quite unsatisfied by my last program on stars point, i have then done an another one with more accuracy. it includes a display menu with several stars, classified in constellation files. to write it i have used an astrophysics master file findable on the net (http://userpages.irap.omp.eu/jleborgne/astrometrie.pdf), but then i have considered all the corrections (precession, nutation, Bradley's aberration, parallax, refraction, skyline depression). to add a display menu i have used data on a webpage : https://openclassrooms.com/courses/creation-de-barres-de-menus-sur-ti89-92-et-v200.
If you want to add stars or constellations into the menu it's easy you just have to follow this useful tuto. to check if it works i have found a pdf which is called « exercices astro 2016-navigation astronomique ». it works for the exercice but it will be needful to test it in real one or two times in good conditions.
Here you have it, i guess and hope i have not let to much mistakes and bullshits when i have copied it (i have read it several times so it could be ok !) :
If it's not working well, don't hesitate to let me a little message!
PS:a website visitor has noticed two mistakes in the program above:
– local hour angle (variable ahl) : mod(tslo+xxx-lone,360)
– tslo (variable tslo) : gg / 36525 must be replaced by ((gg-h/24-m/1440-s/86400)-2451545)/36525
Relative to the stars, it's still more interesting to calculate position with the sun; during the day the most of time the skyline is clear and easier findable, making sextant's use easier and more accurate, with a wider range of practices's time . As the program does with the stars, we will first search to “transform” all data of our time measurement (hours, minutes, seconds, month, years) in one lonely number. this last one will help us to unroll calculations reminder which is no more than functions woking with time. For this beginning we could have used the same as the stars program, but just for fun, This time we will take account of the Gregorian calendar, and get rid of julian calendar!
1) First part: how to calculate time by using gregorian calendar
Recall: in julian calendar the average year lasts 365.25 days, so a leap year comes each 4 years (2008, 2012, 2016, 2020…). on the calendar it means that we add up a day at the end of february each 4 years (29 days instead of 28).
Summary, on the way to have a calendar year period close to reality, in fact a tropic year period (nearly 365,2422 days), the calculations method has been improved to reduce slowly in time the julian calendar size. With the gregorian calendar an average year lasts 365,2425 days. to reach that point, we keep the julian's calendar features , but each secular year we consider it as a bissextile year only if it could be divided by 400. for our programmation we will then consider a year as bissextile if it is 4 years after an another bissextile's year. Then for secular's years which could be divided by 100 we select and we will accept them as bissextile if they are dividable too by 400. for exemple according to gregorian's calendar 2000 and 2400 are bissextile, but 2500 and 2600 are not bissextile.
here you will find the beginning:
we use the loop “while” and "int" function(). Loop “while” asks to the program to calculate something until it reaches one or several conditions, and function “int()” ne garde que la partie entière d’un nombre, for exemple int(2.7)=2. The number at the end of the page 2457388.5 is the julian's day 01/01/2016 at 00:00.
2) second part: how to calculate with time declination and hour angle 's sun.
As we have seen in “programming a TI89 for navigation”, there are many ways to calculate those parameters, more or less accurate on a short or long period, with more or less difficulties…to have accuracy in results on a very long period, it needs a lot of parameters to consider, and in addition planets have a chaotic behavior on wide periods of time…we will then go ahead all this and keep just all we need for hour needs, a program usefull and relevant for a life period and enough accurate to make a point in navigation and not too bothering to write!
a) hour angle:
if we consider things on the easiest side, we follow the fact that earth is spinning in 24 hours, giving illusion of sun's motion around earth moving at 360°/24H=15°per hour, with sun's passage at greenwich's meridian at 12H UTC. By keeping this writting AHG=15*(l*24) would be enough to calculate greenwich hour angle(AHG), with a greenwich's meridian noon as origin data for time. to find our local hour angle (AHL), only substract our dead longitude from AHG would be enough(G).
however it's inadequate, with this simple method at the end we find high gaps more and more 3 wide regards to the truth! in fact apparent sun's motion regards to earth has not a steadily speed because of the elliptic shape of the earth's way (remember the first and second Kepler's laws), and because of the rotation's earth axis inclined too. Fortunately, Gaps reiterates almost the same years by years, allowing quite easily to calculate good results each time. For this we will work with the time's equation. as our friend wikibobby says this equation matches for a precised time difference between average solar time and true solar time. the average solar time is calculation written in the paragraph just above (sun is moving with a constant velocity) and the true solar time will be calculated with those equations given by wikibobby (take care of earth elliptic's way and earth obliquity). Now in adequation with our program's beginning it gives us this:
The mod function() here is used to get a result always being between 0 and 360 degrees (This function provides the euclidian division's remainder depending on dividend and divider choosen), otherwise we would end up with huge angles without meaning. We could also use the loop “for” but for that the calculator would take more time to calculate. There is 180 after mod() because the program begins at midnight 1/1/2016 (see prog's first part), so this time the average hour angle of the sun with greenwich is 180 degrees (if the prog would have begun at noon, we wouldn't have added up 180 but nothing instead).
For the equations of time, coefficients used are easily noteworthy:
357.5291= average anomaly 01/01/2000 to 12h.
0.98560028= it's the angular's velocity of the average's rotation of earth around the sun between the passage of 2 perihelias, so it's 360 divided by the anomalistic year.
2451545= it's the julian's day of 01/01/2000 to 12h.
1.9148; 0.02; 0.0003it's only a coefficent 's change of the center's equation in degrees(we just have to multiply the first coefficients by 180/π). wiki explains how to get the center's equation.
280.4665= it's the ecliptic's longitude value the 01/01/2000 to 12h.
0.98564736= it's the average angular's velocity about earth around the sun on a tropic year, So we find this value by calculating 360/365.2421898.
-2.46569; 0.053; 0.0014= as the center's equation it's a switch from the first coefficients to coefficients in degrees.
b) declination angle
As we have seen in “programming a TI89 for navigation”, there is a lot of several ways to get sun declination; by using methods with less or more parameters, thus less or more accuracy and less or more difficulties to program it too… not easy. So we will remove "useless" data for our case. we will not take care about planets and moon leverage in our calculations, but Kepler's laws used on the earth/sun system only, it will be not bad at all to begin.
the easiest formula to have declination is the formula which have a simple periodic function's framework (sinus, cosinus) like Desmond Fletcher's equation:
D=23.45*sin(2π/365(284+j)) avec j jours juliens. Here 23.45 answers to the sinus function range, thus to the highest value reachable by declination (during solstices). 2π/365 matches to the sinus function's period, here it will repeat itself every 365 days. 284 is a function's shift. this formula doesn't have in charge the earth ellipticity's trajectory and consider its motion as a cercle's shape. the easiest method but really inaccurate.
for declination an another way to do would be to write a periodic function but by parts, each part matches for a season, thus a function with 4 parts (see “programming a TI89 for navigation”). it works well, easy to do because all we have to know is solstices and equinoxes dates for the following year, however its weakness is that it works just for one year. in fact from a year to an another equinoxes and solstices hours change by following an average value, but the truth is that there are gaps with ranges of few minutes, and at the end this differences cumulation brings…highest differences!
on internet it's easy to find formulas for declination given by IMCCE, efficient for a long or short period. we could also deduce it from the time's equation because we know that:
Why ? because , to prove it it's almost like method used for the rectangular and spheric triangle (allow us to find formulas for azimut and calculated height for a star, planet...), as i am lazy and it's very well explained on the following website page : https://sites.google.com/site/astronomievouteceleste/4—mouvement-du-soleil
from all this we could deduce declination's formula to write after our program's beginning:
So for declination, 3 programmation lines are enough when we know AHG. before going ahead we will check efficiency thanks to the IMCCE website, just to notice if it's accurate enough.
Test
by taking 10 several values we get that:
the 15/12/2019 at 2:22 p.m. : -23°16’13.91”/36°44’27.88”→-23°16'/17h31
the 2/11/2043 2h : -14°40’9.90”/214°6’50.43”→-14°40'/14h29
the 28/2/2055 at 4:47 p.m.: -7°47’20”/68°38’28.54”→-7°47'/22h46
the 12/5/2128 at 5h06: 18°28’20.7”/257°22’25.4”→18°13'/3h18
the 7/07/2184 at 21h12: 22°17’33.11”/136°34’51.76”→22°23'/7h12
the 18/10/2079 23h10: -9°56’58.8”/171°15’1.75”→-9°57/13h35
the 13/6/2046 at 12h11: 23°14’12.71”/2°42’14.74”→23°14'/5h28
the 6/09/2033 at 18h55: 6°6’8.13”/104°12’36.45”→6°6'/11h03
the 1/03/2085 at 4h22: -7°19’47.17”/242°27’31.56”→-7°20'/22h51
the 19/02/2074 at 3h18: -11°10’21.23”/226°4’8.9”→-11°10'/22h12
on the left it's our prog results for declination and greenwich hour angle, and on the right it's the IMCCE website's results “calcul des éphémérides” they give them in declination and right ascension. not to bad it seems, we begin to lose accuracy after 2080 on declination so we have time… we know that right ascension is the angle between vernal point's meridian and a planet's one, measured from east and written in hours, and it's the 360 degrees complement of verse ascension. but we have already found a formula able to calculate vernal point's meridian in our prog to get our position with stars :
100+5.5/60+(l-2451545-5843.5)*(360/365.2421898+360)-0.0524*16/(24*3600*36525)-0.0524*(l-2451545-5843.5)/(24*3600*36525)=vernal point's angle. thus by putting it after in our prog it gets that:
when we compare it with right ascensions given by IMCCE it's not too bad. we could say that until 2080 roughly it's accurate, and after it's decreasing in accuracy. we could consider that our ephemeris part is OK for navigation. we will be able to challenge the next step that's cool. for this part we could have been more meticulous by integrate some stable parameters used like variables like excentricity, duration of tropic and anomalistic year, and inclination angle's earth on ecliptic. however for our needs it will be enough i guess.
3) intercept and true azimut calculations
now we have declination and AHL calculated at watch time, the next level will be to define the calculated height (trigo formula like stars), true height (it's the instrumental height with corrections), and thus intercept and the true sun's azimut.
corrections to add up with instrumental height:
– Refraction of light (depends on the height above the horizon line).
– parallax (depends on the radius of earth, earth/sun distance and height above the horizon's line).
– sun's half diameter (roughly 16 minutes).
– eye's watcher height above the sea's level.
– collimation.
our program draft gives this:
this program allows to calculate meridians too, you just have to push 1 or 2 if you are back to the south or north when you are focusing the Sun.
Attenzione: test !
i have checked the device with navastro's webpages (“navastro.free.fr/meridienne.htm” to do a meridian and “navastro.free.fr/exos/exercices.htm” for straight heights). results are not totally the same but it remains "not bullshit" enough. A little disappointed, I dig it and notices an error in the program above; at the AHL calculation line (ahl) instead of having:
: mod(qq-rrr, 360)→ ahl in fact it's :mod(qq+rrr,360)→ahl
And for intercept instead of having:
: hv-hc→in in fact it's (hv-hc)*60→in
The final program that works is that:
By making these changes, this time we find almost the same results compared with navastro! well, lets Haddock's captain concludes with philosophy…
Stars point is working a little like a navigation point made with three landmarks, but in that case bearings are changed by position lines coming from stars. To have an accurate stars point it's needfull to choose stars the most rejected (the best way is with 120 degrees between each star), and do sextant's watch a little before or after the sun's falling to be able to still notice horizon line. It's important to avoid a choice on stars to close to the horizon too because of light refraction (see Snell's law). It's difficult when you are using sextant to”go down the star on horizon's line” (It's easy to confound a star with an another close to its), That's why it's easier to use sextant with head upside down (scale above), On few words “rise horyzon's line on the choosen star” choosen. The most of time it's only possible to use stars close to horyzon's point where the sun disapears, Beyond this it's hard to see clearly horyzon's line to read an accurate angle.
To write this little program i have often used an astrometry master (introduction aux coordonnées célestes, astrométrie, Jleborgne) from Toulouse university, easy to find on the web, and others things grabbed on the lane.
It's working with equatorial coordinates; we will locate stars with their positions against celestial equator which is finally just equator projection on celestial's sphere. In fact it's almost the same method than how to place something on earth; to locate a star we will have:
–Right ascension the most of time symbolysed by alpha letter, it's the angle read on celestial equator between vernal point and the perpendicular line with celestial equator cuting the watched object. the original point is vernal point and the angle is read from west to east, This angle is the most of time written in hours but is easily convertible in degrees (1 hour=15 degrees, 1 degree=60 minutes, 1 minute=60 secondes), in hours it's changing from 0 to 24 hours (or 0 to 360 degrees ).
– declination angle often symbolized by delta letter, angle read between the object seen and celestial equator. It can rise until 90 degrees (-90 at south pole 90 at north pole)and it can be at least 0 degrees (equator). declination angle and latitude have a lot of common points, It's positive on north hemisphere and negative on south hemisphere.
–Vernal point most of time written with gamma letter or just "g", is more difficult to understand. on celestial sphere ecliptic plan (plan with both earth and sun) and celestial equator cross their way in two points , one cross when the sun is going from north to south hemisphere, and the another one when sun is going from south to north hemisphere, so when it's spring equinox (between the 19 and the 21 March).we will see that vernal point is not motionless. equinoxes moves about 50” per year along ecliptic from east to west. vernal point's time angle (or sidereal time) is read from greenwich meridien and is written ahso or ahgy. vernal point is almost moving with constant speed, this speed is about 15.04166 degrees . we will see that to perform our little software we will need more accuracy.
For celestial coordinates J2000.0 coordinates are the origin, in fact measurements read the first of january 2000 at noon in gregorian calendar. If we write a star's name in wikipedia for exemple, the most of time numbers noticed on the right of the page are those read at J2000.0, To have the good coordinates depending of measurement's moment, it's essential to add corrections.
b) Julian calendar, Gregorian calendar
Important to understand for our program; gregorian calendar is a little like a better julian calendar's version, the origin is the same, monthes and weeks system too. The difference is about bissextial year's choices. For julian calendar, there is a bissextial year each 4 years (2008, 2012, 2016…) and for gregorian calendar, more accurate there is a bissextial year if you find other conditions more. In our case we will only use julian calendar. In calendar ther is 12 monthes filled up alternatively by 31 (january, March, mai, july, august, october, December) or 30 days (april, June, September, november). There is one exception for february with 28 days for one year “normal” and 29 days for a bissextial year, so we have 365 days for a classical year and 366 days for a bissextial year, in fact an average annual number of 365.25 days with the Julian calendar. However this duration's time doesn't fit perfectly with a real year's time duration…So we will have to cope with this mess to find the best way to make a good program ! a funny work ! or not...….
c) What does it mean “a year” ?
A year in the Julian calendar has 365.25 days. But there are several options to decide what a year is:
–sideral year : earth revolution relative to a fixed point. It takes about 365 days, 6 hours, 9 minutes and 10 seconds.
–tropical year : time between two successive passages of the sun in the direction of the vernal point. This period will be very important for our program, including the calculation of the vernal point's hour angle. It takes about 365 days, 5 hours, 48 minutes and 45.26 seconds at J2000.0. For our century it decreases about 5.23 secondes each 100 years. There is a formula useful to calculate the average tropical year duration easy to find on the web, It's :
Average tropical year =365.24219052-61.56*10^(-6)*T-0.068*10^(-6)*T^2 263*10*(-9)*T^3 3.2*10^(-9)*T^4. In 2000 it's equal to 365.2421898 days and in 2100 it's equal to 365.2421291582 days. T is time in Julian centuries since J2000.0.
–anomalistic year : time between two successive passages of the Earth at perihelion. It's about 365 days, 6 hours, 13 minutes and 53 seconds. Perihelion is the moment where earth is the closest to the sun (Kepler's laws), so it's when earth is the fastest in its motion around the sun.
d) Motions able to change equatorial coordinates
Unfortunately equatorial coordinates will never keep same data (it would be so easy), They will change depending on 2 main periodic motions called precession and nutation. Precession is because of tide effects on earth rotation. It's a slow motion with a period of 26000 years roughly but with a high range because it can increase until 23.5 degrees roughly (able to change a star's declination until 23.5 degrees ). Nutation (here because of tide too) has a short period of 18.6 years with a weak range compaired to precession : 9 seconds only. So for our programmation, and also to simplify our work it's possible to forget nutation, we lose some accuracy at the end but it will be enough…
We will use those formulas for our work :
–General precession=p=50.290966 0.0222226T” per year, T elapsed time in julian centuries (36525 days) since J2000.0.
-p=m*cos(E)+n*sin(E), E average ecliptic's inclination (23°26’21.448” at J2000.0). m is precession for right ascension, n is precession for declination.
-p=5029.0966 2.22226T-0.000042T^2” per century.
-m=4612.4362 2.79312T-0.000278T^2” per century.
-n=2004.3109-0.85330T-0.000217T^2” per century.
Precession causes on earth a kind of motion “spinned ball with effect” compared with sun. Precession causes a polar star change too; currently it's polaris which indicates north but it will not be always that case.
d) Other things which bring a lot of corrections
We will find atmospheric refraction of course ( Snell-Descartes laws), and Bradley's aberration too, parallax, watcher's height compared to the sea level, the half-diameter's mistake (of our watched object). half-diameter's star (see right height) is forgettable because a star is very little seen from our position on earth; we can forget parallax too because of the very high distance between stars and earth, so parallax is very weak here. we can forget Bradley's aberration too…Fortunately at least we will keep the other things to have some results !
Bradley's aberration : This phenomenon exists because of limited light's speed (roughly 300000 km/s in vacuum) and earth's rotation around the sun (roughly 30 km/s , depends on its elliptic course). It makes in 1 year a slight rotation of all the stars around their “average position” on the celestial sphere; In fact a star is not motionless but turns around a little ring or ellips. Difference between the average position and apparent position is 20.5 seconds… It's not a high difference that's why for our work we don't care about it.
This difference is explained with those draws just above : tan(alpha)=v/c ; but on earth v=30 km/s roughly and c=300000 km/s. So tan(alpha)=0.0001, so alpha=0.00573°=20.6265”.
Little stories :
.It's difficult to see this phenomenon in his mind, so “a little draw is sometimes better to grab every slightest detail of it”, an analogy of situation can be handy. There is Bradley's analogy himself due to his navigations on Tamise, watching a flag onboard : he has noticed that flag wasn't nor in wind direction, , nor in boat's course but in a vectorial addition of them 2. It's the same for a star's position depending on earth 's speed around the sun and light's speed.
. Parallax can be used like an efficient tool to know approximatively distances between earth and planets, stars. For the closest stars parallax is under 1 second…So we will store and forget that it's not really important for us.
Atmospheric refraction's correction it changes depending on atmospheric pressure and temperature ; on wikipedia we find a formula to use when we are on the sea level, with a steadily atmospheric pressure of 1010 mbar with a temperature of 10 °C. This formula, is the same as the formula's master (it's the same but simplified) is written like this :
R=(1.02/tan(Hv 10.3/(Hv 5.11)))/60 ; R is refraction angle in degrees, Hv is the true height in meters.
apparent depression line, due to the watcher's height compared to the sea level is easily findable on a lot of web pages, written like this :
D=(1.77*√(E))/60 , D horizon's decreasing in degrees and E watcher eyes rising in meters.
e) How can i measure a star's height ?
It's the same thing than the height line with the sun. we have to use the vernal's hour angle , with it we will be able to win the star's hour angle ahl :
ahl = ahgy – g + AV (g is estimated point's e longitude, AV is verse ascension).
verse ascension it's the angle between vernal point's angle and star's watched meridian measured like a hour angle, from east to west and on 360 degrees ; it's used on astronomical navigation to point a star.
Right ascension it's the angle beetween vernal point's meridian and star's meridian, measured from west to east and written in hours. it completes 360 degrees of verse ascension. for exemple verse ascension in january 2016 for Betelgeue is 270.985 degrees . so right ascension will be equal to 360-270.985=89.015 degrees. the most of time when we are seeking about a star's coordinates on the web it's given with declination and right ascension, at J2000.0.
Clike the right height with the sun (see right height), formulas to handle for our calculations are those ones :
-sin (hc) = sin (L) * sin (D) + cos (L) * cos (D) * cos (AHL) ; hc is star's height calculated, L is estimated point's e latitude, D is star's declination, and AHL is local hour's angle.
-cos (z^) = (sin (D) – sin (L) * sin (hc)) / (cos (L) * cos (hc)).
– the star's bearing calculated is equal to z^ if 180°<=AHL<360°, and 360-z^ if 0<=AHL<180°.
-Intercept is difference between star's true height seen with sextant, and star's calculated height. Intercept = hv-hc. Intercept is written in miles.
For people who have resist until now in spite of this mess of formulas and unbearable calculations , a little funny story before the next part :
2) Our program
This program works for Dubhé which belongs to the great bear, but it's also able to work for any other star. you have just to replace values t (right ascension of the star at 01/01/16 at 00:00 utc) and u ( star's declination the 01/01/16 at 00:00 utc). this program works from 01/01/2016 at 00h00 utc to the year 2100 which is not a bissextile year (but this program considers it like this, and that is wrong).
1) part 1, change the date and hour in one workable number (line 1/line 36).
Until the line finished by , g (line 36), program will change hour, minutes, seconds, months and years of the watch in a single number called g. to find bissextile years we use a cosinus fonction (a periodic sinus fonction would have been the same); this fonction is equal to 1 every 4 years. When this fonction here called i= 1, we can see that e (day's number depending on the month) changes. Part with “elseif” find the ellapsed day's number in the month since 01/01 of watch's year until the moment of the watch.
int(365.25*f) find year's number between measurement's moment and 01/01/2016. int() is a fonction able to keep the integer part of a number, for exemple int(17.8)=17.
line 35 carves all our parameters in the same unit (the day), They are added up to find g. 2457388.5 (=2451545 5843.5) is the Julian day the 01/01/16 at 00:00.
2) part 2, calculate vernal point's hour angle ( line 37 / line 45).
Line 37, 100+5.5/60 is vernal point's hour angle ahso the 01/01/16 at 00:00 utc.
Lines 37 and 38, with g, we are able to calculate ahso when it has been measured . We know that the vernal point runs in 24 hours more than 360 degrees ; to have its average speed with accuracy we have to divide the distance ran (360 degrees of revolution bitween 2 equinoxes in 1 year) with time made to vernal's point to do it. This duration is the tropical's average year equal to 365.2421898 days. So (360/(365.2421898)+360)/24=15.0410686… In fact the average vernal speed per hour (for 2000. Elle decreases slowly every year).
Currently the average tropical's year decreases about 5,24 seconds per century, -0.0524*16/(24*3600*36525) is calculation for this decreasing time between 01/01/16 and 2016.
-0.0524*(g-2451545-5843.5)/(24*3600*36525) is calculation for this decreasing time between 2016 and the watch's moment.
Between lines 38 and 43, our program gives z (ahso) correct, because before this step z will be quickly higher than 360.
z>DMS changes z writting from decimal to sexagesimal shape (degrees , minutes, seconds).
3) Part 3, (line 46/line 76) calculated watch and intercept.
At the beginning on this step we change sexagesimal data of estimated point and measured angle in decimal data workable for the next calculations. In this program, i could have insert collimation and sextant's excentricity too but i haven't done it (it's a mack 3 model…). Lines 56 and 57 it's right ascension and Dubhé's declination the 01/01/16. Line 58 to 61 we use precession's equations to get Dubhé's celestial coordinates at the measurement's moment. line 62 caculates star's verse ascension at measurement's moment, and…for people who are still reading this article, still able to stand that a little break they deserve it ! (if it could also attract more visitors to the blog…)
So where are we now… well yes ;
program's rest uses formulas seen at the beginning. if fonction uses between lines 70 and line 72 gets true azimut. and now, the test !
3) Test
to manage a star's point we need several things :
-clear horyzon.
-An adjusted clock on UTC.
-A TI89, a programmable calculator or ephemeris.
-A sextant or a tool able to measure an angle.
-An offshore chart.
For me day of the test horyzon wasn't really clear, and i've used an old mack 3's sextant 3 so it wasn't very accurate. besides this it has been done on the english channel, so range of tide could have grown this lack of accuracy.
conditions have let me the allowance to do 3 measures, First on Dubhé, second on Polaris and third on Alkaid (handle's end on the great bear's shape). so finally we stay on 3 close's stars only, but at least it gives a little idea after all. estimated point e (chosen by any chance on the chart) used is 49°50′ North, and 1°5′ East. so e coordinates are positive (negative latitude if it's south, negative longitude if it's west).
Résults:
.For Dubhé, true height is 35°52′ watched at 22h 32 min 52 seconds (local hour, to get it utc we substract 2 hours) the 15/08/16. Program calculates that :
So when we use the map, we go from our estimated point e, then we draw our intercept ; from e to the star if it's positive, and from e to the same direction but the other sense if intercept is negative. then we can easily draw height lines, perpendicular with intercepts to define our triangular position. So our get point locate us at 80 km southward of Fecamp city (my chart was to small to draw my height lines)… and measurements has been made at Pourville next to Dieppe city… make this again with a clearest horyzon and a better sextant maybe, with more distant stars i guess.
This program works well for clestial coordinates corrections, for vernal's hour angle it's accurate at /- 1° (it makes a mistake's range of 0.3 % roughly). It works with an average tropical's year but it changes every year depending on a lot of parameters , like moon influence, the other planets. vernal's point speed is not regular but changes a little.
To improve it we could include collimation, sextant's excentricity, also doing things to get directly coordinates…have a better accuracy on vernal's point and take care about nutation, Bradley,captain's age…
Something practical to avoid spending too much time with calculations for navigation, especially when the sea is rough and when we have a heavy meal inside our stomach, ready to leave it, probably on the map to draw a new iland, or in the companion way : if we program a calculator before, we get more time instead of waste it inside the boat, it's less boring and the most of time we avoid some seasick problems.
on internet "l'almanach du marin breton" has published programs ready to be included in a TI83 calculator for the height line, the sun's height range, and capable arcs. For a TI89 those programs work very well, there are just one or two small things to do:
On the file, "E" symbol is not the exponential fonction. it's a kind of scientific abreviation : E^2=100, E^3=1000…And when we use it we don't write the "^"on the calculator's screen. In a program we always have to indicate every time the program's end “IF” by a “ENDIF” otherwise the calculator will score “syntax error” something like that. that's why on the line 2 of paragraph “celestial mechanics in the Julian calendar” the sentence does not end with “End” but “Endif”. in the program useful to get the sun's height range we have to add up “Endif” Missing for the program validity.
when we are looking about those programs, we are able to guess others like a meridian program, the orthodromic distance, orthodromic's angle, or a program able to give our distance from a seamark with a known height:
For the calculation of the solar declination, marin breton's program take care of the elliptic nature of earth motion around the sun (part: “celestial mechanics in the Julian calendar”) which gets involved in declination's evolution all the year, and begins from an accurate sun's point in the time. Nevertheless there is a simplified formula to get sun's declination which doesn't care about the elliptic shape of earth's motion but coniders it like a circle (in our reality it's almost a circle, its elliptic shape is very weak with an elliptic excentricity roughly equal to 0.017. a circle's excentricity is 0 to have a little idea). This formula, less accurate than the first one displayed at the beginning, is still efficient even useful and in the same time very simple!
This formula is a precious tool to make a little sun's declination program depending on the day, the month, the hour (for 2015 when we compare it with marin breton's program difference is not overtaking 1 degree around mid-October, for the rest it's pretty accurate) without having ephemeridis on board. A second program allows us to have a better result (because we consider the elliptic shape in that way) :
For those declination's formulas we use a sinus fonction because if we write year days and declination's values ina same coordonate system, we get a sinus shape. with a cosinus fonction instead it would be possible. for sinus fonction parameters, in fact how to cope with it to change its period, its range and its departure point it's quite simple : we will use this exemple : A*sin(B(x-H)). "A" parameter adjusts the range's fonction (maxi and mini value it can reach), "B" adjusts period because sinus fonction's period is equal to 360/B ( absolute value). "H" parameter adjusts the phase, For exemple the fonction y=sin x is an impair fonction with a symetry center which is o origin, so if x=0 then y=0. if now we have y=sin (x-3), then y=0 if x=3 so we will have shifted sinus fonction about 3 on the right on absciss line, we have phased it about 3 finally.
To learn about celestial navigation, we think it is essential to buy a sextant. The first prize for this instrument is the mark 3 which costs around sixty euros. For more accurate measurements of its position, price increases quickly…and inversely proportional to the motivation to learn which can quickly decline. Indeed, why bother to spend money and then do calculations if with the same price you can have a good quality gps? The following small method not spend money more than ten euros, but it also provides a measure of star heights without the need to distinguish the horizon.
1)Equipment
Just have on hand a cheap plastic protractor, a long string (one meter, one meter and a half it's enough)and thin, something with weight(one or more large nuts for example)and a piece of wood or long plastic , straight and square. A small tube of glue will allow us to stick the protractor on the piece of wood or plastic as follows:
we have to be careful, we must stick the straight edge of the protractor in line with the straight piece of wood or plastic.
Then the wire is passed through the small ground hole of the protractor, we make a figure of eight knot and at the other end we link the weight stuff(nut, keys, stoolbox...)(for the photo I put a key, As there is a little weight the rest does not matter).
That's done! It is used as follows:
The more the wooden stuff is long, straight, and the wire is thin and pulled, the more we gain in efficience. You can also install the unit on a tripod with a swiveling axis to make more accurate reading without tremor during measurement.
The height's measurment of the star is fund by removing 90 °:
2)Advantages / disadvantages
This method has the distinct advantage of measuring the height of a star without the need to distinguish the horizon, so it can be done in town for example, view of the star is enough. However onboard the sea must be like a mirror and the boat without way, if the weight stuff moves then we will not be able to measure the height. To make a more elaborate sextant you can read the page of the site on how to make an electronic sextant with arduino.
The meridian is easy to implement and does not require a lot of calculations, however the summer line of position can become a big mess with a forbidden step or a mistake. But this method has an advantage on the meridian: we can guess our position at any time of the day, we have just to make two observations of the sun's height(at least two hours of gap for accuracy). In case of overcast sky, the summer line's method is the only choice if at local noon the sun is hidden by clouds.
1)Main idea
Measuring a height of the sun at any time of the day puts us on a circle on the globe's surface:
This circle called C, by the time we are on it show us the sun under a specific height. This height is called calculated height(Hc) it's calculated, it depends on our position(which we estimate, for example by taking as estimated point the last written in the logbook)and the time of day(the foot of the star will not have the same position)when we have caught Hv . Then we compare Hc and Hv, the true height is measured by the sextant(Hv). If Hv>Hc, then reports the point to the sun, otherwise the inverse. The difference between Hv and Hc is called intercept.
For the summer line there are borderline cases:
20°<Hv<80°
Intercept<30 Miles
-60°<observation's latitude<60°
On the chart, Circle C where we see the sun on the calculated height is drawn as a straight line because on the globe it's a very large circle, reducible on a chart with a lower scale. The real circle where we are(where the sun is observed with the true height Hv)is fund with intercept we call it C2. C2 is also drawn like a straight line, it is the summer line of position.
I is the intercept, in fact it is a tranfer's line. If Hv>Hc, we translate the C line on the other direction of the sun in the form of the C2 line, otherwise the inverse. In practice we do not draw C, the intercept is drawn from the estimated point on the chart.
So for a single observation / measurement of the height of the sun at any time of the day, we can position ourselves on a straight line only. To have a position in latitude / longitude a second observation is required. When we have drawn the second straight line found we will have to translate C2 according to our distance over ground between our observations, final intersection give us our position at the time of the second observation.
2)Matter's heart
The calculation of Hc requires a little trigonometry(see "trigo") and a hint of scalar calculations, it is more suitable to keep the formula in a programmable calculator like TI89 . Otherwise the method to find it is the following: Considering the earth as a sphere of radius 1, and included in a Cartesian coordinate system where the origin is the center of this sphere(ho no, do not turn on the GPS yet it's gone a be ok!). B is our estimated position, A is the position of the star's foot when Hv has been caught, P is the North Pole.
Three spherical segments are got, a b and c. a=90°-D(declination of the sun at the time of Hv measurement), b=90°-Le(Latitude of our estimated point), c=90°-Hc.
AHL = AHG-G(Hour Angle of the Sun, angle between the foot of the sun and Greenwich horary circle, and we subtract the longitude of our estimated point, in fact AHL is the angle between sun's foot meridian and estimated point's meridian).
The purpose will be to put it all into an equation to determine Hc. For this we will make a scalar multiplication of 0A and OB vectors(pfffiouuuuu…..).
We will use our Cartesian coordinate system to define these vectors.
0A: On the X axis, OA = 0. On the Y axis, OA = sin(a) and Z-axis OA = cos(a).
0B: On the X axis, OB=sin(b)*sin(AHL), on Y-axis OB=sin(b)*cos(AHL) and axis Z OB = cos(b).
The scalar product of two vectors in a Cartesian coordinate system is done by multiplying their coordinates and add them, so OA(X,Y,Z) and OB(X’,Y ',Z '):
That's why it is better to note the formula directly somewhere…do that on the chart table in rough sea, after having eaten a good heavy meal, it can become a bit difficult!
Troubles are not over, we have Hv, Hc so we have our intercept. It's nice, but how to draw our line C on the chart? something is missing, azimuth. The azimuth is the angle between the noth pole's vertical and the sun's vertical. we call it Z(basically it's the exact bearing of the sun relative to our place of observation). To find it first calculates Z '.
To find Z angle’ we take the formula seen before(circular permutation formula, I don't know the name in English i'm not sure sorry): cos(c)= sin(a)*sin(b)*cos(AHL)+cos(a)*cos(b). What interests us is the angle Z ', so: cos(a)= sin(b)*sin(c)*cos(Z ')+cos(c)*cos(b)
so, cos(90-D)= sin(90-the )*sin(90-Hc)*cos(Z ')+cos(90-Hc)*cos(90-the ), and now:
sin(D)= Cos(the )*cos(Hc)*cos(Z ')+sin(Hc)*sin(the ) and at the end cos(Z ')=(sin(D)-sin(the )*sin(Hc))/(cos(the )*cos(Hc))
Once Z’ determined, to find Z(true azimuth, , the sun's bearing calculated )it's simple:
Z = Z’ if it is morning(if the sun is on the East), and Z = 360 °-Z’ if it is after noon(if the sun is in the West).
Once we have the intercept and azimuth, then we are able to draw our summer line of position from our estimated point. The azimuth which is the calculated sun's bearing give us direction to draw intercept(intercept will be perpendicular to the summer line of position), the intercept gives us the distance between the summer line and the estimated point, and also the right side to draw summer line compared with estimated point(towards the sun, away from the sun).
In the example, Pe is the estimated starting point, Zv is the true azimuth. In this case Hv<Hc, so the intercept is on the opposite side of the sun.
The best method for this is to have a programmable calculator instrument texas like(TI89 it's efficient), for a time the almanac included a Breton sailor sheet program to return to the calculator handy and despite a couple of errors to be corrected. , I think there are also astro Special calculators designed for that. The Breton sailor book explains very well this method.
This method gives us our position with the sextant, a calculator, a stopwatch and graph paper. However it requires special conditions:
.a clear sky(no joke…)at the time of passage of the sun at local meridian. When the sun will be highest in the sky(half an hour before to half an hour after)in fact.
.The angle between the sun and the horyzon(the true height Hv)should be between 20 ° and 80 °, otherwise we increase imprecision because of the refraction of light(see Snell's law). The more light rays pass through a substantial layer of atmosphere, the more they will be more likely to be diverted, and the more Hv is short, the more the layer's width will increase.
1)Sextant
Contrary to its predecessors(kamal, astrolabe), its set of mirrors that reflects light allows a much more precise calculation of Hv, not disturbed by sea motions. The sextant could be understood as a kind of protractor capable of measuring an angle(Sun / observer / horyzon angle).
Vertical lead line indicates the height of the sun or a star Hv. Obviously at sea with wave motions it is less efficient!
A sextant is as follows:
The large mirror reflects the light on the little mirror. The small mirror is fixed and the largest is moving with the alidade. In the little telescope we must see the bottom of the star in line with horyzon, for that we move the alidade. In fact we move the largest mirror to get them in line. After that we have just to read the height written on the graduated scale("limbe" in French). Height read here is the instrumental height, for the true height there are some things to add on , we will see them further. The less sophisticated sextants have no drum("tambour" in french ), instead of it just a ...("vernier" in French, i'm sorry I don't know its name in English)(for example mack 3), the measurement is read as a caliper in that case. There is a wide range of sextant prices, in plastic, in braw…of course the price rises with precision.
2)Method
First we adjust our watch with accuracy as much as possible on UT hour(Greenwich hour). It is best to have a watch assigned just for UT time. then we will have to find roughly sun passage hour on our navigation area. The simplest method is to take the hand bearing compass and catch the Sun. Why ? because , we know that the sun indicates the South when it is at our local zenith(Northern Hemisphere)and north in the southern hemisphere. When the bearing designated approximates more and more than 180 °(Northern Hemisphere), we begin to make measurements of sun height at regular intervals.
On each measurement of height we write instrumental height and exact time that corresponds to it.
When the sun reaches its zenith(highest height), height does not change and it will stabilize for a while. When it is stable we already adjust the sextant on the last measurement noticed when sun was still rising. Then we look out the sun through the telescope of the sextant , we will see it gradually falling, at the same time we start the stopwatch(write UT time when it's started). When we see the bottom of the Sun “touching” horizon, we stop the stopwatch, we deduce UT found at this moment, and we write it. We will do the same for the before last height seen when the sun was risen, etc.…
The height of the sun at its zenith is easy to notice because the star make a break at this time. This is the height that will allow us to find our latitude, However all the mess seen above will help us to find our longitude.
Calculate the true height Hv
Instrumental height indicated by the "limbe" of our sextant is not exactly true height, we must add corrections to get Hv. There is "collimation" in french( instrumental error, it is provided in the sextant's box), the four angles correction(view angle corrections)found in the astro ephemeris books and calendar correction for the Sun(changes every month, also visible in the ephemeris). The difference between the instrumental height and true height is not huge, but better accuracy, better it is!
Find our latitude
Hv is taken when the sun was highest in the sky, and we calculate Dz. It's easy, Dz=90°-Hv. The latitude is then easy to determine, L=D /-(90°-Hv). This is + if the observation is done with North Pole in the back, and – with the South Pole in the back(D is the declination of the sun at the time of measurement, see “the celestial sphere”).
Determine longitude
The longitude is theoretically easy to guess; with earth's rotations the sun moves from east to west around the globe(earth reference)at a rate of about 15 ° / hour(the earth rotates on itself in 24 hours and has a circumference of 360 °). So knowing the exact time when the sun is at its zenith, and knowing UT time we can easily find it. But when you look at the sun it makes a break when it culminates, so it is difficult to find our longitude precisely. To find it we watch our little graph made at the beginning, on where we have written each height measurement taken with the corresponding time when the sun was risen, and when sun came down. We take an angle measurment(preferably the less close to the zenith time to be more accurate), and the corresponding hours(call them T1 and T2) at the time of the rise and descent for the same measurement height. The sun hour culmination is then equal to T = (T1 T2)/2. Our longitude is then calculated as follows: Longitude =(T-Tpass)*15°/hour(ephemeris give us the sun passage speed which can slightly change depending on the date, see V “the celestial sphere”). Tpass is the time passage of the sun at Greenwich meridian.
Notice: use sextant colored filters to avoid eye injuries. we can remove steadily some filters to have a better view of the sun's edges. The best place to make the measurements is close to the mast, this is where we find some stability if we are shaked by the waves.
Notice: Instrumental error can also be determined by simply watch through the telescope horizon, we make in line in the telescope horizon(the part reflected by the mirror and the part seen through the glass, and we get this instrumental error.
In the old days the celestial vault was thought as a solid sphere where stars were hung on it, and that sphere turned around the earth which were inside like Russian dolls. We now know that this solid sphere does not exist, and celestial bodies motions in the sky from east to west are due to the earth's rotation on itself. We will still keep the idea of that sphere to locate ourselves in the sky.
1)Celestial lines
They correspond to lines placed on the globe, it's only their projection. The celestial equator is the projection in the sky of the earth's equator, for the celestial poles it's the same principle. The ecliptic on the celestial sphere is the apparent path of the sun in its annual course.
As we have seen in the short paragraph “position and orientation without gps or calculator”, the sun's path in a year varies between the Cancer and Capricorne's tropic about(23°26’N the 21 June, 23°26’S the 22 December). For an observer in Paris it means a higher sun on horizon in summer and lower in winter, the ecliptic is movable along the year and is the main cause of the seasons, shorter or longer days.
2)Landmarks used
Star's declination(star, sun…) corresponds to its latitude on the earth sphere. The right ascension corresponds to its longitude. These coordinates are taken from the foot of the star, intersection of the line which pass through the center of the earth and the sun, and the tangent to the sphere.
The declination ranges from 90 ° South to 90 ° north passing through the equator(O°). North values are preceded by a + and South values -. The right ascension runs along the celestial equator anticlockwise, it is counted in hours, minutes, seconds.
3)For navigation
For navigation, the term right ascension changes, we rather speak about hour angle. This is the angle between the meridian of origin(Greenwich)and the horary circle of the sun.
we find sun's celestial coordinates, moon's celestial coordinates, full of stars and planets coordinates in nautical almanachs. For all days and hours of the year we are able to know the position of a star.
4)Ephemeris
.The sun
it looks like this:
Ephemeris is based on the Greenwich time(U.T). The small letter d is the declination changes per hour, AHv0 is the hour angle of the sun at 0 hours U.T, V is the angular speed of the earth's rotation on itself relative to the sun( 15 °per hour is an average number), T pass is the passage time(U.T) of the sun at Greenwich meridian. Sund's declination given is at 0 hours U.T(we will call it D0).
D0 is positive from spring to autumn(because above the celestial equator)and negative from autumn to spring(because below the celestial equator). d is positive from spring to summer and from autumn to winter(because in these two cases D0 increases), and when D0 decreases d is negative(from summer to autumn and from winter to spring). AHvp angle is the sun's hour angle at the observation time, so it has to be calculated.
Knowing that, to know the position of the sun at the time of observation(t)we simply apply these formulas:
Declination = D0 d * t
AHvp=AHv0 V*t
Notice: when we are watching this ephemeris page we notice that declination is almost zero the 23 September, it's the autumnal equinox. at this moment the foot of the sun is on the celestial equator.
The rule of the four 21: at equinoxes(20 March, 23 September roughly), D0 = 0 roughly. at Solstices(21 June and 22 December roughly), D0 = 23 ° 26′ and -23 ° 26′ respectively.
.Stars
It's almost the same thing, but for the hour angle origin will change. This time the hour angle of the vernal point is used, its speed is constant and equal to 15 ° 2′,5. The vernal point is the intersection point between ecliptic and celestial equator, the point where the sun passes from south hemisphere to the north hemisphere(ascending node). The hour angle of the star relative to the Greenwich meridian is found by this way:
hour angle=AHGy AV
AHGy is the hour angle of the vernal point measured from the Greenwich meridian.
AV is the verse ascension of the star, measured from the vernal point to the West.
5)What is found on the celestial sphere
As we have seen in “position and locate yourself…” , according to the latitude where we are the celestial sphere will move differently. In the northern hemisphere the stars will rotate anticlockwise around Polaris, in the southern hemisphere clockwise around the south celestial pole identifiable with to the Southern Cross. At the poles stars follow parallels ways relative to horyzon and are circumpolar(do not rise, do not fall). At the equator the two celestial poles are on the horyzon and all the stars rise and fall. At intermediate latitudes some stars are circumpolar, other not.
For a star's declination D and an observer's latitude L:
(90°-L)<D, star visible all the night
D<-(90°-L), the star is not visible
-(90°-L)<D<(90°-L), the star is visible on a limited period(not circumpolar)
During the night, the ecliptic is noticeable with the twelve zodiacal's constellations( we can add up Ophiuchus actually thirteen in fact)and also with the solar system planets(because they are also included in the ecliptic plan). The zodiac constellations are Aries, Taurus, Gemini, Cancer, Lion, Libra, Virgin, Aquarius, Capricorn, Sagittarius, the Scorpion, Pisces. Visible planets are Venus, Mars and Jupiter.
The celestial equator can be identifiable through the constellation of Orion, Mintaka star(between the red giant Betelgeuse and blue giant Rigel) is located almost on the celestial equator.
P.A.M.S mean Pleiades, aldébaran, the 3 magi and Sirius because they are almost in line.
The stars sparkle, planets don't. The brightest star is Sirius in the constellation of the Great Dog. The brightest planet is Venus.
Planets: Venus and Jupiter are the brightest objects after the Sun and the Moon, they move compared with the rest of the background star. The planets move through the zodiac constellations. Venus is visible up to three hours before or after sunset(because the planet is close to him, Mercury is not visible because it is too close to the Sun). Venus is never higher than 46 ° relative to the horyzon.
Today with the use of GPS navigation has been greatly simplified, we are able to know every time our position about few meters. But it is always fun and entertaining to know our position with the sky too, provided that of course not to be bothered by clouds!
1)The sun
The sun's path relative to the earth is easy, during the day it rises at the east side andfall at the West side roughly(it rises in the east and falls in the west with the best accuracy at the equinoxes, the rest of the time it is not exactly correct), and during the year the sun oscillates between 2 tropics. At the summer's solstice the sun is at the zenith of the Cancer's tropic(at a latitude of about 23 ° 26'N)and at the winter solstice it is almost above the Capricorn's tropic(latitude 23°26’S). At equinoxes the sun is near the zenith of the equator. Small diagram!
(this is not really a work of art, but well…)
During the day the sun describes an inverted parabola in the sky, it peaks at noon(the local noon of the observer). For example, if an observer has a watch adjusted in UT time(hours of Greenwich)he will normally see the sun peaking at noon if he is at Greenwich or on Greenwich meridian. If the same observer is placed eastward from Greenwich the sun will peak sooner, and if he goes westward from Greenwich this observation will come later.
We can therefore deduce a method to find the South or the North at 5 ° or 10 ° of accuracy, it's valid in temperate areas(between 23 ° 26′ and 66 ° 34′ South or North):
.Adjust your watch in time UT.
.In the Northern Hemisphere we point hours needle at the sun and the south will be indicated by bisector's angle made between this needle and twelve hour's direction 12. In the southern hemisphere the principle is the same but the number 12 of the watch has to be pointed at the sun, North is located halfway between the 12 and the hour hand(In both cases we don't use minutes hand).
The watch serves a little here as a compass (if you have a watch with a digital display it's the mess).
Notice: in temperate areas at local noon the sun indicates the south in the northern hemisphere and the north in south hemisphere.
2)The North Star(Polaris )
In the northern hemisphere there is a very useful star to find the north and our latitude, it's Polaris. On the celestial vault it's the only one which is motionless. It is found easily because it is located between the great bear and Cassiopeia. Small diagram!
When you keep an eye on the great bear's pan, if you use the edge which is opposed to the pan's handle(the star Dubhe and Merak), you have just to multiply by 5 the distance between these two stars while maintaining alignment to fall on Polaris, so the North.
Use of a sextant and making a measurement of Polaris height above the horyzon, and you will know your latitude on earth(after several attempts the margin of error between my real latitude and latitude found with this method has never exceeded 30 minutes, this margin may vary up to 1 °I think). Small diagram!
Here, Beta angle = 90 °-L, so Alpha=180°-(90° Bêta). So Alpha=L=latitude. Polaris is very far from us(can be considered almost to infinity), its light rays received from the land are parallel.
3)The moon
If the moon rises after midnight(when it decreases), side illuminated by the sun will be at the East; if it rises before the sunset its lighted side will show the West(when it grows). A little mnemonic: “in the northern hemisphere, the moon lies. In the southern hemisphere it does not lie.” It simply means that in our latitudes when the Moon cgrows it has the shape of a D, and when it dshrinks it forms a C. Attenzione, draw!
(I want to thank the kindergarten which has provided this drawing!)
Notice: when the Moon is shaped like a croissant, if the alignment is extended from both ends of this croissant with a straight line, the intersection of this line with the horyzon gives the South direction roughly.
4)Stars
As noted above in the sky only the polar star is relatively motionless(in the northern hemisphere). Earth turning on itself, so we have the illusion in our earthly referential that it's all the celestial vault which is turning around Polaris anticlockwise(the vault rotates clockwise in the Southern Hemisphere). We can deduce that in the Northern Hemisphere:
.If you see a star move to the right it tells us about the South
.If you see a star moving to the left it is the North
.If the star rises it's east
.If it falls it is the West
(in the Southern Hemisphere it is the contrary)
In fact during the night stars will turn around Polaris like this:
