[Halc]

What do you mean by: "2nd rule about only the matter"?
Is it Newton's second law?

The first rule you quoted was Newton's Shell Theorem concerning only uniform spherical objects.
The 2nd rule you quoted was the one from the ohio-state site concerning the mass that has net effect on an orbit and the mass that doesn't.
I don't know the name of that rule or who came up with it, but you quoted it, and then you misrepresented what it means.

The "second rule" is just a direct outcome from the "first rule" which is the Newton's Shell Theorem.
With all the respect to ohio-state (and I have a respect...), they can't just invent new rules for gravity.
So, the second statement is not a second rule, it is just a logical outcome from Newton's Shell Theorem.

I'm not disputing the rule.  It is a known thing, yes.  I just don't know the name of it, or how it is derived from Newton's sphere law.

Hence, by Kepler law we can calculate the total mass which is requested to meet the orbital velocity of the sun around the galaxy, while Newton’s First Theorem tell us that: "M is the total mass in a sphere of radius a, centered on the galactic center.

You're applying the 2nd rule there, not the first theorem.  The galaxy is not a sphere nor is it spherically symmetric, so the first theorem does not immediately apply.  The nameless law does however, so we're good.

Newton’s First Theorem - If the density distribution of a ball of mass M is spherically symmetric, then the size of the force between the ball and a point mass m, that lies outside the interior of the ball, is given by the left-hand side of (1), where r is the distance between the point and the center of the ball."

Slightly generalized version, and thus a better one, yes.  It doesn't require a sphere, but merely something spherically symmetric.  But the galaxy isn't, so this law doesn't directly apply.   The galaxy is roughly modeled as a disk, and the force vector of a point out of the plane of a disk does not point to the center of mass of the disk like Newton's theorem says it must if it were spherically symmetrical.

[Halc]

This law is very relevant
It actually confirms that the shape of the object is none relevant for its central point of mass.

I didn't say otherwise.  The shape of Earth for instance, with its bulges, has no effect on Earth's center of mass.  I agree with that.

As long as all the masses in the object are fully connected the center of mass of this object is none relevant with its shape. (Spanner, dog, cat or even elephant).

All the others, yes, and even a spraying garden hose, but not the cat, which has properties outside physics.

http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/n2ext.html
"A set of masses connected by springs will follow a path such that its center of mass moves along the same path that a point mass of the same total mass would follow under the influence of the same net force."

OK.  That one is pretty obvious too, since it derives directly from the computation of center of mass.  The springs are not even necessary.  It can be a collection of 13 random stars from nowhere in particular, and that collection of 13 unrelated stars will have a center of mass that will follow this rule.  What it doesn't mean is that an object at that center of mass will follow the same path.
What it especially doesn't mean is that any force exerted by that collection of spring-attached objects is going to be the same as if all the mass was concentrated at their center of gravity.  This is very easy to demonstrate.

So, even if that spanner has an offset, it won't set any extra thrust.

The rules don't say that at all.  They talk about net force and those net forces acting on the center of gravity of each object in question.  So if the spanner puts a net force on an object that is anything but perpendicular to its motion, it will be exerting thrust to it.

The Earth/Sun system could be considered one such object which is not a sphere, and thus it puts massive thrust on another object (the moon), all because of its offset.  According to your assertions, that should not happen, and the moon should wander off around the sun on its own in a nice circular orbit around the center of mass of the combined Earth/Sun object.

In our calculation we just need to focus on its center of mass.

Wrong!!  Center of mass has no angular momentum, and this tidal thrust effect is all about net forces resulting from transfer of angular momentum.  None of the laws above describe angular effects on the tumbling spanner and such.

I throw a rapidly spinning pool noodle, and it is spinning far slower before it hits the ground.  The net forces on the noodle do indeed determine the path of its center of gravity, but do not in any way describe the loss of spin.  That rules is inadequate for the situation being described.

The orbit about the galaxy is less about angular momentum and forces since there is no significant transference going on.  You can treat a lot of things as point masses on that scale, but not the galaxy as a whole since it is not spherically symmetrical.

Conclusion -
Based on the following laws (each one by itself):
1. Newton's Shell Theorem
2. Newton's second law

You can't take laws by themselves.  Let's take these two laws.  Consider me on Earth with the sun directly overhead, and Mercury and Venus don't exist.  In a sphere of radius R where R is the distance from the sun to me, only the mass in that radius determines my orbit.  It is the Sun which is spherically symmetric.  So the first law applies.
Now let's apply the 2nd law.  I am going too slow to maintain my orbit at that radius (because Earth's spin velocity cancels some of its orbital velocity at noon), and the mass outside the radius R has no effect on my orbit, therefore I should drop in towards the sun, getting sucked into the sky.  The Earth cannot hold me down because it is outside R when the sun is directly overhead, and thus does not contribute to my orbit, per the 2nd law.

This is the sort of proofs I'm seeing from you.

[Dave Lev (OP)]

Thanks Halc.
I really appreciate all your efforts.
However, please try to use some solid evidences (mathematics, Newton/kepler.. laws) in order to prove your statements.

The rules don't say that at all.  They talk about net force and those net forces acting on the center of gravity of each object in question.  So if the spanner puts a net force on an object that is anything but perpendicular to its motion, it will be exerting thrust to it.

Can you please prove that a spanner can put a net force on an object just by pointing to some offset?

You can't take laws by themselves.  Let's take these two laws.  Consider me on Earth with the sun directly overhead, and Mercury and Venus don't exist.  In a sphere of radius R where R is the distance from the sun to me, only the mass in that radius determines my orbit.  It is the Sun which is spherically symmetric.  So the first law applies.
Now let's apply the 2nd law.  I am going too slow to maintain my orbit at that radius (because Earth's spin velocity cancels some of its orbital velocity at noon), and the mass outside the radius R has no effect on my orbit, therefore I should drop in towards the sun, getting sucked into the sky.  The Earth cannot hold me down because it is outside R when the sun is directly overhead, and thus does not contribute to my orbit, per the 2nd law.

That example is not clear to me.
If the sun is just above me while I am waking on Earth, than based on Newton second law, I'm an integrated mass of Earth.
Therefore, the Sun has no impact at all about my location.
If there is no Earth or moon, and it is all about me and the Sun, than my orbital velocity must be a direct outcome of gravity force based on R. In this case, the Earth and the Moon are not there to have any impact on my orbital cycle around the Sun.

With regards to the following message:
"I am going too slow to maintain my orbit at that radius (because Earth's spin velocity cancels some of its orbital velocity at noon)"
This is a severe mistake.
If I have to orbit around the Sun, there is no way to slow down due to that Earth spin velocity cancelation.
So, you don't have to maintain your orbital velocity. The Sun gravity force works for you. This is a key element.
Newton didn't specify even one word about the impact of spinning velocity.
This is a new idea which had not been confirmed. It seems to me as a new idea which we have invented just in order to offer a nice answer for: Why the Moon and the earth are drifting outwards?
If you believe that there is a possibility to slow down the earth orbital velocity "because Earth's spin velocity cancels some of its orbital velocity at noon", than please prove it.

Wrong!!  Center of mass has no angular momentum, and this tidal thrust effect is all about net forces resulting from transfer of angular momentum.  None of the laws above describe angular effects on the tumbling spanner and such.

I throw a rapidly spinning pool noodle, and it is spinning far slower before it hits the ground.  The net forces on the noodle do indeed determine the path of its center of gravity, but do not in any way describe the loss of spin.  That rules is inadequate for the situation being described.

The orbit about the galaxy is less about angular momentum and forces since there is no significant transference going on.  You can treat a lot of things as point masses on that scale, but not the galaxy as a whole since it is not spherically symmetrical

We do not discuss about rapidly spinning pool of noodle.
We discuss about Sun/earth/Moon orbital cycles.

If the angular momentum or the revolving speed of the Moon or the Earth can slow down the orbital velocity of the Erath or the Moon - than please prove it.
I still can't understand how any sort of offset in bulges can set any sort of thrust.
Please use the spanner as an example. you are more than welcome to force the spanner in any sort of offset as you wish. Try to prove why by doing so we shall get extra trust on the orbital object.
Please try to prove this idea by mathematics.

[Dave Lev (OP)]

So far we have just discussed about the impact of the orbital cycles of Planets and Moons.
However, what about the stars? What about our star - the Sun?
Why all our scientists are positively sure that during all his life time the Sun had to keep the same orbital radius?
How could it be that all the moons and Planets are drifting outwards, (or inwards based on unproved tidal idea) while the sun is fixed at the same radius?
How could it be that in one hand our scientists claim that the SMBH increases its mass by eating stars and gas clouds, while on the other hand they don't consider an option that stars must migrate/drift inwards in order to supply the requested food for the SMBH monster?
If the SMBH has 4 x 10^6 sun mass, (while our scientists believe that this mass had been taken from the stars in the galaxy) than somehow 4 x 10^6  stars had to drift inwards.
So why our scientists are so sure that the Sun was always at the same distance from the center of the galaxy???
What makes our star so unique that it had to stay so far away from the monster at the galaxy center?

[Halc]

Can you please prove that a spanner can put a net force on an object just by pointing to some offset?

I didn't say it could do it by pointing to an offset.
The spanner is tumbling, and that tumblingis slowing measurably.  It couldn't do that if it was treated as a point mass as you are attempting to do.  You can't put torque on a point.

You can't take laws by themselves.  Let's take these two laws.  Consider me on Earth with the sun directly overhead, and Mercury and Venus don't exist.  In a sphere of radius R where R is the distance from the sun to me, only the mass in that radius determines my orbit.  It is the Sun which is spherically symmetric.  So the first law applies.
Now let's apply the 2nd law.  I am going too slow to maintain my orbit at that radius (because Earth's spin velocity cancels some of its orbital velocity at noon), and the mass outside the radius R has no effect on my orbit, therefore I should drop in towards the sun, getting sucked into the sky.  The Earth cannot hold me down because it is outside R when the sun is directly overhead, and thus does not contribute to my orbit, per the 2nd law.

That example is not clear to me.
If the sun is just above me while I am waking on Earth, than based on Newton second law, I'm an integrated mass of Earth.

You're not integrated with Earth since you are completely detached from it. Seatbelts are cheating.  I'm just considering the gravity of the sun on your orbit.  According to that 2nd law, or at least according to the way you are using it, nothing outside that R sphere has a net effect on my motion, therefore, since Earth is entirely outside that sphere, its gravity doesn't affect me.

I know it's wrong, but that's how you're interpreting the rule: by oversimplifying and not considering deviations from the uniformity.  The Earth is a huge deviation.  Deviations matter.  The tidal bulges are deviations, and the change the direction of the force vector acting on the moon just as Earth changes the force vector acting on me at high noon.  But you cherry pick which deviations matter and which do not, depending on what purpose you desire.  That's very fallacious reasoning.

I haven't figured out your purpose in all this.  Clearly not here to learn.  You seem bent on finding contradiction in simple orbital mechanics, but not sure why.

Therefore, the Sun has no impact at all about my location.
If there is no Earth or moon, and it is all about me and the Sun, than my orbital velocity must be a direct outcome of gravity force based on R. In this case, the Earth and the Moon are not there to have any impact on my orbital cycle around the Sun.

So rule #2 suddenly doesn't matter when you find it inconvenient?

With regards to the following message:
"I am going too slow to maintain my orbit at that radius (because Earth's spin velocity cancels some of its orbital velocity at noon)"
This is a severe mistake.
If I have to orbit around the Sun, there is no way to slow down due to that Earth spin velocity cancelation.

My velocity relative to the sun varies daily, fastest at midnight and slowest at noon.  Somewhere in between is proper orbital speed at this radius.  At midnight the speed is too high, and at noon it is too low, as per Kepler's 3rd law.  The speeds are easy to compute:  About 30 km/s ± about 0.5 km/s speed when the sun is directly below and above me respectively.

So, you don't have to maintain your orbital velocity. The Sun gravity force works for you. This is a key element.
Newton didn't specify even one word about the impact of spinning velocity.
This is a new idea which had not been confirmed.

What, that your speed relative to the sun is lower at noon?  That's pretty obvious I'd think.

If you believe that there is a possibility to slow down the earth orbital velocity

My speed, not Earth's speed.  I'm the one that suppose to fall closer to the sun because I'm moving slower than Kepler says I should.
Of course Earth speed is slowest every full moon, for the same reason.  The difference isn't as much as ±0.5 m/sec.

We do not discuss about rapidly spinning pool of noodle.

OK, you don't know what a pool noodle is.  It's like the spanner, but more susceptible to angular forces.

If the angular momentum or the revolving speed of the Moon or the Earth can slow down the orbital velocity of the Erath or the Moon - than please prove it.

That was demonstrated many posts ago.  I should the vector math quite a ways back.  The force vector on the moon is not perpendicular to its motion, therefore there is work being performed (energy transfer).

I still can't understand how any sort of offset in bulges can set any sort of thrust.

Understanding it would cause your argument to fall apart, therefore nothing I can say will make you understand it.  I'm fine with that.

Please use the spanner as an example. you are more than welcome to force the spanner in any sort of offset as you wish. Try to prove why by doing so we shall get extra trust on the orbital object.
Please try to prove this idea by mathematics.

OK, there is a spanner suspended exactly through its center of gravity on a frictionless axis.  It should be able to spin freely, and be balanced.
I give it a permanent stationary offset of 45°.  The spanner is 40 cm long, 20 on each side of the axis.  The lower side is 6372000 meters from the center of earth and has a force of GmM/40602384000000 acting on it.  The upper end is a tiny bit further away from earth and has a force of around GmM/40602384382320 resulting in a 1e-6 % difference in force. It will start to rotate to vertical because the bottom is attracted more than the top, being closer to Earth.  That is a torque force being put on an object despite its center of gravity never moving or the force on that center of gravity ever changing.
You can get rid of the axis suspending the spanner if you just put it in orbit and let its speed hold it up there.

The numbers get more significant if the spanner is a bit longer, like say over 12000 km long, even if the 12000 km spanner has an offset far less than 45°

[Dave Lev (OP)]

Thanks
Great example

I give it a permanent stationary offset of 45°.  The spanner is 40 cm long, 20 on each side of the axis.  The lower side is 6372000 meters from the center of earth and has a force of GmM/40602384000000 acting on it.  The upper end is a tiny bit further away from earth and has a force of around GmM/40602384382320 resulting in a 1e-6 % difference in force.

However, I really don't understand why there is a difference in forces.
You discuss about 20 cm.
Let's set the whole Earth at the shape of spanner.
So, the Earth will look like an extended object with all of its mass while the length of each side is 10,000 Km. 
Based on Newton's Second Law for an Extended Object
http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/n2ext.html
"The motion of any real object may involve rotations as well as linear motion, but the motion of the center of mass of the object can be described by an application of Newton's second law in the following form:
F = M a
In order to understand the calculation, we need to look at the Following "Newton's Second Law for a System of Particles"
http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/n2ext.html#c2
"Newton's Second Law for a System of Particles:
The form of Newton's second law for a system of particles will be developed with the understanding that the result will apply to any extended object where the particles are in face connected to each other.
The center of mass of a system of particles can be determined from their masses and locations."
So, Our dear Newton set a complicated calculation in order to get the outcome of:
M a = F
Newton actually tells us that the shape of the object and its offset can't contribute any extra force as long as all the particles of the object are connected.
So, if you still think that there is an error in Newton calculation, please offer the updated calculation to prove why each side can contribute different force (while all the particles are fully connected).

[Halc]

However, I really don't understand why there is a difference in forces.

I modeled the spanner as two masses separated by 30 cm (I did not put all the mass at the ends of the 40 cm spanner), and applied Newton's F=GMm/r² to get the force, and then F=ma to get the acceleration of spanner that sets it in motion.
In addition, if the Earth under it was entirely stationary and even perfectly spherical at first, it too begins to rotate by application of Newton's conservation of angular momentum law.  The moon does not do this to the Earth because unlike the spanner, it has no net offset and thus exerts no net torque directly on Earth.  All Earth's torque is due to friction, and our spanner example does not involve friction.  We can put some air around it so it stops rotating after a while.

Let's set the whole Earth at the shape of spanner.
So, the Earth will look like an extended object with all of its mass while the length of each side is 10,000 Km. 
Based on Newton's Second Law for an Extended Object
http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/n2ext.html
"The motion of any real object may involve rotations as well as linear motion, but the motion of the center of mass of the object can be described by an application of Newton's second law in the following form:
F = M a

Yes, the motion of the center of mass will follow that law, just like it says.  I never said otherwise.

In order to understand the calculation, we need to look at the Following "Newton's Second Law for a System of Particles"

We need to come up with different names for all these laws, because you're calling them all Newton's second law of something, which is confusing.  Newton's second law typically refers to the second law of motion, which is F=ma.  So lets just call the one above the 'extended object law' and this one below the 'law for a system of particles'.

http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/n2ext.html#c2
"Newton's Second Law for a System of Particles:
The form of Newton's second law for a system of particles will be developed with the understanding that the result will apply to any extended object where the particles are in face connected to each other.
The center of mass of a system of particles can be determined from their masses and locations."

That's not the law you quoted before.  It's just a statement of the fact that the center of mass of any set of particles (connected or not) can be computed.  This lacks any description of how that computation is done (which isn't hard), and it doesn't mention net forces like the last time you mentioned it.  Maybe this is a different law.
Anyway, this wording just says there is a COM and that is it computable.  Hardly a revolutionary statement, and not something that I would call a 'law'.
The 'law' does not say that the set of particles will exert gravitational force as if they were all at the center of mass, or that the center of mass will follow the same path as the object with all its mass concentrated there.  This is very easy to demonstrate with just a two-particle object.
And yet I find you here misrepresenting these laws and attempting to do just that.

So, Our dear Newton set a complicated calculation in order to get the outcome of:
M a = F
Newton actually tells us that the shape of the object and its offset can't contribute any extra force as long as all the particles of the object are connected.

He said no such thing. That does not follow from any of the laws you've quoted.

So, if you still think that there is an error in Newton calculation, please offer the updated calculation to prove why each side can contribute different force (while all the particles are fully connected).

Particles are never connected.  Two things cannot touch.  None of Newtons laws about sets of objects require connectivity for this reason.  The center-of-mass of the Earth-moon system follows an ellipical path around the sun, which the center of mass of just Earth does not because the moon is yanking back and forth each month.  So treating the two as one unit gives a far smoother curve, even though the two are not physically connected.

This is not always the case.   Earth and Venus could be connected with a thin spidery thread and the center-of-mass of the single object would very much obey that law you quote (you quoted a much better version of it in a prior post), but that COM does not follow the same path as would an object that was actually all at that spot, and does not generate a gravitational field that is identical to a single mass at the COM.

Point is, the connection is not necessary.

[Dave Lev (OP)]

Particles are never connected.  Two things cannot touch.  None of Newtons laws about sets of objects require connectivity for this reason.

This is a sever mistake.
All particles on earth are fully connected due to gravity.
In the following law it is stated clearly:
http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/n2ext.html#c2
"The form of Newton's second law for a system of particles will be developed with the understanding that the result will apply to any extended object where the particles are in face connected to each other."
It is also stated:
"Since nothing we have done addressed whether the particles are connected or not, this result generally applies to a system of discrete particles or to an extended object consisting of connected mass elements."
So please - why don't you agree with those simple and clear evidences???

Let's set the whole Earth at the shape of spanner.
So, the Earth will look like an extended object with all of its mass while the length of each side is 10,000 Km. 
Based on Newton's Second Law for an Extended Object
http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/n2ext.html
"The motion of any real object may involve rotations as well as linear motion, but the motion of the center of mass of the object can be described by an application of Newton's second law in the following form:
F = M a

Yes, the motion of the center of mass will follow that law, just like it says.  I never said otherwise.

So, you agree that the center of mass of any kind of object at any shape (even if it is 20,000 Km spanner) must follow that law.
If so, you agree that there is only one center point of mass for any kind of object.
Therefore, there in nothing to disagree.
With only one center point of mass there is no way to get any extra thrust.
I hope that you agree with that.

Conclusions:
I didn't invent those laws.
You are more than welcome to call them at any name. However, those laws prove that there is one center of mass to any shape of object and at any offset.
Therefore, the assumption that there is an extra thrust due to offset contradicts those fundamental laws for gravity.
You didn't offer any alternative mathematic calculation in order to reject those laws and support the hypothetical idea of extra thrust due to bulges offset.
Hence, our scientists must look for better explanation why all the planets and moons are drifting outwards.
I'm specifically using the words "drifting" as I have proved that there is no extra thrust that can push them outwards.
I have no more questions about tidal friction.

With your permission, I would like to focus now on the Sun orbital cycle.

[Dave Lev (OP)]

I hope that you agree with the following:
1. The spanner has a center point of mass.
2. If we won't set an external force of the spanner there is no extra thrust.
3. In our case the spanner represents the Earth.
If you agree with all the above let's move to the following:

You claim that:

If I put force on the spanner, or net force on the system of completely disconnected objects, that will move the center of gravity, which is thrust.

Now
1. Let's use the idea that all the particles on Earth are connected.
In this case, we will assume that the tidal is so strong that it coverts totally the shape of the earth from ball shape into spanner. (Instead of just two bulges)
So, what?
The Earth in a spanner shape has exactly the same center point of mass as the Earth in ball shape (as all the particles are connected).
So, that external force (tidal) didn't change at all the location of the center point of mass (although it changed the earth shape).
Therefore, there is no way to set extra thrust on the moon.
Do you agree with that?

2. Let's assume that the particles on earth aren't connected.
So, the tidal is so strong that it split the earth into two totally separated objects.
Each object is represented by center point of mass. We can set those two center points of mass exactly where the bulges are located.
In this case I agree that the gravity force between each point to the moon is not equal. and the sum of their gravity force is different from just one central point of mass.
However, even in this case, there is no extra thrust.
(Assuming that we can hold those two points of center of mass at a constant distance and offset with regards to the moon, than the moon will orbit around those points without getting any sort of extra thrust.

3. We will split the Earth into infinite separated points of mass while we hold them at the same spot and the same offset from the moon.
Even in this case there is no thrust on the moon. However, the moon will orbit around infinite number of center of mass points instead of only one or two.

Maybe it is my limited understanding, but somehow I really don't see any way to transfer any thrust from the Earth to the moon due to tidal.
With your permission, let's move on.

[Dave Lev (OP)]

Thanks halc

Let's agree that we don't agree on the idea that there is a constant thrust on the moon due to tidal bulges offset on the Earth.

[Dave Lev (OP)]

With regards to the Sun

Why are we so sure that during all of his life time the Sun had to keep the same orbital radius?
How could it be that all the moons and Planets are drifting outwards, (or inwards based on the tidal idea) while the sun is fixed at the same radius?
How could it be that in one hand our scientists claim that the SMBH increases its mass by eating stars and gas clouds, while on the other hand they don't consider an option that stars must migrate/drift inwards in order to supply the requested food for the SMBH monster?
If the SMBH has 4 x 10^6 sun mass, (while our scientists believe that this mass had been taken from the stars in the galaxy) than somehow 4 x 10^6  stars had to drift inwards.
So why our scientists are so sure that the Sun was always at the same distance from the center of the galaxy???
What makes our star so unique that it had to stay so far away from the monster at the galaxy center?

[Halc]

With regards to the Sun

Why are we so sure that during all of his life time the Sun had to keep the same orbital radius?

Well not at the exact same radius since it is under the effect of all the local stars and such.  But it is moving around the galaxy in a pretty good imitation of a circle (having done around 20 laps so far), and if it had been seriously deflected by some large passing object, it would very unlikely be deflected onto this fairly perfect orbit.  So it seems to have been free of that sort of thing the whole time.  That's a large reason why we're here: We've had a nice stable environment that wouldn't have existed if that large mass had gone through and disrupted the solar system.

How could it be that all the moons and Planets are drifting outwards, (or inwards based on the tidal idea) while the sun is fixed at the same radius?

We have no proof that the radius has been fixed all this time.  The clean orbit tells us that we've never had a large close encounter.  The stars that have had one typically become halo objects.
As for tides, you can't raise a tide on a black hole, and I don't think the tides raised by Sgr-A on the sun have any significant impact on its orbital radius.  Tidal forces follow a inverse-cube law, and that means Sgr-A probably has less tidal effect on the sun than I do.

Tides cannot pull our sun in.  The solar system has positive angular momentum (inclination +63°), so if any of that is affected by Sgr-A induced tides, it will push us further out.  Even if we had negative angular momentum, there isn't enough of it to drop us anywhere close to Sgr-A.

How could it be that in one hand our scientists claim that the SMBH increases its mass by eating stars and gas clouds, while on the other hand they don't consider an option that stars must migrate/drift inwards in order to supply the requested food for the SMBH monster?

4 million stars in 15 billion years is one every 4000 years.  Hardly a hungry monster.
Spacetime is bent near the SMBH, and things cannot orbit at all near them.  You need to move at lightspeed to orbit at a radius of 1.5 the radius of the event horizon.  Anything inside that is dragged in (or would need thrust to stay out), and stuff outside that but close still spirals in.  These are relativistic effects, not Newtonian effects.

If the SMBH has 4 x 10^6 sun mass, (while our scientists believe that this mass had been taken from the stars in the galaxy) than somehow 4 x 10^6  stars had to drift inwards.

A lot of that mass was already there before it formed into stars, but I suppose it all was outside the event horizon at some distant point, just like our solar system was all spread out before gravity pulled all that matter close enough together to make a star.

Things at the center of the galaxy are relatively congested.  Two stars pass close to each other and one goes faster and the other loses kinetic energy.  The slow star falls closer to the SMBH and the fast one shoots out and becomes a halo object.  Few things have a clean orbit that close in.

So why our scientists are so sure that the Sun was always at the same distance from the center of the galaxy???

They're not.  There is no history we can consult.  We seem to be on the road (a clean orbit) still, so that's evidence that we've not left it, because it is hard to get back if you fall off the road, or at least it takes a lot more than 20 orbits. That's pretty strong evidence, but not proof.

What makes our star so unique that it had to stay so far away from the monster at the galaxy center?

Most of the stars out this far are like that.  We're in the suburbs, and we're quite ordinary, not unique at all.  The big ones burn up faster, and the little ones burn much slower.  Scientists are not claiming that all the stars out here are heading for Sgr-A except us.
Look near Cassiopeia constellation for our doom.  That's the close encounter that will likely fling us out of our nice circular orbit in less time that it took life to evolve a human.

[Dave Lev (OP)]

There is no history we can consult.  We seem to be on the road (a clean orbit) still, so that's evidence that we've not left it, because it is hard to get back if you fall off the road, or at least it takes a lot more than 20 orbits. That's pretty strong evidence, but not proof.

The solar system has positive angular momentum (inclination +63°),

The Sun has currently positive angular momentum (inclination +63°). In other words, it is moving upwards from the Galactic Disc lane while it orbits around the center of the galaxy.
It is expected that once it gets to the pick it should get down and cross the galactic lane downwards.
Hence, the Sun is Bobbing up and down while it orbits the center of the galaxy.
This is not unique for the Sun. Actually all the stars in the galaxy bobbing up/down, in/out or in between.
If I understand it correctly, the Sun should move up and down at least four times before it set one complete orbital cycle.
So, how can we call it "clean orbit"?
Do you think that Newton or keler would accept that bobbing orbital activity as a "clean orbit"?
Can you please find one planet or moon that is bobbing up and down while it orbits around its main host (four times per cycle)?
How can we ignore that incredible positive angular momentum???
Actually you have already offered the answer for that bobbing activity"

Well not at the exact same radius since it is under the effect of all the local stars

So, you even claim that "it is under the effect of all the local stars".
Hence, why don't we accept the idea that what we see is due to the gravity impact of all the local stars?
We all know that gravity works locally.
As we stay on Earth, we are under the gravity force of the earth (Not the Sun, Not the moon and not under the gravity force of the whole galaxy ).
The moon also works under the gravity of the Earth (Although the Sun gravity on the Moon is stronger by at least twice)
The Earth (or actually - the Earth/moon center of mass) works under the gravity of the Sun (While it ignore the gravity of the whole galaxy, and so on.)
So, why the Sun doesn't orbit under the effect of all the local stars???
In other words, what is the real host of the Sun?
If the Sun goes up and down, could it be that it actually orbits around some sort of a center of mass which is the equivalent center of all the local stars? So, could it be that it doesn't directly orbit around the galaxy, but it orbits around a local center of mass while this local center of mass orbits around the galaxy.
It is the same phenomenon as the Moon orbits around the Sun (12 times per cycle).
If we shut down the light at the Earth, we might see that the moon orbits around the Sun while it's bobbing inwards and outwards. (As the Moon orbits almost horizontally to the earth/sun disc). If the Moon was orbiting vertically around the earth, we would see that it is bobbing up and down as it orbits around the Sun. Almost identical to the Sun bobbing activity.
So, why do we reject the idea that the bobbing activity shows clearly that the Sun doesn't orbit directly around the center of the galaxy, but around some local center of mass which had been set by the "effect of all the local stars"?

If you don't agree with this idea:
Would you kindly show the formula of gravity which can support that strange bobbing activity or "clean orbit" of the Sun?
I assume that you might claim that it is bobbing due to the gravity of the galactic disc lane. If so, please prove this hypothetical idea by real mathematics based only on Newton and kepler.

In my opinion, this bobbing activity is the smoking gun which we are looking for.
I really can't understand how can we ignore so important activity.
Once we agree that the Sun orbits around a local center of mass (Which is "under the effect of all the local stars" - as the Moon/sun orbit), we will get a clear answer for the Spiral galaxy enigma.

[Halc]

The solar system has positive angular momentum (inclination +63°),

The Sun has currently positive angular momentum (inclination +63°). In other words, it is moving upwards from the Galactic Disc lane while it orbits around the center of the galaxy.

Inclination is the tilt angle and has nothing to do with where we are going.  0° means the axis of the solar system is the same as the axis of the galaxy.  It isn't, it is off by 63°, which means it is more on its side than not.  Any inclination over 90° would give that solar system a negative angular momentum relative to the galactic axis.

What I said has nothing to do with us moving towards the center.  As I've said, our path is on average somewhat circular: the eccentricity is low enough to put us in the disk somewhere at a fairly narrow range of radius, but is otherwise pretty meaningless.  We take a wobbly path around the galaxy and have a speed somewhere between 225 and 250 km/sec most of the time.  The forces that change that are not tidal.

It is expected that once it gets to the pick it should get down and cross the galactic lane downwards.

I know of nobody (except you perhaps) that expects this sort of thing to ever happen.

Hence, the Sun is Bobbing up and down while it orbits the center of the galaxy.
This is not unique for the Sun. Actually all the stars in the galaxy bobbing up/down, in/out or in between.
So, how can you call it "clean orbit"?

You're right, its not very clean.  But we're in the plane and moving generally at the velocity needed to be at this current radius, so that's clean enough.

Do you think that Newton or keler would accept that orbit as a "clean orbit"?

The orbit is described as being non-Keplarian, so no.

If I understand it correctly, it should move up and down at least four times before it set one complete cycle.
Can you please find one planet or moon that is bobbing up and down while it orbits around its host (four times per cycle)?

Earth bobs up and down around 13 times each lap.  4 times?  No, I cannot think of one with that cycle.  I'd not heard of it for the galactic orbit.  Is there something nearby that we go around every 50 million years or so?  Is that a known thing?

How can you ignore that incredible positive angular momentum???

Same way I ignore the moon's incredible change in positive solar angular momentum every 14 days.  It all averages out in the end.

Actually you have already offered the answer for that bobbing activity"

Well not at the exact same radius since it is under the effect of all the local stars

So, you even claim that "it is under the effect of all the local stars".

Yes, that's right.  We don't exactly orbit Sgr-A, which is tiny compared to the mass of the galaxy inside our orbital radius.  Not at all like the Earth, dominated by that one huge mass in the middle with nothing but small objects giving minor but regular deflections from that perfect elliptical path.

The Earth works under the gravity of the Sun (While it ignore the gravity of the center of the galaxy, and so on.)

Earth accelerates due to the galaxy exactly as much as does the sun.  Gravity is never ignored.  We are inside the hill radius of the sun, so the galaxy in general isn't going to separate the two of us.  A single passing object might.

So, why the Sun doesn't orbit under the effect of all the local stars???

The sun is within the hill radius of no 'local' dominant mass.  Perhaps there is a cluster that holds together with mutual attraction, but then the orbit of the cluster still goes around the galaxy.  Earth has also made 20 trips around the galaxy, despite the continuous change in velocity.

In other words, what is the real host of the Sun?

Don't know.  Don't know if there is one before the galaxy as a whole.  Something must hold the galactic arms together, but that's not really an orbit.

If the Sun goes up and down, could it be that it actually orbits around some sort of a center of mass which is the equivalent center due to all the stars/SMBH in the galaxy?

That would be orbiting the galaxy, and it wouldn't go up and down due to that, except for normal orbital eccentricity just like Earth and every other planet.  Mercury radius changes an awful lot each orbit, but that is no indication that it is orbiting some mass other than the sun.

If there is some local mass around which all the nearby stars cling, then what really counts is the orbital speed and radius of the center of mass of that local group, and not the individual velocity of any particular object that is part of that group.

So, could it be that it is not directly orbits around the galaxy, but it orbits around a local center of mass while this local center of mass orbits around the galaxy.

Exactly.

It is the same phenomenon as the Moon orbits around the Sun (12 times per cycle).
If we shut down the light at the Earth, we might see that the moon orbits around the Sun while it's bobbing inwards and outwards. (As the Moon orbits almost horizontally to the earth/sun disc).

Right, but that bobbing in and out doesn't mean the moon has any chance of falling into the sun.  OK, it will, but only because the sun will come out here at grab it, not because the moon will fall in.

So, why do we reject the idea that the bobbing activity shows clearly that the Sun doesn't orbit directly around the center of the galaxy, but around some local center of mass which is under direct "effect of all the local stars"?

I didn't assert that one way or the other.  The local masses are pretty well known, and I don't know if our neighbors are always our neighbors, or if they're just nearby right now.  A good text would say.  I'm pretty sure this sort of thing is known. 
Not sure if our motion relative to this local mass would be considered 'orbital' since there is no central mass to stabilize it.  It is sort of a 3-body problem with similar masses: the movements of the masses is chaotic, not at all following the rules of orbital mechanics.  Any one of the bodies might get flung away from the group, bringing the remaining members of the group closer together.  New masses might come from outside and lose enough speed to become new members of the group.

If you don't agree:
Would you kindly show the formula of gravity which can support that strange bobbing activity or "clean orbit" of the Sun?

A=GM/r²
That describes the acceleration of any body given any number of masses spaced around it.

Prove your hypothetical idea for that bobbing activity by real mathematics based only on Newton and kepler.

Integrate the formula above and you get the bobbing activity.  Kepler doesn't much come into play here.  It's all that simple Newton formula.

In my opinion, this bobbing activity is the smoking gun which we are looking for.

Actually, I'm not really sure what you're looking for here.
Some single central mass (closer than SGr-A) around which we orbit?  No, we're not within any object's hill radius, so our local motion is not really best described as 'orbital'.

Once we agree that the Sun orbits around a local center of mass (Which is "under the effect of all the local stars" - as the Moon/sun orbit), we have got the answer for the Spiral galaxy enigma.

Enigma?  The 'bobbing' is quite expected.  There are masses nearby, but none close enough for an orbit.  A local center of gravity may well come into play, loosely holding a group of sister stars together, but our motion around that is chaotic, not orbital.
I call them sister stars because most of the the local ones are probably born of the same parent supernova, which explains their somewhat similar mutual velocity as a group.  We're a 2nd generation solar system.  The 1st generation ones lack planets like our inner ones.

[Dave Lev (OP)]

So, could it be that it is not directly orbits around the galaxy, but it orbits around a local center of mass while this local center of mass orbits around the galaxy.

Exactly.

Wow

If we agree that the sun orbits around a local center of mass while this local center of mass orbits around the galaxy, than we have to agree that all the stars in the spiral galaxy has a similar orbital motion.
Based on that, now we can get better understanding about the Spiral galaxy.
For example -
When we look at the nearby stars, we might see that each one is moving at a different direction and velocity.
We might think that very soon they all will get out from the spiral arm.
This is incorrect. all the nearby stars will stay with us. Each one of them orbits around it's unique center of mass, while the center of mass is moving with the arm. and that center of mass is a direct product of all the other nearby stars...
Let's look at S2.
We think that it orbits around the SMBH. That is big mistake.
If we look carefully, we also might find that it's orbital cycle isn't clean.
In other words, it also orbit around some center of mass with this center of mass orbits around the SMBH.
This is a very important issue.
Let's take the example of the Moon/sun orbital cycle.
If we ignore the earth, than just by monitoring the orbital cycle of the moon and its mass estimated, we might think that the Sun mass is very low.
So, we can't really extract the real mass of the sun directly from the moon orbit.
We must first find the mass of the earth, and then extract the Sun mass.
In the same token, if we want to extract the real mass of the SMBH we must first find the estimated mass of the S2 center of mass point and based on that data we can extract the real value of the SMBH.
We might find that the mass of the SMBH is significantly higher than our faulty calculation.
Do you agree with that?
 

[Dave Lev (OP)]

Orbital speed is a function of 'reduced mass', which for any relationship except a binary star, is approximated by the mass of the primary.
V = √(Gμ/R) where G is the gravitationl constant, μ is the reduced mass of the pair of objects, and R is the radius.

Reduced mass μ for two objects is: μ = (m1^-1 + m2^-1)^-1 which is pretty much the mass of the sun for Earth's orbit, and SGr-A (plus a couple hundred dark objects) for S2. 

That means that the moon would orbit at about the same speed as Earth if it was by itself, and S2 would orbit at the same speed even if it was the only visible part of some larger object that it orbited.  None of these objects have enough mass to affect their respective reduced masses.

Yes.
You are absolutely correct.
Sorry for my mistake.
However, with regards to the SMBH:
Why do we ignore the accretion disc?
Based on our measurements, the velocity of the plasma in that disc is 0.3 c (speed of light)
If we know the radius of the accretion disc (at the verified velocity), we can easily extract the real mass of the SMBH.
So, why do we insist on S2 which doesn't have a clean orbit instead of the accretion disc which has a perfect cycle orbit?

[Dave Lev (OP)]

There is a lot of research going into dark matter, and these sorts of things help them estimate the MACHO component of dark matter (things that don't show light like Mars).

O.K.
As you have mentioned that issue - Dark Matter

Why Newton didn't add that brilliant idea of dark matter in his formulas?
Why he didn't say that any orbital system is based on dark matter?
Our science communities try to prove their hypothetical ideas by using Newton laws. As they have failed to explain the orbital velocity of our Sun, they came with this idea.
Now they even try to set more effort in order to show that this none realistic idea is realistic.
Let me give you an example -
Let's assume that we want to swim in a pool without water.
So, I will tell you that there is water, but the water is dark water.
You can't see it, you can't feel it, you can't smell it or drink it, but it is there because I said that it there. Would you believe me???

So, this is my personal opinion:
Dark matter is a solid proof for the failure of our scientists to show how spiral galaxy really works.
As they have failed to understand the real impact of none "clean orbit", as they have failed to understand the real impact of the Earth/moon "Drifting outwards", as they have failed to understand the real impact of the Ultra high magnetic power around the accretion disc, they couldn't explain how the gravity really works in Spiral galaxy.
I can do it.
I can explain how the whole universe works without any need for dark matter, for dark energy and for any sort of dark magic.
Based on Newton law I can easily explain:
How spiral galaxy really works?
Why there are spiral arms in spiral galaxy?
Why the stars in spiral arms have almost constant orbital velocity at any radius?
Why we see that all the far end galaxies are moving away from us at almost the speed of light, while our observable universe seems to be full with matter.
If we will come back 100,000 billion years (or even 10^Billion years) from now, we will see a similar universe with same density and almost the same numbers of galaxies per observable universe. We might not find our solar system, but there will be many similar.
Again - no need for dark matter or dark energy - just Newton law and simple common sense.
However, in order to do it I need the following basic understanding:
1. "Drifting" outwards
It's very difficult to verify a "drifting" outwards of few cm per cycle while the orbital radius is 1 Arc.  However, all the objects (assuming that they are far enough) are drifting outwards from their host. That is correct for all the objects including: Moons, planets, stars... There is no need for tidal support to explain this Phenomenon.
Therefore, any orbital cycle is by definition spiral shape cycle. So, even if it drifts 1 micro meter each cycle, it is still has spiral orbital shape.
2. Clean orbital cycle.
All the orbital cycles in the Universe must be clean. There is no room in our universe for none clean cycle (unless there is an interruption.)
So, the Sun must set a very clean orbital cycle around its center of mass. That must be correct for any star in the universe even if it is very massive star or object (unless there is no interruption).
3. The magnetic power around the accretion disc, push any matter upwards (or downwards) at a speed of 0.8 c (speed of light) that prevents from any gas cloud, star or even atom to drift inwards in order to be eaten by the SMBH monster.

That's all I need.
Based on those three elements, I can easily explain how our universe works without any sort of dark magic.
If you agree I will show how it works.

[Dave Lev (OP)]

Dark Matter

Newton didn't need to.  His formulas do not apply only to matter that emits light.  Jupiter will orbit the future white dwarf that the sun will become at the same radius as the black dwarf it will be even later on.  The latter is dark matter.

Based on Wiki:
https://en.wikipedia.org/wiki/Dark_matter
"Dark matter is a hypothetical form of matter that is thought to account for approximately 85% of the matter in the universe, and about a quarter of its total energy density."
"Many experiments to directly detect and study dark matter particles are being actively undertaken, but none has yet succeeded."
This is my opinion about the dark matter:
The galaxy rotation problem - We don't know how to explain that problem based on our current hypothesis/understanding how spiral galaxy works.
Hence, instead of reconsider our hypothetical ideas about spiral galaxies and some other issues, we have found a brilliant idea of dark matter.
However, unfortunately we didn't find any evidence for that unrealistic idea: "Many experiments to directly detect and study dark matter particles are being actively undertaken, but none has yet succeeded."
Therefore, we have two options:
1. Continue to hold our none realistic ideas about spiral galaxies and continue to look for that unrealistic dark matter.
2. Open our mind to different ideas which can perfectly explain the galaxy rotation problem and all the other unsolved issues without any need to dark matter.

However, if I understand it correctly - Our science community reject the idea that they might have a problem with understanding how spiral galaxy really works. Based on their point of view, they only need to find the evidence for that dark matter.
So, how long do we have to wait until they will claim - "Sorry, there is a chance that if we don't find it - it is not there?
One year, 10 Year? Or one billion years?
Why you don't even give a chance to look for a different concept about our universe?

[Dave Lev (OP)]

So is there a model supplied by such open-minded thinking?  Trust me, such a thing would be warmly greeted if it worked.

Trust me.
There is a model that works by 100%.
However, I need your cooperation and good willing.
For just one moment try to forget all the wrong understanding/hypothetical ideas that we have about the galaxy (Including: Age of the Star/galaxy/Universe, dark matter/dark energy and so on).

There are three elements which are needed in this model.

1. The impact of gravity force due to local mass - The impact of gravity force due to local mass is stronger than the impact of gravity force of a very far end object. Therefore, the moon had selected to orbit around the Earth instead around the Sun, while the gravity force of the Sun/Moon is stronger by at least twice than the Earth/Moon gravity.
So, even if we shut down the light on Earth , the moon will continue to orbit around the Earth, while their mutual point of mass orbits around the Sun.
The outcome is as follow:
The moon orbits around a virtual host point (as we can't see the earth) while this virtual host point orbits around the Sun. So, the orbital cycle of the moon sets a clean orbital cycle around a virtual host point (Earth - which we can't see).
In the same token:
Every star in the galaxy must orbit around some host Point. It might be something that we see or something that we can't see. However, any star (at any size) must set a clean orbital cycle (in ellipse shape or a perfect cycle). If we can't see that host point, let's call it virtual host point. So, while the star orbits around that virtual host point, the host point orbits around the center of the galaxy. Therefore, we might think that the orbital cycle of a star around the galaxy is not clean.
With regards to our Sun
Please look at the following motion of the solar system in the galaxy:
http://www.biocab.org/Motions_of_the_Solar_System.jpg
http://www.biocab.org/coplanarity_solar_system_and_galaxy.html


The broken gray line shows exactly the virtual host point of the Sun while it orbits around the galaxy.
So, the Sun sets a clean orbital cycle around its virtual host point. This host point is a direct outcome of the local mass gravity (Due to the nearby mass in the Orion spiral arm).
So, the total orbital motion of the Sun is 217 Km/sec however, it also moves locally upwards/downwards to the galactic disc lane at 5-7 km/sec while it moves inwards/outwards at 20 km/sec.
Each star in the spiral arm has a similar orbital motion. Therefore, when we look at the nearby stars we see the following:
http://www.basicknowledge101.com/photos/2015/perception-of-time-rotation-of-galaxy.png
We might think that they move randomly relative to each other, but in reality, each one orbits around its virtual host point, while all their host points stay together in the arm and set a nice orbital motion of about 217 Km/sec around the galaxy.

Summery -
The orbital cycle around a virtual host point is a key element in my explanation.
Please try to accept this idea as is.
If you totally can't agree with that, you are more than welcome to prove it by mathematics.
However, please don't tell a story why this idea isn't logical based on our current understanding about the Galaxy
Agree?
If you have no objection - we will set the next element.

[Halc]

So is there a model supplied by such open-minded thinking?  Trust me, such a thing would be warmly greeted if it worked.

Trust me.
There is a model that works by 100%.
However, I need your cooperation and good willing.

You don't need that at all if you have a good model that works, and not just a set of assertions that all things behave in a manner that they don't.  I see why you're denying the tidal forces and such.  It apparently conflicts with your assertions.

For just one moment try to forget all the wrong understanding/hypothetical ideas that we have about the galaxy (Including: Age of the Star/galaxy/Universe, dark matter/dark energy and so on).

We must forget all of Newton's laws as well.  Shall we go back to 'impetus'?
You've supplied no laws to replace those, so all I see are requirements for this dream model of yours, but no actual model that meets those requirements.

In the same token:
Every star in the galaxy must orbit around some host Point. It might be something that we see or something that we can't see. However, any star (at any size) must set a clean orbital cycle (in ellipse shape or a perfect cycle). If we can't see that host point, let's call it virtual host point.

This for instance violates Newton's laws, since F=GMm/r² doesn't work anymore.  Force actually goes down as distance from that host point decreases.  Sometimes it goes up.  Depends where you are.  That's what Newton's laws say anyway, but you seem to assert that these virtual host points have force of their own, not the objects themselves.  What makes some objects contribute to the force of a host point and other not?

 So, while the star orbits around that virtual host point, the host point orbits around the center of the galaxy. Therefore, we might think that the orbital cycle of a star around the galaxy is not clean.

Summery -
The orbital cycle around a virtual host point is a key element in my explanation.
Please try to accept this idea as is.

I can accept that, sure.

If you totally can't agree with that, you are more than welcome to prove it by mathematics.

On the contrary, you need to prove, with mathematics, that it works.  So far I've seen no model, just a list of requirements.

However, please don't tell a story why this idea isn't logical based on your our current understanding about the Galaxy
Agree?

No problem.  It seems so far not to defy its own rules, so not illogical.  Current understanding of the galaxy has been discarded, as requested.  But don't go quoting that understanding then.  That would be mathematically unsound.

If you have no objection - we will set the next element.

Have at it!