Celestial mechanics for dummies?
The purpose of this document is to try to explain in simple language what is involved in order to understand the movement of objects (i.e. stars and planets in the sky) with out going into much technical details.
The question mark is there on purpose. To understand requires to understand the details and the details are not simple.
The study of celestial mechanics involves a contradiction. On one side it is based on human observations. While at the other side the results i.e. the laws that describe the movements including the evolution of the universe are completely independent from any human perspective. This is an important point to consider.
- Light as we observe it are photons of a specific frequency. The evolution of the Universe and the laws of nature are completely independent of light, except if we want to understand the behaviour of light itself. As such instead of (speed of) light we should consider electromagnetic radiation in general i.e. photons of all frequencies (energies)
- There are two different ways to study the Universe: Local or Global.
Local implies centred from our point of view, from the earth, from the Milky way. That is not correct.
Global implies independent from any point. Global implies the total universe, visible and not visible. That is the correct approach.
- When you compare example 1 and example 2 you have the problem of superluminal motion. This problem accepts that locally nothing can move faster than the speed of light but globally the speed can be much larger.
What you have is an observer at a distance from you which accepts that in his local environment (rest frame) nothing can move faster than the speed of light. In your local environment this observer can not move faster than the speed of light. Combining the two gives you superluminal motion.
The true issue is what is physical possible from every point of view (observer).
1. The movement of two objects in space.
Let us start by studying the movement of two objects in space. With this I mean at short distance an apple that falls on the surface of the earth or two stars from an extremely large distance which move toward each other.
In both cases, in order to calculate the positions of objects with mass m1 and m2 you can use Newton's law. Newton's law is based on the concept that within a "closed system" the sum of all the forces acting on all objects is zero. In this case with two objects there are only two forces.
Figure 1 shows this configuration. F12 is the force from m1 towards m2. F21 is the force from m2 towards m1. Because both forces are equal but in opposite directions the sum of all forces is zero.
Accordingly to Newton's Law the two forces F12 and F21 are equal and can be described by the following mathematical equation:
Figure 1 - Example 1: 2 objects
That means the force is equal to the mass m1 of object 1 times mass m2 of object 2 divided by the distance between the two (times a constant G). Because this law is symmetric the forces are equal.
F12 = F21 = G * (m1*m2)/r^2
As a result of the two forces both objects will start to move towards each other.
This movement is described by the following 2 mathematical equations, one for each object:
r^2 is the squared distance r between the two objects.
F12 = m1 * a1 = G * (m1*m2)/r^2
F21 = m2 * a2 = G * (m1*m2)/r^2
When you remove m1 in Equation 2a and when you set G = 1 you get the following 2 equations which describe the acceleration for each object:
What both equations show is that the acceleration (and the speed) increases in time as the two objects approach each other, because the distance r and r^2 decreases.
F12 = a1 = m2/r^2
F21 = a2 = m1/r^2
What (Equation 3a) shows is that when the mass of object 2 is light compared to object 1 the speed and acceleration of object 1 is small because object 1 is heavy.
What (Equation 3b) shows that under these same conditions the speed and acceleration of object 2 is large because object 2 is light
2. The movement of two objects in space in equilibrium
The most important observation of the above example is that the two objects will collide. That means that this system is not stable or finite.
For the system to become stable (in equilibrium or infinite) the objects should not collide. For a "closed system" to be in equilibrium the sum of all the forces acting on each object (for example) should be zero. That means that beside the force F12 which acts on object m1 you need a different force in opposite direction. The same for the force F21 which acts on object m2.
Figure 2 describes this new situation.
The difference between Figure 1 and Figure 2 is that each object is involved with two forces. One attracting gravitational force towards the other object and one equal force in opposite direction. This second force comes into play (for each object) when you give each object a speed such that they start rotating around each other, around point C.
<---m1---> C <---m2--->
| F12 F21
Figure 2 - 2 objects
The point C is what is called the center of Gravity.
In order to calculate this speed the following equations are used:
r^2 is the squared distance between the two objects.
a1 = m2/r^2 = v1^2/r1
a2 = m1/r^2 = v2^2/r2
r1 is the distance between object 1 (m1) and the center of gravity C.
r2 is the distance between object 2 (m2) and the center of gravity C.
v1^2 is the squared speed v1 of object 1
v2^2 is the squared speed v2 of object 2
The importance of (Equation 4a) is that if you know v1 and r1 of object 1 and r than you can calculate the mass of object 2.
3. The movement of three objects
Let us now study the movement of three objects or stars.
In this case we get for object 1 with mass m1:
Equation 5 consists of three terms. You have to add more terms if more objects are considered.
m1 * a1 = (m1*m2)/r^2 + (m1*m3)/r^2
The purpose of the equation 5 is to calculate the acceleration a1 acting on mass m1 for a small period of time dt. This is simply done by dividing through m1. You get:
- The first term m1 * a1 is the resulting force of object 1 induced by the other objects. a1 is the resulting acceleration.
- The second term (m1*m2)/r^2 is the force from object 2 on object 1. r^2 is the square of the distance between object 2 and object 1. In reality Newton's law also include a factor G. In this case G is considered 1.
- The third term (m1*m3)/r^2 is the force from object 3 on object 1. r^2 is the square of the distance between object object 3 and object 1.
If you know the speed v1 of that moment the new speed becomes: v1 = v1 + a1 * dt.
a1 = m2/r^2 + m3/r^2
If you know the position p1 of that moment the new position becomes: p1 = p1 + v1 * dt
For object 2 a similar methodology exists:
In this case we get for object 2 with mass m2:
m2 * a2 = (m2*m1)/r^2 + (m2*m3)/r^2
a2 = m1/r^2 + m3/r^2
Along the same line as above using the speed v2 a new speed can be calculated. The same for the position p2.
- The first term m2 * a2 is the resulting force of object 2. a2 is the resulting acceleration.
- The second term (m2*m1)/r^2 is the force from object 1 on object 2. r^2 is the square of the distance between object 1 and object 2.
- The third term (m2*m3)/r^2 is the force from object 3 on object 2. r^2 is the square of the distance between object 2 and object 3.
It should be remembered that p1, v1, and a1 are vectors. That means they have an x, y and z component in space. The same for p2, v2 and a2.
In order to calculate the path of the stars the above set of calculations has to be performed in what is called a loop. Before calculating this loop first the initial conditions have to be calculated. In summary the whole procedure can be described by the following set of steps.
- Define the number of objects n you want to simulate.
- Calculate the step size dt.
The following steps are each performed for all the n objects.
- Calculate the initial positions: p0(t)
- Calculate the initial velocities: v0(t)
- Calculate the masses: m(n)
Here after the loop starts
- Calculate the accelerations.
- Calculate the (new) velocities. v(t) = v(t) + a(t)*dt
- Calculate the (new) positions. p(t) = p(t) + v(t)*dt
- Repeat with step 6
4 Real life situation
The next important aspect is to try to simulate a real life situation. A simple case is the Sun and two planets. To make things simple we start with the Sun at a fixed position.
- The first thing to do is to calculate the postion each of the two planets a certain moment.
- In order to calculate the speed you need at least the positions of two moments.
The speed is than the difference between the two positions p1 and p2 divided by dt i.e. v(t) = (p2-p1)/dt
For object 2 at time t2 the speed v22 = (p22-p21)/dt (See below).
- The next step is to calculate the masses of the three objects. The simplest way is to assign the mass of the sun equal to 1.
The first lesson to learn is that in fact you have to use Newton's Law to calculate the masses of the planets. To be more precise the masses of the objects are not so much a physical parameter but much more a mathematical parameter.
- In order to calculate the masses you first have to calculate the resulting accelerations for each of the objects caused by the other objects.
Each acceleration calculation requires 3 positions at regular intervals. For object 2 this are p21, p22 and p23. p21 means the position of object 2 at time t1 etc.
The acceleration of object 2 at time t3 = a23 = (v23 - v22)/dt = (p23-2*p22 + p21) / dt
With (Equation 8) and with m1 = 1, the mass of object 3 i.e. m3, can be calculated. In a similar matter m2 can be calculated.
4.1 The movement of two objects in space simulation.
To perform an actual simulation of two objects select the following link: Visual Basic 5.0 Two Body Program: "VB Two_body"
The purpose of this program is two fold:
The important point is that the simulation is done solely based on observations. In this case these are 3 positions of m2 and two positions of m1.
- First to calculate the masses m1 and m2 of each object.
- Secondly to perform a simulation of the two objects.
When one of these observations is changed the result of the simulation:
- shows different mass values for m1 and m2
- the trajectories will change
5 Initial Conditions
One of the most important points to calculate the positions of the stars or of a total galaxy are the initial conditions. If you want that you simulation starts at present or at any other moment than you must know the positions, velocities and accelerations for all the objects involved at present.
This raises a serious problem because you can observe the stars and all the galaxies at present but not in the present state (position, velocity acceleration) but only in the past. That means when you observe the Andromeda galaxy at present you will observe the galaxy in a state (position, velocity, acceleration) 2.5 million years ago. This observed speed is 110 km/sec towards us. That means you have to do some arithmetic to calculate its present position. For more information about Andromeda Galaxy Wikipedia
The average distance from Pluto to the Sun is roughly 6*10^9 km. It takes roughly 2 hours for light to travel from that distance. That means on average we observe the planet Pluto 2 hours in the past.
When you want to do an accurate simulation this complexity has to be taken into account.
The basic idea behind Newton's Law is that the sum of all the forces between all objects in a closed system is zero. This implies the forces induced by all the objects in the entire universe. From a practical point this are all the stars in our Galaxy. All those forces for one object (our Sun) don't cancel. The resulting force to make Our Galaxy stable (in equilibrium) is the force that drives our Sun around the center of Our Galaxy.
However there is one more complication. Consider two stars of equal mass which move in exact circle around each other. Consider a third much smaller which moves in a large circle in the same plane around the center of gravity of the two large stars.
Figure 1 shows this situation, with m1=m2 and m3 is small
The above situation shows the situation where the distance r13 between m1 and m3 and the distance r23 between m2 and m3 are the same. That means that all the forces F13 (F31) and F23 (F32) are also the same (in absolute sense). F13 is the attracting force from m1 on m3. F23 is the attracting force from m2 on m3. That means m3 is attracted to both at equal strength. That is the situation when forces act instantaneous.
. C . m3
. | v3
Figure 3 - 3 objects
In general the situation is more complex. This is discussed in the next paragraph.
6 Speed of gravitational field
In real that is not the case. In reality it takes time for the forces to propagate.
That means in the drawn situation the force towards m1 at present, which (m1) moves towards us, comes from a distance further away than at present and is smaller than the force based on the present position.
The force towards m2 comes from a closer distance and is larger.
The next question is what is the speed of gravity (of the gravitational field or of gravity waves). Accordingly to General Relativity the speed is equal to the speed of light.
IMO Special Relativity nor General Relativity make any claim what the value of the speed of light c is. This value should be calculated based on observations and is supposed to be constant. The name is also misleading. What we are discussing is in general the speed of an electro-magnetic field not the speed of light.
In the above examples we have studied the movement of two or three objects. In reality the objects could be planets, stars, galaxies but also black holes. All these objects move accordingly to Newton's Law if the objects are considered point masses.
What all these objects do they create a gravitational field. Such a field transmits information by means of hypothetical particles called gravitons. For Black holes this is important because as the name implies black holes can not physical be observed because they don't transmit photons (light) but they can be gravitational be detected. Within our Galaxy there is a Black hole because we can observe stars with rotate around the center of our Galaxy which is "invisible". Using those stars the mass of the Black hole can be calculated.
What that means is that gravitons behave completely different as photons, implying that the speed which these particles propagate (gravitons versus photons) can be different.
6.1 Speed of gravitational field simulation.
To perform an actual simulation of two objects select the following link: Visual Basic 5.0 Newton C Program: "VB Newton C"
The purpose of this program is to study the influence of a moving gravitational field on the behaviour of a system consisting of three objects.
Two examples are studied:
The result of the simulations indicate that a simulation based on actual observations, is much more realistic in the case of "the Sun and the planets" in our Solar system than a simulation based on two Black Holes and a third star.
- Two Black holes and a star.
- The Sun, the planet Mercury and Jupiter.
The problem with the two Black Holes simulation is that the masses are very difficult to calculate solely based on what is observed. In this case the third star. This makes Lisa type projects very difficult.
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First Release: 6 October 2014
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