Twilights , STARS, PZX TRIANGLE, CIRCUMPOLAR BODY

Twilights 

 Twilight is the period of semi-darkness before sunrise and after sunset. Twilight is important to navigators because it is a suitable period to make observations with a sextant as the horizon, stars and planets are visible at that time. During daytime, stars and planets are not visible. During night time, the horizon is not visible. Twilight is of three types – civil, nautical, and astronomical.

In the evening, all the three twilights commence at sunset. Civil twilight ceases when the sun is 6° below the horizon, nautical twilight ceases when the sun is 12° below the horizon, and astronomical twilight ceases when the sun is 18° below the horizon.

In the morning, astronomical twilight commences when the sun is 18° below the horizon, nautical twilight commences when the sun is 12° below the horizon,andcivil twilight commences when the sun is 6° below the horizon. All three twilights cease at sunrise. 

During civil twilight (when the sun is between 0° and 6° below the horizon), the horizon in the direction of the sun is very bright. Stars and planets cannot be seen during this time as the sky is too bright.

Between civil and nautical twilight (when the sun is between 6° and 12° below the horizon), stars and planets and the horizon are visible for observation by sextant.

Between nautical twilight and astronomical twilight (when the sun is between 12° and 18° below the horizon), the horizon is too dim for observations by sextant. The direction of the sun still has a lighter shade of darkness. 

When the sun is below 18° below the horizon, there is equal darkness in all directions.

Magnitude of Stars

Stellar magnitude

Greek astronomers of ancient times estimated that about 6500 stars are visible to man’s naked eye (NOT naked man’s eye!), meaning eyes unaided by artificial visual means such as telescopes, etc. They devised a system of magnitude numbers to denote the brightness of the stars seen in the sky. This system, fine-tuned, is in use by navigators today.

The faintest stars that can be seen by naked eye are classed as magnitude 6. Stars of magnitude 1 are 100 times brighter than stars of magnitude 6. This means that each magnitude number is approximately 2.5 times brighter than the next. So a star of magnitude 1 is 2.5 times brighter than a star of magnitude 2, and so on. The star Vega has magnitude of about 0.1& the Polestar, about 1.97.

Consequently, some celestial bodies are so bright that negative values of magnitudes are needed to fall in line with the system. Hence the brightest star is Sirius, which has magnitude of about -1.6. 

The brightest planet is Venus with magnitude of about -4. Just for reference, on the same scale, the sun has magnitude of about -26 and the moon, about -13.  

A simple formula enables calculation of the relative brightness of a star:

n = 2.5^m1~m2

Where m1 & m2 are the magnitudes of the two chosen celestial bodies & n is the number of times one of them appears brighter than the other.

Example

Star Caph has apparent magnitude of 2.4&star Sirius, -1.6. Which star is brighter and by how many times? 

Sirius is brighter than Caph because it has a lower apparent magnitude. 

Note:

m1~m2 = 2.4 ~ (-1.6) = 4. 

        n = 2.5^m1~m2 =  = 39

 

Sirius appears 39 times brighter than Caph. 

First magnitude stars: are stars brighter than magnitude 1.5. There are 23 of them.

Second-magnitude stars: are stars of magnitudes from 1.5 to 2.5.

The Nautical Almanac lists 57 stars (excluding the Pole star) of 1st & 2nd magnitude, that are suitable for navigational observations.

Star identification

Constellations

A constellation is a group of stars which appear to form a pattern in the sky, fired by the imagination of astronomers and officially named after mythological characters, creatures or objects. Each constellation is given a name, usually Greek, like UrsaMajor (Big Bear), Cassiopeia (a queen in Greek mythology), Orionis (Hunter), Cygnus (Swan), Andromeda (Greek princess, daughter of Andromeda), etc.

Asterisms

An asterism is a clearly visible star-pattern that is not officially classed as a constellation. It may consist of only some conspicuous stars of the same constellation, or a group of stars of different constellations, seeming to form a pattern. One famous example of an asterism is the ‘Big Dipper’ (see coloured diagrams) which is a part the constellation called Ursa Major or Great Bear.

 

Note: A dipper is a utensil like a frying pan. Looking at the sky during watchkeeping in open sea, you may like to form your own asterisms for star identification. Star gazing can be a very interesting hobby.

 

Distinct identities of stars

Each star in a constellation is given a Greek alphabet identity –  (alpha),  (beta),  (gamma),  (delta),  (epsilon), etc. The star is the brightest star of that constellation. Where the star is sufficiently bright for stellar observation by a sextant, it is given a number from 1 to 57 in descending order of its SHA, and a regular name like Dubhe ( Ursa Major), Vega ( Lyrae), Sirius ( CanisMajoris), Canopus ( Carinae), etc. 

Note: Polaris is not one of the 57 stars because its declination is over 89° and hence does not have its SHA tabulated in the Nautical Almanac.

To find the full identity of a star

Where the name of a star is known, its details may be found as illustrated in the worked example below: 

Worked example: 

Find the full identity of Star Bellatrix.

Step 1: Obtain SHA & dec of the star.

Open to any daily page of the NA & obtain the SHA & dec of the star in full degrees, 278° & 6°N in this case. This is easy as the stars listed there are in alphabetical order.

Step 2: Locate English name of star

Open to the star table which is located soon after the daily pages of the Nautical Almanac. Look at the right side page against the SHA, 278° in this case, and note that the star Bellatrix has number 13 allotted to it.

Step 3: Obtain the star’s details

Look at the same line on the left hand page and note that the identity of star Bellatrix is  Orionis. This means that it is star  of the constellation of Orionis. You also note that its stellar magnitude is 1.6. 

Note: Stellar magnitude is an indication of the brightness of a star. The lower the number, the brightest it is. The brightest star in the sky is Sirius which has a stellar magnitude of minus 1.6. The higher the number, the less bright it is.

Step 4: Answer.

Star Bellatrix

SHA about 278°, dec about 6°N (Note: Full degrees without rounding off are sufficient).

Allotted number 13

Stellar magnitude is 1.6.

 star of the constellation of Orionis.

Step 5: Identify the star in its constellation

Open to the star charts in the NA which are just after the daily pages.

Look at the left page “Northern Stars’ because the dec of Bellatrix is N.

Look at SHA 278° & dec 6°N. You cannot find Bellatrix as this chart shows only dec 10°N & above.

Look at the bottom chart spread across both pages. These are stars are near the celestial equator. 

Look at SHA 278° & dec 6°N & locate Bellatrix.

To identify the main stars nearby, hold the star chart above your head against the sky and you can see the stars correctly oriented.

Use of star finder:

One type of star finder consists of a white non-transparent plastic base plate with stars and degree markings on it. One side is marked as N for use in the northern hemisphere and the other side marked S for the southern hemisphere. It has a central hole. (See adjoining pictures)

Seven plastic transparent discs are provided, each marked with latitudes at ten degree intervals from 5° to 65°. Each disc has N on one side and S on the other. The adjoining picture shows the sheet for 35° with the north side up.

 

Example on how to use it:

Given DR latitude: 33°N, LHA Aries at given time: 240°

Hold the base plate N side up.

Insert the centre pin provided - a match-stick would do.

The transparent disc with the latitude nearest the given latitude of 33° is 35°.

Place the disc marked 35° on the base plate north side up so that the centre pin passes through its hole. 

Rotate the transparent disc until the arrow coincides with the required LHA (240° in this case) on the base plate.

The approximate altitude and azimuth of all stars visible at that time can be read off.

Star selection

Stars suitable for observation in the twilight period

Example:

On the evening of 1st December 1992, in DR  27'N   E, which stars of the first magnitude and planets, will be available for observation within  of hour angle from the observer’s meridian?

LMT civil twilight = 16h 53m 00s

LIT (E) (-)          = 06h 15m 20s

GMT                  = 10h 37m 40s

GHA  for 10h =  

Incr. for 37m 40s (+) =  

GHA =  

Long. (E) (+)         =   

LHA =

Using a star identifier: 

Keep the N side of the base plate upwards and place the transparent sheet of 45 N on it. Rotate the transparent sheet until its arrow points to the required LHA . Read off the suitable stars above about 20º altitude and their approximate azimuths.

Check with the star pages of the almanac (shown nest page) and chose those whose magnitudes are 1.0 and less.

Celestial bodies on observer’s meridian will have LHA * = 

SHA * = LHA * - LHA  =  -  =  

SHA * within limits of  on either side of observer’s meridian will be ---

 12.7' ±  i.e. from   to  

Following celestial bodies fall within the above limits ---

Star Altair (Mag. 0.9 and SHA  )

Star Vega (Mag. 0.1 and SHA  )

Planet Venus (Mag. – 4.1 and SHA  )

Planet Saturn (Mag. 0.7 and SHA  )

PZX triangle

The PZX triangle, also known as the Rational Horizon diagram is a projection of the earth's system of position fixing on to the celestial sphere.

Projection of PZX Celestial Spherical Triangle on the Plane of Rational Horizon

This projection is essential to understand, explain and solve all problems dealing with the celestial spherical triangle PZX. In the triangle:

• P is the North or South Pole, same as the hemisphere in which the observer is located.
• Z is the Zenith of the observer.
• X is the position of the body.

Construct the following figure on paper using the scale of 1 cm. 

• Draw a circle of radius 4.5 cm. representing the Rational horizon.
• Mark the center point of the circle as Zenith (Z).
• Through Z draw two vertical and horizontal lines such that the angle between them is exactly 90^o
• Vertical line represents the Principal Vertical circle or Observer’s meridian. Mark the end points of this line as North (N) and South (S) points of the rational horizon.
• Horizontal line represents the Prime Vertical circle. Mark the end points of this line as East (E) and West (W) points of the rational horizon.
• According to the scale ZN = ZS = ZE = ZW = 90^o.
• Any diameter of this circle represents a Vertical circle.
• Convert the Latitude of the observer to cm according to the given scale.
• If Latitude is North, mark a point Q on the Principal vertical circle below Z, such that ZQ represents the Latitude in cm. If Latitude is South, mark the point Q above Z.
• If the Latitude is North, mark a point P, representing the Pole, on the Principal vertical circle below N, such that NP also represents the Latitude in cm. If Latitude is South, mark the point P above S.
• Hence the distance PQ = 90^o and PZ co-Lat.
• Draw an arc passing through E, Q and W. This arc is drawn by placing the point of the compass at some position on the Principal vertical circle, selected by trial and error. The arc EQW represents half of the Equinoctial although it is not a semi-circle.
• Covert the Declination of the body to cm. according to the given scale.
• If Declination is North, mark a point Y on the Principal vertical circle above Q, regardless of whether the Latitude as marked is North or South, such that the distance QY represents the Declination in cm. If Declination is South mark the point Y below Q. 
• Finally the point Y may lie above or below Z depending on the values of Latitude and Declination.
• Through Y draw a free-hand arc concentric to the Equinoctial but not parallel to it, to meet the Rational horizon at points D and D' The arc DYD' represents part of the Declination circle of the body. D and D' represent the positions of the body at theoretical rising and setting.
• If the Azimuth of the body is known, then through Z draw a straight line (vertical circle) such that the angle between it and the Principal vertical circle is the Azimuth measured from the N- end in Easterly (clockwise) direction in 3-figure notation. The intersection between this vertical circle and the Declination circle represents the position of the body (X) at the time of observation.
• If the Azimuth is not known but the True altitude of the body is known, then calculate the True zenith distance (TZD) = 90^o - T. Alt., and convert it to cm. according to the given scale. 
• With Z as center and radius of TZD, draw an arc to cut the Declination circle at X, which represents the position of the body. Join ZX by a straight line, which represents the vertical circle of the body.
• Draw a free-hand arc to join PX, which should not be a straight line. This represents the celestial meridian of the body. PX = Polar distance.
• The PZX triangle drawn on paper will be a true representation of the celestial spherical triangle PZX, depending on how accurate is the construction.

Correction of Altitudes

Observed Altitude, Dip, Apparent Altitude

The angle obtained after correcting for the index error is known as the observed altitude.

Dip is caused by the observer's height of eye above the sea and is always subtracted to bring the sight to sea level.

The altitude obtained after applying dip is the apparent altitude.

Dip

Dip is the angle at the observer's eye contained between the visible horizon and the sensible horizon. Value of dip increases with the height of eye of the observer above the sea level. It is given in the Nautical Almanac.

 

Observed altitude

Observed altitude is the angle at the observer's eye contained between the visible horizon and the celestial body corrected for the instrument error.

• Obs. Alt. = Sextant Alt.  Index error
• Sign is –ve if I.E. is on the arc ; Sign is +ve if I.E. is off the arc.

Apparent altitude

Apparent altitude is the angle at the observer's eye contained between the sensible horizon and the celestial body.

App. Alt. = Obs. alt. – Dip

True altitude

• It is the arc of the vertical circle of a body or the angle at the centre of the celestial sphere contained between the rational horizon and the centre of the body.
• T. Alt. = A. Alt. – Refraction  Semi-diameter (Sun or Moon) + Parallax (Mars, Venus, Sun or Moon)
• Sign for Semi-diameter is–ve if upper limb of Sun or Moon is used; Sign is +ve if lower limb is used.

Circumpolar Bodies

The passage of a celestial body across the observer’s lower (or inferior) meridian is called lower meridian passage, as shown in diagram A.

All celestial bodies (abbreviated here to CB) cross the observer’s upper and lower meridians in their apparent daily orbit around the observer. The point of interest is when the CB is visible at lower meridian passage.

This happens under certain conditions (of latitude and declination), explained later herein. 

A celestial body may remain above the rational horizon - does not set - and is then visible at all times, even during lower meridian passage. Under such conditions, the body is said to be circumpolar.

In the case of stars and planets, when circumpolar, it is possible that during one of these meridian passages, they may not be practically visible due to the presence of strong sunlight.

In diagram A: 

P is the geographic pole. 

The circle is the equator.

X is the celestial body on the observer’s inferior meridian.

In diagram B:

The observer in north latitude.X_R, X & X_S are the positions of the CB when rising, culminating and setting.

In diagram C:

The latitude is the same as before but declination here has been increased. Note that the CB’s time above the Rational Horizon has increased. In a 24-hour day, the CB would theoretically be visible about 18 hours or so in this diagram, and be invisible only for about 6 hours.

In diagram D:

The latitude is the same as before but declination here has been increased even more. Note that the CB does not go below the Rational Horizon. The CB does not set. Y is the CB at lower meridian passage (mer pass). The CB is now circumpolar. This is the critical stage when a CB becomes circumpolar. At lower mer pass, the CB is just on the Rational Horizon. Further increase of declination would keep the CB above the Rational Horizon.

In diagram E:

The latitude is the same as before but declination here has been increased even more. Note: The CB is well above the Rational Horizon at lower mer pass as indicated by Y.

Time interval between upper and lower meridian passages

The time interval between the upper and lower meridian passages of the Sun is 12 hours and that of a star, is 11h 58m 02s (half a sidereal day). During upper meridian passage, LHA = 0° or 360° and during lower meridian passage, LHA = 180°.

The conditions for a CB to be circumpolar

In diagram D:

NP = ZQ = lat.

PX = PY = Polar Distance of the CB.

Therefore, for a CB to be circumpolar:

(1) Lat and dec must be of the same name (i.e., both N or both S).

(2) NP should be  PY

or lat  polar dist

or lat  90 - dec

or lat + dec  90°

Diagrams F to I cover all possible combinations of circumpolar bodies.

Diagram J below gives a three-dimensional black and white side-elevation of diagram H.

Further explanation for calculations – See diagrams F & H (both NH)

At upper mer pass	At lower mer pass
ZX - zenith distance	ZY - zenith distance
PX - polar distance	PY - polar distance
NX - true altitude	NY - true altitude

(1) NX - NY = XY = 2PD   or   

     Alt above pole – alt below pole = 2 Polar distance

(2) NP = NY + PY   or 

     Lat = Lower mer alt + PD

Further explanation for calculations – See diagrams G &I (both SH)

At upper mer pass	At lower mer pass
ZX - zenith distance	ZY - zenith distance
PX - polar distance	PY - polar distance
SX - true altitude	SY - true altitude

(1) SX - SY = XY = 2PD   or 

      Alt above pole – alt below pole = 2 Polar distance

(2) SP = SY + PY   or 

      Lat = Lower mer alt + PD

If the CB considered is the sun

• It will be theoretical sunrise or sunset, as the case may be, when the sun’s centre is on the rational horizon.
• So if lat + dec  90  + 6º, between sunset and sunrise there will be civil twilight but no night.
• If lat + dec  90  + 12º, between sunset and sunrise there will be nautical twilight but no night.
• If lat + dec  90  + 18º, between sunset and sunrise there will be astronomical twilight but no full darkness.

The following media shows the calculation of observer's latitude by using the upper and lower transit meridian altitudes: