "clock time" is measured, is a maximum at the solstices, when the yearly movement of the Sun is parallel to the equator and appears as a change in right ascension, and is a minimum at the equinoxes, when the Sun moves in a sloping direction and appears mainly as a change in declination, leaving less for the component in right ascension, which is the only component that affects the duration of the solar day. As a consequence of that, the daily shift of the shadow cast by the Sun in a sundial, due to obliquity, is smaller close to the equinoxes and greater close to the solstices. At the equinoxes, the Sun is seen slowing down by up to 20.3 seconds every day and at the solstices speeding up by the same amount.

The equation of time was mean minus apparent solar time in the British Nautical Almanac and Astronomical Ephemeris. Earlier, all times in the almanac were in apparent solar time because time aboard ship was determined by observing the Sun. In the unusual case that the mean solar time of an observation was needed, the extra step of adding the equation of time to apparent solar time was needed. Since 1834, all times have been in mean solar time because by then the time aboard most ships was determined by marine chronometers. In the unusual case that the apparent solar time of an observation was needed, the extra step of adding the equation of time to mean solar time was needed, requiring all differences in the equation of time to have the opposite sign.

As the daily movement of the Sun is one revolution per day, that is 360° every 24 hours, and the Sun itself appears as a disc of about 0.5° in the sky, simple sundials can be read to a maximum accuracy of about one minute. Since the equation of time has a range of about 30 minutes, the difference between sundial time and clock time cannot be ignored. In addition to the equation of time, one also has to apply corrections due to one's distance from the local time zone meridian and summer time, if any.

Why is Celestial Navigation still a test subject for Merchant Marine Officers

Celestial navigation is still included on license exams for ocean routes for a number of reasons.

First, celestial navigation is among the required competencies in the applicable part of the International Convention on Standards of Training, Certification and Watchkeeping for Seafarers, 1978, as amended (STCW). For example, the minimum standard of competence for an officer in charge of a navigational watch includes the "ability to use celestial bodies to determine the ship's position." The STCW is undergoing a comprehensive review and celestial navigation is among the areas receiving attention.

While it is too early to tell the outcome of this review, the position of the United States is that while the role of celestial navigation has significantly diminished, it should not be eliminated entirely. Celestial navigation performs an important function as a backup means of navigation in the event that other navigation modes fail.

Second, the use of either azimuths or amplitudes of a celestial body is the only way to determine accurately a ship's compass error when operating outside of the visual range of terrestrial objects. The United States supports limiting the celestial navigation requirements to those necessary to perform its backup navigation role and in order to perform compass error corrections.

It is worth noting that although they have not eliminated celestial navigation from the license examinations, they have made changes that reflect its diminished use in everyday watchkeeping. In early 2002, they reduced the minimum passing grade for celestial navigation exam modules from 90 percent to 80 percent. They believe this reduction is consistent with the reduced (but not eliminated) role celestial navigation plays in modern watchkeeping.

Notwithstanding their agreement that the role of celestial navigation has diminished, its use in prudent navigation has not been entirely eliminated and the Coast Guard does not have any immediate plans to eliminate celestial navigation from its license examinations through the amendment of their regulations found at 46 CFR 10.910.

Azimuths Circles

When taking an azimuth of a celestial body, the azimuth circle is used. An azimuth circle is a nonmagnetic metal ring sized to fit on a 7 inch compass bowl or on a gyro repeater. The inner lip is graduated in degrees from 000° to 360° in a counterclockwise direction for the purpose of taking relative bearings. Two sighting vanes the forward or far van containing a vertical wire, and the after or near vane containing a peep sight will help when making observations of bearings and azimuths. Two finger lugs are used to position the instrument while aligning the vanes. A hinged reflector vane mounted at the base and beyond the forward vane is used for reflecting stars and planets when observing azimuths. Below the forward vane there is a reflecting mirror and a extended vertical wire are mounted, this helps the navigator to read the bearing or azimuth from the reflected portion of the compass card.

For observing azimuths of the Sun, an additional reflecting mirror and housing are mounted on the ring, each midway between the forward and after vanes. The Sun's rays are reflected by the mirror to the housing where a vertical slit admits a line of light. This admitted light passes through a 45° reflecting prism and is projected on the compass card from which the azimuth is directly read. In observing both bearings and azimuths there are two spirit levels, which are attached must be used to level the instrument. Azimuth and bearing are the same in meaning, the horizontal angle that a line drawn from your position to the object sighted makes with a line drawn from your position to true north. The word azimuth applies only to bearings of heavenly bodies. Lets say for example, it is not the bearing, but the azimuth of the Sun, and not the azimuth, but the bearing of a lighthouse. A bearing circle is a nonmagnetic metal ring equipped with sighting devices that is fitted over a gyro repeater or magnetic compass. The bearing circle is used to take bearings of objects on earth's surface. The azimuth circle is a bearing circle equipped with additional attachments for measuring azimuths of celestial bodies. You can either take bearings or azimuths with a azimuth circle.

When taking a bearing lets say that you are getting a bearing on a lighthouse. Install either a bearing or azimuth circle on the gyro repeater, and make sure that the circle rotates freely. Train the vanes on the lighthouse so the lighthouse appears behind the vertical wire in the far vane. Drop your gaze to the prism at the base of the far vane, then read the bearing indicated by a hairline in the prism.

Taking a Azimuth

The azimuth circle can be used in two ways to measure the azimuth of a celestial body. The first method is used with a brilliant body such as the Sun. At the upper center you will see a concave mirror, and at the lower center, a prism attachment. Sight with the mirror nearest you, and the prism toward the observed body. Light from that body is reflected from the concave mirror into the prism. The prism, in turn throws a thin beam on the compass card. This beam strikes the graduation that indicates the azimuth. The second method is used for azimuths of bodies whose brightness is insufficient to throw such a distinct beam. Behind the far vane on the azimuth circle is a dark glass that can be pivoted so as to pick up celestial bodies at various altitudes. When a body is sighted, its reflection appears behind the vertical wire in the far vane, and its azimuth may be read under the hairline in the prism.

The inner lip of the azimuth circle is graduated counterclockwise in degrees making it possible, to obtain relative bearings of objects by training the vanes on an object, then reading the graduation on the inner circle alongside the lubber’s line on the pelorus or repeater. Each of the far vanes contains a spirit level to indicate when the circle is level. Bearings taken when the azimuth or bearing circle is not on an even keel are inaccurate.

Telescopic Alidades

Another means of taking bearings is by using an alidade, which, like the bearing circle, is mounted on a repeater. The telescopic alidade is a bearing circle with a small telescope attached to it. The image is magnified, making distant objects appear larger to the observer. A series of prisms inside the low power telescope enables the bearing taker to read the bearing directly from the compass card without removing the eye from the eye piece. Bearings and azimuths can be true, per gyrocompass (PGC), magnetic, or per steering compass

Celestial Navigation Glossary

Altitude - the arc of a vertical circle between the horizon and a point or body on the celestial sphere. Altitude as measured by a sextant is called sextant altitude (hs). Sextant altitude corrected only for inaccuracies in the reading (instrument, index, and personal errors) and inaccuracies in the reference level (principally dip) is called apparent altitude (ha). After all corrections are applied, it is called corrected sextant altitude or observed altitude (Ho). An altitude taken directly from a table is called a tabular or tabulated altitude (ht). Tabular altitude as interpolated for declination, latitude, and LHA increments as required are called computed altitude (Hc).

Altitude Difference (d) - the first difference between successive tabulations of altitude in a latitude column of these tables.

Argument - one of the values used for entering a table or diagram.

Assumed Latitude (aL), Assumed Longitude (aLo) - geographical coordinates assumed to facilitate sight reduction.

Assumed Position (AP) - a point at which an observer is assumed to be located.

Azimuth (Zn) - the horizontal direction of a celestial body or point from a terrestrial point, the arc of the horizon, or the angle at the zenith, between the north part of the celestial meridian or principal vertical circle and a vertical circle through the body or point, measured from 000° at the north part of the principal vertical circle clockwise through 360°.

Azimuth Angle (Z) - the arc of the horizon, or the angle at the zenith, between the north part or south part of the celestial meridian, according to the elevated pole, and a vertical circle through the body or point, measured from 0° at the north or south reference eastward or westward through 180° according to whether the body is east or west of the local meridian. It is prefixed N or S to agree with the latitude and suffixed E or W to agree with the meridian angle.
Celestial Equator - the primary great circle of the celestial sphere, everywhere 90° from the celestial poles, the intersection of the extended plane of the equator and the celestial sphere. Also called Equinoctial.

Celestial Horizon - that circle of the celestial sphere formed by the intersection of the celestial sphere and a plane through the center of the Earth and perpendicular to zenith-nadir line.
Celestial Meridian - on the celestial sphere, a great circle through the celestial poles and the zenith. The expression usually refers to the upper branch, that half from pole to pole which passes through the zenith.

Course Angle - course measured from 0° at the reference direction clockwise or counterclockwise through 180°, It is labeled with the reference direction as a prefix and the direction of measurement from the reference direction as a suffix. Such as course angle S21° E is 21° East of South, or true course 159°.

Course Line - the graphic representation of a ship's course.

Declination (Dec.) - angular distance north or south of the celestial equator, the arc of an hour circle between the celestial equator and a point on the celestial sphere, measured northward or southward from the celestial equator through 90°, and labeled N or S (+ or -) to indicate the direction of measurement.

Declination Increment (Dec. Inc.) - in sight reduction, the excess of the actual declination of a celestial body over the integral declination argument.

Double-Second Difference (DSD) - the sum of successive second differences. Because second differences are not tabulated in these tables, the DSD can be formed most by subtracting, algebraically, the first difference immediately above the tabular altitude difference (d) corresponding to the entering arguments from the first difference below. The result will always be a negative value.

Ecliptic - the apparent annual path of the Sun among the stars, the intersection of the plane of the Earth's orbit with the celestial sphere. This is a great circle of the celestial sphere inclined at an angle of about 23° 27' to the celestial equator.

Elevated Pole (Pn or Ps) - the celestial pole above the observer's horizon, agreeing in name with the observer's latitude.

First Difference - the difference between successive tabulations of a quantity.

First Point of Aries (Y) - that point of intersection of the ecliptic and the celestial equator occupied by the Sun as it changes from south to north declination on or about March 21. Also called Vernal Equinox.

Geographical Position (GP) - the point where a line drawn from a celestial body to the Earth's center passes through the Earth's surface.

Great Circle - the intersection of a sphere and a plane through its center.

Great Circle Course - the direction of the great circle through the point of departure and the destination, expressed as angular distance from a reference direction, usually north, to the direction of the great circle. The angle varies from point to point along the great circle. At the point of departure it is called Initial Great Circle Course.

Greenwich Hour Angle (GHA) - angular distance west of the Greenwich celestial meridian, the arc of the celestial equator, or the angle at the celestial pole, between the upper branch of the Greenwich celestial meridian and the hour circle of a point on the celestial sphere, measured westward from the Greenwich celestial meridian through 360°.

Hour Circle - on the celestial sphere, a great circle through the celestial poles and a celestial body or the vernal equinox. Hour circles are perpendicular to the celestial equator.

Intercept (a) - the difference in minutes of arc between the computed and observed altitudes (corrected sextant altitudes). It is labeled T (toward) or A (away) as the observed altitude is greater or smaller than the computed altitude, Hc greater than Ho, intercept is away (A), Ho greater than Hc, intercept is toward (T ).

Line of Position (LOP) - a line indicating a series of possible positions of a craft, determined by observation or measurement.

Local Hour Angle (LHA) - angular distance west of the local celestial meridian, the arc of the celestial equator, or the angle at the celestial pole, between the upper branch of the local celestial meridian and the hour circle of a celestial body or point on the celestial sphere, measured westward from the local celestial meridian through 360°.

Meridian Angle (t) - angular distance east or west of the local celestial meridian, the arc of the celestial equator, or the angle at the celestial pole, between the upper branch of the local celestial meridian and the hour circle of a celestial body, measured eastward or westward from the local celestial meridian through 180°, and labeled E or W to indicate the direction of measurement.
Nadir (Na) - that point on the celestial sphere 180° from the observer's zenith.

Name - the labels N and S which are attached to latitude and declination are said to be of the same name when they are both N or S and contrary name when one is N and the other is S.

Navigational Triangle - the spherical triangle solved in computing altitude and azimuth and great circle sailing problems. The celestial triangle is formed on the celestial sphere by the great circles connecting the elevated pole, zenith of the assumed position of the observer, and a celestial body. The terrestrial triangle is formed on the Earth by the great circles connecting the pole and two places on the Earth, the assumed position of the observer and geographical position of the body for celestial observations, and the point of departure and destination for great circle sailing problems. The term astronomical triangle applies to either the celestial or terrestrial triangle used for solving celestial observations.

Polar Distance (p) - angular distance from a celestial pole, the arc of an hour circle between a celestial pole, usually the elevated pole, and a point on the celestial sphere, measured from the celestial pole through 180°.

Prime Meridian - the meridian of longitude 0°, used as the origin for measurement of longitude.

Prime Vertical - the vertical circle through the east and west points of the horizon.

Principal Vertical Circle - the vertical circle through the north and south points of the horizon, coinciding with the celestial meridian.

Respondent - the value in a table or diagram corresponding to the entering arguments.

Second Difference - the difference between successive first differences.

Sidereal Hour Angle (SHA) - angular distance west of the vernal equinox, the arc of the celestial equator, or the angle at the celestial pole, between the hour circle of the vernal equinox and the hour circle of a point on the celestial sphere, measured westward from the hour circle of the vernal equinox through 360°.

Sight Reduction - the process of deriving from a sight (observation of the altitude, and sometimes also the azimuth, of a celestial body) the information needed for establishing a line of position.

Small Circle - the intersection of a sphere and a plane which does not pass through its center.
Vertical Circle - on the celestial sphere, a great circle through the zenith and nadir. Vertical circles are perpendicular to the horizon.

Zenith (Z) - that point on the celestial sphere vertically overhead.

Zenith Distance (z) - angular distance from the zenith, the arc of a vertical circle between the zenith and a point on the celestial sphere.

Navigation General Nautical Astronomy

These are just some of the questions on the Coast Guard exam for Navigation General. This is just one of subjects required for a merchant marine license.

1. The GHA of a star ?
A. Increases at a rate of approximately 15° per hour
B. Increases at a rate of approximately 4° per hour
C. Decreases at a rate of approximately 15° per hour
D. Decreases at a rate of approximately 4° per hour

2. In the celestial equator system of coordinates what is equivalent to the longitude of the Earth system of coordinates?
A. Zenith distance
B. Azimuth angle
C. Declination
D. Greenwich hour angle

3. In the celestial equator system of coordinates what is the equivalent to the meridians of the Earth system of coordinates?
A. Horizon
B. Hour circles
C. Vertical circles
D. Parallel of declination

4. In the celestial equator system of coordinates what is equivalent to the colatitude of the Earth system of coordinates?
A. Coaltitude
B. Zenith distance
C. Polar distance
D. Declination

5. If the right ascension of a body is 9 hours, it also?
A. Is 135°
B. Corresponds to an SHA for the body of 45°
C. Means that the GP of the body is in the western hemisphere
D. All of the above

6. The GHA of the first point of Aries is 315° and the GHA of a planet is 150°. What is the right ascension of the planet?
A. 7 hours
B. 11 hours
C. 19 hours
D. 23 hours

7. The angle measured eastward from the vernal equinox along the celestial equator often expressed in time units?
A. Greenwich sidereal time
B. Right ascension
C. Local sideral time
D. Sideral hour angle

8. Right ascension is primarily used by the navigator for?
A. Calculating amplitudes
B. Calculating great circle sailing by the Agiton method
C. Entering the Air Navigation Tables (Selected Stars) Pub. 249
D. Plotting on star finders

9. The ecliptic is?
A. The path the Sun appears to take among the stars
B. The path the Earth appears to take among the stars
C. a diagram of the zodiac
D. a great circle on a gnomonic chart

10. The navigator is concerned with three systems of coordinates. Which system is not of major concern?
A. Terrestrial
B. Ecliptic
C. Celestial horizon
D. Celestial equator

11. The equator is?
A. The primary great circle of the Earth perpendicular to the axis
B. The line to which all celestial observations are reduced
C. The line from which a celestial body's altitude is measured
D. All of the above

12. The angle at the pole measured through 180° from the prime meridian to the meridian of a point is known as?
A. The departure
B. The polar arc
C. Longitude
D. Greenwich hour angle

13. A plane perpendicular to the polar axis will never form what line on the Earth's surface?
A. Great circle
B. Equator
C. Small circle
D. Meridian

14. A plane that cuts the Earth's surface at any angle and passes through the center will always form?
A. the equator
B. a great circle
C. a small circle
D. a meridian

15. A plane that cuts the Earth's surface and passes through the poles will always form?
A. the equator
B. a loxodromic curve
C. a small circle
D. a meridian

16. When correcting apparent altitude to observed altitude, you do NOT apply a correction for?
A. The equivalent reading to the center of the body
B. The equivalent reading from the center of the Earth
C. The bending of the rays of light from the body
D. Inaccuracies in the reference level

17. When applying a dip correction to the sighted sextant angle (hs), you always subtract the dip because you are correcting?
A. hs to the visible horizon
B. hs to the sensible horizon
C. hs to the celestial horizon
D. Ho to the celestial horizon

18.The height of eye correction is smaller than geometrical dip because of?
A. The angle between the horizontal and the line of sight to the visible horizon
B. Index error
C. Parallax
D. Terrestrial refraction

19. A semidiameter correction is applied to observations of ?
A. Mars
B. The Moon
C. Jupiter
D. All of the above

20.The diameter of the Sun and Moon as seen from the Earth varies slightly but averages about?
A. 1'
B. 52'
C. 32'
D. 15.5'

Celestial Navigation and Time Diagrams

Figure 1

Figure 2

Figure 3

Sometimes the various angles on the celestial sphere are interrelated and can be easier to understand by drawing a time diagram. Time diagrams are used to understand the relationships between the values of the various hour angles.

In a time diagram you are viewing the celestial sphere from the celestial south pole. The center represents the celestial south pole, "Ps", the outer circumference represents the celestial equator, and a line drawn from the center to the edge represents a celestial meridian or an hour circle containing a body. Most of the time, the line drawn from the center to the 12 o'clock position represent either the Greenwich celestial meridian, labeled "G", or the local celestial meridian, labeled "M". Small "g" and "m" represent the lower branches of their meridians.

In figure 1 this is a time diagram where GHA of the sun is. The upper branch of the sun’s hour circle is shown as a solid line. The angle or arc of the celestial equator between the Greenwich meridian and the sun’s hour circle is 90°. The GHA of the sun at this instant is 90°. GHA is always measured westward from G. Local hour angle (LHA) is the name given to the angle of arc (expressed in degrees, minutes, and tenths of minutes) of the celestial equator between the celestial meridian of a point on the celestial sphere and the hour circle of a heavenly body. It is always measured westward from the local meridian through 360°.

Let’s try an example problem of LHA on a time diagram. Say you are at longitude 135° from M toward Greenwich which means that Greenwich will be shown east of M. Think about this for a minute, you are to the west of Greenwich, which means Greenwich is to the east of you. Now that you know where Greenwich is and where you are, let’s figure the LHA of the sun as it is shown above in figure 1.

Figure 2 shows you that the sun is 90° west of Greenwich. We know that the LHA is always measured westward from your location meridian (M) to the hour circle of the body (in this example, the sun). The LHA here is the whole 360° around minus the 45° between the sun’s hour circle and M. This 45° can be found by looking at figure 2 or by subtracting 90° from 135°. Let’s think this over, you are 135°W of Greenwich, G is135° clockwise from you. The sun is 90° W or counterclockwise from G, the difference is 45°. Subtract this 45° from 360° and you will get 315°, the LHA. Look at figure 2. You can see, the sun is east (clockwise on the diagram) of your local meridian (M). Now suppose that you are at the same longitude (135° W), but the GHA of the sun is 225° instead of 90°.

The time diagram will appear as shown in figure 3. The sun is now west of your meridian (M). The LHA is always measured westward from the local celestial meridian to the hour circle of the body. Now the LHA is the 90° from M to the sun’s hour circle. Here are two general rules that will help you in finding the LHA when the GHA and longitude are known:

1. LHA = GHA - W (used when longitude is west)

2. LHA = GHA + E (used when longitude is east)

When in west longitude you might have to add 360° to the GHA before the subtraction can be made. In east longitude, 360° is subtracted from the LHA if it exceeds this amount. You can check your work by referring to a time diagram. It gives you a graphic means of obtaining the data you need.

The GHA of a star is measured in the same direction from Greenwich to the star but because the SHA enters the picture here, your method of locating a star on the time diagram is a little bit different. First, you have to locate the vernal equinox by its tabulated GHA. Let’s say the GHA of the vernal equinox for the time of your observation is 45°. You locate the vernal equinox 45° W from Greenwich. From the Nautical Almanac you find the SHA of the star in your looking for. You already know that the SHA is measured to the west from the vernal equinox (first point of Aries). All you have to do here is find the SHA of this star and measure the SHA westward from the vernal equinox, you then have the star located on your time diagram.

Note: Meridian angle "t", this value is measured either eastward or westward from the local meridian to the hour circle of the body. Its value cannot exceed 180 degrees. If LHA exceeds 180 degrees "t" will be equal to 360 degrees minus LHA.

Try this example problem:

You are in longitude 104° east. The Greenwich hour angle of a star is 211°. The meridian angle "t" of the star is?

A. 45 degrees east

B. 45 degrees west

C. 315 degrees east

D. 315 degrees west

Step 1: Draw a basic time diagram as with the local celestial meridian "M" up. The solid part of the line is the upper branch of your meridian (the part you are on), and the dotted line represents the lower branch. Next put in the known values. Place "G," the meridian of Greenwich, 104 degrees west of "M" because you are east of Greenwich.

Step 2: Next measure westward from Greenwich (counterclockwise) the 211 degrees of GHA and draw in the hour circle of the star.

Note: You do not have to use exact measuring methods, a close approximation will be good enough.

Step 3: LHA is the measurement westward between the local celestial meridian and the hour circle of the body, from 0 degrees to 360 degrees. Draw the LHA on your time diagram. Look at the angles you have drawn, you should see that the LHA of the star is equal to the longitude plus the GHA of the star (104 + 211 = 315).

Step 4: Remember the note above about the definition of meridian angle, "t". The meridian angle is the measurement between the local celestial meridian and the hour circle of the body, its value cannot exceed 180 degrees, and it is labeled east or west. In short, "t" is the shortest angular distance between "M" and the body's hour circle. In this problem you can see that meridian angle, "t", is equal to 360 degrees minus the LHA determined above and is east (360 - 315 = 45° East).

Nautical Astronomy Questions

These are Coast Guard exam question that you may find on the Navigation General exam.

1. The spinning motion of a body around its axis is called?
A. revolution
B. rotation
C. orbit
D. space motion

2. The precession of the equinoxes occurs in a?
A. easterly direction
B. westerly direction
C. northerly direction
D. southerly direction

3. The ecliptic is?
A. the path the sun appears to take among the stars
B. the path the earth appears to take among the stars
C. a diagram of the zodiac
D. a great circle on a nomadic chart

4. Which condition exist during summer solstice in the northern hemisphere?
A. The north polar regions are in continual darkness
B. The northern hemisphere is having short days and long nights
C. The southern hemisphere is having winter
D. The sun shines equally on both hemispheres

5. A first magnitude star is?
A. 2.5 times as bright as a second magnitude star
B 3 times as bright as a second magnitude star
C. 5 times as bright as a second magnitude star
D. 10 times as bright as a second magnitude star

6. What is the length of the lunar day?
A. 24h 50m 00s
B. 24h 00m 00s
C. 23h 56m 04s
D. 23h 03m 56s

7. After Venus passes the point of greatest elongation east in its orbit, the first position at which elongation will be zero is?
A. superior conjunction
B. inferior conjunction
C. opposition
D. none of the above

8. The planet Mars will have its greatest magnitude when at?
A. conjunction
B. opposition
C. east quadrature
D. west quadrature

9. Inferior conjunction is possible for?
A. Mars
B. Venus
C. Saturn
D. Jupiter

10. In low latitudes first quarter moon will always rise at about?
A. sunrise
B. 1200 local time
C. sunset
D. 2400 local mean time

11. The sun is closest to the earth in what month?
A. October
B. July
C. April
D. January

12. The dimmest stars that could reasonably be used for navigational purposes are of what madnitude?
A. First
B. Third
C. Sixth
D. Tenth

13. Aphelion is the point where the sun?
A. moon , and earth form a right angle
B. moon, and earth are in line
C. crosses the celestial equator
D. is farthest from the earth

14. When a superior planet is at 90 degrees elongation, it is also at?
A. conjunction
B. opposition
C. quadrature
D. transit

15. The moon is nearest the earth at?
A. perigee
B. the vernal equinox
C. the new moon
D. the full moon

Nautical Astronomy, The Celestial Sphere

The Earth and the Moon
The earth's major satellite the moon, has motions similar to those of the earth. With respect to the sun, the moon rotates on its axis once in a period of about. 29.5 days, and revolves in its orbit around the earth in exactly the same period of time. Like the earth, the moon's orbit is not a perfect circle. The point of the orbit where the moon is closest to earth is called perigee, and when it is at its greatest distance it is at apogee.

The orbit of the moon is inclined about 5° to the ecliptic, and its axis of rotation is inclined 6.5 degrees to the plane of its orbit. Like the earth, the moon's axis is moving due to precession. As the moon goes through one orbit around the earth, completing one lunar month, its appearance changes. And as it does so, the moon is said to go through a series of phases. Since the moons orbit is only inclined a small amount from the ecliptic it appears to follow the sun through the sky, but falls further behind each day.

At the start of the lunar month (new moon), the moon is between the earth and sun. Each day thereafter the moon appears to have moved about 12 degrees to the east of the sun due to movement in its orbit. This causes the moon to rise about 50 minutes later each day relative to the sun. After about three days, a crescent moon is visible from earth with the points or cusps pointed away from the sun. Because the visible moon is in the process of getting bigger, it is called a waxing crescent. The line dividing the illuminated side from the shadow is called the terminator. When the lunar month is a week old, half of the moon is visible (first quarter). The next phase is called gibbous which is followed by full moon.

At full moon the moon is again in line with earth and sun, but this time the earth lies in the middle. After full moon, the visible moon begins to shrink, or wane, going through gibbous, last quarter, and waning crescent phases back to new moon. Because the moon's orbit is inclined to the ecliptic, the earth, moon, and sun rarely line up perfectly to cause either the moon or earth to pass through the shadow of the other. When they do an eclipse occurs.

The Celestial Sphere and Apparent Motion
Use of heavenly bodies for celestial navigation requires the idea that was accepted as the truth hundreds of years ago and later proved totally false. To use celestial navigation's mathematical principles, you must assume the following to be true.

1. The center of the earth is the center of a great celestial sphere.
2. This sphere has a radius (distance from the center of the earth to the surface of the celestial sphere) of infinite length.
3. All the celestial bodies used in navigation are located on the surface of the celestial sphere at the same distance from the earth.
4. The earth is stationary and the surface of the celestial sphere rotates around the earth, causing the celestial bodies to appear to move across the sky.

What you see from earth, and use in your navigation calculations, is the apparent motion of these bodies on the sphere. It appears to observers on earth that the bodies rise in the east, move in a westerly direction across the sky, and set in the west each day. It also appears that the sun, moon and planets move north, then south, and then north again on the celestial sphere. All of this motion described above is called apparent motion.

Apparent motion is the result of the real motions of the earth, which in some cases are combined with the real motions of the other body. Actual motions effect each celestial bodies apparent motion on the celestial sphere follows:

Sun - The apparent motion of rising and setting results from the earth's rotation. Movement of the sun north and south in the sky with the seasons results from revolution and the inclination of earths axis.

Stars - Rising and setting results from earths rotation, other motions of the stars happen very slowly as a result of precession of the earth's axis.

Planets - Rising and setting and other apparent motions of the planets result from a combination of the earth's rotation and revolution, and the revolutions of the planets.

Moon - The apparent rising and setting and the movement of the moon north or south results from a combination of earth's rotation and revolution, taken together with the moon's rotation, revolution, and precession.

Astronomy and Celestial Navigation

This is about the the celestial sphere and apparent motion as it applies to celestial navigation.
The earth has several separate motions in space including three major motions and some minor motions. The terms "major" and "minor" refer to their effect in celestial navigation and the calculations used to solve navigation related problems.

The major motions are rotation, revolution, and precession. Rotation refers to the earth's rotation on its axis once each day relative to the sun and other celestial bodies. The rising and setting of the sun and the passage of the stars, planets, and moon across the sky during the hours of darkness are results from rotation.

The second major motion, revolution, refers to the motion of the earth around the sun once each year. The earth's path or orbit along with the orbits of the other planets in the solar system is an ellipse. The sun is not at the center of the ellipse. If the earth's orbit is used to define a plane, the orbits of all the other planets, except Pluto would lie in the same plane. This plane in which all the planets revolve is called the Ecliptic. The earth's axis of rotation is inclined at an angle of 23.5° to the ecliptic, which also is the plane of the equator being inclined 23.5° to the ecliptic. This inclination causes the changes in the seasons on earth. As the earth revolves the light rays of the sun changes, being greatest in the northern hemisphere in June, when the northern hemisphere is titled toward the sun, and in the southern hemisphere in December when the southern hemisphere is titled toward the sun.

The points in the orbit where the angle of is greatest are called the summer and winter solstices. The points in the orbit when the sun's light shines equally over both hemispheres occurs in March and September of each year. These points are called the vernal equinox in March, and the autumnal equinox in September. With the sun off center, the earth reaches its closest point to the sun in January, after the winter solstice. This point is called Perihelion. The furthest point in the orbit from the sun is reached in July, after the summer solstice, and this is called Aphelion.

The third major motion, precession of the equinoxes, is caused by the gravitational force of the sun, moon and the other heavenly bodies acting together to deflect the axis of the rotating earth. This force causes the earth to act like a spinning top. The axis slowly traces a circle while remaining inclined at the same angle to the plane of the ecliptic. The earth's axis will travel through a circle over a period of 25,800 years. In time, precession will cause the axis to move away from the star Polaris, which we call the pole star.

Stars
The earth is located in a system which circles a mid sized star, our sun. The sun is located between the center and the edge of cluster of millions of stars circling the center of what is called a galaxy. This formation is one of other galaxies in the universe. The nearest other star in our galaxy is 4.3 light years away. Considering that light travels 186,000 miles in one second, the distance it would travel in 4.3 years is a long way. The light from the sun reaches earth in about 8.33 minutes. The distances are so great to the stars that the light rays which radiate from them are parallel lines when they reach our solar system. Over these distances the stars appear in the same place relative to the other stars from any point on the earth's orbit around the sun.

The stars are categorized by their brightness, or magnitude, with the brightest called first magnitude and the dimmest which can be seen with the unaided eye categorized as sixth magnitude. A first magnitude star is 2.5 times brighter than a second magnitude, and 6.3 (2.5 x 2.5) times brighter than a third magnitude. Each star is given a magnitude number by astronomers, with the first magnitude stars having numbers of 1.5 or less. Some of the brightest stars are so bright they have been given minus numbers. Examples are stars like Sirius (-1.6) and Canopus (- 0.9). The planets are also given numbers which vary with their positions in their orbits. Sometimes the planet Venus is so bright that it can be seen by the unaided eye in the daytime. As an interesting note, the sun's magnitude number is - 26.7.

Stars appear in different colors in the sky, and this color difference is caused by differences in their internal temperature. Astronomers group stars as red, white, or yellow giants, dwarfs, or mid-size stars depending on their stage of evolution. Only a small number of stars are used for navigational purposes. These are stars with magnitudes of 3 or brighter. There are 57 navigation stars listed in the daily pages of the Nautical Almanac.

Solar System
The sun is the central body of a system of nine principal planets including the earth. Satellites (moons) of the planets, and thousands of minor planets, comets and meteors are included in this system. All of these bodies, encircling the sun, are in motion relative to the stars. This motion is called space motion. The other planets like the earth, revolve around the sun in their orbits. Planets whose orbit lies outside the earth's orbit are called superior planets. The two planets with orbits smaller than the earth, Venus and Mercury, are called inferior planets. Because they are closest to earth, the planets used for navigation are Venus, Mars, Jupiter, and Saturn.
The orbits of all the planets except Pluto, lie in the same plane of the ecliptic as the earth's orbit. This makes it appear that the planets are either preceding or following the sun in its path through the sky as seen from earth. Whether a planet is preceding or following the sun depends on where the planet and earth are in their own orbits. Where a planet is located relative to the sun, when seen from earth at a given time, is known as planetary configuration.

When the sun lies directly between the earth and an inferior planet, the planet is at superior conjunction. When the inferior planet is directly between the earth and sun it is at inferior conjunction. When an inferior planet, while moving in its orbit from superior toward inferior conjunction reaches the point where the angle between the planet and sun as seen from the earth reaches its greatest value, it is said to be at greatest elongation east. After passing inferior conjunction, the inferior planet, continuing in its orbit, will appear west of the sun. When at its greatest angle west of the sun it is at greatest elongation west.

Superior planets are at conjunction when they are at the point in their orbits where the earth, sun and the planet are directly in line with the sun between the earth and planet. When a superior planet is aligned with the earth and sun with the earth in the middle, the planet is at opposition. The points in a superior planet's orbit where the difference in bearing between the planet and the sun, as seen from the earth, is 90 degrees is called quadrature. It is at east quadrature when the planet is east of or following the sun in the sky, and west quadrature when west of or preceding the sun.