Finding North and time by stars in the tropics

Finding North and time by stars in the tropics

by tonytran2015 (Melbourne, Australia).

Click here for a full, up to date ORIGINAL ARTICLE and to help fighting the stealing of readers’ traffic.

#find North, #finding North, #direction, #by stars, #Mercator, #sky map, #star map, #declination, #right ascension, #tropic.

This posting gives a method of using stars in the tropics under adverse effects of high skyline and bright sky. It is applicable whenever more than 30% of the night sky is seen.

It is based on the traditional tropical Oriental country methods, with the added improvement by using accurate data on dates and angular distances between stars for their identification. It is useful when the sky is unclear and restricted such as in cities with high skyline and bright sky.

1. Identifying bright stars in the tropics.

BrightStars20Plus2

Figure 1: Table of 20 brightest stars plus two additional easily identifiable stars for navigation in the tropics.

People in the tropics learned using stars differently from people in the temperate zones.

In the tropics it is difficult to see the polar and circumpolar stars to start identifying stars. Traditional (not influenced by Western astronomical knowledge) country methods by tropical people ignore the polar stars and rely on the dates of tropical stars and their order of succession in the sky to identify them.

Any tropical star is visible nightly (either from Sunset to its setting or from its rising to Sunrise) for more than 10 months each year. The visibility cycle for each star begins with the star seen rising near the Eastern horizon few minutes before Sunrise. On subsequent days, the star rises earlier and earlier, it travels gradually towards the West and remains for longer and longer duration in the night sky until one day it stays for the whole night. The star is therefore called a star of that date. After that day, the star is seen setting in the West in the night. On subsequent days, its lead on the Sun gradually increases and it sets on the West at earlier and earlier time. Near the end of the cycle, the star is visible above the Western horizon for only few minutes after Sunset. It then sets on the West. At the end of this cycle, the star is too close to the Sun to be visible in the sky. The cycle then repeats from the beginning.

Mercator star map

Figures 2,3: The Mercator map of the sky for inhabitants of Tropical Zone. North direction is on its top. 24hr of R.A. is near the center and R.A. increases towards the left (East) of the map. The map is to be read South side up in the Southern hemisphere. Click to enlarge figure.

The table of this step shows the stars in their order of appearance in the year. The date of a star is the night when the star attains its highest elevation at mid-night (when the hidden Sun has the most negative elevation) and it is visible for that whole night.

2. Identifying stars positively using patterns in the map.

mercator8gc30.jpg

Figure 1: Sky map for the tropic. It is a Mercator map of the central strip of the night sky for printed side down reading. An observer looking up will see that the continuous strip made up of this map and its identical copies slowly and repeatingly moves from its left (aligned to rise on the East of the sky line) to its right (aligned to set on the West of the sky line).

The bright stars for the date near to the current date are then joined to its neighbours of similar brightness to reveal their relative directions and distances. These directions and distances form shapes and sizes to positively identify the stars.

As only the 20 brightest stars are used with large unique patterns in the sky, there is no possibility of having two similar patterns with equal sizes. A user of this method matches the observed shapes in the sky and compare them with those given in the map to identify stars positively.

The map of this step (for printed side down reading) shows the stars in a central strip of 120 degree width going across the sky starting from the Eastern horizon and ending at the Western horizon. An observer lying on his back, looking upwards vertically will see a part of a long continuous strip made up of the map in this step joined by its identical copies slowly rises from the Eastern horizon, moves across the sky (the right hand side of the map leads, left hand side trails) then sets on the Western horizon. He can only see those stars of this map within a window of 12 hours width (in the direction of left to right on the map) when there is no sunlight. The window remains stationary while the continuous strip, being the map here followed by its identical copies, moves across the sky (the right hand side of the map leads and left hand side trails). This is illustrated in the following four maps of the tropical mid-nights and their four approximate zenithal maps to be found at the end of this section.

Merc12

Figure 2: Reading the Mercator sky map (with a grid of azimuth and elevation lines) at midnight Dec. 21st from the equator.

Merc03

Figure 3: Reading the Mercator sky map (with a grid of azimuth and elevation lines) at midnight Mar. 21st from the equator.

merc0621b

Figure 3: Reading the Mercator sky map (with a grid of azimuth and elevation lines) at midnight Jun. 21st from the equator.

Merc09

Figure 3: Reading the Mercator sky map (with a grid of azimuth and elevation lines) at midnight Sep. 23rd from the equator.

The users of this method should keep in their minds that some bright planets (especially the outer planets Mars, Jupiter, Saturn) may wander on the ecliptic (drawn on the map) and cross some area under observation to confuse the identification of stars and constellations. The positions of those such slowly moving bright planets should be noted when observation condition is favourable.

Example:

For May 15th, the star to use is Antares (of May 29th) in the Scopii. The nearby stars to use are

Spica (201.3 deg R.A., -11.1 deg decl.),

Arcturus (213.9 deg R.A., +19.2 deg decl.),

Antares (247.3 deg R.A., -26.4 deg decl.),

Vega (279.2 deg R.A., +38.8 deg decl.).

3. Mercator maps and the distances on them.

Zenith12

Figure: Approximate zenithal map of the tropical midnight sky for December 21st. Positions of stars near the horizon are not accurate in these maps. The maps are obtained by placing circular windows of 12h width on the Mercator map.

Zenith03

Figure: Approximate zenithal map of the tropical midnight sky for March 21st. Positions of stars near the horizon are not accurate in these maps. The maps are obtained by placing circular windows of 12h width on the Mercator map.

Zenith06

Figure: Approximate zenithal map of the tropical midnight sky for June 21st. Positions of stars near the horizon are not accurate in these maps. The maps are obtained by placing circular windows of 12h width on the Mercator map.

Zenith09

Figure: Approximate zenithal map of the tropical midnight sky for September 23rd. Positions of stars near the horizon are not accurate in these maps. The maps are obtained by placing circular windows of 12h width on the Mercator map.

The map in the preceding step uses a Mercator projection to preserve angles. This projection specifies that

(Vertical distance to equator on Mercator map)/(dist. to equator by rectilinear scale) =

(180deg/3.14159rad) × ln((1+tan(0.5×decl.))/(1-tan(0.5×decl.))) / declination.

The projection makes small shapes look similar to the original shapes on the Celestial sphere. Equatorial shapes on the Celestial sphere are faithfully represented. However shapes near the polar regions of the Celestial sphere are enormously overstretched by this type of maps. The distortion can be easily seen by comparing the Mercator map given here and the two polar maps given in reference [2].

The equator line of a Mercator map can be used as a scale to measure distances between stars near to the equator. The vertical lines for of Right Ascension divides it into hours. Each hour of R.A. corresponds to 15 degrees on the Equator line.

The distance between two close stars anywhere on the map is their distance measured on the map multiplied by the cosine of the declination angle of their midpoint.

When the two stars are widely separated, the great circle arc joining them is divided into (about 3) small segments and their distances are added together to give the total distance.

Example:

The distance between Sirius and Canopus on the map of step 2 is measured (using the equator line as a scale) to be nearly 47 degrees in length.

Their midpoint is nearly at 40 degrees (in absolute value) in declination.

Their great circle distance on the Celestial sphere is therefore nearly

47deg X cos(40deg) =36 deg.

This is reasonably close to the actual distance of 36 degrees deduced from the values obtain from their declination and R.A. values given in the table of step 1

4. Measuring the angle between any two stars.

The actual angular distance between any two stars in the sky can be measured using a compass divider with each leg pointing to a star and their angle measured on a protractor.

The two legs of the compass divider can be substituted by two stretched fingers on one hand. The protractor can be substituted by a 12h clock face with each hour marking representing 30 degree angle separation.

(Instrumental navigators use sextants for highly accurate measurements of angles between stars and they can quickly decide on the identity of stars.)

5. Finding time with stars.

On its date, a star reaches its meridian at midnight. Every month before/after that the star reaches its meridian plane two hours later/earlier.

The user of the method should add the difference between zonal time and local time (when noon is at 12am) to obtain the zonal time.

6. Conclusions.

1/- I have tested this method in Saigon, a city with unclear sky and high-rise buildings and found it to be applicable in 80% of the non-rainy nights.

2/- Users of this method should take care not to mistake planets for dimmer stars along the ecliptic.

Reference.

[1]. tonytran2015, Finding North direction and time by stars, survivaltricks.wordpress.com, https://survivaltricks.wordpress.com/2015/08/28/finding-north-and-time-by-stars/ , posted on August 28, 2015.

Added after 2018 July 20:

[2]. https://misfitsandheroes.wordpress.com/2012/08/28/ancient-navigators/

[3]. http://www.ancient-wisdom.com/zodiac.htm

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Finding direction, distance and navigating to a distant base by stars, fine reading of latitude (Part 2).

Finding direction, distance and navigating to a distant base by stars, fine reading of latitude (Part 2)

by tonytran2015 (Melbourne, Australia).

Click here for a full, up to date ORIGINAL ARTICLE and to help fighting the stealing of readers’ traffic.

#find North, #finding North, #direction, #time, #star, #sky map, #sky disk, #declination, #right ascension, #fine reading, #celestial, #distance, #find, #latitude, #navigation, #no instrument, #polynesian, #zenith,
This is applicable to navigation in an ocean or in a large desert with clear, flat horizontal skyline. It uses the complementary stars touching the horizon instead of stars traveling directly over the zenith of the navigator. It is more suitable for sea travel with readily available horizon but unsteady travel platform. It is a useful trick to return to a base (e.g. a Polynesian island) when having no measuring instrument.

Step 1: Basis of the method.

BStarsN20Vega8C

wpid-30naugplrnc-.jpg.jpeg

wpid-30naugplrsc.jpg

Figure: The trajectory of the complementary star touches or nearly touches the horizon. Figures: Horizon for an example latitude of 30degrees North projected onto North and South Celestial hemispheres respectively.

Stars travel along constant declination circles drawn on the Celestial sphere. If the base city is at latitude L then the constant declination circle of 90°-L on its same (North or South) hemisphere will be seen touching the horizon and the lowest position of the complementary star will be right on the horizon and in the principal Northern/Southern direction. When the (complementary) stars of declination 90°-L is at its lowest point near the horizon, unaided human eyes can easily tell its elevation accurate to 1/4 Moon’s diameter (1/8 of a degree).

If bright complementary stars are unavailable for any latitude, users of this method have to identify some constellations having dim complementary stars for that latitude and use these stars instead.

Step 2: Preparation at base for this method.

BrightStars0b

polrnorthqrefc60.jpg

polrsouthq3c60.jpg

Figures: 20 brightest stars and their positions in the sky represented in Northern and Southern 3/4 spheres. Dimmer stars beyond this list may have to be used by this method for traveling to any arbitrarily given latitude.

1. Work out the latitude of the chosen city.
2. Work out the complementary angle for that latitude.
3. Use a list of bright stars (in reverse order of brightness) to choose a star or stars having declinations being equal or greater than the complementary angle by less than 2 degrees (the difference is less than 2degrees or 4 Moon’s diameters). The less bright stars may have their declinations closer to required values but their poor visibility may make them unsuitable. The chosen star may slightly dive under the horizon but its neighbouring stars can indicate how far it has dived.
4. Practice identifying the complementary stars in all imaginable conditions.

Step 3: Field application

5. Travel North or South until the lowest position of the complementary star touching or slightly above the horizon by the so determined adjustment of less than 4 diameters of the Moon.
6. On attaining that latitude, only travel along a parallel circle to maintain the latitude.

Step 4: Examples.

BStarsN20Vega8C2.jpg

Figure: The trajectory of the complementary star for London touches or nearly touches the horizon when viewed at the latitude of London.

London is at (0°5′ longitude, 51°32′ latitude), choose Vega (18hr 37 RA, +38.8deg declination). Around midnight of Dec. 25th, the star Vega travels to its lowest point on a circle glancing the horizon. Its distance from horizon is 51°32 + 38.8° – 90° = 0.3°.
This angle is half the diameter of the Moon and can be judged accurately by unaided eyes.

Berlin is at (13°25′ longitude, 52°30 latitude), choose Vega (18hr 37 RA, +38.8deg declination). Around midnight of Dec. 25th, the star Vega travels to its lowest point on a circle glancing the horizon. Its distance from horizon is 52°32 + 38.8° – 90° = 1.3°.
This angle is 3 diameters of the Moon and can be judged accurately by unaided eyes.
Manila (120°57′ longitude, 14°35′ latitude), choose a dim star Beta Ursae Minoris, (Kochab, 14hr51RA, +74.3deg declination). Around midnight of Nov. 07th, the star Kochab travels to its lowest point on a circle glancing the horizon. Its distance from horizon is 14°35 + 74.21° – 90° = -1.3° (under the horizon by 1.3degrees. This angle is 3 diameters of the Moon and cannot be seen but its visible neighbouring stars in the Ursa Minoris group can indicate how far this star is below the horizon.).
Mecca(39°45 longitude, 21°29 latitude) choose Gamma Ursae Minoris (Pherkad Major, 15hr 21RA, +71.8° declination). Around midnight of Nov. 16th, the star Kochab travels to its lowest point on a circle glancing the horizon. Its distance from horizon is 21°29 + 71.8° – 90° = +3.3°. This angle is 7 diameters of the Moon and can be judged accurately by unaided eyes using fingerwidths on a stretched arm.

Tonga Capital city is Nukuʻalofa (175°12′W = 184°48′ longitude, 21°08′S latitude). Choose the star Beta Carinae (Miaplacidus 09hr 13 RA -69.7decl). Navigators may have to identify the constellation Carina containing the bright star Canopus in order to identify a not quite bright Beta Carinae. Around midnight of Aug. 10th, the star Beta Carinae travels to its lowest point on a circle glancing the horizon. Its distance from horizon is 21°08′ + 69.7° – 90° = +0.8°. This angle is 1 and 1/2 diameters of the Moon and can be judged accurately by unaided eyes.

The Northern tip of Iceland is at 66°30′ (see the map from viking ships , [2]). Choose the Sun at its June 21st solstice. Around midnight of Jun. 21st, the center of the Sun travels to its lowest point on a circle glancing the horizon. Its center is exactly on the horizon when the navigator is on the latitude of the Northern tip of Iceland. The upper rim of the Sun is just touching the horizon on Jun. 21st when the navigator is on the latitude of Northern Iceland. Keeping this latitude brings the navigator to Iceland on a journey of 900km from Norway.

Step 5: Notes on terminal homing of journeys.

Near to the end of his journey, an ocean navigator may release island spotting birds.
If the birds can attain a height of 800m, they can spot land (even without using cloud features) at distance of 110km away (60 nautical miles, or 1 degree of arc or 2 Moon’s diameters).
If the birds can attain a height of 250m, they can spot land (even without using cloud features) at distance of 55km away (30 nautical miles, or 0.5 degree of arc or 1 Moon’s diameter).
If the birds can attain a height of 62m, they can spot land (even without using cloud features) at distance of 28km away (15 nautical miles, or 0.25 degree of arc or 0.5 Moon’s diameter).

Alternatively the navigator may note the presence of nautical birds from the island ( viking ships , [2]). The navigator can also use currents, winds and even smells in this phase.
The error of this navigation method is thus well within the operational range provided by the spotting birds.

References

[1]. tonytran2015, Finding direction, distance and navigating to a distant base by stars (Part 1). Additional Survival tricks, wordpress.com,
Posted on January 27, 2016.

[2]. viking ships , http://www.hurstwic.org, http://www.hurstwic.org/history/articles/manufacturing/text/norse_ships.htm

Added after 2018 July 20:

[3]. https://misfitsandheroes.wordpress.com/2012/08/28/ancient-navigators/

[4]. https://www.abc.net.au/news/science/2021-09-23/polynesia-settlement-pacific-islands-genome-dna-rapa-nui-history/100471492

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Finding direction, distance and navigating to a distant base by stars (Part 1)

Finding direction, distance and navigating to a distant base by stars (Part 1)

by tonytran2015 (Melbourne, Australia).

Click here for a full, up to date ORIGINAL ARTICLE and to help fighting the stealing of readers’ traffic.

#find North, #finding North, #direction, #time, #star, #sky map, #sky disk, #declination, #right ascension, #celestial, #distance, #find, #latitude, #navigation, #no instrument, #polynesian, #zenith

polrnorthqrefc60.jpg

Figure: The sky map for use in Northern hemisphere.

The method here uses stars for finding base city at a distance between 1000km to 9000km and for traveling to that base using constant latitude for final path.
It uses the stars passing overhead the base city and accurate time to tell those moments. Direction and distance to that city are directly observed from the stars. If no current longitudinal information is available and longer travel distance is acceptable then the user can also use this method to aim for base city on the final constant latitude part of the travel.
This is a back up or emergency method for people who may need to find out their direction to base and how to arrive there even when having no landmarks for current position. This is the case for:
A. People who are lost in the ocean or in a large desert with no reliable landmark. They need some method of orientation using minimal number of tools.
B. People drifted to an isolated island in the ocean after a tsunami !
C. Installers of long distance or satellite communication antennas wishing to aim their devices when not having any map.
Required information:
1. Selected star(s) for the chosen base city with the target point(s) underneath the star(s) reasonably close to the base city.
2. Longitude of the base city and time signal announcing GMT time to show the time at base city. The time signal may come from Broadcast or Marine Band Weather Radio.
3. or an accurate watch that allows determination of the true (not zonal) time of base city (Each minute earlier or later than intended time may cause a longitudinal error of 0.25 degree that is about 27km near the equator.).

Step 1: Preparation before expedition

polrsouthq3c60.jpg

Figure: The sky map for use in Southern hemisphere.
1. Search from the list of brightest stars (in descending order) for the brightest identifiable star that can closely pass overhead the base city (with acceptable error distance) and the approximate date for it to be seen in at midnight.
Examples:
London is at (0°5′ longitude, 51°32′ latitude). In June, choose Eltanin (Gamma Draconis, 17hr 57′, +51.5° declination) target point underneath the star is 0km from base.
Berlin is at (13°25′ longitude, 52°30 latitude). In June, choose Eltanin (Gamma Draconis, 17hr 57′, +51.5°declination), target point underneath the star is nearly 110km South of the city while the zenith of the city is 2 Moon’s diameter from the star and toward the Celestial North . In December choose Gamma Persei (03hr05RA +53.5degrees declination, app magn 2.91) target point underneath the star is nearly 110km North of the city while the zenith of the city is 2 Moon’s diameter from the star and toward the Celestial South.
Mecca(39°45 longitude, 21°29 latitude) choose ArcturusBoote (213.9RA, 19.2° declination) nearby location underneath the star is 230km South of the city while the zenith of the city is 4 Moon’s diameter from the star and toward the Celestial North.
Manila (120°57′ longitude, 14°35′ latitude) choose Regulus, (Alpha Leonis, 10hr08’RA +12.0°declination) nearby location underneath the star is 280km South of the city while the zenith of the city is 5 Moon’s diameter from the star and toward the Celestial North.
2. Work out the day for the star to be highest at midnight. The day is the same for all locations. It is almost Sep23rd plus the RA of the star multiplied by (365.25days/360°).
Example:
Gamma Persei is nearly overhead at midnight of
Sep23 + 3hr05*(365.25days/24hr) =
Sep23 + 46.92days = Oct23 + 17d = Nov 09.
3. Learn by heart how to identify in the sky the stars associated with the base city. The accuracy and speed of this ability is essential to avoid mistakes under adversed circumstances. Users should not confuse between stars near the ecliptic and wandering planets nearby.
4. Practice determining the time when the star passes the vertical North-South plane at the base city on that date. It is midnight minus the local advance on GMT, which is equal to longitude multiplied by (24hr/360°).
Example:
Gamma Persei passes near Berlin (longitude 13°24′) on that date ahead of mid – night GMT by (13°24′)/(15°/hr) = 0.89hr, that is at
24hrGMT – 0.89hr = 23.11hrGMT = 23hr07GMT.
5. Every day later/earlier than that date, the star passes the location (60minx24/365.25) = 3.942 min of time earlier/later. This earliness is observable at all locations including your current one. When observing the star on another day, the earliness adjustment is needed.
6. If the Sun crosses the North-South vertical plane earlier/later than at base, the chosen star also crosses the North-South vertical plane earlier/later than at base by the same amount of time.
Step 2: Field application

BrightStars0b

List of 20 brightest stars. Additional, dimmer stars are also needed to travel closer to any arbitrarily given latitudes.

7. Identify the star and obtain the time signal from GMT. Work out the instant the star is overhead the base. (Alternatively, the moment the chosen star passes overhead the base can also be determined with an accurate watch from the time it passes the North-South vertical plane of current location and the advancement or retardment of local Noon relative to Noon at base.)
8. At that moment, the star is above the nearby spot close to the base. Every degree from your zenith is 111km distance from you. The direction to the star projected onto the ground gives direction to the chosen nearby location. To obtain more accurate direction to your base when the star does not pass its zenith, you can imagine another star at some diameters of the Moon on either North or South side of the RA circle from the chosen star and use it instead. Alternatively you can add some adjustment based on the differentials on a spherical surface to obtain the exact direction to your base.
Step 3: Navigating by only stars.
9. To travel to the target location, aim for a location on the same latitude but more in the North-South direction of the current point. This makes the travel distance longer but ensures that the target is not missed in the final part of the travel. When arriving at that target latitude, aim at the target location. Keeping the selected star on the East West line when it has highest altitude will ensure that the traveler does not miss the target.
This method suggests a possible way used by desert travelers and an alternative for refinement of Polynesian method of navigation.

4. How to find the zenith point.

The navigator has to hang a long plumbing line from a point higher than his eye level, stand away from it and look at the projection of the line onto the sky. The projection is a great circle arc through the zenith.

Looking at the plumbing line from many directions gives many great circle arcs intersecting at the zenith point in the sky. The navigator may have to note its relative distances to familiar stars and draw it and the stars on a piece of paper for future reference and cross checking.

This method requires a steady plumbing line and is suitable for ground travelers when resting at night.

5. How to locate any chosen bright star in the sky.
1. Find out its position relative to the 20 brightest stars by plotting it on the star maps here from its RA and declination.
2. Work out steps starting from identifiable top 10 brightest stars to positively identify it through progressively nearer, easily identifiable, bright neighbours .
3. Use the sky maps here to practice finding it in the sky.
4. Examples.
4.1 Locating Eltanin:
Eltanin is found from star charts and the sky maps here as the brightest star near to the point of one third of the way from Vega to Dubhe (There is no brighter star in the vicinity.).
4.2 Locating Regulus:
The broom shaped group of stars (Sirius, Canopa, Orion-Rigel, Betelgueuse, Procyon) identifies their elements. Betelgueuse-Pollux forms the hypotenus of the isoceles right triangle (Procyon, Betelgueuse, Pollux, Procyon, counter-clockwise) with Procyon at the right angle. Regulus is then one distant vertex of the rhombus (Procyon, Betelgueuse, Pollux. Regulus, Procyon, counter-clockwise).

6. Notes.
1. The local true time at the base city has the Sun crossing the North-South vertical plane at 12am. The zonal time (broadcasted by local radio and TV stations in the winter) is the true time advanced or retarded so that it differs from GMT by a whole number of hours.
2. The Sun crosses the North-South vertical plane before or after 12am zonal time by the difference between the local true time and zonal time. This amount is due to the excess or shortage of longitude to the nearest multitude of 15 degrees chosen for zonal time.
3. With an accurate watch still showing the zonal time at the base city, the longitudinal increment from that of base city can be worked out by the increment in the earliness of the crossing of the North-South vertical plane by the Sun. Each increment of 15 degrees in longitude corresponds to 60 minutes advancement in noon time.
4. Near to the end of the journeys, overland navigators may apply terminal homing using mega-features such as familiar city silhouettes, mountain peaks, rivers, rock and soil formation, permanent cloud formations, existing or ancient tracks, vegetation boundaries or even smell from plants. Some traditional land travelers may even release trained eagles to home on prairies while some traditional ocean travelers may release islands spotting birds to home on islands.
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Finding North and time with unclear sky

Finding North and time with unclear sky

by tonytran2015 (Melbourne, Australia).

Click here for a full, up to date ORIGINAL ARTICLE and to help fighting the stealing of readers’ traffic.

Blog post No. 09

#find North, #finding North, #direction, #time, #bright stars, #unclear sky, #sky map, #by stars, #sky disk, #declination, #right ascension,

Finding North by stars with unclear sky requires determining the Celestial poles mostly from the 10 brightest stars. They are, in descending order of brightness, Sirius, Canopus, Alpha Centauri, Arcturus, Vega, Capella, Rigel, Procyon, Achernar and Betelgeuse.
The projection to the ground of the Celestial axis gives the terrestrial North – South axis.
The method described here uses only brightest stars with high elevations and is suitable for people living in areas with naturally hazy skies, with brightened skies such as in cities or with high horizons such as in valleys.

1. Locating the Celestial poles.

Figure: Finding a Celestial pole using two chosen known stars.

The traditional method uses easily identifiable group of stars such as the Big Dipper or Cassiopeia to locate the next group of star, Little Dipper, which straddles a Celestial pole. One of the stars of this group of star, Little Dipper, is fortuitously quite close to the Northern Celestial pole and is used as that Celestial pole.

This traditional method is quite good for Northern polar and temperate zones but is not applicable to the other zones. In the Southern hemisphere, there is no group of stars straddling the Southern Celestial pole while in the tropical zone, the visibility of the Celestial poles are usually obstructed on the horizon.

Here two additional novel methods of finding North are also used. The first method is my method of “Finding North direction and time from the Sun using bare hands” [2], with the star replacing the Sun. The second method is based on geometry and is my generalization of the traditional method (which is applicable only to groups of stars directly overhead users) used by tropical people who pay little attention to neither Polaris nor Southern Cross.

In the second method (illustrated in the figure), two identifiable bright stars are chosen, one of them is called the pivoting star of the method. A flat cardboard is then used to see the great circle through the pair. The card board is then rotated around the line of sight of the pivoting star by some angle to become the plane of the great circle through the pivoting star and two Celestial poles. The declination of the star determines the directions of the Celestial poles. The Celestial axis is then projected onto the ground to give terrestrial North direction. The error of this method is minimal when the pivoting star has the same elevation as the upper Celestial pole.

Example A:

A1. Choose the pair of brightest stars OrionRigel (pivoting star) and Betelgueuse of the Orion group. Their identifying features are three regularly spaced Orion belt stars in a short straight line bisecting the line joining them .

A2. A flat cardboard is then used to see the great circle through the pair. The card board is then rotated 30 degrees clockwise (this angle is easily read from the sky maps) around the line of sight of OrionRigel to become the plane of the constant RA plane through OrionRigel.

A3. The North and South Celestial poles are respectively 90+8 and 90-8 degrees from OrionRigel.

A4. The error in this example is minimal when OrionRigel has the same elevation as the upper Celestial pole.

Example B:

B1. Choose the pair of very bright stars ArcturusBoote (pivoting star) and Spica. They are the pair of brightest stars 35degrees apart, straddling the Celestial equator, attaining their highest elevation in April.

B2. A flat cardboard is then used to see the great circle through the pair. The card board is then rotated 30 degrees clockwise (this angle is easily read from the sky maps) around the line of sight of ArcturusBoote to become the plane of the constant RA plane through ArcturusBoote.

B3. The North and South Celestial poles are respectively 90-19 and 90+19 degrees from ArcturusBoote.

B4. The error in this example is minimal when ArcturusBoote has the same elevation as the upper Celestial pole.

2. In the Northern hemisphere, over 40 degrees North. (outside tropical zone)

image

Figure 1: Bright stars about Northern Celestial pole.

BrightStarsDatesF

Figure 2: List of brightest stars.

About Northern Celestial pole there is a quadrilateral of bright stars (Vega(0.03), 25 degrees distance, Deneb(1.25), 75 degrees, Capella(0.08), 50 degrees, Dubhe(1.79), 65 degrees, Vega(0.03), in clockwise order). The vertices (accompanied by apparent magnitudes in brackets) are cited with their distances between them. This quadrilateral rotates in the counter-clockwise direction with time.

The quadrilateral has almost the shape of a trapezium with the long base being Capella – Dubhe and the short base being Vega – Deneb. Dubhe is the least bright of the four stars. It is the bright Pointer star (aUMa) of the Big Dipper, close to the dimmer Pointer star (Merak, bUMa) which is on the mid-point of the line Capella – Arcturus Boote.
The North Celestial pole is nearly of equal distances to the three long sides of the quadrilateral, also on the bisector of (Vega, Altair, Deneb), on the extension of Ori Rigel – Capella and almost on the bisector of (Dubhe, ArcturusBoote, Vega). The line Vega-Celestial pole is 12 degrees clockwise from and 60% of the length of the 95 degrees long line Vega Capella. Extending the line OriRigel – Capella by an additional 80% gives the great circle arc through three points OriRigel-Capella-Celestial pole.

Note.

There is a very large, right triangle of brightest stars (Vega(0.03), 90 degree distance, Capella(0.08), 105 degrees, Arcturus Boote (-0.04), 55 degree, Vega, in clockwise order). The vertices (accompanied by apparent magnitudes in brackets) are cited with their distances between them. It could be thought that the triangle would allow easy identification of its three vertices and consequently nearby stars. However the triangle is too large for most observation locations and the whole of it can be seen continuously only from locations above 75 degrees N and can be seen for fractions of 24 hours from locations above 15 degrees N. Therefore identifying Northern stars has to rely on less bright stars forming smaller polygons.

Traditional method for clear sky.

The Big Dipper is a group of 6 mid-bright stars and 1 low-bright star that outlines the corners of a dipper of 30 degrees long. It spreads between 30 and 40 degrees from the Celestial pole. During the nights of May, the Big Dipper stands upright (its deep cup opening pointing upright with the vertical handle on its right) while the dim Little Dipper stands almost upside down on the tip of its curved handle and the deep cup opening pouring outwardly to the right. The tip of the handle of the Little Dipper is the mid-bright star Polaris which is right on the Northern Celestial pole and is in line with the Pointers (of the Big Dipper) and 30 degrees from the Pointer stars (The Pointers of the Big Dipper are 5.5 degrees distance apart and they point from the dimmer to the brighter star towards the Northern Celestial pole.). Both Dippers are in the sky all year round but only the Big Dipper is easily visible.

3. In the Southern hemisphere, over 40 degrees South (outside tropical zone).

image

Figure 1: Bright stars about Southern Celestial pole.

BrightStars20Plus2

Figure 2: Table of 20 brightest +2 stars in order of appearance.

About Southern Celestial pole there is a triangle of bright stars (aCen(-0.01), 63 degrees distance, Achernar(0.46), 30 degrees, Canopus alpha(-0.72), 60 degrees, aCen(-0.01), in counter-clockwise order).The vertices (accompanied by apparent magnitudes in brackets) are cited with their distances between them. This triangle rotates in the clockwise direction with time.

The South Celestial pole is inside the triangle, nearly of equal distance to the 3 vertices and at 2 degrees distance to the mid-point of the line aCen – Acherna. It is also at the mid point of bCen-Achernar, on the bisector of the angle (aCen, Altair, Acherna) and is the reflection of Sirius across Canopus alpha on the extension of the line Sirius – Canopus alpha (Canopus alpha is almost the mid-point of the 75 degree long line Sirius-Celestial pole.).

Traditional method for clear sky.

The very bright Pointers and very bright Acherna are both about 30 degrees from the Celestial pole. During the nights of April, the Pointers lies horizontally with the very bright star (alpha Centauri) trailing the bright star (beta Centauri) by 4.5 degrees. The dimmer Southern Cross group of stars ahead of them is used for their identity confirmation. Southern Celestial pole is on the bisector line of the Pointer stars, on the right of the Pointers’ direction (from very bright pointer to bright pointer) and 30 degrees distance from it.
Southern Cross is group of stars (group of 3 mid-bright and 1 low-bright stars forming the 4 extreme points of a Christian cross, with the low-bright star at the extremity in the leading direction). The shaft length of this Cross is about 6 degrees, cross bar length 4 degrees.

4. Between 40 degrees North and 40 degrees South

image

Figure 1: Example of Northern sky map (Celestial pole above horizon) for 30 degrees North latitude in August.

image

Figure 2: Example of Southern sky map (Celestial pole below horizon) for 30 degrees North latitude in August.

The method described here assumes that observer’s view is obstructed below both Celestial poles. An observer needs to use bright stars with elevation of more than 10 degrees.

An observer in this zone see only slightly more than half of one sky map and slightly less than half of the other sky map. The division line are nearly straight circular arcs going close to the poles on the sky maps. If the observer see a circular disc centered on one pole he will not see the disk of the same size centered on the opposite pole. Any tropical star is visible nightly (either from Sunset to its setting or from its rising to Sunrise) in the tropical zone for more than 11 months each year.

When not able to see the poles the observer has to use identifiable bright stars around the Celestial poles in turns as they circle around the poles when the sky maps rotate.

About Northern Celestial pole there is a quadrilateral of bright stars (Vega, Deneb, Capella, Dubhe, Vega, in clockwise order).

About Southern Celestial pole there is a triangle of bright stars (aCen, Achernar, Canopus alpha, aCen, in counter-clockwise order)

In March there are Big Dipper pointing to N. Celestial pole and (Southern) Pointers giving S. Celestial pole.

Vega is brightest, reaches its highest elevation at mid-night of July 1st. Vega-Deneb is horizontal at midnight of August 1st. The triangle (Vega, 25 deg, Deneb, 32 deg, Altair, 35 deg, Vega, in counter-clockwise order) is known as the Summer triangle and is highly visible in Northern Summer .

From May, use Vega and Deneb to locate Northern Celestial pole. The line Deneb – NCelestial pole is 45 degrees long and 90 degrees counter-clockwise from Deneb-Vega, 150 degrees counter-clockwise from Deneb – Altair, 30 degrees clockwise from Deneb – Capella.

From September, use bright, setting Altair and very bright Acherna to locate S. Celestial pole. The line aCen-Acherna is 63 degrees long and is 105 (180-75) degrees in the clockwise direction from the Southern Pointers’ direction. The line Acherna-Celestial pole is 30 degrees long, originating from Acherna and is 90 degrees clockwise from Acherna-Altair, 70 degrees anti-clockwise from Acherna-(very bright) aCanopus.

When Vega has set, the angled line Deneb – Capella (Goat Star) -Orion Rigel is used to locate the North Celestial pole. The angle (Deneb, Capella, Orion Rigel) is 150degrees clockwise. The line Deneb – Capella is 75 degrees long, originating from Deneb and is 120 degrees anti-clockwise from Deneb – Vega .

The line Capella-Celestial pole is 45 degrees long and 30 degrees counter-clockwise from Capella – Deneb, 50 degrees clockwise from Deneb – Arcturus Boote. The line Capella-Orion Rigel points in the opposite direction, away from the North Celestial pole.

The line Capella-Boote Arcturus is 100 degrees long, originating from Capella and is 70 degrees anti-clockwise from Capella-Deneb. Half-way on this line (at 50 degrees distance from Capella) is the dimmer (Merak) of the two Pointer stars Dubhe and Merak of the Big Dipper. They point at Polaris, 110 degrees in clockwise direction from the line Capella – Pointers.

The line Dubhe – Vega (bright Pointer – Vega) is 65 degrees long, originating from Dubhe and is 45 degrees anti-clockwise from Pointers’ direction. The three stars (Vega, 55 degrees distance, Arcturus Boote, 50degrees distance, Dubhe, 65degrees distance, Vega) are the vertices of an almost equilateral triangle.

The line Vega-Celestial pole is 50 degrees long and 30 degrees counter-clockwise from Vega – Dubhe, 50 degrees clockwise from Vega – Deneb.

The bisector from Altair of the Summer triangle (Vega, Deneb, Altair) goes very near to the North Celestial pole. The pole is 82 degrees from Altair.

When Altair sets on the Western (at about (270+8) degrees) horizon Ori Rigel has risen from the opposite direction, at (90+8) degrees East and the Orions group is already high in the sky.

Near to Orion group, on its South-Trailing (South-Eastward) side, is Sirius, the brightest star in the sky. Sirius simplifies the identification of its 4 neighbours which are in the top 10 brightest stars. Sirius is at the center of a broom shape, surrounded by 4 neighbours outlining the extremities of the broom. Canopa is at the end of the handle and then, in the counter-clockwise direction, are 3 bright stars Orion-Rigel, Betelgueuse, Procyon almost equally spaced on a 120 degree circular arc of 25degree radius centered on Sirius. The handle line Canopa-Sirius is 30degrees clockwise from the line Sirius-Procyon and 30degrees anticlockwise from Sirius-Betelgueuse.

(Sirius, Ori Rigel, Betelgueuse, Procyon, Sirius, in anticlockwise direction) are the vertices of a rhombus. The line Sirius-Procyon joining two very bright stars is also oriented 30 degrees anticlockwise from its RA arc.

When the Orion constellation begins to set in the Western direction the whole Big Dipper near to Northern Celestial pole and Arcturus Boote and the two Southern (Centuri) Pointers near the Southern Celestial pole have all risen for 3 hours. The method is continued with these stars taking their turns.

5. In tropical zone.

mercator8gc30.jpg

Figure 1: Mercator map of brightest stars and great circle arcs to their neighbours.

image

Figure 2: Northern polar (inversion) map of brightest stars and great circle arcs to their neighbours.

image

Figure 2: Southern polar (inversion) map of brightest stars and great circle arcs to their neighbours.

The horizon of a tropical navigator looks like a straight arc going near to the center of each polar sky map. Stars near the Celestial equator rise and set 12 hours apart. Initially any star is first seen setting at the beginning of the night; it rises earlier each subsequent night to appear for the whole night; finally it is so early that it is seen setting at the beginning of the night; it is then invisible for about one month and can be seen again setting at the beginning of the night. The whole cycle takes exactly one year. Their rising and setting time for any day of the year can be read directly from the edge of the sky map and can be used to identify them.

When the star reaches its highest elevation, the angle between the star and the navigator’s zenith reaches a minimum being the difference between its declination and his latitude.

Navigators in the tropical zone can advantageously identify bright equatorial stars by their rising time and their highest elevations and they do not have to bother with polar stars. (Country people and fishermen in Vietnam have been using this method since ancient time).

Examples:

Altair (+9 degrees declination) reaches its highest elevation on midnight (and rises and sets at 18 and 06 hours) on July 15th. It is seen rising before sunrise about 5 months before July and seen setting right after sunset about 5 months after July.

Orion Rigel (-8 degrees declination) reaches its highest elevation on midnight (and rises and sets at 18 and 06 hours) on December 15th.

Navigators can work out the angle from any identified bright star to the lower Celestial pole by remembering its declination (the required angle is equal to 90 degrees plus or minus its declination). Using that only star, navigators can locate the Celestial pole using the method of “Finding North direction and time from the Sun using bare hands” [2], with the star replacing the Sun.

In tropical zone, pairs of stars are useful for finding North.

Boote Arcturus and Vega are two brightest stars in the Northern Celestial hemisphere in May and both are brighter than any other star within 150 degrees distance from both of them. The great arc from Boote Arcturus to Vega is 55 degrees long, attains highest elevation around May 23rd and is 60 degrees in the trailing direction (anti-clockwise) from the intersecting constant RA arc pointing North. Boote Arcturus leads and has only one less bright star Spica close to it (within 35degrees distance). Vega follows and has two less bright stars Deneb and Altair close to it (within 35degrees distance).

In the equatorial sky, Boote Arcturus is at the tip of a V shape formed by (Spica, Boote Arcturus, Antares). This V shape points almost at the Northern Celestial pole.

Any tropical midnight in September has no bright star near to the zenith. Navigators have to use bright stars on the West (including the pair Fomalhaut-Deneb) early in the night and then switch to bright stars on the East late in the night. Fomalhaut and Deneb are high in the sky two hours before midnight.

The star Aris Hamal (of 24 deg. N. declination) of Oct 24th is a 51st brightest star but it is identifiable in this dark area of the Celestial sphere and is often used in this September time.

At midnight of September 7th, a relatively bright Fomalhaut reaches its highest elevation, 30degrees South of the Celestial equator. Deneb and Fomalhaut are separated by about 75 degrees distance, straddling the Celestial equator. The polygonal line (Deneb, 75degrees separation, Fomalhaut, 40degrees, Acherna, 40degrees, Canopa) joining 4 of top 20 brightest stars is almost a great circle arc (straight line) and this unique line can be used to identify these 4 stars. The direction Fomalhaut to Deneb is 30degrees in the leading direction (clockwise) from the North pointing constant R.A. arc.

In October, navigators may have to use a 90degrees long arc joining the bright stars Fomalhaut on the South-West and Aldebaran on the North-East. Fomalhaut-Aldebaran is 60degree in the trailing direction (anti-clockwise) from the North pointing RA arc.

At mid-nights in December, navigators can use the line joining the 1st and 3rd brightest stars in that sky (Sirius and Capella). It is 70degrees long, straddling the Celestial equator. Rotating this line by 15degrees anti-clockwise gives a RA great circle going through the two Celestial poles.

In the nights of December, use Sirius, Canopa and Orion group of stars.

At midnight of January 1st, Sirius, the brightest star of the sky reaches its highest elevation, 16degree South of the Celestial equator..Sirius and Canopa are two brightest in the sky, 36degrees apart and the line Sirius-Canopa points to Southern Celestial pole.The North and South Celestial poles are respectively (90+17)degrees and (90-17)degrees from Sirius.

The line Ori Rigel – Capella is a R.A. arc going through both Celestial poles. The North Celestial pole is 45 degrees from Capella and 98 degrees from Ori Rigel.

The line joining the two brightest stars of Orion (Ori Rigel and Betelgeuse) are about 25 degrees long, has its center on the equator and is oriented 30 degrees anticlockwise from its RA arc. The two shoulder stars of Orion is along a constant declination circle.

In March, before mid-night, use the equilateral triangle (Sirius, Betelgeuse, 25degrees, Procyon, Sirius, in counter-clockwise order). Its center of gravity is less than 2 degrees South of the Celestial equator and its base Betelgeuse-Procyon is a constant declination arc. After mid-night use Spica and Arcturus Boote.

At mid-nights in April, there are one very bright star Arcturus Boote and one bright star Spica. They are 35degrees apart with their mid-point on the 5degrees declination circle. The line from Spica to Boote is 30degrees anticlockwise from the intersecting RA great arc pointing North.

SUMMARY

The stars to use are:

Nov: Aldebaran, O.Rigel (8 deg. S declination), Betelgeuse.

Dec: Capella, O.Rigel, Betelgeuse, Sirius, Canopus.

Feb: Betelgeuse, Procyon, Sirius.

Apr: Spica, Boote Arcturus, Antares.

Aug: Altair (9 deg. N declination), Vega, Deneb, Fomalhaut.

6. Finding time from sky maps.

When a star is used in its date of the year, the Sun leads it by exactly 12 hours. For every month after that, the lead by the Sun is reduced by 2 hours.

Example:

Sirius is a star of Jan 7th. On Jan 7th the time determined by Sirius is 12 hours behind the time by the Sun. On April 7th, the time determined by Sirius is 6 (=12-3*2) hours behind the time by the Sun.

The positions given in the maps here are for mid-night of September 23rd (Autumn equinox time). The maps rotate once every (365/366)*day and the midnight maps rotate once every year. The rotation is counter-clockwise for Northern and clockwise for Southern hemispheres.

Difference in orientation of actual sky and the map gives the time from mid-night of the locality.

In my actual nightly field testings at few suburbs of Melbourne in winter time, it is found that traditional clear sky method is applicable in less than 10% of the times while this bright star method is applicable in about 60% of the times.

7. Preparation for worsening visibility.

divider43.jpg

Figure 1: Aligning the divider along sun rays and the layout of the compass points.

DirectionTimeByStars

Figure 2: Summary of steps for Finding North by any known bright star.

A user has to anticipate which star may remain last visible when visibility worsens. He has to quickly work out its angle to the Celestial pole (which is equal to 90 degrees plus or minus its declination). With only that single visible star, it is still possible to locate the Celestial pole using the method of “Finding North direction and time from the Sun using bare hands” [2], with the star replacing the Sun.

At that moment, the user should also bring out his magnetic compass to check its magnetic declination before relying on it when the last star disappears. Even a button sized compass, provided it is well made, can be quite helpful when Celestial navigation is disabled.

References.

[1]. tonytran2015, Finding North and time by stars, https://survivaltricks.wordpress .com /2015/08/28/finding-north-and-time-by-stars /, Posted on August 28, 2015.

[2]. tonytran2015, Finding North direction and time using the Sun and a divider, http://www. survivaltricks.wordpress.com/, 06 May 2015.

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Finding North and time by stars

Finding North and time by stars.

by tonytran2015 (Melbourne, Australia).

Click here for a full, up to date ORIGINAL ARTICLE and to help fighting the stealing of readers’ traffic.

Blog No. 008

#find North, #finding North, #direction, #time, #by stars, #sky map, #sky disk, #Mercator sky map, #declination, #right ascension,

Finding North direction and time by stars is more than just identifying the Polaris and the Little Dipper around the Northern Celestial pole or the Southern Cross around the Southern pole ! It requires knowledge in representing spherical plates on flat maps, in making star maps and in identifying stars and groups of stars.

1. THE HOLLOW CELESTIAL SPHERICAL SHELL

Stars are considered attached to a hollow Celestial spherical shell centered on the Earth, with very large radius and rotating around the rotation axis of the Earth. This spherical shell has coordinates just like the spherical surface of the Earth has. The poles of the Celestial spherical shell are the intersections of the rotation axis and the spherical shell itself. The great semi-circular arcs connecting the two poles are called the semi-circle of right ascension (R.A.) while the circles on the shell of constant distance to the poles are called the circles of declination (they all have a common axis being the Celestial axis). The arcs of right ascension and circles of declination define the position of each star on the Celestial shell.

The declination circles on the Celestial spherical shell are similar to the latitude circles on Earth while the Right Ascension arcs on the Celestial spherical shell are similar to the terrestrial longitude arcs on Earth. They are marked in hours; 360 degrees correspond to 24 hours. The 0 hr Right Ascension is the R.A. of the Sun when it crosses the Celestial Equator at Spring Equinox on March 21st of every year. On that night the 0 hr ascension arc is (invisibly) lowest while 12 hr ascension arc is highest in the sky at 24 hr.

The terms Celestial North, trailing direction, Celestial South, leading direction refer to the directions drawn on the Celestial sphere at the position of a star, respectively along the Northern direction of the RA semicircle, the trailing side of the circle of constant declination, the Southern direction of the RA semicircle and the leading side of the circle of constant declination . Non-equatorial people should note that the Leading direction of a star points to the West when it is above the Celestial axis and East when below the Celestial axis.

Stars are identified by their angular distances to known features on the Celestial sphere. Their order of brightness cannot be used as a primary mean of identification as relative brightnesses are hard to distinguish and they are often affected by weather conditions.

On the other hand, angular distances can be accurately measured using only a compass divider having two straight legs and a protractor (or a clock face; each complete revolution of the hour hand sweeps 360 degrees, other angles are derived from that). Solar and Lunar diameters can also be conveniently used as small units of 0.5 degree for angles measurement. Most hikers can readily estimate angles by the widths of their fingers and hands when their corresponding arms are fully stretched (10 degrees corresponds to a slope of 0.174, and 6 degrees to 0.10).

The upper (either Northern or Southern) Celestial pole can be found by observing star motions relative to any aiming line on the tops of two sticks planted into the ground. From the known position of the upper Celestial pole the latitude of the observer is known.

The RA line of 0 hr is the midnight meridian on the equinox night of September 23rd. The RA of any star is the time it crosses the meridian line on that night.

The stars to be used as the reference marks on the sky are Polaris (a bright star nearly right on the Celestial North pole), Vega (the second brightest star in the sky, 39 degrees from Polaris, with RA of 18 hr 36 m), Sirius (the brightest star in the sky, 106 degrees from Polaris, 74 degrees from the Southern Celestial pole, RA of 7 hr.).

Starting from Polaris and Vega it is easy to identify Altair, Deneb. Starting from Polaris, Vega and Sirius it is easy to identify Orion group of stars.

These stars can then be used to identify any other stars in the sky.

2. REPRESENTING A SPHERICAL CIRCULAR SHELL HAVING STARS BY FLAT SHEETS OF MAPS.

Mapping a Sphere onto two polar caps and one cylinder

Figure of a sphere touching two caps and one cylinder wall. The flat sheets will be used to represent the spherical shell

We want to observe the stars and record their positions on flat sheets of paper.

We don’t want to project images of the stars from the center of the Celestial sphere onto the flat sheets touching it. Doing so will require the observer to have his eyes at predetermine distance from the flat sheet maps to keep the shapes of star constellation correct.

Six areas in the directions of the faces of a cube can reasonably represent the area of the Celestial sphere. However we will not use them here but use a tin-can representation having only the top and bottom circular caps and the optional cylinder wall. The caps will have Inversion projection and the cylinder wall will be rolled out with a Mercator projection.

3. Visibility of stars.

Any tropical star is visible nightly (either from Sunset to its setting or from its rising to Sunrise) in the tropical zone for more than 11 months each year. The visibility cycle for each star begins with the star seen rising on the Eastern horizon few minutes before Sunrise. On subsequent days, the star rises earlier and earlier, its position shifts gradually towards the West and the star remains for longer and longer duration in the night sky until one day it stays for the whole night. The star is therefore called a star of that date. After that day, the star is seen setting in the West in the night. On subsequent days, its lead on the Sun gradually increases and it sets on the West at earlier and earlier time. Near the end of the cycle, the star is visible above the Western horizon for only few minutes after Sunset. It then sets on the West. At the end of this cycle, the star is too close to the Sun to be visible in the sky.

The cycle then repeats from the beginning.

An arbitrary non-tropical star rises and sets North or South from the principal East and West directions and is visible in the sky for more (in Winter) or less (in Summer) than 12 hours.

Example:

Sirius is called the star of January 1st as it is visible for that whole night and reaches its highest elevation at that mid-night every year.

Astronomers locate stars on the Celestial sphere by their Right Ascensions, ordinary people identify them by their dates of being most useful.

As the duration of the days is nearly constant, the date and the RA of any star can be easily converted based on four principal sets of values that can be seen on the star maps. They are:

Date of maximum stay —–>> Right Ascension.

23 Sep Autumn Equinox —–>> 0 deg (0 hr) RA

21 Dec Winter Solstice ——->> 90 deg (6 hr) RA

21 Mar Spring Equinox —–>> 180 deg (12 hr) RA

21 Jun Summer Solstice —>> 270 deg (18 hr) RA.

4. TECHNIQUE OF MEASURING CELESTIAL ANGLES

Angles are main features used in star identifying. Angles can be estimated by comparing to the angle sustained by the width of one finger, or the combined width of the index and middle fingers, or the width of the closed fist on a fully stretched arm away from the aiming eye. All these rough angles can also be easily checked by aligning the stretched arm along a vertically tiled wall. As sin (10 deg) = 0.174 we have sin(6 deg)= 0.10, and any angle with a slope of 0.1 (=10%) is nearly 6 degrees.

These angles depend on the characteristic shape of an observer and should be individually determined for each observer. Easily available reference angles are:

1 Lunar diameter is slightly less than and Solar diameter is slightly more than 0.5 degree.

2x Lunar diameters are nearly 1 degree.

3x Lunar diameters are 1.5 . degrees. This may approximate the angle sustained by an index finger.

The angle sustained by a closed fist, usually of more than four finger widths may correspond to about 10 degrees. This varies with individual observers.

Large angle can be determined by aligning one index finger to the first star and neighbour middle finger to the second star. The whole hand is then kept unchanged. The angle between fingers is then placed against a clock face and read.

When angles sustained by hands are used near the zenith, the angle may increase by up to 30% due to the reduction in distance between the raised hand and the aiming eye.

Frequent checks against known reference angles are necessary. Long observation of Celestial objects may produce the “zooming in” illusion where angles seems to be much bigger than their real sizes. This phenomenon is well-known with people observing Sunsets, Sunrises, Moonsets and Moonrises.

Stars cannot be identified by only their relative brightnesses. Their brightnesses are hard to distinguish and may change with time and with Earth’s weather. Stars are positively identified by their relative brightnesses combined with their angular distances to nearby stars.

>>> When the identity of a.bright star is in doubt, its angular distances to positively identified, bright, neighbour stars will positively identify it. Frequent checks against reference known angles are necessary.

5. INVERSION PROJECTION AND THE NORTHERN HEMI-SPHERE.

Sky map Northern hemisphere

Figure: Sky map of the Northern Celestial hemisphere (Inversion map, this map shows only 20 brightest stars of the sky)

Sky map Northern 3/4 sphere

 

Figure: Sky map (Inversion type) of the Northern Celestial 3/4-sphere showing only 20 brightest stars and some constellations.

This method let a plane touch the sphere and project all but one points on the sphere onto the plane from the point on the sphere most distant from the plane. The distance of the projection (image) of a point from the center of the projection (the contact point) is proportional to the tangent value of half the arc from the original point to the center of the projection (the contact point). This method preserves the angles of small local features so that users can readily identify the features.

This method makes meridian curves and latitude curves become orthogonal curves. Long exposure photography near to the Upper Celestial pole shows projection of latitude curves onto the film plate. The images are almost the inversion projection. Star paths form family of circles orthogonal to family of meridians.

The concentric circles on the polar map represent the declination circle on the Celestial sphere. They are placed at angles of 30, 60, 90, 120 degrees from the Celestial pole. The non-concentric circle represent the circular path of the Sun on the Celestial sphere. Solar path is called the ecliptic.

Polar maps are simple to understand. Stars move around Celestial poles 366.25/365.25 turns every 24 hours. Viewed from the Earth, they circle in the anticlockwise direction around Northern Celestial pole and they circle in the clockwise direction around Southern Celestial pole . The Sun also rotates in the same direction but with a slightly lower rate of 1 whole turn everyday. So the stars moves ahead of the Sun by 1/365.25 days. The stars and the Sun will return to their relative position after 365.25 days, that is exactly after one year.

Identifying Northern polar stars

A dipper is a liquid transfer tool made of an open cylindrical container (a tubular container, a cup) with a long, nearly vertical handle and is used to take liquids out of their deep, long necked jars. An ordinary spoon or ladle would not be able to take liquids out of such long necked jars as the steep exit angle would spill the liquids back. Each of the dippers outlined by the stars here has a deep container (a cup) with opening diameter equal to only half its depth.

The Big and Little Dipper groups of stars look like a pair of such dippers. The top halves of their handles are nearly parallel and point in opposite direction while their containers open outwardly away from each other, in directions 120 degrees apart (The handle of the Little Dipper curves backward by 60 degrees). Each Dipper has seven stars, four close stars at the vertices of a quadrilateral, or quadrangle, outlining the deep liquid container and three stars outlining the long vertical handle.

The Big Dipper has a total length of 30 degrees, standing upright (deep container standing upright) in May, with the North Celestial pole at 30 degrees distance on its right (East). It is easy to spot as it has all 7 brightest stars (3 bright and 4 half-bright) in its circular neighbourhood of 35 degrees radius. The two bottom stars of the Big Dipper furthest from the tip of its handle are also called the Pointers. They point along the bottom towards Polaris on the other side of the handle line. Polaris is at a 30 degrees past the handle line.

The Little Dipper has a total length of 20 degrees. The handle of the Little Dipper curves away from the axis of the container by 60 degrees. The Little Dipper does not stand out in the sky as it has only 3 half-bright stars and 4 dim stars. One half-bright star (Polaris) is at the end of the curved handle, the other 2 half-bright stars are at the bottom of the container. The half-bright star Polaris is nearly right on the Northern Celestial pole.

(The word “dipper” is hard pressed to have Little Dipper outlining a sauce pan shaped dipper.)

The Little and Big Dippers are also known as Little and Big Bears. In this visualization, each same group of 7 stars is the outline of the upwardly curving tail (3 stars in line) and the hind part (4 close stars) of a bear.

The front (anterior) part of each Bear is well camouflaged and requires over-imagination to see. The word Dipper may be easier for description.

At mid-nights in March the Big Dipper lies horizontally with the handle below the container and parallel to the horizon. The Celestial North pole is 30 degree below its two Pointers. Alternative visualizations are: 1. A Plough in upright position, pointing to the East. 2. An upside down sauce pan (sauce pan shaped dipper). 3. The hind part and the tail of a Big Bear.

At mid-nights in June the Big Dipper rotates to its upright position with a vertical handle on the right of the container. The Celestial North pole is to its right at a 30 degrees distance (on a great circle). The Little Dipper stands on the tip of its handle (Polaris) with its container 60 degrees past the upside down direction.

Little Dipper (Little Bear) with Polaris is useful to people in Northern temperate zone since its high elevation to them (greater than 23 degrees) makes it visible. However, only the Big Dipper (the Plough/ Big Bear) is useful to equatorial people in the six months centered on March.

6. Positively identifying Northern stars.

Boote Arcturus and Summer Triangle

Figure 1: Bootes Arcturus and the Summer Triangle formed by Vega, Deneb and Altair. The star map is from the Inversion map for Northern Celestial hemisphere, aligned for May 21st .

Boote Arcturus and Vega

Figure 2: Mid-point of Bootes Arcturus and Vega at the center of a hemi-spherical sky view.

The features for uniquely identifying the main stars are described below.

Sirius and Vega are brightest stars in the sky. As the RA of Sirius (brightest star) and Vega (2nd brightest) differs by 12 hrs, they can never be seen together therefore cannot be distinguished only by their relative brightness. They are identified mainly based on their distances to their identifiable neighbors.

Bootes Arcturus and Vega are two brightest stars in the Northern Celestial hemisphere and both are brighter than any other star within 150 degrees distance from both of them. The great arc from Bootes Arcturus to Vega is 55 degrees long, attains highest elevation around May 23rd and is 60 degrees anti-clockwise from the intersecting constant RA arc pointing North. Bootes Arcturus leads and has only one less bright star Spica close to it (within 35 degrees distance). Vega follows and has two less bright stars Deneb and Altair close to it (within 35degrees distance).

The line Bootes Arcturus-Vega turn right by 30 degrees at Vega and continue as the line Vega-Deneb. Deneb is the next brightest star within 25 degrees from Vega. Besides Deneb, Altair is the next brightest star within 45 degrees from Vega. The line Vega-Altair is 70 degrees in the anti-clockwise direction from the line Vega-Deneb.

The triplet of the last three forms a triangle of bright stars (Vega(0.03), 25 degrees distance, Deneb(1.25), 32 degrees, Altair(0.77), 35 degrees, Vega(0.03), in anticlockwise order) known as the Summer Triangle. The vertices (accompanied by apparent magnitudes in brackets) are cited with their distances between them.

Only this triplet of three stars (Vega, Deneb and Altair) can satisfy the above descriptions and it is unique.

Arcturus alpha in Bootes (19 degrees, 14 hr), Vega (39deg,18hr36m), Deneb (45deg, 20hr41m) and Altair (9 degrees, 19.8 hr) have been identified.

Vega is closer to the Northern Celestial pole than to the Southern. If its trajectory is more than a half circle it travels anticlockwise around the upper, Northern Celestial pole, if less than, it travels anticlockwise around the now lower, below horizon, Northern Celestial pole. It reaches its highest elevation at midnight of July 1st, Vega helps identifying stars in the sky.

The distant vertex Altair of the short based Summer Triangle points away from the Northern Celestial pole.

Besides Altair, Polaris is the next brightest star within 50 degrees from both Vega and Deneb and on the other side of the line Vega-Deneb. The line Vega-Polaris is 50 degrees long and is 75 degrees in the clockwise direction from the line Vega-Deneb.

Polaris (90 degrees, any RA) has thus been identified. Its identity should be verified by using an aiming line over the tips of two sticks planted in the ground. Polaris should be seen stationary on any such aiming line.

As Polaris is almost the reflection of Altair across the short base Vega-Deneb of the isosceles Summer Triangle , the three vertices and Polaris make the vertices of a rhombus. In the counter-clockwise direction they are respectively Vega, Polaris, Deneb and Altair. This rhombus of these 4 stars is also called Weaver Shuttle by some navigators.

7. THE SOUTHERN HEMI-SPHERE

Sky map Southern hemisphere

Figure: Sky map of the Southern Celestial hemisphere (Inversion map, this map shows only 20 brightest stars of the sky)The small circle on the top half, on the 24hr RA, of this map is of the same size as a great circle of the Celestial Sphere used in this Inversion projection. The circular arcs or circles centered on the 18hr RA radius are the images of some inclined great circles on the Celestial sphere.

Sky map Southern 3/4 sphere

Figure: Sky map (Inversion type) of the Southern Celestial 3/4-sphere showing only 20 brightest stars and some constellations.

Identifying Southern polar stars

The Southern Celestial pole has no noticeable bright star. Near to it (or not so near, at 30 degrees distance!) there are two bright Pointer stars (Rigil Kent, or alpha Centauri, is the trailing but very bright star and Agena, or beta Centauri, is the leading but less bright star, leading by a 5 degrees distance). Both are on the circle of 60 degrees of declination. Near to the opposite side of this circle there is also another bright star Achernar.

The Pointer stars point (into the leading direction, from a-Cen to b-Cen) at the top of a Southern Cross group of stars (group of 3 not as bright and 1 half-bright stars forming the 4 extreme points of a Christian cross, with the half-bright star at the extremity in the leading direction). The shaft length of this Cross is about 6 degrees, cross bar length 4 degrees.

During mid-nights of March the Southern Cross is upright and high in the Southern sky with two brighter Pointers trailing it on its East. Even people in the Northern tropical zone may see this upright Cross in March on their Southern horizon. However only people living well in the South, at latitude of more than 30 degrees South, can see these stars all year round. The Cross may hide under the horizon, be yearly invisible and unusable to people in tropical zone for a long period around September.

The Southern Celestial pole is of equal distances (of about 30 degrees) to the two Pointer stars and the cross-bar of the Cross. The Southern Celestial pole is thus about 4 and 1/2 extended length of the shaft below the bottom of the Cross.

Before the Pointers go lower than the Celestial pole, they point vertically downwards with the Celestial pole 30 degrees on their left. On the other side of the pole, at the same distance of 30 degrees there is a nearly as bright star Achernar (meaning “the end of the river”).

The line of alpha Centauri-Achernar is 63 degrees long and is 105 (180-75) degrees in the clockwise direction from the Pointers’ direction. Achernar is the brightest star in the disk of 30 degrees radius centered on it. Achernar can be unambiguously identified and it is still in the sky for another 11hr after a-Centauri goes lower than the Southern Celestial pole.

Achernar (meaning “the end of the river”, also named alpha Eri, 01h 37mRA, 57° degrees South , 0.46 apparent magnitude) has been identified.

The mid-point of aCen-Achernar is close to Southern Celestial pole. Rotation of this line by 10degree counter-clockwise around aCen or clockwise around Achernar places the mid-point on the Southern Celestial pole.

The line of Achernar-Southern Celestial pole is 33 degrees long and is 90 (180-90) degrees in the clockwise direction from the Achernar-Altair direction.

8. Positively identifying Southern stars.

Sirius, Canopus, Orion Rigel, Betelgeuse and Procyon

Figure: Sirius and 4 surrounding stars (in anti-clockwise order) Canopus, Orion Rigel, Betelgeuse and Procyon.

Sirius is the brightest star in the sky. Sirius is closer to the Southern Celestial pole than to the Northern. If its trajectory is more than a half circle it travels clockwise around the upper Southern Celestial pole, if less than, it travels clockwise around the now lower, below horizon, Southern Celestial pole. It reaches its highest elevation at mid-nights of Jan 1st , Sirius helps identifying stars in the sky.

Sirius (-16 deg, 7 hr) has been identified.

As the RA of Sirius (brightest star) and Vega (2nd brightest) differs by 12 hrs, they can never be seen together for direct brightness comparison.

Canopus Alpha is the second brightest star after Sirius within 90 degrees from Sirius. Canopus is the third brightest star in the sky after Sirius and Vega. The line Sirius-Canopa is 37 degrees long .

Canopa (-53deg, 6.4hr) has been identified.

Orion Rigel is the brightest star within 30 degrees from Sirius. The line Sirius-Orion Rigel is 30 degrees long and 115 degree in the anticlockwise direction from the line Sirius-Canopa. There is a slightly dimmer Canis minor (5 degree,7.5 hr) near to Sirius; the line Sirius-Canis minor is 32 degrees long and is 225 degrees anticlockwise from Sirius-Canopus)

Orion Rigel (-8deg, 5.2hr) has been identified.

Only the triplet of three stars (Sirius, Canopa and Orion Rigel) can satisfy the above descriptions and it is unique.The triplet forms a triangle (Sirius, 37 deg, Canopa, 50 deg, Orion Rigel, 30 deg, Sirius). The vertices are cited in the anti-clockwise direction, begining and ending with the brightest star. The triplet makes a flattened nearly isosceles triangle.

Doubling the length of Sirius-Canopa and rotate it clockwise by 3 degrees gives the line Sirius-Southern Celestial pole.

The line Sirius-Southern Celestial pole is 74 degrees long and 3 degrees from in the clockwise direction from the line Sirius-Canopus and 103 degrees in the clockwise direction from the line Sirius-Orion Rigel.

From Sirius and Canopus, it is easy to identify the Pointer Stars (alpha and beta Centauri) near the Southern Celestial pole. The line Sirius-alpha Centauri(Rigil Kent) is 90 degrees long and 20 degrees in the clockwise direction from Sirius-Canopus while Sirius-beta Centauri(Agena) is 87 degrees and 22 degrees.

Both of alpha Centauri (Rigil Kent ) and beta Centauri (Agena, dimmer and leading it by 4.5 degree distance) are on the 60 degrees South declination circle. Their difference in RA is 36 minutes and their angular distance on the Celestial sphere is nearly 4.5 degrees (= 2arcsin(2cos60sin(8.97/2)) =4.481deg.). This is about 4 finger widths or nearly the width of a closed fist on a stretched arm.

They are two bright stars near the Southern Celestial pole and point to the Southern Cross group of stars. Extending the pointer line alpha-beta Centauri (Rigil Kent-Agena) to 17 degrees (to 3.5 times its original length) from alpha Centauri (Rigil Kent) reaches the top star of Southern Cross (group of 3 bright and 1 dim stars forming the 4 extreme points of a Christian cross, with a shaft length of 6 degrees, cross bar length of about 4 degrees) . The Southern Celestial pole is of equal distances (of about 30 degrees) to the two Pointer Stars and the intersection of the Cross. It is 90 degrees in the clockwise direction from the Pointers’ direction. The Southern Celestial pole is thus about 4 and 1/2 extended length of the shaft below the bottom of the Cross.

During the nights of March the Southern Cross is upright and high in the Southern sky and its two bright Pointer stars are on its left (trailing side).

The Southern Celestial pole should be verified by checking that all stars move on circles centered on it.

The line alpha Centauri (Rigil Kent)-Canopus is 50 degrees long and is 40 degrees in the clockwise direction from the pointer direction alpha-beta Centauri (Rigel Kent-Agena) .The line alpha Centauri (Rigel Kent)-Sirius is 85 degrees long and is 20 degrees in the clockwise direction from the pointer direction alpha-beta Centauri (Rigel Kent-Agena) .

The line alpha Centauri (Rigil Kent)-Altair is 95 degrees long and is 172 degrees in the counter-clockwise direction from the pointer direction alpha-beta Centauri (Rigel Kent-Agena). Within 30 degrees distance from it, Altair is the brightest star. It is thus easy to identify the tropical Altair using the Southern Cross Pointer Stars in order to use it before the Pointers hide themselves under the Southern horizon.

9. USEFUL BRIGHT STARS

Bright Stars Dates F

Figure 1: List of 20 brightest stars.

dippers

Figure 2: A real dipper and insets for Big Dipper and Little Dipper.

So people have to use a number of different identifiable stars for navigation. The star maps (or star charts) help them find suitable stars. For me, the easy to use bright stars are:STAR or CONSTN Decl(deg) RA (hr), Sun RA, Time fr EQUIN MIdnight high on

Polaris Const 90 15, ==> 3, 3×0.5=1.5month Mar21+1.5m=May10th

SCross Const -60 12, ==> 0, 0x0.5=0month Mar21

Rigel Kent (alpha Centauri), (-61deg, 7.7 hr)

Agena (the knee, Hadar, beta Centauri) (-61 deg, 7 hr)

Vega star 39deg 18hr36m, July 1st (from table)

Deneb star 45deg 20hr41m

Altair 9 19.8, ==> 7.8×0.5=3.9month Mar21+3.9month Jul 18th or Jul 20th (from table)

Boote brgt star 19 14, Apr 25th (from table)

Sirius brtest star -16 7, Jan 1st (from table)

Canopus (-53deg, 6.4hr)

Orion Rigel (-8deg, 5.2hr), Dec 19th (from table)

Orion belt star 0 5.5 17.5 17.5×0.5=9month Mar21+9m=Dec21st

10. Notes on identifying stars.

sky map mercator

Figure 1: Mercator sky-map, for upside down reading, of brightest stars and great circle arcs to their neighbours.

Bright Stars 20 Plus 2

Figure 2: Table of 20 brightest +2 stars in order of appearance.

The term “distant vertex” of a triangle denotes the vertex facing the shortest side of that triangle.

The stars (objects outside the Solar system) to be used as the reference marks on the sky are Polaris (a bright star nearly right on the Celestial North pole), Vega (the second brightest star in the sky, 39 degrees from Polaris, with RA of 18hr36m), Sirius (the brightest star in the sky, 106 degrees from Polaris, 74 degrees from the Southern Celestial pole, RA of 7hr.).

Starting from Polaris and Vega it is unambiguous to identify Deneb and Altair while starting from Polaris, Vega and Sirius it is unambiguous to identify Orion group of stars.

These stars can then be used to identify any other stars in the sky.

Polaris (for finding North and latitude) is the brightest star of the constellation Little Bear (alternatively called Little Dipper). The Little Dipper stand inverted on its handle at midnights of May.

Southern Cross Constellation and its two brighter, trailing Pointer Stars (beta Centauri and then the brighter alpha Centauri, on a straight line with the center of the Cross) are all at 30 degrees distance from Southern Celestial pole in the sky. The Southern Cross constellation becomes an upright Christian cross planted above Southern Celestial pole at midnights of March.

Southern Cross and its two Pointer stars are always visible to navigators in Southern temperate zone.

Around September 23rd, Southern Cross and its two trailing Pointers are lowest in the sky and navigators in the Southern tropical zone cannot see them. However, they can instead use Achernar (opposite the Pointers across the Southern Celestial pole) or either of or both of the setting, bright tropical star Altair (9deg, 19hr50) and the rising, bright tropical star Orion Rigel (-8deg, 5.2hr) for navigation. At least one of these two are 12 degrees above the horizon at that time. Bright, late rising mid-declination Canopus (-53deg, 6.4hr) may also be used.

Vega (of July 1st) and Bootes Arcturus (of April 25th) are two brightest stars in the Northern Celestial hemisphere, being 55 degrees apart in the sky and both are brighter than any other star within 150 degrees distance from both of them. Vega has two bright stars Deneb and Altair close to it (within 35degrees distance) whereas Bootes Arcturus has only one bright star Spica close to it (within 35degrees distance). Therefore within a 55 degrees distance around Vega, one of the two brightest stars in the Northern hemisphere, the three other brightest stars are Bootes, Deneb and Altair. They surround Vega and form a slender anti-clockwise triangle with Bootes being the distant vertex and Vega being near to its center of gravity.

Within the distance of 35 degrees around Bootes there is a uniquely identifiable bright star on the other side of the Celestial equator, that is Spica (-11.1deg decl., 201.3deg 7hr25 RA or April 12th). The line Bootes to Spica is at 30 degrees counter-clockwise from line Bootes to Southern Celestial pole.

Within 25 degrees distance around Vega, the next brightest star is Deneb (45 deg, 20 hr 41 m), a star 70 degrees on the trailing side of its Celestial North. Within the second larger distance of 35 degrees around Vega there is another uniquely identifiable bright star, which is Altair, at 40 degrees on the trailing side of its Celestial South.

Altair(9 deg, 19hr 50)’s name has foreign origins (Al Nesr Al Tair, in Arabic; Vultur volan, in Latin). People in the tropic can use Altair around July 20th and Sirius (brightest star) around Jan 1st for identifying neighbouring stars and for navigation. Altair and Sirius stars are 13 hr apart in R.A. so at least one of them will be visible in any tropical or temperate sky except for 1 hr in any continuous 24 hr hours period.

The small, short based, nearly isosceles , anti-clockwise triangle formed by Deneb, Altair and Vega is known as the Summer triangle in the Celestial sphere with its pointed, distant vertex Altair pointing away from the Northern Celestial pole.

As Polaris is almost the reflection of Altair across the short base Vega-Deneb of the isosceles Summer Triangle, the three vertices of this triangle and Polaris make four vertices of a rhombus. The vertices are respectively Vega, Polaris, Deneb and Altair in the counter-clockwise direction . The stars here are Vega and its 3 brightest nearby stars within a distance of 65 degrees. The diagonals of this rhombus have sizes of 81 degrees (Polaris to Altair, along the Celestial North South arc) and 21 degrees (Vega to Deneb, leading to trailing directions). This rhombus of these 4 stars is also called Weaver Shuttle by some navigators from Northern hemisphere.

Sirius (meaning glowing, scorching; at -16 degrees, 7 hr), is the brightest star, or more accurately binary star system, in the sky. It is in the Dog group of stars near to the trailing foot of Orion. It is a useful star for identifying its neighbours. People in the tropic can use Altair around July 20th or Orion-Rigel around Dec 10th or Vega (very bright star in the North) around July 1st or Sirius (the brightest in the South) around Jan 1st and for identifying neighbouring stars.

Sirius simplifies the identification of its 4 neighbours which are in the top 10 brightest stars. Sirius is at the center of a broom head, surrounded by 4 neighbours outlining the extremities of the broom. Canopa is the second brightest star and is at the end of the broom handle. The angled line (Orion-Rigel, 18 degrees distance, Betelgeuse, 25 degrees distance, Procyon) outlines the extremity of the broom head. The vertices are 3 bright stars on a 100 degree counter-clockwise circular arc of 25 degree radius, centered on the brightest star Sirius. The bisector of the 60 degrees anti-clockwise angle (Betelgeuse, Sirius, Procyon) points to the North Celestial pole while Sirius-Canopa points in its opposite direction, at the Southern Celestial pole.The trailing half of the broom head is an equilateral triangle (Sirius, Betelgeuse, 26 degrees distance, Procyon, Sirius, in counter-clockwise order). Its center of gravity is less than 2 degrees South of the Celestial equator and its base Betelgeuse-Procyon is a constant declination arc.

Orion’s central belt star (0 degree, 5.5hr) is right on the Celestial equator. It readily gives the accurate North direction by my method [1]. It is a star of December. However it is not the brightest star in the Orion group. The brightest star of Orion is Rigel, at Orion’s leading (Westward) foot and the second brightest is Betelgeuse, at Orion’s trailing (Eastward) shoulder; all other stars of Orion are much dimmer. The line joining these two brightest stars of Orion is about 25 degrees long and forms with the line of 3 regularly spaced (much dimmer) belt stars of Orion the distinctive shape of two diagonals of a slender rhombus. The line of the 3 belt stars has 1/6 the length of and is at right angle to the line Rigel-Betelgeuse; its extension also goes through Sirius. The line Rigel-Betelgeuse is oriented 30 degree anticlockwise from its North pointing RA semi-circle. The two shoulder stars of Orion are almost right along the Leading-Trailing direction. The dagger stars of Orion are right along the RA semicircle.

Procyon is right behind Orion group of stars and is a mid-January equatorial star. The next bright equatorial stars are Spica and Bootes Arcturus, they are behind Procyon by 6 hours.

Before losing the two Southern Pointer stars to their ever earlier setting time, navigators in Southern tropical zone can easily identify the bright tropical star Altair to replace these Pointers. The line of alpha Centauri-Altair (Rigel Kent, brighter Pointer-Altair) is 95 degrees long and is 170 degrees in the counter-clockwise direction from the direction brighter to dimmer pointers (alpha Centauri, Rigel Kent-beta Centauri, Agena). Within 30 degrees distance from it, Altair is the brightest star. Altair can be unambiguously identified and it is high in the sky.

Before Altair (9 deg, 19.8 hr) sets nearly at 279 (=270+9) degrees in the West there is already Orion Rigel (-8 deg, 5.2 hr) rising at nearly 98 (90+8) degrees in the East (The two stars are almost setting and rising at opposite directions.). This Orion Rigel is only 24 hr +5.2hr -19.8hr = 9.4hr = 141 degrees RA behind Altair. So at least one in this pair (Altair, Orion Rigel) are above 12 degrees (=0.5*(180 – 141)* cos 30) in elevation to tropical navigators to be used as navigational stars.

When the Big Dipper is standing upright (with the dimmer Little Dipper standing on the tip of its handle and its container 60 degrees past the upside down direction) the Summer Triangle (Vega, 25 deg, Deneb, 32 deg, Altair, 35 deg, Vega) is already well above the horizon. When the two Pointers go below the Northern Celestial pole the Summer Triangle can be used to find North.

The line Deneb – Celestial pole is 45 degrees long and is 90 (=180-90) degrees in the counter-clockwise direction from the line Deneb – Vega.

The line Deneb – Capella is 80 degrees long and is 120 (=180-60) degrees in the counter-clockwise direction from the line Deneb – Vega. Rotation of this line by 30 degree or clockwise around Deneb or counter-clockwise around Capella places the mid-point on the Southern Celestial pole. This pole is of equal 45 degrees distances from both stars.

Before Vega goes lower than the Northern Celestial pole, the Orion constellation has risen in the Eastern direction. This constellation has two bright stars Orion-Rigel and Betelgeuse. The line joining these two is the short diagonal of a 3-to-1 rhombus (Sirius, Orion-Rigel, Aldebaran, Betelgeuse, Sirius, in counter-clockwise direction).

The tropical September mid-nights have no bright star near to the zenith. Navigators have to use bright stars on the West (including the pair Fomalhaut-Deneb) early in the night and then switch to bright stars on the East late in the night. Fomalhaut, 30degrees South declination, is a September star. Fomalhaut and Deneb are separated by about 75 degrees distance, straddling the Celestial equator. The North and South Celestial poles are respectively 90+30 and 90-30 degrees from Fomalhaut.

11. USING A POLAR MAP OF STARS

using star map

Figure: Using a star map on the 21st of July, the selected date is on the rim.

Select the current date in the year on the rim of the polar map. The Sun should be located at a point on the ecliptic nearest to this date on the rim. The Sun should be on the radial (meridional) line connecting the date on the rim to the Celestial pole at the center of the map. Verify that the intersection of this radial Right Ascension line and the ecliptic gives a position of the Sun corresponding to the season in the year (for example, around March, the Sun travels on the ecliptic, crossing the Celestial equator, to come to the Northern hemisphere from the Southern hemisphere; Northern Spring equinox occurs on 21st March).

Align the center of the circular map to the corresponding Celestial pole by first aligning its axis to the terrestrial North or South (a magnetic compass is good enough this job!) then raise it to the higher Celestial pole by an angle equal to the latitude value of your place.

Align the map so that the Sun on the map be on the bottom radius (or Right Ascension) line. The line of Right Ascension 12 hr from it is the meridional line for midnight of that day. Stars on this RA line should attain their highest elevations at midnight.

Stars on the RA few hours before or after that will ascend few hours before or after midnight.

The actual stars will be in the radial direction of the stars displayed on the circular map.

A night navigator will select identifiable stars whose declinations are nearly equal to his latitude. They will pass by his zenith. Stars distant from his zenith may not be visible due to terrestrial obstructions. (For example, the Polaris star may not be visible to many people in tropical zone since it is too close to the horizon line which often has obstructions) .Therefore a navigator in non-polar regions has to choose bright stars with various RA but with declination values close to his latitude so that at any time there is at least one of them being close to the zenith and these stars are used cyclically through different times for continuous navigation.

12. FINDING NORTH AND TIME USING THE DECLINATION OF A STAR AND LATITUDE

Find North by a divider - Selection rule

Figure: Finding North direction and time by any star is similar to that by the Sun but with the STAR INSTEAD of the Sun.

Find North Direction and Time By a Star

Figure: Finding North and time by any bright star.

1a/- Once a star is identified, its constant declination is obtained, the Instructables “Finding North direction and time from the Sun using bare hands” [1] can be applied but with the star instead of the Sun. The time from the method is with respect to the star.
When a star is used on its date in the year, the Sun leads it by exactly 12 hours. For every month after that, the lead by the Sun is reduced by 2 hours.

1b/- When the date of the star is not readily available, its RA can be used instead.

Lead on time by the Sun = Solar Time – Time Relative to star = RA star – RA sun.

The difference star RA – solar RA must be ADDED to it to give local time with respect to the Sun (a star with larger RA will rise later than a star with smaller RA).

Example:

Sirius is a star of Jan 1st. On Jan 1st the time determined by Sirius is 12 hours behind the time by the Sun. On April 1st, the time determined by Sirius is 6 (=12-3*2) hours behind the time by the Sun.

2- Alternatively, the shapes of identifiable constellations can tell how the Celestial sphere is orientated, and this may easily give information on both direction and time to people who watch the Celestial sphere nightly and have learned it by heart (Traditional sea fishermen in Vietnam have such memory for positions at various times for bright equatorial stars/constellations in the Celestial sphere).

References

[1]. tonytran2015, Finding North direction and time using the Sun and a divider, http://www.survivaltricks.wordpress.com/, 06 May 2015.

[2]. tonytran2015, Finding North direction and time using the hidden Sun via the Moon,https://survivaltricks.wordpress.com/2015/07/06/finding-north-direction-and-time-using-the-hidden-sun-via-the-moon/, posted on July 6, 2015

[3]. tonytran2015, Finding North direction and time accurately from the horn line of the Moon. https://survivaltricks.wordpress.com/category/moon-horn-line/
posted on August 12, 2015

[4]. tonytran2015, Finding North direction and time using the Moon surface features , https://survivaltricks.wordpress.com/2015/07/01/finding-north-direction-and-time-using-the-moon-surface-features/ , posted on July 1st, 2015

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Finding North direction and time accurately from the horn line of the Moon.

Finding North direction and time accurately from the horn line of the Moon.

by tonytran2015 (Melbourne, Australia).

Click here for a full, up to date ORIGINAL ARTICLE and to help fighting the stealing of readers’ traffic.

 

#find North, #finding North, #direction, #time, #Sun, #hidden Sun, #navigation, #survival, #Moon, #phase, #horn line, #half Moon,
Here I add two more steps to the common horn line method for the Moon to obtain more accurate North direction and time from its horn line. The modified method uses position of the Moon, shape (phase) of the Moon, solar declination and user’s latitude to work out North direction and rough local time.

1. Basic information on the Moon for navigation.

mooncrescent.jpg

moonshapesnangles4c.jpg

The Moon is a satellite of the earth. Everyday Moon-rise and Moon-set time is retarded by about 50 minutes. This allows the Sun to travel further on its journey every subsequent night. Therefore after full moon the partial bright side stays on the East (trailing) side and dark crescent appears on the West and dark area gets fatter daily until the whole moon is dark. Similarly, from new Moon a bright crescent appears on the West and grows fatter and bright area gets fatter daily until full Moon is reached. From the shape of the Moon, it is easy to say how late the Moon is trailing the Sun (new Moon trails by 0 degree and 0 hour, new half-Moon by 90 degree and 6 hours, full-Moon by 180 degrees and 12 hours, and late half-Moon by 270 degrees and 18 hours .). The shape and the position of the Moon allow some guessing of its trajectory for the night.

The Moon completes its orbit in space in 27.321 days, and it completes one full revolution on the Celestial sphere in that time. Its angular velocity on that sphere is 1/27.321 (rev/day) = 0.036601 (rev/day) .

The Sun apparent travel on the ecliptic takes 365.256 days. Its angular velocity on the elliptic is 1/365.256 (rev/day). The Moon revolves on the Celestial sphere faster than the Sun by an angular velocity of 1/((1/27.321)-(1/365.256)) = 1/29.530 (rev/day). When the Sun is one full turn ahead of it, the Moon will catch up, and they will be both in the same direction again on the Celestial sphere after a period of about 29.530 days. So the intervals between consecutive full Moons will be something like a pattern of (30days, 29days).

By keeping records of previous full Moon nights people know it is a WAXING or WANING Moon.

The simple Waxing-Waning rule is that the bright side of the Moon is on the West for waxing and East for waning Moon.

It is natural for people to desire to use the horn line, which is the line connecting the two horns of the Moon, to draw the North South direction. However it has been found that the intersection between the horn line and the horizon does not accurately give the North direction.

Here we find out the reasons for that inaccuracy and show a more accurate method of using Moon’s horn-line.

It may be easier for some readers to first read step 5 then come back to read steps 2, 3 and 4.

CAUTION: The horn line of the partial Moon can point far away from the terrestrial principal North or South directions.

2. Estimating the current solar declination

wpid-divider10l.jpg

Estimating the current declination.

The whole Celestial spherical shell rotates around its two Celestial poles. The Sun moves slowly on that sphere on a great circle called the ecliptic. Its distance to the two opposite Celestial poles varies periodically, and its distance to the Celestial equator is call the declination of the Sun.

You can make a rough sketch of this declination from the principal values and estimate the declination for the current day.

3. The horn line is not easily transformed to the ground meridian line.

image

Figure: Panoramic view of the travel of a partial Moon in the sky. The Celestial axis is always at right angle to the path. This picture is for the winter, with the Sun in the other hemisphere and the bright side of the Moon tilts toward the ground. In the summer, it tilts toward the sky.

The horn line is at right angle to the plane containing the very slender triangle formed by the Earth, the Moon and the Sun but the Celestial axis is at angle of (90-23.5) degrees to that plane. So the horn line usually form an angle of that size to the Celestial axis.

Near to half-moons the horn line is easily defined, and it is also easy to see that the projection of the Celestial axis onto the half-moon makes with it an angle equal to solar declination.

On top of those complication, the Moon also has its own declination and an observer has additional difficulty working out the direction of the horn line as it is usually not at right angle to his line of view.

4. Twisting the horns of the partial Moon.

MoonShapesNAngles5C

Figure 1: Moon phase chart for a Solar declination of (-20) deg (South).

At half Moon, it is possible to twist the horn line about the line of view to generate a line KL parallel to the Celestial axis. The amount of twisting is opposite to the declination angle of the Sun.
Similarly, when the angle Sun-Moon-Earth is about 90°+/-30° (=120° or 60°) the amount of required twisting is about 0.85*declination of the Sun.

The required twisting is varied with the phase of the Moon as in the following:

Half Moon requires twisting by full Solar declination,

15% or 85% bright area requires twisting by half Solar declination,

full or no-Moon requires no twisting.

Remember that if the Sun is into your hemisphere (in your summer) the bright side of the half Moon has to be twisted downwards (toward the Celestial equator) by an angle equal to the solar declination angle. The opposite should be done in your winter.

This correction here already causes a difference between the results from the old horn line method and the current method. There will be another difference caused by drawing the “spear line” parallel to Celestial axis in the next section.

Note:

The required twisting on the horns of the Moon varies sinusoidally with time and peaks to the values of minus/plus Solar declination when the Moon is half-full.

5. Finding North direction from a Celestial North South line seen on the Moon.

image

Observing a Celestial axis KML drawn on the Moon: U is observer, E center of the Earth, N terrestrial North pole, S terrestrial South pole, M Moon, UV local vertical, KL a line parallel to Celestial axis, EQ normal to plane UKL, UP a line parallel to Celestial axis. The red circle through U is the intersection between the plane UKL and the Earth’s surface.

Suppose that there is on the Moon M a line KL parallel to the Celestial axis, as illustrated in the figure. We draw a plane through the observer containing the line KL. On that plane UKL draw a line PU nearly in the direction of KL, descending into the ground at latitude angle.That is

(|angle /(PU,KL)| <90° ) and (angle /VUP = 90 deg. – latitude angle).

The line UP is then parallel to the Celestial axis, and its projection on the ground gives the local North South (meridian) direction.

When the Moon is high in the sky and the plane UKL is steeply inclined to the local horizontal, the last condition is the satisfied by

( angle /MUV < angle /MUP ) and (angle /VUP = 90 deg. – latitude angle).

The line UP is then parallel to the Celestial axis, and its projection on the ground gives the local North South (meridian) direction.

For Southern latitudes draw UP’ close to the direction of LK (PUP’ is a straight line).

6. On the ground view of the horn line method.

image

Figure: Finding out North direction more accurately using horn line.

On half-Moon nights, when the angle Sun-Moon-Earth is 90°, twist the horn line by the declination of the Sun to generate on the Moon the line KL parallel to the Celestial axis, and similarly, when the angle Sun-Moon-Earth is about 90°+/-30° (=120° or 60°) the amount of required twisting is about 0.85*declination of the Sun.

When properly carried out, the plane UKL always makes with the horizontal plane an angle not less than the latitude angle. The line KL is the observer’s view of the Celestial axis. All lines co-planar to UKL generate the same view to the observer! Draw a “spear line” PU co-planar to UKL and spearing the ground at an angle equal to latitude angle. The spear line PU so obtained is then pointing exactly along the Celestial axis, towards the lower Celestial pole. The projection of PU onto the ground gives the terrestrial North-South line. North direction is then found.

When the Moon is high in the sky and the plane UKL is steeply inclined to the local horizontal, the lower end U of the spear line is nearer to the line KL than its upper end P.

Note:

The intersection line between the plane PKL and the ground surface is generally NOT in the North-South direction unless the observer is on the terrestrial equator circle. The drawing of the spear line PU cannot be by-passed in this method.

7. Summary of steps in this method.

moonsungreatcircle.jpg

wpid-divider10l.jpg

MoonShapesNAngles5C

wpid-wp-1439376905855.jpeg

To apply this method, the navigator has to go through all the following steps:

a. Working out the ROUGH (+/- 45 degrees) principal North-Sout-East-West directions, from the Waxing-Waning rule and the knowledge that the horn line direction is being close to an RA arc,

b. Estimating the Solar declination for the current time of the year,

c. Working out the proportion of Solar declination used for twisting the horns of the Moon from the Moon phase,

d. Twisting the horns of the Moon by the required angle and waving a stretched arm along the adjusted horn line to establish the plane through the Earth, Moon containing the direction of the Celestial axis,

e. Drawing a spear line inside that plane, inclined by latitude angle, aligned roughly to the ROUGH direction of the lower Celestial pole to establish its exact direction, and

f. Projecting the Celestial axis onto the ground surface to obtain the terrestrial North-South direction.

New users to this method should initially only use it as a double check for another method using the position of the hidden Sun, given earlier as part 1, in a previous Instructables article [2]. The agreement between the two methods would make users confident on their results.

8. Avoiding hasty and perilous conclusions when using the horn line.

image

Figure: A rare (or not so rare?) failure of the simplistic, traditional horn line method. The horn line intercepts the horizon near to the terrestrial North (wrong by 180 degrees!) in a place in Northern hemisphere. The traditional horn line rule really requires amendments.

At latitude less than 28 degrees, the horn line may point to Northern skyline at high Moon although most of the times it points to Southern skyline ! There is a peril of being sent astray by 180 degrees for unreserved navigators.

At high Moon the horn line only gives North South direction near to equinox times ! The error can be due East or West by the declination value of the Sun.

If the steps for using this method seem to be too complex, the Moon navigators may have to accept reduced accuracy from the Moon and use only Waxing-Waning rule in combination with the stars to navigate !

There is a third part to this topic, with the title “Finding North direction and time using the Moon p3- Moon surface features”.

Conclusions:

It is possible to work out accurately North direction from the horn line of the Moon.

The twisting of the horn line and the drawing of the spear line PU are two additional steps of this modified method.

References

[1]. Tristan Gooley, How to navigate using the Moon, the natural navigator,
http://www.naturalnavigator.com/find-your-way-using/moon, accessed 2015aug12

[2]. tonytran2015, Finding North direction and time using the hidden Sun via the Moon, https://survivaltricks.wordpress.com/2015/07/06/finding-north-direction-and-time-using-the-hidden-sun-via-the-moon/, posted on July 6, 2015

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Finding North direction and time using the hidden Sun via the Moon

Finding North direction and time using the hidden Sun via the Moon.

by tonytran2015 (Melbourne, Australia).

Click here for a full, up to date ORIGINAL ARTICLE and to help fighting the stealing of readers’ traffic.

(Blog No.006).

#find North, #finding North, #compass, #direction, #time, #Sun, #hidden Sun, #navigation, #survival, #Moon, #phase,

Finding North direction from the Moon cannot not be as accurate as from the Sun. There are many causes for this:
1/- The Moon does not always rise on the principal East (at 90 degree of the compass rose).
2/- We cannot work out by heart the Moon’s declination (up to +/- 5.1 degrees to the ecliptic, and 23.5+5.1 degrees to the Celestial equator ).
3/- We cannot easily work out when the Moon reaches its highest elevation angle at its meridian time. The Moon does not often cast strong shadows for shadow sticks to work.
Here I describe my new method to find out North direction and time with improved accuracy. The method uses shape and position of the Moon, solar declination and user latitude to work out the position of the hidden Sun, then work out North direction and approximate local time with an accuracy of 30 minutes. Literally, the user can work out North direction and the local time with his bare hands.
I have field tested this method and I have relied on it for many years.

1. Basic information on the Moon for navigation.

image

image

Figure 1: Moon phase chart. Figure 2: A crescent Moon may not align itself to the terrestrial East or West horizon points (see texts).

The Moon is a satellite of the earth. Everyday Moon-rise and Moon-set time is retarded by about 50 minutes. This allows the Sun to travel further on its journey every subsequent night. Therefore after full moon the partial bright side stays on the East (trailing) side and dark crescent appears on the West and dark area gets fatter daily until the whole moon is dark. Similarly, from new Moon a bright crescent appears on the West and grows fatter and bright area gets fatter daily until full Moon is reached. However the bright and dark sides of a partial Moon rarely point accurately to East or West directions.

Figure 2 of this section shows a crescent Moon on the Celestial sphere. The horizontal great circle represents the horizon of the observer. The inclined great circle represent the Celestial equator and the arrow through the center of the sphere represents the Celestial axis. The circle parallel to the Celestial equator is a constant declination circle being the trajectory of the Moon during the hours. The two intersection points of the two great circles are the terrestrial East and West points of the horizon. This picture shows that the crescent Moon may point its bright to dark line far away from the terrestrial East or West points when there is a combination of high declinations of both the Moon and the Sun on the same side of the Celestial equator.

Each Lunar (Moon) cycle begins with the Moon being visible as a thin bright arc in the sky (called a New Moon), trailing the Sun by less than one hour. After Sunset this thin Moon is seen bright on the West until it sets. On subsequent days, the Moon is more and more behind the Sun, its position shifts gradually towards the East and the Moon remains for longer and longer duration in the night sky till full Moon day. After full Moon day, the Moon (now called a Late Moon) becomes thinner and thinner and is seen risen in the East in the night, it remains visible in the sky after Sunrise, and travels ahead of the Sun. On subsequent days, its lead on the Sun gradually reduces. Near the end of the cycle, the Moon is visible as only a thin bright arc rising in the East for less than one hour before Sunrise and after Sunrise it can still be seen leading the Sun by that same amount of time. At the end of the Lunar cycle, the Moon sends no reflection of Sunlight to Earth and is too close to the Sun to be visible in the day sky.

Keeping diaries of past days of full and new Moon helps people know where their time is in the current cycle, and so they know whether the leading (West) side of the Moon should be bright (new to full Moon) or dark (full to new Moon). Fortunately, the users of my method described here do not have to refer to any such records of the Moon.

It is interesting to note that Buddhist East Asians use lunar calendars and observe fasting at new and full Moon. From their calendar and their fasting festivals , they already know whether the Moon is waxing or waning. This may help explaining why they are good at finding North direction using the Moon.

CAUTION 1: The bright to dark line of the partial Moon can point far away from the terrestrial principal East or West directions.

CAUTION 2: The horn line of the partial Moon can point far away from the terrestrial principal North or South directions.

2. Moon shapes giving Moon-Earth-Sun alignment.

The various shapes of the Moon under various angles of lighting by the Sun are given in the illustration picture. The Moon goes through this cycle every 29.5 days. The picture is drawn for the principal values of the angle of Moon-Earth-Sun. The picture allows determination of the direction of the Sun from the shape of the Moon.
The angle Moon-Earth-Sun will be more accurately known if the navigator is in the habit of directly measuring and recording it before Sunset (few hours earlier) whenever the Moon is seen during day light.

3: Direction of the Sun from the Moon

image

image

The line joining the two horns of the Moon is always at right angle to the plane of Sun, Earth and the Moon. Draw a half-line from you to the Moon and extending far past the Moon. Imagine the Sun is at the far end of this half-line. Swing this half-line in the direction of the bright side (at right angle to the line joining the two horns) of the Moon to have the angle of Moon, Earth and Sun giving a matching shape for the brightened part of the Moon. The half-line then gives the direction of the Sun.
Alternatively, you can think of placing a sphere between you and the Moon, and a torch is is used to shine on the sphere and the torch is placed in various directions until it gives a partially brightened sphere similar to the current Moon shape.

4. Finding North direction and time.

image

image

image

image

image

image

divider43.jpg

With the direction of the Sun known, the technique given by my previous blogpost “Finding North direction and time using the Sun and a divider” [1] can be applied to find North direction and local time.
The selection rule of right or left hand placement of CA in “Finding North direction and time ising the Sun and a divider” has been generalized.

The generalization is:
(Northern latitudes with rising Sun or Southern lat. with setting Sun) ==> CA on the left of CB,
(Northern latitudes with setting Sun or Southern lat. with rising Sun) ==> CA on the right of CB.
The time for rising Sun here is from 0hr to 12hr (AM) and time for setting Sun here is from 12hr to 24hr (PM).
The rest of that method applies to the hidden Sun to give North direction as well as time.

I have tested and found that this method gives direction accurately and easily. The additional benefit is that it also gives approximate time.

Reference
[1]. tonytran2015, Finding North direction and time using the Sun and a divider, http://www.survivaltricks.wordpress.com/, 06 May 2015.

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Finding North direction and time accurately from the horn line of the Moon. posted on August 12, 2015. This is my novel technique.

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Finding North direction and time using the Moon surface features

Finding North direction and time using the Moon surface features.

by tonytran2015 (Melbourne, Australia).

Click here for a full, up to date ORIGINAL ARTICLE and to help fighting the stealing of readers’ traffic.

#find North, #finding North, #compass, #direction, #time, #Moon, #surface features, #natural compass rose, #navigation, #survival

This article shows how to use the Moon for finding direction and time.
The surface features of the Moon can be used as a compass rose for Earth inhabitants.

1. An upside down natural compass rose

Near to full Moon the phase (waxing-waning) and horn-line methods are not accurate. Right at full moon they are not applicable. However at those times we may obtain directional information from the global map of the Moon using the colour and shade of its surface (soil) features. Since moonlight is only reflected light from the Sun and is not intense and we may look at the Moon’s surface for the features.

We have to identify the features of the Moon associated with Lunar own rotational poles, so that the Moon can be placed and aligned on an upside down compass rose aligned for the rotation of the Earth.

Each of us may have have different individual visualization (or a simplified picture) of the Moon to orientate its poles on such compass rose. My own visualization for the shades on the Moon is a small lion licking the face of a kneeling monkey and it is drawn on the Moon in the title figure.

image

Figure: The surface features of the Moon is used as the core of a compass rose.

2. An oscillating core of the compass rose !

image

Figure 1: Moon as an oscillating core of a compass rose.

image

Figure 2: Moon as an oscillating core of a compass rose.

image

Figure 3: Moon as an oscillating core of a compass rose.

Consider that compass rose an UNDERNEATH view of a normal compass rose and you can use it for finding directions when it is high in the sky. (The leading side of the Moon, with a lion visualization, is on our West and its trailing side, with a monkey visualization, East). However the North of the Moon is the North pole of Lunar rotation axis and it makes an angle with the axis of the Earth, the angle sometimes reaches 23.5 + 1.5 = 25 degrees. Imagine that you can walk on the Celestial equatorial plane and the Lunar axis is planted on it at an angle of 90-25 degrees and you go around it once every 27.3 days.

Looking at that inclined Lunar axis, you will see that axis alternately tilted to your right hand then to your left hand. Looking from the earth, the axis of the Moon appears to oscillate clockwise and anti-clockwise (with amplitude equal to the lunar orbit angle, which requires complex calculations, and can be up to up to 25 degrees ) as the Moon orbits around the Earth. This compass rose only gives correct orientation when the Moon is made oscillating inside it ! The North of the Moon is aligned to 0 degree only when the Moon goes through its maximum or minimum value of Lunar declination (at furthest distance to the Celestial equator). When the Moon crosses the Celestial equator, the angle between Moon axis and Celestial axis is highest in absolute value..

So we have a natural compass rose but we must remember that Moonscape features does not easily give accurate direction and the Moon oscillate inside our Earth aligned compass rose between up to +25 and -25 degrees as well as tilling its poles toward or away from us. The title figure of this article is made for the reference, mean orientation of the Moon, when its axis is at right angle to the line of view and its equator is aligned to 90-270 degree marks of the graduation ring. Users intending to use the compass rose on any full Moon should check the orientation of the surface features against the East West directions (given by Waxing-Waning rule and by adjusted horn line method) two or three nights prior to the full Moon. Otherwise an uncertainty of up to 25 degrees should be allowed with this compass rose.

3. Lunar navigation needs a combination of methods.

MoonShapesNAngles5C

Figure 1: Moon phase chart for a Solar declination of (-20) deg (South).

tiltedhornlinec2.jpg

Figure 2: Panoramic view of the travel of the Moon.

MoonRosePath
Figure 3: Panoramic view of the travel of the Moon.

Navigating by the Moon becomes easier when we do it nightly on consecutive nights and keep records from previous nights. At half-Moon times we can use the Waxing-Waning rule and my improved horn-line method (given in p2) to draw the Celestial axis line on the Moon then record the position of the horn-line on the featured surface. At full Moon times we use Lunar surface features with the angle for the Moon obtained previously from the 3/4 Moon nights.We have to remember that the horn-line rotates almost steadily about each full-Moon.

Alternatively, the if we form the habit (when we have to navigate) of daily recording the direct measurements of Lunar declination, from the Moon and the Celestial pole (by either stars at night or the Sun before Sunset), we have accurate values of Lunar declination. The Moon and its declination can then replace the Sun in my method of determining direction and time (reference [3]). The accuracy is further improved if we combine the knowledge of our latitude, the phase and elevation angle of the Moon to predict its trajectory for the night (therefore we already have had an initial estimation of the North-South direction).

After the North-South direction has been found it is easy to tell time from a full Moon as the Moon is trailing the Sun by about 12 hours.The estimation is more accurate if we apply extrapolation to our own records of Moon rises and Moon sets on previous nights. When there is no Moon, we have to use stars and that will open new topics.

With lots of switchings among methods, the navigators may find that finding direction and time via the hidden Sun as given in reference [1] the simplest.

References

[1]. tonytran2015, Finding North direction and time using the hidden Sun via the Moon,https://survivaltricks.wordpress.com/2015/07/06/finding-north-direction-and-time-using-the-hidden-sun-via-the-moon/, posted on July 6, 2015

[2]. tonytran2015, Finding North direction and time accurately from the horn line of the Moon. https://survivaltricks.wordpress.com/category/moon-horn-line/
posted on August 12, 2015

[3]. tonytran2015, Finding North direction and time using the Sun and a divider, http://www.survivaltricks.wordpress.com/, 06 May 2015.

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Finding accurate directions using a watch

Method for finding accurate directions by a common analogue watch.

by tonytran2015 (Melbourne, Australia).

Click here for a full, up to date ORIGINAL ARTICLE and to help fighting the stealing of readers’ trafficby tonytran2015 (Melbourne, Australia).

#find North, #finding North, #compass, #direction, #by Sun, #bisector, #using watch, #with watch, #tilted watch, #inclined watch, #navigation, #without compass

This method uses a common 12-hour watch with analogue face for finding directions. Unlike the traditional method of using the hour hand of a flat lying watch, my method uses a watch tilted from the vertical and gives better accuracy for both North and South hemispheres including tropical zones. When applied to the arctic and antarctic regions, the watch is tilted by more than 67 degrees and lies almost flat on the ground; it becomes the traditional method using flat lying watch.
This method use the position of the Sun, time and known latitude angle to determine directions and Sun declination (therefore estimation of current month of the year).
The method for Northern latitudes is described below.

Method for Northern latitudes.
The word “bisector” here is used to mean the bisector of the angle between the midnight/midday marking and the hour hand.

DirectionBySun_12N

The red line is the bisector. The line CB is drawn on a card representing the half-plane to enable accurate alignment to the Sun

WatchCompass_22NL

The bisector is in the opposite direction of a corresponding 24 hr hand on a 24 hr dial

1/- Hold the watch so that its AXIS rises above the horizontal plane by an angle equal to the latitude of the region. That is its face points to somewhere in the sky and its back is angled downwards into the ground.
2N/- Determine the half-plane limited by the axis of the watch and containing the bisector. This half plane revolves clockwise about the axis of the watch once every 24 hour and goes through the mid-day marking at noon.
3N/- Hold the watch in such composure and rotate your whole body around your vertical axis by your feet until the Sun lies in the above half-plane.
4N/- Alternative to step 3N, observer can determine on the semi-plane a half-line CB from the centre C of the watch dial, forming with the watch axis an angle equal to the angle between the direction to the Sun and the Celestial axis. The half-line CB starts from the center of the dial and is nearly in the direction of the bisector. It rises above the dial toward the glass and points through the glass of the watch during summer time and dives below the dial into the movement compartment of the watch and points through the movement of the watch during winter time. This half-line always points to the Sun if this watch displays the local time and the face of the watch and its axis point to the North Star. Instead of trying to have the half-plane containing the Sun, observer can try to have CB pointing to the Sun. This gives better accuracy.
5N/- At that position, the watch face and its AXIS are POINTING to the North Star. Tilt the watch further, until it lies horizontally. In this horizontal position, the mid-day marking is pointing South and the 6 o’clock marking is pointing North.

watchcompassJ

Figure: Summary of finding North by a watch. Red hand is the bisector of 0 hr direction and the hour hand; green hand is its reflection across the (6-12) axis. Axis C-BN for Northern hemisphere is parallel to red hand at equinox days and is (raised above)/(dipped below) the watch dial by 23 degrees at local summer/winter solstice. Axis C-BS for Southern hemisphere is parallel to green hand at equinox days and is (raised above)/(dipped below) the watch dial by 23 degrees at local summer/winter solstice. Green drawing marks are for Southern hemisphere and are the mirror reflection of red drawing marks.

Method for Southern latitudes.
The word “left-right flip of bisector” here is used to mean the the bisector the bisector of the angle between the midnight/midday marking and the hour after being flipped left-to-right, that is after being reflected across the line mid-day to 6 o’clock on the dial.

1/- Hold the watch so that its AXIS rises above the horizontal plane by an angle equal to the latitude of the region. That is its face points to somewhere in the sky and its back is angled downwards into the ground.
2S/- Determine the half-plane limited by the axis of the watch and containing the left-right flip of bisector . This half plane revolves anti-clockwise about the axis of the watch once every 24 hour and goes through the mid-day marking at noon.
3S/- Hold the watch in such composure and rotate your whole body around your vertical axis by your feet until the Sun lies in the above half-plane.
4S/- Alternative to step 3S, observer can determine on the semi-plane a half-line CB from the centre C of the watch dial, forming with the watch axis an angle equal to the angle between the direction to the Sun and the Celestial axis. The half-line CB starts from the center of the dial and is nearly in the direction of the bisector. It rises above the dial toward the glass and points through the glass of the watch during summer time and dives below the dial into the movement compartment of the watch and points through the movement of the watch during winter time. This half-line always points to the Sun if this watch displays the local time and the face of the watch and its axis point to the Southern Celestial pole. Instead of trying to have the half-plane containing the Sun, observer can try to have CB pointing to the Sun. This gives better accuracy.
5S/- At that position, the watch face and its AXIS are POINTING to the Southern Celestial pole. Tilt the watch further, until it lies horizontally. In this horizontal position, the mid-day marking is pointing North and the 6 o’clock marking is pointing South.

No ambiguity in equatorial latitudes.
The watch is placed almost vertically in equatorial latitudes by both methods. Methods for both Northern and Southern latitudes gives exactly the same outcomes.

Extension application for both hemispheres.

6/ This method applies equally well to the Moon when its declination as well as lateness relative to the Sun is known. If the Moon can be seen in day light, a navigator should continue from the so determined direction of the Celestial axis to take the declination of the Moon as well as its lateness (and its angular distance, which can be accurately measured using the divider) relative to the Sun for that day. He can then continue his accurate determination of Celestial axis during the Moon lit part of that night by replacing the unseen Sun by the Moon together with its value of declination and its lateness supplied by himself. (Remember that the Moon increases its lateness relative to the Sun by a further 50 minutes in every 24 hours).

Figure: Summary of finding North by a watch.

Actual field test.
The author has tested these methods and found them to be applicable, easy and accurate to much better than 30 degrees for latitudes from 0 to 40 degrees. The accuracy is better than 10 degrees when the Sun has low altitude.

Explanation notes.
N1/- The word “watch” here applies to any watch or clock.
N2/- When a watch or a clock dial is hung on a vertical wall, its midnight marking is at the highest position. If the hour hand of a watch completes one revolution in 24 hours the watch is called a 24-hour watch; if it completes in 12 hour the watch is called a 12-hour watch. Most watches and domestic clocks are 12-hour ones. The bisector of the midnight marking and the hour hand of any 12-hour watch complete one revolution in 24 hour. It moves like an imaginary 24-hour hand on that watch.
N3/-The axis of the watch is the oriented line (Note that it is more than “the oriented half-line”.) going through the center of the watch at right angle to its dial disc and is parallel to the rotation axes of both the minute pointer (or “minute hand”) and the hour pointer (or “hour hand”). The direction chosen on the line is from the back to the front face of the watch.
N4/- A watch display local time when it shows 12 o’clock when the Sun reaches its highest point in the sky.
N5/- The angle between the North Star and the Sun varies like a sine wave with amplitude of 23.5 degrees; it should be 90 degree during Spring and Autumn equinoxes and 90-23.5 degree at Northern Summer solstice (21st June) and 90+23.5 degree at Northern Winter solstice (21st of December).
N6/- To tilt the watch accurately as required by step1, we can carry out the following steps:
1a/- Note that hour markings on 12hr watch dials are separated by 30 degrees. Other angles can be similarly worked out.
1b/- Hold the watch verticaly with 0hr at highest position.
1c/- Rotate the watch (either left or right, it does not matter) by angle lamda, keeping its dial plane unchanged. The line 0hr-6hr now makes an angle lamda with the vertical line.
1d/- Keep the axis 0hr-6hr fix in space, rotate the watch around it until the dial is pointing upwards evenly. The watch dial is now tilted upward by the angle lamda.

Relevant to this topic is also a method of finding North and time using neither watch nor compass [1].

Reference

[1]. tonytran2015, Finding North direction and time using the Sun and a divider, https://survivaltricks.wordpress.com/2015/05/06/finding-directions-and-time-using-the-sun-and-a-dividing-compass/
posted on May 06th, 2015.

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Finding accurate directions by a watch .

Method for finding accurate directions by a watch in any latitude.

by tonytran2015 (Melbourne, Australia).

Click here for a full, up to date ORIGINAL ARTICLE and to help fighting the stealing of readers’ traffic.

#find North, #finding North, #compass, #direction, #bisector, #using watch, #with watch, #tilted watch, #inclined watch, #navigation, #without compass

This method uses a watch with analogue face for finding directions. Unlike the traditional method of using the hour hand of a flat lying 24-hour watch, my method uses a 24-hour watch tilted from the vertical and gives better accuracy for both North and South hemispheres including tropical zones. When applied to the arctic and antarctic regions, the watch is tilted by more than 67 degrees and lies almost flat on the ground; it becomes the traditional method using flat lying 24-hour watch.

The method assumes an analogue 24-hour watch is in use. For any analogue 12-hour watch, the bisector between its midnight marking and its 12-hour hand can serve as an imaginary 24-hour hand. From the latitude of the place, the position of the Sun in the sky and the local time shown on the watch, the method gives out the Cardinal directions and declination of the Sun (therefore an estimation of the date and month in the year.

The method for Northern latitudes is described first and is followed by the method for Southern latitudes.

Method for Northern latitudes:

WatchCompass_22NL

A 24-hour watch shown only with hour hand

1/- Hold the watch so that its AXIS rises above the horizontal plane by an angle equal to the latitude of the region. That is its face points to somewhere in the sky and its back is angled downwards into the ground.

2N/- Determine the half-plane limited by the axis of the watch and the backward pointing direction of the 24-hr pointer (the hour hand of the 24hr watch). This half-plane will contain the Sun if this watch displays the local time and the face of the watch and its axis points to the North Star.

3N/-Determine ON THIS SEMI-PLANE a half-line CB from the centre C of its dial, forming with the watch axis an angle equal to the angle between the direction to the Sun and the Northern Star. The half-line CB starts from the centre of the dial and is nearly in the opposite direction of the 24-hour hand (pointer). It rises above the dial toward the glass and points through the glass of the watch during summer time and dives below the dial into the movement compartment of the watch and points through the movement of the watch during winter time. This half-line always points to the Sun if this 24-hr watch displays the local time and the face of the watch and its axis point to the North Star.

4N/- Hold the clock in such composure and rotate your whole body around your vertical axis by your feet until the above half-line CB points towards the Sun (Therefore the Sun lies in the half-plane limited by the watch axis and the backward pointing direction of the 24-hour pointer). At that position, the watch face and its AXIS are POINTING to the North Star.

5/- The projection of the Celestial axis onto the horizontal ground is then the terrestrial Northern-South direction.

The method for determining the North-South direction in the Southern hemisphere is different but is very similar to this method for the North. Paragraphs 2N, 3N and 4N are appropriately replaced by 2S, 3S and 4S for Southern latitudes as in the following.

Method for Southern latitudes:

2S/- The UP-DOWN REFLECTION OF THE HOUR HAND of a 24-hour watch is its imaginary hour hand going anti-clockwise, pointing downwards at midnight and upwards at midday. It is the reflection of the hour hand of a vertically hung 24-hour watch through any water surface below it.

Determine the half-plane limited by the axis of the watch and the up-down reflection of the hour hand. This half-plane will contain the Sun if this 24-hr watch displays the local time and the face of the watch and its axis point to the Southern Celestial pole, while the back of the watch points through the ground to the North Star.

3S/-Determine ON THIS SEMI-PLANE a half-line CB from the centre C of its dial, forming with the watch axis an angle equal to the angle between the direction to the Sun and the Southern Celestial pole. The half-line CB starts from the centre of the dial and is nearly in the direction of the up-down reflection of the 24-hour hand. It rises above the dial toward the glass and points through the glass of the watch during Southern Hemisphere’s summer and dives below the dial into the movement compartment of the watch and points through the movement of the watch during the Southern Hemisphere’s winter. This half-line always points to the Sun if this 24-hr watch displays the local time and the face of the watch and its axis point to the Southern Celestial pole.

4S/- Hold the clock in such composure and rotate your whole body around your vertical axis by your feet until the above half-line CB points towards the Sun (Therefore the Sun lies in the half-plane limited by the watch axis and the up-down reflection of the 24-hour pointer). At that position, the watch face and its AXIS are POINTING to the Southern Celestial pole while the back of the watch points through the ground to the North Star.

No ambiguity in equatorial latitudes.

The watch is placed almost vertically in equatorial latitudes by both methods. Methods for both Northern and Southern latitudes give exactly the same outcomes.

Adaptation for use with any common 12 hr watch.

The method is easily modified for application to any common 12 hr watch. In the following figure, the red hand (the bisector of the 0hr direction and the hour hand of a common 12hr watch ) is in the opposite direction of the hour hand of a 24hr watch.

WatchCompassG

Figure: Summary of finding North by a watch. Red hand is the bisector of 0 hr direction and the hour hand; green hand is its reflection across the (6-12) axis. Axis C-BN for Northern hemisphere is parallel to red hand at equinox days and is (raised above)/(dipped below) the watch dial by 23 degrees at local summer/winter solstice. Axis C-BS for Southern hemisphere is parallel to green hand at equinox days and is (raised above)/(dipped below) the watch dial by 23 degrees at local summer/winter solstice. Green drawing marks are for Southern hemisphere and are the mirror reflection of red drawing marks.

Figure: Summary of finding North by a watch.

Actual field test

The author has tested these methods and found them to be applicable, easy and accurate to within 30 degrees for latitudes from 0 to 40 degrees.

Explanation notes:

N1/- The word “watch” here applies to any watch or clock.

N2/- When a watch or a clock dial is hung on a vertical wall, its midnight marking is at the highest position. If the hour hand of a watch completes one revolution in 24 hours the watch is called a 24-hour watch; if it completes in 12 hour the watch is called a 12-hour watch. Most watches and domestic clocks are 12-hour ones. The bisector of the midnight marking and the hour hand of any 12-hour watch complete one revolution in 24 hour. It moves like an imaginary 24-hour hand on that watch.

N3/-The axis of the watch is the oriented line (Note that it is more than “the oriented half-line”.) going through the centre of the watch at right angle to its dial disc and is parallel to the rotation axes of both the minute pointer (or “minute hand”) and the hour pointer (or “hour hand”). The direction chosen on the line is from the back to the front face of the watch.

N4/- A watch display local time when it shows 12 o’clock when the Sun is highest in the sky.

N5/- The angle between the North Star and the Sun varies like a sine wave with amplitude of 23.5 degrees; it should be 90 degree during Spring and Autumn equinoxes and 90-23.5 degree at Northern Summer solstice (21st June) and 90+23.5 degree at Northern Winter solstice (21st of December).

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