Display of the gravitational forces of the solar system


It is often very difficult to explain in words the simplest things or the structure of one or another mechanism. But usually, understanding comes easy enough if you see them with your eyes, and even better and twist in your hands. But some things are invisible to our sight and even being simple are very difficult to understand.
For example, what is an electric current - there are many definitions, but none of them describes its mechanism exactly, without ambiguity and uncertainty.
On the other hand, electrical engineering is a fairly well-developed science, in which any electrical processes are described in detail using mathematical formulas.
So why not show these processes using these very formulas and computer graphics.
But today we consider the action of a simpler process than electricity - the force of gravity. It would seem that it’s complicated, because the law of gravity is studied at school, but nonetheless ... Mathematics describes the process as it goes under ideal conditions, in a kind of virtual space where there are no restrictions.
In life, usually this is not so, and many different circumstances are constantly imposed on the process under consideration, imperceptible or insignificant at first glance.
To know the formula and understand its effect are slightly different things.
So, let's take a small step towards understanding the law of gravity. The law itself is simple - the force of gravity 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 gravity, so to speak, in complete solitude, which of course is not true, but in this case it is permissible, since this is just a way to show the invisible.
And yet, the article has JavaScript code, i.e. all the drawings are actually drawn using Canvas, so the whole article can be taken here .

Displaying the possibilities of gravity in the solar system

In the framework of classical mechanics, gravitational interaction is described by Newton’s law of universal gravitation, which states that the force of gravitational attraction F between two material points of mass m 1 and m 2 separated by a 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 × 10 -11 N × m 2 × kg -2 .
But I would like to see a picture of the change in gravity throughout the solar system, and not between two bodies. Therefore, we take the mass of the second body m 2 equal to 1, and the mass of the first body is denoted simply by m . (That is, we represent objects in the form of a material point - one pixel in size, and measure the attractive force relative to another virtual object, call it a “test body”, with a mass of 1 kilogram.) In this case, the formula will look like:

Now, instead of m, we substitute the mass of the body of interest, and instead of r, we sort out all the distances from 0 to the orbit of the last planet and get a change in the gravitational force depending on the distance.
When applying forces from different objects, we select the largest in magnitude.
Further, we express this power not in numbers, but in the corresponding shades of color. This will give a clear picture of the distribution of gravity in the solar system. That is, in the physical sense, the color cast will correspond to the weight of a body weighing 1 kilogram at the corresponding point in the solar system.
It should be noted that:
  • gravity is always positive, has no negative values, i.e. mass cannot be negative
  • gravity cannot be zero, i.e. the object either exists with some kind of mass, or does not exist at all
  • gravity cannot be shielded or reflected (like a ray of light by a mirror).

(in fact, these are all the restrictions imposed by physics on mathematics in this matter).
Let’s now look at how to display the gravitational forces of color.

To show numbers by color, you need to create an array in which the index is equal to the number, and the value is the color value in the RGB system.
Here's a color gradient from white to red, then yellow, green, blue, purple and black. A total of 1786 shades of color.

The number of colors is not so large, they simply are not enough to display the entire spectrum of gravitational forces. We restrict ourselves to the forces of gravity from the maximum - on the surface of the Sun and the minimum - in the orbit of Saturn. That is, if the force of attraction on the surface of the Sun (270.0 N) is indicated 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 well beyond 1700. So that all the same, the colors are not enough to uniformly express the values ​​of gravity.
In order for the most interesting places in the displayed gravitational forces to be clearly visible, it is necessary that magnitudes of attraction less than 1H correspond to large color changes, and from 1H and above, the correspondences are not so interesting - it can be seen that the gravitational force, say Earth, is different from the attraction of Mars or Jupiter yes. 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 same hyperbola known since high school, only the square root has been previously extracted from the argument. (Taken purely “from the lantern”, only to reduce the ratio between the largest and smallest values ​​of the force of attraction.)
Look at how the colors are distributed depending on the attraction of the Sun and planets.

As you can see on the surface of the Sun, our test body will weigh about 274 N or 27.4 kG, since 1 N = 0.10197162 kgf = 0.1 kgf. And on Jupiter, almost 26N or 2.6 kgf, on Earth, our test body weighs about 9.8 N or 0.98 kgf.
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 attractive force into the corresponding color values.
So, the table shows that the maximum value of the force of attraction is 274N, and the minimum is 0.00006N. That is, they differ by more than 4.5 million times.

It is also seen that all the planets turned out to be almost the same color. But it doesn’t matter, it’s important that the boundaries of gravity of the planets will be clearly visible, since the attractive forces of small values ​​change 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, cosmic scales, in the literal sense of these words, do not allow you to see the whole picture. We look, Saturn is located 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 you take on this scale one shade of color is equal to one pixel) approximately 822.8 thousand kilometers. And the radius of the Earth is approximately 6371 kilometers, i.e. diameter 12742 kilometers, somewhere 65 times less than one pixel. Here's how to keep proportions.
We will go the other way. Since we are interested in the gravity of near-planet 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 121 and 127 pixels, respectively. Quite comfortable size.
So, we make a picture with a size of 600 by 600 pixels, determine the value of the force of attraction to the Sun in the orbit of Mercury 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.)
So 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 attraction to the Sun and paint it white.
Then in the center of the picture we put a circle corresponding to the diameter of Mercury in scale (48 pixels) and fill it with a color corresponding to the force of attraction to the planet on its surface.
Next, we fill with the gradient from the planet 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, make this pixel black and stop painting further.
Thus, we get some form of visible change in the attractive forces of the planet and the Sun with a clear black border between them.
(I wanted to do just that, but ... it didn’t work, I couldn’t make a pixel-by-pixel comparison of the two layers of the image.)

Over a distance of 600 pixels are 60 thousand kilometers (that is, one pixel is 100 km).
The force of attraction to the Sun in the orbit of Mercury and near it changes only in a small range, which in our case is indicated by one shade of color.

So, Mercury and gravity in the vicinity of the planet.
It should be noted right away that eight subtle rays are defects from drawing circles in Canvas. They have nothing to do with the issue under discussion and they should simply not be noticed.
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, is the planet itself and its color corresponds to the weight of our kilogram test body on the planet’s surface - about 373 grams. A thin circle of blue shows the boundary between the surface of the planet and the zone in which the force of gravity to the planet exceeds the force of gravity to the Sun.
Further, the color gradually changes, becomes more and more red (i.e., the weight of the test body decreases), and finally, it becomes equal to the color corresponding to the force of attraction to the Sun in this place, i.e. in orbit of Mercury. The border 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 life has a slightly different picture. For example, in this and all other images, the Sun is on the left, which means that, in fact, the area of ​​attraction of the planet should be slightly “flattened” on the left and extended on the right. And the image shows a circle.
Of course, the best option would be a pixel-by-pixel comparison of the area of ​​attraction to the Sun and the area of ​​attraction to the planet and the selection (display) of the larger one. But neither I, as the author of this article, nor JavaScript are capable of such feats. Working with multidimensional arrays is not a priority for this language, but its work can be shown in almost any browser, which solved the issue of application.
Yes, and in the case of Mercury, and all other planets of the earth's group, the change in the force of attraction to the Sun is not so great as to display it with an existing set of shades of color. But when considering Jupiter and Saturn, the change in the force of attraction to the Sun is very noticeable.


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

Earth + Moon

To show the Earth and the Moon changing the scale by 10 times (as in the case of Venus) is not enough, you need to increase the size of the picture (the radius of the Moon’s orbit is 384,467 thousand km). The image will be 800 by 800 pixels in size. The scale is 1 thousand kilometers in one pixel (we well understand that the error of the picture 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 are a system of two equivalent planets with different masses.

Mars with Phobos and Deimos

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

Here, you can immediately see the fundamental difference between Mars and its satellites from Earth with 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 the distant satellite - Deimos. Zoom in 10 times for better viewing. In one pixel 100 kilometers.

These "creepy" rays from Canvas spoil the picture quite a lot.
The sizes of Phobos and Deimos are disproportionately increased by 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" everything. Therefore, the color of their surfaces should be equal to the color in their orbits, but only in order to be better visible, the satellites disks are colored in the color of gravity in the absence of gravity to Mars.
These satellites should be simply monolithic. In addition, since there is no attractive force on the surface, it means they could not have formed in this form, that is, both Phobos and Deimos used to be parts of some other, larger object. Or, at least, they were in a different place, with less gravity than in the zone of attraction of Mars.
For example, here is Phobos . Scale - in one pixel 100 meters.
The surface of the satellite is indicated by a blue circle, and the attractive force of the entire mass of the satellite by 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 gravitational force of the satellite mass. The closer to the surface of the planet, the lower the force of attraction.
(Inaccuracy is again made here. Actually, the white circle is the boundary 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 satellite’s surface, but the color transition shown should be inside the white circle, but then nothing will be visible at all.)

It turns out, as it were, a sectional drawing of the planet.
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. That's even the Hubble Space Telescope "spotted" a similar case.

Asteroid decay P / 2013 R3
Asteroid decay P / 2013 R3

Asteroid decay P / 2013 R3, which is located more than 480 million kilometers from the Sun (in the asteroid belt, further Ceres). The diameter of the four largest fragments of the asteroid reaches 200 meters, their total mass is about 200 thousand tons.
And this is Deimos . All the same as in Phobos. Scale - in one pixel 100 meters. Only the planet is smaller and accordingly lighter, and it is also further from Mars and the force of attraction to Mars is smaller (the background of the picture is darker, i.e. more red).


Well, Ceres does not represent anything special, except for coloring. The force of attraction to the Sun is less, so the color is appropriate. Scale - in one pixel 100 kilometers (the same as in the picture with Mercury).
The small blue circle is the surface of Ceres, and the large blue is the boundary where the force of attraction to the planet becomes equal to the force of attraction to the Sun.


Jupiter is very large. Here is a 800x800 pixel image. Scale - in one pixel 100 thousand kilometers. This is to show the area of ​​gravity of the planet as a whole. The planet itself is a small dot in the center. Satellites not shown.
Only the orbit (outer white circle) of the farthest satellite is shown - S / 2003 J 2.

Jupiter has 67 satellites. The largest are Io, Europe, 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, mass is about 1.5 × 10 13 kg. As you can see, it goes far beyond the sphere of gravity of the planet. This can be explained by errors in the calculations (nevertheless, quite a lot of averaging, rounding, and discarding of some details has 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 semimajor axis of the ellipse and the mass of Jupiter, and M sun is the mass of the Sun. Thus, the radius is rounded to 52 million km. S / 2003 J 2 moves away in an eccentric orbit up to 36 million km from Jupiter
. Jupiter also has a system of rings of 4 main components: a thick internal torus of particles, known as the "halo ring"; relatively bright and thin “main ring”; and two wide and weak outer rings - known as "spider rings", called by the material of the satellites - which form them: Amaltheas and Thebes.
Halo ring with an internal radius of 92,000 and an external of 122500 kilometers.
The main ring is 122500–129000 km.
Amalthea's spider ring 129,000-182,000 km.
Thebes' cobweb ring 129,000-226,000 km.
Enlarge the image 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 boundaries of the rings - the inner border of the halo ring, the outer border of the halo ring and it is the inner border of the main ring, etc.
The small circle in the upper left corner is the area where the gravitational force of Jupiter’s satellite Io becomes equal to the gravitational force of Jupiter in the orbit of Io. The satellite itself on this scale is simply not visible.

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


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

Saturn has 62 known satellites. The largest of them are Mimas, Enceladus, Tethius, Dion, Rhea, Titan and Iapetus.
The farthest satellite is Forniot (interim designation S / 2004 S 8). Also referred to as Saturn XLII. The average radius of the satellite is about 3 kilometers, mass 2.6 × 10 14 kg, semi-major axis 25146000 km.
Rings at 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 remoteness from the Sun.
But look further, a couple of rings surround the asteroid Hariklo (10199 Chariklo) (the diameter of the asteroid is about 250 kilometers), which revolves around the Sun between Saturn and Uranus.
Habr article about an asteroid with rings
Wikipedia about asteroid Hariklo
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. Hariklo is the smallest object to have rings open.
Nevertheless, the force of gravity has only an indirect relation to the rings.
In fact, the rings appear from the destruction of satellites, which consist of material of insufficient strength, i.e. not stone monoliths like Phobos or Deimos, but pieces of rock, ice, dust and other space debris frozen in one piece.
Here the planet drags him with its gravity. A similar satellite that does not have its own attraction (or rather has a force of own attraction less than the force of attraction to the planet in its orbit) flies in orbit, leaving behind itself a loop of destroyed material. So the ring is formed. Further, under the action of gravity to the planet, this debris 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 fragments increases, and the ring disappears.


The purpose of the article’s publication is that maybe someone with programming knowledge will be interested in this topic and make a better model of gravitational forces in the Solar System (yes, three-dimensional, with animation.
Or maybe even make the orbits not fixed, and also calculated - it’s also possible, the orbit will be a place where gravity will be compensated by centrifugal force.
It will turn out almost like in life, like a real Solar system. (This is where it will be possible to create a space shooter, with all that space navigation in the asteroid belt, taking into account the forces acting according to real physical laws, and not among hand-drawn graphics.)
And this will be a wonderful physics textbook, which will be interesting to study.
PS The author of this article is an ordinary person:
not a physicist,
not an astronomer,
not a programmer,
has no higher education.

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