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The mapping of the gravitational forces of the solar system

Foreword

It is often very difficult to explain with words the simplest things or the structure of a mechanism. But usually, understanding comes easily enough, if you see them with your eyes, and even better to twist in your hands. But some things are invisible to our vision and even being simple are very difficult to understand.
For example, what an electric current is is a set of definitions, but none of them describes its mechanism exactly, without ambiguity and uncertainty.
On the other hand, electrical engineering is a rather well-developed science, in which any electrical processes are described in detail using mathematical formulas.
So why not show such processes using these very formulas and computer graphics.
But today we consider the effect of a simpler process than electricity — the force of aggression. It would seem that there is a complex, because the law of the world is studied in school, but nevertheless ... Mathematics describes the process as it takes place in ideal conditions, in a kind of virtual space, where there are no restrictions.
In life, everything is usually not the case and the process in question is continuously superimposed on many different circumstances, imperceptible or insignificant at first glance.
Knowing the formula and understanding its action is a little bit different things.
So, we will take a small step towards understanding the law of aggression. The law itself is simple - the strength of the force is directly proportional to the masses and inversely proportional to the square of the distance between them, but the difficulty lies in the unimaginable number of interacting objects.
Yes, we will consider only the strength of the power, so to speak, in complete solitude, which of course is wrong, but in this case is permissible, since this is just a way to show the invisible.
And yet, the article has JavaScript code, i.e. All pictures are actually drawn using Canvas, so the whole article can be found here .

Showing gravity in the solar system


In the framework of classical mechanics, gravitational interaction is described by Newton's universal law, which says that the force of gravitational attraction F between two material points of mass m 1 and m 2 , separated by distance r , is proportional to both masses and inversely proportional to the square of the distance - that is:

where G is the gravitational constant equal to about 6.67384 Ă— 10 -11 N Ă— m 2 Ă— kg -2 .
But I would like to see a picture of the change in the strength of the force across the solar system, and not between two bodies. Therefore, the mass of the second body m 2 is taken equal to 1, and the mass of the first body is simply m . (That is, we represent objects as a material point - one pixel in size, and we measure the force of attraction relative to another, virtual object, let's call it “test body”, with a mass of 1 kilogram.) The formula will look like this:

Now, instead of m, we substitute the mass of the body of interest, and instead of r we go through all the distances from 0 to the value of the orbit of the last planet and obtain the change in the strength of the force depending on the distance.
When applying forces from different objects, we choose the largest one.
Further, we express this force not in figures, but in shades of color corresponding to them. In this case, you get a visual picture of the distribution of gravity in the solar system. That is, in a physical sense, the shade of color will correspond to the weight of a body weighing 1 kilogram at the corresponding point of the solar system.
It should be noted that:

(actually, that's all the limitations imposed by physics on mathematics in this matter).
Let's now look at how to display the magnitude of the force of color.

To show the numbers in color, you need to create an array in which the index would be equal to the number, and the value would be the color value in the RGB system.
Here is a gradient of color from white to red, then yellow, green, blue, violet and black. Total turned out 1786 shades of color.

The number of colors is not so great, they simply are not enough to display the whole range of forces of aggression. We confine ourselves to the forces of the maximum - on the surface of the sun and the minimum - on the orbit of Saturn. That is, if the force of attraction on the surface of the Sun (270.0 N) is designated by the color in the table under the index 1, then the force of attraction to the Sun in the orbit of Saturn (0.00006 N) will be indicated by color, with an index far beyond 1700. So that all the same colors are not enough for the uniform expression of the magnitude of the force of aggression.
In order to clearly see the most interesting places in the displayed forces of attraction, it is necessary that the magnitudes of the force of attraction less than 1H correspond to large color changes, and from 1H and higher, the correspondences are not so interesting - it can be seen that the force of attraction, say, Earth, differs from that of Mars or Jupiter and oh well. That is, the color will not be proportional to the magnitude of the force of attraction, otherwise we will "lose" the most interesting.
To bring the value of the force of attraction to the index of the color table, we use the following formula:

Yes, this is the very hyperbole known since high school, only the square root is extracted from the argument. (Taken purely "from the lantern", only in order to reduce the ratio between the largest and smallest values ​​of the force of attraction.)
See how the colors are distributed depending on the attraction of the sun and the planets.

As you can see, on the surface of the Sun, our test body will weigh about 274N or 27.4 kgf, since 1N = 0.10197162 kgf = 0.1 kgf. And on Jupiter, almost 26N or 2.6 kgf, on Earth, our test body weighs about 9.8N or 0.98kgs.
In principle, all these figures are very, very approximate. For our case, this is not very important, we need to turn all these values ​​of the force of attraction into their corresponding color values.
So, from the table it is seen that the maximum value of the force of attraction is equal to 274, and the minimum value is 0.00006. That is, they differ by more than 4.5 million times.

You can also see that all the planets turned out almost the same color. But it does not matter, it is important that the boundaries of attraction of the planets will be clearly visible, since the forces of attraction of small values ​​vary quite well in color.
Of course, the accuracy is small, but we just need to get a general idea of ​​the forces of gravity in the solar system.
Now we “arrange” the planets in places corresponding to their distance from the Sun. To do this, you need to attach some kind of distance scale to the resulting color gradient. The curvature of the orbits, I think, can be ignored.
But as always, the cosmic scale, in the literal sense of these words, does not allow to see the whole picture. We look, Saturn is approximately 1430 million kilometers from the Sun, the index corresponding to the color of its orbit is 1738. That is it turns out in one pixel (if we take in this scale one color shade is equal to one pixel) approximately 822.8 thousand kilometers. And the Earth's radius is approximately 6371 km, i.e. diameter 12742 kilometers, about 65 times less than one pixel. Here and how to observe the proportions.
We will go the other way. Since we are interested in the gravity of the near-planetary space, we will take the planets separately and color them and the space around them with a color corresponding to the gravitational forces from themselves and the Sun. For example, take Mercury - the radius of the planet is 2.4 thousand km. and equate it to a circle with a diameter of 48 pixels, i.e. in one pixel will be 100 km. Then Venus and Earth will be respectively 121 and 127 pixels. It is quite comfortable size.
So, we make a picture 600 by 600 pixels in size, we determine the value of the force of attraction to the Sun in Mercury's orbit plus / minus 30,000 km (so that the planet is in the center of the picture) and paint the background with a gradient of color shades corresponding to these forces.
At the same time, to simplify the task, we paint over not with arcs of the corresponding radius, but with straight, vertical lines. (Roughly speaking, our “Sun” will be “square” and will always be on the left side.)
To ensure that the background color does not shine through the image of the planet and the zone of attraction to the planet, we determine the radius of the circle corresponding to the zone where the attraction to the planet is greater than the attraction to the Sun and paint it white.
Then in the center of the picture we place a circle corresponding to the diameter of Mercury on a scale (48 pixels) and fill it with a color corresponding to the force of attraction to the planet on its surface.
Further, from the planet, we paint over the gradient in accordance with the change in the force of attraction to it and at the same time constantly compare the color of each point in the layer of attraction to Mercury with a point with the same coordinates, but in the layer of attraction to the Sun. When these values ​​become equal, we make this pixel black and stop further painting.
Thus, we obtain a certain form of visible change in the gravitational force of the planet and the Sun with a clear boundary between them in black.
(I wanted to do just that, but ... it didn’t work out, I couldn’t make a pixel-by-pixel comparison of two image layers.)
')
The distance of 600 pixels is 60 thousand kilometers (i.e., one pixel is 100 km).
The force of attraction to the Sun on orbit of Mercury and near it varies only in a small range, which in our case is denoted by a single shade of color.

So, Mercury and the power of the planet.
Immediately it should be noted that the eight barely visible rays are defects from drawing circles in the Canvas. They have nothing to do with the subject under discussion and should simply be overlooked.
The dimensions of the square are 600 by 600 pixels, i.e. this space is 60 thousand kilometers. The radius of Mercury is 24 pixels - 2.4 thousand km. The radius of the zone of attraction is 23.7 thousand km.
The circle in the center, which is almost white in color, is the planet itself and its color corresponds to the weight of our kilogram test body on the surface of the planet - about 373 grams. The thin blue circle shows the boundary between the surface of the planet and the area in which the force of the force to the planet exceeds the force of the sun.
Then the color gradually changes, becomes more red (i.e., the weight of the test body decreases) and finally becomes equal to the color corresponding to the force of attraction to the Sun in this place, i.e. in orbit of Mercury. The boundary between the zone where the force of attraction to the planet exceeds the force of attraction to the Sun is also marked by a blue circle.
As you can see, there is nothing supernatural.
But in life, a slightly different picture. For example, on this and all other images, the Sun is on the left, which means, in fact, the region of gravity of the planet should be slightly "flattened" on the left and stretched out on the right. And on the image - a circle.
Of course, the best option would be a pixel-by-pixel comparison of the region of attraction to the Sun and the region of attraction to the planet and the choice (display) of the larger of them. But neither I, as the author of this article, nor JavaScript are capable of such exploits. Working with multidimensional arrays is not a priority for this language, but its work can be shown in almost any browser, which decided the question of application.
And in the case of Mercury, and all the rest of the terrestrial planets, the change in the force of attraction to the Sun is not so great as to display it with an existing set of color shades. But when considering Jupiter and Saturn, the change in the force of attraction to the Sun is very noticeable.

Venus

Actually, everything is the same as that of the previous planet, only the size of Venus and its mass is much larger, and the force of attraction to the Sun in the orbit of the planet is smaller (the color is darker, or rather, more red), and the planet is of greater mass, therefore the color of the planet's disk is more light coloured.
In order for a planet with a zone of attraction of a test body with a mass of 1 kg to fit in the picture 600 by 600 pixels, we will reduce the scale by 10 times. Now in one pixel one thousand kilometers.


Earth + Moon

To show the Earth and the Moon to change the scale 10 times (as in the case of Venus) is not enough, you need to increase the size of the picture (the radius of the orbit of the Moon is 384.467 thousand km). The image will be 800 by 800 pixels. The scale is 1 thousand kilometers in one pixel (we are well aware that the picture error will increase even more).

The picture clearly shows that the zones of attraction of the Moon and the Earth are separated by the zone of attraction to the Sun. That is, the Earth and the Moon is a system of two equivalent planets with different mass.

Mars with Phobos and Deimos

Scale - in one pixel 1 thousand kilometers. Those. like Venus, and Earth with the Moon. Remember that distances are proportional, and the display of gravity is non-linear.

Here, you can immediately see the fundamental difference between Mars and its moons from Earth and the Moon. If the Earth and the Moon are a system of two planets and, despite different sizes and masses, act as equal partners, then the satellites of Mars are in the zone of gravity of Mars.
The planet itself and the satellites are practically "lost." The white circle is the orbit of a distant satellite, Deimos. Zoom in 10 times for better viewing. In one pixel 100 kilometers.

These “creepy” rays from Canvas spoil the picture quite strongly.
The sizes of Phobos and Deimos are disproportionately increased 50 times, otherwise they are not visible at all. The color of the surfaces of these satellites is also not logical. In fact, the force of attraction on the surfaces of these planets is less than the force of attraction to Mars in their orbits.
That is, from the surfaces of Phobos and Deimos, the attraction of Mars "blows away" everything. Therefore, the color of their surfaces should be equal to the color in their orbits, but only in order to be better seen, the satellite disks are painted in the color of the force of attraction in the absence of the force of attraction to Mars.
These satellites should be simply monolithic. In addition, since there is no attraction force on the surface, it means that they could not form in this form, that is, Phobos and Deimos were formerly parts of some other, larger object. Well, or, at a minimum, they were located in another place, with a smaller force of attraction than in the zone of attraction of Mars.
For example, here is Phobos . The scale is 100 meters in one pixel.
The surface of the satellite is indicated by a blue circle, and the force of attraction of the entire mass of the satellite is a white circle.
(In fact, the shape of the small celestial bodies of Phobos, Deimos, etc., is far from spherical)
The color of the circle in the center corresponds to the force of attraction of the satellite’s mass. The closer to the surface of the planet, the less the force of attraction.
(Here again, an inaccuracy is made. In fact, the white circle is the border, where the force of attraction to the planet becomes equal to the force of attraction to Mars in the orbit of Phobos.
That is, the color outside of this white circle should be the same as the outside of the blue circle denoting the surface of the satellite. But the color transition shown should be inside the white circle. But then nothing will be visible at all.)
It turns out like a picture of the planet in the section.
The integrity of the planet is determined only by the strength of the material of which Phobos consists. With less strength, Mars would have rings like Saturn, from the destruction of satellites.

And it seems that the collapse of space objects is not such an exceptional event. Here, even the Hubble Space Telescope "spotted" a similar incident.

The collapse of the asteroid P / 2013 R3
The collapse of the asteroid P / 2013 R3

The collapse of the asteroid P / 2013 R3, which is located at a distance of more than 480 million kilometers from the Sun (in the asteroid belt, further than Ceres). The diameter of the four largest asteroid fragments reaches 200 meters, their total mass is about 200 thousand tons.
And this is Deimos . All the same as that of Phobos. The scale is 100 meters in one pixel. Only the planet is smaller and lighter, respectively, and is also farther from Mars and the force of attraction to Mars is smaller (the background of the picture is darker, that is, more red).


Ceres


Well, Ceres is nothing special, except for coloring. The force of attraction to the Sun is less here, so the color is appropriate. The scale is 100 kilometers in one pixel (the same as in the picture with Mercury).
The small blue circle is the surface of Ceres, and the big blue is the border, where the force of attraction to the planet becomes equal to the force of attraction to the Sun.


Jupiter

Jupiter is very large. Here is an 800 by 800 pixel image. Scale - in one pixel 100 thousand kilometers. This is to show the region of gravity of the planet entirely. The planet itself is a small point in the center. Satellites are not shown.
Only the orbit (outer circumference of white) of the farthest satellite, S / 2003 J 2, is shown.

Jupiter has 67 satellites. The largest are Io, Europa, Ganymede and Callisto.
The farthest satellite, S / 2003 J 2, makes a complete revolution around Jupiter at an average distance of 29,541,000 km. Its diameter is about 2 km, its mass is about 1.5 Ă— 10 13 kg. As you can see, it goes far beyond the sphere of the planet. This can be explained by errors in calculations (after all, quite a lot of averaging, rounding and discarding of some details have been done).
Although there is a way to calculate the boundary of the gravitational influence of Jupiter, defined by the Hill sphere , the radius of which is determined by the formula

where a jupiter and m jupiter are the semi-major axis of the ellipse and the mass of Jupiter, and M sun is the mass of the Sun. Thus, a radius of 52 million km is obtained. S / 2003 J 2 moves in eccentric orbit up to 36 million km from Jupiter
Jupiter also has a ring system of 4 main components: a thick inner particle torus, known as a “ring-halo”; relatively bright and subtle "main ring"; and two wide and weak outer rings - known as "spider rings", called by the material of the satellites - that form them: Amalthea and Thebes.
Ring-halo with an inner radius of 92,000 and an outer 1,22500 kilometers.
The main ring 122500-189000 km.
Amalthea Spider Ring 129000-1000000km.
Thebes web ring 129000—226000 km.
Increase the picture 200 times, in one pixel 500 kilometers.
Here are the rings of Jupiter. The thin circle is the surface of the planet. Next come the borders of the rings - the inner border of the ring-halo, the outer border of the ring-halo and the inner boundary of the main ring, etc.
The small circle in the upper left corner is the area where the force of attraction of the satellite of Jupiter Io becomes equal to the force of attraction of Jupiter in the orbit of Io. The satellite itself is simply not visible on this scale.

In principle, large planets with satellites should be considered separately, since the difference in gravitational force is very large, as are the sizes of the area of ​​gravity of the planet. As a result, all the interesting details are simply lost. And to consider a picture with a radial gradient does not make much sense.

Saturn

The image is 800 by 800 pixels. Scale - in one pixel 100 thousand kilometers. The planet itself is a small point in the center. Satellites are not shown.
Clearly visible change in the force of attraction to the Sun (remember that the Sun is on the left).

Saturn has 62 known satellites. The largest of them are Mimas, Enceladus, Tethys, Dion, Rhea, Titan and Japet.
The farthest satellite is Forniot (temporary designation S / 2004 S 8). Also referred to as Saturn XLII. The average radius of the satellite is about 3 kilometers, the mass is 2.6 Ă— 10 14 kg, and the semi-major axis is 25146000 km.
Rings of the planets appear only at a considerable distance from the Sun. The first such planet is Jupiter. Having a mass and size larger than that of Saturn, its rings are not as impressive as the rings of Saturn. That is, the size and mass of the planet for the formation of rings are less important than the distance from the Sun.
But look further, a pair of rings surrounds the asteroid Chariklo (10199 Chariklo) (the diameter of the asteroid is about 250 kilometers), which revolves around the Sun between Saturn and Uranus.
Article on Habré about the asteroid with rings
Wikipedia about asteroid Chariklo
The ring system consists of a dense inner ring 7 km wide and an outer ring 3 km wide. The distance between the rings is about 9 km. The radii of the rings are 396 and 405 km, respectively. Chariklo is the smallest object whose rings were open.
However, the force of the rings is only indirectly related to the rings.
In fact, the rings appear from the destruction of satellites, which consist of a material of insufficient strength, i.e. not stone monoliths of the type of Phobos or Deimos, but pieces of rock, ice, dust, and other space debris, frozen together as one.
So the planet drags it with its own. Such a satellite, which does not have its own attraction (or rather, has its own attraction force less than the force of attraction to the planet in its orbit) flies in orbit, leaving behind it a train of destroyed material. So the ring is formed. Further, under the action of the force of attraction to the planet, this detrital material approaches the planet. That is, the ring expands.
At some level, the force of attraction becomes large enough so that the rate of fall of these debris increases, and the ring disappears.

Afterword

The purpose of the publication of the article - perhaps someone with knowledge of programming, will be interested in this topic and will make a higher-quality model of gravitational forces in the solar system (yes, three-dimensional, with animation.
Or maybe even make the orbits are not fixed, and also calculated - this is also possible, the orbit will be a place where the force of gravity will be compensated by centrifugal force.
It will turn out almost as in life, as the most real Solar system. (This is where you can create a space shooter, with all the subtleties of space navigation in the asteroid belt. Given the forces acting according to real physical laws, and not among hand-drawn graphics.)
And it will be a great physics textbook that will be interesting to study.
PS The author of the article is an ordinary person:
not a physicist,
not an astronomer,
not a programmer
does not have a higher education.

Source: https://habr.com/ru/post/227127/

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