Introduction: ​Finding North and Time With Unclear Sky

Picture of ​Finding North and Time With Unclear Sky

by tonytran2015 (Melbourne, Australia)

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.

Step 1: ​Locating the Celestial Poles.

Picture of ​Locating the Celestial Poles.

Figure: Finding a Celstial pole using two chosen 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 stradling 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.

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.

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.

Step 2: In the Northern Hemisphere, Over 40 Degrees North (outside Tropical Zone):

Picture of In the Northern Hemisphere, Over 40 Degrees North (outside Tropical Zone):

Figure 1: Polar (inversion) map of 3/4 of the sky centered on the Northern Celestial pole. Figure 2: Table of 20 brightest stars in reverse order of brightness.

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.

Step 3: ​In the Southern Hemisphere, Over 40 Degrees South (outside Tropical Zone):

Picture of ​In the Southern Hemisphere,  Over 40 Degrees South (outside Tropical Zone):

Figure 1: Polar (inversion) map of 3/4 of the sky centered on the Southern Celestial pole. 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.

Step 4: ​Between 40 Degrees North and 40 Degrees South

Picture of ​Between 40 Degrees North and 40 Degrees South

Figure 1: Example of Northern skymap (Celestial pole above horizon) for 30 degrees North latitude in August. Figure 2: Example of Southern skymap (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) --OrionRigel is used to locate the North Celestial pole. The angle (Deneb, Capella, OrionRigel) 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-OrionRigel 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 head, surrounded by 4 neighbours outlining the extremities of the broom. Canopa is the sevond brightest star and is at the end of the broom handle. The angled line (Orion-Rigel, 18degrees distance, Betelgueuse, 25degrees distance, Procyon) outlines the extremity of the broom head. The vertices are 3 bright stars on a 100 degree counter-clockwise circular arc of 25degree radius, centered on the brightest star Sirius. The bisector of the 60degrees anti-clockwise angle (Betelgueuse, Sirius, Procyon) points to the North Celescial pole while Sirius-Canopa points in its opposite direction, at the Southern Celestial pole.

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.

Step 5: In Tropical Zone.

Picture of In Tropical Zone.

Figures: Mercator and polar (inversion) maps 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 advatageously 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. declinination) 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.

Step 6: Finding Time From Skymaps.

Picture of Finding Time From Skymaps.

Figure 1: The rule applied to the chosen star is similar to the rule for the Sun illustrated here. Figures 2: Summary of steps to find North direction and time using a 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.

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.

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.

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 70% of the times. When tested in Saigon, with restricted sky view due to multi-storey buildings, the traditional clear sky method is applicable in less than 10% of the times while this bright star method is applicable in about 80% of the times.

Step 7: Preparation for Worsening Visibility.

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 firstly notice any distant land mark such as mountains, sea shores, transmission towers, direction of power transmission lines, railway lines, etc... and use them as as temporary directional clues while Celestial navigation is disabled. Secondly, he should also bring out his magnetic compass to check its magnetic declination before relying on it when the last star disappears and visibility is further reduced. Even a button sized compass, provided it is well made, can be quite helpful when Celestial navigation is disabled.

References

[1]. tonytran2015, Finding North direction and time by stars, Instructables,

https://www.instructables.com/id/Find-North-directi..., posted on Jul 24, 2015

[2].tonytran2015, Finding North direction and time from the Sun using bare hands, instructables,

https://www.instructables.com/id/Find-North-directi..., posted 2015 may29.

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