Are there variations of dark matter concentration on earth based on months or seasons of the year?
Hard to say since no dark matter has ever been detected directly and the density in our solar system is very low even by the standards of dark matter. As I recall, the general estimate is that in the entire solar system there is enough dark matter to make up the mass equivalent of a modest sized asteroid.
The sun is the most massive object in the solar system and the most gravity.. does it mean all the dark matter has gravitated into the sun?
No, it's spread out. Even if it fell into the sun, it would just pass through and go out the other side
You mean the sun gravity is not enough to attract any dark matter even those with stronger coupling to matter of our universe?
No, he means that dark matter falls into the Sun and passes right through, emerging on the other side with the same speed that it started with. This is because the speed it gains falling into the Sun is then given back up as it moves further out of the Sun's gravity well. Imagine a pendulum. Lift it up and then drop it and the pendulum first accelerates downwards, gaining speed until it reaches the bottom before decelerating as it begins to swing upwards. If it weren't for friction and air resistance, the pendulum would oscillate forever, moving back and forth to the same height on each upswing. Dark matter would do the same thing except that it is already likely moving too fast to be captured into any sort of orbit or oscillatory motion by the Sun. It still accelerates while falling towards the Sun and decelerates while moving away, but this results in zero net change in speed.
you mean only a black hole can really capture dark matter? or how many sun masses before it can capture dark matter or put it in orbit?
Dark matter is evidently no better at interacting with other dark matter than ordinary matter. See, for example, the bullet cluster. Only a black hole has the ability to actually entrap dark matter. Even then it can only capture particles that fall directly in. DM cannot be slowed allowing it to spiral into a BH as does ordinary matter Simulations suggest only a tiny fraction of BH mass can be attributed to DM..
Does Sun leave a tail moving through dark matter?
Sun doesn't move through dark matter. Dark matter moves through sun. No tail.
That's hard to say, as both dark matter and the Sun are moving with respect to each other and with respect to the galaxy overall. I'm sure that the Sun moves through dark matter that's in the process of falling in towards the center of the galaxy, but I don't think this would form a tail at all. However I admit that my knowledge of this topic is extremely limited.
Dark matter didn't form a rotating disk like the regular matter. It is more like a big cloud of gas with dark matter particles moving around randomly. The Sun moves through this cloud, but as the particle motion is random and nearly uniform before, it is random and nearly uniform afterwards as well.
Due to Earth's orbit, we move through the cloud at different speeds for different times of the year. In principle this could be measurable as yearly variation of the detection rate - assuming we find dark matter in the lab in the first place.
Does not follow.
Suppose that the nearly uniform random motion were slow relative to movement of Sun relative to dark matter. In that case, the dark matter ahead of Sun could be approximated as an array of particles stationary with respect to dark matter frame.
Now, the particles that move directly the centre of Sun are not deflected. They accelerate as they fall into Sun, then decelerate after passing Sun, to the exact same speed and direction as before.
But particles that move past Sun are deflected. Since they have no way to confer energy to Sun in a frame where Sun is stationary, nor angular momentum, they follow hyperbolic orbits (if outside Sun). Their speed after passing the Sun is the same relative to Sun but in a different direction.
Which means that in the dark matter frame 1) the particles that passed near Sun acquire a substantial crisscrossing velocity they did not previously have, and 2) they also acquire a component of velocity towards Sun´s movement they also did not previously have.
Also: since Sun is conferring momentum to dark matter, Sun must itself be slowed down.
Meaning that the mass distribution as perturbed by passage of Sun must contain some sort of tail attracting Sun from behind.
The particles are not slow relative to the Sun, they have a similar speed.
Yes they get some momentum on average, but the number of particles getting a significant momentum change is tiny.
do all kinds of dark matter interact with gravity? Is there no dark matter that doesn't interact with gravity?
Everything that has a mass interacts with gravity.
is there no dark matter that has no mass that doesn't interact with gravity? yet say it only interacts with matter...
or are there no dark matter that interacts with matter yet doesn't have any mass nor gravity?
All dark matter particles are massive, as far as I know (WIMPs). And gravity and weak interaction are the only interactions via which dark matter interacts.
So now let's take again your questions, one by one:
(Answers: as far as I know, and based on what I stated in these two posts of mine)
(but your question is not syntactically correct [it has two negations])
(again, the first part of the question is not syntactically correct)
(but again no need of the first "no" in your question)
So the definition of dark matter is it has mass and interacts with gravity...
If there is a substance that interacts with matter but has no mass and doesn't interact with gravity.. then what is this substance called? And what laws of physics makes it not possible to exist?
Not the definition. Dark matter gets its name from the fact that it is not seen (thus doesn't interact electromagnetically), but still has a mass to interact gravitationally. You can kind of call that as the definition. (For more see e.g. https://en.m.wikipedia.org/wiki/Dark_matter)
It's just that the prevailing theory of DM claims that it is composed of Weakly Interacting Massive Particles (WIMPs) ...
"The most widely accepted hypothesis on the form for dark matter is that it is composed of weakly interacting massive particles (WIMPs) that interact only through gravity and the weak force."
(from the above source)
But note that WIMPs (and thus Dark Matter) haven't been directly observed yet. Just inferred by their gravitational effects on the galaxies ... (etc.). Experiments are still working on it.
While you take a look at those, I will address your other question (which is a legitimate one).
Separate names with a comma.