Circle 7 class

Head Varvara Alekseevna Kosorotova
2009/2010 academic year

Lesson 13. Graphs

Basic concepts

Under count we will understand the set of points ( peaks), some of which are connected by segments ( ribs).
Vertex Degree graph is the number of edges leaving it (or, what is the same, entering it) (they also say: the number of edges, incidental given vertex). The top of the graph is called even, if its degree is even, and odd otherwise.
Some part of the vertices of the given graph is called connectivity component , if from any of its vertices it is possible to “walk” to any other, moving along the edges.

In some cases, the “direction of movement” is chosen on the edges of the graph (for example, when highway one-way traffic is introduced). This results in directed graph . (If the direction of movement along the edges is not defined, then the graph is called unoriented ). In a directed graph, there are positive and negative degree each vertex (that is, the number of edges, respectively, entering and leaving it). Two vertices can also be connected by several edges, the directions of movement along which are opposite (“two-way road”). The concept of a connected component is changed: now each “route” from one vertex to another must take into account the direction of movement along the edges.

Tasks

Between the nine planets of the solar system is established space communication. Flight rockets fly on the following routes: Earth - Mercury, Pluto - Venus, Earth - Pluto, Pluto - Mercury, Mercury - Venus, Uranus - Neptune, Neptune - Saturn, Saturn - Jupiter, Jupiter - Mars and Mars - Uranus. On each route, rockets fly in both directions. Is it possible to fly on regular rockets from Earth to Mars? 10. The king has 19 vassals. Can it turn out that each vassal has 1, 5 or 9 neighbors?

Decision. Let's make the segments the vertices of the graph and connect the edges of those that intersect with each other. By the condition of item b), such a graph has an odd number of odd vertices, which contradicts Theorem 2. For item a), this argument is not suitable, but this does not mean that it is possible to make such a drawing. To prove it, you need to draw it. To do this, it is enough to come up with a drawing with 4 segments, each of which intersects exactly three others, and then draw two such drawings side by side. (It is important here that it is segments that appear in the condition, and not straight lines.)

• A space communication has been established between the nine planets of the solar system. Regular rockets fly on the following routes:

• Earth-Mercury

• Pluto - Venus

• Earth - Pluto

• Pluto - Mercury

• Mercury - Venus

• Uranus - Neptune

• Neptune - Saturn

• Saturn - Jupiter

• Jupiter - Mars

• Mars - Uranus

• Is it possible to fly on regular rockets from Earth to Mars?

Decision:

The concept of "Graph" A graph is a diagram consisting of points and segments connecting these points Points - graph vertices Segments - graph edges

Task 2

• Arkady, Boris, Vladimir, Grigory and Dmitry shook hands at the meeting (each shook hands with each one once).

• How many handshakes were made in total?

Decision:

• Let each of the young people correspond to a point on the plane, named after the first letter of the name, and the handshakes produced are a segment or a curved line that will connect the points corresponding to the names.

• (zero graph)

(incomplete graph)

(full graph)

Science based on knowledge GRAPH THEORIES:

• The medicine

• Cybernetics

• Informatics

• Chemistry

• Physics

• Transport

• Construction

• Applied Mathematics

• Economy

Leonard Euler

Draw figures in one stroke

Designate the intersection points, and write in brackets how many lines come out of one or another intersection point.

• In what case is it possible to draw figures without lifting the pencil from the paper and without drawing a single line twice, and in which case not?

Conclusion:

• Conclusion:

• - If all the vertices of the graph are even, then it is possible to draw a figure, and one can start from any vertex.

• -If two of these vertices are odd, then you can draw a figure, but you only need to start at one of these two odd vertices, and end at the second odd vertex.

Is it possible to make a walk by crossing each bridge only once?

Task

• A fly got into a sugar jar. The jar is in the shape of a cube. Will the fly be able to sequentially go around all 12 edges of the cube without passing twice along one edge. Jumping and flying from place to place is not allowed.

It is worth noting that no one has ever seen the ninth planet itself. In early 2016, astrophysicists Mike Brown and Konstantin Batygin from the California Institute of Technology announced that they had found it thanks to the analysis of those found in the outer part solar system disturbances. The location of cosmic bodies in the so-called Kuiper belt - the region beyond Neptune - made it possible to assume the impact on them of a large planet that passed by a cluster of relatively small objects. The mass of the hypothetical planet is ten times the mass of the Earth.

According to the version put forward by Brown and Batygin, the planet could have formed in the solar system, and then it was pushed into a more distant orbit under the influence of the gravitational force of Jupiter or Saturn.

They also calculated that, moving around the Sun, the ninth planet maximally moves away from it more than 1000 times further than the Earth. And even at the closest point in the orbit, the distance between mysterious planet and the Sun at least 200 times the average distance from the Earth to the Sun. And it makes one revolution around the star within 10-20 thousand years.

Versions of the "orphan planet"

Based on these data, James Vesper and Paul Mason of the University of New Mexico built 156 computer models of the behavior of a large celestial body. In 40% of cases, the planet, as its discoverers suggested, was fixed far beyond the orbit of Pluto and began to revolve around the Sun. In 60%, it passed through the solar system and continued its movement through space.

It is believed that "orphan planets", or, as they are also called, "wanderer planets", are formed outside of planetary systems. Among them are those formed in other star systems, which they then left, for example, under the influence of the gravity of other planets. From time to time, breaking into outer space, "orphans" can even take with them a "companion" - small planets, which they drag with their gravity.

The discovery of "space orphans" was predicted in the early 1990s. They began to find them already in the 21st century. The number of planets wandering in the universe is amazing: it is assumed that their number can reach 400 billion. The difficulty of detecting them is due to the lack of research methods. So, if exoplanets are mainly found by periodic oscillations and dimmings of the stars around which they revolve, then with wanderer planets everything is somewhat more complicated. So far, scientists can detect only those “wanderers” that are not inferior in size to the largest in the solar system, Jupiter or Saturn.

Presented at the meeting of the American Astronomical Society, the report by Vesper and Mason has not yet had time to pass peer review. However, according to Konstantin Batygin, the version of his colleagues from the University of New Mexico is quite plausible. “Without knowing the exact orbit, it is difficult to unambiguously confirm or reject this version, but there is definitely a possibility that the ninth planet is indeed an object captured by the solar system,” Batygin said in an interview with Gizmodo. Other scientists made a similar statement in the summer of 2016.

Physicists from the University of Lund in Sweden have calculated that 4.5 billion years ago, the Sun and its planets could have approached another planetary system. As a result, our star probably "stole" one of the planets.

The unusual location of the hypothetical ninth planet prompted them to carry out the calculations. It seemed incredible that it formed at such a distance from our star and continued to revolve around it.

It is worth noting that a number of scientists are skeptical about the very hypothesis of the presence of another planet in the solar system, but Batygin is confident in its existence. “The number of seemingly unrelated mysteries in the life of the solar system, which are solved by the assumption of a ninth planet, is too large for this to be a mere coincidence,” he insists. Immediately after the joint discovery with Brown, Batygin predicted that the work to detect the ninth planet using telescopes could take up to eight years.

The planets of the solar system are in order in the following sequence:
1 - Mercury. The smallest of the real planets in the solar system
2 - Venus. The description of hell was taken from her: terrible heat, sulfur evaporation and eruptions of many volcanoes.
3 - Earth. The third planet in order from the Sun, our home.
4 - Mars. The most distant of the planets terrestrial group solar system.
Then the main asteroid belt is located, where dwarf planet Ceres and the minor planets Vesta, Pallas, etc.
Next in order are the four giant planets:
5 - Jupiter. The largest planet in the solar system.
6 - Saturn with its famous rings.
7 - Uranus. The coldest planet.
8 - Neptune. It is the furthest "real" planet in order from the Sun.
And here's what's interesting:
9 - Pluto. A dwarf planet that is usually listed after Neptune. But, Pluto's orbit is such that it is sometimes closer to the Sun than Neptune. For example, this was the case from 1979 to 1999.
No, Neptune and Pluto cannot collide :) - their orbits are such that they do not intersect.
The location of the planets of the solar system in order in the photo:

How many planets are in the solar system

How many planets are in the solar system? This is not so easy to answer. For a long time it was believed that there were nine planets in the solar system:
Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune and Pluto.

But, on August 24, 2006, Pluto ceased to be considered a planet. This was caused by the discovery of the planet Eris and other small planets of the solar system, in connection with which it was necessary to clarify - which celestial bodies can be considered as planets.
Several signs of "real" planets were identified and it turned out that Pluto does not fully satisfy them.
Therefore, Pluto was transferred to the category of dwarf planets, which include, for example, Ceres - the former asteroid No. 1 in the Main asteroid belt between Mars and Jupiter.

As a result, when trying to answer the question of how many planets are in the solar system, the situation is even more confused. Because in addition to the "real" now there are also dwarf planets.
But there are also small planets, which were called large asteroids. For example Vesta, asteroid number 2 in the mentioned Main asteroid belt.
AT recent times the same Eris, Make-Make, Haumea and several other small planets of the solar system, data on which is insufficient and it is not clear what to consider them - dwarf or small planets. Not to mention that some small asteroids are mentioned in the literature as minor planets! For example, the asteroid Icarus, which is only about 1 kilometer in size, is often referred to as a minor planet...
Which of these bodies should be taken into account when answering the question "how many planets are there in the solar system"???
In general, "we wanted the best, but it turned out as always."

Curiously, many astronomers and even simple people act "in defense" of Pluto, continuing to consider it a planet, sometimes arrange small demonstrations and diligently promote this idea on the Web (mainly abroad).

Therefore, when answering the question "how many planets in the solar system" it's easiest to say "eight" briefly and not even try to discuss something ... otherwise it will immediately turn out that there is simply no exact answer :)

The giant planets are the largest planets in the solar system.

There are four giant planets in the solar system: Jupiter, Saturn, Uranus and Neptune. Since these planets are located outside the main asteroid belt, they are called the "outer" planets of the solar system.
In size, two pairs clearly stand out among these giants.
The largest giant planet is Jupiter. Saturn is quite a bit inferior to him.
And Uranus and Neptune are sharply smaller than the first two planets and they are located farther from the Sun.
Look at comparative dimensions giant planets relative to the sun:

Giant planets protect inner planets Solar system from asteroids.
Without these bodies in the solar system, our Earth would be hundreds of times more likely to be hit by asteroids and comets!
How do the giant planets protect us from the fall of intruders?

You can learn more about the largest planets in the solar system here:

terrestrial planets

The terrestrial planets are four planets in the solar system that are similar in size and composition: Mercury, Venus, Earth, and Mars.
Since one of them is the Earth, all these planets were assigned to the terrestrial group. Their sizes are very similar, and Venus and the Earth are generally almost the same. The temperatures are relatively high, which is explained by the proximity to the Sun. All four planets are formed by rocks, while the giant planets are gas and ice worlds.

Mercury is the closest planet to the Sun and the smallest planet in the solar system.
It is generally accepted that Mercury is very hot. Yes, it is, the temperature on the sunny side can reach +427°С. But, there is almost no atmosphere on Mercury, so on the night side it can be up to -170 ° С. And at the poles, because of the low Sun, a layer of underground permafrost is generally assumed ...

Venus. For a long time, it was considered the "sister" of the Earth, until Soviet research stations landed on its surface. It turned out to be a real hell! Temperature +475°C, pressure of almost a hundred atmospheres and an atmosphere of toxic compounds of sulfur and chlorine. To colonize it - you have to try very hard ...

Mars. The famous red planet. It is the most distant of the terrestrial planets in the solar system.
Like Earth, Mars has moons: Phobos and Deimos
Basically it is a cold, rocky and dry world. Only at the equator at noon can it get warmer up to +20°C, the rest of the time - a fierce frost, up to -153°C at the poles.
The planet does not have a magnetosphere and cosmic radiation irradiates the surface mercilessly.
The atmosphere is very rarefied and not suitable for breathing, however, its density is enough to occasionally cause powerful dust storms on Mars.
Despite all the shortcomings. Mars is the most promising planet for colonization in the solar system.

Read more about the terrestrial planets in the article The largest planets in the solar system

The largest planet in the solar system

The largest planet in the solar system is Jupiter. This is the fifth planet from the Sun, its orbit is beyond the main asteroid belt. Look at the size comparison of Jupiter and Earth:
Jupiter is 11 times the diameter of Earth and 318 times its mass. Due to the large size of the planet, parts of its atmosphere rotate at different speeds, so Jupiter's belts are clearly visible in the image. Below, on the left, you can see Jupiter's famous Great Red Spot, a huge atmospheric vortex that has been observed for several centuries.

The smallest planet in the solar system

Which planet is the smallest planet in the solar system? This is not such a simple question...
Today it is generally accepted that the smallest planet in the solar system is Mercury, which we mentioned a little above. But, you already know that until August 24, 2006, Pluto was considered the smallest planet in the solar system.

More attentive readers may recall that Pluto is a dwarf planet. And there are five known. The smallest dwarf planet is Ceres, with a diameter of about 900 km.
But that's not all...

There are also so-called minor planets, the size of which starts at only 50 meters. Both the 1-kilometer Icarus and the 490-kilometer Pallas fall under this definition. It is clear that there are many of them, and it is difficult to choose the smallest one due to the complexity of observations and calculation of sizes. So, when answering the question "what is the name of the smallest planet in the solar system", it all depends on what exactly is meant by the word "planet".

 or tell your friends:

This is a system of planets, in the center of which is a bright star, the source of energy, heat and light - the Sun.
According to one theory, the Sun was formed along with the solar system about 4.5 billion years ago as a result of the explosion of one or more supernovae. Initially, the solar system was a cloud of gas and dust particles, which, in motion and under the influence of their mass, formed a disk in which arose new star The sun and our entire solar system.

At the center of the solar system is the Sun, around which nine large planets revolve in orbits. Since the Sun is displaced from the center of the planetary orbits, then during the cycle of revolution around the Sun, the planets either approach or move away in their orbits.

There are two groups of planets:

Terrestrial planets: and . These planets are small in size with a rocky surface, they are closer than others to the Sun.

Giant planets: and . These are large planets, consisting mainly of gas, and they are characterized by the presence of rings consisting of ice dust and many rocky pieces.

And here does not fall into any group, because, despite its location in the solar system, it is located too far from the Sun and has a very small diameter, only 2320 km, which is half the diameter of Mercury.

Planets of the solar system

Let's start a fascinating acquaintance with the planets of the solar system in order of their location from the Sun, and also consider their main satellites and some other space objects (comets, asteroids, meteorites) in the gigantic expanses of our planetary system.

Rings and moons of Jupiter: Europa, Io, Ganymede, Callisto and others...
The planet Jupiter is surrounded by a whole family of 16 satellites, and each of them has its own, unlike other features ...

Rings and moons of Saturn: Titan, Enceladus and more...
Not only the planet Saturn has characteristic rings, but also on other giant planets. Around Saturn, the rings are especially clearly visible, because they consist of billions of small particles that revolve around the planet, in addition to several rings, Saturn has 18 satellites, one of which is Titan, its diameter is 5000 km, which makes it the largest satellite in the solar system ...

Rings and moons of Uranus: Titania, Oberon and others...
The planet Uranus has 17 satellites and, like other giant planets, thin rings encircling the planet, which practically do not have the ability to reflect light, therefore they were discovered not so long ago in 1977 quite by accident ...

Rings and moons of Neptune: Triton, Nereid and others...
Initially before the exploration of Neptune spacecraft Voyager 2 was aware of two satellites of the planet - Triton and Nerida. Interesting fact that the Triton satellite has a reverse direction of orbital motion, strange volcanoes were also discovered on the satellite that erupted nitrogen gas like geysers, spreading a dark mass (from liquid to vapor) for many kilometers into the atmosphere. During its mission, Voyager 2 discovered six more satellites of the planet Neptune...