Celestial Mechanics from Start to Finish

The purpose of Science - part 2

The purpose of this document is to study science only with experiments which the aid of (free floating) objects. No light signals are used. In fact the universe is considered dark.
The behaviour of light signals is studied in this document: The purpose of Science - part 1. In that document objects are not used and gravitation is not considered.

Contents

Reflection

1. Introduction

This document is a follow up on two documents:
  1. The purpose of Science
    This document services as a guideline to perform science. The three steps are from start to finish:
    1. Observations. All science starts from observations of the physical reality surrouding us. Part of it we understand, part of we don't. This can be because we observe things which are in conflict which previous observations or observations which are completly new.
    2. Experiments. Experiments are the primary tool to learn more of what is the case in a controlled environment. That means we start which a type of basic experiment and in principle modify one parameter observe how this change influences the result.
    3. Laws, Mathematics and Models. The third step is to unravel the mathematical relations discovered in step 2.
  2. Book Review Einsteins Theory Of Relativity This is a Review of the Book "Einsteins Theory Of Relativity" by Max Born.
    This book raises certain questions:
    1. One of the mayor issues of the book is the Equivalence principle discussed at the pages 44 and 312. This principle claims that the inertial mass and the gravitational mass are the same. This raises the issue if you can study physics without this principle. IMO this is possible. You don't need the law of inertia.
    2. In this document the emphasis is on the movement of individual objects free floating into space. I should emphasize in clusters and in a a maximum of three objects. This is done using experiments. The emphasis in these experiments is Newton's law. The question to what extend Einstein's Special or General Relativity has to be metioned is not adressed.
    3. A high portion of Born's Book and in all most all books about Stellar Mechanics inertial frames are discussed. This is not the case in this document. In fact the logical conclusion of this document is that only one coordination system should be used, from which point all the position of all the objects studied should be considered.

2. Experiments to discus gravitational mass.

The purpose of these experiments is to learn more about the behaviour of physical objects, specific related to the concept of gravitational mass

2.1 Experiment 1 - one object

This experiment involves when there is only object or mass is considered. The shape of the mass is round.
The object considered is assumed to consists of one chemical element. All the objects in the next experiments are build of the same element. As part of an experiment extra objects can be involved, but they are considered massless. Examples are: human beings, measurement devices, engines and clocks. However each of those requires a proper definition.

When our universe only consists of one object it is not possible to learn anything about the laws of physics. The only thing we can do is to build a (square) construction with has a certain height. Such a construction does not change the mass of our object.

2.1.1 Visible Observations.

When there is only one object there is not much to observe.
The only special experiment you can perform is to measure the length of the circumference using a rope.

3. Experiment 2 - two objects.

This experiment involves two objects. A large one and a small one.
With 2 objects we can perform 5 type of experiments:
  1. we can eject the second object straight into the air at point A. In that case the object falls straight down.
  2. we can eject the second object under an angle into the air. In that case the object falls down at point B, a certain distance from point A.
  3. we can eject the second object into the sky such that it starts circulating in an ellipse around object 1.
  4. we can eject the second object into the sky such that it starts circulating in a circle around object 1.
  5. we can eject the second object into the sky such that it will be ejected into outer space.
I have used the word eject because in each case you need a certain engine which performs this task. However in each case you need a different engine. The difference is in the power or force produced by the engine.

3.1 Visible Observations.

Visible observations are only of interest for the two experiments type 3 and type 4
In both these experiments the observations are the same if you are a visitor of object 1 or object 2. In experiment type 3, if you have the right position, you will observe (as an observer on object 1) that a light source (object 2) will change in intensity or magnitude in the sky. In a type 4 experiment the intensity or magnitude will be constant. above your head,
It is also possible that the source of light (object 1 or object 2) draws a straight line in the air when it passes at a distance. This depends if the objects itself are rotating.
More details will be discussed in the next paragraph when 3 objects are involved.

4. Experiment 3 - three objects.

Experiment 3 is a refinement of a experiment 2. Specific: type 3 and type 4.
In this experiment we have four types:
  1. Type 1 resembles a Sun with two planets configuration.
    We have a large object, the Sun, a smaller object, the planet Jupiter circulating around the Sun and an even smaller object, our Earth, also circulating around the Sun.
    They are all build from the same material. The shape of the trajectory of each planet is roughly a circle.
  2. Type 2 also resembles a Sun with two planets configuration.
    We have a large object, the Sun, a smaller object, our earth circulating around the Sun and an even smaller object, the planet Mercury, also circulating around the Sun.
    This type of experiment is interesting because the trajectory of Mercury has the shape of an ellipse
  3. Type 3 resembles a Sun with one planet and one Moon configuration.
    We have a large object, the Sun, a smaller object, our earth circulating around the Sun and an even smaller object, our Moon, circulating around the Earth.
    Both last 2 objects are circulating around each other. This is called: a binary system. .
  4. Type 4 is a follow up of type 3.
    Starting point is a large object the Sun. In this case the size and the mass of the Earth and the Moon will be the same (but much smaller as the Sun) and revolving around each other.
Experiments with 3 objects are much more interesting to study than experiments with two objects. The main reason is that from a vissible point of view from our own perspective always the two other objects can be observed. We can learn something from the interaction of both these objects and if we are lucky this can also say something about the object from which we make these observations.
More detail in the next paragraph.

4.1 Visible Observations between type 3 and type 4

Between the experiments of type 3 and type 4 there is one major difference:
  1. The size of both binary objects is different i.e. different mass. (was type 3)
  2. The size of both binary objects is the same i.e. the same mass. (was type 4)
In the first case what we observe that one object revolves around a point that coincides with the largest object.
In the second case what we observe that both masses revolve around a point half way between both objects.
What we can also observe by performing more experiment that the position of this common revolution point, is a function of the size of the two objects i.e. the mass of both objects.

What is important that in the first case the smallest object moves through space around the largest object. In the second case that both objects move around each other. The best way this can be observed if you place 'a light' on top of each object.

4.2 Outlay how a specif experiment is performed

This are the individual steps how a specific experiment is performed:
  1. You start with a first large object and from which a small second object is ejected such that it comes in orbit in a circle.
  2. From this second object a third object is ejected which also comes in orbit in a circle.
  3. In reality neither these steps (the ejection of an object) can be done in one go, but has to be done in smaller steps because this new object has to be build from the material of the parent object.
  4. Observations will reveal during the building of the third object that the point of revolution will slowly change. The assumption is that the trajectories of both objects of both objects are circular shaped. The observations also reveal that neither objects are at rest.
  5. And now we come to a very important conclusion: Also during the building of the second object there will be a point of revolution and this point of revolution will slowly change. .
    This can also expressed in a mathematical equation: m1*r1 = m2*r2 with r1 being the distance from m1 towards the point of rotation and r2 being the distance from m2 towards the point of rotation.
    When m2 is very small we get r1 = 0. That means the point of rotation coincides with object m1.
    When m1 = m2 we get r1 = r2. That means the point of rotation is half way between object m1 and m2.
    This leads to an overall conclusion that neither any of the three objects is at rest.

4.3 Visible Observations related to type 1 and type 2

What is common between both experiments that there are two objects rotating around the Sun: object 2 is the nearest (this is an inner planet) and object 3 is the fartest (this is an outer planet)
Between the experiments of type 1 and type 2 there is one difference:
  1. The type 1 experiment consists of three objects: A Sun sized object, a jupiter sized object with a smaller mass (circulating around the Sun) and an earth sized object with an even smaller mass (also circulating around the Sun). The detail of the experiment are discussed in 4.2 Outlay how a specif experiment is performed
    In this description object 2 is also ejected from object 1. That is the same in the type 1 experiment. As a result of this experiment the point of revolution will very slightly away from object 1 towards object 2
    In this description object 3 is also ejected from object 2. In the type 1 experiment object 3 is ejected from object 1. As a result of this experiment the point of revolution will very slightly move towards object 3
  2. The type 2 experiment is almost the same. The major difference with the type 1 experiment is that the shape of object 3 is not circular but elliptical. For the rest there is no difference. The point of revolution is slightly different from the type 1 experiment and does not coincide with the position of the Sun.

To study the vissible point of view we assume that the Sun and all the planets move in one plane. For a type 1 experiment we can consider 3 cases:
  1. When the observer is on the largest object (the Sun) both the inner and the outer planet will move in a circle around the Sun. To detect this movement a fire can be placed on each planet. What the fire will show that the number of rotations of each of the planets will be different. The closer to the Sun more rotations, the faster the speed of the planet. This behaviour is called
  2. When the observer is on the inner planet the same behaviour will be observed, but now of a movement of the Sun (illusionary) and the outer planet.
  3. When the observer is on the outer planet this is equivalent as his view from the earth of venus and the Sun. In that case both venus and the sun are moving, but in both cases is rather illusionary.
What makes type 2 interesting is that if all the 3 objects are moving in one plane the directing of the long axis of the inner planet (Mercury) will not be fixed but will move in the direction of the rotation of the outer planet. The reason is because the forces that act on this axis are not symmetrical but asymmetrical. This is a case of gravity assist.

4.4 Example of type 2 experiment: The Star s2 around the BH Sagittarius A*

In paragraph 4.3 Visible Observations related to type 1 and type 2 a type 2 experiment is discussed with the Sun, the planet Mercury and a planet revolving around both.
The article: "Detection of the Schwarzschild precession in the orbit of the star S2 near the Galactic centre massive black hole" https://www.aanda.org/articles/aa/pdf/2020/04/aa37813-20.pdf a similar situation is discussed
However the case is simpler: there are not 3 objects but 2. This is a binary system consisting of a Black hole (Sagittarius A*) and the star S2. The star S2 resembles the planet Mercury and shows a precession. The mass of the BH is roughly 4*10^6 M0. The mass of S2 is roughly 14 M0. That is heavy

However in reality the situation is not that simple: At least 6 stars are involved, all revolving around the BH. To get an idea please select: https://en.wikipedia.org/wiki/Sagittarius_A*
Beside S2 there are 5 more stars involved: S1, S8, S12, S13 and S14. This means this situation resembles much more our solar system, where all the planets influence Mercury. In a comparable situation this als can also mean that 5 stars can influence the precession of S2.

This example is in detail studyied of what is called the Sagittarius A* Project which involves a simulation of 10 small BH's around the primary BH. For more detail select this link: Sagittarius A* Project

5. Experiment 3 - three rotating objects

In 4. Experiment 3 - three objects three objects are involved in different experiments. Each experiment starts with 2 objects (#2 being ejected from #1) which are circulating around each other. Next a third object is ejected either from object 1 or from object 2 (being of different size) which will also start to circulate around each other. The general lesson is, that each object ejected in space, will also effect the state of the other objects. This is what is called: action reaction.

As part of 3. Experiment 2 - two objects. two experiments are discussed: To eject an object straight into the air and two eject an object under a certain angle. Both these experiments show no permanent effects because there is no net mass loss. In the next experiment where an object is ejected under an angle, there is a permanent exchange of material between object 1 (a loss) and object 2 (a gain). Because this is an action in the forward direction this will cause a counter force in the backward direction. As a result object 1 will start to rotate.
It is reasonable to assume that object 2, while it grows in size and collects material, will be influenced the same and also will start to rotate.

6 Experiment 4 - 3 objects and a balance.

Experiment 3 always start with one object and a fixed amount of material. Using this material two other objects were created which were ejected (in small increments) into the air.
In Experiment 4 we also start from 1 large object (a round object, our earth) and we make four small round objects, with each both exactly the same size i.e. m0. At the same time keeping object 1 round.
We also in some way or another get a balance (with adjustable arms) See: en.wikipedia.org/wiki/Weighing_scale
The purpose of this experiment is to test the law: m1*r1 = m2*r2 In all cases the results are in accordance with the law: m1*r1 = m2*r2

7 Experiment 5 - 2 falling objects

In Experiment 5 we also start from 1 large object 1 (a round object, our earth) and we make two small round objects of equal size m1 and m2 and one large round object m3. At the same time keeping object 1 round.

Reflection 1 - Dark and Light

The experiments discussed, specific the objects involved, all consists of the same material. In the text I sometimes identify an object as the Sun. The reason is because it is the largest. A Sun shines. The largest object used does not shine, does not emit photons. In fact the Universe discussed is completely dark. What this also means that you can consider each object a black hole.
Does all of this, the fact that we assume that there are no photons involved, have any influence on the behaviour of the objects considered? The answer is: No.
One of objects of the experiments is to study what is happening. We humans do that by performing observations with our eyes. In our everyday world we use sun light (photons). In the experiments discussed the objects involved don't shine (emit photons). One way to solve that is to place a fire on each object. The physical characteristic of photons is, that they have a speed and as such photons show a retarded image of the physical reality. In stead what we imagine is virtual light. The physical characteristic of virtual light (virtual photons) is that it gives an instantaneous image of what is physical happening.
As said above light (photons) and all electromagnetic processes have (almost) no influence on the behaviour of the objects nor on the possible mathematical equations that describe this behaviour.
This is different for the mathematical equations that describe the measurements of the positions, in so far these measurements involve light (photons)

Reflection 2 - The purpose of science

How do we explain the behaviour of objects. The behaviour of objects that travel through the universe?
May be you have to answer one question first: What makes an object an object. How does a star form?
We don't have to discuss black holes because a black hole is nothing more than a very heavy star compressed to a 'point', any way compressed to a 'point' much smaller than an ordinary sky of the same weight.
The answer on that question is not that simple. One approach is that original all the material in the star was more or less a cloud of material particles which started to rotate and as such... But that answer does not really explain how a star formed; it is much more an assumption.
A different answer is the rotating star started to compress under the influence of gravity.
That is also an answer but this transfers the problem to a new question what is gravity. Maybe that is a question impossible to answer. Also here you can answer: that is the way it is. But this is also not a very satisfactoring answer.
Gravity is explained by introducing particles called gravitons that is almost as detailed you can go.
A different concept is a field. But a field is typical a calculated quantity, using a mathematical description of the process under consideration. One mathematical description to describe the behaviour of stars is Newton's Law. This law uses the concept of gravitational forces to describe the behaviour, the atraction, between moving stars. A unit of force per volume (point particle) you can call a field. How ever such a field also explains nothing.

Reflection 3 - Conclusion

The general theme of this document is stellar mechanics. Starting point is an experiment with one object, then two and finally three.
My objective is to keep each experiment simple, give enough information and try to describe everything as clear as possible.
As such experiment 2 is a collection of 5 experiments, each slightly different.
In the first one of this sequence of 5, a small object is ejected straight in the sky. In the second one again a small object but now under an angle. In the third one a much larger in order to test a much larger object. That is physical not possible. In stead what I do subdivide this experiment in small steps, and I use smaller objects to slowly build up a larger object in space. How you actually want to do that I leave that open for the imagination of the reader.

As already mentioned in Reflection 2 - The purpose of science the guiding law to describe the movement of the objects in space is Newton's Law.
The simplest way IMO is one reference frame (or grid) which by definition you call at rest. Moving clocks are not involved. When you do that the positions of all the objects are instantaneous available.

The biggest problem with Newton's Law is that it assumes that all the forces act instantaneous. To solve that (at least that is an option) the speed of gravity has to be introduced. What that means is that the forces that act on an object are not a function of the distance between these objects that are inanciated by the present position of these objects but by the distance in the past.

Reflection 3 - Conclusion

The general theme of this document is stellar mechanics. Starting point is an experiment with one object, then two and finally three.
My objective is to keep each experiment simple, give enough information and try to describe everything as clear as possible.
As such experiment 2 is a collection of 5 experiments, each slightly different.
In the first one of this sequence of 5, a small object is ejected straight in the sky. In the second one again a small object but now under an angle. In the third one a much larger in order to test a much larger object. That is physical not possible. In stead what I do subdivide this experiment in small steps, and I use smaller objects to slowly build up a larger object in space. How you actually want to do that I leave that open for the imagination of the reader.

As already mentioned in Reflection 2 - The purpose of science the guiding law to describe the movement of the objects in space is Newton's Law.
The simplest way IMO is one reference frame (or grid) which by definition you call at rest. Moving clocks are not involved. When you do that the positions of all the objects are instantaneous available.

The biggest problem with Newton's Law is that it assumes that all the forces act instantaneous. To solve that (at least that is an option) the speed of gravity has to be introduced. What that means is that the forces that act on an object are not a function of the distance between these objects that are inanciated by the present position of these objects but by the distance in the past.

Reflection 3.1 - Conclusion - What is not discussed

This document only discusses celestial mechanics. That means the movement of 'free-floating' object through space. Certain issues are not discussed. Let us make a list.
  1. Inertial mass. Only gravitational mass is discussed. Inertial mass is typically related to the movement of objects on the surface on Earth. Friction (resistance) is here the primary extra force. Immediate when you throw an object in the air the driving force becomes gravitational and the masses gravitational. That is why inertial mass is not considered.
  2. Inertial movement.
    Inertial movement is considered the movement of objects, which undergo no extra forces and which move in a straight line. Again here there are two different situations.
    • Movements on the surface of the earth. In all these cases always gravitational forces are involved. As a result, because of friction the moving objects involved always will travel a certain distance and come to 'rest'.
    • Movements in space. In principle, in a thought experiment, these objects can travel in a straight line. That means you can perform thought experiments using clocks which travel in straight lines, in order to mimic the behaviour of a clock.
      In reality, always gravitational forces are involved from different directions which influence the behaviour of different objects (in relatively close contact).
  3. Different reference frames. IMO it does not make sense to study the behaviour of different objects which influence each other in different reference frames.
    By definition, this reference frame is considered at rest.
  4. Moving clocks. A moving clock is a clock which position (x,y,z coordinates) changes in time (defined based on what humans experience).
    The problem with a moving clock is, which internal functioning involves light signals, that the clock count is less than the clock count of a clock at rest.
  5. The two concepts absolute versus relative.
    It does not make sense to make a difference when there exists no clear difference in usage.
    Positions are always measured in a reference frame and are by definition always based upon the origin of that reference frame. They are measured by the three coordinates x,y,z and by a time coordinate.
    Velocities are always calculated based on three x,y,z coordinates (a distance) and two time-coordinates (a duration).
    Accelerations are always calculated based on two velocities or three positions.
  6. The concept of Lorentz Transformations.
    Lorentz Transformations are based on the usage of different reference frames. In Celestial Mechanics, only one reference frame is considered as such the Lorentz Transformations have no usage.
    If the reader assumes that more reference frames are required than the importance can be reconsidered.
  7. The speed of light versus the speed of gravity. Photons are not involved in celestial mechanics. Gravitons are. The behaviour of each is very different.
  8. Schwarzschild Geometry The concept of a test particle moving in the Schwarzschild geometry, described by the line element: See par 25.3 in the book Gravitation MTW or Chapter 14 in the book Introducing Einstein's Relativity by Ray d'Inverno.
    The concept of test particles is not used in any of the experiments discussed in: 4. Experiment 3 - three objects. . Instead, large objects should be studied. In that case, all objects involved will influence each other and have to be studied using the full Einstein's Theory and not the PPN approximation.

Reflection 3.2 - Conclusion - Two reference frames

The purpose of this paragraph is primarily the issues when two reference frames or coordination systems are used.
First, we discuss what are the issues and what is involved in one reference frame in a rather general way.
Before we start with a reference frame we first define local space (part of the universe) which contains n objects. We define the concepts time (human experience), light, and that light has a speed. We also define that each object has a position and that the positions of each object changes (in time) relative towards the other objects. We also define a clock which gives clock counts and also shows the time

A reference frame is a coordinate system consisting of three-axis x, y and z.
The coordinate system has an origin O. The origin O1 is linked to an object. In this case, this is (the centre of) object 1. This is also the place where the clock resides. The clock shows the time t.
Using this coordinate system we can measure positions of each object and time t. A position is of each object is defined as: r(t) = (x,y,z,t)
Along the same line, we can calculate the velocity v(t) and the acceleration a(t) of each object as a function of coordination system and time t

Without going into more detail let us define a second reference with its own three-axis x',y' and z'.
The main difference is that this frame has a different origin O2, is linked to a different object (object 2) and has its own clock and uses time t'. The position of a object is defined as r'(t') = (x',y',z',t'). The speed is v'(t') and acceleration a'(t').

The first question to answer is: what is the relation between the two reference frames.
The answer is, this depends on the relative movement between the two objects 1 and 2 or more generally between the origins O1 and O2.
Let us study this by means of an Example.
  1. In a simple case we can consider that y and z-axis are the same and that both objects move along the same line in the x or x' axis with a constant speed. That means the distance increases linearly. Calculated in reference frame 1 the origin O2 has a speed v(t) and calculated in reference frame 2 the origin O1 has than a speed v'(t').
  2. The first question to answer is: How is the speed v(t) calculated?
    To do that you need a ruler and clocks along the x-axis, which run in sync with the clock at O1.
    Next, you can write down the position and time when object 2 is at 2 distinct positions. This allows you to calculate the speed v2(t) of object 2 or O2 in reference frame 1. The same can you do to get the speeds vn(t) for all n objects.
  3. To calculate the speed v't' in reference frame you can do the same with clocks along the x' in sync with the clock at O2. Finally, you can get the speeds v'n(t') for all n objects in reference frame 2.
  4. The next question is: do both clocks t and t' measure the same time.
    To answer that question you have to perform the same experiment in both reference frames.
    • What you need is a ruler along the x-axis and clocks at equal distance synchronized with the clock at O1 (time t) In this case, you have to compare the clock at O2 with the clocks along the x-axis and write down the clock readings when they meet. The readings of the clock at O2 and the clock along the x-axis will be different.
      For more detail see: The purpose of Science
    • What you also need is a ruler along the x' axis and clocks at equal distance synchronized with the clock at O2. (time t') In this case, you have to compare the clock at O1 with the clocks along the x' axis and write down the clock readings when they meet. The readings of the clock at O1 and the clock along the x' axis will be different.
What this tells you is that to compare the speeds vn(t) with v'n(t'), because they depend on different clocks and these clock readings are not the same, can be complicated.
GR tells you (Book subtle is the Lord... by A Pais) that "1. The laws of physics take the same form in all inertial frames.". That may be true, but it does not tell you what this form is.
The overall lesson is to use only one reference frame, which you consider at rest.

Reflection 3.3 - The Two Most Important Lessons

  1. Observations is the most important part to study Celestial Mechanics.
    The most important reason is that without observations and measurements you cannot perform Celestial Mechanics i.e. the study of free-floating objects through space. The second important reason is that if you want to demonstrate that you understand what is involved by making predictions, you again must compare these predictions with observations.
  2. The second most important reason is if you want to study Celestial Mechanics you must study at least three objects. By preference a system including Jupiter and its Moons. However, if you count carefully, that system already includes 4 objects.
    Such a system can be studied that each object is a point but each object should have a mass.
  3. The third lesson is, but less important, that within the strict context of Celestial Mechanics you cannot perform any experiment.
    Experiments to test the behaviour of clocks are not part of Celestial Mechanics.
    Experiments to drop objects on the earth are not accurate enough to reveal any mathematics (laws) involved.
  4. The fourth lesson is, but also less important, that the physical laws that describe the behaviour of the objects, should as much as possible mimic the physical behaviour involved.

Nooit geschoten, Nooit geraakt
Never fired, Never hit
Created: 12 April 2020

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