Children and parents
21.06.2023

Double stars: A forgotten pleasure. Photometric double stars Double stars in a telescope

Not much attention has always been paid to the observation of double and multiple stars. Even in the old days of the abundance of good astronomy literature, this topic was often avoided, and you are unlikely to find much information on it. The reason for this may lie in the low scientific significance of such observations. It is no secret that the accuracy of amateur measurements of the parameters of double stars is, as a rule, significantly lower than that of professional astronomers who have the opportunity to work with large instruments.

However, almost all astronomy lovers have at least for some time short period Since then, they have been obligated to observe double stars. The goals they pursue can be completely different: from checking the quality of optics or purely sporting interest to conducting truly scientifically significant measurements.


It is also important to note that, among other things, observing double stars is also an excellent training for the eyes of an amateur astronomer. Looking at close pairs, the observer develops the ability to notice the most insignificant, small details of the image, thus maintaining himself in good shape, which in the future is sure to be reflected in observations of other celestial objects. A good example is when one of my colleagues spent several weekends trying to resolve a pair of stars with 1" separation using a 110mm reflector, and finally succeeded. In turn, after a long break, in observations I had to give up in front of this pair with a much larger instrument.

Telescope and observer

The essence of observing a double star is extremely simple and consists in dividing the stellar pair into individual components and determining their relative position and the distance between them. However, in practice, everything turns out to be far from so simple and unambiguous. During observations, various kinds of third-party factors begin to appear, which do not allow you to achieve the result you need without some tricks. You may already know about the existence of such a thing as the Davis limit. This value determines the ability of some optical system to separate two closely spaced point light sources, in other words, it determines the resolution p of your telescope. The value of this parameter in arcseconds can be calculated using the following simple formula:

ρ = 120"/D


where D is the diameter of the telescope lens in millimeters.

In addition to the diameter of the lens, the resolution of the telescope also depends on the type of optical system, on the quality of the optics, and, of course, on the state of the atmosphere and the skills of the observer.

What do you need to have in order to start observing? The most important thing, of course, is the telescope. And the larger the diameter of its lens, the better. In addition, you will need an eyepiece (or Barlow lens) that provides high magnification. Unfortunately, some amateurs do not always correctly use Davis's law, believing that only it determines the possibility of resolving a close double pair. Several years ago, I met with a novice amateur who complained that for several seasons he could not separate a pair of stars located at a distance of 2" from each other in his 65 mm telescope. It turned out that he was trying to do this, using only 25x magnification, arguing that at this magnification the telescope has better visibility. Of course, he was right that a low magnification significantly reduces the harmful effects of air currents in the atmosphere. However, he did not take into account that at such a low magnification the eye is simply not able to distinguish between two closely located light sources!

In addition to a telescope, you may also need measuring instruments. However, if you are not going to measure the positions of the components relative to each other, then you can do without them. Let's say, you may be quite satisfied with the very fact that you managed to separate nearby stars with your instrument and make sure that the stability of the atmosphere today is suitable or your telescope gives good results, and you have not yet lost your former skills and dexterity.

For more serious problems, it is necessary to use a micrometer to measure the distances between stars and a dial scale to determine positional angles. Sometimes these two instruments can be found combined in one eyepiece, at the focus of which a glass plate is installed with scales printed on it, which allow the corresponding measurements to be made. Similar eyepieces are produced by various foreign companies (in particular, Meade, Celestron, etc.); some time ago they were also manufactured at the Novosibirsk enterprise Tochpribor.

Taking measurements

As we have already said, measuring the characteristics of a binary star comes down to determining the relative position of its constituent components and the angular distance between them.

Position angle. In astronomy, this quantity is used to describe the direction of one object relative to another for confident positioning on the celestial sphere. In the case of binary stars, the term position angle involves determining the position of the fainter component relative to the brighter one, which is taken as a reference point. Position angles are measured from the direction north (0°) and further towards the east (90°), south (180°) and west (270°). Thus, two stars with the same right ascension have a position angle of 0° or 180°. If they have the same declination, the angle will be either 90° or 270°.

Before measuring the position angle, it is necessary to correctly orient the measuring scale of the micrometer eyepiece. By placing the star in the center of the field of view and turning off the clock mechanism (the polar axis of the mount should be set to the celestial pole), we will force the star to move in the field of view of the telescope from east to west. The point at which the star will go beyond the boundaries of the field of view is the point of direction to the west. If now, by rotating the eyepiece around its axis, we align the star with a value of 270° on the micrometer hour scale, then we can assume that we have completed the required setting. You can evaluate the accuracy of the work done by moving the telescope so that the star just begins to appear from beyond the line of sight. This point of appearance should coincide with the 90° mark on the hour scale, after which the star, in the course of its daily movement, should again pass the center point and leave the field of view at the 270° mark. If this does not happen, then the micrometer orientation procedure should be repeated.



If you now point the telescope at the star pair you are interested in and place the main star in the center of the field of view, then mentally drawing a line between it and the second component, we will obtain the required value of the position angle by taking its value from the micrometer hour scale.

Separation of components. In truth, the hardest part of the job has already been done. All we have to do is measure the distance between the stars on a linear micrometer scale and then convert the result obtained from a linear measure to an angular one.

Obviously, to carry out such a translation we need to calibrate the micrometer scale. This is done as follows: point the telescope at a star with well-known coordinates. Stop the telescope clock mechanism and note the time it takes for the star to travel from one extreme division of the scale to the other. Repeat this procedure several times. The obtained measurement results are averaged, and the angular distance corresponding to the position of the two extreme marks on the eyepiece scale is calculated using the formula:

A = 15 x t x cos δ


where f is the transit time of the star, δ is the declination of the star. By then dividing the value of A by the number of scale divisions, we obtain the value of a micrometer division in angular measure. Knowing this value, you can easily calculate the angular distance between the components of a double star (by multiplying the number of scale divisions that fit between the stars by the value of the division).

Observing close couples

Based on my experience, I can say that the separation of stars with a distance close to the Davis limit becomes almost impossible, and this becomes more pronounced the greater the difference in magnitude between the components of the pair. Ideally, Davis' rule works if the stars have the same brightness.

Looking at a relatively bright star through a telescope at high magnification, you will notice that the star does not look like just a luminous point, but like a small disk (Erie disk), surrounded by several bright rings (the so-called diffraction rings). It is clear that the number and brightness of such rings directly affects the ease with which you can separate a close couple. If there is a significant difference in the brightness of the components, it may happen that the faint star simply “dissolves” in the diffraction pattern of the main star. It is not for nothing that such well-known bright stars as Sirius and Rigel, which have faint satellites, are very difficult to separate in small telescopes.



In the case of a large difference in the color of the components, the task of separating the double, on the contrary, is somewhat simplified. The presence of color anomalies in the diffraction pattern becomes more noticeable, and the observer's eye notices the presence of a faint satellite much faster.

It is believed that the maximum useful magnification provided by a telescope is approximately equal to twice the diameter of the objective lens in mm, and using higher magnification does not achieve anything. This is not the case with double stars. If the atmosphere is calm on the night of observation, then using 2x or even 4x maximum magnification may help you see some "disturbance" in the diffraction pattern, which will indicate to you the presence of the source of this "interference". Of course, this can only be done with a telescope with good optics.

To determine the magnification at which you can begin to separate a close pair, you can use the following simple formula:

X = 240"/S"


where S is the angular distance between the binary components in arcseconds.

To separate close stars, we can also recommend using a simple device that fits onto the telescope tube and turns the round shape of the aperture, say, into a regular hexagon. Such aperture slightly changes the distribution of light energy in the image of the star: Airy's central disk becomes somewhat smaller in size, and instead of the usual diffraction rings, several bright peak-shaped bursts are observed. If you rotate such a nozzle, you can ensure that the second star appears between two adjacent bursts and thus “allows” its presence to be detected.

Observing double stars- an extremely interesting and fascinating activity, to which astronomy lovers have recently paid undeservedly little attention. This is a special, traditional area of ​​amateur observational practice, combining several principles at once. This is both scientific - the desire to study an object, to advance our knowledge about it, and technical - the desire to improve your telescope and then “squeeze” the maximum out of it. There is also a sporting element in this activity - the desire to achieve the maximum of one’s capabilities, training one’s abilities, overcoming the difficulties that arise in this process, but there is also an aesthetic element - simply looking at these unusual, unearthly pictures, and among thousands and thousands of doubles, no two are identical, and Sometimes among them there are real masterpieces of nature, which you can admire endlessly. Of course, recently, after the launch of ultra-precise satellites into orbit, which measured almost all the bright stars in the sky and received unprecedented information about double stars, amateur scientific measurements lost their relevance, but all other motives remained...

In addition, happy is the astronomer who is lucky enough to become interested in observation. double. He always has something to occupy himself and his telescope on a full moon, on a night with haze, and even if he lives in the city center, there will always be objects that will attract him, inviting him to find something new for himself or just admire another beautiful picture.

From time to time double stars, especially close ones, observed. Almost all amateur astronomers. As a rule, for the purpose of testing the optics of their telescopes (and it is difficult to find a better test than a close double). Of course, no one will refuse to admire famous pairs like Albireo, - γ Cygnus, or - γ Andromeda, but to specifically hunt for beautiful ones, for example, those in which there is a significant difference in color - few people do this, which is a pity: this is very interesting and an area that promises a lot of surprises. Differences in gloss and close proximity of components can cause an increase in visible color contrast, change the shades of components, or even change their color completely. And even observing the same pair through different telescopes can significantly change the already familiar picture and prepare surprises.

It is unnecessary to remind you that when viewing and photographing double stars, you should strive to use a telescope of maximum quality, because observations should be carried out with maximum magnifications, such as 1.50 and even more (apochromats allow you to increase the magnification to 2 and even 30). Of course, attention to the eyepiece should be no less than to the telescope itself; it is worth remembering the old truth: “A good telescope with a bad eyepiece is a bad telescope.”

In this picture from " Larousse Encyclopedia of Astronomy"The colors of the stars are greatly enhanced, more so than they appear in telescopes. However, the contrast in visual pairs can sometimes be just as impressive, especially when observed through small telescopes. All stars are depicted on approximately the same scale, south is at the top, east is on the right. Only ξ Boötes, whose positional angle is now about 320°, has seen a noticeable change in the position of the stars in the almost 50 years since its publication.

Observing double stars



The topic of observing double and multiple stars has somehow always been gently ignored in domestic amateur publications, and even in previously published books on observing double stars by amateur means you are unlikely to find an abundance of information. There are several reasons for this. Of course, it is no longer a secret that amateur observations of binaries are worth little from a scientific point of view, and that professionals have discovered most of these stars, and those that have not yet been discovered or studied are as inaccessible to ordinary amateurs as the latter’s flight to Mars. The accuracy of amateur measurements is significantly lower than that of astronomers working with large and precise instruments, which determine the characteristics of star pairs, sometimes even beyond the limits of visibility, using only mathematical apparatus to describe such systems. All these reasons cannot justify such a superficial attitude towards these objects. My position is based on the simple fact that most amateurs, for some period of time, are necessarily engaged in the simplest observations of double stars. The goals they pursue can be different: from testing the quality of optics, sporting interest, to more serious tasks such as observing with their own eyes changes in distant star systems over several years. Another way observation can be valuable is observer training. By constantly studying double stars, the observer can keep himself in good shape, which can later help in observing other objects and increases the ability to notice minor and minor details. An example is the story when one of my colleagues, after spending several days off, tried to resolve a couple of stars at 1" using a 110mm reflector, and, in the end, achieved a result when I, in turn, had to give up with a larger 150mm Perhaps all these goals are not the primary goals of amateurs, but, nevertheless, such observations are carried out, as a rule, periodically, and therefore this topic needs additional disclosure and some ordering of previously collected known material.

Looking at a good amateur star atlas, you will probably notice that a very large part of the stars in the sky have their own satellite or even a whole group of satellite stars, which, obeying the laws of celestial mechanics, make their entertaining movement around a common center of mass for several hundred years, thousands, or even hundreds of thousands of years. As soon as they have a telescope at their disposal, many immediately point it at a well-known beautiful double or multiple system, and sometimes such a simple and uncomplicated observation determines a person’s attitude to astronomy in the future, forms a picture of his personal attitude to the perception of the universe as a whole. I remember with emotion my first experience of such observations and I think that you too will find something to tell about it, but that first time, when in distant childhood I received a 65 mm telescope as a gift, one of my first objects, which I took from a book Dagaev "Observations of the starry sky", there was a beautiful double system Albireo. When you move your small telescope across the sky and there, in the outlined circle of the field of view, hundreds and hundreds of stars of the Milky Way float by, and then a beautiful pair of stars appears, which stand out in such contrast relative to the rest of the main mass that all those words that formed in your mind to sing the magnificence of the beauties of the sky disappear at once, leaving you only shocked, from the realization that the grandeur and beauty of the cold space is much higher than those banal words that you almost uttered. This is certainly not forgotten, even after many years have passed.
Telescope and observer
To reveal the basics of observing such stars, you can literally use only a couple of general expressions. All this can be simply described as the angular separation of two stars and the measurement of the distance between them for the current era. In fact, it turns out that everything is far from being so simple and unambiguous. When observing, various kinds of third-party factors begin to appear that do not allow you to achieve the result you need without some tricks. It is possible that you already know about the existence of such a definition as the Davis limit. This is a long-known quantity that limits the limit of the ability of some optical system to separate two closely located objects. To put it another way, using another telescope or spotting scope, you will be able to separate (resolve) two more closely located objects, or these objects will merge into one, and you will not be able to resolve this pair of stars, that is, you will see only one star instead of two. This empirical Davis formula for a refractor is defined as:
R = 120" / D (F.1)
where R is the minimum resolvable angular distance between two stars in arcseconds, D is the diameter of the telescope in millimeters. From the table below (Tab.1) you can clearly see how this value changes with increasing entrance aperture of the telescope. However, in reality, this value can vary significantly between two telescopes, even with the same lens diameter. This may depend on the type of optical system, on the quality of manufacturing of the optics, and, of course, on the state of the atmosphere.

What you need to have in order to start observing. The most important thing, of course, is the telescope. It should be noted that many amateurs misinterpret the Davis formula, believing that only it determines the possibility of resolving a close double pair. It is not right. Several years ago, I met with an amateur who complained that for several seasons now he had been unable to separate a pair of stars with a 2.5-inch telescope that were only 3 arcseconds apart. In fact, it turned out that he tried to do this using a low magnification of 25x, arguing that with such magnification he had better visibility. Of course, he was right in one thing, a smaller increase significantly reduces the harmful effects of air currents in the atmosphere, but the main mistake was that he did not take into account another parameter that affects the success of the separation of a close couple. I'm talking about a value known as "resolution magnification".
P = 0.5 * D (F.2)
I have not seen the formula for calculating this quantity as often in other articles and books as the description of the Davis limit, which is probably why people have such a misconception about the ability to resolve a close pair with minimal magnification. True, one must clearly realize that this formula gives an increase when it is already possible to observe the diffraction pattern of stars, and, accordingly, the closely located second component. Once again I emphasize the word observe. Since to carry out measurements, the value of this magnification must be multiplied by at least 4 times, if atmospheric conditions allow.
A few words about the diffraction pattern. If you look at a relatively bright star through a telescope at the highest possible magnification, then you will notice that the star does not appear as a point, as it should in theory when observing a very distant object, but as a small circle surrounded by several rings (the so-called diffraction rings ). It is clear that the number and brightness of such rings directly affects the ease with which you can separate a close couple. It may happen that the weak component will simply be dissolved in the diffraction pattern, and you will not be able to distinguish it against the background of bright and dense rings. Their intensity depends directly on both the quality of the optics and the screening coefficient of the secondary mirror in the case of using a reflector or catadioptric system. The second value, of course, does not make serious adjustments to the possibility of resolving a certain pair in general, but with increasing screening, the contrast of the weak component relative to the background decreases.

In addition to the telescope, of course, you will also need measuring instruments. If you are not going to measure the position of the components relative to each other, then, in general, you can do without them. Let's say you may be quite satisfied with the very fact that you managed to resolve nearby stars with your instrument and make sure that the stability of the atmosphere today is suitable or your telescope gives good results, and you have not yet lost your former skills and dexterity. For deeper and more serious purposes it is necessary to use a micrometer and a dial scale. Sometimes such two devices can be found in one special eyepiece, in the focus of which a glass plate with thin lines is installed. Typically, the marks are applied at certain distances using a laser in a factory setting. A view of one such industrially produced eyepiece is shown nearby. Not only are marks made there every 0.01 microns, but also an hour scale is marked along the edge of the field of view to determine the position angle.


Such eyepieces are quite expensive and you often have to resort to other, usually homemade, devices. It is possible to design and build a homemade wire micrometer over a period of time. The essence of its design is that one of two very thin wires can move relative to the other if the ring with divisions applied to it rotates. Through appropriate gears, it is possible to ensure that a complete rotation of such a ring gives a very slight change in the distance between the wires. Of course, such a device will require a very long calibration until the exact value of one division of such a device is found. But it is available in production. These devices, both the eyepiece and the micrometer, require some additional effort on the part of the observer for normal operation. Both work on the principle of measuring linear distances. As a consequence, there is a need to connect two measures (linear and angular) together. This can be done in two ways, by empirically determining from observations the value of one division of both devices, or by calculating theoretically. The second method cannot be recommended, since it is based on exact data on the focal length of the optical elements of the telescope, but if this is known with sufficient accuracy, then the angular and linear measures can be related by the relation:
A = 206265" / F (F.3)
This gives us the angular magnitude of an object located at the main focus of a telescope (F) and a size of 1 mm.. To put it simply, then one millimeter at the main focus of a 2000mm telescope will be equivalent to 1.72 arcminutes. The first method often turns out to be more accurate, but requires considerable time. Place any type of measuring instrument on the telescope and look at a star with known coordinates. Stop the clock mechanism of the telescope and note the time it takes for the star to travel from one division to another. The several results obtained are averaged and the angular distance corresponding to the position of the two marks is calculated using the formula:
A = 15 * t * COS(D) (F.4)
Taking measurements
As already noted, the tasks that are posed to the observer of double stars come down to two simple things - separation into components and measurement. If everything described earlier serves to help solve the first task, determine the possibility of performing it and contains a certain amount of theoretical material, then this part discusses issues directly related to the process of measuring a stellar pair. To solve this problem, you only need to measure a couple of quantities.
Position angle


This quantity is used to describe the direction of one object relative to another, or for confident positioning on the celestial sphere. In our case, this involves determining the position of the second (weaker) component relative to the brighter one. In astronomy, position angle is measured from a point pointing north (0°) and then towards east (90°), south (180°) and west (270°). Two stars with the same right ascension have a position angle of 0° or 180°. If they have the same declination, the angle will be either 90° or 270°. The exact value will depend on the position of these stars relative to each other (which is to the right, which is higher, and so on) and which of these stars is chosen as the reference point. In the case of double stars, this point is always taken to be the brighter component. Before measuring the position angle, it is necessary to correctly orient the measuring scale according to the cardinal directions. Let's look at how this should happen when using a micrometer eyepiece. By placing the star in the center of the field of view and turning off the clock mechanism, we force the star to move in the field of view of the telescope from east to west. The point at which the star will go beyond the boundaries of the field of view is the point of direction to the west. If the eyepiece has an angular scale at the edge of the field of view, then by rotating the eyepiece it is necessary to set the value of 270 degrees at the point where the star leaves the field of view. You can check the correct installation by moving the telescope so that the star just begins to appear from beyond the line of sight. This point should coincide with the 90 degree mark, and the star, during its movement, should pass the center point and begin to leave the field of view exactly at the 270 degree mark. After this procedure, it remains to deal with the orientation of the north-south axis. It is necessary, however, to remember that a telescope can produce both a telescopic image (the case of a completely inverted image along two axes) and an inverted one along only one axis (in the case of using a zenith prism or a deflecting mirror). If we now focus on the star pair we are interested in, then placing the main star in the center, it is enough to take readings of the angle of the second component. Such measurements are of course best carried out at the highest possible magnification for you.
Measuring angles


In truth, the hardest part of the work has already been done, as described in the previous section. All that remains is to take the results of measuring the angle between the stars from the micrometer scale. There are no special tricks here and the methods for obtaining the result depend on the specific type of micrometer, but I will reveal the general accepted principles using the example of a homemade wire micrometer. Point a bright star at the first wire mark in a micrometer. Then, by rotating the marked ring, align the second component of the star pair and the second line of the device. At this stage, you need to remember the readings of your micrometer for further operations. Now, by rotating the micrometer 180 degrees, and using the telescope's precise movement mechanism, again align the first line in the micrometer with the main star. The second mark of the device should accordingly be away from the second star. Having twisted the micrometer disk so that the second mark coincides with the second star and, taking a new value from the scale, subtract from it the old value of the device to obtain double the angle. It may seem incomprehensible why such an intricate procedure was carried out when it could have been simpler by taking readings from the scale without turning the micrometer over. This is certainly easier, but in this case the measurement accuracy will be slightly worse than in the case of using the double angle technique described above. Moreover, the zero marking on a homemade micrometer may have somewhat dubious accuracy, and it turns out that we are not working with a zero value. Of course, in order to obtain relatively reliable results, we need to repeat the process of measuring the angle several times to obtain an average result from numerous observations.
Other measurement techniques
The principles outlined above for measuring the distance and positional angle of a close pair are essentially classical methods, the use of which can also be found in other branches of astronomy, for example, selenography. But often amateurs do not have access to an accurate micrometer and have to be content with other available means. Let's say, if you have an eyepiece with a crosshair, then simple angular measurements can be made with it. For a very close pair of stars it will not work quite accurately, but for wider ones you can use the fact that a star with declination d per second of time, based on formula F.4, travels a path of 15 * Cos(d) arcseconds. Taking advantage of this fact, you can detect the period of time when both components intersect the same line of the eyepiece. If the position angle of such a star pair is 90 or 270 degrees, then you are lucky, and there is no need to perform any further computational actions, just repeat the entire measurement process several times. Otherwise, you have to use cunning methods to determine the position angle, and then, using trigonometric equations to find the sides in a triangle, calculate the distance between the stars, which should be the value:
R = t * 15 * Cos(d) / Sin(PA) (F.5)
where PA is the position angle of the second component. If you make measurements in this manner more than four or five times, and have a time (t) measurement accuracy of no worse than 0.1 seconds, then using an eyepiece with the highest possible magnification, you can reasonably expect to obtain a measurement accuracy of up to 0.5 arcseconds or even better. It goes without saying that the crosshair in the eyepiece must be positioned exactly at 90 degrees and be oriented according to directions to different cardinal directions, and that at position angles close to 0 and 180 degrees, the measurement technique must be slightly changed. In this case, it is better to slightly deflect the crosshair by 45 degrees, relative to the meridian, and use the following method: by noticing two moments when both components intersect one of the crosshair lines, we obtain the times t1 and t2 in seconds. During time t (t=t2-t1) the star travels a path of X seconds of arc:
X = t * 15 * Cos(delta) (F.6)
Now knowing the position angle and the general orientation of the crosshair measuring line in the eyepiece, we can supplement the previous expression with a second one:
X = R * | Cos(PA) + Sin(PA) | (for SE-NW orientation) (F.7)
X = R * | Cos(PA) - Sin(PA) | (for orientation along the NE-SW line)
It is possible to place a very distant component in the field of view in such a way that it does not enter the field of view of the eyepiece, being located at its very edge. In this case, also knowing the position angle, the time of passage of another star through the field of view and this value itself, you can begin calculations based on calculating the length of the chord in a circle with a certain radius. You can try to determine the position angle by using other stars in the field of view, the coordinates of which are known in advance. By measuring the distances between them with a micrometer or stopwatch, using the technique described above, you can try to find the missing values. Of course, I won’t give the formulas themselves here. Their description may take up a significant part of this article, especially since they can be found in geometry textbooks. The truth is somewhat more complicated with the fact that ideally you will have to solve problems with spherical triangles, and this is not the same as triangles on a plane. But if you use such tricky measurement methods, then in the case of binary stars, when the components are located close to each other, you can simplify your task by forgetting about spherical trigonometry altogether. The accuracy of such results (already inaccurate) cannot be greatly affected by this. The best way to measure the position angle is to use a protractor, such as is used in schools, and adapt it for use with an eyepiece. It will be quite accurate, and most importantly, very accessible.
Among the simple measurement methods, we can mention another, rather original one, based on the use of diffraction nature. If you put a specially made grating (alternating parallel strips of an open aperture and a screened one) on the entrance aperture of your telescope, then when you look at the resulting image through the telescope, you will find a series of fainter “satellites” around the visible stars. The angular distance between the “main” star and the “closest” twin will be equal to:
P = 206265 * lambda / N (F.8)
Here P is the angular distance between the double and the main image, N is the sum of the width of the open and shielded sections of the described device, and lambda is the wavelength of light (560nm is the maximum sensitivity of the eye). If you now measure the three angles using the type of position angle measuring device available to you, you can rely on the formula and calculate the angular distance between the components, based on the phenomenon described above and the position angles:
R = P * Sin | PA1 - PA | / Sin | PA2 - PA | (F.10)
The value of P was described above, and the angles PA, PA1 and PA2 are defined as: PA is the position angle of the second component of the system relative to the main image of the main star; PA1 - position angle of the main image of the main star, relative to the secondary image of the main star plus 180 degrees; PA2 is the position angle of the main image of the second component, relative to the secondary image of the main star. As the main disadvantage, it should be noted that when using this method, large losses in the brightness of stars are observed (more than 1.5-2.0m) and works well only on bright pairs with a small difference in brightness.
On the other hand, modern methods in astronomy have made it possible to make a breakthrough in the observation of binaries. Photography and CCD astronomy allow us to take a fresh look at the process of obtaining results. With both a CCD image and a photograph, there is a method of measuring the number of pixels, or the linear distance, between a pair of stars. After calibrating the image, by calculating the magnitude of one unit based on other stars whose coordinates are known in advance, you calculate the desired values. Using CCD is much preferable. In this case, the measurement accuracy can be an order of magnitude higher than with the visual or photographic method. High-resolution CCD can record very close pairs, and subsequent processing with various astrometry programs can not only facilitate the entire process, but also provide extremely high accuracy down to several tenths, or even hundredths, of fractions of an arcsecond.

> Double stars

– features of observation: what it is with photos and videos, detection, classification, multiples and variables, how and where to look in Ursa Major.

Stars in the sky often form clusters, which can be dense or, on the contrary, scattered. But sometimes stronger connections arise between stars. And then it is customary to talk about double systems or double stars. They are also called multiples. In such systems, stars directly influence each other and always evolve together. Examples of such stars (even with the presence of variables) can be found literally in the most famous constellations, for example, Ursa Major.

Discovery of double stars

The discovery of double stars was one of the first advances made using astronomical binoculars. The first system of this type was the Mizar pair in the constellation Ursa Major, which was discovered by the Italian astronomer Riccoli. Since there are an incredible number of stars in the Universe, scientists decided that Mizar could not be the only binary system. And their assumption turned out to be completely justified by future observations.

In 1804, William Herschel, a famous astronomer who had been making scientific observations for 24 years, published a catalog detailing 700 double stars. But even then there was no information about whether there was a physical connection between the stars in such a system.

A small component "sucks" gas from a large star

Some scientists have taken the view that double stars depend on a common stellar association. Their argument was the heterogeneous shine of the components of the pair. Therefore, it seemed that they were separated by a significant distance. To confirm or refute this hypothesis, measurements of the parallactic displacement of stars were required. Herschel took on this mission and, to his surprise, found out the following: the trajectory of each star has a complex ellipsoidal shape, and not the appearance of symmetrical oscillations with a period of six months. In the video you can observe the evolution of double stars.

This video shows the evolution of a close binary pair of stars:

You can change the subtitles by clicking on the "cc" button.

According to the physical laws of celestial mechanics, two bodies connected by gravity move in an elliptical orbit. The results of Herschel's research became proof of the assumption that there is a gravitational force connection in binary systems.

Classification of double stars

Binary stars are usually grouped into the following types: spectral binaries, photometric binaries, and visual binaries. This classification gives an idea of ​​the stellar classification, but does not reflect the internal structure.

Using a telescope, you can easily determine the duality of visual double stars. Today there is evidence of 70,000 visual binary stars. Moreover, only 1% of them definitely have their own orbit. One orbital period can last from several decades to several centuries. In turn, building an orbital path requires considerable effort, patience, precise calculations and long-term observations in an observatory.

Often, the scientific community has information about only some fragments of orbital movement, and they reconstruct the missing sections of the path using a deductive method. Do not forget that the orbital plane may be inclined relative to the line of sight. In this case, the apparent orbit is seriously different from the real one. Of course, with high accuracy of calculations, it is possible to calculate the true orbit of binary systems. To do this, Kepler's first and second laws are applied.

Mizar and Alcor. Mizar is a double star. On the right is the Alcor satellite. There's only one light year between them

Once the true orbit is determined, scientists can calculate the angular distance between the binary stars, their mass, and their rotation period. Often, Kepler's third law is used for this, which helps to find the sum of the masses of the components of the pair. But to do this you need to know the distance between the Earth and the double star.

Double photometric stars

The dual nature of such stars can be learned only from periodic fluctuations in brightness. As they move, stars of this type take turns blocking each other, which is why they are often called eclipsing binaries. The orbital planes of these stars are close to the direction of the line of sight. The smaller the area of ​​the eclipse, the lower the brightness of the star. By studying the light curve, the researcher can calculate the inclination angle of the orbital plane. When two eclipses are recorded, there will be two minima (decreases) in the light curve. The period when 3 successive minima are observed in the light curve is called the orbital period.

The period of double stars lasts from a couple of hours to several days, which makes it shorter in relation to the period of visual double stars (optical double stars).

Spectral dual stars

Through the method of spectroscopy, researchers record the process of splitting spectral lines, which occurs as a result of the Doppler effect. If one component is a weak star, then only periodic fluctuations in the positions of single lines can be observed in the sky. This method is used only when the components of the binary system are at a minimum distance and their identification using a telescope is complicated.

Binary stars that can be studied through the Doppler effect and a spectroscope are called spectrally dual. However, not every double star has a spectral character. Both components of the system can approach and move away from each other in the radial direction.

According to the results of astronomical research, most of the double stars are located in the Milky Way galaxy. The percentage ratio of single and double stars is extremely difficult to calculate. Working through subtraction, one can subtract the number of known double stars from the total stellar population. In this case, it becomes clear that binary stars are in the minority. However, this method cannot be called very accurate. Astronomers are familiar with the term “selection effect.” To fix the binarity of stars, their main characteristics must be determined. Special equipment will be useful for this. In some cases, it is extremely difficult to detect double stars. Thus, visually, double stars are often not visualized at a significant distance from the astronomer. Sometimes it is impossible to determine the angular distance between stars in a pair. To detect spectroscopic binaries or photometric stars, it is necessary to carefully measure wavelengths in spectral lines and collect modulations of light fluxes. In this case, the brilliance of the stars should be quite strong.

All this sharply reduces the number of stars suitable for study.

According to theoretical developments, the proportion of double stars in the stellar population varies from 30% to 70%.

Problem excess weight makes itself felt not only in the summer on the beach. Every day, looking into the mirror, you have to sadly observe a double chin, jowls and blurry contours. Fortunately, all this can be disguised if you master makeup for a full face with all its nuances.

Peculiarities

For plump girls, make-up artists offer make-up, the main task of which is to elongate the face and make it visually thinner. To solve this, techniques such as contouring (to make the outlines clearer) and vertical shading are used.

Tone and relief

  1. Without a foundation that models contours and visually stretches them, makeup is impossible.
  2. A light foundation (primer) highlights the oval, a darker one - everything else (don’t forget about the neck and décolleté).
  3. Concealers should be matte and dense in texture.
  4. It's important to highlight your eyes, so be sure to cover the dark circles underneath with concealer.
  5. The powder is compact and not shiny.
  6. Apply blush with a soft brush, moving from top to bottom. Ideal shades - beige, bronze.

Eyes and eyebrows

  1. Give preference to lengthening mascara.
  2. Limit pearlescent shadows.
  3. Carefully shade all shade transitions.
  4. The inner corners need to be lightened, the outer corners need to be darkened.
  5. All lines should be directed upwards.
  6. It is better to shade the ends.
  7. Eyebrows should not be too thin or too wide. The bend is moderate.

Lips

  1. There is no need to add extra volume to your lips.
  2. Lip contouring is also excluded.
  3. Young girls can use unobtrusive glitter.
  4. After 35, it is better to give preference to matte lipstick - coral or pink.

If you have a full face, don't worry. Usually girls with this defect have very beautiful eyes, smooth, clear skin and no wrinkles. Try to highlight your strengths and disguise your faded features as much as possible with skillful make-up.

Match your eye color

In this type of makeup, it is necessary to take into account the color of the eyes, since it is recommended to focus on them.

For green eyes

  1. To highlight green eyes on a full face, you will need shadows in shades such as turquoise, green, yellow, and blue.
  2. Unlike makeup for blue-eyed beauties, this will require a multi-layer technique. So don't be afraid to apply multiple layers of shadow.
  3. The main thing is to remember to shade everything thoroughly. A full face does not tolerate contrasts.
  4. Choose the color of the eyeliner to match the shadows: it should be a little richer.
  5. Raise the arrows up so that horizontal lines do not make the face even fuller.
  6. For daytime make-up, use blue or green mascara. For festive, evening wear - black or brown.
  7. To make your lips more prominent, take lipstick or gloss with shimmer. The recommended shade is bright cherry or coral.

For blue-eyed people

  1. Recommended eyeshadow palette: silver, pink, gold, pearl, purple, lilac, sea green, turquoise. If you do, you can take black and brown.
  2. For blue eyes you need to use the easiest techniques. Multilayering is excluded. So the shadows can be applied in 1-2 layers, but no more.
  3. It's the same with mascara. Don't overdo it: 1 application will be enough. Recommended colors - gray, brown (for daytime), black (for evening).
  4. Lipstick and lip gloss can be in a pink tone, but taking into account age. After 35 it is better to use cream or burgundy. The main thing is without moisture and volume.
  5. Makeup artists suggest using these same color solutions for gray-eyed girls.

For brown-eyed people

  1. Makeup for a full face with brown eyes begins with the right selection. Choose beige or apricot shades - they visually lengthen your features.
  2. To add definition to your cheekbones, apply a lilac-pink blush on them. Move the terracotta ones away - they will make them flat.
  3. The shadow palette should open your eyes. The colors in your palette are blue, purple, bronze, gold, chestnut, beige, honey, pink.
  4. The liner can be blue, golden, purple, chestnut, black - to match the color of the shadows. It is better to twist the arrows up.
  5. For eyelashes you will need lengthening mascara in black, blue, brown or purple.
  6. The eyebrow shape must be correct. Avoid straight horizontal lines and too pronounced flirty curves.
  7. Lipstick and lip gloss can be in the following colors: ripe cherry, warm nude, pink neon, coral.

The choice of makeup color scheme may also depend on hair color. But it is the eyes that play a decisive role in this matter.

Step-by-step instruction

Different style make-up options for obese women allow them to feel attractive and beautiful both in everyday life and on holidays. Basic (and) must be mastered.

Day

  1. To lengthen a full face, use liquid foundation without silicone. Pay special attention to masking the wings of the nose and the sides of the cheeks.
  2. To even out the tone, it is better to take matte powder.
  3. To make the contours of the face clearer and more prominent, they need to be darkened, and the center (nose, forehead, chin) should be lightened as much as possible. To do this, you can work with the corrector directly on top of the powder.
  4. You can apply sand blush to your cheekbones.
  5. The upper eyelids are painted in 1 layer with mother-of-pearl. Silver color is better.
  6. Very thin arrows along the upper eyelids are drawn in anthracite and curved upward.
  7. We do not work with the lower part of the eyes during daytime makeup.
  8. We open our eyes with gray lengthening mascara in 1 layer.
  9. For lips, take a glossy gloss in a natural shade.

Evening

  1. Pink concealer allows you to draw out the contour of your face.
  2. To ensure a flawless make-up, pay special attention to camouflage your neckline.
  3. Coral bright blush will elongate the cheekbones.
  4. The shadows fall on the upper eyelid in layers: black, anthracite, emerald. The main thing is to shade everything well so as not to create contrasts.
  5. The lower eyelids are shaded with a shade of wet asphalt.
  6. The black arrows should follow the shape of the eye and connect at the top, leading the lines to the temples.
  7. The outer corners can be highlighted with a white liner or shadows.
  8. Mascara in 2 layers - black lengthening.
  9. It is better not to use glitter and shimmer.
  10. Matte coral lipstick and clear gloss will complete your evening makeup.

If internal complexes are the cause, you have only two ways to solve the problem. The first is to lose weight. But it is long and requires considerable strength and patience. The second is to learn the correct makeup for a full face, which will make it visually thinner. Do not neglect the advice of makeup artists in such a situation - they will make you look much better.
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