Synonyms: sky, horizon, sky, sky, sunset of the sky, with an eye, view, curtain, close, mischief, ovid, look around.
Distance to the visible horizon
- If visible horizon define as the boundary between heaven and earth, then calculate geometric range the visible horizon, using the Pythagorean theorem:
Where d and h in kilometers or
Where d in kilometers, and h in meters. Below is the distance to the horizon when viewed from different heights:
| Height above the surface of the Earth h | Distance to the horizon d | Example of an observation site |
|---|---|---|
| 1.75 m | 4,7 km | standing on the ground |
| 25 m | 17.9 km | 9-storey building |
| 50 m | 25.3 km | Ferris wheel |
| 150 m | 43.8 km | Balloon |
| 2 km | 159.8 km | mountain |
| 10 km | 357.3 km | plane |
| 350 km | 2114.0 km | spaceship |
To facilitate calculations of the horizon range depending on the height of the observation point and taking into account refraction, tables and nomograms have been compiled. Actual values of the visible horizon range can differ significantly from the tabulated values, especially at high latitudes, depending on the state of the atmosphere and underlying surface. Raising (lowering) the horizon refers to the phenomena associated with refraction. When positive refraction the visible horizon rises (expands), geographic range the visible horizon increases in comparison with geometric range, objects are visible, usually hidden by the curvature of the Earth. Under normal temperature conditions, the horizon is uplifted by 6-7%. With an increase in temperature inversion, the visible horizon can rise to the true (mathematical) horizon, the earth's surface straightens out, becomes flat, the visibility range becomes infinitely large, and the radius of curvature of the beam becomes equal to the radius of the globe. With an even stronger temperature inversion, the visible horizon will rise above the true one. It will seem to the observer that he is at the bottom of a huge depression. Objects that are far beyond the geodesic horizon will rise from the horizon and become visible (as if to float in the air). In the presence of strong temperature inversions, conditions are created for the appearance of upper mirages. Large temperature gradients are created when the earth's surface is strongly heated by the sun's rays, often in deserts and steppes. Large gradients can occur in middle and even high latitudes on summer days in sunny weather: over sandy beaches, over asphalt, over bare soil. Such conditions are favorable for the occurrence of inferior mirages. When negative refraction the visible horizon decreases (narrows), even those objects that are visible in normal conditions are not visible.
By the way: Space horizon(the horizon of particles) is also a mentally imaginary sphere with a radius equal to the distance that light has traveled during the existence of the Universe, and the entire set of points of the Universe located at this distance. Visibility range
In the figure on the right, the visibility range of the object is determined by the formula
,
Where - visibility range in kilometers,
and - the heights of the observation point and the object in meters.
For an approximate calculation of the visibility range of objects, the Struisky nomogram is used (see Fig.): On the two extreme scales of the nomogram, points are marked corresponding to the height of the observation point and the height of the object, then a straight line is drawn through them and at the intersection of this straight line with the middle scale, the visibility range of the object is obtained.
On nautical charts, in directions and other navigational aids, the range of visibility of beacons and lights is indicated for an observation point height of 5 m. If the height of the observation point is different, then an amendment is introduced.
Horizon on the moon
It must be said that distances on the Moon are very deceiving. Due to the lack of air, distant objects are seen more clearly on the moon and therefore always seem closer.
Artificial horizon- the device used to determine the true horizon.
For example, the true horizon is easy to determine if you bring a glass of water to your eyes so that the water level is visible as a straight line.
Horizon in philosophy
The concept of the horizon is introduced into philosophy by Edmund Husserl, and Gadamer defines it as follows: "The horizon is a field of view that encompasses and embraces everything that can be seen from any point."
see also
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Notes (edit)
- .
- Article "Horizon" in the Great Soviet Encyclopedia
- Ermolaev G.G., Andronov L.P., Zoteev E.S., Kirin Yu.P., Cherniev L.F. Marine navigation / under the general editorship of the captain of the long voyage G. G. Ermolaev. - 3rd edition, revised. - Moscow: Transport, 1970 .-- 568 p.
- . Interpretations of the expression "visible horizon". .
- . Horizon. Space and astronomy. .
- Dal V.I. Explanatory Dictionary of the Living Great Russian Language. - M .: OLMA Media Group, 2011 .-- 576 p. - ISBN 978-5-373-03764-8.
- Veruzhsky N.A. Nautical Astronomy: Theoretical Course. - M .: RKconsult, 2006 .-- 164 p. - ISBN 5-94976-802-7.
- Perelman Ya.I. Horizon // Interesting geometry. - M .: Rimis, 2010 .-- 320 p. - ISBN 978-5-9650-0059-3.
- Calculated by the formula "distance = 113 roots of height", thus, the influence of the atmosphere on the propagation of light is not taken into account and it is assumed that the Earth has the shape of a ball.
- Nautical tables (MT-2000). Adm. No. 9011 / editor-in-chief K. A. Yemets. - SPb: GUN and O, 2002 .-- 576 p.
- . Calculation of distance to the horizon and line of sight online. .
- . What is the horizon next?. .
- Lukash V.N., Mikheeva E.V. Physical cosmology. - M .: Physical and mathematical literature, 2010 .-- 404 p. - ISBN 5922111614.
- D. Yu. Klimushkin; Grablevsky S.V. . Space horizon (2001). .
- . Chapter VII. Navigation.
- . Visible horizon and visibility range. .
- . Were Americans on the Moon?. .
- . Interpretations of the expression "true horizon". .
- Zaparenko Victor. Great encyclopedia of drawing by Viktor Zaparenko. - M .: AST, 2007 .-- 240 p. - ISBN 978-5-17-041243-3.
- Truth and Method. P. 358
Literature
- Vitkovsky V.V.// Encyclopedic Dictionary of Brockhaus and Efron: in 86 volumes (82 volumes and 4 additional). - SPb. , 1890-1907.
- Horizon // Great Soviet Encyclopedia: [in 30 volumes] / Ch. ed. A.M. Prokhorov... - 3rd ed. - M. : Soviet Encyclopedia, 1969-1978.
Excerpt from Horizon
- What's the matter with you, Masha?
“Nothing… I felt so sad… sad about Andrei,” she said, wiping her tears on her daughter-in-law's knees. Several times, during the morning, Princess Marya began to prepare her daughter-in-law, and each time she began to cry. These tears, of which the little princess did not understand the reason, alarmed her, no matter how observant she was. She said nothing, but looked around uneasily, looking for something. Before dinner, the old prince, whom she had always feared, entered her room, now with a particularly restless, angry face, and without saying a word, left. She looked at Princess Marya, then pondered with that expression in the eyes of an inward-looking attention that is experienced by pregnant women, and suddenly burst into tears.
- Did you get anything from Andrey? - she said.
- No, you know that the news could not come yet, but mon is worried, and I am afraid.
- Oh nothing?
“Nothing,” said Princess Marya, looking firmly at her daughter-in-law with radiant eyes. She decided not to tell her and persuaded her father to hide the terrible news from her daughter-in-law until her permission, which was supposed to be the other day. Princess Marya and the old prince, each in his own way, wore and hid their grief. The old prince did not want to hope: he decided that Prince Andrei was killed, and despite the fact that he sent an official to Austria to look for the trace of his son, he ordered him a monument in Moscow, which he intended to erect in his garden, and told everyone that his son was killed. He tried not to change to lead the old way of life, but his strength betrayed him: he walked less, ate less, slept less, and every day he became weaker. Princess Marya hoped. She prayed for her brother as if she were alive, and every minute she waited for the news of his return.- Ma bonne amie, [My good friend,] - said the little princess on the morning of March 19 after breakfast, and her sponge with a mustache rose up out of old habit; but as in all not only smiles, but the sounds of speeches, even the gaits in this house from the day of receiving the terrible news, there was sadness, even now the smile of the little princess, who succumbed to the general mood, although she did not know its reason, was such that she even more reminiscent of general sadness.
- Ma bonne amie, je crains que le fruschtique (comme dit Fock - chef) de ce matin ne m "aie pas fait du mal. [Dear friend, I'm afraid that the current frishtik (as the chef Fock calls him) won't make me feel bad. ]
- What about you, my soul? You are pale. Oh, you are very pale, - said Princess Marya in dismay, running up to her daughter-in-law with her heavy, soft steps.
- Your Excellency, should you send for Marya Bogdanovna? - said one of the maids who were here. (Marya Bogdanovna was a midwife from a district town who had lived in Lysyh Gory for another week.)
“And in fact,” Princess Marya said, “maybe, exactly. I will go. Courage, mon ange! [Fear not, my angel.] She kissed Lisa and wanted to leave the room.
- Oh, no, no! - And besides pallor, on the face of the little princess, a childish fear of inevitable physical suffering was expressed.
- Non, c "est l" estomac ... dites que c "est l" estomac, dites, Marie, dites ..., [No, this is a stomach ... tell Masha that this is a stomach ...] - and the princess cried childishly suffering, capricious and even somewhat feigned, breaking their little hands. The princess ran out of the room after Marya Bogdanovna.
- Mon Dieu! Mon Dieu! [Oh my God! Oh my god!] Oh! She heard from behind her.
Rubbing full, small, white hands, the midwife was already walking towards her, with a considerably calm face.
- Marya Bogdanovna! It seems to have begun, ”said Princess Marya, looking with fearful open eyes at her grandmother.
“Well, thank God, princess,” said Marya Bogdanovna without adding a step. “You girls shouldn't know about this.
- But how has the doctor not arrived from Moscow yet? - said the princess. (At the request of Liza and Prince Andrei, by the time they were sent to Moscow for an obstetrician, and they waited for him every minute.)
“Nothing, princess, don't worry,” said Marya Bogdanovna, “and everything will be fine without the doctor.
Five minutes later the princess heard from her room that they were carrying something heavy. She looked out - the waiters were carrying a leather sofa, which was in Prince Andrey's office, into the bedroom for some reason. There was something solemn and quiet on the faces of the people who were carrying them.
Princess Marya sat alone in her room, listening to the sounds of the house, occasionally opening the door when they passed by, and looking closely at what was happening in the corridor. Several women with quiet steps passed there and from there, looked back at the princess and turned away from her. She did not dare to ask, shut the door, returned to her room, and then sat down in her chair, then took up the prayer book, then knelt in front of the icon case. To her misfortune and surprise, she felt that prayer did not calm her excitement. Suddenly the door of her room opened quietly and her old nanny Praskovya Savishna, tied with a handkerchief, appeared on her threshold, almost never, due to the prince's prohibition, who did not enter her room.
- With you, Mashenka, I came to sit, - said the nanny, - but here's the prince's wedding candles in front of the saint brought light, my angel, - she said with a sigh.
- Oh, how glad I am, nanny.
- God is merciful, dove. - The nanny lit candles wrapped in gold in front of the icon case and sat down by the door with a stocking. Princess Marya took the book and began to read. Only when footsteps or voices were heard did the princess fearfully, questioningly, and the nanny looked reassuringly at each other. In all parts of the house, the same feeling was poured out and possessed by everyone that Princess Marya felt, sitting in her room. According to the belief that the fewer people know about the suffering of the parturient woman, the less she suffers, everyone tried to pretend not to know; no one spoke about this, but in all people, except for the usual degree and respectfulness of good manners that reigned in the prince's house, one could see some kind of common concern, a softened heart and the consciousness of something great, incomprehensible, happening at that moment.
There was no laughing in the big girl's room. In the waiter's room, all the people sat and were silent, ready for something. Torches and candles were burned on the yard and did not sleep. The old prince, stepping on his heel, walked around the office and sent Tikhon to Marya Bogdanovna to ask: what? - Just tell me: the prince ordered to ask what? and come tell me what she has to say.
“Report to the prince that labor has begun,” said Marya Bogdanovna, looking significantly at the messenger. Tikhon went and reported to the prince.
- Well, - said the prince, closing the door behind him, and Tikhon did not hear the slightest sound in the study. A little later, Tikhon entered the office, as if to fix the candles. Seeing that the prince was lying on the sofa, Tikhon looked at the prince, at his upset face, shook his head, silently approached him and, kissing him on the shoulder, left without straightening the candles and without saying why he had come. The most solemn sacrament in the world continued to be performed. The evening has passed, the night has come. And the feeling of expectation and softening of the heart before the incomprehensible did not fall, but rose. Nobody slept.It was one of those March nights when winter seemed to want to take its toll and pour out its last snows and blizzards with desperate malice. To meet the German doctor from Moscow, who was expected every minute and for whom a set-up was sent to the main road, to the turn to the country road, horsemen with lanterns were sent to escort him through bumps and jams.
Princess Marya had long since left the book: she sat silently, her radiant eyes fixed on the wrinkled face of the nanny, familiar to the slightest detail: on a lock of gray hair that had emerged from under her kerchief, on a hanging bag of skin under her chin.
Nanny Savishna, with a stocking in her hands, in a low voice, told, she herself did not hear and did not understand her words, told hundreds of times about how the deceased princess in Chisinau gave birth to Princess Marya, with a Moldavian peasant woman, instead of her grandmother.
- God have mercy, you never need a doctor, - she said. Suddenly a gust of wind fell on one of the exposed frames of the room (at the behest of the prince, one frame in each room was always exhibited with larks) and, knocking off a badly closed latch, rattled the damask curtain, and, smelling of cold, snow, blew out the candle. Princess Marya shuddered; the nanny, putting down her stocking, went to the window and leaning out began to catch the thrown frame. The cold wind ruffled the ends of her handkerchief and gray strands of hair that had strayed out.
- Princess, mother, someone is going on the prospect! She said, holding the frame and not closing it. - With lanterns, must be a doctor ...
- Oh my god! Thank God! - said Princess Marya, - we must go to meet him: he does not know Russian.
Princess Marya threw on a shawl and ran to meet those who were riding. When she passed the front, she saw through the window that some kind of carriage and lanterns were standing at the entrance. She went out onto the stairs. A tallow candle stood on the rail of the railing and flowed in the wind. The waiter Philip, with a frightened face and with another candle in his hand, stood below, on the first landing of the stairs. Even lower, around the bend, up the stairs, footsteps in warm boots could be heard moving. And some familiar voice, as it seemed to Princess Marya, was saying something.
- Thank God! The voice said. - And father?
- We lay down to rest, - answered the voice of the butler Demyan, who was already below.
Then a voice said something else, and Demian answered something, and steps in warm boots began to approach faster along an invisible turn of the stairs. "This is Andrey! - thought Princess Marya. No, it cannot be, it would be too extraordinary, "she thought, and at the same moment as she thought it, on the platform on which the waiter stood with a candle, the face and figure of Prince Andrei in a fur coat with a collar sprinkled snow. Yes, it was him, but pale and thin, and with a changed, strangely softened, but anxious expression on his face. He entered the stairs and hugged his sister.
- Didn't you receive my letter? - he asked, and without waiting for an answer, which he would not have received, because the princess could not speak, he returned, and with the obstetrician who entered after him (he gathered with him at the last station), with quick steps again entered the stairs and hugged his sister again. - What a fate! - he said, - Masha is dear - and, throwing off his fur coat and boots, he went to the princess's half.The little princess was lying on pillows in a white cap. (Suffering had just released her.) Black hair curled in strands around her sore, sweaty cheeks; a ruddy, charming mouth with a sponge covered with black hair was open, and she smiled happily. Prince Andrew entered the room and stopped in front of her, at the foot of the sofa on which she was lying. Shining eyes, looking childish, frightened and worried, stopped on him, without changing their expression. “I love you all, I didn’t do any harm to anyone, why am I suffering? help me, ”her expression said. She saw her husband, but did not understand the significance of his appearance now in front of her. Prince Andrey walked around the sofa and kissed her on the forehead.
“My darling,” he said, a word that he had never spoken to her. - God is merciful. She looked at him questioningly, childishly reproachful.
- I expected help from you, and nothing, nothing, and you too! - said her eyes. She was not surprised that he came; she did not understand that he had arrived. His arrival had nothing to do with her suffering and relief. The agony began again, and Marya Bogdanovna advised Prince Andrey to leave the room.
The midwife entered the room. Prince Andrew went out and, meeting Princess Marya, again went up to her. They spoke in a whisper, but the conversation fell silent every minute. They waited and listened.
- Allez, mon ami, [Go, my friend,] - Princess Marya said. Prince Andrew went to his wife again, and sat down in the next room, waiting. Some woman left her room with a frightened face and was embarrassed when she saw Prince Andrew. He covered his face with his hands and sat there for several minutes. Pathetic, helplessly animal moans were heard from outside the door. Prince Andrew got up, went to the door and wanted to open it. Someone was holding the door.
- You can't, you can't! - said a frightened voice from there. - He began to walk around the room. The screams stopped, and a few more seconds passed. Suddenly a terrible cry - not her cry, she could not scream like that - rang out in the next room. Prince Andrew ran to the door; the cry fell silent, the cry of a child was heard.
“Why did they bring the child there? thought for the first second Prince Andrew. Child? What? ... Why is there a child? Or was it a baby born? " When he suddenly understood all the joyful meaning of this cry, tears strangled him, and he, leaning with both hands on the windowsill, sobbing, wept like children cry. The door opened. The doctor, with his shirt sleeves rolled up, no frock coat, pale and with a trembling jaw, left the room. Prince Andrew turned to him, but the doctor looked at him in confusion and, without saying a word, walked by. The woman ran out and, seeing Prince Andrey, hesitated on the threshold. He entered his wife's room. She lay dead in the same position in which he had seen her five minutes ago, and the same expression, in spite of the fixed eyes and the pallor of her cheeks, was on this lovely, childish face with a sponge covered with black hair.
"I love you all and have never done anything wrong to anyone, and what have you done to me?" spoke her lovely, pitiful, dead face. In the corner of the room something small, red grunt and squeaked in Marya Bogdanovna's shaking white hands.Two hours later, Prince Andrei entered his father's study with quiet steps. The old man already knew everything. He stood at the very door, and as soon as it opened, the old man silently, with his old, stiff hands, like a vice, clasped his son's neck and sobbed like a child.
Three days later, the funeral service for the little princess was performed, and, saying goodbye to her, Prince Andrei ascended the steps of the coffin. And in the coffin was the same face, albeit with closed eyes. "Oh, what have you done to me?" it kept saying, and Prince Andrew felt that something had come off in his soul, that he was guilty of guilt, which he could not correct and forget. He couldn't cry. The old man also entered and kissed her wax pen, lying calmly and high on the other, and her face said to him: "Oh, what and why did you do this to me?" And the old man turned away angrily when he saw that face.
Five days later, the young Prince Nikolai Andreich was baptized. Mother held the diapers with her chin, while the priest smeared the wrinkled red palms and steps of the boy with a goose feather.
The godfather grandfather, afraid to drop, shuddering, carried the baby around the crumpled tin font and handed it over to the godmother, Princess Marya. Prince Andrew, dying with fear that the child would not be drowned, was sitting in another room, waiting for the end of the sacrament. He glanced happily at the child when his nanny carried him out, and nodded his head approvingly when the nanny informed him that the wax with hairs thrown into the font had not drowned, but swam through the font.Rostov's participation in Dolokhov's duel with Bezukhov was hushed up by the efforts of the old count, and Rostov, instead of being demoted, as he expected, was appointed an adjutant to the Moscow governor-general. As a result, he could not go to the village with the whole family, and remained in his new position all summer in Moscow. Dolokhov recovered, and Rostov became especially friends with him at this time of his recovery. Dolokhov was sick with his mother, who passionately and tenderly loved him. The old woman Marya Ivanovna, who fell in love with Rostov for his friendship with Fedya, often told him about her son.
What is the distance to the horizon for an observer on the ground? The answer - the approximate distance to the horizon - can be found using the Pythagorean theorem.
To carry out approximate calculations, we will make the assumption that the Earth has the shape of a ball. Then a person standing upright will be a continuation of the earth's radius, and the line of sight directed to the horizon will be tangent to the sphere (the surface of the Earth). Since the tangent is perpendicular to the radius drawn to the point of tangency, the triangle (center of the Earth) - (point of tangency) - (observer's eye) is rectangular.
Two sides of it are known. The length of one of the legs (the side adjacent to the right angle) is equal to the radius of the Earth $ R $, and the length of the hypotenuse (the side lying opposite the right angle) is equal to $ R + h $, where $ h $ is the distance from the ground to the observer's eyes.
According to the Pythagorean theorem, the sum of the squares of the legs is equal to the square of the hypotenuse. Hence, the distance to the horizon is
$$
d = \ sqrt ((R + h) ^ 2-R ^ 2) = \ sqrt ((R ^ 2 + 2Rh + h ^ 2) -R ^ 2) = \ sqrt (2Rh + h ^ 2).
$$ The value of $ h ^ 2 $ is very small compared to the term $ 2Rh $, therefore, the approximate equality
$$
d \ sqrt (2Rh).
$$
It is known that $ R 6400 $ km, or $ R 64 \ cdot10 ^ 5 $ m. We will assume that $ h 1 (,) 6 $ m. Then
$$
d \ sqrt (2 \ cdot64 \ cdot10 ^ 5 \ cdot 1 (,) 6) = 8 \ cdot 10 ^ 3 \ cdot \ sqrt (0 (,) 32).
$$ Using the approximate value $ \ sqrt (0 (,) 32) 0 (,) 566 $, we find
$$
d 8 \ cdot10 ^ 3 \ cdot 0 (,) 566 = 4528.
$$ Received answer - in meters. If we translate the found approximate distance from the observer to the horizon into kilometers, we get $ d 4.5 $ km.
In addition, there are three microplots related to the considered problem and the calculations performed.
I. How is the distance to the horizon related to the change in the height of the observation point? The formula $ d \ sqrt (2Rh) $ gives the answer: to double the distance $ d $, the height of $ h $ must be quadrupled!
II. In the formula $ d \ sqrt (2Rh) $, we had to extract the square root. Of course, the reader can take a smartphone with a built-in calculator, but, firstly, it is useful to think about how the calculator solves this problem, and secondly, it is worthwhile to feel mental freedom, independence from an “all-knowing” gadget.
There is an algorithm that reduces the extraction of the root to simpler operations - addition, multiplication and division of numbers. To extract the root from the number $ a> 0 $, consider the sequence
$$
x_ (n + 1) = \ frac12 (x_n + \ frac (a) (x_n)),
$$ where $ n = 0 $, 1, 2,…, and any positive number can be taken as $ x_0 $. The sequence $ x_0 $, $ x_1 $, $ x_2 $, ... converges very quickly to $ \ sqrt (a) $.
For example, when calculating $ \ sqrt (0.32) $, you can take $ x_0 = 0.5 $. Then
$$
\ eqalign (
x_1 & = \ frac12 (0.5+ \ frac (0.32) (0.5)) = 0.57, \ cr
x_2 & = \ frac12 (0.57+ \ frac (0.32) (0.57)) 0.5657. \ cr)
$$ Already at the second step we received the correct answer in the third decimal place ($ \ sqrt (0.32) = 0.56568… $)!
III. Sometimes algebraic formulas can be so clearly represented as ratios of elements of geometric figures that the whole "proof" consists in a drawing with the caption "Look!" (in the style of ancient Indian mathematicians).
It is possible to explain geometrically the used formula of "abbreviated multiplication" for the square of the sum
$$
(a + b) ^ 2 = a ^ 2 + 2ab + b ^ 2.
$$ Jean-Jacques Rousseau wrote in his Confession: “When I first discovered by calculating that the square of a binomial is equal to the sum of the squares of its terms and their double product, I, despite the correctness of the multiplication I have made, did not want to believe it until until he drew the figures. "
Literature
- Perelman Ya. I. Entertaining geometry in the open air and at home. - L .: Time, 1925. - [And any edition of the book by Ya. I. Perelman "Entertaining geometry"].
The spherical radius Bc of the small circle with c¹c²c³ is called the theoretical range of the visible horizon.
The value of the spherical radius depends on the height of the observer's eye above sea level.
So, if the eye of the observer is located at point A1 at an altitude of BA¹ = e¹ above sea level, then the spherical radius of Bc will be greater than the spherical radius of Bc.
To determine the relationship between the observer's eye height and the theoretical range of his visible horizon, consider the right-angled triangle AOc:
Ac² = AO² - Os²; AO = OB + e; OB = R,
Then AO = R + e; Os = R.
Due to the insignificance of the observer's eye height above sea level in comparison with the dimensions of the Earth's radius, the length of the tangent Ac can be taken equal to the value of the spherical radius of Bc and, denoting the theoretical distance of the visible horizon through D T, we obtain
D 2T = (R + e) ² - R² = R² + 2Re + e² - R² = 2Re + e²,
Fig. eight
Considering that the observer's eye height e on ships does not exceed 25 m, a 2R = 12 742 220 m, the ratio e / 2R is so small that it can be neglected without sacrificing accuracy. Hence,
since e and R are expressed in meters, then Dt will also turn out in meters. However, the actual range of the visible horizon is always greater than the theoretical one, since the ray coming from the observer's eye to a point on the earth's surface is refracted due to the unequal density of atmospheric layers in height.
In this case, the ray from point A to c goes not along the straight line Ac, but along the curve ASm "(see Fig. 8). Therefore, the observer sees the point c as visible in the direction of the tangent AT, that is, raised by the angle r = L TAc The angle d = L HAT is called the inclination of the visible horizon. And in fact, the visible horizon will be a small circle m ", m" 2, mz ", with a slightly larger spherical radius (Bm"> Bc).
The magnitude of the angle of terrestrial refraction is not constant and depends on the refractive properties of the atmosphere, which vary with the temperature and humidity of the air, the amount of suspended particles in the air. It also changes depending on the time of the year and the date of the day, so the actual range of the visible horizon in comparison with the theoretical one can increase up to 15%.
In navigation, an increase in the actual range of the visible horizon in comparison with the theoretical one is taken as 8%.
Therefore, denoting the actual, or, as it is also called, the geographical, range of the visible horizon through D e, we get:
To get De in nautical miles (assuming R and e in meters), the radius of the earth R, as well as the height of the eye e, is divided by 1852 (1 nautical mile equals 1852 m). Then
To get the result in kilometers, enter a factor of 1.852. Then
to facilitate calculations to determine the range of the visible horizon in table. 22-a (MT-63) gives the range of the visible horizon, depending on e, in the range from 0.25 to 5100 m, calculated by formula (4a).
If the actual eye height does not coincide with the numerical values indicated in the table, then the range of the visible horizon can be determined by linear interpolation between two values close to the actual eye height.
Visibility range of objects and lights
The range of sight of the object Dn (Fig. 9) will be the sum of two ranges of the visible horizon, depending on the height of the observer's eye (D e) and the height of the object (D h), ie.It can be determined by the formula
where h is the height of the landmark above the water level, m.
To facilitate the determination of the range of visibility of objects, use the table. 22-v (MT-63), calculated by the formula (5a): To determine from this table, from what distance the object will open, it is necessary to know the height of the observer's eye above the water level and the height of the object in meters.
The range of visibility of an object can also be determined by a special nomogram (Fig. 10). For example, the height of the eye above the water level is 5.5 m, and the height h of the setting sign is 6.5 m, in order to determine D n, a ruler is applied to the nomogram so that it connects the points corresponding to h and e on the extreme scales. The point of intersection of the ruler with the middle scale of the nomogram will show the desired range of visibility of the object D n (in Fig. 10 D n = 10.2 miles).
In navigational manuals - on charts, in directions, in descriptions of lights and signs - the range of sight of objects DK is indicated at an observer's eye height of 5 m (on English charts - 15 feet).
In the case when the actual height of the observer's eye is different, it is necessary to introduce a correction AD (see Fig. 9).
Fig. nine
Example. The range of visibility of the object, indicated on the map, DK = 20 miles, and the height of the observer's eye e = 9 m. Determine the actual range of visibility of the object D n using table. 22-a (MT-63). Decision.
At night, the visibility range of a fire depends not only on its height above water level, but also on the strength of the light source and on the discharge of the lighting device. Typically, the lighting apparatus and the strength of the light source are calculated so that the range of visibility of the fire at night corresponds to the actual range of visibility of the horizon from the height of the fire above sea level, but there are exceptions.
Therefore, the lights have their own "optical" visibility range, which may be greater or less than the horizon visibility range from the height of the fire.
In navigation manuals, the actual (mathematical) range of visibility of lights is indicated, but if it is greater than the optical range, then the latter is indicated.
The visibility range of coastal signs of the navigable situation depends not only on the state of the atmosphere, but also on many other factors, which include:
A) topographic (determined by the nature of the surrounding area, in particular, the predominance of a particular color in the surrounding landscape);
B) photometric (brightness and color of the observed sign and the background on which it is projected);
B) geometric (distance to the sign, its size and shape).
Fig. 4 Main lines and planes of the observer
For orientation in the sea, a system of conventional lines and planes of the observer is adopted. In fig. 4 shows the globe, on the surface of which at the point M the observer is located. His eye is at a point BUT... By letter e the height of the observer's eye above sea level is indicated. The ZMn line drawn through the observer's position and the center of the globe is called a plumb or vertical line. All planes drawn through this line are called vertical, and perpendicular to it - horizontal... The horizontal plane HH / passing through the eye of the observer is called plane of the true horizon... The vertical plane VV / passing through the observer's place M and the earth's axis is called the plane of the true meridian. At the intersection of this plane with the Earth's surface, a large circle PnQPsQ / is formed, called true meridian of the observer... The straight line obtained from the intersection of the plane of the true horizon with the plane of the true meridian is called true meridian line or the midday line N-S. This line defines the direction to the north and south points of the horizon. The vertical plane FF /, perpendicular to the plane of the true meridian, is called plane of the first vertical... At the intersection with the plane of the true horizon, it forms the E-W line, perpendicular to the N-S line and defining the directions to the east and west points of the horizon. Lines N-S and E-W divide the plane of the true horizon into quarters: NE, SE, SW and NW.
Fig. 5. Horizon visibility range
In the open sea, the observer sees the water surface around the vessel, limited by a small circle CC1 (Fig. 5). This circle is called the visible horizon. The distance De from the position of the vessel M to the line of the apparent horizon SS 1 is called range of the visible horizon... The theoretical range of the visible horizon Dt (segment AB) is always less than its actual range De. This is due to the fact that due to the different density of the layers of the atmosphere along the height, the light beam propagates in it not in a straight line, but along the AC curve. As a result, the observer can see in addition some part of the water surface located behind the line of the theoretical visible horizon and limited by a small circle CC 1. This circle is the line of the visible horizon of the observer. The phenomenon of refraction of light rays in the atmosphere is called terrestrial refraction. Refraction depends on atmospheric pressure, temperature and humidity. In one and the same place on the Earth, refraction can change even over the course of one day. Therefore, the calculations take the average value of refraction. Formula for determining the distance of the visible horizon:
As a result of refraction, the observer sees the horizon line in the direction AC / (Fig. 5), tangent to the arc AC. This line is raised at an angle r above the direct ray AB. Angle r also called terrestrial refraction. Angle d between the plane of the true horizon НН / and the direction to the visible horizon is called inclination of the visible horizon.
VISIBILITY RANGE OF OBJECTS AND LIGHTS. The range of the visible horizon makes it possible to judge the visibility of objects at the water level. If the object has a certain height h above sea level, then the observer can detect it at a distance:
On nautical charts and in navigation aids, a pre-calculated range of visibility of beacon lights is given Dk from an observer's eye height of 5 m.From this height De equals 4.7 miles. When e other than 5 m, a correction should be made. Its value is equal to:
Then the visibility range of the beacon Dn is equal to:
The range of visibility of objects, calculated according to this formula, is called geometric, or geographical. The calculated results correspond to a certain average state of the atmosphere during the daytime. In fog, rain, snowfall or foggy weather, the visibility of objects is naturally reduced. On the contrary, in a certain state of the atmosphere, the refraction can be very large, as a result of which the visual range of objects turns out to be much greater than the calculated one.
Visible horizon range. Table 22 MT-75:
The table is calculated using the formula:
De = 2.0809 ,
Entering the table. 22 MT-75 with object height h above sea level, get the visibility range of this item from sea level. If we add to the obtained range the range of the visible horizon, found in the same table according to the height of the observer's eye e above sea level, the sum of these ranges will make the object's visibility range, without taking into account the transparency of the atmosphere.
To obtain the range of the radar horizon Dp taken selected from the table. 22, increase the visible horizon range by 15%, then Дp = 2.3930 . This formula is valid for standard atmospheric conditions: pressure 760 mm, temperature + 15 ° C, temperature gradient - 0.0065 degrees per meter, relative humidity, constant with height, 60%. Any deviation from the accepted standard atmospheric state will cause a partial change in the range of the radar horizon. In addition, this range, that is, the distance from which the reflected signals can be seen on the radar screen, largely depends on the individual characteristics of the radar and the reflective properties of the object. For these reasons, use the coefficient 1.15 and the data in Table. 22 should be used with caution.
The sum of the ranges of the radar horizon of the antenna Ld and the observed object of height A will represent the maximum distance from which the reflected signal can return.
Example 1.
Determine the detection range of the beacon with height h = 42 m from sea level from the height of the observer's eye e = 15.5 m.
Decision. From table. 22 choose:
for h = 42 m..... . Dh= 13.5 miles
for e= 15.5 m. . . . . . De= 8.2 miles
therefore, the detection range of the beacon
Дп = Дh + Дe = 21.7 miles.
The range of visibility of the object can also be determined by the nomogram placed on the insert (Appendix 6). MT-75
Example 2.
Find the radar range of an object with a height of h = 122 m, if the effective height of the radar antenna Hd = 18.3 m above sea level.
Decision. From table. 22 selects the object and antenna visibility ranges from sea level 23.0 and 8.9 miles, respectively. By summing these ranges and multiplying them by a factor of 1.15, the object, under standard atmospheric conditions, is likely to be detected from a distance of 36.7 miles.
Chapter VII. Navigation.
Navigation is the foundation of the science of navigation. The navigational method of navigation is to navigate the ship from one place to another in the most advantageous, shortest and safest way. This method solves two problems: how to direct the ship along the chosen path and how to determine its place in the sea by the elements of the ship's movement and observations of coastal objects, taking into account the effect on the ship of external forces - wind and current.
To be sure of the safety of your vessel's movement, you need to know the position of the vessel on the map, which determines its position relative to the dangers in a given navigation area.
Navigation develops the basics of navigation, it studies:
Dimensions and surface of the earth, ways of depicting the earth's surface on maps;
Methods of reckoning and laying the path of the vessel on nautical charts;
Methods for determining the position of a vessel at sea by coastal objects.
§ 19. Basic information about navigation.
1. Basic points, circles, lines and planes
Our earth has the shape of a spheroid, which has a semi-major axis OE equals 6378 km, and the semi-minor axis OR 6356 km(fig. 37).
Fig. 37. Determining the coordinates of a point on the earth's surface
In practice, with some assumption, the earth can be considered a ball rotating around an axis that occupies a certain position in space.
To determine points on the earth's surface, it is customary to mentally divide it by vertical and horizontal planes, forming lines with the earth's surface - meridians and parallels. The ends of the imaginary axis of rotation of the earth are called poles - north, or north, and south, or south.
Meridians are large circles passing through both poles. Parallels are small circles on the earth's surface parallel to the equator.
The equator is a large circle, the plane of which passes through the center of the earth perpendicular to the axis of its rotation.
Both meridians and parallels on the earth's surface can be imagined innumerable. The equator, meridians and parallels form a grid of geographic coordinates for the earth.
Any point location BUT on the earth's surface can be determined by its latitude (f) and longitude (l) .
The latitude of a place is the meridian arc from the equator to the parallel of the given place. Otherwise: the latitude of a location is measured by the central angle between the plane of the equator and the direction from the center of the earth to that location. Latitude is measured in degrees from 0 to 90 ° from the equator to the poles. In the calculations, it is assumed that the northern latitude f N has a plus sign, and the southern latitude f S has a minus sign.
The latitude difference (f 1 - f 2) is the meridian arc between the parallels of these points (1 and 2).
The longitude of a place is the arc of the equator from the prime meridian to the meridian of a given place. Otherwise: the longitude of a place is measured by the arc of the equator, enclosed between the plane of the prime meridian and the plane of the meridian of the given place.
The difference in longitudes (l 1 -l 2) is the equatorial arc between the meridians of the given points (1 and 2).
The zero meridian is the Greenwich meridian. Longitude is measured from it in both directions (east and west) from 0 to 180 °. West longitude is measured on the map to the left of the Greenwich meridian and is taken with a minus sign in calculations; east - to the right and has a plus sign.
The latitude and longitude of any point on earth are called the geographic coordinates of that point.
2. Dividing the true horizon
The mentally imagined horizontal plane passing through the eye of the observer is called the plane of the observer's true horizon, or the true horizon (Fig. 38).
Suppose that at the point BUT is the observer's eye, the line ZABC- vertical, HH 1 is the plane of the true horizon, and the line P NP S is the axis of rotation of the earth.
Of the many vertical planes, only one plane in the drawing will coincide with the earth's axis of rotation and a point BUT. The intersection of this vertical plane with the surface of the earth gives on it a large circle P N BEP SQ, called the true meridian of place, or the meridian of the observer. The plane of the true meridian intersects with the plane of the true horizon and gives on the last line of north-south NS. Line OW, perpendicular to the line of true north-south, called the line of true east and west (east and west).
Thus, the four main points of the true horizon - north, south, east and west - occupy a well-defined position anywhere on earth, except for the poles, so that different directions along the horizon can be determined relative to these points.
Directions N(north), S (south), ABOUT(East), W(west) are called the main points. The entire circumference of the horizon is divisible by 360 °. Division is made from point N clockwise.
The intermediate directions between the main points are called quarter points and are called NO, SO, SW, NW. Major and quarter points have the following meanings in degrees:
Fig. 38. Observer's true horizon
3. Visible horizon, range of the visible horizon
The water space visible from the vessel is limited by a circle formed by the apparent intersection of the celestial vault with the surface of the water. This circle is called the visible horizon of the observer. The distance of the visible horizon depends not only on the height of the observer's eyes above the water surface, but also on the state of the atmosphere.
Fig. 39. Object visibility range
The navigator should always know how far he can see the horizon in different positions, for example, standing at the helm, on deck, sitting, etc.
The distance of the visible horizon is determined by the formula:
d = 2.08
or, approximately, for the observer's eye height less than 20 m to formula:
d = 2,
where d is the distance of the visible horizon in miles;
h - the height of the observer's eye, m.
Example. If the observer's eye height is h = 4 m, then the range of the visible horizon is 4 miles.
The range of visibility of the observed object (Fig. 39), or, as it is called, the geographical range D n , is the sum of the distances of the visible horizon from the height of this object H and the height of the eye of the observer A.
Observer A (Fig. 39), located at a height h, from his ship can see the horizon only at a distance d 1, that is, to point B of the water surface. If the observer is placed at point B of the water surface, then he could see the lighthouse C , located from it at a distance d 2 ; therefore, the observer at the point BUT, will see the beacon from a distance equal to D n :
D n = d 1 + d 2.
The visibility range of objects located above the water level can be determined by the formula:
D n = 2.08 (+).
Example. Beacon height H = 1b, 8 m, observer's eye height h = 4 m.
Decision. D n = l 2.6 miles, or 23.3 km.
The range of visibility of the object is also determined approximately according to the Struisky nomogram (Fig. 40). Applying a ruler so that one straight line connects the heights corresponding to the eye of the observer and the observed object, the visual range is obtained on the middle scale.
Example. Find the range of sight of an object with an altitude of 26.2 m at an observer's eye height above sea level of 4.5 m.
Decision. D n= 15.1 miles (dotted line in Figure 40).
On maps, directions, in navigation aids, in the description of signs and lights, the visibility range is given for an observer's eye height of 5 g from the water level. Since on a small boat the observer's eye is located below 5 m, for him, the visibility range will be less than indicated in the manuals or on the map (see table. 1).
Example. The map shows the visibility range of the beacon at 16 miles. This means that the observer will see this beacon from a distance of 16 miles, if his eye is at an altitude of 5 m above sea level. If the observer's eye is at a height of 3 m, then the visibility will accordingly decrease by the difference in the horizon visibility range for heights of 5 and 3 m. Horizon visibility range for height 5 m equal to 4.7 miles; for height 3 m- 3.6 miles, difference 4.7 - 3.6 = 1.1 miles.
Consequently, the visibility range of the beacon will not be equal to 16 miles, but only 16 - 1.1 = 14.9 miles.
Fig. 40. Struisky's nomogram
