Astronomical navigation

Taking stock at sea

As long as navigators keep their eyes on the coast, navigation is quite easy: you just need to find your bearings in relation to remarkable points on earth (cape, island, mountain peak, etc.). Even today, most boaters sail this way. But what to do on the high seas, far from the coast?

In the Western world, the question was resolved in two stages: latitude, then longitude.

Chinese and Arab navigators seem to have rarely taken their eyes off the coast, and to have used the compass (which they invented) from the 10th century.

For their part, the Oceanians (Polynesians, Melanesians, etc.) seem to have had navigation techniques on the high seas using the stars for much longer. Given their oral tradition, it is a fairly poetic mode: songs, poems or fables describe the “star paths” and “pillars” to follow. Please note, the Polynesians not only used the stars to find their way on the high seas, but also the swell, birds, clouds, etc.

In the Western world, it was the Vikings who seem to have been the first to leave the coast of sight, for their crossings to Iceland, then Greenland.

Knowing the latitude

Knowing the latitude is quite simple and it is believed that sailors have almost always known how to navigate at constant latitude. Thus, the great Viking crossings were probably carried out in this way. Indeed, we note that Iceland and Greenland are approximately at the same latitude. If you go up the coast of Norway to the right latitude, you then just have to sail due west on the way out and due east on the return. Similarly, for a long time, Atlantic crossing was carried out by starting from the Canary Islands and heading west. On the way back, we went up to the latitude of the Azores and headed east. The first major voyages were made at constant latitude.

How to travel at constant latitude?

The latitude has been known since Greek antiquity. The Greeks knew well that when traveling south, the northern stars lower on the horizon while the southern stars rise. Latitude is used extremely brilliantly in Eratosthenes’ experiment, who used it to measure the Earth (see here).

In the Northern Hemisphere, the simplest thing is to rely on the North Star, which is currently located less than one degree from the “true” North Pole. Precisely 39 arc seconds, or almost two thirds of a degree (39/60 = 65%). In other words, the polar star is a very good approximation of the North Pole. In these conditions, it’s simple: the height of the Polar gives the latitude. It is measured with an astrolabe, or more precisely a Jacob’s staff.

Not having a Jacob’s staff, the Vikings are said to have used a “sun shadow board” to navigate at constant latitude (see article).

Could the ancients trust the North Star?

The precession of the equinoxes means that, over a long period, the Earth’s axis of rotation is not oriented in the same direction at all times. Like a spinning top that oscillates slowly while spinning very quickly, the axis of rotation of the Earth also swings. It describes a circle in 26,000 years, or a displacement of 1.38° per century. So the star that is polar today will not always be so. In 12,000 years, it will be Vega (Alpha Lyra) which will in turn be polar. So, what was the North Star in the time of the Vikings or Christopher Columbus?

At the time of the Vikings, the precision of measurements, of the order of a few degrees, meant that the polar star could undoubtedly be trusted as good approximation. The size of Iceland makes it an achievable target with this level of precision: it measures a little over 200 km from north to south, which represents around 2 degrees.

At the time of Christopher Columbus, the Polar was 3.7 degrees away from the true North Pole. It was therefore necessary to make some corrections. Do do this, one must first have a model of the ratation of the Polar star around the North pole (“polar regiment”). Or on can use a Man-Pole Wheel.

From 1471, the Portuguese crossed the equator. The North Star is no longer visible. The reference star for knowing the latitude must now be the sun. Indeed, during the day, or if the Northern star is no longer visible, we can also determine the latitude using the height of the Sun at noon. However, an adjustment must be made, since the sun is only on the celestial equator twice a year, at the equinox. You must therefore know the date and have a table giving the declination of the sun for a given day.

The declination tables are developed by astronomers (and kept secret, initially). The first simplified publication dates from 1509, under the name “Regimento do estrolabio e do quadrante” (work also known as the “Munich Regiment”).

From this time, tables also give the position of the Southern Cross according to the date to know the direction of the South, despite the fact that the Southern Cross is not as “polar” as the Polar.

In conclusion, the Polar, the Sun or the South Cross make it quite easy to know the latitude, but knowing the longitude is another story, much more difficult.

The longitude problem

In principle, it should not be very complicated: if we know the time where the boat is (for example noon), it is enough to know the time it is at that moment in another place including we know the longitude. By difference, we know the longitude of the boat. Indeed, longitude is nothing other than a time difference . However, it is quite easy to know that it is noon in local time: we see this by observing that the sun passes the meridian, if we know the direction of South, or by noting when it is at its highest in the sky.

But in practice, this was not always easy: before the 20th century, sailors did not have a radio to know the time of the reference location (Greenwich, of course) and before the 18th century, there was no marine chronometer precise enough to know the time of the place of departure: after a few days at sea, the humidity of the air and the movements of the boat had disrupted all the clocks…

Solving the problem of longitude was vital for mastering the seas and was the subject of intense scientific discussions. It is (in particular) to resolve this that Louis XIV founded the Paris observatory and that England founded that of Greenwich. Among the disasters linked to poor knowledge of longitude, one is often cited: the shipwreck in 1707 of the English squadron returning from Gibraltar, after having attempted to take Toulon, which ran aground against the Sorlingues archipelago (at the West off the coast of Cornwall, Scilly in English). In 1714, the British government decided to take the necessary measures and passed the Longitude Act , with a prize of 20,000 pounds to anyone who solved the problem in a practical way. The prize will be awarded to anyone who proposes a method allowing one to locate oneself with an accuracy of half a degree or 30 nautical miles. A prize of 10,000 pounds is also expected for an accuracy of one degree (60 miles).

The price offered was considerable (20,000 pounds sterling at the time would represent around 10 million euros today) and gave rise to many unrealistic, or downright magical, proposals. Among the unrealistic solutions, we can cite the idea of positioning boats all along the route, which would receive a signal (a cannon shot at noon) and communicate it from far to far. Thus, boats sailing nearby would know the time of noon… Among the magical solutions, we can cite “sympathy powder”, the effect of which at a distance would make it possible to make an injured dog whine at a fixed time for this purpose…

A theoretically valid solution, but impossible to put into practice, is based on magnetic declination. From the moment the gap between geographic North and magnetic North was well known, it became possible to draw up maps. It would then be enough for a navigator to measure the difference between the polar and the compass and read his position on a map. But in practice, the compass does not give a sufficiently precise North and the magnetic declination is not stable enough to make these calculations.

As he traveled across the Atlantic, Christopher Columbus and his crew noticed the difference between the North of the compass and that of the Polar. Nobody then knew about the phenomenon of magnetic declination. Fearing a mutiny, he explains that the Polaire moved, to calm the fears of its crew: there was still a month to sail before “discovering America”…

We will only cover here the serious methods for knowing the reference time:

• Observation of astronomical events (eclipse, star occultation, position of Jupiter’s satellites, etc.).
• The lunar distance method
• The watch (the marine chronometer).

Ultimately, it was the least astronomical method that was chosen: knowing the reference time using a watch. The precision of half a degree, required by law, was achieved in 1761 by Harrison’s marine chronometer method. But the prize would not be awarded to his son until years later, astronomers having always preferred a purely astronomical method.

Observation of astronomical events

Astronomical events occur at a certain time, which can be accurately predicted by astronomers. Two distant observers, located on different continents (or at sea), can observe them at the same time. A lunar eclipse is visible from half the surface of the Earth and the moment when the Moon enters the cone of the Earth’s shadow is the same for everyone. Knowing the start time of the eclipse in Greenwich, and knowing the local time of this same event, we know the time difference. It is then enough to travel with tables giving the time of astronomical events.

But these events are rare (eclipse) and/or local (star occultations may only be visible from certain locations). The navigator must be at sea when the event occurs, navigate in the area where the phenomenon is visible and the sky must be observable at that precise moment (no clouds). Furthermore, for a star occultation by the Moon, there is a parallax problem: the Moon being relatively close to the Earth, two distant observers on the Earth’s surface do not see it at the same angle, and therefore not at the same time.

This is why Newton was convinced that the solution lay in the observation of Jupiter’s satellites, whose movement his equations made it possible to precisely predict. But this solution is not universal: Jupiter is only visible at night. Additionally, for about three months a year, Jupiter is not visible at all: when Jupiter is in the sky, it is daytime.

Jupiter is not visible when the Sun and Jupiter are in the same direction to the observer, because the Sun is too bright. This happens when the Earth is moving opposite the Sun (in other words, the Sun is positioned between the Earth and Jupiter. Please note: traditionally, in astronomy, we call opposition the moment when the Sun is opposite of a planet, which corresponds to the best time to observe it. In the geocentric system, the Earth being fixed, the Sun and the planet can be “in opposition”. In the heliocentric system, the Earth and the planet must be on the same side of the Sun.

Finally, even when Jupiter is easily spotted in the sky, observing its satellites to obtain usable information about the time requires an instrument with high magnification, practically impossible to handle on board a boat at sea. It is only by setting foot (and telescope) on land that sailors could use the method of Jupiter’s satellites. Thus, they could determine the longitude of the stopover and reset their clocks, which improved the quality of the next point at sea. The method was used by Captain Cook to determine the longitude of points on the coast of New Zealand, which he mapped.

The lunar distance method

The principle is as follows: the Moon moves against the background of the sky in a way that is known and predictable. So, if we know where it is in the sky at a given moment, we know the time by comparing what we observe with the forecast from an almanac.

The method was theorized and described as early as 1524 but was only really used from the end of the 18th century to the mid-19th century. In fact, it requires two things:

• An instrument for measuring angles precisely. However, the octant was only invented in 1731 and the sextant in 1757; And
• A precise almanac, to compare observation and prediction. However, Newton’s formulas were only published in 1687, and even afterward, the Moon’s orbit still posed serious problems, due to its irregularity.

Thus, the British Navy only published the almanac necessary for its use from 1767. It did so until 1906, even if, in practice, the method was abandoned in favor of using of marine chronometers around the middle of the 19th century. At the end of the 18th century, the lunar distance method was used to check the time of marine chronometers, until they were considered truly reliable.

Please note: the Moon is not visible, around the new Moon and the method cannot be used for a few days.

Relative to the background of the sky, the Moon moves eastward by half a degree per hour (the size of its own diameter). In other words, one minute of angle on the position of the Moon (the maximum precision that can be obtained on the position of the Moon with a sextant) corresponds to two minutes on a clock. Consequently, an error of one minute of angle in the position of the Moon corresponds to an error of 2 minutes on the watch. However, the Earth rotates 360 degrees in 24 hours, or 15 degrees per hour, or 4 degrees per minute. So a 2 minute error on the clock corresponds to 0.5 degrees, or 30 minutes of angle, or 30 nautical miles.

This precision was therefore sufficient to allow the designers of the method to receive the prize of 20,000 pounds.

The famous Captain Cook, happy to finally have a reliable method, is said to have declared that 30 nautical miles was “all the precision a sailor would ever need.” We can’t help but think of the famous phrase attributed to Bill Gates in 1981: “640k (of memory) is all anyone needs.” The quote is likely apocryphal, and Bill Gates denies saying it.

In practice, you must first measure the angle between the Moon and a reference star with a sextant. Then, you have to make several adjustments, in particular, parallax. Finally, we look for the time corresponding to this angle in an almanac, which gives the angles between the Moon and several reference stars, including the Sun, every three hours. We thus know the time in Greenwich with an accuracy of around 2 minutes and we can therefore take stock. The process is estimated to take approximately 4 hours. If no error occurs in these 4 hours of calculation, we have located the ship with an accuracy of 30 miles…

Astronomical navigation with stopwatch

There are actually several ways to take stock at sea if you have a chronometer precise enough to know the reference time.

In theory, the simplest method would be to note the time at which the Sun peaks in the sky. It is then local noon. By difference with the reference time on the stopwatch, we know the time difference. And as the height of the Sun at noon also gives (after correction for declination) the latitude, we have both data: latitude and longitude. But this method is almost never used in practice, because precisely measuring the time of the meridian crossing is not very easy at sea. Observing the Sun at its highest point does not give very reliable information, since it remains at its highest point for quite a long time. We therefore know midday to within a few minutes, which is not ideal for determining longitude. We can do better.

In addition, the browser cannot be satisfied with a method that only works once a day.

Until the middle of the 19th century, one estimated one’s position with a single point. In 1843, Captain Sumner, of the American merchant marine, published a booklet in which he explained that one could estimate one’s position on a straight line, by doing the calculations several times, by changing the latitude by a few degrees. In 1873, the French naval officer Marcq Saint-Hilaire published the so-called height line (or intercept) method, which is today the reference method, taught throughout the world. Both methods are briefly described below.

The point with a single point

The method requires having an almanac which gives the coordinates of a series of stars easy to observe: Sun, Moon, Jupiter, Mars and Venus, as well as a list of some remarkable stars (57 for the 2021 almanac of the United States Air Force). These coordinates are given in universal time (UT), formerly Greenwich Mean Time (GMT). The forecasts cover the whole year, in intervals of a few minutes. The observer arrives at the figures corresponding to the precise time of his observation by interpolation. Interpolations can be calculated, or read directly from interpolation tables, also provided.

Of course, since the stars are fixed against the background of the sky, it is not necessary to give the positions of each of them in 10 minute increments. We simply give their celestial coordinates (right ascension and declination) and the position of the vernal point at Greenwich.

NB: the right ascension is given from the vernal point towards the East, while the longitude is measured towards the West. As a result, the almanac does not give the right ascension of the stars, but the angle measured in the other direction, called SHA, Sidereal Hour Angle, which is the complement to 360. In French, it is called ascent pours. We therefore have SHA + AD = 360°.

For each star, the almanac actually gives the coordinates of the point on the earth which, at that moment, sees the star directly above it (height 90°). This point is called the “ground point” (noted GP), or substellar point.

If we measure the height of this star at the same time from another point on the globe, we know how far from this point we are. This distance is equal to the difference between 90 degrees and the apparent height of the star. Indeed, an angle in the sky corresponds to a distance on Earth. This is the definition of the nautical mile: two observers who see an object located at infinity at an angle of one minute of arc are one nautical mile apart. The set of points which “see” the star at the same angle therefore defines a circle, the center of which is the ground point and the radius the distance that we have just measured, it is the circle of height. Someone who observes the star from this angle at this moment is somewhere on this circle.

If he knows his latitude, the observer knows that he is at the intersection of the height circle and the latitude.

Another possibility: measure the height of another star at the same time (or the same star a moment later) and identify the intersection of the two height circles, which corresponds to the position of the observer.

But it is not easy to draw a height circle on a map, since the ground point is rarely found on the map we use. For example, if we observe the Sun at 45° on the horizon, we are 2,700 nautical miles from the ground point, or 5,000 km. A nautical chart does not cover an area of 5,000 km on each side, or it is very imprecise. If we use a nautical chart of a reasonable size and we are on the same map as the ground point, we are very close to it. In this case, the height of the star is close to 90°, which makes the measurement unreliable, which has the same effect as using a gigantic map.

The Marcq de Saint-Hilaire method (so-called “line of position” or “intercept” method)

In 1873, Marcq de Saint-Hilaire published a much more practical method based on an observation and a tip.

The observation: the height circle is so large that it can be confused on the map with a straight line (its tangent). It is called the line of height (LOP, line of position). This approximation is possible if the star is less than 65° on the horizon and if we trace the line for less than 60 miles.

The tip: instead of trying to position yourself in relation to the ground point, we will do it in relation to a point chosen to facilitate calculations and close to where we think we are.

First step: observation. With a sextant, we note the height of the chosen star, noting precisely the time of observation. This is the observed height, noted Ho (Hauteur Observée). Warning: it is necessary to correct this height, for reasons linked to the instrument and/or the measurement: the systematic error of the sextant, the height of the observer on the horizon and atmospheric refraction. (See at the end of this article the note on the sextant’s limitations).

Second step: calculations. We choose a point on the map, fairly close to the position we think we occupy, and for which the calculations will be simple. This point is called the estimated position. We then calculate the height below which we would see the star if we were in this place at the time of the observation. This is the “calculated height”. We also calculate the azimuth of the star in this place (the direction in which we would see the star, if we were there).

Third step: plotting on the map. We compare the calculated height and the observed height. If the observed height is greater than the calculated height, it is because we are in reality closer to the ground point; if the height is lower, it means we are further away. This difference is called the intercept, it is measured in degrees, minutes (and therefore in nautical miles). On the map, we draw a straight line passing through the estimated position, in the direction of the azimuth of the ground point. On this line, at the distance from the intercept, we trace its perpendicular: it is the height line. The observer is somewhere on this right. If we repeat the calculations with another star, we draw another line and we know that we are at the intersection of the two.

The calculation step consists of determining the distance between two points on the sphere (the estimated position and the ground point), knowing their coordinates. This amounts to determining the length of a spherical triangle (the third point being the North Pole), of which we know two sides.

Spherical trigonometry allows us to solve the problem, thanks to the cosine formula, which says that if we know a and b we have:

cos(c)=sin(Lat_a).sin (Lat_b)+cos(Lat_a).cos(Lat_b).cos(Long_b−Long_a)

Depending on the technology available, the problem can be solved in the following ways:

• Fully automatic: enter the time (UT) and its estimated position, the program calculates the position of the ground point observed and the distance between the ground point and the estimated position, it deduces the intercept and the azimuth, which allows to trace the height line. Obviously, this is the simplest method in practice, but it requires having a computer or a programmable calculator, and therefore batteries, electronics etc., which sailors may not have on board, in case of a hard blow.
• Entirely manual: determine the position of the ground point using an almanac and solve the spherical triangle using tables, called Sight reduction tables. The most famous are those published by the American Navy, called HO-229 tables (6 tables of more than 300 pages). The HO-249 tables are intended for the Air Force (in “only” 3 volumes). The British Navy also publishes a version in a single volume, in order to weigh less on board. Naturally, the less the tables are capacious, the less they are accurate (or practical).
• Semi-automatic: only use a calculator or a computer for its trigonometric functions, for the calculation of the third side of the spherical triangle (calculation of the distance to the ground point) and its angle (the azimuth of the star at the estimated point). Determining the position of the ground point using ephemeris remains manual.  

Using “sight reduction” tables

The table gives the distance, directly in the form of a sextant height, denoted Hc (Hauteur Calculée), and the angle, in the form of an azimuth angle.

The input data is:

• Local hour angle (LHA)
• Latitude of the estimated point
• Declination of the ground point

Tables are incremented by degree. This is why, in order to avoid having to make numerous interpolations, it is necessary to choose an estimated point so that its latitude is a whole degree and the local hour angle is also a whole degree.

The only number that cannot be reduced to a whole number of degrees is declination. This interpolation can be carried out with a calculator, but the tables also provide a means of interpolating between two whole degrees.

The HO-229 tables are in 6 volumes, in 30 degree intervals:

In order to reduce the size of the work, identical and/or symmetrical calculations are not reproduced. This allows you to print only a quarter of the necessary tables, a saving of three quarters. To do this, the tables are divided into two: cases where the two points are in the same hemisphere and cases where they are in two opposite hemispheres. We avoid specifying whether it is the ground point or the estimated position. We therefore see in the title of each result table: “Latitude same name as declination”, or “Latitude name contrary to declination”. Likewise, for the local hour angle (LHA), a single page contains two angles whose sum makes 360°, since they are symmetrical. The top of the page, however, reminds that the azimuth will be read directly in the table for hour angles greater than 180°, and that it is necessary to adjust the azimuth read for hour angles less than 180°: the real azimuth being equal to 360° minus the angle read in the table. In case the two are not in the same hemisphere, the azimuth read must always be adjusted: if the LHA is greater than 180°, the azimuth is 180° minus the angle read. If LHA is less than 180°, the azimuth is equal to 180° plus the angle read. All these corrections are recalled in the corner of the tables.

The modern method

In practice, the modern method consists of four steps which are detailed here:

1. Observation
   
   • Measure the sextant height of a star at a specific time (Hs)
   • Adjust the raised height for:
     
     • The height of the eye relative to the horizon (dip)
     • Refraction due to the height of the star on the horizon
     • The possible intrinsic error of the sextant
     • The radius of the star (semi-diameter) if we have noted the edge of the Sun or the Moon
   • The height thus corrected is noted Ho
2. Search for the ground point at the time of observation (almanac or computer)
   
   • Find the date and time of the observation in the almanac (Hs)
   • Note the GHA and the declination corresponding to the time closest to the observed time
   • Add an interpolation for the precise time of the observation (minutes and seconds)
3. Calculation/finding the calculated height and azimuth (Hc and z) for a reference point (estimated position)
   
   • Semi-automatic method (calculations):
     
     • Use the spherical triangle formula to define the length Hc from (1) the latitude and longitude coordinates of the location where you estimate yourself to be and (2) the ground point
     • Deduce the angle z
   • Manual method: round the latitude of the location where you estimate yourself to be to the nearest degree and choose a longitude such that the local hour angle (LHA) is a whole degree (reminder: LHA = GHA – longitude)
     
     • Search the tables for the calculated height and azimuth corresponding to the data
     • We start by opening the LHA page, we find the latitude column and we read the two figures sought (make the necessary adjustments if LHA is less than 180° and/or if the latitude is not of the same sign as the declination)
     • An interpolation is necessary for the distance Hc on the basis of minutes and tenths of minutes attached to the declination noted in the almanac. You can calculate the interpolation, or read it in tables (which allows you to only do additions – no calculator is used)

4. Representation on the map
   
   • Locate on the map the point of the estimated position (latitude to a whole degree and longitude such that the difference with GHA is a whole degree)
   • On this point, trace the direction given by the angle z
   • Compare Hc, the calculated height and Ho, the observed height. If Ho is greater than Hc, it is because the real position is closer to the ground point. If Ho is smaller, it is because the real position is further away from the ground point. The Ho-Hc distance is called intercept.
   • Using a compass, mark the intercept from the estimated position, in the direction of z
   • Draw a line perpendicular to z at this point. It is called height straight. It is the local approximation of the height circle
   • The position is on this right. A second measurement, preferably on another star and/or at a different time, will confirm the position which will then be at the intersection of the two lines.

The limits of the sextant:

• One must see both the star and the horizon. If it’s night, you can’t see the horizon; if it’s daytime, you can’t see the stars. One can therefore only aim for the stars and planets at dusk, or when the Moon is full enough to make the horizon visible. At dusk, to have time to point out several stars (wait until it’s dark enough to see the stars, but before it’s too dark to see the horizon), you must start in the East and end in the West. Indeed, in the evening the East is already dark enough to see the stars, while the West is still too bright; in the morning, you must hurry to observe East, which will become too bright, while you still have time for the West, which will remain in the twilight a little longer.
• Refraction: depending on the angle at which a star is observed, its light must pass through a greater or lesser amount of atmosphere, and therefore it is more or less curved by the refraction of the atmosphere. It is therefore necessary to correct the observed height of a star by the refractive index corresponding to this height. The correction is zero for a star at 90° height (its light passes through the atmosphere at right angles and there is no refraction). The lower the star is observed on the horizon, the greater the layer to be crossed, at a steep angle, and the more it is necessary to correct this curvature. The almanac gives the correction to be made.
• Parallax: especially for the Moon, which is proportionally much closer to Earth, parallax becomes significant and must also be corrected. This is of course only the parallax in longitude, called horizontal parallax. A parallax correction must also be made for Mars and Venus, which, depending on the month, may be so close to Earth that an adjustment is necessary.
• We cannot aim at a star too low on the horizon, because the atmosphere becomes too thick and distorts the viewing angle by refraction-diffraction. In practice, it is good not to pick up a star below 15°.
• We cannot aim at a star too high, because in this case we lose sight of the horizon. It is good not to raise a star above 70°.
• The sextant error: the sextant is a mechanical device, it can go wrong. It normally contains adjustment screws, but you cannot always act on them. To identify the systematic error of the sextant, it is enough to fix the horizon; if one does not read zero, what one reads is the error of the sextant that must be taken into account in the measurement. It is called the index.
• Eye height (dip): it must be taken into account that the sextant is not at sea level, but at the eye level of the observer. If it is on the deck of a container ship or cargo ship, it may be 30 meters above sea level. This angle should be corrected. The almanac gives a correction table.
• Center or edge of the star? When we point to the Sun or the Moon, it is not easy to find the center of the star, and it is much easier to make the edge of the star coincide with the horizon. However, the almanac is calculated for the center of the star, so we must subtract half of the apparent diameter of the Moon or the Sun. And this correction is not a constant, since the Earth does not describe a circle around the Sun, but an ellipse. The apparent diameter of the Sun is therefore more or less large throughout the year. It is the same for the Moon, which also describes an ellipse around the Earth. The almanac therefore gives values for the half-diameter of the Sun or the Moon, depending on the position of the Earth in its orbit around the Sun and that of the Moon around the Earth. They are denoted SD, for semi-diameter.

A brief history of instruments for measuring angles

NB: We often find on the internet that the sextant was invented in 1731. It is the octant which was invented on this date.

For more, you can read my article, published in L’Astronomie magazine (in french): here.